Chiellini-Integrable Cosmologies with Phantom Divide Crossing

TL;DR

The paper embeds the Chiellini integrability condition into General Relativity to obtain exactly solvable cosmologies driven by a minimally coupled scalar with a Higgs-like self-interaction. The Klein–Gordon dynamics reduce to a damped Ermakov–Painlevé II system, yielding closed-form solutions for the scalar field and a Hubble function $H(a)$ that can be inverted analytically via the Lambert–$W$ function; a rational Padé expansion around $a=1$ provides a practical, ΛCDM–like parametrization with controlled departures. Bayesian analysis incorporating CC, BAO, CMB, DESI, and Pantheon+SH0ES data delivers a best-fit $H_0$ around $73$ km s$^{-1}$ Mpc$^{-1}$ when late-time distance-ladder data are included, while still supporting a smooth phantom-divide crossing in the effective dark-energy EOS without instabilities. The work thus offers a mathematically robust alternative to phenomenological dark-energy parametrizations, linking observational constraints directly to an exact nonlinear dynamical framework and suggesting pathways to extensions in scalar-tensor or modified gravity theories.

Abstract

We investigate exact cosmological solutions with a massive scalar field minimally coupled to the Einstein-Hilbert action in General Relativity. For an extended Higgs-like scalar self-interaction, we find that the resulting field equations belong to the damped Ermakov-Painlevé II class and construct novel analytical solutions within the framework of the Chiellini integrability condition. We analyze whether the expanding branch of the solutions can describe a late-time cosmic acceleration, using a combined statistical analysis of BAO, CMB, cosmic chronometer and Pantheon+SHOES supernova datasets. A crucial outcome of this exercise is the analytical emergence of a smooth phantom divide crossing in the dark energy equation of state, achieved without introducing any pathological instabilities. The reconstruction yields a present-day Hubble parameter $H_0 \gtrsim 70 \,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, with a reduced tension relative to the $Λ$CDM cosmology. The results indicate that Chiellini-integrable scalar cosmologies are capable of providing a robust and analytically controlled framework for modeling late-time cosmic acceleration and phantom divide crossing, offering a viable alternative to phenomenological dark-energy parametrizations.

Figures (6)

• Figure 1: Plot of Hubble function (top) and $\ddot{a}$ (bottom) as a function of time
• Figure 2: Two-dimensional marginalized posterior distributions and one-dimensional marginalized constraints for the six-parameter cosmological model $\{H_0,\,\Omega_m^{\rm eff},\,r_1,\,r_2,\,s_1,\,s_2\}$ obtained from the combined CC + BAO + Pantheon+SH0ES + DESI + CMB dataset (Set 1). Contours correspond to the $68\%$ and $95\%$ confidence levels. The inclusion of late-time distance-ladder data leads to a preference for a higher value of $H_0$ relative to $\Lambda$CDM.
• Figure 3: Reconstructed cosmological evolution from the full CC+BAO+Pantheon+SH0ES+DESI+CMB data combination (Set 1). The top panel shows the rescaled Hubble parameter $H(z)/(1+z)$, and the bottom panel shows the effective equation-of-state parameter $\omega_{\rm eff}(z)$. The shaded regions denote the $1\sigma$ and $2\sigma$ confidence intervals inferred from the MCMC posterior.
• Figure 4: Two-dimensional marginalized posterior distributions and one-dimensional marginalized constraints for the six-parameter cosmological model $\{H_0,\,\Omega_m^{\rm eff},\,r_1,\,r_2,\,s_1,\,s_2\}$ obtained from the CC + BAO + DESI + CMB dataset, excluding Pantheon+ (Set 2). Contours correspond to the $68\%$ and $95\%$ confidence levels. In the absence of late-time distance-ladder measurements, the inferred value of $H_0$ shifts toward lower values, consistent with Planck 2018 constraints.
• Figure 5: Reconstructed cosmological evolution from the full CC+BAO+DESI+CMB data combination (Set 2). The top panel shows the rescaled Hubble parameter $H(z)/(1+z)$, and the bottom panel shows the effective equation-of-state parameter $\omega_{\rm eff}(z)$. The shaded regions denote the $1\sigma$ and $2\sigma$ confidence intervals inferred from the MCMC posterior.