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Introduction
To date, General Relativity (GR) remains the best available theory to describe gravitational phenomena. Its many successes include, at cosmological scales, those achieved by the Λ \Lambda Λ Λ \Lambda Λ Λ \Lambda Λ [1 , 2 ] — to the largest cosmic scales — where the list of explained phenomena includes, but is not limited to, the formation of the large-scale structure [3 , 4 , 5 ] , the abundances of light elements [6 , 7 ] , the anisotropies of temperature fluctuations in the cosmic microwave background (CMB) [8 , 9 ] and the measured amplitude of the weak gravitational lensing effect [10 ] . Without CDM, all of these observations would be difficult to explain in the context of a universe governed by GR. With CDM, the Λ \Lambda Λ [9 , 11 ] and the evolution of the expansion rate [12 , 13 ] . The measurement of dynamical observables enables to infer the amount of DM that must be present around a specific object, like a galaxy or a galaxy cluster, in order to solve the missing mass problem [14 , 15 , 16 ] . Numerical simulations in the context of the Λ \Lambda Λ [17 ] .
Despite this alleged success, over the years, Λ \Lambda Λ H 0 H_{0} H 0 [12 , 18 ] significantly differs from the best-fit value derived from the analysis of the CMB spectrum [13 ] . Another important cosmological tension concerns the observed and predicted abundance of 7 L i {}^{7}\mathrm{Li} 7 Li [9 , 19 , 20 ] . In addition, there exists a strong tension between the observed value of the cosmological constant Λ \Lambda Λ Λ \Lambda Λ [21 , 22 ] . Recent measures seem to suggest that Λ \Lambda Λ [23 ] . On smaller scales, the numerical results obtained from N-body simulations of structure formation in Λ \Lambda Λ [24 ] , the dwarf galaxy problem [25 , 26 ] , the too-big-to-fail problem [27 ] and the plane of satellite problem [28 ] .
An additional relevant weakness of the model concerns several phenomenological relations observed in galactic dynamics. The model does not naturally account for the radial acceleration relation (RAR) — which relates the measured acceleration of stars in disk galaxies and the acceleration due to the baryonic mass component of the galaxy [29 ] — and for the baryonic Tully-Fisher relation (BTFR) — the tight power-law correlation between a galaxy’s total baryonic mass and its rotation velocity [30 , 31 ] . The DM model can explain some of these problematic aspects, but only at the cost of a careful fine-tuning of the parameters governing galaxy formation [1 ] : for instance, simulations can reproduce the BTFR by assuming very precise relations between the DM and the baryonic content of the halo [32 , 33 ] . Still, these relations possess a statistical scatter that is larger than the very tight observed scatter [29 , 34 , 35 , 36 ] . These discrepancies may be explained by assuming a CDM halo distribution different from the classical Navarro-Frenk-White profile [37 ] , or by invoking complicated "gastrophysical" feedback effects [38 , 39 ] .
A final challenge to the DM paradigm is the fact that, despite decades of efforts, no clear indication of the existence of a DM particle has been obtained yet. Weakly interactive massive particles (WIMPs) have long been considered the most likely candidate for DM, and more recently many other types of candidate have been studied, including axions, sterile neutrinos, primordial black holes, and hidden sector DM [1 , 40 , 41 ] . All of these possibilities are currently under scrutiny, with improving bounds on their mass and cross-section, but all still await a confirmation [42 , 43 , 44 ] .
A number of experimental facts, most famously the rotational velocity of stars in the outskirts of disk galaxies, demonstrate that the Newtonian theory of gravitation is unable to correctly predict the internal dynamics of galaxies. The theory of Newton seems to predict "less gravity" than what is needed to explain observations, thus we speak of a "missing gravity" problem [45 ] . DM is usually invoked to explain these discrepancies. The idea of Modified Gravity (MG) constitutes an alternative way to deal with the inadequacies of the Newtonian prediction: these inadequacies could derive from our fundamental lack of understanding of the theory of gravity, rather than from the failure to account for a new, exotic kind of matter. MOdified Newtonian Dynamics (MOND) [46 ] constitutes the archetype of MG theories and has proven to be capable of correctly describing and even predicting the dynamics of objects on galactic scale [29 , 34 , 45 , 47 , 48 ] . MOND is a research program including an extended family of non-relativistic and relativistic theories [49 ] . The latter are essential to study the cosmological predictions of the theory [41 ] . Even as the landscape of relativistic theories expanded, it has been extremely challenging to provide a covariant theory capable of reproducing the observational successes of Λ \Lambda Λ [41 , 50 ] . For an extensive review of current observational constraints on various MG models, we refer the reader to [51 ] .
In our previous work [52 ] , we considered a MG theory belonging to the class of the Brans–Dicke theories, with the aim of studying whether such a theory was capable of reproducing the expected behavior for the growth factor of density fluctuations in an expanding universe, without the need of introducing a DM component. Brans–Dicke theories are a class of scalar-tensor theories, where a new degree of freedom is introduced in the form of a scalar field, with an associated kinetic term and a scalar potential. In that first paper, by fixing the form of potential and kinetic term, we selected a specific Brans–Dicke theory with the requirement that, in its weak-field limit, the theory reduces to Refracted Gravity (RG). RG is a non-relativistic theory of gravity that appears to reproduce the observed dynamics of galaxies and galaxy clusters [53 , 54 , 55 , 56 ] . Specifically, RG solves the missing mass problem through the introduction of a gravitational permittivity, a monotonically increasing function ϵ ( ρ ) \epsilon(\rho) ϵ ( ρ ) ρ \rho ρ [57 ] . In addition, an effective approach can show that the scalar field is responsible for the accelerated expansion of the universe [57 ] .
Thus, RG could, in principle, explain both dark energy effects on cosmological scales and DM effects on galactic scales. Our first work [52 ] , however, showed that this Brans–Dicke covariant formulation of RG fails to correctly reproduce the growth of cosmological structure. The theory predicts a very delayed formation of structure, at redshifts z < 1 z < 1 z < 1 z > 10 z > 10 z > 10 [58 ] , although even Λ \Lambda Λ [59 , 60 ] . This result might be linked to the specific choice of the scalar potential. Here, we check whether it is possible, with a different choice of the potential — that still makes the theory reduce to RG in the weak-field limit — to obtain a result for structure formation in agreement with the observations, or if the Brans–Dicke theory is in fact unable to correctly account for structure formation with any potential. In fact, we aim to study an even wider generalization, namely the whole class of Horndeski theories, of which Brans–Dicke theories constitute a sub-class.
In § 2 we show that a specific family of Horndeski theories, which we dub generalized CRG (gCRG), reduces to the classical RG formulation in the weak-field regime, i.e. on galactic scales. The covariant theory previously studied in [52 ] constitutes a single example within this wider class. In § 3 we study the background evolution of a gCRG universe, while in § 4 we develop linear perturbation theory and consider structure formation. § 5 explains how we numerically solve the perturbation equations using the quasi-static approximation (QSA). In § 6 we consider whether curvature might have any effect on structure formation in gCRG. We conclude in § 7 .
Horndeski theory and generalized Covariant Refracted Gravity
The Horndeski theory represents the most general scalar-tensor theory that still admits second order equations of motion [61 , 62 ] . The action of this theory can be divided into a matter part and a gravity part as follows
S = S m + S g r a v = S m + ∫ d 4 x − g L ( g μ ν ) , L = ∑ i = 2 5 L i , (1) S = S_{\mathrm{m}} + S_{\mathrm{grav}} = S_{\mathrm{m}} + \int d^{4}x\sqrt{-g}\mathcal{L}(g_{\mu\nu}), \quad\mathcal{L} = \sum_{i=2}^{5}\mathcal{L}_{i}, \tag*{(1)} S = S m + S grav = S m + ∫ d 4 x − g L ( g μν ) , L = i = 2 ∑ 5 L i , (1) where each of the four terms of the Lagrangian density L \mathcal{L} L G i ( φ , X ) G_i(\varphi, X) G i ( φ , X ) φ \varphi φ X X X
L 2 ≡ G 2 ( φ , X ) , L 3 ≡ G 3 ( φ , X ) □ φ , L 4 ≡ G 4 ( φ , X ) R − 2 G 4 X ( φ , X ) [ ( □ φ ) 2 − ( ∇ μ ∇ ν φ ) 2 ] , L 5 ≡ G 5 ( φ , X ) G μ ν ∇ μ ∇ ν φ + 1 3 G 5 X ( φ , X ) [ ( □ φ ) 3 − 3 □ φ ( ∇ μ ∇ ν φ ) 2 + 2 ( ∇ μ ∇ ν φ ) 3 ] . (2) \begin{aligned}
\mathcal{L}_2 &\equiv G_2(\varphi, X),\\
\mathcal{L}_3 &\equiv G_3(\varphi, X)\Box\varphi,\\
\mathcal{L}_4 &\equiv G_4(\varphi, X)R-2G_{4X}(\varphi, X)\left[(\Box\varphi)^2-(\nabla_\mu\nabla_\nu\varphi)^2\right],\\
\mathcal{L}_5 &\equiv G_5(\varphi, X)G_{\mu\nu}\nabla^\mu\nabla^\nu\varphi+\frac{1}{3}G_{5X}(\varphi, X)\left[(\Box\varphi)^3-3\Box\varphi(\nabla_\mu\nabla_\nu\varphi)^2+2(\nabla_\mu\nabla_\nu\varphi)^3\right]. \tag*{(2)}
\end{aligned} L 2 L 3 L 4 L 5 ≡ G 2 ( φ , X ) , ≡ G 3 ( φ , X ) □ φ , ≡ G 4 ( φ , X ) R − 2 G 4 X ( φ , X ) [ ( □ φ ) 2 − ( ∇ μ ∇ ν φ ) 2 ] , ≡ G 5 ( φ , X ) G μν ∇ μ ∇ ν φ + 3 1 G 5 X ( φ , X ) [ ( □ φ ) 3 − 3 □ φ ( ∇ μ ∇ ν φ ) 2 + 2 ( ∇ μ ∇ ν φ ) 3 ] . (2) Here, X = g μ ν ∇ μ φ ∇ ν φ X = g^{\mu\nu}\nabla_\mu\varphi\nabla_\nu\varphi X = g μν ∇ μ φ ∇ ν φ G G G φ \varphi φ X X X G 4 X = G 5 = 0 G_{4X}=G_5=0 G 4 X = G 5 = 0 [58 , 63 , 64 ] . When G 4 = 1 / 2 G_4=1/2 G 4 = 1/2 G G G φ \varphi φ G 4 ( φ ) ∝ φ G_4(\varphi)\propto\varphi G 4 ( φ ) ∝ φ G 2 G_2 G 2
G 2 ( φ , X ) = ( − W ( φ ) φ ∇ α φ ∇ α φ − 2 V ( φ ) ) , (3) G_2(\varphi, X)=\left(-\frac{W(\varphi)}{\varphi}\nabla_\alpha\varphi\nabla^\alpha\varphi-2V(\varphi)\right), \tag*{(3)} G 2 ( φ , X ) = ( − φ W ( φ ) ∇ α φ ∇ α φ − 2 V ( φ ) ) , (3) the theory is a Brans–Dicke theory [65 , 66 ] . In Brans–Dicke theories, G 3 = 0 G_3=0 G 3 = 0
The field equations, obtained by variation of the action with respect to the metric and setting G 4 X = G 5 = 0 G_{4X}=G_5=0 G 4 X = G 5 = 0 [67 ] :
G 2 X ∇ μ φ ∇ ν φ − 1 2 G 2 g μ ν + G 3 X □ φ ∇ μ φ ∇ ν φ − ∇ ( μ G 3 ∇ ν ) φ + 1 2 g μ ν ∇ λ G 3 ∇ λ φ + G 4 G μ ν + g μ ν ( G 4 φ □ φ + X G 4 φ φ ) − G 4 φ ∇ μ ∇ ν φ − G 4 φ φ ∇ μ φ ∇ ν φ = 4 π G T μ ν , (4) G_{2X}\nabla_\mu\varphi\nabla_\nu\varphi-\frac{1}{2}G_2g_{\mu\nu}+G_{3X}\Box\varphi\nabla_\mu\varphi\nabla_\nu\varphi-\nabla_{(\mu}G_3\nabla_{\nu)}\varphi+\frac{1}{2}g_{\mu\nu}\nabla_\lambda G_3\nabla^\lambda\varphi+G_4G_{\mu\nu}+g_{\mu\nu}(G_{4\varphi}\Box\varphi+XG_{4\varphi\varphi})-G_{4\varphi}\nabla_\mu\nabla_\nu\varphi-G_{4\varphi\varphi}\nabla_\mu\varphi\nabla_\nu\varphi=4\pi GT_{\mu\nu}, \tag*{(4)} G 2 X ∇ μ φ ∇ ν φ − 2 1 G 2 g μν + G 3 X □ φ ∇ μ φ ∇ ν φ − ∇ ( μ G 3 ∇ ν ) φ + 2 1 g μν ∇ λ G 3 ∇ λ φ + G 4 G μν + g μν ( G 4 φ □ φ + X G 4 φφ ) − G 4 φ ∇ μ ∇ ν φ − G 4 φφ ∇ μ φ ∇ ν φ = 4 π G T μν , (4) where we have left implicit the ( φ , X ) (\varphi, X) ( φ , X ) G 2 G_2 G 2 G 3 G_3 G 3 φ \varphi φ G 4 G_4 G 4 T μ ν T_{\mu\nu} T μν S m S_{\mathrm{m}} S m [67 ] :
∇ μ [ ( 2 G 2 X − 2 G 3 X □ φ − 2 G 3 ∂ G 3 φ ∂ X − 2 G 3 φ ) ∇ μ φ − G 3 X ∇ μ X ] = G 2 φ − ∇ λ G 3 φ ∇ λ φ + G 4 φ R . (5) \nabla^\mu\left[\left(2G_{2X}-2G_{3X}\Box\varphi-2G_3\frac{\partial G_{3\varphi}}{\partial X}-2G_{3\varphi}\right)\nabla_\mu\varphi-G_{3X}\nabla_\mu X\right]=G_{2\varphi}-\nabla_\lambda G_{3\varphi}\nabla^\lambda\varphi+G_{4\varphi}R. \tag*{(5)} ∇ μ [ ( 2 G 2 X − 2 G 3 X □ φ − 2 G 3 ∂ X ∂ G 3 φ − 2 G 3 φ ) ∇ μ φ − G 3 X ∇ μ X ] = G 2 φ − ∇ λ G 3 φ ∇ λ φ + G 4 φ R . (5) We now ask under which conditions this rather general theory can describe, in its non-relativistic weak-field limit, the small-scale dynamics of galaxies and galaxy clusters according to Refracted Gravity (RG) [53 , 57 ] . The dynamics of RG is ruled by a modified version of the Poisson equation
∇ ⋅ ( ϵ ∇ Ψ ) = 4 π G ρ , (6) \nabla\cdot(\epsilon\nabla\Psi)=4\pi G\rho, \tag*{(6)} ∇ ⋅ ( ϵ ∇Ψ ) = 4 π Gρ , (6) where Ψ \Psi Ψ G G G ϵ = ϵ ( ρ ) \epsilon=\epsilon(\rho) ϵ = ϵ ( ρ ) G 2 G_2 G 2 G 3 G_3 G 3 G 4 G_4 G 4 (4) -(5) corresponds to Eq. (6) .
In the weak field limit, the metric is:
d s 2 = − ( 1 + 2 Ψ ) d t 2 + ( 1 + 2 Φ ) δ i j d x i d x j (7) ds^2 = -(1 + 2\Psi)dt^2 + (1 + 2\Phi)\delta_{ij}dx^i dx^j \tag*{(7)} d s 2 = − ( 1 + 2Ψ ) d t 2 + ( 1 + 2Φ ) δ ij d x i d x j (7) and we consider a static non relativistic fluid for the right-hand side of the Einstein field Eqs. (4) . Thus, we assume ϕ ˙ = Φ ˙ = Ψ ˙ = 0 \dot{\phi} = \dot{\Phi} = \dot{\Psi} = 0 ϕ ˙ = Φ ˙ = Ψ ˙ = 0 T 00 = ρ T_{00} = \rho T 00 = ρ (7) up to first order in Φ \Phi Φ Ψ \Psi Ψ (4) and the linearized KG Eq. (5) , which can be combined to obtain an expression that relates the Laplacian of the gravitational potential Φ \Phi Φ ρ \rho ρ (6) . The details of this calculation can be found in Appendix A. The end result is that the conditions required to have the Horndeski weak-field equations reduce to the RG Eq. (6) are
{ G 3 = 0 ; G 2 X X = 0 ; G 2 − G 4 G 2 ϕ G 4 ϕ + 2 G 4 G 2 ϕ X G 4 ϕ X − 2 G 4 ϕ ϕ X = 0 ; G 4 G 2 X G 4 ϕ = G 4 ϕ . (8) \begin{cases}
G_3 = 0; \quad G_{2XX} = 0;\\
G_2 - \frac{G_4G_{2\phi}}{G_{4\phi}} + \frac{2G_4G_{2\phi X}}{G_{4\phi}}X - 2G_{4\phi\phi}X = 0;\\
\frac{G_4G_{2X}}{G_{4\phi}} = G_{4\phi}. \tag*{(8)}
\end{cases} ⎩ ⎨ ⎧ G 3 = 0 ; G 2 X X = 0 ; G 2 − G 4 ϕ G 4 G 2 ϕ + G 4 ϕ 2 G 4 G 2 ϕX X − 2 G 4 ϕϕ X = 0 ; G 4 ϕ G 4 G 2 X = G 4 ϕ . (8) Now we need to find suitable functions G 4 G_4 G 4 G 2 G_2 G 2 (8) we get G 2 X = G 4 ϕ 2 / G 4 G_{2X} = G_{4\phi}^{2}/G_4 G 2 X = G 4 ϕ 2 / G 4 G 4 G_4 G 4 X X X
G 2 = X G 4 ϕ 2 G 4 + V ( ϕ ) , (9) G_2 = X\frac{G_{4\phi}^{2}}{G_4} + V(\phi), \tag*{(9)} G 2 = X G 4 G 4 ϕ 2 + V ( ϕ ) , (9) The other relevant derivatives of G 2 G_2 G 2
G 2 ϕ X = − G 4 ϕ 3 G 4 2 + 2 G 4 ϕ G 4 ϕ ϕ G 4 ; G 2 ϕ = ( − G 4 ϕ 3 G 4 2 + 2 G 4 ϕ G 4 ϕ ϕ G 4 ) X + V ϕ ( ϕ ) . (10) \begin{aligned}
G_{2\phi X} &= -\frac{G_{4\phi}^{3}}{G_4^{2}} + \frac{2G_{4\phi}G_{4\phi\phi}}{G_4};\\
G_{2\phi} &= \left(-\frac{G_{4\phi}^{3}}{G_4^{2}} + \frac{2G_{4\phi}G_{4\phi\phi}}{G_4}\right)X + V_{\phi}(\phi). \tag*{(10)}
\end{aligned} G 2 ϕX G 2 ϕ = − G 4 2 G 4 ϕ 3 + G 4 2 G 4 ϕ G 4 ϕϕ ; = ( − G 4 2 G 4 ϕ 3 + G 4 2 G 4 ϕ G 4 ϕϕ ) X + V ϕ ( ϕ ) . (10) Substituting these expressions for G 2 G_2 G 2 (8) we find V ϕ / V = G 4 ϕ / G 4 V_{\phi}/V = G_{4\phi}/G_4 V ϕ / V = G 4 ϕ / G 4 V = 2 Ξ G 4 V = 2\Xi G_4 V = 2Ξ G 4 Ξ \Xi Ξ
G 2 ( ϕ , X ) = G 4 ϕ 2 G 4 X + 2 Ξ G 4 ; G 4 ( ϕ ) = ϵ ( ϕ ) , (11) G_2(\phi, X) = \frac{G_{4\phi}^{2}}{G_4}X + 2\Xi G_4; \quad G_4(\phi) = \epsilon(\phi), \tag*{(11)} G 2 ( ϕ , X ) = G 4 G 4 ϕ 2 X + 2Ξ G 4 ; G 4 ( ϕ ) = ϵ ( ϕ ) , (11) give a Horndeski theory that reduces to RG in the weak-field limit. In our previous work [52 ] , we considered a specific case of this general result, where the relation between the permittivity and the scalar field was assumed to be linear: if we fix G 4 = ϵ = ϕ / 2 G_4 = \epsilon= \phi/2 G 4 = ϵ = ϕ /2 G 2 = X / 2 ϕ + Ξ ϕ G_2 = X/2\phi+ \Xi\phi G 2 = X /2 ϕ + Ξ ϕ W ( ϕ ) = − 1 W(\phi) = -1 W ( ϕ ) = − 1 V ( ϕ ) = − Ξ ϕ V(\phi) = -\Xi\phi V ( ϕ ) = − Ξ ϕ [57 ] and used in [52 ] .1 (11) reduces to RG for any choice of the permittivity as a function of the scalar field, not just the linear relation ϵ ( ϕ ) ∝ ϕ \epsilon(\phi) \propto\phi ϵ ( ϕ ) ∝ ϕ
Background universe in gCRG
Here we study the equations for the background evolution of a universe governed by a Horndeski theory in which the choices in Eqs. (11) have been made. We follow the same derivations of our previous work and generalize them. As was shown in [52 ] , the background evolution of the scalar field can have a great impact on the evolutions of linear perturbations, therefore it is an important preliminary step in our study of the growth factor equation. With Eq. (11) , the 00 and r r rr r r (4) and Eq. (5) become, respectively,
G 4 R μ ν − 1 2 ( G 4 φ 2 G 4 X + 2 Ξ G 4 + G 4 R − 2 G 4 φ □ φ − 2 X G 4 φ φ ) g μ ν − G 4 φ ∇ μ ∇ ν φ + ( G 4 φ 2 G 4 − G 4 φ φ ) ∇ μ φ ∇ ν φ = 4 π G T μ ν , (12) G_{4}R_{\mu\nu}-\frac{1}{2}\left(\frac{G_{4\varphi}^{2}}{G_{4}}X+2\Xi G_{4}+G_{4}R-2G_{4\varphi}\Box\varphi-2XG_{4\varphi\varphi}\right)g_{\mu\nu}-G_{4\varphi}\nabla_{\mu}\nabla_{\nu}\varphi+\left(\frac{G_{4\varphi}^{2}}{G_{4}}-G_{4\varphi\varphi}\right)\nabla_{\mu}\varphi\nabla_{\nu}\varphi=4\pi G T_{\mu\nu}, \tag*{(12)} G 4 R μν − 2 1 ( G 4 G 4 φ 2 X + 2Ξ G 4 + G 4 R − 2 G 4 φ □ φ − 2 X G 4 φφ ) g μν − G 4 φ ∇ μ ∇ ν φ + ( G 4 G 4 φ 2 − G 4 φφ ) ∇ μ φ ∇ ν φ = 4 π G T μν , (12) 2 G 4 φ 2 G 4 □ φ + ( − G 4 φ 3 G 4 2 + G 4 φ G 4 φ φ G 4 ) X = 2 Ξ G 4 φ + G 4 φ R . (13) 2\frac{G_{4\varphi}^{2}}{G_{4}}\Box\varphi+\left(-\frac{G_{4\varphi}^{3}}{G_{4}^{2}}+\frac{G_{4\varphi}G_{4\varphi\varphi}}{G_{4}}\right)X=2\Xi G_{4\varphi}+G_{4\varphi}R. \tag*{(13)} 2 G 4 G 4 φ 2 □ φ + ( − G 4 2 G 4 φ 3 + G 4 G 4 φ G 4 φφ ) X = 2Ξ G 4 φ + G 4 φ R . (13) By substituting Eq. (13) into Eq. (12) , we obtain
G 4 R μ ν − G 4 φ ∇ μ ∇ ν φ + ( G 4 φ 2 G 4 − G 4 φ φ ) ∇ μ φ ∇ ν φ = 4 π G T μ ν . (14) G_{4}R_{\mu\nu}-G_{4\varphi}\nabla_{\mu}\nabla_{\nu}\varphi+\left(\frac{G_{4\varphi}^{2}}{G_{4}}-G_{4\varphi\varphi}\right)\nabla_{\mu}\varphi\nabla_{\nu}\varphi=4\pi G T_{\mu\nu}. \tag*{(14)} G 4 R μν − G 4 φ ∇ μ ∇ ν φ + ( G 4 G 4 φ 2 − G 4 φφ ) ∇ μ φ ∇ ν φ = 4 π G T μν . (14) Contracting this result with g μ ν g^{\mu\nu} g μν (13) yields
G 4 φ □ φ − 2 Ξ G 4 + G 4 φ φ X = 4 π G T , (15) G_{4\varphi}\Box\varphi-2\Xi G_{4}+G_{4\varphi\varphi}X=4\pi G T, \tag*{(15)} G 4 φ □ φ − 2Ξ G 4 + G 4 φφ X = 4 π GT , (15) where T T T
d s 2 = − d t 2 + a ( t ) 2 ( d r 2 1 − k r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 ) , (16) \mathrm{d}s^{2}=-\mathrm{d}t^{2}+a(t)^{2}\left(\frac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\right), \tag*{(16)} d s 2 = − d t 2 + a ( t ) 2 ( 1 − k r 2 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 ) , (16) where a ( t ) a(t) a ( t ) k k k t t t ( r , θ , ϕ ) (r,\theta,\phi) ( r , θ , ϕ ) H ( a ) H(a) H ( a ) φ \varphi φ G 4 [ φ ( a ) ] G_{4}[\varphi(a)] G 4 [ φ ( a )] § 11 ). The final results are the following expressions
H ( a ) = − 1 2 Ω 0 H 0 + 1 4 Ω 0 2 + 2 Ω Ξ 0 H 0 2 [ ( C + C 2 − 1 ) a ± 3 + ( C + C 2 − 1 ) − 1 a ∓ 3 ] , (17) H(a)=-\frac{1}{2}\Omega_{0}H_{0}+\sqrt{\frac{1}{4}\Omega_{0}^{2}+2\Omega_{\Xi0}}\frac{H_{0}}{2}\left[\left(C+\sqrt{C^{2}-1}\right)a^{\pm\sqrt{3}}+\left(C+\sqrt{C^{2}-1}\right)^{-1}a^{\mp\sqrt{3}}\right], \tag*{(17)} H ( a ) = − 2 1 Ω 0 H 0 + 4 1 Ω 0 2 + 2 Ω Ξ0 2 H 0 [ ( C + C 2 − 1 ) a ± 3 + ( C + C 2 − 1 ) − 1 a ∓ 3 ] , (17) G 4 [ φ ( a ) ] = H 0 G 4 , 0 H ( a ) a 3 . (18) G_{4}[\varphi(a)]=\frac{H_{0}G_{4,0}}{H(a)a^{3}}. \tag*{(18)} G 4 [ φ ( a )] = H ( a ) a 3 H 0 G 4 , 0 . (18) The parameter Ω 0 ≡ Ω m 0 / G 4 , 0 ≡ 8 π G ρ m 0 / 3 H 2 G 4 , 0 \Omega_{0}\equiv\Omega_{\mathrm{m}0}/G_{4,0}\equiv8\pi G\rho_{\mathrm{m}0}/3H^{2}G_{4,0} Ω 0 ≡ Ω m 0 / G 4 , 0 ≡ 8 π G ρ m 0 /3 H 2 G 4 , 0 1 / G 4 1/G_{4} 1/ G 4 Λ \Lambda Λ G 4 G_{4} G 4 G 4 G_{4} G 4
The contribution of dark matter in the Λ \Lambda Λ Ω Ξ 0 ≡ Ξ / 3 H 0 2 \Omega_{\Xi0} \equiv\Xi/3H_0^2 Ω Ξ0 ≡ Ξ/3 H 0 2 Λ \Lambda Λ Ω Λ 0 \Omega_{\Lambda0} Ω Λ0 Ξ \Xi Ξ H 0 H_0 H 0 G 4 , 0 = G 4 ( φ 0 ) G_{4,0} = G_4(\varphi_0) G 4 , 0 = G 4 ( φ 0 ) φ 0 \varphi_0 φ 0 (17) coincides with the results already obtained in our previous work [52 ] , while Eq. (18) is a generalized version of our previous result.
We note that the background evolution of the boost factor, G 4 ( φ ) G_4(\varphi) G 4 ( φ ) G 4 G_4 G 4 Figure 1 of [52 ] for the scalar field: the boost factor decreases sharply as a function of the scale factor. This result, which turned out to be a pivotal criticality for the Brans-Dicke theory previously analysed, could potentially prove determinant also for the more general case of Horndeski theories. In the next section, we derive the linear perturbation theory for gCRG, in order to check if this is the case.
Figure 1. Evolution of the density contrast δ \delta δ H H H G 4 G_{4} G 4 [74 ] . The smaller panel on the right depicts a zoomed-in region of the H H H φ 0 \varphi_{0} φ 0 G 4 , 0 G_{4,0} G 4 , 0 Ω Ξ 0 \Omega_{\Xi0} Ω Ξ0 (33) , following the results of [52 ] and [57 ] . The dashed red line in the top and middle panels represents the Λ \Lambda Λ
Linear perturbation theory and growth factor equation in gCRG
In this section we develop the linear perturbation theory associated with our proposed Horndeski model, with the aim to apply such a theory to the study of structure formation in a gCRG universe. Scalar and tensor perturbations to the metric can be considered separately, since, at first perturbative order, the resulting evolution equations do not contain terms that mix the two kinds of perturbations. The scalar perturbations describe the formation of the large-scale cosmic structure, whereas the tensor perturbations are potentially useful to study the propagation of gravitational waves.
