#### Cosmic acceleration in the nonlocal approach to the cosmological constant problem

Eur. Phys. J. C
Cosmic acceleration in the nonlocal approach to the cosmological constant problem
Ichiro Oda 0
0 Department of Physics, Faculty of Science, University of the Ryukyus , Nishihara, Okinawa 903-0213 , Japan
We have recently constructed a manifestly local formulation of a nonlocal approach to the cosmological constant problem which can treat with quantum effects from both matter and gravitational fields. In this formulation, it has been explicitly shown that the effective cosmological constant is radiatively stable even in the presence of the gravitational loop effects. Since we are naturally led to add the R2 term and the corresponding topological action to an original action, we make use of this formulation to account for the late-time acceleration of expansion of the universe in case of the open universes with infinite space-time volume. We will see that when the “scalaron”, which exists in the R2 gravity as an extra scalar field, has a tiny mass of the order of magnitude O(1meV ), we can explain the current value of the cosmological constant in a consistent manner.
1 Introduction
The recent development of observational cosmology has
made it possible to tell that our universe might have
undergone two phases of cosmic acceleration [
1–4
]. The first stage
of the acceleration is usually called inflation and is believed
to have happened prior to the radiation-dominated era [
5–
8
]. The inflation is needed not only to resolve the flatness
and horizon problems but also to account for the almost flat
spectrum of temperature anisotropies observed in the cosmic
microwave background. An attractive feature of the inflation
is that it provides a mechanism to generate the small scale
fluctuations which are required to seed galaxy formation.
These fluctuations are nothing but quantum fluctuations due
to the zero-point energy of a generic quantum field, which
get pushed into the classical regime by the large expansion of
the universe. This accelerated expansion must end to connect
to the radiation-dominated universe via a period of reheating,
so the pure cosmological constant should not be responsible
for inflation.
The fact that we live in a large and old universe with a
large amount of entropy can also be regarded as a primary
consequence of inflation in the early universe. The standard
paradigm of inflation is that the accelerated expansion of
the universe is realized when a scalar field, which is usually
dubbed as the inflaton, slowly rolls over its potential down to
a global minimum, which is identified with the true vacuum,
in a time scale longer than the Hubble time. However, the
origin of inflaton and its potential is quite vague, which is
one important issue to be clarified in future.
Actually, the slow-roll inflation is not the only inflation
scenario studied so far, and there are a few alternative
theories. Among them, what we call, the Starobinsky model
of inflation does not require introduction of the inflaton by
hand and the scalar field degree of freedom, which is
specially called “scalaron”, emerges from the higher-derivative
curvature term R2 with a gravitational origin [
9
]. Unlike the
inflationary models such as “old inflation”, the Starobinsky
model is not plagued by the graceful exit problem since the
accelerated expansion of the universe is triggered by the R2
term whereas the followed radiation-dominated epoch with a
transient matter-dominated phase is caused by the
EinsteinHilbert term.
On the other hand, in order to explain the second stage of
the acceleration, it might be possible to make use of the
cosmological constant since the acceleration in “recent” years
does not have to end and could last forever. However, at this
point we encounter one of the most serious contradictions in
modern physics, the cosmological constant problem, which
implies the enormous mismatch between the observed value
of cosmological constant and estimate of the contributions of
elementary particles to the vacuum energy density [
10,11
].
