10
2 Modifications of the Pure Gravitational Sector
The solution of (2.7) was explicitly obtained in [13] where it was found that the de
Sitter-like solution is possible, with the scale factor given by a(t) = H
−1 cosh Ht, or
a(t) = a 0 exp Ht, or a(t) = H
−1 sinh Ht, for closed, flat and open Universe respectively. So, we see that accelerating solution is possible in this theory, just as in the
presence of the cosmological term. Moreover, it is clear that a wide class of models
involving higher orders in curvatures will admit accelerated solutions as well. This
result called interest to f (R) gravity displaying it to be a possible candidate for a
consistent explanation of cosmic acceleration. Afterwards, many cosmological solutions for various versions of the function f (R) were obtained and observationally
tested, some of these results will be discussed in the next section.
Now, let us discuss the problem of degrees of freedom in R
2 -gravity. First of
all, we note that there is a common difficulty characteristic for higher-derivative
theories, either gravitational or not. Indeed, in any Lorentz-invariant theory with
four derivatives, the propagator will be proportional to the momentum depending
factor looking like:
f (k) =
1
k 2 −
k 4
M 2
,
(2.8)
where M
2 is the energy scale at which the higher derivatives become important. It
is clear that we can rewrite this factor as
f (k) =
1
k 2 −
1
k 2 − M 2 .
(2.9)
Therefore we see that this propagator actually describes two distinct degrees of
freedom, the massive and the massless one. Moreover, these two contributions to
the propagator have opposite signs (otherwise, if signs of these contributions are the
same, the UV behavior of the propagator is not improved). Clearly it means that the
Hamiltonian describing these two degrees of freedom is composed by two terms with
opposite signs:
H =
1
2
(π
2
1 + ∂ i φ 1 ∂ i φ 1 ) −
1
2
(π
2
2 + ∂ i φ 2 ∂ i φ 2 + M
2
φ
2
2 ).
(2.10)
We see that the energy is not bounded from below, hence, we cannot define a vacuum
in the theory consistently, i.e. one can take energy from the system without any
limitations, as from a well without a bottom. Moreover, actually it means that the
spectrum of the theory describes free particles with negative energy which seems to
be nonsense from the viewpoint of the common sense. Actually this is the simplest
example of the so-called Ostrogradsky instability plaguing higher-derivative field
theory models except of special cases, see a detailed discussion of this example and
similar situations in [14]; a profound discussion of difficulties arising within the
Hamiltonian formulation of these theories is given also in [15]. Moreover, in some
cases the higher-derivative theories involve not only ghosts but even tachyons, for
2 Modifications of the Pure Gravitational Sector
The solution of (2.7) was explicitly obtained in [13] where it was found that the de
Sitter-like solution is possible, with the scale factor given by a(t) = H
−1 cosh Ht, or
a(t) = a 0 exp Ht, or a(t) = H
−1 sinh Ht, for closed, flat and open Universe respectively. So, we see that accelerating solution is possible in this theory, just as in the
presence of the cosmological term. Moreover, it is clear that a wide class of models
involving higher orders in curvatures will admit accelerated solutions as well. This
result called interest to f (R) gravity displaying it to be a possible candidate for a
consistent explanation of cosmic acceleration. Afterwards, many cosmological solutions for various versions of the function f (R) were obtained and observationally
tested, some of these results will be discussed in the next section.
Now, let us discuss the problem of degrees of freedom in R
2 -gravity. First of
all, we note that there is a common difficulty characteristic for higher-derivative
theories, either gravitational or not. Indeed, in any Lorentz-invariant theory with
four derivatives, the propagator will be proportional to the momentum depending
factor looking like:
f (k) =
1
k 2 −
k 4
M 2
,
(2.8)
where M
2 is the energy scale at which the higher derivatives become important. It
is clear that we can rewrite this factor as
f (k) =
1
k 2 −
1
k 2 − M 2 .
(2.9)
Therefore we see that this propagator actually describes two distinct degrees of
freedom, the massive and the massless one. Moreover, these two contributions to
the propagator have opposite signs (otherwise, if signs of these contributions are the
same, the UV behavior of the propagator is not improved). Clearly it means that the
Hamiltonian describing these two degrees of freedom is composed by two terms with
opposite signs:
H =
1
2
(π
2
1 + ∂ i φ 1 ∂ i φ 1 ) −
1
2
(π
2
2 + ∂ i φ 2 ∂ i φ 2 + M
2
φ
2
2 ).
(2.10)
We see that the energy is not bounded from below, hence, we cannot define a vacuum
in the theory consistently, i.e. one can take energy from the system without any
limitations, as from a well without a bottom. Moreover, actually it means that the
spectrum of the theory describes free particles with negative energy which seems to
be nonsense from the viewpoint of the common sense. Actually this is the simplest
example of the so-called Ostrogradsky instability plaguing higher-derivative field
theory models except of special cases, see a detailed discussion of this example and
similar situations in [14]; a profound discussion of difficulties arising within the
Hamiltonian formulation of these theories is given also in [15]. Moreover, in some
cases the higher-derivative theories involve not only ghosts but even tachyons, for