The weakly non-linear density-velocity relation

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Abstract
We rigorously derive up to third order in perturbation theory the weakly non-linear relation between the cosmic density and velocity fields. The density field is described by the mass density contrast, δ. The velocity field is described by the variable θ proportional to the velocity divergence, θ = −f(Ω)−1H0−1 ∇ · v, where f (Ω) ≃ Ω0.6,Ω is the cosmological density parameter and H0 is the Hubble constant. Our calculations show that mean δ given θ is a third-order polynomial in θ, ⟨δ ⟩∣θ= α1θ + α2(θ2 − σθ2) + α3θ3. This result constitutes an extension of the formula ⟨δ ⟩∣θ = θ + α2(θ2 − σθ2) found by Bernardeau which involved second-order perturbative solutions. Third-order perturbative corrections introduce the cubic term. They also, however, cause the coefficient α1 to depart from unity, in contrast with the linear theory prediction. We compute the values of the coefficients αp for scale-free power spectra, as well as for standard cold dark matter (CDM), for Gaussian smoothing. The coefficients obey a hierarchy α3 ≪ α2 ≪ α1, meaning that the perturbative series converges rapidly. Their dependence on Ω. is expected to be very weak. The values of the coefficients for CDM spectrum are in qualitative agreement with the recent results of N-body simulations by Ganon et al. The results provide a method for breaking the Ω-bias degeneracy in comparisons of cosmic density and velocity fields such as IRAS-potent.

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