Combining expansion, growth and lensing tests

Expansion, matter motion and light test different responses

A background expansion history describes the homogeneous scale factor. Structure growth also depends on the force response and matter conservation, while light responds to a combination of metric potentials.

Agreement in one of these quantities need not enforce agreement in the others. A coherent gravity test therefore follows their linked predictions under the same assumptions.

Use a deliberately fixed positive flat-dust GR reference background H = H₀a^(−3/2) and ρ_m = [3H₀²/(8πG)]a⁻³, with supplied H₀>0 in s⁻¹ and mass density in kg/m³.

The three response choices below are phenomenological diagnostics on that background. They are not demonstrated covariant gravity theories or claims that arbitrary response functions automatically satisfy a complete set of field equations.

Keep potential normalization and approximations explicit

Consider one linear pressureless mode with conserved minimally coupled matter, negligible relativistic anisotropic stress and a quasistatic sub-Hubble approximation. Use dimensionless cosine-mode amplitudes δ,Ψ,Φ and positive comoving k in m⁻¹.

Supply −c²k²Ψ = 4πGa²μρ_mδ and Φ = ηΨ. Here μ changes the matter-acceleration response and η is the potential ratio; it is not a synchronous-gauge variable. Define Σ = μ(1+η)/2.

Choose ck/H₀ = 100 and 1/4≤a≤1. The scale ratio ck/(aH) = 100√a is at least 50; this diagnoses the selected sub-Hubble domain without providing a rigorous bound on omitted relativistic corrections.

The equations use proper-time H and restore c² for dimensionless potentials. They omit relativistic metric-time terms and horizon-scale effects; those terms cannot be imported from a conformal-time convention without conversion.

Both initial amplitude and derivative select a mode

Matter conservation and the supplied force law give δ̈+2Hδ̇−4πGμρ_mδ = 0 in this approximation. With N = ln a and Ḣ/H² = −3/2, it becomes δ_NN+(1/2)δ_N−(3/2)μδ = 0.

The positive characteristic exponent is p = [−1+√(1+24μ)]/4. Select the growing solution with δ = 0.01 and δ_N = pδ at a = 1/4 separately for each case.

Then δ(a) = 0.01(4a)^p and f = d ln δ/d ln a = p. Equal initial amplitudes at this finite epoch do not assert a common preceding primordial history; their selected derivatives differ when p differs.

Use A:(μ,η) = (1,1), B:(4/3,1/2) and C:(1,7/5). A and C have p = 1; B has p = (√33−1)/4 ≈ 1.186141.

The light response also contains the evolved density

Define dimensionless Fourier Weyl proxy W = −c²k²(Φ+Ψ)/(3H₀²). The supplied background and response give W = Σδ/a. This is a mode proxy, not an observed convergence map or angular power spectrum.

A and B share Σ = 1 but have different growth. At a = 1, A has δ = W = 0.04, while B has δ = W ≈ 0.051776.

A and C have identical density growth but different Σ. Case C therefore has W = 0.048 throughout this interval, while A has W = 0.04.

The velocity-growth proxy fδ is another defined combination. At a = 1 it is 0.04 in A and C, and about 0.061414 in B; no measured redshift-space distortion or smoothed variance is supplied.

Response functions need physical and observational completion

The example demonstrates two useful distinctions: the same expansion need not fix the chosen growth response, and equal Σ need not fix a light proxy if the density has evolved differently.

It supplies neither a radiation epoch nor nonlinear evolution, galaxy bias, survey selection or projected lensing geometry. Each is relevant when comparing actual observations with a proposed theory.

A full theory must derive its background, perturbation responses and initial conditions consistently, and assess its degrees of freedom and stability. Choosing μ and η algebraically does not establish those properties.

A real joint test asks whether the same physical model explains expansion, growth and light responses across applicable regimes. The finite curves here expose dependencies to check without supplying an empirical ranking of gravity proposals.

Worked reasoning

Try the reasoning

1. For C at a = 1/2, find δ and W.

   Show the answer and reasoning

   δ = 0.02 and W = 0.048.

   Its p = 1 gives δ = 0.02; Σ = 6/5 then gives W = Σδ/a.

2. Must two cases with matching Σ have the same W?

   Show the answer and reasoning

   No.

   W also depends on the evolved δ/a; full lensing additionally needs projection and source information.

3. Do the constant μ,η pairs establish full viable covariant theories?

   Show the answer and reasoning

   No.

   Their consistent dynamics, background origin, stability and broader observational responses have not been derived.

What to carry forward

• Background expansion and perturbation responses are distinct ingredients.
• A growth solution requires both amplitude and derivative initial conditions.
• A coherent gravity test connects matter growth, light response and the physical model across regimes.