A primer on problems and prospects of dark energy

M. Sami1
1
Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India
This review on dark energy is intended for a wider audience, beginners as well as experts. It
contains important notes on various aspects of dark energy and its alternatives. The section on
Newtonian cosmology followed by heuristic arguments to capture the pressure effects allows us
to discuss the basic features of physics of cosmic acceleration without actually resorting to the
framework of general theory of relativity. The brief discussion on observational aspects of dark
energy is followed by a detailed exposition of underlying features of scalar field dynamic relevant
to cosmology. The review includes pedagogical presentation of generic features of models of dark
energy and its possible alternatives.
arXiv:0904.3445v3 [hep-th] 26 Sep 2009

PACS numbers: 98.80 Cq

I. INTRODUCTION direction). Late time acceleration can be fueled either

by an exotic fluid with large negative pressure dubbed
dark energy[4, 5, 6, 7, 8, 9, 10] or by modifying the grav-
Twentieth century has witnessed remarkable develop- ity itself[11]. The simplest candidate of dark energy is
ments in the field of cosmology. The observation of red- provided by cosmological constant Λ[12, 13, 14], though,
shift of light emitted by distant objects and the discovery there are difficult theoretical issues associate with it. Its
of microwave background in 1965 have revolutionized our small numerical value leads to fine tuning problem and
thinking about universe. The hot big bang model then we do not understand why it becomes important today
received the status of standard model of universe. How- a la coincidence problem.
ever, in spite of the theoretical and observational suc- Scalar fields provide an interesting alternative to cos-
cesses, cosmology remained confined to a rather narrow mological constant[16, 17]. To this effect, cosmological
class of scientist; others considered it as the part of a dynamics of a variety of scalar fields has been investi-
respectable philosophy of science. Cosmology witnessed gated in the literature(see review[6] for details). They
the first revolution in 1980 with the invent of cosmo- can mimic cosmological constant like behavior at late
logical inflation making it acceptable to the larger com- times and can provide a viable cosmological dynamics
munity of physicists. Since then it goes hand in hand at early epochs. Scalar field models with generic features
with high energy physics. The scenario envisages that are capable of alleviating the fine tuning and coincidence
universe has gone through a phase of fast accelerated ex- problems. As for the observation, at present, it is abso-
pansion at early epochs. Inflation is a beautiful paradigm lutely consistent with Λ but at the same time, a large
which can resolve some of the in built inconsistencies of number of scalar field models are also permitted. Future
the hot big bang model and provides a mechanism for data should allow to narrow down the class of permissible
generation of primordial fluctuations needed to seed the models of dark energy.
structure we see in the universe today. In the past two As an alternative to dark energy, the large scale mod-
decades, observations have repeatedly confirmed the pre- ifications of gravity could account for the current accel-
dictions of inflation. However, its implementation is ad eration of universe. We know that gravity is modified at
hoc and requires support from a fundamental theory of short distance and there is no guarantee that it would not
high energy physics. As inflation takes place around the suffer any correction at large scales where it is never ver-
Planck epoch, the needle of hope points towards string ified directly. Large scale modifications might arise from
theory − a consistent theory of quantum gravity. extra dimensional effects or can be inspired by fundamen-
The second revolution cosmology witnessed in 1998, is tal theories. They can also be motivated by phenomeno-
related to late time cosmic acceleration[1, 2]. The ob- logical considerations such as f (R) theories of gravity.
servations of high redshift supernovae reveals that uni- However, any large scale modification of gravity should
verse is accelerating at present. The phenomenon is indi- reconcile with local physics constraints and should have
rectly supported by data of complimentary nature such potential of being distinguished from cosmological con-
as CMB, large scale structure, baryon acoustic oscillation stant. To the best of our knowledge, all the schemes of
and weak lensing. It is really interesting that the thermal large scale modification, at present, are plagued with one
history of our universe is sandwiched between two phases or the other problems.
of accelerated expansion. In the Newtonian language, The review is organized as follows: After introduction
cosmic repulsion can be realized by supplementing the and a brief background, we present cosmology in Newto-
Newtonian force by a repulsive term on phenomenological nian framework in section III and mention efforts to put
grounds. The rigorous justification of the phenomenon it on the rigorous foundations in the domain of its valid-
can only be provided in the frame work of general the- ity. In section IV, we put forward heuristic arguments
ory of relativity (see Ref.[3] for early attempts in this to incorporate Λ, in particular and pressure corrections,
 2

in general, in the evolution equations and describe the emitted by a source receding from the observer appears
broad features of cosmological dynamics in presence of shifted towards red end of spectrum and the redshift is re-
cosmological constant. In section V, we provide a short lated to the velocity of recession v as, z ≃ v/c for v << c.
introduction to relativistic cosmology and discus issues In the beginning of the last century, astronomers could
associated with cosmological constant. After a brief sub- measure the distances to a number of distant galaxies.
section on observational aspects of cosmic acceleration, Hubble carried out investigations of recession velocities
we proceed to highlight the generic features of scalar field and plotted them against the distances to galaxies. He
dynamics relevant to cosmology and mention the current concluded in 1929 that there is a linear relation between
observational status of dynamics of dark energy. In the recession velocity of the galaxies and the distance to them
last section before summary, we present a discussion on − the so called Hubble Law.
the current problems of alternatives to dark energy. The observational conclusion that universe expands is
Last but not the least, a suggestion for the follow up of based upon the redshift of radiation emitted by distant
this review is in order. At present, there exist, a number galaxies. Can we have another explanation for the red-
of excellent reviews on dark energy[4, 5, 6, 7, 8, 9, 10] and shift? It might look surprising that photons from larger
cosmological constant[12, 13, 14] which focus on different distances emitted from galaxies reach us redshifted due
aspects of the subject. Four recent and very interesting to the recession of galaxies and nothing else happens to
reviews[7, 8, 9, 10] which try to address the theoretical them. They travel through intergalactic medium and
and observational aspects of late time cosmic acceleration could be absorbed by matter present there and then emit-
are highly recommended. Humility does not allow to say ted loosing part of their energy in this process thereby
that Ref.[6] is the most comprehensive theoretical review leading to their redshift without resorting to expansion of
on dark energy with pedagogical exposition. universe. This apprehension can be refuted by a simple
argument. As for the absorption, the underlying process
is related to the scattering of photons by the particles
II. THE SMOOTH EXPANDING UNIVERSE of intergalactic medium. If true, the source should have
appeared blurred which is never observed. Other efforts
assuming the exotic interactions of photons could not
Universe is clumpy at small scales and consists of very
account for the observed redshift. Thus the only viable
rich structure of galaxies, local group of galaxies, clusters
explanation of the phenomenon is provided by the ex-
of galaxies, super clusters and voids. These structures
pansion of universe[22].
typically range from kiloparsecs to 100 megaparsecs. The
study of large scale structures in the universe shows no If we imagine moving backward in time, universe was
evidence of new structures at scales larger than 100 mega- smaller in size, the temperature was higher and there was
parsecs. Universe appears smooth at such scales which an epoch when the universe was vanishingly small with
leads to the conclusion that universe is homogeneous and infinitely large energy density and temperature-the be-
isotropic at large scales which serves as one of the fun- ginning of universe dubbed as the big bang. The matter
damental assumptions in cosmology known as cosmolog- was thrown away with tremendous velocity, since then
ical principal[18]. Homogeneity tells us that universe the universe is expanding and cooling. At early times it
looks the same observed from any point whereas isotropy was extremely hot and consisted of a hot plasma of el-
means that lt looks same in any direction. In gen- ementary particles, there were no atoms and no nuclei.
eral these are two independent requirements. However, Roughly speaking, at temperatures higher than the bind-
isotropy at each point is stronger assumption which im- ing energy of hydrogen atom, the photons were freely
plies homogeneity also. Cosmological principal presents scattering on electrons and atoms could not form. As
an idealized picture of universe which allows us to un- the universe cools below the temperature characterized
derstand the background evolution. The departure from by the binding energy of hydrogen atom, electrons com-
smoothness can be taken into account through perturba- bine with protons to form hydrogen atom leading to the
tions around the smooth background. Observations con- decoupling of radiation from matter. This was an im-
firm the presence of tiny fluctuations from smoothness portant epoch in the history of universe known as re-
in the early universe. According to modern cosmology, combination. The decoupled radiation since than is just
these small perturbations via gravitational instability are expanding with the expanding universe and cooling. The
believed to have grown into the structures we see today discovery of microwave background, the relic of the big
in the universe[19, 20, 21, 22, 23, 24, 25, 26, 27]. bang, in 1965 confirms the hypothesis of hot big bang.
One of the most remarkable discoveries in cosmology
includes the expansion of universe and its beginning from
the big bang. The analysis of radiation spectrum emitted III. THE HOMOGENEOUS AND ISOTROPIC
from distant galaxies shows that wavelengths of spectral NEWTONIAN COSMOLOGY
lines are larger than the actually emitted ones, the phe-
nomenon is known as redshift of light. The redshift is Newtonian theory of gravitation allows us to under-
quantified by symbol z defined as, z = (λob − λem )/λem . stand the expansion of a homogeneous isotropic universe
According to the Doppler effect, the wavelength of light in a simple way. Newtonian description is valid pro-
 3

vided the matter filling the universe is non-relativistic It can easily be verified that Hubble law holds at any
and scales associated with the problem are much smaller point. If we move from O to O′ , we can write
than the Hubble radius. For instance, at early epochs,
the universe was hot dominated by radiation. Hence v′ (r′p ) = Hrp − Hro′ = H(t)r′p (6)
early universe strictly speaking, should be treated by rel- The Hubble’s law gives the most general form of veloc-
ativistic theory. The general theory effects are also cru- ity field permissible by the homogeneity and isotropy of
cial at super Hubble scales. Despite its limitations, New- space.
tonian cosmology provides a simple and elegant way of Hubble law tells us how distance between any two
understanding the expansion of universe[18, 22, 27, 28]. points in space changes with time provided we know the
expansion rate given by H(t),
Rt
A. Hubble law as a consequence of homogeneity H(t)dt
r(t) = xe 0 , x ≡ r(t = 0) (7)
and isotropy
The law of expansion depends upon how the Hubble pa-
Using the Newtonian notions of physics, let us show rameter H varies with time. Eq. (7) shows how distances
that the Hubble law is a natural consequence of homo- in a homogeneous and isotropic universe scale with the
geneity and isotropy. Let us choose a coordinate system scale factor a(t),
with origin O such that matter is at rest there and let Rt ȧ
H(t)dt
us observe the motion of matter around us from this co- a(t) ≡ e 0 or H(t) = (8)
a
ordinate system. The velocity field, i.e., the velocity of
r(t) = a(t)x (9)
matter at each point p around us at an arbitrary time,
depends upon the radius vector r and time t. We should The complete information of dynamics of a homogeneous
now look for the most general velocity field in a homo- and isotropic universe is contained in the scale factor; we,
geneous and isotropic universe. Let us assume another thus, need evolution equation to determine a(t). In case
observer located at point O′ with radius vector ro′ and H is independent of time we have exponentially expand-
moving with velocity v(ro′ ) with respect to the observer ing universe dubbed de-Sitter space. In what follows
O. If we denote the velocity of point p relative to O and we shall confirm that constant Hubble rate is allowed
O′ at time t by v(rp ) and v′ (r′p ), we have, in relativistic cosmology provided the energy density of
matter in the universe is constant. It is believed that
universe has passed through an exponentially expanding
r′p = rp − ro′ (1) phase known as inflation at early times.
v ′
(r′p ) = v(rp ) − v(ro′ ) (2) According to Hubble law, in a homogenous and
isotropic universe, all the material particles move away
where rp and r′p denote the radius vectors of point p radially from the observer located at any point in the uni-
with respect to O and O′ respectively. The cosmological verse. This motion is refereed to as Hubble flow. Indeed,
principal tells that the velocity field should have the same any freely moving particle in such a background would ul-
functional form at any point, timately follow the Hubble flow. Motion over and above
the Hubble flow is called peculiar motion which can only
v(r′p ) = v(rp ) − v(ro′ ) (3) arise in a perturbed universe. It often proves convenient
to change a coordinate system dubbed comoving which
which clearly implies that the velocity field is a linear expands with expanding universe. Matter which follows
function of its argument r, the Hubble flow will be at rest in the comoving coordi-
nate system, i.e. matter filling a homogeneous isotropic
v(r, t) = T (t)r (4) universe is at rest with respect to the comoving observer.
Both the frames are physically equivalent. Let us clarify
where T is a 3 × 3 matrix. The matrix can always be that universe does not appear homogeneous and isotropic
diagonalized by choosing a suitable coordinate system. to any observer; for instance if an observer is moving with
Isotropy then reduces it to kronecker symbol (Ti,j = a large velocity say towards a particular galaxy, universe
H(t)δi,j ) leading to looks different to him/her. A physical coordinate system
is the system in which matter is at rest at the origin and
v(r, t) = H(t)r (5) moves away radially at other points. The radius vector
r of any point in this system called physical, changes
where H is known as the Hubble parameter. In gen- with time whereas its counterpart x in the comoving sys-
eral, a velocity field can always be decomposed into ro- tem is constant. This means that physical distance be-
tational part, inhomogeneous part and isotropic part at tween any two points in the expanding universe is given
each point. It is not then surprising that the homoge- by the comoving distance multiplied by a factor which de-
neous and isotropic velocity field has the form (5) known pends upon time which is precisely expressed by Eq.(7)
as Hubble law. or equivalently by Eq.(9).
 4

continuity equation to have the usual form,

∂ρb (t)
+ 3Hρb = 0 (14)
∂t
which formally integrates to,
 a 3
(0) 0
ρb (t) = ρb (15)
a

