primordial black holes as dark matter through higgs
TRANSCRIPT
YITP-19-51
Primordial Black Holes as Dark Matter through Higgs Criticality
Samuel Passaglia,1, ∗ Wayne Hu,1 and Hayato Motohashi2
1Kavli Institute for Cosmological Physics, Department of Astronomy & Astrophysics,Enrico Fermi Institute, University of Chicago, Chicago, IL 60637
2Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan(Dated: December 6, 2019)
We study the dynamics of a spectator Higgs field which stochastically evolves during inflationonto near-critical trajectories on the edge of a runaway instability. We show that its fluctuationsdo not produce primordial black holes in sufficient abundance to be the dark matter, nor do theyproduce significant second-order gravitational waves. First we show that the Higgs produces largerfluctuations on CMB scales than on PBH scales, itself a no-go for a viable PBH scenario. Then wetrack the superhorizon perturbations nonlinearly through reheating using the δN formalism to showthat they are not converted to large curvature fluctuations. Our conclusions hold regardless of anyfine-tuning of the Higgs field for both the Standard Model Higgs and for Higgs potentials modifiedto prevent unbounded runaway.
I. INTRODUCTION
Bottom-up naturalness suggests that new physicsshould appear at the TeV scale to stabilize the Higgsmass [1]. Unfortunately, no physics beyond the Stan-dard Model (SM) has yet been found by the LHC, norhave weakly interacting massive particles been found indark matter direct detection experiments.
If naturalness is abandoned as a guiding principle,then there may be no new physics intervening before thePlanck scale. If this possibility is taken seriously, thenthe SM exhibits the unusual property that the Higgs ef-fective potential turns over at large field values [2] andour electroweak vacuum is metastable, with a lifetimelonger than the age of our Universe [3].
This near-criticality of the Higgs has cosmological con-sequences [4–7], and it may also be that cosmology canhelp elucidate its origins and provide the solution to theHiggs naturalness problem [8–10].
In particular, Ref. [11] proposed that Higgs criticalitycould explain the existence of the dark matter within theSM as primordial black holes (PBHs). Furthermore thisscenario provides a potential anthropic reason why theHiggs should be near critical [12, 13]. Here the Higgsfield is a spectator during inflation and not the infla-ton itself if its coupling to gravity is not large (see, e.g.,[14–17]). Large Higgs fluctuations are formed during in-flation, and they are potentially converted to large su-perhorizon curvature perturbations after inflation due toHiggs criticality. Once they reenter the horizon duringradiation domination, PBHs of horizon scale mass formfrom their gravitational collapse (see Ref. [18] for a re-cent review) and in certain specific mass ranges providea viable dark matter candidate [19, 20].
To form PBHs in sufficient abundance to be the darkmatter, curvature perturbations must be amplified to∼ 10−1 on small scales while not violating the ∼ 10−5
constraints on CMB scales [21, 22]. We will focus in thispaper on both these conditions and carefully assess theviability of this scenario.
This paper is organized as follows: in §II we provide ageneral overview of the Higgs instability and the PBHproduction mechanism; in §III we compute the Higgspower spectrum produced during inflation on all observa-tionally relevant scales; in §IV we track these Higgs fluc-tuations nonlinearly through reheating to compute thecurvature fluctuations on which the PBH abundance de-pends; and we summarize our results in §V.
Throughout this paper, a prime ′ denotes a derivativewith respect to the e-folds N , where N = 0 marks the endof inflation, and an overdot ˙ denotes a derivative withrespect to the conformal time η. We assume a spatiallyflat background metric throughout and work in units inwhich MPl ≡ 1/
√8πG = 1 unless otherwise specified.
II. HIGGS INSTABILITY MECHANISM
In this section, we review the general features of theHiggs instability mechanism for producing PBH darkmatter and the principles governing the spectrum of per-turbations it generates.
The Higgs field acts as a spectator during inflation,which is driven by an inflaton field, but converts its quan-tum fluctuations into curvature fluctuations once it de-cays after inflation. If these quantum fluctuations areamplified into large enough curvature fluctuations by theHiggs instability, they will form PBHs when they re-enterthe horizon in the radiation dominated epoch.
In particular, after inflation the spatial metric on auniform total energy density hypersurface,
gij = a2e2ζδij , (1)
possesses a curvature perturbation ζ that can be decom-posed as
ζ =
(1− ρ′h
ρ′tot
)ζr +
ρ′hρ′tot
ζh, (2)
arX
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912.
0268
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in linear theory. Here ζr is the curvature perturbationon hypersurfaces of uniform energy density ρr of reheatproducts from the inflaton, and ζh is the curvature per-turbation on hypersurfaces of uniform Higgs energy den-sity ρh. Here ρtot ≡ ρr + ρh is the total energy density.
For this mechanism to succeed observationally, thevariance per logarithmic interval in k of the Fourier mod-efunction ζk,
∆2ζ(k) ≡ k3
2π2|ζk|2, (3)
must be small on the large scales probed by anisotropiesin the CMB,
∆2ζ(kCMB) ∼ 10−9, (4)
while on small scales associated with primordial blackholes the power spectrum must reach
∆2ζ(kPBH) ∼ 10−2, (5)
so that enough regions are over the collapse thresholdζc ∼ 1 to form PBHs in sufficient abundance to be thedark matter [21, 22]. Moreover, the large-scale modesshould be sourced predominantly by a single degree offreedom in order to comply with isocurvature constraintsfrom the CMB. The working assumption for a successfulmodel is that inflaton perturbations lead to a ζ whichdominates on CMB scales while Higgs fluctuations pro-duce one which dominates on PBH scales. We use asuperscript k to denote relations that are exclusively inFourier space as opposed to real space quantities or linearrelations that apply to both.
Although the Higgs field h is a spectator during infla-tion, the mechanism works by enhancing its impact onthe total ζ by exploiting the unstable, unbounded natureof the Higgs potential V (h) at large field values h > hmax
in the Standard Model. In particular, the effective Higgspotential at field values far larger than its electroweakvacuum expectation value can be approximated as [6, 12]
V (h) =1
4λh4, (6)
with λ = λSM and
λSM ' −b ln
(h2
h2max
√e
), (7)
where hmax is the location of the maximum of the Higgspotential which separates the familiar metastable elec-troweak vacuum from the unstable region, and b controlsthe flatness of the potential around the maximum. hmax
and b are computable given the parameters of the SM,and in this work, we choose to fix them at representativevalues hmax = 4 × 1012 GeV and b = 0.09/(4π)2, corre-sponding to a top quark mass Mt ' 172 GeV, followingRefs. [11–13] to facilitate comparisons. The Higgs insta-bility exists for Mt & 171 GeV [8], which includes therange from the most recent constraints by the Tevatron
2 3 4 5 6
h/H
−1.0
−0.5
0.0
0.5
1.0
V/V
max
UnstableStable hmax
FIG. 1. Beyond the field value hmax, the Standard ModelHiggs effective potential turns over and decreases rapidly, withan unbounded true vacuum of negative energy density. In thePBH scenario, the spectator Higgs rolls down the unstablepotential, amplifying its stochastic field fluctuations.
and the LHC [23–26]. Here we have neglected an effec-tive mass term for the Higgs generically generated by anonminimal Higgs coupling to the Ricci scalar, since atthe level expected from quantum corrections it does notchange the qualitative features of the mechanism [12].
We show the potential just around its maximum inFig. 1, and across a wider range of scales in Fig. 2 onthe unstable side in order to illustrate the field valuesthat will be crucial to the Higgs instability phenomenol-ogy detailed in the following sections. Specifically, fora representative choice for the Hubble scale during in-flation H = 1012 GeV which we employ throughout forillustration, the potential maximum is at
hmax = 4H. (8)
When h < hmax, the minimum corresponds to our famil-iar electroweak vacuum, while for h > hmax, the potentialdecreases and is unbounded from below.
On the unstable side of the potential, the Higgs at firstrolls slowly relative to the Hubble rate before acceleratingas it rolls down the instability. The location where theHiggs’ roll in one e-fold becomes comparable to H/2πdefines the classical roll scale hcl, which we shall defineprecisely in §III. For our parameter choices, it lies at
hcl ' 8.3H. (9)
At the scale kcl which crosses the horizon at Ncl ≡N(hcl), the power spectrum of ζh at horizon crossing be-comes order unity,
∆2ζh
(kcl) ∼ 1. (10)
Our working assumption is that the classical-roll scalekcl will be the one to produce primordial black holes,kcl = kPBH. We will show in §III that this implies thatthe Higgs is in fact on the unstable side of the potentialduring all phases of inflation relevant for observation. In
3
−10
−8
−6
−4
−2
0λ×
103
hm
ax
h60
hcl
hend
hre
scue
ms
101 102 103
h/H
−60
−40
−20
0
N
Ncl
SM BSM
FIG. 2. Quartic coupling λ and e-foldsN corresponding to theHiggs field position h, with marked special values as computedin §III and §IV: hmax, the maximum of the Higgs potential;h60, its position 60 e-folds before the end of inflation; hcl, itsclassical roll position; hend, its position at the end of inflation;hrescue, beyond which the SM Higgs cannot be “rescued” byreheating so that it rolls back and oscillates around the ori-gin after inflation. The BSM Higgs adds a coupled scalar ofsuitable mass ms to eliminate the runaway instability.
particular, the CMB scales left the horizon during in-flation a few e-folds after our Hubble patch crossed thehorizon, which we will assume is 60 e-folds before the endof inflation. At this time, the Higgs is on the unstableside of the potential at
h60 ' 5.8H, (11)
if we take the field value at the end of inflation to be
hend ' 1200H, (12)
which we will see below is approximately the largestvalue possible. This also leads to Ncl ∼ −20. If PBHsare produced on that scale then they have a small massMPBH ' 10−15M� at formation, and after mergers andaccretion could today lie in the region MPBH ' 10−12M�where all the dark matter could be in the form of PBHs[19, 20].
