General Relativity and Gravitation, VoL 14, No. 5, 1982. Cosmology without Singularity and Nonlinear. Gravitational Lagrangians. RICHARD KERNER.
General Relativity and Gravitation, VoL 14, No. 5, 1982
Cosmology without Singularity and Nonlinear Gravitational Lagrangians RICHARD KERNER Dbpartement de Mbcanique, Universitk Pierre et Marie Curie, 4, Place Jussieu- 75005 Paris, France Received March 9, 1981 A bs tract We investigate the generalized Einstein equations derived from the Lagrangian which is an arbitrary function of R. The importance of the saturation phenomenon is underlined, which may replace the role of a cosmological constant. The spherically symmetric homogeneous model is analyzed in more detail, and an approximate solution without singularity is constructed using the method of matched asymptotic expansions.
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Introduction
The notion of the cosmological constant was first introduced by Einstein [1] in order to obtain a solution of the equations of the general relativity which would describe a homogeneous, isotropic, and stationary Universe. The original set of equations deduced by Einstein, Ruv-
1 8riG -~ guvR = - c 4 Tuv
(1)
did not admit any solution with the aforementioned properties. This is why Einstein added what he called the cosmological term, modifying (1) into 1 Ruv - -~ g~vR -
87rG c4 Tuv + Aguv
(2)
The factor A being constant, this new term does not modify the covariant divergence of both sides o f the equation, which remains null. 453 0001-7701182]0500.-0453503.00]0
9 1982 Plenum Publishing Corporation
454
KERNER
In the simplest solution without matter, when Tt~v - O, the scalar curvature is constant:
R =-
A 4
(3)
The Einstein and de Sitter solutions are of this type; the difference between them is in the sign of the cosmological constant A, which is positive in Einstein's solution and negative in de Sitter's one. By being introduced as a mathematical tool in order to avoid a singularity, it is no wonder that the constant A does not have a direct physical meaning. The form of the term A g u v , which is added to the energy-momentum tensor, suggests an interpretation in terms of a constant pressure. This pressure would be responsible for avoiding the cosmological collapse even in the case of the nonzero density of matter. But then it becomes necessary to explain the existence of such universal pressure by some microscopic phenomena. A number of such explanations has been proposed by Sakharov [2], Wheeler [3], Landau [4], Pomeranchuk [5], and others. In their papers, in which the authors evaluate the constant A as being of the order of 10 -s6 cm -2, the value of A along with its physical interpretation is deduced by supposing that the vacuum is endowed with very high elasticity (of order A -1 ). This elasticity is due to the quantum fluctuations of energy in the vacuum. The idea of the space possessing its own elasticity even in the absence of any matter in it goes back to Clifford [6]. This elasticity should explain the fact that the Euclidian (or Minkowskian) geometry is observed among so many other Riemannian geometries. However, if we admit such a possibility, we run into ugly trouble of a philosophical nature. Assoon as we begin to speak of any physical properties of the vacuum, we are de facto replacing it with the concept of aether. If vacuum is really nothingness, i.e., the absence of anything, then it is a contradictio in adiecto to speak not only of elasticity of the vacuum, but of any physical property of the vacuum as well. In the case when the vacuum is conceived as some basic or equilibrium state of something that exists, e.g., like what is assumed in the theory of the excited states of a crystal lattice (phonons), then, of course, such a contradiction does not appear. In contrast, the universal vacuum, conceived as the absence of anything that exists, should not have any known physical property at all-except that of "non-being." Here we come very close to the concepts put forth by Parmenides [7]. According to what we know about his ideas, he started with defining the Being as everything that exists, while the non-Being, or Nothingness, was what did not exist, by opposition to the Being. From this principle it was easy to deduce that the Being (the Universe, as we would say today) is unique, and does not have either a beginning or an end. As a
COSMOLOGY WITHOUT SINGULARITY
455
