8.3 Non-linear matter perturbations

So far we have discussed the evolution of linear perturbations relevant to the matter spectrum for the scale −1 k ≲ 0.01– 0.2h Mpc. For smaller scale perturbations the effect of non-linearity becomes important. In GR there are some mapping formulas from the linear power spectrum to the non-linear power spectrum such as the halo fitting by Smith et al. [540Jump To The Next Citation Point]. In the halo model the non-linear power spectrum P (k) is defined by the sum of two pieces [169]:
2 P (k) = I1(k) + I2(k )PL (k), (8.118 )
where P (k) L is a linear power spectrum and
∫ ( )2 ∫ ( )2 dM-- -M-- --dn--- 2 dM-- -M-- --dn--- I1(k) = M ρ(0) d ln M y (M, k), I2(k) = M ρ (0) d lnM b(M )y(M, k).(8.119 ) m m
Here M is the mass of dark matter halos, ρ(m0) is the dark matter density today, dn ∕dln M is the mass function describing the comoving number density of halos, y (M, k) is the Fourier transform of the halo density profile, and b(M ) is the halo bias.

In modified gravity theories, Hu and Sawicki (HS) [307Jump To The Next Citation Point] provided a fitting formula to describe a non-linear power spectrum based on the halo model. The mass function dn ∕dln M and the halo profile ρ depend on the root-mean-square σ(M ) of a linear density field. The Sheth–Tormen mass function [535] and the Navarro–Frenk–White halo profile [449] are usually employed in GR. Replacing σ for σGR obtained in the GR dark energy model that follows the same expansion history as the modified gravity model, we obtain a non-linear power spectrum P (k) according to Eq. (8.118View Equation). In [307Jump To The Next Citation Point] this non-linear spectrum is called P ∞(k). It is also possible to obtain a non-linear spectrum P0 (k) by applying a usual (halo) mapping formula in GR to modified gravity. This approach is based on the assumption that the growth rate in the linear regime determines the non-linear spectrum. Hu and Sawicki proposed a parametrized non-linear spectrum that interpolates between two spectra P ∞ (k ) and P (k ) 0 [307Jump To The Next Citation Point]:

P0-(k) +-cnlΣ2(k)P-∞(k-) P (k) = 1 + c Σ2(k) , (8.120 ) nl
where cnl is a parameter which controls whether P (k) is close to P0(k ) or P ∞(k). In [307] they have taken the form Σ2(k) = k3PL (k)∕(2π2).
View Image

Figure 6: Comparison between N-body simulations and the two fitting formulas in the f (R) model (4.83View Equation) with n = 1∕2. The circles and triangles show the results of N-body simulations with and without the chameleon mechanism, respectively. The arrow represents the maximum value of k(= 0.08h Mpc −1) by which the perturbation theory is valid. (Left) The fitting formula by Smith et al. [540Jump To The Next Citation Point] is used to predict Pnon− GR and PGR. The solid and dashed lines correspond to the power spectra with and without the chameleon mechanism, respectively. For the chameleon case cnl(z) is determined by the perturbation theory with cnl(z = 0) = 0.085. (Right) The N-body results in [479Jump To The Next Citation Point] are interpolated to derive Pnon−GR without the chameleon mechanism. The obtained P non−GR is used for the HS fitting formula to derive the power spectrum P in the chameleon case. From [371Jump To The Next Citation Point].

The validity of the HS fitting formula (8.120View Equation) should be checked with N-body simulations in modified gravity models. In [478479Jump To The Next Citation Point529] N-body simulations were carried out for the f (R) model (4.83View Equation) with n = 1∕2 (see also [562379] for N-body simulations in other modified gravity models). The chameleon mechanism should be at work on small scales (solar-system scales) for the consistency with local gravity constraints. In [479Jump To The Next Citation Point] it was found that the chameleon mechanism tends to suppress the enhancement of the power spectrum in the non-linear regime that corresponds to the recovery of GR. On the other hand, in the post Newtonian intermediate regime, the power spectrum is enhanced compared to the GR case at the measurable level.

Koyama et al. [371Jump To The Next Citation Point] studied the validity of the HS fitting formula by comparing it with the results of N-body simulations. Note that in this paper the parametrization (8.120View Equation) was used as a fitting formula without employing the halo model explicitly. In their notation P0 corresponds to “Pnon−GR” derived without non-linear interactions responsible for the recovery of GR (i.e., gravity is modified down to small scales in the same manner as in the linear regime), whereas P∞ corresponds to “PGR” obtained in the GR dark energy model following the same expansion history as that in the modified gravity model. Note that cnl characterizes how the theory approaches GR by the chameleon mechanism. Choosing Σ as

( 3 )1 ∕3 Σ2 (k,z) = k--PL (k,z) , (8.121 ) 2π2
where P L is the linear power spectrum in the modified gravity model, they showed that, in the f (R) model (4.83View Equation) with n = 1∕2, the formula (8.120View Equation) can fit the solutions in perturbation theory very well by allowing the time-dependence of the parameter cnl in terms of the redshift z. In the regime 0 < z < 1 the parameter cnl is approximately given by cnl(z = 0) = 0.085.

In the left panel of Figure 6View Image the relative difference of the non-linear power spectrum P (k ) from the GR spectrum PGR (k) is plotted as a dashed curve (“no chameleon” case with cnl = 0) and as a solid curve (“chameleon” case with non-zero cnl derived in the perturbative regime). Note that in this simulation the fitting formula by Smith et al. [540] is used to obtain the non-linear power spectrum from the linear one. The agreement with N-body simulations is not very good in the non-linear regime (k > 0.1h Mpc −1). In [371] the power spectrum Pnon−GR in the no chameleon case (i.e., cnl = 0) was derived by interpolating the N-body results in [479]. This is plotted as the dashed line in the right panel of Figure 6View Image. Using this spectrum Pnon −GR for cnl ⁄= 0, the power spectrum in N-body simulations in the chameleon case can be well reproduced by the fitting formula (8.120View Equation) for the scale −1 k < 0.5h Mpc (see the solid line in Figure 6View Image). Although there is some deviation in the regime −1 k > 0.5h Mpc, we caution that N-body simulations have large errors in this regime. See [530] for clustered abundance constraints on the f (R) model (4.83View Equation) derived by the calibration of N-body simulations.

In the quasi non-linear regime a normalized skewness, 3 2 2 S3 = ⟨δm ⟩∕ ⟨δm ⟩, of matter perturbations can provide a good test for the picture of gravitational instability from Gaussian initial conditions [79]. If large-scale structure grows via gravitational instability from Gaussian initial perturbations, the skewness in a universe dominated by pressureless matter is known to be S3 = 34 ∕7 in GR [484]. In the ΛCDM model the skewness depends weakly on the expansion history of the universe (less than a few percent) [335]. In f (R) dark energy models the difference of the skewness from the ΛCDM model is only less than a few percent [576], even if the growth rate of matter perturbations is significantly different. This is related to the fact that in the Einstein frame dark energy has a universal coupling √-- Q = − 1∕ 6 with all non-relativistic matter, unlike the coupled quintessence scenario with different couplings between dark energy and matter species (dark matter, baryons) [30].

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