Nonsingular Ekpyrotic Cosmology with a Nearly Scale-Invariant Spectrum of Cosmological Perturbations and Gravitational Waves

PHYSICAL REVIEW D 101, 063522 (2020)

Nonsingular ekpyrotic cosmology with a nearly scale-invariant spectrum of cosmological perturbations and gravitational waves

† Robert Brandenberger * and Ziwei Wang Department of Physics, McGill University, Montr´eal, Quebec City, H3A 2T8, Canada

(Received 6 February 2020; accepted 3 March 2020; published 20 March 2020)

We propose a mechanism borrowed from string theory which yields a nonsingular transition from a phase of ekpyrotic contraction to the expanding phase of standard big bang cosmology. The same mechanism converts the initial vacuum spectrum of cosmological fluctuations before the bounce into a scale-invariant one, and also changes the spectrum of gravitational waves into an almost scale-invariant one. The scalar and tensor tilts are predicted to be the same, in contrast to the predictions from the “string gas cosmology” scenario. The amplitude of the gravitational wave spectrum depends on the ratio of the string scale to the Planck scale and may be in reach of upcoming experiments.

DOI: 10.1103/PhysRevD.101.063522

I. INTRODUCTION scalar field φ to serve as an inflaton, its potential energy VðφÞ has to be very flat in order that the potential energy The inflationary universe scenario [1] has become the dominates over the kinetic energy.1 However, general standard paradigm of early universe cosmology. It is based arguments from string theory lead to the swampland on the assumption that there was a period of almost constraint [5] (see [6] for reviews) exponential expansion during a time period in the very early universe. Inflation provides a solution of the horizon V0 c > ð2Þ and flatness problems of standard big bang cosmology, and V m provides a causal mechanism for producing cosmological pl perturbations and microwave background anisotropies for slowly rolling scalar fields whose energy density based on the assumption that all fluctuation modes start dominates the universe, where c is a constant of the order our in their vacuum state inside the Hubble radius at early 1, mpl is the four space-time dimensional Planck mass, and times [2]. The spectrum of curvature perturbations is a prime indicates the derivative with respect to the field φ. predicted to be almost scale-invariant, with a slight red tilt. Effective field theory models which violate this constraint Inflation also produces [3] an approximately scale-invariant are said to lie in the swampland and are not consistent with spectrum of gravitational waves, again with a slight red tilt. string theory.2 Single scalar field models of slow-roll Inflation is usually obtained by working in the context of inflation are thus [8] in the swampland.3 Einstein gravity and assuming that there is a new scalar A second constraint on inflationary cosmology comes field, the inflaton field φ, whose stress-energy tensor has an from the recently proposed trans-Planckian censorship equation of state conjecture (TCC) [10] which states that during cosmological evolution no scales whose wavelengths were smaller ≡ p ≃−1 ð Þ w ρ 1 than the Planck length ever exit the Hubble horizon. This conjecture can be viewed as analogous to Penrose’s cosmic (where p and ρ are pressure and energy density, respec- censorship hypothesis [11] which states that timelike tively) which leads to accelerated expansion. singularities must be hidden by horizons. If the TCC is Recently, however, inflationary cosmology has come satisfied, then trans-Planckian modes are hidden from the under some pressure. First of all, recall that in order for a classical region.4 The TCC also shields the classical region

*[email protected] 1Warm inflation [4] provides an avenue of relaxing this constraint. † [email protected] 2These considerations also have implications for scalar field models of dark energy [7]. Published by the American Physical Society under the terms of 3Once again, models of warm inflation avoid this constraint [9]. the Creative Commons Attribution 4.0 International license. 4Fluctuation modes oscillate on sub-Hubble scales, but Further distribution of this work must maintain attribution to become squeezed states and classicalize on super-Hubble the author(s) and the published article’s title, journal citation, scales [12] (see [13] for reviews of the theory of cosmological and DOI. Funded by SCOAP3. perturbations).