matter of fact, if there were another separate Being, then there should be a nonBeing between the two; but the non-Being does not exist, therefore anything that is between the two Beings is also part of Being, just because it exists. Q.E.D. About 2300 years later Hegel [8] tried to include the vacuum (nothingness) among the categories of reality by remarking that nothing when defined as a contrary of something that exists becomes no less real and acquires an autonomous existence. To take an example from physics, a positron can be considered as the absence of an electron ("a hole") in the continuum of electrons with negative energy. Such an interpretation is very successful in quantum electrodynamics. However, it seems to be clear that Hegel's affirmation loses all its sense when applied to the Universe as a whole. One can consider some properties of nothingness as long as Being exists to be opposed to; but if no such possibility is left, nothingness (the vacuum) ceases to exist even in a relative sense. Mach [9] translated these philosophical problems into physical terms. We shall not betray his thought if we formulate a statement that it is impossible to define an inertial mass of an object which would be a unique thing existing in the Universe. In the same way, it should be impossible to measure its momentum, angular momentum, acceleration, etc., which has been remarked on already by Berkeley [ 10]. We think that Mach's principle, formulated in this way, should apply also to the constant energy density of the vacuum, represented in the variational principle by the cosmological term Igll/2A added to Igll/2R. If we enlarge Mach's principle in order to include the inertia (energy) of the vacuum, we conclude that the energy of the vacuum can differ from zero only if some matter is present in the Universe. Thus we conclude that A ought not to be constant, but should be a function of the state of the Universe, namely, of the mass-energy distribution, or the curvature, which is the same by virtue of Einstein's equations (2). In other words, we should replace the cosmological term by a (nonlinear) function of the curvature. w
Gravitation and Nonlinearity
Although the differential equations of Einstein's theory of gravitation are highly nonlinear, some relations they imply between several important measurable quantities are linear nevertheless. The mathematical description of a linear relation between some physical phenomena can often include nonlinear equations, especially if we admit a sufficiently large class of coordinate transformations. Even Newton's second law will be represented by a nonlinear equation if we admit an arbitrary (though nonsingular) change of coordinates (x, y, z, t) into some nonlinear functions depending on all these variables. Still, the intrinsic physical meaning of Newton's second law will remain unchanged: the acceleration of a point mass is a linear function of the force.
456
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The situation is somewhat more subtle in Einstein's theory. If we want to find the components of the metric tensor guy for given Tuv, the relation is highly nonlinear and not even one-to-one. Let us remark, however, that neither the components ofg~v nor even the components of the Christoffel connection P~'~v represent physically measurable quantities. The components of the Riemann tensor R~v ~x, which are measurable physically, also cannot be considered to be linear functions of the source represented by Tuv. It is well known that outside of a massive body, where Tuv = O, the components of R~v~x are different from zero; moreover, the RuvKx generated by two different massive bodies are not a sum of the components of Riemann tensors that each body alone would produce. However, if we are interested only in the components of the Ricci tensor R~v, or just the scalar curvature R, then the fundamental relations are linear: v I 6~R = R u - ~-
8rrG ca T~
(4a)
or
4~G R=-'~ - T
(4b)
In the framework of a homogeneous, isotropic, and spherically symmetric cosmology, when the metric tensor takes on the Robertson-Walker form, and when only one unknown function, namely, the scale factor a(t), remains relevant, only one linear relation R = (4~rG/c4) Twill survive. In this sense we may call the spherically symmetric, homogeneous Einstein cosmology linear. We know, however, that, although for different reasons, the Newtonian mechanics as well as Maxwell's electrodynamics are linear approximations of some more complicated nonlinear theories. For instance, the linear relation between the current (source) and the field in MaxweU's theory becomes nonvalid on the microscopic scale because of the vacuum polarization caused by quantum fluctuations. Even without the exact knowledge of the way in which these phenomena occur and affect the theory, one can quite successfully take them into account on a purely phenomenological level, usually by modifying the Lagrangian of the theory. It has been done for the electrodynamics by Mie [1 t] and Born and Infeld [12]. Their theory, known under the name of nonlinear electrodynamics, was based on the modified ("effective") Lagrangian, which in the simplest case has been chosen as
