Convergence of Scalar-Tensor theories toward General Relativity and Primordial Nucleosynthesis

21 Pages·0.24 MB·English

by  A. Serna

Convergence of Scalar-Tensor theories toward
General Relativity and Primordial Nucleosynthesis
A. Serna†, J.-M. Alimi‡, and A. Navarro¶
2
Dept. F´ısica y Computacio´n, Universidad Miguel Hern´andez, , E03202-Elche,Spain
0 †
0 LAEC (CNRS-UMR 8631),Observatoire de Paris-Meudon,F92195-Meudon,France
‡
2 Dept. F´ısica, Universidad de Murcia, E30071-Murcia,Spain
¶
n
a
Abstract. In this paper, we analyze the conditions for convergence toward General
J
Relativity of scalar-tensor gravity theories defined by an arbitrary coupling function
5
1 α (in the Einstein frame). We show that, in general, the evolution of the scalar field
(ϕ) is governedby twoopposite mechanisms: anattractionmechanismwhichtends to

1
drivescalar-tensormodels towardEinstein’s theory,anda repulsionmechanismwhich
v
9 has the contrary effect. The attraction mechanism dominates the recent epochs of
4 the universe evolution if, and only if, the scalar field and its derivative satisfy certain
0
boundary conditions. Since these conditions for convergence toward general relativity
1
depend on the particular scalar-tensortheory used to describe the universe evolution,
0
2 the nucleosynthesis bounds on the present value of the coupling function, α0, strongly
0 differ from some theories to others. For example, in theories defined by α ϕ
/ ∝|−19 |
c analytical estimates lead to very stringent nucleosynthesis bounds on α0 (< 10 ).
q By contrast, in scalar-tensor theories defined by α ϕ much larger lim∼its on α0
r- (<10−7) are found. ∝
g
∼
:
v
i
X PACS numbers: 04.50.+h, 04.80.+z,98.80.Cq, 98.80.Hw
r
a
Convergence of Scalar-Tensor theories 2
1. Introduction
Scalar-tensor (ST) gravity theories [1, 2, 3] have become a focal point of interest in
many areas of gravitational physics and cosmology. They provide the most natural
generalizations of General Relativity (GR) by introducing an additional scalar field, φ,
the dynamical importance of which is determined by an arbitrary coupling function
ω(φ). Indeed, most recent attempts at unified models of fundamental interactions, i.e.,
string theories [4] predict the existence of scalar partners to the tensor gravity of GR,
and would have ST gravity as their low energy limit. In addition, ST theories are
important in cosmology because they provide a natural (non-fine tuned) way of exiting
the inflationary epoch.
Solar system experiments (time delay in the Viking Mars data, Lunar Laser
Ranging, etc.) can put limits on the present deviation of ST theories with respect
to GR. It is admitted [5] that such experiments impose the constraint ω > 500 to the
0
present value of the coupling function and, therefore, that ST theories are at present
very close to General Relativity. This limit on ω does not necessarily imply that the
0
universe evolution is, at any time, very close to that found in GR. It has been shown
[6, 7] that, in some ST theories, the cosmological evolution drives the scalar field toward
a state indistinguishable from GR. Within this ’attraction mechanism’, the scalar field
canplayanimportantroleonlyinearlycosmologybecause, afterwards, itevolvestoward
a state with a vanishingly small scalar contribution.
The dynamical importance of the scalar field in the early universe can be checked
by means of the observed abundance of light elements, which has to be explained as an
outcome of the primordial nucleosynthesis process (PNP). Contrary to the weak field
limit, there is not a commonly accepted PNP constraint on ω . Some authors [8, 9]
0
have recently found that, in the framework of some ST theories, ω > 107 is required
0
∼
to obtain the observed primordial abundance of light elements. Other authors [10, 11]
have instead found that the PNP test typically imposes the limit ω > 1020.