We remark that in the model under scrutiny there is no dark matter nor dark energy, since both of these effects should be accounted for by the single scalar field introduced in the Horndeski theory. Our approach is similar but substantially different from that of the so-called Unified Dark Matter (UDM) models, in which the scalar field acts as an effective dark fluid, that, by clustering, provides the gravitational potential wells where the baryonic matter falls [68 , 69 ] . In our case there is no clustering of the scalar field, but the clustering of baryonic matter is expected to be driven by the boosted gravitational force, as computed in Eq. (6) . Therefore, it is imperative that the boost factor, which, in the case of a Horndeski theory, corresponds to the inverse of the function G 4 ( φ ) G_4(\varphi) G 4 ( φ )
We introduce scalar perturbations on a flat FLRW background metric in the Newtonian gauge
d s 2 = a 2 ( η ) [ − ( 1 + 2 Ψ ) d η 2 + ( 1 + 2 Φ ) δ i j d x i d x j ] , (19) \mathrm{d}s^2 = a^2(\eta)\left[-(1+2\Psi)\mathrm{d}\eta^2 + (1+2\Phi)\delta_{ij}\mathrm{d}x^i\mathrm{d}x^j\right], \tag*{(19)} d s 2 = a 2 ( η ) [ − ( 1 + 2Ψ ) d η 2 + ( 1 + 2Φ ) δ ij d x i d x j ] , (19) where η \eta η d η = d t / a ( t ) \mathrm{d}\eta= \mathrm{d}t/a(t) d η = d t / a ( t ) Ψ \Psi Ψ Φ \Phi Φ [52 ] , we can obtain the growth factor equation in the case of the Horndeski theory under study (for a detailed derivation, see Appendix § 12 .§ 1 )
δ ′ ′ + ( 2 + H ′ H ) δ ′ − 8 π G G 4 ρ m 0 H 2 a 3 δ − ∇ 2 δ φ a 2 H 2 G 4 φ G 4 + 2 Ξ H 2 δ φ G 4 φ G 4 = − 6 ( 1 + H ′ H ) Ψ \delta'' + \left(2+\frac{H'}{H}\right)\delta' - \frac{8\pi G}{G_4}\frac{\rho_{m0}}{H^2a^3}\delta- \frac{\nabla^2\delta\varphi}{a^2H^2}\frac{G_{4\varphi}}{G_4} + \frac{2\Xi}{H^2}\delta\varphi\frac{G_{4\varphi}}{G_4} = -6\left(1+\frac{H'}{H}\right)\Psi δ ′′ + ( 2 + H H ′ ) δ ′ − G 4 8 π G H 2 a 3 ρ m 0 δ − a 2 H 2 ∇ 2 δ φ G 4 G 4 φ + H 2 2Ξ δ φ G 4 G 4 φ = − 6 ( 1 + H H ′ ) Ψ − 3 Ψ ′ + 3 G 4 φ G 4 ( 1 + H ′ H ) δ φ + G 4 φ G 4 ( 6 Ψ φ ′ − 3 Φ ′ φ ′ − 3 δ φ ′ ) − 3 G 4 φ φ G 4 φ ′ δ φ − ( 2 G 4 φ G 4 φ φ G 4 2 − G 4 φ 3 G 4 3 ) ( φ ′ ) 2 δ φ − 2 ( G 4 φ 2 G 4 2 − G 4 φ φ G 4 ) φ ′ δ φ ′ . (20) -3\Psi' + 3\frac{G_{4\varphi}}{G_4}\left(1+\frac{H'}{H}\right)\delta\varphi+ \frac{G_{4\varphi}}{G_4}\left(6\Psi\varphi' - 3\Phi'\varphi' - 3\delta\varphi'\right) - 3\frac{G_{4\varphi\varphi}}{G_4}\varphi'\delta\varphi
-\left(\frac{2G_{4\varphi}G_{4\varphi\varphi}}{G_4^2}-\frac{G_{4\varphi}^3}{G_4^3}\right)(\varphi')^2\delta\varphi- 2\left(\frac{G_{4\varphi}^2}{G_4^2}-\frac{G_{4\varphi\varphi}}{G_4}\right)\varphi'\delta\varphi'. \tag*{(20)} − 3 Ψ ′ + 3 G 4 G 4 φ ( 1 + H H ′ ) δ φ + G 4 G 4 φ ( 6Ψ φ ′ − 3 Φ ′ φ ′ − 3 δ φ ′ ) − 3 G 4 G 4 φφ φ ′ δ φ − ( G 4 2 2 G 4 φ G 4 φφ − G 4 3 G 4 φ 3 ) ( φ ′ ) 2 δ φ − 2 ( G 4 2 G 4 φ 2 − G 4 G 4 φφ ) φ ′ δ φ ′ . (20) In this equation δ = δ ρ / ρ \delta=\delta\rho/\rho δ = δ ρ / ρ N = ln a N=\ln a N = ln a
δ φ ′ ′ + ( 3 + H ′ H ) δ φ ′ − 4 π G G 4 φ ρ m 0 H 2 a 3 δ − ∇ 2 δ φ a 2 H 2 + 2 Ξ H 2 δ φ = 2 Ψ φ ′ ′ + 2 Ψ ( 1 + H ′ H ) φ ′ + 4 φ ′ Ψ + φ ′ ( Ψ ′ − 3 Φ ′ ) − G 4 φ φ G 4 φ δ φ [ φ ′ ′ + ( 3 + H ′ H ) φ ′ ] − G 4 φ φ φ G 4 φ ( φ ′ ) 2 δ φ − 2 G 4 φ φ G 4 φ φ ′ δ φ ′ + 2 G 4 φ φ G 4 φ Ψ ( φ ′ ) 2 . (21) \begin{aligned}
\delta\varphi'' + \left(3+\frac{H'}{H}\right)\delta\varphi' - \frac{4\pi G}{G_{4\varphi}}\frac{\rho_{m0}}{H^2a^3}\delta- \frac{\nabla^2\delta\varphi}{a^2H^2} + 2\frac{\Xi}{H^2}\delta\varphi
&= 2\Psi\varphi'' + 2\Psi\left(1+\frac{H'}{H}\right)\varphi' \\
&\quad+ 4\varphi'\Psi+ \varphi'\left(\Psi'-3\Phi'\right) - \frac{G_{4\varphi\varphi}}{G_{4\varphi}}\delta\varphi\left[\varphi''+\left(3+\frac{H'}{H}\right)\varphi'\right] \\
&\quad- \frac{G_{4\varphi\varphi\varphi}}{G_{4\varphi}}(\varphi')^2\delta\varphi- 2\frac{G_{4\varphi\varphi}}{G_{4\varphi}}\varphi'\delta\varphi' + 2\frac{G_{4\varphi\varphi}}{G_{4\varphi}}\Psi(\varphi')^2 . \tag*{(21)}
\end{aligned} δ φ ′′ + ( 3 + H H ′ ) δ φ ′ − G 4 φ 4 π G H 2 a 3 ρ m 0 δ − a 2 H 2 ∇ 2 δ φ + 2 H 2 Ξ δ φ = 2Ψ φ ′′ + 2Ψ ( 1 + H H ′ ) φ ′ + 4 φ ′ Ψ + φ ′ ( Ψ ′ − 3 Φ ′ ) − G 4 φ G 4 φφ δ φ [ φ ′′ + ( 3 + H H ′ ) φ ′ ] − G 4 φ G 4 φφφ ( φ ′ ) 2 δ φ − 2 G 4 φ G 4 φφ φ ′ δ φ ′ + 2 G 4 φ G 4 φφ Ψ ( φ ′ ) 2 . (21) which describes the evolution of the scalar field perturbations, δ φ \delta\varphi δ φ φ \varphi φ
φ ′ ′ + ( 3 + H ′ H ) φ ′ − 4 π G G 4 φ ρ m 0 H 2 a 3 + 2 Ξ H 2 G 4 G 4 φ = G 4 φ φ G 4 φ ( φ ′ ) 2 . (22) \varphi''+\left(3+\frac{H'}{H}\right)\varphi' - \frac{4\pi G}{G_{4\varphi}}\frac{\rho_{m0}}{H^2a^3} + \frac{2\Xi}{H^2}\frac{G_4}{G_{4\varphi}} = \frac{G_{4\varphi\varphi}}{G_{4\varphi}}(\varphi')^2 . \tag*{(22)} φ ′′ + ( 3 + H H ′ ) φ ′ − G 4 φ 4 π G H 2 a 3 ρ m 0 + H 2 2Ξ G 4 φ G 4 = G 4 φ G 4 φφ ( φ ′ ) 2 . (22) These three equations are the generalizations of the growth factor and KG equations we obtained in [52 ] , specifically Eq. (4.9), Eq. (4.5e) and Eq. (3.13), respectively. In particular, the source term in Eq. (20) , that in the case of the Brans–Dicke theory appeared divided by the scalar field, is now divided by the function G 4 G_4 G 4 G 4 G_4 G 4 § 3 , how the background evolution of the boost factor in our Horndeski theory corresponds to the evolution of the scalar field in the Brans–Dicke case. Here, again, the role formerly played by φ \varphi φ G 4 ( φ ) G_4(\varphi) G 4 ( φ ) [52 ] we have shown how, due to the background evolution of φ \varphi φ [52 ] to the larger class of Horndeski theories: since G 4 G_4 G 4 φ \varphi φ § 12 .§ 2 .
QSA numerical solution
As discussed in Appendix § 12 .§ 1 , it is difficult to solve the whole system of coupled differential equations in Eqs. (20) , (21) and (22) , because there is no physically-motivated way to set the initial conditions on the scalar field and its derivatives. This situation is different from the one previously considered in [52 ] , because in that case an analytical solution for φ ( a ) \varphi(a) φ ( a ) φ \varphi φ φ ′ \varphi' φ ′ z = 10 − 3 z=10^{-3} z = 1 0 − 3 [52 ] , we can resort to the quasi-static approximation (QSA) to understand the qualitative behavior of Eq. (20) and provide a more straightforward way to integrate it.
In QSA [70 , 71 , 72 , 73 ] , it is assumed that the time derivatives of the metric scalar perturbations can be neglected, because they are of the order of the Hubble-Lemaître parameter. Such terms are negligible on sufficiently small scales, i.e. sub-horizon scales. If we neglect these terms, our set of equations considerably simplifies, and we can find a solution for the evolution of the density contrast δ \delta δ
The basic approach of QSA is the following: one can write three Poisson-like equations, which relate the potentials Ψ \Psi Ψ Φ \Phi Φ φ \varphi φ G G G [72 ] for the complete definitions):
M 2 = 2 G 4 , (23) M^2 = 2G_4 , \tag*{(23)} M 2 = 2 G 4 , (23) α M = 1 H M 2 d M 2 d t = G ˙ 4 H G 4 , (24) \alpha_M = \frac{1}{HM^2}\frac{\mathrm{d}M^2}{\mathrm{d}t} = \frac{\dot{G}_4}{HG_4} , \tag*{(24)} α M = H M 2 1 d t d M 2 = H G 4 G ˙ 4 , (24) α K = 1 M 2 H 2 2 X G 4 φ 2 G 4 = − G ˙ 4 2 H 2 G 4 2 , (25) \alpha_K = \frac{1}{M^2H^2}2X\frac{G^2_{4\varphi}}{G_4} = -\frac{\dot{G}_4^2}{H^2G_4^2} , \tag*{(25)} α K = M 2 H 2 1 2 X G 4 G 4 φ 2 = − H 2 G 4 2 G ˙ 4 2 , (25) α B = 1 2 M 2 H d M 2 d t = 1 2 α M , (26) \alpha_B = \frac{1}{2M^2H}\frac{\mathrm{d}M^2}{\mathrm{d}t} = \frac{1}{2}\alpha_M , \tag*{(26)} α B = 2 M 2 H 1 d t d M 2 = 2 1 α M , (26) α T = α H = 0. (27) \alpha_T = \alpha_H = 0 . \tag*{(27)} α T = α H = 0. (27) Here, we consider the two effective gravitational constants μ Y \mu_Y μ Y μ Z \mu_Z μ Z
∇ 2 Ψ = 4 π G μ Y a 2 δ ρ m , ∇ 2 Φ = − 4 π G μ Z a 2 δ ρ m . (28) \nabla^2\Psi= 4\pi G\mu_Y a^2\delta\rho_m,\qquad\nabla^2\Phi= -4\pi G\mu_Za^2\delta\rho_m . \tag*{(28)} ∇ 2 Ψ = 4 π G μ Y a 2 δ ρ m , ∇ 2 Φ = − 4 π G μ Z a 2 δ ρ m . (28) One can also define the combination μ W = Σ = ( μ Y − μ Z ) / 2 \mu_W = \Sigma= (\mu_Y-\mu_Z)/2 μ W = Σ = ( μ Y − μ Z ) /2 μ Y \mu_Y μ Y μ W \mu_W μ W μ Z \mu_Z μ Z μ Y \mu_Y μ Y Ψ \Psi Ψ [73 ] , we can compute μ Y \mu_Y μ Y
μ Y = μ Y , ∞ = α c s , δ φ 2 ( 1 + α T ) + 2 [ α B ( 1 + α T ) + α T − α M ] 2 α c s , δ φ 2 M ˉ 2 , (29) \mu_Y = \mu_{Y,\infty} = \frac{\alpha c^2_{s,\delta\varphi}(1+\alpha_T)+2[\alpha_B(1+\alpha_T)+\alpha_T-\alpha_M]^2}{\alpha c^2_{s,\delta\varphi} \bar{M}^2} , \tag*{(29)} μ Y = μ Y , ∞ = α c s , δ φ 2 M ˉ 2 α c s , δ φ 2 ( 1 + α T ) + 2 [ α B ( 1 + α T ) + α T − α M ] 2 , (29) where α = α K + 6 α B 2 \alpha= \alpha_K+6\alpha_B^2 α = α K + 6 α B 2 c s , δ φ 2 c^2_{s,\delta\varphi} c s , δ φ 2 [73 ]
c s , δ φ 2 = − 2 ( 1 + α B ) [ α B ( 1 + α T ) − ( α M − α T ) + H ˙ H 2 ] + 2 α ˙ B H − 1 α − ρ m + P m α M 2 H 2 = 1. (30) c^2_{s,\delta\varphi}=-\frac{2(1+\alpha_B)\left[\alpha_B(1+\alpha_T)-(\alpha_M-\alpha_T)+\frac{\dot{H}}{H^2}\right]+2\dot{\alpha}_BH^{-1}}{\alpha}-\frac{\rho_m+P_m}{\alpha M^2H^2}=1 . \tag*{(30)} c s , δ φ 2 = − α 2 ( 1 + α B ) [ α B ( 1 + α T ) − ( α M − α T ) + H 2 H ˙ ] + 2 α ˙ B H − 1 − α M 2 H 2 ρ m + P m = 1. (30) Inserting the above relations into Eq. (29) yields μ Y = 1 / G 4 \mu_Y=1/G_4 μ Y = 1/ G 4 G 4 G_4 G 4 [52 ] .
Knowing μ Y \mu_Y μ Y
δ ′ ′ + ( 2 + H ′ H ) δ ′ − 4 π G G 4 ρ m 0 H 2 a 3 δ = 0. (31) \delta^{\prime\prime}+\left(2+\frac{H^{\prime}}{H}\right)\delta^{\prime}-\frac{4\pi G}{G_4}\frac{\rho_{m0}}{H^2a^3}\delta=0. \tag*{(31)} δ ′′ + ( 2 + H H ′ ) δ ′ − G 4 4 π G H 2 a 3 ρ m 0 δ = 0. (31) This simplified differential equation can be solved without assuming any specific initial condition on the scalar field. We note here that this equation can also be obtained starting from the more general results obtained in Appendix § 12 .§ 2 : in fact, combining Eq. (C.14a) with Eq. (C.14b) multiplied by G 4 ϕ / G 4 G_{4\phi}/G_4 G 4 ϕ / G 4
δ ′ ′ + ( 2 + H ′ H ) δ ′ − 4 π G G 4 ρ m 0 H 2 a 3 δ − G 4 ϕ G 4 δ ϕ ′ ′ − H ′ H G 4 ϕ G 4 δ ϕ ′ + 3 G 4 ϕ G 4 ( 1 + H ′ H ) δ ϕ + G 4 ϕ ϕ G 4 δ ϕ ( ϕ ′ ′ + H ′ H ϕ ′ ) + ( G 4 ϕ ϕ ϕ G 4 − 2 G 4 ϕ G 4 ϕ ϕ G 4 2 + G 4 ϕ 3 G 4 3 ) ( ϕ ′ ) 2 δ ϕ − 2 ( G 4 ϕ 2 G 4 2 − G 4 ϕ ϕ G 4 ) ϕ ′ δ ϕ ′ = 0. (32) \begin{aligned}
\delta^{\prime\prime}+\left(2+\frac{H^{\prime}}{H}\right)\delta^{\prime}
-\frac{4\pi G}{G_4}\frac{\rho_{m0}}{H^2a^3}\delta
-\frac{G_{4\phi}}{G_4}\delta\phi^{\prime\prime}
-\frac{H^{\prime}}{H}\frac{G_{4\phi}}{G_4}\delta\phi^{\prime}
+3\frac{G_{4\phi}}{G_4}\left(1+\frac{H^{\prime}}{H}\right)\delta\phi\\
+\frac{G_{4\phi\phi}}{G_4}\delta\phi\left(\phi^{\prime\prime}+\frac{H^{\prime}}{H}\phi^{\prime}\right)
+\left(\frac{G_{4\phi\phi\phi}}{G_4}-\frac{2G_{4\phi}G_{4\phi\phi}}{G_4^2}+\frac{G_{4\phi}^3}{G_4^3}\right)(\phi^{\prime})^2\delta\phi
-2\left(\frac{G_{4\phi}^2}{G_4^2}-\frac{G_{4\phi\phi}}{G_4}\right)\phi^{\prime}\delta\phi^{\prime}=0. \tag*{(32)}
\end{aligned} δ ′′ + ( 2 + H H ′ ) δ ′ − G 4 4 π G H 2 a 3 ρ m 0 δ − G 4 G 4 ϕ δ ϕ ′′ − H H ′ G 4 G 4 ϕ δ ϕ ′ + 3 G 4 G 4 ϕ ( 1 + H H ′ ) δ ϕ + G 4 G 4 ϕϕ δ ϕ ( ϕ ′′ + H H ′ ϕ ′ ) + ( G 4 G 4 ϕϕϕ − G 4 2 2 G 4 ϕ G 4 ϕϕ + G 4 3 G 4 ϕ 3 ) ( ϕ ′ ) 2 δ ϕ − 2 ( G 4 2 G 4 ϕ 2 − G 4 G 4 ϕϕ ) ϕ ′ δ ϕ ′ = 0. (32) However, we can neglect all the terms containing δ ϕ \delta\phi δ ϕ ∝ k 2 δ ϕ \propto k^2\delta\phi ∝ k 2 δ ϕ k δ ϕ k\delta\phi k δ ϕ δ ϕ \delta\phi δ ϕ H δ ϕ H\delta\phi H δ ϕ H 2 δ ϕ H^2\delta\phi H 2 δ ϕ (31) . The result of the integration of Eq. (31) is depicted in the first panel of Figure 1 . The other two panels depict the analytical solutions for H H H (17) , and G 4 G_4 G 4 (18) .
In Figure 1 , different curves represent different choices for the free parameters ( φ 0 , G 4 , 0 ) (\varphi_0, G_{4,0}) ( φ 0 , G 4 , 0 ) [52 ] , the parameters ρ m 0 \rho_{m0} ρ m 0 H 0 H_0 H 0 (17) ] are considered known and are taken to be equal to their current Λ \Lambda Λ ρ m 0 \rho_{m0} ρ m 0 ρ m 0 \rho_{m0} ρ m 0 Ω m 0 \Omega_{m0} Ω m 0 Ω m 0 = 0.05 \Omega_{m0}=0.05 Ω m 0 = 0.05 [8 ] . We also adopt H 0 = 67 H_0=67 H 0 = 67
On the other hand, the parameters Ω Ξ 0 \Omega_{\Xi0} Ω Ξ0 φ 0 \varphi_0 φ 0 G 4 , 0 G_{4,0} G 4 , 0 Ξ \Xi Ξ Ω 0 \Omega_0 Ω 0 Ω Ξ 0 \Omega_{\Xi0} Ω Ξ0 Ω 0 \Omega_0 Ω 0
Ω 0 > 0 , 0 < Ω Ξ 0 ≤ Ω 0 + 1 2 . (33) \Omega_0>0,\qquad0<\Omega_{\Xi0}\le\frac{\Omega_0+1}{2}. \tag*{(33)} Ω 0 > 0 , 0 < Ω Ξ0 ≤ 2 Ω 0 + 1 . (33) However, it has been shown that the values of Ω Ξ 0 \Omega_{\Xi0} Ω Ξ0 (33) provide the best-fit for measurements of the Hubble-Lemaître parameter coming from both a SNe dataset [57 ] and a cosmic chronometers dataset [52 , 75 ] . Since the value of Ω 0 \Omega_0 Ω 0 G 4 , 0 G_{4,0} G 4 , 0 G 4 , 0 G_{4,0} G 4 , 0 Ω Ξ 0 \Omega_{\Xi0} Ω Ξ0 ( Ω 0 + 1 ) / 2 (\Omega_0+1)/2 ( Ω 0 + 1 ) /2 ( φ 0 , G 4 , 0 ) (\varphi_0,G_{4,0}) ( φ 0 , G 4 , 0 ) (20) can be numerically integrated. Initial conditions on δ \delta δ δ ∝ a \delta\propto a δ ∝ a δ ( z = 1000 ) = 10 − 3 \delta(z=1000)=10^{-3} δ ( z = 1000 ) = 1 0 − 3
The second panel of Figure 1 also reports several data points for the Hubble-Lemaître parameter as a function of redshift. The smaller zoomed-in panel on the right provides a better visual inspection of the data points and their error bars. In the literature, there exist two
main data sets for H ( z ) H(z) H ( z ) [76 , 77 , 78 ] : the cosmic chronometer data set [79 , 80 ] , and the baryonic acoustic oscillations (BAO) dataset [81 ] . As already done in [52 ] , we decide to employ the values reported in [74 ] and only use data points based on cosmic chronometers. Our choice is motivated as follows: the cosmic chronometer approach, contrary to the BAO measurement, is not dependent on the choice of a specific cosmological model, because it estimates the age of a galaxy based on evolution models of stellar populations [74 , 75 , 82 ] . Estimates based on the BAO scale are computed by solving the Boltzmann equations, an operation that must necessarily assume a cosmological model (usually Λ \Lambda Λ
In Figure 1 , two families of solutions are represented: those that maximize growth (blue, orange, green and red curves) and those that successfully fit the background evolution of the Hubble-Lemaître parameter (grey, pink, brown and purple curves). One can immediately see that there is no overlap between the two families: a given choice of free parameters can either provide a substantial growth of the cosmic structure in linear perturbation theory, or it can properly describe the background expansion rate.
Solutions that provide a considerable growth (i.e. those that reach at least, δ ∼ 1 \delta\sim1 δ ∼ 1 G 4 , 0 G_{4,0} G 4 , 0 G 4 , 0 G_{4,0} G 4 , 0 δ \delta δ G 4 , 0 = 10 − 10 G_{4,0}=10^{-10} G 4 , 0 = 1 0 − 10 G 4 − 1 G_4^{-1} G 4 − 1 φ 0 \varphi_0 φ 0 φ 0 = 10 − 10 \varphi_0=10^{-10} φ 0 = 1 0 − 10
On the other hand, the solutions that can fit the H ( z ) H(z) H ( z ) G 4 , 0 G_{4,0} G 4 , 0 G 4 , 0 G_{4,0} G 4 , 0 G 4 , 0 G_{4,0} G 4 , 0 H ( z ) H(z) H ( z ) G 4 , 0 G_{4,0} G 4 , 0
Even disregarding the problem so far illustrated, the growth predicted by the family of curves that maximize δ \delta δ z < 1 z<1 z < 1 z ∼ 14 z\sim14 z ∼ 14 [83 ] . The Λ \Lambda Λ [27 , 84 ] .
We can also pinpoint precisely, in our equations, the origin of this delay in structure formation: it is due to the coefficient of the source term in Eq. (20) : δ Γ p 0 / ( G 4 H 2 a 3 ) \delta\Gamma_{\rm p0}/(G_4H^2a^3) δ Γ p0 / ( G 4 H 2 a 3 ) G 4 G_4 G 4 H − 1 H^{-1} H − 1 Figure 1 shows that the Hubble-Lemaître parameter decreases with a − 3 a^{-\sqrt{3}} a − 3 Λ \Lambda Λ
Our numerical analysis has confirmed the expectations illustrated at the end of § 4 . We have thus extended the results of our previous work [52 ] to the broader class of Horndeski theories considered in this work.
Horndeski theory in a curved universe
As a further test and possible generalization of our original Brans-Dicke approach in [52 ] , we considered the possibility that the space curvature of the universe might influence structure formation. We start from the metric for a universe with curvature K \mathcal{K} K
d s 2 = − a 2 ( 1 + 2 Ψ ) d η 2 + a 2 1 + 2 Φ ( 1 + K r 2 / 4 ) 2 δ i j d x i d x j (34) \mathrm{d}s^2=-a^2(1+2\Psi)\mathrm{d}\eta^2+a^2\frac{1+2\Phi}{(1+\mathcal{K}r^2/4)^2}\delta_{ij}\mathrm{d}x^i\mathrm{d}x^j \tag*{(34)} d s 2 = − a 2 ( 1 + 2Ψ ) d η 2 + a 2 ( 1 + K r 2 /4 ) 2 1 + 2Φ δ ij d x i d x j (34) where r r r r 2 = x i x i r^2=x^ix^i r 2 = x i x i § 13 .§ 1 , we derive the equations for the evolution of linear perturbations in a general Horndeski model, and we later specialize it to the theory defined by the choices in Eqs. (11) . This calculation yields the following differential equation, that describes the evolution of the growth factor in a curved universe:
δ ′ ′ + ( 2 + H ′ H ) δ ′ − 8 π G G 4 ρ m 0 H 2 a 3 δ − ∇ 2 δ φ γ 2 a 2 H 2 G 4 φ G 4 + K 2 γ x j ∂ j δ φ a 2 H 2 G 4 φ G 4 + 2 Ξ H 2 δ φ G 4 φ G 4 = − 6 ( 1 + H ′ H ) Ψ − 3 Ψ ′ + 3 G 4 φ G 4 ( 1 + H ′ H ) δ φ + G 4 φ G 4 ( 6 Ψ ′ φ ′ − 3 Φ ′ φ ′ − 3 δ φ ′ ) − 3 G 4 φ φ G 4 φ ′ δ φ − ( 2 G 4 φ G 4 φ φ G 4 2 − G 4 φ 3 G 4 3 ) ( φ ′ ) 2 δ φ − 2 ( G 4 φ 2 G 4 2 − G 4 φ φ G 4 ) φ ′ δ φ ′ . (35) \begin{aligned}
\delta''+\left(2+\frac{H'}{H}\right)\delta'-\frac{8\pi G}{G_4}\frac{\rho_{\mathrm{m}0}}{H^2a^3}\delta-\frac{\nabla^2\delta\varphi}{\gamma^2a^2H^2}\frac{G_{4\varphi}}{G_4}+\frac{\mathcal{K}}{2\gamma}\frac{x^j\partial_j\delta\varphi}{a^2H^2}\frac{G_{4\varphi}}{G_4}+2\frac{\Xi}{H^2}\delta\varphi\frac{G_{4\varphi}}{G_4}
&=-6\left(1+\frac{H'}{H}\right)\Psi-3\Psi'+3\frac{G_{4\varphi}}{G_4}\left(1+\frac{H'}{H}\right)\delta\varphi+\frac{G_{4\varphi}}{G_4}\left(6\Psi'\varphi'-3\Phi'\varphi'-3\delta\varphi'\right)\\
&\quad-3\frac{G_{4\varphi\varphi}}{G_4}\varphi'\delta\varphi-\left(\frac{2G_{4\varphi}G_{4\varphi\varphi}}{G_4^2}-\frac{G_{4\varphi}^3}{G_4^3}\right)(\varphi')^2\delta\varphi-2\left(\frac{G_{4\varphi}^2}{G_4^2}-\frac{G_{4\varphi\varphi}}{G_4}\right)\varphi'\delta\varphi'. \tag*{(35)}
\end{aligned} δ ′′ + ( 2 + H H ′ ) δ ′ − G 4 8 π G H 2 a 3 ρ m 0 δ − γ 2 a 2 H 2 ∇ 2 δ φ G 4 G 4 φ + 2 γ K a 2 H 2 x j ∂ j δ φ G 4 G 4 φ + 2 H 2 Ξ δ φ G 4 G 4 φ = − 6 ( 1 + H H ′ ) Ψ − 3 Ψ ′ + 3 G 4 G 4 φ ( 1 + H H ′ ) δ φ + G 4 G 4 φ ( 6 Ψ ′ φ ′ − 3 Φ ′ φ ′ − 3 δ φ ′ ) − 3 G 4 G 4 φφ φ ′ δ φ − ( G 4 2 2 G 4 φ G 4 φφ − G 4 3 G 4 φ 3 ) ( φ ′ ) 2 δ φ − 2 ( G 4 2 G 4 φ 2 − G 4 G 4 φφ ) φ ′ δ φ ′ . (35) This equation is identical to its counterpart, Eq. (20) , in a flat universe K = 0 \mathcal{K}=0 K = 0 (35) , with its curved equivalent, ∇ 2 → ∇ 2 / γ 2 − ( K / 2 γ ) x j ∂ j \nabla^2\rightarrow\nabla^2/\gamma^2-(\mathcal{K}/2\gamma)x^j\partial_j ∇ 2 → ∇ 2 / γ 2 − ( K /2 γ ) x j ∂ j
We can exclude the possibility that such a small modification of the equation could lead to a significant change in the behavior of the growth factor. In fact, as it was shown in [52 ] , the terms involving the perturbations of the scalar field turn out to be negligible when compared with the other terms of Eq. (35) . Moreover, the new term containing the curvature, ( K x j ∂ j δ φ G 4 φ ) / ( 2 γ a 2 H 2 G 4 ) (\mathcal{K}x^j\partial_j\delta\varphi G_{4\varphi})/(2\gamma a^2H^2G_4) ( K x j ∂ j δ φ G 4 φ ) / ( 2 γ a 2 H 2 G 4 ) ∝ k 2 \propto k^2 ∝ k 2 k k k ∝ k \propto k ∝ k
Another way to see that no significant deviation is expected is to proceed as in § 5 : since the specific Horndeski model is not modified by the presence of curvature, we obtain again Eq. (31) under the QSA assumption. The difference in the Laplacian operator is neglected because the QSA is only valid on sub-horizon scales, as discussed above.
The study of QSA can also be performed for a general Horndeski theory with spatial curvature using the field equations presented in [85 ] . Taking their Eqs. (8)–(12) written in configuration space, to apply the QSA we move to the Fourier space. However, it is not possible to perform a simple Fourier transform, because the covariant derivative D 2 D^2 D 2 − k 2 -k^2 − k 2 χ ( x ⃗ ) = ∑ X χ q Q l m n \chi(\vec{x}) = \sum X_\chi q Q_{lmn} χ ( x ) = ∑ X χ q Q l mn D 2 Q l m n = − ( q 2 − 3 K ) Q l m n D^2 Q_{lmn} = -(q^2 - 3\mathcal{K})Q_{lmn} D 2 Q l mn = − ( q 2 − 3 K ) Q l mn § 13 .§ 2 for details). Thus, terms like D 2 + 3 K D^2 + 3\mathcal{K} D 2 + 3 K − q 2 -q^2 − q 2 K ≡ k / a H \mathcal{K} \equiv k/aH K ≡ k / a H Q = q / a H Q = q/aH Q = q / a H K \mathcal{K} K ( i j ) (ij) ( ij ) π \pi π § 13 .§ 2 . The application of QSA implies that Q ≫ 1 Q \gg1 Q ≫ 1 K \mathcal{K} K M p l = ( 8 π G ) − 1 / 2 M_{\mathrm{pl}} = (8\pi G)^{-1/2} M pl = ( 8 π G ) − 1/2
H 2 + K a 2 = 1 3 M p l 2 ( ρ m + ρ d e ) , (36) H^2 + \frac{\mathcal{K}}{a^2} = \frac{1}{3M_{\mathrm{pl}}^2}(\rho_{\mathrm{m}} + \rho_{\mathrm{de}}), \tag*{(36)} H 2 + a 2 K = 3 M pl 2 1 ( ρ m + ρ de ) , (36) this quantity is of order H 2 H^2 H 2 [73 ] , with the replacement K 2 → Q 2 \mathcal{K}^2 \to Q^2 K 2 → Q 2
Conclusions
In this work we generalized our previous study of a Brans–Dicke theory of gravity without dark matter [52 ] to the full class of Horndeski theories, reducing to Refracted Gravity (RG) in the weak-field limit. Our goal was determining whether a broader scalar-tensor framework could succeed where the specific Brans–Dicke realization had failed in reproducing the observed growth of cosmic structure. We began by deriving, in § 2 and Appendix A, the general conditions on the Horndeski functions G 2 G_2 G 2 G 3 G_3 G 3 G 4 G_4 G 4 (11) , in which the function G 4 ( ϕ ) G_4(\phi) G 4 ( ϕ ) ϵ ( ϕ ) \epsilon(\phi) ϵ ( ϕ )
Studying the background evolution of a flat, matter-dominated universe in gCRG (§ 3 and Appendix § 11 ), we showed that the boost factor G 4 [ ϕ ( a ) ] G_4[\phi(a)] G 4 [ ϕ ( a )] H ( a ) H(a) H ( a ) [52 ] . This result is a direct consequence of the structural similarity imposed by the RG-matching conditions, and it implies that any success or failure of the background cosmology found previously carries over to the more general Horndeski setting.