2470-0010=2020=101(6)=063522(9) 063522-1 Published by the American Physical Society ROBERT BRANDENBERGER and ZIWEI WANG PHYS. REV. D 101, 063522 (2020) of cosmology from nonunitarities associated with setting Among bouncing scenarios, the ekpyrotic scenario [23] up quantum field theory in an expanding background [14]. (see also [24] for a cyclic version) has a number of Since during inflation the physical wavelength of fluc- attractive features. The ekpyrotic scenario assumes that tuation modes increases almost exponentially, while the the contracting phase has an equation of state parameter Hubble radius remains almost constant, the TCC clearly provides severe constraints on inflationary cosmology. In w ≫ 1: ð5Þ [15] it was shown that, assuming that the post-inflationary cosmology is like in standard big bang cosmology, and that This can be realized if matter is given by a scalar field φ the potential energy during inflation is approximately with negative exponential potential constant, the potential energy is constrained to obey the pffiffiffiffiffiffi upper bound − 2=pφ=m VðφÞ¼−V0e pl ð6Þ V1=4 < 3 × 109 Gev; ð3Þ with V0 > 0 and 0

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[37], and converting the entropy fluctuations to curvature vacuum power spectrum with a small red tilt δn produced perturbations.7 It has also been shown in [39] that nontrivial during the ekpyrotic phase of contraction is converted into a matching conditions of fluctuations on the spacelike sur- scale-invariant one with the same small red tilt δn. The face separating the contracting phase from the expanding spectrum of gravitational is enhanced by the same mecha- one can convert the scale-invariant spectrum of φ to that of nism when passing through the bounce. Thus, unlike in the curvature perturbations.8 The spectrum of gravitational other approaches to ekpyrotic cosmology [37], we obtain a waves, on the other hand, remains close to vacuum, and roughly scale-invariant spectrum of gravitational waves.11 hence a negligible amplitude of such waves on cosmo- Our discussion is at the level of an effective field theory, logical scales is predicted. but we have in mind a setting coming from string theory. In this paper we suggest a way of simultaneously We will work in units where the speed of light and the obtaining a cosmological bounce and obtaining a scale- Planck and Boltzmann constants are set to 1. Space-time invariant spectrum of both curvature fluctuations and indices are denoted by Greek symbols, spatial indices by gravitational waves.9 Our mechanism is based on the fact lower case Latin ones. The cosmological scale factor is that (in the context of string theory), at the string scale, denoted by aðtÞ, and HðtÞ is the Hubble expansion rate. G enhanced symmetries in the low energy effective action are is Newton’s gravitational constant, related to the reduced expected to appear (see, e.g., [43]): a tower of string states 8π ¼ −2 Planck mass mpl via G mpl . Since spatial curvature is which has string scale mass in a Minkowski space-time not important in the early universe we will set it to zero. background becomes massless and has to be included in the low energy effective action.10 In the low energy effective action, there is thus an extra term which appears at the time II. S-BRANE AND NONSINGULAR BOUNCE ¼ t tB when the density reaches the string scale density. It We are working with a four-dimensional effective action is a delta function term localized on the spacelike hyper- S of the form ¼ surface t tB (as will be discussed in Sec. III we are Z working in the uniform density gauge), and we hence call pffiffiffiffiffiffi 1 ¼ 4 − þ ∂ φ∂μφ − ðφÞ this term an S brane. In the same way that a D-brane has S d x g R 2 μ V negative pressure (equals positive tension) in the spatial Z 4 pffiffiffi directions along the brane and vanishing pressure in the − d xκδðτ − τBÞ γ; ð9Þ normal direction, an S-brane has vanishing energy density and negative pressure. This is discussed in detail in [45] in where R is the Ricci scalar of the four-dimensional space- the context of a specific string model with thermal duality time metric gμν with determinant g, φ is the scalar field with in the Euclidean temporal direction. The S-brane hence ðφÞ γ yields a contribution to the effective energy-momentum negative exponential potential V (6), ij is the induced ¼ γ κ tensor which violates the null energy condition and hence, metric on the hypersurface t tB with determinant , and as shown in [45] can meditate a transition from contraction is the tension of the S-brane. One of the conditions on our to expansion. As we show here, the effect of the S-brane on coordinates (this will be important when studying fluctua- the cosmological perturbations and gravitational waves tions in the next section) is that the constant t surfaces converts initial vacuum fluctuations before the bounce to correspond to constant density. The time tB is the time scale-invariant ones after the bounce. when the density reaches the critical value when the extra In the following section we discuss the origin of the tower of string states becomes massless and when hence the S-brane and how this object mediates the transition S-brane appears. between contraction and expansion. Then, in Sec. III we As discussed in detail in [45], the S-brane induces a ρ study the coupling of cosmological perturbations and localized stress-energy tensor with energy density B and gravitational waves to the S-brane and show that the slope pressure pB given by of the power spectrum of curvature perturbations changes −2 ρ ¼ 0 ð Þ by a factor of k , k being comoving momentum. Thus, the B ; 10

