Diego Saez-Gomez

Cosmological constraints on modified 

gravity theories 

Diego Sáez-Gómez 

Astrophysics, Cosmology and Gravity Center 

University of Cape Town 

and 

Department of Theoretical Physics 

University of the Basque Country 

August 3, 2013 

Kobayashi-Maskawa Instiute 

University of Nagoya

Index 

Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies in 

F(R) gravity 

The cosmological evolution in viable F(R) gravities, and their 

scalar-tensor counterpart. Future singularities and Little Rip. 

Testing F(R) gravity with SNe Ia data: a simple model 

F(R,T) gravity and cosmological perturbations

Brief Article 

The Author 

April 6, 2012 

FLRW cosmologies 

1 

w>−1, quintessence fluid , 

w = −1, cosmological constant , 

Cosmological Principle: assumes that the Universe is the same in every point (at large 

scales), what from w

General Relativity: Big Bang model 

R0 

0 = 3ä 

a , (5) 

(ä 

Rj i = 

a + 2ȧ2 

a 2 + 2K ) 

a 2 δj i , (6) 

(ä 

R = 6 

a + ȧ2 

a 2 + K ) 

a 2 , (7) 


where a dot denotes a derivative with respect to t. 

Let us consider an ideal perfect fluid as the source of 

the energy momentum tensor T µ ν .Inthiscasewehave 

T ν µ =Diag(−ρ,p,p,p) , (8) 

B. The evolution of the universe filled with a 

where ρB. andThe p are evolution theperfect energy of the density fluid universe and the filledpressure 

with a 

density of the fluid, respectively. perfect Then fluid Eq. (4) gives the 

FLRW equations in General two Relativity independent equations 

Let us consider the evolution of the universe filled with 

B. The Let usevolution consider (ȧ the ofevolution the universe of the universe filled filledawith 

abarotropicperfectfluidwithanequationofstate 

ion of homogeneity and isotropy of the 

pproximately true on large scales. The 

m homogeneity at early epochs played 

oleinthedynamicalhistoryofourunil 

density perturbations grew via gravy 

into the structure we see today in 

temperature anisotropies observed in 

wave Background (CMB) are believed 

from quantum fluctuations generated 

nary stage in the early universe. See 

3, 74, 75, 76] for details on density pered 

by inflationary cosmology. In this 

eview the main features of the homopic 

cosmology necessary for the subseic 

is given by [70, 77, 78, 79] 

) 2 

[ 

dr 2 

] 


H 2 ≡ perfect = fluid 

8πGρ − K 

) 

1 − Kr 2 + r2 (dθ 2 +sin 2 θdφ 2 ) , 

a 3 a 2 , (9) 

w = p/ρ , 

(1) 

Ḣ = −4πG(p + ρ)+ K a 2 , (10) (17) 

Let us consider the evolution of the universe filled with 


where w is assumed to be constant. Then by solving the 

where 

factorwithcosmictimet. Thecoordire 

known as comoving For a perfect coordinates. fluid with 

Einstein H is the equations Hubble given parameter, in Eqs. ρ(9) and and p denote (10) with the K to-=0tal 

energy density and pressure of all the species present 

A wean inobtain 

equation we obtainof state, w = p/ρ , (17) 

the universe at a given epoch. 

icle comes to rest in these coordinates. 

The energy momentum tensor is conserved 2 by virtue of 

a purely kinematic statement. In this where w is assumed toH be= 

constant. 2 Then by solving the 

the Bianchi identities, leading to the continuity equation 

ics is associated with the scale factor– 

H = 3(1 + w)(t − t 0 ) , (18) 

Einstein equations given in Eqs. (9) and 

2 

(10) with K =0, 

ations allow us to determine the scale we obtain 

a(t) ∝ (t − t 0 ) 

3(1+w) , (19) 

2 

e matter content of the universe is spect 

K in the metric (1) describes the ge- 

2 

a(t) ˙ρ +3H(ρ ρ ∝(t a −3(1+w) −+ p) t 0 ) =0. , (11)(20) 

tial section of space time, with closed, Equation (11) can H be = 

where t derived from Eqs. (9) and (10), 

3(1 + w)(t − t 

erses corresponding to K =+1, 0, −1, which means that two of Eqs. (9), (10) 0 ) , (18) 

0 is constant. We note that the above solution 

is valid for w ≠ −1. The radiation and (11) are independent. 

Eliminating a(t) ∝ the (t −K/a t 2 2 dominated universe 

0 ) 

3(1+w) term from , Eqs. (9) and 

which contains an initial singularity corresponds (and toperhaps w = 1/3, a future whereas one..). the dust dominated (19) 

enient to write the metric (1) in the (10), universe we obtain to w 

ρ=0. ∝ aInthesecaseswehave 

−3(1+w) , (20) 

ä 

t) [ dχ 2 + fK 2 (χ)(dθ2 +sin 2 θdφ 2 ) ] universe toRadiation w =0. a = Inthesecaseswehave 

: − a(t) 4πG ∝ (ρ(t +3p) − t 0 ) . (12) 

where t 3 

, (2) 

0 is constant. We note that 1/2 , ρ ∝ a above −4 , (21) 

solution 

is valid Hencefor thewaccelerated ≠Dust −1. 

: 

The 

a(t) 

expansion radiation 

∝ (t − t 0 ) 2/3 

occurs dominated 

, ρ ∝ a −3 

for +3p


12 

3 

No Big Bang 

(54) 

2 

Supernovae 

t 0 → 

ig. 3 

verse 

(0) 

Λ = 

t 0 = 

s the 

tellar 

s the 

to go 

a flat 

rized 

n the 

vacuum energy density 

(cosmological constant) 

1 

0 

-1 

Clusters 

Maxima 

SNAP 

Target Statistical Uncertainty 

CMB 

Boomerang 

open 

expands forever 

recollapses eventually 

flat 

closed 

0 1 2 3 

mass density 

arget 

the 

MB 

AP 

f prithe 

G. Aldering [SNAP Collaboration], “Future Research 

FIG. 4: The Direction Ω (0) 

m -Ω (0) 

Λand confidence Visions for regions Astronomy”, constrained Alan M. from 

the observations Dressler, of SN editor, Ia, Proceedings CMB and of the galaxy SPIE, Volume clustering. 4835, We 

also show the pp. expected 146-157 [arXiv:astro-ph/0209550]. 

confidence region from a SNAP satellite 

for a flat universe with Ω (0) 

m =0.28. From Ref. [106]. 

We need something Clearly the flat else: universe Dark with- 

energy or modified gravity....but how is the Equation of State 

the “cosmic triangle”). 

out a cosmological constant is ruled out. The compilation 

of three different cosmological data sets strongly 

reinforces the need for a dark energy dominated universe

α = 

Dark energy equation of state 

= H 0 +1− Einstein H 0 (H 0 + gravity 10) + 1 

4 


Brief Article 

1 

w>−1, 

w = −1, 

w


Future singularities 

BRIEF ARTICLE 

THE AUTHOR 

• Type I (“Big Rip”): For t → t s , a →∞and ρ →∞, |p| →∞. 

• Type II (“Sudden”): For t → t s , a → a s and ρ → ρ s , |p| →∞. 

• Type III: For t → t s , a → a s and ρ →∞, |p| →∞. 

• Type IV: For t → t s , a → a s and ρ → ρ s , p → p s but higher derivatives of Hubble 

parameter diverge. 

J. D. Barrow, G. J. Galloway and F. J. Tipler, Mon. Not. Roy. Astr. Soc. 223, 835-844 (1986); R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, 

Phys. Rev. Lett. 91 071301 (2003), astro-ph/0302506; S. Nojiri, S. Odintsov, S. Tsujikawa, Phys. Rev. D 71, 063004 (2005); arXiv:hep-th/0501025; 

M. P. Dabrowski and T. Denkieiwcz, Phys. Rev. D 79, 063521 (2009) [arXiv:0902.3107 [gr-qc]]....

1 

aside from the gravity part, also contains I. FORMULAE 

a matter contribution, namely 

F(R) gravity 

 

aside from the gravity part, also contains a matter contribution, namely 

S = d 

 

4 x √ −g f(R)+2κ 2 

L m . (2.1) 

c 1 (R/R HS ) n 

f(R) =R − R HS 

c 

Action, 

2 (R/R HS ) n +1 , =R + Rn (aR n − b) 

S = d 4 x √ −g f(R)+2κ 2 

L m . 

1+cR n . (2.1) (1) 

Here the coupling constant is given by κ 2 = 8πG, while L m stands for the Lagrangian 

I. FORMULAE 

corresponding to the matter that fills the 

Here the coupling constant 

S EH = d 4 x √ is 

−g given by κ 

R +2κ 2 

2 particular system to be studied. Note that the 

= 8πG, while L 

L m → S = d 4 x √ m stands 

−g for the 

f(R)+2κ 2 Lagrangian 

Hilbert-Einstein action is recovered for f(R) =R in (2.1), The field equations corresponding 

corresponding to the matter that fills the particular system to be studied. LNote m . that the (2) 

to action (2.1) are obtained by the variation of this action with respect to the metric tensor 

Hilbert-Einstein action is recovered for f(R) =R in (2.1), The field equations corresponding 

g 

c 1 (R/R HS ) n 

f(R) =R − R 

to action (2.1) are obtained by the variation of this action with HS 

respect c 2 (R/R to HS 

the ) n metric +1 , f(R) tensor 

=R + Rn (aR n − b) 

µν , what yields 

1+cR n . 

g Field µν , what equations, yields 

R µν f R (R) − 1 

R µν f R (R) − 1 2 g S 

µνf(R)+g µν f R (R) EH = d 

−∇ 4 x √ −g 

2 g µνf(R)+g µν f R (R) −∇ µ ∇ ν f R (R) =κ 2 T µν (m) . (2.2) 

R +2κ 

µ ∇ ν f R (R) =κ 2 2 

L 

T µν (m) m → S = d 

. (2.2) 

4 x √ −g f(R)+2κ 2 

L m . 