For later convenience, within the context of quantum field
theory (QFT), let us explain the cosmological constant
problem by using a real scalar field of mass m with λφ4
interaction, which is minimally coupled to the classical gravity
[
12
]1
S =
b − 21 gμν ∂μφ∂ν φ
where MPl is the reduced Planck mass defined as MPl =
√81π G (G is the Newton constant), and b is the bare
cosmological constant which is in principle divergent. Using the
dimensional regularization, the 1-loop effective potential can
be calculated to be
V φ,1−loop
i
≡ 2 tr log
δ2 S
i
− δφ2
1
= 2
= − (8π )2
m4
2
(d24πk)E4 log(k2E + m2)
+ f i ni t e ,
(2)
− m22 φ2 − 4λ! φ4 ,
where μ is the renormalization mass scale. In order to
cancel the divergence associated with a simple pole 2 , we are
required to choose the bare cosmological constant at the
1loop level
φ,1−loop
b
m4
= (8π )2
2
+ log
μ2
M 2
,
where M is an arbitrary subtraction mass scale where the
measurement is carried out. (In cosmology, the physical
meaning of M is not so clear as in particle physics since
M may be identified with the Hubble parameter or any other
cosmological energy scale related to the evolution of the
universe [
14
].) Then, by summing up the two contributions, the
1-loop renormalized cosmological constant is of form
rφe,n1−loop = V φ,1−loop +
m4
= (8π )2
log
φ,1−loop
b
m2
M 2
− f i ni t e .
The cosmological observation requires us to take rφe,n1−loop
∼ (1meV )4. If the particle mass m is chosen to the
electro-weak scale, we have V φ,1−loop ∼ (1T eV )4 =
1060(1meV )4. Thus, the measurement suggests that the finite
contribution to the 1-loop renormalized cosmological
constant is cancelled to an accuracy of one part in 1060 between
V φ,1−loop and bφ,1−loop. This big fine tuning is sometimes
called the cosmological constant problem.2 Following the
lore of QFT, at this stage of the argument, we have no issue
1 We follow notation and conventions of the textbook by Misner et al
[
13
].
2 There is a loophole in this argument. If the particle mass m is taken
to be 1meV like neutrinos, we have V φ,1−loop ∼ (1meV )4, which is
equivalent to the observed value of the cosmological constant, so in this
(1)
(3)
(4)
with this fine tuning since we have just replaced the
renormalized quantity depending on the arbitrary scale M with the
measured value.
However, the issue arises when we further consider higher
loops. For instance, at the 2-loop level, V φ,2−loop is
proportional to λm4 (In general, at the n-loop level, V φ,n−loop ∝
λn−1m4). Then, the consistency between the measurement
and the perturbation theory requires us to set up an equality
(5)
(1meV )4 =
b + V φ,1−loop
ren =
+V φ,2−loop + V φ,3−loop + · · · .
The problem is that each V φ,i−loop(i = 1, 2, . . .) has almost
the same and huge size compared to the observed value
(1meV )4. Thus, even if we fined tune the cosmological
constant at the 1-loop level, the fine tuning would be spoilt at the
2-loop level so that we must retune the finite contribution in
the bare cosmological constant term to the same degree of
accuracy. In other words, at each successive order in
perturbation theory, we are required to fine tune to extreme accuracy!
This problem is called “radiative instability”, i.e., the need to
repeatedly fine tune whenever the higher loop corrections are
included, which is the essence of the cosmological constant
problem.
In order to solve the problem of the radiative instability
of the cosmological constant, some nonlocal formulations
have been advocated [
15–27
], but many of them have been
restricted to the semiclassical approach where only radiative
corrections from matter fields are taken account of whereas
the gravity is regarded as a classical field merely serving
for detecting the vacuum energy. Recently, an interesting
approach, which attempts to deal with the graviton loop
effects by using the topological Gauss-Bonnet term, has been
proposed [
22
].
More recently, we have proposed an alternative
formulation for treating with the graviton loops where the higher
derivative term R2 plays an important role [
12
]. In the article
[
12
], we have also pointed out a connection between the
formulation and the R2 inflation by Starobinsky [
9
]. Thus far,
there are a few articles dealing with inflation in the closed
universes with finite space-time volume by a similar formalism
[
18–20,28
], but these papers are written in the framework of
the semiclassical approach, so there is indeed radiative
instability in the gravity sector. In this article, we would like to
apply our formulation [12], which is completely free from the
issue of radiative instability, to the late-time acceleration of
expansion of the universe in the open universes with infinite
space-time volume. The reasons why we consider the open
universes are two-fold. Current cosmological data show as
a whole good agreement with a spatially flat universe with
Footnote 2 continued
case there is no cosmological constant problem. Actually, we will meet
such a situation later in discussing the scalaron mass.