FIG. 1: Particle of mass m on the surface of a sphere of radius where the subscript ’0’ denotes the quantities at the
r(t) in an expanding universe with uniform matter density. present epoch. The evolution of matter density of non-
relativistic fluid has a simple meaning that mass of fluid
in a co-moving volume remains constant.
B. Evolution equations Though the Eq.(12) formally resembles the evolution
equation of relativistic cosmology, its derivation pre-
We now turn to the evolution equation for the scale sented above is defective. The expression for the potential
factor. Thank to isotropy, we can employ spherical sym- energy is written with an assumption that gravitational
metry to derive the evolution equation. At a given time potential can be chosen zero at infinity which is not true
t called the cosmic time, let us consider a sphere cen- in an infinite universe. Since the mass density ρb is con-
tered at O with radius r(t). Let ρb (t) be the density stant in space, the total mass of universe diverges as r3 .
of matter in the homogeneous isotropic space referred to As a result, the potential −4πGρb r2 /3 can not be nor-
as background space hereafter. We assume that the net malized to zero at r = ∞. One could try to circumvent
gravitational force on a particle of mass m situated on the the problem by assuming that ρb vanishes for a given
surface of the sphere due to matter out side the sphere large value of r but it would conflict with the underlying
is zero which means that matter inside the sphere alone assumption of homogeneity. Therefore, conservation of
can influence the motion of the particle. The total energy energy is difficult to understand in an infinite universe
of the particle on the surface of the sphere (see Fig.1) at with uniform matter density.
any time is constant given by the expression[29], We can also derive the evolution equations using the
Newtonian force law[18]. The force on the unite mass sit-
uated on the surface of homogeneous sphere with radius
1 2 4π r is given by
ET ot = mṙ − mGρb r2 (10)
2 3
4πG
This equation can be cast in the following convenient F=− ρr (16)
form, 3
 2 The Euler’s equation
ṙ(t) 8π 2ET ot
H2 ≡ = Gρb (t) + (11)
r(t) 3 mr2 (t) ∂v ∇Pb
+ (v.∇)v = − +F (17)
∂t ρb
which readily translates into an evolution equation for
a(t) (see, Eq.(9)) known as Friedmann equation, in a homogeneous isotropic background simplifies to
 2 F = (Ḣ + H 2 )r (18)
2 ȧ 8π K 2ET ot
H ≡ = Gρb (t) − 2 , K=− 2 (12)
a 3 a x m where F is the force per unit mass on the fluid element
where K can be zero, negative or positive depending how given by Eq.(16). We have used the fact that pressure
kinetic energy compares with the potential energy. gradients are absent in a homogeneous isotropic back-
In order to solve the evolution equation for a(t), we ground and the velocity field is given by the Hubble
need to know how matter density ρb (t) changes with time, law. It should also be noted that the pressure Pb = 0
i.e., we need the conservation equation in the expanding for the non-relativistic background fluid under consid-
universe. For non-relativistic fluid, the continuity equa- eration. Using expressions (16) & (18), we obtain the
tion that gives us the evolution of matter density of the equation for acceleration,
fluid is, 1 d2 a 4πG
2
=− ρb (t) (19)
∂ρb (t) a dt 3
+ (∇.ρb v) = 0 (13)
∂t which could also be obtained directly from Eq.(16).
Remembering that the matter density of the background Equation (19) can easily be integrated to give the Fried-
fluid is independent of the coordinates and the fluid ve- mann equation. Indeed, by multiplying the above equa-
locity is given by the Hubble law (6), we transform the tion by ȧ and using the evolution of mass density allows
 5

us to write which is permitted by the Friedmann equation (22) but

8πG K not allowed by the equation for acceleration (24). It is
H2 = ρb (t) − 2 (20) really remarkable that Newtonian cosmology gives rise to
3
 a  an evolving universe. It is an irony that the discovery of
8πGρ0 expansion of universe had to wait the general theory of
K ≡ a20 − H02 (21)
3 relativity. This is related to the commonly held percep-
tions of static universe which was prevalent before Fried-
The above derivation is also problematic as it assumes
mann discovered the non-static cosmological solution of
that mass out side the sphere, used while writing Eq.(16),
Einstein equations. So much so that Einstein himself did
can be neglected which is not true for infinite universe
not believe in the Friedmann solution in the beginning
with constant mass density.
and tried to reconcile his theory with static universe by
The problem can be circumvented by using the geo-
introducing cosmological constant which he later with-
metric reformulation of Newtonian gravity in the lan-
drew.
guage of Cartan. According to Cartan’s formulation, or-
bits of particles are assumed to be the geodesics of an
affine space and gravity is then described by the curva- C. The past, the future and how old are we?
ture of the affine connection(see Ref.[30] and references
therein). According to Ref.[30], no pathology in cosmol-
ogy associated with Newton’s force law then occurs and The general features of solutions of evolution equations
the evolution equations of Newtonian cosmology, can be understood without actually solving them. What
can we say about the past and the fate of universe? The
8πG K equation for acceleration tells us that ä < 0 for standard
H2 = ρb (t) − 2 (22)
3 a  form of matter. This means that a(t) as a function of
8πGρ0b time is concave downward. We need input regarding ȧ

2
K ≡ a0 − H02 (23) at present to make important conclusion about the past.
3
Observation tells us that ȧ(t) > 0 at present. Thus a(t)
1 d2 a 4πG monotonously decreases as t runs backward. It is there-
=− ρb (t) (24)
a dt2 3 fore clear that there was an epoch in the history of uni-
∂ρb (t) verse when a(t) vanishes identically. Without the loss of
+ 3Hρb (t) = 0 (25)
∂t generality we can take t = 0 corresponding to a(t) = 0.
As for the fate of universe, the problem is similar to
can be put on rigorous foundations. Eqs.(22), (24), &
that of escape velocity, namely, if K > 0, the kinetic en-
(25) are identical to evolution equations of Friedmann
ergy is less than the potential energy. In this case a(t)
cosmology for non-relativistic fluid filling the universe.
would increase to a maximum value where ȧ(t) = 0, it
Whether or not one adopts the formulation presented
would start decreasing thereafter till it vanishes and uni-
in Ref.[30], Newtonian cosmology is nevertheless elegant
verse ends itself in big crunch. In case, K < 0, scale fac-
and simple.
tor would go on increasing for ever; K = 0 represents the
Let us point out an important feature of Newtonian
critical case. Three different possibilities, K = 0, K > 0
cosmology. We note that the expression of K/a2 remains
or K < 0 correspond to critical, closed and open universe
unchanged under the scale transformation a(t) → Ca(t),
respectively. We should emphasize that the fate of uni-
C being constant. As a result, the evolution equations
verse also crucially depends upon the nature of matter
(24) & (22) also respect the scale invariance. This in-
filling the universe. In some case, the universe may end
variance is a characteristic of specially flat Friedmann
itself in a singular state or the cosmic doomsday.
cosmology. The Newtonian cosmology can mimic all
Which of the three possibilities is realized in nature?
three topologies of relativistic cosmology corresponding
To answer this question, let us rewrite Eq.(12) in a con-
to K = 0, ±1 in spite of the fact that the underlying ge-
venient form,
ometry in Newtonian cosmology is Euclidean. Let us note
that the scale factor in Newtonian cosmology can always K ρb (t)
Ωb (t) − 1 = , Ωb (t) = (26)
be normalized to a convenient value at the present epoch. (aH)2 ρc (t)
This is related to a simple fact that the Friedmann equa-
tion (22) does not change if we re-scale the scale factor where the critical density is defined as, ρc (t) =
which leaves the normalization of a arbitrary. The often 3H 2 (t)/8πG. Specializing the expression (26) to the
used normalization fixes the scale factor a(t) = 1 at the present epoch, we find that,
present epoch, i.e, a0 = 1. In case of relativistic cosmol- (0) (0)
Ωb > 1 (ρb > ρ(0)
c ) ⇒ K > 0 → closed universe,
ogy, the latter can only be done in case of K = 0 whereas
(0) (0)
in case of K = ±1, the numerical value of the scale factor Ωb = 1 (ρb = ρ(0)
c ) ⇒ K = 0 → critical universe,
a0 depends upon the matter content of universe. (0) (0)
Ωb < 1 (ρb < ρ(0)
c ) ⇒ K < 0 → open universe.
The second important feature of Newtonian cosmol-
ogy is that it leads to an evolving universe. Indeed, we where the super script ′ 0′ designates the correspond-
could ask for a static solution given by ȧ(t) and ä(t) = 0 ing physical quantities at the present epoch. Since we
 6

(0)
know the observed value of ρc , one of the three types where Λ is known as cosmological constant which is pos-
of universe we live in, depends upon how matter density itive in the present context. It is interesting to note
(0) that there are only two central forces namely, the inverse
in universe compares with ρc . Observations on Cosmic
Microwave background (CMB) indicate that universe is square force and the linear force which give rise to stable
critical to a good accuracy or K ≃ 0 which is consistent circular orbits.
with inflationary paradigm. Our discussion of cosmological constant is heuristic
Let us come to the solution of Newtonian cosmology and the cheap motivation here is to incorporate the re-
in case of K = 0. Substituting ρb (t) from Eq.(15) in pulsive effect in the evolution equations. We rewrite the
Eq.(22), we find that, ȧ2 ∼ a−1 which easily integrates modified force law (32) as an equation of acceleration
giving rise to using the comoving coordinates,
 2/3
t 1 d2 a 4πG Λ
a(t) = (27) 2
=− ρb (t) + (33)
t0 a dt 3 3
 2 which shows that a positive Λ term contributes to accel-
(0) t0
ρb (t) = ρb (28) eration as it should. The integrated form of Eq.(33) is
t given by,
21
H(t) = (29) 8πG K Λ
3t H2 = ρb (t) − 2 + (34)
3 a 3
The above solution is known as Einstein-de-Sitter solu-
tion. We can estimate the age of universe using Eq.(29), where the integration constant K can be formally written
again through physical quantities defined at the present
2 1 epoch. The modified force law (32) was proposed much
t0 = (30)
3 H0 before Einstein’s general theory of relativity by Neumann
Interestingly, if gravity were absent, universe would ex- and Seeliger in 1895-96[31, 32].
pand with constant rate given by H0 . Using Hubble law Let us note that adding cosmological constant to New-
we would then find, tonian force is equivalent to adding a constant matter
density ρΛ = Λ/8πG to the background matter density
1 ρb which does not to go well with the continuity equation
t0 = (31)
H0 (14). Since the acceleration equation also gets modified
which is the maximum limit for the age of universe in the in presence of Λ, we should check whether the modified
hot big bang model (2H0−1 /3 ≤ t0 < H −1 ). The presence evolution equations allow this possibility. If we differenti-
of standard matter always leads to deceleration thereby ate Eq.(34) with respect to time and respect the modified
leading to smaller time taken to reach the present Hubble acceleration equation, we find that constant matter den-
rate of expansion. The presence of cosmological constant sity is permissible in the expanding universe. As for the
or any other exotic form of matter can crucially alter this continuity Eq.(14), it is valid for a perfect non-relativistic
conclusion. fluid. The cosmological constant does not belong to this
category, the pressure corresponding to constant energy
density is not zero. The continuity equation should take
D. Cosmological constant a la Hooke’s law the note of pressure and get appropriately modified. As
pointed out earlier our present discussion of cosmological
We have seen that Newtonian cosmology gives rise to constant here is qualitative. Rigorously speaking, we are
evolving universe but for the historical reasons, cosmol- trying to get the right thing in the wrong place! We shall
ogy had to wait the general theory of relativity to dis- come back to this point after we incorporate the pressure
cover it. The fact that Newtonian cosmology leads to corrections in the evolution equations.
non-stationary solution was known before general theory Evolution equations (34) and (33) admit a static solu-
was discovered but it could receive attention as it con- tion (a = const = a0 ) in case of K > 0. Static Einstein
flicted with perception of static universe. Attempts were universe ( ȧ = 0 and ä = 0) is possible provided that Λ
then made to modify Newtonian gravity to reconcile it has definite numerical value
with the static universe. Clearly, the modification should Λ = Λc = 4πGρb
(0)
(35)
be such that it becomes effective at large scales leaving
local physics unchanged. Looking at the Newton’s force We shall observe after a short while that the static Ein-
law (16), it is not difficult to guess that static solution stein universe is unstable under small fluctuations.
is possible provided that we add a repulsive part propor- The qualitative features of solutions of evolution equa-
tional to the radius vector r in Eq.(16). Newton’s law of tions can be understood without actually solving them.
gravitation should therefore be supplemented by linear Eq.(33) can be thought as an equation of a point particle
force law[18, 22, 31, 32] in one dimension[4, 33],
4πG 1 ∂V
F=− ρb r + Λr (32) ä = − (36)
3 3 ∂a
 7

