From Eq. (9.28) the perturbation
can be expressed by the matter perturbation
, as
Let us consider the perturbation equations in Fourier space. We choose the Longitudinal gauge
() with
and
. In this case Eq. (9.26
) gives
The parameter is a crucial quantity to characterize the evolution of perturbations. This quantity can
be estimated as
, which is much larger than
for sub-horizon modes (
). In
the regime
the matter perturbation evolves as
. Meanwhile the evolution of
in the
regime
is completely different from that in GR. If the transition characterized by
occurs
before today, this gives rise to the modification to the matter spectrum compared to the GR
case.
In the regime , let us study the evolution of matter perturbations during the matter
dominance. We shall consider the case in which the parameter
(with
evolves as
When , the growing mode solution to Eq. (9.38
) is given by
When , the perturbations show a damped oscillation:
The f (R) models can be consistent with observations of large-scale structure if the universe does not
enter the regime by today. This translates into the condition [597
]
If we use the observational constraint of the growth rate, [418, 605, 211], then the
deviation parameter
today is constrained to be
-
for the model
(
) as well as for the models (4.83
) and (4.84
) [597
]. Recall that, in
metric f (R) gravity, the deviation parameter
can grow to the order of 0.1 by today. Meanwhile
f (R) dark energy models based on the Palatini formalism are hardly distinguishable from the
CDM
model [356, 386, 385, 597]. Note that the bound on
becomes even severer by considering
perturbations in non-linear regime. The above peculiar evolution of matter perturbations is associated with
the fact that the coupling between non-relativistic matter and a scalar-field degree of freedom is very strong
(as we will see in Section 10.1).
The above results are based on the fact that dark matter is described by a cold and perfect fluid with no
pressure. In [358] it was suggested that the tight bound on the parameter
can be relaxed by
considering imperfect dark matter with a shear stress. Although the approach taken in [358] did not aim to
explain the origin of a dark matter stress
that cancels the
-dependent term in Eq. (9.35
), it will be
of interest to further study whether some theoretically motivated choice of
really allows the
possibility that Palatini f (R) dark energy models can be distinguished from the
CDM
model.
http://www.livingreviews.org/lrr-2010-3 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |