Fresh Look at Shift Symmetric Modified Gravity

Prepared for submission to JCAP

Introduction

Dark energy from a non-interacting scalar field, whether with a canonical kinetic structure like quintessence or a noncanonical one like k-essence, must evolve with an equation of state ratio $w$ wholly on one side or the other of the cosmological constant value $w=-1$. This boundedness is referred to as the phantom divide. One can cross $w=-1$ by modifying gravity away from general relativity, through a scalar-tensor theory.

Horndeski gravity is a general scalar-tensor allowing noncanonical kinetic structure, nonminimal coupling to matter, and braiding between the scalar and tensor aspects. The terms associated with these effects can be both field and field derivative dependent. Nonminimal coupling – often problematic to fit observations [1, 2, 3, 4, 5, 6] – relies on field dependence. Moreover, field dependent terms run into issues with high energy (early universe) quantum loop corrections [7, 8, 9]. Therefore there is great interest in terms that rely only on field derivatives, and hence have protective shift symmetry. However, common wisdom in the literature is that an unbroken shift symmetric Horndeski theory that is both ghost free and perturbatively stable cannot cross the phantom divide [10, 11, 12, 13, 14], as observations appear to prefer [15, 16, 17, 18].

We revisit this issue and show that quite simple shift symmetric theories can in fact cross the phantom divide (though we do not here attempt to fit the data in detail). Taking a dynamical systems approach detailed in § 2, we identify the regions of phase space free of ghosts and instabilities in § 3, together with the fixed points and attractor structure. Overlaying the $w=-1$ condition reveals the analytic conditions, discussed in § 4, needed for crossing the phantom divide. § 5 presents two mechanisms for crossing and presents the full numerical evolution solutions. § 6 concludes.

Dynamical System for Shift Symmetry

The Lagrangian for shift symmetric Horndeski gravity with a luminal speed of tensor perturbation propagation is

where $X = (1/2)\dot{\phi}^{2}$ and we set $M_{\mathrm{Pl}}^{2}=1$. Note that there is no potential: we do not allow a cosmological constant because, while shift symmetric, it suffers similar quantum correction issues.

The Friedmann and scalar field evolution equations are

where a subscript X denotes a derivative with respect to $X$, we write $g \equiv XG_{3X}$, and we identify the effective dark energy density and pressure as

We define a phase space using the variables representing kinetic and braiding contributions to the dark energy density,

here writing $M_{\mathrm{Pl}}^{2}$ explicitly to show these are dimensionless quantities, and

and $n=XK_{X}/K$, while

A prime denotes $d/d\ln a$, while $m\equiv Xg_{X}/g$ and we will generally take $n_{X}=0$ (i.e. a monomial $K\sim X^{n}$) and $\Omega_{r}=0$ (focus on the matter dominated era and later) unless otherwise stated. The $\alpha_{i}$ quantities are often referred to as property functions, with $\alpha_{K}$ the kineticity and $\alpha_{B}$ the braiding parameter [19].

The equations of motions – Eqs. (2), (3), (4) – become a dynamical coupled system of autonomous first order differential equations in a $x_{k}$–$x_{g}$ phase space:

In the presence of shift symmetry, a conserved charge quantity appears and there is a fixed relation $m=(2n-1)/4$ as discussed in [20], and this is indicated by the right arrow. There is a subtlety if $n$ and $m$ are not constants, but unless otherwise stated (see § 5 below) we take them to be constant, i.e. $K\sim X^n$ and $G_3\sim X^m$. This gives a particularly simple theory with few free parameters, and will be our main focus.

Phase Space Analysis

The first thing to note is that the lines $x_k=0$ and $x_g=0$ are invariant; the quantities $x_k$, $x_g$ cannot flip sign during evolution and so the trajectories must stay within the initial quadrant of the phase space.

To check the health of the theory, we require the no ghost and stability conditions, respectively

Dynamical system analysis provides the fixed points and eigenvalues of the coupled evolution equations. Table 1 presents the results. We immediately see that $n<0$ will give a ghost. Since the dark energy equation of state at early times is $w=1/(2n-1)$ due to charge conservation [20, 21], this means that initial conditions with $w\in(0,1)$ cannot give a viable theory with late time dark energy domination within this model. For $0<n<1/2$ – a phantom early dark energy equation of state – we can have a healthy theory, with for example a de Sitter attractor. The case of $n>1/2$ is problematic since then the early dark energy equation of state is $w>0$ and this would dominate over matter at early times (and over radiation for $1/2<n<2$). Furthermore, $n>1$ has no stable attractor. Therefore we focus on $0<n<1/2$, with phantom behavior at early times, as in [20].

