A smooth exit from eternal inflation?

Journal of High Energy Physics, Apr 2018

S. W. Hawking, Thomas Hertog

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A smooth exit from eternal inflation?

HJE A smooth exit from eternal in ation? S.W. Hawking 0 1 2 Thomas Hertog 0 1 2 3 0 tum Gravity , Spacetime Singularities 1 Celestijnenlaan 200D , 3001 Leuven , Belgium 2 Wilberforce Road, CB3 0WA Cambridge , U.K 3 Institute for Theoretical Physics, University of Leuven The usual theory of in ation breaks down in eternal in ation. We derive a dual description of eternal in ation in terms of a deformed Euclidean CFT located at the threshold of eternal in ation. The partition function gives the amplitude of di erent geometries of the threshold surface in the no-boundary state. Its local and global behavior in dual toy models shows that the amplitude is low for surfaces which are not nearly conformal to the round three-sphere and essentially zero for surfaces with negative curvature. Based on this we conjecture that the exit from eternal in ation does not produce an in nite fractal-like multiverse, but is nite and reasonably smooth. AdS-CFT Correspondence; Gauge-gravity correspondence; Models of Quan- 1 Introduction 3 Discussion 1 Introduction 2 A holographic measure on eternal in ation 2.1 2.2 2.3 2.4 Setup Local measure: perturbations around S3 Global measure: squashed three-spheres Global measure: general metric deformations Eternal in ation [ 1 ] refers to the near de Sitter (dS) regime deep into the phase of in ation in which the quantum uctuations in the energy density of the in aton are large. In the usual account of eternal in ation the quantum di usion dynamics of the uctuations is modeled as stochastic e ects around a classical slow roll background. Since the stochastic e ects dominate the classical slow roll it is argued eternal in ation produces universes that are typically globally highly irregular, with exceedingly large or in nite constant density surfaces [2{5]. However this account is questionable, because the dynamics of eternal in ation wipes out the separation into classical backgrounds and quantum uctuations that is assumed. A proper treatment of eternal in ation must be based on quantum cosmology. In this paper we put forward a new quantum cosmological model of scalar eld driven eternal in ation by using gauge-gravity duality [6{8]. We de ne the Euclidean dual theory on the threshold surface of eternal in ation, which therefore describes the transition from the quantum realm of eternal in ation towards a classical universe, in line with the original vision behind in ation [9]. The subsequent evolution is assumed to be classical. A reliable theory of eternal in ation is important to sharpen the predictions of slow roll in ation. This is because the physics of eternal in ation speci es initial conditions for classical cosmology. In particular a quantum model of eternal in ation speci es a prior over the so-called zero modes, or classical slow roll backgrounds, in the theory. This in turn determines its predictions for the precise spectral properties of CMB uctuations on observable scales. Our starting point remains the no-boundary quantum state of the universe [10]. This gives the ground state and is heavily biased towards universes with a low amount of ination [11]. However we do not observe the entire universe. Instead our observations are limited to a small patch mostly along part of our past light cone. Probabilities for local observations in the no-boundary state are weighted by the volume of a surface f of constant measured density, to account for the di erent possible locations of our past light cone [12]. { 1 { associated with an in ationary universe. The saddle point action includes an integral over time from the no-boundary origin or South Pole (SP) to its endpoint on f . Di erent contours for this give di erent geometric representations of the saddle point, each giving the same amplitude for the nal real con guration (hij(~x); (~x)) on f . The interior saddle point geometry along the nearly vertical contour going upwards from the SP consists of a regular, Euclidean, locally AdS domain wall with a complex scalar pro le. Its regularized action speci es the tree-level probability in the no-boundary state of the associated in ationary, asymptotically de Sitter history. Euclidean AdS/CFT relates this to the partition function of a dual eld theory yielding (1.1). This transforms the probability distribution for the amount of in ation and leads to the prediction that our universe emerged from a regime of eternal in ation [12, 13]. Thus we