General scalar-tensor theories for induced gravity inflation (*)(**)
H. BOUTALEB-JOUTEIand A. L. MARRAKCHI(***)
Laboratoire de Physique Théorique (LPT/ICAC), Faculté des Sciences B.P. 1014, Rabat, Morocco
(ricevuto il 7 Gennaio 1997; approvato il 25 Giugno 1997)
Summary. — The fine-tuning problem is a common puzzle of both the new and chaotic scenarios in all known inflationary models. A general scalar-tensor theory for induced gravity (or extended) inflation with a decreasing gravity is proposed. It proves to be free from this problem. The model exhibits all the good features of inflationary cosmology, it includes most known interesting inflationary models and special cases. A Ginzburg-Landau potential is used for illustration. The results are investigated in the case of a scalar-tensor theory of the Barker type which proves to be, from our viewpoint, not viable.
PACS 04.50 – Gravity in more than four dimensions, Kaluza-Klein theory, unified field theories; alternative theories of gravity.
PACS 98.80 – Cosmology.
1. – Introduction
Inflation [1] provides cosmology and particle physics with a sort of peace, even while it is not yet a fully complete theory. It has been assigned to solve some puzzles of the standard hot big-bang model [2], the flatness or large entropy problem and the causality or horizon problem among others [1]. Guth’s original idea [3], now known as the old inflation, is based on a first-order phase transition that drives the universe, in the early times of its evolution, to a supercooling and then to a reheating period. The old inflation proved to overcome not only the flatness and causality problems but also the primordial monopoles one. However, as has been realized by Guth [3] himself, if the old inflation had happened it would be eternal and the universe would be presently much inhomogeneous. A new inflationary scenario has been proposed independently by Linde [4] and Albercht and Steinhardt [5]. It is based on the SU(5) second-order phase transition by means of the Coleman-Weinberg potential and was aimed to overcome
(*) The authors of this paper have agreed to not receive the proofs for correction.
(**) This is a revised and expanded version of the ICTP internal report IC/92/132 (Trieste, Italy, July 1992).
(***) Permanent address: Département de Physique, Faculté des Sciences, Dhar El Mehraz, B.P. 1796, Atlas, Fes, Morocco.
such a difficulty with the old scenario. The new inflation suffers, however, from its own problems; let us emphasize the two most ambiguous ones. The Coleman-Weinberg potential does not satisfy all the requirements [1] the potential energy must meet in order for the model to be acceptable. Furthermore, in order to make the model not in contradiction with cosmological observations [6], one is led to impose unnatural initial conditions to various parameters in the theory; this is the so-called fine-tuning problem [1]. The chaotic inflation has been introduced by Linde [7] in order to solve these problems. There, as in the new scenario, the scalar field driving inflation, the inflaton, rolls down slowly enough [1] towards the spontaneous symmetry-breaking phase. While in the new inflation the inflaton field starts off near zero correspondingly to a metastable vacuum, in the chaotic inflation it takes in some regions of the universe (chaotic inflationary miniuniverses) values larger than its vacuum expectation value (the Planck mass in general); no phase transition is needed in such a picture of inflation. Moreover, there are no more restrictions on the choice of the potential; a quartic or even a quadratic term is sufficient for the model to be reasonable. These are the important features of this scenario also known as the standard model of inflationary universe [1]. Despite its attractiveness, naturalness and simplicity, the chaotic inflation knows some fine-tuning problems.
Many variations of the idea of inflation have been proposed during the last few years [1]; we shall list hereby only those of interest to us, extended inflation and induced gravity inflation. Extended inflation was proposed by La and Steinhardt [8] in order to solve the problems of old inflation. This is also a supercooled first-order phase transition with the difference that gravity is described by Brans-Dicke’s theory [9] instead of general relativity. This in turn leads to a power law expansion rather than the exponential expansion. Recently, it has been shown [10] that extended inflation knows some problems and especially of the fine-tuning type. It is claimed in ref. [10] that one must abandon the pure Brans-Dicke theory and work with models which have a potential to fix the present value of the Brans-Dicke scalar field to the Planck mass squared [11]. This is in fact contained in even earlier works on induced gravity inflation [11-13] where one considers the new and chaotic pictures rather than the old one. These models know some fine-tuning problems as well. The chaotic inflation itself is shown to be improved within non-minimal coupling in general [14]. In particular, in the framework of the Zee [15] induced gravity, it is found to be more attractive than the new inflation which requires higher fine-tuning of the coupling constants [13]. This improves much the situation in ref. [12]. However, because of the fine-tuning problem from which all inflationary models suffer, many physicists have been led to conclude that such models are not viable despite the solutions they provide to most puzzles of standard cosmology.
Brans-Dicke’s theory is only a special case of a large family known as scalar-tensor gravitational theories [16, 17]. In a general scalar-tensor theory as in Brans-Dicke’s theory, the gravitational coupling is time-dependent. This fact, in addition to the wish to incorporate Mach’s principle in cosmology, has been the starting point for Brans and Dicke in the scheme to obtain a G-varying gravitational theory, as has been suggested by several physicists and especially by Dirac (see refs. [9, 17-19]). Our interest in these theories resides in the fact that one can expect the effective gravitational coupling to behave as a decreasing function of cosmic time. This would realize our main viewpoint on gravity and cosmology. Gravitation could be strong in the very early stages of the evolution of the universe and would physically evolve to its present weakness. Obviously, the form of the coupling function between the scalar and the tensor fields
and the vacuum expectation value of the scalar field are of great importance to this end [16]. These facts among others (see the introduction to sect. 2) motivated us to investigate the cosmological predictions of the model of our previous work [16] on general scalar-tensor theories for induced gravity and especially to try to extend and even improve the results of ordinary induced gravity inflation [12, 13].