(5)
COSMOLOGY WITHOUT SINGULARITY
457
Here E is the electric field, H is the magnetic field, b is the value of the cutoff, i.e., the maximum strength the expression (H 2 - E2) 1/2 can take on by hypothesis. Such a bagrangian enabled the authors to construct an electrostatic soliton with the f'mite energy-the price to pay for it was the loss of relativistic invariance. The Lagrangian (4) has been proposed by analogy with special relativity and by no means pretented to describe the real behavior of the hypothetical cutoff. As the quantum electrodynamics has shown, the actual cutoff in the field strength is rather of logarithmic type than of the form given in (4). However, the most essential feature of the modified nonlinear Lagrangian remains the phenomenon of saturation: as the field strength approaches the limiting value b, it takes more and more energy to make the field stronger, just as it took an infinite amount of energy to accelerate a mass to the velocity of light o -+ c in special relativity. It may be argued that a similar kind of cutoff should occur for the curvature tensor when the values of its components approach the order of magnitude at which the quantum fluctuations become nonnegligible. This should happen when -R . ~ . R p ~ lfi 2 ,~ 1066 cm -2
(6a)
where the Planck length l? is given by the following function of universal constants: [Gh~ 1t2 ~ 10 -33 l/, = ~c 3 ] cm
(6b)
Such a cutoff is by no means in contradiction with Einstein's general relativity, which has been verified experimentally with wonderful accuracy. Let us observe, however, that all those experiments have been carried out at very low values of the gravitational field or the matter density. As the order of magnitude of R? suggests, the only reasonable stage for the influence of such a cutoff will be the cosmological collapse. Many nonlinear modifications have been proposed already by different authors, namely, Lanczos [13], Weyl [ 14], Stelle [15 ], Ruzmaikina and Ruzmaikin [ 16], Nariai [ t 7], and others. In all these works the proposed Lagrangians were finite polynomials in R, R~v, and Rlzv~x. In what follows, we shall limit our study to the simplest case of a nonlinear Lagrangian depending only on R and not on R~v or Rt, v~x. As we want to investigate in more detail the vicinity of the cosmological collapse, without chang. ing the standard Friedmann model outside that relatively short time region, we think that at that primary stage of evolution only the simplest features of matter remain to be important. These reduce probably to a very small number, such as the energy density or pressure, maybe the total charge, the total angular momen-
458
KERNER
tum, and not many more. Therefore the number of the unknown functions seems to be too big in R,v or in R,u Kx, especially in the case of the spherical symmetry. Any nonlinearity [i.e., replacing R by some function f(R) in the original Einstein Lagrangian] modifies the content of the theory by introducing higher than second-order derivatives; nevertheless, it turned out to be difficult to obtain the saturation phenomenon independent of the initial conditions with a LagrangJan being a finite polynomial inR, R,v, or R,~Kx (cf. [16, 17]). Roughly speaking, whatever polynomial we choose, it will tend to infinity when the components of the Riemann tensor grow infinitely, in contrast to what happens with the similar relation between the Lagrangian and the field strength in (6), and there is little hope in stopping the collapse. Moreover, any polynomial implicitly contains some constants, corresponding to the coefficients of the different powers of R it contains; these constants have also to be explained somehow, whether we feel that the saturation phenomenon should be ruled by only one constant [equations (6)]. We conclude therefore that the polynomial Lagrangians considered in [13] -[17] represent but the Taylor series expansions of the Lagrangian f(R) for R ~ 0, and are a very bad approximation when R ~ R / , or when R goes to infinity. Different possibilities of introducing the saturation phenomenon should be considered, e.g., f ( R ) ~ oo for R ~ RI,, f(R) ~ 0 for R ~ RI, , f(R ) --~const for R ~ ~, etc. In the following paragraphs we examine closer the last choice, showing also that the behavior of the cosmological solution near the collapse depends on the interplay between f(R) and its first and second derivatives.