M P2l Gμν + cgμν = Tμν .
Taking the trace of this equation, we can describe the constant
c as
1
c = 4 (M P2l R + T ).
1
c = 4 (M P2l R + T ).
In general, the right-hand side (RHS) is not a constant, so
to make the RHS be a constant as well, we will take the
space-time average of the both sides:
Here, for a generic space-time dependent quantity Q(x ), the
operation of taking the space-time average is defined as
Q(x ) =
d4x √−g Q(x )
d4x √−g
,
where the denominator V ≡ d4x √−g denotes the
spacetime volume. Finally, inserting Eq. (11) to Eq. (9), we arrive
at the desired equation of motion for the gravitational field:
1 1
M P2l Gμν + 4 M P2l Rgμν = Tμν − 4 T gμν .
It is easy to show that this new type of Einstein equation
has a close relationship with unimodular gravity [
30
]. To do
so, let us first take the trace of Eq. (13):
non-vanishing cosmological constant and cold dark matter,
which suggests that our universe might be open. Moreover,
the scenario of eternal inflation leads to space-times with
infinite volume [
29
].
2 Gravitational field equation with a high-pass filter
It is an interesting direction of research to explore a
possibility of constructing an effective theory with nonlocal
properties in order to address the cosmological constant problem.
The fundamental physical idea that we wish to implement in
the present study is to render the effective Newton constant
depend on the frequency and wave length such that it works as
a filter shutting off oscillating modes with high energy. Since
we want to keep the fundamental physical principles such as
locality and unitarity, we will attempt to simply modify
general relativity at the level of not the action but the equations
of motion. Because of it, we can present a solution to the
cosmological constant problem without losing all the successes
of general relativity. As seen later in the present article, this
idea is only effective for making the vacuum energy
density stemming from matter fields be radiatively stable, so
we need an additional mechanism for shutting off the high
energy modes of gravitational field. We will present such an
additional mechanism by incorporating the R2 term and the
corresponding topological term in the next section.
In Ref. [
17
], such a filter mechanism has been already
proposed so we will make use of this mechanism, but the
obtained equation in the present article is different from that
in Ref. [
17
] since we proceed along a slightly different path
of arguments. The filter mechanism advocated in [
17
] is to
start with the following modified Einstein equation:s
M P2l 1 + F (L2 ) Gμν = Tμν ,
where Gμν = Rμν − 21 gμν R is the well-known Einstein
tensor, Tμν is the energy-momentum tensor, and the filter
function F (L2 ) is assumed to take the form; F (x ) → 0
for x 1 whereas F (x ) 1 for x 1. Here L is the
length scale where gravity is modified, and ≡ gμν ∇μ∇ν
is a covariant d’Alembertian operator. It is a natural
interpretation that we regard the above Einstein equation (6) as the
Einstein equation with the effective Newton constant G˜ e f f
or the effective reduced Planck mass M˜ e f f defined as
1
M˜ e2f f ≡ 8π G˜ e f f ≡ M P2l 1 + F (L2 ) .
As considered in Ref. [
17
], let us take the infinite length
limit L → ∞ to pick up the “zero mode” of Gμν such that
Gμν = gμν . In this limit we obtain
F (L2 )Gμν = F (0)gμν ≡ cgμν ,
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
where we have defined the constant c by c = F (0). Then,
substituting Eq. (8) into Eq. (6) leads to the equation
M P2l R + T = M P2l R + T .
Substituting Eq. (14) into Eq. (13) gives rise to the equation
of motion in unimodular gravity
1
Rμν − 4 gμν R =
1
M P2l
1
Tμν − 4 gμν T .
Actually, operating the covariant derivative ∇μ on Eq. (15)
and using the Bianchi identity ∇μGμν = 0 and the
conservation law ∇μTμν = 0, we have the equation
∇μ(M P2l R + T ) = 0,
M P2l R + T = 4 ,
from which we can conclude that M P2l R + T is a mere
constant. If we define this integration constant as
and insert this equation to Eq. (15), we can obtain the standard
Einstein equation with a cosmological constant 3
M P2l Gμν +
gμν = Tμν .