Indeed, we discover, from the beginning, that the statement that the chaotic inflation is more natural than the new one in ordinary induced gravity inflation is in fact related to the increase of the effective gravitational “constant” with cosmic time in the former and to its decrease in the latter. In a general scalar-tensor theory for induced gravity inflation with—for instance—a Ginzburg-Landau–like potential that is minimized by an inflaton value of order the Planck mass, the demand of a decreasing gravitational interaction during the evolution of the early universe allows one to have, in the early universe, a coupling function of order unity within the new scenario and forces one to have it much smaller than unity within the chaotic scenario. Hence, one can claim that this sort of fine tuning is not an unnatural restriction but it can be seen as an intrinsic feature of inflationary cosmology. By use of a coupling function rather than a coupling constant, the very problem of fine tuning does not have any existence in the sense that, even if the initial or final value of this function must be adjusted to some special small or large numerical value, one can understand this behavior by attributing it to the variation of the considered coupling with cosmic time. For instance, within inflationary cosmology, one can understand the fact of having a very large cosmological constant in the early times and a small vanishing one in the present times of the evolution of the universe due only to a special behavior of a certain potential energy density in the course of cosmic time.
In sect. 2, we review briefly the fundamentals of a general scalar-tensor theory for induced gravity [16] and apply it to cosmology, being more interested in the vacuum-dominated regime. In sect. 3, we study the fields equations in the slow-rolling approximation and discuss the new and chaotic scenarios of this inflationary model. Our results reduce to those of refs. [12] and [13] in the limit where the coupling function
e(W) tends to a constant e or j , respectively, which are such that the new inflation can
work as well as if not better than the chaotic one. Section 4 is devoted to the application of scalar-tensor gravitational theories to a specific case that satisfies the strong equivalence principle. There, the effective gravitational function can vary only in a narrow band and it is increasing instead of decreasing during inflation in such a way that this model appears to be not viable from the point of view developed in the present paper. We end the paper by giving some comments and concluding remarks in sect. 5.
2. – General scalar-tensor theory for induced gravity inflation
The development of various metric theories for gravitation [17] has been basically motivated by the successful agreement of the Brans-Dicke [9] scalar-tensor theory with observation at large scales. As the Brans-Dicke coupling constant goes to infinity, this theory tends just to the general theory of relativity. Unfortunately, at the microscopic scale, neither the former nor the latter theory do provide us with a satisfactory description of quantum gravity [20].
Since the idea of induced gravity has been emitted by Sakharov [21] and Zel8dovich [22], many tentatives have been approached [23]. However, the spontaneous
realization of Sakharov’s idea. There again, it is supposed that the gravitational coupling is an effective function of the scalar field; it reduces to Newton’s constant only at the broken symmetric phase.
In a previous work [16], we have extended the Zee model to a general scalar-tensor theory for induced gravity. There, not only the gravitational constant is not a true constant but also the scalar coupling does depend on the four-position through the gravitational scalar field. Our model of ref. [16] reproduces the Zee one easily as the coupling function becomes constant. Moreover, it improves it essentially in the sense that in the former the spontaneous symmetry breaking may happen at any epoch of the evolution of the universe. In particular, with an appropriate choice of the coupling function one can solve, even partially, the hierarchy problem by breaking a grand-unified theory into strong and electroweak interactions and the previous model into Einstein’s theory by the same mechanism and at the same energy scale. Furthermore, the induced cosmological constant can be made as small as expected by observation.
The fact of having a coupling function rather than simply a coupling constant gives the possibility that this model is free from problems with ordinary extended inflation [8] like those mentioned in ref. [10]. More precisely, one can better understand here how the gravitational coupling has evolved from a large value in the early past to a small one in our present times. In this way, the model can provide us, in a natural way, with a good description of both the early universe and the present-day universe. This, in addition to the motivations mentioned above in the introduction, drove us to look for the cosmological and especially the inflationary implications of our model. This is the aim of the present work.
Our theory for gravitation in the presence of matter is based on the action
A 4
d4x kgk
21 2 e(W) W 2 R 2 1 2g mn ¯mW¯nW 2V(W)1 LMl
. (1)Here, g is the negative of the determinant of gmn, W is a gravitational scalar field, e(W) is a dimensionless coupling function assumed to be differentiable, V(W) the potential energy density associated with the field W, R the scalar curvature, and kg LM the
contribution of matter fields to the Lagrangian density. In order to make a metric gravitational theory from this model, we simply assume that LMis W-independent, i.e.
W does not interact directly with matter, and, cosmologically speaking, one does not
consider the reheating process. We are using the Weinberg [2] conventions in the metric, the connection, the curvature and the natural units system.
Let us observe that the present theory can be seen as general relativity coupled to a scalar field in addition to usual matter fields with the effective gravitational coupling
Geff(W) 4
1 8 pe(W) W2 , (2)
which reduces presently to the Newton constant
GN4 MPL224 Geff(W0) 4
1 8 pe(W0) W20
, (3)
where MPL represents the Planck mass and W0 denotes the present value of W. It is
dimension of a squared mass, is an extreme specific element of a class of coupling functions allowing many good properties of the model [16]. Note that, in order to avoid the notion of antigravity, e(W) must be positive.
