The sound of dynamical dark energy and modified gravity

Technical Summary: The Sound of Dynamical Dark Energy and Modified Gravity

Problem Statement
Recent cosmological datasets (DESI, Pantheon+, Planck) have indicated a preference for dynamical dark energy (DE) over a cosmological constant ($\Lambda$), often characterized by a phantom equation of state ($w_{DE} < -1$) in the past transitioning to non-phantom at low redshifts. While the $w_0 w_a$ parametrization captures this background expansion history, it does not distinguish between fundamental theories (e.g., quintessence, k-essence, Horndeski) which may predict identical background evolutions but distinct perturbation behaviors. A key differentiator is the effective sound speed of dark energy, $c_s^2$. In General Relativity (GR), varying $c_s^2$ has a negligible impact on observables. However, in Modified Gravity (MG) scenarios, specifically those involving kinetic gravity braiding (KGB) or scalar-tensor theories, the sound speed is intrinsically linked to modifications in the clustering of matter. The challenge lies in exploring these correlations without relying on specific, often unstable, functional forms for the underlying MG functions.

Methodology
The authors propose a phenomenological framework that treats the dark energy sound speed, $c_s^2$, as a fundamental free parameter to drive modifications in gravity, rather than specifying the MG functions ($\alpha_i$) directly.

1. Theoretical Framework:

   • The study utilizes Horndeski theory (specifically the Effective Field Theory of Dark Energy formalism) where deviations from GR are characterized by time-dependent functions $\alpha_K$ (kineticity) and $\alpha_B$ (braiding).
   • Instead of prescribing $\alpha_B$, the authors invert the relation between the sound speed and these functions (derived from the linear perturbation equations) to solve for $\alpha_B$ given a chosen $c_s^2$ and a parametrized $\alpha_K$.
   • This approach ensures stability ($c_s^2 > 0$ and $D_{kin} > 0$) even when crossing the phantom divide ($w_{DE} = -1$), a regime where standard minimally coupled scalar fields often fail.
   • Four parametrizations for $\alpha_K$ are tested: proportional to $\alpha_B$, proportional to the dark energy abundance ($\Omega_{DE}$), "k-essence-like," and "Cubic Galileon-like."
   • The modified Einstein equations are implemented in a version of CAMB, calculating the modified Poisson equation parameters $\mu$ and $\Sigma$ which govern the growth of structure and light propagation.

2. Data Analysis:

   • The analysis combines multiple datasets: Planck PR4 CMB (primary spectra, lensing, and ISW), DESI DR2 BAO, Pantheon+ and DES-Dovekie Type Ia Supernovae (SN), and DES-Y3 Cosmic Shear (CS).
   • Two dataset combinations are analyzed:
     • CMB+BAO+SN: Probes background expansion and large-scale perturbations.
     • CMBL+BAO+SN+CS: Includes small-scale structure growth via cosmic shear and CMB lensing.
   • The authors sample cosmological parameters ($\Omega_m, H_0, S_8, w_0, w_a$) alongside the sound speed ($c_s^2$) and the MG amplitude parameter ($\lambda_{MG}$) using MCMC methods (Cobaya/CosmoLike).

Key Contributions

• Phenomenological Inversion: The paper introduces a method to constrain MG models by treating $c_s^2$ as the primary varying quantity, deriving the necessary braiding function $\alpha_B$ dynamically. This avoids the instability issues associated with phantom crossing in standard scalar field models.
• Stability Enforcement: The framework explicitly enforces stability conditions ($c_s^2 > 0, D_{kin} > 0$) during numerical integration, allowing for the exploration of phantom regimes without unphysical divergences.
• Correlation Analysis: It establishes a direct link between the preference for dynamical dark energy in current data and deviations from GR in the growth of structure, mediated by the sound speed.

Results

• CMB+BAO+SN Constraints:

  • There is a significant preference for dynamical dark energy ($w_0 \approx -0.78, w_a \approx -0.66$) over $\Lambda$CDM.
  • In MG scenarios, this preference correlates with deviations from GR. Specifically, the parameter $\mu$ (modifying the Poisson equation) deviates from unity ($\mu > 1$) at redshifts $z < 2$ with over 95% confidence (approx. $2\sigma$).
  • The mean value of $\mu_0$ (at $z=0$) is found to be approximately 1.1 for most parametrizations.
  • This deviation is driven by the phantom nature of the preferred DE equation of state, not by an improved fit to the CMB temperature spectrum.
  • The preference for $\mu > 1$ persists even when assuming superluminal ($c_s^2 > 1$) or dynamical sound speeds, but vanishes if the equation of state is fixed to a cosmological constant ($w_0=-1, w_a=0$).
  • Constraints on $c_s^2$ and $\lambda_{MG}$ remain weak (mostly flat posteriors), indicating current data cannot tightly constrain the sound speed alone without the MG coupling.

• Impact of Cosmic Shear and Lensing (CMBL+BAO+SN+CS):

  • The inclusion of DES-Y3 cosmic shear and CMB lensing data significantly alters the constraints.
  • The deviations from GR ($\mu > 1$) are suppressed; the mean $\mu_0$ shifts closer to unity (e.g., $\approx 1.02$ for $\alpha_K \propto \Omega_{DE}$).
  • The constraints on $w_0$ and $w_a$ shift back towards the cosmological constant ($\Lambda$CDM) values.
  • This suggests a tension between the background expansion preference for dynamical DE and the growth of structure data, which favors GR-like behavior when small-scale probes are included.

Significance and Claims
The authors claim that their framework provides a physically meaningful way to test modified gravity models using the dark energy sound speed as a free parameter. They demonstrate that the current statistical preference for dynamical dark energy is intrinsically linked to deviations from GR in the growth of matter perturbations at low redshifts ($z < 2$) within their specific MG scenarios. However, they modestly conclude that this preference is not robust against the inclusion of large-scale structure data (cosmic shear), which pulls the constraints back toward GR and a cosmological constant. The work highlights the critical interplay between background expansion history and perturbative growth in distinguishing between dark energy and modified gravity, noting that future large-scale structure surveys will be essential to resolve these tensions. The paper does not claim to have definitively detected modified gravity or dynamical dark energy, but rather establishes a methodology to explore these correlations and demonstrates the sensitivity of current constraints to the inclusion of small-scale structure data.