Notation	$(x_k,x_g)$	$\Omega_{\rm de}$	$w$	$f$	$\alpha$	$c_s^2$	eigenvalues	type
M	$(0,0)$	$0$	—	—	$0$	—	$\frac{-3}{2n-1}$ (twice)	repeller
K	$\left(\frac{1}{2n-1},0\right)$	$1$	$\frac{1}{2n-1}$	$0$	$6n$	$\frac{1}{2n-1}$	$\frac{3}{2n-1},\frac{-3}{2(2n-1)}$	saddle
B	$(0,1)$	$1$	$\frac{1}{2n}$	$1$	$3n$	$\frac{8n+3}{12n^2}$	$\frac{3}{2n}$ (twice)	repeller
dS	$(-1,2n)$	$1$	$-1$	$2n$	$3n$	$\frac{2(1-n)}{3}$	$-3$ (twice)	attractor

Figure 1 and Figure 2 show the phase space trajectories for $n=1/8$, $n=1/4$, $n=3/8$. Note how the ghost and instability regions shift across the phase space, but a clear healthy path always lies between the initial matter dominated state and the final de Sitter state. Also, since $x_g=f\Omega_{\rm de}$ and $x_k=\Omega_{\rm de}(1-f)/(2n-1)$, the map $(x_k,x_g)\leftrightarrow(\Omega_{\rm de},f)$ is a bijection, and lines through the origin are curves of constant $f$. One can show directly from $x'_g$ that $f'\sim\mathcal{O}(\Omega_{\rm de})$, i.e. that $f$ becomes constant at early times. Conditions on this early time $f$ limit for a healthy theory were derived in [20]. The right panel of Figure 1 illustrates that view of the phase space.

Analytic Conditions for Crossing the Phantom Divide

While we have established a class of healthy theories, and the end states of their evolution, we have not yet addressed the crossing of the phantom divide. The trajectories can end up at $w=-1$ but can they cross $w=-1$ before reaching the de Sitter state?

Consider the quantity $\alpha(\Omega_{\mathrm{de}}+w\Omega_{\mathrm{de}})$. This will be zero whenever $w=-1$. Using Eq. (11) one finds

For $\Omega_{\mathrm{de}}\ne0$, $w_{\mathrm{de}}=-1$ is therefore the phase space parabola $Q=0$, which passes through $M$ and through de Sitter, $dS=(-1,2n)$. Wherever $\alpha>0$ then $\operatorname{sign}(1+w_{\mathrm{de}})=\operatorname{sign}Q$.

Calling the value of $x_g$ where the trajectory hits $Q=0$ by $s$, we have there

Again this is only ghost free for $n>0$.

The question “does $w_{\mathrm{de}}$ cross $-1$?” is the question “does the trajectory cross the curve $Q=0$?”. The parabola splits the plane into two sides, $Q>0$ and $Q<0$, which are exactly $w_{\mathrm{de}}>-1$ and $w_{\mathrm{de}}<-1$ wherever $\alpha>0$. To see whether, and in which direction, a trajectory passes from one side to the other, follow the value of $Q$ along it. Writing $Q(N)\equiv Q\left(x_k(N),x_g(N)\right)$, with time coordinate $N\equiv\ln a$, the chain rule gives

Since $\nabla Q$ is normal to the curve $Q=0$, $Q'$ evaluated on the curve measures the component of the flow velocity that points across it. There are two possibilities at each point of the curve:

• $Q'=0$: the velocity is tangent to the curve. If this held at every point, a trajectory starting on the curve would stay on it. The curve would then be made of orbits (an invariant curve), and nothing could cross it.

