Quantum Cosmic No-Hair Theorem and Inflation

PHYSICAL REVIEW D 99, 103514 (2019)

Quantum cosmic no-hair theorem and inflation

† Nemanja Kaloper* and James Scargill Department of Physics, University of California, Davis, California 95616, USA

(Received 11 April 2018; published 13 May 2019)

We consider implications of the quantum extension of the inflationary no-hair theorem. We show that when the quantum state of inflation is picked to ensure the validity of the effective field theory of fluctuations, it takes only Oð10Þ e-folds of inflation to erase the effects of the initial distortions on the inflationary observables. Thus, the Bunch-Davies vacuum is a very strong quantum attractor during inflation. We also consider bouncing universes, where the initial conditions seem to linger much longer and the quantum “balding” by evolution appears to be less efficient.

DOI: 10.1103/PhysRevD.99.103514

I. INTRODUCTION As we will explain, this is resolved by a proper Inflation is a simple and controllable framework for application of EFT to fluctuations. First off, the real describing the origin of the Universe. It relies on rapid vacuum of the theory is the Bunch-Davies state [2]. This cosmic expansion and subsequent small fluctuations follows from the quantum no-hair theorem for de Sitter described by effective field theory (EFT). Once the space, which selects the Bunch-Davies state as the vacuum slow-roll regime is established, rapid expansion wipes with UV properties that ensure cluster decomposition – out random and largely undesirable initial features of the [3 5]. The excited states on top of it obey the constraints – Universe, and the resulting EFT of fluctuations on the arising from backreaction, to ensure that EFT holds [6 14]. expanding background replaces them with small, nearly The initial state need not be the vacuum, but rather scale-invariant spectra of scalar and tensor fluctuations. The some deformation of it with some (quantum, as well as beginning of inflation might be described by some of the classical) memory of the initial conditions, whatever those existing theories, such as no-boundary, tunneling from may be. nothing, a preinflationary origin, or eternal inflation in the Starting with this, we will quantify explicitly how multiverse.1 The exact details are largely irrelevant for the quickly inflationary evolution wipes out the initial excitations and evolves the initial state to the point where it is last 60 e-folds when the observable features are generated, practically indistinguishable from the Bunch-Davies vac- except for the description of the initial state. “Common uum. In other words, we will calculate the rate of the wisdom” dictates that the initial state is taken to be the “thermalization” process induced by inflation on the Bunch-Davies vacuum, which readily yields a spectrum of quantum state of the Universe. This will generically require scale-invariant perturbations. Yet picking this state “by an additional Oð10Þ e-folds, during which the initial choice” can lead to confusion, and a critic might even excitations (random or entangled) will be reduced to be object that this is a tuning, i.e., putting in the answer by subleading to the intrinsic inflationary fluctuations in the handpicking the initial state. Further, there are concerns that Bunch-Davies vacuum that generate the cosmic microwave the fluctuation modes seem to appear out of nowhere in the background (CMB) anisotropies and serve as the seeds for Bunch-Davies vacuum, with apparently negligible initial structure formation. Moreover, we will also see that the backreaction. same thermalization dynamics of the quantum vacuum reduces the initial nonlinearities, implying that the initial *[email protected] † non-Gaussianities also diminish in the course of inflation, [email protected] and that the non-Gaussianities which survive the initial ∼10 1These ideas have been discussed so much in the literature that even identifying the proper original references at this point is e-folds are really due to the nonlinear effects in the inflaton difficult, to say the least. We shall only mention one of the earliest sector rather than initial conditions. Our results will suggestions about a quantum, uncertain, and cosmic origin [1]. explicitly show how the Bunch-Davies state is dynamically realized by inflation, and determine the point after which Published by the American Physical Society under the terms of the standard calculations apply. Hence, there is a price to the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to pay for using the Bunch-Davies state for the computation of the author(s) and the published article’s title, journal citation, δρ=ρ. The good news is that it is acceptable. Further, since and DOI. Funded by SCOAP3. the standard EFT is valid throughout this regime, there is no

2470-0010=2019=99(10)=103514(12) 103514-1 Published by the American Physical Society NEMANJA KALOPER and JAMES SCARGILL PHYS. REV. D 99, 103514 (2019) trans-Planckian problem whatsoever: it is merely a mirage where H is the (nearly constant) Hubble parameter during that follows from inconsistent assumptions. inflation. The latter case describes the inflationary freeze- A similar argument used for the vacuum state employed out of perturbations. The former case describes the evolu- in bouncing cosmologies generically shows that imposing tion of fluctuations at subhorizon scales, ignoring their the Bunch-Davies vacuum there is much costlier, since one interactions. The normalization coefficient A can be calcu- must impose that the quantum state is initially homo- lated using EFT and declaring A to be the expectation value geneous over many more orders of magnitude. Bouncing of the inflaton’s propagator in the Bunch-Davies state. cosmology models therefore need to include a mechanism Now, the statement of the “trans-Planckian problem” which explains how the quantum vacuum was achieved. [15] is that if one traces the fluctuations back in time, from the horizon crossing to the earlier stages of inflation, one II. TRANS-PLANCKIAN ADDLING AND finds that the wavelength shrinks exponentially, and before CIS-PLANCKIAN SANGFROID one blinks it will become shorter than the Planck length. Indeed, one can plot this as in Fig. 1. This is then taken to As a starting point, we briefly review the “standard” mean that in order to really retain the predictivity of description of the genesis of inflationary perturbations with inflation as a means for determining the late-time amplitude a particular focus on the so-called “trans-Planckian prob- of fluctuations, one must specify the theory all the way to lem” [15]. We then point out a simple, logical resolution of arbitrarily short lengths. Further, this might make one the alleged problem, consistent with the framework of the suspect that the inflationary results—specifically, the scale EFT of quantum fluctuations of the inflaton and the cosmic invariance of A—might be a consequence of “fine-tunings” no-hair theorem. hidden in the choice of the Bunch-Davies vacuum for the Indeed, imagine an inflationary theory in the slow-roll EFT of fluctuations, which “evidently” must be sensitive to regime, and truncate the theory of the fluctuations to only “trans-Planckian” physics. the Gaussian sector. Focusing for simplicity only on the This conclusion is faulty because it ignores the evolution scalar field, in the longitudinal gauge one finds of the amplitude of fluctuations for a given physical wavelength. Before horizon crossing, Φk ∼ 1=a ∼ 1=λ (as ds2 ¼ a2ð−ð1 − 2ΦÞdη2 þð1 þ 2ΦÞdx⃗2Þ; also plotted in Fig. 2). This is just the virial theorem, since _ λ 1 _ 2 _ a for < =H the d.o.f. of the scalar behave just like free ϕ ¼ ϕ0ðηÞþδϕðη; x⃗Þ; ϕ0δϕ ¼ −2M Φ þ Φ ; Pl a harmonic oscillators in a cavity. Hence, if one extrapolates λ back in time, one finds that its amplitude grows large! If ð1Þ one takes the scale of inflation as the highest allowed that might fit the data, by taking into account the bounds on where a is the scale factor in a flat Friedmann-Robertson- primordial tensor modes, H<1013 GeV, and fixing the Walker (FRW) universe, and Φ is its scalar, Newtonian −5 amplitude Φ to 10 when λ ≃ 1=H one finds that Φ ∼ 1 perturbation generated by the inflaton perturbation δϕ for λ > 10l . because of the mixing induced by the time derivative of Pl This means that linear perturbation theory in the Bunch- ϕ_ η the inflaton background 0ð Þ. Clearly, the system has only Davies vacuum cannot be extrapolated to trans-Planckian δϕ Φ one degree of freedom (d.o.f.), since once is given, is scales. When λ is short, gravitational effects become completely fixed. Turning to the dynamics of this d.o.f., it is strong and the fluctuations distort the background dramati- extremely convenient to use Mukhanov’s curvature pertur- _ cally. The linear approximation fails, and one cannot pass φ δϕ − aϕ0 Φ bation variable ¼ a a=a_ , which obeys a free-field equation in the FRW background [16]. In momentum space, 2 φ̈ ⃗2 − þ φ 0 k þ k η2 k ¼ ; ð2Þ where the ellipsis denotes the slow-roll corrections. Since roughly Φ ∼ φ=a, we can follow the time evolution of the fluctuations by using the solutions of the free-field equation (2). Using the physical wavelength of the modes, λðηÞ¼λ0aðηÞ¼aðηÞ=k, it is easy to see that

ˆ A cosðkη þ δÞ; for λ < 1=H; Φ ≃ a ð3Þ k B FIG. 1. Wavelength of a fluctuation as a function of time during A 3 ; λ > 1=H; þ a for slow-roll inflation.

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freeze. Clearly, the fluctuations are never really subject to any significant “trans-Planckian” influences. Simply put, they never really probe physics beyond the Planck cutoff in a significant manner, thanks to decoupling. This picture will lay the foundation for our analysis that follows. Let us outline it: we will start with some generic initial quantum state that is a nontrivial excitation of the Bunch-Davies state, which is selected by the cosmic no- hair theorems as the vacuum state of the EFT. We will then quantify the magnitude of the excitation by computing the backreaction in this state, using the method of boundary actions as a clear-cut means to locally parametrize the initial deviations from Bunch-Davies. These are bounded FIG. 2. Amplitude of a fluctuation as a function of its wave- so that the weakly coupled EFT is valid. This, by default, length (and thus time) during slow-roll inflation. includes all those initial states where inflation can begin. Prior to this stage, the dynamics in the excited state was in through this regime at one’s whim [17]. The large blueshift strong coupling, and so it is not directly calculable without factors enhance EFT interactions and the irrelevant oper- details of the full theory. Some states might not survive for ators cannot be ignored any more. In fact, even the long enough to allow inflation to set in; however, some will. background metric (1) does not really make sense for The precise statement of the survival probability of a these fluctuations: Φ is really an expectation value of the completely generic state is beyond the scope of this work. metric perturbation in the quantum state of inflation, and its We will simply analyze the dynamics in the states that did dispersion will be too large. The dynamics at short survive, as those indeed support the weakly coupled EFT distances yields all kinds of background distortions satu- and are covered by our analysis. We will then study how rated with the formation and evaporation of small black long the initial deviations from Bunch-Davies survive, and holes, which play the role of the dynamical cutoff. Hence, show that in generic states they become negligible after at the shortest scales some regions will be perturbed so Oð10Þ e-folds. We will also find the magnitude of their much that they collapse and inflation never even starts corrections to the inflationary perturbations. Finally, we there. In other regions of space, however, high-energy will consider similar processes in bouncing cosmologies. dynamics would be less destructive and inflation can begin [18]. These regions would initially not be as smooth and III. BOUNDARY ACTION AND INITIAL STATES flat, but inflation would iron them out as it goes, at both the classical and quantum level. So a part of the trans- Here we first review the formalism for implementing the Planckian confusion is the misidentification of the quantum quantum boundary conditions on the state of inflation expectation value of the operator Φˆ in an arbitrary state with which ensures the validity of the EFT description of the classical mode function Φ in the Bunch-Davies state. fluctuations of the inflaton. This provides a systematic While this is correct in the Bunch-Davies state, it is not true tool to parametrize the deviations of the initial state from in a general state where fluctuations are large. the Bunch-Davies vacuum. Following Ref. [6], the basic Instead, the curvature fluctuations come about from idea is to supplement the action for the field whose particle production in an external field [19], analogously perturbation we are considering with a term evaluated on a space-like boundary which encodes the initial conditions to particle production in background fields in quantum 2 electrodynamics, where pairs can be created by a large for the field. In particular, for a massless scalar one has electric field in a parallel plate capacitor. Pair creation will Z 1 pffiffiffiffiffiffi discharge the plates and decrease the field. Inflationary 4 − ∂ ϕ∂μϕ S ¼ 2 d x g μ perturbations are similar, starting as the quanta of the Z 1 pffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffi inflaton placed on shell by the background fields and 3 3 þ d xd y γðxÞ γðyÞϕðxÞκðx; yÞϕðyÞ; ð4Þ perturbing the metric by generating density perturbations a 2 Σ long time after inflation, when the curvature perturbations of the geometry yield the perturbations of the matter density where γ is the induced metric on the space-like surface Σ, generated at reheating. The production rate will be slow and κðx; yÞ encodes information about the initial state of ϕ. thanks to a flat potential with a value significantly below For a translation-invariant state one has κ ¼ κðjx − yjÞ. the Planck scale, and slightly inhomogeneous because of Variation of the action with respect to ϕ (demanding, of the fluctuations. As the fluctuations evolve in the background, they will decohere, and soon after the fluctuation 2A good leading-order approximation for the fluctuations of a length becomes ∼1=H they will cross the horizon and light inflaton in slow roll.

103514-3 NEMANJA KALOPER and JAMES SCARGILL PHYS. REV. D 99, 103514 (2019) course, that the variation is not set to zero on the boundary, “interaction” of the modes in the Bunch-Davies state is because the state initially differs from Bunch-Davies) given by yields the usual bulk equation of motion, along with the boundary condition − 2η κ k 0 BD ¼ : ð7Þ 1 − ikη0 ð∂n þ κÞϕ ¼ 0; ð5Þ μ We will treat the corresponding quantities in a general state where ∂n ≡ n ∂μ is the derivative normal to the boundary. Fourier transforming, and expanding the field in terms of as a perturbation of the Bunch-Davies operators. This creation and annihilation operators, yields Z d3k 1 ϕ η φ η φ η † − ik·x κ ¼ κ þ δκ; φ ¼ ðφ þ bφ−Þ: ð8Þ ðx; Þ¼ 3 ð −ð ÞAðkÞþ þð ÞA ð kÞÞe ; ð6Þ BD b 1 − 2 þ ð2πÞ jbj φ the mode function þ obeys Eq. (5). Reference [6] Using Eq. (5) and the Klein-Gordon normalization of the emphasized that the physics is encoded in the mode φ ∂ φ − φ ∂ φ i “ mode functions þ n − − n þ ¼ a3, the Bogoliubov functions, and so the precise location of the boundary is rotation” b is given by arbitrary, so long as one also changes κ so as to ensure that

Eq. (5) is still satisfied. Thus, we can fix the boundary at the κφ ∂ φ 3δκφ2 η ∂ ∂ φ − þ þ n þ ia0 þ conformal time 0, and so n ¼ η=a. Let us now use to b ¼ ¼ 3 2 : ð9Þ κφ− þ ∂ φ− 1 − ia δκ φ denote the mode functions in the Bunch-Davies vacuum, n η¼η0 0 j þj η¼η0 and denote their counterparts in more general states by φb. The massless scalar field modes are the Hankel func- Then, a straightforward calculation shows that with the ∓ η tions, φ ¼ pffiffiffiffiffiffiffiffiH ð1 ikηÞe ik , and so the effective modes defined by Eq. (8) the Green function is 2k3=2

; η η ϕ η ϕ η φ η φ η Gbðk 1; 2Þ¼hbj bðk; 1Þ bðk; 2Þjbi¼ bð 1Þ bð 2Þ 4 3 0 H a ð Þ 0 2 −ikð2η0−η1−η2Þ ¼ G − Im½δκð1 þ ikη0Þ ð1 − ikη1Þð1 − ikη2Þe b 2k6 H6a6 0 2 2 2 2 −ikðη1−η2Þ þ ½jδκj ð1 þ k η Þ ½ð1 − ikη1Þð1 þ ikη2Þe þ 2ðη1 ↔ η2Þ 8k9 0 2 2 2 2 −ikð2η0−η1−η2Þ 3 − 2Reðδκ ð1 þ k η0Þð1 − ikη0Þ ð1 − ikη1Þð1 − ikη2Þe Þ þ Oðδκ Þ; ð10Þ

ð0Þ where Gb is the Green function in the Bunch-Davies state. state as a leading-order measure of backreaction; this follows from Ehrenfest’s theorem. The nonzero value of 0 A. Backreaction δhbjT0jbi in the excited state must not overwhelm the The above mathematical framework can be used to background sources and impede inflation. The boundary estimate the backreaction of the excitations in the initial action framework above then allows us to compute this state on the evolution. Specifically, the backreaction must quantity unambiguously and in a generic excitation of be subleading to the background effects in controlling the Bunch-Davies. evolution. A simple way to proceed is to consider the For a scalar field, the stress-energy tensor is Tμν ¼ 1 λ 3 expectation value of the stress-energy tensor in the excited ∂μϕ∂νϕ − 2 gμν∂λϕ∂ ϕ, and so

Z 1 3 0 i d k 2 δhbjT0jbi¼−δhbjT jbi¼− ð∂η ∂η þ k ÞδG ðk; η1; η2Þjη η η: ð11Þ i a2 ð2πÞ3 1 2 b 1¼ 2¼

3Note that since the Feynman two-point function is the time-ordered version of Eq. (10), there is an extra factor of 2 in the integral for δT.

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− 2 M2 2 A natural IR cutoff regulating the integral is given by aH, yielding a factor e k = a0 in the integrand in Eq. (11).In since comoving momenta smaller than this have already that case, one can consider larger expectation values of the exited the horizon [20]. “bare” number operators, with the physical expectation The UV cutoff may be directly implemented at the values nevertheless suppressed by the explicit nonlocal leading order by considering the states with finite occu- cutoff. pation numbers relative to the Bunch-Davies vacuum. The terms in Eq. (10) involving e−ikð2η0−η1−η2Þ will Alternatively, one can impose the cutoff by hand, as approximately integrate to zero, and so the leading back- e.g., in Ref. [7], where the high-energy states are cut off reaction effect is actually given by the second-order 2 ∂2 M2 2 2 4 by regulating the momenta according to ∂ → e = a0 ∂ , corrections to the state:

Z 3π2 6 6 0 a0H 2 −5 2 2 2 2 2 −k2=M2a2 δhbjT0jbi¼− dkjδκj k ð1 þ k η0Þ ð5 þ 2k η Þe 0 2ð2πÞ3a2 2 Z 3a0 2 − 2 M2 2 ≈− dkjδκj ke k = a0 ; ð12Þ 8πa4 where the last line is of course valid only if δκ is sufficiently what follows. In any case, one sees that the backreaction blue so that the integral is dominated by the UV contri- redshifts away like aγ, 2 ≤ γ ≤ 4. butions. The explicit cutoff has been retained for com- This represents a finite renormalization of the stress- pleteness and illustrative purposes. One sees that the energy tensor which corrects the background via its backreaction is UV finite if δκ ∼ k−ð1þϵÞ or more sup- gravitational effects. Specifically, in this state the effective pressed for large momenta k. In this case, we can drop the Friedman equation with the backreaction included is 2 1 ρð0Þ δ 0 H ¼ 3 2 ð þ hbjT0jbiÞ. Using it, we can derive three exponential cutoff in Eq. (12), essentially by taking the MPl limit M → ∞. If on the other hand the function δκ has less constraints on the scale of backreaction: intrinsic suppression for large momenta k, it is essential to (1) Demanding perturbativity yields keep the cutoff M finite. This scaling behavior, as we will 0 0 2 2 Δ ≡ δhbjT jbi ≪ ρð Þ ≈ 3H M : ð13Þ see below, is crucial to ensure the perturbativity of the 0 Pl theory in the covariant local limit. We will return to this in (ii) Requiring validity of the slow roll leads to

H_ ρ_ Δ_ 3 ϵ − − ≈ ϵ ∼ ϵ ⇒ Δ ≲ H2M2 ϵ : 14 ¼ 2 ¼ 2 ρ 6 3 2 þ measured measured 2 Pl measured ð Þ H H H MPl

afewe-folds. Again, the boundary action formalism leads (iii) Finally, retaining the scale invariance of the scalar quickly to these results. power spectrum implies B. Correcting observables ρ̈ Δ̈ ≈ þ ϵ ðϵ þ η Þ Now we can check the scaling of the corrected power H2ρ 3H4M2 measured measured measured Pl spectrum in an excited state. The power spectrum of 3 ⇒ Δ ≲ 2 2 ϵ ϵ η fluctuations is H MPl measuredð measured þ measuredÞ: 16 k3 P ðkÞ¼ lim G ðk; η; ηÞ ð15Þ b η→0− 2π2 b 3 2 2 ¼ P ð1 − 2a ðδκφ ðη0Þ ÞþOðδκ ÞÞ The third constraint yields the strongest bounds on the BD 0 þ a3H2 initial state. However, this constraint really only applies 0 2 2 −2ikη0 ¼ P 1 þ Imðδκð1 − k η þ 2ikη0Þe Þ when modes we can observe on the CMB leave the BD k3 0 horizon, and becomes equivalent to the second one within þ Oðδκ2Þ : ð16Þ 4 The first-order change considered by Porrati [8,9] is localized – to the boundary, and so, as explained in Ref. [7], there are In Refs. [6 10,13,14] the change of a state was due to an renormalization ambiguities, in addition to it being negligible in irrelevant boundary operator, which in turn was due to in- the bulk. tegrating out unknown high-energy physics. This suggests