In order for the effective gravitational coupling to decrease with cosmic time,
G
n
effE 0 ,
(4)
we will distinguish, as can be shown from eq. (2), only two cases: i ) j(W) D0, Wn D 0 , “new inflation” (5) and ii ) j(W) E0, Wn E 0 , “chaotic inflation” , (6)
W being considered always positive. Indeed, it follows from eq. (2) with W a spatially
homogeneous field that depends, as required by the cosmological principle
(
see eqs. (15)-(18))
, only on the cosmic time t:G n eff Geff 4 2 2 Wj(W) W n , (7)
where dots and primes stand for differentiation with respect to t and W , respectively, and j(W) is defined by j(W) 411 1 2 e 8(W) W e(W) . (8)
Also, in order to meet the requirement (4), one is led to conclude from the very beginning that in the new inflationary scenario e(W) is not constrained to any sort of fine tuning, while in the chaotic scenario it must be much smaller than unity at least during the slow-rolling period of the quasi-de Sitter phase. Indeed, one sees from eq. (2) that if Geff is to be initially c GN and presently AGN, then e(W) W2 must be
initially b M2
PL and presently AMPL2 . Hence, within a Ginzburg-Landau potential that
reaches its minimum for W AMPL, e(W) can be presently of order unity in both the new
and chaotic scenarios; but it must be initially much smaller than unity in the framework of the chaotic scenario, while it can be of order unity in the framework of the new one. Within a Ginzburg-Landau potential with a minimum at W AMGUT, e(W) has to be
presently of order 1010in both inflationary scenarios, while it can initially be chosen of order unity in the framework of new inflation and necessarily of order or smaller than unity in the framework of chaotic inflation. This will be excluded since, as will be shown below
(
see (33)-(34))
, in this case e(W) would have to vary very fastly as compared to W in a very short time, namely during the slow-rolling period.In this context, the chaotic scenario is, in principle, not the best candidate for inflationary universe cosmology; it could be the new scenario that will play this crucial role. This contradicts the statements made by the authors of refs. [12] and [13] within ordinary induced gravity inflation. Our aim is to pursue these notions and to check the limits of their validity.
The problem of fine-tuning a coupling constant is rather a hard one, it is known almost everywhere in modern physics: a coupling constant has to have excessively small or large values in order to overcome various difficulties within the same model. By use of a coupling function rather than just a coupling constant, this problem does
not have any very existence; that is, even if the initial or final value of this function must be adjusted to some special small or large size, one can understand this behavior by attributing it to the variation of the considered coupling with cosmic time. For instance, within inflationary cosmology, one can understand the fact of having a very large cosmological constant in the early times and a small vanishing one in the present times of the evolution of the universe due only to a special behavior of a potential energy density with cosmic time.
The tensor field equations read
e(W) W2
g
Rmn2
1
2gmnR
h
4 2 Tmn, (9)where Tmn is an energy-momentum tensor which includes both contributions of the W and matter fields; it is given by [16]
Tmn4 TmnM1 [e(W) W2], m ; n2 gmnpc[e(W) W2] 2 (10) 2
g
1 2g ab ¯aW¯bW 1V(W)h
gmn1 ¯mW¯nW ,TmnMbeing the usual matter energy-momentum tensor and the covariant D’Alembertian of any scalar function f is defined as pcf 4gmnf, m ; n. Let us note that Tmn2 TmnM can be thought of as an improved energy-momentum tensor for the scalar field coupled to gravity as in eq. (1). The equation of motion of the scalar field reads [16]
pcW 4e(W) j(W) WR1V 8(W) . (11)
By taking the trace of eqs. (9), one easily evaluates the Ricci scalar:
e(W) W2
R 4TM
2 4 V(W) 2 gmn¯mW¯nW 23pc[e(W) W2] , (12)
TMbeing the trace of TM
mn, and by eliminating R in eqs. (11) and (12), one rewrites the W field equation in a more convenient form:
pcW 4 (13) 4 j(W) W[ 1 16e(W) j2(W) ]
y
T M 1 V 8(W) W j(W) 2 4 V(W) 2 [ 1 1 3(
e(W) W 2)
9 ] gmn¯mW¯nWz
. Note that 1 16e(W) j2(W) is always strictly positive. The computation of the covariantdivergence TM
mn; n shows via the Bianchi identities and the fields equations that the
energy-momentum tensor of matter is conserved:
TMmn; n4 0 ,
(14)
which implies that our theory is a true metric gravitational one [17].