Here the subscript R denotes derivatives with respect to R. We assume a flat FLRW metric, 

Here the subscript R denotes derivatives with respect to 3 R. We assume a flat FLRW metric, 

where as usual the energy-momentum tensor is defined as: T µν (m) = −√ 2 δ ( √ −gL m ) 

ds 2 = −dt 2 + a(t) 2 dx i2 . −g δg 

3 

µν (2.3) 

ds 2 = −dt 2 + a(t) 2 i=1 

dx i2 . (2.3) 

Spatially Then, for flat this FLRW metric, equations, modified FLRW equations i=1 are obtained through the 00 and ij components 

of the field equations (2.2), 

Then, for this metric, modified FLRW equations are obtained through the 00 and ij components 

of the field equations (2.2), 

H 2 = 1 

κ 

H 2 = 1 

2 ρ m + Rf 

R − f 

− 

3f R 

κ 2 ρ m + Rf 2 

3HṘf RR , 

 

R − f 

− 

3f R 2 

3HṘf RR , 

−3H 2 − 2Ḣ = 1 

κ 

−3H 2 − 2Ḣ = 1 

2 p m + 

f Ṙ2 f RRR +2HṘf RR + ¨Rf RR + 1 

R 

κ 2 p m + 

f Ṙ2 f RRR +2HṘf RR + ¨Rf RR + 1 2 (f − Rf R) , (2.4) 

 

R 2 (f − Rf R) , (2.4) 

where dots denote derivatives with respect the time, Hubble parameter is defined as usual 

H(t) =ȧ/a, and Ricci scalar for the metric (2.3) isR =6(2H2 +Ḣ). Here, we are interested 

where dots denote derivatives with respect the time, Hubble parameter is defined as usual 

in studying a subclass of modified gravities, the so-called 

H(t) =ȧ/a, and Ricci scalar for the metric (2.3) isR =6(2H2 viable f(R) gravities, as they 

+Ḣ). Here, we are interested 

accomplish some indispensable conditions to be considered realistic candidates to describe

Viable f(r) gravity 

Viability conditions for F(R) gravity 

1. f RR > 0 for hight curvature, which ensures the existence of stable high 

curvature regimes, as the matter dominated epoch. 

2. 1 + f R > 0, which avoids the appearance of a negative effective gravitational 

coupling G eff = G/(1 + f R ), and the anti-gravity regime, implying also 

the avoidance of the graviton to turn into a ghost. 

3. f R → 0 as R →∞. GR has to be recovered at early times. Together 

with f RR > 0, implies f R < 0, and consequently −1

L DARK ENERGY MODEL AND BEHAVIOUR OF ITS FRW SOLUTIONS 

(1 + x 2n 

1 

λ = 

)2 

g from (1) can be written 

x 2n−1 

, (17) 

1 (2 

in 

+ 

the 

2x 2n 

1following − 2n) 

Einsteinian form (though gravity itself is not the 

where x 1 = R 1 /R c . From the stability condition 0 < 

Rµ ν − 1 ( 

Viable 

) 

f(r) gravity 

m(r = −2) ≤ 1weobtain 

2 δν µR = −8πG Tµ(m) ν + T µ(DE) 

ν (2) 

2x 4n 

1 − (2n − 1)(2n +4)x 2n 

1 +(2n − 1)(2n − 2) ≥ 0 .(18) 

1 

When n = 1, 

≡ F ′ (R)Rµ ν − 1 for example, 

2 

Some F (R)δν µ + ( we have x 1 ≥ √ 3 and 

λ ≥ 8 √ 3/9. Under Eq. (18) one 

∇ µ ∇ ν can 

−I. show 

δµ∇ ν FORMULAE that 

ρ ∇ ρ) F ′ the condition 

(4) is satisfied. This situation is similar to the 

(R) , F(R) ≡ f(R) − R (3) 

Starobinsky’s model (12), 

viable 

see Ref. 

f(R) 

[15] 

gravity 

for details. 

models 1 

riation We can of Lextend m and the satisfies abovethe twogeneralized models to the conservation more gen-laeral FORMULAE 

andform 

the right-hand side of Eq. c 1 

Tµ;ν(m) ν =0separately(since 

) (3) (R/R satisfythiscondition,too).Thereexistsasubtletyin 

HS ) n 

discussed below. f(R) The =R trace − R HS ofc 2 

Eq. (R/R (2)reads HS ) n +1 , f(R) =R + Rn (aR n − b) 

W. Hu and I. Sawicki, Phys. 

1+cR n . Rev. D 76 064004 (2007), arXiv:0705.1158[astro-ph] 

(1) 

f(R) =R − ξ(R), ξ(0) = 0, ξ(R ≫ R c ) → const. (19) 

3∇ µ ∇ µ f ′ − Rf ′ +2f =8πGT m . (4) 

RThe HS ) n conditions 

ns (de Sitter ones for R>0) are roots of the algebraic equation Rf ′ =2f. 

S) n +1 , f(R) (4) =R translate + Rn (aR into n − b) 

1+cR n . (1) 

following3-parametricform: 

ξ ,R < 1 , ξ ,RR < 0 , for R ≥ R 1 . (20) 

( 

In order to satisfy LGC, we require ( that ξ(R) ) −n approaches 

aconstantrapidlyasR f(R) =R + λR grows 0 1+ in the R2 

R0 

2 region− R1) 

≫ R c A. Starobinsky, JETP Lett. 86 157 (2007),(5) 

arXiv:0706.2041 [astro-ph] 

(such as ξ(R) ≃ constant − (R c /R) 2n discussed above). 

Another model to meet these requirements is 

the order of the presently observed 

( 

effective cosmological constant. Thenf(0) = 0 (the 

appears’ in flat space-time) and Rµ ν ) R =0isalwaysasolutionofEq. (2)intheabsence 

gative – flat 

f(R) 

space-time 

=R − 

is 

λR 

unstable. c tanh 

For 

, 

|R| ≫ R 0 , 

(21) S. Tsujikawa, Phys. Rev. D 77 023507 (2008), arXiv:0709.1391[astro-ph] 

R c 

f(R) =R − 2Λ(∞) where the highctive 

cosmological constant is Λ(∞) =λR 0 /2. The equation for de Sitter solutions having 

1 where > 0canbewrittenintheform 

λ and R c are positive constants. A similar model 

was proposed by Appleby and Battye [16], although it is 

different from (21) in the sense x 1 (1 + that x 2 

λ = 

1) ξ(R) n+1 canbenegative 

for R < In Rsome limit, 2((1+x GR is 2 1 )n+1 recovered, − 1 − (n the +1)x local 2 1 violations ) . are avoided, and the non-linear (6) 

1 . In the region R ≫ R c the model (21) 

part become important for large 

scales, also avoids matter instabilities...but this kind of models usually cross the phantom barrier, which may lead to 

themaximal root of Eq. (6). So, instead of specifying λ, onemaytakeanyvalueofx 

future singularities. 

1 

rresponding value of λ. It follows from the structure of Eq. (6) that x 1 < 2λ. Thus, the 

tant at the de Sitter solution Λ(R 1 )=R 1 /4 < Λ(∞). On the other hand, x 1 → 2λ in both 

1andx 1 ≫ 1, n fixed. In these cases the Universe evolution becomes indistinguishable 

odel. 

S. Nojiri and S.D. Odintsov, Phys. Rev. D 77 026007 (2008), arXiv:0710.1738[hep-th]

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 


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 


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

e quite different, since the universe may I. enter FORMULAE in a phantom phase [19], or may produce 

Cosmological 

oscillations along the cosmological evolution 

[27]. We will explore 

in 

the evolution, 

viable 

and 

the crossing of the phantom barrier in viable f(R) gravity in next sections. 

1 

F(R) 

1 

Hence, under these circumstances, the cF 1 (R/R (R) term HS ) n is able to reproduce both accelerating 


epochs of the universe evolution. Here c 2 (R/R we will HS ) n focus +1 , on f(R) the =R study + Rn (aR n − b) 

I. FORMULAE 

of1+cR two n . (1) 

I. FORMULAE 

models of this 

kind of viable gravity, proposed by Hu and Sawicki in Ref. [15], and Nojiri 1and Odintsov in 

 

Ref. [16], whose actions are given by, 

S EH = d 4 x √ −g 

R +2κ 2 

L m → S = d 4 x √ −g 1 

f(R)+2κ 2 

I. FORMULAE c 1 (R/R HS ) n 

L m . (2) 

Hu-Sawicki model 


c 1 c HS ) n 

F HS (R) =−R HS 

c 2 (R/R HS ) n +1 , F NO(R) = Rn (aR n 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n − b) 

c 1+cR − b) 

n . (1) 

1 (R/R 

f(R) =R − R 

I. FORMULAE 

HS ) n 

HS 

c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n − b) 

1+cR n . (2.10) 

T µν (m) = 2 δ ( √ 1+cR n . (1) 

 

1 

−gL m ) 

√ 

Here, {c 1 ,c 2 ,n} are free parameters where and R HS κ 2 ρ −g 0 m according δg µν (3) 

S EH = d 4 x √ −g 

c 

R +2κ 2 

L m → S = d 4 x √ to −gRef. 1 (R/R HS ) n 

f(R)+2κ [15], while 2 

f(R) =R − R 

 

L m {a, . b, c, n} (2) 

I. FORMULAE 

are free parameters 

S EH 

for 

= 

the 

d 4 second 

x √ −g 

HS 

model 

R +2κ 2 

in(2.10). 

L m → 

The 

S 

first 

= 

model 

d 4 x √ −g in(2.10) 

f(R)+2κ 

has 2 

c c 1 (R/R HS ) f(R) =R − R HS 

been 

L m . 

studied (2) 

c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n − b) 

1+cR n . (1) 

− b) 

1+cR n . (1) 

in Ref. [7], where it was proven andthat then =1, universe c 1 

evolution =2, c 2 = goes 1 through two different De 

(4) 

Sitter points, being one of them stableT and (m) the other one unstable, which can be identified to 

µν = −√ 2 δ ( √ −gL m ) 

 

the current accelerated f(R) era=R and−to the inflationary c 1 (R/R HS −g ) n 

epoch, respectively. Similarly, the second 

Initial conditions: 

HS 

model in(2.10) was studied in Ref. [18], c 2 

also (R/Rwith HS ) n the +1 , δg µν (3) 

T f(R) =R + Rn (aR n − b) 

µν (m) = − 2 δ ( √ −gL m ) 

S EH = d 4 x √ −g 

R +2κ 2 

L m 

√→ −g 

S = 

δg µν d 4 x √ −g f(R)+2κ 2 

L m 

presence of an extra1+cR field, n . . (2) (3) (1) 

and was shown 

n =1, c 1 =2, c 2 = 1 (4) 

 

1. Assuming "CDM model at z=0, 

S EH = d 4 x √ −g n =1, 

R +2κ 2 c 1 =2, c 2 = 1 

L m → S = d 4 x √ −g f(R)+2κ 2 (4) 

T µν (m) = − 2 δ ( √ −gL m ) 

√ L m . (2) 

−g 

H(z = 0) = H 0 = 100 hkms −1 δg 

Mpc µν (3) 

−1 ,h=0.71 ± 0.03 (5) 

H(z = 0) = H 0 = 100 –4– hkms −1 Mpc −1 

T µν (m) = −√ 2 δ ( √ ,h=0.71 ± 0.03 (5) 

n =1, c −gL m ) 

−g δg µν (3) 

H (z) = κ2 ρ 1 =2, c 2 = 1 

m 

2 H 0 (1 + z)h(z) , → H (0) = κ2 

ρ 0 m = 3 (4) 

2H 0 2 Ω0 m . (6) 

H(z = 0) = H 0 = 100 hkms −1 Mpc −1 ,h=0.71 ± 0.03 (5) 

n =1, c 1 =2, c 2 = 1 (4) 

2. Assuming a phantom expansion at z=0, 

ρ m 

H (z) = κ2 

H(z = 2 0) H = H 0 = 100 hkms −1 Mpc −1 0 (1 + z)h(z) , → H (0) = κ2 

ρ 0 m = 3 ,h=0.71 2H 0 ± 20.03 Ω0 m . (6) 

(5) 

H (0) 0 (7)

Cosmological evolution 

in viable f(R) 

Hu-Sawicki and Nojiri-Odintsov models 

1. "CDM initial conditions, 

Cosmological evolution in viable modified gravity 8 

EoS parameter w(z) 

0.2 

w HS z 

0.996 

w HS z 

w NO z 

0.998 

w NO z 

0.4 

1.000 

0.6 

1.002 

1.004 

0.8 

1.006 

1.0 

1.0 0.5 0.0 0.5 1.0 

z 

(a) 

The EoS parameter will cross the phantom barrier in the future. 