moving in potential field

4πGρb a2 Λa2
 
V (a) = − + , (37) V(a)
3 6

where we have used the fact that ρb ∼ a−3 . The Hub-

ble equation acquires the form of the total energy of the
mechanical particle
o a
C
ȧ2
E= + V (a) (38)
2
where E = −K/2. In order to make the mechanical
analogy transparent, let us compute the minimum of the
kinetic energy. If the minimum exists, it should obviously A B
correspond to the numerical value of the scale factor that
gives rise to the maximum of the effective potential V (a).
It is easy to see that the kinetic energy is minimum if
a = am , FIG. 2: Plot of the effective potential V (a) versus the scale
factor a. Configurations (A) & (B) correspond to motion of
am = (A/Λ)1/3 (39) system beginning from a = 0 and a = ∞ respectively. (C)
corresponds to static solution unstable under small fluctua-
 2
ȧ 1  2/3 1/3 
= A Λ −K , (40) tions.
2 m 2
(0)
where A = 4πGρb a30 . Note that V (a) is maximum at • (b) Bouncing universe: If the potential barrier is
a = am . From Eq.(40), we infer that the kinetic energy approached from the right side with a = ∞, the scale
of the system at the top of the potential is, factor first decreases and reaches a minimum value and
 2 then bounces to expanding phase as the kinetic energy
ȧ K3
≥ 0 if Λ ≥ Λc ≡ 2 (41) is not enough to overcome the barrier.
2 m A • (c) Einstein static universe : This configuration
In case Λ = Λc , the system barely makes to the hump of corresponds to the maximum of the potential with
the potential (ȧ = 0) corresponding to am = a0 where ȧ = 0 and ä = 0, possible for a particular value of Λ,
ä = 0 as it should be (see Eq.(36) which is nothing obtained earlier. Clearly, static universe corresponding
but the Einstein’s static solution. We are now ready to point particle sitting on the hump of the potential,
to provide the qualitative description of solutions of is not stable. Small perturbations would derive it to ei-
evolution equations. For Λ < Λc , the kinetic energy ther contracting (a → 0) or expanding (a → ∞) universe.
is formally negative for a = am which means that it
vanishes before the particle reaches the maximum of 2. Λ > Λc : The kinetic energy is sufficient to overcome
the potential. In Fig.2 we have displayed the plot of the barrier for this choice of Λ. As a result, motion first
V (a) versus the scale factor a. We show three possible decelerates till the system reaches the top of the potential
configuration of interest: (A) Corresponds to motion and then slides down the hill with acceleration. Scale fac-
starting from the left of the barrier with a = 0. (B) tor exhibits the point of inflection at a(t) = am < a0 . If Λ
Depicts the situation in which the potential barrier is slightly exceeds its critical value, an interesting possibil-
approached beginning from the right with a large value ity dubbed loitering universe can be realized. The scale
of the scale factor. (C) Represents the possibility of first increases as it should, approaches a0 and remains
static solution. nearly frozen for a substantial period before entering the
phase of acceleration. Such a scenario has important im-
We first analyse the case of K > 0 or E < 0 which plications for structure formation.
gives rise to a variety of interesting possibilities. For K ≤ 0 or E ≥ 0, the system always has enough
1. Λ < Λc : In this case, the kinetic energy is insufficient kinetic energy to surmount the barrier allowing the scale
to overcome the potential barrier giving rise to the factor to increase from a = 0 to large values as time
following interesting solutions. increases. This case is similar to the one with K > 0 and
• (a) Oscillating solution: In this case, motion starts Λ > Λc .
from a = 0 with insufficient kinetic energy to reach For any given value of Λ, the scale factor exhibits the
(0)
the hump of the potential. In this situation, the scale point of inflection at a = am = (4πGρb a30 /Λ)1/3 . This
factor increases up to a maximum value where ȧ = 0 also clear from Eq.(32) & (33) in which the first term is
for a < am marking the turning point followed by the of attractive character and dominates in the beginning
contraction to a = 0. leading to deceleration. However, as the scale factor in-
 8

creases and reaches a particular value, the repulsive term We can now present cosmological constant as a per-
takes over; the scale factor exhibits the point of inflection fect fluid with constant energy density. The continuity
and the expansion becomes accelerating thereafter. Eq.(44) then implies that ρΛ = −PΛ . Next, we claim
Observations should tell us when deceleration changed that the correct equation of acceleration in case of back-
into acceleration. This crucially depends upon how ground fluid with energy density ρb and pressure Pb is
(0) given by
4πGρb compares with Λ or how ρΛ compares with
(0)
ρM /2. The transition from deceleration to acceleration ä 4πG Λ
should have taken place around the present epoch. Had =− (ρb + 3Pb ) + (45)
a 3 3
it happened much earlier it would have obstructed struc-
To verify, let us multiply Eq.(45) left right by ȧ
ture formation[34]. We shall come back to this point to
confirm that cosmic acceleration is indeed a recent phe- 1 d 2
4πG Λ
ȧ = − a (ρb ȧ + 3Pb ȧ) + aȧ (46)
nomenon. 2 dt 3 3
Using the continuity equation, we can express the term
containing pressure Pb in Eq.(46) through ρb , ρ˙b and ȧ
IV. BEYOND NEWTONIAN PHYSICS:
PRESSURE CORRECTIONS
 
1 d 2
4πG d 2 Λ 2
ȧ = ρb a + a (47)
2 dt 3 dt 6
The formalism of Newtonian cosmology is not appli-
which can put in form of Friedmann equation in the pres-
cable to relativistic fluids. Relativistic fluids essentially
ence of matter with non-zero pressure.
have non-zero pressure. For instance, radiation is a rel-
ativistic fluid with pressure Pb = ρb c2 /3. The cosmolog- 8πG K Λ
H2 = ρb (t) − 2 + (48)
ical constant also belongs to the category of relativistic 3 a 3
systems. In general theory of relativity, pressure appears We again observe that pressure corrects the energy den-
on the same footing as energy density. Here we present sity. Positive pressure adds to deceleration where as
heuristic arguments to capture the pressure corrections the negative pressure contributes towards acceleration,
in the evolution equations (see Ref.[22]). see Eq.(45). It looks completely opposite to our intu-
Let us consider a unit comoving volume in the expand- ition that highly compressed substance explodes out with
ing universe and assume the expansion to be adiabatic. tremendous impact whereas in our case pressure acts in
The first law of thermodynamics then tells that the opposite direction. It is important to understand
that our day today intuition with pressure is related to
dE + Pb dV = 0 (42)
pressure force or pressure gradient. In a homogeneous
where Pb (t) is the pressure of background fluid. The universe pressure gradients can not exist. Pressure is a
first law of thermodynamics applies to any system, be relativistic effect and can only be understood within the
it relativistic or non-relativistic, classical or quantum − frame work of general theory of relativity. Pressure gra-
thermodynamics is a great science. dient might appear in Newtonian frame work in the in-
The energy density of the fluid can always be expressed homogeneous universe but pressure can only be induced
through the mass density, by relativistic effects. Strictly speaking, it should not
appear in Newtonian cosmology. This applies to Λ also
4π 3 2 with negative pressure which we introduced in Newto-
E= a ρb c (43) nian cosmology by hand. Eqs.(44),(45) & (48) coincide
3
with the evolution equations of relativistic cosmology.
Substituting (43) into (42), we obtain the continuity Their derivation presented here is heuristic. The rigorous
equation in the expanding universe, treatment can only be given in the framework of general
  theory of relativity where cosmological constant appears
Pb naturally.
ρ˙b + 3H ρb + 2 = 0 (44)
c In order solve the evolution equations, we need a rela-
tion between the energy density and pressure known as
Thus the continuity equation responds to pressure cor- equation of state. In case of barotropic fluid the equa-
rections: ρb → ρb + Pb /c2 . For a non-relativistic fluid, tion of state is given by wb = Pb /ρb . Dust and radiation
rest energy density dominates over pressure and the sec- correspond to wb = 0, 1/3 respectively. Assuming that
ond term in the parenthesis can be neglected. For in- universe is filled with perfect fluid with constant equa-
stance, for dust, Pb ≃ 0. At early times, universe was tion of state parameter wb , we find from Eqs.(44) & (34)
hot and was dominated by radiation. Hence the early in case of K = 0,
universe should be treated by relativistic theory; Newto-
nian description becomes valid at late times when matter ρb ∝ a−3(1+w) (49)
dominates. For the sake of convenience, we shall use the a(t) ∝ t
2
3(1+w) , (w > −1) (50)
unit c = 1. With this choice, relativistic mass density √Λ
t
and energy density are same. a(t) ∝ e 3 (w = −1) (51)
 9

In case of radiation, wb = 1/3 and as a result ρb ≡ ρr ∝ gravity is attractive (provided universe is filled with mat-
a−4 . In contrast to the case of dust dominated universe, ter of non-negative pressure), its roll is to decelerate the
the radiation energy density decreases faster with the expansion. What caused big bang, has no satisfactory
expansion of universe. The positive radiation pressure answer. The big bang is a physical singularity which
adds to energy density making the gravitational attrac- should be treated by quantum gravity. The inflationary
tion stronger. Consequently, the Hubble damping in the paradigm can mimic big bang without singularity but
conservation equation increases allowing the energy den- in that case, we do not know what caused inflation! In
sity decrease faster than dust in expanding universe. This the cosmic history, there was an epoch when matter took
can also be understood in a slightly different way, if we over leading to matter dominate era. It turns out that
assume that radiation consists of photons. As universe it took around 105 years for radiation energy density to
expands, the number density of photons scales as a−3 as equalize with energy density of matter. The age of uni-
usual. But since any length scale in the expanding uni- verse, i.e., the time elapsed since the big bang till the
verse grows proportional to the scale factor, the energy present epoch given by (30) changes insignificantly, if we
of a photon, hc/λ decreases as 1/a leading to ρr ∼ a−4 consider the universe filled with radiation and dust both.
and a(t) ∝ t1/2 . It is clear that radiation dominated at This is because the time taken from the big bang till radi-
early epochs as ρM ∼ a−3 for dust. ation matter equality is negligibly small as compared to
Let us make an important remark on the dynamics the actual age of universe which is around 14 Gyr. Thus
in the early universe which was dominated by radiation the age given by (30) is a reliable theoretical estimate.
(for simplicity, we ignore here other relativistic degrees Unfortunately, the age given by Eq.(30) falls short than
of freedom). As ρr ∼ a−4 , the first term on the RHS of the age of some very old objects found in the universe.
evolution of Hubble equation dominates over the curva- This is one of the old problems of hot big bang model.
ture term K/a2 ; obviously, cosmological constant plays We shall discuss its possible remedy in the dark energy
no role in the present case. We therefore conclude that dominated universe.
all the models effectively behave as K = 0 model at early
times,
A. Dark energy
!1/4
(0)
ȧ2 8πG (0) a40 a(t) 32πGρr
2
= ρr 4 → = t1/2 (52) Eqs.(48) and (46) tell us that the positive cosmologi-
a 3 a a0 3
cal constant Λ contributes positively to the background
energy density and negatively to pressure. It can be
We next assume that radiation was in thermal equilib-
thought as a perfect barotropic fluid with,
rium characterized by the black body distribution,
Λ Λ
ρr = bT 4 , (53) ρΛ = , PΛ = − (56)
8πG 8πG
where b is the radiation constant. From Eqs.(52) & (53), which corresponds to wΛ = −1. In general, we find from
we find how temperature scales with the expansion of Eq.(45) that expansion has the character of acceleration
universe, for large negative pressure,
1/4 ä 4πG
ρ0r =− (ρb + 3Pb ) (57)

a0
T = (54) a 3
b a ρb
ä > 0 ⇒ Pb < − : Dark energy.
3
which on using Eq.(52) tells us how early universe cooled
with time, where we have included Λ in the background fluid. Thus
we need an exotic fluid dubbed dark energy to fuel the ac-

32πG
1/4 celerated expansion of universe. The various data sets of
T = t−1/2 (55) complimentary support the late time acceleration of uni-
3b
verse. The simplest candidate of dark energy is provided
At t = 0, both the radiation density and temperature be- by the cosmological constant with wΛ = −1 Observations
come infinitely large; all the physical quantities diverged at present do not rule out the phantom dark energy with
at that time referred to as big bang. The big bang singu- w < −1 corresponding to super acceleration. In this case
larity is not the artifact of homogeneity and isotropy. It is the expanding solution takes the form,
a generic feature of any cosmological model based upon a(t) = (ts − t)n , (n = 2/3(1 + w) < 0) (58)
classical general theory of relativity. Classical physics n
breaks down as big bang is approached. In the frame- H= (59)
ts − t
work of classical general relativity, the big bang is taken
to be the beginning of our universe. Universe was thus where ts is an integration constant. It is easy to see that
born in a violent explosion like event throwing away cos- phantom dominated universe will end itself in a singu-
mic matter and giving rise to expansion of universe. Since larity, in future known as big rip or cosmic doomsday as
 10

t → ts . Clearly, as t → ts , both the Hubble parameter

(Age of the universe) x (H0/72) (Gyr)

and the background energy density diverge[36]. 18

-1.0 -2.0
B. Age crisis and its possible resolution 16

Apart from the cosmic acceleration, dark energy has -0.75 Globular clusters
14
important implications, in particular, in relation to the
age problem. In any cosmological model with normal w=-0.5
form of matter, the age of universe falls short compared 12
to the age of some known objects in the universe. Since
the age of universe crucially depends upon the expansion
history, it can serve as an important check on the model 10
0 0.1 0.2 0.3 0.4 0.5 0.6
building in cosmology. In order to appreciate the prob- ΩM = 1 - ΩDE
lem, let us first consider the case of flat dust dominated
Universe (ΩM = 1) in which case as shown earlier, FIG. 3: Plot of age of Universe versus ΩM (at present epoch)
2 1 for a flat universe with matter and dark energy with constant
t0 = (60) equation of state parameter w, from Ref.[10]
3 H0
The observational uncertainty of H0 gives rise to the fol-
lowing estimate, matter dominated era and we, therefore, have omitted Ωr
in Eq.(63). In case dark energy is cosmological constant
H0−1 = 9.8h−1 Gyr (61) (wΛ = −1), we get the analytical expression for the age
0.64 < <
∼ h ∼ 0.8 → t0 = (8 − 10)Gyr (62) of Universe,
 