The Higgs continues to roll to larger field values untilthe end of reheating when interactions with the thermalbath lift the effective Higgs potential. If the Higgs lieswithin a maximum rescuable distance hrescue,
hrescue ' 2400H, (13)
which we compute in §IV, then after inflation it rollssafely back to the metastable electroweak vacuum,oscillating in a roughly quadratic potential with atemperature-dependent mass ∝ T ∝ 1/a until it decaysto radiation on a uniform Hubble surface.
The behavior of Higgs perturbations through reheatingis complicated. However, all relevant physical scales areat this stage far outside the horizon. So long as a gra-dient expansion holds, under which such perturbationscan be absorbed into an approximate FLRW backgroundfor a local observer, then the curvature field after horizoncrossing evolves locally, with no explicit scale-dependence[27]. Therefore the Higgs contribution to the fully nonlin-ear curvature field ζ on a uniform total density slice afterthe Higgs decays is related to the curvature ζh on con-stant Higgs slices on superhorizon scales during inflationby
ζ|decay = R(ζh)× ζh|inflation , (14)
with all of the complicated physics of reheating absorbedinto a local remapping R(ζh). More generally, this localremapping would also involve inflaton curvature fluctua-tions but these are statistically independent and can becalculated separately in the usual way. In linear theory,the mapping becomes a simple rescaling factor
R ≡ limζh→0
R(ζh). (15)
The power spectrum after Higgs decay is then relatedmode-by-mode to the power spectrum during inflation ina scale-independent fashion
∆2ζ(k)
∣∣decay
= R2 ∆2ζh
(k)∣∣inflation
, (16)
in which the number R encodes all of the details of re-heating.
It is therefore important to emphasize here that ifthe Higgs instability mechanism is successful, such that∆2ζ ∼ 10−2 on PBH scales after the Higgs decays, then
in linear theory R must reach at least 0.1 for the or-der unity Higgs perturbations (10) to be converted tocurvature perturbations with the correct amplitude (5)to form sufficient PBHs and the details of how this isachieved through reheating and Higgs decay are irrele-vant for the prediction of the linear power spectrum onother scales. In particular, its value on CMB scales de-pends solely on the inflationary ∆2
ζh(k). This form is
controlled by the Higgs potential itself and by the evolu-tion of the Hubble rate during inflation. Therefore, theviability with respect to CMB anisotropies of the PBHformation scenarios introduced in the literature [11–13],which all assume mode evolution can be calculated lin-early through reheating, can be assessed independentlyof the details of the reheating model.
On the other hand, we shall see that the nonlinear na-ture of the Higgs instability plays an important role in themapping between ζh and ζ in Eq. (14). Here, though themapping remains local in that a given value of ζh(~x) at a
4
given position ~x is mapped into a specific value of ζ(~x),Fourier modes no longer evolve independently. Instead,we will compute the mapping using the nonlinear δNformalism. This mapping does depend on the specifics ofhow inflation ends, but is independent of physical scale.CMB scale fluctuations are in principle calculable fromthe spatial field ζh(~x) determined by modes that frozeout during inflation.
To illustrate these concepts, we make a few simplifyingassumptions about how inflation and reheating proceed.We show in §III that the most optimistic case for thescenario occurs when H is effectively constant throughinflation (see (44)). Therefore rather than introducinga specific inflaton potential we assume that inflation oc-curs at a fixed H and ends after an appropriate numberof e-foldings. The constant Hubble scale during inflationH and the position hend of the Higgs at the end of infla-tion then together control the number of e-folds betweenthe classical-roll scale hcl and the end of inflation, andtherefore they control the physical scales on which PBHsare formed.
We then assume that at the end of inflation, the in-flaton decays instantly into radiation and that the Higgslater also suddenly decays into radiation, as in the modelproposed in Ref. [11]. Maximizing R in linear theoryrequires that the position of the Higgs at the end ofinflation, hend, is as close as possible to the maximumrescuable distance hrescue. This criticality requirementmotivates the various choices of scale in Eqs. (8)-(12),following Ref. [11]. Once hend is set in this way, the valueH = 1012 GeV is chosen to give a certain mass scale toPBHs by fixing Ncl ∼ −20.
However, evolving the Higgs on the unstable side ofits potential during inflation is dangerous, and the re-quired proximity of hend to hrescue aggravates the situa-tion beyond linear theory. Due to quantum fluctuationsof the Higgs during inflation, there are regions in whichthe local Higgs value at the end of inflation exceeds thebackground value, overshoots hrescue, and cannot be re-stored by reheating to the metastable electroweak vac-uum created thermally. Such vacuum decay bubbles,with infinitely growing |ρh|, expand even after the endof inflation and eventually engulf our current horizon.These quantum fluctuations occur independently in thee120 causally disconnected regions at Ncl ∼ −20 whichmake up our current horizon, and therefore avoiding thevacuum decay bubbles requires extreme fine-tuning [12].In §IV D, we will cast this fine-tuning in terms of a break-down in linear theory at the end of inflation, and we willshow using the nonlinear δN formalism that fine-tuningaway the vacuum decay bubbles directly tunes away thePBH abundance.
Vacuum decay bubbles can be avoided by stabilizingthe Higgs at some large field value between hend andhrescue. By adding a singlet heavy scalar of mass ms
with appropriate couplings to the theory, a threshold ef-fect can be exploited to lift the Higgs effective poten-tial during inflation and induce a new true minimum at
h ∼ ms, preventing unbounded runaway [13, 28]. For thepurposes of this mechanism this Higgs potential beyondthe Standard Model can be modeled as
λBSM ' λSM +δλ
2
[1 + tanh
(h−ms
δ
)], (17)
such that for h � ms the potential is as in the SM,given in (7), while for h� ms the potential is increasedby δλ(h4/4). The step height δλ should be such thatthe Higgs potential is stabilized, the step position ms
should be close to hrescue, and the step width δ sufficientlynarrow to not interfere with hend. In Fig. 2, we plot thispotential with the representative choices {δλ,ms, δ} ={0.02, 2000H, 100H}. While the BSM potential doesnot suffer from vacuum decay bubbles, it still experiencesa breakdown in linearity at the end of inflation. We willtherefore also use the nonlinear δN formalism to computethe conversion of ζh to ζ in this case.
Despite this difference at the end of inflation, the SMand BSM potentials are identical until large field val-ues and therefore ∆2
ζhduring inflation is the same in
both potentials. Fluctuations at this stage can be locallyremapped onto ∆2
ζ . For a successful PBH model, this
remapping must still achieve ∆2ζ ∼ 10−2 in both cases.
We will therefore focus on the SM potential until we be-gin discussing nonlinear effects at the end of inflation in§IV.
III. INFLATIONARY HIGGS SPECTRUM
In this section, we compute the power spectrum of theHiggs fluctuations from their production inside the hori-zon through to a common epoch when all modes relevantfor observation are superhorizon in scale.
In §III A, we present the equation of motion forthe Higgs and describe the local competition betweenstochastic kicks and classical roll which governs its evolu-tion. In §III B, we argue that a well-defined backgroundfor the Higgs exists during the inflationary epochs rele-vant for observations, and that Higgs fluctuations dur-ing inflation can be computed by linearizing around thisbackground mode-by-mode. In §III C, we follow this pro-cedure and compute the Higgs power spectrum duringinflation at all scales relevant for observations, regardingthe Higgs field as a spectator and hence dropping metricperturbations. In App. A, we show that our results areconsistent with the creation of our background by super-horizon stochastic kicks. In App. B we discuss the roleof metric perturbations and nonadiabatic pressure in theevolution of Higgs fluctuations.
