1. ON MODIFIED GRAVITY
AND COSMOLOGY
Branko Dragovich
http://www.ipb.ac.rs/˜ dragovich
dragovich@ipb.ac.rs
Institute of Physics
Belgrade, Serbia
Balkan Workshop BW 2013
25 – 29. 04. 2013
Vrnjaˇcka Banja – SERBIA
BW - 2013 B. Dragovich Balkan Workshop BW2013 1/15
3. 1. Introduction: Einstein theory of gravity (ETG)
Einstein Theory of Gravity (1915)
4
8
2
R G
R g T g
c
μν μν μν μν
π
− = − Λ
2
2
0
d x dx dx
d d d
μ α β
μ
αβ
τ τ τ
+ Γ =
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4. 1. Introduction: Einstein theory of gravity (ETG)
ETG is the simplest self-consistent theory of gravity
It is General Relativity (GR) and contains Newton theory of
gravity
Its predictions are confirmed mainly in Solar System
It gives possibility to understand gravitational phenomena
from laboratory scale to cosmological scales
ETG predicts existence of Dark Energy (DE) and Dark
Matter (DM)
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5. 1. Introduction: Some problems of Einstein theory
of gravity
General Relativity is non-renormalizable quantum field
theory.
It predicts Dark Energy and Dark Matter which are
mysterious and without other evidence.
General Relativity has not been tested and confirmed at
large cosmic scales, hence its application for the Universe
as a whole is questionable.
Cosmological solutions of GR contain Big Bang singularity.
All these problems serve as motivation to investigate a
Modified Gravity, which is a generalization of ETG.
There are many modifications motivated by different
reasons.
To get nonsingular bounce cosmological solutions we
consider a Nonlocal Modified Gravity.
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6. 2. Nonlocal Modified Gravity: Relevant references
T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, “Modified
gravity and cosmology”, Phys. Rep. 513 (1), 1–189 (2012).
[arXiv:1106.2476v2 [astro-ph.CO]].
T. Biswas, T. Koivisto, A. Mazumdar, “Towards a resolution
of the cosmological singularity in non-local higher
derivative theories of gravity”, JCAP 1011 (2010) 008
[arXiv:1005.0590v2 [hep-th]].
A. S. Koshelev, S. Yu. Vernov, “On bouncing solutions in
non-local gravity”, Phys. Part. Nuclei 43, 666–668 (2012)
[arXiv:1202.1289v1 [hep-th]].
I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, “On
modified gravity”, to appear in Springer Proc. in Math. and
Statistics 36 (2013) [arXiv:1202.2352v2 [hep-th]].
I. Dimitrijevic, B. Dragovich, J. Grujic, Z. Rakic, “New
cosmological solutions in nonlocal modified gravity”, to
appear in Romanian J. Physics (2013) [arXiv:1302.2794
[gr-qc]].
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7. 2. Nonlocal Modified Gravity
Nonlocal gravity action without matter (Biswas et al.)
S = d4
x
√
−g
R − 2Λ
16πG
+
C
2
RF(2)R
F(2) =
∞
n=0
fn2n
, 2 = µ
µ =
1
√
−g
∂µ
√
−ggµν
∂ν
Equations of motion
C 2RµνF(2)R − 2( µ ν − gµν2)(F(2)R) −
1
2
gµνRF(2)R
+
∞
n=1
fn
2
n−1
l=0
gµν gαβ
∂α2l
R∂β2n−1−l
R + 2l
R2n−l
R
− 2∂µ2l
R∂ν2n−1−l
R =
−1
8πG
(Gµν + Λgµν).
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8. 2. Nonlocal Modified Gravity
Trace and 00-component
62(F(2)R) +
∞
n=1
fn
n−1
l=0
∂µ2l
R∂µ
2n−1−l
R + 22l
R2n−l
R
=
1
8πGC
R −
Λ
2πGC
C 2R00F(2)R − 2( 0 0 − g002)(F(2)R) −
1
2
g00RF(2)R
+
∞
n=1
fn
2
n−1
l=0
g00 gαβ
∂α2l
R∂β2n−1−l
R + 2l
R2n−l
R
− 2∂02l
R∂02n−1−l
R =
−1
8πG
(G00 + Λg00)
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9. 3. Nonsingular bounce cosmological solutions
We use FLRW metric
ds2
= −dt2
+a2
(t)
dr2
1 − kr2
+r2
dθ2
+r2
sin2
θdφ2
, k = 0, ±1
We use ansatz
2R = rR + s
2n
R = rn
(R +
s
r
), n ≥ 1, F(2)R = F(r)R +
s
r
(F(r) − f0)
We look for a solution of the form (Dimitrijevic, B.D., Grujic,
Rakic)
a(t) = a0(σeλt
+ τe−λt
), 0 < a0, λ, σ, τ ∈ R
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14. 3. Nonsingular bounce cosmological solutions
Equations of motion are satisfied when λ = ± Λ
3 , as well
as Q1 = Q2 = Q3 = 0 and R1 = R2 = R3 = 0. There are
three cases of solutions.
Case 1.
F 2λ2
= 0, F 2λ2
= 0, f0 = −
1
64πGCΛ
Case 2.
3k = 4a2
0Λστ
Case 3.
F 2λ2
=
1
96πGCΛ
+
2
3
f0, F 2λ2
= 0, k = −4a2
0Λστ
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15. 4. Concluding Remarks
We have considered a nonlocal gravity model with
cosmological constant Λ and without matter.
Using ansatz 2R = rR + s we found three types of
nonsingular bouncing solutions for cosmological scale
factor in the form a(t) = a0(σeλt + τe−λt ).
Solutions exist for all three values of spatial curvature
constant k = 0, ±1.
All these solutions depend on cosmological constant Λ,
which is here an arbitrary positive parameter.
Nonsingular bounce solution a(t) = a0e
1
2
Λ
3
t2
for k = 0
was found by A. Koshelev and S. Vernov.
Nonsingular bounce solution a(t) = a0 cosh Λ
3 t for
k = 0 and some radiation was found by T. Biswas, T.
Koivisto, A. Mazumdar and W. Siegel.
There are also some other ans¨atze which can be also
useful in finding cosmological solutions.
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