Phenomenology of Modified Gravity at Recombination

TL;DR

This work develops a phenomenological framework for modified gravity that decouples early- and late-time effects via $μ(a,k)$ and $γ(a,k)$ and embeds it in CAMB to explore recombination-era signatures. The authors derive superhorizon initial conditions and analyze both analytic and numerical perturbation evolution, showing that early-time modifications imprint phase shifts on the CMB acoustic peaks and alter lensing. They demonstrate that a larger early-time gravitational constant $μ_inite$ can partially relieve tensions between Planck CMB data, local $H_0$ measurements, and weak lensing, though BAO constraints limit this resolution under a fixed background. Overall, the study broadens the set of gravity tests accessible to current and upcoming surveys by exploiting the full constraining power of primary CMB peaks and their lensing analogs.

Abstract

We discuss the phenomenological imprints of modifications to gravity in the early universe with a specific focus on the time of recombination. We derive several interesting results regarding the effect that such modifications have on cosmological observables, especially on the driving and phasing of acoustic oscillations, observed in the CMB and BAO, as well as the weak gravitational lensing of the CMB and of galaxy shapes. This widens the pool of measurements that can be used to test gravity with present and future surveys, in particular realizing the full constraining power of the structure of the primary peaks of the CMB spectrum. We investigate whether such a phenomenology can relax tensions between cosmological measurements and find that a modification of the gravitational constant at recombination would help in reconciling measurements of the CMB with local measurements of the Hubble constant.

Figures (21)

• Figure 1: The comparison of the Weyl potential evolution between our MG example models and GR. The four panels represent models with $\mu_\infty=1.2$, $\gamma_\infty=1.2$, $\mu_0=1.2$ and $\gamma_0=1.2$ respectively. The three vertical dashed lines indicate, from left to right respectively, matter-radiation equality, the transition of the MG parameters (here $z=30$, see definition in Sec. \ref{['sec:step']}), and $\Lambda$-matter equality. Before horizon crossing, $\mu$ has a limited effect on the evolution of Weyl potential due to the small fraction of neutrino energy density, while a larger $\gamma$ decreases its amplitude in both radiation and matter dominated epochs and slows the potential decay in the acceleration epoch. When crossing the horizon during radiation epoch, a larger $\mu$ delays the potential decay significantly while the same change in $\gamma$ leads to a small effect. After horizon crossing, a larger $\mu$ increases the amplitude of the potential due to a larger effective gravitational constant $\mu G$, while $\gamma$ does not affect the subhorizon evolution. For details of the early and late time behaviors and the effect of the transition, see the discussion in Sec. \ref{['sec:FullNumSolution']}.
• Figure 2: The fractional change in transfer functions relative to their GR values at redshift $z=0$ due to MG of (a) the Weyl potential and (b) the synchronous gauge matter density perturbations. Different colors represent different example models as in Fig. \ref{['fig:Weyl_all']}, as shown in legend. The vertical lines show the scales, $(k_{\rm DE},k_{\rm T},k_{\rm eq})$, corresponding to the modes crossing the horizon at $\Lambda$-matter equality, transition in the MG functions, and matter-radiation equality respectively.
• Figure 3: The CMB anisotropy source functions in $k$-space in units of amplitude of primordial comoving curvature perturbation in two MG example models with $\mu_\infty=1.2$ and $\gamma_\infty=1.2$ and GR. Different lines correspond to different physical effects and models, as shown in figure and legend. The vertical dashed line shows mode that crosses the horizon at recombination $(z_*)$.
• Figure 4: The fractional change in the unlensed CMB temperature spectrum in two MG example models with $\mu_\infty=1.2$ and $\gamma_\infty=1.2$ relative to the GR spectrum. The vertical solid lines indicate the angular position of the GR peaks of the unlensed CMB spectrum. Notice that the variations are mainly out of phase with the peaks.
• Figure 5: The CMB lensing potential power spectrum in the harmonic domain. Different colors correspond to different models as shown in legend.