A sub-horizon framework for probing the relationship between the cosmological matter distribution and metric perturbations

TL;DR

The paper addresses how to test gravity on cosmological scales by relating metric perturbations to nonrelativistic matter via a sub-horizon expansion in $aH/k$ with coefficient functions. It develops an ansatz that reproduces Poisson behavior on small scales and captures possible scale-dependent departures for several theories. The authors compute the coefficient functions for GR+ΛCDM, scalar-tensor theories, quintessence, $f(R)$ gravity, and DGP, highlighting a unique linear-in-$aH/k$ signature in DGP. They discuss observational routes, including growth measurements, weak lensing, and CMB lensing, to constrain the coefficient functions in a scale-dependent manner. This framework offers a streamlined, model-agnostic way to distinguish gravity theories with upcoming surveys.

Abstract

The relationship between the metric and nonrelativistic matter distribution depends on the theory of gravity and additional fields, providing a possible way of distinguishing competing theories. With the assumption that the geometry and kinematics of the homogeneous universe have been measured to sufficient accuracy, we present a procedure for understanding and testing the relationship between the cosmological matter distribution and metric perturbations (along with their respective evolution) using the ratio of the physical size of the perturbation to the size of the horizon as our small expansion parameter. We expand around Newtonian gravity on linear, subhorizon scales with coefficient functions in front of the expansion parameter. Our framework relies on an ansatz which ensures that (i) the Poisson equation is recovered on small scales (ii) the metric variables (and any additional fields) are generated and supported by the nonrelativistic matter overdensity. The scales for which our framework is intended are small enough so that cosmic variance does not significantly limit the accuracy of the measurements and large enough to avoid complications from nonlinear effects and baryon cooling. The coefficient functions provide a general framework for contrasting the consequences of Lambda CDM and its alternatives. We calculate the coefficient functions for general relativity with a cosmological constant and dark matter, GR with dark matter and quintessence, scalar-tensor theories, f(R) gravity and braneworld models. We identify a possibly unique signature of braneworld models. Constraining the coefficient functions provides a streamlined approach for testing gravity in a scale dependent manner. We briefly discuss the observations best suited for an application of our framework.

Figures (5)

• Figure 1: The ratio of the physical size of the perturbation to the size of the horizon is used as an expansion parameter in our anzatz. We plot this ratio, $(aH/k)$, as a function of $a$ from last scattering to the present for the concordance model (yellow region). The upper and lower bounds of the yellow region are determined by considering scales that are small enough so that cosmic variance does not dominate the errors and at the same time large enough so that nonlinear evolution and baryon cooling are not a significant factor. Most of the observations in the next decade will yield information in the range $10^{-1}\lesssim a\lesssim 1$. If we are interested in observations that only care about a smaller range of the scale factor, then the allowed range of $H_0/k$ increases. We also plot lines of constant multipole $l\sim k d(a)$, which provides a rough estimate of the relationship between $k$ and angular scales at different redshifts.
• Figure 2: The dimensionless coefficient functions characterizing the relationship between the metric perturbations and matter distribution are shown above for $F\Lambda$CDM(dashed lines) and the scalar-tensor theory (STT) (solid lines). The STT model is chosen so that its expansion history is consistent with observations. In the case of $\Lambda$CDM $\beta_0=\gamma_0=1$, $\beta_1=\gamma_1=0$ and $\beta_2=\gamma_2$. At early time (matter domination) $\beta_2=\gamma_2=-3$ with the cosmological constant causing a departure from this value at late times. The variation of $\beta_0$ with the scale factor in the STT can be interpreted as a variation of Newton's constant "$G\beta_0$" as far as growth of perturbations is concerned. Also note that for STT, $\beta_0\ne\gamma_0$ and $\beta_2\ne\gamma_2$. For STT, the difference in the coefficient functions is due to $\Phi-\Psi=-\alpha(\varphi)\delta\varphi\ne0$. Note that $\beta_1=\gamma_1=0$ in STT as well as $\Lambda$CDM. We remind the reader that in the ansatz (\ref{['ansatz']}) the coefficients $\beta_2$ and $\gamma_2$ are multiplied by $(aH/k)^2$, whose magnitude is shown in Figure 1, making them accessible at large scales only.
• Figure 3: The dimensionless coefficient functions characterizing growth of structure are show above for $\Lambda$CDM(dashed lines) and the scalar-tensor theory (STT) (solid lines). The STT model is chosen so that its expansion history is consistent with observations. $\delta_0$ is the usual growth function on small scales, whereas $\delta_2$ characterizes the departures as we move to larger scales. For $\Lambda$CDM and STT, $\delta_1=0$. We note that $\delta_2$ is the coefficient of $(aH/k)^2$, which is small withing the scales of interest (see Figure 1). The initial conditions for $\delta_0$ and $\delta_2$ are chosen at $a_i\sim10^{-2}$ and are consistent with growth of structure in a matter dominated era.
• Figure 4: The dimensionless coefficient functions characterizing the relationship between the metric perturbations and matter distribution are show above for $\Lambda$CDM(dashed lines) and DGP braneworld model (solid lines). The variation of $\beta_0$ with the scale factor in DGP can be interpreted as a variation of Newton's constant "$G\beta_0$" as far as growth of perturbations is concerned. Also note that for DGP, $\beta_0\ne\gamma_0$. In contrast to all the other examples considered, the coefficients of $aH/k$: $\beta_1,\gamma_1\ne0$. This is due to the junction conditions on the brane. The linear $aH/k$ term provides an intriguing signature of braneworld models.
• Figure 5: The dimensionless coefficient functions characterizing growth of structure are show above for $\Lambda$CDM(dashed lines) and DGP braneworld model (solid lines). $\delta_0$ is the usual growth function on small scales, whereas $\delta_1$ characterizes the departures as we move to larger scales. In contrast to $\Lambda$CDM, for the DGP case $\delta_1\ne 0$. This could provide a distinct signature of braneworld models.