must understand eternal in ation in order to understand the observational implications of the no-boundary wave function. However the standard saddle point approximation of the no-boundary wave function breaks down in eternal in ation. We therefore turn to gauge-gravity duality or dS/CFT [6{8], which gives an alternative form of the wave function evaluated on a surface f in the large three-volume limit. In this, the wave function is speci ed in terms of the partition function of certain deformations of a Euclidean CFT de ned directly on f . Euclidean AdS/CFT generalized to complex relevant deformations implies an approximate realisation of dS/CFT [14{19]. This follows from the observation [18] that all no-boundary saddle points in low energy gravity theories with a positive scalar potential V admit a geometric representation in which their weighting is fully speci ed by an interior, locally AdS, domain wall region governed by an e ective negative scalar potential V . We illustrate this in gure 1. Quantum cosmology thus lends support to the view that Euclidean AdS/CFT and dS/CFT are two real domains of a single complexi ed theory [6, 14, 20{23]. In the large three-volume limit this has led to the following proposal for a holographic form of the semiclassical no-boundary wave function [18] in Einstein gravity, ~ NB[hij ; ] = ZQF1 T [hij ; ~] exp(iSst[hij ; ]=~) : (1.1) Here the sources (h~ij ; ~) are conformally related to the argument (hij ; ) of the wave function, Sst are the usual surface terms, and ZQF T in this form of dS/CFT are partition functions of (complex) deformations of Euclidean AdS/CFT duals. The boundary metric ~ hij stands for background and uctuations. { 2 { The holographic form (1.1) has led to a fruitful and promising application of holographic techniques to early universe cosmology (see e.g. [15, 24{28]). No eld theories have been identi ed that correspond to top-down models of realistic cosmologies where in ation transitions to a decelerating phase. However we nd that many of the known AdS/CFT duals are ideally suited to study eternal in ation from a holographic viewpoint. This is because supergravity theories in AdS4 typically contain scalars of mass m2 = 2lA2dS with a negative potential for large . In the context of (1.1) such scalars give rise to (slow roll) eternal in ation in the dS domain of the theory that is governed e ectively by V . In fact the Breitenlohner-Freedman bound in AdS corresponds precisely to the condition for eternal in ation in dS. Here we use (1.1) to study eternal in ation holographically in toy-model cosmologies of this kind in which a single bulk scalar drives slow roll eternal in ation. We take the dual to be de ned on a global constant density surface f at the threshold (or exit) of the regime of scalar eld driven eternal in ation. The bulk scalar driving in ation corresponds to a source ~ that turns on a low dimension scalar operator in the dual. Hence we use holography to excise the bulk regime of eternal in ation and replace this by eld theory degrees of freedom on a kind of `end-of-the-world' brane. This is somewhat analogous to the holographic description of vacuum decay in AdS [29], although the interpretation here is di erent. Conventional wisdom based on semiclassical gravity asserts that surfaces of constant scalar eld in eternal in ation typically become highly irregular on the largest scales, developing a con guration of bubble-like regions with locally negative curvature. Holography provides a new perspective on this: the dependence of the partition function on the conformal geometry hij of f in the presence of a constant source ~ 6= 0 speci es a holographic measure on the global structure of constant density surfaces in eternal in ation. We analyse various properties of this measure and nd that the amplitude of surfaces with conformal structures far from the round one is exponentially small, in contrast with expectations based on semiclassical gravity. We also argue on general grounds that the amplitude is zero for all highly deformed conformal boundaries with a negative Yamabe invariant. This raises doubt about the widespread idea that eternal in ation produces a highly irregular universe with a mosaic structure of bubble like patches separated by in ationary domains. 