The central contribution of this work is the derivation of the linear perturbation theory and the growth factor equation for gCRG, both in a flat universe (§ 4 and Appendix § 12 .§ 1 ) and in a universe with non-zero spatial curvature (§ 6 and Appendix § 13 .§ 1 ). We showed that curvature effects introduce no qualitatively new behavior in the regime relevant for structure formation, since the additional curvature terms in the perturbed field equations scale linearly with the Fourier wavenumber k k k k 2 k^2 k 2 ϕ 0 \phi_0 ϕ 0 G 4 , 0 G_{4,0} G 4 , 0 Ω Σ 0 \Omega_{\Sigma0} Ω Σ0 § 5 ). Our numerical results, summarized in Figure 1 , show a clear tension within the model: parameter choices that correctly reproduce the observed evolution of H ( z ) H(z) H ( z ) G 4 , 0 G_{4,0} G 4 , 0 δ \delta δ G 4 , 0 ≪ 1 G_{4,0} \ll1 G 4 , 0 ≪ 1 H ( z ) H(z) H ( z ) z < 1 z < 1 z < 1 z ∼ 14 z \sim14 z ∼ 14 [83 ] .
We traced the reason of this behavior directly to the structure of the source term of the growth factor equation, Eq. (20) , where the standard 4 π G ρ m 4\pi G\rho_m 4 π G ρ m G 4 G_4 G 4 G 4 ( a ) G_4(a) G 4 ( a ) [52 ] — the source term is strongly suppressed at high redshift, delaying the onset of gravitational instability well past the epoch required by observations. Since this feature is not tied to any particular choice of the potential or kinetic function, but it follows generically from the conditions in Eqs. (8) that any Horndeski theory must satisfy in order to reduce to RG in the weak-field limit, we conclude that the delayed structure formation obtained in our earlier Brans–Dicke study is not an artifact of that specific model. Rather, it appears to be a structural obstruction shared by the entire class of Horndeski theories capable of reproducing RG phenomenology on galactic scales.
This conclusion suggests that reconciling a scalar-tensor “gravity-only” explanation of both dark matter and dark energy phenomenology with the observed timeline of structure formation might require relaxing one of the assumptions underlying the weak-field correspondence with RG derived here. We leave the exploration of this possibility to future work.
Acknowledgments
We sincerely thank Lorenzo Fatibene for enlightening discussions. We acknowledge partial support from the grant InDark of the Italian National Institute of Nuclear Physics (INFN). This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
Weak field limit of a Horndeski theory
In this appendix, we derive the conditions, reported in Eq. (8) , under which a generic Horndeski theory reduces to RG in the weak-field limit. We start by deriving the expression for the Ricci scalar: from Eq. (5) , we see that
R = 1 G 4 φ { ∇ μ [ ( 2 G 2 X − 2 G 3 X □ φ − 2 G 3 φ ) ∇ μ φ + 2 G 3 X ∇ μ X ] − G 2 φ + ∇ λ G 3 φ ∇ λ φ } = 2 G 2 X Δ φ G 4 φ − 4 G 2 X Δ φ Φ G 4 φ − 2 G 3 X ( Δ φ ) 2 G 4 φ + 8 G 3 X ( Δ φ ) 2 Φ G 4 φ + 2 G 3 X Δ Φ ( ∇ φ ) 2 G 4 φ − G 2 φ G 4 φ \begin{aligned}
R = \frac{1}{G_{4\varphi}}\left\{\nabla^\mu\left[(2G_{2X}-2G_{3X}\Box\varphi-2G_{3\varphi})\nabla_\mu\varphi+2G_{3X}\nabla_\mu X\right]-G_{2\varphi}+\nabla_\lambda G_{3\varphi}\nabla^\lambda\varphi\right\} \\
&= \frac{2G_{2X}\Delta\varphi}{G_{4\varphi}}-\frac{4G_{2X}\Delta\varphi\Phi}{G_{4\varphi}}-\frac{2G_{3X}(\Delta\varphi)^2}{G_{4\varphi}}+\frac{8G_{3X}(\Delta\varphi)^2\Phi}{G_{4\varphi}}+\frac{2G_{3X}\Delta\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{G_{2\varphi}}{G_{4\varphi}}
\end{aligned} R = G 4 φ 1 { ∇ μ [ ( 2 G 2 X − 2 G 3 X □ φ − 2 G 3 φ ) ∇ μ φ + 2 G 3 X ∇ μ X ] − G 2 φ + ∇ λ G 3 φ ∇ λ φ } = G 4 φ 2 G 2 X Δ φ − G 4 φ 4 G 2 X Δ φ Φ − G 4 φ 2 G 3 X ( Δ φ ) 2 + G 4 φ 8 G 3 X ( Δ φ ) 2 Φ + G 4 φ 2 G 3 X ΔΦ ( ∇ φ ) 2 − G 4 φ G 2 φ − G 3 φ Δ φ G 4 φ + 2 G 3 φ Δ φ Φ G 4 φ + 2 G 2 φ X ( ∇ φ ) 2 G 4 φ − 4 G 2 φ X Φ ( ∇ φ ) 2 G 4 φ − 2 G 3 φ X Δ φ ( ∇ φ ) 2 G 4 φ + 8 G 3 φ X Δ φ Φ ( ∇ φ ) 2 G 4 φ − G 3 φ φ ( ∇ φ ) 2 G 4 φ + 2 G 3 φ φ Φ ( ∇ φ ) 2 G 4 φ − 2 G 3 X φ i j φ i j G 4 φ + 8 G 3 X Φ φ i j φ i j G 4 φ \begin{aligned}
-\frac{G_{3\varphi}\Delta\varphi}{G_{4\varphi}}+\frac{2G_{3\varphi}\Delta\varphi\Phi}{G_{4\varphi}}+\frac{2G_{2\varphi X}(\nabla\varphi)^2}{G_{4\varphi}}-\frac{4G_{2\varphi X}\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{2G_{3\varphi X}\Delta\varphi(\nabla\varphi)^2}{G_{4\varphi}} \\
&+\frac{8G_{3\varphi X}\Delta\varphi\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{G_{3\varphi\varphi}(\nabla\varphi)^2}{G_{4\varphi}}+\frac{2G_{3\varphi\varphi}\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{2G_{3X}\varphi_{ij}\varphi_{ij}}{G_{4\varphi}}+\frac{8G_{3X}\Phi\varphi_{ij}\varphi_{ij}}{G_{4\varphi}}
\end{aligned} − G 4 φ G 3 φ Δ φ + G 4 φ 2 G 3 φ Δ φ Φ + G 4 φ 2 G 2 φX ( ∇ φ ) 2 − G 4 φ 4 G 2 φX Φ ( ∇ φ ) 2 − G 4 φ 2 G 3 φX Δ φ ( ∇ φ ) 2 + G 4 φ 8 G 3 φX Δ φ Φ ( ∇ φ ) 2 − G 4 φ G 3 φφ ( ∇ φ ) 2 + G 4 φ 2 G 3 φφ Φ ( ∇ φ ) 2 − G 4 φ 2 G 3 X φ ij φ ij + G 4 φ 8 G 3 X Φ φ ij φ ij − 4 G 3 X φ i i j φ j G 4 φ + 16 G 3 X Φ φ i i j φ j G 4 φ + 4 G 2 X X φ i φ i j φ j G 4 φ − 16 G 2 X X Φ φ i φ i j φ j G 4 φ -\frac{4G_{3X}\varphi_{iij}\varphi_j}{G_{4\varphi}}+\frac{16G_{3X}\Phi\varphi_{iij}\varphi_j}{G_{4\varphi}}+\frac{4G_{2XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{16G_{2XX}\Phi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}} − G 4 φ 4 G 3 X φ iij φ j + G 4 φ 16 G 3 X Φ φ iij φ j + G 4 φ 4 G 2 X X φ i φ ij φ j − G 4 φ 16 G 2 X X Φ φ i φ ij φ j − 4 G 3 X X Δ φ φ i φ i j φ j G 4 φ + 24 G 3 X X Δ φ Φ φ i φ i j φ j G 4 φ − 4 G 3 φ X φ i φ i j φ j G 4 φ + 16 G 3 φ X Φ φ i φ i j φ j G 4 φ -\frac{4G_{3XX}\Delta\varphi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}+\frac{24G_{3XX}\Delta\varphi\Phi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{4G_{3\varphi X}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}+\frac{16G_{3\varphi X}\Phi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}} − G 4 φ 4 G 3 X X Δ φ φ i φ ij φ j + G 4 φ 24 G 3 X X Δ φ Φ φ i φ ij φ j − G 4 φ 4 G 3 φX φ i φ ij φ j + G 4 φ 16 G 3 φX Φ φ i φ ij φ j − 4 G 3 X X φ i j φ i k φ j φ k G 4 φ + 24 G 3 X X Φ φ i j φ i k φ j φ k G 4 φ − G 3 Δ φ Ψ i G 4 φ φ i − G 3 φ ( ∇ φ ) 2 Ψ i G 4 φ φ i + 2 G 2 X φ i Ψ i G 4 φ − 4 G 3 X Δ φ φ i Ψ i G 4 φ − G 3 φ φ φ i Ψ i G 4 φ − 2 G 3 φ X ( ∇ φ ) 2 φ i Ψ i G 4 φ − 6 G 3 X φ i j φ j Ψ i G 4 φ \begin{aligned}
-\frac{4G_{3XX}\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}+\frac{24G_{3XX}\Phi\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}-\frac{G_3\Delta\varphi\Psi_i}{G_{4\varphi}\varphi_i}-\frac{G_{3\varphi}(\nabla\varphi)^2\Psi_i}{G_{4\varphi}\varphi_i} \\
&+\frac{2G_{2X}\varphi_i\Psi_i}{G_{4\varphi}}-\frac{4G_{3X}\Delta\varphi\varphi_i\Psi_i}{G_{4\varphi}}-\frac{G_{3\varphi\varphi}\varphi_i\Psi_i}{G_{4\varphi}}-\frac{2G_{3\varphi X}(\nabla\varphi)^2\varphi_i\Psi_i}{G_{4\varphi}}-\frac{6G_{3X}\varphi_{ij}\varphi_j\Psi_i}{G_{4\varphi}}
\end{aligned} − G 4 φ 4 G 3 X X φ ij φ ik φ j φ k + G 4 φ 24 G 3 X X Φ φ ij φ ik φ j φ k − G 4 φ φ i G 3 Δ φ Ψ i − G 4 φ φ i G 3 φ ( ∇ φ ) 2 Ψ i + G 4 φ 2 G 2 X φ i Ψ i − G 4 φ 4 G 3 X Δ φ φ i Ψ i − G 4 φ G 3 φφ φ i Ψ i − G 4 φ 2 G 3 φX ( ∇ φ ) 2 φ i Ψ i − G 4 φ 6 G 3 X φ ij φ j Ψ i − 4 G 3 X X φ i φ j φ k φ k j Ψ i G 4 φ + G 3 φ Ψ i i j G 4 φ φ i 2 − G 3 φ Ψ i j G 4 φ φ i − 2 G 3 X φ i φ j Ψ i j G 4 φ -\frac{4G_{3XX}\varphi_i\varphi_j\varphi_k\varphi_{kj}\Psi_i}{G_{4\varphi}}+\frac{G_{3\varphi}\Psi_{iij}}{G_{4\varphi}\varphi_i^2}-\frac{G_{3\varphi}\Psi_{ij}}{G_{4\varphi}\varphi_i}-\frac{2G_{3X}\varphi_i\varphi_j\Psi_{ij}}{G_{4\varphi}} − G 4 φ 4 G 3 X X φ i φ j φ k φ k j Ψ i + G 4 φ φ i 2 G 3 φ Ψ iij − G 4 φ φ i G 3 φ Ψ ij − G 4 φ 2 G 3 X φ i φ j Ψ ij − G 3 Δ φ Φ i G 4 φ φ i − G 3 φ ( ∇ φ ) 2 Φ i G 4 φ φ i + 2 G 2 X φ i Φ i G 4 φ − 4 G 2 X X ( ∇ φ ) 2 φ i Φ i G 4 φ + 4 G 3 X X Δ φ ( ∇ φ ) 2 φ i Φ i G 4 φ -\frac{G_3\Delta\varphi\Phi_i}{G_{4\varphi}\varphi_i}-\frac{G_{3\varphi}(\nabla\varphi)^2\Phi_i}{G_{4\varphi}\varphi_i}+\frac{2G_{2X}\varphi_i\Phi_i}{G_{4\varphi}}-\frac{4G_{2XX}(\nabla\varphi)^2\varphi_i\Phi_i}{G_{4\varphi}}+\frac{4G_{3XX}\Delta\varphi(\nabla\varphi)^2\varphi_i\Phi_i}{G_{4\varphi}} − G 4 φ φ i G 3 Δ φ Φ i − G 4 φ φ i G 3 φ ( ∇ φ ) 2 Φ i + G 4 φ 2 G 2 X φ i Φ i − G 4 φ 4 G 2 X X ( ∇ φ ) 2 φ i Φ i + G 4 φ 4 G 3 X X Δ φ ( ∇ φ ) 2 φ i Φ i − G 3 φ X φ i Φ i G 4 φ + 2 G 3 φ X ( ∇ φ ) 2 φ i Φ i G 4 φ + 8 G 3 X X ( ∇ φ ) 2 φ i φ i j Φ i G 4 φ + 2 G 3 X φ i j φ j Φ i G 4 φ -\frac{G_{3\varphi X}\varphi_i\Phi_i}{G_{4\varphi}}+\frac{2G_{3\varphi X}(\nabla\varphi)^2\varphi_i\Phi_i}{G_{4\varphi}}+\frac{8G_{3XX}(\nabla\varphi)^2\varphi_i\varphi_{ij}\Phi_i}{G_{4\varphi}}+\frac{2G_{3X}\varphi_{ij}\varphi_j\Phi_i}{G_{4\varphi}} − G 4 φ G 3 φX φ i Φ i + G 4 φ 2 G 3 φX ( ∇ φ ) 2 φ i Φ i + G 4 φ 8 G 3 X X ( ∇ φ ) 2 φ i φ ij Φ i + G 4 φ 2 G 3 X φ ij φ j Φ i − 4 G 3 X X φ i φ j φ k φ k j Φ i G 4 φ + G 3 φ Φ i i j G 4 φ φ i 2 − G 3 φ Φ i j G 4 φ φ i − 2 G 3 X φ i φ j Φ i j G 4 φ . (37) -\frac{4G_{3XX}\varphi_i\varphi_j\varphi_k\varphi_{kj}\Phi_i}{G_{4\varphi}}+\frac{G_{3\varphi}\Phi_{iij}}{G_{4\varphi}\varphi_i^2}-\frac{G_{3\varphi}\Phi_{ij}}{G_{4\varphi}\varphi_i}-\frac{2G_{3X}\varphi_i\varphi_j\Phi_{ij}}{G_{4\varphi}}. \tag*{(37)} − G 4 φ 4 G 3 X X φ i φ j φ k φ k j Φ i + G 4 φ φ i 2 G 3 φ Φ iij − G 4 φ φ i G 3 φ Φ ij − G 4 φ 2 G 3 X φ i φ j Φ ij . (37) In this expression a latin subscript indicates partial differentiation with respect to a Cartesian spatial direction, while Δ = ∂ i ∂ i \Delta=\partial_i\partial_i Δ = ∂ i ∂ i ( ∇ φ ) 2 = ∂ i φ ∂ i φ (\nabla\varphi)^2=\partial_i\varphi\partial_i\varphi ( ∇ φ ) 2 = ∂ i φ ∂ i φ Φ \Phi Φ Ψ \Psi Ψ (4) ), one gets, again to first order in Φ \Phi Φ Ψ \Psi Ψ
∇ 2 Ψ G 4 + ∇ 2 Φ ( ∇ φ ) 2 G 4 G 3 X G 4 φ − Φ [ − ( ∇ φ ) 2 G 3 φ + 2 ∇ 2 φ G 4 G 2 X G 4 φ − 4 ( ∇ 2 φ ) 2 G 4 G 3 X G 4 φ \nabla^2\Psi G_4+\frac{\nabla^2\Phi(\nabla\varphi)^2G_4G_{3X}}{G_{4\varphi}}-\Phi[-(\nabla\varphi)^2G_{3\varphi}+\frac{2\nabla^2\varphi G_4G_{2X}}{G_{4\varphi}}-\frac{4(\nabla^2\varphi)^2G_4G_{3X}}{G_{4\varphi}} ∇ 2 Ψ G 4 + G 4 φ ∇ 2 Φ ( ∇ φ ) 2 G 4 G 3 X − Φ [ − ( ∇ φ ) 2 G 3 φ + G 4 φ 2 ∇ 2 φ G 4 G 2 X − G 4 φ 4 ( ∇ 2 φ ) 2 G 4 G 3 X − ∇ 2 φ G 4 G 3 φ G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 4 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φ X G 4 φ − ( ∇ φ ) 2 G 4 G 3 φ φ G 4 φ -\frac{\nabla^2\varphi G_4G_{3\varphi}}{G_{4\varphi}}-2\nabla^2\varphi G_{4\varphi}+\frac{2(\nabla\varphi)^2G_4G_{2\varphi X}}{G_{4\varphi}}-\frac{4\nabla^2\varphi(\nabla\varphi)^2G_4G_{3\varphi X}}{G_{4\varphi}}-\frac{(\nabla\varphi)^2G_4G_{3\varphi\varphi}}{G_{4\varphi}} − G 4 φ ∇ 2 φ G 4 G 3 φ − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − G 4 φ 4 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φX − G 4 φ ( ∇ φ ) 2 G 4 G 3 φφ − 2 ( ∇ φ ) 2 G 4 φ φ − 4 G 4 G 3 X φ i j 2 G 4 φ − 8 G 4 G 3 X φ i i j φ j G 4 φ − 4 G 3 X φ i φ i j φ j + 8 G 4 G 2 X X φ i φ i j φ j G 4 φ -2(\nabla\varphi)^2G_{4\varphi\varphi}-\frac{4G_4G_{3X}\varphi_{ij}^2}{G_{4\varphi}}-\frac{8G_4G_{3X}\varphi_{iij}\varphi_j}{G_{4\varphi}}-4G_{3X}\varphi_i\varphi_{ij}\varphi_j+\frac{8G_4G_{2XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}} − 2 ( ∇ φ ) 2 G 4 φφ − G 4 φ 4 G 4 G 3 X φ ij 2 − G 4 φ 8 G 4 G 3 X φ iij φ j − 4 G 3 X φ i φ ij φ j + G 4 φ 8 G 4 G 2 X X φ i φ ij φ j − 12 ∇ 2 φ G 4 G 3 X X φ i φ i j φ j G 4 φ − 8 G 4 G 3 φ X φ i φ i j φ j G 4 φ − 12 G 4 G 3 X X φ i j φ i k φ j φ k G 4 φ ] -\frac{12\nabla^2\varphi G_4G_{3XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{8G_4G_{3\varphi X}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{12G_4G_{3XX}\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}] − G 4 φ 12 ∇ 2 φ G 4 G 3 X X φ i φ ij φ j − G 4 φ 8 G 4 G 3 φX φ i φ ij φ j − G 4 φ 12 G 4 G 3 X X φ ij φ ik φ j φ k ] @page 17/35 starts_mid=0 ends_mid=1
Weak field limit of a Horndeski theory
In this appendix, we derive the conditions, reported in Eq. (8) , under which a generic Horndeski theory reduces to RG in the weak-field limit. We start by deriving the expression for the Ricci scalar: from Eq. (5) , we see that
R = 1 G 4 φ { ∇ μ [ ( 2 G 2 X − 2 G 3 X □ φ − 2 G 3 φ ) ∇ μ φ + 2 G 3 X ∇ μ X ] − G 2 φ + ∇ λ G 3 φ ∇ λ φ } = 2 G 2 X Δ φ G 4 φ − 4 G 2 X Δ φ Φ G 4 φ − 2 G 3 X ( Δ φ ) 2 G 4 φ + 8 G 3 X ( Δ φ ) 2 Φ G 4 φ + 2 G 3 X Δ Φ ( ∇ φ ) 2 G 4 φ − G 2 φ G 4 φ \begin{aligned}
R = \frac{1}{G_{4\varphi}}\left\{\nabla^\mu\left[(2G_{2X}-2G_{3X}\Box\varphi-2G_{3\varphi})\nabla_\mu\varphi+2G_{3X}\nabla_\mu X\right]-G_{2\varphi}+\nabla_\lambda G_{3\varphi}\nabla^\lambda\varphi\right\} \\
&= \frac{2G_{2X}\Delta\varphi}{G_{4\varphi}}-\frac{4G_{2X}\Delta\varphi\Phi}{G_{4\varphi}}-\frac{2G_{3X}(\Delta\varphi)^2}{G_{4\varphi}}+\frac{8G_{3X}(\Delta\varphi)^2\Phi}{G_{4\varphi}}+\frac{2G_{3X}\Delta\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{G_{2\varphi}}{G_{4\varphi}}
\end{aligned} R = G 4 φ 1 { ∇ μ [ ( 2 G 2 X − 2 G 3 X □ φ − 2 G 3 φ ) ∇ μ φ + 2 G 3 X ∇ μ X ] − G 2 φ + ∇ λ G 3 φ ∇ λ φ } = G 4 φ 2 G 2 X Δ φ − G 4 φ 4 G 2 X Δ φ Φ − G 4 φ 2 G 3 X ( Δ φ ) 2 + G 4 φ 8 G 3 X ( Δ φ ) 2 Φ + G 4 φ 2 G 3 X ΔΦ ( ∇ φ ) 2 − G 4 φ G 2 φ − G 3 φ Δ φ G 4 φ + 2 G 3 φ Δ φ Φ G 4 φ + 2 G 2 φ X ( ∇ φ ) 2 G 4 φ − 4 G 2 φ X Φ ( ∇ φ ) 2 G 4 φ − 2 G 3 φ X Δ φ ( ∇ φ ) 2 G 4 φ + 8 G 3 φ X Δ φ Φ ( ∇ φ ) 2 G 4 φ − G 3 φ φ ( ∇ φ ) 2 G 4 φ + 2 G 3 φ φ Φ ( ∇ φ ) 2 G 4 φ − 2 G 3 X φ i j φ i j G 4 φ + 8 G 3 X Φ φ i j φ i j G 4 φ \begin{aligned}
-\frac{G_{3\varphi}\Delta\varphi}{G_{4\varphi}}+\frac{2G_{3\varphi}\Delta\varphi\Phi}{G_{4\varphi}}+\frac{2G_{2\varphi X}(\nabla\varphi)^2}{G_{4\varphi}}-\frac{4G_{2\varphi X}\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{2G_{3\varphi X}\Delta\varphi(\nabla\varphi)^2}{G_{4\varphi}} \\
&+\frac{8G_{3\varphi X}\Delta\varphi\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{G_{3\varphi\varphi}(\nabla\varphi)^2}{G_{4\varphi}}+\frac{2G_{3\varphi\varphi}\Phi(\nabla\varphi)^2}{G_{4\varphi}}-\frac{2G_{3X}\varphi_{ij}\varphi_{ij}}{G_{4\varphi}}+\frac{8G_{3X}\Phi\varphi_{ij}\varphi_{ij}}{G_{4\varphi}}
\end{aligned} − G 4 φ G 3 φ Δ φ + G 4 φ 2 G 3 φ Δ φ Φ + G 4 φ 2 G 2 φX ( ∇ φ ) 2 − G 4 φ 4 G 2 φX Φ ( ∇ φ ) 2 − G 4 φ 2 G 3 φX Δ φ ( ∇ φ ) 2 + G 4 φ 8 G 3 φX Δ φ Φ ( ∇ φ ) 2 − G 4 φ G 3 φφ ( ∇ φ ) 2 + G 4 φ 2 G 3 φφ Φ ( ∇ φ ) 2 − G 4 φ 2 G 3 X φ ij φ ij + G 4 φ 8 G 3 X Φ φ ij φ ij − 4 G 3 X φ i i j φ j G 4 φ + 16 G 3 X Φ φ i i j φ j G 4 φ + 4 G 2 X X φ i φ i j φ j G 4 φ − 16 G 2 X X Φ φ i φ i j φ j G 4 φ -\frac{4G_{3X}\varphi_{iij}\varphi_j}{G_{4\varphi}}+\frac{16G_{3X}\Phi\varphi_{iij}\varphi_j}{G_{4\varphi}}+\frac{4G_{2XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{16G_{2XX}\Phi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}} − G 4 φ 4 G 3 X φ iij φ j + G 4 φ 16 G 3 X Φ φ iij φ j + G 4 φ 4 G 2 X X φ i φ ij φ j − G 4 φ 16 G 2 X X Φ φ i φ ij φ j − 4 G 3 X X Δ φ φ i φ i j φ j G 4 φ + 24 G 3 X X Δ φ Φ φ i φ i j φ j G 4 φ − 4 G 3 φ X φ i φ i j φ j G 4 φ + 16 G 3 φ X Φ φ i φ i j φ j G 4 φ -\frac{4G_{3XX}\Delta\varphi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}+\frac{24G_{3XX}\Delta\varphi\Phi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{4G_{3\varphi X}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}+\frac{16G_{3\varphi X}\Phi\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}} − G 4 φ 4 G 3 X X Δ φ φ i φ ij φ j + G 4 φ 24 G 3 X X Δ φ Φ φ i φ ij φ j − G 4 φ 4 G 3 φX φ i φ ij φ j + G 4 φ 16 G 3 φX Φ φ i φ ij φ j − 4 G 3 X X φ i j φ i k φ j φ k G 4 φ + 24 G 3 X X Φ φ i j φ i k φ j φ k G 4 φ − G 3 Δ φ Ψ i G 4 φ φ i − G 3 φ ( ∇ φ ) 2 Ψ i G 4 φ φ i + 2 G 2 X φ i Ψ i G 4 φ − 4 G 3 X Δ φ φ i Ψ i G 4 φ − G 3 φ φ φ i Ψ i G 4 φ − 2 G 3 φ X ( ∇ φ ) 2 φ i Ψ i G 4 φ − 6 G 3 X φ i j φ j Ψ i G 4 φ \begin{aligned}
-\frac{4G_{3XX}\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}+\frac{24G_{3XX}\Phi\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}-\frac{G_3\Delta\varphi\Psi_i}{G_{4\varphi}\varphi_i}-\frac{G_{3\varphi}(\nabla\varphi)^2\Psi_i}{G_{4\varphi}\varphi_i} \\
&+\frac{2G_{2X}\varphi_i\Psi_i}{G_{4\varphi}}-\frac{4G_{3X}\Delta\varphi\varphi_i\Psi_i}{G_{4\varphi}}-\frac{G_{3\varphi\varphi}\varphi_i\Psi_i}{G_{4\varphi}}-\frac{2G_{3\varphi X}(\nabla\varphi)^2\varphi_i\Psi_i}{G_{4\varphi}}-\frac{6G_{3X}\varphi_{ij}\varphi_j\Psi_i}{G_{4\varphi}}
\end{aligned} − G 4 φ 4 G 3 X X φ ij φ ik φ j φ k + G 4 φ 24 G 3 X X Φ φ ij φ ik φ j φ k − G 4 φ φ i G 3 Δ φ Ψ i − G 4 φ φ i G 3 φ ( ∇ φ ) 2 Ψ i + G 4 φ 2 G 2 X φ i Ψ i − G 4 φ 4 G 3 X Δ φ φ i Ψ i − G 4 φ G 3 φφ φ i Ψ i − G 4 φ 2 G 3 φX ( ∇ φ ) 2 φ i Ψ i − G 4 φ 6 G 3 X φ ij φ j Ψ i − 4 G 3 X X φ i φ j φ k φ k j Ψ i G 4 φ + G 3 φ Ψ i i j G 4 φ φ i 2 − G 3 φ Ψ i j G 4 φ φ i − 2 G 3 X φ i φ j Ψ i j G 4 φ -\frac{4G_{3XX}\varphi_i\varphi_j\varphi_k\varphi_{kj}\Psi_i}{G_{4\varphi}}+\frac{G_{3\varphi}\Psi_{iij}}{G_{4\varphi}\varphi_i^2}-\frac{G_{3\varphi}\Psi_{ij}}{G_{4\varphi}\varphi_i}-\frac{2G_{3X}\varphi_i\varphi_j\Psi_{ij}}{G_{4\varphi}} − G 4 φ 4 G 3 X X φ i φ j φ k φ k j Ψ i + G 4 φ φ i 2 G 3 φ Ψ iij − G 4 φ φ i G 3 φ Ψ ij − G 4 φ 2 G 3 X φ i φ j Ψ ij − G 3 Δ φ Φ i G 4 φ φ i − G 3 φ ( ∇ φ ) 2 Φ i G 4 φ φ i + 2 G 2 X φ i Φ i G 4 φ − 4 G 2 X X ( ∇ φ ) 2 φ i Φ i G 4 φ + 4 G 3 X X Δ φ ( ∇ φ ) 2 φ i Φ i G 4 φ -\frac{G_3\Delta\varphi\Phi_i}{G_{4\varphi}\varphi_i}-\frac{G_{3\varphi}(\nabla\varphi)^2\Phi_i}{G_{4\varphi}\varphi_i}+\frac{2G_{2X}\varphi_i\Phi_i}{G_{4\varphi}}-\frac{4G_{2XX}(\nabla\varphi)^2\varphi_i\Phi_i}{G_{4\varphi}}+\frac{4G_{3XX}\Delta\varphi(\nabla\varphi)^2\varphi_i\Phi_i}{G_{4\varphi}} − G 4 φ φ i G 3 Δ φ Φ i − G 4 φ φ i G 3 φ ( ∇ φ ) 2 Φ i + G 4 φ 2 G 2 X φ i Φ i − G 4 φ 4 G 2 X X ( ∇ φ ) 2 φ i Φ i + G 4 φ 4 G 3 X X Δ φ ( ∇ φ ) 2 φ i Φ i − G 3 φ X φ i Φ i G 4 φ + 2 G 3 φ X ( ∇ φ ) 2 φ i Φ i G 4 φ + 8 G 3 X X ( ∇ φ ) 2 φ i φ i j Φ i G 4 φ + 2 G 3 X φ i j φ j Φ i G 4 φ -\frac{G_{3\varphi X}\varphi_i\Phi_i}{G_{4\varphi}}+\frac{2G_{3\varphi X}(\nabla\varphi)^2\varphi_i\Phi_i}{G_{4\varphi}}+\frac{8G_{3XX}(\nabla\varphi)^2\varphi_i\varphi_{ij}\Phi_i}{G_{4\varphi}}+\frac{2G_{3X}\varphi_{ij}\varphi_j\Phi_i}{G_{4\varphi}} − G 4 φ G 3 φX φ i Φ i + G 4 φ 2 G 3 φX ( ∇ φ ) 2 φ i Φ i + G 4 φ 8 G 3 X X ( ∇ φ ) 2 φ i φ ij Φ i + G 4 φ 2 G 3 X φ ij φ j Φ i − 4 G 3 X X φ i φ j φ k φ k j Φ i G 4 φ + G 3 φ Φ i i j G 4 φ φ i 2 − G 3 φ Φ i j G 4 φ φ i − 2 G 3 X φ i φ j Φ i j G 4 φ . (38) -\frac{4G_{3XX}\varphi_i\varphi_j\varphi_k\varphi_{kj}\Phi_i}{G_{4\varphi}}+\frac{G_{3\varphi}\Phi_{iij}}{G_{4\varphi}\varphi_i^2}-\frac{G_{3\varphi}\Phi_{ij}}{G_{4\varphi}\varphi_i}-\frac{2G_{3X}\varphi_i\varphi_j\Phi_{ij}}{G_{4\varphi}}. \tag*{(38)} − G 4 φ 4 G 3 X X φ i φ j φ k φ k j Φ i + G 4 φ φ i 2 G 3 φ Φ iij − G 4 φ φ i G 3 φ Φ ij − G 4 φ 2 G 3 X φ i φ j Φ ij . (38) In this expression a latin subscript indicates partial differentiation with respect to a Cartesian spatial direction, while Δ = ∂ i ∂ i \Delta=\partial_i\partial_i Δ = ∂ i ∂ i ( ∇ φ ) 2 = ∂ i φ ∂ i φ (\nabla\varphi)^2=\partial_i\varphi\partial_i\varphi ( ∇ φ ) 2 = ∂ i φ ∂ i φ Φ \Phi Φ Ψ \Psi Ψ (4) ), one gets, again to first order in Φ \Phi Φ Ψ \Psi Ψ
