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Home , False vacuum decay, Inflation (cosmology), Paul Steinhardt

D 91, 083527 (2015) after false decay: Observational prospects after

† ‡ Raphael Bousso,1,2,* Daniel Harlow,3, and Leonardo Senatore4,5,6, 1Center for Theoretical and Department of Physics, University of California, Berkeley, California 94720, USA 2Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Princeton Center for Theoretical Science, , Princeton, New Jersey 08540, USA 4Stanford Institute for , , Stanford, California 94306, USA 5Kavli Institute for Particle and , Stanford University and SLAC, Menlo Park, California 94025, USA 6CERN, Division, 1211 Geneva 23, Switzerland (Received 1 December 2014; published 22 April 2015) We assess two potential signals of the formation of our by the decay of a false vacuum. Negative spatial curvature is one possibility, but the window for its detection is now small. However, another possible signal is a suppression of the cosmic microwave background (CMB) power spectrum at large angles. This arises from the steepening of the effective potential as it interpolates between a flat inflationary plateau and the high barrier separating us from our parent vacuum. We demonstrate that these two effects can be parametrically separated in angular scale. Observationally, the steepening effect appears to be excluded at large l; but it remains consistent with the slight lack of power below l ≈ 30 found by the WMAP and Planck collaborations. We give two simple models which improve the fit to the Planck data; one with observable curvature and one without. Despite , we argue that future CMB polarization and most importantly large-scale structure observations should be able to corroborate the Planck if it is real. If we further assume the specific theoretical setting of a landscape of metastable vacua, as suggested by , we can estimate the probability of seeing a low-l suppression in the CMB. There are significant theoretical uncertainties in such calculations, but we argue the probability for a detectable suppression may be as large as Oð1Þ, and in general is significantly larger than the probability of seeing curvature.

DOI: 10.1103/PhysRevD.91.083527 PACS numbers: 98.80.Es, 98.80.Cq

I. INTRODUCTION AND SUMMARY is generic. One could argue that some tuning of both the fundamental theory and the initial conditions is required: Inflation [1–4] is a powerful framework for understand- (i) Theory parameters: For the observed structure to be ing the early universe, in particular the spectrum of density seeded, there must be a hierarchy between the perturbations that led to and to anisot- fundamental scale and the scale of inflation, and also ropies in the cosmic microwave background (CMB) [5–9]. sufficient flatness over a large enough range. Measurement of these [10–14] out to l ≈ 1000 (ii) Initial conditions: For inflation to begin, vacuum were crucial in establishing the standard ΛCDM model of must dominate in a region of order the cosmology. Very strong additional evidence for this model associated Hubble scale. If initial conditions can has recently been provided by the Planck collaboration be tuned to give rise to such a patch, why not solve [15–18], which reported measurements of temperature the horizon, , and flatness problems anisotropies out to l ≈ 2500. Their results for the power directly by tuning? Moreover, why should the field spectrum above l ¼ 50 are shown in Fig. 1. There are some basic facts in cosmology which are not start off high enough on the potential to inflate but explained by the ΛCDM model. Chief among these is the not so high as to overshoot the inflationary region? smallness of the observed value of the cosmological It is difficult to quantify such concerns except within a constant. Moreover, it is not obvious that slow-roll inflation specific and well understood theoretical setting. A setting in which one could aspire to do this is the string landscape of metastable vacua [22–25], populated by *[email protected] † – [email protected] [26 28]. In this setting one can in principle ‡ [email protected] derive statistical predictions for the parameters of ΛCDM and inflation. This determines which regions of parameter Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- are typical and which are not. bution of this work must maintain attribution to the author(s) and In practice, our understanding of the string landscape is the published article’s title, journal citation, and DOI. too primitive to derive prior distributions from the top

1550-7998=2015=91(8)=083527(22) 083527-1 Published by the American Physical Society RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015)

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FIG. 2 (color online). The decay of our parent vacuum followed 0 l 0 500 1000 1500 2000 2500 by slow-roll inflation. The dashed line indicates a tunneling event from the high energy false vacuum, after which the field lðlþ1Þ μ 2 FIG. 1 (color online). Planck data for 2π Cl,inð KÞ , for classically rolls down the hill. There is an initial period of l ≥ 50. The red theory curve is ΛCDM with the Planck best-fit curvature domination, where a ≈ t, during which the field does parameters, as computed using CLASS [19–21]. 1 ϕ not travel very much. This ends when a2 becomes of order Vð Þ, after which inflation begins. Eventually inflation exits and the system reheats into our vacuum, with small cosmological con- down. For a few parameters the prior can be derived with stant. For later convenience we choose ϕ ¼ 0 where modes of some confidence. In particular, the wavelength equal to our horizon size today are just exiting the should have a flat distribution near Λ ¼ 0 if inflationary horizon. is broken [29]. Along with the assumption that observers require galaxies, this can explain its smallness [22,29–32]. If inflation was preceded by , there (With the causal patch measure, specific anthropic assump- are two generic features which lead to potentially observ- tions can be eliminated and, in addition, the coincidence able signatures1: that is comparable to the present (i) The negative spatial curvature inside the bubble density can be explained [33].) Other parameters can also could be seen if it is not diluted by too much be treated using simple phenomenological assumptions inflation. The current observational bound is −2 −4 about their distributions [34–38]. ΩK ≲ 10 ,butΩK as small as 10 should even- In the string landscape, our universe would emerge tually be distinguishable from cosmic variance. through the decay of a metastable de Sitter vacuum, as a (ii) During the earliest part of slow-roll inflation, the Coleman-De Luccia bubble [39]. This setting goes a long potential steepens as it interpolates from the infla- way toward solving the initial conditions problem for tionary plateau to the potential barrier. This causes inflation. The SOð3; 1Þ invariance of the bubble ensures the to roll faster, which suppresses the a homogeneous, isotropic Friedmann-Robertson-Walker P ∼ H4 primordial power spectrum ϕ_ 2 at low angular (FRW) universe inside. Because neighboring vacua typi- parameter l [51]. Depending on the strength of this cally have very different cosmological constant, the poten- effect and on the value of l at which it turns on, it tial energy after decay can be large enough for slow-roll could be detectable in the CMB and in large scale inflation and reheating to follow. And the spatial curvature structure surveys. of the universe inside is always negative, which creates The first of these effects was discussed in detail in the wide- Hubble friction that prevents the field from overshooting ranging analysis of [36]. The possibility of a signature from the inflationary plateau [36]. the steepening of the potential was also raised, without Motivated by this theoretical setting, but without com- claiming that this generically leads to power suppression mitting specifically to the string landscape, we will con- sider the phenomenological consequences of a first-order 1 phase transition followed by inflation. The relevant type of These are not the only possible signatures associated to potential [36,40] is shown in Fig. 2. In the first sections inflation as originating after vacuum decay. Two noteworthy others are bubble collisions [41] and tensor modes from the of the paper we examine possible signatures of this type of nucleation process [42]. The likelihood of the former is quite potential using standard inflationary techniques. Only in difficult to estimate, and in any event it has recently been strongly the final sections will we make assumptions about the constrained by the optimal analysis [43,44] (see also [45,46]). distribution of potentials in the landscape; and subject to The latter only affects modes whose wavelength is of order the radius of curvature, so we can simply think of it as another these assumptions, we will estimate the probability that an manifestation of the open curvature that we discuss here. The imprint of false vacuum decay will be detected in future same can be said for the recent discussion of [47]. Other more experiments. exotic possibilities include [48–50].

083527-2 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) relative to ΛCDM. The idea that a steeper potential leads to (iv) The confidence in a suppression of power at alowl power suppression was noted in [51], outside of the low l can be increased substantially, if such an context of bubble nucleation. effect is real. Polarization measurements in the In this paper, we will argue that in some respects the CMB will be of some help in this direction, but power suppression from the steepening of the potential is a a more useful tool is large scale structure and more promising signature to look for than negative curva- possibly 21 cm measurements. We ture. To do so, we need to answer the following questions: argue that such observations may eventually (i) Both the power suppression and the curvature have the potential to increase the significance of ∼5 − 6σ originate from the same feature, the potential barrier the anomaly to as much as . at the beginning of inflation. Since curvature has (v) With a plausible prior for the distribution of −2 the steepening feature among landscape vacua been bounded at the level of jΩKj ≲ 10 , is there still hope of observing steepening? with slow-roll inflation, the probability for an (ii) Any suppression of the power would have to happen observable effect to lie in the visible region can 1 2 below l ≈ 50, since the running is strongly con- be as high as Oð Þ. This is higher than the strained at smaller scales. For such large angles, will probability for seeing curvature under similar 10% cosmic variance prevent us from ever being able to assumptions, which is Oð Þ. – confirm this feature with high confidence? (b) Outline In Secs. II IV, we assume only that our (iii) If false vacuum decay induces a steepening feature, universe was produced by the decay of a metastable what is the probability that the associated power vacuum, followed by slow-roll inflation. In Sec. II we suppression begins between l ¼ 2 and l ¼ 50,the derive the perturbative effect of a steepening feature in range in which it may be detectable in the future? Why the inflationary potential on the power spectrum. not in the invisible region (l ≪ 1), or at higher l? Section III discusses the observational status of l ≳ 100 We will be able to analyze the first two questions without this effect. We find it to be excluded for l ≲ 30 any specific assumptions about the underlying theory. The but allowed, and slightly favored, for . This anomaly has little significance for now, but we argue last will require a quantitative discussion of plausible prior that future observations have the potential to eliminate distributions in the string landscape, as well as a consid- or corroborate it. In Sec. IV, we illustrate key features eration of anthropic boundaries. of the steepening effect using two models. We show Our theoretical understanding of both the prior and the that the onset of curvature can be parametrically catastrophic boundary is limited, so our analysis of this separated from the steepening feature. third question should be considered preliminary. In the final two sections we assume the existence of a (a) Summary of results Below we list our key findings: large landscape of metastable vacua. Sec. Vargues that (i) It is possible for steepening to be observable a steepening feature near the beginning of inflation is even if inflation lasted long enough to wipe out statistically favored in the landscape, whereas flat- all observable traces of curvature. More gener- tening is disfavored. In particular, this disfavors ally, it will typically be the case that one should scenarios such as slow-roll eternal inflation and see the suppression before seeing curvature. Hawking-Moss tunneling. In Sec. VI, we estimate (ii) One might worry that competing effects could the probability that the steepening feature lies in the make it uncertain whether potential steepening observable region. We consider a range of plausible will result in a suppression or an enhancement prior distributions for its location in the potential. We of power [36]. We show that suppression is the argue for a catastrophic boundary near l ∼ Oð104.5Þ larger effect in the parametric limit of small and find that the probability for observable steepening ϵ η slow roll parameters and . We therefore can be as large as Oð1Þ. After observational exclusion conclude that suppression is the generic signal. of a feature for l ≳ 102, we find that the probability We also show that there should be no analogous for a feature at lower l ranges between 10% and suppression in the tensor power spectrum. 40%, depending on our assumptions about the prior (iii) The Planck collaboration has reported a sup- distribution. pression of power below l ≈ 30 [16], albeit 2 5 − 3σ (c) Related work Our work builds on the seminal with low significance ( . ). This is such a analysis of Freivogel, Kleban, Martinez, and low level of statistical significance that it does Λ Susskind [36]. Other work with some overlap in- not demand a modification of CDM. How- cludes Refs. [40,42,51–55]. We believe that each of ever, motivated by the theoretical case for a steepening feature, we exhibit two models that parametrize its inclusion. Both are found to 2Throughout, by a probability p ∼ Oð1Þ we mean that improve the fit to the CMB power spectrum p=ð1 − pÞ is of order unity; that is, p is neither very close to without violating the constraints on curvature. 0 nor very close to 1.

