On the initial conditions for super-exponential inflation

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ON T H E I N I T I A L C O N D I T I O N S FOR SUPER-EXPONENTIAL INFLATION M.D. POLLOCK Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400005, India Received 15 December 1987

In a higher-dimensional theory of gravity containing higher-derivative terms and a cosmological constant, a period of superexponential inflation of the physical spaeetime is possible, during which the Hubble parameter H increases with time, as discovered by Shaft and Wenerich. This means that the initial value Ho must be small H0< 6 × 10-Smp. We discuss the meaning of this initial condition from the standpoint of quantum cosmology.

Inflation [ 1 ] offers a solution to many cosmological paradoxes, as reviewed in ref. [2 ]. Virtually all inflationary models so far proposed, however, require the existence of some very small dimensionless parameter 2 < 10-~°, or some small mass m < 1 0 - 5rnp, where mp----G- 1/2 is the Planck mass. The reason for this is that density perturbations are generated during the phase of exponential (or quasi-exponential) expansion, and the observational constraint 5p/p< 10 -4 implies a restriction on the Hubble parameter H. We assume a FriedmannRobertson-Walker spacetime ds2=dt2-a2(t)

d × 2,

( 1)

where a ( t ) is the scale factor of the Universe and H--h/a, and the dot denotes differentiation with respect to comoving time t. Setting H(t~ ) =HI, where tl is the time when a fluctuation crosses outside the de Sitter causal horizon H - J, if it is to re-enter the causal horizon of the subsequent Friedmann universe around the time at which matter and radiation decouple, then we must arrange that [ 3 ]

H~/mp<6× 10 -5 .

(2)

A similar inequality, weaker by only a factor ~ 2n, follows from consideration of the anisotropy of the cosmic microwave background radiation [4]. Now, for most models of inflation,/:/~< 0. Hence, if inflation effectively ceases at some time tf> t~, then we shall require that

Hr/m,, < 6 × 10-sY~,

(3)

where Yj ~< 1, so that (3) is at least as strong an inequality as (2). This is the origin of the requirement of finetuning of some parameter. On the other hand, if the theory originates in D > 4 dimensions, and also if it contains both a cosmological constant .4 and some combination of higher-derivative terms, 2 ~ t~ I 1~ 2 _~ (~2J~IBI~,,tlI.~ ~3t~AB(,DI~ABC'D,

(4)

where the indices A run from 0 to D - 1, then it is possible to construct a model of inflation for w h i c h / / > 0, provided that the arbitrary constants ~i are suitably chosen, as discovered by Shaft and Wetterich [ 5 ], hereafter called paper A. (The possibility of obtaining inflation in a higher-dimensional gravitational theory was also pointed out in ref. [ 6 ]. ) This means that the inequality (3) is replaced by

Hf/mp < 6 × 10-5Y2,

(5)

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where Y2> 1, which is accordingly a weakerinequality than (2). By adjusting the parameters, it is possible to set Y2 >> 1. The ratio Hr/mp is itselfa free parameter, which need not necessarily be much less than unity. This has led to the statement in A that an inflationary model can be constructed without excessive fine-tuning of parameters, by taking Y2~ 102, Ht-/mp..~10 -2 and certain other dimensionless ratios fine-tuned only at the level ~ 10 -2. This is a significant result, because it looks as if the otherwise generic need for fine-tuning as a prerequisite to inflation has finally been evaded. But it should be realized that now the initial value H~ at which inflation commences must be much smaller than rap,

H~/ml, ~<6× 10-5 Y3,

(6)