0
∼
In this paper we will first reexamine the attraction mechanism of Refs. [6, 7] and,
then, we will investigate the reason for the enormous discrepancy (thirteenth orders of
magnitude) in the PNP bound on ω obtained by different authors. We will show that,
0
in general, the evolution of the scalar field is governed by two opposite mechanisms:
an attraction and a repulsion mechanism. The attraction mechanism dominates the
recent epochs of the universe evolution only if the scalar field and its derivative satisfy
certain boundary conditions which depend on the particular scalar-tensor theory used
to describe the universe evolution. We will also apply this generalized formalism to
the theories considered in Ref. [11], where the coupling function was restricted to be a
monotonic function of time. We will show that the nucleosynthesis bounds numerically
obtained in [11] (ω > 1020) are in close agreement with the analytical estimates for
0
∼
these theories. When the same arguments are applied to other ST theories (as those of
Refs. [6, 7]), one obtains much less stringent bounds onω . Consequently, the particular
0
ST theory used to describe the universe evolution has a crucial importance on the PNP
Convergence of Scalar-Tensor theories 3
limit on ω and, therefore, it is not possible to establish a general and unique limit for
0
all ST models.
The plan of the paper is as follows. We begin outlining the scalar-tensor theories
as well as the two frames usually considered to build up cosmological models in their
framework (Sec. 2). An autonomous evolution equation for the scalar field is then
obtained in Sec. 3 both for the Jordan and the Einstein frame. Using this equation,
and the particular family of scalar-tensor theories specified in Sec. 4, we then analyze
the evolution of the scalar field both in the radiation-dominated epoch (Sect. 5) and
the matter-dominated epoch (Sect. 6). Our results are then applied to estimate the
nucleosynthesis bounds on ω for this family of theories (Sec. 7). Finally, conclusions
0
and a summary of our results are given in Sec. 8.
2. Scalar-Tensor Gravity Theories
2.1. The Jordan frame
We consider a scalar-tensor gravity theory in which the gravitational interaction is
carried by the metric g and an additional massless scalar field, φ. Since these fields
µν
are measured through laboratory clocks and rods, which are made of atoms and are
essentially based on a non-gravitational physics, the action describing a scalar-tensor
gravity theory must keep unaltered the basic laws of non-gravitational interactions (as,
e.g., statistical physics and electrodynamics). When this fact is taken into account,
the resulting space-time units are termed as ”atomic”, ”Jordan” or ”physical” units
[12, 13, 14, 15].
Using the Jordan frame, the most general action describing a massless scalar-tensor
theory of gravitation is [1, 2, 3]
1 ω(φ)
S = (φ φ φ,µ)√ gd4x+S (1)
,µ M
16π R− φ −
Z
where is the curvature scalar of the metric g , g det(g ), φ is the scalar field, and
µν µν
R ≡
ω(φ) is an arbitrary coupling function.
The variation of Eq. (1) with respect to g and φ leads to the field equations:
µν
1 8π ω 1
g = T (φ φ g φ φ,α)
Rµν − 2 µνR − φ µν − φ2 ,µ ,ν − 2 µν ,α
1
(φ g φ) (2a)
,µ;ν µν
− φ − ⊓⊔
dω
(3+2ω) φ = 8πT φ φ,α (2b)
,α
⊓⊔ − dφ
which satisfy the usual conservation law
Tµν = 0 (3)
;ν
where Tµν is the energy-momentum tensor and φ gµνφ .
,µ;ν
⊓⊔ ≡
Convergence of Scalar-Tensor theories 4
For a homogeneous andisotropic universe, theline-element hasa Robertson-Walker
form and the energy-momentum tensor corresponds to that of a perfect fluid. The field
equations (2a) and (2b) then become
8π c2K R˙2 ωΦ˙2 R˙Φ˙
ρ = + + (4a)
3Φ R2 R2 − 6 Φ2 RΦ
8πG R¨ R˙ Φ˙ Φ¨ 2ωΦ˙2
(ρ+3P/c2) = 2 + + + (4b)
− 3Φ R RΦ Φ 3 Φ2
R˙ 1 dω
Φ¨ +3 Φ˙ = [8πG(ρ 3P/c2) Φ˙2] (4c)
R (3+2ω) − − dΦ
where K = 0, 1, Φ Gφ, R(t) is the scale factor, ρ and P are the energy-mass density
± ≡
and pressure, respectively, and dots mean time derivatives. In addition, we have the
usual conservation equation:
d(ρR3)+(P/c2)dR3 = 0 (5)
which ensures that the standard laws of non-gravitational interactions are not modified
by the presence of a scalar field.