p ¼ −κδðt − t Þ: ð11Þ 7An almost scale-invariant spectrum can also be obtained [38] B B by making use of a rapidly changing equation of state before the phase of ekpyrotic contraction. Integrating the Friedmann equations across the bounce 8 In the case of a singular bounce between an ekpyrotic time tB yields the following change of the Hubble constant: contracting phase and an expanding phase the transfer of fluctuations was studied in [40] (see also [41] for a study of the transfer in δH ≡ limHðt þ ϵÞ − Hðt − ϵÞ¼4πGκ: ð12Þ a holographic cosmology setup.) ϵ→0 B B 9See [42] for attempts at obtaining a nonsingular ekpyrotic bounce using a cubic Galileon Lagrangian. 10This is the same physics discussed under the name distance 11Note that the anamorphic scenario of [29] also produces an conjecture in the recent superstring literature [44]. approximately scale-invariant spectrum of gravitational waves.

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Hence, provided that the energy density just before the field energy density is constant on constant time surfaces bounce obeys the constraint (on large scales where the spatial gradient energy is pffiffiffi negligible). We can impose a second gauge condition 3 and choose it to be E ¼ 0 for computational ease. ρðt Þ1=2 −1κ ð Þ B < 2 mpl ; 13 We wish to obtain the effects of the S-brane on the equation of motion for the fluctuations. To this end, we then the S-brane will induce a cosmological bounce. We insert the above ansatz for the perturbated metric and expect the S-brane to appear at the string energy scale ηs, perturbed matter into the full action (9) and expand to 4 and hence ρðtBÞ ∼ ηs. We expect κ to be given by second order in the fluctuation variables. Note that the terms linear in the fluctuations vanish if the background 3 κ ∼ Nηs ; ð14Þ satisfies the background equations of motion. The contribution of the bulk term in the action to the second order where N is an integer given by the number of string states action for scalar fluctuations is which become massless at the enhanced symmetry point Z η 00 (where the brane appears). Hence, provided that the N s is 1 z Sð2Þ ¼ d4x v02 − v v þ v2 ; ð18Þ larger than the Planck mass, the S-brane will induce a 2 ;i ;i z cosmological bounce. We will assume that small-scale interactions between the where v is the Mukhanov-Sasaki variable [46] which is enhanced symmetry states and the degrees of freedom of given by the Standard Model lead to a bath of radiation after the bounce. This is the analog of the reheating process at the ¼ ζ ð Þ end of inflation. With this assumption, the bounce will v z ; 19 induce a transition between an ekpyrotic phase of contraction and a radiation phase of expansion after the where ζ is the curvature fluctuation in comoving gauge, and bounce. zðτÞ ∝ aðτÞð20Þ III. FLUCTUATIONS PASSING THROUGH THE S-BRANE BOUNCE if the equation of state of the background is time-independent. A prime in the above equations denotes a Let us now consider how metric fluctuations couple to derivative with respect to conformal time. the S-brane. In this section it is convenient to use conformal During the phase of ekpyrotic contraction we have time τ in terms of which the homogeneous and isotropic [see (7)] background metric takes the form