In order to discuss the cosmological implications of the model, we consider the Friedmann-Robertson-Walker metric: ds2 4 2 gmndxmdxn4 dt22 a2(t)
y
dr2 1 2kr2 1 r 2( du2 1 sin2u dc2)z
, (15)where t is the cosmic time, a(t) is the scale factor of the universe, (r , u , c) are the polar coordinates, r being normalized in such a way that the space curvature parameter k takes only the values 21, 0 or 11 for open, spatially flat or closed universe. This choice is a straight consequence of the observed homogeneity and isotropy of the universe [2, 6]. It implies that W is a spatially homogeneous scalar that depends only on cosmic time:
¯rW 4¯uW 4¯cW 40 . (16)
This is also strongly related to the very physical meaning of the scalar gravitational
field in the framework of both scalar-tensor gravity and induced gravity
models [17, 25]. Furthermore, one is led to choose the perfect-fluid model for the matter content of the universe [2]. In this case
TM
mn4 (r 1 p) umun1 pgmn, (17)
where r and p are the energy density and the pressure of cosmic matter, respectively, and um are the covariant components of the proper four-velocity:
um4 2 d0m. (18)
In these conditions, there are only two independent fields equations; the time-time component of eq. (9) H2 (t) 1 k a2(t) 4 r(t) 1V(W)1Wn2 (t) O2 3 e(W) W2(t) 2 2 j(W) H(t) Wn (t) W(t) , (19)
and the reduced W field equation, eq. (13) (20) Wn n (t) 13H(t) Wn (t) 4 4 j(W) W(t)[ 1 16e(W) j2(W) ]
{
r(t) 23p(t)14V(W)2 W(t) V 8(W) j(W) 2 [ 1 1 3[e(W) W 2 ]9] Wn2 (t)}
. In these equations, we have explicitly written the arguments of various functions and we defined—as usual—the expansion rate byH(t) 4 a
n
(t)
a(t) .
(21)
Actually, H(t) depends on the age of the universe; it is a true constant only in a de Sitter phase of the Friedmann-Robertson-Walker geometry. The time component of eq. (14) gives
rn
13 H(r 1 p) 4 0 . (22)
In the case of a non-metric (non-conservative) theory as that described in ref. [25], LMdepends on W , eq. (14) no longer holds and eq. (22) is replaced by
(Geffr)Q 13H(r1p) Geff4 0 .
(228)
Then, the expansion of the universe will be, in the framework of the present model, as in the standard model of cosmology, an adiabatic isentropic process. Thus, as has been suggested in the framework of the standard model of cosmology, one needs to
introduce the notion of inflation in the context of the model described here in order to explain the present large entropy in the universe.
In the fields equations (19) and (20), when r is much larger than V(W) either the radiation—an almost ultra-relativistic perfect fluid: p 4r/3—or the matter—an almost pressureless dust matter: p 40—dominates the matter content of the universe. These are, respectively, the conditions of the early stages and the present times of the
evolution of the universe. The inflation takes place when the universe is
vacuum-dominated, i.e. whenever r becomes negligibly small with respect to
V(W)—almost constant.
In order for eq. (19) to behave, in the limit of a radiation-dominated universe, as the Einstein-Friedmann equation, W and e(W) must satisfy the constraint (3). This implies that, during the radiation epoch, W remains constant, j(W) vanishes and eq. (20) holds just as a trivial identity provided that the constant value of W does minimize the potential; and then one is back to the standard model, obviously in the absence of a cosmological constant.
In the vacuum-dominated era, the energy density and the pressure of radiation can be neglected with respect to the potential energy of the scalar field and eqs. (19) and (20) reduce to H2 (t) 1 k a2(t) 4 V(W) 1Wn2 (t) O2 3 e(W) W2(t) 2 2 j(W) H(t) Wn (t) W(t) (23) and (24) Wn n (t) 13H(t) Wn (t) 4 4 j(W) W(t)[ 1 16e(W) j2(W) ]
{
4 V(W) 2 W(t) V 8(W) j(W) 2 [ 1 1 3[e(W) W 2 ]9] Wn2 (t)}
. These equations reduce clearly, in the case of a constant scalar coupling, to those of induced gravity inflation à la Zee [12, 13]. They will be the starting point of our discussion about inflation in the framework of a general scalar-tensor theory for induced gravity; the scalar gravitational field being, as in most extended inflation models [8, 11], taken for the inflaton, i.e. the field driving inflation in the universe.It is well known that soon after the first few e-foldings in the quasi-de Sitter phase,
a grows so much that, independently of the initial value of k (either +1 or 21), k/a2can be neglected with respect to V(W) /3 e(W) W2or H2during practically the whole period of
inflation. On the other hand, since the kinetic energy of the inflaton is negligible in the vacuum-dominated era with respect to its potential energy,
Wn2
bV(W) , (25)
eqs. (23) and (24) read during inflation
H2 4 V(W) 3 e(W) W2 2 2 j(W) H Wn W , (26) Wn n 13 H Wn 4 j(W) W[ 1 16e(W) j2(W) ]
y
4 V(W) 2 WV 8(W) j(W)z
(27)provided that one excludes the extreme case [e(W) W2
]9c1 , (28)
which has nothing to do with a strong restriction on the choice of the coupling function form.
3. – Slow-rolling approximation
The slow-rolling approximation characterizing both the new and the chaotic scenarios of inflationary universe is defined by the set of conditions [1]
N WnN/W b H , (29) N Wn n / Wn N b H , (30) N H n N b H2. (31)
The last condition, as well as condition (25), is in fact related to the very concept of inflation [1]; that is, to exhibit a quasi-de Sitter phase in the early universe, a theory must contain a potential energy density which behaves, during those times, as a cosmological constant. Let us note that eq. (7) shows that the notion of slow rolling is in fact related to the slow variation [18], during inflation, of the effective gravitational function and its first derivative.
To order zero in NWn
N/HW, one has from eq. (26)
H2C V(W)
3 e(W) W2 ,
(32)
provided that the extreme situation
Ne 8(W) NW
e(W) c1
(33)
is excluded. Note that this exclusion is far from being restrictive because condition (33) simply means that
N en (W) N e(W) 4 Ne 8(W) NW e(W) N WnN W (34)
can be of order H, which in turn contradicts the notion of slow rolling.