2 2.1 Setup A holographic measure on eternal in ation For de niteness we start with the well known consistent truncation of M-theory on AdS4 S7 down to Einstein gravity coupled to a single scalar with potential HJEP04(218)7 in units where = 3 and hence lA2dS = 1. The scalar has mass m2 = 2. Therefore in the large three-volume regime it behaves as V ( ) = 2 cosh(p2 ) ; (~x; r) = (~x)e r + (~x)e 2r + { 3 { (2.1) (2.2) (2.3) V~ hold, where r is the overall radial coordinate in Euclidean AdS, with scale factor er. The Fe erman-Graham expansion implies that in terms of the variable r the asymptotically (Lorentzian) dS domain of the theory is to be found along the vertical line = r + i =2 in the complex -plane [18]. This is illustrated in gure 1 where r changes from real to imaginary values along the horizontal branch of the AdS contour from xA to xT P . This also means that in the dS domain the original potential (2.1) acts as a positive e ective potential V~ ( ) = V = 2 + cosh(p2 ) : This is a potential for which the conditions for in ation and eternal in ation where V~;2 =V~ 2, for a reasonably broad range of eld values around its minimum. This close connection between AdS supergravity truncations and eternal in ation in the framework of the no-boundary wave function (1.1) stems from the fact that the BreitenlohnerFreedman stability bound on the mass of scalars in AdS corresponds precisely to the condition for eternal in ation in the de Sitter domain of the theory. Bulk solutions with 1 initially are at all times dominated by the cosmological constant and eternally in ate in a trivial manner. By contrast, solutions with 1 initially have a regime of scalar eld driven eternal in ation, which eventually transitions into a -dominated phase. The wave function (1.1) contains both classes of histories. We are mostly interested in the latter class and in particular in the amplitude of di erent (conformal) shapes of the constant scalar eld transition surface1 f between these two regimes. The variance of the semiclassical wave function of inhomogeneous uctuation modes in the bulk is of order V~ = , evaluated at horizon crossing. In eternal in ation V~ . Hence the uctuation wave function spreads out and becomes broadly distributed [ 5 ]. This is a manifestation of the fact that the universe's evolution, according to semiclassical gravity, is governed by the quantum di usion dynamics of the uctuations and their backreaction on the geometry rather than the classical slow roll [2{5]. It is usually argued that the typical individual histories described by this wave function develop highly irregular constant density surfaces with a con guration of bubble-like regions with locally negative curvature. Below we revisit this from a holographic viewpoint. We conclude this discussion of our setup with a few technical remarks. The argument (hij ; ) of the wave function evaluated at in gure 1 is real. This means that in saddle points associated with in ationary universes, the scalar eld must become real along the vertical dS line in the -plane. The expansion (2.2) shows this requires its leading coe cient to be imaginary, which in turn means that the scalar pro le is complex along the entire interior AdS domain wall part of the saddle points. But the bulk scalar sources a deformation by an operator O of dimension one with coupling in the dual ABJM theory. Hence the holographic measure in this model involves the AdS dual partition function on deformed three-spheres in the presence of an imaginary mass deformation i ~. We are primarily interested in the probability distribution over h~ij for su ciently large defor1A realistic cosmology of course involves an intermediate radiation and matter dominated phase before the cosmological constant takes over. However since we are concerned with the structure of the universe at the exit from scalar eld eternal in ation this toy-model setup su ces. { 4 { mations , since these correspond to histories with a scalar eld driven regime of eternal in ation. Finally whilst we formally de ne our dual on the exit surface f from scalar eld eternal in ation, at in gure 1, we might as well take ! 1 because the classical, asymptotic -phase amounts to an overall volume rescaling of the boundary surface which preserves the relative probabilities of di erent conformal bopundary geometries [18]. 2.2 Local measure: perturbations around S3 We rst recall the general behavior of partition functions for small perturbations away from the round S3. Locally around the round sphere, the F-theorem and its extension to spin-2 deformations provide a general argument that the round sphere is a local minimum S3 background [32, 33]. The coupling of the energy-momentum tensor of the CFT to the curved background metric triggers a spin-2 deformation. The fact that the free energy is a local maximum for the round sphere is essentially equivalent to the positive de niteness of the stress tensor two-point function. Applied to the holographic no-boundary wave function (1.1) these results imply that the pure de Sitter history in the bulk is a local maximum of the holographic probability distribution, in contrast with expectations based on semiclassical bulk gravity in eternal in ation. 