Observational constraints on scalar-nonmetricity cosmology: the role of the connection

Introduction

The observed late-time acceleration of the Universe constitutes one of the main challenges of modern cosmology and has stimulated an extensive investigation of theories beyond the standard $\Lambda$CDM paradigm [1, 2]. A natural possibility is to modify the gravitational sector itself, and a large variety of extensions of General Relativity (GR) have been developed along this direction. Within the standard curvature formulation, one may generalize the Einstein-Hilbert action by introducing nonlinear or higher-order curvature invariants, leading, among others, to $f(R)$ gravity [3, 4, 5], $f(G)$ gravity [6, 7], cubic gravity [8], and Lovelock gravity [9, 10]. An alternative and particularly important class is provided by scalar-tensor constructions, including Horndeski gravity [11, 12, 13] and generalized Galileon theories [14]. On the other hand, since gravity admits an equivalent torsional formulation, one can construct modified theories starting from teleparallel gravity instead. This leads to $f(T)$ gravity [15, 16, 17], $f(T,T_G)$ gravity [18, 19], $f(T,B)$ gravity [20, 21], scalar-torsion theories [22], as well as the Teleparallel Equivalent of Horndeski gravity [23, 24, 25, 26, 27].

A third, geometrically equivalent, description of gravity is provided by symmetric teleparallel gravity, originally discussed by Nester [28]. In this formulation both curvature and torsion vanish, while the gravitational interaction is entirely encoded in nonmetricity. This alternative geometrical representation has opened a new direction for constructing modified gravity theories, with $f(Q)$ gravity being its simplest and most extensively studied extension [29, 30]. In this case the nonmetricity scalar $Q$ appearing linearly in the symmetric teleparallel equivalent of GR is promoted to a general function $f(Q)$. The resulting theory exhibits a rich phenomenology and has been investigated in a variety of cosmological and astrophysical applications, including background cosmology, observational constraints, and static spherically symmetric configurations [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. For a review of nonmetricity-based gravity and its extensions, see [51].

Following the same logic that leads from General Relativity to scalar-tensor gravity, or from teleparallel gravity to scalar-torsion theories, one may extend symmetric teleparallel gravity by introducing a scalar field nonminimally coupled to the nonmetricity scalar [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71]. Scalar-nonmetricity gravity therefore provides the natural nonmetricity counterpart of the familiar scalar-curvature and scalar-torsion constructions. Apart from enlarging the space of viable cosmological behaviors, such a framework has an additional feature which is absent in the usual metric scalar-tensor theories. Namely, the affine connection is an independent geometrical quantity, and cosmological symmetry does not select a unique symmetric-teleparallel connection. For a spatially flat Friedmann-Robertson-Walker geometry there exist three inequivalent connection branches, which lead to different expressions for the nonmetricity scalar and, in general, to different background cosmological equations [72, 73, 74, 75, 76, 77, 78, 79]. Hence, the choice of connection is not merely a technical issue, but may leave observable signatures in the cosmic expansion history.

An additional ingredient of considerable interest is the form of the scalar-field potential. Although the exponential potential is frequently employed because of its mathematical simplicity, more general potentials can lead to a substantially richer evolution [80]. In particular, potentials that become steeper than the standard exponential form naturally introduce a varying effective slope and may generate nontrivial transitions during the cosmological evolution [81, 82]. At first sight, steep potentials might appear unfavorable for late-time acceleration, since a rapidly varying potential tends to enhance the scalar-field evolution. Nevertheless, this conclusion can be significantly modified in the presence of a nonminimal gravitational coupling [83, 84]. In scalar-nonmetricity gravity the interplay between the steepness of the potential, the scalar-nonmetricity coupling, and the affine connection may therefore lead to viable late-time cosmologies. Determining whether such scenarios are actually supported by observations, and whether the three allowed connection branches are observationally distinguishable, is thus an important question.

In the present work we address this question by confronting scalar-nonmetricity cosmology with current observational data. We consider a steep scalar potential and a power-law nonminimal coupling and analyze separately all three symmetric-teleparallel connection branches. The dynamical properties of scalar-nonmetricity cosmologies with steep potentials have recently been investigated in [85], while a data-driven reconstruction of the dynamical connection function in $f(Q)$ cosmology was performed in [86]. However, a systematic observational comparison of the three symmetric-teleparallel connection branches in scalar-nonmetricity cosmology has not yet been performed. In the present work we fill this gap by confronting all three branches with current observational data.

In particular, we derive the background evolution equations in a form suitable for numerical integration and perform a Markov Chain Monte Carlo analysis using combinations of the Pantheon Plus and DES Year 5 Type Ia supernova samples, DESI DR2 baryon acoustic oscillation measurements, and compressed cosmic microwave background information. We obtain constraints on the cosmological and model parameters, reconstruct the evolution of the matter and effective dark-energy density parameters and of the effective equation of state, and investigate the correlations between the parameters. Additionally, we compare the statistical performance of the three connection branches with that of $\Lambda$CDM.

The manuscript is organized as follows. In Sec. II we present the scalar-nonmetricity framework and derive the cosmological equations for the three symmetric-teleparallel connection branches. In § 3 we rewrite the corresponding background equations in terms of dimensionless variables and redshift in the form required for the observational analysis. In Sec. IV we introduce the observational datasets and methodology, present the resulting parameter constraints and reconstructed cosmological evolution, and perform the posterior and model-comparison analysis. Finally, in § 5 is devoted to the conclusions.

Scalar-tensor symmetric teleparallel cosmology

In this section we present the scalar-tensor extension of symmetric teleparallel gravity considered in the present work and derive the corresponding cosmological equations. We first introduce the general scalar-nonmetricity framework and obtain the gravitational, scalar-field, and connection field equations. We then specialize to a spatially flat Friedmann-Robertson-Walker geometry and examine the three independent symmetric-teleparallel connection branches allowed by cosmological symmetry.

Scalar-nonmetricity gravity

In symmetric teleparallel gravity, the gravitational interaction is encoded in nonmetricity, while both curvature and torsion of the affine connection vanish. More specifically, alongside the metric $g_{\mu\nu}$ one considers an independent torsion-free and curvature-free affine connection $\Gamma^{\alpha}_{\mu\nu}$. The latter can be decomposed as

where $\mathring{\Gamma}^{\alpha}_{\mu\nu}$ is the Levi-Civita connection,

and $L^{\alpha}_{\mu\nu}$ is the disformation tensor. The nonmetricity tensor is defined as

and the disformation reads

Introducing the two nonmetricity vectors $Q_{\mu}\equiv Q_{\mu\ \nu}^{\ \nu}$ and $\widetilde{Q}_{\mu}\equiv Q_{\nu\mu}^{\ \ \nu}$, the nonmetricity conjugate is

while, with the conventions adopted in the present work, the nonmetricity scalar is

A scalar-tensor extension of symmetric teleparallel gravity is obtained by allowing a scalar field to couple nonminimally to the nonmetricity scalar [53]. We consider the action

with $\kappa$ the gravitational constant, and where $f(\phi)$ determines the nonminimal coupling, $h(\phi)$ specifies the kinetic sector, and $U(\phi)$ is the scalar-field potential, while $S_{\mathrm{m}}$ accounts for the matter action. This construction constitutes the symmetric-teleparallel counterpart of curvature-based scalar-tensor gravity. In particular, the choice $f(\phi)=\phi$ and $h(\phi)=\omega/\phi$ leads to the nonmetricity analogue of Brans-Dicke gravity. Moreover, in the vanishing-kinetic-term limit the theory reproduces $F(Q)$ gravity through the usual auxiliary-field representation, with $\phi=F_{Q}$ and $U(\phi)=F_{Q}Q-F(Q)$.

Variation of (7) with respect to the metric gives

where $\mathring{G}_{\mu\nu}$ is the Einstein tensor constructed from the Levi-Civita connection. Variation with respect to the scalar field yields

Finally, the independent affine connection provides an additional field equation. For matter minimally coupled to the metric, for which the hypermomentum vanishes, it may be written in the form

Hence, although the symmetric teleparallel connection is curvature-free and torsion-free, its allowed cosmological form is not arbitrary: it must satisfy (10) together with the metric and scalar-field equations.

In the following we take $h(\phi)=h_{0}=\mathrm{const}$, which, for a nonvanishing kinetic sector of fixed sign, can be achieved through an appropriate scalar-field redefinition. Furthermore, motivated by the possibility of obtaining a richer scalar-field evolution than that associated with the standard exponential potential, we consider

where $U_{0}$, $\alpha$, $\beta$, $m$, and $n$ are constants. The standard exponential potential is recovered for $n=1$, while $n>1$ corresponds to the steep-potential case considered in the present work. In what follows we set the Planck mass as $M_{\mathrm{Pl}}^{2}\equiv1/\kappa=1$.

FRW cosmology and connection branches

We proceed to the cosmological application of the theory. We consider a spatially flat, homogeneous and isotropic Friedmann-Robertson-Walker (FRW) geometry,

where $a(t)$ is the scale factor and $H\equiv\dot{a}/a$ is the Hubble parameter. The scalar field is homogeneous, $\phi=\phi(t)$, while the matter sector is described by a perfect fluid,

Unlike in the metric formulation of gravity, specifying the FRW metric does not uniquely determine the affine connection. Cosmological symmetry must be imposed independently on the flat and torsion-free connection. For a spatially flat FRW geometry, the most general symmetric-teleparallel connection compatible with homogeneity and isotropy separates into three branches [72, 73, 74, 75, 76, 77, 78, 79]. In Cartesian coordinates its nonvanishing time-dependent components can be parametrized as

Imposing the vanishing-curvature condition then leads to the three families

where $\gamma(t)\ne0$ for Connections II and III.

It is useful to clarify the role of $\gamma(t)$. It parametrizes the residual freedom of the flat and torsion-free affine connection and should not, by itself, be regarded as an additional matter degree of freedom. Its evolution is constrained by the connection equation (10). Moreover, the three branches do not affect the cosmological evolution in the same way. For Connection I the function $\gamma(t)$ drops out from the background equations, and the resulting background cosmology is equivalent to that obtained in the usual coincident gauge. On the contrary, for Connections II and III the connection function enters both the nonmetricity scalar and the background field equations, and therefore contributes nontrivially to the cosmological evolution.

For the parametrization (15), the nonmetricity scalar takes the form

Using Eqs. (2.16–2.18), one obtains respectively

Thus, Connection I reproduces the standard FRW expression of the coincident-gauge formulation, whereas Connections II and III induce explicit connection-dependent contributions.

For later convenience, the two independent metric equations can be written for all three branches in the compact form

Similarly, the homogeneous scalar-field equation becomes

For minimally coupled perfect-fluid matter, the connection equation leads to the standard conservation relation

Equations (23)–(26), supplemented by the corresponding connection equation, form the background cosmological system. Since the three allowed connections lead to different expressions for $Q$ and, for Connections II and III, to an explicit dependence on $\gamma(t)$, they define three distinct cosmological branches of the same underlying scalar-nonmetricity theory. In the next section we formulate the corresponding background equations in a form suitable for their subsequent confrontation with observations.

Background equations for observational analysis

The dynamical properties of scalar-nonmetricity cosmology with the steep potential and power-law coupling considered here have been investigated in detail in Ref. [85], including the finite critical points, their stability, the asymptotic structure at infinity, and the non-hyperbolic solutions through center-manifold techniques. In the present work we do not repeat this analysis. Instead, our purpose is to formulate the background cosmological equations in the form required for the subsequent confrontation with observations.

We restrict ourselves to pressureless matter, $p=0$, and set $h_0=1$. For all three connection branches we introduce the dimensionless quantities

For Connections II and III an additional variable associated with the affine connection will be introduced below. Notice that $\widetilde{\Omega}_m$ is normalized by the nonminimal coupling $f(\phi)$ and therefore does not coincide in general with the standard matter density parameter.