3 Precisely speaking, the conventional Einstein equation with a
cosmological constant 0 is described as Gμν + 0gμν = M1P2l Tμν . In this
article, we have defined the cosmological constant as = MP2l 0,
which is nothing but the vacuum energy density usually written as ρ.
In the above “derivation” of the equation of motion (13),
we have considered the large length limit L → ∞, by which
the filter excises only the infinite wave length and period
fluctuations from dynamics [
17
]. This prescription of taking the
limit L → ∞ corresponds to taking the Fourier transform
at vanishing frequency and momenta, which is in essence
equivalent to taking the space-time average of dynamical
observables. Comparing Eq. (9) with Eq. (18), we find that
the constant c corresponds to the cosmological constant .
Since the cosmological constant is a global variable, which
is a parameter of a system of codimension zero, and its
measurement requires separating it from all the other long wave
length modes in the universe, it takes a detector of the size of
the universe to measure the value of the cosmological
constant [
18–20
]. In this sense, taking the space-time average of
the constant c in Eq. (11) makes sense as well.
The equation of motion obtained in Eq. (13) has a
remarkable property in that it sequesters the vacuum energy from
matter fields at both classical and quantum levels. To show
this fact explicitly, let us write the energy-momentum tensor
Tμν = −Vvac gμν + τμν ,
where Vvac denotes the vacuum energy density involving
a classical cosmological constant and quantum fluctuations
of the matter fields while τμν is the local excitation
involving radiation and non-relativistic matters etc. Because of the
structure of the RHS of Eq. (13), the vacuum energy
density Vvac completely decouples from the modified Einstein
equation Eq. (13) since we have
1 1
M P2l Gμν + 4 M P2l Rgμν = τμν − 4 τ gμν .
As a result, we have a residual effective cosmological
constant given by
1 1
e f f ≡ 4 M P2l R + 4 τ .
Note that this quantity is stable under radiative corrections
and would in general take a small value in a large and old
universe.
To close this section, we should comment on the
operation of taking the space-time average in more detail.
Obviously, the definition of this operation is very sutble in case
of the open universes with infinite space-time volume, V ≡
d4x √−g = ∞. Since the residual cosmological constant
includes R and τ in the definition (21), we will focus on the
two quantities in what follows.4 First of all, let us notice that
in any space-time with infinite volume, when d4x √−g R
is finite, R takes the vanishing value. This situation includes
the physically interesting space-times where the universe
is described in terms of a radiation or matter-dominated
(19)
(20)
(21)
1
e f f ≡ 4 M P2l R∞.
(22)
Friedmann-Robertson-Walker (FRW) universe. As a
different example, let us consider the universe which begins with
the big bang and then is approaching the Sitter or Minkowski
space-time asymptotically. Then, both the numerator and
denominator in R are completely dominated by the infinite
contributions from the asymptotic space-time region in the
future, so R is plausibly supposed to be equal to its
asymptotic value R = R∞ [
17
].
Next, as for τ , in the proccess of expansion of the universe,
it is natural to assume that the matter density is gradually
dilued in such a way that in the open universes τ associated
with the local excitations would be asymptotically
vanishing, τ = 0. Consequenly, in these space-times the residual
cosmological constant would take the value
Thus, in space-times with R = 0 or R∞ ∼ 0 like an
asymptotically Minkowskian space-time, the late-time acceleration
cannot be explained in the theory at hand. Moreover, it is
highly improbable that Eq. (22) happens to provide precisely
the current value ∼ (1meV )4 under a situation without
physically plausible reasons. It therefore seems that we need to
find a more sophisticated mechanism into consideration in
order to account for the late-time acceleration of the universe
when the the formula (22) does not match the current value
of the cosmological constant or takes the vanishing result.
We wish to present such a new mechanism in Sect. 4 on the
basis of the formulation explained in the next section. Before
doing it, let us briefly review our formulation of the nonlocal
approach to the cosmological constant [
12
].
3 Review of the R2 model
In this section, we will review a manifestly local and
generally coordinate invariant formulation [
23–26
] for a nonlocal
approach to the cosmological constant problem, in particular,
our most recent work [12].5
A manifestly local and generally coordinate invariant
action for our nonlocal approach consists of two parts
S = SG R + ST op,
where the gravitational action SG R with a bare cosmological
constant as well as a generic matter Lagrangian density
Lm = Lm (gμν , ) ( is a generic matter field), and the
topological action ST op are respectively defined as
SG R =
d4x √−g η(x )(R − 2 ) + Lm
(23)
4 The physical meaning of R has been already mentioned in Ref. [
17
].
5 See the related papers [
31–33
].
and
ST op =
where η(x ) is a scalar field of dimension of mass squared
and f (x ) is a function which cannot be a linear function. Let
us note that the scalar field η(x ) has no local degrees of
freedom except the zero mode because of the gauge symmetry
of the 4-form strength [
34
]. Moreover, Fμνρσ and Hμνρσ are
respectively the field strengths for two 3-form gauge fields
where the square brackets denote antisymmetrization of
enclosed indices. Finally, ε˚μνρσ and ε˚μνρσ are the Levi-Civita
tensor density defined as
and they are related to the totally antisymmetric tensors εμνρσ
and εμνρσ via relations
εμνρσ = √−g ε˚μνρσ , εμνρσ = (28)
Now let us derive all the equations of motion from the
action (23). First of all, the variation with respect to the
3form Bμνρ gives rise to the equation for a scalar field η(x ):
√−gε˚μνρσ .
ε˚μνρσ f ∂σ η(x ) = 0,
where f (x ) ≡ d df(xx) . From this equation, we have a classical
solution for η(x ):
η(x ) = η,
√−g(R − 2 ) + 41 ε˚μνρσ f Hμνρσ = 0.
!
Since we can always set Hμνρσ to be
Hμνρσ = c(x )εμνρσ = c(x )√−gε˚μνρσ ,
with c(x ) being some scalar function, Eq. (31) can be
rewritten as
R − 2
− f c(x ) = 0.
In order to take account of the cosmological constant
problem, let us take the space-time average of this equation, which
provides a constraint equation:
R = 2
+ f
η
where η is a certain constant. Next, taking the variation of
the scalar field η(x ) leads to the equation:
1 1
M P2l Gμν + 4 M P2l Rgμν = Tμν − 4 T gμν ,
(24)
(25)
(26)
(27)
(29)
(30)
(31)
(32)
(33)
(34)
where we have used Eq. (30).
The equation of motion for the 3-form Aμνρ gives the
Maxwell-like equation:
∇μ Fμνρσ = 0.
As in Hμνρσ , if we set
Fμνρσ = θ (x )εμνρσ = θ (x )√−gε˚μνρσ ,
with θ (x ) being a scalar function, Eq. (35) requires θ (x ) to
be a mere constant
θ (x ) = θ ,
where θ is a constant.
Finally, the variation with respect to the metric tensor
yields the gravitational field equation, i.e., the Einstein
equation:
where the energy-momentum tensor is defined by Tμν =
− √2−g δ(√δ−gμgνLm ) . In deriving Eq. (38), we have again used
Eq. (30). Furthermore, using Eqs. (36) and (37), this equation
can be simplified to be the form
where we have chosen η = M2P2l . Note that this gravitational
equation precisely coincides with the equation of motion
(13), which was obtained from the filter mechanism. In other
words, the action (23) provides us with a concrete realization
of the gravitational model with a high-pass filter obtained in
a rather phenomenological manner in Sect. 2.
As done in Sect. 2, if we separate the energy-momentum
tensor Tμν into two parts as in Eq. (19), we have the
equation of motion for the gravitational field, Eq. (20), with the
residual effective cosmological constant (21). The first term
in the RHS of Eq. (21) is radiatively stable since R is so.
Actually, as seen in Eq. (34), is a mere number and c(x )
is proportional to the flux of the 4-form which is the infrared
(IR) quantity, thereby implying that R is a radiatively
stable quantity. The second term 41 τ in the RHS of Eq. (21)
is obviously radiatively stable. Thus, our cosmological
constant e f f is a radiatively stable quantity so it can be fixed
by the measurement in a consistent manner.
The only disadvantage of this nonlocal approach to the
cosmological constant problem is that we confine ourselves
to the semiclassical approach where the matter loop effects
are included in the energy-momentum tensor while the
quantum gravity effects are completely ignored. In this context,
note that gravity is a classical field merely serving the
purpose of detecting the vacuum energy.
Next, let us therefore turn our attention to quantum gravity
effects [
12
]. From the 1-loop calculation, the dimensional
analysis and general covariance, it is easy to estimate the
loop effects from both matter and the gravitational fields.
For instance, the renormalization of the Newton constant and
the cosmological constant amounts to adding the following
effective action to the total action (23) up to the logarithmic
divergences which are so subdominant that they are irrelevant
to the argument at hand [
22
]:
Sq =
where M is a cutoff and the coefficients ai , bi (i = 0, 1, 2, . . .)
are O(1). As shown in Ref. [
12
], when we start with the action
S = SG R + ST op + Sq , we find that the effective
cosmological constant e f f in Eq. (21) is not radiatively stable because
of the presence of β(η), which means that the graviton loop
effects render the effective cosmological constant be
radiatively unstable.
To remedy this situation, we have added the
higherderivative R2 term and the corresponding topological term
[
12
]. Then, the total effective action constituting of four parts
is given by
ε˚μνρσ fˆ ∂σ ω(x ) = 0,
S = SG R + ST op + Sq + SR2 ,
where the last action SR2 is defined as
SR2 =
d4x √−gω(x )R2
1
4!
+
d4x
ε˚μνρσ M P2l fˆ(ω)Hˆμνρσ ,
where Hˆμνρσ ≡ 4∂[μ Bˆνρσ ].
As before, the variation of the total action (43) with respect
to the 3-form Bˆμνρ produces
which gives us a classical solution for ω(x ):
ω(x ) = ω,
where ω is a certain constant. Next, taking the variation of
the scalar field ω(x ) yields the field equation:
√−g R2 + 41! ε˚μνρσ M P2l fˆ Hˆμνρσ = 0.
Setting Hˆμνρσ again to be
Hˆμνρσ = cˆ(x )εμνρσ = cˆ(x )√−gε˚μνρσ ,
R2 − M P2l fˆ cˆ(x ) = 0.
R2 = M P2l fˆ (ω)cˆ(x ).
with cˆ(x ) being some scalar function, Eq. (47) can be cast to
Then, the space-time average of this equation leads to a new
constraint equation:
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
Note that R2 is radiatively stable since both fˆ (ω) and cˆ(x )
are radiatively stable.
With the help of Eqs. (30), (36), (37) and (46), the variation
with respect to the metric tensor yields the Einstein equation:
Taking the trace of this equation, one obtains
from which one can express the scalar curvature as
Note that this expression reduces to Eq. (55) up to a surface
term when one takes the space-time average, which
guarantees the correctness of our derivation.
Taking the square of Eq. (58) and then the space-time
average makes it possible to describe ( )2 in terms of R2
and τ
(
)2 =
Me4f f R2
16
1
− 16
τ 2 − τ 2
1 ⎛
− 16 ⎝
Me2f f τ ⎠
− 12ω
(57)
(58)
This expression clearly shows that is radiatively
stable since Me2f f turns out to be radiatively stable, R2 is also
radiatively stable as shown in Eq. (50), and the remaining
terms involving τ are radiatively stable as well as long as
is not equal to M12e2ωff . The radiative stability of ensures
that the effective cosmological constant e f f in Eq. (54) is
also radiatively stable even when the gravitational radiative
corrections are taken into account.
