12
2 Modifications of the Pure Gravitational Sector
V ( r ) =
d
3 k
(2π) 4
e
i
k·· r
k 4
∝ || r|.
(2.13)
So, this theory has only one difficulty—it does not yield Einstein–Hilbert limit which
was tested through many observations. Many aspects of the pure R
2 -gravity are
discussed in [19], see also references therein.
2.3 f (R)-Gravity
Clearly, the natural development of the idea of R
2 gravity will consist in the suggestion that the classical action can involve not only second but any degree (involving
negative!) of the scalar curvature. Thus, the concept of f (R) gravity was introduced.
Its action is given by (2.1), with f (R) = R + γ R
n .
First of all, we can discuss the renormalizability of this theory along the same
lines as in the previous section. It is easy to see that the term proportional to R
N (or,
which is similar, to N th degree or Riemann or Ricci tensors) is characterized by the
degree of divergence ω, in the four-dimensional space-time given by
ω = 4L − 2n(P − V ) − 2N = (4 − 2n)L + 2n − 2N .
(2.14)
Immediately we see that now discussion of the renormalizability is more involved
than for n = 2 (the similar situation occurs for Horava–Lifshitz-like theories where
increasing of the critical exponent z implies in growing not only of degree of momentum in the denominator of the propagator but also of numbers of derivatives in vertices). Actually, for any n > 2 one should classify possible divergences with various
values of N for the given n. Many examples of quantum calculations in theories for
various n, as well as in other higher-derivative gravity theories, including studies of
one-loop divergences and running couplings are presented in [6], see also references
therein. It is clear that the ghosts will arise for any polynomial form of f (R) just as in
the case of R
2 -gravity, so, conceptually the quantum calculations for n = 2 and for
n > 2 do not differ essentially (for discussion of renormalizability aspects of f (R)
gravity, see also [20]).
The main line of study of f (R) gravity consists in a detailed investigation of its
classical, especially cosmological aspects. The modified Einstein equations in this
case look like
f
(R)R μν −
1
2
g μν f (R) + (g μν ∇ λ ∇
λ
− ∇ μ ∇ ν ) f
(R) = 8πGT μν . (2.15)
It is evident that the dS/adS spaces will be vacuum solutions of these equations
yielding f (R) = bR
2
+ , with b being a constant. Then, to study the cosmological
aspects, we can use the expressions for components of the Ricci tensor and the scalar
2 Modifications of the Pure Gravitational Sector
V ( r ) =
d
3 k
(2π) 4
e
i
k·· r
k 4
∝ || r|.
(2.13)
So, this theory has only one difficulty—it does not yield Einstein–Hilbert limit which
was tested through many observations. Many aspects of the pure R
2 -gravity are
discussed in [19], see also references therein.
2.3 f (R)-Gravity
Clearly, the natural development of the idea of R
2 gravity will consist in the suggestion that the classical action can involve not only second but any degree (involving
negative!) of the scalar curvature. Thus, the concept of f (R) gravity was introduced.
Its action is given by (2.1), with f (R) = R + γ R
n .
First of all, we can discuss the renormalizability of this theory along the same
lines as in the previous section. It is easy to see that the term proportional to R
N (or,
which is similar, to N th degree or Riemann or Ricci tensors) is characterized by the
degree of divergence ω, in the four-dimensional space-time given by
ω = 4L − 2n(P − V ) − 2N = (4 − 2n)L + 2n − 2N .
(2.14)
Immediately we see that now discussion of the renormalizability is more involved
than for n = 2 (the similar situation occurs for Horava–Lifshitz-like theories where
increasing of the critical exponent z implies in growing not only of degree of momentum in the denominator of the propagator but also of numbers of derivatives in vertices). Actually, for any n > 2 one should classify possible divergences with various
values of N for the given n. Many examples of quantum calculations in theories for
various n, as well as in other higher-derivative gravity theories, including studies of
one-loop divergences and running couplings are presented in [6], see also references
therein. It is clear that the ghosts will arise for any polynomial form of f (R) just as in
the case of R
2 -gravity, so, conceptually the quantum calculations for n = 2 and for
n > 2 do not differ essentially (for discussion of renormalizability aspects of f (R)
gravity, see also [20]).
The main line of study of f (R) gravity consists in a detailed investigation of its
classical, especially cosmological aspects. The modified Einstein equations in this
case look like
f
(R)R μν −
1
2
g μν f (R) + (g μν ∇ λ ∇
λ
− ∇ μ ∇ ν ) f
(R) = 8πGT μν . (2.15)
It is evident that the dS/adS spaces will be vacuum solutions of these equations
yielding f (R) = bR
2
+ , with b being a constant. Then, to study the cosmological
aspects, we can use the expressions for components of the Ricci tensor and the scalar