083527-3 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015) the points we listed in our “summary of results” The tensor to scalar ratio is above is original to this paper. Also the day before P this paper was posted, [56,57] appeared on the arXiv, ≡ t ≈ 16ϵ l r : ð5Þ which give stringy realizations of a low- power Ps suppression in the CMB from a steepening potential. A preliminary version of our results was reported by In the slow-roll approximation these power spectra are “ ” D. H. at the Primordial Cosmology workshop at close to a power-law spectrum, which for Ps is often Kavli Institute for Theoretical Physics (KITP) on May parametrized as 23rd, 2013, and are available online. This work was “ n −1 although discussed further by D. H. at the Open 3 2 k s ” k P ¼ 2π A : ð6Þ QuestionsinanOpenUniverse workshop in Istanbul s s k at Bogazici University, on August 12th 2013. piv

Here kpiv is some reference wave number often taken to be II. LARGE SCALE POWER SUPPRESSION FROM −1 :05 Mpc , and ns is called the spectral index. The Planck POTENTIAL STEEPENING −9 best-fit parameters are As ¼ 2.215 × 10 and ns ¼ :9624 In this section we study the effects of a steepening [17]. For slow-roll inflation we have feature in the potential on the scalar and tensor − 1 ≈ 2η − 6ϵ power spectra of the CMB. Our working hypothesis is that ns : ð7Þ if our universe was created through a bubble nucleation, then the inflaton potential is expected to be steeper at the To find the power spectra as functions of k we need to relate beginning of slow-roll inflation, as in Fig. 2. This can leave ϕ and k, by solving a detectable imprint, as we now show.3 Z ϕ ϕ 1 We work in the context of slow roll inflation, where the k Hh d log ≡ ΔN ≈ pffiffiffiffiffi ; ð8Þ parameters kh HðϕÞ 0 MP 2ϵ 2 0 2 MP V where we used the condition that ka0 ¼ aH at horizon ϵ ≡ ; ð1Þ 5 2 V crossing. Here and throughout, a subscript h indicates the value of a parameter at the when scales comparable to V00 our current horizon today were exiting the inflationary η ≡ M2 ; ð2Þ P V horizon. Thus the wave number of modes which were horizon sized at that time (and therefore today) is k ¼ H0. 4 h are small compared to unity. We use the reduced Planck We shift ϕ to set ϕ ¼ 0. It will often be convenient to use −2 ≡ 8π h mass, MP G. the approximate relationship The scalar and tensor spectra are related to the dynamical ϕ_ 0 variables (H, ) and to the potential parameters (V, V ), l ≈ kDls; ð9Þ evaluated at the time when modes with wave number k exit the horizon during inflation (see for example [60]): which relates angular scale on the CMB to the wave number of modes which contribute strongest to the Cl’s 1 4 1 3 1 1 3 H V V at that scale. Here D is the proper distance to the last- k P ¼ ≈ ¼ ; ð3Þ ls s 2 _ 2 6 6 02 12 4 ϵ ϕ MP V MP scattering surface today, given by Z 2 4 1 1 dx 3P 4 H ≈ V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k t ¼ 2 4 : ð4Þ Dls ¼ 2 −2 −3 −4 3 H0 1 1 Ω Ω Ω Ω MP MP =ð þzlsÞ x Λ þ Kx þ Mx þ Rx 3 1 ≈ . 3 : ð10Þ Imprints from the steepening feature could be evaded if the H0 Coleman-de Luccia decay deposits the field not on the steep slope but further along on the inflationary plateau. We argue To include a steepening feature, we write the potential in in Appendix B that such potentials are not generic. 4 the form Our results could also be derived in the of inflation [58,59]. This framework is largely motivated by the fact that all observations apart from the curvature are related to 5We mostly follow the conventions of [60], where both the the perturbations themselves, and not to the background solution. a and the wave vector k~ are dimensionful. The However, here we wish to exhibit the connection between a dimensionless canonical conjugate to the dimensionless FRW perturbative signal (power suppression) and a property of the ≡ ~ background solution (steepening). Therefore, we use a formalism coordinate x~ is q~ a0k. UnlikeR [60] however, we define the ≡ 3 −ik~·x~ that makes the background solution explicit. Fourier transform as fk~ d xe fðx~Þ.

083527-4 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) P V0 s ≈ 1 − 2γ R ; P 0 s;S VS P t ≈ 1: ð16Þ Pt;S

This demonstrates the claimed effect: to leading order in the slow-roll parameters and γ~, the scalar power spectrum is suppressed (never enhanced) by the steepening perturba- tion. In this limit we can regard only the slope of the potential as perturbed, not the height; thus, the tensor spectrum is unaffected at this order. This means that low-l suppression should be seen in the TT, TE, and EE FIG. 3 (color online). Our decomposition of the potential into a correlation functions in the CMB, which are dominated flat piece VS, that fits ΛCDM, and a steepening perturbation γVR. by Ps, but not in the BB correlation function, which is controlled by Pt. ϕ V ¼ VS þ γVR: ð11Þ We have expressed the power spectra as functions of . What is actually measured is the power as a function of k, Here S and R stand for “slow” and “rapid.” VS is a slowly- but this does not change our result at leading nontrivial varying potential whose predicted values for ns, V=ϵ, and r order. To see this, we must solve (8), keeping track of are consistent with the Planck best-fit parameters [17]. The corrections proportional to γ. Parametrizing the exact steepening perturbation VR is a positive, monotonically solution as decreasing function of ϕ that is small compared to VS at ϕ ϕ δϕ large ϕ but becomes order VS as ϕ approaches 0 from the ðkÞ¼ SðkÞþ ðkÞ; ð17Þ right. γ is a small parameter, which we can make unam- biguous by normalizing VR such that one has 0 0 1 V ½ϕ þ γδϕ3 VS½ ¼VR½ ; ð12Þ P ðϕðkÞÞ ≈ S S s;S 6 6 0 ϕ γδϕ 2 MP VS½ S þ ϕ 0   remembering that by definition, h ¼ corresponds to the pffiffiffiffiffiffiffi η δϕ present horizon scale. This decomposition is illustrated in ≈ P ðϕ ðkÞÞ 1 þ 3 2ϵ − 2 pffiffiffiffiffiffiffiS : s;S S S 2ϵ M Fig. 3. We expand in γ ≪ 1. S P The full slow-roll parameter ϵ can be expanded as ð18Þ   0 To understand the size of these corrections, we need to find VR VR 2 pffiffiffi ϵ ¼ ϵ 1 þ 2γ − þ Oðγ Þ : ð13Þ Hh S 0 δϕ. Since ¼ 1 þ Oð ϵÞ, Eq. (8) can be approximated as VS VS H Z   γ k ϕ dϕ 1 V0 Crucially, the first Oð Þ correction will generically be larger log ≈ pffiffiffiffiffiffiffi 1 − γ R : ð19Þ ϕ 0 0 2ϵ 0 than the second near ¼ . This is because VR is kh ϕh MP S VS 0 significantly larger than VS , which is proportional to pffiffiffiffiffi γ ϵS. This can be captured systematically by organizing Solving this equation perturbatively in shows that the expansion in terms of V0 pffiffiffiffiffiffiffi δϕðkÞ ≈ γ R 2ϵ logðk=k Þ; ð20Þ γ V0 S h γ~ ≡ pffiffiffiffiffi : ð14Þ S ϵS so the corrections coming from this effect are smaller We will therefore demand not just that γ ≪ 1 but also that than the suppression term in (16) by factors of either ϵ η γ~ ≪ 1, so that perturbative corrections to ϵ are small. S logðk=khÞ or S logðk=khÞ. It is important here that we Expanding to first order in γ~, we find are interested only in logðk=khÞ of order one, since the effect due to VR decays at higher k. In practice, this means ϕ V ≈ V ; that we can just use the solution S in (8). S 0 ϵ ≈ ϵ 1 2γ VR S þ 0 ; ð15Þ III. OBSERVATION VS In this section, we consider the observational prospects and thus of the power suppression from steepening.

083527-5 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015) l l 1 Cl l 2 models of potential steepening can improve the fit at low . 2500 We will further argue in Sec. VI that the probability for the onset of such a feature is distributed smoothly over log l 4.5 2000 with support mainly in the range 1 ≲ l ≲ 10 , rendering a potential onset near l ∼ 30 quite natural.

1500 Given this theoretical motivation, it will be important to corroborate or eliminate any anomaly at large scales. Can future measurements of cosmological data improve the 1000 significance of the Planck anomaly, possibly at the level of discovery? 500 B. Future sensitivity 0 l 01020304050In estimating the possible sensitivity of future experi- lðlþ1Þ 2 ments an essential point is that, due to statistical rotational FIG. 4 (color online). Planck data for Cl,inðμKÞ , for 2π invariance, having access to more modes with the same l < 50. As in Fig. 1, the red curve is ΛCDM with the new best fit parameters [17]. wave number allows a more precise measurement of the power spectrum PsðkÞ. Let us briefly review how this applies to the CMB. One expands the observed temperature A. The Planck anomaly as From Fig. 1 it is clear that the Planck data is very well X Δ ˆ ≡ ˆ − ˆ described by ΛCDM for l ≳ 100. The situation at low l is TðnÞ TðnÞ T0 ¼ almYlmðnÞ; ð21Þ l;m less clear. From Fig. 4 it appears that for l ≲ 30 most of the R ’ 1 2 ˆ ˆ Cl s are below the best fit. where T0 is the monopole T0 ¼ 4π d nTðnÞ. The TT Cl’s It is not clear that this suppression should be taken are then defined as the two-point function seriously. If the feature were localized in the middle of the ≡ δ δ power spectrum, it would be easily attributable to the look- halmal0m0 i ll0 m;−m0 Cl; ð22Þ elsewhere effect. That it affects the lowest l’s makes it where the average is taken over the ensemble of realizations more special, but its significance is still decreased by the a of the universe. posteriori choice of l ∼ 30 for the onset of suppression. We only get to measure the CMB once, so we cannot Its significance was reported by the Planck collaboration observe this average directly. We can mitigate this limita- [16] as 2.5 − 3σ depending on the choice of estimator.6 In tion by defining an estimator Appendix A we give a crude method for estimating the 1 X ˆ ≡ significance, which is in basic agreement with this result. Cl 2l 1 almal;−m; ð23Þ Thus, even assuming that the Planck anomaly will with- þ m stand improved analysis of the Planck temperature data, we which has the property that on average it is equal to Cl. do not argue that the anomaly is serious enough to require We can evaluate this estimator on the observed set of alm ’s, an explanation. and this represents our “best guess” for the true Cl. We can Rather, we have advanced a theoretical argument for a get a sense of the accuracy of this guess by computing the l suppression of the CMB spectrum at low . If slow-roll variance inflation was preceded by false vacuum decay, as would be   ˆ 2 natural in a landscape setting, it becomes more plausible Cl − Cl 2 ¼ ; ð24Þ that the Planck anomaly is indicative of a real suppression Cl 2l þ 1 of the primordial power spectrum. We will illustrate this quantitatively in Sec. IV, by showing how two simple toy where we have now assumed that the distribution for the alm’s is Gaussian in the sense that higher-point correlation functions can be computed by Wick contraction. Thus the 6A similar suppression was already seen in WMAP, at even ˆ typical deviation of the estimator Cl from the true average lower significance. With current data analysis by the Planck team pffiffiffi [16], the measurement of the low Cl’sbyPlanck is not Cl falls like 1= l. ˆ appreciably more precise than that of WMAP; both are essentially We can heuristically understand this as follows: Cl is limited by cosmic variance. However, the inclusion of the new 2lþ1 l Λ roughly an average over 2 independently fluctuating measurements at high- by Planck gives CDM less flexibility to 2 fit the low Cl’s. In other words, the data points have not gone quantities almal;−m, each with mean Cl and variance Cl, ^ down, but the best fit curve has gone up. This has the practical so the standard deviation of the distribution of values of Cl effect of increasing the significance of any suppression. Fur- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi falls like 1= number of modes, which for the CMB is just thermore, the better frequency coverage of Planck allows for a pffiffi more reliable exclusion of galactic foreground contamination. ∼1= l. The estimator in (23) is optimal, in the sense that

083527-6 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) no other unbiased estimator can have smaller variance than The new term is correct only when l ≫ l , but for l ≳ pffiffiffiffiffiffi z z that. This cosmic variance permanently limits the precision 2π this will basically be the case before it begins to with which the Cl’s can be determined by CMB observa- compete with the first term. We can then approximate the tions alone. new, smaller, cosmic variance of the power spectrum, However, if different observations gave us access to repackaged as the Cl’s, as more modes of a given l, we could reduce cosmic variance. We now argue that in fact future measurements of large 2 2 2 σl ≈ Cl ð26Þ scale structure (LSS) have the potential to improve the Nl situation considerably. The key point is that the CMB ’ provides only a two-sphere s worth of information about for all l > lz without any large error. The factor of 2, as in l ∼ the primordial power spectrum PsðkÞ, so at fixed kDls Eq. (24), arises because we are measuring a two-point there are only ∼l modes available to average over. By ~ ~ function that is symmetric under k → −k.Forl < lz LSS contrast, the large-scale distribution of galaxies, and perhaps does not provide any new information, so we will continue 21 cm radiation, can provide three-dimensional information to use the error bars from just the CMB. about the power spectrum PsðkÞ. For a given k, one then has In this paper we have in mind a particular, theoretically ∼ 2 k modes to average over. Compared to the CMB alone, motivated feature at low l: a monotonically decreasing this would decrease the cosmic variance on PsðkÞ for each suppression of power. It is interesting to ask to what extent measured wave number k by an additional factor propor- pffiffiffi including LSS would allow confirmation of such an effect. 1 l l l ∼ tional to = , where and k are related by kDls. As we mentioned in the previous subsection, there is Determining the precise cosmic variance for PsðkÞ after already a hint of such an effect in the current analysis of the inclusion of both the CMB and LSS is beyond the the Planck data, and in the following section we will present scope of this work, but we can get a rough idea by the two models which improve the fit to the current data. following construction. Imagine that in the future we There is a simple way to estimate how sharply future are able to measure the density of matter inside a cube LSS experiments could distinguish these models from centered on the , with the center of each face at ΛCDM. Suppose that the second of our models, the 2 z. The linear size of the cube is Dz, where Dz is determined power-law model of Sec. IV B, is correct. We can then from Eq. (10). Imposing periodic boundary conditions, the generate future “data” for the Cl’s independently for each l, number of modes with wave number between k and k þ dk, from a Gaussian ensemble centered on this model. The 3 Dz 2 l l for sufficiently large k, is approximately 4πð π Þ k dk.To standard deviations are taken as (26) for > z and (24) for make contact with the rest of our paper, we can reexpress k in l < lz. Remember that here lz should be thought of as terms of l via l ¼ kDls, and we can also reexpress Dz in setting the largest scale out to which we can measure 3D Dls information from LSS. terms of an effective l as lz ≡ π . The total number Dz For l ¼ 50 the “data” we generate should resemble independent of modes between l and l þ 1 accessible from z Fig. 4, while for lower values of l it should move closer to the CMB and LSS is then approximately determined by z our exponential model. We illustrate this in Figs. 5, 6, adding the number of modes measured in the CMB to this where we plot some fairly typical instances of “data” for estimate: lz ¼ 50; 6.7; 5.5, and 3.6. Finally, we can use our crude 2 statistical techniques from Appendix A to estimate the l “ ” Λ Nl 2l 1 4π : probability that this data arises from CDM, again with ¼ þ þ l3 ð25Þ z the error bars appropriately reduced thanks to the LSS

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lðlþ1Þ FIG. 5 (color online). “Data” for the 2π Cl’s generated from the power-law model. The red curve is ΛCDM, while the green curve is the model we introduce in Sec. IV B. On the left we include no measurements of large scale structure, while on the right we include LSS measurements only down to lz ¼ 6.7. The “data” on the left differs from ΛCDM at 2.4σ, while that on the right differs at 3.4σ.

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0 0 0 1020304050 0 1020304050

FIG. 6 (color online). More “data” from the power-law model. On the left LSS is included down to lz ¼ 5.5, while on the right it is included down to lz ¼ 3.6. The left side “data” differs from ΛCDM at 3.8σ, while the right differs at 5.2σ.

“data.” To focus on the low l region we use the diagnostic E are completely uncorrelated. However, some of the l 40 with max ¼ . difference in the visibility function comes from the con- We have chosen the numbers 6.7,5.5, and 3.6 for lz tribution to polarization from the epoch of . because the first two are roughly what might be expected This unfortunately contaminates the low-l signal with from upcoming EUCLID-like galaxy surveys, while the contribution from higher wave numbers than the ones third is what might come out of 21 cm measurements. More contributing to the same l’s at recombination. Removing explicitly, lz ¼ 6.7 corresponds to z ≈ 3, while lz ¼ 5.5 this effect would increase the TE correlation, decreasing corresponds to z ≈ 5. If we imagine that 21 cm measure- the potential improvement to the statistical significance ments can map gas out to z ≈ 50 then this gets us down to of the Planck anomaly from EE measurements, so we expect lz ¼ 3.6. Generating the data 20-30 for each choice the benefit of these measurements to be limited, although of lz, we find that with lz ¼ 50 the significance of the non-negligible, and we do not study them in detail here deviation from ΛCDM is typically of order 2 − 3σ. This is (for a discussion of many of the relevant issues see [61]).7 As in rough agreement with what Planck has found. Including we already discussed however, the potential steepening LSS with lz ¼ 6.7, the typical significance increases to does lead to power suppression directly in the TE and about 3 − 4σ.Forlz ¼ 5.5 it increases to about 3.5 − 4.5σ, EE CMB, and this should eventually be detectable with at while for lz ¼ 3.6 it is typically in the vicinity of 5 − 6σ. least some significance. More optimistically, if BB is ever Thus, we find that future LSS surveys may have the measured than eventually it might be possible to see that it is statistical power to strongly corroborate or exclude a low-l not suppressed at low l, as predicted in Sec. II. power suppression of the type suggested by the Planck anomaly. Our statistical methods have admittedly been IV. TWO MODELS rather rough, but a more precise analysis that takes into account the details of the experiments is quite possible. Our We will now illustrate our general discussion with two significance results are rather sensitive to the order one simple models. In the first, the steepening of the potential is l2 rapid enough that the CMB power suppression is closely coefficient in front of l3 in Eq. (25), so it will be very z related to the onset of inflation, when ΩK ≃ 1. As expected, important to compute this more carefully. Of course, in an one finds that this model is already fairly tightly con- actual LSS survey, there will also be systematic challenges strained; in models of this type, a future discovery of to face in order to approach these asymptotic estimates. curvature is likely. The second model illustrates a more Although we have focused on a single model, this gradual steepening feature which allows parametric sepa- analysis shows that future LSS measurements may in ration of the onset of CMB power suppression and the general allow strong constraints on models that predict beginning of inflation. It is thus much less constrained. In deviation from ΛCDM at low l. If the actual inflationary this model, observation of the CMB suppression is possible potential differs from LambdaCDM more than our model, even if curvature is never found. We show that both models future observations could rule out ΛCDM at even greater are able to improve the fit to the Planck data at low l. levels of confidence. We hope we have made it clear how Both models are toy models. They represent the two important this effort might be, and the potential discoveries extreme possibilities for the suppression of the curvature it could lead to. Finally, let us comment on polarization measurements. 7To get a rough idea of size, if we imagine we get 4=3 as many Given that E and T modes are quite uncorrelated, due to independent modes from polarization this should allow us to difference in the visibility function, measurement of the EE multiplypffiffiffiffiffiffiffiffi the existing significance from the TT anomaly by power spectrum will also give us access to more indepen- 4=3 ≈ 1.15. We then might expect increases of order a few dent modes, in fact twice as many in the limit where T and tenths of a σ.

083527-8 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) while exhibiting power suppression at low l. We expect 2500 that realistic inflationary models with these characteristics can be constructed, but we will not attempt this here. 2000

A. Exponential steepening 1500 We first take pffiffiffiffiffiffiffi ϕ 1000 V ¼ V 1 − 2ϵ S i S M P 500 − ϕ VR ¼ Vie M: ð27Þ 0 This model has two parameters, γ and M. It is convenient to 0 1020304050 define 6000 pffiffiffiffiffiffiffi M 5000 n ≡ 2ϵ P ; ð28Þ S M and instead work in terms of γ and n. Equations (8), (16) 4000 then give   3000 n n kh PsðkÞ¼Ps;SðkÞ 1 − γ þ … : ð29Þ 2000 ϵS k We need to take n ∼ 1 in order to fit the low l power 1000 suppression in Planck. With this constraint, we need to 0 0 500 1000 1500 2000 2500 take γ small enough so that the correction to ϵS is still ϕ 0 perturbative for > . This model is plotted against the lðlþ1Þ −3 FIG. 7 (color online). Comparison of Cl for the model data in Fig. 7 for n ¼ :7 and γ ¼ 3.5 × 10 .8 Using the 2π (29) in green, ΛCDM in red, and the Planck data. Theory curves crude statistical measure from the Appendix, we find that – l 30 l are computed using CLASS [19 21]. for max ¼ the significance of the low- anomaly is 2 4σ 1 0σ l 49 decreased from about . to . , while for max ¼ it and then evaluate Ω a H −2. General properties of 1 8σ 36σ K ¼ð h hÞ decreases from . to : . From the figure it seems that the solutions of these equations are discussed in detail for γ we could fit even better by increasing both and n a little example in [62]; the initial conditions from the CDL bit, but as we now argue the exponential growth of this [39] are that ϕ_ ¼ 0, a ¼ 0, a_ ¼ 1, and ϕ is some potential allows so little inflation for ϕ < 0 that doing so negative number such that jϕj is significantly greater than would produce observable curvature. ϕ ≪ 4 γ ϕ 0 M but Vð Þ MP. and n are taken from the quoted When < in this model, the potential rapidly approaches −9 4 − ϕ values above, and we take Vi ¼3×10 Mp and ϵS ¼ 0.006 Viγe M. When this happens, the slow-roll parameters are both 2 2 to match the Planck data using Eqs. (3), (7). A rough analytic proportional to ðMPÞ ¼ n , so the slow roll approximation M 2ϵS treatment is possible, but it is more convenient to simply fails almost immediately. To compute the size of curvature in solve them numerically. For the quoted values of parameters the model we need to solve the full FRW equations this gives ΩK ¼ 0.011, which is just barely consistent   with the current 2σ bound, −0.028 < Ω < 0.008, from 1 1 1 K H2 ϕ_ 2 V ϕ (Planck þ lensing þ WP þ highL) [17]. Some plots of the ¼ 2 þ 3 2 2 þ ð Þ a MP solution are shown in Fig. 8. We show some values of ΩK as n γ Planck ϕ̈¼ −3Hϕ_ − V0ðϕÞ; ð30Þ a function of , with chosen to approximately fit the anomaly, in Fig. 9. The simplicity of the exponential potential in this model 8This plot is slightly dishonest for the following reason; the produces an interesting tension between curvature and simple VS from Eq. (27) was chosen to make our discussion cosmic variance. From Fig. 9 we see that getting enough Λ simple, but strictly speaking it does not quite match CDM with inflation to dilute curvature requires n<1, but since the the Planck parameters since it predicts r ≈ :1 which is a bit large. deviation of the power spectrum from ΛCDM falls off like This has the effect of slightly enhancing the Cl’satlowl,byof 1 l 5 order a few percent, so to be consistent with Planck in Fig. 7 we ln at large we need n>: for this to fall off faster than have set r ¼ 0 for both curves. This can be justified by slightly cosmic variance. Above we chose n ¼ :7, which satisfies η modifying VS by making the slow roll parameter nonvanishing, both of these constraints, but not by much. This tension can and therefore to suppress r, but since our models are really just ϕ heuristic illustrations anyway we have not done so. This comment be relaxed either by allowing VR to fall off faster at large also applies to Fig. 10 below. or making it a little less steep for ϕ < 0.

083527-9 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015)

H M 1 H2 1.0 t 1 2 3 4 0.8

1 0.6

2 0.4

3 0.2 t 4 1234

ϕ H_ ϕ −4 FIG. 8 (color online). Plots of =M and H2 from a numericalqffiffiffiffiffiffiffi solution of (30). The field is dropped at =M ¼ , and the time is Vi measured in units of the Hubble constant on the plateau, 2 . The second plot shows that inflation does not really begin until soon 3Mp before horizon crossing at t ≈ 2.9.

The correction turns off for kD > l , which in the CMB K ls c 0.10 corresponds to l > lc. It is quite easy to fit the data by varying ς, γ, and lc. For example for ς ¼ 2.3, γ ¼ :08, and 0.08 lc ¼ 65 using the method from the Appendix we find the 2 4σ 29σ l 30 significance decreases from . to : for max ¼ and 0.06 1 8σ 35σ l 49 from . to : for max ¼ . A comparison of this model to standard ΛCDM and the data is shown in Fig. 10. 0.04 It may appear that the improvement of fit for this model compared to the previous one came from having three 0.02 parameters as opposed to two, but lc does not really have n too much effect on the curve in the relevant region. The 0.6 0.7 0.8 0.9 1.0 1.1 important difference is that in this model it is easy to inflate Ω away the curvature, so we are free to take a larger value for FIG. 9 (color online). K as a function of n. A rough analytic γ Ω −2.5 without running into the observational bound on K. treatment suggests approximating this function as e n . To see that this model indeed allows inflation for ϕ < 0, we observe in this region the potential is essentially just ϕ ς B. Power law steepening ∼γV ð Þ . The slow roll parameters are both proportional i MP MP 2 The previous model was an example of a potential that to ð ϕ Þ , which are quite small for jϕj ≫ MP. The number of steepens rapidly enough that it is very difficult to have an ϕ 2 e-foldings prior to ϕ ¼ 0 will thus be approximately ð i Þ , observable suppression of power without observable cur- MP ϕ vature. We now consider a more gentle model, with the where i is the point where curvature domination ends. The number of e-foldings in the past is limited by the onset of same VS as before but now with slow roll eternal inflation, as in that case we have ς 3P ∼ Ω ∼ 1 Vi ϕc − ϕ k sðkÞ K , and so, having inflation in that regime V ¼ Θðϕ − ϕÞ ; with ς > 1: ð31Þ 9 R c ς M does not help diluting curvature anymore. To avoid slow P ϕ ς 2 γ i þ ≪ 4 roll eternal inflation we need Við Þ MP, so we can Here Θ is the Heaviside theta function; this potential is MP get at most a number of e-foldings before horizon entry of nonanalytic at ϕ , but that is no issue for the illustrative c order purposes of our toy model. For future convenience we parametrize ϕc as 9Slow-roll eternal inflation is a situation where the scalar field pffiffiffiffiffiffiffi is rolling slowly enough that fluctuations carry it up the lc ϕc ≡ 2ϵS log : ð32Þ potential as often as classical rolling takes it down [28]. In this DlsH0 situation, inflation lasts forever globally, but in each region of space sooner or later by chance the field fluctuates down the Equation (16), at leading order in slow roll and γ, then gives potential enough that it then rolls smoothly down. This allows a  fairly homogeneous inflating region to be created in a manner that l 2γ pffiffiffiffiffi l ς−1 is crudely similar to bubble nucleation. Instead modes that exited c ffiffiffiffiffiffiffi c PsðkÞ¼Ps;SðkÞ 1 − Θ − k p 2ϵlog the horizon during the eternal inflation period induce density D 2ϵ D k 2 _  ls S ls perturbations H =ϕ of order one. While for many years a quantitative and sharp understanding of slow roll eternal inflation þ … ð33Þ was lacking, recently relevant progress in this direction has been made [63–65].