103514-5 NEMANJA KALOPER and JAMES SCARGILL PHYS. REV. D 99, 103514 (2019) the following parametrization, involving an explicit cutoff M of the previous section. Note that demanding momentum dependence which suppresses the deformations n<−1 ensures the finiteness of the stress-energy tensor in in the UV in the construction of the initial state: the excited state even without the nonlocal cutoff, as we have seen in Eq. (12). We will mostly work with this k n requirement, although we will reflect on the situation where δκ ¼ βnM : ð17Þ a0M the suppression of δκ is weaker and the explicit nonlocal 2 2 2 cutoff expð∂ =a0M Þ is used. The correction to the power β Here M is the UV regulator, and n is a dimensionless spectrum is coefficient. In principle M could be different from the

δP H n−1 k n−3 2k 2k ¼ Reβn sin þ Imβn cos P M a0H a0H a0H BD k n−2 2k 2k þ 2 Reβn cos þ Imβn sin a0H a0H a0H k n−1 2k 2k − Reβn sin þ Imβn cos : ð18Þ a0H a0H a0H

The powers of a0 above show that the correlation excitations are completely irrelevant in the UV. However, functions at a later time receive only a small correction our analysis will establish that this arises naturally by the compared to their values in the Bunch-Davies state. They evolution of the initial state towards the Bunch-Davies are redshifted as momenta due to de Sitter expansion and vacuum, and is not an arbitrary assumption. covariance. Since the power spectrum essentially measures Let us confirm this by computing the occupation the modes as they leave the horizon, the correction to numbers in an excited state. The number of particles per modes that were deeply subhorizon at η0 are greatly unit volume, compared to the Bunch-Davies vacuum, is suppressed. From Eq. (18) one has Z 3 Z 3 2 d k † d k jbj δ n−1 N ¼ 3 hbjA ðkÞAðkÞjbi¼ 3 2 2 ≫ ⇒ P ∼ β a H ð2πÞ ð2πÞ ð1 − jbj Þ k a0H n ; ð19Þ Z P a0 M 3 3 2 2 BD k¼aH d k 1 − ia jφ j δκ a6 δκ 2 φ 4 0 þ : ¼ 2π 3 0j j j þj 1 2 3 φ 2 δκ ð20Þ which is already exponentially (in time) suppressed [by ð Þ þ a0j þj Imð Þ powers of a=a0 ≃ expðHtÞ] for n<1. As noted above, the condition n<−1 guarantees the finiteness of the corrected The large k-behavior of the integrand is stress-energy tensor. j δκj2ð1 þ j δκj2Þ; if jImðδκÞj ≪ jReðδκÞj; This conforms with the fact that the standard approach to ∼ k k Nk δκ 2 ð21Þ quantum field theory fails in states which are separated j k j ; otherwise: from the Bunch-Davies vacuum by infinite occupation α – n 1 numbers, such as vacua [21 24]. Some of the problems In either case we see that δκ ∼ k with n<− 2 ensures that were discussed in Refs. [25–28]. If the initial state is an the number of particles in the UV is finite. Thus, physically excitation of the Bunch-Davies vacuum described by a low- realistic cases do indeed obey the no-hair scaling, and a energy theory with heavy states and higher-dimension, nonlocal cutoff is not required. Note that this condition is irrelevant operators integrated out, the UV completeness of weaker than the requirement that the extra energy density is the theory implies that in a general excited state the finite, n<−1, following from Eq. (12). However, it is still occupation numbers are finite, and the perturbation theory stronger than the requirement that the change to the power is meaningful [29,30]. These are the states with n<−1 on spectrum decays for large k (n<1). The main reason for which we have been mainly focusing. Alternatively, using this is that the backreaction and particle number involve 2 2 2 the nonlocal cutoff expð∂ =a0M Þ [7], one may be able to integrals of δκ which smear out the momentum depend- sharply suppress the UV effects regardless of their origin. ence, whereas the local operator for the power spectrum This shows that the seeming enhancements for n ≥−1 are does not. The power spectrum is thus the least sensitive merely an illusion, since they are unphysical. If the cutoff is observable to the UV distortions of the theory, and one can explicitly inserted, the exponential suppression due to it easily err by picking for initial states such configurations completely overwhelms any power-law enhancement with where the power spectrum might not be affected too much, ≥−1 M 2 k for n , for

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C. Excitations as squeezed Bunch-Davies excitations and a direct second-quantized framework, So far we have been referring to the non-Bunch-Davies yielding immediately the systematic normalization of the initial states as “Bunch-Davies excitations” in a somewhat latter. heuristic way, basically taking it on faith that the identi- We do this by actually constructing these states as fication is correct. The observables computed in these states normalizable deformations of Bunch-Davies. To this end, indeed support this. Further, this provides a direct link we deploy the general formalism found in Ref. [31], with between the boundary action parametrization of initial the result

( P n=2 i h0jbi fðk1;k2Þfðkn−1;knÞ;neven; P hk1; …;knjbi¼ ð22Þ 0;nodd:

n! The summation is over the 2 ways of forming pairs from k1;k2; k . In terms of the Bogoliubov coefficients 2n= ðn=2Þ! f ng which relate the operators which annihilate jbi to those which annihilate j0i, one has Z fðq; pÞ¼−i d3kα−1ðp; kÞβðk; qÞ: ð23Þ