2.3 Global measure: squashed three-spheres We now turn to large deformations. The dual of our bulk model is the ABJM SCFT. Hence to evaluate (1.1) we are faced with the problem of evaluating the partition function of supersymmetry breaking deformations of this theory. We do not attempt this here. Instead we rst focus on a simpli ed model of this setup where we consider an O(N ) vector model. This is conjectured to be dual to higher-spin Vassiliev gravity in four dimensions [34]. Higher-spin theories are very di erent from Einstein gravity. However, ample evidence indicates that the behavior of the free energy of vector models qualitatively captures that of duals to Einstein gravity when one restricts to scalar, vector or spin 2 deformations [35{37]. This includes a remarkable qualitative agreement of the relation between the vev and the source for the particular scalar potential (2.1) [38]. We therefore view these vector models in this section as dual toy models of eternal in ation and proceed to evaluate their partition functions for a speci c class of large deformations. We return to Einstein gravity and a general argument in support of our conjecture below in section 2.4. Speci cally we consider the O(N ) vector model on squashed deformations of the three-sphere, r 2 0 4 ds2 = ( 1)2 + 1 1 + A ( 2)2 + 1 1 + B ( 3) 2 ; (2.4) where r0 is an overall scale and i, with i = 1; 2; 3, are the left-invariant one-forms of SU(2). Note that the Ricci scalar R(A; B) < 0 for large squashings [ 36 ]. We further turn on a mass deformation O with coupling . This is a relevant deformation which in our dual O(N ) vector toy model induces a ow from the free to the critical O(N ) model. The coe cient { 5 { is imaginary in the dS domain of the wave function as discussed above. Hence we are led to evaluate the partition function, or free energy, of the critical O(N ) model as a function of the squashing parameters A and B and an imaginary mass deformation m~ 2. The key question of interest is whether or not the resulting holographic measure (1.1) favors large deformations as semiclassical gravity would lead one to believe. The deformed critical O(N ) model is obtained from a double trace deformation f ( )2=(2N ) of the free model with an additional source f m~ 2 turned on for the single trace operator O ( ). By taking f ! 1 the theory ows from its unstable UV xed point, where the source has dimension one, to its critical xed point with a source of dimension two [34]. To see this we write the mass deformed free model partition function as where Ifree is the action of the free O(N ) model Here a is an N -component eld transforming as a vector under O(N ) rotations and R is the Ricci scalar of the squashed boundary geometry. Introducing an auxiliary variable m~ 2 = mf2 + O yields which can be written as with Zfree[m2] = Z D Dm~ 2e Ifree+N R d3xpgh fm~ 2O f2 O2 21f (m2 fm~ 2)2i ; Zfree[m2] = Dm~ 2e 2Nf R d3xpg(m2 fm~ 2)2 Zcrit[ m~2] ; Zcrit[ m~2] = D e Ifree+N R d3xpg[ fm~ 2O f2 O2] : the transformation from critical to free in [39]. We compute Zcrit for a single squashing A 6= 0 and m~2 6= 0 by rst calculating the partition function of the free mass deformed O(N ) vector model on a squashed sphere and then evaluate (2.10) in a large N saddle point approximation.2 Evaluating the Gaussian integral in (2.5) amounts to computing the following determinant log Zfree = F = log det N 2 " r 2 + m2 + R8 #! 2 ; 2The generalization to double squashings A; B 6= 0 yields qualitatively similar results but requires extensive numerical work and is discussed in [38]. { 6 { 0.5 -0.5 0.6 0.4 eigenvalues of the operator in (2.11) can be found in closed analytic form [40], n;q = n2 + A(n 1 2q)2 1 4(1 + A) + m2 ; q = 0; 1; : : : ; n 1; n = 1; 2; : : : (2.12) To regularize the in nite sum in (2.11) we follow [ 36, 37 ] and use a heat-kernel type regularization. Using a heat-kernel the sum over eigenvalues divides in a UV and an IR part. The latter converges and can readily be done numerically. By contrast the former contains all the divergences and should be treated with care. We regularize this numerically by verifying how the sum over high energy modes changes when we vary the energy cuto . From a numerical t we then deduce its non-divergent part which we add to the sum over the low energy modes to give the total renormalized free