1 1− ds2 ¼ a2ðτÞðdτ2 − dx2Þ: ð15Þ τðtÞ¼− ð−tÞ p; ð21Þ 1 − p We will consider scalar metric fluctuations and gravita- and hence tional waves separately.12 The metric for scalar fluctuations takes the form (see, e.g., [13] for reviews of the theory of ðτÞ ∼ τp=ð1−pÞ ð Þ cosmological perturbations) z ; 22 1 þ 2Φ − 2 B;i from which it follows that gμν ¼ a ðτÞ ; ð16Þ −B;i ð1 þ 2ΨÞδij þ E;ij z00 pð2p − 1Þ ¼ τ−2; ð23Þ where the fluctuation variables Φ, Ψ, B, and E are functions z ð1 − pÞ2 of space and time. The scalar field is which implies that an initial vacuum spectrum remains φðτ; xÞ¼φ0ðτÞþδφðx; τÞ; ð17Þ almost vacuum on super-Hubble scales, acquiring only a small red tilt proportional to p. After the bounce in the where φ0 is the background scalar field and δφ is the field radiation phase of expansion we have fluctuation. Not all of the fluctuation variables are independent. We z00 δφ ¼ 0 ¼ 0; ð24Þ will work in comoving gauge in which the scalar z

12We postpone the discussion of vector modes to a later study. as is well known.

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In the gauge we are using the induced metric γij is after the bounce. This is the main result of our analysis. If the spectrum before the bounce is a vacuum spectrum 2 γij ¼ a ð1 þ 2ΨÞδij; ð25Þ modulated by a slight red tilt (as it is in the case of ekpyrotic contraction), the power spectrum after the bounce will be a ð2Þ scale-invariant one modulated by the same red tilt. and hence the contribution SB of the brane term in the action to quadratic order in Ψ is Let us now study the coupling of gravitational waves to Z the S-brane. If we consider a gravitational wave traveling in ð2Þ 3 the x direction, then the induced metric γij is S ¼ d4xκa3 1 þ 3Ψ þ Ψ2 δðτ − tτ Þ: ð26Þ B 2 B 10 2 γij ¼ a ðτÞ ; ð33Þ Note that the sum of the terms linear in the fluctuations 01þ hϵab cancel since we are expanding about a solution of the background equations. where ϵab is the polarization tensor of gravitational waves In the gauge we are using Ψ is proportional to the Sasaki- in the y=z plane. Hence, to leading order in the amplitude h2 Mukhanov variable v: pffiffiffi 3 1 2 −1 γ ¼ a 1 − h : ð34Þ Ψ ¼ z v; ð27Þ 2 where z is a function of the background cosmology which The canonical variable for gravitational waves is depends both on the geometry and on the matter (its form will be discussed below), and hence the brane contribution u ≡ ahmpl; ð35Þ to the second order action can be rewritten as where the factor of mpl is important to have the right Z 3 ð2Þ 1 a dimensions. The bulk action for u is S ¼ d4x 3κδðτ − τ Þv2; ð28Þ B 2 z2 B Z 1 a00 Sð2Þ ¼ d4x u02 − u u þ u2 ; ð36Þ and thus the full second order action for v is 2 ;i ;i a Z 00 3 ð2Þ 1 4 02 z a 2 and the brane contribution is S ¼ d x v − v v þ þ 3 κδðτ − τ Þ v ; 2 ;i ;i z z2 B Z ð2Þ 1 ð Þ S ¼ d4 κ δðτ − τ Þ 2 −2 ð Þ 29 B 2 x a B u mpl ; 37 and the resulting equation of motion for v is Hence, the equation of motion for uk becomes 00 3 00 þ 2 − z − 3 a κδðτ − τ Þ ¼ 0 ð Þ a00 v k 2 B v : 30 u00 þ k2 − − κam−2δðτ − τ Þ u ¼ 0: ð38Þ z z a pl B 00 ¼ 0 00 Since z after the bounce and z =z is proportional to p This equation is analogous to the equation for scalar modes, and hence very small before the bounce, we can to first except that the coefficient of the delta function term is approximation neglect this term. different. In analogy with what was shown above for the As shown in the Appendix, the δ function contribution to scalar modes, we find that the tensor modes are enhanced the mass in the equation of motion (30) leads to an by the factor enhancement of the mode functions by a factor β ¼ κ −2 −1 ð Þ m k ampl k : 39 β ¼ ; ð31Þ k k Our results (32) and (39) show that the passage through where m is the coefficient of the delta function. From (30) it the S-brane leads to the conversion of an initial vacuum follows that spectrum for the scalars and tensors to a scale-invariant one for both scalars and tensors. This is very different from a 2 what is obtained in previous realizations of ekpyrotic β ¼ 3κa ðτ Þk−1: ð32Þ k z B cosmology where the tensors retain their vacuum form. The tensor to scalar ratio r can be read off of the This factor leads to a conversion of the vacuum power amplitude ratio of the Bogoliubov coefficients βk.To spectrum of v before the bounce to a scale-invariant one evaluate this ratio, recall [13] that