083527-10 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) 2500 quantitative and falsifiable predictions. At present, the landscape provides the only viable explanation of the 2000 smallness of the cosmological constant [22,30]. The land- scape can also explain other fine-tuning or coincidence 1500 problems. The increasingly appears to be of this type, as no evidence for naturalness of the weak

1000 scale has so far been found at the LHC or by any other experiment. Any theory that contains at least one long-lived meta- 500 stable vacuum with positive cosmological constant (such as, apparently, our own [66,67]) leads to eternal inflation: 0 0 1020304050globally, the universe grows faster than it decays. Thermal

lðlþ1Þ effects and vacuum decays give rise to infinite recurrences FIG. 10 (color online). Comparison of 2π Cl for the model Λ and obstruct the computation of relative probabilities. (31) in orange, CDM in red, and the Planck data. The question of how to regulate these infinities is known as the measure problem of eternal inflation; see, e.g., 4 2 M ςþ2 Refs. [68,69] for reviews. Any viable proposal must N ≈ P : ð34Þ γV reproduce the standard probabilities for the outcome of i laboratory experiments and closely related physical proc- 10 This can be quite large. We thus view this model as an esses. Perturbative effects on the CMB spectrum are of extreme case where curvature and steepening can be this type [76], so we will be able to ignore the measure separated almost arbitrarily. problem for the purposes of our analysis. The main idea that will go into our discussion of the V. STATISTICAL TUNING IN A genericity of potentials is that having flat directions is LARGE LANDSCAPE statistically tuned. By statistically tuned we do not mean that it is radiatively unstable in the way that the Higgs In the former sections, we described predictions for boson mass is (without a new symmetry at the weak scale). observables, given a certain potential. We discussed how to A potential which is approximately flat at tree level will observationally recognize an inflationary potential with a also be flat at higher order in , since in steepening at large scales, as in Fig. 2. We argued the limit of zero slope there is a shift symmetry which qualitatively that such potentials are natural if slow-roll emerges. This type of argument cannot explain however inflation followed the decay of a false vacuum. why the tree-level potential was flat in the first place, and it In this and the next section, we will turn to a quantitative is reasonable to ask what type of tree-level potentials are question: what is the likelihood that our own inflationary typical. history contains a potential with steepening, and what is the The question of genericity of potentials goes beyond the probability that this feature is observable? This question usual particle-physics notion of tuning. But in a large can only be posed if many possible potentials exist, that is, landscape, the question is well defined. Its answer depends if there exists a large landscape of effective potentials. In on the statistical distribution of potentials, which can at order to answer it, we would need to compute the statistical least in principle be derived from the underlying theory. To distribution of potentials. For the landscape of string theory, emphasize the distinction from the standard notion of this task lies beyond our current understanding. However, tuning, we will refer to the atypicality of a feature in the we can make progress by considering a range of plausible landscape as statistical tuning. Atypicality in the prior distributions and investigating their implications. distribution need not conflict with observation. But if an We stress that the conclusions of the previous sections observed feature is statistically tuned even after are not affected by the following discussion. They do not depend on the assumptions we are about to make. 10This requirement turns out to be a powerful constraint (which A. The landscape of string theory is fortunate, since no fundamental derivation of the correct measure is known). Measure proposals that survive this constraint The landscape of string theory provides a setting in have proven remarkably successful phenomenologically. In which low energy parameters can take on many different particular, the causal patch measure allows for a direct explan- values. Their statistical distribution is controlled by the ation of the Why Now coincidence [33] and other cosmological underlying unique fundamental theory. If this prior dis- coincidences [37,38,70,71]. This has improved on the predictions of classic arguments (e.g., [29,31,32,34,35,72]) quantitatively, tribution can be computed or at least constrained, proba- while eliminating their specific anthropic assumptions. A small bility distributions over observed values of parameters can set of closely related, phenomenologically interesting proposals be obtained by conditioning on observers. This allows for also remain viable [73–75].

083527-11 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015) conditioning on observers, then it rules out the theory at the lower in , which causes this number to increase corresponding level of confidence. quadratically in the energy scale. We emphasize that we are not claiming that it is impossible to have slow-roll eternal B. Steepening vs flattening inflation somewhere in the landscape, but only that we can argue that our immediate ancestor is most likely to be a As a first application of this framework, we will now false vacuum. motivate our assumption that our region of the universe is This argument can also be applied to the question of inside a bubble nucleated in a metastable false vacuum. whether we should have included other field directions near Why could the era of standard slow-roll inflation not the inflationary plateau in Fig. 2. In fact, multifield inflation instead be preceded by slow-roll eternal inflation? In this is even more disfavored from this point of view, since situation there would be no potential barrier of the type V −12 during the observable period of inflation, 4 ≲ 10 .(Of shown in Fig. 2, so there would be no generic argument for Mp a steepening feature. We suggest however that this type of course, even if there were other fields whose masses are eternal inflation requires significantly more statistical tun- comparable to Hubble on the inflationary plateau of Fig. 2, ing of the potential than the false vacuum eternal inflation one would still expect the field trajectory to steepen near a we have been considering so far. Consider an extremum in potential barrier.) a multifield landscape, where the potential obeys C. Coleman-De Luccia vs Hawking-Moss decay ∂V 0 ∀ As a second application of the landscape framework, we ∂ϕ ¼ n: ð35Þ n will explain why we believe we are justified in taking the barrier to be quite sharp.12 By contrast, one could imagine a A point where either false-vacuum or slow-roll eternal potential of the form shown in Fig. 11, in which the false inflation is happening should satisfy this to a very good 2 vacuum is separated by a broad barrier. This construction approximation. Let us now consider the eigenvalues mn of ∂2V however requires the top of the barrier to be quite flat. More the second-derivative matrix ∂ϕ ∂ϕ . In order to be able to n m quantitatively we can argue that we must have η ≪ 1 at the neglect quantum and in order to have inflation, we top, by noting that as long as V0 < 0 we have roughly need pffiffiffiffiffi d 2ϵ d2 η − 2ϵ 2 2 log V ¼ − ; log V ¼ ; ð37Þ − H − V mn 1 ∀ ϕ ϕ2 2 ¼ 4 < 2 < n: ð36Þ d MP d MP Mp 3Mp Mp and thus that The first inequality ensures that the field will not roll down Z a tachyonic direction before inflating, and is equivalent to pffiffiffiffiffi ϕ dϕ η > −1. The second inequality ensures that the masses are 2ϵ ¼ − ðη − 2ϵÞ: ð38Þ ϕ M not Planckian. The point however is that the lower bound peak P on each m2 is much smaller in absolute value than the upper n We then observe that any Oð1Þ negative value of η near bound, so typically we can expect all of the eigenvalues to ϕ would need to be offset by an Oð1Þ positive value of η be positive and significantly larger than H2. In this case we peak at some later time over a comparable field range in order to have a false-vacuum, which decays via bubble nucleation get ϵ to be small again during the period of observable as in Fig. 2. More quantitatively, let us imagine that the 10−1 inflation. This would lead to the kind of potential with energy scale of the potential is of order in Planck steeping that we have considered so far. One could try to units. We get to raise this to the fourth power in Eq. (36),so 2 avoid this conclusion by asking instead that the field if we imagine a uniform distribution for each m over the excursion from ϕ to the beginning of inflation be small 10−4 peak allowed range then the probability is roughly per in Planck units, shrinking the range of the integration in direction that we get slow-roll eternal inflation. This is not (38) and thus allowing large values of η near the top. Doing quite correct, since because of eigenvalue repulsion the this would require η to get from Oð1Þ to a small value quite different eigenvalues are not all independent. This problem quickly and then stay there, which would need fairly has been studied quantitatively for the Gaussian matrix precise manipulation of the higher derivatives of V. Once ensemble in [77], and applying their results we find that for we have jηj ≪ 1 at the top of the hill, this requires ð V Þ1=4 ∼ 10−1 we need at least of order 102 fields to have a Mp additional statistical tuning beyond the one needed to get decent chance of even a single tachyonic direction.11 More inflation. In general this “wide-barrier” scenario seems to realistically we should probably demand the energy be even require extra statistical tuning compared to the potential in

11We thank Timm Wrase for help in understanding and using 12This discussion is inspired by conversations with Enrico their results. Pajer.