2.2. The Einstein frame
When the metric is assumed to be measured through purely gravitational clocks and
rods, the space-time units are termed as ”Einstein” or ”spin” units. In this frame, the
general action describing a massless scalar-tensor theory can be obtained from Eq. (1)
by a conformal transformation
g = A2(ϕ)g (6a)
µν µ∗ν
A2(ϕ) = (Φ) 1 (6b)
−
where A(ϕ) is an arbitrary function related to ω(Φ) by
dlnA
α = (3+2ω) 1/2 = (7)
−
dϕ
Using Eqs. (1), (6a), (6b) and (7), one obtains
c4 d4x
S = ( 2ϕ ϕ,µ)√ g +S (8)
∗ 16πG R∗ − ,µ − ∗ c M∗
Z
∗
where G is Newton’s constant and asterisks denote quantities expressed in Einstein
∗
units. Since our measures are based on non-purely gravitational rods and clocks,
quantities written in the Einstein frame are not observable. Comparison between theory
and observations must be then performed by using the Jordan frame.
From the action (8), the Einstein field equations are:
1
= 2ϕ ϕ +8πG (T T g ) (9a)
R∗µν ,µ ,ν ∗ µ∗ν − 2 ∗ µ∗ν
ϕ = 4πG α(ϕ)T (9b)
⊓⊔∗ − ∗ ∗
1/2
where Tµ∗ν = 2g− δSm/δgµ∗ν is the energy-momentum tensor in Einstein units.
∗
Convergence of Scalar-Tensor theories 5
If we consider a homogeneous and isotropic universe, the field equations (9a)-(9b)
become
1 d2R dϕ 2
3 ∗ = 4πG (ρ +3P )+2 (10a)
− R dt2 ∗ ∗ ∗ dt !
∗ ∗ ∗
2 2
1 dR K dϕ
3 ∗ +3 = 8πG ρ + (10b)
R2 dt ! R2 ∗ ∗ dt !
∗ ∗ ∗ ∗
d2ϕ 1 dR dϕ
+3 ∗ = 4πG α(ϕ)(ρ 3P ) (10c)
dt2 R dt dt − ∗ ∗ − ∗
∗ ∗ ∗ ∗
and the ’conservation’ equation is modified to
d(ρ R3)+P d(R3) = (ρ 3P )R3da(ϕ) (11)
∗ ∗ ∗ ∗ ∗ − ∗ ∗
Since the mass-energy is not conserved in Einstein units, the basic laws of non-
gravitational physics are modified in this frame (see [16, 17, 18] for a reformulation of
nuclear reaction rates and thermodynamics in Einstein units).
3. Decoupled evolution of the Jordan scalar field
In the form given above, the time evolution of the scale factor and the scalar field are
coupled both in the Jordan and the Einstein frame. Previous works [6, 7] have found
that, byintroducing anappropriatechangeofvariables, itispossibletofindanevolution
equation for the Einstein scalar field which is independent of the cosmic scale factor.
We will now show that it is also possible to find such a decoupled evolution equation
for the Jordan scalar field.