∇ 2 Ψ G 4 + ∇ 2 Φ ( ∇ φ ) 2 G 4 G 3 X G 4 φ − Φ [ − ( ∇ φ ) 2 G 3 φ + 2 ∇ 2 φ G 4 G 2 X G 4 φ − 4 ( ∇ 2 φ ) 2 G 4 G 3 X G 4 φ \nabla^2\Psi G_4+\frac{\nabla^2\Phi(\nabla\varphi)^2G_4G_{3X}}{G_{4\varphi}}-\Phi[-(\nabla\varphi)^2G_{3\varphi}+\frac{2\nabla^2\varphi G_4G_{2X}}{G_{4\varphi}}-\frac{4(\nabla^2\varphi)^2G_4G_{3X}}{G_{4\varphi}} ∇ 2 Ψ G 4 + G 4 φ ∇ 2 Φ ( ∇ φ ) 2 G 4 G 3 X − Φ [ − ( ∇ φ ) 2 G 3 φ + G 4 φ 2 ∇ 2 φ G 4 G 2 X − G 4 φ 4 ( ∇ 2 φ ) 2 G 4 G 3 X − ∇ 2 φ G 4 G 3 φ G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 4 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φ X G 4 φ − ( ∇ φ ) 2 G 4 G 3 φ φ G 4 φ -\frac{\nabla^2\varphi G_4G_{3\varphi}}{G_{4\varphi}}-2\nabla^2\varphi G_{4\varphi}+\frac{2(\nabla\varphi)^2G_4G_{2\varphi X}}{G_{4\varphi}}-\frac{4\nabla^2\varphi(\nabla\varphi)^2G_4G_{3\varphi X}}{G_{4\varphi}}-\frac{(\nabla\varphi)^2G_4G_{3\varphi\varphi}}{G_{4\varphi}} − G 4 φ ∇ 2 φ G 4 G 3 φ − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − G 4 φ 4 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φX − G 4 φ ( ∇ φ ) 2 G 4 G 3 φφ − 2 ( ∇ φ ) 2 G 4 φ φ − 4 G 4 G 3 X φ i j 2 G 4 φ − 8 G 4 G 3 X φ i i j φ j G 4 φ − 4 G 3 X φ i φ i j φ j + 8 G 4 G 2 X X φ i φ i j φ j G 4 φ -2(\nabla\varphi)^2G_{4\varphi\varphi}-\frac{4G_4G_{3X}\varphi_{ij}^2}{G_{4\varphi}}-\frac{8G_4G_{3X}\varphi_{iij}\varphi_j}{G_{4\varphi}}-4G_{3X}\varphi_i\varphi_{ij}\varphi_j+\frac{8G_4G_{2XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}} − 2 ( ∇ φ ) 2 G 4 φφ − G 4 φ 4 G 4 G 3 X φ ij 2 − G 4 φ 8 G 4 G 3 X φ iij φ j − 4 G 3 X φ i φ ij φ j + G 4 φ 8 G 4 G 2 X X φ i φ ij φ j − 12 ∇ 2 φ G 4 G 3 X X φ i φ i j φ j G 4 φ − 8 G 4 G 3 φ X φ i φ i j φ j G 4 φ − 12 G 4 G 3 X X φ i j φ i k φ j φ k G 4 φ ] + Ψ [ G 2 − ( ∇ φ ) 2 G 3 φ + 2 ∇ 2 φ G 4 G 2 X G 4 φ − 2 ∇ 2 φ 2 G 4 G 3 X G 4 φ − 2 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 X G 4 φ − G 4 G 2 φ G 4 φ − ∇ 2 φ G 4 G 3 φ G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φ X G 4 φ − ( ∇ φ ) 2 G 4 G 3 φ φ G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ − 2 G 4 G 3 X φ i j 2 G 4 φ − 4 G 4 G 3 X φ i i j φ j G 4 φ − 2 G 3 X φ i φ i j φ j + 4 G 4 G 2 X X φ i φ i j φ j G 4 φ − 4 ∇ 2 φ G 4 G 3 X X φ i φ i j φ j G 4 φ − 4 G 4 G 3 φ X φ i φ i j φ j G 4 φ − 4 G 4 G 3 X X φ i j φ i k φ j φ k G 4 φ + G 3 G 4 φ j φ i i j G 4 φ φ i 2 − G 3 G 4 φ j Ψ i i j G 4 φ φ i 2 + 1 2 [ ∇ 2 φ G 3 G 4 G 4 φ φ i − ( ∇ φ ) 2 G 4 G 3 φ G 4 φ φ i + 2 G 4 G 2 X φ i G 4 φ − 4 ∇ 2 φ G 4 G 3 X φ i G 4 φ − G 4 G 3 φ φ i G 4 φ − 2 ( ∇ φ ) 2 G 4 G 3 φ X φ i G 4 φ − 6 G 4 G 3 X φ i j φ j G 4 φ − 4 G 4 G 3 X X φ i φ j φ k φ k j G 4 φ ] Ψ i + 1 2 [ G 3 G 4 φ j G 4 φ φ i − 2 G 4 G 3 X φ i φ j G 4 φ ] Ψ i j − 1 2 [ ∇ 2 φ G 3 G 4 G 4 φ φ i + ( ∇ φ ) 2 G 4 G 3 φ G 4 φ φ i − 2 ( ∇ φ ) 2 G 3 X φ i − 2 G 4 G 2 X φ i G 4 φ + 4 G 4 G 2 X X φ i G 4 φ − 4 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 X X φ i G 4 φ + G 4 G 3 φ φ i G 4 φ + 2 G 4 φ φ i − 2 ( ∇ φ ) 2 G 4 G 3 φ X φ i G 4 φ − 8 G 4 G 3 X X φ i φ i j G 4 φ − 2 G 4 G 3 X φ i j φ j G 4 φ + 4 G 4 G 3 X X φ i φ j φ k φ k j G 4 φ ] Φ i − 1 2 [ G 3 G 4 φ j G 4 φ φ i + 2 G 4 G 3 X φ i φ j G 4 φ ] Φ i j + 1 2 [ G 2 − ( ∇ φ ) 2 G 3 φ + 2 ∇ 2 φ G 4 G 2 X G 4 φ − 2 ( ∇ 2 φ ) 2 G 4 G 3 X G 4 φ − G 4 G 2 φ G 4 φ − ∇ 2 φ G 4 G 3 φ G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φ X G 4 φ − ( ∇ φ ) 2 G 4 G 3 φ φ G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ − 2 G 4 G 3 X φ i 2 G 4 φ − 4 G 4 G 3 X φ i j φ j G 4 φ − 2 G 3 X φ i φ i j φ j + 4 G 4 G 2 X X φ i φ i j φ j G 4 φ − 4 ∇ 2 φ G 4 G 3 X X φ i φ i j φ j G 4 φ − 4 G 4 G 3 φ X φ i φ i j φ j G 4 φ − 4 G 4 G 3 X X φ i j φ i k φ j φ k G 4 φ + G 3 G 4 φ j Φ i i j G 4 φ φ i 2 − G 3 G 4 φ j Ψ i i j G 4 φ φ i 2 ] = 4 π G ρ . (39) \begin{aligned}
-\frac{12\nabla^2\varphi G_4G_{3XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{8G_4G_{3\varphi X}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}-\frac{12G_4G_{3XX}\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}] \\
&+\Psi[
G_2-(\nabla\varphi)^2G_{3\varphi}
+\frac{2\nabla^2\varphi G_4G_{2X}}{G_{4\varphi}}
-\frac{2\nabla^2\varphi^2G_4G_{3X}}{G_{4\varphi}}
-\frac{2\nabla^2\varphi(\nabla\varphi)^2G_{4}G_{3X}}{G_{4\varphi}}
-\frac{G_4G_{2\varphi}}{G_{4\varphi}}
-\frac{\nabla^2\varphi G_4G_{3\varphi}}{G_{4\varphi}}
-2\nabla^2\varphi G_{4\varphi}
+\frac{2(\nabla\varphi)^2G_4G_{2\varphi X}}{G_{4\varphi}}
-\frac{2\nabla^2\varphi(\nabla\varphi)^2G_4G_{3\varphi X}}{G_{4\varphi}}
-\frac{(\nabla\varphi)^2G_4G_{3\varphi\varphi}}{G_{4\varphi}}
-2(\nabla\varphi)^2G_{4\varphi\varphi}
-\frac{2G_4G_{3X}\varphi_{ij}^2}{G_{4\varphi}}
-\frac{4G_4G_{3X}\varphi_{iij}\varphi_j}{G_{4\varphi}}
-2G_{3X}\varphi_i\varphi_{ij}\varphi_j
+\frac{4G_4G_{2XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4\nabla^2\varphi G_4G_{3XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4G_4G_{3\varphi X}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4G_4G_{3XX}\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}
+\frac{G_3G_4\varphi_j\varphi_{iij}}{G_{4\varphi}\varphi_i^2}
-\frac{G_3G_4\varphi_j\Psi_{iij}}{G_{4\varphi}\varphi_i^2}
+\frac{1}{2}[
\frac{\nabla^2\varphi G_3G_4}{G_{4\varphi}\varphi_i}
-\frac{(\nabla\varphi)^2G_4G_{3\varphi}}{G_{4\varphi}\varphi_i}
+\frac{2G_4G_{2X}\varphi_i}{G_{4\varphi}}
-\frac{4\nabla^2\varphi G_4G_{3X}\varphi_i}{G_{4\varphi}}
-\frac{G_4G_{3\varphi}\varphi_i}{G_{4\varphi}}
-\frac{2(\nabla\varphi)^2G_4G_{3\varphi X}\varphi_i}{G_{4\varphi}}
-\frac{6G_4G_{3X}\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4G_4G_{3XX}\varphi_i\varphi_j\varphi_k\varphi_{kj}}{G_{4\varphi}}
]\Psi_i
+\frac{1}{2}[
\frac{G_3G_4\varphi_j}{G_{4\varphi}\varphi_i}
-\frac{2G_4G_{3X}\varphi_i\varphi_j}{G_{4\varphi}}
]\Psi_{ij}
-\frac{1}{2}[
\frac{\nabla^2\varphi G_3G_4}{G_{4\varphi}\varphi_i}
+\frac{(\nabla\varphi)^2G_4G_{3\varphi}}{G_{4\varphi}\varphi_i}
-2(\nabla\varphi)^2G_{3X}\varphi_i
-\frac{2G_4G_{2X}\varphi_i}{G_{4\varphi}}
+\frac{4G_4G_{2XX}\varphi_i}{G_{4\varphi}}
-\frac{4\nabla^2\varphi(\nabla\varphi)^2G_4G_{3XX}\varphi_i}{G_{4\varphi}}
+\frac{G_4G_{3\varphi}\varphi_i}{G_{4\varphi}}
+2G_{4\varphi}\varphi_i
-\frac{2(\nabla\varphi)^2G_4G_{3\varphi X}\varphi_i}{G_{4\varphi}}
-\frac{8G_4G_{3XX}\varphi_i\varphi_{ij}}{G_{4\varphi}}
-\frac{2G_4G_{3X}\varphi_{ij}\varphi_j}{G_{4\varphi}}
+\frac{4G_4G_{3XX}\varphi_i\varphi_j\varphi_k\varphi_{kj}}{G_{4\varphi}}
]\Phi_i
-\frac{1}{2}[
\frac{G_3G_4\varphi_j}{G_{4\varphi}\varphi_i}
+\frac{2G_4G_{3X}\varphi_i\varphi_j}{G_{4\varphi}}
]\Phi_{ij}
+\frac{1}{2}[
G_2-(\nabla\varphi)^2G_{3\varphi}
+\frac{2\nabla^2\varphi G_4G_{2X}}{G_{4\varphi}}
-\frac{2(\nabla^2\varphi)^2G_4G_{3X}}{G_{4\varphi}}
-\frac{G_4G_{2\varphi}}{G_{4\varphi}}
-\frac{\nabla^2\varphi G_4G_{3\varphi}}{G_{4\varphi}}
-2\nabla^2\varphi G_{4\varphi}
+\frac{2(\nabla\varphi)^2G_4G_{2\varphi X}}{G_{4\varphi}}
-\frac{2\nabla^2\varphi(\nabla\varphi)^2G_4G_{3\varphi X}}{G_{4\varphi}}
-\frac{(\nabla\varphi)^2G_4G_{3\varphi\varphi}}{G_{4\varphi}}
-2(\nabla\varphi)^2G_{4\varphi\varphi}
-\frac{2G_4G_{3X}\varphi_i^2}{G_{4\varphi}}
-\frac{4G_4G_{3X}\varphi_{ij}\varphi_j}{G_{4\varphi}}
-2G_{3X}\varphi_i\varphi_{ij}\varphi_j
+\frac{4G_4G_{2XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4\nabla^2\varphi G_4G_{3XX}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4G_4G_{3\varphi X}\varphi_i\varphi_{ij}\varphi_j}{G_{4\varphi}}
-\frac{4G_4G_{3XX}\varphi_{ij}\varphi_{ik}\varphi_j\varphi_k}{G_{4\varphi}}
+\frac{G_3G_4\varphi_j\Phi_{iij}}{G_{4\varphi}\varphi_i^2}
-\frac{G_3G_4\varphi_j\Psi_{iij}}{G_{4\varphi}\varphi_i^2}
]=4\pi G\rho. \tag*{(39)}
\end{aligned} − G 4 φ 12 ∇ 2 φ G 4 G 3 X X φ i φ ij φ j − G 4 φ 8 G 4 G 3 φX φ i φ ij φ j − G 4 φ 12 G 4 G 3 X X φ ij φ ik φ j φ k ] + Ψ [ G 2 − ( ∇ φ ) 2 G 3 φ + G 4 φ 2 ∇ 2 φ G 4 G 2 X − G 4 φ 2 ∇ 2 φ 2 G 4 G 3 X − G 4 φ 2 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 X − G 4 φ G 4 G 2 φ − G 4 φ ∇ 2 φ G 4 G 3 φ − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − G 4 φ 2 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φX − G 4 φ ( ∇ φ ) 2 G 4 G 3 φφ − 2 ( ∇ φ ) 2 G 4 φφ − G 4 φ 2 G 4 G 3 X φ ij 2 − G 4 φ 4 G 4 G 3 X φ iij φ j − 2 G 3 X φ i φ ij φ j + G 4 φ 4 G 4 G 2 X X φ i φ ij φ j − G 4 φ 4 ∇ 2 φ G 4 G 3 X X φ i φ ij φ j − G 4 φ 4 G 4 G 3 φX φ i φ ij φ j − G 4 φ 4 G 4 G 3 X X φ ij φ ik φ j φ k + G 4 φ φ i 2 G 3 G 4 φ j φ iij − G 4 φ φ i 2 G 3 G 4 φ j Ψ iij + 2 1 [ G 4 φ φ i ∇ 2 φ G 3 G 4 − G 4 φ φ i ( ∇ φ ) 2 G 4 G 3 φ + G 4 φ 2 G 4 G 2 X φ i − G 4 φ 4 ∇ 2 φ G 4 G 3 X φ i − G 4 φ G 4 G 3 φ φ i − G 4 φ 2 ( ∇ φ ) 2 G 4 G 3 φX φ i − G 4 φ 6 G 4 G 3 X φ ij φ j − G 4 φ 4 G 4 G 3 X X φ i φ j φ k φ k j ] Ψ i + 2 1 [ G 4 φ φ i G 3 G 4 φ j − G 4 φ 2 G 4 G 3 X φ i φ j ] Ψ ij − 2 1 [ G 4 φ φ i ∇ 2 φ G 3 G 4 + G 4 φ φ i ( ∇ φ ) 2 G 4 G 3 φ − 2 ( ∇ φ ) 2 G 3 X φ i − G 4 φ 2 G 4 G 2 X φ i + G 4 φ 4 G 4 G 2 X X φ i − G 4 φ 4 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 X X φ i + G 4 φ G 4 G 3 φ φ i + 2 G 4 φ φ i − G 4 φ 2 ( ∇ φ ) 2 G 4 G 3 φX φ i − G 4 φ 8 G 4 G 3 X X φ i φ ij − G 4 φ 2 G 4 G 3 X φ ij φ j + G 4 φ 4 G 4 G 3 X X φ i φ j φ k φ k j ] Φ i − 2 1 [ G 4 φ φ i G 3 G 4 φ j + G 4 φ 2 G 4 G 3 X φ i φ j ] Φ ij + 2 1 [ G 2 − ( ∇ φ ) 2 G 3 φ + G 4 φ 2 ∇ 2 φ G 4 G 2 X − G 4 φ 2 ( ∇ 2 φ ) 2 G 4 G 3 X − G 4 φ G 4 G 2 φ − G 4 φ ∇ 2 φ G 4 G 3 φ − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − G 4 φ 2 ∇ 2 φ ( ∇ φ ) 2 G 4 G 3 φX − G 4 φ ( ∇ φ ) 2 G 4 G 3 φφ − 2 ( ∇ φ ) 2 G 4 φφ − G 4 φ 2 G 4 G 3 X φ i 2 − G 4 φ 4 G 4 G 3 X φ ij φ j − 2 G 3 X φ i φ ij φ j + G 4 φ 4 G 4 G 2 X X φ i φ ij φ j − G 4 φ 4 ∇ 2 φ G 4 G 3 X X φ i φ ij φ j − G 4 φ 4 G 4 G 3 φX φ i φ ij φ j − G 4 φ 4 G 4 G 3 X X φ ij φ ik φ j φ k + G 4 φ φ i 2 G 3 G 4 φ j Φ iij − G 4 φ φ i 2 G 3 G 4 φ j Ψ iij ] = 4 π Gρ . (39) We want this long equation to reduce to the much simpler classical RG equation, i.e.
∇ ⋅ [ ϵ ( φ , X ) ∇ Ψ ] = ∇ ϵ ⋅ ∇ Ψ + ϵ ∇ 2 Ψ = ϵ φ ∇ φ ⋅ ∇ Ψ + ϵ X ∇ X ⋅ ∇ Ψ + ϵ ∇ 2 Ψ = 4 π G ρ , (40) \nabla\cdot[\epsilon(\varphi,X)\nabla\Psi]=\nabla\epsilon\cdot\nabla\Psi+\epsilon\nabla^2\Psi=\epsilon_\varphi\nabla\varphi\cdot\nabla\Psi+\epsilon_X\nabla X\cdot\nabla\Psi+\epsilon\nabla^2\Psi=4\pi G\rho, \tag*{(40)} ∇ ⋅ [ ϵ ( φ , X ) ∇Ψ ] = ∇ ϵ ⋅ ∇Ψ + ϵ ∇ 2 Ψ = ϵ φ ∇ φ ⋅ ∇Ψ + ϵ X ∇ X ⋅ ∇Ψ + ϵ ∇ 2 Ψ = 4 π Gρ , (40) where ϵ \epsilon ϵ ∇ X \nabla X ∇ X X = ( 1 − 2 Φ ) ( ∇ φ ) 2 X=(1-2\Phi)(\nabla\varphi)^2 X = ( 1 − 2Φ ) ( ∇ φ ) 2
∂ i X = − 2 ∂ i Φ ( ∇ φ ) 2 + 2 ( 1 − 2 Ψ ) ∂ j ∂ i φ ∂ j φ , (41) \partial_iX=-2\partial_i\Phi(\nabla\varphi)^2+2(1-2\Psi)\partial_j\partial_i\varphi\partial_j\varphi, \tag*{(41)} ∂ i X = − 2 ∂ i Φ ( ∇ φ ) 2 + 2 ( 1 − 2Ψ ) ∂ j ∂ i φ ∂ j φ , (41) and then Eq. (A.3) becomes
ϵ φ ∇ φ ⋅ ∇ Ψ + 2 ϵ X ∂ j ∂ i φ ∂ j φ ∂ i Ψ + ϵ ∇ 2 Ψ = 4 π G ρ , (42) \epsilon_\varphi\nabla\varphi\cdot\nabla\Psi+2\epsilon_X\partial_j\partial_i\varphi\partial_j\varphi\partial_i\Psi+\epsilon\nabla^2\Psi=4\pi G\rho, \tag*{(42)} ϵ φ ∇ φ ⋅ ∇Ψ + 2 ϵ X ∂ j ∂ i φ ∂ j φ ∂ i Ψ + ϵ ∇ 2 Ψ = 4 π Gρ , (42) where we discarded second order terms. To have Eq. (A.2) reduce to Eq. (A.5) it is sufficient for the coefficients of the various terms in the two equations to be equal term by term: for instance, the coefficients multiplying ∇ φ ⋅ ∇ Φ \nabla\varphi\cdot\nabla\Phi ∇ φ ⋅ ∇Φ ∇ 2 Φ \nabla^{2}\Phi ∇ 2 Φ ∇ 2 Ψ \nabla^{2}\Psi ∇ 2 Ψ Φ \Phi Φ Ψ \Psi Ψ X X X Φ \Phi Φ X X X G G G Φ \Phi Φ Φ \Phi Φ Φ \Phi Φ G G G
Eq. (A.2) contains many more terms than its RG counterpart, and we thus have many constraints: the first and most notable one comes from the term multiplying ∇ 2 Ψ \nabla^{2}\Psi ∇ 2 Ψ ϵ = G 4 \epsilon= G_{4} ϵ = G 4 G 4 G_{4} G 4
The second thing to notice is that we do not have any second derivative of Ψ \Psi Ψ ∇ 2 Ψ \nabla^{2}\Psi ∇ 2 Ψ Ψ i j \Psi_{ij} Ψ ij G 3 = 0 G_{3}=0 G 3 = 0
∇ 2 Ψ G 4 + G 4 G 2 X φ i G 4 φ Ψ i − 1 2 [ − 2 G 4 G 2 X φ i G 4 φ + 4 G 4 G 2 X X X φ i G 4 φ + 2 G 4 φ φ i ] Φ i − Φ [ 2 ∇ 2 φ G 4 G 2 X G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ + 8 G 4 G 2 X X φ i φ i j φ j G 4 φ ] + Ψ [ G 2 + 2 ∇ 2 φ G 4 G 2 X G 4 φ − G 4 G 2 φ G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ + 4 G 4 G 2 X X X φ i φ i j φ j G 4 φ ] + 1 2 [ G 2 + 2 ∇ 2 φ G 4 G 2 X G 4 φ − G 4 G 2 φ G 4 φ − 2 ∇ 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ + 4 G 4 G 2 X X X φ i φ i j φ j G 4 φ ] = 4 π G ρ . (43) \begin{aligned}
\nabla^{2}\Psi G_{4} &+ \frac{G_{4}G_{2X}\varphi_{i}}{G_{4\varphi}}\Psi_{i} - \frac{1}{2}\left[-\frac{2G_{4}G_{2X}\varphi_{i}}{G_{4\varphi}}+\frac{4G_{4}G_{2XX}X\varphi_{i}}{G_{4\varphi}}+2G_{4\varphi}\varphi_{i}\right]\Phi_{i} \\
&-\Phi\left[\frac{2\nabla^{2}\varphi G_{4}G_{2X}}{G_{4\varphi}}-2\nabla^{2}\varphi G_{4\varphi}+\frac{2(\nabla\varphi)^{2}G_{4}G_{2\varphi X}}{G_{4\varphi}}-2(\nabla\varphi)^{2}G_{4\varphi\varphi}+\frac{8G_{4}G_{2XX}\varphi_{i}\varphi_{ij}\varphi_{j}}{G_{4\varphi}}\right] \\
&+\Psi\left[G_{2}+\frac{2\nabla^{2}\varphi G_{4}G_{2X}}{G_{4\varphi}}-\frac{G_{4}G_{2\varphi}}{G_{4\varphi}}-2\nabla^{2}\varphi G_{4\varphi}+\frac{2(\nabla\varphi)^{2}G_{4}G_{2\varphi X}}{G_{4\varphi}}-2(\nabla\varphi)^{2}G_{4\varphi\varphi}\right.\\
&\qquad\left.+\frac{4G_{4}G_{2XX}X\varphi_{i}\varphi_{ij}\varphi_{j}}{G_{4\varphi}}\right]+\frac{1}{2}\left[G_{2}+\frac{2\nabla^{2}\varphi G_{4}G_{2X}}{G_{4\varphi}}-\frac{G_{4}G_{2\varphi}}{G_{4\varphi}}-2\nabla^{2}\varphi G_{4\varphi}+\frac{2(\nabla\varphi)^{2}G_{4}G_{2\varphi X}}{G_{4\varphi}}\right.\\
&\qquad\left.-2(\nabla\varphi)^{2}G_{4\varphi\varphi}+\frac{4G_{4}G_{2XX}X\varphi_{i}\varphi_{ij}\varphi_{j}}{G_{4\varphi}}\right]=4\pi G\rho. \tag*{(43)}
\end{aligned} ∇ 2 Ψ G 4 + G 4 φ G 4 G 2 X φ i Ψ i − 2 1 [ − G 4 φ 2 G 4 G 2 X φ i + G 4 φ 4 G 4 G 2 X X X φ i + 2 G 4 φ φ i ] Φ i − Φ [ G 4 φ 2 ∇ 2 φ G 4 G 2 X − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − 2 ( ∇ φ ) 2 G 4 φφ + G 4 φ 8 G 4 G 2 X X φ i φ ij φ j ] + Ψ [ G 2 + G 4 φ 2 ∇ 2 φ G 4 G 2 X − G 4 φ G 4 G 2 φ − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − 2 ( ∇ φ ) 2 G 4 φφ + G 4 φ 4 G 4 G 2 X X X φ i φ ij φ j ] + 2 1 [ G 2 + G 4 φ 2 ∇ 2 φ G 4 G 2 X − G 4 φ G 4 G 2 φ − 2 ∇ 2 φ G 4 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − 2 ( ∇ φ ) 2 G 4 φφ + G 4 φ 4 G 4 G 2 X X X φ i φ ij φ j ] = 4 π Gρ . (43) Next, equating the coefficients of Ψ i \Psi_{i} Ψ i G 4 G 2 X / G 4 φ = G 4 φ G_{4}G_{2X}/G_{4\varphi}=G_{4\varphi} G 4 G 2 X / G 4 φ = G 4 φ
∇ 2 Φ G 4 + Ψ [ 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ + 8 G 4 G 2 X X X φ i φ i j φ j G 4 φ ] + Φ [ G 2 − G 4 G 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ + 4 G 4 G 2 X X X φ i φ i j φ j G 4 φ ] + 1 2 [ G 2 − G 4 G 2 φ G 4 φ + 2 ( ∇ φ ) 2 G 4 G 2 φ X G 4 φ − 2 ( ∇ φ ) 2 G 4 φ φ + 4 G 4 G 2 X X X φ i φ i j φ j G 4 φ ] + G 4 G 2 X φ i G 4 φ Φ i + 2 G 4 G 2 X X X φ i G 4 φ Ψ i = 4 π G ρ . (44) \begin{aligned}
\nabla^{2}\Phi G_{4} &+\Psi\left[\frac{2(\nabla\varphi)^{2}G_{4}G_{2\varphi X}}{G_{4\varphi}}-2(\nabla\varphi)^{2}G_{4\varphi\varphi}+\frac{8G_{4}G_{2XX}X\varphi_{i}\varphi_{ij}\varphi_{j}}{G_{4\varphi}}\right]\\
&+\Phi\left[G_{2}-\frac{G_{4}G_{2\varphi}}{G_{4\varphi}}+\frac{2(\nabla\varphi)^{2}G_{4}G_{2\varphi X}}{G_{4\varphi}}-2(\nabla\varphi)^{2}G_{4\varphi\varphi}+\frac{4G_{4}G_{2XX}X\varphi_{i}\varphi_{ij}\varphi_{j}}{G_{4\varphi}}\right]\\
&+\frac{1}{2}\left[G_{2}-\frac{G_{4}G_{2\varphi}}{G_{4\varphi}}+\frac{2(\nabla\varphi)^{2}G_{4}G_{2\varphi X}}{G_{4\varphi}}-2(\nabla\varphi)^{2}G_{4\varphi\varphi}+\frac{4G_{4}G_{2XX}X\varphi_{i}\varphi_{ij}\varphi_{j}}{G_{4\varphi}}\right]\\
&+\frac{G_{4}G_{2X}\varphi_{i}}{G_{4\varphi}}\Phi_{i}+\frac{2G_{4}G_{2XX}X\varphi_{i}}{G_{4\varphi}}\Psi_{i}=4\pi G\rho. \tag*{(44)}
\end{aligned} ∇ 2 Φ G 4 + Ψ [ G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − 2 ( ∇ φ ) 2 G 4 φφ + G 4 φ 8 G 4 G 2 X X X φ i φ ij φ j ] + Φ [ G 2 − G 4 φ G 4 G 2 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − 2 ( ∇ φ ) 2 G 4 φφ + G 4 φ 4 G 4 G 2 X X X φ i φ ij φ j ] + 2 1 [ G 2 − G 4 φ G 4 G 2 φ + G 4 φ 2 ( ∇ φ ) 2 G 4 G 2 φX − 2 ( ∇ φ ) 2 G 4 φφ + G 4 φ 4 G 4 G 2 X X X φ i φ ij φ j ] + G 4 φ G 4 G 2 X φ i Φ i + G 4 φ 2 G 4 G 2 X X X φ i Ψ i = 4 π Gρ . (44) Now, paying attention to the fact that, to first order, X ≈ ( 1 − 2 Φ ) ( ∇ φ ) 2 ⇒ ( ∇ φ ) 2 ≈ ( 1 + 2 Φ ) X X \approx(1-2\Phi)(\nabla\varphi)^{2} \Rightarrow(\nabla\varphi)^{2} \approx(1+2\Phi)X X ≈ ( 1 − 2Φ ) ( ∇ φ ) 2 ⇒ ( ∇ φ ) 2 ≈ ( 1 + 2Φ ) X Φ \Phi Φ Φ \Phi Φ Φ \Phi Φ Φ i \Phi_i Φ i G 2 X X = 0 G_{2XX}=0 G 2 X X = 0
G 2 − G 4 G 2 ϕ G 4 ϕ + 2 G 4 G 2 ϕ X G 4 ϕ X − 2 G 4 ϕ ϕ X = 0. (45) G_2-\frac{G_4G_{2\phi}}{G_{4\phi}}+\frac{2G_4G_{2\phi X}}{G_{4\phi}}X-2G_{4\phi\phi}X=0. \tag*{(45)} G 2 − G 4 ϕ G 4 G 2 ϕ + G 4 ϕ 2 G 4 G 2 ϕX X − 2 G 4 ϕϕ X = 0. (45) Thus, we have obtained the whole set of conditions given in Eqs. (8) .