1.008 

1.00 0.95 0.90 0.85 0.80 

z 

(b) 

K. Bamba, C. Q. Geng and C. C. Lee, JCAP 1011, 001 (2010) 

DSG, Class.Quant.Grav. 30 095008 (2013)

0.8 

1.006 

1.0 



1.0 0.5 0.0 0.5 1.0 

Hu-Sawicki and Nojiri-Odintsov model 

z 

(a) 

1.00 0.95 0.90 0.85 0.80 

Figure 1: Evolution of the EoS parameter (12) as a function of the redshift z for the models F HS (R) 

and F NO (R). Herewehaveassumedtheinitialconditionsh(0) = 1 and h 

2. Phantom initial conditions 

(0) = 0.4, accordingto(14). 

The panel 1a shows the evolution from the past to the future, where the phantom barrier is crossed for 

negative redshifts in EoS both parameter models. The panel 1b shows in more detail the range −1



Hu-Sawicki and Nojiri-Odintsov model 

Deceleration parameter 

Cosmological evolution in viable modified gravity 9 

q = −aä/ȧ 2 DSG, Class.Quant.Grav. 30 095008 (2013) 

0.4 

0.2 

0.0 

HS 

NO 

CDM 

0.5 

0.0 

HS 

NO 

CDM 

0.2 

0.4 

0.5 

0.6 

0.8 

1.0 

1.0 

1.0 0.5 0.0 0.5 1.0 

z 

"CDM initial (a) conditions 

1.0 0.5 0.0 0.5 1.0 

z 

Phantom (b) initial conditions 

Figure 3: Evolution of the deceleration parameter q = −aä/ȧ 2 for the models F HS (R) and F NO (R)

Moreover, lations along the the election cosmological of phantom evolution, conditions a fact atpointed z = 0 gives out before an anomalous for this behavior kind of models of the 

Scalar-tensor 

cosmological in Ref. [27]. parameters In general, for we both have models shown 

representation 

that whenthe z>0. transition to the phantom epoch occurs. 

Moreover, the election of phantom conditions z = 0 gives an anomalous behavior of the 

f(r) 

Then, Then, 

4cosmological Scalar-tensor by bya a 

gravity: particular 

parameters f(R) f(R) 

representation for action, action, 

bothwe presence 

we 

models can can 

of reconstruct 

when f(R) z>0. gravity its its equivalent 

of scalar-tensor 

a sudden 

theory(4.1), 

so sothat thatwe wecan canstudy studythe theviable viablemodels modelsstudied studiedabove aboveby byanalyzing analyzingtheir their scalar-tensorcoun- 

terpart. terpart. It 

coun- 

singularity 

4 is Scalar-tensor well Moreover, known that by byvarying representation f(R) gravity the the action(4.1) is equivalent of f(R) withrespecttothemetricg togravity 

a kind of Brans-Dicke µν µν theory , , field fieldequations 

with equations a null 

are are kinetic obtained, term, and a scalar potential, (see for example, [6, 23] and references therein), 

It is wellCosmological known that evolution f(R) gravity is equivalent to a kind of Brans-Dicke theory with a null 

kinetic term, and a scalarS potential, d 4 x √ (see −g for Rexample, − V (φ)+2κ [6, 23] 2 

and references therein), 

Action, 

R L m . (4.1) 

µν 1 2 g µνR κ2 

φ T µν (m) 

1 

S = d 4 x √ φ (∇ µ∇ ν φ − g µν φ) − 1 µν − 1 in viable modified 

2 g µνR = κ2 

φ T µν (m) + 1 gravity 

−g φ (∇ µ∇ ν φ − g µν φ) − 1 12 

22 g µνV g 0.92 

µνV 

(φ) (φ), 

, 

2.0 

Φz 

φ R − V (φ)+2κ 2 Φz 

By varying the action with respect to the scalar field φ, theLrelation m . between both theories 

0.90 

(4.1) 

is obtained, and f(R) action(2.1) 3φ 3φ = κisrecovered, 

κ 2 T 2 T (m) (m) φ dV dV (φ) (φ) 1.9 

+ φ −− 2V 2V (φ) (φ), , (4.4) (4.4) 

By varying the action with respect to the scalar dφ dφ 

field φ, the relation between both theories 

0.88 

R = V 

is obtained, and f(R) 1.8 

(φ) → φ = φ(R) , ⇒ f(R) =φ(R)R − V (φ(R)) . (4.2) 

where wherethe thesecond secondequation equation action(2.1) is isthe thetrace trace isrecovered, of ofthe thefield fieldequations equationsin(4.4). in(4.4). Hence, Hence, for fora aflat flatFLRW 

metric metric 

While(2.3), the 

(2.3), 

scalar the the Friedmann 

field 

R = V and the equations equations 

potential yield, yield, 

0.86 

are related with 1.7 the particular f(R) action by, 

FLRW equations, 

(φ) → φ = φ(R) , ⇒ f(R) =φ(R)R − V (φ(R)) . (4.2) 

φ = f 

While the scalar field and the R (R) , V(φ(R)) = f 

potential are related R (R) R − f(R) . (4.3) 

3H 2 1 κ 2 ρ m 3H with the particular f(R) action by, 

φ 

˙φ + 1 

3H 2 = 1 κ 2 ρ m − 3H 

φ 

˙φ + 1 

22 V V (φ) (φ) 0.84 

1.6 

, , 

φ = f R (R) , V(φ(R)) = f R (R) R − f(R) . (4.3) 

3H 2 2Ḣ 1 κ 2 p m φ 

¨φ +2H ˙φ − 1 

− 3H 2 − 2Ḣ = 1 κ 2 p m + 

φ 

¨φ +2H ˙φ − 1 

1.5 

1.0 0.5 0.0 0.5 1.0 

1.0 0.5 

–9– 22 V V (φ) (φ) 0.0 0.5 1.0 

z 

z. . (4.5) (4.5) 

(a) 

(b) 

Trace while while equation, the thetrace traceequation equationin(4.4) in(4.4) is isgiven givenby, 

by, 

Figure 7: Evolution of the scalar field φ(z) 

–9– 

for the HS model (7a) and the NO model (7b). In both 

cases, the initial conditions 

33 ¨φ ¨φ are: = κκ 2 (ρ 2 h(0) = 1, andh 

(ρ m m −p m 

p m ) ) − (0) 

φV φV = 

+2V 3Ω 

+2V 0 m/2. 

−− 9H 9H ˙φ ˙φ . . (4.6) (4.6) 

In vacuum, Let’s Let’s reconstruct 

the For phase simplicity, space the the corresponding 

can webe assume described the scalar-tensor 

FLRW by, equations theory theory 

(19) for forthe in 

the 

vacuum, particular particular 

so models by 

models 

combining considered 

the 

in inthe theprevious equations section. section. 

(19)-(20), By By 

the the 

following expressions(4.3), 

system results, the the correspondingrelation relationφ(R) φ(R) for forthe 

the 

models(2.10) are aregiven givenby, 

by, 

Ḣ = −2H 2 + 1 

c 1 

cc 1 2 

c 2 R 

φφ HS =1+ 2 c 1 

cR(aR − HS 1+ c 2R 1+ c , φ 

2R NO =1− 

(1 + cR) 2 + aR 

1+cR + aR − b 

HS =1+ 2 

+ c 6 V 1 

(φ) , ˙φ = 

−3H 2 φ + 1 

1 

3H cR(aR −2b) 

V (φ) 

R 1+cR , (4.7) 

HS 1+ c 2R 1+ c , φ 

2R NO =1− 

(1 + cR) 2 + aR . 

1+cR + aR − b (25) 

Note that this is a dynamical 1+cR , (4.7) 

R 

Rsystem R 

R HS 

whose critical points are given by Ḣ = ˙φ =0, which 

HS HS 

corresponds to HS de Sitter solutions, as mentioned in the previous sections as a feature of 

where wef(R) havegravity. assumed Then, a power the critical of n = points 1 for areboth the solutions models(2.10). of the algebraic Then, the equation, scalar poten- 


8 

− 3H 2 − 2Ḣ = 1 κ 2 p m + 

φ 

¨φ +2H ˙φ − 1 

Let’s reconstruct the corresponding Cosmological 

100 scalar-tensor 

2 V (φ) evolution . theory in viable for the (4.5) particular modified models gravityconsi 

6 

in the previous section. By the expressions(4.3), the corresponding relation φ(R) fo 

while the 4 trace equation in(4.4) is given by, 

Scalar-tensor representation of 

models(2.10) are given by, 

f(r) gravity 

3 ¨φ = κ 2 (ρ m − p m 

c) 1 −c 2 φV R +2V − 9H ˙φ . cR(aR(4.6) 

− b) 

2 

φ HS =1+ 2 

+ c 1 

R 

Let’s reconstruct the corresponding scalar-tensor HS 1+ c 2R 1+ c , φ 

2R NO =1− 

(1 + cR) 2 + aR 

1+cR + aR − b 

Hence, 1+cR , 

0 the R 

Rtheory for the 

HS 

scalar-tensor representation for both mod 

0.0 0.2 0.4 0.6 0.8 1.0 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

HS particular models considered 

Φ 

potentials exhibit 

Φtwo branches due to the particu 

in the previous section. By the expressions(4.3), the corresponding relation φ(R) for the 

(a) where we have assumed 

models(2.10) are given by, 

considered 

a power of n 

here. 

= 1 for (b) Inboth addition, 

models(2.10). 

both models 

Then, the 

introduce 

scalar p 

tials(4.3) yield, 

c 1 c 2 R 

φ HS =1+ 2 

+ c 1 

cR(aR − b) 

R HS 1+ c 2R 1+ c , φ 

2R NO =1− V 

(1 + cR) 2 + aR 

1+cR + aR − b 

HS (φ) = 1+c 

Figure 7: Scalar potentials V (φ) in terms of H0 2 ,oftheHSmodelwithn of the scalar 

1 −field, φ =1,c ± 2being c 12 =2,c c1 

φ 

(1 2 

< = − 

1+cR , 1φ) 

(4.7) 

(figure in the7a), 

HS model, an 

and of the NO model with n =1,a =0.1/H R HS , 

0 2 , b =1,c =0.05/H limit where 0 2 ,fig.7b. both Both branches potentials 

c 2 

of arethe not uniquely scalar potentials co 

defined, where each branch behaves veryR different, 

R HS converging to the boundary of the 

HS 

V 

where we have assumed a power of n = 1 for NO (φ) = 2 a + c (1 + b − φ) ± 2 scalar field φ. 

