Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation: implications for primordial black hole production
Daniel G. Figueroa, Sami Raatikainen, Syksy Rasanen, Eemeli Tomberg
HHIP-2020-32/TH
Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation:implications for primordial black hole production
Daniel G. Figueroa, Sami Raatikainen, Syksy R¨as¨anen, and Eemeli Tomberg Instituto de F´ısica Corpuscular (IFIC), CSIC-Universitat de Valencia, Spain University of Helsinki, Department of Physics and Helsinki Instituteof Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland Laboratory of High Energy and Computational Physics, National Instituteof Chemical Physics and Biophysics, R¨avala pst. 10, 10143 Tallinn, Estonia (Dated: December 14, 2020)We consider quantum diffusion in ultra-slow-roll (USR) inflation. Using the ∆ N formalism, wepresent the first stochastic calculation of the probability distribution P ( R ) of the curvature per-turbation during USR. We capture the non-linearity of the system, solving the coupled evolutionof the coarse-grained background with random kicks from the short wavelength modes, simultane-ously with the mode evolution around the stochastic background. This leads to a non-Markovianprocess from which we determine the highly non-Gaussian tail of P ( R ). Studying the production ofprimordial black holes in a viable model, we find that stochastic effects during USR increase theirabundance by a factor ∼ compared to the Gaussian approximation. Introduction.–
Compelling evidence [1] supports aphase of accelerated expansion, inflation, as the leadingframework for the early universe [2–15]. In the simplestmodels, a scalar field – the inflaton – rolls down its poten-tial with the Hubble friction and potential push balanced.This is known as slow-roll (SR). However, if the potentialhas a very flat section or a shallow minimum, the poten-tial push becomes negligible, and the inflaton velocityfalls rapidly due to Hubble friction. This is called ultra-slow-roll (USR). While SR generates almost Gaussianand close to scale-invariant perturbations, as observed inthe cosmic microwave background (CMB), USR can pro-duce perturbations that are highly non-Gaussian and farfrom scale-invariant. This implies that the inflaton can-not be in USR when the observed CMB perturbationsare generated. However, if the inflaton enters USR after-wards, large perturbations can be created on small scales,potentially seeding primordial black holes (PBH) [16–28],a longstanding dark matter candidate [29–36].During inflation, initially sub-Hubble ( k (cid:29) aH ) quan-tum fluctuations are amplified and stretched to super-Hubble scales ( k (cid:28) aH ), where k is the comovingwavenumber, a is the scale factor and H ≡ ˙ a/a is theHubble rate. Once modes reach super-Hubble scales,they can be coarse-grained, contributing stochastic noiseto the evolution of the background formed by long wave-length modes, which are squeezed and ’classicalized’ [37–42]. This is described by the formalism of stochastic in-flation [43–75]. Stochastic effects can be particularly rel-evant during USR for two reasons: i ) the classical pushfrom the potential is negligible, so the inflaton velocitydecays rapidly and the background evolution is more sen-sitive to stochastic kicks, ii ) the perturbations are largerand hence give stronger kicks. Stochastic effects on the power spectrum P R ( k ) of thecurvature perturbation R generated during USR havebeen studied in [25–27, 73, 75–78] (see [73] for highermoments). It was demonstrated in [79], however, thatstochastic effects lead to an exponential tail in the prob-ability distribution P ( R ), which overtakes the lineartheory Gaussian tail. Calculating the power spectrum P R ( k ) is therefore not enough to determine the PBHabundance today, Ω PBH , which is exponentially sensi-tive to the shape of the tail of P ( R ). In this Letter wepresent the first calculation of the non-Gaussian tail of P ( R ) due to stochastic effects during USR. We solve si-multaneously the evolution of the background dynamicswith stochastic kicks from the small wavelength modes,and the evolution of the small wavelength modes thatlive in this stochastic background. As a working exam-ple, we consider a scenario where the Standard ModelHiggs is the inflaton [80, 81], exploiting the renormali-sation group running to create a shallow minimum thatleads to USR [23] (see also [17, 18, 22, 82, 83]). We adjustthe SR part of the potential to fit the CMB observations,while the USR part is tuned to produce PBHs with mass M PBH ∼ − M (cid:12) , with an abundance significantly con-tributing to dark matter in the Gaussian approximation. Stochastic formalism.–
We consider a spatially flatFriedmann–Lemaˆıtre–Robertson–Walker (FLRW) back-ground metric with scalar perturbations, which we splitinto long and short wavelength modes. Correspondingly,the inflaton is decomposed as φ = ¯ φ ( t, (cid:126)x ) + δφ ( t, (cid:126)x ),where ¯ φ = (2 π ) − / (cid:82) k
We consider an inflaton potential V ( ¯ φ ) where the CMB perturbations are generated at aplateau, and there is a shallow local minimum at smallerfield values, as shown in Fig. 1. The inflaton starts in SR,enters USR as it rolls over the minimum, and then returnsto SR until the end of inflation. The potential is inspiredby a Higgs inflation model where the local minimum isproduced by quantum corrections [23]. It is tuned to pro-duce PBHs with mass M PBH ∼ − M (cid:12) with an abun-dance that roughly agrees with the observed dark matterdensity in the Gaussian approximation. Contrary to [23],here the plateau is adjusted by hand to fit CMB obser-vations [1]. At the CMB pivot scale k ∗ = 0 .
05 Mpc − the spectral index is n s = 0 .
966 and the tensor-to-scalarratio is r = 0 . Squeezing and classicalisation.–
For the stochastic for-malism to be valid, the perturbations must be classicalby the time they join the background. Classicality can becharacterized by squeezing of the mode wave functions.A squeezed state can be written as [38, 86] | ψ (cid:105) = exp (cid:20) (cid:0) s ∗ ˆ a − s ˆ a † (cid:1)(cid:21) | (cid:105) , (5)where s = re iϕ is the squeezing parameter, and ˆ a, ˆ a † are standard ladder operators that satisfy (cid:2) ˆ a, ˆ a † (cid:3) = 1.They determine the vacuum state, ˆ a | (cid:105) = 0, with respectto which the squeezing is measured. The amplitude r indicates how squeezed the state is, and the phase ϕ givesthe squeezing direction in phase space.Choosing Q (cid:126)k = √ kaδφ (cid:126)k and P (cid:126)k = a Hδφ (cid:48) (cid:126)k / √ k for thecanonical variables that define the vacuum leads to theBunch–Davies vacuum for the sub-Hubble modes. Thecorresponding operators are related to the ladder opera-tors in the usual way, and we have (cid:10) ψ (cid:126)k (cid:12)(cid:12) ˆ Q (cid:126)k + ˆ P (cid:126)k (cid:12)(cid:12) ψ (cid:126)k (cid:11) = cosh(2 r k ) . (6)The value of r k is then a proxy for classicalization. Forthe Bunch–Davies vacuum, the mode initially has theminimum uncertainty wave packet, for which r k = 0, and r k grows as the phase space probability distribution getssqueezed. Large r k implies that the probability distribu-tion covers a large region in phase-space, where the ex-pectation value of the commutator [ ˆ Q (cid:126)k , ˆ P (cid:126)k (cid:48) ] = iδ ( (cid:126)k − (cid:126)k (cid:48) )is negligible compared to expectation values such as (cid:10) ψ (cid:126)k (cid:12)(cid:12) ˆ Q (cid:126)k ˆ P (cid:126)k (cid:48) + ˆ P (cid:126)k ˆ Q (cid:126)k (cid:48) (cid:12)(cid:12) ψ (cid:126)k (cid:11) . Thus, all relevant expectationvalues can be reproduced by a classical probability dis-tribution. Squeezing makes the operators ˆ Q (cid:126)k and ˆ P (cid:126)k highly correlated, so the field and momentum kicks be-come approximately proportional to each other. Notethat r k (cid:29) σ has to be small enough to ensure that the modeprobability distribution is sufficiently classical. However,the larger the value of σ , the more interactions betweenthe short and long wavelength modes we capture. Wechoose the value σ = 0 .
01 for which all modes satisfycosh(2 r k ) >
100 when they exit the coarse-graining scale.
Gauge-dependence.–
The perturbation equation of mo-tion (4) is in the spatially flat gauge, which is conve-nient for calculating the mode functions, whereas thestochastic equations (1), (2) for the background are inthe uniform- N gauge, as N does not receive kicks. Itwas shown in [77] that the correction to the mode func-tions when changing from the flat gauge to the uniform- N gauge is small both in SR and USR. We have checked nu-merically that in our calculation this holds at all times,including during transitions between SR and USR, so thegauge difference has negligible impact on our results.∆ N formalism.– We aim to calculate the coarse-grained comoving curvature perturbation R in a givenpatch of space, since this determines whether the patchcollapses into a PBH. We use the ∆ N formalism [85, 87–89], where R is given by the difference between the num-ber of e-folds N of the local patch and the mean numberof e-folds ¯ N , measured between an initial unperturbedhypersurface with fixed initial field value ¯ φ i and a finalhypersurface of constant field value ¯ φ f , R = N − ¯ N ≡ ∆ N . (7)When we solve the stochastic equations, we follow a patchof size determined by the coarse-graining scale k c = σaH ,which changes in time. The patch size at the end of thecalculation gives the PBH scale we probe; we fix this tothe value k PBH , which we discuss below. To ensure that k PBH gives the final patch size, we stop the time evolutionof k c once k c = k PBH . After this, no modes from δφ contribute to ¯ φ , so the stochastic noise is switched off, and modes with larger k do not give kicks. This makessense, since perturbations with wavelengths smaller thanthe size of the collapsing region should not affect PBHformation; they behave as noise that is averaged out inthe coarse-graining process. We continue to evolve thelocal background without kicks until the field reaches ¯ φ f .We record the final value of N for each simulation, andbuild statistics over many runs to find the probabilitydistribution P ( N ). Iterative Algorithm.–
We consider a discrete grid ofmodes with modulus evenly distributed on a logarithmicscale as ln( k i +1 ) = ln( k i )+0 . k = αaH , with α = 100 (the results are insensitive to mak-ing α larger). The longest wavelength mode we considercorresponds to the CMB pivot scale k ∗ , and its evolutionstarts immediately at the onset of each simulation. Foreach realization, the code executes the algorithm below: Algorithm 1:
Evolution for each runSet initial values for N , H , ¯ φ , ¯ π . while ¯ φ > ¯ φ f do Evolve H , ¯ φ , ¯ π one time step (without kicks). for k ∈ { k , k , . . . } doif k = αaH then Set initial values for δφ (cid:126)k , δφ (cid:48) (cid:126)k . if σaH < k < αaH then Evolve δφ (cid:126)k , δφ (cid:48) (cid:126)k one time step.Add stochastic kick to ¯ φ , ¯ π from the mostrecent mode with k < σaH , unless k > k PBH .We use an explicit Runge–Kutta method of order 4with fixed time step d N = 0 . − ¯ φ (cid:48) ) H = 2 V ( ¯ φ ), which is verified in eachsimulation up to a maximum relative error of order 10 − . PBH production.–
When a perturbation of wavenum-ber k re-enters the Hubble radius during the radiation-dominated phase after inflation, it may collapse into ablack hole of mass M = 43 πγH − k ρ k ≈ . × γ (cid:18) kk ∗ (cid:19) − M (cid:12) , (8)where M (cid:12) ≈ × g, γ ≈ . H k and ρ k are, respectively,the background Hubble parameter and energy density atHubble entry. We assume standard expansion history.Collapse occurs if the perturbation exceeds the thresh-old R c , which is of order unity [91–93]. We adopt R c = 1.The fraction of simulations where R > R c gives the ini-tial PBH energy density fraction β . Since PBHs behaveas matter, this fraction grows during radiation domina-tion, and today isΩ PBH ≈ × γ β (cid:18) MM (cid:12) (cid:19) − . (9)It is often assumed that R follows a Gaussian distribution[92, 94], with variance σ R = (cid:82) k PBH k IR d(ln k ) P R ( k ), where k IR is a cutoff corresponding roughly to the size of thepresent Hubble radius, and whose precise value makes nodifference to our results. The Gaussian approximationgives β = 2 (cid:90) ∞R c d R √ πσ R e − R σ R ≈ √ σ R √ π R c e − R c σ R . (10)Our example model is fine-tuned to give a substantialPBH abundance in the Gaussian approximation. Wewant to capture all the strong perturbations generatedduring USR, so we choose k PBH = e . k ∗ , which exitsthe Hubble radius at the end of USR, and correspondsto M = 1 . × g = 7 . × − M (cid:12) . PBHs of thismass can constitute all of the dark matter [95, 96]. Inthe Gaussian approximation we obtain σ R = 0 .
015 and β = 2 . × − . We then obtain from (9) the abun-dance Ω PBH = 0 .
13. However, we will see below thatthis Gaussian approximation severely underestimates thetrue PBH abundance.In reality, all PBHs will not have exactly the samemass. The mass distribution could be estimated by vary-ing k PBH . However, USR produces a sharp peak in theperturbations, corresponding to a strongly peaked distri-bution of PBH masses. To keep the discussion simple,we stick to the value M ∼ − M (cid:12) . Results.–
We have run 256 million simulations to findthe distribution P ( N ) of the number of e-folds betweenthe CMB pivot scale and the end of inflation, shown inFig. 2. The red solid line is the numerical result, andthe dotted black line is the Gaussian fit. The deviationfrom Gaussianity is evident for | ∆ N | (cid:38) .
5. Althoughstochastic kicks can either slow down or speed up thefield, the field is more likely to spend more time in theUSR region than to spend less time, so ∆ N is skewedtowards positive values. The Gaussian fit has variance σ R = 0 . β = 2 . × − .Our data reaches up to about ∆ N = 0 .
9, though theinterval ∆ N = 0 . . . . . N = 51 .
62. We estimate that resolving the tail of the dis-tribution beyond ∆ N = 1 would require 10 − timesmore simulations. To determine the PBH abundance, wefit an exponential to a resolved part of the tail and ex-trapolate. The black dashed line in Fig. 2 shows the best-fit P ( N ) = e A − BN to the data between ∆ N = 0 . N = 0 .
8. A jackknife analysis where we divide our datainto 20 subsamples gives the mean values and error esti-mates A = 1476 ± B = 28 . ± .
2. The mean and thebest-fit are very close. The extrapolated PBH abundance 51 51 . . − − − N P ( N ) DataGaussian fitExponential fit − . . N FIG. 2. The probability distribution for the number of e-folds.The bottom label shows the number of e-folds until the end ofinflation, the top label the deviation from the mean. The redsolid line is the numerical stochastic result, the black dottedline is a Gaussian fit to all points, and the black dashed line isan exponential fit to the tail. The collapse threshold ∆ N = 1is marked. is β = (cid:82) ∞ ¯ N +1 d N P ( N ) = B − e A − B ( ¯ N +1) = 1 . × − ,which corresponds to Ω PBH = 5 . × . Varying A and B to the edges of the error estimates changes these numbersby less than one order of magnitude. The difference fromthe Gaussian approximation for the PBH abundance to-day is a factor ∼ . Conclusions.–
Applying the ∆ N formalism to a work-ing model, we find that stochastic effects in USR generatean exponential tail in the probability distribution P ( R )of the curvature perturbation, as generally expected [79].Considering a model tailored to fit CMB observationsand to give roughly the observed dark matter abundancein PBHs (of mass M ∼ − M (cid:12) ) in the Gaussian ap-proximation, we find that stochastic effects during USRincrease the PBH abundance today by a factor of ∼ .Our results demonstrate that when considering PBHsseeded during USR, it is crucial to calculate the shapeof the tail of the probability distribution P ( R ), insteadof simply using the power spectrum P R based on the as-sumption that P ( R ) is Gaussian. Our calculation servesas a proof of concept that the Gaussian approximationcan underestimate the PBH abundance by orders of mag-nitude. A similar qualitative behaviour is expected in anyUSR scenario. The quantitative effect depends on howfar into the tail of the distribution the PBHs sample,growing with smaller PBH mass and abundance.Our results are sensitive to the value of σ , which givesan offset between the time a mode exits the Hubbleradius, and the time it is coarse-grained (when it ’kicks’the local background). In SR, modes freeze to an almostscale-invariant spectrum at super-Hubble scales, so thestochastic results are insensitive to the value of σ aslong as it is sufficiently small that modes have stoppedevolving [43] (but not too small [44, 46, 48, 72]). In USRthis is not the case, because the near scale-invarianceis lost and super-Hubble perturbations can also evolvelonger. The validity of our choice of σ (more generally,the form of the stochastic equation) should be checkedwith a first principle derivation of the separation betweensystem and environment in quantum field theory. Whilesuch derivations exist for stochastic inflation, none ofthe ones with explicit Langevin equations apply toUSR [45–47, 50–56, 59–70, 73, 74]. The dependenceon σ may suggest that USR is a more sensitive probeof decoherence and the quantum nature of inflationaryperturbations than SR.We thank Nuutti Auvinen for participation in test-ing our code, David Alonso for advice on the jack-knife analysis, and Vincent Vennin for discussions.DGF (ORCID 0000-0002-4005-8915) is supported by aRam´on y Cajal contract with Ref. RYC-2017-23493,and by the grant “SOM: Sabor y Origen de la Ma-teria” under no. FPA2017-85985-P, both from Span-ish Ministry MINECO. DGF also acknowledges hos-pitality and support from KITP in Santa Barbara,where part of this work was completed, supported bythe National Science Foundation Grant No. NSF PHY-1748958. This work was supported by the Estonian Re-search Council grants PRG803 and MOBTT5 and bythe EU through the European Regional DevelopmentFund CoE program TK133 “The Dark Side of the Uni-verse”. This work used computational resources pro-vided by the Finnish Grid and Cloud Infrastructure(urn:nbn:fi:research-infras-2016072533). [1] Y. Akrami et al. (Planck), Astron. Astrophys. , A10(2020), arXiv:1807.06211 [astro-ph.CO].[2] A. A. Starobinsky, Phys. Lett. , 99 (1980).[3] D. Kazanas, Astrophys. J. , L59 (1980).[4] A. H. Guth, Phys. Rev. D23 , 347 (1981).[5] K. Sato, Mon. Not. Roy. Astron. Soc. , 467 (1981).[6] V. F. Mukhanov and G. V. Chibisov, JETP Lett. , 532(1981), [Pisma Zh. Eksp. Teor. Fiz.33,549(1981)].[7] A. D. Linde, Second Seminar on Quantum GravityMoscow, USSR, October 13-15, 1981 , Phys. Lett. ,389 (1982).[8] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. ,1220 (1982).[9] S. W. Hawking and I. G. Moss, Phys. Lett. , 35(1982).[10] G. V. Chibisov and V. F. Mukhanov, Mon. Not. Roy.Astron. Soc. , 535 (1982).[11] S. W. Hawking, Phys. Lett. , 295 (1982).[12] A. H. Guth and S. Y. Pi, Phys. Rev. Lett. , 1110(1982).[13] A. A. Starobinsky, Phys. Lett. , 175 (1982). [14] M. Sasaki, Prog. Theor. Phys. , 1036 (1986).[15] V. F. Mukhanov, Sov. Phys. JETP , 1297 (1988), [Zh.Eksp. Teor. Fiz.94N7,1(1988)].[16] J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. , 47 (2017), arXiv:1702.03901 [astro-ph.CO].[17] J. M. Ezquiaga, J. Garcia-Bellido, and E. Ruiz Morales,Phys. Lett. B776 , 345 (2018), arXiv:1705.04861 [astro-ph.CO].[18] K. Kannike, L. Marzola, M. Raidal, and H. Veerm¨ae,JCAP , 020 (2017), arXiv:1705.06225 [astro-ph.CO].[19] C. Germani and T. Prokopec, Phys. Dark Univ. , 6(2017), arXiv:1706.04226 [astro-ph.CO].[20] H. Motohashi and W. Hu, Phys. Rev. D96 , 063503(2017), arXiv:1706.06784 [astro-ph.CO].[21] Y. Gong and Y. Gong, JCAP , 007 (2018),arXiv:1707.09578 [astro-ph.CO].[22] G. Ballesteros and M. Taoso, Phys. Rev.
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