This model is certainly in trouble as its prediction for −1
2 H0
1/2
1 + ΩΛ 
age of Universe fails to meet the constrain following t0 = ln  (65)
3 Ω1/2 (0) 1/2
from the study of ages of old stars in globular clusters: Λ ΩM
12Gyr < <
∼ t0 ∼ 15Gyr[35]. One could try to address the
(0)
problem by invoking the open model with ΩM < 1. In For dark energy other than the cosmological constant,
this case the age of universe is expected to be larger the integral in Eq.(64) should be computed numerically.
than the flat dust dominated Universe − for less amount In Fig.3, we have plotted the age of universe versus the
(0)
of matter, it would take longer for gravitational attrac- ΩM for various possibilities of dark energy including the
tion to slow down the expansion rate to its present value. phantom one. The age constraint can be met by flat dark
(0)
Looking at Eq.(22), it is not difficult to guess that in energy models provided that −2 < <
∼ w ∼ −0.5 for ΩM
(0) lying between 0.2 and 0.3, see Ref.[10]. It is remarkable
this case, H0 t0 → 1 for ΩM → 0 which is a substan-
tial improvement. However, this model is not viable for that hot big bang model can be rescued by introducing
several reasons. In particular, the study of large scale the dark energy component. Interestingly, cosmological
structure and its dynamics constrain the matter density: constant was invoked to address the age problem before
(0)
0.2 < ΩM < 0.3 and observations on CMB un-isotropy the invention of cosmic acceleration. The observation of
reveal that universe is critical to a good accuracy. cosmic acceleration in 1998 was a blessing in disguise for
The age problem can be resolved in a flat universe cosmological constant.
dominated by dark energy. Let us rewrite the Friedmann
equation in a convenient form,
C. The discovery of cosmic acceleration and its
 2   3  a 3(1+w)  confirmation
ȧ (0) a0 (0) 0
= H02 ΩM + ΩDE (63)
a a a
The direct evidence of current acceleration of universe
which allows us to write the expression of t0 in the closed is related to the observation of luminosity distance by
form high redshift supernovae by two groups independently in
1
Z ∞
dz 1998. The luminosity distance for critical universe domi-
t0 = i1/2 nated by non-relativistic fluid and cosmological constant
H0 0 h
(0) (0) is given by
(1 + z) ΩM (1 + z)3 + ΩDE (1 + z)3(1+w)
(64) (1 + z) z
Z
dz ′
(0) dL =
where ΩM is the contribution of dark matter and (1 + H0
q
(0) 3 (0)
0 ΩM (1 + z ′ ) + ΩDE (1 + z ′ )3(1+w)
z) ≡ a0 /a, z being the redshift parameter. The domi-
nant contribution to the age of universe comes from the (66)
 11

Eq.(66) is the expanding universe generalization of abso-

(i)
lute luminosity Ls of a source and its flux F at a distance
d given by F = Ls /(4πd2 ). It follows from Eq.(66) that (ii)
DL ≃ z/H0 for small z and that
(iii)
 
(0)
dL = 2 1 + z − (1 + z)1/2 H0−1 , ΩM = 1 (67)
(0)
dL = z(1 + z)H0−1 , ΩDE = ΩΛ = 1 (68)

which means that luminosity distance at high redshift

is larger in universe dominated by cosmological constant
which also holds true in general for an arbitrary equa- (i)
tion of state w corresponding to dark energy. Therefore
supernovae would appear fainter in case the universe is (ii)
dominated by dark energy. The luminosity distance can (iii)
be used to estimate the apparent magnitude m of the
source given its absolute magnitude M
 
dL
m − M = 5 log + 25 (69)
M pc

Let us consider two supernovae 1997ap at redshift z = FIG. 4: Plot of the luminosity distance H0 dL versus the red-
0.83 with m = 24.3 and 1992p at z = 0.026 with M = shift z for a flat cosmological model. The black points come
16.08 respectively. Since the supernovae are assumed to from the “Gold” data sets by Riess et al. [38], whereas the
be the standard candles, they have the same absolute red points show the recent data from HST. Three curves show
(0)
magnitude. Eq.(69) then gives the following estimate the theoretical values of H0 dL for (i) ΩM = 0, ΩΛ = 1, (ii)
(0) (0)
ΩM = 0.31, ΩΛ = 0.69 and (iii) ΩM = 1, ΩΛ = 0. From
H0 dL ≃ 1.16 (70) Ref. [39].

Then theoretical estimate for the luminosity distance is

given by yet another independent probe of dark energy. The com-
bined analysis of data of complimentary nature demon-
(0) (0) (0)
dL ≃ 0.95H0−1, ΩM = 1 (71) strate that ΩDE ≃ 0.7 and ΩM ≃ 0.3, see Fig.5. The
dL ≃ 1.23H0−1,
(0)
ΩM = 0.3, ΩΛ = 0.7 (72) constraint on the equation of state parameter w and
(0)
ΩM shows that w is restricted to a narrow strip around
where we have used the fact that, dL ≃ z/H0 for small wΛ = −1 (Fig.6). It is clear from the figure that the
z. The above estimate lands a strong support to the combined analysis allows super-negative values of w cor-
hypothesis that late time universe is dominated by dark responding to phantom energy. Let us now confirm that
energy (see, Fig4). the transition from deceleration to cosmic acceleration
The observations related to CMB and large scale struc- took place in the recent past. Indeed, observations allow
ture (LSS) provide an independent confirmation of dark to estimate the time of transition from deceleration to ac-
energy scenario. The acoustic peaks of angular power celeration. Let us rewrite Eq.(38) through dimensionless
spectrum of CMB temperature anisotropies contains im- density parameters,
portant information. The location of the major peak tells !
(0)
us that universe is critical to a good accuracy which fixes ȧ2 H02 ΩM a30 2
= + ΩΛ a (74)
for us the cosmic energy budget. Specializing the Fried- 2 2 a
mann Eq.(63) to the present epoch (a = a0 ), we have
(0) (0) (0)
Using Eq(74), we can find out the numerical value of
Ωb = ΩM + ΩDE (73) (a/a0 ) corresponding to the minimum of kinetic energy
(ȧ2 /2) which precisely gives the transition from deceler-
The contribution of radiation to total fractional energy ation to acceleration,
(0)
density Ωb is negligible at present. The study of large
!1/3 !1/3
scale structure and its evolution indicate that nearly 
a
 (0)
ΩM 2ΩΛ
30% of the total energy content is contributed by non- = ⇒ ztr = (0)
− 1 ≃ 0.67
a0 2ΩΛ ΩM
luminous component of non-barionic nature with dust tr
like equation of state popularly known as dark matter. (75)
(0)
The missing component which is about 70% is dark en- for the observed values of density parameters (ΩM ≃ 0.3;
ergy. The recent data on baryon acoustic oscillation is ΩΛ ≃ 0.7) and this confirms that the contribution of
 12

2.0
Pressure in cosmology is a relativistic effect which can
No Big Bang
be consistently understood in the frame work of general
theory of relativity. Einstein equations are complicated
1.5
non-linear equations which do admit analytical solutions
in presence of symmetries. Homogeneity and isotropy
of universe is an example of a generic symmetry of space
time. The assumption of homogeneity and isotropy forces
ΩΛ

1.0
the metric to assume the FRW form
SNe

dr2
 
2 2 2 2 2 2 2
ds = −dt + a (t) + r (dθ + sin θdφ )
1 − Kr2
0.5 K = 0, ±1 (76)
CM
B
where a(t) is scale factor. Coordinates (r,θ, φ) are the
BAO comoving coordinates. A freely moving particle comes to
F
la
t
0.0 rest in these coordinates.
0.0 0.5 1.0
Ωm
Eq.(76) is purely a kinemetic statement. The infor-
mation about dynamics is contained in the scale factor
FIG. 5: Figure shows the best fit regions in the (ΩΛ , ΩM ) a(t). Einstein equations allow to determine the scale fac-
plane obtained using the CMB, BAO and supernovae data, tor provided the matter contents of universe is specified.
from Ref.[37] Constant K in the metric (76) describes the geometry of
the spatial section of space time. K = 0, ±1 corresponds
to spatially flat, sphere like and hyperbolic geometry re-
spectively.
0.0
The differential equation for the scale factor follows
from Einstein equations
BAO
1
Gµν = Rµν − gµν R = 8πGTµν (77)
2
-0.5 B
CM where Gµν is the Einstein tensor, Rµν is the Ricci tensor.
The energy momentum tensor Tµν takes a simple form
w reminiscent of ideal perfect fluid in FRW cosmology

Tµν = Diag(−ρb , Pb , Pb , Pb ) (78)

-1.0
Note that pressure in general theory of relativity appears
on the same footing as energy density. In the FRW back-
SNe ground, the components of Gµν can easily be computed

3 1
-1.5 G00 = − ȧ2 + K , Gji = 2 2aä + ȧ2 + K



(79)
0.0 0.1 0.2 0.3 0.4 0.5 a2 a

ΩM Other components of Gµν are identically zero. Einstein

equations then give rise to the following two independent
equations
FIG. 6: Constraints on the dark energy equation of w and
ΩM obtained from CMB, BAO and supernovae observations,
8πG K
from Ref.[37] H2 = ρb − 2 (80)
3 a
ä 4πG
=− (ρb + 3Pb ) (81)
Λ to cosmic dynamics became important at late times a 3
such that the cosmic acceleration is indeed a recent phe- We remind that ρb designates the total energy density
nomenon. of all the fluid components present in the universe. The
continuity equation ρ˙b +3H(ρb +Pb ) = 0 can be obtained
by using Eqs.(80) & (81) which also follows naturally
V. RELATIVISTIC COSMOLOGY from the Bianchi identity. As mentioned earlier, we can
normalize the scale factor to a convenient value at the
In the last section we presented heuristic arguments to present epoch in case of specially flat geometry. In other
capture the pressure effects in the evolution equations. cases, it should be determined from the relation a0 H0 =
 13