Before closing this section, we would like to comment on
two remarks. One of them is that in the high energy limit,
( )2 reduces to
( )2 → M1e46f f R2 + 116 τ 2, (60)
which is manifestly radiatively stable. The other remark is
that we see that using Eqs. (55) and (59), R is also radiatively
stable whose situation should be contrasted with the case
without the R2 term, for which R is not so.
4 Cosmic acceleration
One might be concerned that the higher-derivative term ω R2,
which was introduced in the action (44), would generate new
radiative corrections that also depend on ω, thereby
inducing new radiative corrections to the vacuum energy density
and consequently breaking its radiative stability. We will first
show that this is not indeed the case explicitly when the mass
of the scalaron is around 1meV . Nevertheless, it turns out
that the existence of the scalaron enables the cosmological
constant to have the currently observed value in the open
universes.
For this purpose, it is convenient to move from the
Jordan frame to the Einstein one since the higher-derivative R2
gravity is conformally equivalent to Einstein gravity with an
extra scalar field called “scalaron” in the Einstein frame. Of
course, one should not confuse the conformal metric with the
original physical metric since they generally describe
manifolds with different geometries and the final results should be
interpreted in terms of the original metric. However, as seen
shortly, it turns out that our conformal factor connecting with
the two metrics depends on the scalar curvature which does
not change significantly during the late-time acceleration, so
we have same results in the both frames.
With the help of Eqs. (30), (36), (37) and (46), the total
action (43) reads
S =
d4x √−g η(R − 2 ) + Lm − 21 θ 2
R − ˆ + ω R2 + Lm ,
+ α(η)R + β(η) + ω R2
=
In this regime, the scalaron φ decouples in the matter
Lagrangian:
e−2 23 Meφf f Lm (e− 3 Mef f g¯μν , ) ∼ Lm (g¯μν , ).
2 φ
(67)
Furthermore, the potential V (φ) can be expanded in the
Taylor series as
V (φ) = ˆ − 2
ˆ
2
3 Me f f
φ +
4 ˆ Me2f f
3 Me2f f + 24ω
φ2
2 8 ˆ
− 3
Me f f
9 Me3f f + 24ω
φ3 + O(φ4).
(68)
It has been recently shown in Ref. [
14
] that the
compatibility between the acceleration of the expansion rate of the
universe, local tests of gravity and the quantum stability of
the theory converges to select the relation
ω ∼
Then, using the formula (4), the 1-loop renormalized
cosmological constant up to a finite part is easily calculated to
be
φ,1−loop
ren
m4
= (8π )2 log
m2
M2
Me4f f
∼ (96π )2ω2 log
We have therefore obtained the current value of the
cosmological constant. The important point is that owing to the
6 Here we neglect the chameleon mechanism [
36
], which is irrelevant
to the present discussion.
7 We have selected M to be the current Hubble constant, i.e., M ∼
10−30 meV.
form of the potential (70), the higher-order contributions to
the cosmological constant are strictly suppressed by more ω1
factors, so these contributions are very tiny compared to the
1-loop result, thereby implying that the cosmological
constant in the theory at hand is radiatively stable even if the R2
term and its radiative corrections are incorporated.
5 Discussions
In our previous work [
12
], we have constructed a
nonlocal approach to the cosmological constant problem which
includes both gravity and matter loop effects. In this
approach, we are naturally led to incorporate the
higherderivative R2 term in the action in order to guarantee the
radiative stability of the cosmological constant under the
gravitational loop effects. The presence of such a
higherderivative term generally generates additional radiative
corrections which depend on the coefficient, that is, the coupling
constant in front of the higher-derivative term. It was
suggested in the previous paper [
12
] that such a renormalizable
term would keep the radiative stability when the mass of the
scalaron is very tiny. In the present paper, we have
explicitly shown this fact by moving from the Jordan frame to the
Einstein one, calculate the 1-loop renormalized
cosmological constant, and then examine higher-loop corrections via
the analysis of interaction terms in the low curvature regime.