where Y3 < 1. If the classical universe is supposed to come into being from some pre-existing quantum gravitational state, however, then one might naively expect that H~ ~ me, since this is the typical magnitude of spacetime fluctuations at the Planck epoch. Thus, it appears that fine-tuning of the parameters of the theory may have been traded in for some kind of fine-tuning in the initial conditions of the expansion of the universe. Various aspects of the model A have been further analyzed in refs. [7-9 ], to which we refer the reader for details. There are two distinct potentials, Vand W, whose combined action is to cause H first to increase from Hi u p tO some maximum value Hma x ~ Hf, and then to decrease again. (For some choices of the parameters &i, a subsequent period of sub-exponential inflation may occur, as for example when ~ 2 has the Gauss-Bonnet form [9 ]. But this inflation would be driven by the term .3 alone, and we should return to the original finetuning requirement. ) Thus, one might guess that the classical evolution would most likely start from the point H ...... and that the phase of super-exponential inflation would be absent. One should then ask how probable it is for the universe to begin its evolution from H, instead. This question is relevant also because super-exponential inflation appears to be the only way of reconciling inflation with the production of cosmic strings, if these are to lead to galaxy formation [ 10], for then a higher reheating temperature Tr is required than is normally possible [ 11 ] (see also ref. [ 12 ] ). The properties of superexponential inflation have been studied in refs. [ 13,14 ] and the possibility of raising TF without limit was mentioned in ref. [ 14 ], the results of which apply when V= - W. The purpose of this letter is to examine the issue of initial conditions from the point of view of quantum cosmology. It is a continuation of the discussion in ref. [ 9 ], hereafter called paper B. It was suggested in A that the evolution of the universe envisaged there might occur subsequent to an initial quantum creation process. Now, in four-dimensional spacetime, the theory of quantum cosmology has been developed considerably, following the paper by Hartle and Hawking [ 15 ] - see refs. [ 16-18 ] and references therein. A full treatment would require solution of the Wheeler-De Witt equation for the wave function of the universe [ 19 ], together with specification of the boundary conditions. There is no real obstacle to extending the approach of refs. [ 15-18 ] to theories formulated in more than four dimensions, as discussed by Halliwell [ 20 ]. According to the generalized version of the prescription [ 15 ], the quantum state of the Universe is defined by a path integral over compact D-dimensional metrics bounded by some specified ( D - 1 )-dimensional hypersurface. In four dimensions, the three-dimensional compact hypersurfaces are unrestricted, whereas it may be necessary to require the equivalent ( D - 1 )-dimensional hypersurface to be disconnected, as a consequence of cobordism theory, in order to obtain a non-vanishing wave function ~P [20]. We shall ignore this technicality here, and assume that it is sufficient to integrate over the internal dimensions and work from the effective four-dimensional action. This should be adequate in the mini-superspace approximation assumed. The starting point, then, is the ( D = 4+N)-dimensional theory of A, whose action function we write in the form (B25), S = (167rG) -~ 636

fdOx x/~ ( / ~ - 2 / 1 -

~2),

(7)

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where ~ is the determinant of the metric ~AB, /~ is the Ricci scalar, ~2 is given by expression (4), and the quantities ~ and A are constants. We write the metric in the form (B27),

~u~=(gU(xk)g22(Xk) 0

0 ) go~p(yl,) e2UtXk) ,

(8)

where g~j(x k) is the metric of spacetime, g,a(y ¢') describes the internal space, with Greek indices running from 4 to D - 1, and the function u(x k) is to be determined. The conformal factor g2(x k) is chosen so as to produce a constant coefficient of R in the effective four-dimensional action obtained by integrating over the internal space. This is given by expression (B32), m~,

s4= 1--~ f d n x ~ - ~ ( R - V( u) - ( &lR2 + &2RkiRki + &sRkh,,,,RJ"l'"") eNu + {½N(N+2) - [ (3N2+ 2N) dq

+N2&2 +N2&3I R e u"} U:kU:k

+ [ (N+ 2) ( N + 4 ) &2 - 2 ( N 2 - 4 N - 8 )

&3]

e NuRklU:kU:I--N(261~ +&2 + 2 ~ 3 ) eNuR:ku;k).

(9)

In writing expression (9), which is derived in B, we have used expression (B30), ~Q2~ e - IN+2)",

(10)

and have ignored all the terms in g2 with four derivatives. The potential V(u) is defined via expression (B33) as V(u) =2/1

e-N"+A e-~U+2"+B

( 11 )

e -~N+4)u,

where the constants A and B are defined in (B28),

When the radius of the internal space is very large, implying that u >> 1, then expression ( 11 ) can be approximated as

V(U) "~2~e-u".

(13)

Let us now assume a quasi-exponential expansion of the spacetime ( 1 ), together with the slow-rolling approximation for the field u, so that//<< 3Hfi, and also the condition (B61 ) on the parameters &~, that 2&~ +&: +2des =0.

