6.3 Classical Solutions in Nonlocal Gravity Models
59
K 1 =
∞
n=1
f n
n−1
l=0
l R
n−l R; F() =
∞
n=0
f n
n
.
It is important to note that in two last lines
l acts only to the adjacent R.
Now, the natural problem is finding some solutions of these equations. In [116],
the following ansatz has been proposed, with r 1 , r 2 are some real numbers:
R − r 1 R − r 2 = 0
(6.12)
which implies (here f 0 is zeroth order in expansion of F() in series)
F()R = F(r 1 )R +
r 2
r 1
(F(r 1 ) − f 0 ).
(6.13)
This allows to reduce the order of equations to at maximum second. It is clear that
constant curvature makes the equation trivial, just this situation occurs for Gödel-type
solutions.
One can find nontrivial cosmological solutions for this theory. In particular, bouncing solutions, for r 1 > 0, are possible:
a(t) = a 0 cosh
r 1
2
t
.
(6.14)
Let us give more details for cosmology. Indeed, if we substitute the FRW metric
(1.6) to (6.11), and suggest that, as usual in cosmology, ρ = ρ 0 (
a 0
a
)
4 , we have from
(6.12), with r 1 = 0:
d
3 H
dt 3 + 7H ¨
H + 4 ˙
H
2
− 12H
2 ˙
H = −2r 1 H
2
− r 1 ˙
H −
r 2
6
,
(6.15)
whose solution is H =
r 1
2
tanh(
r 1
2
t) which just implies hyperbolic dependence
of a(t) (6.14). It is well known that namely such a scenario (decreasing of scale
factor changing then to increasing) is called bouncing scenario. We also introduce
h 1 = ¨
H /M
3 .
The density can be found as well: if we use G = M
−2
P , and redefine F() →
F(/M
2
), with M is the characteristic nonlocality scale, we find
ρ 0 =
3(M
2
P r 1 − 2λ f 0 r 2 )(r 2 − 12h 1 M
4
)
12r
2
1 − 4r 2
.
(6.16)
Let us discuss possible implications of the Eq. (6.15). The cosmological constant
turns out to be equal to = −
r 2 M
2
P
4r 1
, and there are three scenarios for evolution of
the Universe:
59
K 1 =
∞
n=1
f n
n−1
l=0
l R
n−l R; F() =
∞
n=0
f n
n
.
It is important to note that in two last lines
l acts only to the adjacent R.
Now, the natural problem is finding some solutions of these equations. In [116],
the following ansatz has been proposed, with r 1 , r 2 are some real numbers:
R − r 1 R − r 2 = 0
(6.12)
which implies (here f 0 is zeroth order in expansion of F() in series)
F()R = F(r 1 )R +
r 2
r 1
(F(r 1 ) − f 0 ).
(6.13)
This allows to reduce the order of equations to at maximum second. It is clear that
constant curvature makes the equation trivial, just this situation occurs for Gödel-type
solutions.
One can find nontrivial cosmological solutions for this theory. In particular, bouncing solutions, for r 1 > 0, are possible:
a(t) = a 0 cosh
r 1
2
t
.
(6.14)
Let us give more details for cosmology. Indeed, if we substitute the FRW metric
(1.6) to (6.11), and suggest that, as usual in cosmology, ρ = ρ 0 (
a 0
a
)
4 , we have from
(6.12), with r 1 = 0:
d
3 H
dt 3 + 7H ¨
H + 4 ˙
H
2
− 12H
2 ˙
H = −2r 1 H
2
− r 1 ˙
H −
r 2
6
,
(6.15)
whose solution is H =
r 1
2
tanh(
r 1
2
t) which just implies hyperbolic dependence
of a(t) (6.14). It is well known that namely such a scenario (decreasing of scale
factor changing then to increasing) is called bouncing scenario. We also introduce
h 1 = ¨
H /M
3 .
The density can be found as well: if we use G = M
−2
P , and redefine F() →
F(/M
2
), with M is the characteristic nonlocality scale, we find
ρ 0 =
3(M
2
P r 1 − 2λ f 0 r 2 )(r 2 − 12h 1 M
4
)
12r
2
1 − 4r 2
.
(6.16)
Let us discuss possible implications of the Eq. (6.15). The cosmological constant
turns out to be equal to = −
r 2 M
2
P
4r 1
, and there are three scenarios for evolution of
the Universe: