#### A unified universe

Eur. Phys. J. C
A unified universe
Alessandro Codello
Rajeev Kumar Jain 1
0 Origins, Centre for Cosmology and Particle Physics Phenomenology, University of Southern Denmark , Campusvej 55, 5230 Odense M , Denmark
1 Present Address: Department of Physics, Indian Institute of Science , Bangalore 560012 , India
We present a unified evolution of the universe from very early times until the present epoch by including both the leading local correction R2 and the leading non-local term R 12 R to the classical gravitational action. We find that the inflationary phase driven by R2 term gracefully exits in a transitory regime characterized by coherent oscillations of the Hubble parameter. The universe then naturally enters into a radiation dominated epoch followed by a matter dominated era. At sufficiently late times after radiation-matter equality, the non-local term starts to dominate inducing an accelerated expansion of the universe at the present epoch. We further exhibit the fact that both the leading local and non-local terms can be obtained within the covariant effective field theory of gravity. This scenario thus provides a unified picture of inflation and dark energy in a single framework by means of a purely gravitational action without the usual need of a scalar field.
1 Introduction
Cosmological observations strongly support the idea that the
universe underwent an early period of accelerated expansion
called inflation [
1,2
]. Besides, local supernovae
measurements [
3,4
] also suggest that the universe is experiencing a
phase of acceleration at the present epoch, caused by dark
energy [
5
]. Whether there exists a deeper and fundamental
connection between the two (or not), an interesting question
to ask is if it is possible to unify both these epochs in a single
framework with minimal degrees of freedom. Such scenarios
have been explored by using matter fields (scalar fields with
an appropriate potential) as well as by modifying gravity.
Within General Relativity (GR) and in the absence of a
cosmological constant, it is not possible to explain either
inflation or dark energy without adding extra degrees of freedom
and therefore, these two epochs are very novel consequences
of physics beyond classical GR described by the Einstein–
Hilbert (EH) action. The framework of the covariant Effective
Field Theory (EFT) of quantum gravity, developed in [
6–
8
], predicts both local and non-local correction terms which
then become natural candidates for driving inflation and dark
energy thereby allowing us to construct a unified picture of
the universe. The key advantage of using EFT methods is that
they allow to compute such corrections from first principles
even in the absence of a complete theory of quantum gravity
[
9–12
].
In addition to scalar fields, an inflationary epoch in the
early universe can also be driven by higher order curvature
corrections to the EH action, notably by a R2 term – a
scenario first proposed by Starobinsky [
13
] and further discussed
in [
14–16
]. This scenario is one of the simplest and oldest
model of inflation based on purely gravitational corrections.
Albeit the existence of a large number of inflationary
models, Starobinsky inflation should be one of the most probable
among all from an Occam’s razor point of view and also
turns out to be the most preferred model with the highest
Bayesian evidence in the recent datasets [
2,17
]. The presence
of higher order curvature terms (including the derivatives)
can be understood by means of one-loop quantum
corrections to the EH action as suggested in [13] and extended, for
example, in [
18
]. In the covariant EFT of gravity, we found
that a R2 term naturally arises as a leading local correction to
the EH action and becomes responsible for driving inflation
at early times without the need of additional matter fields.
Thus, inflation in this set-up is entirely a feature of the
leading corrections in the gravitational sector [
7
].
Within the framework of GR, cosmic acceleration at the
present epoch can be achieved by adding a cosmological
constant and is indeed the simplest possibility. Recently, it has
been realized that specific curvature square non-local terms
can also drive the current acceleration of the universe. In
particular, the non-local term R 12 R leads to a very interesting
phenomenology as it effectively behaves like a cosmological
constant as R/ → 1 at late times. This term has been argued
as a consistent IR modification of GR and the theory together
with the EH action remains free of any propagating ghost-like
degree of freedom or other instabilities [
19
]. This scenario
provides a viable alternative to dynamical dark energy and
has been greatly studied including the study of
cosmological perturbations and other observable imprints [
20, 21
]. In
order to understand the origin of such non-local terms in a
consistent framework, we have recently shown that various
non-local terms appear at the second order in a curvature
expansion [
6, 7
]. However, the R 12 R term among others is
the most relevant one for dark energy due to being roughly a
constant at late times. Lately, modifications of GR including
the Weyl-square term, also predicted by the covariant EFT
[
6
], have also been studied. Such terms do not contribute to
the background expansion but only to the evolution of
cosmological perturbations [
22
].
In this Letter, we present a unified evolution of the universe
from very early times until the present epoch by including
both the leading local correction R2 and the leading non-local
contribution R 12 R to the EH action together with radiation
and matter. We find that the initial inflationary epoch induced
by the R2 term exists into a transitory regime with coherent
and damped oscillations of the Hubble parameter and could
be characterized as reheating in this scenario. This phase then
naturally enters into a radiation dominated epoch followed
by a matter dominated era. At sufficiently late times, the
non-local term becomes dominant and leads to a dark energy
dominated universe with equation of state w ≤ −1. This
setup provides a unified picture of our universe from earliest
epochs until today thereby unifying both inflation and dark
energy in a single framework by means of purely
gravitational corrections to the EH action. Later we will show that
these corrections are naturally present in the covariant EFT
of gravity.
2 A unified scenario
We are interested in studying the evolution of the universe in
a scenario wherein the leading local quadratic term is
combined with the most relevant IR non-local terms. The total
effective action including the matter part Sm is then given by
=
d4 x √−g
MP2l R − ξ1 R2 + m4 R
2
where ξ and m are phenomenological parameters which must
be fixed by observations. The normalization of CMB
spectrum fixes ξ ∼ 1.2 × 10−9 and the dark energy density
today leads to m ∼ 0.3√ H0 MPl. The specific non-local
term R 12 R has been chosen since it is the simplest that
can effectively emulate a cosmological constant term since
R 12 R ∼ 1 as R/ → 1 at late times. This term as an
alternative to dark energy was first proposed in [
19
]. The
effective action in (1) is the minimal unified action (with
deviations from GR of purely gravitational character)
capable of describing inflation and dark energy together in a
single framework sans a cosmological constant was proposed
in [
7, 21, 23
].
In order to understand the implications of this effective
action for our universe, we work in a (3 + 1)-dimensional,
spatially flat, FRW spacetime described by the line element
ds2 = −dt 2 + a2(t )dx2 where a(t ) is the scale factor.
Einstein’s equations can now be written as Gμν + Gμν =
Tμν /MP2l, where Gμν corresponds to the correction terms
arising from the local and non-local terms in (1). It is evident
that Gμν is covariantly conserved and its explicit form can
be found in [
7
]. The modified Friedmann equations of motion
(EOM) can be obtained by varying this action
H 2 − M122
2 H H¨ + 6 H 2 H˙ − H˙ 2
4 m4
− MP2l
1 1
2 H 2 S + H S˙ + 2 H˙ S − 6 U˙ S˙
U¨ + 3 H U˙ − 6 2 H 2 + H˙
ρ
= 3MP2l
= 0
S¨ + 3 H S˙ − U = 0 ,
(2)
where H = a˙ /a is the Hubble parameter and the two
auxiliary fields U and S are defined as, U = −1 R and S = 12 R,
respectively. We also define the mass scale characterizing
inflation as M 2 ≡ ξ MP2l.
The modified Friedmann EOM must be solved together
with the continuity equations for radiation and matter which
are given by ρ˙i + 3 H (1 + wi)ρi = 0, where i = {r, m} with
wr = 1/3 and wm = 0. At early times when local corrections
are most relevant, our scenario reduces to the Starobinsky
model and naturally describes inflation. To achieve sufficient
e-folds of inflation, the time derivate of the Hubble
parameter at an initial time can be appropriately tuned. The CMB
normalization instead determines the value of the coupling ξ .
