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.

• The paper derives a logistic activation profile, β(a)=β₀aⁿ/(aⁿ+a_cⁿ), with the index fixed by the scalar potential’s local restoring order as n=3/p.
• The paper finds no statistically significant preference for interacting dark energy over ΛCDM, with Δlog Z≈−1.4 at most and growth deviations limited to roughly 1%.
• The paper shows that fixing the canonical n=3 value compresses posteriors toward prior boundaries without improving fit quality, while the model leaves H₀ unchanged and has negligible correlation with S₈.

Overview and motivation

This work examines a class of interacting dark energy (IDE) models in which the dark matter–scalar coupling is not imposed phenomenologically but emerges from density-driven spontaneous symmetry breaking (SSB) in a conformally coupled scalar–tensor theory. The central structural claim is that the redshift dependence of the effective coupling follows a logistic activation profile whose index $n$ is fixed by the local curvature of the symmetry-breaking potential, specifically $n = 3/p$, where $p$ is the order of the first non-vanishing derivative of $V'(\phi)$ at the broken minimum. The authors confront this structurally constrained activation with late-time data — Planck 2018 CMB lensing reconstruction, $f\sigma_8(z)$ redshift-space distortion (RSD) measurements from eBOSS, and Pantheon+SH0ES supernovae — within a controlled perturbative regime in which the background remains $\Lambda$CDM to within $\epsilon_\star = 10^{-2}$.

The paper's principal empirical finding is a null result: no statistically significant preference for interaction over $\Lambda$CDM across all dataset combinations ($\Delta\log Z \approx -1.4$ at most). Its principal structural finding is that fixing the activation index to the canonical quartic value $n=3$ compresses the posterior manifold toward prior boundaries without improving goodness-of-fit, whereas allowing $n = 3/p$0 to vary preserves an extended degeneracy direction. The analysis also delivers a clear negative result regarding cosmological tensions: by construction the model leaves $n = 3/p$1 untouched, and the inferred clustering amplitude $n = 3/p$2 is uncorrelated with the interaction parameters (Pearson coefficients $n = 3/p$3), so this class cannot address either tension.

Covariant framework and the perturbative regime

The starting point is a canonical scalar $n = 3/p$4 in the Einstein frame with a conformal coupling function $n = 3/p$5 acting only on the dark matter sector, inducing a field-dependent DM mass $n = 3/p$6 and dimensionless coupling $n = 3/p$7. Variation of the action yields the standard coupled quintessence exchange terms between $n = 3/p$8 and $n = 3/p$9, with baryons minimally coupled throughout. The scalar potential carries an additive constant tuned to reproduce the observed dark energy density at its late-time minimum.

The key theoretical device is the adiabatic tracking reduction: when the field sits near a density-dependent minimum of the effective potential

$p$0

the algebraic balance condition $p$1 defines a density-controlled trajectory $p$2. For a constant microscopic coupling and a $p$3-symmetric polynomial family $p$4, finite DM density induces a linear tilt that explicitly breaks the degeneracy and selects the positive branch. The deformation parameter $p$5 decreases with expansion, driving the minimum monotonically toward the vacuum value.

The magnitude of background deformation is quantified by $p$6, which in the tracking limit scales as $p$7. Imposing $p$8 over $p$9 guarantees that cumulative deviations from standard $V'(\phi)$0 dilution remain sub-percent, leaving the expansion history perturbatively equivalent to $V'(\phi)$1CDM while permitting growth-sector modifications through $V'(\phi)$2. This bound is enforced as a post-chain filter rather than a likelihood constraint; roughly half of the six-parameter posterior volume survives the filter (acceptance fractions 0.485–0.492 for free $V'(\phi)$3, dropping to 0.220–0.333 for fixed $V'(\phi)$4), indicating that the perturbative restriction removes substantial parameter space and that reported constraints refer only to the surviving subset.

Attractor structure and logistic normal form

The dynamical core of the paper establishes that the approach to the density-controlled attractor is universal within a well-defined class. Linearizing around a smooth analytic minimum where $V'(\phi)$5, the displacement obeys

$V'(\phi)$6

so the broken minimum is a hyperbolic fixed point of the reduced flow with eigenvalue $V'(\phi)$7. Introducing the inverse order parameter $V'(\phi)$8 compactifies the physical domain to $V'(\phi)$9 and yields, to leading order near the fixed point,

$f\sigma_8(z)$0

i.e., the logistic normal form with solution $f\sigma_8(z)$1. This result depends only on the local analytic structure of the potential and holds provided the tracking hierarchy is valid, the Taylor expansion exists, and no additional time-dependent sources modify the dilution law — assumptions the authors state explicitly.

