12.3
gravity
Let us consider the theory (12.7) in the presence of matter, i.e.
where we have recovered
. For the matter action
we consider perfect fluids with an equation of
state
. The variation of the action (12.9) leads to the following field equations [178
, 383
]
where
is the energy-momentum tensor of matter. If
, then it is clear that the theory reduces
to GR.
12.3.1 Cosmology at the background level and viable
models
In the flat FLRW background the
component of Eq. (12.10) leads to
where
and
are the energy densities of non-relativistic matter and radiation, respectively. The
cosmological dynamics in
dark energy models have been discussed in [458
, 165, 383
, 188
, 633
, 430].
We can realize the late-time cosmic acceleration by the existence of a de Sitter point satisfying the
condition [458]
where
and
are the Hubble parameter and the GB term at the de Sitter point, respectively. From
the stability of the de Sitter point we require the following condition [188
]
The GB term is given by
where
is the effective equation of state. We have
and
during both
radiation and matter domination. The GB term changes its sign from negative to positive during the
transition from the matter era (
) to the de Sitter epoch (
). Perturbing
Eq. (12.11) about the background radiation and matter dominated solutions, the perturbations in the
Hubble parameter involve the mass squared given by
[188
]. For the stability of
background solutions we require that
, i.e.,
. Since the term
in Eq. (12.11) is
of the order of
, this is suppressed relative to
for
during the radiation and
matter dominated epochs. In order to satisfy this condition we require that
approaches
in the
limit
. The deviation from the
CDM model can be quantified by the following quantity [182
]
where a prime represents a derivative with respect to
. During the radiation and matter eras we
have
and
, respectively, whereas at the de Sitter attractor
.
The GB term inside and outside a spherically symmetric body (mass
and radius
) with a
homogeneous density is given by
and
, respectively (
is a
distance from the center of symmetry). In the vicinity of Sun or Earth,
is much larger than the
present cosmological GB term,
. As we move from the interior to the exterior of the star, the GB term
crosses 0 from negative to positive. This means that
and its derivatives with respect to
need to
be regular for both negative and positive values of
whose amplitudes are much larger than
.
The above discussions show that viable
models need to obey the following conditions:
-
1.
and its derivatives
,
, …are regular.
-
2.
for all
and
approaches
in the limit
.
-
3.
at the de Sitter point.
A couple of representative models that can satisfy these conditions are [188
]
where
,
and
are positive constants. The second derivatives of
in terms of
for the
models (A) and (B) are
and
, respectively.
They are constructed to give rise to the positive
for all
. Of course other models can be
introduced by following the same prescription. These models can pass the constraint of successful expansion
history that allows the smooth transition from radiation and matter eras to the accelerated
epoch [188
, 633]. Although it is possible to have a viable expansion history at the background level, the
study of matter density perturbations places tight constraints on these models. We shall address this issue
in Section 12.3.4.
12.3.2 Numerical analysis
In order to discuss cosmological solutions in the low-redshift regime numerically for the models (12.16) and
(12.17), it is convenient to introduce the following dimensionless quantities
where
. We then obtain the following equations of motion [188
]
where a prime represents a derivative with respect to
. The quantities
and
can be expressed by
and
once the model is specified.
Figure 12 shows the evolution of
and
without radiation for the model (12.16) with
parameters
and
. The quantity
is much smaller than unity in the deep matter
era (
) and it reaches a maximum value prior to the accelerated epoch. This is followed by the
decrease of
toward 0, as the solution approaches the de Sitter attractor with
. While the
maximum value of
in this case is of the order of
, it is also possible to realize larger maximum
values of
such as
.
For high redshifts the equations become too stiff to be integrated directly. This comes from the fact that,
as we go back to the past, the quantity
(or
) becomes smaller and smaller. In fact, this also
occurs for viable f (R) dark energy models in which
decreases rapidly for higher
. Here we show
an iterative method (known as the “fixed-point” method) [420, 188
] that can be used in these cases,
provided no singularity is present in the high redshift regime [188
]. We define
and
to be
and
, where the subscript “0” represents present values. The models (A) and
(B) can be written in the form
where
and
is a function of
. The modified Friedmann equation reduces to
where
(which represents the Hubble parameter in the
CDM model). In
the following we omit the tilde for simplicity.