Combined, these arguments will show that during in-flation the Higgs power spectrum at CMB scales is largerthan the Higgs power spectrum at the primordial blackhole scales
∆2ζh
(kCMB) > ∆2ζh
(kPBH). (18)
5
After the conversion of these superhorizon Higgs fluc-tuations during inflation into curvature fluctuations afterinflation through Eq. (14), this leads to
∆2ζ(kCMB) > ∆2
ζ(kPBH), (19)
in linear theory. Accounting for nonlinearity, we shall seethat a similar relation between the scales holds as longas the mapping between the Higgs and curvature fluctua-tions is local. Therefore the first conclusion of the presentpaper is that in the Higgs vacuum instability scenario,a large amplitude of the power spectrum on small scalesgenerating PBHs is ruled out by the CMB normalization.Conversely, if one chooses a different set of parameters inthis scenario in order to satisfy the CMB normalization,one ends up with a small-scale power spectrum of at mostO(10−9) which fails to form PBHs.
A. Classical Roll vs Stochastic Kicks
The equation of motion for the position- and time-dependent Higgs field h(~x,N) is the Klein-Gordon equa-tion
�h(~x,N) =∂V
∂h
∣∣∣∣h(~x,N)
≡ V,h|h(~x,N) , (20)
where here and throughout we denote partial derivativeswith comma subscripts for compactness.
An important scale in this equation is the classical-rollscale hcl, defined as follows. Every e-fold, the potentialderivative leads h(~x,N) to roll by
∆h ' − 1
3H2V,h|h(~x,N) . (21)
Meanwhile, if one splits the field into a piece averagedon scales larger than a fixed proper distance ∼ 1/H andsmall scale modes which continually cross the averagingscale, the small scale modes can be viewed as providing alocal stochastic noise term to the equation for the coarse-grained superhorizon field [29, 30]. The rms of this noiseterm each e-fold is
〈∆h2〉 12 ' H
2π. (22)
In the language of perturbation theory, this is the pere-fold rms of the free field fluctuation δh and leads to astochastic behavior of h(~x,N). There are no subtletiesinvolved in usingN as a time coordinate since the numberof e-folds is not a stochastic quantity so long as the Higgsremains a spectator.
The location hcl in the potential where the roll contri-bution and the stochastic contribution are equal,
− 1
3H2V,h|hcl
=H
2π, (23)
defines the ‘classical roll’ scale hcl beyond which the clas-sical term dominates the evolution of h(~x,N). We showthis scale in Fig. 2, where it lies at hcl ' 8.3H.
The classical-roll scale is important because in slow-roll, Higgs modes which cross the horizon when the back-ground satisfies Eq. (23) generically have a large powerspectrum. In particular, the Higgs power spectrum atthe scale kcl which crosses the horizon at Ncl = N(hcl) isorder one at horizon crossing,
∆2ζh
(kcl) ∼〈∆h2〉(∆h)2
∼ 1. (24)
If these Higgs fluctuations are converted to large curva-ture fluctuations, they can satisfy the requirements of §IIsuch that kcl = kPBH.
B. Background and Linearization
We split h(~x,N) equation into a background and per-turbations
h(~x,N) = h(N) + δh(~x,N). (25)
Here we define the background to be the part of the fieldrepresenting the spatial average over our Hubble patch.Therefore at N ∼ −60, the spatial average for the per-turbation vanishes, 〈δh(~x,−60)〉 = 0. The fluctuationsare then generated by kicks from quantum fluctuationsat N > −60.
To evolve the Higgs field under Eq. (20) we need toevaluate its position on the potential after N = −60,as established by its classical roll or quantum kicks. Todo this, we need to establish whether the perturbationsδh(~x,N) are linear around the background h(N).
For the mechanism to work, there should be a well-defined classical roll to the Higgs field at Ncl, and we canlinearize the Higgs fluctuations at that epoch as usual[11, 12]. Between −60 . N . Ncl, there is a competitionbetween the local stochastic kicks and the bulk classicalroll, and we need to check whether the kicks destabilizethe average field in our Hubble patch.
In linear theory, stochastic kicks at the same ~x butsubsequent times evolve independently from each other.In particular, the potential term in the Klein-Gordonequation controls their interactions. When the Higgs is aspectator field, and we can expand this term around thehomogeneous piece as
V,h|h(~x,N) = V,h|h(N) + δh(~x,N) V,hh|h(N) + . . . (26)
where ‘. . .’ contains terms higher order in δh(~x,N). If weneglect the higher order terms, then each subsequent kickevolves as a free field and, as previously mentioned, hasrms H/2π at horizon crossing. The higher order termsthen are suppressed relative to the linear term by
1
2
V,hhhV,hh
√〈δh2〉 ' 1
2
V,hhhV,hh
H
2π, (27)
in which we approximate all modes by their value at hori-zon crossing, which is appropriate while both the inflaton
6
101 102 103
h/H
10−5
10−4
10−3
10−2
10−11 2V,hhh
V,hh
H 2π
hm
ax
h60
hcl
hend
FIG. 3. Interaction strength for modes of the typical horizoncrossing amplitude H/2π from (27). Since it is far less thanone at the relevant scales h60 and hcl, modes evolve indepen-dently and the cumulative background roll (31) dominatesover the stochastic displacement (28). Linear perturbationtheory holds until δh grows much larger than H/2π, nearhend (see §IV D).
and the Higgs fields are slowly rolling. We plot this quan-tity for the Standard Model Higgs potential in Fig. 3 andshow that it is less than one at all scales in the unstableregion.
Therefore, the distance traveled due to stochastic kicksbetween N = −60 and Ncl accumulates as a randomwalk. For Ncl = −20, they therefore lead to a displace-ment
|∆h|stochastic =H
2π×√
40 ' H. (28)
This is significantly less than the distance between theclassical-roll scale hcl = 8.3H and the maximum of thepotential hmax = 4H. This means that stochastic kicksdo not, over 40 e-folds, kick our horizon into the otherside of the potential. Therefore our whole Hubble volumewas on the unstable side of the potential when it crossedthe inflationary horizon at N = −60.
Moreover, the total displacement from stochastickicks (28) is less than the amount the background fieldrolls in these 40 e-folds as we shall now see. For a homo-geneous field h(N), Eq. (20) becomes
h′′ + (3− εH)h′ +V,hH2
= 0, (29)
where εH = −H ′/H is the first Hubble slow-roll param-eter, which is zero during the exact de Sitter inflation inour fiducial model.
The initial conditions h60 and h′60 for this equationshould be such that the Higgs reaches the desired field po-sition hend at the end of inflation close enough to hrescue
such that R is maximized. With one constraint andtwo initial values, a range of h′60 and h60 can lead toh(N = 0) = hend. Assuming attractor initial conditions
for the Higgs, we choose
h′60 = − 1
(3− εH)H2V,h|h60
, (30)
making the initial field position given by Eq. (11) h60 '5.8H, when hend is set by Eq. (12).
Therefore the classical roll from −60 to Ncl is
|∆h|roll = |h60 − hcl| = 2.5H, (31)
which is significantly larger than the stochastic displace-ment (28) but is nonetheless safely on the unstable sideof the potential. This occurs despite the fact that eachstochastic kick is larger than the per e-fold roll becausethe roll is coherent across our Hubble volume while thekicks are random.
Therefore we have a consistent picture where if we be-gin with an average field in our horizon volume aroundh60 ∼ 5.8H, then our local background will reach hcl atN ∼ −20, unspoiled by stochastic kicks. Between thesescales perturbations are linear, thanks to Fig. 3, and sub-dominant over the background roll. Our background willthen continue to roll to hend, where the Higgs will beuplifted. We plot this background in the lower panel ofFig. 2. In App. A we show that this picture is consistentwith the creation of our background from superhorizonstochastic fluctuations.
We can now use this background to solve for δh modeby mode during inflation for all relevant observationalscales as in linear theory. This linearization depends onignoring the interaction of Higgs fluctuations rather thanthe full machinery of linear perturbation theory for themetric and the matter, and in particular its validity doesnot assume |ζh| � 1. Higgs nonlinearities become im-portant in the last e-fold of inflation and beyond as thefield fluctuations are amplified by the Higgs instability.Such nonlinear effects will affect the superhorizon CMBand PBH modes equally as we shall show in §IV.
C. Higgs Power Spectrum
The linearized Klein-Gordon equation for the Fouriermode δhk(N) of δh(~x,N) in spatially flat gauge is(
d2
dη2+ 2
a
a
d
dη+ k2
)δhk + a2δV k,h ' 0, (32)
where δV k,h = V,hhδhk during inflation. Here we have
dropped metric perturbations, which are suppressedwhen the Higgs is a spectator; we restore these in App. Bfor completeness (see (B4)).
The Klein-Gordon equation can then be convenientlyexpressed in terms of the auxiliary variable uk ≡ aδhk,
uk +
(k2 − a
a+ a2V,hh
)uk = 0, (33)
7
which holds at all orders of background and Higgs slowroll parameters. To order O(ε2H) and O(εHηH), where ηHis the second Hubble roll parameter (see, e.g., Ref. [31]),but fully general in terms of the Higgs roll, we can write
uk +
(k2 − z
z
)uk = 0, (34)
where
z ≡ Hh. (35)
This equation, of the Mukhanov-Sasaki type, is conve-niently solved in the variable s ≡ ηend − η, the positivedecreasing conformal time to the end of inflation (see,e.g., Refs. [32, 33]).
