Structural Analysis of a Scalar-Tensor Realization of Interacting Dark Energy

Abstract

We investigate a class of interacting dark energy (IDE) models arising from density-driven spontaneous symmetry breaking in a conformally coupled scalar-tensor framework. In this construction, the dark matter-scalar interaction is dynamically activated as the cosmological density evolves, and the redshift dependence of the coupling follows a logistic profile whose steepness is determined by the local curvature of the symmetry-breaking potential. Working in the controlled adiabatic tracking regime, we implement the resulting epoch-dependent interaction in a perturbative background close to $Λ$CDM and confront the model with late-time cosmological data, including Planck 2018 CMB lensing reconstruction, redshift-space distortions, and Pantheon+SH0ES supernova data. We analyze realizations in which the activation index is allowed to vary and compare them with a restricted realization in which it is fixed to the canonical quadratic minimum value, thereby probing the structural role of the activation profile. We find no statistically significant preference for interaction over $Λ$CDM; current observations constrain the model to a hierarchical regime in which the scalar remains heavier than the Hubble scale at activation and background deformations remain perturbatively small. Allowing the activation index to vary preserves an extended degeneracy direction in parameter space, whereas fixing it removes this freedom and leads to a contraction of the allowed posterior region once geometric and growth data are combined. Our results delineate the viable parameter regime of symmetry-breaking IDE and clarify the structural distinction between microphysically motivated scalar-tensor realizations and phenomenological interacting models.

Figures (7)

• Figure 1: Dimensionless effective potential $V_{\mathrm{eff}}(x)$ for the polynomial class \ref{['eq:poly_potential']} with $m=2,3,4$ (left to right). Each curve shows representative epochs corresponding to decreasing deformation parameter $\xi(a)=\xi_0 a^{-3}$. At early times ($\xi\gg1$), the density-induced linear term lifts the vacuum degeneracy and selects a unique minimum at $x>1$. As $\xi(a)$ decreases with expansion, the deformation weakens and the minimum approaches the vacuum value $x=1$. The progressive flattening of the minimum for larger $m$ illustrates how the local restoring structure controls the width of the density-driven crossover.
• Figure 2: Late-time attractor dynamics for the polynomial potentials \ref{['eq:poly_potential']}. Left: Displacement from the broken minimum, $\delta(a)=x(a)-1$, for $m=2,3,4$. At late times the solutions converge to the universal power-law scaling $\delta \propto a^{-3/(m-1)}$, in agreement with Eq. \ref{['eq:poly_late_scaling_clean']}. Right: Instantaneous scaling exponent $d\ln\delta/d\ln a$ for the same models. The curves asymptote to the constant eigenvalues $-3$, $-3/2$, and $-1$, confirming that the broken minimum is a hyperbolic fixed point with eigenvalue $\lambda_p=-3/(m-1)$.
• Figure 3: Redshift evolution of the linear growth observables $f\sigma_8(z)$ and $f(z)$ in the IDE (6p) model for three dataset combinations: (a) RSD only, (b) Planck+RSD, and (c) Planck+RSD+SN. Solid curves denote posterior mean predictions, with shaded bands showing $68\%$ credible regions. Across all datasets, deviations from $\Lambda$CDM remain at the percent level at $z=0$ and within current observational uncertainties.
• Figure 4: Posterior-averaged dependence of $S_8=\sigma_8\sqrt{\Omega_m/0.3}$ on the interaction amplitude $\beta_0$ in the IDE (6p) model. The weak slope and small correlation coefficient indicate near-orthogonality between the interaction strength and late-time clustering normalization, implying minimal degeneracy in the combined Planck+RSD+SN fit.
• Figure 5: Supernova Hubble-diagram residuals $\mu_{\rm th}-\mu_{\rm obs}$ for the Planck+RSD+SN best-fit cosmologies in the flexible IDE (6p) and fixed-index ($n=3$) realizations. The 6p model yields $\chi^2_{\rm SN}=1809$ for $N_{\rm SN}=1701$ ($\chi^2/{\rm dof}\simeq1.06$), statistically comparable to $\Lambda$CDM. The fixed-index realization produces a slightly larger $\chi^2_{\rm SN}$, corresponding to a small change in the average supernova residual.