Moreover, the evolution(a of + the bc)(a scalar + c −field, φ) φ(z) =f 

both models(2.10). Then, the c 2 . 

scalar potentials(4.3) 

0.92 yield, Hence, the scalar-tensor 

16 

V HS (φ) = 1+c representation 2.0 

for both models is not uniquely defined, bu 

potentials exhibit 1 − φ ± two 2 cbranches, 2 c1 (114 

−which φ) in principle do not affect V Φ the cosmological evolu 

Φz 

R HS 

Φz , 

V 

but it may influence cthe 2 

V NO (φ) = 2 a + c (1 + b − φ) ± 2 behavior of the phase space. In addition, 

Φ 

200 

0.90 

both models intro 

1.9 

12 

a boundary condition on the value of the scalar field, being φ < 1 for the HS model 

(a + bc)(a + c − φ) 

φ

H(z = 0) = H 0 = 100 hkms −1 Mpc −1 ,h=0.71 ± 0.03 (5) 

H (z) = κ2 

2 


ρ m 

f(r) gravity 

H 0 (1 + z)h(z) , → H (0) = κ2 

ρ 0 m = 3 2H 0 2 Ω0 m . (6) 

H (0) 0 (7) 


Ω m (z) , Cosmological Ω f(R) (z) evolution in viable modified(8) 

gravity 

Phase space 

1.0 

V + (φ) V − (φ) (9) 

0.9 

1.6 

1.4 

0.8 

1.2 

H 2 H 0 

2 

0.7 

0.6 

H 2 H 0 

2 

1.0 

0.8 

0.5 

0.6 

0.4 

0.4 

0.3 

0.90 0.92 0.94 0.96 0.98 1.00 

Φ 

(a) 

This branch of the potential makes the scalar field to tend to the boundary, where a 

sudden singularity occurs. 

2.0 2.2 2.4 2.6 

Figure 9: Phase space H(φ) for the HS model (panel 9a), where different init 

considered. The right panel corresponds to the NO model. For both models, the po 

been assumed. As observed, all the curves in both models reach DSG, Class.Quant.Grav. the bounding 30 095008 (2013) of the sc 

(b) 

Φ

(z = 0) = H 0 = 100 hkms −1 Mpc −1 ,h=0.71 ± 0.03 (5) 

)= κ2 

2 


f(r) gravity 

ρ m 

H 0 (1 + z)h(z) , → H (0) = κ2 

ρ 0 m = 3 2H 0 2 Ω0 m . (6) 


H (0) 0 (7) 

Phase space 

Cosmological evolution in viable modified gravity 

Ω m (z) , Ω f(R) (z) (8) 

1.2 

V + (φ) V − (φ) (9) 

2.0 

1.1 

1.0 

1.5 

H 2 H 0 

2 

0.9 

0.8 

H 2 H 0 

2 

1.0 

0.7 

0.5 

0.6 

0.5 

0.65 0.70 0.75 0.80 0.85 0.90 

There is a an asymptotic focus, which corresponds to a stable de Sitter solution. 

(a) 

Φ 

0.0 

0.0 0.5 1.0 

(b) 

Φ 

DSG, Class.Quant.Grav. 30 095008 (2013) 

Figure 8: Phase space of the HS model (panel 8a), and the NO model (8b) 

G. Cognola, E. Elizalde, S. D. Odintsov, P. Tretyakov, and S. Zerbini, Phys. Rev. D 79 044001 (2009)

H(z = 0) = H 0 = 100 hkms −1 Mpc −1 ,h=0.71 ± 0.03 (5) 


H (z) = κ2 

2 

ρ m 

f(r) gravity 

H 0 (1 + z)h(z) , → H (0) = κ2 

ρ 0 m = 3 2H 0 2 Ω0 m . (6) 

Nojiri-Odintsov model 

H (0) 0 (7) 

Ω m (z) , Ω f(R) (z) (8) 

n in viable modified Phase space gravity 15 

V + (φ) V − (φ) (9) 

1.6 

1.4 

1.2 

H 2 H 0 

2 

1.0 

0.8 

0.6 

.94 0.96 0.98 1.00 

(a) 

Φ 

0.4 

2.0 2.2 2.4 2.6 2.8 3.0 

The scalar field reaches its boundary, where (b) a sudden singularity occurs. 

Φ 


(z = 0) = H 0 = 100 hkms −1 Mpc −1 ,h=0.71 ± 0.03 (5) 

)= κ2 

2 


f(r) gravity 

ρ m 

H 0 (1 + z)h(z) , → H (0) = κ2 

ρ 0 m = 3 2H 0 2 Ω0 m . (6) 

Nojiri-Odintsov model 

H (0) 0 (7) 

n in viable Ωmodified m (z) , Ω f(R) gravity (z) (8) 13 

Phase space 

2.0 

V + (φ) V − (φ) (9) 

1.5 

H 2 H 0 

2 

1.0 

0.5 

5 0.80 0.85 0.90 

(a) 

Φ 

0.0 

0.0 0.5 1.0 1.5 2.0 

Φ 

(b) 

The Hubble parameter tends to a an asymptotically stable dS point. 

of the HS model (panel 8a), and the NO model (8b). In both models, the 


R 2 ˜f(R) (24) 

The Little Rip 

INERTIAL FORCE INTERPRETATION OF THE LITTLE RIP 

As the universe expands, the relative acceleration between two points separated by a comoving 

As the universe expands, the relative acceleration 

 

between two points separated 

 

by a comoving distance l is given 

distance l is affected by the acceleration of the expansion. An observer located at a comoving 

byH(t) lä/a, =H wherea 

distance 0 [cos(ωt)] is the scale 2 2ωµ 

+ factor. An observer a comoving distance l away from a mass m will measure an inertial 

force on the mass 

l away 

of 

from a 1mass − 3λ m +3µ will measure tan(ωt) 2µ 

1 − 

an inertial 1 −force 3λ +3µ on the [sin(ωt)]2 mass of 

F iner = mlä/a = ml 

(Ḣ ) (29) 

+ H 

2 

. (1) 

P (φ) =P 

Let us assume the two particles are bound 0 (cos(ωφ)) − 2(1−3λ) 

1−3λ+3µ (30) 

 

by a constant force F 0 . If F iner is positive and greater than F 0 ,thetwo 

particles become unbound. This is the “rip” produced by the accelerating expansion. Note that equation (1) shows 

If the two particles are bounded 

that a rip always occurs when 

S 

either 

= dby H 4 xa √ constant 

diverges 

−gβ(φ)Lforce, or Ḣ m 

when 

diverges (assuming Ḣ>0). 

(31) 

The first case corresponds to a 

“big rip” [13], while if H is finite, but Ḣ diverges with Ḣ>0, wehaveaTypeIIor“suddenfuture”singularity[5–7], 

The binding system is broken and a 

which also leads to a rip. However, as F iner noted>F in 0 Ref. [11], it is possible for H, andtherefore,F (32) 

“Rip”of system occurs. iner ,toincreasewithout 

bound and yet not produce a future singularity at a finite time; thisisthelittlerip. Boththebigripandlittlerip 

are characterized by F iner →∞;thedifference is that F iner →∞occurs at a finite time for a big rip and as t →∞ 

for theThis littleusually rip. occurs for singularities of the type of Big Rip (H diverges) and Type II singularities (H’ 

An interesting diverges). case occurs when H is finite and Ḣ diverges but is negative. In this case, even though the universe 

is expanding, all structures are crushed rather than ripped. Anexampleisgivenby 

However, a break of the bounded system H may = H 0 

occur + H 1 

with (t c −no t) α future . singularity, which is called the (2) 

Little Rip. 

Here H 0 and H 1 are positive constants and α is a constant with 0 < α < 1. 

By using It could theoccur FRWeven equations in cyclic cosmologies, where the Hubble parameter presents a bound, but whose 

strength during the accelerating phase 3 may be strong 

κ 2 H2 = ρ , − 1 ( enough to make a Little Rip to occur. 

κ 2 2Ḣ +3H2) = p, (3) 

we may rewrite (1) in the following form: 

φ = t, H= g (φ) =g (t) (25) 

P (φ) =P 0 (cos ωφ) 4 (26) 

H(t) =H 0 (cos ω t) 2 +2ω(2 sin ω t − tan ω t) (27) 

 

II. 

S = dtd 3 x g (3) NF( ˜R) (28) 

P. Frampton, K. Ludwick, R. Scherrer, Phys. Rev. D 84: 063003,2011, arXiv:1106.4996 

2

effective repulsive force 12h 

R ∼ reproduced 0 

+6h 0 by the F (R) terms . and compare (19) with the binding forces of 

for example the Solar ⎪⎩ System). However, to ensure that a little rip is reproduced, one has to stu 

acceleration but free of future singularities, which may have a stronger growth in time than thos 

big rip singularities. Let us use the reconstruction technique using Eqs. (12-21). As an exam 

iods okingof at super-accelerating some of the models expansion reconstructed and it does in not the contain previous any sections, future 

(t s 

singularity. we can see To for 

that find β 

the 

=1 

−t) 

out 

2 Hubble parameter (25) 

a 

ay 

periodic 

lead to 

function 

a little rip, 

that 

one 

reproduces 

has to determine 

a cyclic 

the 

Universe. 

duration 

From 

of each 

the 

cycle6βh effective 

and 0 

EoS 

the strength 

parameter 

of the 

(t s 

for β < 1 

−t) (41), it is clear that this 

force 

odel exhibits 

reproduced 

periods 

by the 

of 

F 

super-accelerating 

(R) terms and compare 

expansion 

with the 

and 

binding 

it does 

forces β+1 

not contain 

of coupled 

any 

systems 

future 

(as 

singularity. To find out 

lar System). However, to ensure that a little rip is reproduced, 

the model (25) may lead to a little rip, one hasfunction, 

one has to f( study 

to determine the duration ˜R) ∝ ˜R models n n = 2 of eternal 

(20) 

of each cycle and the strength of the 

ee of future singularities, which may have a stronger 

The 

growth 

Little 

in time 

 

than 

ective repulsive force reproduced by the F (R) terms and compare d 

2 those models containing 

with the binding forces of coupled systems (as 

s. Let us use the reconstruction technique using Eqs. (12-21). −3n As an example, we consider the U(φ) =e −βeαφ , 

r example the Solar System). However, to ensure that a little rip is dt 2 reproduced, +3H d Rip 

