Accelerated expansion of the Universe
filled up with the scalar gravitons
Yu. F. Pirogov
Theory Division, Institute for High Energy Physics, Protvino,
RU-142281 Moscow Region, Russia
Abstract
The concept of the scalar graviton as the source of the dark matter and dark energy
of the gravitaional origin is applied to study the evolution of the isotropic homo-
geneous Universe. A realistic self-consistent solution to the modified pure gravity
equations, which correctly describes the accelerated expansion of the spatially flat
Universe, is found and investigated. It is argued that the scenario with the scalar
gravitons filling up the Universe may emulate the LCDM model, reducing thus the
true dark matter to an artefact.
1 Introduction
According to the present-day cosmological paradigm our Universe is fairly isotropic,
homogeneous, spatially flat and experiences presently the accelerated expansion.
The conventional description of the latter phenomenon is given by the model with
the Λ-term and the cold dark matter (CDM).1 Nevertheless, such a description may
be just a phenomenological reflection of a more fundamental mechanism. A realistic
candidate on such a role is presented in the given paper.
In a preceding paper [2], we proposed a modification of the General Relativity
(GR), with the massive scalar graviton in addition to the massless tensor one.2 The
scalar graviton was put forward as a source of the dark matter (DM) and the dark
energy (DE) of the gravitational origin. In ref. [4], this concept was applied to study
the evolution of the isotropic homogeneous Universe. The evolution equations were
derived and the plausible arguments in favour of the reality of the evolution scenario
with the scalar gravitons were presented.
In the present paper, we expose an explicit solution to the evolution equations
in the vacuum, which gives the correct description of the accelerated expansion of
the spatially flat Universe. It is shown that the emulation of the LCDM model can
indeed be reached as it was anticipated earlier [4]. In Section 2, we first briefly
remind the evolution equations in the vacuum filled up only with the scalar gravi-
tons. Then the master equation for the Hubble parameter is presented. Finally, a
self-consistent solution of the latter equation, possessing the desired properties, is
found and investigated. In the Conclusion, the proposed solution to the DM and
DE problems is recapitulated.
1Hereof, the LCDM model. For a review on cosmology, see, e.g., ref. [1].
2For a brief exposition of such a modified GR, see ref. [3].
http://arxiv.org/abs/0704.1638v1
2 Accelerated expansion
Evolution equations We consider the isotropic homogeneous Universe without
the true DM. Besides, we neglect by the luminous matter missing thus the initial
period of the Universe evolution. Then, the vacuum evolution equations look like3
(ρs + ρΛ),
(ps + pΛ), (1)
with a(t) being the dynamical scale factor of the Universe, t being the comoving
time and ȧ = da/dt, etc. In the above, κ2 is proportional to the spatial curvature,
with κ2 = 0 for the spatially flat Universe. The parameter mP is the Planck mass.
On the r.h.s. of eq. (1), ρΛ and pΛ are the energy density and the pressure
corresponding to the cosmological constant Λ: ρΛ = −ρΛ = m2PΛ ≥ 0. Likewise, ρs
and ps are, respectively, the energy density and pressure of the scalar gravitons:
ρs = f
σ̇2 + 3
σ̇ + σ̈
+m2PΛs(σ),
ps = f
σ̇2 − 3
σ̇ − σ̈
−m2PΛs(σ). (2)
Here, fs = O(mP) is a constant with the dimension of mass entering the kinetic
term of the scalar graviton field σ. The latter in the given context looks like
σ = 3 ln
, (3)
with ã(t) being a nondynamical scale factor given a priori. The σ-field is defined up
to an additive constant. Without any loose of generality, we can fix the constant
by the asymptotic condition: σ(t) → 0 at t → ∞.