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φ_ 1 0 pffiffiffiffiffi zðtÞ¼aðtÞ ; ð40Þ vk;0 ¼ : ð48Þ H 2k φ ð Þ where 0 t is the background scalar field. For the The dimensionless power spectrum is given by ekpyrotic potential of (6), this background is given by pffiffiffiffiffiffi ð Þ¼ 3j j2 ð Þ pffiffiffiffiffiffi Pv k k vk : 49 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV0 φ0ðtÞ¼ 2pmpl log − t ; ð41Þ mpl pð1 − pÞ According to the definition of the scalar tilt ns we have

n −1 and hence PðkÞ ∼ k s : ð50Þ pffiffiffiffiffiffiffiffi φ_0 Combining (46), (47), (48), (49) and (50) we find that the ¼ 2=pm : ð42Þ H pl tilt of the scalar modes before the bounce is given by

Hence, comparing (39) and (32) and making use of (42) we ns ¼ 3 þ 2α ≃ 3 þ 2p: ð51Þ find that the tensor to scalar ratio r becomes Since the passage through the S-brane changes the spectral 72 103 r ¼ 16π ∼ : ð43Þ tilt by -2, the tilt after the bounce is p3 p3 1 − ns ¼ −2α ≃ −2p: ð52Þ The factor of 16π stems from the different normalizations of the scalar and tensor power spectrum, the other term is Thus, we predict a blue tilt of the scalar spectrum with the the square of the ratio of the Bogoliubov coefficients (39) deviation of the tilt from that of scale-invariance whose to (32). magnitude is 2p. The tensor spectrum has the same tilt. We find that the amplitude of the tensor spectrum is larger Note that the tensor index nt is defined via than the amplitude of the scalar spectrum, a result which is obviously inconsistent with observations. This is related to PðkÞ ∼ knt : ð53Þ the fact that the analog of the inflationary slow-roll parameter ϵ is proportional to 1=p ≫ 1 in the case of ekpyrotic Our new ekpyrotic scenario hence predicts roughly scale contraction. Hence, in our model we need to invoke a invariant scalar and tensor spectra with tilts which obey the separate mechanism to boost the scalar modes, as considered consistency relation in previous work on the ekpyrotic scenario [37,39]. ¼ − 1 ð Þ Let us now consider the corrections to the spectrum due nt ns : 54 to the z00=z term in the mode equation. Since The amplitude of the spectrum of gravitational waves z00 pð1 − 2pÞ 1 1 after the bounce can be found from (39). The power ¼ − ≡ −αðpÞ ; ð44Þ spectrum of the canonical variable u is z ð1 − pÞ2 τ2 τ2 ð τÞ ≃ 2κ2 −4ð τ Þ2p ð Þ the dominant mode in for the v and u modes on super- Pu k; a mpl k B ; 55 Hubble wavelengths scales as and hence the power spectrum of gravitational waves vðτÞ ∼ τþα ð45Þ becomes ð τÞ ≃ κ2 −6ð τ Þ2p ð Þ and similarly for u. Hence we find for the Fourier modes of Ph k; mpl k B ; 56 v and u on super-Hubble lengths κ Recall from Sec. II that the tension is expected to be given τ ð Þ −α by the string scale η , and thus the amplitude A of the ðτÞ ≃ H k ð Þ s vk vk;0 τ ; 46 power spectrum (dropping the last factor above which represents the small blue tilt) will be τ ð Þ where H k is the time of Hubble radius crossing of the kth η 6 1 A ≃ 2 s ð Þ mode and is given by N 2 : 57 mpl ð2π Þ p τ ðkÞ¼ k−1: ð47Þ H ð1 − pÞ If the string scale is the one preferred by particle physics considerations in the early textbook [47] on string theory, 17 If we use vacuum initial conditions then namely ηs ∼ 10 GeV, then the amplitude is