For the model functions

one obtains

Hence, for $m \ne2$, these two quantities are not independent. Introducing

we obtain

Thus, the background evolution can be formulated without treating $\lambda$ as an independent variable. In the following we restrict ourselves to the positive-field branch, $\phi> 0$, and consider $n > 1$, $m > 2$, $\alpha> 0$, and $\beta> 0$, corresponding to the steep-potential sector employed in the observational analysis.

For each connection we introduce an effective dark-energy density parameter according to

while the total effective equation-of-state parameter is

For the numerical integration it is convenient to employ the redshift $z$, with $1+z=a^{-1}$. Therefore

Finally, in the following, a subscript 0 denotes the present-day value of the corresponding quantity, namely its value at $z=0$.

Connection I

For Connection I the affine function $\gamma(t)$ drops out from the background equations and the nonmetricity scalar reduces to $Q=-6H^2$. The first Friedmann equation gives

and hence

The second Friedmann equation yields

from which we deduce

Since matter is pressureless, the corresponding effective dark-energy equation of state can also be written as

whenever $\widetilde{\Omega}_{\mathrm{DE}}\ne0$.

Using Eq. (31) to eliminate $\lambda$, the background system to be integrated in redshift takes the compact form

Together with Eqs. (35) and (36), these equations completely determine the background evolution of Connection I and provide the system used in the observational analysis of Sec. IV.

Connection II

For Connection II the affine connection contributes explicitly to the background evolution. In addition to the variables introduced in Eq. (27), we define

The first Friedmann equation then gives

while for pressureless matter, the second Friedmann equation yields

Hence, the total effective equation-of-state parameter is

Since the matter component is pressureless, the effective dark-energy equation of state is simply $w_{\mathrm{DE}}=w_{\mathrm{eff}}/\widetilde{\Omega}_{\mathrm{DE}}$ whenever $\widetilde{\Omega}_{\mathrm{DE}}\ne0$. Using the model-specific relation $\lambda=\mathcal{K}\zeta^{p_\zeta}$ from Eq. (31), the background equations can be written directly in terms of the redshift as

Notice that the term $\lambda/\zeta$ appearing in the unreduced formulation is now replaced by $\mathcal{K}\zeta^{p_\zeta-1}$. Since $m>2$ and $n>1$ imply $p_\zeta>1$, this quantity remains regular in the limit $\zeta\to0$.

Equations (49)–(53), together with the constraint (45), constitute the system used for the numerical background evolution and the observational analysis of Connection II.

Connection III

For Connection III we introduce $\mathring{\gamma}\equiv\gamma/a^2$ and define the dimensionless connection variable

The first Friedmann equation then gives

while for pressureless matter, the second Friedmann equation becomes

Consequently, the total effective equation-of-state parameter is

Since the matter sector is pressureless, the corresponding effective dark-energy equation of state is $w_{\mathrm{DE}}=w_{\mathrm{eff}}/\widetilde{\Omega}_{\mathrm{DE}}$ whenever $\widetilde{\Omega}_{\mathrm{DE}}\ne0$.

For compactness, we further introduce

Using the model-specific relation $\lambda=\mathcal{K}\zeta^{p_\zeta}$ from Eq. (31), the background equations can be written directly in terms of the redshift as

The system is well defined for $\Xi\ne0$, and the numerical integrations performed below are restricted to this regular domain. Together with the constraint (55), Eqs. (60)–(64) constitute the background system used for the observational analysis of Connection III.

Observational constraints and cosmological implications

In this section we confront the three scalar-nonmetricity connection branches with current cosmological observations and investigate the extent to which the choice of symmetric-teleparallel connection affects the inferred background evolution. We first describe the observational datasets, likelihoods, and statistical methodology employed in the analysis. We then present the resulting parameter constraints and reconstruct the cosmological evolution for each branch. Finally, we compare the statistical performance of the three connections with that of the standard $\Lambda$CDM scenario.

Observational datasets and methodology

We constrain the model parameters through a Markov Chain Monte Carlo (MCMC) analysis implemented within the COBAYA framework [87]. The background evolution equations derived in § 3 are incorporated through CosmoDS [88], which provides an external theory component for COBAYA suitable for the numerical integration of nonstandard cosmological backgrounds. The resulting posterior samples are analyzed and visualized using the publicly available GetDist package.

Type Ia supernovae

Type Ia supernovae (SN-Ia) constitute one of the main probes of the late-time expansion history, since their standardized luminosities provide measurements of the luminosity distance over a broad redshift interval [89]. In the present analysis we consider separately the Pantheon Plus compilation [90, 91, 92] and the DES Year 5 supernova sample [93]. The two compilations are based on different surveys, calibration procedures, and analysis pipelines, and therefore provide complementary determinations of the distance-redshift relation.

DESI DR2 BAO

We additionally employ the baryon acoustic oscillation (BAO) measurements from the 2025 DESI Data Release 2 (DR2) [94, 95]. BAO observations provide a standard ruler through the sound horizon at the drag epoch, $r_d$, and constrain both radial and transverse cosmological distances. The relevant quantities are

where $D_H$ and $D_M$ denote the Hubble and transverse comoving distances, respectively. For isotropic BAO measurements we use

Throughout the analysis, $r_d$ is treated as a free parameter rather than being fixed by a specific early-time cosmological model.

Compressed CMB likelihood

In order to complement the low-redshift probes with information from the early Universe, we employ the compressed cosmic microwave background (CMB) likelihood described in Appendix A of Ref. [94]. The CMB information relevant for the present background analysis is encoded in a Gaussian likelihood involving the acoustic angular scale $\theta_*$ and the physical baryon and matter densities, $\omega_b \equiv\Omega_bh^2$ and $\omega_{bc} \equiv(\Omega_b+\Omega_c)h^2$. These quantities are tightly constrained by CMB observations and depend only weakly on the details of the late-time cosmological evolution once secondary contributions are appropriately marginalized over [96].