In eq. (2), we put
Vs + ∂Vs/∂σ ≡ m2P
Λs(σ) + Λ
, (4)
where Vs(σ) is the scalar graviton potential. More particularly, we put
Vs = V0 +
s (σ − σ0)2 +O
(σ − σ0)3
, (5)
with σ0 being a constant, fs(σ−σ0) the physical field of the scalar graviton and ms
the mass of the latter. By their nature, Λs and Λ are quite similar. To make the
division onto these two parts unambiguous we normalize Λs by an additive constant
so that Λs(0) = 0. Clearly, we get from eq. (2) that ρs + ps = f
2. Here, the
contribution of Λs exactly cancels what is quite similar to the relation ρΛ + pΛ = 0.
So, the contribution of Λs is a kind of the dark energy. In what follows, we put
σ0 = 0 and V0 = m
PΛ, with σ → 0 at t → ∞ becoming the ground state.
The nondynamical functions Vs and ã being the two characteristics of the vacuum
are not quite independent. More particularly, adopting the isotropic homogeneous
ansatz for the solution of the gravity equations, with only one dynamical variable a,
we tacitly put a consistency relation between ã and Vs. As a result, only one
combination of the two lines of eq. (1) is the true equation of evolution, with the
second independent combination giving just the required consistency condition.
3We refer the reader to ref. [4] for more details.
Master equation In what follows, we restrict ourselves by the case of the spa-
tially flat Universe, κ = 0. Subtracting the first line of eq. (1) from the second one
and accounting for eq. (2) we get the relation
Ḣ = −
ασ̇2, (6)
where H ≡ ȧ/a is the Hubble parameter and
α = 2
. (7)
We assume that α = O(1). Substituting σ̇ given by eq. (6) into the first line of
eq. (1) we get the integro-differential master equation for the Hubble parameter:
H2 = −1
Ḧ + 6HḢ
Λs(σ) + Λ
. (8)
where it is to be understood
−Ḣ(τ) dτ. (9)
Remind that we assume σ(t) → 0 at t → ∞. Equations (8) and (6) supersede the
pair of the original evolution equations (1).
Self-consistent solution Let us put in what follows Λs ≡ 0. This will be
justified afterwards. Iterating eq. (8), with Λ considered as a perturbation, we can
get the solution with any desired accuracy. In particular, substituting into the r.h.s.
of eq. (8) the solution H = α/t from the zeroth approximation (Λ = 0) we get the
first approximation as follows
O(1) at t → 0 ,
O(1/t3) at t → ∞, (10)
or otherwise
(tΛ/t)
2 + 1
α2/t2 at t/tΛ < 1,
α2/t2Λ at t/tΛ > 1,
being the characteristic time of the evolution of the Universe. Numerically, tΛ ∼
1010yr is of order the age of the Universe. Equation (11) is the basis for the quali-
tative discussion in what follows.
Integrating eq. (11) we get the scale factor as follows:
+ 1− ln
α ln(t/tΛ) at t/tΛ < 1,
αt/tΛ at t/tΛ > 1,
where a0 is an integration constant.
4 Explicitly, the scale factor looks like
(t/tΛ)
α at t/tΛ < 1,
exp(αt/tΛ) at t/tΛ > 1,
4To phenomenologically account for the effect of the initial inflation we could formally shift the origin
of time: t → t+ t0, with t0 > 0.
with tΛ bordering thus the epoch of the the power law expansion from the epoch of
the exponential expansion. Equation (13) gives the two-parametric representation
for the scale factor of the acceleratedly expanding Universe after the initial period.
With account for eq. (9) the σ-field behaves as
σ = −
(t/tΛ)
ξ(1 + ξ)1/4
2 ln(t/tΛ) at t/tΛ < 1,
tΛ/t at t/tΛ > 1.
Note that at tΛ → ∞ or, equivalently, Λ → 0 the integral above diverges and the
σ-field can not be normalized properly. Λ 6= 0 is thus necessary as a regulator in
the theory. Now, the consistency condition looks like
ã = a exp (−σ/3) ∼
(t/tΛ)
α−2/3 at t/tΛ < 1,
exp (αt/tΛ) at t/tΛ > 1.
Clearly, Λ should be already presupposed in ã. Note that in the case α = 2/3, the
parameter ã is approximately constant at t/tΛ < 1.