(0)
|Ω0b − 1| where Ωb defines the total energy content of


this procedure of throwing out the vacuum energy is ad-
universe at the present epoch. hoc, one might try to cancel it by introducing the counter
Let us note that the Einstein equations (77) with the terms. The later, however requires fine tuning and may
energy momentum tensor of standard fluid with positive be regarded as unsatisfactory. The divergence is related
pressure can not lead to accelerated expansion. The re- to the modes of very small wavelength. As we are ig-
pulsive effect can be captured either by supplementing norant of physics around the Planck scale, we might be
the energy momentum tensor (on right hand side of Ein- tempted to introduce a cut off around the Planck length
stein equations) with large negative pressure or by mod- Lp and associate with this a fundamental scale. Thus
ifying the geometry itself, i.,e. the left hand side of Ein- we arrive at an estimate of vacuum energy ρvac ∼ Mp4
stein equations. We can ask for a consistent modification 1/4
(corresponding mass scale- Mvac ∼ ρvac ) which is away
of Einstein equations (equation of motion should be of by 120 orders of magnitudes from the observed value of
second order with the highest derivative occurring lin- this quantity which is of the order of 10−48 (GeV )4 . The
early so that the Cauchy problem is well posed) in four vacuum energy may not be felt in the laboratory but
space time dimensions within the classical frame work. plays important role in GR through its contribution to
Under the said conditions, the only admissible modifica- the energy momentum tensor as
tion is provided by the cosmological constant. Thus we
can add a term Λgµν on the left hand side of Eq.(77) < Tµν >0 = −ρvac gµν , ρvac = Λ/8πG (85)
which we can formally carry to the right hand side and
interpret it as the part of energy momentum tensor of a and appears on the right hand side of Einstein equations.
perfect fluid[12](see also Refs.[40] for a different approach The problem of zero point energy is naturally resolved
to cosmological constant), by invoking supersymmetry which has many other re-
1 markable features. In the supersymmetric description,
Gµν = Rµν − gµν R = 8πGTµν − Λgµν (82) every bosonic degree of freedom has its Fermi counter
2
part which contributes zero point energy with opposite
Such a modification is allowed by virtue of Bianchi iden- sign compared to the bosonic degree of freedom thereby
tity. It is remarkable that cosmological constant does not doing away with the vacuum energy. It is in this sense the
need adhoc assumption for its introduction; it is always supersymmetric theories do not admit a non-zero cosmo-
present in Einstein equations. It could be considered as a logical constant. However, we know that we do not live in
fundamental constant of classical general theory of rela- supersymmetric vacuum state and hence it should be bro-
tivity at par with Newton’s constant G. It is also interest- ken. For a viable supersymmetric scenario, for instance
ing to note that model based upon cosmological constant if it is to be relevant to hierarchy problem, the supper-
is consistent with all the observational findings in cos- symmetry breaking scale should be around Msusy ≃ 103
mology at present. However, there are deep theoretical GeV. We are still remain away from the observed value
problems related to cosmological constant. by many orders of magnitudes. We do not know how
Planck scale or SUSY breaking scales is related to the
observed vacuum scale!
A. Theoretical issues associated with Λ At present there is no satisfactory solution to cosmo-
logical constant problem. One might assume that there
There are important theoretical issues related to cos- is some way to cancel the vacuum energy. One can then
mological constant. Cosmological constant can be as- treat Λ as a free parameter of classical gravity similar to
sociated with vacuum fluctuations in the quantum field Newton constant G. However, the small value of cosmo-
theoretic context[12, 13, 14]. Though the arguments are logical constant leads to several puzzles including the fine
still at the level of numerology but may have far reaching tuning and coincidence problems. The energy density in
consequences. Unlike the classical theory, the cosmologi- radiation at the Planck scale is of the order of Plank en-
cal constant in this scheme is no longer a free parameter ergy density ρP ≃ 1072 GeV 4 and the observed value of
(0)
of the theory. Broadly the line of thinking takes the the dark energy density, ρΛ ≃ 0.7 × ρc ≃ 10−48 GeV 4
following route. The ground state energy dubbed zero which implies that ρΛ /ρP ∼ 10 −120
. Thus ρΛ needs to
point energy or vacuum energy ρvac of a free quantum be fine tuned at the level of one part in 10−120 around
field with spin j given by the Plank epoch, in order to match the current universe.
Z ∞ 3 p Such an extreme fine tuning is absolutely unacceptable
1 2j d k at theoretical grounds. Secondly, the energy density in
ρvac = (−1) (2j + 1) k 2 + m2 (83)
2 0 2π 3 cosmological constant is of the same order as matter en-
2j ergy density at the present epoch. The question what
(−1) (2j + 1) ∞
Z p
2
= dkk k 2 + m2 (84) causes this coincidence has no satisfactory answer.
4π 2 0
Efforts have recently been made to understand Λ
is ultraviolet divergent. This contribution is related the within the frame work of string theory using flux com-
ordering ambiguity of fields in the classical Lagrangian pactification. String theory predicts a very complicated
and disappears when normal ordering is adopted. Since landscape of about 10500 de-Sitter vacua[14]. Using An-
 14

log(r) log(r)
rf rb rf
rb

Present epoch
rf
rf Present epoch log(a)
log(a) FIG. 8: Evolution of ρφ and ρb in absence of scaling regime in
case of overshoot and undershoot. The field remains trapped
FIG. 7: Cosmologically viable evolution of field energy density in the locking regime till its energy density becomes compara-
versus the scale factor. The dotted line shows the evolution ble to that of the background component. It then starts evolv-
of background (matter/radiation) energy density. The field ing slowly and overtakes the background to become dominant
energy density ρφ (with different initial conditions) joins the at late times.
scaling regime and mimics the background. At late times
it exits the scaling matter regime to become the dominant
component and to account for the late time acceleration. these system viable to cosmology.

thropic principal, we are led to believe that we live in one

of these vacua! A. Quintessence
A novel approach to cosmological constant problem is
provided in Ref.[15]. The line of thinking takes follow- A standard scalar field (minimally coupled to gravity)
ing route: In the conventional framework, the equations capable of accounting for the late time cosmic accelera-
of motion for matter fields are invariant under the shift tion is termed as quintessence. Its action is given by
of the matter Lagrangian by a constant while gravity √
Z  
√
1 µν
Z
4
breaks this symmetry. Thus, one cannot obtain a sat- S = L −gd x = − g ∂µ φ∂ν φ + V (φ) −gd4 x
isfactory solution to the cosmological constant problem 2
until the gravity is made to respect the same symmetry. (86)
An effective action suggested by Padmanbhan in Ref.[15] The energy momentum tensor corresponding to this ac-
is explicitly invariant under the ”shift symmetry”. In tion is given by
his approach, the observed value of the cosmological con-
 
1 αβ
stant should arise from the energy fluctuations of degrees Tµν = ∂µ φ∂ν φ − gµν g ∂α φ∂β φ + V (φ) (87)
2
of freedom located in the boundary of a spacetime region.
which gives rise the following expression for energy den-
sity and pressure in FRW background
VI. SCALAR FIELD DYNAMICS RELEVANT 1 2 1
TO COSMOLOGY ρφ = φ̇ + V (φ), Pφ = φ̇2 − V (φ) (88)
2 2
The Euler-Lagrangian equation
The fine tuning problem associated with cosmo-
√ √
logical constant led to the investigation of cosmo- δ ( −gL) δ ( −gL)
logical dynamics of a variety of scalar field systems ∂α − =0 (89)
δ∂ α φ δφ
such as quintessence, phantoms, tachyons and K- √
−g = a3 (t) (90)
essence[16, 17, 41, 42, 43](see review[6] for details).
Scalar fields can easily mimic dark energy at late times for the action (86) in FRW background acquires the form
and posses rich dynamics in the past. We should note
that scalar fields models do not address the cosmological dV
φ̈ + 3H φ̇ + =0 (91)
constant problem, they rather provide an alternative dφ
way to describe dark energy. The underlying dynamics which is formally equivalent to the continuity equation
of these systems has been studied in great detail in the and can put in the form
literature. Scalar fields naturally arise in models of high  Z 
energy physics and string theory. It is worthwhile to 0 da
ρφ = ρφ exp − 3(1 + w(φ)) , (92)
bring out the broad features their dynamics that make a
 15

where w(φ) = Pφ /ρφ . Eq.(88) tells us that for a steep tions,

potential φ̇2 >> V (φ), the equation of state parameter ρφ
approaches the stiff matter limit, w(φ) → 1 where as = const. (95)
ρb
w(φ) → −1 in case of a flat potential, φ̇2 << V (φ).
Hence the energy density scales as ρφ ∼ a−n , 0 ≤ n ≤ 6. The steep exponential potential V (φ) ∼ exp(λφ/MP )
Let us note that while the field rolls along the steep part with λ2 > 3(1 + wb ) in the frame work of standard GR
of the potential, its energy density ρφ scales faster than gives rise to scaling solutions
ρr . √ whereas the shallow expo-
nential potential with λ ≤ 2 leads to a field dominated
From Eq.(81) we find that solution (Ωφ = 1). Nucleosynthesis further constraints λ.
The introduction of a new relativistic degree of freedom
ä > 0 → ρb + 3Pb < 0 ⇒ φ̇2 < V (φ) (93) at a given temperature changes the Hubble rate which
crucially effects the neutron to proton ratio at temper-
which means that we need nearly flat potential to account ature of the order of one MeV when weak interactions
for accelerated expansion of universe such that freeze out. This results into a bound on λ, namely[6],
2
Ωφ ≡ 3(1 + wb )/λ2 < >

1 V,φ V,φφ
<< 1, << 1 (94) ∼ 0.13 ⇒ λ ∼ 4.5. (96)
V V V2
In this case, for generic initial conditions, the field ul-
In case of field domination regime, the two conditions in timately enters into the scaling regime, the attractor of
Eq.(94) define the slow roll parameters which allow to the dynamics, and this allows to alleviate the fine tuning
neglect the φ̈ term in equation of motion for φ. In the problem to a considerable extent. The same holds for the
present context, unlike the case of inflation, the evolution case of undershoot, see Fig.7.
of field begins in the matter dominated regime and even Scaling solutions, however, are not accelerating as they
today, the contribution of matter is not negligible. The mimic the background (radiation/matter). One therefore
traditional slow roll parameters can not be connected to needs some late time feature in the potential. There are
the conditions on slope and curvature of potential which several ways of achieving this: (1) The potential that
essentially requires that Hubble expansion is determined mimics a steep exponential at early epochs and reduces
by the field energy density alone. Thus the slow roll to power law type V ∼ φ2p at late times gives rise to
parameters are not that useful in case of late time accel- accelerated expansion for p < 1/2 as the average equation
eration, though, Eq.(94) can still be helpful. of state < w(φ) >= (p − 1)/(p + 1) < −1/3 in this
The scalar field model aiming to describe dark energy case[44, 45]. (ii) The steep inverse power law type of
should possess important properties allowing it to allevi- potential which becomes shallow at large values of the
ate the fine tuning and coincidence problems without in- field can support late time acceleration and can mimic
terfering with the thermal history of universe. The nucle- the background at early time[46].
osynthesis puts an stringent constraint on any relativistic The solutions which exhibit the aforesaid features are
degree of freedom over and above that of the standard referred to as tracker solutions. For a viable cosmic evo-
model of particle physics. Thus a scalar field has to sat- lution we need a tracker like solution. However, on the
isfy several important constraints if it is to be relevant basis of observations, we can not rule out the non-tracker
to cosmology. Let us now spell out some of these fea- models at present.
tures in detail, see Ref.[6, 16] for details. In case the In the second class of models where trackers are absent,
scalar field energy density ρφ dominates the background there are two possibilities. First, if ρφ scales faster than
(radiation/matter) energy ρb , the former should redshift ρb in the beginning, it then overshoot the background
faster than the later allowing radiation domination to and enter the locking regime. In case of the undershoot,
commence which in tern requires a steep potential. In the field is frozen from the beginning due to large Hubble
this case, the field energy density overshoots the back- damping. In both the cases, for a viable cosmic evolu-
ground and becomes subdominant to it. This leads to tion, models parameters are chosen such that ρφ ∼ ρΛ
the locking regime for the scalar field which unlocks the during the locking regime. Hence at early times, the
moment the ρφ is comparable to ρb . The further course field gets locked (w(φ) = −1) and waits for the matter
of evolution crucially depends upon the form the scalar energy density to become comparable to field energy den-
potential. For the non-interference with thermal history, sity which is made to happen at late times. The field then
we require that the scalar field remains unimportant dur- begins to evolve towards larger values of w(φ) starting
ing radiation and matter dominated eras and emerges out from w(φ) = −1(see Fig.8). In this case one requires to
from the hiding at late times to account for late time ac- tune the initial conditions of the field. The two classes of
celeration. To address the issues related to fine tuning, scalar fields are called freezing and thawing models[7, 42].
it is important to investigate the cosmological scenarios In case of tracker (freezing) models, one needs to tune the
in which the energy density of the scalar field mimics the slope of the field potential. Nevertheless, these are supe-
background energy density. The cosmological solution rior to thawing models as they are capable addressing
which satisfy this condition are known as scaling solu- both the fine tuning and the coincidence problems.
 16

Before we proceed further, we should make an honest energy may arise in this case as a transient phenomenon.
remark about scalar field models in general. These mod- (ii) V (φ) → 0 slower then 1/φ2 for φ → ∞ ; these models
els lack predictive power: for a give cosmic history, it is give rise to dark energy as late time attractor. The two
always possible to construct a field potential that would classes are separated by V (φ) ∼ 1/φ2 which is scaling
give rise to the desired evolution. Their merits should potential with w(φ) = const. These models suffer from
therefore be judged by the generic features which arise the fine tuning problem; dynamics in this case acquires
in them. For instance tracker models deserve attention dependence on initial conditions.
for obvious reasons. Scalar fields inspired by a funda-
mental theory such as rolling tachyons are certainly of
interest. C. Phantom field