0 ð3Þ 0 0 ð3Þ 0 In this case the Bogoliubov transformation is “diagonal,” with αðk; k Þ¼αkδ ðk − k Þ and βðk; k Þ¼βkδ ðk þ k Þ. Reading off the coefficients from Eq. (8), we find Z X∞ 1 3 3 b 0 b d k1 d k b k1 b k k1; −k1;k2; −k2; …;k2 ; −k2 ; j i¼h j i 2n ! n ð Þ ð nÞ j n ni 0 n n¼ Z 1 0 3 † † − 0 ¼h jbi exp 2 d kbðkÞ a ðkÞa ð kÞ j i; ð24Þ where the second line identifies the operator which turns IV. QUANTUM INFLATION the Bunch-Davies vacuum into the modified state. Note that Inflation in its standard form is a mechanism to get rid this is precisely a squeezed state on top of the Bunch- of initial inhomogeneities and anisotropies. Once the Davies vacuum, akin to the α vacua [21–24], but with an cosmological constant-like source starts to dominate explicit UV suppression manifest in the function bðkÞ. This the expansion, (almost) everything else dilutes away ensures that the occupation number is finite, and that the [32–37]. Intuitively—by invoking Ehrenfest’s theorem transformation (24) is normalizable. Indeed, using Eqs. (17) —this must also happen with initial quantum excitations and (9) we see that for n<1 of the Bunch-Davies vacuum. Beside proving this, we will also compute precisely how many e-folds it takes to 8 suppress the effect of the initial excitations on the −1 −2i k −2i k − i β k n a0H < a0H ≪ 2 nð Þ e e ;ka0M; observable predictions of inflation to below their intrinsic ≈ a0M ≈ b −1 1−n 1 − i β k n : i a0M dynamical value, obtained by the computation in the 2 nð Þ − β ;k≫ a0M: a0M 2 n k Bunch-Davies state. In other words, we can precisely “ ” ð25Þ state the cost of the choice of Bunch-Davies as the initial state of inflation in terms of the number of e-folds it takes prior to the last 60 e-folds in order to iron out Not surprisingly by now, we see yet again that the integrals generic initial excitations. in Eq. (24) will be finite for n<−1, as evidenced by the So let us take as the initial state a deformation given by −1 suppression of b in the last line of Eq. (25). Without Eq. (17), with n< so that the quantum correction of the such behavior we would have to explicitly cut off the stress-energy tensor is finite. In this case we can drop the momentum integral in Eq. (24) to keep the transformation exponential cutoff factor from Eq. (12). Then, direct normalizable. computation yields

103514-7 NEMANJA KALOPER and JAMES SCARGILL PHYS. REV. D 99, 103514 (2019) −1 2 2 3 H n 5 a0 scalar perturbations by less than a percent, at which point ΔðaÞ¼ jβj H4 cðn − 1Þ 16π M 2 a they are completely negligible. 4 Remarkably, this completely confirms the intuition that a0 þ cðnÞ ; ð26Þ the erasure of the initial distortions of the quantum vacuum a during inflation takes about 10 or so e-folds [25,26].We should still verify that the third of the backreaction 1 2 1 β where cðnÞ¼n−1 þ n þ nþ1. Let the parameters and M be constraints is obeyed. However, this is straightforward. such that at a ¼ a0 the extra stress-energy contribution is From Eq. (15) one sees that this will be satisfied when just large enough to disrupt slow roll. Clearly this is the ϵ −3 ΔðaÞ has decreased by ∼ 8 ∼ 10 which happens after ∼3 maximal value allowed at the beginning of the slow-roll e-folds, i.e., even sooner. phase, by the fact that the slow roll is an attractor for a Note that the cost of preparation of the initial state as sufficiently flat inflationary potential. Equation (14) then measured by the number of e-folds that it takes to erase the yields excitations increases as the scale of inflation is lowered. rffiffiffiffiffiffiffiffi The reason is that while the background energy density is H n−1 M 8πϵ ∼ 2 2 Δ ∼ 4 jβj ¼ Pl ; ð27Þ H MPl, the perturbation to the energy density is H , M H c˜ because it arises from relativistic inflaton fluctuations at subhorizon scales. Therefore, the requirement that the 5 where c˜ ¼j2 cðnÞþcðn − 1Þj. Using this we can calculate perturbation is just about to disrupt slow roll when a ¼ the change to the power spectrum at a later time. Using a0 means that the size of the perturbation as measured by β Eq. (19), we find or M must be larger for smaller H. In other words, a rffiffiffiffiffiffiffiffi relatively larger initial excitation is allowed. This is why it δ 1−n 8πϵ 1−n pffiffiffiffiffiffiffiffi takes longer for this perturbation to wash away. P ≃ a0 MPl a0 MPl 8πϵ ˜ < : PBD k¼aH a H c a H A. Non-Gaussianities ð28Þ The no-hair theorems in de Sitter space [3–5] proscribe The inequality holds for n ≳ −13. For larger negative all hair—not just that endowing the two-point function. values of n the dimensionless factor can be larger, but This also includes non-Gaussian signatures, which is 1−n the redshift suppression ∼ða0=aÞ is so efficient that indeed why they are small during inflation [38]. these cases are essentially ignorable. Nevertheless, it is interesting from a phenomenological Indeed, while the contribution of the initial excitations to point of view to consider whether the effects induced by the scalar power spectrum (28) may be very large initially, initial excitations on higher-point correlators might be more MPl ≳ 105 resilient to inflationary washout. In other words, does one since H , the correction dilutes at least as quickly as 1=a2 because n<−1. The number of e-folds to reduce the need more e-folds than Eq. (30) to bleach nonlinear hairs? initial ratio to a desired value is To test this, we turn to the three-point function of the perturbation field. The effect on the three-point function 1 M pffiffiffiffiffiffiffiffi δP −1 from non-Bunch-Davies initial states has been calculated in N ¼ ln Pl 8πϵ : ð29Þ Refs. [11,39]. We can translate the result of Ref. [39] into 1 − n H P our language, and interpret it in the context of the no-hair Thus, to suppress the contribution from the initial condition theorem. The relative change in the three-point function of −2 the scalar curvature perturbation ζ, in the high-frequency to δP to be below ∼10 of the Bunch-Davies value P , BD limit where gravity is decoupled, yields using the slow-roll parameter ϵ ∼ 0.01 which allows O 100 ð Þ e-folds of inflation, one has δhζ ζ ζ i δ ≡ k1 k2 k3 3 ζ ζ ζ 1 h k1 k2 k3 i ≃ 4 2 MPl N þ log10 : ð30Þ 2 −2 η ˜ η 1 − X iki 0 iki 0 n H kt 3 2 ð1þikiη0Þ e ð1−e Þδκ þc:c: ≃ P a0H ; 4 k−2 k5k˜ ∼ 10−5 i i i i i For the highest scale inflation, H=MPl , and the most UV sensitive distribution of excitations, n ∼−1, this yields ð32Þ P ˜ − 2 ˜ N ≃ 7: ð31Þ where kt ¼ iki, and ki ¼ kt ki. Due to ki in the denominator, δ3 is maximized in the so-called “folded ˜ For lower scale inflation, N should be larger, but generi- limit” defined by taking ki → 0 for some i. This case is cally it will not take much more than N ∼ Oð10Þ e-folds to interesting as other effects which alter the three-point reduce the initial excitations to the level where they affect function do not display this signature (occurring instead