For the polynomial family, $f\sigma_8(z)$2: the quartic Mexican-hat potential ($f\sigma_8(z)$3) has a non-degenerate minimum with $f\sigma_8(z)$4 giving the canonical index $f\sigma_8(z)$5, while $f\sigma_8(z)$6 yields tree-level flat minima with progressively slower attractor convergence ($f\sigma_8(z)$7 for sextic, $f\sigma_8(z)$8 for octic). Coleman-Weinberg and axion-like potentials both possess locally quadratic minima and therefore fall into the $f\sigma_8(z)$9 class, so the mapping from potential type to attractor eigenvalue is tabulated generically. An important consequence follows: the phenomenological activation index used in IDE parametrizations acquires a microphysical interpretation as the inverse local restoring order, so scanning over $\Lambda$0 corresponds to scanning over curvature classes of the underlying scalar theory.

The effective coupling inherits the activation profile as $\Lambda$1, with $\Lambda$2 set by the condition $\Lambda$3. The authors are explicit that the logistic form is a local normal-form approximation valid near the late-time fixed point, not an exact global solution, and that far from the fixed point higher-order terms and departures from strict adiabatic tracking modify the trajectory.

Numerical implementation

The three nested models share identical priors and likelihoods: a six-parameter IDE realization $\Lambda$4 with $\Lambda$5 treated continuously despite discrete microphysical values; a five-parameter variant with $\Lambda$6 fixed; and the $\Lambda$7CDM limit $\Lambda$8. Baryon density, spectral index, and optical depth are fixed to Planck 2018 best-fit values, and $\Lambda$9 is mapped internally to $\epsilon_\star = 10^{-2}$0. The model is implemented in CLASS with the interaction entering only through linear perturbations as a covariant momentum-conserving energy transfer aligned with the CDM four-velocity; the background is held at its $\epsilon_\star = 10^{-2}$1CDM form by construction. Gauge consistency was verified in both Newtonian and synchronous gauges, and the code reproduces standard spectra to machine precision as $\epsilon_\star = 10^{-2}$2. Inference uses emcee with uniform priors, followed by post-chain filtering on $\epsilon_\star = 10^{-2}$3 over $\epsilon_\star = 10^{-2}$4.

A methodological caveat deserves note: because geometric observables are unchanged by construction, supernova distances constrain the interaction sector only indirectly through the baseline parameters, and the linear perturbation treatment restricts validity to $\epsilon_\star = 10^{-2}$5, which the conservative RSD $\epsilon_\star = 10^{-2}$6-cuts accommodate.

Observational results

Across RSD-only, Planck+RSD, and Planck+RSD+SN combinations, the flexible six-parameter model yields $\epsilon_\star = 10^{-2}$7 relative to $\epsilon_\star = 10^{-2}$8CDM, with information criteria penalizing the extra parameters as expected ($\epsilon_\star = 10^{-2}$9; $\Lambda$0 up to 22.4 for the full dataset) and Bayesian evidence differences of $\Lambda$1 corresponding to weak evidence against IDE. The coupling amplitude $\Lambda$2 is consistent with zero for every combination, while the KL divergences show that SN data primarily inform $\Lambda$3 (KL ≈ 0.99 nats) and contribute little to the interaction parameters. Posterior-allowed growth modifications remain at the percent level: fractional shifts in $\Lambda$4 of −0.53% (RSD only), +1.44% (Planck+RSD), and +0.11% (Planck+RSD+SN), with the sign following the posterior preference for $\Lambda$5.

The fixed-$\Lambda$6 realization behaves comparably under RSD-only and Planck+RSD, but the combined Planck+RSD+SN constraints produce a qualitatively different posterior geometry. The supernova fit degrades modestly ($\Lambda$7 versus 1809, i.e., $\Lambda$8 versus 1.06 over $\Lambda$9), but the dominant effect is contraction of the marginalized distributions of $\Delta\log Z \approx -1.4$0, $\Delta\log Z \approx -1.4$1, and $\Delta\log Z \approx -1.4$2 toward lower prior boundaries:

Parameter	Model	Median	Std	% near lower boundary
$\Delta\log Z \approx -1.4$3	IDE 6p	0.360	0.019	0.00%
$\Delta\log Z \approx -1.4$4	IDE $\Delta\log Z \approx -1.4$5	0.200	0.014	87.26%
$\Delta\log Z \approx -1.4$6	IDE 6p	0.705	0.085	4.52%
$\Delta\log Z \approx -1.4$7	IDE $\Delta\log Z \approx -1.4$8	0.550	0.058	70.74%
$\Delta\log Z \approx -1.4$9	IDE 6p	0.808	0.115	4.70%
$n=3$0	IDE $n=3$1	0.605	0.034	91.97%

