tokens: a with σ = 0, Ω m = 0.26, and Ω r = 8.4 × 10 −5 . We see that k cr decreases in time and reaches the minimum at a M = [Ω m /2(1 − Ω m )] 1/3 ≃ 0.56. The super-interaction effect appears in a very small region (large k ) in the very early time of the universe. In addition, the larger the ν is, the wider the influences of dark energy to the matter density perturbation will be. Consequently, the size of the super-interaction regime enlarges in the cosmological evolution and finally reaches the scale , 2.8 × 10 −2 , and 2.8 × 10 −1 ( h Mpc −1 ) at a = a M for ν = 10 −2 , 10 −4 , and 10 −6 , respectively. Figure 1. The critical scale factor a cr as a function of the wavenumber k with σ = 0 and ν = 10 −2 (solid line), 10 −4 (dashed line), and 10 −6 (dotted line), where the boundary conditions of Ω m = 0.26 and Ω r = 8.4 × 10 −5 are used. 4 EVOLUTIONS OF MATTER DENSITY PERTURBATION AND POWER SPECTRUM The Hubble radius d H ≡ H −1 increases during the evolution of our universe. More and more density perturbation modes of δ k with the wavenumber k enter the horizon, in which k 2 = a 2 H 2 with a = a k . Explicitly using Ω m = 0.26, Ω r = 8.4 × 10 −5 , and H 0 = 70 km s −1 · Mpc, we find that k = 10 −3 and 0.25 ( h Mpc −1 ) go to the horizon at a k ≃ 0.03 and 1.24 × 10 −5 , respectively. Following a similar procedure to that discussed in the literature (Borges et al. 2008a ), we can recover P ( k ) in the ΛCDM model with the BBKS transfer function within 10 per cent accuracy by taking the initial density perturbation to be the scale invariance when all modes are located at the super-horizon scale, i.e. at a ≪ a k , where n s is the spectral index, given from the inflationary epoch. From the above discussion, it is reasonable to choose ln a = −18 to be the initial time in our work. We note that from equation ( 26 ), the super-interaction effect appears at the scale deep inside the horizon , indicating that the ΛCDM initial condition is applicable to the RVM. In this section, we analyse the evolution of δ m from equation ( 25 ) for two cases: (i) σ = 0 and (ii) σ ≠ 0, and numerically solve δ m ( a ) and P ( k ). (i) σ = 0. At the super-interaction regime ( k ≫ k cr ), the friction in equation ( 25 ) plays the most important role in the evolution history of δ m . Thus, with 0 ν ≪ 1, the growth of δ m is frozen with , and the matter power spectrum is suppressed by the dark energy perturbation. From equations ( 17 ) and ( 18 ), one can derive that and θ m ≃ 0 at k ≪ k cr , showing that δρ Λ = ρ Λ δ Λ 0. When entering the super-interaction scale, the second term on the right-hand side of equation ( 21 ) dominates, and the momentum perturbation of θ m becomes of the order of , implying that the RVM heats up the cold dark matter for a large value of k . As a result, the particles propagate inside the super-interaction divide, and the growth of δ m is frozen. In Fig. 2 (a), we show the evolution of as a function of the scale factor a with k = 0.25 ( h Mpc −1
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