In modified gravity theories, Hu and Sawicki (HS)  provided a fitting formula to describe a non-linear power spectrum based on the halo model. The mass function and the halo profile depend on the root-mean-square of a linear density field. The Sheth–Tormen mass function  and the Navarro–Frenk–White halo profile  are usually employed in GR. Replacing for 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 according to Eq. (8.118). In  this non-linear spectrum is called . It is also possible to obtain a non-linear spectrum 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 and : they have taken the form .
The validity of the HS fitting formula (8.120) should be checked with -body simulations in modified gravity models. In [478, 479, 529] -body simulations were carried out for the f (R) model (4.83) with (see also [562, 379] for -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  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.  studied the validity of the HS fitting formula by comparing it with the results of -body simulations. Note that in this paper the parametrization (8.120) was used as a fitting formula without employing the halo model explicitly. In their notation corresponds to “” 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 corresponds to “” obtained in the GR dark energy model following the same expansion history as that in the modified gravity model. Note that characterizes how the theory approaches GR by the chameleon mechanism. Choosing asf (R) model (4.83) with , the formula (8.120) can fit the solutions in perturbation theory very well by allowing the time-dependence of the parameter in terms of the redshift . In the regime the parameter is approximately given by .
In the left panel of Figure 6 the relative difference of the non-linear power spectrum from the GR spectrum is plotted as a dashed curve (“no chameleon” case with ) and as a solid curve (“chameleon” case with non-zero derived in the perturbative regime). Note that in this simulation the fitting formula by Smith et al.  is used to obtain the non-linear power spectrum from the linear one. The agreement with -body simulations is not very good in the non-linear regime (). In  the power spectrum in the no chameleon case (i.e., ) was derived by interpolating the -body results in . This is plotted as the dashed line in the right panel of Figure 6. Using this spectrum for , the power spectrum in -body simulations in the chameleon case can be well reproduced by the fitting formula (8.120) for the scale (see the solid line in Figure 6). Although there is some deviation in the regime , we caution that -body simulations have large errors in this regime. See  for clustered abundance constraints on the f (R) model (4.83) derived by the calibration of -body simulations.
In the quasi non-linear regime a normalized skewness, , of matter perturbations can provide a good test for the picture of gravitational instability from Gaussian initial conditions . 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 in GR . In the CDM model the skewness depends weakly on the expansion history of the universe (less than a few percent) . In f (R) dark energy models the difference of the skewness from the CDM model is only less than a few percent , 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 with all non-relativistic matter, unlike the coupled quintessence scenario with different couplings between dark energy and matter species (dark matter, baryons) .
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