Substituting equations (11) and (6) into the first line of eq. (2) we can explicitly
verify that
. (17)
This is to be anticipated already from the relation ρΛ/m
P = Λ, as well as eq. (10)
and the first line of eq. (1). Clearly, ρs is positive. At t/tΛ < 1 we get from
equations (14) and (17) that ρsa
3 ∼ t3α−2. In the case α = 2/3, we have ρs ∼ 1/a3
as it should be for the true CDM. On the other hand, the pressure of the scalar
gravitons is as follows
ps = −
(tΛ/t)
(tΛ/t)2 + 1
3α(2/3 − α)/t2 − α/t2Λ at t/tΛ < 1,
(2α/t2Λ)(tΛ/t)
3 at t/tΛ > 1.
At the same conditions as before, the pressure is ps/m
P = −Λ/2, being near constant
though not zero as it should be anticipated for the true CDM. Nevertheless, we see
that the value α = 2/3 is exceptional in many respects. Conceivably, such a value
is distinguished by a more fundamental theory.
Introducing the critical energy density ρc = 3m
2, we get for the partial energy
densities Ωs = ρs/ρc and ΩΛ = ρΛ/ρc, respectively, of the scalar gravitons and the
Λ-term the following:
1 + (t/tΛ)2
1− (t/tΛ)2 at t/tΛ < 1,
(tΛ/t)
2 at t/tΛ > 1,
with ΩΛ = 1 − Ωs. Note that Ωs = ΩΛ = 1/2 at t/tΛ = 1. Presently, we have
Ωs/ΩΛ ≃ 1/3 and thus the respective time t, in the neglect by the effect of the
initial inflation, is somewhat larger tΛ.
Finally, the condition Λs = 0 adopted earlier can be justified as follows. First
of all, Λs is indeed negligible at t → ∞ due to σ → 0 and Λs(0) = 0. On the other
hand, at t ∼ tΛ we have |σ| ∼ 1 and hence Λs ∼ m2s . For Λs to be negligible in this
region, too, we should require ms ≤
Λ ∼ 1/tΛ. Nevertheless, in the early period
of evolution when |σ| > 1 the contribution of Λs may be significant. The parameter
α = 2/3 being fixed the theory may be terminated just by two mass parameters:
the ultraviolet mP and the infrared t
Λ or, otherwise, ms.
3 Conclusion
To conclude, let us recapitulate the proposed solution to the DM and DE problems
in the context of the evolution of the Universe. According to the viewpoint adopted,
there is neither true DM nor DE in the Universe (at least, in a sizable amount).
Instead, the field σ of the scalar graviton serves as a common source of both the DM
and DE of the gravitational origin. DM is represented by the derivative contribution
of σ, with DE being reflected by the derivativeless contribution. In this, the constant
part of the latter contribution corresponds to the conventional Λ-term, while the
σ-dependent part corresponds to DE. The latter is less important than the Λ-term
at present, becoming conceivably more crucial at the early time.
The self-consistent evolution of the Universe may be considered as the transition
of the “sea” of the scalar gravitons, produced in the early period, from the excited
state with |σ| > 1 to the ground state with σ = 0. The ground state is characterized
by the cosmological constant Λ which, in turn, predetermines the characteristic
evolution time of the Universe, tΛ ∼ 1/
Λ. The scenario is in the possession to
naturally describe the accelerated expansion of the spatially flat Universe, correctly
emulating thus the conventional LCDM model. The more complete study of the
scenario, the initial period of the evolution including, is in order.
The author is grateful to O. V. Zenin for the useful discussions.
References
[1] M. Trodden and S. M. Carroll, astro-ph/0401547.
[2] Yu. F. Pirogov, Phys. Atom. Nucl. 69, 1338 (2006) [Yad. Fiz. 69, 1374 (2006)];
gr-qc/0505031.
[3] Yu. F. Pirogov, gr-qc/0609103.