Derivation of the Equations for the Robertson-Walker Metric
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Let us first consider a Lagrangian density depending smoothly on R: the variation principle can be written as 6 [lgl 1/2 [f(R) + Lm] d4x = 0
(7)
where Lrn means the Lagrangian of matter and all other fields. This variational principle yields the following equations (see also Buchdahl, [18] ): f'R~
-
1
~ g~f(R) +f"(V~ V~R - g~gxp Vx VpR) +f ' ( V u R V u R
guvgXOVxRVoR) + -s~. a
r.~ = 0 (8)
Here by definition
8~rC T ~ = "8(tgll/~Lm) Igl 1/2 c ~ ~g~V
(9)
COSMOLOGY WITHOUT SINGULARITY
459
Obviously, when f ( R ) = R, the system (8) reduces to Einstein's equations, because f ' = 1, f " = f " - O. The equations (8) are self-consistent whatever function f ( R ) we choose, because the covariant divergence of the left-hand side of (8) vanishes by virtue of the Bianchi identities VURu~ = 89%R
(10)
We can go further without specifying the function f(R). Let us consider the homogeneous, spherically symmetric Universe. It is easy to prove that also with a generalized Lagrangian (7) the only metric compatible with such assumptions is the Robertson-Walker metric, for which ds 2 = - d r 2 + a s ( t ) (
d r 2 . + r 2 d 0 2 + r 2sin 20d~02/ k l - kr 2 /
(11)
(k constant). The metric tensor is diagonal: _
gtt = - 1,
grr
a2(t) 1 - kr 2 '
goo = a2(t) r 2,
g ~ = sin 20goo
(12)
This metric involves one unknown function of time [the scale factor a(t)], and one constant k which can be determined by the initial conditions; k may be positive, negative, or null. The Universe is closed when k > 0, open when k ~< 0. The nonvanishing components of Ruv are the following: 3/i Rtt = y ,
R/I = - (aa + 2a 2 + 2k) a2 gq
(13)
here h = da/dt, etc., i, ] = 1,2, 3. Therefore R = gUVRuv = - 6(a//+aS fi2 + k)
(14)
and gUVVuTvR=_l~+ g u , V u R V u R = _/~2
3~k)
(15) (16)
The equations (8) contain, via the expression (15), third and fourth derivatives ofa(t). As in general relativity, with such a restrained form of the metric only two differential equations can be obtained from the system (8); we can choose for example, the tt component and the trace obtained by the contraction of (8) with the guy. This yields the following two equations:
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KERNER
. /
Ls
'R
- ~ - - T-- 0
+ 1
3 " a / ~ + 8rrG T
,, 7s- s, a
-7-,,=
(17)
0
(with T = g"VT.v ). These equations are not independent: one can be obtained from another by differentiation, using the covariant conservation of the T~v. and the Bianchi identities. Therefore we can from now on study only the last equation, which we rewrite as
3~ f , _ I f + f , a
31tR a
=
87rG c4
(18)
Ttt
where 6(a//+ h 2 + k) R =-
a2
The equation (18) contains only the third- and lower-order derivatives of
a(t). If the energy-momentum tensor depends on more than one function of t, e.g., the density p(t), the pressure p(t), etc., we need one or more equations of state in order to solve completely the equation (18). After expressing R in terms of a, fi,//, and "a, we can write down the equation for ~i':
.... aa =
2a 2k + 2a 4 - aaci
f'aa
a2f
a2
6f"
36f"
4zrGa 2 gc4f "
ztt
(19)
Even if the second derivative f " never vanishes, we still cannot determine ~" in the neighborhood of a = 0. In what follows, we shall consider the simplest equations of state for the matter, which are the following: p =~ 0, p -= 0 for the matter-dominated stage; p 4=O, p = p/3 for the radiation-dominated stage. In the first case Ttt= Pola a , and in the second case T t t = poac/a 4. Here P0 means the mean density of the matter at present, when we also gauge a = 1; ae is the effective value of a at which the radiation becomes dominant. One can also consider the "hardest" equation of state for which the energy is still positive, p = p, discussed by Zeldovich and Novikov [19]. This would lead to Ttt = C/a ~, C being some appropriate constant. From now on we shall denote the right-hand side of (18) by CN/a ~v, where CN is the appropriate constant, and N is equal to 3, 4, or 6, depending on the case considered. We shall now proceed with choosing some plausible form of the function f(R) and trying to construct some approximate solutions.