First, let us focus on the evolution in the superhori-zon regime. In that limit, the analytic solution for theMukhanov-Sasaki equation (34) is given by
uk
z= c0 + c1
∫dη
z2, (36)
where c0 and c1 are constants. So long as the secondmode is decaying, we therefore have that on superhorizonscales
δhk
h′= c0H
2 +O(εH , ηH). (37)
In linear theory, the curvature perturbation on uniformHiggs density slices is obtained by gauge transformationas
ζkh = −δρkh
ρ′h' −δh
k
h′, (38)
where first the approximate equality indicates that whenthe Higgs is slowly rolling, uniform Higgs density anduniform Higgs field slicing coincide to order εH (seeApp. B). More generally ζh is defined as the change ine-folds from a spatially flat surface to a constant densityHiggs surface. This linear approximation holds so longas ρ′′h/ρ
′2h δρh � 1, as it is here (see Eq. (59)).
Using the superhorizon evolution equation (37), wetherefore have that if H evolves during inflation, the cur-vature on uniform Higgs density slices is not conservedon superhorizon scales and in particular decays accordingto
ζkh′
ζkh= −2εH , (39)
at leading order in εH . This estimate of superhorizonevolution assumes only background slow roll.
This superhorizon evolution is due to a pressure per-turbation on uniform density slices for the Higgs, in otherwords a nonadiabatic pressure, induced because the uni-form Higgs density slicing is not a uniform Hubble slic-ing when εH 6= 0. We study this phenomenon in detailin App. B. Conversely, if H is constant, then the fact
that the Higgs field evolves onto an attractor solutionimplies that nonadiabatic stress vanishes thereafter andζh is conserved nonlinearly. In this case, much like single-field slow-roll inflation, the Higgs field supplies the onlyclock and field perturbations are equivalent to changingthat clock on the background trajectory.
Next, let us focus on the evolution from subhorizonscales to the superhorizon regime. This evolution can betracked by solving the Mukhanov-Sasaki equation (34)with Bunch-Davies initial conditions deep inside the hori-zon
uk(s) =1√2k
(1 +
i
ks
)eiks. (40)
For analytic estimates, we can assume slow-roll evolutionof z, in which case we can take the de Sitter modefunction(40) to the superhorizon limit, and find that each fieldfluctuation crosses the horizon with amplitude
δhk ' iH√2k3
, (41)
and the field fluctuation power spectrum at that time is
∆2δh(k) =
k3
2π2
∣∣δhk∣∣2 ' (H2π
)2
. (42)
Using the gauge transformation Eq. (38) with the fieldfluctuation Eq. (41) and the field velocity from the slow-roll solution of Eq. (29), the curvature perturbation onuniform Higgs density hypersurfaces at horizon crossingis
ζkh(ηk) ' − iH√2k3
1
h′
∣∣∣∣ηk
' − iH√2k3
(3− εH)H2
−V,h
∣∣∣∣ηk
. (43)
To lowest order in Higgs- and background-slow-roll ηkis chosen to be the epoch of horizon-crossing kηk =1, but to next order can be optimized to kηk =exp [7/3− ln 2− γE ], with γE the Euler-Mascheroni con-stant [32, 33].
We can now estimate the relative amplitude of ∆2ζh
on CMB and PBH scales. Choosing some comparisontime η∗ once both scales have exited the horizon but farenough from the end of inflation that slow-roll parame-ters are still small, we have
∆2ζh
(kCMB)
∆2ζh
(kPBH)
∣∣∣∣∣η∗
'(HPBH
HCMB
)4 k3CMB
∣∣∣ζkCMB
h
∣∣∣2ηk
k3PBH
∣∣∣ζkPBH
h
∣∣∣2ηk
'(HCMB
HPBH
)2(V,h|PBH
V,h|CMB
)2
, (44)
where we have used at horizon crossing the Higgs slow-roll expression (43) and outside the horizon the Hubble
8
10−3 101 105 109 1013 1017
k
10−1
100
101
102∆
2 ζh
kPBH
kCMBExact
Optimized Slow Roll
FIG. 4. The Higgs power spectrum during inflation com-puted as described in §III C on a uniform Higgs energy den-sity slice by exact solution of the Mukhanov-Sasaki equation(34) (solid blue) and by the optimized slow-roll approximation(43) (dashed red), at some time η∗ after the relevant modeshave crossed the horizon in the optimistic scenario where His constant until η∗. The Higgs power spectrum is larger onCMB scales than on the classical-roll scales.
slow-roll expression (37). Thus in the generic situationwhere H is decreasing and the Higgs rolls downhill, wefind that ∆2
ζh(kCMB)/∆2
ζh(kPBH) > 1. The most opti-
mistic case for the scenario is therefore the one where His strictly constant between the CMB and PBH scales,and it still results in ∆2
ζh(kCMB)/∆2
ζh(kPBH) > 1.
We show in Fig. 4 the Higgs power spectrum computedby solving the Mukhanov-Sasaki equation exactly in theoptimistic case where H is constant between CMB andclassical-roll scales, evaluated at the convenient time η∗when all relevant modes have crossed the horizon. Wecompare this exact solution to the slow-roll expression(43) with the optimized freeze-out epoch. For decreasingH, the ratio ∆2
ζh(kCMB)/∆2
ζh(kPBH) would be larger than
the one estimated from Fig. 4.These Higgs fluctuations at η∗ will be converted to
a total curvature fluctuations after Higgs decay by thefactorR(ζh)2 which we discuss in §IV. The key is that thismapping affects all mode contributions to ζh uniformly.For example in linear theory R is a constant whose valuemust be ∼ 0.1 for successful PBH formation at kPBH.Eq. (44) determines the total curvature power relative tothis scale. In particular, the power spectrum at CMBscales is an order of magnitude larger than the powerspectrum at the classical-roll scale. It is simply a featureof the Higgs potential that the field slope increases as theHiggs goes farther into the unstable region, and thereforethat the Higgs fluctuation shrinks as k increases. Thus, amodel with ∆2
ζh(kCMB)/∆2
ζh(kPBH) > 1 that forms PBHs
at kPBH will necessarily violate CMB constraints.The results of this section hold equally for the SM and
BSM potential. The difference between the two poten-tials enters only into R(ζh) which converts these resultsinto the final curvature perturbation after Higgs decay.More generally as long as this mapping depends only on
field amplitude and not on k explicitly, PBHs cannot beformed from the Higgs instability without violating CMBconstraints.
∗ ∗ ∗
In summary, we have shown that during inflation
∆2ζh
(kCMB) > ∆2ζh
(kPBH). (45)
As we argued in §II, the conversion of the inflationary ζhto the final ζ depends only on the amplitude of ζh andthus all the information about reheating can be encodedin a scale-independent function R(ζh).
This means that in linear theory, where R is a constant,if primordial black holes are produced on small scalesthen on large scales
∆2ζ(kCMB) > ∆2
ζ(kPBH) & 10−2, (46)
which is incompatible with measurements of the CMB.Nonlinearly, when CMB and PBH modes cannot be
tracked independently through the final e-folding of in-flation and reheating, the ∆2
ζh(k) results in this section
provide the superhorizon initial conditions which can bemapped to the final ζ. Given that this local mappingR(ζh) does not distinguish between Higgs fluctuationsof different physical scales, we will argue in §IV thateven nonlinearly, Higgs induced curvature fluctuationson CMB scales will be larger than those on PBH scales.
We now study in §IV the specific values taken by theconversion function R(ζh) itself. This will allow us to de-termine whether or not the Higgs fluctuations computedhere can be transferred into large enough curvature per-turbations to form PBHs, regardless of the compatibilitywith the CMB. Moreover, given that we have producedlarge Higgs fluctuations on CMB scales, this will allowus to determine under which conditions Higgs critical-ity is incompatible with the small curvature fluctuationsobserved in the CMB.
IV. CURVATURE FLUCTUATIONS FROMREHEATING
We now track the curvature perturbations ζh on uni-form Higgs density slices during inflation, computed in§III, through the end of inflation, reheating, and Higgsdecay to compute the final curvature perturbations ζ onuniform total density hypersurfaces relevant for PBH for-mation.
In §IV A, we discuss how the Higgs evolves near theend of inflation and present the basic features of the in-stantaneous reheating model proposed by Ref. [11].
In §IV B, we discuss how to use the nonlinear δN for-malism to convert ζh to ζ for any local reheating sce-nario, and we discuss jump conditions which much besatisfied during instantaneous reheating. We also presentlinearized δN formulae which yield results corresponding
9
−1.0 −0.5 0.0 0.5 1.0
N
10−11
10−9
10−7
10−5
10−3−ρh/(1
2H
2)
SM
BSM
FIG. 5. The (negative) Higgs energy density evolution duringinflation, with H = const. N = 0 corresponds to the endof inflation for the background, and N > 0 shows how theHiggs energy would evolve in local regions in advance of thebackground. The SM Higgs (dashed blue) at N = 0 is closeto saturating the rescue condition (51) (blue star). The BSMHiggs (solid red) hits a wall during inflation and thereforeis always rescued. Due to the Higgs attractor behavior, theexact value of ζ(0+) for any given local shift δN = ζh can beread off using Eq. (57).
to those of linear perturbation theory, allowing an im-portant crosscheck of the computation.