In investigating the evolution of W(t) during inflation, one must obviously expand eq. (27) not to zeroth but to first order in NWn
N/HW. One obtains Wn C
(
4 j(W) /W)
V(W) 2V 8(W) 3 H[ 1 16e(W) j2(W) ] , (35)or equivalently, by use of eq. (32),
Wn
C e(W) WH 4 j(W) 2V 8(W) W/V(W) 1 16e(W) j2(W) .
Before going ahead in looking for the implications of the slow-rolling approximation of eqs. (26) and (27), we would like to investigate whether or not the present model can exhibit an inflationary structure at all. The condition for this is expressed by (31). By eliminating H2or V(W) between eq. (23) and either the trace of eq. (9) or its space-space
component in conditions (15)-(18), or equivalently, by eliminating the potential terms in eq. (23) and eq. (10) adjusted to the same conditions as above and using the conservation law (22), one evaluates H
n
; the result reads in the vacuum-dominated regime H n 4 2 j(W) W n n W 2 1 2 e(W)] 1 1 [e(W) W 2 ]9( W n2 W2 1 j(W) H Wn W . (37)
This reduces in the approximation (30) to
H n 4 2 1 2 e(W)] 1 1 [e(W) W 2 ]9( W n2 W2 1 j(W) H Wn W . (38)
Conditions (29) and (31) give from eqs. (32) and (38), again provided that Ne8(W)NW/e(W) is not too large, the constraint
3 2] 1 1 [e(W) W 2 ]9( Wn2 bV(W) (39)
which is compatible with condition (25) for any non-infinite value of [e(W) W2
]9; this shows the self-consistency of the exclusions of the cases (28) and (33). Hence, in principle, no actually strong restriction is to be imposed on the choice of the coupling function in order for the model to exhibit an inflationary, quasi-de Sitter, phase.
Until now, we did not specify the explicit form of the potential. The introduction of the Coleman-Weinberg potential has illustrated some advantages of the model [16] on that of Zee [15]; namely, the induced gravitational and cosmological constants at finite temperature can be adapted to the cosmological observations. However, since we are mostly interested in the improvements the present model might give on the predictions of ordinary induced gravity inflation of refs. [12] and [13], we are going to consider the same potential used therein, the Ginzburg-Landau potential
V(W) 4 l
8(W
2
2 v2)2, (40)
where l is a dimensionless constant and v the vacuum expectation value of W(t). When
W 4v, i.e. when V(W) reaches its minimum, the model breaks down into general
relativity with an induced gravitational constant
Gind4
1 8 pe(v) v2
(41)
and a vanishing induced cosmological constant. One can choose for the value of v either
MGUTA 1014GeV or MPLA 1019GeV ; we will prefer the latter choice unless one permits
Using the explicit expression (40) for the potential in eqs. (32) and (35), one obtains, respectively, H C
o
l 24 e(W) WN
1 2 v2 W2N
(42) and Wn C 6o
2 l 3 e(W) W2 1 16e(W) j2(W)N
g
1 2 v2 W2h
j(W) 21N
. (43)Here, the plus-minus sign stands, respectively, for the new and chaotic scenarios of inflation. Let us note that one easily recovers the results of ordinary induced gravity inflation [12, 13] when e(W) turns to be a pure coupling constant.
It is necessary to verify the self-consistency of the above results. To do so, let us start by taking the expressions of H and NWn
N from eqs. (42) and (43) and check the validity of condition (29). We indeed obtain the constraint
4 e(W) 1 16e(W) j2(W)
N
j(W) 2 WV 8(W) 4 V(W)N
4 4 e(W) 1 16e(W) j2(W)N
j(W) 2 1 1 2v2/W2N
b1 . (44)In what follows, we will analyze the cases of the new and chaotic scenarios of inflationary universe. In the former case, W evolves from a small value Wito its vacuum
expectation value v so that before the accelerating regime W/v remains negligibly small. The constraint (44) reads
4 e(W) j(W) 1 16e(W) j2(W) b1
(45)
to first order in W/v
(
j(W) being positive as in (5))
, or, equivalently, 4 j(W)1 16j2(W) b1
(46)
by taking into account the fact that e(W) Ae(Winitial) A1. This can be verified for small
and large values of j(W) or of e 8(W) but not for j(W) A1, i.e. not in the case of a nearly constant e(W). One can easily verify that a j(Winitial) A1021–1022or a j(Winitial) A10–100
is enough for condition (46) to be fulfilled. This shows the self-consistency freely from any strong constraint on initial conditions of the new inflationary version of the present model. When the coupling function turns out to behave as a constant e , j(W) tends to unity and condition (46) is not satisfied at all. If now in these conditions one permits the constant e to take values much smaller than unity, then condition (45) is satisfied. This is a strong fine tuning to impose to the coupling constant for the new inflation to proceed in such a limit and this is what happens in the Zee [15] induced gravity inflation [12, 13].