The compressed likelihood therefore provides a high-redshift anchor for the expansion history while avoiding the computational cost and additional model dependence associated with a full CMB power-spectrum analysis. In particular, it constrains the matter sector and breaks degeneracies that remain when BAO and supernova data are considered alone. As shown in Ref. [94], for late-time dark-energy analyses the resulting constraints closely reproduce those obtained with the corresponding full CMB likelihood.

Dataset combinations and priors

Having introduced the three observational probes, we combine their likelihoods assuming statistical independence. Thus, for the combinations including supernovae,

Observational constraints and cosmological evolution

the total chi-square is

while for the CMB+DESI DR2 combination the supernova contribution is omitted.

We consider the following three combinations of observational datasets:

1. Pantheon Plus + CMB + DESI DR2,

2. DES Y5 + CMB + DESI DR2,

3. CMB + DESI DR2.

The first two combinations allow us to assess separately the impact of the Pantheon Plus and DES Y5 supernova compilations, while the third provides a supernova-independent constraint based on BAO and CMB information.

For each connection branch we sample the present-day values of the corresponding background variables together with $H_0$, $r_d$, $m$, and $n$. The parameters $\alpha$ and $\beta$ are fixed to unity throughout the analysis. Flat priors are adopted within the ranges summarized in Table I.

Parameter	Connection I	Connection II	Connection III
$H_0$	$[60, 80]$	$[60, 80]$	$[60, 80]$
$x_0$	$[-1, 1]$	$[-0.3, 0.3]$	$[-0.3, 0.3]$
$y_0$	$[10^{-4}, 2]$	$[0.5, 2]$	$[0.5, 2]$
$\eta_0$	–	$[0.05, 0.5]$	$[0.05, 0.5]$
$\zeta_0$	$[0.01, 1]$	$[0.15, 0.35]$	$[0.15, 0.35]$
$m$	$[2.01, 3.5]$	$[2.01, 3.5]$	$[2.01, 3.5]$
$n$	$[1.01, 2]$	$[1.01, 2]$	$[1.01, 2]$
$r_d$	$[130, 180]$	$[130, 180]$	$[130, 180]$
$\alpha$	$1$	$1$	$1$
$\beta$	$1$	$1$	$1$

We now present the observational constraints on the three connection branches and examine the corresponding reconstructed cosmological evolution. The marginalized constraints obtained from the three dataset combinations are summarized in Table II, Table III, and Table IV for Connections I, II, and III, respectively. In addition to the directly sampled parameters, we report the derived present-day quantities $\lambda_0$, $\tilde{\Omega}_{m0}$, and $w_{\mathrm{eff},0}$. In particular, $\lambda_0$ is calculated through Eq. (31) and is therefore not treated as an independent MCMC parameter. In all cases the quoted constraints correspond to the marginalized posterior distributions, with the uncertainties referring to the 68% credible intervals.

Datasets: Pantheon Plus + CMB + DESI DR2
Model	$H_0$	$\Omega_{m0}$	$r_d$	$x_0$	$y_0$	$\zeta_0$	$m$	$n$	$\lambda_0$	$w_{\mathrm{eff},0}$
Connection I	$69.40^{+0.51}_{-0.45}$	$0.29^{+0.004}_{-0.005}$	$146.91^{+0.303}_{-0.261}$	$-0.16^{+0.094}_{-0.050}$	$1.34^{+0.067}_{-0.019}$	$0.15^{+0.029}_{-0.122}$	$2.29^{+0.090}_{-0.091}$	$1.84^{+0.076}_{-0.092}$	$1.47^{+1.46}_{-1.44}\times10^{-9}$	$-0.69^{+0.014}_{-0.021}$
$\Lambda$CDM	$69.53^{+0.35}_{-0.36}$	$0.29\pm0.004$	$146.92^{+0.24}_{-0.25}$	$-0.70\pm0.004$
Datasets: DES Y5 + CMB + DESI DR2
Connection I	$67.34^{+0.50}_{-0.54}$	$0.31^{+0.005}_{-0.005}$	$147.10^{+0.255}_{-0.252}$	$-0.25^{+0.040}_{-0.041}$	$1.24^{+0.047}_{-0.043}$	$0.10^{+0.041}_{-0.067}$	$2.30^{+0.099}_{-0.101}$	$1.84^{+0.104}_{-0.095}$	$3.83^{+3.83}_{-3.81}\times10^{-10}$	$-0.60^{+0.033}_{-0.022}$
$\Lambda$CDM	$68.63^{+0.42}_{-0.40}$	$0.30\pm0.004$	$147.16^{+0.27}_{-0.23}$	$-0.69\pm0.004$
Datasets: CMB + DESI DR2
Connection I	$68.23^{+0.89}_{-0.71}$	$0.30^{+0.007}_{-0.008}$	$147.14^{+0.220}_{-0.266}$	$-0.35^{+0.046}_{-0.051}$	$1.13^{+0.079}_{-0.073}$	$0.41^{+0.065}_{-0.047}$	$2.27^{+0.092}_{-0.106}$	$1.87^{+0.098}_{-0.099}$	$4.48^{+3.13}_{-4.45}\times10^{-8}$	$-0.68^{+0.023}_{-0.031}$
$\Lambda$CDM	$69.01^{+0.43}_{-0.45}$	$0.29^{+0.004}_{-0.005}$	$147.09\pm0.25$	$-0.70^{+0.004}_{-0.005}$

