ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 12, с. 2198-2201
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
MINIMAL METAGRAVITY VERSUS DARK MATTER AND/OR DARK ENERGY
© 2007 Yu. F. Pirogov
Institute for High Energy Physics, Protvino, Russia Received December 12, 2006
The minimal metagravity theory, explicitly violating the general covariance but preserving the unimodular one, is applied to study the evolution of the isotropic homogeneous Universe. The massive scalar graviton, contained in the theory in addition to the massless tensor one, is treated as a source of the dark matter and/or dark energy. The modified Friedmann equation for the scale factor of the Universe is derived. The question whether the minimal metagravity can emulate the LCDM concordance model, valid in General Relativity, is discussed.
PACS:04.20.Cv, 95.35.+d, 98.80.-k
1. INTRODUCTION
According to the present-day cosmological paradigm, given by General Relativity (GR) and the standard cosmology, the reasonable description of our Universe in total is achieved in the so-called LCDM concordance model (for a review of cosmology, see, e.g., [ 1]). In accordance with the model, the Universe is spacially flat, fairly isotropic and homogeneous being filled predominantly with the dark energy, accounted for by the A term, as well as with the cold dark matter in the energy proportion roughly 3 : 1. The energy fraction of the luminous matter is almost negligible. The nature of the dark energy and the dark matter seems to be the main puzzle of the contemporary physics.
Thus, all the sources of the dark substances, including the indirect ones, are to be investigated. With this in mind, we study in the present paper whether the above substances (or the parts of them) can be mimicked by a modification of GR, namely, the minimal metagravity theory proposed earlier [2]!). Due to the explicit violation of the general covariance (GC) to the unimodular covariance (UC), such a theory describes the massive scalar graviton in addition to the massless tensor one. The idea is to try and associate the scalar graviton with the dark matter and/or dark energy. In Section 2, the compendium of the minimal metagravity theory is given. In Section 3, the evolution of the isotropic homogeneous Universe in the framework of such a theory is considered, and the modified Friedmann equation for the scale factor of the Universe is derived. It is argued then in
Resume that the minimal metagravity is not explicitly inconsistent with the LCDM concordance model motivating thus for the further study.
2. MINIMAL METAGRAVITY
To begin with, let us present the short compendium of the metagravity theory. Under the latter, we understand generally the effective field theory of the metric revealing, due to the explicit GC violation, the extra physical degrees of freedom contained in the metric besides those for the massless tensor graviton. In the case of the minimal violation, to be used in what follows, the metagravity preserves the residual UC and describes for this reason only one additional particle, the massive scalar graviton. The generic action of such a minimal metagravity is as follows:
5 =
У (yLg{g^v)
) + La {g^v ,a) +
(1)
+ Lm
where g^v is the dynamical metric, is the generic matter field, and
1, 9 a = - m -. 2 9
(2)
1)For a brief exposition of the theory, see [3].
In the above, g = det g^v and g is an absolute (non-dynamical) scalar density of the same weight as g. Depending on the ratio of the two similar scalar densities, a is the scalar and thus can serve as the Lagrangian field variable. This field corresponds to the metric compression waves and is to be treated as representing the scalar graviton.
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MINIMAL METAGRAVITY VERSUS DARK MATTER
2199
In Eq. (1), Lg is the generally covariant La-grangian for the tensor graviton being chosen conventionally in the extended Einstein—Hilbert form:
Lg = ~^mlR{gßV) + A,
(3)
La = \flda ■ da - Va{a).
Here, f a is a constant with the dimension of mass and Va is the potential producing the mass for the scalar graviton. The dependence of a on g explicitly violates the part of GC, namely the local scale covariance, still preserving the residual UC. A priori, one expects f a = O(mP). Also, one expects that Va, though being nonzero, is suppressed. Finally, Lm is the matter Lagrangian possessing, generally, only the residual UC, too.
Varying the action (1), with respect to g^v, g being fixed, one arrives at the minimal metagravity equation:
1
with
The term T^V^ can be treated as the scalar-graviton contribution to the dark matter and/or dark energy. Finally,
PA5V , (10)
T (a) = _
ßv
where R = R^v is the Ricci scalar, with R^v being the Ricci curvature, and A is the cosmologi-cal constant. Also, mP = (8nGN)-l/2 is the Planck mass, with GN being the Newton's constant. La is the scalar-graviton Lagrangian looking in the lowest order on the derivatives as follows:
with pA + pA = 0 and pa = mP A, is the vacuum contribution to the dark energy. Under A > 0, it produces the negative pressure. Due to the Bianchi identity, = 0, and the property V\g^v = 0,
the energy—momentum of the matter and the scalar gravitons is conserved covariantly:
(4)
vM(Tmv+Tßv ) = o,
(11)
whereas the energy—momentum of the matter alone,
/n(m) j
T^v , ceases to conserve.