COSMOLOGY WITHOUT S I N G U L A R I T Y
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461
Approximate Solution o f the Simplest Model
It is obvious that when R is small compared to lt52 , f ( R ) ~ R. What we have to modify is the dependence o f f on R when [RI ~ oo. According to general principles exposed in the first paragraph, we would like to impose a finite limit on f(_+o~). A closer look at equation (18) suggests that the important features of the behavior of solutions are contained in f ' and f " . When the density [righthand side of (18)] becomes very high, it should be more and more difficult to make the curvature rise; this means that f ' and f " should go to zero when IR[ ~ oo. We shall choose a very simple possibility for such kind of saturation, without pretending that our choice is anything more than a qualitative guess. Let us put f(R)-
R 1 - liaR
(20)
(we remind the reader that at present R < 0). When IRI goes to infinity, f ( R ) tends to -1} 2 . Of course, when liar ~ 1, we can develop f ( R ) in powers of R, obtaining as first terms R + liar z + 9 9 but the validity of such an approximation may be lost in the neighborhood of the collapse. Many other choices of "saturating" functions are possible; let us show some of them: e ~'n - 1 f(R) = - - - , f(R) =
f(R) =-
log (1 - XR) ~[1 + log (I - XR)]
tanh ;~R ~. etc.
(21)
(we put k = l~). We are unable to tell what is the actual form of the nonlinearities involved; let us just remark that if the nonlinearity prevents the collapse early enough, i.e., before ]Rt attains the values comparable with l~ 2, then all these functions are equivalent to the homographic function (20). If we develop our differential equation (19) into powers of l~, we shall set that the zeroth order coincides with Friedmann's equation, which contains only and a, whereas the next terms contain the higher derivatives of a; therefore the perturbing terms nonlinear in R modify the order of the differential equation. The situation we are facing here has been studied by the mathematicians. It is known under the name of the singular perturbation problem, and the technique used to solve it is called the method of matched asymptotic expansions. (One of the best reviews available is the paper by R. E. O'Malley Jr. [20] .) A singular perturbation occurs when the term proportional to the small parameter modifies the order of the differential equation, e.g., ey" +ay' + b y = O ,
O 0, a2 > 0. Consequently, a(t) = as t + aa t 2 + a4 t 3 + - - . 2! " 3!
(24a)
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463
WITHOUT SINGULARITY tz4 t 2
~/(t) = a2 + aa t +
R(t)
+-..
2
(24b)
t2
+ Rxt + R2 -~ +" "
=Ro
(24c)
with Ro -
6(a2ao + k) a~ 6a3
R1 = - - -
a0
,
(25a)
R2 =
12ka2ao
- 6a~ -
6a4ao
a2
(25b)
Let us denote f[t=o = f(Ro) =fo,
f'[t=o = f ' ( R o ) = f ~ , etc.