Datasets: Pantheon Plus + CMB + DESI DR2
Model	$H_0$	$\Omega_{m0}$	$r_d$	$x_0$	$y_0$	$\eta_0$	$\zeta_0$	$m$	$n$	$\lambda_0$	$w_{\mathrm{eff},0}$
Connection II	$68.67^{+0.26}_{-0.27}$	$0.30^{+0.004}_{-0.003}$	$147.24^{+0.238}_{-0.228}$	$-0.00010^{+0.00069}_{-0.00067}$	$1.39^{+0.007}_{-0.008}$	$0.20^{+0.038}_{-0.028}$	$0.26^{+0.043}_{-0.028}$	$2.30^{+0.026}_{-0.030}$	$1.40^{+0.038}_{-0.034}$	$5.94^{+3.84}_{-5.90}\times10^{-9}$	$-0.69^{+0.004}_{-0.003}$
$\Lambda$CDM	$69.53^{+0.35}_{-0.36}$	$0.29\pm0.004$	$146.92^{+0.24}_{-0.25}$	$-0.70\pm0.004$
Datasets: DES Y5 + CMB + DESI DR2
Connection II	$68.09^{+0.29}_{-0.30}$	$0.30^{+0.004}_{-0.004}$	$147.27^{+0.243}_{-0.232}$	$-0.00008^{+0.00084}_{-0.00080}$	$1.38^{+0.008}_{-0.009}$	$0.17^{+0.030}_{-0.016}$	$0.23^{+0.009}_{-0.024}$	$2.18^{+0.015}_{-0.005}$	$1.37^{+0.023}_{-0.022}$	$2.44^{+1.73}_{-2.22}\times10^{-15}$	$-0.69^{+0.004}_{-0.004}$
$\Lambda$CDM	$68.63^{+0.42}_{-0.40}$	$0.30\pm0.004$	$147.16^{+0.27}_{-0.23}$	$-0.69\pm0.004$
Datasets: CMB + DESI DR2
Connection II	$68.15^{+0.30}_{-0.30}$	$0.30^{+0.003}_{-0.004}$	$147.25^{+0.256}_{-0.216}$	$0.00007^{+0.00069}_{-0.00066}$	$1.38^{+0.008}_{-0.007}$	$0.17^{+0.042}_{-0.052}$	$0.26^{+0.011}_{-0.014}$	$2.28^{+0.012}_{-0.013}$	$1.42^{+0.017}_{-0.020}$	$1.51^{+1.57}_{-1.22}\times10^{-10}$	$-0.69^{+0.004}_{-0.004}$
$\Lambda$CDM	$69.01^{+0.43}_{-0.45}$	$0.29^{+0.004}_{-0.005}$	$147.09\pm0.25$	$-0.70^{+0.004}_{-0.005}$

Model	$H_0$	$\widetilde{\Omega}_{m0}$	$r_d$	$x_0$	$y_0$	$\eta_0$	$\zeta_0$	$m$	$n$	$\lambda_0$	$w_{\mathrm{eff},0}$
Datasets: Pantheon Plus + CMB + DESI DR2
Connection III	$68.73^{+0.14}_{-0.14}$	$0.29^{+0.017}_{-0.016}$	$147.20^{+0.214}_{-0.177}$	$-0.04624^{+0.04128}_{-0.02415}$	$1.41^{+0.033}_{-0.031}$	$0.19^{+0.018}_{-0.019}$	$0.25^{+0.039}_{-0.023}$	$2.31^{+0.011}_{-0.021}$	$1.41^{+0.022}_{-0.011}$	$6.16^{+5.08}_{-4.62} \times10^{-10}$	$-0.72^{+0.022}_{-0.017}$
$\Lambda$CDM	$69.53^{+0.35}_{-0.36}$	$0.29 \pm0.004$	$146.92^{+0.24}_{-0.25}$	$-0.70 \pm0.004$
Datasets: DES Y5 + CMB + DESI DR2
Connection III	$68.63^{+0.17}_{-0.15}$	$0.27^{+0.025}_{-0.014}$	$147.28^{+0.213}_{-0.250}$	$-0.06734^{+0.06618}_{-0.04147}$	$1.42^{+0.033}_{-0.036}$	$0.17^{+0.053}_{-0.057}$	$0.26^{+0.060}_{-0.033}$	$2.34^{+0.033}_{-0.024}$	$1.41^{+0.040}_{-0.041}$	$4.74^{+4.13}_{-4.54} \times10^{-9}$	$-0.72^{+0.019}_{-0.027}$
$\Lambda$CDM	$68.63^{+0.42}_{-0.40}$	$0.30 \pm0.004$	$147.16^{+0.27}_{-0.23}$	$-0.69 \pm0.004$
Datasets: CMB + DESI DR2
Connection III	$68.56^{+0.15}_{-0.14}$	$0.27^{+0.045}_{-0.030}$	$147.25^{+0.198}_{-0.207}$	$0.01023^{+0.14472}_{-0.14292}$	$1.41^{+0.050}_{-0.051}$	$0.25^{+0.032}_{-0.061}$	$0.24^{+0.0632}_{-0.0473}$	$2.29^{+0.028}_{-0.025}$	$1.40^{+0.021}_{-0.022}$	$1.94^{+1.82}_{-1.88} \times10^{-10}$	$-0.68^{+0.014}_{-0.047}$
$\Lambda$CDM	$69.01^{+0.43}_{-0.45}$	$0.29^{+0.004}_{-0.005}$	$147.09 \pm0.25$	$-0.70^{+0.004}_{-0.005}$

For Connection I, the Pantheon Plus+CMB+DESI DR2 combination yields $H_0=69.40^{+0.51}_{-0.45}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $\tilde{\Omega}_{m0}=0.29^{+0.004}_{-0.005}$, while the corresponding sound-horizon scale is $r_d=146.91^{+0.303}_{-0.261}\ \mathrm{Mpc}$. The present effective equation of state is $w_{\mathrm{eff},0}=-0.69^{+0.014}_{-0.021}$. Replacing Pantheon Plus with DES Y5 leads to a lower value, $H_0=67.34^{+0.50}_{-0.54}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$, together with $\tilde{\Omega}_{m0}=0.31\pm0.005$ and $w_{\mathrm{eff},0}=-0.60^{+0.033}_{-0.022}$. On the other hand, the CMB+DESI DR2 combination gives $H_0=68.23^{+0.89}_{-0.71}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $w_{\mathrm{eff},0}=-0.68^{+0.023}_{-0.031}$. Hence, Connection I displays a noticeable sensitivity to the adopted supernova compilation, particularly in the inferred value of $H_0$ and in the present effective equation of state. This sensitivity is considerably weaker for the other two connection branches, as we discuss below.