The scalar field models discussed above lead to w(φ) ≥

B. Tachyon field as source of dark energy −1 and can not give rise to super acceleration correspond-
ing to phantom dark energy with w(φ) < −1 permitted
Next we shall be interested in the cosmological dy- by observations, see Fig6. The simplest possibility of
namics of tachyon field which is specified by the Dirac- getting phantom energy is provided by a scalar field with
Born-Infeld (DBI) type of action given by (see Ref.[6] and negative kinetic energy. Phantom field is nothing but
references therein), the Hoyle-Narlikar’s creation field (C-field) which was
introduced in the steady state theory to reconcile the
model with the perfect cosmological principle. Though
√
Z
−V (φ) 1 − ∂ µ φ∂µ φ −gd4 x the quantum theory of phantom fields is problematic, it
p
S= (97)
is nevertheless interesting to examine the cosmological
consequences of these fields at classical level. Phantom
where on phenomenological grounds, we shall consider
field is described by the following action
a wider class of potentials satisfying the restriction that
V (φ) → 0 as φ → ∞. In FRW background, the pressure √
Z  
1 µν
and energy density of φ are given by S= g ∂µ φ∂ν φ − V (φ) −gd4 x (103)
2
q
Its corresponding equation of state parameter is given by
Pφ = −V (φ) 1 − φ̇2 (98)
1 2
2 φ̇ + V (φ)
w(φ) = 1 2
(104)
V (φ) 2 φ̇ − V (φ)
ρφ = q (99)
1 − φ̇2 which tells us the w(φ) < −1 for φ̇2 /2 < V (φ). An
unusual equation of motion for φ follows from (103)
The equation of motion which follows from (97) is dV
φ̈ + 3H φ̇ − =0 (105)
V′ dφ
2
φ̈ + 3H φ̇(1 − φ̇ ) + (1 − φ̇2 ) = 0 (100)
V It should be noted that evolution equation of phantom
field is same as that of the ordinary scalar field but with
where H is the Hubble parameter
inverted potential allowing the field with zero kinetic en-
1 ergy to rise up the hill. As mentioned earlier, phantom
H2 = (ρφ + ρb ) (101) energy is plagued with big rip singularity which is charac-
3Mp2
terized by divergence of the Hubble parameter and cur-
Tachyon dynamics is very different from that of the vature of space time after a finite interval of time. In
quintessence. Irrespective of the form of its potential such a situation, quantum effects become important and
one should include higher curvature corrections to gen-
w(φ) = φ̇2 − 1 ⇒ − 1 ≤ w(φ) ≤ 0 (102) eral theory of relativity which can crucially modify the
structure of the singularity. To the best of our knowl-
The investigations of cosmological dynamics shows that edge, the big rip singularity can be fully resolved in the
in case of tachyon field, there exists no solution which frame work of loop quantum cosmology[47]. Big rip can
can mimic scaling matter/radiation regime. These mod- also be avoided at the classical level in a particular class
els necessarily belong to the class of thawing models. of models in which potential has maximum. In this case,
Tachyon models do admit scaling solution in presence of the field rises to the maximum of the potential and ulti-
a hypothetical barotropic fluid with negative equation of mately settle on the top of the potential to give rise to
state. Tachyon fields can be classified by the asymptotic de-Sitter like behavior.
behavior of their potentials for large values of the field: For a viable cosmic history, the phantom energy den-
(i) V (φ) → 0 faster then 1/φ2 for φ → ∞. In this case sity similar to the case of rolling tachyon should be sub-
dark matter like solution is a late time attractor. Dark dominant at early epochs. The field then remains frozen
 17

till late times before its energy density becomes compara- E. Quintessential inflation on brane: A beautiful
ble to matter energy density. Its evolution begins there- model that does not work
after. Clearly, dark energy models based upon phantom
fields belong to the category of thawing models. Quintessential inflation refers to attempts to describe
inflation and dark energy with a single scalar field. The
unifications of the two phases of accelerated expansion
D. Late time evolution of dark energy could be realized in the framework of Randall-Sundrum
(RS) brane worlds[45, 46]. In order to achieve this, the
In the preceding subsections, we have described the field potential should be flat during inflation but steep in
cosmological dynamics of quintessence, phantoms and radiation and matter dominated eras such that ρφ could
rolling tachyon. These scalar field models fall into two mimic the background energy density at early epochs. At
broad categories: (i) Tracker or freezing models in which late times, it should become flat so as to allow the cur-
the field rolls fast at early stages such that it mimics rent acceleration of universe. Since the potential does not
the background with wb = 0. At late times, w(φ) starts exhibit minimum, the conventional reheating mechanism
deviating from dust like behavior and becomes negative does not work in this scenario. One could employ al-
moving towards de-Sitter phase as the field rolls down its ternative mechanisms such as reheating via gravitational
potential. (ii) Non-tracker or thawing models are those particle production or instant preheating. It is not real-
in which the field is trapped in the locking regime due to istic to have a potential which changes from flat to steep
large Hubble damping such that w(φ) = −1. And only and back to flat at late times(see, Fig.9). However, it is
at late times, as ρφ becomes comparable to the back- generic to have a potential which is steep and allows to
ground energy density, the field begins to evolve towards track the background at early epochs and gives rise to a
larger values of w(φ). As demonstrated by Caldwell and viable late time cosmic evolution.
Linder [42], these models occupy narrow regions in the In case of a steep potential, the field energy density
(w′ ≡ dw/d ln(a), w) plane, scales faster that radiation energy density leading to the
commencement of radiative regime. But a steep poten-
3w(1 + w) < w′ < 0.2w(1 + w) F reezing models.
tial can not support inflation in FRW cosmology. This
1 + w < w′ < 3(1 + w) T hawing models. is precisely where the brane assisted inflation comes to
where the upper and the lower bounds are obtained using our rescue. In RS brane world model, the Friedmann
analytical arguments and numerical analysis of generic equation is modified to,
 
models belonging to both the classes of models. As 8πG ρb
pointed out earlier (see Fig.6), combined analysis of dif- H2 = ρb 1 + (106)
3 2λB
ferent observations reveal that dark energy equation of
state parameter lies in the narrow strip around wΛ = −1. where λB is the brane tension. The presence of quadratic
The observational resolution between the two classes of density term in the Friedmann equation changes the dy-
the model which is of the order of 1 + w is therefore a namics at early epochs in crucial manner. Consequently,
challenge to future observations. the field experiences greater damping and rolls down its
As mentioned earlier, the phantom and the tachyon potential slower than it would during the conventional in-
dark energy models belong to the class of thawing mod- flation. This effect is reflected in the slow-roll parameters
els. In this case, we can simplify the dynamics around the which have the form,
present epoch by using the approximation that |1+w| << 1 + V /λB
1 and that the slope of the potential is small. The validity ǫ = ǫF RW (107)
(1 + V /2λB )2
of the second approximation can be verified numerically
η = ηF RW (1 + V /2λB )−1 (108)
in each case. In this scheme of a plausible approxima-
tion, one arrives at an amazing result: All the differ- where ǫF RW and ηF RW are the standard slow-roll pa-
ent dynamical systems, thawing quintessence, phantom, rameters in absence of brane corrections. The influence
tachyon and phantom tachyon follow a unique evolution- of brane corrections becomes specially important when
ary track. The distinction between the four classes of V /λB >> 1. In this case, we have,
scalar field systems and the distinction between differ-
ǫ ≃ ǫF RW (V /λB )−1 , η ≃ 2ηF RW (V λB )−1 (109)
ent models within each class is an effect of higher order
than |1 + w|[43] which certainly throws a great challenge which tells us that slow-roll (ǫ, η << 1)is possible when
to future generation experiments! Indeed, a recent ex- V /λB >> 1 even if the potential is steep (ǫF RW , ηF RW >
amination of observational data including 397 Type Ia 1). As the field rolls down its potential, the high energy
supernovae at redshifts 0.015 ≤ z ≤ 1.55 has shown that brane correction to Friedmann equation disappears giv-
evolving dark energy models provide a slightly better fit ing rise to the natural exit from inflation.
to the data than the cosmological constant [48]. If future It is possible to choose potentials suitable to
data confirms this result then it could mean that cosmic quintessential inflation and fine tune the model pa-
acceleration is currently slowing down which may have rameters such that the model respects nucleosynthe-
important consequences for dark energy model building. sis constraints and leads to observed late time cosmic
 18

acceleration[45, 46]. However, the problem occurs on the tive action,

other side. Recent measurements of CMB anisotropies Z
√ h 1
place fairly strong constraints on inflationary models. S = d4 x −g R − (1/2)g µν ∂µ φ ∂ν φ −
The tensor to scalar ratio of perturbations turns out to 16πG
i
be lager than its observed value in case of steep brane 2
− V (φ) − f (φ)RGB + Sm (110)
world inflation. Clearly, the brane world unification of
2
inflation and dark energy is ruled out by observation. where RGB is the Gauss-Bonnet term,
2
RGB ≡ R2 − 4Rµν Rµν + Rαβµν Rαβµν (111)
Brane
V(f) Damping The dilaton potential V (φ) and its coupling to curvature
f (φ) are given by,
V (φ) ∼ e(αφ) , f (φ) ∼ e−(µφ) (112)
The cosmological dynamics of system (110) in FRW back-
ground was investigated in Ref.[51, 52]. It was demon-
strated that scaling solution can be obtained in this case
provided that µ = α. In case µ 6= α, the de-Sitter so-
lution is a late time attractor. Hence, the string cur-
vature corrections under consideration can give rise to
late time transition from matter scaling regime. Un-
f fortunately, it is difficult to reconcile this model with
nucleosynthesis[51, 52]constraint.
FIG. 9: A desired form of potential for quintessential infla-
tion. It is generic to have a steep potential at early times with
brane corrections helping the slow-roll of the field. B. DGP model

In DGP model, gravity behaves as four dimensional

at small distances but manifests its higher dimensional
VII. MODIFIED THEORIES OF GRAVITY AND effects at large distances. The modified Friedmann equa-
LATE TIME ACCELERATION
tions on the brane lead to late time acceleration. The
model has serious theoretical problems related to ghost
The second approach to late time acceleration is re- modes and superluminal fluctuations. The combined
lated to the modification of left hand side of Einstein observations on background dynamics and large angle
equations or the geometry of space time. It is perfectly anisotropies reveal that the model performs much worse
legitimate to investigate the possibility of late time accel- than ΛCDM [60]. However, generalized versions of DGP
eration due to modification of Einstein-Hilbert action In can be ghost free and can give rise to transient accelera-
the past few years, several schemes of large scale modifi- tion as well as a phantom phase[49].
cations have been actively investigated. Some of these
modifications are inspired by fundamental theories of
high energy physics where as the others are based upon C. f(R) theories of gravity
phenomenological considerations. In what follows, we
shall briefly describe the modified theories of gravity and On purely phenomenological grounds, one could seek
their relevance to cosmology. a modification of Einstein gravity by replacing the Ricci
scalar in Einstein-Hilbert action by f (R). The action of
f (R) gravity is given by[11],
A. String curvature corrections
√
Z  
f (R)
S= + Lm −g d4 x, (113)
It is interesting to investigate the string curvature cor- 16πG
rections to Einstein gravity amongst which the Gauss- The modified Einstein equations which follow from (113)
Bonnet correction enjoys special status[50, 51, 52, 53, have the form,
54, 55, 56, 57, 58]. These models, however, suffer from  
several problems. Most of these models do not include ′ ′ ′ 1
f Rµν − ∇µ ∇ν f + f − f gµν = 8πGTµν . (114)
tracker like solution and those which do are heavily con- 2
strained by the thermal history of universe. For instance,
which are of fourth order for a non-linear function f(R).
the Gauss-Bonnet gravity with dynamical dilaton might
Here prime denotes the derivatives with respect to R.
cause transition from matter scaling regime to late time
The Ricci scalar in FRW background is given by
acceleration allowing to alleviate the fine tuning and co-
incidence problems. Let us consider the low energy effec- R = 12H 2 + 6Ḣ (115)
 19

which tells us that the modified Eq.(114) contains de- has dynamics. It is convenient to define scalar function
Sitter space time as a vacuum solution provided that φ as,
f (4Λ) = 2Λf ′ (4Λ). The f (R) theories of gravity may in-
deed provide an alternative to dark energy. To see this, φ ≡ f ′ − 1, (122)
let us write the evolution equations which follow from
(114) in a convenient form which is expressed through Ricci scalar once f (R) is spec-
ified. We can write the trace equation (Eq.(121)) in the
8πG terms of V and T as
H2 = ρR (116)
3f ′ dV 8πG
ä 4πG φ = + T. (123)
= − ′ (ρR + 3PR ) (117) dφ 3
a f
which is a Klein-Gordon equation in presence of a deriv-
where ρR and PR are energy density and pressure con- ing term. Thus φ is indeed a scalar degree of freedom
tributed by curvature modification which controls the curvature of space time.
Rf ′ − f The effective potential can be evaluated using the fol-
ρR = − 3H Ṙf ′′ (118) lowing relation
2
1 dV dV dφ 1
PR = 2H Ṙf ′′ + R̈f ′′ + (f − f ′ R) + f ′′′ Ṙ2 (119) = = (2f − f ′ R) f ′′ . (124)
2 dR dφ dR 3
ρR and PR identically vanish in case of Einstein-Hilbert
Models which satisfy the stability conditions belong to
action, f (R) = R as it should be. As an example of f (R)
two categories: (1) Either they are not distinguishable
model let us consider, f (R) = R − αn /Rn , where αn is
from ΛCDM or are not viable cosmologically. (ii) Models
constant for given n. In case of a power law solution
with disappearing cosmological constant: In these mod-
a(t) ∼ tn , the effective equation of state parameter can
els, f (R) → 0 for R → 0 and they give rise to cosmologi-
be computed as
cal constant in regions of high density and differ from the
2 Ḣ 2(n + 2) latter otherwise. In principal, these models can be dis-
wR = −1 − 2
= −1 + (120) tinguished from cosmological constant. Models belong-
3H 3(2n + 1)(n + 1)
ing to the second category were proposed by Hu-Sawicki
Choosing a particular value of n, we can produce a de- and Starobinsky [59, 62](see also Ref.[63] on the simi-
sired equation of state parameter for dark energy. lar theme). The functional form of f (R) in Starobinsky
The functional form of f (R) should satisfy certain re- parametrization is given by,
quirements for the consistency of the modified theory " −n #
of gravity. The stability of f (R) theory would be en- R2
sured provided that, f ′ (R) >0 and f ′′ (R) > 0 which f (R) = R + λR0 1+ 2 −1 . (125)
R0
means that graviton is not ghost and scalar degree is
not tachyon. We can understand the stability conditions Here n and λ are positive. And R0 is of the order of
heuristically without entering into their detailed investi- presently observed cosmological constant, Λ = 8πGρvac .
gations. From evolution equations (116) & (116), we see The model satisfies the stability conditions quoted above.
that the effective gravitational constant Gef f = G/f ′ In the Starobinsky model, the scalar field φ, in the
which should be positive or f ′ > 0 in order to avoid the absence of matter, is given by
pathological situation. As for the second condition, V.
Faraoni has given an interesting interpretation[61]: let 2nλR
φ(R) = − 2 . (126)
us consider the opposite case when f ′′ < 0 which means R0 (1 + R
R2
)n+1
0
that G′ef f = −f ′′ G/f ′2 > 0. This implies that gravita-
tional constant increases for increasing value of R mak- Notice that R → ∞ for φ → 0. For a viable late time cos-
ing the gravity stronger. In view of Einstein equations, it mology, the field should be evolving near the minimum
leads to yet larger value of curvature and so on which ul- of the effective potential. The finite time singularity in-
timately leads to a catastrophic situation. Thus we need herent in the class of models under consideration severely
f ′′ to be positive to avoid the catastrophe. constrains dynamics of the field.
Let us note that f (R) gravity theories apart from a
spin two object necessarily contain a scalar degree of
freedom. Taking trace of Eq.(114) gives the evolution The curvature singularity and fine tuning of
equation for the scalar degree of freedom, parameters