energy. The resulting determinant after heat-kernel regularization captures all modes with energies lower than the cuto . The contribution of modes with eigenvalues above the cuto is exponentially small. For more details on this procedure we refer to [ 36, 38 ]. To evaluate the holographic measure we must substitute our result for Zfree[A; m2] in (2.10) and compute the integral in a large N saddle point approximation. The factor outside the path integral in (2.10) diverges in the large f limit. We cancel this by adding the appropriate counterterms. The saddle point equation then becomes : (2.13) We are interested in imaginary m~2 as discussed above. This means we need Zfree[A; m2] for complex deformations m2. Numerically inverting (2.13) in the large f limit we nd a saddle point relation m2( m~2). This is shown in gure 2, where the real and imaginary parts of m2 are plotted as a function of im~ 2 for three di erent values of A. Notice that Re(m2) R(A)=8. This re ects the fact that the determinant (2.11), which is a product over all eigenvalues of the operator operator has a zero eigenvalue. Since the lowest eigenvalue of the Laplacian r2 is always zero, the rst eigenvalue 1 of the operator in (2.11) is zero when R=8 + m2 = 0. In the r2 + m2 + R=8, vanishes when the { 7 { a function of the coupling of the mass deformation m~2 that is dual to the bulk scalar, and the squashing A of the future boundary that parameterizes the amount of asymptotic anisotropy. The distribution is smooth and normalizable over the entire con guration space and suppresses strongly anisotropic future boundaries. region of con guration space where the operator has one or more negative eigenvalues the Gaussian integral (2.5) does not converge, and (2.11) does not apply. This in turn means that the holographic measure Zcr1it[A; m~ 2] is zero on such boundary con gurations, as we now see. Inserting the relation m2( m~2) in (2.10) yields the partition function Zcrit[A; m~ 2]. We show the resulting two-dimensional holographic measure in gure 3. The distribution is well behaved and normalizable with a global maximum at zero squashing and zero deformation corresponding to the pure de Sitter history, in agreement with the F-theorem and its spin-2 extensions. When the scalar is turned on the local maximum shifts slightly towards positive values of A. However the total probability of highly deformed boundary geometries is exponentially small as anticipated.3 We illustrate this in gure 4 where we plot two one-dimensional slices of the distribution for two di erent values of m~2. 2.4 Global measure: general metric deformations It is beyond the current state-of-the-art to evaluate partition functions, be it of vector models or ABJM or duals to other models, for general large metric deformations. However, the above calculation implies a general argument suggesting that the amplitude of large deformations of the conformal boundary geometry is highly suppressed in the holographic measure both in higher-spin and in Einstein gravity. This is because the action of any dual CFT includes a conformal coupling term of the form R 2. For geometries that are close to the round sphere this is positive and prevents the partition function from diverging. On 3The distribution has an exponentially small tail in the region of con guration space where the Ricci scalar R(A) is negative and Zfree diverges. We attribute this to our saddle point approximation of (2.10). { 8 { A 0.12 0.10 2.0 1.5 the other hand the same argument suggests that the conformal coupling likely causes the partition function to diverge on boundary geometries that are far from the round conformal structure [ 41 ]. These include in particular geometries with patches of negative curvature or, more accurately, a negative Yamabe invariant. The Yamabe invariant Y (h~) is a property of conformal classes. It is essentially the in mum of the total scalar curvature in the conformal class of h~, normalized with respect to the overall volume. It is de ned as Y (h~) inf! I(!1=4h~) ized average scalar curvature of !1=4h~, where the in mum is taken over conformal transformations !(x) and I(!h~) is the normalI(!1=4h~) = R M R M !6ph~ d3x ph~ d3x 1=3 : (2.14) (2.15) There always exists a conformal transformation !