Let us define the functions
1
ψ lnΦ (12a)
≡ 2
P/c2
γ (12b)
≡ ρ
3c2KΦ
ǫ (12c)
≡ 8πGρR2
W (3+2ω)/3 (12d)
≡
The Jordan evolution equations of R and Φ then become:
R¨ R˙ 4πGρ
+ ψ˙ +ψ¨+2Wψ˙2 = (1+3γ) (13a)
R R − 3Φ
8πGρ R˙ 2
(1 ǫ) = +ψ˙ Wψ˙2 (13b)
3Φ − R ! −
R˙ 8πGρ 1 dW
2ψ¨+4ψ˙2 +6 ψ˙ = (1 3γ) ψ˙2 (13c)
R 3ΦW − − W dψ
In order to obtain a decoupled evolution equation for ψ, we will define a ’time’
parameter, p, as:
R˙
˙
dp = h dt; h = +ψ (14)
c c
R
Convergence of Scalar-Tensor theories 6
In terms of these variables, and denoting f df/dp, Eqs. (13a)-(13c) reduce to:
′
≡
h 4πGρ
′c = ψ 1 2Wψ2 (1+3γ) (15)
′ ′
h − − − 3Φh2
c c
8πGρ
h2(1 Wψ2) = (1 ǫ) (16)
c − ′ 3Φ −
h
ψ + ′cψ ψ2 +3ψ =
′′ ′ ′ ′
h −
c
4πGρ1 3γ 1 dW
= − ψ2 (17)
′
3Φh2 W − 2W dψ
c
and, using Eqs. (16) and (15) to eliminate h and h , respectively, we finally obtain:
c ′c
2(1 ǫ)
− ψ + (3 3γ 4ǫ)ψ =
′′ ′
1 Wψ2 − −
′
−
1 3γ 1 ǫ 1 dW
= − − ψ2 (18)
′
W − 1 Wψ′2! W dψ
−
which is an evolution equation for ψ = (1/2)lnΦ independent of the evolution of the
cosmic scale factor. This equation is the analogous, in the Jordan frame, to Eq. (3.15)
of [6]. Therefore, we findthat a decoupled evolution ofthe scalar field is not anexclusive
feature of the Einstein frame.
In order to find the well-known decoupled evolution equation for the Einstein scalar
field, we will first multiply Eq. (18) by W1/2(1 Wψ2)/2(1 ǫ). We obtain:
′
− −
2(1 ǫ)
− (W1/2ψ ) =
′ ′
1 Wψ2
′
−
1 3γ 4
= − 3(1 γ ǫ)W1/2ψ (19)
′
W1/2 − − − 3
We can now change to the Einstein frame by performing the conformal
transformation (6a)-(6b) with
dϕ
= √3W = α 1 (20)
−
dψ − −
Eq. (19) then becomes:
2(1 ǫ) 4
− ϕ +(1 γ ǫ)ϕ = α(1 3γ) (21)
′′ ′
3 ϕ2 − − 3 − −
′
−
which agrees with Eq. (3.15) of [6].
We also note that, by defining H and ρ through
∗ ∗
Φ1/2H = h ; ρ = Φ2ρ , (22)
c
∗ ∗
Eq. (16) yields
8πGρ 3 ϕ2
′
∗ = − (23)
H2 1 ǫ
−
∗
From this equation we see that, when ǫ = 0, the local positivity of the energy
density implies that
ϕ2 3. (24)
′
≤
Convergence of Scalar-Tensor theories 7
Most papers analyzing the convergence toward General Relativity of scalar-tensor
theories are based on the Einstein frame. In order to make easier the comparison of our
results with those founds in previous works, we will use hereafter the Einstein frame.
Nevertheless, it must be noted that most of our conclusions can also be found by using
the Jordan evolution equation (18).
4. The coupling function
The time evolution of scalar-tensor theories can be studied only after specifying a
functional form of the coupling function. Barrow and Parsons [19] have noted that
most expressions used in the literature for (3+2ω) can be classified into three different
families of theories:
1
a) Theories-1: 3+2ω = (δ > 1/2) (25a)
B Φ 1 δ
1
| − |
1
b) Theories-2: 3+2ω = (δ > 1/2) (25b)
B lnΦ δ
1
| |
1
c) Theories-3: 3+2ω = (δ > 0) (25c)
B Φδ 1
1
| − |
where B is an arbitrary positive constant.
1
The three theories defined by Eqs. (25a)-(25c) imply very different behaviours of
the early universe, which have been analyzed in detail by Barrow and Parsons [19]. The
first class of theories has also been studied by Garc´ıa-Bellido and Quir´os [20], Serna and
Alimi [21], Comer et al. [22], and Navarro et al. [23]. However, it is important to note
that all these theories have similar behaviours in the limit close to General Relativity
(Φ 1). As a matter of fact, since lnΦ Φ 1 and Φδ 1 δlnΦ, we can take
→ ≃ − − ≃
1 1
3+2ω = = (δ > 1/2) (26)
B lnΦ δ B 2ψ δ
1 1
| | | |
to represent the way in which these three types of theories approach the limit of GR.
