Parametrized Post-Friedmann Signatures of Acceleration in the CMB

TL;DR

This work expands the parametrized post-Friedmann (PPF) framework to a fully covariant, multicomponent setting that includes relativistic species and spatial curvature, enabling consistent evolution of metric perturbations and compatibility with Einstein-Boltzmann codes. It introduces three core functions—$f_\zeta(\ln a)$, $f_G(\ln a)$, and $g(a,k)$—and a transition scale $c_\Gamma$, along with a dynamical variable $\Gamma$, to bridge superhorizon and quasi-static regimes in a unified description of modified gravity effects. The authors probe the impact on the integrated Sachs-Wolfe (ISW) effect in the CMB, deriving how horizon-scale metric evolution modifies the low-$\ell$ spectrum and extracting constraints from WMAP via both direct parameter limits and a principal component approach (notably $g_{\rm eff}= -0.12 \pm 0.27$). They demonstrate that modified gravity scenarios such as DGP and certain $f(R)$ models are testable within this framework, and discuss designer-like models that could selectively alter ISW features, while acknowledging caveats around initial power spectra and expansion history assumptions. Overall, the framework provides a general, gauge-consistent tool to study late-time gravity on the largest scales and to explore connections to lensing and galaxy correlations in upcoming data.

Abstract

We extend the covariant, parametrized post-Friedmann treatment of cosmic acceleration from modified gravity to an arbitrary admixture of matter, radiation, relativistic components and spatial curvature. Explicit expressions in the comoving, Newtonian and synchronous gauges facilitate the adaptation of Einstein-Boltzmann codes for solving CMB and matter perturbations in the linear regime. Using a comoving gauge code, we study the effect of metric evolution on the CMB through the integrated Sachs-Wolfe effect. Modified gravity can alter the low multipole spectrum, including lowering the power in the quadrupole. From a principal component description of the primary metric ratio parameter, we obtain general constraints from WMAP on modified gravity models of the acceleration.

Figures (8)

• Figure 1: CMB temperature power spectrum as a function of the amplitude of the metric ratio parameter today $g_0$ given the evolution described by Eqn. (\ref{['eqn:gevolution']}). $\Lambda$CDM corresponds to $g_{0}=0$. Increasing $|g_{0}|$ in the positive direction monotonically increases the ISW effect at low multipoles whereas in the negative direction it first decreases and then increases the effect. The other parameters have been fixed at $c_{g}=0.01$, $c_{\Gamma}=1$ and $w_{e}=-1$.
• Figure 2: Effect of changing the scale at which metric ratio deviations occur through $c_g$. Once $c_{g}\gtrsim 0.1$ deviations have been suppressed for $k \gtrsim 10^{{-3}}$Mpc$^{-1}$ where the ISW effect at the lowest multipoles peak. Other parameters have been fixed to $g_{0}=1$, $c_{\Gamma}=1$ and $w_{e}=-1$.
• Figure 3: Effect of changing the scale below which the quasistatic Poisson equation (\ref{['eqn:qspoisson']}) holds through $c_\Gamma$. At a fixed metric ratio $g$, the ISW effect is enhanced by lowering $c_{\Gamma}$ and delaying the onset of the quasistatic dynamics. The other parameters have been fixed to $g_{0}=1$, $c_{g}=0.01$ and $w_{e}=-1$.
• Figure 4: Effect of changing the expansion history through $w_e$ is small compared with cosmic variance and the effect of the PPF parameters for small deviations $-1.1 \le w_{e} \le -0.9$. The other parameters have been fixed to $g_{0}=1$, $c_{g}=0.01$ and $c_{\Gamma}=1$.
• Figure 5: WMAP likelihood of models as a function of $g_{0}$ for various $c_{\Gamma}$ with $c_{g}=0.01$ and other parameters fixed. Taking $c_{\Gamma}=1$ provides conservative constraints. Normalization is arbitrary.