In §IV C, we follow the assumption of Refs. [11–13, 34]that linear theory holds through reheating and we com-pute explicitly the conversion of the inflationary ζh tothe final ζ. We show that energy conservation at reheat-ing, neglected in previous works, prevents the model fromachieving the required R = 0.1 for both the SM and BSMpotentials and thus PBH are not produced in sufficientquantities to be the dark matter in linear theory.
In §IV D we show that linear theory is in fact violatedat the end of inflation and we explicitly compute the fullnonlinear conversion R(ζh) for the SM and BSM Higgseffective potentials. We show that PBH DM is neverformed, second-order gravitational waves are suppressed,and only for a special class of criticality scenarios canobservable perturbations be produced on CMB scales.
A. Instantaneous Reheating
As the Higgs travels farther and farther on the unstableside of the SM potential, it rolls faster and faster and ifinflation never ended its energy density would diverge infinite time. In Fig. 5, we show this ρh during this lastphase of the instability as a function of the number ofe-folds. Of course inflation does end and for our chosenexample this occurs at N = 0 and a field value hend inthe background. In this case, N > 0 then shows howthe Higgs energy would continue to evolve if inflation didnot end at N = 0. This range around N = 0 will also beuseful when we consider perturbations that can be aheadof or behind the background value.
As we can see, the fiducial position of the backgroundSM Higgs we have chosen is near-critical. Its energy isincreasing rapidly and if inflation lasts much longer it willgain sufficient energy such that it is no longer a spectator.On this edge, the background Higgs field experiences thesame evolution in the SM and BSM cases by construction(see Fig. 5).
Once inflation ends, the process of reheating transfersthe inflaton’s energy to radiation, which in turn transfersenergy to the Higgs by uplifting the Higgs potential to[6]
V T = V +1
2M2
Th2e−h/2πT , M2
T ' 0.12T 2, (47)
with a thermal bath temperature
T =
(30ρr
π2g∗
)1/4
, (48)
where g∗ = 106.75 is the number of degrees of freedom inthe Standard Model and ρr is the radiation density withthe usual equation of state
ρ′r = −4ρr. (49)
If reheating is instantaneous, as proposed by Ref. [11],then the total energy is conserved,
(ρφ + ρh) (0−) = (ρr + ρh) (0+). (50)
Since 3H2(0−) = (ρφ + ρh)(0−), neglecting the Higgs’energy density contribution to H during inflationbut including it after entails a dynamically negligibleO(ρh/ρφ) ∼ O(ρh/3H
2) discontinuity in the Hubble rateat the end of inflation Hend (see Fig. 5). On the otherhand, strict energy conservation (50) at reheating is im-portant because we will evaluate perturbations on con-stant density surfaces.
The new thermal term in Eq. (47) will rescue the Higgsfrom the unbounded SM minimum so long as
Max [h] < hrescue, (51)
where Max [h] is the maximum displacement of the Higgsfield and hrescue is the peak of the uplifted potential V T .Due to the nonzero kinetic energy of the Higgs at the endof inflation, Max [h] is larger than the field displacementat the end of inflation hend. We mark the maximal pointwhich saturates this bound with a star in Fig. 5.
Neglecting the exponential term, we find that the peakof the uplifted potential at reheating is
h(0)rescue =
MT√|λSM|
, (52)
a solution which can be iterated to account for the expo-nential term, yielding
h(1)rescue = h(0)
rescuee−h(0)
rescue/4πT ∼ 1.6√HendMPl, (53)
10
where we have used |λ| ∼ 0.007 and which with Hend =1012 GeV evaluates to ∼ 2500H. The value of the up-lifted Higgs potential at this approximate maximum is
V T (h(1)rescue) ∼ 0.02H2
endM2Pl. (54)
These scalings are in good agreement with the exact cal-
culation for h(0)rescue/4πT � 1 and they serve to highlight
the dependence of the results with parameter choices. Forour fiducial parameter set, the exact calculation yieldshrescue ' 2400Hend, V T (hrescue) ' 0.02H2
endM2Pl.
If the Higgs is rescued, then it oscillates in its up-lifted temperature-dependent potential, redshifting as ra-diation on the cycle-average up to corrections from thenon-quadratic components of its potential, until it decayson the e-fold timescale on constant Hubble surfaces. Therescue point is therefore relevant even for the BSM po-tential. In our example shown in Fig. 5, we set the ms
barrier close to hrescue to maximize the instability whileensuring that the field returns to the electroweak vacuumafter reheating.
We now describe in §IV B how to track perturbationsthrough the end of inflation and this instantaneous re-heating epoch.
B. Nonlinear Curvature Evolution
The PBH abundance depends on the probability thatthe local horizon averaged density field exceeds some col-lapse threshold δc. We approximate this by the Gaussianprobability that the curvature on uniform total densityslices ζ lies above some threshold ζc. We therefore needto compute ζ after the Higgs decays.
Nonlinearly in the Higgs field perturbations, the trans-formation of the curvature on uniform Higgs density slic-ing during inflation ζh to the curvature on uniform totaldensity slicing after inflation ζ can be performed in theδN formalism [35–38], which allows us to evolve super-horizon perturbations by counting the number of e-foldsof expansion from an initial flat slice at some convenientinitial time Ni to a uniform total density slice at a finaltime N ,
ζ(N,~x) = N (ρα(Ni, ~x); ρtot(N))−N (ρα(Ni); ρtot(N)),(55)
where N is the local number of e-folds of expansion fromthe initial flat hypersurface at Ni on which any fields αhave energy densities ρα(Ni, ~x) = ρα(Ni) + δρα(Ni, ~x) toa final surface of uniform total density ρtot(N), and Nis the corresponding expansion of the unperturbed uni-verse. Using the separate universe assumption, N can becomputed in terms of background FLRW equations for auniverse with the labeled energy contents.
If we chose Ni to be some time during inflation whenwe know the superhorizon density fluctuation δρh (or ζh)from our computation in §III, then since we are consider-ing only perturbations sourced by the Higgs we can make
the inflaton density at Ni implicit and keep only the de-pendence on the initial δρh. We will perform this fullygeneral calculation in §IV D.
To understand these results, it is also useful to have asimple analytic approximation for the impact of reheatinggiven conditions just before reheating. In this case we canset the initial time just after reheating at Ni = 0+. TheHiggs energy density is a small component of the totalenergy budget, and therefore to leading order in ζ wecan evaluate the δN formula assuming that ρtot ∝ a−4
to find
ζ(0+, ~x) ' 1
4ln
(ρtot(0
+, ~x)
ρtot(0+)
)' ρtot(0
+, ~x)− ρtot(0+)
12H2end
. (56)
Conservation of energy at reheating (50) then implies theN = 0 jump condition
ζ(0+, ~x) =ρh(0−, ~x)− ρh(0−)
12H2end
. (57)
Note that this condition applies to nonlinear Higgs den-sity fluctuations |(ρh − ρh)/ρh| � 1 so long as |(ρh −ρh)/ρtot| � 1.
We can further simplify this condition by noting thatto the extent that H is constant during inflation, whichwe have shown by (44) is the most optimistic scenario forPBH production, the shift in efolds to a constant Higgsenergy density δNh = ζh is conserved nonlinearly.*1
Therefore ρh(0−, ~x) = ρh(−δNh), with this Higgs den-sity computed as though inflation did not end at N = 0as in Fig. 5. We can therefore read off ζ(0+, ~x) for a givenζh from ρh(N) as
ζ(0+, ~x) =ρh(−ζh)− ρh(0−)
12H2
∣∣∣inf, (58)
where |inf denotes this convention of evaluating the back-ground as if inflation never ends.
Before using these nonlinear formulae (55) and (58)in §IV D, we will in §IV C perform the calculation usinglinear perturbation theory. To validate the linear theorycalculations, below we derive linear approximations tothe δN formulae.
The full δN formula (55) can be linearized in δρhi≡
δρh(Ni, ~x) to obtain
ζ ' ∂N (ρhi; ρtot)
∂ρhi
δρhi' −∂N (ρhi
; ρtot)
∂ρhi
ρ′hiζhi, (59)
where ρhiand the ρ′hi
are Higgs density and its deriva-tive at Ni and ζhi
is the Higgs curvature at that time.
*1 If H evolves, the leading order effect will be simply to shrink theHiggs δNh.
11
Likewise, a linear Taylor expansion of the jump condition(58) is given by
ζ(0+, ~x) ' −ζh(0−)× ρ′h(0−)
12H2. (60)
The ratio −ρ′h(0−)/12H2 is the rescaling factor R(0+) ifH is constant through to the end of inflation.
As we shall see below, these linear δN formulae providean important point of contact between the nonlinear δNand linear perturbation theory approaches.