103514-8 QUANTUM COSMIC NO-HAIR THEOREM AND INFLATION PHYS. REV. D 99, 103514 (2019) in the equilateral and squeezed limits). Thus, any unusual scenarios suffer from severe fine-tuning problems related to behavior in the folded limit might indicate the significance their classical initial conditions (e.g., see Refs. [41–44]). of a non-Bunch-Davies initial state. These prompted the proposal for cyclic universes where the Let us now write δκ as in Eq. (17), and use the classical fine-tuning problems are addressed by repeated backreaction constraint to eliminate the β and M para- incarnations of the Universe (see Refs. [45,46] for some meters via Eq. (27). As before we take generic phases details of the idea and its critique). Some aspects of the of β, assuming ReðβÞ ∼ ImðβÞ ∼ jβj. In the folded limit of quantum mechanics of fluctuations in the bouncing ana- 1 1 Eq. (32), setting k2 ¼ k3 ¼ 2 k1 ¼ 2 k, the dominant con- logue of the Bunch-Davies state have been analyzed (see, tribution is e.g., Refs. [47,48]); however, a general discussion of how rffiffiffiffiffiffiffiffi the Bunch-Davies state is attained appears to have been 1 8πϵ n 1 δ overlooked to date. Without getting into a detailed dis- MPl k k P δ3 ≃ ≃ : ð33Þ cussion of the bounce dynamics (and what may or may not 18 H c˜ a0H 18 a0H P BD cause it), we wish to merely point out the difference For a mode leaving the horizon at k ¼ aH, this is larger between quantum state selection and evolution in inflation than the fractional change in the power spectrum by a factor and in bouncing universes. While in inflation the Bunch- of a=a0. However, the constraints on the primordial Davies state is an attractor, it is far less clear whether this is contribution to the three-point function are much less true in bouncing cosmologies. Hence, such models need to severe than those on the two-point function. So a value be explored in more detail to see if a quantum attractor δP of δ3 which is larger than is still practically irrelevant. In mechanism exists, that would justify the calculations of the PBD perturbations in the Bunch-Davies state. particular, using Eq. (29), we can write To make things more precise, we consider a flat FRW p rffiffiffiffiffiffiffiffi p 1− − 1 universe with scale factor a ¼ a0ðt=t0Þ ¼ a0ðη=η0Þ p, 1 ˜ δ 1−n δ δ ≃ c H P P with η ∈ ð−∞; 0Þ and 0

5 “ ” where the ellipsis refers to more terms of a roughly similar This information suppression might be avoided by fully n three-dimensional observation, such as large-scale structure form. Note that the integral is UV finite for δκ ∼ k with surveys. n<−1, just as in the inflationary case, thanks to the