(14)

(The restriction (14) includes the special case &j = &3 = - 6~2/4 for which ~ z is proportional to the Eulernumber density, and the theory is free of ghosts, but super-exponential inflation is not then possible [ 9 ]. ) We can rescale the field u via (B58),

u~O=

k/~Nmpu,

(15)

where the constant k is given by expression (B46),

k = N ( N + 2 ) - [ 16N(N+ 1) &, + 2 ( N 2 - 6 N - 8 )

a2 + 4 ( N 2 - 4 N - 8 )

&3],

(16)

in which we have introduced the dimensionless parameters ~ , defined as &i=&i -~,

(17) 637

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so that there is a kinetic term in the canonical form ½0:k0:~ in the expression for the lagrangian. It then turns out that the Hubble parameter H and the field 0 evolve according to equations (B 10) and (B 11 ), respectively,

H2~-(8rc/3m~,)V

(18)

and

3HO_~-014:/00.

(19)

The potentials V and W are related by the formula (B42, B43, B63 )

OV/OW 1 - ~ / ~ - X = N2(86z ' +2&2 + ~ 6 ! 3 -

(20) 1) '

where X must be positive if super-exponential inflation is to occur. This is possible, as discussed in B. Returning to quantum cosmology, we observe that the theory (7) contains both higher-derivative terms and an unspecified number of extra dimensions. This immediately suggests that an analytic expression for the wave function Twill be very difficult to obtain, even in the mini-superspace approximation, because at least two new fields are now present, in addition to the radius function a(t) of the four-dimensional spacetime. This can also be appreciated from the complexity of expression (9). But here we are really interested in the question of initial conditions, rather than in the evolution of T, and hence the central issue is that of boundary conditions. The proposal [ 15 ] is that the Universe "has no boundary". This leads to a time-symmetric wave function expressible as the path integral

T(h.~, ~)=fd[g,j]

d[q~] e -sEte"j ,

(21)

where h,/~ is the metric on some compact three-manifold Z, the euclidean functional integral is to be taken over all riemannian four-metrics gi/and matter fields q~ on manifolds M whose only boundary is Z, such that g~/ induces h,/j on Z, and SE [gi/] is the euclidean Einstein action. It can be shown that this wave function automatically satisfies the Wheeler-DeWitt equation. It has been pointed out by Horowitz [21 ], however, that the integral (21) is not obviously well defined, because Re (SE) is not bounded from below. In order to get round this difficulty, Horowitz quantized the classical theory 5 ° - - x / ~ f l R 2, where fl is a constant, for which SE>~0. Working with conformal time t/, so that dt 2= a 2(t/) dtf, and using a - In a and a ' - da/dt/as canonical variables, he obtained an analogue of the WheelerDeWitt equation for T which contains only a single "time" derivative 0/0a, in place of the double derivative 02/0a 2 which normally occurs, and hence resembles a Schr6dinger equation rather than a wave equation. This approach ~ has been generalized in ref. [23] to theories of the form 5 e = ~ - g [½ez2R+ V(X) +fiR2], where Z is a scalar field, and hence may be of some use in dealing with the quantum cosmology of the model A. Whatever solution we obtain, however, the physical picture - in particular, the probability density P for creation of the Universe - depends crucially upon the choice of boundary conditions, as recently stressed by Goncharov et al. [ 24 ]. For the time-symmetric boundary condition [ 15 ], it turns out that

Pvce -sE

(22)

whereas for the alternative "tunnelling" boundary condition [25-27 ], we find that

Poce +s~.