The corresponding reconstructed histories are shown in Figure 1. All three dataset combinations recover a cosmological evolution in which the normalized matter contribution increases towards the past and the effective dark-energy component becomes dominant at late times. Correspondingly, $w_{\mathrm{eff}}(z)$ evolves from values close to zero at higher redshift to negative values at the present epoch, yielding the transition to accelerated expansion. Although the overall behavior is common to the three reconstructions, their late-time evolution is not identical. In particular, the DES Y5 combination favors a less negative present value of $w_{\mathrm{eff}}$ and allows a visibly different late-time evolution, while the Pantheon Plus and CMB+DESI DR2 reconstructions remain closer to each other. Thus, for Connection I the choice of the supernova dataset has a non-negligible impact on the inferred recent expansion history.

For Connection II, the three observational combinations lead to a much more uniform set of constraints. We obtain $H_0 = 68.67^{+0.26}_{-0.27}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ from Pantheon Plus+CMB+DESI DR2, $H_0 = 68.09^{+0.29}_{-0.30}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ from DES Y5+CMB+DESI DR2, and $H_0 = 68.15 \pm0.30\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ from CMB+DESI DR2. The nor-

malized matter density remains close to $\widetilde{\Omega}_{m0} \simeq0.30$ in all three cases, while $r_d \simeq147.2$ Mpc. Moreover, the present scalar kinetic variable is strongly constrained around $x_0 \simeq0$, whereas $y_0 \simeq1.38 - 1.39$. The inferred effective equation of state is also remarkably stable, with $w_{\mathrm{eff},0} \simeq-0.69$ and uncertainties of order a few $\times10^{-3}$. Therefore, in contrast to Connection I, the main late-time cosmological quantities in Connection II are only weakly affected by whether Pantheon Plus, DES Y5, or no supernova sample is included.

This behavior is directly reflected in the reconstructed histories shown in Figure 2. The three dataset combinations produce almost overlapping central evolutions for $\widetilde{\Omega}_m(z)$, $\widetilde{\Omega}_{\mathrm{DE}}(z)$, and $w_{\mathrm{eff}}(z)$, with comparatively narrow credible regions. The evolution again interpolates between an approximately matter-dominated regime at higher redshift and the late-time accelerated phase. Hence, although the affine connection enters explicitly in the background equations of this branch, the observationally allowed expansion history is tightly restricted and exhibits only a weak dependence on the particular supernova compilation.

Finally, Connection III also leads to well-constrained late-time cosmological parameters. For Pantheon Plus+CMB+DESI DR2 we find $H_0 = 68.73 \pm0.14\ \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and $\tilde{\Omega}_{m0} = 0.29^{+0.017}_{-0.016}$, while DES Y5+CMB+DESI DR2 gives $H_0 = 68.63^{+0.17}_{-0.15}\ \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and $\tilde{\Omega}_{m0} = 0.27^{+0.025}_{-0.014}$. For CMB+DESI DR2 we obtain $H_0 = 68.56^{+0.15}_{-0.14}\ \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and $\tilde{\Omega}_{m0} = 0.27^{+0.045}_{-0.030}$. In all three cases $r_d$ remains close to 147.2 Mpc. The present effective equation of state lies in the interval $w_{\mathrm{eff},0} \simeq-0.68$ to $-0.72$, while the preferred values of the steep-potential parameters remain concentrated around $m \simeq2.3$ and $n \simeq1.4$. Thus, the inferred value of $H_0$ is particularly stable in this branch, while the matter fraction shows a somewhat stronger dependence on the dataset combination.

The reconstructed evolution for Connection III is displayed in Figure 3. As in the previous two branches, the observationally allowed histories exhibit the transition from a matter-dominated regime towards late-time accelerated expansion. The Pantheon Plus and DES Y5 reconstructions remain relatively close, whereas the CMB+DESI DR2 combination allows somewhat greater freedom in the relative matter and effective dark-energy fractions. Nevertheless, the corresponding $w_{\mathrm{eff}}(z)$ histories remain mutually compatible within their credible regions and approach values close to zero towards higher redshift.

Therefore, the additional geometrical contribution associated with Connection III remains compatible with the combined low- and high-redshift probes considered here, while producing an observational behavior that is distinct from both Connections I and II.

Overall, all three symmetric-teleparallel connection branches provide cosmological histories compatible with a matter-dominated epoch followed by late-time accelerated expansion. However, the degree to which the inferred background evolution depends on the observational dataset is clearly connection dependent. Connection I exhibits the strongest sensitivity to the choice of supernova compilation, most notably through the shifts in $H_0$ and $w_{\mathrm{eff},0}$. Connection II displays the most stable reconstruction, with the three dataset combinations yielding nearly coincident expansion histories, whereas Connection III presents an intermediate behavior, with a very stable value of $H_0$ but somewhat larger freedom in the reconstructed matter fraction. These differences demonstrate that the choice of symmetric-teleparallel connection can leave a measurable imprint on the inferred

background cosmology, even when all three branches remain compatible with the observational probes considered here. A more detailed examination of the multidimensional parameter constraints and their correlations is presented in the following subsection.

Posterior distributions and model comparison

We finally examine the multidimensional posterior distributions and assess the statistical performance of the three connection branches relative to $\Lambda$CDM. The one- and two-dimensional marginalized posterior distributions are presented in Figure 4-Figure 6. They complement the one-dimensional constraints discussed above by displaying the correlations and degeneracies among the cosmological parameters, the scalar-field variables, and, for Connections II and III, the variables associated with the affine connection.