083527-12 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) power spectrum given by (6). Motivated by our discussion of the previous sections, we will further assume that below some k the power is suppressed. At some even lower k, the potential has become steep enough that inflation is no longer possible; this is the beginning of inflation. This model of the power spectrum involves essentially four parameters: the asymptotic value of As and ns, as well as the point of onset of power suppression and of the onset of inflation. Here we will FIG. 11. A potential with a fat barrier. The tunneling path is take the asymptotic parameters A and n to be fixed. We shown as a dashed line. For a broad enough peak the tunneling s s instanton will not exist, and the Hawking-Moss instanton will thus do not need to discuss prior distributions or anthropic dominate, leading to slow-roll eternal inflation at the top of the cuts for these parameters. Of course a more general analysis barrier. We argue that potentials of this type are statistically would do so, but already in fixing them, there is still a chance suppressed in the landscape. that our discussion of the remaining two parameters might be in conflict with observations.13 (If they are not, i.e., if we η ≪ 1 succeed for now, then later work may still falsify the theory Fig. 2, with no added benefit. Incidentally j j at the top by allowing A and n to vary and exhibiting a conflict of the hill changes the physics qualitatively. As reviewed s s between the predicted and observed values. In the present for example in [78], in this situation the Coleman-De context, theory should be understood to include our assump- Luccia instanton no longer exists and the relevant transition tions about the prior distribution, Eqs. (45) below, which is now the Hawking-Moss instanton [79]. This instanton may require modification.) Our project for the rest of this can be interpreted as causing an entire Hubble-sized region section is thus to motivate prior distributions for the location to thermally jump up to the top of the potential barrier of the onset of potential steepening and the beginning of [78,80]. Since jηj ≪ 1 and ϵ ¼ 0 there, slow-roll eternal inflation, identify anthropic cuts on this two-parameter inflation then necessarily ensues. Thus the present issue is space, and then compute the probabilities for observable closely related to the previous subsection. suppression or curvature. Of course the potential in Fig. 2 still has some non- A first worry is that, given that we have excluded generic features: the inflationary plateau is flat, its scale is steepening above l ≈ 100, the remaining window is small low, and the cosmological constant in our vacuum is enough that seeing steepening is very unlikely. In looking small. Unlike the other statistical tunings we have just for features of the inflationary potential in the CMB, a discussed however, these can be anthropically explained. crucial point is that the relationship between ϕ and l is The anthropic argument for the smallness of ρΛ is well logarithmic, as can be seen from Eq. (8) and the discussion known [29,31,32]. The low scale of the inflationary plateau below it. We emphasize this in Fig. 12, where we plot the is needed to avoid over-production of structure, since we Λ ≈ 10−10ϵ power spectrum for CDM and for the two models of need Vinflation in Planck units. The flatness of the Sec. IV against log l. For example, the Cl’s with 2 < l < plateau is needed to get enough e-foldings to dilute 70 contain information about a region of the inflationary curvature [34,36]. This last statistical tuning, the flatness potential which is just as large as the region described by the of the potential, is quite close to the steepening we are Cl’s with 70 < l < 2500. We will argue below that the interested in; it was studied quantitatively in [36], and we probability distribution over the location of the steepening now study the steepening along similar lines. feature, in terms of log ls, is not very steep in this regime. Moreover, we will argue that the entire observable regime is VI. THE PROBABILITY FOR anthropically allowed. Thus, we will find that the proba- OBSERVABLE STEEPENING bilities for finding a steepening feature, say, in the range In this section, we further quantify the notion that flat 2 ≲ l < 70 is not much smaller than finding it in the 70 l 2500 potential regions require statistical tuning. Our goal will be range < s < . to estimate the probability for observing a power suppres- Another question is why a steepening feature should lie sion in the CMB due to steepening of the potential near the in the region 2 < l < 2500 in the first place. Why should it beginning of inflation. We will consider a range of plausible assumptions about the statistical distribution of potentials 13We briefly comment however that it is unclear to what extent in the landscape. Again, as for the former section, it is the prior for As favors larger or smaller values. On the one hand V worth stressing that only the conclusions of the present likes to be large, but on the other hand so does ϵ, so the tendency section depend on these assumptions. of their ratio is unclear. Making As smaller prevents structure In making statistical predictions from the landscape, it is formation, so any tendency in this direction would be cutoff anthropically fairly quickly. Making it larger is more subtle, but it important to clearly state which parameters are being varied has been argued that there is an anthropic cut in this direction as and which are not. In this section we assume that the power well [35]. Scanning the parameter ns is expected to be quite spectrum asymptotes at large k to the standard ΛCDM irrelevant. We leave a detailed discussion of this to future work.

083527-13 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015)

l l 1 Cl 2 which are associated with the mass of a typical galaxy, it 6000 will prevent the formation of galaxies. To see this quantitatively we need to briefly recall how 5000 primordial density perturbations are connected to structure formation. In evolving modes from horizon entry to last 4000 scattering, it is important to know whether or not they were inside the horizon at the time of matter-radiation equality. 3000 To understand which modes we are interested in, we note that the proper distance to the last-scattering surface today, 2000 Eq. (10), is about 14 gigaparsecs. The typical distance

1000 between galaxies is of order a few megaparsecs [60]. From Eq. (9) the relationship between the proper wavelength λ of

0 l a fluctuation today and the angular scale of that fluctuation 2 10 30 50 70 100 200 300 500 70010001500 2500 on the CMB is FIG. 12 (color online). The TT power spectrum for the best fit to ΛCDM in red, our exponential model in green, and our power- 2πD l ≈ ls ; ð39Þ law model in orange, all plotted on a log scale. λ

so the density perturbations important for the formation of be located where it can be observed through the CMB? The galaxies evolved from fluctuations whose angular size on angular modes that leave the horizon during inflation range l ≈ 63 the CMB is of order up to exponentially large values [ reh e for (GUT)-scale inflation]. Moreover, if there l ≈ 104.5 were extra e-folds of inflation beyond what is needed to g : ð40Þ dilute curvature, then there would also be room at “l ≪ 1” for the steepening to happen. This regime is invisible By comparison modes which were entering the horizon at because the corresponding modes are too far outside matter-radiation equality have our horizon. In Sec. VI A, we will describe an anthropic a H boundary which prevents the steepening feature from turn- l EQ EQ ≈ 143 4 EQ ¼ H0Dls ; ð41Þ ing on at l ≫ 10 . a0H0 In Sec. VI B, we will discuss prior distributions for the number of e-folds of inflation, and for the onset of so the modes of interest for galaxy formation entered the steepening. They quantify our assumption that having the horizon long before matter-radiation equality. The cold suppression turn on in the invisible “l ≪ 1” region is density perturbation for such modes at the time statistically suppressed, since it would require a larger total of last scattering is given by14 number of e-foldings of inflation. Combining these assumptions and results in Sec. VI C, δρ 9 l 2 4 l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≈ EQ ffiffiffi l we conclude that the probability for a steepening feature log p Psð =DlsÞ: ρ l 2 l 3 l lying in the observable region can be as much as Oð1Þ, t¼tls;k¼ =Dls LS EQ while the probability for a curvature lying in the observable ð42Þ region is of order Oð10%Þ. This probability is larger than pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ≈ 1 Ω ≈ 59 the probability for observable curvature, because the Here LS ð þ zLSÞ MH0Dls is the angular scale relevant anthropic boundaries correspond to different val- of modes which were entering the horizon at the time of last ues of l; and it remains larger after observational exclu- scattering, and PsðkÞ is the primordial power spectrum. sions are taken into account. This is the density perturbation for an individual mode, to get the typical size of the total mass fluctuation in a box of 2π A. Anthropic bound on steepening linear size k we should sum in quadrature over the modes 2π If flat potentials are rare, as we assume, the steepening in the box whose wavelengths are of order k . This amounts 3=2 l l should begin at very large l. But there could be an to multiplying by a factor of k ; for of order g, this anthropic reason why this cannot be the case. The power spectrum of scalar density perturbations generated during 14This equation follows from a hydrodynamic treatment of the 2 inflation is proportional to H4=ϕ_ evaluated at the time evolution of density perturbations from horizon entry to last l l 2 where the mode of interest is leaving the horizon. A steeper scattering; see Eqs. 6.5.11 and 6.5.15 of [60]. The ð EQ= LSÞ _ factor comes from the linear (in the scale factor) growth of the potential will lead to a larger ϕ and thus to a lower power perturbation after matter domination. The log comes from the spectrum [36,42,51]. If this happens at distance scales growth during radiation domination.

083527-14 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) −3 pgivesffiffiffiffiffiffiffiffiffiffiffiffiffiffi a total mass perturbation of order 10 since anthropic cutoff of the previous section demands 3 −5 S S ≡ 52 k PðkÞ is of order 10 . > g . During the matter dominated era these perturbations Following inflation back in time, steepening indicates −1 15 grow like the scale factor a ∝ ð1 þ zÞ , so since zls is of the transition between a flat slow-roll potential and a steep 3 δρ barrier separating our vacuum from its parent vacuum. order 10 , ρ would become order one and the overdensity Slow-roll inflation can begin at a somewhat more negative would become gravitationally bound at redshift z ≈ 1.But value of ϕ than the location of this feature, but one expects this is when the cosmological constant began dominating that the location of the steepening feature is statistically the evolution of the universe and turning off perturbation distributed in a way very similar to the distribution of the growth. total number of e-folds of inflation, N . For motivation we We then can see that galaxy formation in our universe is now briefly recall this distribution. occurring close to a catastrophic boundary. Suppose that Based on our general discussion of statistical tuning, the the initial density contrast was smaller, even by just a factor prior probability distribution for N in the landscape should of order one. Then only exponentially rare multi-sigma be suppressed at large N . A plausible guess is that at large overdensities would have formed large galaxies, and there N would be exponentially fewer observers (according to all the prior distribution should fall like some power viable measures currently under consideration [81]). dP ∝ N −ν Smaller galaxies are further from the catastrophic boun- N ; ð44Þ dary, but cooling problems impose a lower bound on galaxy d mass [70], and metallicity may be insufficient if hierarchi- with ν > 1 but of order unity [36]. (Indeed for a particular cal structure formation is disrupted too early. parametrization of a linear potential, assuming a uniform The above argument makes the simplifying assumption distribution over parameters implies ν¼4 [36], while ν ≈ 3 that the change of slope, at the onset of steepening, is in a mini-landscape inspired by a brane moving in a immediately so large as to suppress the density contrast by [82].) Exponential suppression is ruled out ex- a factor of order one. A more natural expectation is that the perimentally, since slow-roll inflation would be too rare in steepening sets on more gradually, ramping up from the landscape to explain the observed flatness of the undetectable, to detectable, to order-one suppression of universe even after anthropic conditioning. the power as l decreases. It is difficult to parametrize such A natural first guess for the prior distribution for the features in detail. But it is clear that this effect moves the steepening location S is that at large S it falls like S−ζ for anthropic boundary for the onset of detectable steepening some ζ > 1 of order unity, but this does not quite work l toward even larger values of . We will neglect this shift in since in fact N and S are not independent; the beginning of what follows. This makes our estimates below for the inflation must happen at more negative ϕ than the onset of probability of detecting steepening more conservative, but a steepening. We can implement this as a constraint N ≥ S, larger uncertainty arises in any case from our ignorance of but this means that in fact we need to consider a joint the details of the prior distribution. distribution for both N and S. We thus propose an unnormalized joint distribution16 B. Prior distribution for steepening and curvature dP S ν From an observational viewpoint, the steepening feature S−ðζþ1Þ Θ N − S N S ¼ N ð Þð45Þ is conveniently parametrized by the angular scale ls at d d which it appears in the CMB spectrum. However, from a ν ζ theoretical perspective, the feature appears in the slow-roll with and being order one positive numbers. The first S potential. Thus, its prior distribution is more naturally term in this distribution suppresses large values of ,as parametrized in terms of the number of e-folds before the expected from our general discussion of flatness, while the end of inflation, S, at which the potential steepens notice- second implements the expectation that once the potential ably. This number is related simply to the potential begins to steepen, it typically should not be too much via Eq. (8). longer before it can no longer support inflation. The N The logarithmic relationship (8) between k and ϕ implies parametrization is chosen so that integrating out , without dP ∝ S−ζ imposing anthropic constraints, gives dS ; and inte- l S S ≡ reh ≈ 63 − l grating out , again without anthropic constraints, gives log l log s: ð43Þ dP ∝ N −γ γ ζ − ν N s dN , with ¼ , so the suppression for is smaller than for S. Larger ζ suppresses large S more, Here the index reh indicates reheating, and in the second making observable power suppression more likely. equality we have assumed reheating at the GUT scale. The 16We have also considered some other similar distributions, the 15See for example Sec. 8.2 of [60]. results are comparable to what we present here.