 

R n−1 ∼ (2 − n)R n (21) 

dt one has to study models of eternal 

celeration but free of future singularities, which may have a stronger n = 2growth free of singularities time than those models containing (22) 

g rip singularities. Let us useU(φ) the =e reconstruction technique using Eqs. (12-21). As an example, we consider the 

nction, Little Rip in f(R) −βeαφ , (45) 

n = 

gravity 

1 2 + 3µ 

(23) 

2(1 − 3λ +3µ) 

constants. Then, by the expression (21), the Hubble parameter and the scale factor yield, 

U(φ) =e −βeαφ , R 2 ˜f(R) (24) (45) 

H(t) =h 0 e αt + h 1 , → a(t) =a 0 e 4βeαt +6αt . (46) 

here α and 

Hubble 

β are constants. 

parameter 

φ = t, H= g (φ) =g (t) (25) 

Then, by the expression (21), the Hubble parameter and the scale factor yield, 

d h 1 =6α. It is straightforward to see that the function (46) describes aP Universe, (φ) =P 0 (cos where ωφ) 4 for small 

(26) 

H(t) =h 0 e αt be easily reconstructed by 

+ h 1 , → a(t) =a 0 e 4βeαt +6αt the expressions (15) and (20), 

ubbleparametercanbeapproximatedasaconstant,reproducingadeSittersolution,asinthe 

H(t) =H 0 (cos ω t) 2 +2ω(2 . sin ω t − tan ω t) (27) (46) 

del. For large times, the Universe ends in an eternal phantom phase, where the EoS parameter 

here ithout h 0 big =4αβ rip singularity. and h 1 =6α. Nevertheless, It is straightforward a little rip to (dissolution see that the of bound function S = structures) dtd (46) 3 x describes g (3) NF( might ˜R) occur a Universe, where(28) 

8for small 

ilarly es t to α, the Action theHubbleparametercanbeapproximatedasaconstant,reproducingadeSittersolution,asinthe 

model presented in Ref. [5], as it is pointed out below. The functions P (φ) and Q(φ) 

se structed of ΛCDM by themodel. expressions For large (15) and times, (20), the Universe R = ends 6(2H 2 in+ Ḣ) an and eternal by using phantom expression 

8 

phase, (13), where which yields, 

 

Here the couplings κ i are constants depending on C 1,2 .Hence,forsmallvaluesoftheRicciscalartheactionreduces EoS parameter 

F (R) < 

to−1, the 

but 

action 

without 

for General 

big rip 

Relativity 

singularity. 

plus power-law H(t) =H 

Nevertheless, 

curvature 0 [cos(ωt)] 

a little 

corrections, 2 2ωµ 

+ 

rip (dissolution of bound structures) 

might occur 

a finite time, P (φ) =e similarly 4βeαφ , toQ(φ) the model =−6αpresented 2 (3 + 4βe αφ in)(3 Ref. + 8βe [5], αφ as )e 4βeαφ it is pointed . out below. F (R) = The 

C (47) 1 functions + C 2 4 R 

are constants depending on C 1,2 .Hence,forsmallvaluesoftheRicciscalartheactionreduces 

1 − 3λ as +3µ R 2 , tan(ωt) 2µ 

which is1known − 

1 − 3λ recipe +3µ to [sin(ωt)]2 cure the 

singularities and which has a viable behavior ([13]). R 

12R 

+ 

P + (φ) 75 and e Q(φ) 

25 

0 

ral Relativity plus power-law curvature corrections, as R 2 action (48) represents a viable model where GR can (29) 

be recovered while the curvature scalar corrections remain , which small. is It known is important recipe to that cure such theadditional Rcorrections 

0 

hn be easily reconstructed by the expressions (15) and (20), 

hat hasreproduces abecome viablerelevant behavior the solution close ([13]). to(46) the Hence, little can be rip thecalculated evolution. action (48) Inby order represents inverting to estimate athe viable expression themodel time for where ofthe thelittle GR Ricci rip can scalar induced dissolution of 

e 

nd 

curvature 

by using 

scalar 

the expression 

corrections 

P 

(13), 

remain 

(φ) =e 

which 

small. 

yields, 

It is important that such 

P (φ) 

additional 

=P 

bound structures in a naive way, one might compare the energy-density 4βeαφ , Q(φ) =−6α 2 (3 + 4βe αφ of a 0 (cos(ωφ)) 

corrections 

− 2(1−3λ) 

1−3λ+3µ (30) 

 

bound 

)(3 + 8βe αφ system as the Solar System with 

to the the little density rip evolution. ρ )e 4βeαφ F (R) .Forthemodel(46),suchdensitycanbeapproximatedforlargetimesby 

In order to estimate time little rip induced dissolution . of 

(47) 

 

S = d 

he F (R) action that F (R) reproduces = 

C 4 x √ −gβ(φ)L 

naive way, one might compare the energy-density 

the solution (46) can be calculated by inverting the expression of the Ricci scalar 

1 + C 2 4 R of 

m (31) 

a 

exponential 

R 

bound system as the Solar System with 

12R 

+ 

+ 75 e 

25 

0 16 functions are expanded power series, 

themodel(46),suchdensitycanbeapproximatedforlargetimesby 

terms adjusted with the dark energy density ρ F (R) = nowadays, 

ρ. 0 e 2αt , (48) 

(50) 

= 6(2H 2 + Ḣ) and by using the expression R 

F iner >F 0 (32) 

0 (13), which yields, 

where ρ R 2 R 3 

0 is a constant that 

ρ 

F (R) = 

C 1 + C 2 4 R 

1 = −24e −39/12 R 0 ,andC 2 =2 √ can be set F (R) ∼ κ 1 R + κ 2 + κ 3 

R 0 R 

3R 0 . Hence, we have reconstructed F (R) R model 

12R 

+ 

+ 75 e 

25 

0 

which is able to 

2 + ··· . 

F (R) = ρ 0 e 2αt by imposing that the current value of the energy-density is ρ 

, (50) F (R) (t 0 )= 3 κ 

H 2 0 2 ∼ 

10 −47 GeV 4 , where the age of the Universe is takenwhere 

to be t 0 ∼ 13.73 

ρ F Gyrs, (R) (t 0 

accordingtoRef.[14]. 

)= 3 κ 2 H2 0 ∼ 10 −47 GeV 

Onecansetthetime 

4 (33) 

of the little rip dissolution occurrence when the gravitational coupling of the 0 Sun-Earth 16 

accelerated that can be (eternal) set by imposing phase, that whichthe does current not lead valueto ofa the future energy-density . (48) 

R 

singularity. isNote ρ F (R) that (t 0 )= the 3 system 

action (48) 

0 κ 

H 2 is broken due to the 

Einstein-Hilbert action plus some corrections for small 

here R 0 = α 2 , C 1 = −24e −39/12 R 0 ,andC 2 =2 √ values of the Ricci curvature R, 2 0 ∼ 

cosmological expansion. By assuming a mean density of the Sun-Earth system given by ρ ⊙−⊕ =0.594×10 

age of the Universe is taken to be t 0 ∼ 13.73 Gyrs, accordingtoRef.[14]. Onecansetthetime where the 

−3 kg/m 3 ∼ 

10 −21 GeV 4 ,thetimeforthelittleripdissolutionofboundstructuresis, 

tion ns are occurrence expanded 

Time for 

when in power 

Little 

the gravitational series, 

Rip 

coupling of the 

3R 

Sun-Earth 

0 . Hence, 

system 

we have 

is 

reconstructed 

broken due to the 

F (R) model which is able to 

n. By assuming a mean density of the Sun-Earth 

produce a super-accelerated (eternal) Rphase, 2 tsystem which R 3 given by ρ 

does not lead ⊙−⊕ =0.594×10 

to a future singularity. −3 kg/m 3 LR = 13.73 Gyrs + 29.93 

∼ 

Note that the action (48) 

forthelittleripdissolutionofboundstructuresis, 

α . (51) 

rns out to be the Einstein-Hilbert 

F (R) ∼ κ 1 R + 

action 

κ 2 + 

R 0 

plus 

κ 3 

Rsome 2 + ··· 

corrections 

. 

for small values of the Ricci 

(49) 

curvature R, where the 

Hence, depending on the parameter α, the 0 

ponential functions are expanded in power series, 

t LR = 13.73 Gyrs + 29.93 appearance of the little rip may last shorter or longer. For example, when 

α = 10 −1 Gyrs −1 ,thelittleripoccursattheUniverseageoft 

α . LR = 313.03 Gyrs, while for α (51) ≥ 1Gyrs −1 ,thetimefor 

the decoupling will be much shorter. 

RS. 2 Nojiri, S. ROdintsov AlotofmodelscanbereconstructedinframesofF 3 and DSG, AIP Conf. Proc. 1458, 207 (2011), arXiv:1108.0767. 

he parameter α, the appearance of the little F (R) rip∼ may κ 1 R last + κ(R) shorter 2 

gravity + κ 3 

which + are ··· able . to reproduce a super-accelerating (49) 

R 

or longer. For example, when 

0 

where α and β are constants. Then, by the expression (21), the Hubble parameter and the scale 

H(t) =h 0 e αt + h 1 , → a(t) =a 0 e 4βeαt +6αt . 

where h 0 =4αβ and h 1 =6α. It is straightforward to see that the function (46) describes a Univ 

times t α, theHubbleparametercanbeapproximatedasaconstant,reproducingadeSitte 

case of ΛCDM model. For large times, the Universe ends in an eternal phantom phase, where 

w F (R) < −1, but without big rip singularity. Nevertheless, a little rip (dissolution of bound stru 

in a finite time, similarly to the model presented in Ref. [5], as it is pointed out below. The funct 

P (φ) =e 4βeαφ , Q(φ) =−6α 2 (3 + 4βe αφ )(3 + 8βe αφ )e 4βeαφ . 