In Eq. (12.24) there are derivatives of
in terms of
up to second-order. Then we write
Eq. (12.24) in the form
where
. At high redshifts (
) the models (A) and (B) are
close to the
CDM model, i.e.,
. As a starting guess we set the solution to be
.
The first iteration is then
, where
. The second iteration is
, where
.
If the starting guess is in the basin of a fixed point,
will converge to the solution of
the equation after the
-th iteration. For the convergence we need the following condition
which means that each correction decreases for larger
. The following relation is also required to be
satisfied:
Once the solution begins to converge, one can stop the iteration up to the required/available level of
precision. In Figure 13 we plot absolute errors for the model (12.16), which shows that the iterative
method can produce solutions accurately in the high-redshift regime. Typically this method stops working
when the initial guess is outside the basin of convergence. This happen for low redshifts in which the
modifications of gravity come into play. In this regime we just need to integrate Eqs. (12.19) – (12.22)
directly.
12.3.3 Solar system constraints
We study local gravity constraints on cosmologically viable
models. First of all there is a big
difference between
and f (R) theories. The vacuum GR solution of a spherically symmetric manifold,
the Schwarzschild metric, corresponds to a vanishing Ricci scalar (
) outside the star. In the presence
of non-relativistic matter,
approximately equals to the matter density
for viable
f (R) models.
On the other hand, even for the vacuum exterior of the Schwarzschild metric, the GB term has a
non-vanishing value
[178, 185
], where
is the Schwarzschild
radius of the object. In the regime
the models (A) and (B) have a correction term of the order
plus a cosmological constant term
. Since
does not vanish even in the
vacuum, the correction term
can be much smaller than 1 even in the absence of non-relativistic
matter. If matter is present, this gives rise to the contribution of the order of
to the GB
term. The ratio of the matter contribution to the vacuum value
is estimated as
At the surface of Sun (radius
and mass
), the density
drops down rapidly from the order
to the order
. If we take the value
we
have
(where we have used
). Taking the value
leads to a much smaller ratio:
. The matter density approaches a
constant value
around the distance
from the center of Sun. Even at this
distance we have
, which means that the matter contribution to the GB term can be
neglected in the solar system we are interested in.
In order to discuss the effect of the correction term
on the Schwarzschild metric, we introduce a
dimensionless parameter
where
is the scale of the GB term in the solar system. Since
is of the order of the
Hubble parameter
, the parameter for the Sun is approximately given by
. We can then decompose the vacuum equations in the form
where
is the Einstein tensor and
Here
is defined by
.
We introduce the following ansatz for the metric
where the functions
and
are expanded as power series in
, as
Then we can solve Eq. (12.30) as follows. At zero-th order the equations read
which leads to the usual Schwarzschild solution,
. At linear order one has
Since
and
are known, one can solve the differential equations for
and
. This process
can be iterated order by order in
.
For the model (A) introduced in (12.16), we obtain the following differential equations for
and
[185
]:
where
. The solutions to these equations are
Here we have neglected the contribution coming from the homogeneous solution, as this would
correspond to an order
renormalization contribution to the mass of the system. Although
, the
term in
only contributes by a factor of order
. Since
the largest contributions to
and
correspond to those proportional to
, which are different from the Schwarzschild–de Sitter
contribution (which grows as
). Hence the model (12.16) gives rise to the corrections larger than that in
the cosmological constant case by a factor of
. Since
is very small, the contributions to the
solar-system experiments due to this modification are too weak to be detected. The strongest bound comes
from the shift of the perihelion of Mercury, which gives the very mild bound
[185
]. For the
model (12.17) the constraint is even weaker,
. In other words, the corrections
look similar to the Schwarzschild–de Sitter metric on which only very weak bounds can be
placed.