Using the coupling function given by Eq. (26), integration of Eq. (20) yields
sign(ψ)
ϕ = 2ψ (2 δ)/2 (δ < 2) (27)
−
−(2 δ)B1/2| |
− 1
where we have normalized the integration constant so that ϕ = 0 corresponds to Φ = 1
(or ψ = 0). Note that, according to Eq. (27), sign(ϕ) = sign(ψ).
−
By introducing Eq. (27) into Eq. (26), we obtain the Einstein form of the coupling
function:
δ
α = B2|ϕ|2−δ = κ(ϕ)|ϕ| (28)
where B2 ≡ B11/(2−δ)(2−δ)δ/(2−δ) > 0, and
2(δ 1)
κ(ϕ) = B2|ϕ| 2−−δ (29)
Convergence of Scalar-Tensor theories 8
5. Radiation-dominated evolution
In the radiation-dominated epoch, the state equation is P/c2 = ρ/3 (i.e., γ = 1/3) and
the curvature effects are negligible (ǫ = 0). Consequently, the evolution equation of the
Einstein scalar field (Eq. 21) is well approximated by:
2 2
ϕ + ϕ = 0 (30)
′′ ′
3 ϕ2 3
′
−
which does not depend on the functional form of α(ϕ).
The integration of Eq. (30) gives:
3k2
ϕ2 = (31)
′
e2p +k2
where k is related to the initial (p = 0) velocity ϕ through:
′R
(ϕ )2
k2 = ′R (32)
3 (ϕ )2
− ′R
In terms of k, the solution of Eq. (30) is:
√1+k2e 2p +ke p
ϕ = ϕ √3 sign(k)ln − − (33)
R − " √1+k2 +k #
6. Matter-dominated evolution: The attraction-repulsion mechanism
Let us now analyze the evolution of the scalar field during the matter-dominated era
(γ = 0) of a flat universe (ǫ = 0). The evolution equation for the Einstein scalar field
is, in this case:
2
ϕ +ϕ +α = 0 (34)
′′ ′
3 ϕ2
′
−
As in previous works [6, 8], we will first assume that, at some time (for instance,
at the beginning of the matter-dominated era), the scalar-tensor theory is not very far
from GR so that we can neglect ϕ2 against 3 and, in addition, the coupling function
′
α(ϕ) is represented by Eq. (28).
In this case, the evolution equation (34) reduces to:
2
ϕ +ϕ +σ κ(ϕ)ϕ = 0 (35)
′′ ′ ϕ
3
where
σ = sign(ϕ) (36)
ϕ
When σ = +1, the above expression corresponds to the evolution equation of
ϕ
a damped harmonic oscillator with a variable elastic coefficient κ(ϕ). The first term
(2ϕ /3) represents the total force on a fictitious particle of mass m = 2/3. The second
′′
term (ϕ) corresponds to a friction force proportional to the velocity and, finally, the
′
third term represents an elastic force with a variable coefficient κ(ϕ). The existence,
in these conditions, of a damped oscillatory behaviour of the Einstein scalar field was
Convergence of Scalar-Tensor theories 9
first reported by Damour and Nordvedt [6] and it is usually termed as the ’attraction
mechanism’ toward General Relativity .
We note however that, if σ = 1, the effective elastic coefficient σ κ(ϕ) is negative
ϕ ϕ
−
and, consequently, there exists a ’repulsion’ mechanism instead an attraction one. We
will now analyze the matter-dominated evolution of the scalar field by considering
separately the cases δ = 1 and δ = 1 in Eq. (29).
6
It must be noted that the class of scalar-tensor theories analyzed in [6] correspond
to a positive constant σ and, therefore, they are always attractive.
ϕ
6.1. Case of a constant elastic coefficient (δ = 1)
When δ = 1, the elastic coefficient defined by Eq. (29) becomes a positive constant B .