The F (R) action that reproduces the solution (46) can be calculated by inverting the expressio 

where R 0 = α 2 , C 1 = −24e −39/12 R 0 ,andC 2 =2 √ 3R 0 . Hence, we have reconstructed F (R) mo 

reproduce a super-accelerated (eternal) phase, which does not lead to a future singularity. Note 

turns out to be the Einstein-Hilbert action plus some corrections for small values of the Ricci cur 

One can set the time of the little rip dissolution occurrence when the gravitational coupling of the 

Sun-Earth system is broken due to the cosmological expansion by assuming a density for the f(R) 

A. López-Revelles, 

R0 

2 R. Myrzakulov and DSG, Phys. Rev. D 85, 103521 (2012), arXiv:1201.5647. 

−1 

phase free of singularities. Let us consider, for example, a Hubble parameter that reproduces a phantom Universe 

16 

.

cosmological evolution, 

e of the Solar system, the force of the coupling between Sun and a body of mass m 

aracterized by the Newtonian law, 

F N = G mM ⊙ 

F 

F N = G mM r 2 . (5.2) 

cosm r 3 

= 

The 

⊙ 

H0 

2 Little Rip 

where r 2 . (5.2) 

M⊙ =1.9 

F N 10 30 GM 

kg is Sun the mass of the Sun, and r is the distance between the Sun 

and the body m. Here, in both cases it is assumed that F 

⊙ =1.9 10 30 i = | F i |, as both forces have radial 

directions, kg is thebut mass opposite of thesign. Sun, Hence, and r we is may the distance between the Sun 

ody m. Here, in both cases it is assumed that F 

Little Rip in viable f(R) gravity 

i = | F compare both quantities along the cosmological 

For the case of the Earth-Sun system, 

i |, as 

r 

both 

= 

forces 

149.6 

have 

× 10 

radial 

9 

evolution, 

, but opposite sign. Hence, we mayF compare cosm 

both quantities along the cosmological 

, 

= r3 

F N 

F cosm 

= r3 

GM⊙ H2 0 h(z) 2 − (1 + z)h(z) h (z) , (5.3) 

of the previous sections, h(0) = 1 and a) h 

F N GM⊙ H2 0 h(z) 2 − (1 + z)h(z) h (z) (0) = 3 2 Ω(0) 

of both where forces recall athat z =0is we defined given H(z) by=H 0 h(z) in , section 3, whereH(5.3) 

0 = 100 hkms −1 Mpc −1 

with h = 0.71 ± 0.03. Let’s compare both forces for the Earth-Sun system, where r = 

all that we defined 149.6 × H(z) 10 9 m=H is the 0 h(z) Earth-Sun in section distance, 3, whereH at the 0 = present 100 hkms time, −1 z Mpc = 0, 

−1 

0.71 ± Earth-Sun 0.03. Let’ssystem 

(i) F cosm 

∼ 0.82 × 10 −22 , (ii) F cosm 

compareF N both forces for the Earth-Sun system, where r = 

0 9 m is the Earth-Sun distance, At z=0 at theF cosm present 

=1.4 

time, 

× 10 

z −22 = 0, h(0) 2 − h(0)h (0) , (5.4) 

F N 

F cosm 

 

h(z) 2 − (1 + z)h(z) h (z) , (31) 

where recall that H(z) =H 0 h(z), where H 0 =100hkms −1 Mpc −1 with h =0.71±0.03. 

m, the ratio (31) yields 

F cosm /F N =1.4×10 −22 [h(0) 2 − h(0)h (0)] and by considering the same initial conditions 

m ,andb) h (0) = −0.1, theratio 

F N 

∼ 1.95 × 10 −22 . (32) 

=1.4 × 10 −22 h(0) 2 − h(0)h (0) , (5.4) 

Then, Fby N assuming the initial conditions from section 3, h(0) = 1 and a) h (0) = 3 2 Ω(0) m , and 

b) h (0) = −0.1, 

assuming the initial conditions from section 3, h(0) = 1 and a) h (0) = 3 2 Ω(0) m , and 

−0.1, 

(i) F HS 

NOcosm 

∼ 0.82 × 10 −22 , (ii) F cosm 

∼ 1.95 × 10 −22 . (5.5) 

F N F N 

1.10 22 

1.10 22 

(i) F cosm 

∼ 0.82 × 10 −22 , (ii) F cosm 

Hence, F N 

both values are too small 

∼ 1.95 × 10 −22 . (5.5) 

F N 

to produce any kind of instability on the orbits of the 

1.510 22 z 

1.210 22 z 

8.10 23 

oth values are too small to produce any kind of instability on 5.10 the 23 orbits of the 

6.10 23 

4.10 23 

– 14 – 

– 14 – 

1.0 0.5 0.0 0.5 1.0 

1.0 0.5 0.0 0.5 1.0 

0 

HS 

NO 


Testing F(R) gravity with Sne Ia 

data 

A simple model 

f(R) =a 

R 

R 0 

 

n R 

+ b 

R 0 

Viability conditions 

To ensure the occurrence of a matter dominated epoch at high redshifts: 0.75 −bn 

 

b>0 and n>1 or 

b0.75 

n−1 

R 

R 0 

Fixed point: matter dominated epoch 

a = 

n [2n(1 + q 0 +2δ 1 ) − 3(1 + δ 1 )] 

(n − 1) [3 + n(8n − 13)] (q 0 − 1)(1 + δ 2 ) 

b = k(1 − q 0) −n 2n 2 (q 0 − 1) − 3(1 + δ 1 )+4n(1 + δ 1 ) 

(n − 1) [3 + n(8n − 13)] (1 + δ 2 ) 

A. Cruz-Dombriz, P. Dunsby, DSG, in preparation

Testing F(R) gravity with 

Sne Ia data 

A simple model 

Fixing the initial conditions at z = 2000 to match the ΛCDM model 

50 000 

0.6 

40 000 

0.4 

30 000 

0.2 

H H 0 

q 

0.0 

20 000 

0.2 

10 000 

0.4 

0 

0 500 1000 1500 2000 

z 

0.6 

0 2 4 6 8 10 

z 


Testing F(R) gravity with 

Sne Ia data 

Using the Union2 SNIa Dataset (557 Sne) 

1.0 

"CDM model, 

0.5 

χ 2 min = 542.68 (Ω m =0.27 ± 0.02) 

∆2 

0.0 

f(R) gravity, 

χ 2 min = 546.204 (Ω m =0.3) 

δ 1 = −0.07186, δ 2 =1 

0.5 

1.0 

0.090 0.085 0.080 0.075 0.070 0.065 0.060 

∆1 


tudy bations of the are kept quasi-static [13, 15]. approximation Let us point out for II. that this Θ f(R, 

F(R,T) 

µν 

kind this = T −2T ) approximation of GRAVITY models. 

gravity 

µν − pg µν 

InTHEORIES 

Section . may VI we apply our results to two particu (7 

tion odelsof and thenumerical first-order results perturbed are obtained fields [16, and23] compared and therefore with the requires ΛCDM model. Finally in Section VII we conclu 

us iththe Let theequations us main start conclusions bymotion writingbecome 

of the this general investigation. action for f(R, T ) gravities [4], 

on II, we briefly review the state-of-the-art of f(R, T 

S = 

II. 

S G + 

f(R, 

S m T 

= 1 ) gravity. 

) GRAVITY 2κ 2 d 4 x √ Section 

osmological equations f R R for µν f(R, − 1 2 fg T )=f µν − (g 1 (R) µν +f −∇ 2 (T µ ∇) ν models ) f R = − as 

THEORIES 

−g κwell 2 

−(f(R, fas 

T Tµν T )+L + f Tm pg ) µν , . (8 

omentum conservation for such models. Then, Section IV addressed I. FORMULAE 

ons ere we 

where, 

for have f(R, 

Let us Action, κstart 2 dropped 

=8πG, 

T )=f 1 

by writing R 

(R)+f the explicit 

is the 2 (T 

general Ricci 

) models dependences 

scalar 

while of 

actionand forT Section f in R 

f(R, represents 

Vand deals T . 

T ) gravities the 

with 

trace 

the 

[4], of the energy-momentum tensor, i.e., T = 

tis is 

while 

kind straightforward 

L 

of models. 

m is the matter 

Into Section see that 

Lagrangian. 

VI we theapply usual 

As usual 

our continuity 

the 

results 

energy-momentum 

toequation two particular is not satisfied for the field equations (8), an 

S = S c 1 (R/R HS ) n 


c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n G + S m = 1 

tensor is defined as, 

dsequently comparedthe with covariant the ΛCDM derivative model. ofFinally the energy-momentum in Section VII we 

− b) 

2κ 2 d 4 x √ tensor conclude is not null in general ∇ µ T µν = 0. In order t 

n. ain the modified continuity equation, let us take the covariant−g (f(R, T )+L m ) , 

T 1+cR n . 

µν = 2 δS derivative m 

√ −g δg µν . 

of the equation (8), 

 

here, κ 

Field 2 =8πG, R∇ is µ the f Ricci scalar and T represents the trace of the energy-momentum tensor, i.e., T = T 

equations, R R µν − 1 (R, 

S EH = d 4 x √ −g 

hile Then, T ) GRAVITY 

L m 

byis varying the matter the THEORIES action Lagrangian. with2respect fg µν − (g µν −∇ µ ∇ ν ) f R = − 

κ 2 

+ f T Tµν − f T Θ µν 

As usual to the energy-momentum metric field g µν , the tensor 

R +2κ 2 field is equations defined as, are obtained, 

L m → S = d 4 x √ −g f(R)+2κ 2 

L m 

or f(R, T ) fgravities T µν = √ 2 

R (R, T )R[4], 

µν − 1 δS m 

2 f(R, T )g µν − (g µν −∇ µ ∇ ν ) f R (R, T )=− 

−g δg µν . κ 2 + f T (R, T ) T µν − f T (R, T )Θ µν , 

where the energy-momentum tensor is defined as: T µν (m) = −√ 2 δ ( √ where = 1 → 

the subscripts on the function f(R, T ) mean differentiation −gL m ) 

hen, by varying the action with respect to the metric field g µν with respect to R or T , and the tensor Θ 

2κ , the field equations −g δg µν are obtained, 

defined 2 d 4 x √ f R ∇ µ R µν + R µν ∇ µ f R − 1 2 g µν(f R ∇ µ R + f T ∇ µ T ) − (g µν ∇ µ −∇ µ ∇ µ ∇ ν ) f R = 

−g (f(R, T )+L m ) , (1) 

as, 

f R (R, T )R µν − 1 ∇ µ − κ 2 

+ f 

T represents the trace of 

Divergence of the field 2 f(R, the energy-momentum T )g µν − (g µν −∇ µ ∇ ν ) f R (R, )=− κ 2 + f T (R, T ) T µν − f T (R, T )Θ µν , 

equations Θ µν ≡ g αβ δT tensor, T Tµν − f 

i.e., T T Θ 

= µν . (9 

T µ µ, 

αβ 

δg µν = −2T µν g µν L m +2g αβ δL m 

the energy-momentum tensor is defined as, 

us, using the identities ∇ µ R 

. 