(26)
Then, if we develop the equation (18) into powers of t in the neighborhood of t = 0, we obtain the following relations: 3a2ao f • -
oo
21f ~ = aCN N ,
1+
(for zeroth order)
(~176 :o _ NCN
6aoN ,
(27)
a oo,
(for second order in t)
(28)
The terms linear in t vanish identically on both sides. We see that for the five parameters ao, al, a2, a3, and a4 we have two independent equations-therefore three parameters to fix the initial value problem. We have chosen ax = 0; the choice of ao determines a2 by virtue of (27), whereas a~ will determine a4 via (28), etc. This is our "inner expansion." The "outer expansion" will be supposed to be an entire series in the inverse of t:
a(t) aF(t) ( l+b'lP =
t
b~l~
+--7-
+""
)
(29)
where aF(t ) is the usual Friedmann solution. Relations between the constants b l , b2, ba, etc. can be obtained if we develop the equation (18) with a(t) given by the outer expansion (29) into powers of le. The first relation is independent of the choice o f f ( R ) and gives. b2 = - I b~
(30)
464
KERNER
In principle, if we develop farther the series (2), its domain of validity spreads toward bigger values of t; similarly, if we continue to compute the coefficients b3, b4, etc. in the series (29), its domain of validity spreads toward smaller values of t. We may hope that if the function f(R) is well chosen, the two domains will overlap in some finite region It1, t2 ], and we can match the two approximate solutions at some point B E [tl, t2 ]. In principle, we can determine as many coefficients as we wish, together with B, if we match higher and higher derivatives. Of course we have to stop when the number of equalities becomes bigger than the number of constants to determine; it may happen even before that the equations become incompatible. If this procedure converges, we may be able to find the more and more accurate approximations to the exact solution of the nonlinear equation (18). We shall illustrate this by a simple example, in which f(R) = R/(1 - l~R), and in which the expansion on both sides is not very long. Let us also consider the case k = 0 for the sake of simplicity. In that case Ro
6a2
f o - 1-l~,Ro -
ao +6a21~
1
(31)
ag
f ~ - (1 - l~Ro) 2 = (ao + 6a2 l~o)2
(32)
The equation (27) takes on the following form:
cN
a~ - 18l~aN~ (ao + 6 a 2 / ~ ) 2
(33)
The equation (33) determines a2 as a function of a0. On the ao-a2 plane there is an obvious asymptote corresponding to ao i> amin
=
(2l~CN) I/N
(34)
This form of dependence of a2 on ao, which seems to be reasonable, is altered if we change the asymptotic behavior off(R). To illustrate it by an example, let us generalize (20), replacing it by
f(R) = R(1 - l~R) -v
(35)
with v real (we remind the reader that in our case R is negative all the time). The relation (34) is modified then into 2 2I~CN[1 + 61~a2 '~'+1 a-----~/ = ~aNo '~ ao )
~61~a2/
One can distinguish between three different c~ises: (a) ~ = 1,
(discussed above)
(36)
COSMOLOGY
(b) v < I,
WITHOUT
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SINGULARITY
then t/(R)t -+oo as [RI -~ t~2 --> o0
as
a0 ~
a2 "+ 0
as
ao ~ oo
()
and there is no minimal value for ao; (c) p > l ,
if(R)l -~0
as
tR[ ~ ~ 1 7 6
there is no vertical asymptote and not even a one-to-one relation (two values of a2 for one value ofao); there exists, however, a minimal value ofao. On the other hand, a2 tends to zero when ao goes to infinity. The absolute minimum for ao will have the same value for any "saturating" Lagrangian of (21). The actual value of ami n depends on the choice of the equation of state (N) and the constant CN. Even if we do not take any pressure and any radiation into account, N = 3 and C3 = 87rGpo/C 4 ~ t 0 - s 6 c m - 2 , then we obtain amin ~ 10-41 9 Recalling that we scaled a equal to 1 at present, which corresponds to the radius of the Universe of the order of 1028 cm, we see that ami n .~ 10 -41 . Recalling that we scaled a equal to 1 at present, which correIf we take into account the radiation and its pressure Prad = Prad/3, we can roughly evaluate ami n as being equal to 10 "s cm. We shall match together the truncated developments: near t = 0 we put
a(t) = ao + a2 t 2 + a3 t 3 2 6
(37)
and for larger values of t we have
a(t) = [Jt 1/2 (1 + b t l e
b2L1/~_~
t
(38)
t2 ]
We have chosen file radiation-dominated Friedmann solution 13tlt~; then = (4C4) It4. Matching the two expansions means that at some t = B we have the function
ao + a2B2 ....... 2
a3B3 6
+ - -
= (jB~/2
+
2 2 b~l}~ 3B 2 ]
blip - B
(39)
the first derivative,
B2
3B-1/2
1
bill,
.