Background evolution equations in gCRG
In this section, we derive the explicit time evolution for the background value of the scalar field, ϕ ( a ) \phi(a) ϕ ( a ) H ( a ) H(a) H ( a ) (16) . In this metric, Eqs. (14) and (15) yield the following equations,
( G 4 ϕ 2 G 4 − G 4 ϕ ϕ ) ϕ ˙ 2 − G 4 ϕ ϕ ¨ − 3 G 4 a ¨ a = 4 π G ρ (46) \left(\frac{G_{4\phi}^{2}}{G_4}-G_{4\phi\phi}\right)\dot{\phi}^{2}-G_{4\phi}\ddot{\phi}-3G_4\frac{\ddot{a}}{a}=4\pi G\rho \tag*{(46)} ( G 4 G 4 ϕ 2 − G 4 ϕϕ ) ϕ ˙ 2 − G 4 ϕ ϕ ¨ − 3 G 4 a a ¨ = 4 π Gρ (46) G 4 ϕ H ϕ ˙ + G 4 ( 2 H 2 + a ¨ a ) = 0 (47) G_{4\phi}H\dot{\phi}+G_4\left(2H^2+\frac{\ddot{a}}{a}\right)=0 \tag*{(47)} G 4 ϕ H ϕ ˙ + G 4 ( 2 H 2 + a a ¨ ) = 0 (47) G 4 ϕ ( − ϕ ¨ − 3 H ϕ ˙ ) − 2 Ξ G 4 + G 4 ϕ ϕ ϕ ˙ 2 = − 4 π G ρ (48) G_{4\phi}(-\ddot{\phi}-3H\dot{\phi})-2\Xi G_4+G_{4\phi\phi}\dot{\phi}^{2}=-4\pi G\rho \tag*{(48)} G 4 ϕ ( − ϕ ¨ − 3 H ϕ ˙ ) − 2Ξ G 4 + G 4 ϕϕ ϕ ˙ 2 = − 4 π Gρ (48) where H = a ˙ / a H=\dot{a}/a H = a ˙ / a
Ω ≡ Ω m G 4 ≡ 8 π G 3 H 2 G 4 ρ m , Ω Ξ ≡ Ξ 3 H 2 , (49) \Omega\equiv\frac{\Omega_{\mathrm{m}}}{G_4}\equiv\frac{8\pi G}{3H^2G_4}\rho_{\mathrm{m}},\qquad\Omega_{\Xi}\equiv\frac{\Xi}{3H^2}, \tag*{(49)} Ω ≡ G 4 Ω m ≡ 3 H 2 G 4 8 π G ρ m , Ω Ξ ≡ 3 H 2 Ξ , (49) q ≡ − a ¨ a ˙ 2 = − a ¨ a H 2 , q ϕ ≡ − G 4 ϕ G 4 ϕ ¨ H 2 , ζ ϕ ≡ G 4 ϕ G 4 ϕ ˙ H , ζ ϕ ϕ ≡ G 4 ϕ ϕ G 4 ϕ ˙ 2 H ϕ ˙ , (50) q\equiv-\frac{\ddot{a}}{\dot{a}^{2}}=-\frac{\ddot{a}}{aH^2},\qquad q_{\phi}\equiv-\frac{G_{4\phi}}{G_4}\frac{\ddot{\phi}}{H^2},\qquad\zeta_{\phi}\equiv\frac{G_{4\phi}}{G_4}\frac{\dot{\phi}}{H},\qquad\zeta_{\phi\phi}\equiv\frac{G_{4\phi\phi}}{G_4}\frac{\dot{\phi}^{2}}{H\dot{\phi}}, \tag*{(50)} q ≡ − a ˙ 2 a ¨ = − a H 2 a ¨ , q ϕ ≡ − G 4 G 4 ϕ H 2 ϕ ¨ , ζ ϕ ≡ G 4 G 4 ϕ H ϕ ˙ , ζ ϕϕ ≡ G 4 G 4 ϕϕ H ϕ ˙ ϕ ˙ 2 , (50) the system of equations simplifies to
q + 1 3 ζ ϕ 2 − 1 3 ζ ϕ ϕ + 1 3 q ϕ = 1 2 Ω , 2 + ζ ϕ + 1 3 ζ ϕ 2 − 1 3 ζ ϕ ϕ + 1 3 q ϕ = 1 2 Ω , − 1 3 q ϕ + ζ ϕ + 1 3 ζ ϕ ϕ = 1 2 Ω − 2 Ω Ξ . (51) \begin{aligned}
q+\frac{1}{3}\zeta_{\phi}^{2}-\frac{1}{3}\zeta_{\phi\phi}+\frac{1}{3}q_{\phi}&=\frac{1}{2}\Omega,\\
2+\zeta_{\phi}+\frac{1}{3}\zeta_{\phi}^{2}-\frac{1}{3}\zeta_{\phi\phi}+\frac{1}{3}q_{\phi}&=\frac{1}{2}\Omega,\\
-\frac{1}{3}q_{\phi}+\zeta_{\phi}+\frac{1}{3}\zeta_{\phi\phi}&=\frac{1}{2}\Omega-2\Omega_{\Xi}. \tag*{(51)}
\end{aligned} q + 3 1 ζ ϕ 2 − 3 1 ζ ϕϕ + 3 1 q ϕ 2 + ζ ϕ + 3 1 ζ ϕ 2 − 3 1 ζ ϕϕ + 3 1 q ϕ − 3 1 q ϕ + ζ ϕ + 3 1 ζ ϕϕ = 2 1 Ω , = 2 1 Ω , = 2 1 Ω − 2 Ω Ξ . (51) Subtracting the first equation from the second in the system (B.6), we find
q − ζ = 2 ⟹ d d t ln G 4 = − d d t ( ln a ˙ + 2 ln a ) ⟹ G 4 = H 0 G 4 , 0 H a 3 , (52) q-\zeta=2\quad\Longrightarrow\quad\frac{d}{dt}\ln G_4=-\frac{d}{dt}(\ln\dot{a}+2\ln a)\quad\Longrightarrow\quad G_4=\frac{H_0G_{4,0}}{Ha^3}, \tag*{(52)} q − ζ = 2 ⟹ d t d ln G 4 = − d t d ( ln a ˙ + 2 ln a ) ⟹ G 4 = H a 3 H 0 G 4 , 0 , (52) where H 0 H_0 H 0 G 4 , 0 G_{4,0} G 4 , 0 q = Ω − 2 Ω Ξ − ζ − ζ 2 / 3 q=\Omega-2\Omega_{\Xi}-\zeta-\zeta^2/3 q = Ω − 2 Ω Ξ − ζ − ζ 2 /3
( d d t ln a ˙ − d d t ln a ) 2 = ( d d t ln a ˙ a ) 2 ⟹ d H d t = ± 3 ( H 4 + Ω 0 H 0 H 3 − 2 Ω Ξ 0 H 0 2 H 2 ) 1 / 2 , (53) \left(\frac{d}{dt}\ln\dot{a}-\frac{d}{dt}\ln a\right)^2=\left(\frac{d}{dt}\ln\frac{\dot{a}}{a}\right)^2\Longrightarrow\frac{dH}{dt}=\pm\sqrt{3}\left(H^4+\Omega_0H_0H^3-2\Omega_{\Xi0}H_0^2H^2\right)^{1/2}, \tag*{(53)} ( d t d ln a ˙ − d t d ln a ) 2 = ( d t d ln a a ˙ ) 2 ⟹ d t d H = ± 3 ( H 4 + Ω 0 H 0 H 3 − 2 Ω Ξ0 H 0 2 H 2 ) 1/2 , (53) where Ω 0 \Omega_0 Ω 0 Ω Ξ 0 \Omega_{\Xi0} Ω Ξ0
H ( a ) = − 1 2 Ω 0 H 0 + 1 4 Ω 0 2 + 2 Ω Ξ 0 H 0 2 [ ( C + C 2 − 1 ) a ± 3 + ( C + C 2 − 1 ) − 1 a ∓ 3 ] (54) H(a)=-\frac{1}{2}\Omega_0H_0+\sqrt{\frac{1}{4}\Omega_0^2+2\Omega_{\Xi0}}\frac{H_0}{2}\left[\left(C+\sqrt{C^2-1}\right)a^{\pm\sqrt{3}}+\left(C+\sqrt{C^2-1}\right)^{-1}a^{\mp\sqrt{3}}\right] \tag*{(54)} H ( a ) = − 2 1 Ω 0 H 0 + 4 1 Ω 0 2 + 2 Ω Ξ0 2 H 0 [ ( C + C 2 − 1 ) a ± 3 + ( C + C 2 − 1 ) − 1 a ∓ 3 ] (54) where the inequality
C ≡ 1 + Ω 0 2 1 4 Ω 0 2 + 2 Ω Ξ 0 ≥ 1 (55) C\equiv\frac{1+\frac{\Omega_0}{2}}{\sqrt{\frac{1}{4}\Omega_0^2+2\Omega_{\Xi0}}}\geq1 \tag*{(55)} C ≡ 4 1 Ω 0 2 + 2 Ω Ξ0 1 + 2 Ω 0 ≥ 1 (55) must be satisfied to guarantee that H ( a ) H(a) H ( a )
Linear perturbation theory in gCRG
Growth factor equation
In this section, we want to develop the linear perturbation theory for the Horndeski theory we are studying, in order to obtain the growth factor equation for the density contrast in a gCRG universe. First of all, we perturb the Einstein and KG equations, Eqs. (14) and (15) , obtaining
G 4 φ δ φ R ν μ + G 4 δ R ν μ − G 4 φ [ δ g μ λ ∂ λ ∂ ν φ + g μ λ ∂ λ ∂ ν δ φ − δ g μ λ Γ λ ν ρ ∂ ρ φ − g μ λ δ Γ λ ν ρ ∂ ρ φ − g μ λ Γ λ ν ρ ∂ ρ δ φ ] + ( 2 G 4 φ G 4 φ φ G 4 − G 4 3 G 4 2 − G 4 φ φ φ ) δ φ g μ λ ∂ λ φ ∂ ν φ + ( G 4 2 G 4 − G 4 φ φ ) δ g μ λ ∂ λ φ ∂ ν φ + ( G 4 2 G 4 − G 4 φ φ ) g μ λ ( ∂ λ δ φ ∂ ν φ + ∂ λ φ ∂ ν δ φ ) = 4 π G δ T ν μ , (56) G_{4\varphi}\delta\varphi R^\mu_{\ \nu}+G_4\delta R^\mu_{\ \nu}-G_{4\varphi}\left[\delta g^{\mu\lambda}\partial_\lambda\partial_\nu\varphi+g^{\mu\lambda}\partial_\lambda\partial_\nu\delta\varphi-\delta g^{\mu\lambda}\Gamma^\rho_{\lambda\nu}\partial_\rho\varphi-g^{\mu\lambda}\delta\Gamma^\rho_{\lambda\nu}\partial_\rho\varphi-g^{\mu\lambda}\Gamma^\rho_{\lambda\nu}\partial_\rho\delta\varphi\right]+\left(\frac{2G_{4\varphi}G_{4\varphi\varphi}}{G_4}-\frac{G_4^3}{G_4^2}-G_{4\varphi\varphi\varphi}\right)\delta\varphi g^{\mu\lambda}\partial_\lambda\varphi\partial_\nu\varphi+\left(\frac{G_4^2}{G_4}-G_{4\varphi\varphi}\right)\delta g^{\mu\lambda}\partial_\lambda\varphi\partial_\nu\varphi+\left(\frac{G_4^2}{G_4}-G_{4\varphi\varphi}\right)g^{\mu\lambda}\left(\partial_\lambda\delta\varphi\partial_\nu\varphi+\partial_\lambda\varphi\partial_\nu\delta\varphi\right)=4\pi G\delta T^\mu_{\ \nu}, \tag*{(56)} G 4 φ δ φ R ν μ + G 4 δ R ν μ − G 4 φ [ δ g μ λ ∂ λ ∂ ν φ + g μ λ ∂ λ ∂ ν δ φ − δ g μ λ Γ λ ν ρ ∂ ρ φ − g μ λ δ Γ λ ν ρ ∂ ρ φ − g μ λ Γ λ ν ρ ∂ ρ δ φ ] + ( G 4 2 G 4 φ G 4 φφ − G 4 2 G 4 3 − G 4 φφφ ) δ φ g μ λ ∂ λ φ ∂ ν φ + ( G 4 G 4 2 − G 4 φφ ) δ g μ λ ∂ λ φ ∂ ν φ + ( G 4 G 4 2 − G 4 φφ ) g μ λ ( ∂ λ δ φ ∂ ν φ + ∂ λ φ ∂ ν δ φ ) = 4 π G δ T ν μ , (56) G 4 φ φ δ φ □ φ + G 4 φ [ δ g α β ∂ α ∂ β φ + g α β ∂ α ∂ β δ φ − δ g α β Γ α β λ ∂ λ φ − g α β δ Γ α β λ ∂ λ φ − g α β Γ α β λ ∂ λ δ φ ] − 2 Ξ G 4 φ δ φ + G 4 φ φ φ X δ φ + 2 G 4 φ φ ∂ α φ ∂ α δ φ = 4 π G ( δ g μ ν T μ ν + g μ ν δ T μ ν ) . (57) G_{4\varphi\varphi}\delta\varphi\Box\varphi+G_{4\varphi}\left[\delta g^{\alpha\beta}\partial_\alpha\partial_\beta\varphi+g^{\alpha\beta}\partial_\alpha\partial_\beta\delta\varphi-\delta g^{\alpha\beta}\Gamma^\lambda_{\alpha\beta}\partial_\lambda\varphi-g^{\alpha\beta}\delta\Gamma^\lambda_{\alpha\beta}\partial_\lambda\varphi-g^{\alpha\beta}\Gamma^\lambda_{\alpha\beta}\partial_\lambda\delta\varphi\right]-2\Xi G_{4\varphi}\delta\varphi+G_{4\varphi\varphi\varphi}X\delta\varphi+2G_{4\varphi\varphi}\partial^\alpha\varphi\partial_\alpha\delta\varphi=4\pi G\left(\delta g^{\mu\nu}T_{\mu\nu}+g^{\mu\nu}\delta T_{\mu\nu}\right). \tag*{(57)} G 4 φφ δ φ □ φ + G 4 φ [ δ g α β ∂ α ∂ β φ + g α β ∂ α ∂ β δ φ − δ g α β Γ α β λ ∂ λ φ − g α β δ Γ α β λ ∂ λ φ − g α β Γ α β λ ∂ λ δ φ ] − 2Ξ G 4 φ δ φ + G 4 φφφ X δ φ + 2 G 4 φφ ∂ α φ ∂ α δ φ = 4 π G ( δ g μν T μν + g μν δ T μν ) . (57) The perturbed stress-energy tensor that appears in these expressions is
δ T μ ν = ( δ ρ + δ P ) u μ u ν + ( ρ + P ) [ δ u μ u ν + u μ δ u ν ] + δ P g μ ν + P δ g μ ν + π μ ν , (58) \delta T_{\mu\nu}=(\delta\rho+\delta P)u_\mu u_\nu+(\rho+P)\left[\delta u_\mu u_\nu+u_\mu\delta u_\nu\right]+\delta P g_{\mu\nu}+P\delta g_{\mu\nu}+\pi_{\mu\nu}, \tag*{(58)} δ T μν = ( δ ρ + δ P ) u μ u ν + ( ρ + P ) [ δ u μ u ν + u μ δ u ν ] + δ P g μν + P δ g μν + π μν , (58) Finally, we compute the first order components of the perturbed Christoffel symbols and Ricci tensor from the metric in Eq. (19) and substitute them in Eq. (C.1), which yields the following four field equations (corresponding, respectively, to the 00, 0 i 0i 0 i i j ij ij i j ij ij
G 4 [ 6 H ′ Ψ − 3 Φ ′ ′ + ∇ 2 Ψ − 3 H ( Φ ′ − Ψ ′ ) ] + ( G 4 φ H φ ′ − G 4 φ φ ′ ′ − 3 H ′ G 4 φ ) δ φ + G 4 φ [ 2 Ψ φ ′ ′ − δ φ ′ ′ − 2 H φ ′ Ψ + φ ′ Ψ ′ + H δ φ ′ ] + ( 2 G 4 φ G 4 φ φ G 4 − G 4 3 G 4 2 − G 4 φ φ φ ) δ φ ( φ ′ ) 2 + 2 ( G 4 2 G 4 − G 4 φ φ ) [ − Ψ ( φ ′ ) 2 + φ ′ δ φ ′ ] = 4 π G a 2 δ ρ , (59) G_4\left[6\mathcal{H}'\Psi-3\Phi''+\nabla^2\Psi-3\mathcal{H}(\Phi'-\Psi')\right]+\left(G_{4\varphi}\mathcal{H}\varphi'-G_{4\varphi}\varphi''-3\mathcal{H}'G_{4\varphi}\right)\delta\varphi+G_{4\varphi}\left[2\Psi\varphi''-\delta\varphi''-2\mathcal{H}\varphi'\Psi+\varphi'\Psi'+\mathcal{H}\delta\varphi'\right]+\left(\frac{2G_{4\varphi}G_{4\varphi\varphi}}{G_4}-\frac{G_4^3}{G_4^2}-G_{4\varphi\varphi\varphi}\right)\delta\varphi(\varphi')^2+2\left(\frac{G_4^2}{G_4}-G_{4\varphi\varphi}\right)\left[-\Psi(\varphi')^2+\varphi'\delta\varphi'\right]=4\pi G a^2\delta\rho, \tag*{(59)} G 4 [ 6 H ′ Ψ − 3 Φ ′′ + ∇ 2 Ψ − 3 H ( Φ ′ − Ψ ′ ) ] + ( G 4 φ H φ ′ − G 4 φ φ ′′ − 3 H ′ G 4 φ ) δ φ + G 4 φ [ 2Ψ φ ′′ − δ φ ′′ − 2 H φ ′ Ψ + φ ′ Ψ ′ + H δ φ ′ ] + ( G 4 2 G 4 φ G 4 φφ − G 4 2 G 4 3 − G 4 φφφ ) δ φ ( φ ′ ) 2 + 2 ( G 4 G 4 2 − G 4 φφ ) [ − Ψ ( φ ′ ) 2 + φ ′ δ φ ′ ] = 4 π G a 2 δ ρ , (59) 2 G 4 ( ∂ i Φ ′ − H ∂ i Ψ ) + G 4 φ [ ∂ i δ φ ′ − ∂ i Ψ φ ′ − H ∂ i δ φ ] − ( G 4 φ 2 G 4 − G 4 φ φ ) φ ′ ∂ i δ φ = 4 π G a 2 ( ρ + P ) v i (60) 2G_{4}\left(\partial_i\Phi' - \mathcal{H}\partial_i\Psi\right) + G_{4\varphi}\left[\partial_i\delta\varphi' - \partial_i\Psi\varphi' - \mathcal{H}\partial_i\delta\varphi\right] - \left(\frac{G_{4\varphi}^{2}}{G_{4}} - G_{4\varphi\varphi}\right)\varphi'\partial_i\delta\varphi= 4\pi G a^{2}(\rho+ P)v_i \tag*{(60)} 2 G 4 ( ∂ i Φ ′ − H ∂ i Ψ ) + G 4 φ [ ∂ i δ φ ′ − ∂ i Ψ φ ′ − H ∂ i δ φ ] − ( G 4 G 4 φ 2 − G 4 φφ ) φ ′ ∂ i δ φ = 4 π G a 2 ( ρ + P ) v i (60) G 4 ∂ i ∂ j ( Ψ + Φ ) + G 4 φ ∂ i ∂ j δ φ = − 4 π G a 2 ∂ i ∂ j Π (61) G_{4}\partial^{i}\partial^{j}(\Psi+\Phi) + G_{4\varphi}\partial^{i}\partial^{j}\delta\varphi= -4\pi G a^{2}\partial^{i}\partial^{j}\Pi \tag*{(61)} G 4 ∂ i ∂ j ( Ψ + Φ ) + G 4 φ ∂ i ∂ j δ φ = − 4 π G a 2 ∂ i ∂ j Π (61) G 4 φ δ φ ( H ′ + 2 H 2 ) + G 4 ( − ∇ 2 Φ − 2 H ′ Ψ + Φ ′ ′ − H Ψ ′ + 5 H Φ ′ − 4 H 2 Ψ ) − 2 Φ H G 4 φ φ φ ′ + φ ′ [ 2 H ( Φ − Ψ ) + Φ ′ ] + H δ φ ′ − G 4 3 ∇ 2 ( Ψ + Φ ) − G 4 φ ∇ 2 δ φ 3 φ = 4 π G a 2 δ P (62) G_{4\varphi}\delta\varphi\left(\mathcal{H}' + 2\mathcal{H}^{2}\right) + G_{4}\left(-\nabla^{2}\Phi- 2\mathcal{H}'\Psi+ \Phi'' - \mathcal{H}\Psi' + 5\mathcal{H}\Phi' - 4\mathcal{H}^{2}\Psi\right) - 2\Phi\mathcal{H}G_{4\varphi\varphi}\varphi' + \varphi'\left[2\mathcal{H}(\Phi-\Psi)+\Phi'\right] + \mathcal{H}\delta\varphi' - \frac{G_{4}}{3}\nabla^{2}(\Psi+\Phi) - G_{4\varphi}\frac{\nabla^{2}\delta\varphi}{3\varphi} = 4\pi G a^{2}\delta P \tag*{(62)} G 4 φ δ φ ( H ′ + 2 H 2 ) + G 4 ( − ∇ 2 Φ − 2 H ′ Ψ + Φ ′′ − H Ψ ′ + 5 H Φ ′ − 4 H 2 Ψ ) − 2Φ H G 4 φφ φ ′ + φ ′ [ 2 H ( Φ − Ψ ) + Φ ′ ] + H δ φ ′ − 3 G 4 ∇ 2 ( Ψ + Φ ) − G 4 φ 3 φ ∇ 2 δ φ = 4 π G a 2 δ P (62) In these equations, H = a ′ / a \mathcal{H}=a'/a H = a ′ / a Π \Pi Π π i j = a 2 ( ∂ i ∂ j Π − δ i j ∇ 2 Π / 3 ) \pi_{ij}=a^{2}\left(\partial_i\partial_j\Pi-\delta_{ij}\nabla^{2}\Pi/3\right) π ij = a 2 ( ∂ i ∂ j Π − δ ij ∇ 2 Π/3 )
G 4 φ φ δ φ ( − φ ′ ′ − 2 H φ ′ ) + G 4 φ [ 2 Ψ φ ′ ′ − δ φ ′ ′ + ∇ 2 δ φ + 4 H φ ′ Ψ + φ ′ ( Ψ ′ − 3 Φ ′ ) − 2 H δ φ ′ ] − 2 Ξ G 4 φ a 2 δ φ − G 4 φ φ φ ( φ ′ ) 2 δ φ − 2 G 4 φ φ φ ′ δ φ ′ + 2 G 4 φ φ Ψ ( φ ′ ) 2 = 4 π G a 2 ( 3 δ P − δ ρ ) (63) G_{4\varphi\varphi}\delta\varphi\left(-\varphi''-2\mathcal{H}\varphi'\right)+G_{4\varphi}\left[2\Psi\varphi''-\delta\varphi''+\nabla^{2}\delta\varphi+4\mathcal{H}\varphi'\Psi+\varphi'(\Psi'-3\Phi')-2\mathcal{H}\delta\varphi'\right]-2\Xi G_{4\varphi}a^{2}\delta\varphi-G_{4\varphi\varphi\varphi}(\varphi')^{2}\delta\varphi-2G_{4\varphi\varphi}\varphi'\delta\varphi'+2G_{4\varphi\varphi}\Psi(\varphi')^{2}=4\pi G a^{2}(3\delta P-\delta\rho) \tag*{(63)} G 4 φφ δ φ ( − φ ′′ − 2 H φ ′ ) + G 4 φ [ 2Ψ φ ′′ − δ φ ′′ + ∇ 2 δ φ + 4 H φ ′ Ψ + φ ′ ( Ψ ′ − 3 Φ ′ ) − 2 H δ φ ′ ] − 2Ξ G 4 φ a 2 δ φ − G 4 φφφ ( φ ′ ) 2 δ φ − 2 G 4 φφ φ ′ δ φ ′ + 2 G 4 φφ Ψ ( φ ′ ) 2 = 4 π G a 2 ( 3 δ P − δ ρ ) (63) Finally, perturbing the conservation equation, ∇ μ T ν μ = 0 \nabla_{\mu}T^{\mu}_{\nu}=0 ∇ μ T ν μ = 0
δ ′ + 3 H ( c s 2 − w ) δ + ( 1 + w ) ( θ + 3 Φ ′ ) = 0 (64) \delta' + 3\mathcal{H}(c_{s}^{2}-w)\delta+ (1+w)(\theta+3\Phi')=0 \tag*{(64)} δ ′ + 3 H ( c s 2 − w ) δ + ( 1 + w ) ( θ + 3 Φ ′ ) = 0 (64) ρ ( 1 + w ) ( H θ + θ ′ + ∇ 2 Ψ − 3 w H θ ) + c s 2 ρ ∇ 2 δ + 2 3 ∇ 4 Π = 0 (65) \rho(1+w)\left(\mathcal{H}\theta+\theta'+\nabla^{2}\Psi-3w\mathcal{H}\theta\right)+c_{s}^{2}\rho\nabla^{2}\delta+\frac{2}{3}\nabla^{4}\Pi=0 \tag*{(65)} ρ ( 1 + w ) ( H θ + θ ′ + ∇ 2 Ψ − 3 w H θ ) + c s 2 ρ ∇ 2 δ + 3 2 ∇ 4 Π = 0 (65) where we have defined the equation of state, P = w ρ P=w\rho P = w ρ c s 2 = δ P / δ ρ c_{s}^{2}=\delta P/\delta\rho c s 2 = δ P / δ ρ δ = δ ρ / ρ \delta=\delta\rho/\rho δ = δ ρ / ρ θ = ∂ i v i \theta=\partial_i v^i θ = ∂ i v i
In order to determine whether this theory can describe the formation of cosmic structure, we must derive an equation for the evolution of the density contrast, δ \delta δ w = c s 2 = 0 w=c_{s}^{2}=0 w = c s 2 = 0
δ ′ = − ( θ + 3 Φ ′ ) (66) \delta'=-(\theta+3\Phi') \tag*{(66)} δ ′ = − ( θ + 3 Φ ′ ) (66) θ ′ = − H θ − ∇ 2 Ψ (67) \theta'=-\mathcal{H}\theta-\nabla^{2}\Psi \tag*{(67)} θ ′ = − H θ − ∇ 2 Ψ (67) The first step is to combine these two equations: we switch to a derivative with respect to the variable N = ln a N=\ln a N = ln a
δ ′ ′ + ( 2 + H ′ H ) δ ′ = − 6 Φ ′ + ∇ 2 Ψ ( a H ) 2 − 3 H ′ H Φ ′ − 3 Φ ′ ′ (68) \delta''+\left(2+\frac{\mathcal{H}'}{\mathcal{H}}\right)\delta'=-6\Phi'+\frac{\nabla^{2}\Psi}{(a\mathcal{H})^{2}}-3\frac{\mathcal{H}'}{\mathcal{H}}\Phi'-3\Phi'' \tag*{(68)} δ ′′ + ( 2 + H H ′ ) δ ′ = − 6 Φ ′ + ( a H ) 2 ∇ 2 Ψ − 3 H H ′ Φ ′ − 3 Φ ′′ (68) where now ′ ' ′ N N N φ ′ ′ \varphi'' φ ′′ ∇ 2 Ψ \nabla^{2}\Psi ∇ 2 Ψ (20) . By performing the same change of variable in the perturbed KG equation, Eq. (C.8), and in the background KG equation, Eq. (15) , we obtain Eqs. (21) and (22) .