µν − 1 2 Rg µν 

= 0, and (∇ν − ∇ ν ) f R (R, T )=R δg µν g µν αβ 

∇ µ f R , the covariant derivativ 

the here energy-momentum the subscripts on the function f(R, T ) mean differentiation with respect to R or T , and the tensor Θ µν 

efined Note T µν that as, = √for 2 δS m 

tensor needs to satisfy, 

a regular . f(R, T ) function, in absence of any kind of matter, (2) the corresponding f(R) gravity equa 

−g δg 

µν 

are recovered, and consequently 

∇ µ T 

the corresponding properties and the well-known solutions for f(R) gravity are 

µν = 

f 

T 1 

he satisfied metric by fieldf(R, g µν T , the ) theories field equations inκclassical Θ 2 + f 

µν ≡ gare αβ δT T 2vacuum g µν∇ µ T − (T µν + Θ µν )∇ µ ln f T −∇ µ Θ 

obtained, αβ 

δg µν = −2T (for a review on f(R) 

µν − g µν L m +2g αβ theories, δL µν . (10 

m see [1]). Moreover, here w 

. 

interested to study the behavior of this kind of theories for spatially flatδg FLRW µν g αβ spacetimes, which are express 

nce, comoving for coordinates by the line element, 

−∇ 

In 

a 

ote that µ ∇ 

general, 

perfect 

ν ) f 

for R (R, 

the 

fluid 

T )=− 

divergence 

with an 

κ 

a regular f(R, 2 + 

of 

equation 

f 

the 

T ) function, T (R, 

equations 

T ) of state 

T 

in µν − 

is not 

p 

f 

absence T (R, 

null, 

= wρ, 

T 

which 

beingw 

)Θ 

of any kind µν 

of matter, the corresponding f(R) gravity equati 

re recovered, and consequently the corresponding ds 2 properties = a 2 (η) , 

may give 

a constant, 

(3) 

rise to an 

the 

anomalous 

0−component 

behavior 

of 

of the 

the covarian 

ivative matter (10) turns content. out to become, 

 

dη and 2 −the dxwell-known 2 solutions for f(R) gravity are a 

atisfied ) mean by differentiation f(R, T ) theories with respect in classical to Rvacuum or T , and (for the a review tensoron Θ µν f(R) is 

κ 2 + w − 3 

 

 

f theories, see [1]). Moreover, here we 

terested where a(η) to study is the the scale behavior factor T − (1 + w)Tf 

2 in of conformal TT T ˙ + 3(1 + w) 

H(κ 2 − f 

this kind of time theories η. Then, for spatially the main T ) − 2f 

flatissue TR (4HḢ FLRW arises spacetimes, on + the Ḧ) T =0. (11 

T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, content #Phys. which Rev. D84, are of 024020 the expressed 

(2011) Unive

ere right lethand us remind side (r.h.s.). that T Thus, = T it µ = may ρ −lead 3p. toThe violations last equation of the usual differs evolution from theofusual different continuity species equation in theon th 

,-null T Nevertheless, ) mean rightdifferentiation hand in side the next (r.h.s.). with section Thus, respect 

F(R,T) 

we it focus may to Rour lead orattention T to , and violations 

gravity 

the on atensor model of theΘ that usual keeps evolution the usual of the continuity differentequation 

species in th 

µν is 

iverse. d. Nevertheless, in the next section we focus our attention on a model that keeps the usual continuity equatio 

hanged. 

δT αβ 

δg µν = −2T µν − g µν L m +2g αβ δL m 

. (4) 

III. δg µν 

f 1 g(R)+f αβ 2 (T ) TYPE THEORIES 

III. f 1 (R)+f 2 (T ) TYPE THEORIES 

section, absencewe of any choose kind the ofalgebraic matter, the function corresponding f(R, T ) to f(R) be agravity sum ofequations 

two independent functions 

nonding thisA section, particular properties wecase, 

choose and the thewell-known algebraic function solutionsf(R, for f(R) T ) togravity be a sum areof also two independent functions 

acuum (for a review on f(R) theories, f(R, T see )=f [1]). 1 (R)+f Moreover, 2 (T ) here we are 

d of theories for spatially flat FLRW spacetimes, f(R, T )=f which 1 (R)+f are expressed 2 (T ) in 

(12) 

(12 

(R) and f 2 (T ), respectively depend on the curvature R and the trace T . The generalized Einstein equations 

ere rgy 

yield fdensity, Friedmann 

1 (R) andwith f 

equations, 

2 (T ), respectively depend on the curvature R and the trace T . The generalized Einstein equation 

(8) ds 2 yield = a 2 (η) f 10 ≡ f 1 (R 0 ), f 1R0 ≡ df 1 (R 0 )/dR 0 , f 20 ≡ f 2 (T 0 ), f 2T0 ≡ df 2 (T 0 )/dT 0 and c 2 

dη 2 − dx 2 s = p 0 /ρ 0 . Th 

tinuity equation (11) for Lagrangians given by (12) yields 

(5) 

 

− 3Hf 1R +3H f 1R0 − a2 

2 f 

 

10 = −κ 2 a 2 ρ 0 +(1+c 2 s)ρ 0 a 2 f 2T0 + a2 

l time η. Then, the main ∇ 2 f 20 , (13) 

− 3Hf 1R 0 

+3H f 1R0 − a2 

f 

1R 0 

+ Hf 1R 0 

− (H +2H 2 )f 1R0 + a2 

2 f 2 f 10 = −κ 2 a 2 ρ 0 +(1+c 2 s)ρ 0 a 2 f 2T0 + a2 

µ T µ issue arises on the content of the Universe is 

0 ν = 1 

1 

10 = −κ 2 2 c 2 sρ 0 − a2 

2 f 2 f 20 , (13 

ensor, defined in (2). Since we are κ 2 δ ν µ ∂ µ 

−interested f 2T0 on flat 2 f 20 + c 2 sρ 0 f 2T0 + T µ 

0 ν ∂ µf 2T0 , (15 

FLRW cosmologies, 

less matter, radiation,... ) can be well described 20 . (14) 

f 

1R 0 

+ Hf 1R 0 

− (H +2H 2 )f 1R0 + a2 by 

2 f perfect fluids, whose 

wing explicitly that the energy-momentum tensor is not 

10 = a priori −κ 2 a 2 covariantly c 2 sρ 0 − a2 

2 f conserved 

20 . in f(R, T ) theories. Thus (14 

these theories, the test particles moving in a gravitational 

e prime holds for the derivative with respect to η, H ≡ a field do not follow geodesic lines. By exploring th 

ation /a and the subscript 0 holds for unperturbed 

ere ndT (15) 

quantities: the prime Rholds 0 denotes for the thederivative scalar curvature with respect corresponding to η, Hto ≡the a µν =(ρ 

for 

+ p)u 

ν = 

µ u 

0 component, 

ν − pg µν . 

one gets 

Continuity equation, 

/a unperturbed and (6) the subscript metric, ρ0 holds the unperturbed for unperturbe 

kground quantities: R 0 denotes the scalar curvature corresponding to the unperturbed metric, ρ 0 the unperturbe 

ρ 0 +3Hρ 0 (1 + c 2 1 

s)= 

κ 2 (1 + c 2 

− f 

s)ρ 0 f2T 0 

+ c 2 sρ 0f 2T0 + 1 

2T0 

2 f 20 

. (16 

e that The whether usual continuity f 2 vanishes equation (i.e., is f(R) recovered theories) when, or characterizes a non-running cosmological constant, both f 2T 

vanish, and then the continuity equation in these scenarios becomes 

f 2T0 

f 

ρ 0 +3H 2 (T )=α √ 1+c 2 T + β 

s 

ρ0 =0 . (17 

n order to Lagrangians such as (12) consistent with the standard conservation equation, the r.h.s. of (15) has t 

ish leading to the differential equation 

1 

F. Alvarenga, A. Cruz-Dombriz, S. Houndjo, M. Rodrigues, DSG, to appear in PRD, ArXiv:1302.1866

sence, function the complete represents sethe of equations unique 

F(R,T) 

Lagrangian that describes thatthe satisfies general 

gravity 

thelinear usual perturbations continuity equation for the kind (17) of within models the considere class o 

e, elsf(R, given T )=f by expression 1 (R) +f 

IV. 2 (12). (T ), 

PERTURBATIONS 

have been obtained, which 

IN f(R, 

provides 

T ) THEORIES 

enough information about the behavior of th 

turbations within this class of theories, that can be compared with expected results from ΛCDM model. 

consider the scalar perturbations of a flat FLRW metric in the longitudinal gauge 

IV. PERTURBATIONS IN f(R, T ) THEORIES 

V. Cosmological EVOLUTION perturbations 

OF SUB-HUBBLE MODES AND THE QUASI-STATIC APPROXIMATION 

ds 2 = a 2 (η) (1 + 2Φ)dη 2 − (1 − 2Ψ)dx 2 , (21) 

et us consider the scalar perturbations of a flat FLRW metric in the longitudinal gauge 

≡e Φ(η, are interested x) and Ψ in ≡ the Ψ(η, possible x) are the effects scalar on perturbations. the 

ds 2 = a 2 (η) density contrast The components evolution 

(1 + 2Φ)dη 2 − (1 − 2Ψ)dx 2 once of perturbed the perturbations energy-momentum enter the Hubbl 

ius thisingauge the matter are given dominated by era. In the sub-Hubble limit, i.e., H k, and , after having neglected all the (21 tim 

ivative for the 

re Φ ≡ˆδT Φ(η, 0 x) and Ψ ≡ Ψ(η, x) are the scalar perturbations. The components of perturbed energy-momentum 

0 = ˆδρ 

Bardeen’s potentials Φ and Ψ, the equations (25) and (26) 

i 

= ρ 0 δ , ˆδT j = −ˆδp δ i j = −c 2 sρ 0 δ i 0 

jδ , ˆδT i = −ˆδT i 0 = − can be combined yielding 

⎛ 

⎞ 1+c 2 s 

ρ0 ∂ i v, (22) 

or in this gauge are given by2k2 

f 

1+ 

1R0 R 0 

a 

6 

denotes the potential Ψ = 

ˆδT 0 0 = ˆδρ 

Φ 2 f 1R0 

for the velocity perturbations. Using the model (12), the 

i 

= ρ 0 δ , ˆδT j = −ˆδp δ i j = −c 2 sρ 0 δ i 0 

jδ , ˆδT i = −ˆδT i 0 = − perturbed metric (21) and 

rbed The energy-momentum perturbed equation 1+ 

tensor can 1+c 2 s 4k2 f 1R0 

; Φ = − 1 

4k2 f 

1+ 

1R0 R 0 

a 

⎝ 2 f 1R0 

⎠ 

R 0 

be (22), reduced the first to, 

2k 

order perturbed 2 a ρ0 ∂ i v, (22 

when combined yield 

2 f 1R0 

1+ 3k2 f 1R0 

κ 2 a 2 ρ 0 

− f 2T0 δ . (35 

R 0 

f 

a 2 1R0 

equations f 1R0 reads 

addition, re v denotes the 

 

µ equation the potential for velocity perturbations. Using the model (12), the perturbed metric (21) and 

f 

perturbed 1R0 

ˆδG 

energy-momentum ν +(R µ 

0ν + ∇µ ∇ ν − tensor ν µ ) f 

(22), 1R0 Rthe 0 

ˆδR + 

µα 

ˆδg ∇ 

first order perturbed ν ∇ α δ 

equations ν 

ˆδg 

µ αβ 

δ 3f(34) 2T0 

in the 

 

QS approximation 

+ H 1 − 

∇ 

reads α ∇ β f 1R0 

2(κ 2 δ + k 2 f 2T0 

− f 2T0 ) 2(κ 

 