Note that, as pointed out in [
7
], R + R2 gravity is not exactly
solvable due to the non-linear nature of the equations.
However, at sufficiently early times when R2 is dominant over the
EH term, the theory admits a quasi de-Sitter solution leading
to Starobinsky inflation. This solution can be understood by
neglecting higher derivative terms in H (i.e. | H˙ | H 2) in
the first equation in Eq. (2), leading to H 2 − M722 H 2 H˙ 0,
which clearly admits an exponential solution for the scale
factor, as expected. This behavior is depicted by the constant
H solution on the left in Fig. 1. Note that, the universe in
this scenario gracefully exits from inflation and enters into
an oscillatory regime. We obtain the oscillatory solution for
H
g
o
l
inflation
oscillations
radiation
matter
tend
treh
tde
Fig. 1 A unified evolution of the universe from very early times until
today. We plot log H vs. log t starting from an inflationary stage, passing
through a transitionary phase characterized by strong coherent
oscillations, followed by radiation and matter to finally end in a dark energy
the Hubble parameter numerically but one can also gain an
analytical understanding of their origin by ignoring some
higher order terms in the modified Friedmann EOM [
7
]. We
stress that this transient oscillatory regime is very novel and
generic feature of our scenario which can be considered as
the reheating epoch in this set-up. This could in principle
have interesting observable imprints. We leave the study of
the details of the reheating stage and its possible imprints to
future work.
After the oscillatory phase is over, the universe naturally
enters into the radiation dominated epoch. After radiation–
matter equality, the universe then evolves in a matter
dominated era as clearly shown in Fig. 1. At sufficiently late times
after equality, the non-local term R 12 R which was so far
subdominant starts to become relevant and drives the current
acceleration of the universe, due to the fact that R 12 R → 1
as R/ → 1 at the present epoch. The effects of this
nonlocal term are studied by solving the modified Friedmann
equation numerically, as shown in Fig. 1 and also discussed
in [
19
].
Moreover, we are also interested in understanding the
imprints of the non-local term using an analytical approach
in the background of both radiation and matter. It is useful
to note that in terms of the conformal time τ = dt /a(t ),
the Friedmann equation including only radiation and matter
can indeed be solved exactly and the solution is a(τ )/aeq =
(2√ √
2 + 2) τ /τeq 2 with τeq =
2 − 2) τ /τeq
+ (1 − 2
(2√2 − 2)/aeq 3MP2l/ρeq. Here, ρeq, aeq and τeq are the
energy density, scale factor and conformal time at the epoch
of radiation–matter equality, respectively. Using this
background solution in a closed form, we have solved the
Friedmann EOM corresponding to the non-local term with
initial conditions U = S = 0 and U = S = 0. We have
also treated the contributions from the non-local term when
teq
log t
w
radiation
R
1
−
R
0.5
0.0
0.5
1.0
Λ
15
dominated regime. The critical epochs that separate these phases are
end of inflation tend, reheating treh, radiation–matter equality teq and the
onset of dark energy tde
matter
darkenergy
1
R 2 R
log a
10
5
0
5
10
taken on the right hand side of the EOM as an effective dark
energy density. In Fig. 2, we have plotted the total
equation of state parameter w as a function of log a. It is
evident from the figure that the universe starting from radiation
epoch (w = 1/3) smoothly transits to the matter dominated
regime (w = 0) which then evolves in the present epoch
dominated by the non-local term with w ≤ −1. For later
discussion we have also shown the corresponding behavior
for the leading non-local term R 1 R but found that only
−
R 12 R leads to an accelerating universe today. Moreover, in
Fig. 3, we have plotted the dimensionless energy density ratio
r, m and de as a function of log a which perfectly
corroborates the behavior of Fig. 2. By demanding de 0.68
to be consistent with observations today, we can also obtain
the value of the mass scale m of the non-local term to be
m ∼ 0.3√ H0 MPl ∼ 5.7 × 10−4 eV.
0.8
0.6
0.4
0.2
0.0
15
10
0
5
5
log a
As we have seen, the effective action in (1) is very appealing
from a phenomenological perspective since it can naturally
describe all the different phases through which the universe
has evolved, but can it have a deeper origin or interpretation?
Under which assumptions is it a good low energy description
of quantum gravity? We will discuss here how the model (1)
can be justified from low energy quantum gravity when this
is treated by means of EFT methods.
Since we are interested in studying the evolution of the
whole universe, the EFT approach must be developed in a
covariant way in order to derive an effective action valid on
an arbitrary spacetime and in particular FRW. In our previous
paper [
6
], we have computed the leading order EFT action
to the second order in the curvatures. The final result for the
gravitational part of the effective action is given by
=
d4x √−g
M2P2l R − ξ1 R2 − R F
−
m2
R ,
(3)
where the structure function F is completely determined
once the matter content of the theory is specified. The general
form of the structure function can be found in [
6
]. The action
depends on two parameters ξ and m. The first is free to tune
while the second is in principle related to some mass scale
of the underlying theory. On FRW the Weyl tensor vanishes
identically and therefore we have not reported the Weyl part
of the effective action (3) since it will not contribute to the
background EOM. However, this is not true when dealing
with cosmological perturbations and the contributions due
to these terms must be taken into account which could lead
to very distinct signatures in the cosmological observables
[
22
].
The structure function F is non-local in the low energy
limit m2 − and has the following form [
6
]
F
= α log −m2 + β
log −m2 + δ
(4)
− −
where the coefficients α, β, γ and δ are indeed predictions of
the EFT of gravity which ultimately depend only on the field
content of the theory1 and are listed in [
6
]. We recognize in (4)
the appearance of the non-local term R 12 R assumed in the
1
model (1) if m ≡ |δ| 4 m. As mentioned earlier, for viable
cosmology, we should have m2 ∼ H0 MPl. Since −1 ∼ H0−2 at
the present epoch, the low energy limit m2 − translates
to |δ| 21 MHP0l 1. Although H0 MPl, this inequality can still
be satisfied for a suitable choice of δ or N as δ ∝ N . These
relations also imply that |δ| 21 m2 ∼ H0 MPl ∼ 10−6 (eV)2
and m H0 ∼ 10−33 eV. Therefore, the low energy
expansion can be justified for a suitable value of δ for the structure
function F . Even though there exists specific matter choices
that can make one or more of these coefficients vanish, in
the most general case these are all non-zero and so all other
non-local terms are in principle present in the effective action
[
7
].
Non-local modifications of GR based on one or more of
the operators in (4) have been considered by many authors,
see for example [
24–35
] and references therein. In order to
justify the model (1) we need to show that indeed the operator
R 12 R dominates the late time evolution of the universe.
For this reason in Fig. 2 we have compared the equation of
state parameter for a dark energy fluid generated by the first
two non-local terms R 1 R and R 12 R. As can be seen,
only the latter is capable−of changing the late time evolution
while the first only affects the intermediate regime. The terms
containing the log operators are more delicate to define and
have been discussed in [
6
]. The analysis shows that their
behavior is sub-leading with respect to the relative operator
without the log and so to first approximation can be discarded.
The R log −m2 R term may be relevant at early times but in
our model this epoch is dominated by the local R2. Finally,
due to the smallness of the mass scale m, all higher order
operators will be suppressed by large additional factors and
to first approximation can also be discarded. Of the terms
contained in the expansion (4) the one dominant at late times
is thus the one included in equation (1) and the effective
action (1) is in principle justifiable from first principles via
EFT arguments.
As discussed in [
6
], the effective action also contains many
new non-local operators with three or more curvatures, as for
1 They are of order N where N is the number of particles of a given
species.
example R 1 R 1 R. Within these there is also the
contribution from− the −conformal anomaly. These operators may
have interesting cosmological implications (see for instance
[36, 37] and references therein) and, even if sub-leading when
compared with the two curvature terms considered in this
Letter, their effects should be studied carefully, for a recent
discussion, see [
23, 38
].