2 2 bile
(40)
etc. Let us introduce the dimensionless variables by putting
ao = n[3B 112,
a2 = s3B -3/~,
B = mle,
a3 = p 3 B -s/2
(41)
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KERNER
In terms of the dimensionless variables n, s, m, p, bl, our equations can be rewritten as follows: 6 ~
+
(42)
[instead of ( 3 3 ) ] bl n + s + __p= 1 + m 2 6
(function)
(first derivative)
s + -p = 1 2 2
(second derivative)
1-
b~ 3m 2 b__.!
m
+
rn2]
(43) (44)
1 {1 361 + 5b~ s + P = - -4 ~- - rn --~}
(45)
3 (1 5bl + 3 5 b ~ p= ~ - m -~-m2 ]
(46)
(third derivative) etc.
It is impossible to have a simultaneous solution for the last three equations if
bl/rn is real. We conclude then that it is impossible to match anywhere the two developments including the third derivative. We shall therefore match only the function, its first and its second derivative. This will leave us with a one-parameter family of solutions. If we introduce a parameter x = bl/m, then the three matching equations (43), (44), (45) give us n = 1 ( 3 + 1 0 x - 7 x 2) s = 88 (5 - 7x + 9x z)
(47)
p = - 8 8 ( 3 - 5 x + 7 x 2) For x real s is always positive, p is always negative. By the definition of ao n has to be positive too; thus -
-
7
1, which restrains the possible values of x: 0.14 < x < 1.5. We have put some typical results obtained for different values o f x in Table I. If we assume the numerical values (?4 = (8rrpo Glc4) ae, with Po ~ 10-3~ g/cm3 ,
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Table L Some typical values ofao, 32 and the matching parameters. x
n
0.15
0.724
0.2 0.3 0.5 1.0 1.2 1.4
0.787 0.895 1.042 1.0 0.82 0.547
s
p
m
bl
ao
1.038
-0.602
4.434
0.665
2.2 • 10 -33
15.7 • 1031
0.99 0.928 0.938 1.75 2.39 3.21
-0.57 -0.533 -0.563 -1.25 -1.77 -2.43
5.138 6.008 7.572 14.103 15.5 11.939
1.028 1.802 3.786 14.103 18.6 16.715
2.5 • 10 -33 3.1 X 10 -33 4.1 X 10-33 5.3 X 10-33 4.6 X 10 -33 3.2 X 10-33
12 • 1031 9 • 1031 6.4 • 1031 4.6 X 1031 5.5 X 1031 6.6 x 1031
a2
(cm 2)
ac ~ 10 -1~ and lp = 10 -33 cm, we obtain the values o f a o and a2 which we have included in the table. w
Discussion and Conclusion
We have seen how an approximate solution of the equation (18) with avery simple choice of the function f(R) can be constructed. The equation has been used in fact only via its zeroth-order approximation. We can go farther if we introduce next terms, 34 t4/4 ! in the inner expansion and b3 l~/t 3 in the outer expansion. The most important problem is to know if the too formal series converge, and if their domains of validity overlap. This we are unable to determine at the moment. However, if the function f(R) is chosen in such a way that its derivatives and itself do not change the sign, it is hardly conceivable that starting with the nonsingular initial conditions ao > 0, 32 > O, al = 0 the solution a(t) for t sufficiently big would end in anything other than a Friedmann solution. The orders of magnitude of a0, a2 can depend very much on the character of the nonlinearity o f f ( R ) . This can be investigated by numerical methods and the use of a computer. There is another promising direction in which the properties of the Lagrangian f(R) guaranteeing that the absence of the singular collapse may be investigated. It has been pointed out by Eysseric [21 ] that our fundamental equation (18) is a quasilinear equation of the second order for the unknown function f(R), if we insert some given a(t) and if we can express all the coefficients as the functions of R. In order to see what functions f(R)correspond to the solutions having a regular minimum at t = 0 in a(t), we should replace in the equation (18) the aF(t) of the corresponding Friedmann solution by some a(t) differing from it significantly only in the neighborhood of the collapse, t = 0. Let us illustrate this by a short example. In the case of uniformly distributed dust without pressure Friedmann's equation is 42+1 a:
2 a3
(here we have rescaled our quantities in such a way that k = I and 6"3 = 2).