µ 

f 1R0 

ˆδG ν +(R µ 

0ν + − g αµ γ ∇µ ∇ ν − δ ν µ ) f 1R0 R 0 

ˆδR + 

µα 

ˆδg ∇ ν ∇ α − δ ν 

ˆδg 

0 

ˆδΓ 2 − f 2T0 ) δ + yields, 

 

 

αν − δ ν µ g αβ γ 

0 

ˆδΓ k2 Φ − 3Ψ 3f 2T0 

 

− 3H 1 − 

2(κ 2 Ψ = 0 (34) 

− f 2T0 

βα 

∂ γ f 1R0 = − µ (κ − f 

αβ 2T0 ) ˆδT 

) 

δ 3f 0 

2T 

+ H 1 − 

ν 

µ nce, the complete set of equations that2(κ 

describes 2 − f2T 0 the ) δ + k 2 f2T 

0 

general2(κ linear 2 − perturbations f2T 0 )δ + k2 Φ = 0 (36 

for the ∇ 

α kind ∇ β of f 1R0 models considered 

f(R, In T )=f the sub-Hubble 1 (R) +f 2 (T limit, ), have k Hbeen 1 

 

+ 

− g αµ γ 

0 

ˆδΓ αν − δ ν µ g αβ γ 

2 (1 obtained, − which 

0 

ˆδΓ c2 s)f 2T0 δ ν µ provides 

+(1− 3c 2 enough 

s)(1 + c 

βα 

∂ 2 information 

s)ρ 

γ f 0 f 2T0 1R0 = 

T 

− (κ − f 2T0 ) ˆδT ν 

µ 0 

u µ about the behavior of the 

en, by using the previous result (35) in the equation (36) one gets 

rbations within this class of theories, that can be compared with expected results 

u 

from ν 

ˆδρ 

ΛCDM 

, 

model. 

(23) 

⎡ 

⎛ 

⎞⎤ 

 

 

1 

0R + 

2 (1 − c2 s)f 2T0 δ ν µ +(1− 3c 2 s)(1 + c 2 s)ρ 0 f 2T0 T 0 

u µ 0 

=d δ 2 f 1 (R 0 )/dR0, 2 ∇ 

3f α 2T0 

+ H 1 − ∇ α and ∇ holds for the covariant derivative with respect to the unperturbed u ν 

ˆδρ , metric (23 

3), V. we EVOLUTION have made use 2(κof OF 2 δ + 1 f 1R0 

⎣k 2 f 2T0 

−the SUB-HUBBLE f relation linkingMODES the traceAND to theTHE eneryQUASI-STATIC density, T 0 = ρ 0 −APPROXIMATION 

3p 0 =(1− 3c 2 2T0 ) 2 (κ 2 − f 2T0 ) − (κ2 − f 2T0 ) a2 ρ 0 

⎝ 1+4k2 

R 0 

a 2 f 1R0 

⎠⎦ δ = 0 (37 

f 1R0 1+3 k2 

s)ρ 0 , and 

re f 1R0 R 0 

=d 2 f 1 (R 0 )/dR0, 2 ∇ α ∇ α and ∇ holds for the covariant derivative with respect to the unperturbed metric 

et are caninterested be understood in theas possible the quasi-static effects on equation the density forcontrast f(R, T ) evolution models ofonce the form the perturbations (19). By neglecting enter the in 

In (23), we have made use of the relation linking trace to the enery density, T 0 = ρ 0 − 3p 0 =(1− 3c 2 Hubble (37) th 

ms s)ρ 0 , and 

s inf 2 the (T 0 ), matter i.e., paying dominated attention era. only In the to f(R) sub-Hubble theories, limit, one i.e., recovers H the k, usual and after quasi-static having approximation neglected all the fortime 

thos 

ative ories (see for the for Bardeen’s instance [22], potentials [23] and Φ [24]) and Ψ, the equations (25) and (26) can be combined yielding 

⎛ ⎛ ⎞ ⎞ 

2k2 f 

1+ 

1R0 

f 

R 0 

a 

Ψ = Φ 2 f 1R0 

1+ 4k2 f 1R0 

; Φ = − 1 

1R0 

4k2 f 

1+ 

1R0 R 0 

a 

⎝ 2 f 1R0 

⎠ 

R 0 

2k 2 

a 2 f 1R0 

1+ 3k2 f 1R0 

κ 2 a 2 ρ 

δ + Hδ − ⎝ 1+4k2 

R 0 

a 2 f 1R0 

⎠ κ2 a 2 ρ 0 

0 

− f 2T0 δ . (35) 

R 

1+3 k2 f 1R0 

δ = 0 (38 

R 0 2f 0 

f 

a 2 1R0 

f a 2 1R0 1R0 F. Alvarenga, A. Cruz-Dombriz, S. Houndjo, M. Rodrigues, DSG, to appear in PRD, ArXiv:1302.1866 

f 1R0 

a 2 f 1R0 R 0 

f 1R0

e previous section, we study two particular f(R, T )modelswithf 1 (R 0 ) 

rtional to the Ricci scalar, i.e., f 1 (R 0 )=R 0 . This choice encapsulates a 

function f 2 (T 0 ) introduced in Section 2 through the expression (19). 

. f A (R 0 ,T 0 )=R 0 + αT 1/2 

0 

F(R,T) gravity 

Evolution of the matter perturbations in the quasi-static limit 

tant α according to expression (43), thus possessing the appropriate 

ckground evolution that can be rewritten as 

= Ω 0 ma −1 +(1− Ω 0 m)a 1/2 10 1 

(47) 

k=50 H 0 

k=100 H 0 

k=200 H 0 

k=500 H 0 

k=1000 H 0 

ΛCDM 

Log[δ k ] 

10 2 10 0 

0 0.5 1 1.5 2 2.5 3 

10 -1 

10 -2 

10 -3 

Log[1+z] 

ure 1: δ k evolution for f A (R, T )modelaccordingtothequasi-staticevolutiongivenby(40)andΛCDM given by (39). The depicte 

des are k =50,100,200,500and1000(inH 0 units). The plotted redshift ranged from z =1000toz =0(today). ThevalueofΩ 0 m 

fixed to 0.27 for illustrative purposes. It is seen how whereas the ΛCDM is k-independent, the f A (R, T )modelevolutionsdivergefo 

ll the studied modes and leave the linear region at redshifts z ≈ 100. F. Alvarenga, For larger A. Cruz-Dombriz, k-modes S. Houndjo, (deep M. Sub-Hubble Rodrigues, DSG, modes) to appear in the PRD, ArXiv:1302.1866 

divergence

n at redshift z = 1000 where δ is assumed to behave as in a matter 

k-dependence. In Fig. 1 we have plotted the evolution of the density 

he strong k-dependence of equation (40) renders the evolution of these 

ity contrast evolution provided by the Concordance ΛCDM model and 

100. 



f B (R 0 ,T 0 )=R 0 + α T 1/2 

0 − 2β 

α < 0 

nd in (19), which also satisfies the usual continuity equation in the 

s supplemented with a cosmological constant −2β. The first FLRW 

10 1 

Ω 0 ma −1 − c 1 Ω 0 ma 1/2 + c 2 a 2 , (49) 

order to provide the correct dimensions 

10 0 

to the free constants parameters 

(witha(z = 0) = 1), one gets the constraint, 

Log[δ k ] 

Ω 0 m + c 2 → c 2 =1− Ω 0 m(1 − c 1 ) . (50) 

10 -1 

k=50 H 0 

k=100 H 0 

k=200 H 0 

k=500 H 0 

k=1000 H 0 

ΛCDM 

10 2 

0 0.5 1 1.5 2 2.5 3 

imensionless parameters {c 1 ,c 2 }, where one remains arbitrary. As for 

in the equation (37) leads to an evolution of the matter perturbations 

only a very restricted limit for 10 -2 the free parameter c 1 can avoid such 

10 -3 

Log[1+z] 

igure 2: δ k evolution for f B (R, T )modelaccordingtothequasi-staticevolutiongivenby(37)andΛCDM given by (39). Here we 

ave assumed a value c 1 = −10 −3 .Aspreviously,thedependenceonk leads to a strong growth of the matter perturbations for large 

values of k, whereasthebehaviorissimilartotheΛCDM model for the modes k

en at redshift z = 1000 where δ is assumed to behave as in a matter 


ok-dependence. In Fig. 1 we have plotted the evolution of the density 

the strong k-dependence of equation (40) renders the evolution of these 

sity contrast evolution provided by the Concordance ΛCDM model and 

≈ 100. 


f B (R 0 ,T 0 )=R 0 + α T 1/2 

0 − 2β 

1 

α > 0 

und in (19), which also satisfies 0.9 the usual continuity equation in the 

is supplemented with a cosmological constant −2β. The first FLRW 

0.8 

= Ω 0 ma −1 − c 1 Ω 0 ma 1/2 + c 2 a 2 0.7 

, (49) 

0.6 

order to provide the correct dimensions to the free constants parameters 

(witha(z = 0) = 1), one gets0.5 

the constraint, 

δ k 

1Ω 0 m + c 2 → c 2 =1− Ω 0 0.4 

m(1 − c 1 ) . (50) 

0.3 

dimensionless parameters {c 1 ,c 2 }, where one remains arbitrary. As for 

k in the equation (37) leads to0.2 

an evolution of the matter perturbations 

, only a very restricted limit for the free parameter c 1 can avoid such 

0.1 

k=50 H 0 

k=100 H 0 

k=200 H 0 

k=500 H 0 

k=1000 H 0 

ΛCDM 

1 

0 

-0.1 

0 0.5 1 1.5 2 2.5 3 

Log[1+z] 

igure 3: δ k evolution for f B (R, T )modelaccordingtothequasi-staticevolutiongivenby(37)andΛCDM given by (39). Here we 

have assumed a positive value for the free parameter c 1 =10 −3 ,whichleadstoanoscillatingbehaviorofthematterperturbations, 

which turns out stronger as k is larger, and whose oscillations are observed for large small redshifts. The model mimics the ΛCDM 

F. Alvarenga, A. Cruz-Dombriz, S. Houndjo, M. Rodrigues, DSG, to appear in PRD, ArXiv:1302.1866 

model only those modes small enough k

Summary 

Viable modified gravities reproduce a sudden singularity, which 

may be circumvented in the scalar-tensor picture. 

The transition to a phantom phase probably occurs in the class of 

the so-called viable f(R) gravities, where the EoS parameter 

presents an oscillating behavior, but no future singularity or Little 

Rip occurs. 

The analysis of a simple model of f(R) gravity, and its test with 

the Sne Ia reveals that the f(R) gravities can describe late-time 

acceleration with great accuracy. 

f(R,T) gravities seem to be ruled out, since the non-standard 

coupling among matter and gravity introduce an anomalous 

behavior in the matter perturbations.

Diego Saez-Gomez

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