[4] Yu. F. Pirogov, gr-qc/0612053.
http://arxiv.org/abs/astro-ph/0401547
http://arxiv.org/abs/gr-qc/0505031
http://arxiv.org/abs/gr-qc/0609103
http://arxiv.org/abs/gr-qc/0612053
	Introduction
	Accelerated expansion
	Conclusion

  The concept of the scalar graviton as the source of the dark matter and dark
energy of the gravitaional origin is applied to study the evolution of the
isotropic homogeneous Universe. A realistic self-consistent solution to the
modified pure gravity equations, which correctly describes the accelerated
expansion of the spatially flat Universe, is found and investigated. It is
argued that the scenario with the scalar gravitons filling up the Universe may
emulate the LCDM model, reducing thus the true dark matter to an artefact.

Introduction
According to the present-day cosmological paradigm our Universe is fairly isotropic,
homogeneous, spatially flat and experiences presently the accelerated expansion.
The conventional description of the latter phenomenon is given by the model with
the Λ-term and the cold dark matter (CDM).1 Nevertheless, such a description may
be just a phenomenological reflection of a more fundamental mechanism. A realistic
candidate on such a role is presented in the given paper.
In a preceding paper [2], we proposed a modification of the General Relativity
(GR), with the massive scalar graviton in addition to the massless tensor one.2 The
scalar graviton was put forward as a source of the dark matter (DM) and the dark
energy (DE) of the gravitational origin. In ref. [4], this concept was applied to study
the evolution of the isotropic homogeneous Universe. The evolution equations were
derived and the plausible arguments in favour of the reality of the evolution scenario
with the scalar gravitons were presented.
In the present paper, we expose an explicit solution to the evolution equations
in the vacuum, which gives the correct description of the accelerated expansion of
the spatially flat Universe. It is shown that the emulation of the LCDM model can
indeed be reached as it was anticipated earlier [4]. In Section 2, we first briefly
remind the evolution equations in the vacuum filled up only with the scalar gravi-
tons. Then the master equation for the Hubble parameter is presented. Finally, a
self-consistent solution of the latter equation, possessing the desired properties, is
found and investigated. In the Conclusion, the proposed solution to the DM and
DE problems is recapitulated.
1Hereof, the LCDM model. For a review on cosmology, see, e.g., ref. [1].
2For a brief exposition of such a modified GR, see ref. [3].
http://arxiv.org/abs/0704.1638v1
2 Accelerated expansion
Evolution equations We consider the isotropic homogeneous Universe without
the true DM. Besides, we neglect by the luminous matter missing thus the initial
period of the Universe evolution. Then, the vacuum evolution equations look like3
(ρs + ρΛ),
(ps + pΛ), (1)
with a(t) being the dynamical scale factor of the Universe, t being the comoving
time and ȧ = da/dt, etc. In the above, κ2 is proportional to the spatial curvature,
with κ2 = 0 for the spatially flat Universe. The parameter mP is the Planck mass.
On the r.h.s. of eq. (1), ρΛ and pΛ are the energy density and the pressure
corresponding to the cosmological constant Λ: ρΛ = −ρΛ = m2PΛ ≥ 0. Likewise, ρs
and ps are, respectively, the energy density and pressure of the scalar gravitons:
ρs = f
σ̇2 + 3
σ̇ + σ̈
+m2PΛs(σ),
ps = f
σ̇2 − 3
σ̇ − σ̈
−m2PΛs(σ). (2)
Here, fs = O(mP) is a constant with the dimension of mass entering the kinetic
term of the scalar graviton field σ. The latter in the given context looks like
σ = 3 ln
, (3)
with ã(t) being a nondynamical scale factor given a priori. The σ-field is defined up
to an additive constant. Without any loose of generality, we can fix the constant
by the asymptotic condition: σ(t) → 0 at t → ∞.
In eq. (2), we put
Vs + ∂Vs/∂σ ≡ m2P
Λs(σ) + Λ
, (4)
where Vs(σ) is the scalar graviton potential. More particularly, we put
Vs = V0 +
s (σ − σ0)2 +O
(σ − σ0)3
, (5)
with σ0 being a constant, fs(σ−σ0) the physical field of the scalar graviton and ms
the mass of the latter. By their nature, Λs and Λ are quite similar. To make the
division onto these two parts unambiguous we normalize Λs by an additive constant
so that Λs(0) = 0. Clearly, we get from eq. (2) that ρs + ps = f
2. Here, the
contribution of Λs exactly cancels what is quite similar to the relation ρΛ + pΛ = 0.
