In modified gravity theories, Hu and Sawicki (HS) [307] 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 [535] and the Navarro–Frenk–White halo profile [449] 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 [307
] 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
[307
]:
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 [479
] 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. [371] 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
as
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. [540] 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 [371] the power spectrum
in the no chameleon case (i.e.,
) was
derived by interpolating the
-body results in [479]. 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 [530] 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 [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
in GR [484]. In the
CDM 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
CDM 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
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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