In the case of the chaotic scenario of inflation, W starts off near a large value Wi,
slow-rolling phase. Therefore, the constraint (44) reads 4 e(W)[ 1 2j(W) ] 1 16e(W) j2(W) 4 2 4 e 8(W) W 1 16e(W) j2(W) b1 (47)
to first order in v/W
(
j(W) being negative as in (6))
, or 2e 8(W) b 1 /W b 1 /v (48)since e(W) Ae(Winitial) b 1 and W c v during almost all the inflationary time. This is a
stronger constraint on initial conditions for the chaotic inflation than for the new one to proceed. This shows that within our model the chaotic scenario can work self-consistently but at the price of strong restriction on initial conditions of both the coupling function and its derivative. When e(W) 4e, then j(W) 41 and (47) is satisfied independently of the value of e . This is the case of ordinary induced gravity inflation [12, 13] where, ignoring any conditions for the effective gravitational “constant” to behave correctly during inflation, the chaotic scenario of inflationary universe proves to be more natural and less restrictive than the new one. But this is only a special case of a general scalar-tensor theory for induced gravity inflation where we just gave a proof in favor of the new scenario of inflation compared to the chaotic one. Let us study now the situation as far as the flatness and horizon problems are con-cerned. It is known [1] that the inflation period must be long enough in order to over-come these puzzles of the standard model of cosmology. This drives one to the condition
ti tf H(t) dt 4 Wi Wf H Wn dW N , (49)where the indices i and f stand for the initial and final values of t and W, and N is the number of e-foldings in the quasi-de Sitter regime safely bounded by [1, 6]
60 N 70 , (50)
with an average value of 65 e-foldings. Condition (49) means simply that only after some 65 e-foldings in the supercooling phase, the universe can reheat to high temperature producing enough entropy to solve the flatness and causality problems and then becomes sufficiently described by the standard hot big-bang model. It is clear that Wf is either smaller or larger than v. However, since the accelerating and
reheating periods are known to be much more brief than the friction-dominated one defined by (29)-(31), one can approximate (49) by
Wi v H Wn dW N . (51)From the left-hand side of (44), or, equivalently, from eqs. (42) and (43), one can show that, in the present model, condition (51) reads
Wi v 1 16e(W) j2(W) 4 e(W) 1 j(W) 2 1 1 2v2/W2 dW W N . (52)In the new picture of inflationary universe, one obtains
Wi v 1 16e(W) j2(W) 4 e(W) j(W) dW W N (53)to first order in W/v . However, we saw that in this case e(W) A1 and in view of the slow rolling of W the coupling function and its derivative would have no time to vary much during the quasi-de Sitter phase so, that j(W) remains almost constant
(A
j(Wi))
inthe course of most of the inflationary time. Condition (53) implies then 1 16j2(W) 4 j(W)
Wi v dW W N , (54)which in turn implies
Wiexp
y
2 4 j(Wi) 1 16j2(W i) Nz
v . (55)Since Wiis rather too small and the argument of the exponential function is bounded
between 0 and k2 /3 N and N is bounded as given in (50), condition (55) is easily satisfied only for j(Wi) around unity but self-consistency is then lost. Between
j(Wi) A1021–1022 and j(Wi) A10–100, which is a reasonable bound for
self-consistency, the only value that is compatible with (55) is j(Wi) A1021, which gives
N WnN/HW 1 /10 , and Wiv/10 , allowing the evolution of Geff from 100 GN to GN. If in
ordinary induced gravity inflation e has to be b 1 for self-consistency, then, contrary to what is claimed by the authors of ref. [13], e A1021 is enough to produce sufficient
inflation in its new version. Therefore, except for the case j(W) 40 where the model reduces to inflationary cosmology in Einstein’s theory and which knows hard fine-tuning problems with the new and even with the chaotic scenario, no strange restriction is to be imposed on the initial conditions for the new inflation to solve the causality and flatness problems in the framework of our general model.
In the chaotic version of inflation, condition (51) gives
Wi v 1 16e(W) j2(W) 4 e(W) 1 j(W) 21 dW W 4 Wi v 1 16e(W) j2(W) 2 e 8(W) dW W2 N (56)to first order in v/W , with j(W) neither too large nor unity. Let us recall here that
e(W) Ae(Wi) b 1 during almost the whole slow-rolling phase; also for reasons
analogous to those mentioned before e 8(W) can be considered as varying slowly during this period, so that condition (56) reduces to
1 2 e 8(Wi)
Wi v dW W2 N . (57)Therefore, one is led to a further strong fine tuning expressed by
Wi
v
2 Ne 8(Wi) v 11
. (58)
Indeed, Wimust be c v , e.g. Wiis some ten or hundred times v, then 2 NNe8(Wi) NvA1
and then Ne8(Wi) N must be fine-tuned to 1/2 Nv 10221GeV21. This was in the first
place the condition for self-consistency of the chaotic picture of inflation
(
see (48))
. In the special case where j(W) 41, the above analysis no longer holds; one should go to order two in v/W and obtain the following condition on e:Wi
o
8 e
1 16e N v ,
(59)
which is consistent with Wicv for any non-vanishing value of e because
k
[ 8 e/( 1 16e) ] N is bounded between 0 and k4 N/3. One has WiA 10 v for e verylarge and WiA 5 v for e A 1
(
for comparison, see Linde (1984) in ref. [1])
.Hence, our results can be summarized as follows. In a general scalar-tensor theory for induced gravity, the new picture of inflationary universe works self-consistently and solves the horizon and flatness problems without any need to strongly fine-tune initial conditions. Only in the special case of Zee’s model, self-consistency requires small values of the coupling constant. However, the chaotic version requires fine tuning of initial conditions not only for self-consistency or for the model to solve the horizon and flatness problems but also and in the first place for the gravitational interaction to behave as expected by the very physical meaning of the expansion of the universe and by our present understanding of the evolution of the universe, i.e. for Geff to be a
decreasing function of cosmic time. But this does not represent any problem as it would do if we were dealing with a coupling constant which would be demanded to unnaturally take always a very small value.
