Measuring the metric: a parametrized post-Friedmanian approach to the cosmic dark energy problem

TL;DR

The paper develops a theory-agnostic framework for cosmology in the linear regime, proposing to measure the expansion history through the effective density $ρ_{\rm eff}(z)$ and the linear growth factor $g(z,k)$ without assuming the Einstein equations. By focusing on $ρ_{\rm eff}(z)$, derived from $H(z)$ via $ρ_{\rm eff}(z)=\dfrac{3H(z)^2}{8\pi G_{\rm eff}}$, the authors show how space-time geometry probes (SN Ia, distances, ages) constrain a weighted view of the cosmic density evolution, independent of a specific dark energy model. They derive both a non-parametric reconstruction from current SN1a data and an analytic Fisher-based framework to forecast how precisely SNAP-like data could resolve $ρ_{\rm eff}(z)$ in about a dozen redshift bins, emphasizing the importance of redshift resolution and smoothing choices. The work argues that combining multiple geometric and growth probes will enable robust tests distinguishing dark energy from modified gravity, with significant implications for understanding cosmic acceleration and gravity on large scales.

Abstract

We argue for a ``parametrized post-Friedmanian'' approach to linear cosmology, where the history of expansion and perturbation growth is measured without assuming that the Einstein Field Equations hold. As an illustration, a model-independent analysis of 92 type Ia supernovae demonstrates that the curve giving the expansion history has the wrong shape to be explained without some form of dark energy or modified gravity. We discuss how upcoming lensing, galaxy clustering, cosmic microwave background and Lyman alpha forest observations can be combined to pursue this program, which generalizes the quest for a dark energy equation of state, and forecast the accuracy that the proposed SNAP satellite can attain.

Figures (7)

• Figure 1: Solid curve shows a currently popular model for the evolution of the effective cosmic mean density $\rho_{\rm eff}(z)\propto H(z)^2$. This curve uniquely characterizes the spacetime metric to zeroth order. The horizontal bars indicate the rough redshift ranges over which the various cosmological probes discussed are expected to constrain this function. If the Friedmann equation is correct, then $\rho_{\rm eff}(z)=\rho(z)$. Since the redshift scalings of all density contributions except that of dark energy are believed to be straight lines with known slopes in this plot (power laws), combining into a a simple quartic polynomial, an estimate of the dark energy density $\rho_X(z)$ can be readily extracted from this curve. Specifically, $\rho\propto (1+z)^4$ for the Cosmic Microwave Background (CMB), $\rho\propto (1+z)^3$ for nonrelativistic matter (like baryons and CDM), $\rho\propto (1+z)^2$ for spatial curvature, $\rho\propto (1+z)^0$ for a cosmological constant and $\rho\propto (1+z)^{3(1+w)}$ for quintessence with a constant equation of state $w$. Error bars are for our SNAP SN 1a simulation.
• Figure 2: Crosses show the luminosity distance for 92 SN 1a. From top to bottom, solid curves correspond to models $(\Omega_{\rm m},\Omega_\Lambda)=(0,1)$, $(0.38,0.62)$ and $(1,0)$, respectively. The middle curve is almost indistinguishable from the best fit quartic polynomial (dashed). The density history $\rho_{\rm eff}(z)$ is simply the squared inverse slope of this curve. Scatter increases with $z$ since the relative distance errors are roughly constant.
• Figure 3: Zoom of Figure \ref{['rhoFig']} showing constraints on $\rho_{\rm eff}(z)$ from actual and simulated data, assuming a flat Universe. Solid black curve shows best fit to the 92 SN1a, corresponding to the polynomial fit shown in Figure \ref{['zdFig']}, and yellow/light grey area shows the associated 68% confidence region. Green/dark grey area shows the corresponding 68% confidence region from our SNAP simulation, for a fiducial model with $\Omega_{\rm m}=0.38$, $\Omega_\Lambda=0.62$ (red/grey curve) whose two components are shown as dashed lines. Error bars are for the non-parametric reconstruction of Section \ref{['SNAPsec']}, spaced so that measurements of neighboring bands are uncorrelated, and are identical to those shown in Figure \ref{['rhoFig']}.
• Figure 4: Same as previous figure, but using a quadratic spline instead of a polynomial to fit $\eta(z)$. Bin widths were $\Delta z=0.2$ for the actual data and $\Delta z=0.1$ for SNAP.
• Figure 5: Same as previous figure but using linear interpolation to fit $\eta(z)$, giving a piecewise constant density.