So, the contribution of Λs is a kind of the dark energy. In what follows, we put
σ0 = 0 and V0 = m
PΛ, with σ → 0 at t → ∞ becoming the ground state.
The nondynamical functions Vs and ã being the two characteristics of the vacuum
are not quite independent. More particularly, adopting the isotropic homogeneous
ansatz for the solution of the gravity equations, with only one dynamical variable a,
we tacitly put a consistency relation between ã and Vs. As a result, only one
combination of the two lines of eq. (1) is the true equation of evolution, with the
second independent combination giving just the required consistency condition.
3We refer the reader to ref. [4] for more details.
Master equation In what follows, we restrict ourselves by the case of the spa-
tially flat Universe, κ = 0. Subtracting the first line of eq. (1) from the second one
and accounting for eq. (2) we get the relation
Ḣ = −
ασ̇2, (6)
where H ≡ ȧ/a is the Hubble parameter and
α = 2
. (7)
We assume that α = O(1). Substituting σ̇ given by eq. (6) into the first line of
eq. (1) we get the integro-differential master equation for the Hubble parameter:
H2 = −1
Ḧ + 6HḢ
Λs(σ) + Λ
. (8)
where it is to be understood
−Ḣ(τ) dτ. (9)
Remind that we assume σ(t) → 0 at t → ∞. Equations (8) and (6) supersede the
pair of the original evolution equations (1).
Self-consistent solution Let us put in what follows Λs ≡ 0. This will be
justified afterwards. Iterating eq. (8), with Λ considered as a perturbation, we can
get the solution with any desired accuracy. In particular, substituting into the r.h.s.
of eq. (8) the solution H = α/t from the zeroth approximation (Λ = 0) we get the
first approximation as follows
O(1) at t → 0 ,
O(1/t3) at t → ∞, (10)
or otherwise
(tΛ/t)
2 + 1
α2/t2 at t/tΛ < 1,
α2/t2Λ at t/tΛ > 1,
being the characteristic time of the evolution of the Universe. Numerically, tΛ ∼
1010yr is of order the age of the Universe. Equation (11) is the basis for the quali-
tative discussion in what follows.
Integrating eq. (11) we get the scale factor as follows:
+ 1− ln
α ln(t/tΛ) at t/tΛ < 1,
αt/tΛ at t/tΛ > 1,
where a0 is an integration constant.
4 Explicitly, the scale factor looks like
(t/tΛ)
α at t/tΛ < 1,
exp(αt/tΛ) at t/tΛ > 1,
4To phenomenologically account for the effect of the initial inflation we could formally shift the origin
of time: t → t+ t0, with t0 > 0.
with tΛ bordering thus the epoch of the the power law expansion from the epoch of
the exponential expansion. Equation (13) gives the two-parametric representation
for the scale factor of the acceleratedly expanding Universe after the initial period.
With account for eq. (9) the σ-field behaves as
σ = −
(t/tΛ)
ξ(1 + ξ)1/4
2 ln(t/tΛ) at t/tΛ < 1,
tΛ/t at t/tΛ > 1.
Note that at tΛ → ∞ or, equivalently, Λ → 0 the integral above diverges and the
σ-field can not be normalized properly. Λ 6= 0 is thus necessary as a regulator in
the theory. Now, the consistency condition looks like
ã = a exp (−σ/3) ∼
(t/tΛ)
α−2/3 at t/tΛ < 1,
exp (αt/tΛ) at t/tΛ > 1.
Clearly, Λ should be already presupposed in ã. Note that in the case α = 2/3, the
parameter ã is approximately constant at t/tΛ < 1.
