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Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Scalar perturbations in f(R)-cosmology
Jan Novak, Maxim Eingorn
Mathematical Institute of Academy of Sciences, Prague, Czech republicNorth Caroline Central University, Durham NC, U.S.A.
20.August 2013
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingornNEW TYPE OF ENERGY
or ...
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
MODIFICATION OF THE THEORY OF GRAVITATION
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
S =1
2
g f (R) d4x + SM
SM = g LM d4x
T = 2LMg
+ gLM
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
We want to consider briefly the homogeneous (flat forsimplicity) background problem and establish allnecessary designations:Being based on the review [Antonio de Felice], let us startwith its equations (2.4) and (2.7):
F (R)R 12
f (R)g F (R) + gF (R) = 2T ,(2.4)
3F (R) + F (R)R 2f (R) = 2T , (2.7)
where F (R) = f (R), T = gT andF (R) = (1/
g)(
ggF ).
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
In the case of the spatially flat background spacetime withthe metrics
ds2 = gdxdx = dt2 + a2(t)(
dx2 + dy2 + dz2),
the Hubble parameter H = a/a and the scalar curvature
R = 6(2H2 + H).
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
these equations give the equations (2.15) and (2.16):
3FH2 = (FR f )/2 3HF + 2 , (2.15)
2FH = F HF + 2(+ P) , (2.16)
where the perfect fluid with the energy-momentum tensorcomponents T = diag(,P,P,P) satisfies thecontinuity equation
+ 3H(+ P) = 0 .
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
CONFORMAL NEWTONIAN GAUGE Now let us turn tothe formula (6.1) from [Antonio De Felice], describing theperturbed metric, and without loss of generality present itin the following form:
ds2 = (1 + 2)dt2 + a2(1 + 2)ijdx idx j .
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Not taking into account the possible presence of thescalar field in equations (6.11)-(6.15) from [Antonio DeFelice], substituting = 0 and A = 3
(H
), we get a
particular system of equations. After further substitution = and = we get:
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
a2
+ 3H(
H + )
= 12F
[
(3H2 + 3H +
a2
)F
3H F + 3HF + 3F(
H + )
+ 2] ,
H + =1
2F
(F HF F
),F () = F ,
3(
H + H + )
+ 6H(
H + )
+ 3H +
a2=
=1
2F[3F+3H F6H2FF
a23F 3F
(H +
)
(
3HF + 6F)
+ 2(+ P)] ,
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
F + 3H F Fa2 1
3RF =
132( 3P)+
+F (3H + 3 + ) + 2F + 3HF 13
FR ,
F = F R,
R = 2[3(
H + H + )
+ 12H(
H + )
+
+
a2+ 3H 2
a2] .
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
So, the previous system of equations describes thescalar cosmological perturbations in the case of thenonlinear f (R) theory of gravity.
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingornLet us assume, that they permit the superposition
principle, then, neglecting P, we will solve them twice:1) dropping all terms containing the Laplace operator,
and substituting instead of and,2) substituting instead of and considering only one
point-like mass m, resting in the origin ofcoordinates)
Let us start by item 2), considering the domain r > 0(where = 0) and neglecting also R and, hence, F(that is assuming, that the scalarons mass is largeenough).
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
a2
+ 3H(
H + )
= 12F
[3HF + 3F
(H +
)],
H + =1
2F
(F
),
= 0 ,
3(
H + H + )
+ 6H(
H + )
+ 3H +
a2=
=1
2F[3F 3F (H + )
(3HF + 6F
)] ,
0 = F (3H + 3 + ) + 2F + 3HF ,
0 = 3(
H + H + )
+12H(
H + )
+
a2+3H2
a2.
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
= =
a
F
1a3
F+
3F2
4aF 2
F = 0 .
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
F (R) = 1 + o(1),
f (R) = R 2 + o(R R)
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Thus, in the case of large enough scalarons mass wereproduce the linear cosmology from the nonlinearone, as it should be.
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Taking the scalaron field into account, we need to solvethe following equation with respect to the scalar curvatureperturbation:
R(r) =3F
a2FR(r) +
1F2
a3c ,
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
So we have the solutions
=F
2F
Asinh(
a2F3F
r)
a2F3F
r+ B
cosh(
a2F3F
r)
a2F3F
r+
2
Fa3c
++
a,
= F
2F
Asinh(
a2F3F
r)
a2F3F
r+ B
cosh(
a2F3F
r)
a2F3F
r+
2
Fa3c
++
a,
where = GNm0
r
GNm02r30
r2 +3GNm0
2r0.
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn
Neglecting the influence of the cosmologicalbackground, but not neglecting the scalaronscontribution, we have found the scalar perturbations.They represent the mix of the standard potential andthe additional Yukawa term.
=F
2F
2m12aF exp(
a2F3F
r)
r
2
Fa3c
+ a ,
= F
2F
2m12aF exp(
a2F3F
r)
r
2
Fa3c
+ a .
Scalarperturbations inf(R)-cosmology
Jan Novak, MaximEingorn