083527-15 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015)

Psuppression Pcurvature

0.7 0.25 2 2 3 3 5 0.6 5 0.20 8 8 0.5 0.15

0.4 0.10 0.3 0.05

3 4 5 6 7 8 345678

ζ FIG. 13 (color online). On the left, a plot of Psuppression, the probability for observable power suppression, as a function of , for various ν ν ζ values of . On the right, a plot of Pcurvature, the probability for observable curvature, as a function of , for various values of . These are total probabilities that do not take into account that some signal regions have already been excluded.

ν governs the expected separation between power suppres- in slope is so dramatic as to completely eliminate galaxy ν sion and curvature. If is large, then potentials like our formation, for a steepening feature located at Sg.) As a result, exponential model should be more typical. If ν is small, we are about to find that the probability for observing then our power law model should be more typical. steepening is about twice as large as the probability for observing curvature. Why is the window for steepening C. Conditional probability distributions larger, even though both anthropic boundaries are related to We now turn to the computation of the conditional galaxy formation? probability distributions for the location of the steepening The reason is that curvature, like a positive cosmological feature, S, and the number of e-foldings, N , after con- constant, disrupts the growth of small perturbations. It can ditioning on the existence of galaxies, and later, in addition, do so at any time while perturbations are still in the linear on observational exclusion limits. This will allow us to regime, nearly until virialization. Thus, if curvature domi- estimate the probability for a discovery of either feature in nates the evolution of the scale factor before perturbations on the CMB. In this analysis we will need the corresponding the galactic scale become nonlinear, galaxies will not form. anthropic boundary and exclusion limit for curvature, Steepening, on the other hand, only affects the initial which we now briefly review. strength of the perturbations. Consequently, the anthropic The anthropic constraint that curvature does not bound from curvature is set by the horizon scale at 10 l ∼ 1 disrupt galaxy formation requires aLSHLS > [36]. virialization, which corresponds in the CMB to Oð Þ, Assuming GUT-scale inflation,17 this condition becomes whereas the bound associated with steepening is set by the much smaller comoving scale of the current intergalactic N 61 4.5 > : ð46Þ distance, corresponding to l ∼ lg ¼ 10 . Treating both anthropic cutoffs as sharp, the probability The observational constraint is of course stronger. The distributions conditioned on the existence of galaxies can number of e-foldings required to dilute curvature to the be obtained by removing the forbidden region and renorm- Ω 10−2 N ≈ 64 current exclusion limit, K < ,is . Curvature alizing. We can then compute the probabilities that the Ω ≲ 10−4 becomes unobservable in principle [83] when K , power suppression or curvature lie in their respective which happens for N ≳ 66. observable windows 52 < S < 63 and 61 < N < 66: Note that the window in which the steepening is both R R anthropically allowed and potentially observable, 52 < S < 63 ∞ dS dN dP 63 R52 R61 dSdN , is about twice as wide as the analogous window for Psuppression ¼ 61 N 66 ∞ dS ∞ dN dP curvature, < < . (And the difference would be even R52 R61 dSdN greater, had we not assumed the worst case, that the change ∞ 66 dP 52 dS 61 dN S N P ¼ R R d d ; ð47Þ curvature ∞ S ∞ N dP 52 d 61 d dSdN 17The appearance of the GUT scale here has to do with . The scale is really chosen by taking the highest −10 dP possible scale for inflation that has PðkÞ of order 10 and ϵ at where dSdN is the prior (45). (The probability for seeing most 10−2. Using the quoted parameters below Eq. (6), this gives both curvature and power suppression is nearly as large as reheating ρ ≈ ð2 × 1016 GeVÞ4. It is straightfor- the probability for seeing curvature.) ward to check that making the scale of reheating high is the ν Psuppression and Pcurvature are plotted for some ranges of “worst-case” assumption from the point of view the probabilities ζ for observing steepening and curvature we compute below, but and in Fig. 13. As expected, the probability for seeing the dependence is logarithmic so the scale has to be decreased a power suppression is significantly larger than the proba- lot to improve the odds significantly. bility for seeing curvature. For ν ¼ 3 and ζ ¼ 4 it is about

083527-16 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015)

Psuppression Pcurvature

0.40 2 0.08 2 0.35 3 3 5 5 0.30 8 0.06 8 0.25 0.04 0.20 0.15 0.02 0.10

345678 345678

0 0 FIG. 14 (color online). Plots of Psuppression and Pcurvature, obtained after conditioning on current observational exclusions. These are the probabilities that future experiments will find a signal.

40%, and it can be even larger. It may be somewhat ACKNOWLEDGMENTS surprising that increasing ν at fixed ζ, which makes seeing We would like to thank Julien Lesgourgues for help curvature more likely, makes seeing power suppression less S ν understanding and using his CLASS CMB Boltzman likely. This is because the term ðN Þ in the prior distribution code, and especially for modifying it to include one of N S (45) suppresses a large separation between and , which our models. D. H. would like to thank Xiaowei Yu for S N pushes up towards the anthropic cut for once we help in understanding hypothesis testing. We thank Adam S impose it, even though the cut for is significantly lower. Brown and Alex Dahlen for very helpful comments on 0 We can also compute the probabilities P for future CDL , and Matias Zaldarriaga for pointing detection of power suppression or curvature, given the out a mistake in an earlier version of Sec. 3.2. We are N ≳ 64 Ω ≲ 10−2 current exclusions. is required for K , grateful to Sergei Dubovsky, Daniel Green, Ben while S ≳ 58 is needed for the suppression to begin below Freivogel, , Matt Kleban, , l ≈ 100 as suggested in Sec. III A. We can include these Samuel Lee, Juan Maldacena, Daan Meerburg, Enrico constraints simply by replacing 61 → 64 and 52 → 58 in Pajer, Michael Salem, Uros Seljak, Steve Shenker, Eva (47). We show the results in Fig. 14; as expected they both Silverstein, , , Lenny decrease by about a factor of two. Susskind, Timm Wrase, and Matias Zaldarriaga for These estimates are quite rough. The odds for a future helpful discussions. D. H. and R. B. would like to thank discovery improve if we take into account that the steep- CERN for hospitality during a workshop when this work l ≫ l ening feature may turn on gently at g and slowly ramp was initiated, and D. H. and L. S. would like to thank up. This shifts the anthropic boundary to values below KITP for hospitality and a stimulating environment S 52 g ¼ . On the other hand, the odds for an observable at the “Primordial Cosmology” workshop. D. H. and feature decrease if the steepening is so subtle that it escapes R. B. would also like to thank the KITP for hospitality detection. Moreover, cosmic variance will limit our ability to during the Fuzz or Fire workshop. This work was detect a feature close to l ∼ 1ðS ≈ 63Þ, so it may be supported by the Berkeley Center for Theoretical appropriate to narrow the range of S that can be considered Physics, by the National Science Foundation (Contracts observable. One could also weaken the anthropic bounds No. 1002399, No. 0855653 and No. 0756174), by somewhat by allowing observers to form in smaller galaxies, fqxi Grant No. RFP3-1004, by “New Frontiers in but this does not significantly change our results. S depends and Cosmology”, by the Princeton Center only logarithmically on scale, so decreasing the galaxy scale for Theoretical Science, and by the U.S. Department of even by a factor of 10 gives only a small (3–5%) increase in Energy under Contract No. DE-AC02-05CH11231. the probability of seeing suppression. Moreover this is only Leonardo Senatore is supported by DOE Early Career true for the probability before imposing observational Award No. DE-FG02-12ER41854 and by NSF Grant constraints. Since the observational constraint is already No. PHY-1068380. at lower log l than the anthropic bound, relaxing the latter to larger log l has no effect on the probability for future APPENDIX A: STATISTICS FOR detection. STRING THEORISTS It is worth emphasizing an important point again. Because the probability distribution for the steepening location is There is a simple way to get a rough estimate for the roughly flat not over l but over log l, we should not significance of the suppression we have been discussing. necessarily expect the steepening feature to be close to the The basic idea is to calculate the likelihood of the data 4 5 anthropic boundary, in the sense of l ∼ 10 . .Forexample, assuming that the measured Cl’s are independently assuming an observable steepening feature and ζ ¼ 4,the Gaussian distributed about the red curve in Fig. 4. The probability that the feature lies at l < 50 is about 70%. χ2 variable