• $Q'\ne0$: the velocity has a component across the curve, so the trajectory through that point passes from one side to the other. If a trajectory is on the curve at time $N_1$, then $Q(N_1)=0$ and, to first order, $Q(N)\simeq Q'(N_1)(N-N_1)$. If $Q'(N_1)<0$, the trajectory has $Q>0$ just before $N_1$ and $Q<0$ just after: it crosses from $w_{\mathrm{de}}>-1$ to $w_{\mathrm{de}}<-1$. If $Q'(N_1)>0$, it crosses the other way.

To evaluate $Q'$ on the curve, substitute the flow equations into $Q'$ and use the parametrisation $x_g=s$, $x_k=-s(s+2n)/(8n^2)$. On the curve $\alpha=3s^2/(4n)\ne0$ so the flow is regular there. The result is

It vanishes only at $s = 0$ and $s = 2n$, i.e. at $M$ and dS. That is expected: these are fixed points, where the velocity itself is zero. At every other point of the curve $Q' \ne0$, so the parabola is not an orbit, and every trajectory that reaches it crosses it there. Moreover, the sign of (25) is the same at every point of a given branch, $s > 0$ or $s < 0$. Each branch is therefore a one-way door: it can only be crossed in a single direction.

Finally, $x'_g$ is proportional to $x_g$, so the axis $x_g = 0$ is itself an orbit and no trajectory can cross it. A trajectory therefore keeps the sign of $x_g$ and can only ever meet one branch of the parabola.

Crossing Mechanisms

For a healthy theory giving eventual dark energy domination, Table 1 requires $0 < n \le1$. First consider the case $0 < n < 1/2$. At early times $1 + w = 2n/(2n - 1) < 0$, i.e. phantom, and so $Q < 0$. Moreover for early time stability we require $x_g > 0$ (see [20]). Now suppose the trajectory manages to reach the $Q = 0$ curve, i.e. $w = -1$, before reaching the de Sitter asymptote. At this point $Q' \le0$ by Eq. (25). Thus it cannot cross $Q = 0$ but either bounces off, or sticks to $w = -1$ if it is at the de Sitter fixed point $s = 2n$.

For the case $1/2 < n \le1$ (leaving aside for the moment issues of early dark energy domination), initially $w > 0$ and $Q > 0$. We still have $Q' < 0$, so one can in theory cross from $w > -1$ to $w < -1$. At early times $x_\phi= -1/(2n - 1)$ (as in [20]) and so $x'_g = -3x_g/(2n - 1)$. Thus for $n > 1/2$ we have $x'_g < 0$ and it will not evolve away from a matter dominated $(\Omega_{\rm de} \ll1, x_g \ll1)$ state. However, a loophole is to start from a state with large enough $x_g$, specifically near the braiding fixed point with early dark energy domination.

Figure 3 plots the trajectory for $n = 3/4$, and shows that indeed $w$ crosses from above to below $w = -1$. This demonstrates a healthy phantom crossing in a purely shift symmetric Horndeski gravity model.

This is an important proof of principle. Thus one mechanism for phantom crossing in a healthy shift symmetric theory is to have $1/2 < n \le1$. Of course the drawback is that there is no early matter (or radiation) dominated era so it is not our universe. However, one could imagine starting in the early universe with $0 < n_{\rm nearly} < 1/2$, and having a somewhat more complicated theory than simple monomials that allows $n$ to evolve from $0 < n_{\rm early} < 1/2$ – where the theory is healthy and matter domination holds – to a later epoch of $1/2 < n \le1$ where the theory has the right properties to cross the phantom divide.

Evolution of $n$ was in fact studied in [20, 21] and had the potential to cross the phantom divide. In [20] it was studied both in terms of $n(\Omega_{\rm de})$ and the more fundamental $n(X)$. Adopting the $n(X)$ model of [20] we find that a purely shift symmetric theory with $0 < n(X) < 1/2$ at all times can be healthy and give one or even two phantom crossings – the second mechanism for crossing is thus appropriate variation $n(X)$, and can preserve early matter domination.