In cosmology, it is known that the R2 gravity [
9
] and its
generalization, f (R) gravity [
35
], play a role in various
cosmological phenomena such as inflation, dark energy and
cosmological perturbations etc. In this article, we have therefore
investigated a possibility of applying our nonlocal approach
to the cosmological constant problem having the R2 term
[
12
] for the late-time acceleration of the expansion of the
universe. So far, the similar formulations have been already
applied to inflation in the early universe [
18–20,28
], but these
formulations ignore quantum effects coming from the
graviton loops so that they are not free from the radiative instability
issue. On the other hand, since our formulation includes both
gravity and matter loop effects, the cosmological constant is
completely stable under radiative corrections.
In the nonlocal approach to the cosmological constant
problem [
12,23,24
] or the model of vacuum energy
sequestering [
18–22,25
], the residual effective cosmological
constant is automatically very small in a large and old universe.
The main advantage of this formalism is that all the vacuum
energy corrections except a finite and tiny residual
cosmological constant are completely removed from the gravitational
equation. In particular, in the open universes with infinite
space-time volume, it seems to be natural to conjecture that
the residual cosmological constant is exactly zero or very tiny
compared to the present value of the cosmological constant.
In other words, the present mechanism seems to sequester
away the vacuum energy almost completely in case of the
open universes.
Then, it is a natural question to ask ourselves if the
nonlocal approach to the cosmological constant problem [
12
] is
compatible with inflation at an early epoch or the late-time
acceleration of the universe. In this article, we have answered
this question partially that our formalism can be applied to
the late-time acceleration where the key observation is that
the present mechanism cannot sequester away the vacuum
energy associated with the scalaron existing in the R2
gravity as an extra scalar field. When the mass of the scalaron is
about 1meV , which is strongly supported by the recent
analysis in [
14
], our formalism not only accounts for the current
cosmological constant given by (1meV )4 but also assures the
radiative stability of the vacuum energy density.
It is worthwhile to mention the relation between the
present work and the recent work [
14
]. In Ref. [
14
], it is
explicitly assumed that all matter contributions to the
vacuum energy vanish or contribute in a negligible way to the
vacuum energy at the energy scale μ associated with our
universe.8 Thus, the following tuning to the vacuum energy is
assumed:
2 0 M P2l + ρtrans
+
j
is true that our present work is closely related to these works
in that we rely on nonlocal operation and make use of the
higher derivative terms, but a big difference is that in our
work taking the space-time average of operators plays a
critical role to understand the cosmological constant problem.
Finally, let us comment on a future work. The
important problem is to explain the inflation at the early epoch
within the present framework. In the inflationary scenario,
it is postulated that the universe is dominated by a transient
large vacuum energy. However, our nonlocal approach
usually removes the large vacuum energy and leaves only small
one. At present, there seems to be a difficulty to reconcile
the both results without losing the radiative stability. In any
case, we wish to return to this problem in future.
Acknowledgements This work was supported by JSPS KAKENHI
Grant Number 16K05327.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
(75)
where the first term is the bare cosmological constant, ρtrans
denotes the energy density corresponding to all the phase
transitions such as the electro-weak phase transition, m j is
the mass of a particle of spin j , and the mass scale μ0 is
for instance chosen to be the electron mass, which is the
minimum mass of observed particles thus far, apart from the
neutrinos which might be of the right order of magnitude if
the neutrino masses are in the order 1meV . The appealing
point of our nonlocal approach to the cosmological constant
[
12
] is that this Eq. (75) is derived in a natural manner.
Furthermore, radiative corrections from gravitational field are
also sequestered away.
At this stage, let us refer to recent related works. Recently,
there has been a revival of higher derivative gravity theories
and nonlocal approaches to cosmology. For instance, in the
papers [
37–39
]9, analytic infinite derivative gravity theories
are proposed to resolve unsolved problems such as black hole
and big bang singularities at a classical level as well as
quantum level. Moreover, more recently, a detailed discussion of
conceptual aspects related to nonlocal gravity and its
cosmological consequences has been presented in the paper [
41
]. It
8 For simplicity, we neglect the classical energy density in the gravity
sector.
9 See a review paper [
40
] for further references.
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