The posterior width of $n=3$2 shrinks by more than a factor of three relative to the flexible case, yet the likelihood peak height is comparable and $n=3$3 does not improve. The authors interpret this honestly as prior-truncated degeneracy compression rather than a purely data-driven contraction: the interior degeneracy direction present when $n=3$4 varies extends slightly beyond the adopted prior domain once $n=3$5 is fixed, so the boundary accumulation partly reflects truncation. Structured correlations emerge under the fixed index — most notably between $n=3$6 and $n=3$7 — confirming increased anisotropy of the likelihood surface. The conclusion drawn is structural: current observations favor realizations retaining at least one continuous degree of freedom in the redshift dependence of the coupling, and imposing the canonical quartic value constitutes over-restriction rather than physical detection.

Microscopic positioning of the observational null result

Translating the perturbative bound into microphysical language, evaluating $n=3$8 at the activation epoch where $n=3$9 gives

$n = 3/p$00

so viable solutions occupy the hierarchical regime $n = 3/p$01 with a heavy scalar and perturbatively small coupling. The appendix confirms this consistency empirically: reconstructing $n = 3/p$02 from the full MCMC chains shows all samples satisfy the adiabatic condition at $n = 3/p$03 and the median trajectory remains adiabatic over $n = 3/p$04, though a minority of samples approach the boundary $n = 3/p$05 near $n = 3/p$06. Notably, the early-time asymptotics derived in the appendix show $n = 3/p$07, which vanishes as $n = 3/p$08; the strict tracking hierarchy necessarily fails at sufficiently high redshift, where the system crosses into a Hubble-drag dominated regime. The logistic description is therefore confined to the late-time window, which suffices for the analysis but means the model is not a complete cosmological solution at all epochs.

The reduced continuity equation derived in the appendix identifies the integrated deformation parameter $n = 3/p$09, tying background modification directly to the stability ratio at activation and showing that positive $n = 3/p$10 slows matter dilution relative to standard scaling.

Limitations and open questions

Several limitations qualify these results. First, the entire analysis operates within the controlled perturbative regime enforced by post-chain filtering; behavior outside this regime — including the physically interesting limit $n = 3/p$11 where adiabatic tracking breaks down — is explicitly left unexplored. Second, the absence of primary CMB anisotropy data leaves $n = 3/p$12 weakly constrained ($n = 3/p$1314% width), and conclusions about joint geometric-growth sensitivity may shift once full CMB temperature and polarization data are included. Third, the prior-truncation diagnosis of the fixed-$n = 3/p$14 compression means the reported posterior widths for the restricted model should be read as artifacts of the chosen prior domain as much as of the data; whether the extended degeneracy direction terminates or continues beyond the priors is unresolved by this analysis. Fourth, treating $n = 3/p$15 as continuous obscures the discrete spectrum predicted microphysically; the posterior centered at $n = 3/p$16 maps to restoring orders $n = 3/p$17–$n = 3/p$18 but cannot discriminate among them. Fifth, radiative corrections may lift the tree-level flatness of $n = 3/p$19 minima, potentially altering the attractor classification for flatter potentials, an effect suppressed only within EFT validity below the cutoff. Finally, the model cannot address the $n = 3/p$20 tension — the correlation between $n = 3/p$21 and $n = 3/p$22 is negligible — and the $n = 3/p$23 tension is untouched by construction, so the framework offers no resolution of either discrepancy.

Conclusion

This paper demonstrates that density-driven SSB in a conformally coupled scalar–tensor theory produces a logistic activation of the dark sector coupling whose index encodes the local restoring order of the scalar potential through $n = 3/p$24. Confronted with Planck lensing, eBOSS RSD, and Pantheon+SH0ES data, the scenario yields no detection of interaction ($n = 3/p$25 favoring $n = 3/p$26CDM at most weakly), confines the theory to the hierarchical adiabatic regime $n = 3/p$27, and limits growth deviations to the percent level. The main structural lesson is that functional flexibility in the activation history matters: freeing $n = 3/p$28 preserves an extended posterior region consistent with both geometric and growth data, while fixing it to the canonical quartic value compresses the allowed volume toward prior boundaries without improving the fit — evidence of over-restriction rather than preference. The open question the framework poses for upcoming surveys such as Euclid, Rubin/LSST, and Roman is whether sub-percent precision in $n = 3/p$29 can discriminate among distinct curvature classes $n = 3/p$30 of the underlying scalar potential, and whether dynamics outside the perturbative tracking regime modify these structural conclusions.