(x) such that the metric h~0 = !1=4h~ has constant scalar curvature [42]. The in mum de ning Y is obtained for this metric h~0. The Yamabe invariant is negative in conformal classes containing a metric of constant R < 0. Since the lowest eigenvalue of the conformal Laplacian is negative on such backgrounds one expects that the partition function of a CFT does not converge, thereby strongly suppressing the amplitude of such conformal classes in the measure (1.1). This is born out by the holographic measure speci ed by the partition function of the deformed O(N ) model on squashed spheres evaluated in section 2.3. There the probabilities of large squashings for which R < 0 are exponentially small, which can be traced in the calculation to the divergence of Zfree on such backgrounds. Conformal classes with negative Y (h~) precisely include the highly irregular constant density surfaces featuring in a semiclassical gravity analysis of eternal in ation. This general argument therefore suggests their amplitude will be low in a holographic measure. We interpret this as evidence against the idea that eternal in ation typically leads to { 9 { a highly irregular universe with a mosaic structure of bubble like patches separated by in ationary domains.4 Instead we conjecture that the exit from eternal in ation produces classical universes that are reasonably smooth on the largest scales. 3 Discussion We have used gauge-gravity duality to describe the quantum dynamics of scalar eld driven eternal in ation in the no-boundary state in terms of a dual eld theory de ned on a global constant density surface at the exit from (scalar eld) eternal in ation. Working with the semiclassical form (1.1) of dS/CFT the dual eld theories involved are Euclidean AdS/CFT duals deformed by a complex low dimension scalar operator sourced by the bulk scalar driving eternal in ation. The inverse of the partition function speci es the amplitude of di erent shapes of the conformal boundary at the exit from scalar eld eternal in ation. This yields a holographic measure on the global structure of such eternally in ating universes. We have computed this explicitly in a toy model consisting of a mass deformed interacting O(N) vector theory de ned on squashed spheres. In this model we nd that the amplitude is low for geometries far from the round conformal structure. Second, building on this result we have argued on general grounds that exit surfaces with signi cant patches of negative scalar curvature are strongly suppressed in a holographic measure in Einstein gravity too. Based on this we conjecture that eternal in ation produces universes that are relatively regular on the largest scales. This is radically di erent from the usual picture of eternal in ation arising from a semiclassical gravity treatment. We have considered toy model cosmologies in which a scalar eld driven regime of eternal in ation transitions directly to a -dominated phase. The application of our ideas to more realistic cosmologies that include a decelerating phase requires further development of holographic cosmology (as is the case for all current applications of holographic techniques to early universe cosmology, e.g. [15, 24{28]). It has been suggested that in realistic cosmologies, in ation corresponds to an IR xed point of the dual theory [24] in which case the partition function of the IR theory might specify the amplitude of exit surfaces. Our conjecture strengthens the intuition that holographic cosmology implies a signi cant reduction of the multiverse to a much more limited set of possible universes. This has important implications for anthropic reasoning. In a signi cantly constrained multiverse discrete parameters are determined by the theory. Anthropic arguments apply only to a subset of continuously varying parameters, such as the amount of slow roll in ation. The dual Euclidean description of eternal in ation we put forward amounts to a signi cant departure from the original no-boundary idea. In our description, histories with a regime of eternal in ation have an inner boundary in the past, at the threshold for (scalar eld) eternal in ation. The eld theory on this inner boundary gives an approximate de4This resonates with [13] where we argued that probabilities for local observations in eternal in ation can be obtained by coarse-graining over the large-scale uctuations associated with eternal in ation, thereby e ectively restoring smoothness. Our holographic analysis suggests that the dual description implements some of this coarse-graining automatically. scription of the transition from the quantum realm of eternal in ation, to a universe in the semiclassical domain. For simplicity we have assumed a sharp inner boundary, but of course one can imagine models where this is fuzzy. 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S. W. Hawking, Thomas Hertog. A smooth exit from eternal inflation?, Journal of High Energy Physics, 2018, 147, DOI: 10.1007/JHEP04(2018)147