083527-17 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015) ! l 2 Xmax fdatag ftheoryg parameters, although we believe it suffices for the theo- 2 Cl − Cl χ ≡ ; ðA1Þ retical points we make here. σl l¼2 where we take the standard deviations σl of the Gaussians APPENDIX B: WHERE DOES THE CDL to be the average of the distances of the upper and lower INSTANTON DROP US? error bars in Fig. 4 from the observed value, gives a decent In this appendix we argue that for potentials like that of measure of the absolute deviation of the data away from Fig. 2, the tunneling instanton typically drops the field off the red theory curve.18 l is a parameter which controls max high enough that the subsequent Lorentzian evolution rolls over what range we are looking for an effect. With our through enough of the steepening region to produce a Gaussian probability assumption we can compute the potentially observable power suppression. For simplicity p-value for a deviation at least this large from the → ∞ χ2 19 we first consider the Mp limit, where the cumulative distribution : becomes and the Euclidean instanton is a Z 1 ∞ l −3 solution of the equation 2 max − 2 2 χ ≡   2 x= pχ ð Þ l −1 dxx e : ðA2Þ max l −1 2 2 2 Γ max χ 3 2 ϕ00 ϕ0 0 þ ξ ¼ V ; ðB1Þ In looking for a suppression we must also take into account ξ χ2 where V is some potential of the form shown in Fig. 2 and the sign of the deviations, which the parameter is is Euclidean time. The boundary conditions are that insensitive to. For our Gaussian distribution this sign is ϕ0 0 −1 ϕ ∞ ϕ ð Þ¼ and ð Þ¼ fv. For intuition we can reinter- uncorrelated with the absolute size of the fluctuation, so we pret ξ as a Lorentzian time, in which case this equation can take the probability that at most m points lie above the describes the motion of a particle with potential −V, with a red curve to be given by a binomial distribution friction term. This situation is illustrated in Fig. 15. The X instanton is a trajectory which starts at rest at ξ ¼ 0 and 1 #ðSÞ p ðmÞ¼ ; ðA3Þ ϕ ¼ ϕ0, rolls “down” through the potential barrier, and 2#ðSÞ k k∈S;k≤m then climbs back “up” to come to rest on top of the false vacuum at ξ ¼ ∞. If the friction term were absent then where to make this estimate more robust we have included by it is obvious that we would need l’ l l ’ only s in a set S containing those < max whose Cl s ϕ ϕ ϕ 20 to choose 0 such that Vð 0Þ¼Vfv. Since 0 is also the differ from the theory curve nontrivially. The full p-value starting point for real Lorentzian evolution inside the bubble, for estimating the significance of the suppression is then ≫ 2 and since we expect that Vfv Vinf, where Vinf is the scale ≡ 2 χ p pχ ð ÞpðmÞ, from which we can determine the of inflation, this would ensure that the Lorentzian evolution number of standard deviations by solving of the field begins at a scale much higher than the infla- Z tionary plateau, leaving plenty of room to roll through the 1 ∞ − 2 2 steepening feature on the way to the inflationary plateau. p ¼ pffiffiffiffiffiffi e x = : ðA4Þ 2π n Because of friction however, we need to start the field higher up on minus the potential, which means that the tunneling Applying this method to the model and data in Fig. 4 we event drops the field lower on the potential. 1 8 l 49 2 4 l 30 find n ¼ . for max ¼ and n ¼ . for max ¼ , which is roughly consistent with the much more sophis- ticated analysis of [16]. We emphasize that this method is no substitute for a real Monte Carlo analysis scanning over

18A subtlety which was emphasized to us by Uros Seljak is that most sources of error are proportional to the signal. So in comparing models other than ΛCDM to the data we should rescale the standard deviations σl by a ratio of the Cl for the model over Cl for ΛCDM with the best-fit parameters. We take this into account in the significances reported in Sec. IV, although the effect turns out to not be too large. 19See, for example, the review of probability and statistics from [84], although it is straightforward to derive this from the Gaussian distribution by integrating out the angular directions. 20 − This may seem arbitrary, but in any event all of the six FIG. 15 (color online). Evolution on the potential V.To − excluded l’s below 30 and five out of the seven excluded between overcome friction we must start the field higher than Vfv, after 30 and 50 lie below the curve, so including them would only which it rolls down through the barrier at ϕb and comes up back ϕ ξ ∞ increase the significance of the suppression. up to rest at fv at Euclidean time ¼ .

083527-18 INFLATION AFTER FALSE VACUUM DECAY: … PHYSICAL REVIEW D 91, 083527 (2015) One may worry that because of friction it is actually necessary to have ϕ0 already on the inflationary plateau. If this were generically what happened, then the steepening of the potential would not lead to an observable effect in the CMB since the field would not roll through it inside the bubble. It may appear “obvious” from Fig. 15 that, since ≫ Vfv Vinf , friction should not be important enough to move the starting point all the way from Vfv to something of order Vinf. Adam Brown and Alex Dahlen have emphasized however that this conclusion depends on the form of the potential barrier. In particular if the height of the barrier Vb is − parametrically larger than Vfv Vinf, then from the point of view of solving Eq. (B1) we should think of the false vacuum FIG. 16 (color online). A piecewise potential. For ϕ < ϕb it is and the inflationary plateau as essentially being degenerate, quadratic, and there are three linear segments of decreasing ≪ 1 even if Vinf =Vfv . From Fig. 15 it would then seem quite steepness as ϕ increases. likely that friction would require the field to be dropped off on the inflationary plateau. This limit seems statistically the case. For our second model, for ϕ > ϕb we take the tuned to us, but still this argument suggests that if we take potential to be ≈ − ≈ Vb Vfv Vinf Vfv then we should be right at the edge of VS þ γVR; ðB2Þ this limit and it should be ambiguous whether or not ϕ0 is on the inflationary plateau. with VS, γ, and VR taken from the exponential model we In fact the situation is better than it appears. For anthropic discuss in Sec. IVA, with the parameters the same as were reasons the potential barrier is quite asymmetric; on the left used to fit the Planck anomaly. For ϕ < ϕb we take the side is a high-scale false vacuum whose parameters should potential to be quadratic typically be Planckian, while on the right side it interpolates 1 2 ϕ − ϕ 2 to a very flat region in order to get inflation and structure V ¼ m ð fvÞ þ Vfv; ðB3Þ formation. As long as this interpolation does not happen 2 ϕ ϕ extremely sharply, we expect that the field will be dropped with fv chosen to make the potential continuous at b.In off high enough that it will be able to roll through the last part this model inflation begins at ϕ ≈ 0, which is also where we of the interpolation and thus produce a potentially observ- took horizon crossing to be in Sec. IVA, so as long as we able suppression of power. We now briefly present two find ϕ0 less than zero the field will roll through the models that support this intuition. steepening region that led to CMB power suppression. −4 4 −1 For the first model, we consider the piecewise linear/ We will take VðϕbÞ¼10 Mp, m ¼ 10 Mp, and then ϕ quadratic potential shown in Fig. 16. For this potential the study 0 as a function of Vfv. Solving Eq. (B1) numerically, instanton can be constructed analytically by sewing together 10−1 we find that if we take Vfv ¼ Vb then the field is solutions of (B1) in the different regions. We will not ϕ −9 6 10−2 dropped at 0=M ¼ . , while if we take Vfv ¼ Vb it describe this explicitly, but the main results are as follows. is dropped at ϕ0=M ¼ −3.6. In both cases ϕb=M ≈ 16 and First of all if V > 4V then we must have ϕ0 > ϕ1.Soifwe ϕ ≈ 17 ϕ − ϕ ≪ ϕ b f fv=M , so indeed b fv b and the barrier is had tried to only have two linear segments, meaning that the quite asymmetric. Thus we seem to be ok even in the interpolation was instantaneous, this would realize the ≈ 10−2 somewhat unnatural case where Vfv=Vb . Since the Brown-Dahlen claim that a reasonably high barrier requires fairly steep case of an exponentially growing potential the instanton to drop us on the flat region. If we insert an already works, we should also be fine for our more gentle extra segment, which here we take be in the region power-law steepening model from Sec. IV B. ϕ1 < ϕ < ϕ2, the situation is more interesting. In particular Finally let us consider including dynamical gravity. The ≫ ϕ ≫ ϕ ϕ − ϕ ≪ 1 0 in the regime where Vb Vð 1Þ Vð 2Þ, with b fv ϕ a only change to the equation of motion is to change ξ to a , ϕ1 − ϕ , we find that as long as ϕ2 − ϕ1 is at least of order b with a determined by simultaneously solving the Euclidean ϕ1 − ϕ then ϕ0 is greater than but close to ϕ1. The b FRW equation Lorentzian evolution then rolls through most of the region ϕ1 < ϕ < ϕ2, giving potentially observable steepening. a0 2 1 1 1 ¼ þ ϕ02 − V : ðB4Þ This is perhaps the most natural region of parameter space, a a2 3M2 2 since it represents a more smooth interpolation between the p steep barrier and the flat plateau. Instead of the instanton coming to rest at the false Of course one might worry that this argument requires to vacuum at ξ ¼ ∞, it now comes to rest earlier at a interpolation to be so gradual that the steepening would not point ξc where a linearly returns to zero. It is not hard to be observable. In our second model we show that this is not see that the friction is now less than in the gravity-free

083527-19 RAPHAEL BOUSSO, DANIEL HARLOW, AND LEONARDO SENATORE PHYSICAL REVIEW D 91, 083527 (2015) case we just studied. Indeed observe that the Euclidean slowly turn on gravity, the drop-off point ϕ0 will have to a0 0 − 1 4π ϕ02 Hubble constant H ¼ a obeys H ¼ ða2 þ G Þ,so move up the potential (or down minus the potential). This the positivity of ϕ02 ensures that H is always less than it only makes it easier for the field to roll through the would have been with G ¼ 0. This means that as we steepening feature.

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