One can also see the chaotic scenario as being less restrictive than the new one. However, in the framework of our model, the fine-tuning problem is found, from self-consistency and naturalness considerations, to be much more relaxed in both scenarios. One is then led to compare these scenarios in various situations. For instance, the fact of having e(W) 1 at the beginning of inflation and e(W) c 1 at the end cannot be seen as a constraint on the chaotic scenario, it is rather an advantage of the model since W itself varies in a large band during inflation.
Now, one can wonder whether there exists a coupling function which satisfies our general assumptions and especially which varies, as well as its derivative, slowly enough with W during inflation for the above analysis to be justified. The answer is affirmative, there are at least two well-known cases to which our model tends asymptotically and which verify our results, inflationary models in general relativity [1]
(
j(W) 40)
and in Zee’s induced gravity [12, 13](
j(W) 41)
. A “conformal” theory for extended inflation [26] constitutes a physically more important application of the present-paper ideas. Moreover, we have been inspired by the Barker [27] model of general scalar-tensor theories in providing the case we are going to discuss in the next section.4. – Barker’s model and induced gravity inflation
Barker’s purpose [27] was to provide a general scalar-tensor theory with constant gravitational coupling G, i.e. a theory which satisfies the strong equivalence princi-ple [19]. Within the time-independent weak-gravitational-field approximation [17], one easily obtains an expression for G in any scalar-tensor theory from which it appears
that G is a pure constant if the coupling function takes a special form. He then obtained fields equations which are consistent with any assumed values for the cosmological parameters, the Hubble constant in particular. He also showed that such a theory differs from general relativity only in the two extreme situations, the early universe and the future universe [27]; these are the situations where general relativity is supposed to fail in its predictions [2, 6].
Indeed, in the time-independent weak-field approximation, one considers small perturbations in various gravitational fields:
gmn(x) 4hmn1 hmn(x) , (60)
W(x) 4W01 z(x) ,
(61)
where hmnis the Minkowski metric tensor with signature (2, 1, 1, 1) and W0is the
present value of the scalar field, such that
Nhmn(x) Nb1 , (62) Nz(x) N b W0 (63) and ¯0hmn4 0 , (64) ¯0z 40 . (65)
By substituting the fields by eqs. (60)-(61) and using conditions (64)-(65), it appears from the fields equations expanded to first order in hmnand z that a test particle of mass
M located at the origin of the coordinates creates a gravitational perturbation at a
distance r such that [16, 17, 19]
g00(r) 42 12 2 M r 1 8 pe(W0) W20 1 18e(W0) j2(W0) 1 16e(W0) j2(W0) . (66)
Here, we used the harmonic gauge
¯mhmn4 1 2¯nh m m, (67)
the special relativistic energy-momentum tensor
TM mn4 Md0md0n, (68) and set V(W) 40 , (69)
in order to permit the Newtonian behavior; in particular, this allows the “particle” z to be massless [15, 16]. It is then clear that the Newton constant obtained as a parametrized post-Newtonian (PPN) parameter [17] reads
GN4 1 8 pe(W0) W20 1 18e(W0) j2(W0) 1 16e(W0) j2(W0) , (70)
or, equivalently
(
see eq. (2))
, GN4 Geff(W0) 1 18e(W0) j2(W0) 1 16e(W0) j2(W0) . (71)From this, one sees that the modified gravitational coupling in the present scalar-tensor theory which replaces GN in linearized general relativity is a function of
cosmic time t via W:
G(W) 4 1 8 pe(W) W2 1 18e(W) j2(W) 1 16e(W) j2(W) , (72) or, equivalently, G(W) 4Geff(W) 1 18e(W) j2(W) 1 16e(W) j2(W) . (73)
Now, if one suggests the constancy of G(W),
G
n
(W) 40 , (74)
then one is led to the following equation:
Gn(W) G(W) 4 2
g
[e(W) j2 (W) ]9 [ 1 16e(W) j2(W) ][ 1 18e(W) j2(W) ] 2 j(W) Wh
W n 4 0 . (75)Apart from the trivial solution
j(W) 40
(76)
that leaves not only G(W) but also Geff(W) constant, there is a more general situation for
a t-dependent W , i.e. for Wn
c0, corresponding to [e(W) j2(W) ]9 [ 1 16e(W) j2 (W) ][ 1 18e(W) j2(W) ] 4 [e(W) W2]8 e(W) W2 . (77)
The general solution to the last equation is 1 18e(W) j2(W)
1 16e(W) j2(W) 4 Ke(W) W 2,
(78)
with K an integration constant determined by the demand of the independence of G(W) from t:
K 48pG(W) 48pGN.
(79)
In view of the non-linearity of eq. (78) in e(W), there is unfortunately no systematic way to extract an explicit form of the coupling function other than the general solution of eq. (76),
e(W) 4 C W2 ,
with C another integration constant determined in the same fashion as K: C 4 1 8 p G(W) 4 1 8 p Geff(W) 4 1 8 p GN , (81)
which makes from our scalar-tensor theory one of the Barker type. One can, however, see from eq. (78) that, once found, such a solution is not of great interest to us since it makes Geffvarying during the whole age of the universe only from ( 3 /4 ) GNto GN. This
does not answer our requirement that the gravitational interaction must be very strong in the early past to evolve presently to the weakest force of nature. The reason for this resides in the fact that Geff increases instead of decreasing and this increase can be
made only within a narrow band. This is why we do not deem it fruitful to go farther in the discussion of this case which, after all, can be seen as a wrong example in the way to construct a reasonable inflationary model satisfying our first requirements. As for the case given in eqs. (76), (80) and (81), it is clear that it satisfies our general assumptions
(
exclusion of the extreme situations in eqs. (28) and (33) and assumption of a smooth variation of e(W) during inflation)
and shows that the chaotic scenario of inflationary universe is much more natural and less restrictive than the new one, but Geff does notvary at all! This is an extensively [1] studied case and need no supplementary comments.