(23)

The relative merits of these two expressions, and where each is appropriate, is discussed in ref. [24 ]. This paper "~ By writing the wave function in the Born-Oppenheimer approximation ~u= TO~UM,it is possible to derive the standard Schr6dinger equation for the malter wave function TM [22 ]. 638

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is concerned with the stochastic approach to inflation [28-32] (see also the earlier paper by Vilenkin [ 33 ] ), which yields a Fokker-Planck equation for P directly, without involving the wave function ~u. Also discussed in ref. [ 24 ] is the possibility of constructing an eternally self-reproducing inflationary universe, which can be done provided that the Hubble parameter exceeds some critical value Hc at some moment [ 30]. For the model A, however, this is not what we are immediately seeking, because it would blot out the effects of super-exponential inflation. But it is easy to ensure that Hc > Hmax. The exact criterion is derived in B, where it is shown that H,. is related to the spectral index s, defined from the expression for density perturbations 8,0/ p3cM-~ where M is the mass scale. The relationship is (B77),

H~.lrn~, ~ 3x/s,

(24)

and since Hm,x < rap/3, whilst s > 1/20, then we typically find that Hc > m p > H . . . . which is what we shall assume from here on. (The opposite case Hc < Hmaxwould also be interesting, if we changed the context of the discussion. ) From the above, it is clear that we need an expression for the euclidean action SE. The quasi-de Sitter solution ( 18 ) is assumed in A, so that the Ricci scalar is R E 12HZ~2V,

(25)

where V can be approximated by expression ( 13 ) when u >> 1, which is required for self-consistency. Next, we analytically continue into the euclidean region. The four-volume after euclideanization is •4 ~ 38492R -2. Dropping all the kinetic terms in the reduced action (9), we obtain the approximation to the euclidean action, SE ~ --24~r[m ~,R - l - - m ~ V(u)R - 2 _ 16ha'

eN"],

(26)

where we have introduced the constant a ' defined as c~,- (~, + I ~ 2 + go~3 ~- )mr,/16=A. , -

(27)

In the semi-classical approximation, we can now use the solution (25 ), whereupon expression (26) reads

SE ,~, -- 12nm~,R - ' + 384n2ce ' eU"~ 3nN -2m 2z[ - I X - i eN,,.

(28)

The sign of the constant a ' is indeterminate, but the requirement that X > 0 means that a ' > 0. Selection of the Hartle-Hawking boundary condition [ 15 ] would then suggest that the most probable initial state for the universe is not the one adopted in A: rather, R~ > m 2 and Nu~ < 1, implying that - SE > -- 1. (Essentially the same argument has, however, been used by Baum [34 ] and by Hawking [ 35 ] in an attempt to explain why the cosmological constant vanishes, in the Einstein theory of gravity for which a ' = 0, and the argument can be generalized to a broken-symmetry theory of gravity [ 36 ]. ) On the other hand, if the classical universe is imagined to arise through some type of diffusion process from spacetime foam, then the "tunneling" boundary condition [25-27 ] may be more appropriate [24 ]. This would lead to the opposite expectation, that initially Ri << m 2pand that Nui >> 1, as required. More precisely, expression (22) represents a stationary solution for which the probability flux J from the spacetime foam vanishes. A second solution has been found by Starobinsky [29] for the case of non-vanishing J. The boundary condition is changed, since the time symmetry is now lost. The theory considered in ref. [29] is characterized by a potential V(0), and the solution, valid when Vm b-4 << 1, is

eocJV( d V/ dO )- ',

(29)

where ~ is the "inflaton" field with canonical kinetic term. Now, for the theory A, the equivalent ratio V(d Vide) -' is a constant, because of the exponential dependence of V upon 0 [ see expressions ( 13 ) and ( 16 ) ]. Further, the ratio X is a constant, whenever Nu >> 1. Thus, although the solution (29) was derived for a four-dimensional theory without any higher-derivative terms, it will also hold for the present theory. The validity of the solution (29) has been questioned in ref. [ 24 ] on the grounds that any stationary solution 639

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may be inconsistent. But if we provisionally accept the solution (29), then we do have a quantitative answer to the question posed at the beginning of this paper. For it implies that all initial values of u are equally probable, and hence that all initial values of ln(mp/H) are equally probable, within some range O<~H<~H. . . . It then appears that no very stringent fine-tuning requirement is involved in setting my~Hi/>2 X 104, in significant contrast to what is usually necessary. This conclusion rests on the exponential dependence of V upon 0, but more particularly on our choice of the solution (29) and of the corresponding boundary condition. There does not at the moment seem any way to quantify how special this choice is, but the issue deserves further investigation.

The author would like to thank Professor J.V. Narlikar for hospitality and financial support from the Tata Institute of Fundamental Research at Bombay, and the Royal Society of London for financial (travel) support.

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