For Connection I, shown in Figure 4, the posterior distributions exhibit a noticeable dependence on the adopted supernova sample. This is particularly evident in quantities controlling the late-time expansion, consistently with the shifts in $H_0$, $\tilde{\Omega}_{m0}$, and $w_{\mathrm{eff},0}$ found in Sec. IV B. The Pantheon Plus and CMB+DESI DR2 constraints show substantial overlap in several parameter directions, whereas the inclusion of DES Y5 shifts the preferred region and modifies a number of parameter degeneracies. This behavior confirms that Connection I possesses comparatively large phenomenological freedom at the background level and can accommodate different preferred late-time expansion histories depending on the supernova compilation employed.

The posterior distributions for Connection II are displayed in Figure 5. In this case the three dataset combinations select considerably more similar regions of parameter space. In particular, the late-time cosmological quantities show only moderate shifts when the supernova compilation is changed, in agreement with the nearly overlapping reconstructed histories of Figure 2. The posterior distributions also confirm the strong localization of $x_0$ around zero, while the remaining scalar and connection variables are confined to comparatively narrow regions. Hence, Connection II exhibits a substantially more dataset-independent background phenomenology than Connection I.

Connection III, whose posterior distributions are presented in Figure 6, displays an intermediate behavior. The three observational combinations remain broadly compatible, although the CMB+DESI DR2 case, which does not contain supernova information, allows larger regions for some of the background variables. At the same time, the steep-potential parameters remain concentrated around $m \simeq2.3$ and $n \simeq1.4$, while the preferred values of $H_0$ are remarkably stable among the three dataset combinations. Thus, Connections II and III both exhibit a comparatively robust observational behavior, although their multidimensional posterior structures are different, reflecting the different manner in which the affine connection enters their respective background equations.

In order to quantify the relative statistical performance of the three connection branches, we compare their minimum chi-square values and the Akaike information criterion (AIC), defined as

where $k$ is the number of free parameters of the corresponding model. We further define

and

such that negative values favor the scalar-nonmetricity model relative to $\Lambda$CDM, while positive values favor $\Lambda$CDM. The corresponding results are summarized in Table V.

Model	$\chi^2_{\min}$	$\Delta\chi^2_{\min} = \chi^2_{\min} - \chi^2_{\min,\Lambda CDM}$	AIC	$\Delta AIC = AIC - AIC_{\Lambda CDM}$
Connection I:	1484.20	−1.18	1498.20	6.82
Connection II:	1488.81	3.43	1504.81	13.43
Connection III:	1490.00	4.62	1506.00	14.62
$\Lambda$CDM:	1485.38	0	1491.38	0
Datasets: DES Y5 + CMB + DESI DR2
Connection I:	1645.77	−13.99	1659.77	−5.99
Connection II:	1659.63	−0.13	1675.63	9.87
Connection III:	1660.96	1.20	1676.96	11.20
$\Lambda$CDM:	1659.76	0	1665.76	0
Datasets: CMB + DESI DR2
Connection I:	9.31	−1.00	23.31	7.00
Connection II:	10.65	0.34	26.65	10.34
Connection III:	10.39	0.08	26.39	10.08
$\Lambda$CDM:	10.31	0	16.31	0

For Pantheon Plus+CMB+DESI DR2, Connection I achieves a slightly smaller minimum chi-square than $\Lambda$CDM, with $\Delta\chi^2_{\min} = -1.18$. This improvement in the fit is nevertheless too small to compensate for the additional model parameters, resulting in $\Delta\mathrm{AIC} = 6.82$. Connections II and III give $\Delta\chi^2_{\min} = 3.43$ and $4.62$, respectively, together with $\Delta\mathrm{AIC} = 13.43$ and $14.62$. Hence, for this dataset combination the additional freedom of the scalar-nonmetricity scenarios is not favored by the AIC, and $\Lambda$CDM provides the more economical description of the data.

A markedly different behavior is obtained for the DES Y5+CMB+DESI DR2 combination. In this case Connection I reduces the minimum chi-square by $\Delta\chi^2_{\min} = -13.99$ relative to $\Lambda$CDM. The improvement is sufficiently large to overcome the AIC penalty associated with the additional parameters, yielding $\Delta\mathrm{AIC} = -5.99$. Thus, among all the scalar-nonmetricity cases and dataset combinations considered here, this is the only one for which the AIC favors the modified-gravity scenario over $\Lambda$CDM. Connection II produces only a very small improvement in the minimum chi-square, $\Delta\chi^2_{\min} = -0.13$, which is insufficient to compensate for its additional parameters and gives $\Delta\mathrm{AIC} = 9.87$. Connection III instead yields $\Delta\chi^2_{\min} = 1.20$ and $\Delta\mathrm{AIC} = 11.20$.

For CMB+DESI DR2, the four models achieve very similar minimum chi-square values. In particular, Connections I, II, and III give $\Delta\chi_{\min}^{2} = -1.00$, 0.34, and 0.08, respectively. Consequently, there is little difference in their maximum-likelihood description of these data. Once the additional free parameters are taken into account, however, the corresponding values become $\Delta\mathrm{AIC} = 7.00$, 10.34, and 10.08, respectively. Hence, $\Lambda$CDM is again favored according to the AIC for this dataset combination.

The model comparison therefore reveals an important connection dependence that goes beyond the differences already seen in the reconstructed background histories. All three branches can accommodate the observed latetime expansion, but they do so with different degrees of parameter freedom and different responses to the observational datasets. Connection I is the most sensitive to the adopted supernova compilation and exhibits the largest phenomenological flexibility. This flexibility does not improve its statistical performance for Pantheon Plus or for CMB+DESI DR2 alone, once the additional parameters are penalized. Remarkably, however, for the DES Y5 combination it leads to a reduction of almost fourteen units in $\chi^2_{\min}$ and to $\Delta AIC < 0$, such that the improvement in fit outweighs the complexity penalty. Connections II and III, in contrast, display considerably more stable posterior constraints and reconstructed histories across the three dataset combinations, but this robustness is accompanied by no comparable improvement in the best-fit likelihood.