In ref. [27], Barker is using the modified scalar field
f 48pGNe(W) W2
(82)
and coupling function
v(f) 4 1
4 e(W) j2(W) . (83)
In this formulation, the theory satisfies the strong equivalence principle when
G(f) 4 GN f 4 12v(f) 3 12v(f) 4 Geff(f) 4 12v(f) 3 12v(f) (84)
reduces everywhere to GN. This gives [27], for a t-dependent f , an explicit form of the
coupling function:
v(f) 4 4 23f
2 f 22 (85)
which makes G(f) constant but not Geff(f)
(
the solution for Geff(f) 4const correspondsin this formulation to the extreme situation fn 4 0
)
. The study of Barker’s theory in our formulation from the point of view of the PPN formalism and cosmology is the aim of a work in preparation [28].5. – Discussion
In the present paper we have been dealing with some cosmological predictions of a general scalar-tensor theory for induced gravity. Our results can be seen as a generalization of and even a revolution with respect to those of ordinary induced gravity inflation.
The model is shown to work well in the radiation-dominated era. When attention is drawn to the vacuum-dominated regime, the model is shown to exhibit an inflationary epoch of either the new or the chaotic type. It includes most known inflationary models: formulae for inflationary models within general relativity are found from ours for vanishing j(W), while those for ordinary induced gravity inflation are obtained when
j(W) tends to unity.
We have shown that for the new scenario to be self-consistent and to solve the flatness and causality problems one must require stronger constraints on the coupling function than for the chaotic inflation only in the Zee induced gravity inflation. But this is conditioned by the allowance of a decreasing gravitation in the course of inflation within the new scenario and an increasing one within the chaotic scenario! If Geff is
required to be decreasing, then the chaotic scenario will present an even stronger fine-tuning difficulty. In a generic situation where the scalar coupling to gravity depends on cosmic time through the inflaton in such a way that j(W) c 1 , we showed that the chaotic inflation rather than the new one requires fine tuning on the initial values of the coupling function and its derivative for self-consistency and for the model to overcome the horizon and flatness puzzles of the standard hot big-bang cosmology and in the first place for Geffto evolve from a strong value GS4 [ 8 p e(Wi) W2i]21in the
very early universe to GNin the present-day universe. For all these demands, the new
version does not need any strong restriction on initial values of e(W) and e 8(W). From this point of view, we can claim that inflationary cosmology is more natural and attractive in its new picture than in its chaotic one. On the other hand, one can easily see the improvement that the introduction of a coupling function instead of just a coupling constant gives in general: even if initial conditions are restricted to have a special form or value, one is not constrained to live with unnaturally too small or too large parameters during the whole age of the universe. Thus, fine-tuning initial conditions and/or final conditions could be understood as any natural phenomenon. This is the case of the conditions imposed by inflation or by observation on the early and present values of the cosmological constant, the ratio W/v in the new and chaotic scenarios of inflation, and the mass scales of electroweak and grand-unified theories; there are many other examples.
In spite of the domain walls problem it drives [1], we used a Ginzburg-Landau potential essentially for two reasons. It permitted us to have a direct access to comparison with earlier works in the field on the one hand, and on the other hand it provides us—as expected by observation—with a vanishing cosmological constant in the present universe. Let us note here that the very large cosmological constant attributed to V(W) in the slow-rolling phase would have decayed during an excessively brief period of damped oscillations of the inflaton about its vacuum expectation value
v AMPL, thus reheating the universe to an enough large temperature allowing the
standard model to take place starting from the end of inflation. The choice of a main value of W of order MPL saves the model from a further very strong restriction on the
present value of e(W) in both inflationary versions, or, equivalently, this choice enables us to have a coupling function that varies with time slowly enough as compared with the scale factor. Any other choice of the potential will, in principle, give the same qualitative results as far as the reheating process is not involved.
We called the inflationary epoch in our model a quasi-de Sitter and not a de Sitter phase. The expansion is in principle a “generalized” power law which reduces to the Brans-Dicke power law expansion in the case of a coupling constant e and to the de Sitter exponential expansion in the case where e(W) 4C/W2.
Finally, a scalar-tensor theory of the Barker kind has been studied as an application of the ideas and results of this work. Because of the strong equivalence principle, Geff
increases instead of decreasing and its increase has been shown to be so small that one can discard this type of scalar-tensor theories from consideration in the framework of inflationary cosmology. A more important application of our results is contained in a forthcoming paper [26].
More attractiveness of the ideas underlying this work should be obtained if other aspects of inflationary cosmology were considered. Indeed, forthcoming publications [29] are essentially aimed at the discussion of the energy density perturbations constraints on the constant l of the Ginzburg-Landau potential, at the reheating process and at the use of other forms of the potential, the Coleman-Weinberg potential in particular.
* * *
One of the authors (ALM) would like to thank Prof. A. SALAM, the International
Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, where part of this work was done. He would also like to thank the Arab grant available for the ICTP Associateship scheme.
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