12.3.4 Ghost conditions in the FLRW background
In the following we shall discuss ghost conditions for the action (12.9). For simplicity let us
consider the vacuum case (
) in the FLRW background. The action (12.9) can be
expanded at second order in perturbations for the perturbed metric (6.1), as we have done for the
action (6.2) in Section 7.4. Before doing so, we introduce the gauge-invariant perturbed quantity
This quantity completely describes the dynamics of all the scalar perturbations. Note that for the gauge
choice
one has
. Integrating out all the auxiliary fields, we obtain the second-order
perturbed action [186
]
where we have defined
Recall that
has been introduced in Eq. (12.15).
In order to avoid that the scalar mode becomes a ghost, one requires that
, i.e.
This relation is dynamical, as one requires to know how
and its derivatives change in
time. Therefore whatever
is, the propagating scalar mode can still become a ghost. If
and
, then
and hence the ghost does not appear. The quantity
characterizes the speed of propagation for the scalar mode, which is again dependent on the
dynamics. For any GB theory, one can give initial conditions of
and
such that
becomes negative. This instability, if present, governs the high momentum modes in Fourier space,
which corresponds to an Ultra-Violet (UV) instability. In order to avoid this UV instability
in the vacuum, we require that the effective equation of state satisfies
. At the
de Sitter point (
) the speed
is time-independent and reduces to the speed of light
(
).
Suppose that the scalar mode does not have a ghost mode, i.e.,
. Making the field redefinition
and
, the action (12.41) can be written as
where a prime represents a derivative with respect to
and
. In order to
realize the positive mass squared (
), the condition
needs to be satisfied in the regime
(analogous to the condition
in metric f (R) gravity).
12.3.5 Viability of
gravity in the presence of matter
In the presence of matter, other degrees of freedom appear in the action. Let us take into account a perfect
fluid with the barotropic equation of state
. It can be proved that, for small scales (i.e., for
large momenta
) in Fourier space, there are two different propagation speeds given by [182
]
The first result is expected, as it corresponds to the matter propagation speed. Meanwhile the presence
of matter gives rise to a correction term to
in Eq. (12.43). This latter result is due to the
fact that the background equations of motion are different between the two cases. Recall that
for viable
models one has
at high redshifts. Since the background evolution
is approximately given by
and
, it follows that
Hence the UV instability can be avoided for
. During the radiation era (
) and the
matter era (
), the large momentum modes are unstable. In particular this leads to the
violent growth of matter density perturbations incompatible with the observations of large-scale
structure [383, 182
]. The onset of the negative instability can be characterized by the condition [182]
As long as
we can always find a wavenumber
satisfying the condition (12.49). For
those scales linear perturbation theory breaks down, and in principle one should look for all higher-order
contributions. Hence the background solutions cannot be trusted any longer, at least for small scales, which
makes the theory unpredictable. In the same regime, one can easily see that the scalar mode
is not a ghost, as Eq. (12.44) is satisfied (see Figure 12). Therefore the instability is purely
classical. This kind of UV instability sets serious problems for any theory, including
gravity.
12.3.6 The speed of propagation in more general modifications of gravity
We shall also discuss more general theories given by Eq. (12.8), i.e.
where we do not take into account the matter term here. It is clear that this
function allows more freedom with respect to the background cosmological
evolution,
as now one needs a two-parameter function to choose. However, once more the behavior of perturbations
proves to be a strong tool in order to have a deep insight into the theory.
The second-order action for perturbations is given by
where we have introduced the gauge-invariant field
with
and
. The forms of
,
and
are given explicitly
in [186
].
The quantity
vanishes either on the de Sitter solution or for those theories satisfying
If
, then the modes with high momenta
have a very different propagation. Indeed the frequency
becomes
-dependent, that is [186]
If
, then a violent instability arises. If
, then these modes propagate with a group velocity
This result implies that the superluminal propagation is always present in these theories, and the speed is
scale-dependent. On the other hand, when
, this behavior is not present at all. Therefore, there
is a physical property by which different modifications of gravity can be distinguished. The
presence of an extra matter scalar field does not change this regime at high
[185], because the
Laplacian of the gravitational field is not modified by the field coupled to gravity in the form
.