The sigmoid $n(X)$ of [20], with $n(X) \equiv d\ln K/d\ln X$, is equivalent to replacing the $K \sim X^n$ monomial with almost as simple a form:

where $\Delta n = n_{\rm late} - n_{\rm early}$ gives the transition size, $X_c$ the transition timing, and $\delta$ the transition width. Note that the early time charge balance implies $m = (2n_{\rm early} - 1)/4$, not $m = [2n(X) - 1]/4$. Figure 4 demonstrates that phantom crossing indeed occurs (twice), and the model is healthy throughout its history. For illustration we use the values $n_{\rm early} = 0.35$ and $n_{\rm late} = 0.05$, i.e. $\Delta n = -0.3$. Note that using the early time behavior $X \sim a^{-6w_{\rm early}}$ we have $\ln(X_c/X_i) \approx-6w_{\rm early}\ln(a_c/a_i) \approx30 - 40$; the figure uses $\ln(X_c/X_i) = 34$. For the width $\delta= \Delta\ln X = 6x_\phi\Delta\ln a$ we note that the phantom crossing occurs as $x_\phi$ approaches zero, so for a transition taking an e-fold of expansion we expect $\delta\approx0.1 - 0.2$; the figure uses $\delta= 0.16$.

The phantom crossing can be viewed as a consequence of the rotation of the $w = -1$ parabola as $n$ changes, as seen in Figure 1 and Figure 2. Basically the variation in $n$ sweeps the trajectory across the changing position of $w = -1$ in such a way as to enable crossing. Another way to view it is through Eq. (21). Since for $0 < n < 1/2$ we have $Q < 0$, and $x_k < 0$, then reducing $n$ diminishes the negative contribution from the $8n^2x_k$ term, which effectively raises $Q$ toward zero.

Conclusions

A remarkably simple shift symmetry Horndeski gravity class with monomial kinetic and braiding terms possesses a rich phase space structure. By tracing this structure through dynamical systems analysis we identify the fixed points, attractors, and healthy regions free from ghosts and instability. In particular, dark energy can come to dominate the total energy density and give a de Sitter future state. The shift symmetry protects against troublesome high energy quantum loop corrections that bedevil almost all field dependent (non-shift symmetric) theories.

The curve $w = -1$ is a parabola in the phase space and divides it into phantom and standard regions. Solving for the properties of the parabola and trajectories that intersect it, we determine analytically the conditions for crossing this phantom divide. Contrary to conventional thought, shift symmetric Horndeski gravity can cross $w=-1$ in a healthy manner. Note that many of the arguments against phantom crossing by shift symmetric modified gravity assumed zero charge, justifying it as the state of the de Sitter attractor (recall the charge density $C \propto a^{-3}$ so in the asymptotic future $C \to0$). However, $C=0$ implies $x_g=-2nx_k$ or $Q=x_g(x_g-2n)$ in our notation. This is simply a straight line in phase space between the fixed points $M$ and $dS$. It cannot cross the parabola of $Q=0$, just meet it at $dS$, $x_g=2n$.

We identify two mechanisms for phantom crossing within a healthy shift symmetry theory with no potential or cosmological constant, and confirm their validity through full numerical solutions. One is replacing the $K \sim X^n$ monomial with nearly as simple a form having $n(X)$, and the second is taking initial conditions near the braiding point with $1/2<n\leq1$. The latter does not give an early matter dominated era, but this might be dealt with by an appropriate $n(X)$ model. The first, with $0<n<1/2$, does has a standard early matter dominated era, and dark energy evolves from the phantom to $w>-1$ (as observations appear to prefer), and can even cross a second time to reach $w<-1$ again. Both mechanisms should be regarded here as proofs of principle for phantom crossing, not as models proposed to fit the current observational data.

Finally, another approach to phantom crossing is to add field dependence in such a way as to preserve shift symmetry in the early universe, and hence protection from quantum corrections where it is needed most. This will be explored in future work [22]. We emphasize that a major advantage of starting from the action, i.e. specifying the Horndeski terms, rather than parametrizing the time dependence of property functions, $\alpha_i(a)$, or gravitational coupling strengths, $\mu(a)$ and $\Sigma(a)$, is the dynamical autonomous system approach and its insights into trajectories, fixed points, and crossing $w=-1$, plus the possibility of computing the perturbative sector evolution as well.

Acknowledgments

We thank the Galileo Galilei Institute for hospitality during part of this work. R.C. is funded by the Czech Ministry of Education, Youth and Sports (MEYS) and European Structural and Investment Funds (ESIF) under project number CZ.02.01.01/00/22 008/0004632. E.L. thanks the Simons Foundation for support as a Simons Visiting Scientist at GGI.