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On the Growth of Perturbations as a Test of Dark Energy

Edmund Bertschinger

TL;DR

Long-wavelength cosmological perturbations offer a gravity-agnostic route to test dark energy and gravity theories beyond the Friedmann equation. The paper derives three independent relations among the metric and matter perturbations in the large-scale limit, generalizing Einstein constraints to arbitrary theories with local energy-momentum conservation, and shows that GR yields a dynamical constraint $\Phi=\Psi$ enabling full quadrature reduction. In GR, the growth of perturbations on scales larger than the Jeans length is fully governed by the expansion history via quadratures, while alternative theories modify the relation between $\Phi$ and $\Psi$, opening a path to test gravity with observations of lensing, ISW, and growth, including models like DGP. The work clarifies how geometric and dynamic tests of dark energy complement each other, and provides a practical framework for using upcoming data to distinguish between dark energy as a new form of energy and modifications to gravity.

Abstract

The strongest evidence for dark energy comes presently from geometric techniques such as the supernova distance-redshift relation. By combining the measured expansion history with the Friedmann equation one determines the energy density and its time evolution, hence the equation of state of dark energy. Because these methods rely on the Friedmann equation which has not been independently tested it is desirable to find alternative methods that work for both general relativity and other theories of gravity. Assuming that sufficiently large patches of a perturbed Robertson-Walker spacetime evolve like separate Robertson-Walker universes, that shear stress is unimportant on large scales and that energy and momentum are locally conserved, we derive several relations between long-wavelength metric and matter perturbations. These relations include generalizations of the initial-value constraints of general relativity. For a class of theories including general relativity we reduce the long-wavelength metric, density, and velocity potential perturbations to quadratures including curvature perturbations, entropy perturbations, and the effects of nonzero background curvature. When combined with the expansion history measured geometrically, the long-wavelength solution provide a test that may distinguish modified gravity from other explanations of dark energy.

On the Growth of Perturbations as a Test of Dark Energy

TL;DR

Long-wavelength cosmological perturbations offer a gravity-agnostic route to test dark energy and gravity theories beyond the Friedmann equation. The paper derives three independent relations among the metric and matter perturbations in the large-scale limit, generalizing Einstein constraints to arbitrary theories with local energy-momentum conservation, and shows that GR yields a dynamical constraint enabling full quadrature reduction. In GR, the growth of perturbations on scales larger than the Jeans length is fully governed by the expansion history via quadratures, while alternative theories modify the relation between and , opening a path to test gravity with observations of lensing, ISW, and growth, including models like DGP. The work clarifies how geometric and dynamic tests of dark energy complement each other, and provides a practical framework for using upcoming data to distinguish between dark energy as a new form of energy and modifications to gravity.

Abstract

The strongest evidence for dark energy comes presently from geometric techniques such as the supernova distance-redshift relation. By combining the measured expansion history with the Friedmann equation one determines the energy density and its time evolution, hence the equation of state of dark energy. Because these methods rely on the Friedmann equation which has not been independently tested it is desirable to find alternative methods that work for both general relativity and other theories of gravity. Assuming that sufficiently large patches of a perturbed Robertson-Walker spacetime evolve like separate Robertson-Walker universes, that shear stress is unimportant on large scales and that energy and momentum are locally conserved, we derive several relations between long-wavelength metric and matter perturbations. These relations include generalizations of the initial-value constraints of general relativity. For a class of theories including general relativity we reduce the long-wavelength metric, density, and velocity potential perturbations to quadratures including curvature perturbations, entropy perturbations, and the effects of nonzero background curvature. When combined with the expansion history measured geometrically, the long-wavelength solution provide a test that may distinguish modified gravity from other explanations of dark energy.

Paper Structure

This paper contains 11 sections, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Robertson-Walker Spacetimes and their Perturbations
  3. Curvature and Coordinate Perturbations
  4. Linear Cosmological Perturbations in General Relativity
  5. Entropy Perturbations
  6. General Solution for Long-Wavelength Perturbations in GR
  7. Density and Velocity Perturbations and Einstein Constraints
  8. Multicomponent Fluids
  9. Quadratures for Alternative Theories of Gravity
  10. Comparing Geometric and Dynamic Methods for Testing Gravity and Dark Energy
  11. Summary and Outlook for Cosmological Tests of Gravity

Figures (1)

  • Figure 1: Sensitivity of geometric methods (left panel, characterized by the comoving distance $\chi$ to redshift $z$) and dynamic methods (right panel, characterized by the curvature perturbation $\Psi_+$ at redshift $z$) to the equation of state parameter $w$. A flat model with $\Omega_m=0.3$ was assumed. Curves are shown for $w=-0.6,-0.7,\ldots,-1.1$. Dynamic (perturbation) methods are insensitive to dark energy at high redshift but are more sensitive than geometric methods at low redshift.