(50)
468
KERNER
The solution aF(t) can be parametrized as aF = 1 - cos 0
(51) t = 0 - sin 0
and has a singularity at t = 0. Let us modify slightly this curve in order to avoid the singularity: we shall put a ( t ) = 1 - (1 - e) cos 0
(52)
t = 0 - (1 - e) sin0 Now a ~ e when t ~ 0. We have da _ (t - e)sin 0 dt a
and
R = - 6(//a + h2 + 1) =_ 6 a2
(53)
a3
Inserting the corresponding expressions for a, h,//, R, etc. into (18) and expressing dr/dR as ( d f / d a ) ( d a / d R ) = (a4/18)(df/da), etc., we obtain the following equation out of the equation (18): a[(1-e) 2-(l-a)
d2f+ [ 5 ( 1 - e ) 2 -
2] da 2
5+9a-4a2]
~df
3 f = ---~ 36a
(54) One easily verifies that when e = 0, the corresponding solution is f = -6/a 3 = R . The equation (54) may be solved in principle by the method of analytic series; the initial conditions should be taken from the fact that f h a s to behave as R for small R, i.e., when a ~ ~. Closer investigation of this problem will be the subject of our next papers. We think that it is worthwhile to continue the research along these two complementary lines of approach: trying different types of function f ( R ) with some well-defined asymptotic behavior for R -->Re or R ~ 0% and to look at the implications on the corresponding solutions a(t); and inversely, inserting the functions a(t) which are close to the corresponding Friedmann solutions but have no zeros, into the equation (18), and then solving it for f(R) if possible. We h o p e that more and more generalizing conclusions may be drawn then. To conclude, let us make a few remarks on the experimental checking of the predictions of such a theory. The difference between these predictions and what happens in the singular Einstein-Friedmann cosmological model can be felt by the values of ~, and/i when t becomes very small. As we do not know how to evaluate accurately h/a, let alone 8/ait, it is highly improbable to check any influence of the nonsingular cosmology upon the values of these parameters now. In contrast, the difference can be much greater at the early stage of evolution of the Universe. Probably, the only remaining trace of what happened may be
COSMOLOGY WITHOUT SINGULARITY
469
found in the relict gravitation and neutrino radiation, which survived the passage through the furnace. Unfortunately, their detection cannot be scheduled for tomorrow; maybe even after t o m o r r o w would be too optimistic a guess.
Acknowledgments The author expresses his gratitude to his father, Dr. Samuel Kerner, for transmitting some o f his knowledge about Antiquity. Thanks are due to Dr. J.-P. Duruisseau and Mr. P. Eysseric for many discussions, suggestions, and enlightening remarks. Professor M. Roseau's pointing out the asymptoting expansion m e t h o d as an appropriate tool for treating the problem is gratefully acknowledged. Last but not least, the author is obliged to the referee, whose remarks enabled him to avoid some errors and clarify the argument. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
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