Numerical solution
In this section, we show how to approach the numerical solution of the set of equations given by Eqs. (20) , (21) and (22) , which we report here under the assumptions of the QSA, meaning that we neglect the terms containing the scalar potentials Φ \Phi Φ Ψ \Psi Ψ
δ ′ ′ + ( 2 + H ′ H ) δ ′ − 8 π G G 4 ρ m 0 H 2 a 3 δ − ∇ 2 δ φ a 2 H 2 G 4 φ G 4 + 2 Ξ H 2 δ φ G 4 φ G 4 + 3 G 4 φ G 4 ( 1 + H ′ H ) δ φ = − 3 G 4 φ G 4 δ φ ′ − 3 G 4 φ φ G 4 φ ′ δ φ − ( 2 G 4 φ G 4 φ φ G 4 2 − G 4 3 G 4 3 ) ( φ ′ ) 2 δ φ − 2 G 4 2 G 4 2 φ ′ δ φ ′ ; (69) \begin{aligned}
\delta^{\prime\prime}+\left(2+\frac{H^{\prime}}{H}\right)\delta^{\prime}-\frac{8\pi G}{G_{4}}\frac{\rho_{\mathrm{m}0}}{H^{2}a^{3}}\delta-\frac{\nabla^{2}\delta\varphi}{a^{2}H^{2}}\frac{G_{4\varphi}}{G_{4}}+\frac{2\Xi}{H^{2}}\delta\varphi\frac{G_{4\varphi}}{G_{4}}+3\frac{G_{4\varphi}}{G_{4}}\left(1+\frac{H^{\prime}}{H}\right)\delta\varphi\\
=-3\frac{G_{4\varphi}}{G_{4}}\delta\varphi^{\prime}-3\frac{G_{4\varphi\varphi}}{G_{4}}\varphi^{\prime}\delta\varphi-\left(\frac{2G_{4\varphi}G_{4\varphi\varphi}}{G_{4}^{2}}-\frac{G_{4}^{3}}{G_{4}^{3}}\right)(\varphi^{\prime})^{2}\delta\varphi-2\frac{G_{4}^{2}}{G_{4}^{2}}\varphi^{\prime}\delta\varphi^{\prime}; \tag*{(69)}
\end{aligned} δ ′′ + ( 2 + H H ′ ) δ ′ − G 4 8 π G H 2 a 3 ρ m 0 δ − a 2 H 2 ∇ 2 δ φ G 4 G 4 φ + H 2 2Ξ δ φ G 4 G 4 φ + 3 G 4 G 4 φ ( 1 + H H ′ ) δ φ = − 3 G 4 G 4 φ δ φ ′ − 3 G 4 G 4 φφ φ ′ δ φ − ( G 4 2 2 G 4 φ G 4 φφ − G 4 3 G 4 3 ) ( φ ′ ) 2 δ φ − 2 G 4 2 G 4 2 φ ′ δ φ ′ ; (69) δ φ ′ ′ + ( 3 + H ′ H ) δ φ ′ − 4 π G G 4 φ ρ m 0 H 2 a 3 δ + ∇ 2 δ φ a 2 H 2 + 2 Ξ H 2 δ φ = − G 4 φ φ G 4 φ δ φ [ φ ′ ′ + ( 3 + H ′ H ) φ ′ ] − G 4 φ φ φ G 4 φ ( φ ′ ) 2 δ φ − 2 G 4 φ φ G 4 φ φ ′ δ φ ′ ; (70) \begin{aligned}
\delta\varphi^{\prime\prime}+\left(3+\frac{H^{\prime}}{H}\right)\delta\varphi^{\prime}-\frac{4\pi G}{G_{4\varphi}}\frac{\rho_{\mathrm{m}0}}{H^{2}a^{3}}\delta+\frac{\nabla^{2}\delta\varphi}{a^{2}H^{2}}+2\frac{\Xi}{H^{2}}\delta\varphi\\
=-\frac{G_{4\varphi\varphi}}{G_{4\varphi}}\delta\varphi\left[\varphi^{\prime\prime}+\left(3+\frac{H^{\prime}}{H}\right)\varphi^{\prime}\right]-\frac{G_{4\varphi\varphi\varphi}}{G_{4\varphi}}(\varphi^{\prime})^{2}\delta\varphi-2\frac{G_{4\varphi\varphi}}{G_{4\varphi}}\varphi^{\prime}\delta\varphi^{\prime}; \tag*{(70)}
\end{aligned} δ φ ′′ + ( 3 + H H ′ ) δ φ ′ − G 4 φ 4 π G H 2 a 3 ρ m 0 δ + a 2 H 2 ∇ 2 δ φ + 2 H 2 Ξ δ φ = − G 4 φ G 4 φφ δ φ [ φ ′′ + ( 3 + H H ′ ) φ ′ ] − G 4 φ G 4 φφφ ( φ ′ ) 2 δ φ − 2 G 4 φ G 4 φφ φ ′ δ φ ′ ; (70) φ ′ ′ + ( 3 + H ′ H ) φ ′ − 4 π G G 4 φ ρ m 0 H 2 a 3 + 2 Ξ H 2 G 4 G 4 φ = G 4 φ φ G 4 φ ( φ ′ ) 2 . (71) \varphi^{\prime\prime}+\left(3+\frac{H^{\prime}}{H}\right)\varphi^{\prime}-\frac{4\pi G}{G_{4\varphi}}\frac{\rho_{\mathrm{m}0}}{H^{2}a^{3}}+2\frac{\Xi}{H^{2}}\frac{G_{4}}{G_{4\varphi}}=\frac{G_{4\varphi\varphi}}{G_{4\varphi}}(\varphi^{\prime})^{2}. \tag*{(71)} φ ′′ + ( 3 + H H ′ ) φ ′ − G 4 φ 4 π G H 2 a 3 ρ m 0 + 2 H 2 Ξ G 4 φ G 4 = G 4 φ G 4 φφ ( φ ′ ) 2 . (71) These three equations yield a coupled system for the three functions δ \delta δ δ φ \delta\varphi δ φ φ \varphi φ N = ln a N=\ln a N = ln a H H H G 4 G_{4} G 4
H ( a ) = − 1 2 Ω 0 H 0 + 1 4 Ω 0 2 + 2 Ω Ξ 0 H 0 2 [ ( C + C 2 − 1 ) a ± 3 + ( C + C 2 − 1 ) − 1 a ∓ 3 ] (72) H(a)=-\frac{1}{2}\Omega_{0}H_{0}+\sqrt{\frac{1}{4}\Omega_{0}^{2}+2\Omega_{\Xi0}}\frac{H_{0}}{2}\left[\left(C+\sqrt{C^{2}-1}\right)a^{\pm\sqrt{3}}+\left(C+\sqrt{C^{2}-1}\right)^{-1}a^{\mp\sqrt{3}}\right] \tag*{(72)} H ( a ) = − 2 1 Ω 0 H 0 + 4 1 Ω 0 2 + 2 Ω Ξ0 2 H 0 [ ( C + C 2 − 1 ) a ± 3 + ( C + C 2 − 1 ) − 1 a ∓ 3 ] (72) G 4 [ φ ( a ) ] = H 0 G 4 , 0 H ( a ) a 3 (73) G_{4}[\varphi(a)]=\frac{H_{0}G_{4,0}}{H(a)a^{3}} \tag*{(73)} G 4 [ φ ( a )] = H ( a ) a 3 H 0 G 4 , 0 (73) Since we have the explicit evolution of G 4 G_{4} G 4
G 4 φ = G 4 ′ φ ′ where G 4 ′ = − G 4 , 0 H a − 3 ( H ′ H + 3 ) ; (74) G_{4\varphi}=\frac{G_{4}^{\prime}}{\varphi^{\prime}}\quad\text{where}\quad G_{4}^{\prime}=-\frac{G_{4,0}}{H}a^{-3}\left(\frac{H^{\prime}}{H}+3\right); \tag*{(74)} G 4 φ = φ ′ G 4 ′ where G 4 ′ = − H G 4 , 0 a − 3 ( H H ′ + 3 ) ; (74) G 4 φ φ = G 4 ′ ′ φ ′ 2 − G 4 ′ φ ′ ′ φ ′ 3 where G_{4\varphi\varphi}=\frac{G_{4}^{\prime\prime}}{\varphi^{\prime2}}-\frac{G_{4}^{\prime}\varphi^{\prime\prime}}{\varphi^{\prime3}}\quad\text{where} G 4 φφ = φ ′2 G 4 ′′ − φ ′3 G 4 ′ φ ′′ where G 4 ′ ′ = G 4 , 0 H a − 3 ( H ′ H + 3 ) 2 − G 4 , 0 H a − 3 [ H ′ ′ H − ( H ′ H ) 2 ] ; (75) G_{4}^{\prime\prime}=\frac{G_{4,0}}{H}a^{-3}\left(\frac{H^{\prime}}{H}+3\right)^{2}-\frac{G_{4,0}}{H}a^{-3}\left[\frac{H^{\prime\prime}}{H}-\left(\frac{H^{\prime}}{H}\right)^{2}\right]; \tag*{(75)} G 4 ′′ = H G 4 , 0 a − 3 ( H H ′ + 3 ) 2 − H G 4 , 0 a − 3 [ H H ′′ − ( H H ′ ) 2 ] ; (75) G 4 φ φ φ = G 4 ′ ′ ′ φ ′ 3 − 2 G 4 ′ ′ φ ′ ′ φ ′ 4 − G 4 ′ φ ′ ′ ′ φ ′ 4 + 3 φ ′ ′ 2 G 4 ′ φ ′ 5 − φ ′ ′ G 4 ′ ′ φ ′ 4 where G_{4\varphi\varphi\varphi}=\frac{G_{4}^{\prime\prime\prime}}{\varphi^{\prime3}}-\frac{2G_{4}^{\prime\prime}\varphi^{\prime\prime}}{\varphi^{\prime4}}-\frac{G_{4}^{\prime}\varphi^{\prime\prime\prime}}{\varphi^{\prime4}}+\frac{3\varphi^{\prime\prime2}G_{4}^{\prime}}{\varphi^{\prime5}}-\frac{\varphi^{\prime\prime}G_{4}^{\prime\prime}}{\varphi^{\prime4}}\quad\text{where} G 4 φφφ = φ ′3 G 4 ′′′ − φ ′4 2 G 4 ′′ φ ′′ − φ ′4 G 4 ′ φ ′′′ + φ ′5 3 φ ′′2 G 4 ′ − φ ′4 φ ′′ G 4 ′′ where G 4 ′ ′ ′ = − G 4 , 0 H a − 3 ( H ′ H + 3 ) 3 + 3 G 4 , 0 H a − 3 ( H ′ H + 3 ) [ H ′ ′ H − ( H ′ H ) 2 ] G_{4}^{\prime\prime\prime}=-\frac{G_{4,0}}{H}a^{-3}\left(\frac{H^{\prime}}{H}+3\right)^{3}+3\frac{G_{4,0}}{H}a^{-3}\left(\frac{H^{\prime}}{H}+3\right)\left[\frac{H^{\prime\prime}}{H}-\left(\frac{H^{\prime}}{H}\right)^{2}\right] G 4 ′′′ = − H G 4 , 0 a − 3 ( H H ′ + 3 ) 3 + 3 H G 4 , 0 a − 3 ( H H ′ + 3 ) [ H H ′′ − ( H H ′ ) 2 ] − G 4 , 0 H a − 3 [ H ′ ′ ′ H − 3 H ′ ′ H ′ H 2 − 2 ( H ′ H ) 3 ] . (76) -\frac{G_{4,0}}{H}a^{-3}\left[\frac{H'''}{H}-3\frac{H''H'}{H^2}-2\left(\frac{H'}{H}\right)^3\right]. \tag*{(76)} − H G 4 , 0 a − 3 [ H H ′′′ − 3 H 2 H ′′ H ′ − 2 ( H H ′ ) 3 ] . (76) The third derivation of φ \varphi φ
φ ′ ′ ′ = − 9 4 Ω m a − 3 H 2 φ ′ G 4 ′ + 3 4 Ω m a − 3 H 2 φ ′ ′ G 4 ′ − 3 Ω X H 2 φ ′ − 3 Ω X G 4 H 2 G 4 ′ φ ′ ′ − 1 2 ( H ′ H ) 2 φ ′ − 1 2 H ′ ′ H φ ′ + 1 2 H ′ H G 4 ′ ′ G 4 ′ φ ′ − 5 2 H ′ H φ ′ ′ − 3 H ′ H φ ′ − 3 2 G 4 ′ ′ G 4 ′ φ ′ − 3 2 φ ′ ′ + 1 2 G 4 ′ ′ ′ G 4 ′ φ ′ − 1 2 G 4 ′ ′ G 4 ′ φ ′ ′ , \begin{aligned}
\varphi'''={}&-\frac{9}{4}\frac{\Omega_{\mathrm{m}}a^{-3}}{H^2}\frac{\varphi'}{G'_4}+\frac{3}{4}\frac{\Omega_{\mathrm{m}}a^{-3}}{H^2}\frac{\varphi''}{G'_4}-\frac{3\Omega_{\mathrm{X}}}{H^2}\varphi'-\frac{3\Omega_{\mathrm{X}}G_4}{H^2G'_4}\varphi''-\frac{1}{2}\left(\frac{H'}{H}\right)^2\varphi'\\
&-\frac{1}{2}\frac{H''}{H}\varphi'+\frac{1}{2}\frac{H'}{H}\frac{G''_4}{G'_4}\varphi'-\frac{5}{2}\frac{H'}{H}\varphi''-3\frac{H'}{H}\varphi'-\frac{3}{2}\frac{G''_4}{G'_4}\varphi'-\frac{3}{2}\varphi''+\frac{1}{2}\frac{G'''_4}{G'_4}\varphi'-\frac{1}{2}\frac{G''_4}{G'_4}\varphi'',
\end{aligned} φ ′′′ = − 4 9 H 2 Ω m a − 3 G 4 ′ φ ′ + 4 3 H 2 Ω m a − 3 G 4 ′ φ ′′ − H 2 3 Ω X φ ′ − H 2 G 4 ′ 3 Ω X G 4 φ ′′ − 2 1 ( H H ′ ) 2 φ ′ − 2 1 H H ′′ φ ′ + 2 1 H H ′ G 4 ′ G 4 ′′ φ ′ − 2 5 H H ′ φ ′′ − 3 H H ′ φ ′ − 2 3 G 4 ′ G 4 ′′ φ ′ − 2 3 φ ′′ + 2 1 G 4 ′ G 4 ′′′ φ ′ − 2 1 G 4 ′ G 4 ′′ φ ′′ , In order to compute the time derivatives of G 4 G_4 G 4 H H H
H ′ = 1 4 Ω 0 2 + 2 Ω Ξ 0 H 0 2 [ ± 3 ( C + C 2 − 1 ) a ± 3 ∓ 3 a ∓ 3 C + C 2 − 1 ] , (77) H'=\sqrt{\frac{1}{4}\Omega_0^2+2\Omega_{\Xi0}}\frac{H_0}{2}\left[\pm\sqrt{3}\left(C+\sqrt{C^2-1}\right)a^{\pm\sqrt{3}}\mp\frac{\sqrt{3}a^{\mp\sqrt{3}}}{C+\sqrt{C^2-1}}\right], \tag*{(77)} H ′ = 4 1 Ω 0 2 + 2 Ω Ξ0 2 H 0 [ ± 3 ( C + C 2 − 1 ) a ± 3 ∓ C + C 2 − 1 3 a ∓ 3 ] , (77) H ′ ′ = 1 4 Ω 0 2 + 2 Ω Ξ 0 H 0 2 [ 3 ( C + C 2 − 1 ) a ± 3 + 3 a ∓ 3 C + C 2 − 1 ] , (78) H''=\sqrt{\frac{1}{4}\Omega_0^2+2\Omega_{\Xi0}}\frac{H_0}{2}\left[3\left(C+\sqrt{C^2-1}\right)a^{\pm\sqrt{3}}+\frac{3a^{\mp\sqrt{3}}}{C+\sqrt{C^2-1}}\right], \tag*{(78)} H ′′ = 4 1 Ω 0 2 + 2 Ω Ξ0 2 H 0 [ 3 ( C + C 2 − 1 ) a ± 3 + C + C 2 − 1 3 a ∓ 3 ] , (78) H ′ ′ ′ = 3 H ′ . (79) H'''=3H'. \tag*{(79)} H ′′′ = 3 H ′ . (79) Substituting all of these quantities into the differential system in Eqs. (C.14), we obtain a set of equations in which the only unknowns are δ \delta δ δ φ \delta\varphi δ φ φ \varphi φ φ \varphi φ φ ′ \varphi' φ ′ [52 ] , where we just imposed initial conditions using the analytical solution available for φ ( a ) \varphi(a) φ ( a ) § 4 . Therefore, one may resort to the simplified approach offered by QSA, as was done in § 5 .
Horndeski curved universe
Linear perturbation theory in a Horndeski curved universe
In this section we derive the Einstein and KG equations for a Horndeski curved universe. We start from the metric in Eq. (34) . The resulting Christoffel symbols are, to first order in the perturbations,
Γ 00 0 = H + Ψ ′ , Γ 0 i 0 = ∂ i Ψ , Γ 00 i = ∂ i Ψ γ 2 , Γ j 0 i = ( H + Φ ′ ) δ j i , \begin{aligned}
\Gamma^{0}_{00}&=\mathcal{H}+\Psi',\\
\Gamma^{0}_{0i}&=\partial_i\Psi,\\
\Gamma^{i}_{00}&=\frac{\partial^i\Psi}{\gamma^2},\\
\Gamma^{i}_{j0}&=(\mathcal{H}+\Phi')\delta^i_j,
\end{aligned} Γ 00 0 Γ 0 i 0 Γ 00 i Γ j 0 i = H + Ψ ′ , = ∂ i Ψ , = γ 2 ∂ i Ψ , = ( H + Φ ′ ) δ j i , Γ i j 0 = γ 2 ( H + Φ ′ + 2 H ( Φ − Ψ ) ) δ i j ; \Gamma^{0}_{ij}=\gamma^{2}\left(\mathcal{H}+\Phi'+2\mathcal{H}(\Phi-\Psi)\right)\delta_{ij}; Γ ij 0 = γ 2 ( H + Φ ′ + 2 H ( Φ − Ψ ) ) δ ij ; Γ j k i = − K γ 2 ( x k δ j i + x j δ k i − x i δ j k ) + ∂ k Φ δ j i + ∂ j Φ δ k i − ∂ i Φ δ j k ; (80) \Gamma^{i}_{jk}=-\frac{\mathcal{K}\gamma}{2}\left(x_{k}\delta^{i}_{j}+x_{j}\delta^{i}_{k}-x^{i}\delta_{jk}\right)+\partial_{k}\Phi\,\delta^{i}_{j}+\partial_{j}\Phi\,\delta^{i}_{k}-\partial^{i}\Phi\,\delta_{jk}; \tag*{(80)} Γ j k i = − 2 K γ ( x k δ j i + x j δ k i − x i δ j k ) + ∂ k Φ δ j i + ∂ j Φ δ k i − ∂ i Φ δ j k ; (80) while the Ricci tensor components are
R 0 0 = a − 2 ( 3 H ′ − Δ Ψ γ 2 + K 2 γ ∂ i Ψ x i + 3 Φ ′ ′ − 6 H ′ Ψ + 3 H ( Φ ′ − Ψ ′ ) ) ; R^{0}_{0}=a^{-2}\left(3\mathcal{H}'-\frac{\Delta\Psi}{\gamma^{2}}+\frac{\mathcal{K}}{2\gamma}\partial_{i}\Psi\,x^{i}+3\Phi''-6\mathcal{H}'\Psi+3\mathcal{H}(\Phi'-\Psi')\right); R 0 0 = a − 2 ( 3 H ′ − γ 2 ΔΨ + 2 γ K ∂ i Ψ x i + 3 Φ ′′ − 6 H ′ Ψ + 3 H ( Φ ′ − Ψ ′ ) ) ; R i 0 = a − 2 ( 2 ∂ i Φ ′ − 2 H ∂ i Ψ ) ; R^{0}_{i}=a^{-2}\left(2\partial_{i}\Phi'-2\mathcal{H}\partial_{i}\Psi\right); R i 0 = a − 2 ( 2 ∂ i Φ ′ − 2 H ∂ i Ψ ) ; R 0 i = − a − 2 1 γ ( 2 ∂ i Φ ′ − 2 H ∂ i Ψ ) ; R^{i}_{0}=-a^{-2}\frac{1}{\gamma}\left(2\partial_{i}\Phi'-2\mathcal{H}\partial_{i}\Psi\right); R 0 i = − a − 2 γ 1 ( 2 ∂ i Φ ′ − 2 H ∂ i Ψ ) ; R j i = a − 2 [ ( 2 K + 2 H 2 + H ′ + 8 K Φ γ − 12 K Φ − Δ Φ γ 2 − 2 K 2 r 2 Φ + K γ ( ∂ k Φ x k + 1 2 ∂ k Ψ x k ) + Φ ′ ′ − 4 H 2 Ψ + 5 H Φ ′ − H Ψ ′ − 2 Ψ H ′ ) δ j i − ∂ i ∂ j Φ γ 2 − ∂ i ∂ j Ψ γ 2 + K 2 γ ( ∂ i Φ + ∂ i Ψ ) x j + K γ 2 ( ∂ j Φ + ∂ j Ψ ) x i ] , (81) \begin{aligned}
R^{i}_{j}=a^{-2}\Bigg[&
\left(2\mathcal{K}+2\mathcal{H}^{2}+\mathcal{H}'+\frac{8\mathcal{K}\Phi}{\gamma}-12\mathcal{K}\Phi-\frac{\Delta\Phi}{\gamma^{2}}-2\mathcal{K}^{2}r^{2}\Phi\right.\\
&\left.+\frac{\mathcal{K}}{\gamma}\left(\partial_{k}\Phi\,x^{k}+\frac{1}{2}\partial_{k}\Psi\,x^{k}\right)+\Phi''-4\mathcal{H}^{2}\Psi+5\mathcal{H}\Phi'-\mathcal{H}\Psi'-2\Psi\mathcal{H}'\right)\delta^{i}_{j}\\
&-\frac{\partial^{i}\partial_{j}\Phi}{\gamma^{2}}-\frac{\partial^{i}\partial_{j}\Psi}{\gamma^{2}}+\frac{\mathcal{K}}{2\gamma}\left(\partial^{i}\Phi+\partial^{i}\Psi\right)x_{j}
+\frac{\mathcal{K}\gamma}{2}\left(\partial_{j}\Phi+\partial_{j}\Psi\right)x^{i}\Bigg], \tag*{(81)}
\end{aligned} R j i = a − 2 [ ( 2 K + 2 H 2 + H ′ + γ 8 K Φ − 12 K Φ − γ 2 ΔΦ − 2 K 2 r 2 Φ + γ K ( ∂ k Φ x k + 2 1 ∂ k Ψ x k ) + Φ ′′ − 4 H 2 Ψ + 5 H Φ ′ − H Ψ ′ − 2Ψ H ′ ) δ j i − γ 2 ∂ i ∂ j Φ − γ 2 ∂ i ∂ j Ψ + 2 γ K ( ∂ i Φ + ∂ i Ψ ) x j + 2 K γ ( ∂ j Φ + ∂ j Ψ ) x i ] , (81) where γ = ( 1 + K r 2 / 4 ) − 1 \gamma=(1+\mathcal{K}r^{2}/4)^{-1} γ = ( 1 + K r 2 /4 ) − 1 H = a ′ / a \mathcal{H}=a'/a H = a ′ / a Φ \Phi Φ Ψ \Psi Ψ
− 3 G 4 φ H δ φ ′ a 2 − G 4 2 δ φ ′ φ ′ a 2 G 4 − 6 G 4 H Φ ′ a 2 − 3 G 4 φ φ ′ Φ ′ a 2 + G 4 φ Δ δ φ a 2 γ 2 − 3 G 4 φ H 2 δ φ a 2 − 3 G 4 φ φ H φ ′ δ φ a 2 + G 4 3 φ ′ 2 δ φ 2 a 2 G 4 2 − G 4 φ G 4 φ φ φ ′ 2 δ φ a 2 G 4 + 2 G 4 Δ Φ a 2 γ 2 − 7 G 4 K ∂ k Φ x k 4 a 2 γ − 3 G 4 K ∂ k Ψ x k 4 a 2 γ − G 4 φ K ∂ k δ φ x k 2 a 2 γ − G 4 γ K ∂ k Φ x k 4 a 2 − G 4 γ K ∂ k Ψ x k 4 a 2 − 3 G 4 φ K δ φ a 2 − G 4 φ δ φ Ξ + 18 G 4 K Φ a 2 − 12 G 4 K Φ a 2 γ + 3 G 4 r 2 K 2 Φ a 2 + 6 G 4 H 2 Ψ a 2 + 6 G 4 φ H φ ′ Ψ a 2 + G 4 2 φ ′ 2 Ψ a 2 G 4 = 0 (82) \begin{aligned}
&-\frac{3G_{4\varphi}\mathcal{H}\delta\varphi'}{a^{2}}-\frac{G_{4}^{2}\delta\varphi'\varphi'}{a^{2}G_{4}}-\frac{6G_{4}\mathcal{H}\Phi'}{a^{2}}-\frac{3G_{4\varphi}\varphi'\Phi'}{a^{2}}+\frac{G_{4\varphi}\Delta\delta\varphi}{a^{2}\gamma^{2}}-\frac{3G_{4\varphi}\mathcal{H}^{2}\delta\varphi}{a^{2}}\\
&-\frac{3G_{4\varphi\varphi}\mathcal{H}\varphi'\delta\varphi}{a^{2}}+\frac{G_{4}^{3}\varphi'^{2}\delta\varphi}{2a^{2}G_{4}^{2}}-\frac{G_{4\varphi}G_{4\varphi\varphi}\varphi'^{2}\delta\varphi}{a^{2}G_{4}}+\frac{2G_{4}\Delta\Phi}{a^{2}\gamma^{2}}-\frac{7G_{4}\mathcal{K}\partial_{k}\Phi\,x^{k}}{4a^{2}\gamma}-\frac{3G_{4}\mathcal{K}\partial_{k}\Psi\,x^{k}}{4a^{2}\gamma}\\
&-\frac{G_{4\varphi}\mathcal{K}\partial_{k}\delta\varphi\,x^{k}}{2a^{2}\gamma}-\frac{G_{4}\gamma\mathcal{K}\partial_{k}\Phi\,x^{k}}{4a^{2}}-\frac{G_{4}\gamma\mathcal{K}\partial_{k}\Psi\,x^{k}}{4a^{2}}-\frac{3G_{4\varphi}\mathcal{K}\delta\varphi}{a^{2}}-G_{4\varphi}\delta\varphi\Xi+\frac{18G_{4}\mathcal{K}\Phi}{a^{2}}\\
&-\frac{12G_{4}\mathcal{K}\Phi}{a^{2}\gamma}+\frac{3G_{4}r^{2}\mathcal{K}^{2}\Phi}{a^{2}}+\frac{6G_{4}\mathcal{H}^{2}\Psi}{a^{2}}+\frac{6G_{4\varphi}\mathcal{H}\varphi'\Psi}{a^{2}}+\frac{G_{4}^{2}\varphi'^{2}\Psi}{a^{2}G_{4}}=0 \tag*{(82)}
\end{aligned} − a 2 3 G 4 φ H δ φ ′ − a 2 G 4 G 4 2 δ φ ′ φ ′ − a 2 6 G 4 H Φ ′ − a 2 3 G 4 φ φ ′ Φ ′ + a 2 γ 2 G 4 φ Δ δ φ − a 2 3 G 4 φ H 2 δ φ − a 2 3 G 4 φφ H φ ′ δ φ + 2 a 2 G 4 2 G 4 3 φ ′ 2 δ φ − a 2 G 4 G 4 φ G 4 φφ φ ′ 2 δ φ + a 2 γ 2 2 G 4 ΔΦ − 4 a 2 γ 7 G 4 K ∂ k Φ x k − 4 a 2 γ 3 G 4 K ∂ k Ψ x k − 2 a 2 γ G 4 φ K ∂ k δ φ x k − 4 a 2 G 4 γ K ∂ k Φ x k − 4 a 2 G 4 γ K ∂ k Ψ x k − a 2 3 G 4 φ K δ φ − G 4 φ δ φ Ξ + a 2 18 G 4 K Φ − a 2 γ 12 G 4 K Φ + a 2 3 G 4 r 2 K 2 Φ + a 2 6 G 4 H 2 Ψ + a 2 6 G 4 φ H φ ′ Ψ + a 2 G 4 G 4 2 φ ′ 2 Ψ = 0 (82) G 4 φ ∂ i δ φ ′ a 2 + 2 G 4 ∂ i Φ ′ a 2 − G 4 φ H ∂ i δ φ a 2 − G 4 2 φ ′ ∂ i δ φ a 2 G 4 + G 4 φ φ φ ′ ∂ i δ φ a 2 − 2 G 4 H ∂ i Ψ a 2 − G 4 φ φ ′ ∂ i Ψ a 2 = 0 (83) \frac{G_{4\varphi}\partial_{i}\delta\varphi'}{a^{2}}+\frac{2G_{4}\partial_{i}\Phi'}{a^{2}}-\frac{G_{4\varphi}\mathcal{H}\partial_{i}\delta\varphi}{a^{2}}-\frac{G_{4}^{2}\varphi'\partial_{i}\delta\varphi}{a^{2}G_{4}}+\frac{G_{4\varphi\varphi}\varphi'\partial_{i}\delta\varphi}{a^{2}}-\frac{2G_{4}\mathcal{H}\partial_{i}\Psi}{a^{2}}-\frac{G_{4\varphi}\varphi'\partial_{i}\Psi}{a^{2}}=0 \tag*{(83)} a 2 G 4 φ ∂ i δ φ ′ + a 2 2 G 4 ∂ i Φ ′ − a 2 G 4 φ H ∂ i δ φ − a 2 G 4 G 4 2 φ ′ ∂ i δ φ + a 2 G 4 φφ φ ′ ∂ i δ φ − a 2 2 G 4 H ∂ i Ψ − a 2 G 4 φ φ ′ ∂ i Ψ = 0 (83) − G 4 φ ∇ i ∂ j δ φ a 2 γ 2 − G 4 ∇ i ∂ j Φ a 2 γ 2 − G 4 ∇ i ∂ j Ψ a 2 γ 2 − G 4 φ δ φ ′ ′ a 2 δ j i − 2 G 4 Φ ′ ′ a 2 δ j i − G 4 φ H δ φ ′ a 2 δ j i + G 4 2 φ ′ δ φ ′ a 2 G 4 δ j i − 2 G 4 φ φ δ φ ′ φ ′ a 2 δ j i − 4 G 4 H Φ ′ a 2 δ j i − 2 G 4 φ φ ′ Φ ′ a 2 δ j i + 2 G 4 H Ψ ′ a 2 δ j i + G 4 φ φ ′ Ψ ′ a 2 δ j i + G 4 φ Δ δ φ a 2 γ 2 δ j i − G 4 φ H 2 δ φ a 2 δ j i − G 4 φ φ φ ′ ′ δ φ a 2 δ j i − 2 G 4 φ H ′ δ φ a 2 δ j i − G 4 φ φ H φ ′ δ φ a 2 δ j i − G 4 3 φ ′ 2 δ φ 2 a 2 G 4 2 δ j i + G 4 φ G 4 φ φ φ ′ 2 δ φ a 2 G 4 δ j i − G 4 φ φ φ φ ′ 2 δ φ a 2 δ j i + G 4 Δ Φ a 2 γ 2 δ j i + G 4 Δ Ψ a 2 γ 2 δ j i \begin{aligned}
&-\frac{G_{4\varphi}\nabla^{i}\partial_{j}\delta\varphi}{a^{2}\gamma^{2}}-\frac{G_{4}\nabla^{i}\partial_{j}\Phi}{a^{2}\gamma^{2}}-\frac{G_{4}\nabla^{i}\partial_{j}\Psi}{a^{2}\gamma^{2}}-\frac{G_{4\varphi}\delta\varphi''}{a^{2}}\delta^{i}_{j}-\frac{2G_{4}\Phi''}{a^{2}}\delta^{i}_{j}-\frac{G_{4\varphi}\mathcal{H}\delta\varphi'}{a^{2}}\delta^{i}_{j}\\
&+\frac{G_{4}^{2}\varphi'\delta\varphi'}{a^{2}G_{4}}\delta^{i}_{j}-\frac{2G_{4\varphi\varphi}\delta\varphi'\varphi'}{a^{2}}\delta^{i}_{j}-\frac{4G_{4}\mathcal{H}\Phi'}{a^{2}}\delta^{i}_{j}-\frac{2G_{4\varphi}\varphi'\Phi'}{a^{2}}\delta^{i}_{j}+\frac{2G_{4}\mathcal{H}\Psi'}{a^{2}}\delta^{i}_{j}+\frac{G_{4\varphi}\varphi'\Psi'}{a^{2}}\delta^{i}_{j}\\
&+\frac{G_{4\varphi}\Delta\delta\varphi}{a^{2}\gamma^{2}}\delta^{i}_{j}-\frac{G_{4\varphi}\mathcal{H}^{2}\delta\varphi}{a^{2}}\delta^{i}_{j}-\frac{G_{4\varphi\varphi}\varphi''\delta\varphi}{a^{2}}\delta^{i}_{j}-\frac{2G_{4\varphi}\mathcal{H}'\delta\varphi}{a^{2}}\delta^{i}_{j}-\frac{G_{4\varphi\varphi}\mathcal{H}\varphi'\delta\varphi}{a^{2}}\delta^{i}_{j}\\