Substituting equations (11) and (6) into the first line of eq. (2) we can explicitly
verify that
. (17)
This is to be anticipated already from the relation ρΛ/m
P = Λ, as well as eq. (10)
and the first line of eq. (1). Clearly, ρs is positive. At t/tΛ < 1 we get from
equations (14) and (17) that ρsa
3 ∼ t3α−2. In the case α = 2/3, we have ρs ∼ 1/a3
as it should be for the true CDM. On the other hand, the pressure of the scalar
gravitons is as follows
ps = −
(tΛ/t)
(tΛ/t)2 + 1
3α(2/3 − α)/t2 − α/t2Λ at t/tΛ < 1,
(2α/t2Λ)(tΛ/t)
3 at t/tΛ > 1.
At the same conditions as before, the pressure is ps/m
P = −Λ/2, being near constant
though not zero as it should be anticipated for the true CDM. Nevertheless, we see
that the value α = 2/3 is exceptional in many respects. Conceivably, such a value
is distinguished by a more fundamental theory.
Introducing the critical energy density ρc = 3m
2, we get for the partial energy
densities Ωs = ρs/ρc and ΩΛ = ρΛ/ρc, respectively, of the scalar gravitons and the
Λ-term the following:
1 + (t/tΛ)2
1− (t/tΛ)2 at t/tΛ < 1,
(tΛ/t)
2 at t/tΛ > 1,
with ΩΛ = 1 − Ωs. Note that Ωs = ΩΛ = 1/2 at t/tΛ = 1. Presently, we have
Ωs/ΩΛ ≃ 1/3 and thus the respective time t, in the neglect by the effect of the
initial inflation, is somewhat larger tΛ.
Finally, the condition Λs = 0 adopted earlier can be justified as follows. First
of all, Λs is indeed negligible at t → ∞ due to σ → 0 and Λs(0) = 0. On the other
hand, at t ∼ tΛ we have |σ| ∼ 1 and hence Λs ∼ m2s . For Λs to be negligible in this
region, too, we should require ms ≤
Λ ∼ 1/tΛ. Nevertheless, in the early period
of evolution when |σ| > 1 the contribution of Λs may be significant. The parameter
α = 2/3 being fixed the theory may be terminated just by two mass parameters:
the ultraviolet mP and the infrared t
Λ or, otherwise, ms.
3 Conclusion
To conclude, let us recapitulate the proposed solution to the DM and DE problems
in the context of the evolution of the Universe. According to the viewpoint adopted,
there is neither true DM nor DE in the Universe (at least, in a sizable amount).
Instead, the field σ of the scalar graviton serves as a common source of both the DM
and DE of the gravitational origin. DM is represented by the derivative contribution
of σ, with DE being reflected by the derivativeless contribution. In this, the constant
part of the latter contribution corresponds to the conventional Λ-term, while the
σ-dependent part corresponds to DE. The latter is less important than the Λ-term
at present, becoming conceivably more crucial at the early time.
The self-consistent evolution of the Universe may be considered as the transition
of the “sea” of the scalar gravitons, produced in the early period, from the excited
state with |σ| > 1 to the ground state with σ = 0. The ground state is characterized
by the cosmological constant Λ which, in turn, predetermines the characteristic
evolution time of the Universe, tΛ ∼ 1/
Λ. The scenario is in the possession to
naturally describe the accelerated expansion of the spatially flat Universe, correctly
emulating thus the conventional LCDM model. The more complete study of the
scenario, the initial period of the evolution including, is in order.
The author is grateful to O. V. Zenin for the useful discussions.
References
[1] M. Trodden and S. M. Carroll, astro-ph/0401547.
[2] Yu. F. Pirogov, Phys. Atom. Nucl. 69, 1338 (2006) [Yad. Fiz. 69, 1374 (2006)];
gr-qc/0505031.
[3] Yu. F. Pirogov, gr-qc/0609103.
[4] Yu. F. Pirogov, gr-qc/0612053.
http://arxiv.org/abs/astro-ph/0401547
http://arxiv.org/abs/gr-qc/0505031
http://arxiv.org/abs/gr-qc/0609103
http://arxiv.org/abs/gr-qc/0612053
	Introduction
	Accelerated expansion
	Conclusion