1 8πG
f ′ = (2f ′ − f ′ R) + T. (121) The effective potential has minimum which depends
3 3 upon n and λ. For generic values of the parameters, the
It should be noticed that Eq.(121) reduces to an alge- minimum of the potential is close to φ = 0(see Fig.10)
braic relation in case of Einstein gravity; in general f ′ corresponding to infinitely large curvature. Thus while
 20
VR0

0.4
for densities of the order of nuclear matter density. The
problem deserves further investigation.
0.2
In scenarios of large scale modification of gravity, one
-1.0 -0.5 0.5 1.0
Φ should worry about the local gravity constraints. The
-0.2
f (R) theories are related to the class of scalar tensor
theories corresponding the Brans-Dicke parameter ω = 0
-0.4
or the PPN parameter γ = (1 + ω)/(2 + ω) = 1/2
unlike GR where γ = 1 consistent with observation
FIG. 10: Plot of effective potential for n = 2 and λ = 1.2. (|γ − 1| < −5
∼ 2.3 × 10 ). This conclusion can be escaped
The red spot marks the initial condition for evolution. by invoking the so called chameleon mechanism[72]. In
case, the scalar degree of freedom is coupled to matter,
the effective mass of the field depends upon the matter
the field is evolving towards minimum, it can easily oscil- density which can allow to avoid the conflict with solar
late to a singular point[64, 65]. However, depending upon physics constraints. However, the problem of singularity
the values of parameters, we can choose a finite range of in these models is genuine and should be addressed.
initial conditions for which scalar field φ can evolve to the
minimum of the potential without hitting the singular-
ity. We find that the range of initial conditions allowed VIII. SUMMARY
for the evolution of φ to the minimum without hitting
singularity shrinks as the numerical values of parameters We have given a pedagogical exposition of physics of
n and λ increase. In the presence of matter, the mini- late time cosmic acceleration. Most of the part of the
mum of the effective potential moves towards the origin. review should be accessible to a graduate student. The
In case of the compact objects such as neutron stars, the discussion of Newtonian cosmology is comprehensive and
minimum is extremely near the origin and the singularity reviews the efforts to put the formalism of Newtonian
problem becomes really acute[65, 66]. cosmology on rigorous foundations in its domain of va-
lidity. Heuristic discussion on the introduction of cos-
mological constant and pressure corrections in evolution
Avoiding singularity with higher curvature equations is included. The underlying idea leading to
corrections late time cosmic acceleration is explained without the
use of general theory of relativity. The basic features of
We know that in case of large curvature, the quan- cosmological dynamics in presence of cosmological con-
tum effects become important leading to higher curva- stant is presented in a simple and elegant fashion mak-
ture corrections. Keeping this in mind, let us consider ing it accessible to non-experts. The review also gives
the modification of Starobinsky’s model[67, 68, 69], the glimpses of relativistic cosmology, contains important
" # notes on the dynamics of dark energy and discusses un-
α 2 1 derlying features of cosmological dynamics of a variety of
f (R) = R + R + R0 λ −1 + 2 , (127) scalar fields including quintessence, rolling tachyon and
R0 (1 + RR2
)n
0 phantom. Special emphasis is put on the cosmic viabil-
ity of these models; the cosmological relevance of scaling
then φ becomes solutions is briefly explained. The review ends with a
" # discussion on modified theories of gravity as possible al-
R 2nλ ternatives to dark energy. The treatment is simple but
φ(R) = 2α − R2 n+1
. (128)
R0 (1 + R2) conveys the successes and problems of cosmology in the
0
frame work of modified theories of gravity. Basic fea-
In case |R| is large, the first term which comes from tures of f (R) cosmology are explained avoiding the cum-
αR2 dominates. In this case, the curvature singular- bersome mathematical expressions. The latest develop-
ity, R = ±∞ corresponds to φ = ±∞. Hence, in this ments of f (R) theories with disappearing cosmological
modification, the minimum of the effective potential is constant are highlighted. The problem of singularities in
separated from the curvature singularity by the infinite these models and their possible resolution are discussed.
distance in the φ, V (φ) plane. Though the introduction I hope the review would be helpful to beginners and will
of R2 term formally allows to avoid the singularity but also be of interest to experts.
can not alleviate the fine tuning problem as the min-
imum of the effective potential should be near the in
generic cases. As for the compact objects, Langlois and IX. ACKNOWLEDGEMENTS
Babichev[70](see, Ref.[71] also on the similar theme) have
argued that neutron stars can be rescued from singu- I am indebted to T. Padmanabhan for giving me an
larity if a realistic equation of state for these objects is opportunity to write this review for a special issue of
used though the numerical simulation is yet challenging Current Science. I am thankful to D. Jain, V. Sahni
 21

and I. Thongkool for taking pain in going through the Kirshner, K. Raza and A. A. Sen for useful comments
manuscript and making suggestions for its improvement. and discussion.
I also thank S. A. Abbas, N. Dadhich, S. Jhingan, R. P.

[1] S. Perlmutter el al, Measurements Of Omega And fied Gravity With Ln R Terms And Cosmic Acceleration.
Lambda From 42 High Redshift Supernovae. Astrophys. Gen. Rel. Grav. 36, 1765-1780 (2004) [hep-th/0308176].
J. 517, 565-586 (1999). [12] V. Sahni, A. Krasinski and Ya.B. Zeldovich, Repub-
[2] A. Reiss et al, Observational Evidence from Supernovae lication of: The cosmological constant and the the-
for an Accelerating Universe and a Cosmological Con- ory of elementary particles (By Ya. B. Zeldovich).
stant. Astronomical Journal 116, 1009(1998); A. Reiss Gen.Rel.Grav.40, 1557-1591 (2008), Sov. Phys. Usp. 11
et al, Bv Ri Light Curves For 22 Type Ia Supernovae. 381-393 (1968).
Astron. J. 117, 707-724 (1999). [13] S. Weinberg, The Cosmological Constant Problem.
[3] G. Efstathiou et al, The Cosmological Constant And Rev. Mod. Phys. 61, 1-23 (1989); S. M. Carroll,
Cold Dark Matter Nature 348, 705-707 (1990). J. P. Os- The Cosmological Constant. Living Rev.Rel.4:1,2001
triker and P. J. Steinhardt, The observational case for a [astro-ph/0004075]; T. Padmanabhan, Cosmological
low-density Universe with a non-zero cosmological con- Constant: The Weight Of The Vacuum. Phys. Rept.
stant. Nature 377, 600-602 (1995). J. S. Bagla, T. Pad- 380, 235-320 (2003).
manabhan and J. V. Narlikar, Crisis In Cosmology: Ob- [14] R. Bousso, Tasi Lectures On The Cosmological Constant.
servational Constraints On Omega And H(0). Comments Gen. Rel. Grav. 40, 607-637 (2008) [arXiv:0708.4231].
on Astrophysics, (1996), 18, 275 [astro-ph/9511102]. [15] T. Padmanabhan, Dark Energy and its Implications for
[4] V. Sahni and A. A. Starobinsky, The Case For A Positive Gravity. Adv. Sci. Lett.2, 174(2009)[arXiv:0807.2356]; T.
Cosmological Lambda Term. Int. J. Mod. Phys. D 9, 373- Padmanabhan , Dark Energy and Gravity. Gen. Rel.
444 (2000). Grav. 40, 529-564 (2008) [arXiv:0705.2533].
[5] V. Sahni and A. Starobinsky, Reconstruct- [16] I. Zlatev, L. M. Wang and P. J. Steinhardt, Quintessence,
ing Dark Energy. Int.J.Mod.Phys.D 15, 2105- Cosmic Coincidence, And The Cosmological Constant.
2132(2006)[astro-ph/0610026]; T. Padmanabhan, Phys. Rev. Lett. 82, 896-899 (1999); P. J. Steinhardt,
Dark Energy: Mystery Of The Millennium. AIP L. M. Wang and I. Zlatev, Cosmological Tracking So-
Conf.Proc.861, 179-196 (2006) [astro-ph/0603114]; lutions. Phys. Rev. D 59, 123504 (1999); L. Amendola,
P. J. E. Peebles and B. Ratra, The Cosmological Con- Coupled Quintessence. Phys. Rev. D 62, 043511 (2000).
stant And Dark Energy. Rev. Mod. Phys. 75,559-606 [17] C. Armendariz-Picon, V. Mukhanov and P. J. Stein-
(2003); L. Perivolaropoulos, Accelerating Universe: hardt, A Dynamical Solution To The Problem Of A
Observational Status And Theoretical Implications. Small Cosmological Constant And Late Time Cosmic
AIP Conf.Proc.848, 698-712 (2006) [astro-ph/0601014]; Acceleration. Phys. Rev. Lett. 85, 4438-4441 (2000);
N. Straumann, Dark Energy. arXiv:gr-qc/0311083; C. Armendariz-Picon, V. Mukhanov and P. J. Stein-
J. Frieman, Lectures On Dark Energy And Cosmic hardt, Essentials Of K Essence. Phys. Rev. D 63, 103510
Acceleration. AIP Conf.Proc. 1057, 87-124 (2008) (2001); T. Chiba, T. Okabe and M. Yamaguchi, Kinet-
[arXiv:0904.1832]; M. Sami, Models of dark energy. Lect. ically Driven Quintessence. Phys. Rev. D 62, 023511
Notes Phys. 720, 219-256 (2007). M. Sami, Dark Energy (2000).
And Possible Alternatives. arXiv:0901.0756. [18] H. Bondi, Cosmology, Cambridge University Press, 1960.
[6] E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics Of [19] S. Weinberg, Gravitation and Cosmology, New York,
Dark Energy. Int. J. Mod. Phys., D15 , 1753-1936 (2006) John Wiley and sons, 1972.
[hep-th/0603057]. [20] T. Padmanabhan, Large Scale Structure of the Universe,
[7] E. V. Linder, Mapping The Cosmological Expansion. Cambridge University Press, 1993.
Rept. Prog. Phys. 71, 056901 (2008). [21] J. V. Narlikar, Introduction to Cosmology, Cambridge
[8] R. R. Caldwell and M. Kamionkowski, The Physics Of University Press, 2004.
Cosmic Acceleration. arXiv:0903.0866. [22] Ya. B. Zel’dovich and I. D. Novikov, Structure and Evo-
[9] A. Silvestri and M. Trodden, Approaches To Understand- lution of Universe, Chacago University Press, 1983.
ing Cosmic Acceleration. arXiv:0904.0024. [23] E. W. Kolb and M. S. Turner, The Early Universe, New
[10] J. Frieman, M. Turner and D. Huterer, Dark Energy And York, Addison-Wesley, Publishing Company, 1990.
The Accelerating Universe. Ann. Rev. Astron .Astrophys. [24] T. Padmanabhan, Theoretical Astrophysics, V. III, Cam-
46, 385-432 (2008) [arXiv:0803.0982]. bridge University Press, 2002.
[11] S. Capozziello, Curvature Quintessence. Int. J. Mod. [25] A. R. Liddle and D. H. Lyth, Cosmological Inflation and
Phys. D 11, 483-492 (2002); S. Capozziello, S. Car- Large-Scale Structure, Cambridge University Press, New
loni, and A. Troisi, Quintessence Without Scalar Fields. York, 2000.
Recent Res. Dev. Astron. Astrophys. 1, 625 (2003) [26] S. Dodelson, Modern Cosmolog, Academic Press, 2003.
[astro-ph/0303041]; S. Carroll, V. Duvvuri, M. Trod- [27] V. Mukhanov, Physical Foundations of Cosmology, Cam-
den, and M. Turner, Is Cosmic Speed - Up Due To New bridge University Press, 2005.
Gravitational Physics, Phys. Rev. D 70, 043528 (2004) ; [28] R. T. Cahill, Unravelling The Dark Matter - Dark Energy
T. P. Sotiriou and V. Faraoni, F(R) Theories Of Grav- Paradigm. [arXiv:0901.4140].
ity. arXiv:0805.1726; S. Nojiri and S. Odintsov, Modi- [29] E. Milne, A Newtonian Expanding Universe, Q. J. Math.
 22

