Diego Saez-Gomez

3 Cosmological evolution in viable
evolution
f(R) gravity
in viable
In this Cosmological section, we explore evolution the cosmological in viable f(R) evolution gravity for the models considered above. For
F(R)
Inthat thisreason, section, itwe is convenient explore theto cosmological perform a change evolution of variables for the models in the equations considered(2.4), above. consid-
For
andscalar-tensor a 0 is the value equivalent, of the scale where factor the phase at thespace present willtime be explored. t 0 , such that the current epoch
corresponds 3 Cosmological to z = 0 Then, evolution the time derivative in viableis f(R) transformed gravity as d dt = −(1 + z)H d dz
, and the
first FLRW equation in (2.4) and the continuity equation (2.7) yield,
that
In this
reason,
section, H
it 2 (z)
is
we
convenient
= explore 1
κ 2 the ρ
to m (z)+ cosmological R(z)f R − f
perform a change
evolution +
of
3(1
variables
for + z)H the 2 in
models f RR
the
R equations
(z) considered ,
(2.4),
above. (3.2) For
ering the redshift z 3f as
that reason, it is convenient R the independent variable 2 instead of the cosmological time t, wherethe considering
to perform a change of variables in the equations (2.4), considering
the
redshift the redshift is defined z as the usual, independent variable instead of the cosmological time t, wherethe
Using the redshift isas redshift
defined the independent
z
as
as
usual,
the (1 independent + variable: z)ρ m(z) −variable 3(1 + w m instead of the cosmological time t, wherethe
redshift is defined as usual,
1+z = a )ρ m (z) =0, (3.3)
1+z = a 0
1+z = a 0
where now primes denote derivatives with respect to 0the a(t) ,
a(t) , redshift. Then, the Ricci scalar can (3.1)
be rewritten as R =6 2H 2 (z) − (1 + z)H(z)H (z) a(t) , (3.1)
and a (3.1)
0 is the value of the scale factor at the present , whiletime the equation t 0 , such that (3.3) the cancurrent be easily epoch
solved and corresponds
and
for a 0
a
ais 0 is
constant the to value z =
the value
EoS of 0 Then, the
of the
parameter scale the factor time
scale factor
w m , derivative the present is transformed time t 0 , such as d that the current epoch
at the present time t 0 , such that the current epoch
FLRW equations
corresponds
corresponds yield:
to z = 0 Then, the time derivative is transformed as d dt = −(1 + z)H d
dt
to z = 0 Then, the time derivative is transformed as d = dt = −(1 −(1 + + z)H d dz
, and the
first FLRW equation in (2.4) and the continuity equation (2.7) yield,
z)H dz d , and the
first FLRW equation in (2.4) and ρ(z) the =ρcontinuity 0 (1 + z) 3(1+w equation m) . (2.7) yield,
dz
,(3.4)
and the
first FLRW equation in (2.4) and the continuity equation (2.7) yield,
H 2 (z) = 1
κ 2 ρ m (z)+ R(z)f
H 2 (z) = 1
κ 2 ρ m (z)+ R(z)f
Here ρ 0 is the value
H 2 (z)
of the
= 1
matter
κ 2 ρ
energy m (z)+ R(z)f
R − f
density R −atf
the
+ 3(1
present
+ z)H
epoch 2 f RR R
R − f + 3(1 + z)H 2 z
3f + 3(1 + z)H 2 f RR R =
(z)
0,
,
which can
(3.2)
3f R 2
f RR R (z) , (3.2)
be rewritten as ρ 0 = 3 H
κ 2 0 2 R 2
(z) , (3.2)
3f Ω0 m. Then, we can fit the current values of the cosmological
parameters using the observational R (1 + data z)ρ m(z) 2
(1 + z)ρ [28], −where 3(1 + wH m 0
)ρ= m 100 (z) =0, hkms −1 Mpc −1 with h = (3.3)
0.71 m(z)
(1 + z)ρ − 3(1 + w m )ρ m (z) =0, (3.3)
where ± 0.03now andprimes the matter denote density, derivatives Ω 0 m =0.27
m(z) with − 3(1 respect ±
+
0.04,
w m
to)ρ while the m (z) redshift. the
=0,
matter Then, fluid theis Ricci considered scalar (3.3) can
where pressureless the where last be rewritten equation now(cold primes can as dark R be denote matter =6 easily derivatives with respect to the redshift. Then, the Ricci scalar can
be
where
rewritten
now primes
as R
denote
=6 2Hand solved,
2 (z) baryons), − (1 + z)H(z)H such that (z) w
derivatives with respect to the redshift. Then, Ricci scalar can
be rewritten as R =6 2H 2 − (1 + z)H(z)H (z) m , = while 0. The the equation (3.2) (3.3) iscan a second be easily
order solved differential for a constant equationEoS 2H 2 parameter H(z), so by w m
fixing , the
(z) − (1 + z)H(z)H (z) initial , whileconditions, the equation the(3.3) corresponding can be easily
cosmological solved for evolution a constant can EoS beparameter obtained through w
, while the equation (3.3) can be easily
m , the Hubble parameter in terms of the redshift
for solved both models for a constant considered EoSin parameter previous ρ(z) section, w m =ρ , 0 (1 and + z) explore 3(1+wm) how . the future evolution (with(3.4)
−1