&-\frac{G_{4}^{3}\varphi'^{2}\delta\varphi}{2a^{2}G_{4}^{2}}\delta^{i}_{j}+\frac{G_{4\varphi}G_{4\varphi\varphi}\varphi'^{2}\delta\varphi}{a^{2}G_{4}}\delta^{i}_{j}-\frac{G_{4\varphi\varphi\varphi}\varphi'^{2}\delta\varphi}{a^{2}}\delta^{i}_{j}+\frac{G_{4}\Delta\Phi}{a^{2}\gamma^{2}}\delta^{i}_{j}+\frac{G_{4}\Delta\Psi}{a^{2}\gamma^{2}}\delta^{i}_{j}
\end{aligned} − a 2 γ 2 G 4 φ ∇ i ∂ j δ φ − a 2 γ 2 G 4 ∇ i ∂ j Φ − a 2 γ 2 G 4 ∇ i ∂ j Ψ − a 2 G 4 φ δ φ ′′ δ j i − a 2 2 G 4 Φ ′′ δ j i − a 2 G 4 φ H δ φ ′ δ j i + a 2 G 4 G 4 2 φ ′ δ φ ′ δ j i − a 2 2 G 4 φφ δ φ ′ φ ′ δ j i − a 2 4 G 4 H Φ ′ δ j i − a 2 2 G 4 φ φ ′ Φ ′ δ j i + a 2 2 G 4 H Ψ ′ δ j i + a 2 G 4 φ φ ′ Ψ ′ δ j i + a 2 γ 2 G 4 φ Δ δ φ δ j i − a 2 G 4 φ H 2 δ φ δ j i − a 2 G 4 φφ φ ′′ δ φ δ j i − a 2 2 G 4 φ H ′ δ φ δ j i − a 2 G 4 φφ H φ ′ δ φ δ j i − 2 a 2 G 4 2 G 4 3 φ ′ 2 δ φ δ j i + a 2 G 4 G 4 φ G 4 φφ φ ′ 2 δ φ δ j i − a 2 G 4 φφφ φ ′ 2 δ φ δ j i + a 2 γ 2 G 4 ΔΦ δ j i + a 2 γ 2 G 4 ΔΨ δ j i − 3 G 4 K ∂ k Φ x k 4 a 2 γ δ j i − 3 G 4 K ∂ k Ψ x k 4 a 2 γ δ j i − G 4 ′ γ K ∂ k Φ x k 4 a 2 δ j i − G 4 ′ γ K ∂ k Ψ x k 4 a 2 δ j i − G 4 ϕ K δ ϕ a 2 δ j i − G 4 ϕ δ ϕ Ξ a 2 δ j i + 6 G 4 K Φ a 2 δ j i − 4 G 4 K Φ a 2 γ δ j i + G 4 r 2 K 2 Φ a 2 δ j i + 2 G 4 H 2 Ψ a 2 δ j i + 2 G 4 ϕ ϕ Ψ ϕ ′ ′ a 2 δ j i + 4 G 4 H ′ Ψ a 2 δ j i + 2 G 4 ϕ H ϕ ′ Ψ a 2 δ j i − G 4 ϕ ϕ ′ 2 Ψ a 2 G 4 δ j i + 2 G 4 ϕ ϕ ϕ ′ ′ Ψ a 2 δ j i − G 4 ϕ K ∂ k δ ϕ x k 2 a 2 γ ∂ j δ ϕ x i + G 4 ′ γ K ∂ k Φ x k 2 a 2 ∂ j Φ x i + G 4 ′ γ K ∂ k Ψ x k 2 a 2 ∂ j Ψ x i − G 4 ϕ K ∂ k δ ϕ x k 2 a 2 γ ∂ i δ ϕ x j + G 4 K ∂ k Φ x k 2 a 2 γ ∂ i Φ x j + G 4 K ∂ k Ψ x k 2 a 2 γ ∂ i Ψ x j = 0. (84) -\frac{3G_4\mathcal{K}\partial_k\Phi\,x^k}{4a^2\gamma}\delta^i_j-\frac{3G_4\mathcal{K}\partial_k\Psi\,x^k}{4a^2\gamma}\delta^i_j-\frac{G_4'\gamma\mathcal{K}\partial_k\Phi\,x^k}{4a^2}\delta^i_j-\frac{G_4'\gamma\mathcal{K}\partial_k\Psi\,x^k}{4a^2}\delta^i_j-\frac{G_{4\phi}\mathcal{K}\delta\phi}{a^2}\delta^i_j
-\frac{G_{4\phi}\delta\phi\Xi}{a^2}\delta^i_j+\frac{6G_4\mathcal{K}\Phi}{a^2}\delta^i_j-\frac{4G_4\mathcal{K}\Phi}{a^2\gamma}\delta^i_j+\frac{G_4r^2\mathcal{K}^2\Phi}{a^2}\delta^i_j+\frac{2G_4\mathcal{H}^2\Psi}{a^2}\delta^i_j+\frac{2G_{4\phi\phi}\Psi\phi''}{a^2}\delta^i_j
+\frac{4G_4\mathcal{H}'\Psi}{a^2}\delta^i_j+\frac{2G_{4\phi}\mathcal{H}\phi'\Psi}{a^2}\delta^i_j-\frac{G_{4\phi}\phi'^2\Psi}{a^2G_4}\delta^i_j+\frac{2G_{4\phi\phi}\phi''\Psi}{a^2}\delta^i_j-\frac{G_{4\phi}\mathcal{K}\partial_k\delta\phi\,x^k}{2a^2\gamma}\partial_j\delta\phi\,x^i
+\frac{G_4'\gamma\mathcal{K}\partial_k\Phi\,x^k}{2a^2}\partial_j\Phi\,x^i+\frac{G_4'\gamma\mathcal{K}\partial_k\Psi\,x^k}{2a^2}\partial_j\Psi\,x^i-\frac{G_{4\phi}\mathcal{K}\partial_k\delta\phi\,x^k}{2a^2\gamma}\partial_i\delta\phi\,x_j+\frac{G_4\mathcal{K}\partial_k\Phi\,x^k}{2a^2\gamma}\partial_i\Phi\,x_j
+\frac{G_4\mathcal{K}\partial_k\Psi\,x^k}{2a^2\gamma}\partial_i\Psi\,x_j=0. \tag*{(84)} − 4 a 2 γ 3 G 4 K ∂ k Φ x k δ j i − 4 a 2 γ 3 G 4 K ∂ k Ψ x k δ j i − 4 a 2 G 4 ′ γ K ∂ k Φ x k δ j i − 4 a 2 G 4 ′ γ K ∂ k Ψ x k δ j i − a 2 G 4 ϕ K δ ϕ δ j i − a 2 G 4 ϕ δ ϕ Ξ δ j i + a 2 6 G 4 K Φ δ j i − a 2 γ 4 G 4 K Φ δ j i + a 2 G 4 r 2 K 2 Φ δ j i + a 2 2 G 4 H 2 Ψ δ j i + a 2 2 G 4 ϕϕ Ψ ϕ ′′ δ j i + a 2 4 G 4 H ′ Ψ δ j i + a 2 2 G 4 ϕ H ϕ ′ Ψ δ j i − a 2 G 4 G 4 ϕ ϕ ′2 Ψ δ j i + a 2 2 G 4 ϕϕ ϕ ′′ Ψ δ j i − 2 a 2 γ G 4 ϕ K ∂ k δ ϕ x k ∂ j δ ϕ x i + 2 a 2 G 4 ′ γ K ∂ k Φ x k ∂ j Φ x i + 2 a 2 G 4 ′ γ K ∂ k Ψ x k ∂ j Ψ x i − 2 a 2 γ G 4 ϕ K ∂ k δ ϕ x k ∂ i δ ϕ x j + 2 a 2 γ G 4 K ∂ k Φ x k ∂ i Φ x j + 2 a 2 γ G 4 K ∂ k Ψ x k ∂ i Ψ x j = 0. (84) The fourth and final equation is the perturbed KG equation,
− 2 G 4 ϕ δ ϕ ′ ′ a 2 − 6 G 4 Φ ′ ′ a 2 − 4 G 4 ϕ H δ ϕ ′ a 2 + 2 G 4 ϕ ϕ ϕ ′ δ ϕ ′ a 2 G 4 − 4 G 4 ϕ ϕ ϕ ′ δ ϕ ′ a 2 − 18 G 4 H Φ ′ a 2 − 6 G 4 ϕ ϕ Φ ′ a 2 + 6 G 4 H Ψ ′ a 2 + 2 G 4 ϕ ϕ ′ Ψ ′ a 2 + 2 G 4 ϕ Δ δ ϕ a 2 γ 2 − 6 G 4 ϕ H 2 δ ϕ a 2 − 2 G 4 ϕ ϕ ϕ ′ ′ δ ϕ a 2 − 6 G 4 ϕ H ′ δ ϕ a 2 − 4 G 4 ϕ ϕ H ϕ ′ δ ϕ a 2 − G 4 3 ϕ ′ 2 δ ϕ a 2 G 4 2 + 2 G 4 ϕ G 4 ϕ ϕ ϕ ′ 2 δ ϕ a 2 G 4 − 2 G 4 ϕ ϕ ϕ ′ 2 δ ϕ a 2 + 4 G 4 Δ Φ a 2 γ 2 + 2 G 4 Δ Ψ a 2 γ 2 − 7 G 4 K ∂ k Φ x k 2 a 2 γ − 5 G 4 K ∂ k Ψ x k 2 a 2 γ − G 4 ϕ K ∂ k δ ϕ x k a 2 γ − G 4 ′ γ K ∂ k Φ x k 2 a 2 − G 4 ′ γ K ∂ k Ψ x k 2 a 2 − 6 G 4 ϕ K δ ϕ a 2 − 2 G 4 ϕ δ ϕ Ξ + 36 G 4 K Φ a 2 − 24 G 4 K Φ a 2 γ + 6 G 4 r 2 K 2 Φ a 2 + 12 G 4 H 2 Ψ a 2 + 4 G 4 ϕ ϕ ϕ ′ ′ Ψ a 2 + 12 G 4 H ′ Ψ a 2 + 8 G 4 ϕ H ϕ ′ Ψ a 2 − 2 G 4 2 ϕ ′ 2 Ψ a 2 G 4 + 4 G 4 ϕ ϕ ϕ ′ 2 Ψ a 2 = 0. (85) -\frac{2G_{4\phi}\delta\phi''}{a^2}-\frac{6G_4\Phi''}{a^2}-\frac{4G_{4\phi}\mathcal{H}\delta\phi'}{a^2}+\frac{2G_{4\phi\phi}\phi'\delta\phi'}{a^2G_4}-\frac{4G_{4\phi\phi}\phi'\delta\phi'}{a^2}-\frac{18G_4\mathcal{H}\Phi'}{a^2}-\frac{6G_{4\phi\phi}\Phi'}{a^2}
+\frac{6G_4\mathcal{H}\Psi'}{a^2}+\frac{2G_{4\phi}\phi'\Psi'}{a^2}+\frac{2G_{4\phi}\Delta\delta\phi}{a^2\gamma^2}-\frac{6G_{4\phi}\mathcal{H}^2\delta\phi}{a^2}-\frac{2G_{4\phi\phi}\phi''\delta\phi}{a^2}-\frac{6G_{4\phi}\mathcal{H}'\delta\phi}{a^2}
-\frac{4G_{4\phi\phi}\mathcal{H}\phi'\delta\phi}{a^2}-\frac{G_4^3\phi'^2\delta\phi}{a^2G_4^2}+\frac{2G_{4\phi}G_{4\phi\phi}\phi'^2\delta\phi}{a^2G_4}-\frac{2G_{4\phi\phi}\phi'^2\delta\phi}{a^2}+\frac{4G_4\Delta\Phi}{a^2\gamma^2}+\frac{2G_4\Delta\Psi}{a^2\gamma^2}
-\frac{7G_4\mathcal{K}\partial_k\Phi\,x^k}{2a^2\gamma}-\frac{5G_4\mathcal{K}\partial_k\Psi\,x^k}{2a^2\gamma}-\frac{G_{4\phi}\mathcal{K}\partial_k\delta\phi\,x^k}{a^2\gamma}-\frac{G_4'\gamma\mathcal{K}\partial_k\Phi\,x^k}{2a^2}-\frac{G_4'\gamma\mathcal{K}\partial_k\Psi\,x^k}{2a^2}-\frac{6G_{4\phi}\mathcal{K}\delta\phi}{a^2}
-2G_{4\phi}\delta\phi\Xi+\frac{36G_4\mathcal{K}\Phi}{a^2}-\frac{24G_4\mathcal{K}\Phi}{a^2\gamma}+\frac{6G_4r^2\mathcal{K}^2\Phi}{a^2}+\frac{12G_4\mathcal{H}^2\Psi}{a^2}+\frac{4G_{4\phi\phi}\phi''\Psi}{a^2}+\frac{12G_4\mathcal{H}'\Psi}{a^2}
+\frac{8G_{4\phi}\mathcal{H}\phi'\Psi}{a^2}-\frac{2G_4^2\phi'^2\Psi}{a^2G_4}+\frac{4G_{4\phi\phi}\phi'^2\Psi}{a^2}=0. \tag*{(85)} − a 2 2 G 4 ϕ δ ϕ ′′ − a 2 6 G 4 Φ ′′ − a 2 4 G 4 ϕ H δ ϕ ′ + a 2 G 4 2 G 4 ϕϕ ϕ ′ δ ϕ ′ − a 2 4 G 4 ϕϕ ϕ ′ δ ϕ ′ − a 2 18 G 4 H Φ ′ − a 2 6 G 4 ϕϕ Φ ′ + a 2 6 G 4 H Ψ ′ + a 2 2 G 4 ϕ ϕ ′ Ψ ′ + a 2 γ 2 2 G 4 ϕ Δ δ ϕ − a 2 6 G 4 ϕ H 2 δ ϕ − a 2 2 G 4 ϕϕ ϕ ′′ δ ϕ − a 2 6 G 4 ϕ H ′ δ ϕ − a 2 4 G 4 ϕϕ H ϕ ′ δ ϕ − a 2 G 4 2 G 4 3 ϕ ′2 δ ϕ + a 2 G 4 2 G 4 ϕ G 4 ϕϕ ϕ ′2 δ ϕ − a 2 2 G 4 ϕϕ ϕ ′2 δ ϕ + a 2 γ 2 4 G 4 ΔΦ + a 2 γ 2 2 G 4 ΔΨ − 2 a 2 γ 7 G 4 K ∂ k Φ x k − 2 a 2 γ 5 G 4 K ∂ k Ψ x k − a 2 γ G 4 ϕ K ∂ k δ ϕ x k − 2 a 2 G 4 ′ γ K ∂ k Φ x k − 2 a 2 G 4 ′ γ K ∂ k Ψ x k − a 2 6 G 4 ϕ K δ ϕ − 2 G 4 ϕ δ ϕ Ξ + a 2 36 G 4 K Φ − a 2 γ 24 G 4 K Φ + a 2 6 G 4 r 2 K 2 Φ + a 2 12 G 4 H 2 Ψ + a 2 4 G 4 ϕϕ ϕ ′′ Ψ + a 2 12 G 4 H ′ Ψ + a 2 8 G 4 ϕ H ϕ ′ Ψ − a 2 G 4 2 G 4 2 ϕ ′2 Ψ + a 2 4 G 4 ϕϕ ϕ ′2 Ψ = 0. (85) When we consider the flat universe limit, i.e. K → 0 \mathcal{K}\to0 K → 0 γ → 1 \gamma\to1 γ → 1 § 4 and Appendix § 12 .§ 1 .
The next step is to derive the growth factor equation for the case of a curved universe, i.e one with the metric in Eq. (34) . This derivation is easily performed by perturbing the conservation equation, ∇ μ T ν μ = 0 \nabla_\mu T^\mu_\nu=0 ∇ μ T ν μ = 0
δ ′ + 3 H ( c s 2 − w ) δ + ( θ + 3 Φ ′ ) ( 1 + w ) − 3 K 2 γ x j v j ( 1 + w ) = 0 , (86) \delta'+3\mathcal{H}(c_s^2-w)\delta+(\theta+3\Phi')(1+w)-\frac{3\mathcal{K}}{2\gamma}x_jv^j(1+w)=0, \tag*{(86)} δ ′ + 3 H ( c s 2 − w ) δ + ( θ + 3 Φ ′ ) ( 1 + w ) − 2 γ 3 K x j v j ( 1 + w ) = 0 , (86) and the Euler equation
ρ ( 1 + w ) ( H θ + θ ′ + ∇ 2 Ψ − 3 w H θ ) + c s 2 ρ ∇ 2 δ + 2 3 ∇ 2 ∇ 2 Π − K γ x i ∂ j ∇ 2 Π − 3 4 K 2 γ 2 x i x j π j i = 0 , (87) \rho(1+w)(\mathcal{H}\theta+\theta'+\nabla^2\Psi-3w\mathcal{H}\theta)+c_s^2\rho\nabla^2\delta+\frac{2}{3}\nabla^2\nabla^2\Pi-\mathcal{K}\gamma x^i\partial_j\nabla^2\Pi-\frac{3}{4}\mathcal{K}^2\gamma^2x_ix^j\pi^i_j=0, \tag*{(87)} ρ ( 1 + w ) ( H θ + θ ′ + ∇ 2 Ψ − 3 w H θ ) + c s 2 ρ ∇ 2 δ + 3 2 ∇ 2 ∇ 2 Π − K γ x i ∂ j ∇ 2 Π − 4 3 K 2 γ 2 x i x j π j i = 0 , (87) which, in the case c s 2 = w = Π = 0 c_s^2=w=\Pi=0 c s 2 = w = Π = 0
δ ′ + θ + 3 Φ ′ − 3 K 2 γ x j v j ( 1 + w ) = 0 (88) \delta'+\theta+3\Phi'-\frac{3\mathcal{K}}{2\gamma}x_jv^j(1+w)=0 \tag*{(88)} δ ′ + θ + 3 Φ ′ − 2 γ 3 K x j v j ( 1 + w ) = 0 (88) θ ′ + H θ + ∇ 2 Ψ = 0. (89) \theta'+\mathcal{H}\theta+\nabla^2\Psi=0. \tag*{(89)} θ ′ + H θ + ∇ 2 Ψ = 0. (89) We combine these two equations into
δ ′ ′ + ( 2 + H ′ H ) δ ′ = − 3 Φ ′ ′ − 3 ( 2 + H ′ H ) Φ ′ + ∇ 2 Ψ γ 2 ( a H ) 2 − K x j ∂ j Ψ 2 γ ( a H ) 2 (90) \delta''+\left(2+\frac{H'}{H}\right)\delta'=-3\Phi''-3\left(2+\frac{H'}{H}\right)\Phi'+\frac{\nabla^2\Psi}{\gamma^2(aH)^2}-\frac{\mathcal{K}x^j\partial_j\Psi}{2\gamma(aH)^2} \tag*{(90)} δ ′′ + ( 2 + H H ′ ) δ ′ = − 3 Φ ′′ − 3 ( 2 + H H ′ ) Φ ′ + γ 2 ( a H ) 2 ∇ 2 Ψ − 2 γ ( a H ) 2 K x j ∂ j Ψ (90) which is the curved extension of Eq. (C.13). Now we proceed in an identical fashion as in Appendix C.1: we solve for φ ′ ′ \varphi'' φ ′′ ∇ 2 Ψ \nabla^2\Psi ∇ 2 Ψ (35) .
QSA in a Horndeski curved universe
In this appendix we illustrate the detailed computations related to the application of the QSA to the curved case, as discussed at the end of § 6 . In particular, following [85 ] , we use a set of perturbed equations which includes the field equations and the evolution equation for the scalar field. We then apply the QSA to this set, to show that the presence of curvature does not influence the final result.
The curved metric in the Newtonian gauge is
d s 2 = − ( 1 + 2 Φ ) d t 2 + a 2 ( 1 − 2 Ψ ) γ i j d x i d x j \mathrm{d}s^2=-(1+2\Phi)\mathrm{d}t^2+a^2(1-2\Psi)\gamma_{ij}\mathrm{d}x^i\mathrm{d}x^j d s 2 = − ( 1 + 2Φ ) d t 2 + a 2 ( 1 − 2Ψ ) γ ij d x i d x j with γ i j = δ i j / ( 1 + K r 2 / 4 ) 2 \gamma_{ij}=\delta_{ij}/(1+\mathcal{K}r^2/4)^2 γ ij = δ ij / ( 1 + K r 2 /4 ) 2 Φ \Phi Φ Ψ \Psi Ψ [85 ] . From this metric, the covariant derivatives are:
D i D j = ∂ i ∂ j − K 2 γ [ x i ∂ j + x j ∂ i − δ i j x l ∂ l ] , D_iD_j=\partial_i\partial_j-\frac{\mathcal{K}}{2\gamma}\left[x_i\partial_j+x_j\partial_i-\delta_{ij}x^l\partial_l\right], D i D j = ∂ i ∂ j − 2 γ K [ x i ∂ j + x j ∂ i − δ ij x l ∂ l ] , D 2 Ψ = γ 2 ∇ 2 Ψ − K γ 2 x i ∂ i Ψ , D^2\Psi=\gamma^2\nabla^2\Psi-\frac{\mathcal{K}\gamma}{2}x^i\partial_i\Psi, D 2 Ψ = γ 2 ∇ 2 Ψ − 2 K γ x i ∂ i Ψ , D 2 Φ = γ 2 ∇ 2 Φ − K γ 2 x i ∂ i Φ , D^2\Phi=\gamma^2\nabla^2\Phi-\frac{\mathcal{K}\gamma}{2}x^i\partial_i\Phi, D 2 Φ = γ 2 ∇ 2 Φ − 2 K γ x i ∂ i Φ , D 2 π = γ 2 ∇ 2 π − K γ 2 x i ∂ i π , D^2\pi=\gamma^2\nabla^2\pi-\frac{\mathcal{K}\gamma}{2}x^i\partial_i\pi, D 2 π = γ 2 ∇ 2 π − 2 K γ x i ∂ i π , where γ = ( 1 + K r 2 / 4 ) \gamma=(1+\mathcal{K}r^2/4) γ = ( 1 + K r 2 /4 ) γ i j = δ i j / γ 2 \gamma_{ij}=\delta_{ij}/\gamma^2 γ ij = δ ij / γ 2 π = δ φ / φ ˙ \pi=\delta\varphi/\dot{\varphi} π = δ φ / φ ˙ [85 ] : the (00) component,
6 ( 1 + α B ) H Ψ ˙ + ( 6 − α K + 12 α B ) H 2 Φ − 2 ( 1 + α H ) ( D 2 + 3 K ) a 2 Ψ + ( α K − 6 α B ) H 2 π ˙ + 6 [ ( 1 + α B ) H ˙ + ρ m + p m 2 M 2 + ( 1 − α B ) K a 2 − 1 3 ( α H − α B ) ( D 2 + 3 K ) a 2 ] H π = − δ ρ m M 2 (91) 6(1+\alpha_{\mathrm{B}})H\dot{\Psi}+(6-\alpha_{\mathrm{K}}+12\alpha_{\mathrm{B}})H^2\Phi-2(1+\alpha_{\mathrm{H}})\frac{(D^2+3\mathcal{K})}{a^2}\Psi+(\alpha_{\mathrm{K}}-6\alpha_{\mathrm{B}})H^2\dot{\pi}
+6\left[(1+\alpha_{\mathrm{B}})\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}+(1-\alpha_{\mathrm{B}})\frac{\mathcal{K}}{a^2}-\frac{1}{3}(\alpha_{\mathrm{H}}-\alpha_{\mathrm{B}})\frac{(D^2+3\mathcal{K})}{a^2}\right]H\pi=-\frac{\delta\rho_{\mathrm{m}}}{M^2} \tag*{(91)} 6 ( 1 + α B ) H Ψ ˙ + ( 6 − α K + 12 α B ) H 2 Φ − 2 ( 1 + α H ) a 2 ( D 2 + 3 K ) Ψ + ( α K − 6 α B ) H 2 π ˙ + 6 [ ( 1 + α B ) H ˙ + 2 M 2 ρ m + p m + ( 1 − α B ) a 2 K − 3 1 ( α H − α B ) a 2 ( D 2 + 3 K ) ] H π = − M 2 δ ρ m (91) the (0i) components,
2 Ψ ˙ + 2 ( 1 + α B ) H Φ − 2 H α B π ˙ + ( 2 H ˙ + ρ m + p m M 2 ) π = − ( ρ m + p m ) v m M 2 (92) 2\dot{\Psi}+2(1+\alpha_{\mathrm{B}})H\Phi-2H\alpha_{\mathrm{B}}\dot{\pi}+\left(2\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{M^2}\right)\pi=-\frac{(\rho_{\mathrm{m}}+p_{\mathrm{m}})v_{\mathrm{m}}}{M^2} \tag*{(92)} 2 Ψ ˙ + 2 ( 1 + α B ) H Φ − 2 H α B π ˙ + ( 2 H ˙ + M 2 ρ m + p m ) π = − M 2 ( ρ m + p m ) v m (92) the traceless part of the (ij) components,
( 1 + α H ) Φ − ( 1 + α T ) Ψ + ( α M − α T ) H π − α H π ˙ = − σ m M 2 (93) (1+\alpha_{\mathrm{H}})\Phi-(1+\alpha_{\mathrm{T}})\Psi+(\alpha_{\mathrm{M}}-\alpha_{\mathrm{T}})H\pi-\alpha_{\mathrm{H}}\dot{\pi}=-\frac{\sigma_{\mathrm{m}}}{M^2} \tag*{(93)} ( 1 + α H ) Φ − ( 1 + α T ) Ψ + ( α M − α T ) H π − α H π ˙ = − M 2 σ m (93) and, finally, the trace of the (ij) components,
2 Ψ ¨ + 2 ( 3 + α M ) H Ψ ˙ + 2 ( 1 + α B ) H Φ ˙ + 2 [ H ˙ − ρ m + p m 2 M 2 + ( α B H ) ⋅ + ( 3 + α M ) ( 1 + α B ) H 2 ] Φ − 2 H α B π ¨ + 2 [ H ˙ + ρ m + p m 2 M 2 − ( α B H ) ⋅ − ( 3 + α M ) α B H 2 ] π ˙ + 2 [ ( 3 + α M ) H H ˙ + p ˙ m 2 M 2 + H ¨ ] π − 6 K a 2 [ π ˙ + ( 1 + α M ) H π ] = 1 M 2 ( δ p m + 2 ( D 2 + 3 K ) 3 a 2 σ m ) . (94) 2\ddot{\Psi}+2(3+\alpha_{\mathrm{M}})H\dot{\Psi}+2(1+\alpha_{\mathrm{B}})H\dot{\Phi}
+2\left[\dot{H}-\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}+(\alpha_{\mathrm{B}}H)^{\cdot}+(3+\alpha_{\mathrm{M}})(1+\alpha_{\mathrm{B}})H^2\right]\Phi
-2H\alpha_{\mathrm{B}}\ddot{\pi}+2\left[\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}-(\alpha_{\mathrm{B}}H)^{\cdot}-(3+\alpha_{\mathrm{M}})\alpha_{\mathrm{B}}H^2\right]\dot{\pi}
+2\left[(3+\alpha_{\mathrm{M}})H\dot{H}+\frac{\dot{p}_{\mathrm{m}}}{2M^2}+\ddot{H}\right]\pi
-\frac{6\mathcal{K}}{a^2}\left[\dot{\pi}+(1+\alpha_{\mathrm{M}})H\pi\right]
=\frac{1}{M^2}\left(\delta p_{\mathrm{m}}+\frac{2(D^2+3\mathcal{K})}{3a^2}\sigma_{\mathrm{m}}\right). \tag*{(94)} 2 Ψ ¨ + 2 ( 3 + α M ) H Ψ ˙ + 2 ( 1 + α B ) H Φ ˙ + 2 [ H ˙ − 2 M 2 ρ m + p m + ( α B H ) ⋅ + ( 3 + α M ) ( 1 + α B ) H 2 ] Φ − 2 H α B π ¨ + 2 [ H ˙ + 2 M 2 ρ m + p m − ( α B H ) ⋅ − ( 3 + α M ) α B H 2 ] π ˙ + 2 [ ( 3 + α M ) H H ˙ + 2 M 2 p ˙ m + H ¨ ] π − a 2 6 K [ π ˙ + ( 1 + α M ) H π ] = M 2 1 ( δ p m + 3 a 2 2 ( D 2 + 3 K ) σ m ) . (94) The equation of motion for π \pi π
H 2 α K π ¨ + { [ H 2 ( 3 + α M ) + H ˙ ] α K + ( H α K ) ⋅ } H π ˙ + 6 { ( H ˙ + ρ m + p m 2 M 2 ) H ˙ + H α ˙ B [ H 2 ( 3 + α M ) + H ˙ ] + H ( H α ˙ B ) ⋅ } π + 2 D 2 + 3 K a 2 { H ˙ + ρ m + p m 2 M 2 + H 2 [ 1 + α B ( 1 + α M ) + α T − ( 1 + α H ) ( 1 + α M ) ] + ( H ( α B − α H ) ) ⋅ } π + 6 H α B Ψ ˙ + H 2 ( 6 α B − α K ) Φ ˙ + 6 [ H ˙ + ρ m + p m 2 M 2 + H 2 α B ( 3 + α M ) + ( α B H ) ⋅ ] Ψ ˙ + [ 6 ( H ˙ + ρ m + p m 2 M 2 ) + H 2 ( 6 α B − α K ) ( 3 + α M ) + 2 ( 9 α B − α K ) H ˙ + H ( 6 α ˙ B − α ˙ K ) ] H Φ \begin{aligned}
H^2\alpha_{\mathrm{K}}\ddot{\pi}+\left\{\left[H^2(3+\alpha_{\mathrm{M}})+\dot{H}\right]\alpha_{\mathrm{K}}+(H\alpha_{\mathrm{K}})^{\cdot}\right\}H\dot{\pi} \\
&+6\left\{\left(\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}\right)\dot{H}+H\dot{\alpha}_{\mathrm{B}}\left[H^2(3+\alpha_{\mathrm{M}})+\dot{H}\right]+H(H\dot{\alpha}_{\mathrm{B}})^{\cdot}\right\}\pi\\
&+2\frac{D^2+3\mathcal{K}}{a^2}\left\{\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}+H^2[1+\alpha_{\mathrm{B}}(1+\alpha_{\mathrm{M}})+\alpha_{\mathrm{T}}-(1+\alpha_{\mathrm{H}})(1+\alpha_{\mathrm{M}})]+(H(\alpha_{\mathrm{B}}-\alpha_{\mathrm{H}}))^{\cdot}\right\}\pi\\
&+6H\alpha_{\mathrm{B}}\dot{\Psi}+H^2(6\alpha_{\mathrm{B}}-\alpha_{\mathrm{K}})\dot{\Phi}
+6\left[\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}+H^2\alpha_{\mathrm{B}}(3+\alpha_{\mathrm{M}})+(\alpha_{\mathrm{B}}H)^{\cdot}\right]\dot{\Psi} \\
&+\left[6\left(\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^2}\right)+H^2(6\alpha_{\mathrm{B}}-\alpha_{\mathrm{K}})(3+\alpha_{\mathrm{M}})+2(9\alpha_{\mathrm{B}}-\alpha_{\mathrm{K}})\dot{H}+H(6\dot{\alpha}_{\mathrm{B}}-\dot{\alpha}_{\mathrm{K}})\right]H\Phi
\end{aligned} H 2 α K π ¨ + { [ H 2 ( 3 + α M ) + H ˙ ] α K + ( H α K ) ⋅ } H π ˙ + 6 { ( H ˙ + 2 M 2 ρ m + p m ) H ˙ + H α ˙ B [ H 2 ( 3 + α M ) + H ˙ ] + H ( H α ˙ B ) ⋅ } π + 2 a 2 D 2 + 3 K { H ˙ + 2 M 2 ρ m + p m + H 2 [ 1 + α B ( 1 + α M ) + α T − ( 1 + α H ) ( 1 + α M )] + ( H ( α B − α H ) ) ⋅ } π + 6 H α B Ψ ˙ + H 2 ( 6 α B − α K ) Φ ˙ + 6 [ H ˙ + 2 M 2 ρ m + p m + H 2 α B ( 3 + α M ) + ( α B H ) ⋅ ] Ψ ˙ + [ 6 ( H ˙ + 2 M 2 ρ m + p m ) + H 2 ( 6 α B − α K ) ( 3 + α M ) + 2 ( 9 α B − α K ) H ˙ + H ( 6 α ˙ B − α ˙ K ) ] H Φ − 6 K a 2 [ H 2 α B ( 1 − α M ) + ( H α B ) ⋅ ] π − 2 D 2 + 3 K a 2 [ α H Ψ ˙ + { H ( α M + α H ( 1 + α M ) − α T ) + α ˙ H } Ψ + H ( α H − α B ) Φ ] = 0. (95) -\frac{6\mathcal{K}}{a^2}\left[H^2\alpha_{\mathrm{B}}(1-\alpha_{\mathrm{M}})+(H\alpha_{\mathrm{B}})^{\cdot}\right]\pi
-2\frac{D^2+3\mathcal{K}}{a^2}\left[\alpha_{\mathrm{H}}\dot{\Psi}+\{H(\alpha_{\mathrm{M}}+\alpha_{\mathrm{H}}(1+\alpha_{\mathrm{M}})-\alpha_{\mathrm{T}})+\dot{\alpha}_{\mathrm{H}}\}\Psi+H(\alpha_{\mathrm{H}}-\alpha_{\mathrm{B}})\Phi\right]=0. \tag*{(95)} − a 2 6 K [ H 2 α B ( 1 − α M ) + ( H α B ) ⋅ ] π − 2 a 2 D 2 + 3 K [ α H Ψ ˙ + { H ( α M + α H ( 1 + α M ) − α T ) + α ˙ H } Ψ + H ( α H − α B ) Φ ] = 0. (95) In all the above equations, the α \alpha α [72 ] . At the background level, the two Friedmann equations are as follows