os-5, 64-72 (1934); W. H. McCrea, Newtonian Cosmol- Dark Energy: Can We Distinguish It From Quintessence
ogy, Nature 175, 466 (1955). ? arXiv:0904.1070.
[30] F. J. Tipler, Am. J. Phys. 64, 1311-1315 (1996). [44] V. Sahni and L. Wang, A New Cosmological Model Of
[31] C. Neumann, Uber das newtonische Prinzip der Fern- Quintessence And Dark Matter. Phys. Rev. D62, 103517
wirkung, Leipzig, 1986. (2000).
[32] H. Seeliger, Astr. Nachr. 137, 129 (1895). [45] M. Sami, Inflation With Oscillations. Grav.Cosmol. 8
[33] V. Sahni, H. Feldman and A. Stebbins, Loitering Uni- 309-312 (2003) [gr-qc/0106074]; M. Sami, N. Dad-
verse. Astrophys. J. 385, 1-8 (1992). hich and T. shiromizu, Steep Inflation Followed By
[34] S. Weinberg, Anthropic Bound On The Cosmological Born-Infeld Reheating. Phys. Lett. B 568, 118-126
Constant. Phys. Rev. Lett.59, 2607(1987). (2003)[hep-th/0304187]; M. Sami, V. Sahni, Quintessen-
[35] L. M. Krauss and B. Chaboyer, Age Estimates Of Glob- tial Inflation On The Brane And The Relic Gravity
ular Clusters In The Milky Way: Constraints On Cos- Wave Background. Phys. Rev. D 70, 083513 (2004) [
mology. Science 299, 65-70 (2003). hep-th/0402086]; M. Sami, T. Padmanabhan, A Viable
[36] R. R. Caldwell, Marc Kamionkowski and N. N. Wein- cosmology with a scalar field coupled to the trace of the
berg, Phantom Energy And Cosmic Doomsday. Phys. stress tensor. Phys. Rev. D67 083509 (2003).
Rev. Lett. 91, 071301(2003). [46] V. Sahni , M. Sami and T. Souradeep, Relic Gravity
[37] M. Kowalski et.al, Improved Cosmological Constraints Waves From Brane World Inflation. Phys. Rev. D. 65,
From New, Old And Combined Supernova Datasets. As- 023518 (2002),[gr-qc/01051211].
trophys. J. 686, 749-778 (2008). [47] M. Sami, P. Singh and S. Tsujikawa, Avoidance Of Fu-
[38] A.G. Reiess et al, Type Ia Supernova Discoveries At Z ¿ ture Singularities In Loop Quantum Cosmology. Phys.
1 From The Hubble Space Telescope: Evidence For Past Rev. D 74, 043514 (2006) [gr-qc/0605113].
Deceleration And Constraints On Dark Energy Evolu- [48] A. Shafieloo, V. Sahni and Alexei A. Starobinsky, Is Cos-
tion. Astrophys. J. 607, 665-687 (2004). mic Acceleration Slowing Down? arXiv:0903.5141.
[39] T. R. Choudhury and T. Padmanabhan, Cosmological [49] V. Sahni and Y. Shtanov, Brane World Models Of Dark
parameters from supernova observations: A Critical com- Energy. JCAP 0311, 014 (2003) [astro-ph/0202346]; Y.
parison of three data sets. Astron. Astrophys. 429 807 Shtanov, V. Sahni, A. Shafieloo and A. Toporensky, In-
(2005). duced Cosmological Constant And Other Features Of
[40] D. Stojkovic, G. D. Starkman and R. Matsuo, Dark Asymmetric Brane Embedding. JCAP 0904, 023 (2009)
Energy, The Colored Anti-De Sitter Vacuum, And Lhc [arXiv:0901.3074].
Phenomenology. Phys. Rev. D77,063006(2008); P. Ba- [50] S. Nojiri, S. D. Odintsov and M. Sasaki, Gauss-Bonnet
tra, K. Hinterbichler, L. Hui and D. Kabat, Pseudo- Dark Energy. Phys. Rev. D 71, 123509 (2005).
Redundant Vacuum Energy. Phys.Rev.D78, 043507 [51] T. Koivisto and D. F. Mota, Cosmology And Astro-
(2008) [arXiv:0801.4526]. physical Constraints Of Gauss-Bonnet Dark Energy.
[41] S. Tsujikawa and M. Sami, A Unified Approach To Scal- Phys. Lett. B644, 104-108 (2007) [astro-ph/0606078]; T.
ing Solutions In A General Cosmological Background. Koivisto and D. F. Mota, Gauss-Bonnet Quintessence:
Phys. Lett. B 603, 113-123 (2004) [hep-th/0409212]; Background Evolution, Large Scale Structure And Cos-
E. J.Copeland, Mohammad R.Garousi, M.Sami, Shinji mological Constraints. Phys. Rev. D75, 023518 (2007)
Tsujikawa, What Is Needed Of A Tachyon If It [hep-th/0609155].
Is To Be The Dark Energy? Phys. Rev. D 71, [52] S. Tsujikawa and M. Sami, String-Inspired Cosmology:
043003 (2005) [hep-th/0411192]; M. Sami, N. Savchenko, Late Time Transition From Scaling Matter Era To Dark
A. Toporensky, Aspects Of Scalar Field Dynam- Energy Universe Caused By A Gauss-Bonnet Coupling.
ics In Gauss-Bonnet Brane Worlds Phys.Rev. D 70, JCAP 0701, 006 (2007)[hep-th/0608178].
123528(2004) [hep-th/0408140]; M. Sami, Implement- [53] G. Calcagni, S. Tsujikawa and M. Sami, Dark En-
ing Power Law Inflation With Rolling Tachyon On The ergy And Cosmological Solutions In Second-Order String
Brane. Mod.Phys.Lett. A18 (2003) 691[hep-th/0205146]; Gravity. Class. Quant. Grav. 22, 3977-4006 (2005);
M.R. Setare , Holographic Tachyon Model Of Dark En- M. Sami, A. Toporensky, P. V. Tretjakov and S. Tsu-
ergy Phys. Lett. B653,116-121 (2007); M.R. Setare, In- jikawa, The Fate Of (Phantom) Dark Energy Universe
teracting Holographic Phantom. Eur.Phys.J.C 50, 991- With String Curvature Corrections. Phys. Lett. B 619,
998 (2007); M .R Setare, The Holographic dark energy 193 (2005).
in non-flat Brans-Dicke cosmology. Phys. Lett. B 644, [54] B. M. N. Carter and I. P. Neupane, Dynamical Re-
99-103 (2007). laxation Of Dark Energy: A Solution To Early In-
[42] R.R. Caldwell and E. V. Linder, The Limits Of flation, Late-Time Acceleration And The Cosmological
Quintessence. Phys. Rev. Lett.95, 141301(2005). Constant Problem. Phys. Lett. B 638, 94-99 (2006);
[43] A.A Sen, N.Chandrachani Devi, Cosmological Scal- B. M. N. Carter and I. P. Neupane, Towards In-
ing Solutions With Tachyon:Modified Gravity Model. flation And Dark Energy Cosmologies From Modi-
Phys.Lett.B668, 182-186,2008; R. J. Scherrer and A.A. fied Gauss-Bonnet Theory. JCAP 0606, 004 (2006);
Sen, Thawing Quintessence With A Nearly Flat Po- I. P. Neupane, On Compatibility Of String Effective
tential. Phys. Rev. D77 083515 (2008); A. A. Sen, Action With An Accelerating Universe. Class. Quant.
G. Gupta and Sudipta Das, Non-Minimal Quintessence Grav. 23, 7493-7520 (2006) [hep-th/0602097]; B. M.
With Nearly Flat Potential. arXiv:0901.0173; P. Singh, Leith and I. P. Neupane, Gauss-Bonnet Cosmologies:
M. Sami and N. Dadhich, Cosmological Dynam- Crossing The Phantom Divide And The Transition
ics Of Phantom Field. Phys. Rev. D 68, 023522 From Matter Dominance To Dark Energy. JCAP 0705,
(2003)[hep-th/0305110]; A. Ali, M. Sami and A. A. Sen, 019(2007)[hep-th/0702002]; I.P. Neupane, Simple Cos-
The Transient And The Late Time Attractor Tachyon mological De Sitter Solutions On Ds(4) X Y(6) Spaces.
 23

arXiv: 0901.2568; I. P. Neupane, Extra Dimensions, Dark Energy. Phys. Rev. Lett. 101, 061103 (2008)
Warped Compactifications And Cosmic Acceleration. [arXiv:0803.2500].
arXiv: 0903.4190. [66] T. Kobayashi and K. Maeda, Relativistic Stars In F(R)
[55] G. Calcagni, B. de Carlos and A. De Felice, Ghost Condi- Gravity, And Absence Thereof. Phys. Rev. D78, 064019
tions For Gauss-Bonnet Cosmologies. Nucl. Phys. B 752, (2008) [arXiv:0807.2503].
404-438 (2006). [67] Abha Dev , D. Jain , S. Jhingan , S. Nojiri , M. Sami
[56] S. Nojiri, S. D. Odintsov and M. Sami, Dark Energy Cos- and I. Thongkool, Delicate F(R) Gravity Models With
mology From Higher-Order, String-Inspired Gravity And Disappearing Cosmological Constant And Observational
Its Reconstruction. Phys. Rev. D 74, 046004 (2006); E. Constraints On The Model Parameters. Phys. Rev. D78
Elizalde, S. Jhingan , S. Nojiri , S.D. Odintsov , M. Sami 083515 (2008) [arXiv:0807.3445].
and I. Thongkool, Dark Energy Generated From A (Su- [68] T. Kobayashi and K. Maeda, Can Higher Curvature Cor-
per)String Effective Action With Higher Order Curva- rections Cure The Singularity Problem In F(R) Gravity?
ture Corrections And A Dynamical Dilaton. Eur. Phys. Phys. Rev. D79, 024009 (2009) [arXiv:0810.5664].
J. C53, 447-457 (2008) [arXiv:0705.1211]. [69] M Abdalla, S nojiri and S D Odintsov, Consistent Mod-
[57] S. Tsujikawa, Cosmologies From Higher-Order String ified Gravity: Dark Energy, Acceleration And The Ab-
Corrections. Annalen Phys. 15 302-315 (2006) sence Of Cosmic Doomsday. Class. Quant. Grav. 22, L35
[hep-th/0606040]. (2005) [hep-th/0409177]; K Bamba, S Nojiri and S D
[58] T. Tatekawa and S. Tsujikawa, Second-Order Matter Odintsov, The Universe Future In Modified Gravity The-
Density Perturbations And Skewness In Scalar-Tensor ories: Approaching The Finite-Time Future Singularity.
Modified Gravity Models. JCAP 0809, 009 (2008) JCAP 0810, 045 (2008) [0807.2575]; S. Nojiri and S. D.
[arXiv: 0807.2017]. Odintsov, The Future Evolution And Finite-Time Sin-
[59] W. Hu and I. Sawicki, Models Of F(R) Cosmic Acceler- gularities In F(R)-Gravity Unifying The Inflation And
ation That Evade Solar-System Tests. Phys. Rev. D 76, Cosmic Acceleration. arXiv:0804.3519[hep-th].
064004 (2007) [arXiv:0705.1158]. [70] E. Babichev and D. Langlois, Relativistic Stars In F(R)
[60] R. Durrer and R. Maartens, Dark Energy And Modified Gravity. arXiv:0904.1382.
Gravity. arXiv:0811.4132. [71] S. Nojiri and S. D. Odintsov, Singularity Of Spherically-
[61] V. Faraoni, De Sitter Space And The Equivalence Be- Symmetric Spacetime In Quintessence/Phantom Dark
tween F(R) And Scalar-Tensor Gravity. Phys. Rev. D Energy Universe. arXiv:0903.5231.
75, 067302 (2007). [72] N. Straumann, Problems with Modified Theo-
[62] A. A. Starobinsky, Disappearing Cosmological Constant ries of Gravity, as Alternatives to Dark Energy.
In F(R) Gravity. JETP. Lett. 86, 157 (2007). arXiv:0809.5148. S. Capozziello and S. Tsujikawa, Solar
[63] S. A. Appleby and R. A. Battye, If Gauss-Bonnet In- System And Equivalence Principle Constraints On F(R)
teraction Plays The Role Of Dark Energy. Phys. Lett. Gravity By Chameleon Approach. Phys. Rev. D 77,
B645, 1-5 (2007) [astro-ph/0608104]. 107501 (2008); P. Brax, C. van de Bruck, Anne-Christine
[64] S. A. Appleby and R. A. Battye, Aspects Of Cosmological Davis and D. J. Shaw, f(R) Gravity and Chameleon
Expansion In F(R) Gravity Models. JCAP 0805, 019 Theories. Phys. Rev. D 78, 104021 (2008).
(2008) [arXiv:0803.1081].
[65] A. V. Frolov, A Singularity Problem With F(R)