Growth, Lensing, and Gravitational Slip

Large-scale structure compares two aspects of the scalar metric: nonrelativistic matter accelerates mainly in the Newtonian potential $\Psi$, while weak lensing responds to the Weyl combination $\Phi+\Psi$. Alternative gravity can change either relation and can produce gravitational slip, $\Phi\ne\Psi$, even when the material anisotropic stress is negligible. The useful phenomenological functions follow from a theory’s perturbation equations; they are not substitutes for those equations outside their stated regime.

Helpful background. Einstein Equations supplies the scalar constraint and matter conservation equations; Gravitational Actions and Boundary Terms supplies their derivation from a covariant model; Formulations of General Relativity supplies gauge-invariant initial data; Gravity Beyond General Relativity supplies the additional-field and screening scope; Causality and Global Structure supplies the domain-of-dependence conditions behind a valid approximation; and Gravitational Radiation and Waveforms supplies the complementary tensor-sector observables.

Two metric potentials and three response functions

Use Newtonian gauge and denote conformal time by $\tau$ to distinguish it from the slip function $\eta$:

$$ds^2=a^2(\tau)\left[-(1+2\Psi)d\tau^2 +(1-2\Phi)d\boldsymbol{x}^2\right],$$

and let $\Delta_m$ denote the gauge-invariant comoving matter-density contrast. For a Fourier mode with comoving wavenumber $k$, define

$$-\frac{k^2}{a^2}\Psi =4\pi G\,\mu(a,k)\rho_m\Delta_m,$$ $$\eta(a,k)=\frac{\Phi}{\Psi},$$ $$-\frac{k^2}{a^2}(\Phi+\Psi) =8\pi G\,\Sigma(a,k)\rho_m\Delta_m.$$

These definitions immediately imply

$$\Sigma=\frac{\mu(1+\eta)}{2}.$$

Only two of $\mu$, $\eta$, and $\Sigma$ are independent under these conventions, wherever the defining ratios exist. In GR with pressureless matter and a cosmological constant, with other density and anisotropic-stress sources negligible, $\mu=\eta=\Sigma=1$. Clustering additional species can change a response normalized to matter alone, and free-streaming relativistic species can generate GR slip. Thus “$\eta\ne1$” is not by itself a theory discriminator.

The signs depend on the Fourier and metric conventions. A reliable normalization check is the GR Poisson equation: a positive overdensity must correspond to the attractive Newtonian potential with the sign shown above. The parameterization and its relation to common modified-gravity equations are reviewed in Koyama 2016, § 4.2, Eqs. (137)–(139).

From matter conservation to the growth equation

Assume matter is minimally coupled to the metric above and is pressureless on the scales of interest. Its continuity and Euler equations retain their usual form even if the gravitational field equation changes. In the subhorizon regime, neglecting time derivatives of the potentials relative to spatial gradients gives

$$\ddot\delta_m+2H\dot\delta_m \simeq -\frac{k^2}{a^2}\Psi.$$

Substituting the modified Poisson relation and using $4\pi G\rho_m=3H^2\Omega_m/2$ yields

$$\delta_m''+\left(2+\frac{H'}{H}\right)\delta_m' -\frac32\Omega_m(a)\mu(a,k)\delta_m=0,$$

where primes now mean $d/d\ln a$ and $\Delta_m\simeq\delta_m$ in this limit. Thus growth depends both on the background friction through $H(a)$ and on the force law through $\mu(a,k)$. Two theories with the same expansion history can produce different growth.

Weak lensing instead integrates transverse derivatives of $\Phi+\Psi$ along the photon path. Cross-correlating a lensing tracer with a velocity-sensitive or clustering tracer can therefore probe slip. This clean theoretical separation does not make an observed ratio automatically model independent: galaxy bias, selection, redshift kernels, magnification, and relativistic projection terms belong to the measurement model developed in Tests of Gravity.

When the quasistatic reduction is valid

The quasistatic approximation is stronger than “inside the Hubble radius.” It normally requires all of the following on the selected solution branch:

• $k/a$ is large compared with $H$;
• the scalar response frequency $\omega_s$ is large compared with $H$; for a light scalar this reduces to $c_s k/a\gg H$;
• coefficients and particular solutions vary on the background timescale rather than a rapid transient timescale;
• the mode is not near a kinetic degeneracy or a dynamically excited resonance;
• the physical wavenumber remains below the EFT cutoff.

If a new scalar has effective mass $m$, eliminating it algebraically commonly produces a scale-dependent response with a transition near $c_s k/a\sim m$. Far above and far below that scale, $\mu$ and $\eta$ may approach different constants. Crossing that scale alone does not invalidate quasistatics: for an equation with $\omega_s^2=c_s^2k^2/a^2+m^2$, the relevant requirement is $\omega_s\gg H$ and slow variation of the particular solution, with negligible free oscillations. A heavy scalar can satisfy it even near its Compton transition. Rapid changes of its mass or appreciable homogeneous oscillations instead require the full time-dependent equation. Bertschinger and Zukin show how local metric theories lead to rational scale dependence under controlled assumptions in Bertschinger and Zukin 2008, §§ IV–V, Eqs. (21)–(22) and (27)–(28).

Screening adds another restriction. Linear $\mu(a,k)$ describes perturbations about a cosmological background; it cannot simply be inserted into a nonlinear halo or Solar-System calculation. The local solution may follow a different branch and depend on environmental density, potential depth, or acceleration. Matching between the cosmological and screened regimes is a theory problem, not a universal interpolation formula.

Beyond a pair of algebraic functions

The $\mu$–$\eta$ description is most useful after auxiliary fields have been solved in a stable linear theory. It can fail if extra fields have relevant initial conditions, if their time evolution is nonlocal in the matter source, if multiple fluids exchange energy, or if the equations contain scale-dependent memory. Then response kernels or the full coupled differential system are needed.

A theory prediction should therefore provide $H(a)$, initial conditions, the stable branch, the range of $a$ and $k$, the matter-coupling frame, and the approximation used to obtain $\mu$ and $\eta$. Those ingredients let an empirical analysis construct transfer functions and likelihoods without treating the response functions as valid beyond their derivation.

Exercises

Starting from the definitions of $\mu$ and $\eta$, derive the lensing response $\Sigma$. Then evaluate it for a model with $\mu=4/3$ and $\eta=1/2$ on a stated quasistatic scale.

Solution

Since $\Phi=\eta\Psi$,

$$-\frac{k^2}{a^2}(\Phi+\Psi) =-(1+\eta)\frac{k^2}{a^2}\Psi =4\pi G\mu(1+\eta)\rho_m\Delta_m.$$

Comparing this with the definition containing $8\pi G\Sigma$ gives

$$\Sigma=\frac{\mu(1+\eta)}{2}.$$

For $\mu=4/3$ and $\eta=1/2$,

$$\Sigma=\frac{(4/3)(3/2)}{2}=1.$$

The example has enhanced nonrelativistic clustering but the same algebraic lensing response as GR on that scale. It does not say that the full lensing signal is identical, because the growth history and background geometry may still differ.

References

• Bertschinger, Edmund, and Phillip Zukin. “Distinguishing Modified Gravity from Dark Energy.” Physical Review D 78, 024015 (2008). DOI. Open PDF
• Koyama, Kazuya. “Cosmological Tests of Modified Gravity.” Reports on Progress in Physics 79, 046902 (2016). DOI. Open PDF