H 2 + K a 2 = 1 3 M 2 ( ρ m + ρ d e ) , H ˙ − K a 2 = − 1 2 M 2 ( ρ m + ρ d e + P m + P d e ) , (96) H^2+\frac{\mathcal{K}}{a^2}=\frac{1}{3M^2}(\rho_{\mathrm{m}}+\rho_{\mathrm{de}}),\qquad\dot{H}-\frac{\mathcal{K}}{a^2}=-\frac{1}{2M^2}(\rho_{\mathrm{m}}+\rho_{\mathrm{de}}+P_{\mathrm{m}}+P_{\mathrm{de}}), \tag*{(96)} H 2 + a 2 K = 3 M 2 1 ( ρ m + ρ de ) , H ˙ − a 2 K = − 2 M 2 1 ( ρ m + ρ de + P m + P de ) , (96) and the continuity equation for the dark sector reads
ρ ˙ d e + 3 H ( ρ d e + P d e ) = 3 H M 2 α M ( H 2 + K a 2 ) , (97) \dot{\rho}_{\mathrm{de}}+3H(\rho_{\mathrm{de}}+P_{\mathrm{de}})=3HM^2\alpha_{\mathrm{M}}\left(H^2+\frac{\mathcal{K}}{a^2}\right), \tag*{(97)} ρ ˙ de + 3 H ( ρ de + P de ) = 3 H M 2 α M ( H 2 + a 2 K ) , (97) or, equivalently,
ρ ˙ d e = 3 H ( ρ m + P m ) + 6 H M 2 ( H ˙ − K a 2 ) + 3 H M 2 α M ( H 2 + K a 2 ) . (98) \dot{\rho}_{\mathrm{de}}=3H(\rho_{\mathrm{m}}+P_{\mathrm{m}})+6HM^2\left(\dot{H}-\frac{\mathcal{K}}{a^2}\right)+3HM^2\alpha_{\mathrm{M}}\left(H^2+\frac{\mathcal{K}}{a^2}\right). \tag*{(98)} ρ ˙ de = 3 H ( ρ m + P m ) + 6 H M 2 ( H ˙ − a 2 K ) + 3 H M 2 α M ( H 2 + a 2 K ) . (98) One can also express P ˙ d e \dot{P}_{\mathrm{de}} P ˙ de
P ˙ d e = − P ˙ m − M 2 [ 2 H ¨ + 2 H H ˙ ( 3 + α M ) + 3 H 3 α M − H ( 2 − α M ) K a 2 ] . (99) \dot{P}_{\mathrm{de}}=-\dot{P}_{\mathrm{m}}-M^2\left[2\ddot{H}+2H\dot{H}(3+\alpha_{\mathrm{M}})+3H^3\alpha_{\mathrm{M}}-H(2-\alpha_{\mathrm{M}})\frac{\mathcal{K}}{a^2}\right]. \tag*{(99)} P ˙ de = − P ˙ m − M 2 [ 2 H ¨ + 2 H H ˙ ( 3 + α M ) + 3 H 3 α M − H ( 2 − α M ) a 2 K ] . (99) However, we usually want the dark sector to obey a standard continuity equation. We, therefore, define an effective fluid such that
ρ d e = M ‾ 2 ρ d e e f f + ( M ‾ 2 − 1 ) ρ m , P d e = M ‾ 2 P d e e f f + ( M ‾ 2 − 1 ) P m , (100) \rho_{\mathrm{de}}=\overline{M}^{2}\rho_{\mathrm{de}}^{\mathrm{eff}}+\left(\overline{M}^{2}-1\right)\rho_{\mathrm{m}},\qquad P_{\mathrm{de}}=\overline{M}^{2}P_{\mathrm{de}}^{\mathrm{eff}}+\left(\overline{M}^{2}-1\right)P_{\mathrm{m}}, \tag*{(100)} ρ de = M 2 ρ de eff + ( M 2 − 1 ) ρ m , P de = M 2 P de eff + ( M 2 − 1 ) P m , (100) with M ‾ 2 = M 2 / M p l 2 \overline{M}^{2}=M^{2}/M_{\mathrm{pl}}^{2} M 2 = M 2 / M pl 2 M p l = ( 8 π G ) − 1 / 2 M_{\mathrm{pl}}=(8\pi G)^{-1/2} M pl = ( 8 π G ) − 1/2
H 2 + K a 2 = 1 3 M p l 2 ( ρ m + ρ d e e f f ) , H ˙ − K a 2 = − 1 2 M p l 2 ( ρ m + ρ d e e f f + P m + P d e e f f ) , (101) H^{2}+\frac{\mathcal{K}}{a^{2}}=\frac{1}{3M_{\mathrm{pl}}^{2}}\left(\rho_{\mathrm{m}}+\rho_{\mathrm{de}}^{\mathrm{eff}}\right),\qquad\dot{H}-\frac{\mathcal{K}}{a^{2}}=-\frac{1}{2M_{\mathrm{pl}}^{2}}\left(\rho_{\mathrm{m}}+\rho_{\mathrm{de}}^{\mathrm{eff}}+P_{\mathrm{m}}+P_{\mathrm{de}}^{\mathrm{eff}}\right), \tag*{(101)} H 2 + a 2 K = 3 M pl 2 1 ( ρ m + ρ de eff ) , H ˙ − a 2 K = − 2 M pl 2 1 ( ρ m + ρ de eff + P m + P de eff ) , (101) and
ρ ˙ d e e f f + 3 H ( ρ d e e f f + P d e e f f ) = 0. (102) \dot{\rho}_{\mathrm{de}}^{\mathrm{eff}}+3H\left(\rho_{\mathrm{de}}^{\mathrm{eff}}+P_{\mathrm{de}}^{\mathrm{eff}}\right)=0. \tag*{(102)} ρ ˙ de eff + 3 H ( ρ de eff + P de eff ) = 0. (102) We can now write the effective dark energy density and pressure in terms of the functions G i G_i G i ρ d e e f f \rho_{\mathrm{de}}^{\mathrm{eff}} ρ de eff P d e e f f P_{\mathrm{de}}^{\mathrm{eff}} P de eff H 2 H^{2} H 2 H ˙ \dot{H} H ˙
ρ d e = ρ d e e f f + 3 ( M 2 − M p l 2 ) ( H 2 + K a 2 ) , P d e = P d e e f f − ( M 2 − M p l 2 ) ( 3 H 2 + 2 H ˙ + K a 2 ) , (103) \rho_{\mathrm{de}}=\rho_{\mathrm{de}}^{\mathrm{eff}}+3\left(M^{2}-M_{\mathrm{pl}}^{2}\right)\left(H^{2}+\frac{\mathcal{K}}{a^{2}}\right),\qquad P_{\mathrm{de}}=P_{\mathrm{de}}^{\mathrm{eff}}-\left(M^{2}-M_{\mathrm{pl}}^{2}\right)\left(3H^{2}+2\dot{H}+\frac{\mathcal{K}}{a^{2}}\right), \tag*{(103)} ρ de = ρ de eff + 3 ( M 2 − M pl 2 ) ( H 2 + a 2 K ) , P de = P de eff − ( M 2 − M pl 2 ) ( 3 H 2 + 2 H ˙ + a 2 K ) , (103) with
ρ d e e f f = 2 X G 2 , X − G 2 − X G 3 , ϕ − 6 H ϕ ˙ X G 3 , X − 6 H 2 ( G 4 − 4 X G 4 , X − 4 X 2 G 4 , X X ) − 6 H ϕ ˙ ( G 4 , ϕ + 2 X G 4 , ϕ X ) + 2 H 3 ϕ ˙ X ( 5 G 5 , X + 2 X G 5 , X X ) + 3 H 2 X ( 3 G 5 , ϕ + 2 X G 5 , ϕ X ) + 3 H 2 M p l 2 . (104) \begin{aligned}
\rho_{\mathrm{de}}^{\mathrm{eff}}={}&2XG_{2,X}-G_{2}-XG_{3,\phi}-6H\dot{\phi}XG_{3,X}-6H^{2}\left(G_{4}-4XG_{4,X}-4X^{2}G_{4,XX}\right)\\
&-6H\dot{\phi}\left(G_{4,\phi}+2XG_{4,\phi X}\right)+2H^{3}\dot{\phi}X\left(5G_{5,X}+2XG_{5,XX}\right)+3H^{2}X\left(3G_{5,\phi}+2XG_{5,\phi X}\right)+3H^{2}M_{\mathrm{pl}}^{2}. \tag*{(104)}
\end{aligned} ρ de eff = 2 X G 2 , X − G 2 − X G 3 , ϕ − 6 H ϕ ˙ X G 3 , X − 6 H 2 ( G 4 − 4 X G 4 , X − 4 X 2 G 4 , X X ) − 6 H ϕ ˙ ( G 4 , ϕ + 2 X G 4 , ϕX ) + 2 H 3 ϕ ˙ X ( 5 G 5 , X + 2 X G 5 , X X ) + 3 H 2 X ( 3 G 5 , ϕ + 2 X G 5 , ϕX ) + 3 H 2 M pl 2 . (104) P d e e f f = G 2 + X ( 2 ϕ ¨ G 3 , X − G 3 , ϕ ) + 2 ( 3 H 2 + 2 H ˙ ) G 4 − 4 H ( X ˙ + 3 H X ) G 4 , X − 8 X ( H ˙ G 4 , X + H X ˙ G 4 , X X ) + 2 ( ϕ ¨ + 2 H ϕ ˙ ) G 4 , ϕ − 2 X G 4 , ϕ ϕ + 4 ( ϕ ¨ − 2 H ϕ ˙ ) X G 4 , ϕ X − 2 X ( 2 H 3 ϕ ˙ + 2 H H ˙ ϕ ˙ + 3 H 2 ϕ ¨ ) G 5 , X − 4 H 2 ϕ ˙ X 2 G 5 , X X − 2 H X ( X ˙ − H X ) G 5 , ϕ X − [ 2 ( H X ) ⋅ + 3 H 2 X ] G 5 , ϕ − 2 H ϕ ˙ X G 5 , ϕ ϕ − ( 3 H 2 + 2 H ˙ ) M p l 2 . (105) \begin{aligned}
P_{\mathrm{de}}^{\mathrm{eff}}={}&G_{2}+X\left(2\ddot{\phi}G_{3,X}-G_{3,\phi}\right)+2\left(3H^{2}+2\dot{H}\right)G_{4}-4H\left(\dot{X}+3HX\right)G_{4,X}\\
&-8X\left(\dot{H}G_{4,X}+H\dot{X}G_{4,XX}\right)+2\left(\ddot{\phi}+2H\dot{\phi}\right)G_{4,\phi}-2XG_{4,\phi\phi}+4\left(\ddot{\phi}-2H\dot{\phi}\right)XG_{4,\phi X}\\
&-2X\left(2H^{3}\dot{\phi}+2H\dot{H}\dot{\phi}+3H^{2}\ddot{\phi}\right)G_{5,X}-4H^{2}\dot{\phi}X^{2}G_{5,XX}-2HX\left(\dot{X}-HX\right)G_{5,\phi X}\\
&-\left[2(HX)^{\cdot}+3H^{2}X\right]G_{5,\phi}-2H\dot{\phi}XG_{5,\phi\phi}-\left(3H^{2}+2\dot{H}\right)M_{\mathrm{pl}}^{2}. \tag*{(105)}
\end{aligned} P de eff = G 2 + X ( 2 ϕ ¨ G 3 , X − G 3 , ϕ ) + 2 ( 3 H 2 + 2 H ˙ ) G 4 − 4 H ( X ˙ + 3 H X ) G 4 , X − 8 X ( H ˙ G 4 , X + H X ˙ G 4 , X X ) + 2 ( ϕ ¨ + 2 H ϕ ˙ ) G 4 , ϕ − 2 X G 4 , ϕϕ + 4 ( ϕ ¨ − 2 H ϕ ˙ ) X G 4 , ϕX − 2 X ( 2 H 3 ϕ ˙ + 2 H H ˙ ϕ ˙ + 3 H 2 ϕ ¨ ) G 5 , X − 4 H 2 ϕ ˙ X 2 G 5 , X X − 2 H X ( X ˙ − H X ) G 5 , ϕX − [ 2 ( H X ) ⋅ + 3 H 2 X ] G 5 , ϕ − 2 H ϕ ˙ X G 5 , ϕϕ − ( 3 H 2 + 2 H ˙ ) M pl 2 . (105) To apply the QSA and use the previous equations, we need to leave the configuration space. We cannot perform a simple Fourier transform as D 2 D^{2} D 2 k 2 k^{2} k 2 Ψ ( x ⃗ ) = ∑ q Ψ q Q q l m \Psi(\vec{x})=\sum_q\Psi_qQ_{qlm} Ψ ( x ) = ∑ q Ψ q Q q l m D 2 Q q l m = − ( q 2 − 3 K ) Q q l m D^{2}Q_{qlm}=-(q^{2}-3\mathcal{K})Q_{qlm} D 2 Q q l m = − ( q 2 − 3 K ) Q q l m D 2 + 3 K D^{2}+3\mathcal{K} D 2 + 3 K − q 2 -q^{2} − q 2 π \pi π
6 ( 1 + α B ) H Ψ ˙ + ( 6 − α K + 12 α B ) H 2 Φ + 2 ( 1 + α H ) q 2 a 2 Ψ + ( α K − 6 α B ) H 2 π ˙ + 6 [ ( 1 + α B ) H ˙ + ρ m + p m 2 M 2 + ( 1 − α B ) K a 2 + 1 3 ( α H − α B ) q 2 a 2 ] H π = − δ ρ m M 2 ; (106) \begin{aligned}
6(1+\alpha_{\mathrm{B}})H\dot{\Psi}+(6-\alpha_{\mathrm{K}}+12\alpha_{\mathrm{B}})H^{2}\Phi+2(1+\alpha_{\mathrm{H}})\frac{q^{2}}{a^{2}}\Psi+(\alpha_{\mathrm{K}}-6\alpha_{\mathrm{B}})H^{2}\dot{\pi}\\
+6\left[(1+\alpha_{\mathrm{B}})\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{2M^{2}}+(1-\alpha_{\mathrm{B}})\frac{\mathcal{K}}{a^{2}}+\frac{1}{3}(\alpha_{\mathrm{H}}-\alpha_{\mathrm{B}})\frac{q^{2}}{a^{2}}\right]H\pi=-\frac{\delta\rho_{\mathrm{m}}}{M^{2}}; \tag*{(106)}
\end{aligned} 6 ( 1 + α B ) H Ψ ˙ + ( 6 − α K + 12 α B ) H 2 Φ + 2 ( 1 + α H ) a 2 q 2 Ψ + ( α K − 6 α B ) H 2 π ˙ + 6 [ ( 1 + α B ) H ˙ + 2 M 2 ρ m + p m + ( 1 − α B ) a 2 K + 3 1 ( α H − α B ) a 2 q 2 ] H π = − M 2 δ ρ m ; (106) 2 Ψ ˙ + 2 ( 1 + α B ) H Φ − 2 H α B π ˙ + ( 2 H ˙ + ρ m + p m M 2 ) π = − ( ρ m + p m ) v m M 2 ; (107) 2\dot{\Psi}+2(1+\alpha_{\mathrm{B}})H\Phi-2H\alpha_{\mathrm{B}}\dot{\pi}+\left(2\dot{H}+\frac{\rho_{\mathrm{m}}+p_{\mathrm{m}}}{M^{2}}\right)\pi=-\frac{(\rho_{\mathrm{m}}+p_{\mathrm{m}})v_{\mathrm{m}}}{M^{2}}; \tag*{(107)} 2 Ψ ˙ + 2 ( 1 + α B ) H Φ − 2 H α B π ˙ + ( 2 H ˙ + M 2 ρ m + p m ) π = − M 2 ( ρ m + p m ) v m ; (107) ( 1 + α H ) Φ − ( 1 + α T ) Ψ + ( α M − α T ) H π − α H π ˙ = − σ m M 2 (108) (1+\alpha_{\mathrm H})\Phi-(1+\alpha_{\mathrm T})\Psi+(\alpha_{\mathrm M}-\alpha_{\mathrm T})H\pi-\alpha_{\mathrm H}\dot{\pi}=-\frac{\sigma_{\mathrm m}}{M^2} \tag*{(108)} ( 1 + α H ) Φ − ( 1 + α T ) Ψ + ( α M − α T ) H π − α H π ˙ = − M 2 σ m (108) 2 Ψ ¨ + 2 ( 3 + α M ) H Ψ ˙ + 2 ( 1 + α B ) H Φ ˙ + 2 [ H ˙ − ρ m + p m 2 M 2 + ( α B H ) ⋅ + ( 3 + α M ) ( 1 + α B ) H 2 ] Φ − 2 H α B π ¨ + 2 [ H ˙ + ρ m + p m 2 M 2 − ( α B H ) ⋅ − ( 3 + α M ) α B H 2 ] π ˙ + 2 [ ( 3 + α M ) H H ˙ + p ˙ m 2 M 2 + H ¨ ] π − 6 K a 2 [ π ˙ + ( 1 + α M ) H π ] = 1 M 2 ( δ p m − 2 3 q 2 a 2 σ m ) (109) \begin{aligned}
2\ddot{\Psi}+2(3+\alpha_{\mathrm M})H\dot{\Psi}+2(1+\alpha_{\mathrm B})H\dot{\Phi}
&+2\left[\dot{H}-\frac{\rho_{\mathrm m}+p_{\mathrm m}}{2M^2}+(\alpha_{\mathrm B}H)^{\cdot}+(3+\alpha_{\mathrm M})(1+\alpha_{\mathrm B})H^2\right]\Phi\\
&-2H\alpha_{\mathrm B}\ddot{\pi}+2\left[\dot{H}+\frac{\rho_{\mathrm m}+p_{\mathrm m}}{2M^2}-(\alpha_{\mathrm B}H)^{\cdot}-(3+\alpha_{\mathrm M})\alpha_{\mathrm B}H^2\right]\dot{\pi}\\
&+2\left[(3+\alpha_{\mathrm M})H\dot{H}+\frac{\dot{p}_{\mathrm m}}{2M^2}+\ddot{H}\right]\pi
-6\frac{\mathcal{K}}{a^2}[\dot{\pi}+(1+\alpha_{\mathrm M})H\pi]\\
&=\frac{1}{M^2}\left(\delta p_{\mathrm m}-\frac{2}{3}\frac{q^2}{a^2}\sigma_{\mathrm m}\right) \tag*{(109)}
\end{aligned} 2 Ψ ¨ + 2 ( 3 + α M ) H Ψ ˙ + 2 ( 1 + α B ) H Φ ˙ + 2 [ H ˙ − 2 M 2 ρ m + p m + ( α B H ) ⋅ + ( 3 + α M ) ( 1 + α B ) H 2 ] Φ − 2 H α B π ¨ + 2 [ H ˙ + 2 M 2 ρ m + p m − ( α B H ) ⋅ − ( 3 + α M ) α B H 2 ] π ˙ + 2 [ ( 3 + α M ) H H ˙ + 2 M 2 p ˙ m + H ¨ ] π − 6 a 2 K [ π ˙ + ( 1 + α M ) H π ] = M 2 1 ( δ p m − 3 2 a 2 q 2 σ m ) (109) H 2 α K π ¨ + { [ H 2 ( 3 + α M ) + H ˙ ] α K + ( H α K ) ⋅ } H π ˙ (110) H^2\alpha_{\mathrm K}\ddot{\pi}+\left\{[H^2(3+\alpha_{\mathrm M})+\dot{H}]\alpha_{\mathrm K}+(H\alpha_{\mathrm K})^{\cdot}\right\}H\dot{\pi} \tag*{(110)} H 2 α K π ¨ + { [ H 2 ( 3 + α M ) + H ˙ ] α K + ( H α K ) ⋅ } H π ˙ (110) + 6 { ( H ˙ + ρ m + p m 2 M 2 ) H ˙ + H ˙ α ˙ B [ H 2 ( 3 + α M ) + H ˙ ] + H ( H α ˙ B ) ⋅ } π − 2 q 2 a 2 { H ˙ + ρ m + p m 2 M 2 + H 2 [ 1 + α B ( 1 + α M ) + α T − ( 1 + α H ) ( 1 + α M ) ] + ( H ( α B − α H ) ) ⋅ } π + 6 H α B π ¨ + H 2 ( 6 α B − α K ) Φ ˙ + 6 [ H ˙ + ρ m + p m 2 M 2 + H 2 α B ( 3 + α M ) + ( α B H ) ⋅ ] Ψ ˙ + [ 6 ( H ˙ + ρ m + p m 2 M 2 ) + H 2 ( 6 α B − α K ) ( 3 + α M ) + 2 ( 9 α B − α K ) H ˙ + H ( 6 α ˙ B − α ˙ K ) ] H Φ − 6 K a 2 [ H 2 α B ( 1 − α M ) + ( H α B ) ⋅ ] π + 2 q 2 a 2 [ α H Ψ ˙ + { H ( α M + α H ( 1 + α M ) − α T ) + α ˙ H } Ψ + H ( α H − α B ) Φ ] = 0 (111) \begin{aligned}
&+6\left\{\left(\dot{H}+\frac{\rho_{\mathrm m}+p_{\mathrm m}}{2M^2}\right)\dot{H}+\dot{H}\dot{\alpha}_{\mathrm B}\left[H^2(3+\alpha_{\mathrm M})+\dot{H}\right]+H(H\dot{\alpha}_{\mathrm B})^{\cdot}\right\}\pi\\
&-2\frac{q^2}{a^2}\left\{\dot{H}+\frac{\rho_{\mathrm m}+p_{\mathrm m}}{2M^2}+H^2[1+\alpha_{\mathrm B}(1+\alpha_{\mathrm M})+\alpha_{\mathrm T}-(1+\alpha_{\mathrm H})(1+\alpha_{\mathrm M})]+(H(\alpha_{\mathrm B}-\alpha_{\mathrm H}))^{\cdot}\right\}\pi\\
&+6H\alpha_{\mathrm B}\ddot{\pi}+H^2(6\alpha_{\mathrm B}-\alpha_{\mathrm K})\dot{\Phi}
+6\left[\dot{H}+\frac{\rho_{\mathrm m}+p_{\mathrm m}}{2M^2}+H^2\alpha_{\mathrm B}(3+\alpha_{\mathrm M})+(\alpha_{\mathrm B}H)^{\cdot}\right]\dot{\Psi}\\
&+\left[6\left(\dot{H}+\frac{\rho_{\mathrm m}+p_{\mathrm m}}{2M^2}\right)+H^2(6\alpha_{\mathrm B}-\alpha_{\mathrm K})(3+\alpha_{\mathrm M})+2(9\alpha_{\mathrm B}-\alpha_{\mathrm K})\dot{H}+H(6\dot{\alpha}_{\mathrm B}-\dot{\alpha}_{\mathrm K})\right]H\Phi\\
&-6\frac{\mathcal{K}}{a^2}\left[H^2\alpha_{\mathrm B}(1-\alpha_{\mathrm M})+(H\alpha_{\mathrm B})^{\cdot}\right]\pi\\
&+2\frac{q^2}{a^2}\left[\alpha_{\mathrm H}\dot{\Psi}+\left\{H(\alpha_{\mathrm M}+\alpha_{\mathrm H}(1+\alpha_{\mathrm M})-\alpha_{\mathrm T})+\dot{\alpha}_{\mathrm H}\right\}\Psi+H(\alpha_{\mathrm H}-\alpha_{\mathrm B})\Phi\right]=0 \tag*{(111)}
\end{aligned} + 6 { ( H ˙ + 2 M 2 ρ m + p m ) H ˙ + H ˙ α ˙ B [ H 2 ( 3 + α M ) + H ˙ ] + H ( H α ˙ B ) ⋅ } π − 2 a 2 q 2 { H ˙ + 2 M 2 ρ m + p m + H 2 [ 1 + α B ( 1 + α M ) + α T − ( 1 + α H ) ( 1 + α M )] + ( H ( α B − α H ) ) ⋅ } π + 6 H α B π ¨ + H 2 ( 6 α B − α K ) Φ ˙ + 6 [ H ˙ + 2 M 2 ρ m + p m + H 2 α B ( 3 + α M ) + ( α B H ) ⋅ ] Ψ ˙ + [ 6 ( H ˙ + 2 M 2 ρ m + p m ) + H 2 ( 6 α B − α K ) ( 3 + α M ) + 2 ( 9 α B − α K ) H ˙ + H ( 6 α ˙ B − α ˙ K ) ] H Φ − 6 a 2 K [ H 2 α B ( 1 − α M ) + ( H α B ) ⋅ ] π + 2 a 2 q 2 [ α H Ψ ˙ + { H ( α M + α H ( 1 + α M ) − α T ) + α ˙ H } Ψ + H ( α H − α B ) Φ ] = 0 (111) In these equations, v m v_{\mathrm m} v m σ m \sigma_{\mathrm m} σ m 0 i 0i 0 i i j ij ij [73 ] , and we denote ( ρ m + p m ) v m = q m (\rho_{\mathrm m}+p_{\mathrm m})v_{\mathrm m}=q_{\mathrm m} ( ρ m + p m ) v m = q m α H = 0 \alpha_{\mathrm H}=0 α H = 0
C Ψ 00 Ψ ¨ + C Φ 00 Φ ¨ + C Ψ 00 2 Q 2 Ψ ¨ + C π 00 π ¨ + ( C π 00 + C π 2 00 Q 2 ) π = − δ ρ m M 2 (112) C_{\Psi}^{00}\ddot{\Psi}+C_{\Phi}^{00}\ddot{\Phi}+C_{\Psi}^{00\,2}Q^2\ddot{\Psi}+C_{\pi}^{00}\ddot{\pi}+(C_{\pi}^{00}+C_{\pi2}^{00}Q^2)\pi=-\frac{\delta\rho_{\mathrm m}}{M^2} \tag*{(112)} C Ψ 00 Ψ ¨ + C Φ 00 Φ ¨ + C Ψ 00 2 Q 2 Ψ ¨ + C π 00 π ¨ + ( C π 00 + C π 2 00 Q 2 ) π = − M 2 δ ρ m (112) C Ψ 0 i Ψ ˙ + C Φ 0 i Φ ˙ + C π 0 i π ˙ + C π 0 i π = − q m M 2 (113) C_{\Psi}^{0i}\dot{\Psi}+C_{\Phi}^{0i}\dot{\Phi}+C_{\pi}^{0i}\dot{\pi}+C_{\pi}^{0i}\pi=-\frac{q_{\mathrm m}}{M^2} \tag*{(113)} C Ψ 0 i Ψ ˙ + C Φ 0 i Φ ˙ + C π 0 i π ˙ + C π 0 i π = − M 2 q m (113) C Φ i j , t r l Φ + C Ψ i j , t r l Ψ + C π i j , t r l π = − σ m M 2 (114) C_{\Phi}^{ij,\mathrm{trl}}\Phi+C_{\Psi}^{ij,\mathrm{trl}}\Psi+C_{\pi}^{ij,\mathrm{trl}}\pi=-\frac{\sigma_{\mathrm m}}{M^2} \tag*{(114)} C Φ ij , trl Φ + C Ψ ij , trl Ψ + C π ij , trl π = − M 2 σ m (114) C Ψ i j , t r Ψ ¨ + C Ψ i j , t r Ψ ˙ + C Φ i j , t r Φ ˙ + C Φ i j , t r Φ + C π i j , t r π ¨ + C π i j , t r π ˙ + C π i j , t r π − 6 K a 2 [ π ˙ + ( 1 + α M ) H π ] = 1 M 2 ( δ p m − 2 3 H 2 Q 2 σ m ) (115) \begin{aligned}
C_{\Psi}^{ij,\mathrm{tr}}\ddot{\Psi}+C_{\Psi}^{ij,\mathrm{tr}}\dot{\Psi}+C_{\Phi}^{ij,\mathrm{tr}}\dot{\Phi}+C_{\Phi}^{ij,\mathrm{tr}}\Phi+C_{\pi}^{ij,\mathrm{tr}}\ddot{\pi}+C_{\pi}^{ij,\mathrm{tr}}\dot{\pi}+C_{\pi}^{ij,\mathrm{tr}}\pi\\
-6\frac{\mathcal{K}}{a^2}[\dot{\pi}+(1+\alpha_{\mathrm M})H\pi]=\frac{1}{M^2}\left(\delta p_{\mathrm m}-\frac{2}{3}H^2Q^2\sigma_{\mathrm m}\right) \tag*{(115)}
\end{aligned} C Ψ ij , tr Ψ ¨ + C Ψ ij , tr Ψ ˙ + C Φ ij , tr Φ ˙ + C Φ ij , tr Φ + C π ij , tr π ¨ + C π ij , tr π ˙ + C π ij , tr π − 6 a 2 K [ π ˙ + ( 1 + α M ) H π ] = M 2 1 ( δ p m − 3 2 H 2 Q 2 σ m ) (115) C π π π ¨ + C π π π ˙ + ( C π π + C π 2 π Q 2 ) π + C Ψ π Ψ ¨ + C Φ π Φ ˙ + C Ψ π Ψ ˙ + ( C Φ π + C Φ 2 π Q 2 ) Φ + C Ψ 2 π Q 2 Ψ − 6 K a 2 [ H 2 α B ( 1 − α M ) + ( H α B ) ⋅ ] π = 0 (116) \begin{aligned}
C_{\pi}^{\pi}\ddot{\pi}+C_{\pi}^{\pi}\dot{\pi}+(C_{\pi}^{\pi}+C_{\pi2}^{\pi}Q^2)\pi+C_{\Psi}^{\pi}\ddot{\Psi}+C_{\Phi}^{\pi}\dot{\Phi}+C_{\Psi}^{\pi}\dot{\Psi}+(C_{\Phi}^{\pi}+C_{\Phi2}^{\pi}Q^2)\Phi+C_{\Psi2}^{\pi}Q^2\Psi\\
-6\frac{\mathcal{K}}{a^2}\left[H^2\alpha_{\mathrm B}(1-\alpha_{\mathrm M})+(H\alpha_{\mathrm B})^{\cdot}\right]\pi=0 \tag*{(116)}
\end{aligned} C π π π ¨ + C π π π ˙ + ( C π π + C π 2 π Q 2 ) π + C Ψ π Ψ ¨ + C Φ π Φ ˙ + C Ψ π Ψ ˙ + ( C Φ π + C Φ2 π Q 2 ) Φ + C Ψ2 π Q 2 Ψ − 6 a 2 K [ H 2 α B ( 1 − α M ) + ( H α B ) ⋅ ] π = 0 (116) Here we have defined a set of coefficients C X a C_X^a C X a a a a i j ij ij i j ij ij π \pi π X X X Φ \Phi Φ Ψ \Psi Ψ π \pi π Q 2 = ( q a H ) 2 Q^2 = \left(\frac{q}{aH}\right)^2 Q 2 = ( a H q ) 2 Q ≫ 1 Q \gg1 Q ≫ 1 Q 2 Q^2 Q 2 K \mathcal{K} K H 2 H^2 H 2 σ m \sigma_m σ m
C Ψ 00 Q 2 Ψ + C π 00 Q 2 π = − ρ m Δ m M 2 , (117) C_{\Psi}^{00}Q^2\Psi+C_{\pi}^{00}Q^2\pi=-\frac{\rho_m\Delta_m}{M^2}, \tag*{(117)} C Ψ 00 Q 2 Ψ + C π 00 Q 2 π = − M 2 ρ m Δ m , (117) C Φ i j , t r l Φ + C Ψ i j , t r l Ψ + C π i j , t r l π = 0 , (118) C_{\Phi}^{ij,\mathrm{trl}}\Phi+C_{\Psi}^{ij,\mathrm{trl}}\Psi+C_{\pi}^{ij,\mathrm{trl}}\pi=0, \tag*{(118)} C Φ ij , trl Φ + C Ψ ij , trl Ψ + C π ij , trl π = 0 , (118) C Φ 2 π Q 2 Φ + C Ψ 2 π Q 2 Ψ + ( C π π + C π 2 π Q 2 ) π = 0. (119) C_{\Phi2}^{\pi}Q^2\Phi+C_{\Psi2}^{\pi}Q^2\Psi+\left(C_{\pi}^{\pi}+C_{\pi2}^{\pi}Q^2\right)\pi=0. \tag*{(119)} C Φ2 π Q 2 Φ + C Ψ2 π Q 2 Ψ + ( C π π + C π 2 π Q 2 ) π = 0. (119) The solution of this system is given in Eqs. (A.6) – (A.9) in [73 ] , once we replace K → Q \mathcal{K} \to Q K → Q Q → ∞ Q \to\infty Q → ∞