C. Linear Conversion
We now follow the assumption of Refs. [11–13, 34] thatlinear theory holds through reheating, and we show thatunder this assumption primordial black holes cannot bethe dark matter.
While Higgs field values h and δh and their derivativesh′ and δh′ are all continuous through reheating, the Higgspotential and its slope change instantaneously when theHiggs potential is uplifted. Therefore the Higgs energydensity,
ρh =1
2
h2
a2+ V T , (61)
its derivative
ρ′h = −3h2
a2− V T,TT, (62)
and its perturbation
δρh =1
a2hδh+ V T,h δh+ V T,T δT, (63)
are not continuous with their values at N = 0−. δT hereis any perturbation in the bath temperature correlatedwith the Higgs, which we shall see is generically inducedat reheating. For simplicity, we have omitted here a con-tribution to the perturbed energy density coming fromthe metric lapse perturbation, which we restore in App. B(see (B35)). Its relative contribution is negligible.
The jump in ρ′h
∆ [ρ′h] = −V T,TT, (64)
and in the energy density perturbation
∆ [δρh] = δh(V T,h − V,h
)+ V T,T δT, (65)
imply that the curvature perturbation (38) on constantHiggs energy density slices is discontinuous at reheating.This instantaneous change in ζh is due to an instanta-neous source in the conservation equation from the inter-action of the Higgs with the thermal bath.
However, the instantaneous increase in the Higgs en-ergy density perturbation δρh does not come for free.Conservation of energy, which we imposed at the level of
the background in Eq. (50), also holds locally. It impliesthat the increase in the Higgs energy density is coun-terbalanced by an induced perturbation in the radiationfield
δρr(0+) = −∆ [δρh] , (66)
and therefore that the Higgs and radiation energy densi-ties after uplift are nearly cancelling. In other words, theuplift creates a Higgs-radiation isocurvature fluctuationrather than a net curvature fluctuation. To the extentthat the Higgs fluctuation then redshifts like radiation,the isocurvature mode does not subsequently contributeto the curvature fluctuation.
The conserved curvature on uniform radiation densityslices which corresponds to this induced radiation per-turbation is
ζr = −δρr
ρ′r=δT
T, (67)
and solving for the radiation perturbation (66) using thejump in Higgs energy (65) and Eq. (67), we find
δρr(0+) = −δh
(V T,h − V,h
)(1− V T,T
T
ρ′r
)−1
. (68)
This induced radiation perturbation comes from the di-rect interaction of the Higgs with the radiation during thethermal uplift. It is distinct from radiation perturbationscorresponding to intrinsic inflaton fluctuations, which areuncorrelated and can be computed separately, or to infla-ton perturbations produced by the gravitational influenceof the Higgs perturbations during inflation discussed inApp. B, which are suppressed.
This radiation perturbation was omitted in Refs. [11–13, 34], though it in fact has a large impact on the finalcurvature perturbation ζ. In particular, on a constanttotal density surface, the curvature perturbation is givenby Eq. (2), reproduced here for convenience,
ζ =
(1− ρ′h
ρ′tot
)ζr +
ρ′hρ′tot
ζh. (69)
Immediately after the uplift of the Higgs potential, ζtherefore satisfies
ζ(0+) = ζh(0−)× ρ′h(0−)
ρ′tot(0+)
' −ζh(0−)× ρ′h(0−)
12H2end
, (70)
by virtue of Eq. (66). This equation is nothing but thelinear jump condition (60), this time derived from linearperturbation theory rather than the δN formalism.
To evolve ζ(N) from its value at ζ(0+), we must solvefor the Higgs perturbations after uplift. Although dur-ing inflation we assumed the Higgs is a spectator, afterpotential uplift we jointly solve the Higgs backgroundequation (29), now with V → V T , and the radiation
12
0.00 0.05 0.10 0.15 0.20
N
0.00
0.02
0.04
0.06
0.08
0.10∆ζ(k
PB
H)
without δρr
with δρr
FIG. 6. The curvature perturbation on uniform total energyslices computed in linear perturbation theory with and with-out including radiation perturbations required by energy con-servation at reheating. These induced perturbations suppressthe cycle-averaged curvature by several orders of magnitude(see §IV C).
background equation (49), with the Hubble rate deter-mined by Friedmann equation. For the perturbations wesolve the linearized Klein-Gordon equation (32) for theHiggs, now with a perturbed potential
δV Tk,h = V T,hhδhk + V T,hT δT
k, (71)
which accounts for the effect of temperature perturba-tions. Again the metric terms are negligible and we ex-ploit here that ζr is constant to avoid solving perturba-tion equations for the radiation component; we shall showthat both are good approximations in App. B.
To quantify the importance of the induced radiationperturbation (66), we chose hend such that the post-inflationary Higgs contribution to the total ∆2
ζ(kPBH) is
∼ (0.1)2. This is the calculation performed in the liter-ature which suggests that primordial black holes can beformed at the classical-roll scale. We then add the in-duced radiation contribution and see how ∆ζ is affected.
We show these numerical results in Fig. 6. Here weplot ∆ζ(kPBH) with a phase convention ϕ such that theanalogous superhorizon Higgs fluctuation
∆ζh ≡ eiϕ√
∆2ζh, (72)
is negative real during inflation. Note that ∆ζh changessign at the potential uplift and becomes positive real.After inflation, ∆ζ(kPBH) oscillates between negative andpositive values but stays real.
It is immediately clear from Fig. 6 that the inducedradiation perturbation δρr suppresses the amplitude ofthe total curvature ζ by orders of magnitude, makingit much more difficult to achieve the required R = 0.1in this model. For the fiducial background, which wasclaimed to produce R = 0.1, by taking into account δρr
we instead have R(0+) ' 3× 10−4.
0.00 0.02 0.04 0.06 0.08 0.10
N
−0.004
−0.002
0.000
0.002
0.004
∆ζ(k
PB
H)
Linear δN
Perturbation Theory
FIG. 7. The linear perturbation theory calculation includinginduced radiation fluctuations agrees well with the linearizedδN result based on (59), validating our result that energyconservation suppresses curvature fluctuations (see §IV C).
In Fig. 7, we validate our calculation of ∆ζ(kPBH) byalso computing ζ from the linearized δN equation (59).The δN result relies solely on the behavior of the back-ground equations and thus is an independent check on therather involved perturbation theory calculations. TheδN result agrees closely with our perturbation theorycalculation and confirms that the induced radiation per-turbation is crucial in this mechanism. This test wouldfail if the radiation compensation in Eq. (68) were omit-ted as in Refs. [11–13, 34].
ζ is not conserved after reheating, and in particular itoscillates due to the changing nonadiabatic pressure in-duced as the Higgs oscillates. Even though oscillations inδρh are relatively small, the initial near cancellation be-tween δρh and δρr make them prominent in ζ. Moreover,deviations of the Higgs potential from a simple quadraticwith a temperature-dependent mass make these oscilla-tions grow in time. We discuss these effects in detail inApp. B. So long as the Higgs decay time, generally of or-der an e-fold, is larger than the oscillation timescale, it isthe cycle-averaged ζ that matters. Because of the initialoutgoing trajectory of the Higgs, the instantaneous valueof |ζ| at N = 0+ is always larger than the cycle averageof the first oscillations.
To the extent that the cycle-averaged Higgs energy red-shifts as radiation, the cycle-averaged value of ζ will beconserved. However, the nonquadratic terms in the Higgspotential also cause deviation from this behavior whichleads the near cancellation between the Higgs and radia-tion energy densities gradually to break down.
In particular, Higgs perturbations redshift slightly
13
slower than radiation on the cycle average,*2
〈δρkh〉 ∝ a−4−3∆w, (74)
with ∆w ≡ 〈w〉 − 1/3 ∼ −0.004. Therefore the cancel-lation between the radiation piece and the Higgs piecegradually becomes undone,
ζk = −δρkr + δρkhρ′tot
∼ −δρkr + δρkhρ′r
= ζkr +δρkh4ρr
, (75)
and 〈ζk〉 grows gradually.Once the Higgs piece dominates, the cycle-average be-
comes
〈ζk〉(N) ∼ −3
4
δρkh(0+)
ρr(0+)∆w ×N ∼ 10−3N. (76)
Thus the curvature grows to O(10−1) only on a timescale
∆N ∼ 100 e-folds. (77)
The Higgs must decay to radiation well before this,and therefore the curvature perturbations cannot becomelarge enough in this scenario to form PBHs.
In summary, under the assumption that linear theory isvalid through reheating, the Higgs instability mechanismfalls far short of being able to form PBHs as the darkmatter. Models that were previously thought to achievethe required R = 0.1 in fact produce R . 10−3 oncethe radiation density perturbations required by energyconservation at reheating are properly accounted for.
D. Nonlinear Conversion
In §IV C, we computed curvature fluctuations assum-ing that the Higgs perturbations remain linear throughreheating. In fact, the Higgs instability induces a break-down of linear theory when the background position ofthe Standard Model Higgs at the end of inflation hend isclose to the maximum rescue scale hrescue.