Finally, we can try to answer the question: what is m? In
the case of scalars and fermions, m is the effective mass of
the particle. For these cases the mass required for
observational consistency is m ∼ 0.57 meV which makes it a very
light particle. Photons do not contribute while gravitons can
induce an effective gravitational mass m2grav = 2 V (v)/MP2l
(where V (ϕ) is the scalar potential with v its minimum).
Such a light particle must have been relativistic all its life
during the evolution of the universe and would contribute to
the effective number of relativistic degrees of freedom Neff .
For instance, a light scalar in equilibrium with the same
temperature as neutrinos would lead to a departure of Neff from
the standard value of 4/7 [39]. The presence of such extra
degrees of freedom can therefore be tightly constrained using
Planck and other observations [40]. Furthermore, structure
formation can also be used to constrain the imprints of such
light particles. It has been shown recently that the
difference between the predictions of the non-local model R 12 R
and CDM is small with respect to the present
observational errors [
20
]. This is somewhat expected as neutrinos,
for instance, being very light with mν 0.23 eV induce
only sub-leading corrections at a few percent level to CDM
at small scales and a similar conclusion can be anticipated
for the even lighter particle in our scenario. A more detailed
analysis to judge the consistency of this picture will require
studying the cosmological evolution of the full model (3)
with the exact structure function F before the simplifying
expansion (4) [41]. A different microscopic interpretation of
the mass scale m has been proposed in [
21
].
4 Discussion and outlook
In this Letter we have studied a simple yet elegant scenario
which generalizes GR by including leading local R2 and
leading non-local R 12 R terms to the action. We have derived the
modified Friedman EOM by evaluating the generalized
Einstein’s equations on an FRW background which are
subsequently solved by a combination of analytical and numerical
methods. Our main result is that the solution, depicted in Fig.
1, exhibits a unified and consistent evolution of the universe
that starts from an inflationary regime at very early times and
ends in a dark energy phase at the present epoch.
In particular, we have found that the transition between
the end of inflation to the radiation epoch is characterized by
strong coherent oscillations of the Hubble parameter which
may be interpreted as a concrete realization of the
reheating phase. These oscillations originate from the competition
between R and R2 contributions and are indeed a precise and
distinct feature of our scenario (when the theory is treated
consistently in the Jordan frame). After this transient
oscillatory phase, the universe naturally enters the Einstein regime
with radiation domination followed by a matter dominated
era. Eventually, the non-local term starts to dominate and
drives the evolution of the universe, behaving as a dynamical
and purely gravitational dark energy with w ≤ −1. Thus,
the model (1) is able to describe the evolution of the
universe through all the epochs and is characterized by only two
new parameters ξ and m. The oscillatory phase it predicts
may have characteristic observable imprints which should
be analyzed more carefully.
We emphasize that the unified scenario we studied is not
only phenomenological but can be justified using an effective
approach to low energy quantum gravity. The mass scale m
can either be identified with the mass of a very light species
or with the effective gravitational mass. This connection is
not only satisfactory from a theoretical point of view but it
also provides a way to reduce the number of free parameters
by linking m to the underlying theory. In conclusion, this
scenario represents a unified cosmological model with one
parameter less, and is thus characterized by a higher
predictive power that can in principle be falsified.
An immediate next step in this direction will be to study the
evolution of perturbations in this model and obtain relevant
observables to fully construct a viable cosmological model.
Although the evolution of perturbations has been studied in
the Starobinsky inflation and in the non-local model
respectively, it will be very interesting to study them in this unified
model which we leave for future work.
Acknowledgements We would like to thank the anonymous referee
for his/her suggestions. The CP3-Origins centre is partially funded by
the Danish National Research Foundation, Grant number DNRF90.
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