3. MODIFIED FRIEDMANN EQUATION
In the properly chosen observer's coordinates xß = (t, p, d, p) the Friedmann—Robertson—Walker solution for the interval in the isotropic homogeneous Universe is
(12)
— m2 {rffi + T^P + Tßt^ ) (5)
G^ — R^v — -Rgtw (6)
being the gravity tensor. The r.h.s. of Eq. (5) is the total energy—momentum of the nontensor-graviton origin, produced by the matter and the scalar gravitons, plus the vacuum energy. TjV ^ is the matter energy—momentum tensor including, if required, the real dark-matter contribution, too. For the matter as the continuous medium, of interest in cosmology, one has
T^} = (pm + Pm)U^Uv - Pm9pv, (7)
with pm being the energy density, pm being the
pressure, and u^ being the medium 4-velocity. T^^ is the scalar-graviton contribution:
ds2 = dt2 - a2(t) x
dp2 + p2(d02 + sin2 Odip2)
1 — K2p2
with k being a constant with the dimension of mass. This interval is form-invariant relative to the shift of the origin of the spacial coordinates, reflecting the isotropy and homogeneity of the Universe. Conventionally, one can rescale the unit of mass so that k2 = = k, with k = ±1, 0. The last three cases correspond, respectively, to the spacially closed, open, and flat Universe. In Eq. (12), the spacial factor 1/(1 — K2p2) is the geometrical one, given a priori, while the temporal scale factor a(t) is the dynamical one to be determined by the gravity equations.
Choosing the new radial variable r:
P
1 + k2 r2/4
one gets
ds2 = dt2 -
a2(t)
(1 + k2 r2/4)2
dx2
(13)
(14)
T$ = fa (d^ad^a - ]-da • dag^ + (8) with x* = (x° = t, {xm} = x), m = 1,2,3, and r2 = V 2 ' = x2. In other terms, the metric looks like
+ Vag^v + (f 2V • Va + Vj)g^,
goo = 1, gmo = °
(15)
with V'a = dVa/da, the covariant derivative Vßa = 1
= dßa, and
gmn —
a2(t)
(1 + k2 r2/4)2
5m
a
\TI9
(9)
with y^g = a3/(l + K2r2/4)3. These coordinates will be understood in what follows.
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ftŒPHAfl OH3HKA TOM 70 № 12 2007
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PIROGOV
From Eq. (8), one gets
Tm0 = fadmVV,
where a = da/dt and
a3(t)
a = In ■
(16)
(17)
Hence, generally, T^ = 0. On the other hand due to
the isotropy, there should fulfil Gm0 = 0 and T^ =
= 0. This requires T^ = 0, too, and thus dma = 0. To achieve this the spacial parts of g and g should coincide, so that
a3(t)
(18)
(1 + K2r2/4)3'
with a(t) being a free temporal factor. Altogether one has
a(t)
a(t) = 3ln
a(t) '
(19)
Gn = 3
+
1
mt
-P,
(20)
m
m n ,
with à = d?a/dt2, and p and p being the total energy density and pressure, respectively:
p = Pm + pa + Pa, (21)
P = Pm + Pa + Pa (remind that pA = -pA). The continuous medium is taken to be nonrelativistic: u° = 1, um = 0. The energy density and pressure for the scalar gravitons are formally defined as for the continuous medium:
1
Ta0 = f
+ 3-<7 + •& ) + (Va + v;) = Pa, 2 a
(22)
T m _
+ (Va + va )
¿m = -Pa S.
m n i
with the effective "equation of state"
Pa + Pa = f>2. (23)
Note that pa and Pa are coordinate dependent, in distinction to the scalars pm and Pm. In the above, one has
<7 = 3
- H
(24)
a = ^\a--(a-\2-H
aa
with H = a/a, and use is made of the relation
V • Va = ä + 3-à.
a
With account for Eqs. (22) and (24), the two equations (20) substitute the similar equations valid in GR. The first of equations (20), the modified Friedmann equation, determines the scale factor a(t), with the second equation giving the consistency condition. Introducing the "critical" energy density
oi a pc = 3 mp -a
(26)
one can bring the modified Friedmann equation to the form
,2
= 1 + ^7- (27)
Q = P
Pc
a2'
The minimal metagravity equation (5) results now in the two following equations:
22
Further, differentiating the first of equations (20) and combining the result with the second equation, one can substitute the latter by the continuity condition:
p + 3-(p + p) = 0, a
(28)
with pa + pa = fla2 and pA + Pa = 0.
To really solve these equations one should specify the free functions entering the theory. For the continuous medium, there are conventionally two extreme cases: the dust
Po
Q 5 3
Pm
Pm = 0,
(29)
and the radiation
_ o _ PO Pm — ¿Pm — a ■ a4
(30)
For the scalar graviton, the good starting point would conceivably be the assumption aa = const and thus H = 0.As for the potential Va, little can be said about it a priori, and probably it should be looked for by the trial-and-error method. With these caveats, the equations above are ready for use in working out the cosmological scenarios in the framework of the minimal metagravity.
4. RESUME
The above system of the modified Friedmann equation plus the continuity condition is much more intricate compared
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