This breakdown can be seen immediately from Fig. 5.With δN = ζh ∼ ±1, a typical outwardly perturbed re-gion of the Standard Model Higgs field crosses hrescue
during inflation, gains exceedingly large negative energyand will inevitably backreact on the background trajec-tory. Reheating will be disrupted, the perturbed Higgswill not be rescued from the unbounded vacuum, and ouruniverse will be destroyed. Even for smaller δN which do
*2 The rate at which Higgs perturbations redshift (74) is differentfrom the rate at which the background Higgs redshifts,
〈ρh〉 ∝ a−4−3∆w, (73)
with ∆w ∼ −0.002. This means that there is also internal nona-diabatic stress in the Higgs field itself and so the cycle averagedζkh would also evolve.
not cross hrescue, the perturbed Higgs energy density isnot well represented by the linear Taylor expansion (60)around the background value due to the extremely rapidevolution of ρh.
In terms of field interactions, linear theory itself alsoreveals its own breakdown. At the end of inflation, an or-der unity ζh leads to an RMS Higgs fluctuation of roughly
δhend ' h′end ' −1
3H2λh3
end, (78)
where we have used the Higgs slow-roll approximationthroughout. With hend ∼ 103H and |λ| ∼ 10−2 we have
δhend ' 107H, (79)
which as we have seen is orders of magnitude larger thanthe distance between hend ' 1200H and hrescue ' 2400H.Moreover, the potential interaction ratio (27) is
1
2δhV,hhhV,hh
' δh
4h' 104 � 1, (80)
and thus field fluctuations interact. Nevertheless, thisbreakdown has no effect on our previous computation of∆2ζh
during inflation since ζh is conserved nonlinearly solong as H ∼ const.
Though it was not phrased in terms of a breakdownof linear theory, Ref. [12] noted that the Standard ModelHiggs is generally not rescued in this scenario. It wasargued in Ref. [13] that the background hend can beplaced near hrescue while multiverse and anthropic con-siderations justify tuning the local Higgs field at the endof inflation such that Max [h(~x)] < hrescue everywhere.However, tuning δh(~x) at the end of inflation is equiva-lent to tuning δN = ζh to be small at the end of inflationand so it directly tunes away the ability to form PBHsas we shall now show.
In Fig. 8 we show ζ(0+) as a function of ζh as computedusing the nonlinear δN formalism using Eq. (58) andFig. 5. Linear theory holds for small enough inflationary|ζh| . 10−3, but breaks down for perturbations of thetypical amplitude produced during inflation.
Large inwards perturbations away from the instability,shown on the left-hand side of Fig. 8, saturate to a con-stant ζ(0+) that is independent of ζh. These uphill kicksproduce a local Higgs energy density at N = 0− that hasa much smaller magnitude than its background value ascan be seen in Fig. 5. Using Eq. (57), the left-hand sidesaturation can therefore be written as
ζ inSM(0+) =
−ρh(0−)
12H2' +1.4× 10−6, (81)
which we show as a horizontal dashed line on the left-hand side in Fig. 8.
Outward perturbations of the SM Higgs toward theinstability, shown on the right-hand side of Fig. 8, areenhanced relative to linear theory. This is because theamplitude of the energy density of the Higgs shown in
14
0
2
4
6
8R
(ζh)| 0
+×
104
SM
BSM
−100 −10−2 −10−4 10−4 10−2 100
−ζh
10−3
10−2
10−1
100
|ζ(0
+)|×
104
FIG. 8. The nonlinear mapping of the inflationary ζh to thepost-reheating ζ(0+) = R(ζh)|0+ ζh. Fluctuations outwards,toward the instability, correspond to −ζh > 0. The horizontalaxis scale is linear between ±10−4 and logarithmic elsewhere.R deviates from the linear theory value ' 3×10−4 for fluctu-ations larger than about ±10−3. The horizontal dashed linesindicate analytic saturation values (84) and (85). For the SMHiggs, the maximal ζ satisfying the rescue condition (51) ismarked with a star. For the BSM Higgs, ζ saturates to amaximum. Neither value is large enough to form PBHs
Fig. 5 grows much faster than expected from a linearapproximation. The largest outward perturbations thatsatisfy the rescue condition (51) produce a curvature
ζoutSM(0+) =
ρh(hrescue)− ρh(0−)
12H2' −2.7× 10−6. (82)
Despite the enhancement of the Higgs perturbation rel-ative to linear theory, ζout
SM evolves after inflation muchlike the linear theory ζ computed in §IV C. The cycle av-erage of ζout
SM(N) is smaller than ζoutSM(0+). So long as the
Higgs redshifts like radiation after inflation, this valueis then conserved. Nonlinear evolution does not changethe conclusion of linear theory on PBHs with the SMpotential.
So far in this nonlinear calculation we have kept thebackground trajectory of the Higgs fixed. We might won-der whether a different background Higgs trajectory, atfixed H, can achieve a larger value for the saturating ζ.Moving the background away from the instability sup-presses ρh(0−), and so the largest curvature perturbationwhich can be produced in the SM, maximized over the
choice of background trajectory, is
ζmaxSM (0+) = ζout
SM(0+)− ζ inSM(0+) ' −4.1× 10−6. (83)
which is still insufficient to form PBHs. Therefore sinceζ is uniquely determined by ζh in this way, anthropicallytuning away ζh(~x) or setting it to the edge of rescueabilityforbids the Standard Model Higgs from forming PBHs insufficient abundance to be the dark matter.
Ref. [13] proposed that the mechanism could functionwith a BSM potential derived from the addition of a mas-sive scalar as detailed in §II. The massive scalar adds awall in the potential between the field value where thebackground Higgs ends inflation hend and the maximumrescuable point hrescue, preventing the local Higgs fromreaching parts of the potential from which it cannot berescued by thermal uplift at reheating.
When CMB and PBH modes cross the horizon, theBSM potential behaves like the SM potential and as wehave seen in §III this means that it generates larger in-flationary ζh perturbations on CMB scales than on PBHscales. In addition, so long as the background trajectorynever encounters the wall, this model behaves like the SMin linear theory and yields R� 0.1 as shown in §IV C.
However, whereas typical regions with outward fieldfluctuations were not rescued in the SM, in the BSMcase such regions oscillate in a new minimum of the po-tential during inflation and then can be safely rescued atreheating.
To evolve fluctuations through this highly nonlinearprocess, we again use the nonlinear δN formalism de-scribed in §IV B, just as in the Standard Model case,and the BSM results for ζ(0+) are also shown in Fig. 8.We compute results using the representative parameterset for the BSM scalar described in §II, and we will latershow how our results scale with different choices of modelparameters.
Inward perturbations of the BSM Higgs act just likeinward perturbations in the SM and thus again lead tothe same saturation
ζ inBSM(0+) = ζ in
SM(0+) ' +1.4× 10−6. (84)
Large outward Higgs perturbations hit the BSM po-tential wall, become trapped in the new minimum ath ∼ ms, lose their kinetic energy, and end inflation with apotential dominated Higgs with energy Vmin. This leadsto a saturating curvature
ζoutBSM(0+) =
Vmin − ρh(0−)
12H2end
' −2.4× 10−4, (85)
which we show with a horizontal dashed line on the right-hand side in Fig. 8. This value is still too small to formPBH DM.
For perturbations which do not fully saturate thislimit, ζ(0+) has a stepped behavior and R(0+) an oscilla-tory one as depicted in Fig. 8. These features correspondto the energy density oscillations for the BSM Higgs seen
15
0.00 0.02 0.04 0.06 0.08 0.10
N
−0.004
−0.002
0.000
0.002
0.004ζ
Linear Theory
Nonlinear δN
FIG. 9. Curvature evolution after reheating. The BSM non-linear δN result for the local ζ(~x), from an example infla-tionary ζh(~x) = ∆ζh(kPBH), is compared to the linear theoryapproximation ∆ζ(kPBH) of Fig. 7. The nonlinear evolutionof ζ after reheating is too small to form PBHs (see §IV D).
in Fig. 5, induced because the Higgs has large oscilla-tions around the potential minimum before settling onthe e-fold timescale. Note that the approximate equal-ity of the linear theory R(0+) and the nonlinear R(0+)corresponding to |ζh| ' 0.1 is a coincidence: changes tothe background position change the linear theory R(0+)while leaving R(0+) on this brief plateau fixed.
Once more we might wonder if a different choiceof background trajectory could enhance the saturationvalue, but in fact the maximum over background trajec-tories can again be computed as
ζmaxBSM(0+) = ζout
BSM(0+)− ζ inBSM(0+) ' ζout
BSM(0+), (86)
and therefore PBHs cannot be formed for our fiducialBSM potential no matter the position of the backgroundor the size of initial fluctuation.
Note that so far we have only computed ζ(0+) forBSM. We should check whether ζ evolves significantlyafter N = 0+. We do so again with the δN formalism,using the full Eq. (55). Since inward fluctuations lead toa negligible ζ(0+), we can select a typical outward fieldfluctuation with ζh(~x) = ∆ζh(kPBH) ' −1 as an exam-ple.