103514-9 NEMANJA KALOPER and JAMES SCARGILL PHYS. REV. D 99, 103514 (2019) behavior of Hankel functions forffiffiffi large argument. Indeed, homogeneous FRW metric is invalid. Such initial quantum ix p for large argument, HνðxÞ ∼ e = x which means that each distortions need to be suppressed. In inflation this occurs power of the Hankel function in Eq. (38) contributes a automatically once accelerated expansion starts. While the pffiffiffi power of 1= k to the integrand. Thus at large momenta, the background backreaction in bouncing universes remains integrand behaves as ∼kjδκj2. Also note that Δ involves a under control, the influence of initial excitations on prefactor ∼a−2 and so, unlike in the case of inflation, as observables is more persistent. In the absence of a more time goes on, and the universe shrinks, Δ increases. The efficient mechanism to smooth these excitations away, explicit η dependence in the integral cannot counter this, as bouncing universes seem to require fine-tunings by hand the Hankel functions blow up for small argument, render- to pick the right form of the perturbations from generic ing the mode functions finite. Nevertheless, the background initial conditions. In a way, the Bunch-Davies vacuum state energy density driving the contraction behaves as seems to be a much weaker attractor. ρ ∼ 1=a2=p. Since p<1 this grows more quickly than Note that we have used initial backreaction as a measure Δ of the excitations away from the Bunch-Davies state. One the perturbation , subduing the backreaction. This keeps β control of the collapse at the classical level [47,48]. could try imposing direct limits on n in Eq. (17) instead. In The question is: what happens with the corrections to the such an approach one opens the door to sensitivity to UV observables due to the excitations ensconced in the initial physics, since generically the results will depend explicitly quantum state of collapse, when it is not the Bunch-Davies on the UV cutoff M because the backreaction and the modification of the power spectrum involve positive vacuum? Writing δκ as Eq. (17), at the initial time η0 one has powers of the ratio M=H0. This would therefore seem counterproductive, requiring direct tunings of UV physics 2 2n in ways that conflict with decoupling. Fixing the cutoff M 3π 1 − p 2 2 2 H0 1 − p Δðη0Þ¼ Iðp; nÞ H0β M ; leads to the same problems, as does fixing the ratio M=H0. 128 p M p The bottom line is that in the absence of the inflationary ð37Þ redshifts one needs to develop different mechanisms that R may even have to go beyond standard field theory to justify ∞ 4þ2n ð1Þ 4 ð1Þ 2 where Iðp; nÞ¼ 1 dxx jHν ðxÞj ðjHν ðxÞj þ using the Bunch-Davies state to compute perturbations in ð1Þ 2 bouncing cosmologies. jHν−1ðxÞj Þþ. Using Eq. (37) to constrain β and M in terms of the initial backreaction, we can now examine the change in the power spectrum. We find VI. SUMMARY sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The standard calculation of inflationary perturbations 1 1− 1− 2 − p p 2 δ 32 Δ n p p involves the computation of the spectrum of fluctuations of P ≃ 0 a ð1Þ a0 2 Hν : relativistic fields in the Bunch-Davies vacuum. On the P − η 1 3πIðp; nÞ H0 a0 a BD k ¼ other hand, one can—and should—imagine that the initial ð38Þ quantum state of the Universe in the beginning of inflation is more general, and check the effects of the more general So the fractional change in the power spectrum decreases choices on the observables. After all, inflation is the for modes which leave the horizon at a later time, just as in mechanism for smoothing the initial conditions away using the inflationary case considered above (recall that we are accelerated expansion as the attractor dynamics. considering n<−1 and 0

103514-10 QUANTUM COSMIC NO-HAIR THEOREM AND INFLATION PHYS. REV. D 99, 103514 (2019) inflationary quasi–de Sitter expansion is indeed the mecha- like a very economical scenario since the dilution requires nism which smooths out the Universe at both the classical very long times (see, e.g., Ref. [52]). But if the number of and quantum levels. The “thermalization rate” of Eq. (30) cycles is large… A more careful investigation would seem (in the sense of the initial excitations being reduced below to be warranted to test these issues. the level of the quantum quasi–de Sitter fluctuations) is We note in closing that our analysis does not preclude quite rapid. Note that while it has been colloquially said large effects due to the quantum initial conditions on the that inflation prepares the Bunch-Davies state as the post-inflationary universe. It does make them very unlikely, vacuum of fluctuations, the precise and general details however, if inflation is longer than the minimal 60 e-folds. were lacking in the literature. Our work fills that gap. We If for example we consider large-field slow-roll inflation, nevertheless should note that larger signatures may occur in it can easily last longer than 60 e-folds, by at least Oð10Þ “just-so” models where inflation lasts only Oð60Þ e-folds, e-folds, and as we saw the initial excitations of the quantum but this may require fine-tuning both the theory and the state of the Universe will be suppressed to very small levels. initial state from the model building point of view. Some Similarly, in the case of false vacuum inflation, where the recent examples are Refs. [49,50]. last stage happens after the tunneling out of a false vacuum For bouncing universes, the tuning of the initial quantum (see, e.g., Refs. [53,54]), the quantum state at the onset of state has not previously been discussed in the literature. In the last stage is prepared to be the Bunch-Davies one by the this case the dynamics is different; as a result, the EFT long time the Universe lingered in the false vacuum. Again, description permits much larger initial excitations which the quantum excitations will be grossly suppressed. To are much harder to suppress. The resultant Bunch-Davies avoid this, one needs a “just-so” inflation: the last 60 e- state typically used for computing the perturbations is a folds need to start with a “bang” which provides a large much weaker attractor. Hence, if one starts with generic initial sudden excitation, disrupting the adiabaticity of initial conditions in the collapsing phase, one has to pick cosmic evolution [17,30]. While perhaps possible, such more carefully the right initial state as a function of the dynamics seem to be tuned. duration of collapse to get the observed quantities. By itself, this is a fine-tuning. A dynamical mechanism explaining it would be preferable. Perhaps this could be ACKNOWLEDGMENTS alleviated in cyclic universes [45] where the universe We would like to thank Albion Lawrence for useful undergoes many stages of collapse followed by expansion, discussions. N. K. thanks the CERN Theory Division for with a late long phase dominated by a small cosmological hospitality in the course of this work. This work is “ ” constant. In this case the bleaching of the quantum state supported in part by DOE Grants No. DE-SC0009999 of the universe to a Bunch-Davies vacuum would be done and No. DE-SC0019081. by a long late low-scale inflation [51]. This does not seem

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