Short answer: The coefficient 4 β^3 β,φ in your φ–equation is an artifact of treating the matter/radiation piece as S_m ∼ ∫√−g β^4(φ) ρ and varying it as if ρ were an external constant. That is not correct in GR. The matter action depends on the metric seen by matter, ĝμν = β^2(φ) gμν, and when you vary the action properly you find that φ couples to the trace of the matter stress–energy, not to ρ with a factor 4. In particular, radiation does not source φ (its stress–energy is traceless), while nonrelativistic matter does. The correct source term is proportional to d ln β/dφ, not to 4 β^3 β,φ.

How to do it correctly

1. Write the matter action with the conformal coupling:

• S_m = ∫ d^4x √−ĝ L_m(ψ, ĝμν), with ĝμν = β^2(φ) gμν.

2. Vary with respect to φ at fixed gμν:

• Using the standard definition T̃μν = −2/√−ĝ δS_m/δĝμν, one finds δS_m|φ = −1/2 ∫ d^4x √−ĝ T̃μν δĝμν = − ∫ d^4x √−ĝ β β,φ gμν T̃μν δφ = − ∫ d^4x √−g (β,φ/β) T δφ, where T ≡ gμν Tμν is the trace of the matter stress tensor defined with respect to gμν. In the last step we used T ≡ gμν Tμν = β^4 T̃ and √−ĝ = β^4 √−g.

3. Vary the scalar part:

• δS_φ gives √−g [□φ − V,φ] δφ.

4. Combine and set δS/δφ = 0:

• □φ − V,φ = (β,φ/β) T.

In a spatially flat FRW background (signature −,+,+,+), □φ = −φ̈ − 3H φ̇, so

• φ̈ + 3H φ̇ + V,φ = − (β,φ/β) T.

For a perfect fluid, T = −ρ + 3p. Therefore:

• Radiation (p = ρ/3): T = 0 ⇒ no source term.
• Nonrelativistic matter (p ≈ 0): T ≈ −ρ_M ⇒ φ̈ + 3H φ̇ + V,φ = (β,φ/β) ρ_M.

This is the equation appearing in the cyclic/ekpyrotic literature. If one parametrizes β(φ) = e^{−c φ/M_Pl}, then (β,φ/β) = −c/M_Pl and the source term is −(c/M_Pl) ρ_M (with the expected sign).

Why your “4 β^3 β,φ ρ” is wrong

Writing the matter/radiation piece as √−g β^4(φ)(ρ_M + ρ_R) and varying only the explicit β(φ) factor treats ρ as a fixed external quantity. But in GR, ρ and p are not independent fields; they arise from Tμν, which itself is obtained by varying S_m with respect to the metric that matter sees (ĝμν). Doing the variation properly automatically includes the implicit β–dependence and yields the trace T, not a spurious factor 4. This also explains why radiation does not couple: its T = 0.

Friedmann equations and coefficients

Varying the action with respect to gμν gives the Einstein equations

• Gμν = 8πG (Tφ μν + Tm μν + Tr μν),

with

• Tφ μν = ∂μ φ ∂ν φ − gμν [1/2 (∂φ)^2 + V(φ)].
• The matter/radiation tensors in the Einstein frame are related to the ones in the matter frame by Tμν = β^6 T̃μν. In a homogeneous FRW background this implies ρ_m = β^4 ρ̃_m, p_m = β^4 p̃_m (and similarly for radiation).

Hence the Friedmann equations are

• H^2 = (8πG/3) [ρ_φ + ρ_m + ρ_r],
• Ḣ = −4πG [φ̇^2 + ρ_m + 4/3 ρ_r],

where ρ_φ = 1/2 φ̇^2 + V(φ), p_φ = 1/2 φ̇^2 − V(φ).

Continuity equations (showing the energy exchange) are

• ρ̇_φ + 3H (ρ_φ + p_φ) = + (β,φ/β) ρ_m φ̇,
• ρ̇_m + 3H ρ_m = − (β,φ/β) ρ_m φ̇,
• ρ̇_r + 4H ρ_r = 0.

From these, ρ_m ∝ a^−3 β^−1 and ρ_r ∝ a^−4, consistent with the fact that radiation does not couple to φ while nonrelativistic matter does.

In summary:

• The correct φ equation is φ̈ + 3H φ̇ + V,φ = (β,φ/β) ρ_M (radiation does not appear).
• The incorrect factor 4 arises from an improper variation that ignores the implicit β–dependence of the matter sector.
• The Friedmann equations take their standard form in the Einstein frame with ρ_m, ρ_r the Einstein-frame densities (ρ = β^4 ρ̃), and the coupling shows up in the modified continuity equations.