2
The scalar field evolution equation then reduces to:
3 3
ϕ + ϕ + σ B ϕ = 0 (37)
′′ ′ ϕ 2
2 2
Since the roots of the characteristic equation are:
3 3 8
r = 1 B σ (38)
2 ϕ
± −4 ± 4s − 3
the general solution of Eq. (37) admits four different behaviours, depending on B and
2
σ :
ϕ
a) Damped harmonic motion (σ > 0, B > 3/8)
ϕ 2
ϕ = C1e−34pcos(ω1p+C2) (39)
where
3 8
ω = B 1 (40a)
1 2
4s3 −
ϕ + 3ϕ 2 1/2
C = ϕ ′0 4 0 +1 (40b)
1 0
 ω1ϕ0 ! 
 ϕ + 3ϕ 
C = atan ′0 4 0 (40c)
2
− ω1ϕ0 !
b) Critically damped motion (σ > 0, B = 3/8):
ϕ 2
ϕ = e−43p(C1p+C2) (41)
where
3
C = ϕ + ϕ ; C = ϕ (42)
1 ′0 4 0 2 0
c) Overdamped attraction motion (σ > 0, B < 3/8):
ϕ 2
ϕ = e−43p[C1eβ1p +C2e−β1p] (43)
where
Convergence of Scalar-Tensor theories 10
3 8 3
β = 1 B < (44a)
1 2
4s − 3 4
ϕ +(β +3/4)ϕ
C = ′0 1 0; (44b)
1
2β
1
ϕ +( β +3/4)ϕ
C = ′0 − 1 0 (44c)
2
− 2β
1
d) Overdamped repulsion motion (σ < 0, and any B ):
ϕ 2
ϕ = e−34p[C1eβ2p +C2e−β2p] (45)
where
3 8 3
β = 1+ B > (46a)
2 2
4s 3 4
ϕ +(3/4+β )ϕ
C = ′0 2 0; (46b)
1
2β
2
ϕ +(3/4 β )ϕ
C = ′0 − 2 0 (46c)
2
− 2β
2
Apparently, onlythefirst threesolutions converge towardGR(ϕ = 0) while thelast
one diverges to as p . However, it must be noted that each of such solutions
±∞ → ∞
governs the whole matter-dominated era if, and only if, σ does not change. This is
ϕ
the case of the models analyzed in Refs. [6, 7] where, in Eq. (37), σ is forced to be
ϕ
σ +1 at any time.
ϕ
≡
In the case of the models considered here, where the sign σ of ϕ can be variable in
ϕ
time, the matter-dominated evolution of the scalar field has a much more complicated
behaviour. As a matter of fact, the above four solutions (Eqs. 39–45) admit the
possibility of a change in the sign of ϕ at a finite time p > 0 (this is specially obvious for
the first solution due to the oscillatory behaviour of the cosinus function). Therefore,
an initially convergent model (σ = +1) could finally diverge from GR as a consequence
ϕ
of a change in the sign of ϕ. Reciprocally, an initially divergent model (σ = 1) could
ϕ
−
finally converge toward GR if sign(ϕ) changes. We must then perform a more detailed
analysis to find the conditions for convergence (or divergence) with respect to GR.
According to Eqs. (39)–(46c), ϕ vanishes (and, therefore, σ changes) at a finite
ϕ
p > 0 if, and only if, the initial values of ϕ and ϕ satisfy the following conditions:
′
a) always
b) ϕ < 3ϕ , (ϕ > 0)
 ′0 −4 0 0 (47)
 c) ϕ′0 < −(43 +β1)ϕ0, (ϕ0 > 0)
d) ϕ > (3 +β )ϕ , (ϕ < 0)
When these condi′0tion−s a4re sa2tisfi0ed, an i0nitially attraction behaviour becomes a
repulsion one at p > 0. Reciprocally, an initially repulsion behaviour becomes, at
1
p > 0, an attraction one. The question is now to determine whether these new
1
behaviours, reached at p > p , govern the rest of the matter-dominated evolution.
1