Hence, the observational impact of the symmetric-teleparallel connection is twofold. It affects both the reconstructed expansion history and the statistical response of the model to different combinations of cosmological data. In particular, the fact that the same scalar-nonmetricity action and the same steep potential lead to distinct posterior structures and model comparison results for the three allowed connections demonstrates that the choice of affine connection is not observationally innocuous at the background level. This constitutes one of the main results of the present analysis.

Conclusions

The geometric formulation of gravity based on nonmetricity has emerged as an important alternative framework for exploring gravitational physics beyond General Relativity. In particular, $f(Q)$ gravity has demonstrated that symmetric teleparallel geometry can lead to a rich cosmological phenomenology while preserving a conceptually distinct description of the gravitational

interaction. Scalar-nonmetricity theories provide a further natural extension, in close analogy with scalar-tensor and scalar-torsion gravity, by allowing a scalar degree of freedom to couple nonminimally to the nonmetricity scalar. An especially interesting feature of this framework is that, unlike in the usual metric formulation, cosmological symmetry does not uniquely determine the affine connection. Even for a spatially flat Friedmann-Robertson-Walker background, three inequivalent symmetric-teleparallel connection branches are allowed. Consequently, the choice of connection may acquire physical relevance and, in principle, leave observable imprints on the cosmological evolution.

In the present work we investigated this possibility by performing a systematic observational analysis of scalar-nonmetricity cosmology for all three connection branches. We considered a steep scalar-field potential, $U(\phi) = U_0 \exp(\alpha\phi^n)$, together with a power-law nonminimal coupling, $f(\phi) = \beta\phi^m$. Such a choice is particularly interesting because the steepness of the potential, the nonminimal coupling, and the connection contribution can compete in determining the late-time evolution. After deriving the corresponding background equations in a form suitable for numerical integration, we confronted the three branches with combinations of Pantheon Plus or DES Year 5 Type Ia supernovae, DESI DR2 BAO measurements, and compressed CMB information. This allowed us to constrain the model parameters, reconstruct the cosmological histories, and compare the statistical performance of the three connections with that of $\Lambda$CDM. In this way, our analysis directly addresses whether the freedom associated with the symmetric-teleparallel connection can be probed observationally.

Our results show first of all that all three connection branches can provide viable background cosmological histories. In every case the observationally allowed evolution exhibits the required transition from a matter-dominated regime at higher redshift to an effective dark-energy-dominated phase at late times, accompanied by the onset of accelerated expansion. Nevertheless, the three branches do not respond to the data in the same way. Connection I displays the strongest dependence on the supernova compilation. In particular, the inferred values of $H_0$ and $w_{\mathrm{eff},0}$ change noticeably when Pantheon Plus is replaced by DES Y5, and this behavior is clearly reflected in the reconstructed histories and posterior distributions. Hence, although Connection I is the simplest branch at the level of the background connection structure, it possesses considerable phenomenological freedom in the late Universe.

Connection II exhibits a markedly different behavior. The three dataset combinations lead to very similar values of the main cosmological quantities, with $\Omega_{\mathrm{m}0} \simeq0.30$, $r_{\mathrm{d}} \simeq147.2$ Mpc, and $w_{\mathrm{eff},0} \simeq-0.69$. The reconstructed histories are almost overlapping and the corresponding posterior regions are comparatively narrow. Thus, despite the explicit appearance of the affine connection in the background equations, the observationally allowed cosmology of this branch is highly restricted and remarkably insensitive to the particular supernova compilation.

Connection III presents an intermediate situation. Its inferred value of $H_0$ is particularly stable among the three observational combinations, while somewhat larger freedom is allowed in the reconstructed matter and effective dark-energy fractions. At the same time, the steep-potential parameters are consistently localized around $m \simeq2.3$ and $n \simeq1.4$. These differences between the three branches already demonstrate that the connection choice affects the way in which a given scalar-nonmetricity action is mapped onto the observationally allowed cosmic expansion.

The model comparison strengthens this conclusion. For Pantheon Plus+CMB+DESI DR2, Connection I slightly improves the minimum chi-square relative to $\Lambda$CDM, but the improvement is not sufficient to compensate for its additional parameters, while Connections II and III are also disfavored according to the AIC. A qualitatively different result arises for DES Y5+CMB+DESI DR2. In this case Connection I achieves $\Delta\chi^2_{\min} = -13.99$ and $\Delta\mathrm{AIC} = -5.99$, and is therefore favored over $\Lambda$CDM according to the AIC among the models considered. Neither Connection II nor Connection III exhibits a comparable improvement. Finally, for CMB+DESI DR2 alone the minimum chi-square values of all four cosmologies are very similar, and the AIC consequently favors the more economical $\Lambda$CDM description. Hence, the additional freedom of scalar-nonmetricity gravity is not generically rewarded by the data. However, the fact that one connection branch can exploit this freedom to improve substantially the fit to a specific modern supernova dataset, whereas the other two cannot, is particularly noteworthy. It indicates that the choice of symmetric-teleparallel connection affects not only the form of the cosmological equations, but also the statistical response of the theory to observations.

There are several directions in which the present analysis can be extended. A natural next step is to move beyond the homogeneous background and study linear cosmological perturbations for the three connection branches, including the evolution of matter perturbations, the effective gravitational coupling, gravitational slip, and structure-growth observables. This would allow the models to be tested against redshift-space distortions, weak lensing, large-scale structure, and $S_8$ measurements, and could reveal additional differences between connections that remain partly degenerate at the background level. It would also be important to incorporate the full CMB likelihood, investigate broader classes of scalar-field potentials and nonminimal couplings, and complement the AIC analysis with Bayesian model comparison. Future standard-siren, large-scale-structure, and next-generation supernova and BAO measurements will further improve the ability to distinguish between the different branches. More generally, identifying whether observations can favor a particular symmetric-teleparallel connection may provide valuable guidance for understanding which geometrical realization of nonmetricity gravity is physically relevant. The results obtained here provide a first step in this direction and show that the connection freedom inherent in scalar-nonmetricity gravity can translate into genuinely distinguishable cosmological phenomenology.

References