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−12 A ∼ 10 ; ð58Þ nt ¼ 1 − ns. The predicted tensor tilt is blue, but the scalar tilt is red, unlike in the scenario studied here. which, using the observed amplitude of the scalar spectrum, In this paper we have assumed that the state after the corresponds to bounce is that of radiation, in a similar way that the state after reheating in inflation is that of radiation. The explicit r ∼ 10−3: ð59Þ model of ultraviolet physics that yields the production of radiation across the bounce remains to be explored. In Note, however, that this value depends sensitively on the future work we plan to study the amplitude of non- ratio between the string scale and the Planck scale. Gaussianities generated in our scenario.

IV. CONCLUSIONS AND DISCUSSION ACKNOWLEDGMENTS It is generally expected that at string energy density The research at McGill is supported in part by funds scales a new tower of string states becomes massless. If this from NSERC and from the Canada Research Chair pro- is the case, this tower of states has to be included in the low gram. Z. W. also acknowledges partial support from a energy effective action for cosmological evolution. It will McGill Space Institute Graduate Fellowship. appear as a term in the effective action localized at a particular time, i.e., as an S-brane. This object has zero energy density and negative pressure, and can hence induce APPENDIX the transition from contraction to expansion. Let us consider the equation of motion We have used this S-brane construction to provide a new realization of the ekpyrotic scenario. We have shown that 00 2 X ðτÞþ½k þ mδðτ − τ ÞX ðτÞ¼0: ðA1Þ the coupling of the S-brane to cosmological fluctuations k B k and in particular to gravitational waves leads to a change in The solutions are plane waves for τ < τ and for τ > τ . the spectral index of super-Hubble cosmological perturba- B B If we denote the positive frequency solutions as fk and the tions of δn ¼ −2. This converts a vacuum spectrum into a s negative frequency ones as f , where the indicates scale-invariant one. Our minimal construction leads to a k complex conjugation, then the solution which is pure larger amplitude for the tensor modes than the scalar positive frequency before τ can be written for τ > τ as modes, and hence has to be supplemented by a separate B B source of cosmological perturbations. X ¼ α f þ β f; ðA2Þ The amplitude of the power spectrum of gravitational k k k k k waves depends on the ratio of the string scale η and Planck s where α and β are the Bogoliubov mode matching scale m [see (57)]. The tilt of the gravitational wave k k pl coefficients which obey the relationship spectrum is predicted to be the same as the tilt of the scalar spectrum. jα j2 − jβ j2 ¼ 1: ðA3Þ It is interesting to compare these predictions with k k those obtained in string gas cosmology [20], an emergent By integrating the equation (A1) over time τ against a universe scenario motivated by string theory. String gas test function (a smooth function which decays exponen- cosmology (SGC) also yields a roughly scale-invariant tially at τ → ∞) fðτÞ it can be easily shown that spectrum of scalar and tensor modes, and the amplitude of the power spectrum of cosmological perturbations is given m β ¼ : ðA4Þ by a combination of the string scale and the Planck k k scale, but in this case the fourth power and not the sixth power as here (see, e.g., [48] for detailed discussions). SGC This result implies that the power spectrum of Xk is boosted yields two consistency relations between the four basic by a factor of ðm=kÞ2. This turns a vacuum spectrum into a cosmological observables. In particular, the tensor tilt is scale-invariant one.

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