We show this case in Fig. 9. As expected from Fig. 8,the nonlinearly evolved ζ is small, comparable in ampli-tude to the linear ζ(0+) but not in its evolution. In fact ζevolves negligibly after N = 0+ and we can robustly con-clude that PBHs are generically not formed nonlinearlyin this case.
This lack of nonlinear evolution can be explained bythe difference in the impact of uplift on the perturba-tions. In linear theory the small amplitude of ζ resultedfrom large cancellations between the Higgs and radiationperturbations due to energy conservation and the largeimpact of uplift. Nonlinearly the impact of uplift is muchsmaller, bounded by the BSM modification, and so theHiggs energy density fluctuations after reheating are no
longer as dominated by the uplift contribution. In par-ticular,
ρh(0+, ~x)− ρh(0+)
ρh(0−, ~x)− ρh(0−)� δρh(0+)
δρh(0−)
∣∣∣∣linear
, (87)
where the right-hand side is in linear theory. Thereforethe cancellation with radiation is less dramatic than inlinear theory. As discussed in §IV C, the cancellation andsubsequent decancellation are responsible for the lineartheory oscillations and slow secular drift (76), and there-fore all these effects are suppressed in the nonlinear case.
Finally, the fiducial BSM potential used to computethe results of Fig. 8 was constructed according to thespecifications of Ref. [13]: it uplifts the Standard Modelpotential somewhere between hend and hrescue. We mightwonder whether PBHs could be formed by optimizing theposition of the uplift so that it as close as possible tohrescue, maximizing the criticality of the scenario.
The maximum position ofms will be just before hrescue.This leads to a maximum curvature for this entire sce-nario of
Max[ζmaxBSM] =
V (hrescue)
12H2end
. (88)
Using the approximate maximum rescueable field value(53) and λ = λSM ∼ −0.007, we have
V (h(1)rescue) =
1
4λSM
(h(1)
rescue
)4
= −0.012H2end, (89)
which yields
ζmaxBSM ' −1.0× 10−3, (90)
which depends on H only through the logarithmic evolu-tion of λSM evaluated at hrescue. This estimate is in goodagreement with the computation using the exact value ofhrescue which yields ζmax
BSM = −8.2× 10−4.Therefore no matter the size of inflationary Higgs per-
turbations, the position of the background Higgs, the SMor BSM nature of the Higgs potential, the position of theBSM wall, or the Hubble rate, this mechanism does notproduce perturbations large enough to form PBHs in suf-ficient abundance to be the dark matter.
Along with the non-production of PBH DM from theHiggs field, the largest possible curvature perturbationsproduced by this model, Eq. (90), are so small that thesecond-order gravitational waves predicted by Ref. [34]will be undetectable with LISA.
We can also ask what happens to CMB scale fluctua-tions nonlinearly in this model. We showed in §III thatduring inflation ∆2
ζh(kCMB) > ∆2
ζh(kPBH). In Fig. 10, we
show the effect of the highly nonlinear local transforma-tion of ζh to ζ shown for the BSM potential in Fig. 8 ona cartoon realization of the curvature field on a constantHiggs density surface during inflation.
For visualization purposes, we have generated twomodes apart by a factor of only 20 in scale rather than the
16
0.0 0.2 0.4 0.6 0.8 1.0
x/λL
−5.0
−2.5
0.0
2.5
5.0
Rea
lS
pac
eC
urv
atu
reF
ield
s
ζh during inflation
ζ(0+)× 104
FIG. 10. The real-space post-inflationary curvature field ζ(red) produced by the nonlinear transformation of the in-flationary ζh (blue). Perturbations on small scales are sup-pressed when they occur where the long-wavelength λL modehas saturated the curvature field, and therefore CMB scaleperturbations are larger than PBH scale perturbations fullynonlinearly in this mechanism (see §IV D).
∼ 35 e-folds which separate the CMB and PBH modes.We see that fluctuations on the long-wavelength scalescause a saturation of the short-wavelength fluctuations,and therefore fully nonlinearly we have that perturba-tions are larger on long wavelengths than on small wave-lengths in this model as long as this is true of the in-flationary ζh itself. Therefore if the reheating scenariois changed somehow to achieve large PBH scale fluctua-tions, the CMB scale fluctuations will still be larger thanPBH modes unless the functional form of the transfor-mation shown in Fig. 8 is radically altered.
Beyond the specific motivation of PBH DM formation,we can now return to the question of whether Higgs insta-bility is compatible with the small curvature fluctuationsobserved in the CMB.
For the SM Higgs, suppressing ζoutSM requires placing
the background hend far from hrescue. Specifically, sincethe typical CMB scale perturbation has |ζh| ∼ 5, hend
should be moved at least ∼ 5 e-folds backwards along itstrajectory.
For the BSM potential, with the scalar mass set nearhrescue to ensure that no regions ever fall into the unres-cuable region, there are two situations which are com-patible with the CMB and one which is not.
If hend does not approach hrescue, then we return tothe linear theory, Standard Model result, where Higgsfluctuations on CMB scales do not lead to large curvaturefluctuations since ρh/H
2 decreases sharply in Fig. 5 andpredicts ζ through Eq. (58).
Conversely, if the Higgs travels far down the unstableregion early in inflation, the Higgs becomes uniform inthe potential well during inflation and thus leads to nocurvature perturbations after inflation. In this region aswell the Higgs instability is compatible with the CMB.
It is only in the region of parameter space near our fidu-cial model, where the background Higgs hend approachesbut does not reach the minimum induced by the BSMmassive scalar near hrescue, that Higgs fluctuations onCMB scales can be converted to curvature fluctuationswhich are large enough to disturb the CMB.
To avoid this possibility completely, one should set themass ms of the scalar field to be slightly smaller thanhrescue. In particular to achieve |ζ| . 10−5, using Eq. (88)and Eq. (90), one requires
ms
hrescue.
(1
100
) 14
.1
3. (91)
In summary, it is only a special class of Higgs critical-ity scenarios where the parameters are arranged so thatthe regions of the universe fluctuate near the edge of res-cuable instability which would be testable in the CMBand even that class cannot form PBHs as the majority ofthe dark matter, nor generate second-order gravitationalwaves at an amplitude detectable with LISA.
V. CONCLUSIONS
We have definitively shown that the dark matter is notcomposed of primordial black holes produced by the col-lapse of density perturbations generated by a spectatorHiggs field during inflation.
While a spectator Higgs field evolving on the unsta-ble side of its potential can generate large Higgs fluc-tuations on PBH scales, even larger Higgs fluctuationsare produced on CMB scales. This result is obtained us-ing linear perturbation theory during inflation, which weshow holds even though the CMB modes cross the hori-zon at an epoch when the Higgs’ per e-fold classical roll issmaller than its per e-fold stochastic motion, because thestochastic motion is incoherent and does not backreacton the Higgs background.
Inflation ends well after all relevant modes have crossedthe horizon, and when reheating occurs the Higgs poten-tial is uplifted by the interaction between the Higgs andthe thermal bath. If the Higgs is rescued from the unsta-ble region by this thermal uplift then the Higgs redshiftsas, and eventually decays to, radiation. The CMB andPBH modes are superhorizon at these epochs and evolvein the same way through these processes. Therefore ifthe Higgs fluctuations are converted into sufficiently largecurvature perturbations such that PBHs as dark matterwere produced, CMB constraints would necessarily beviolated.
In fact though, PBHs as dark matter are never pro-duced and CMB constraints are only violated in cases ofnear criticality. We first showed that this is true underthe assumption that linear theory holds through reheat-ing, where we correct an error in local energy conserva-tion made in the literature.
We then showed that linear theory is violated becausethe model requires the Higgs to be as close as possible to
17
the maximum value beyond which it cannot be rescuedat reheating. This criticality condition leads typical per-turbations to evolve nonlinearly, and using the nonlinearδN formalism we also show that the Standard ModelHiggs, regardless of fine-tuning or anthropic arguments,can never produce PBH DM. Modifying the Higgs po-tential at large field values can eliminate fine-tuning oranthropic issues, but cannot enhance curvature pertur-bations significantly enough to produce PBH DM.
ACKNOWLEDGMENTS
We thank Andrew J. Long for fruitful discussions alongwith Peter Adshead, Jose Marıa Ezquiaga, and Lian-TaoWang for helpful comments. SP and WH were supportedby U.S. Dept. of Energy contract DE-FG02-13ER41958and the Simons Foundation. SP was additionally sup-ported by the Kavli Institute for Cosmological Physicsat the University of Chicago through grant NSF PHY-1125897 and an endowment from the Kavli Foundationand its founder Fred Kavli. HM was supported by JapanSociety for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) No. JP17H06359and No. JP18K13565.
Appendix A: Formation of inflationary backgroundby superhorizon stochastic modes
In §III B we showed that the Higgs field fluctuationsare linearizable around a slow-roll background when allobservationally relevant scales crossed the inflationaryhorizon. In §III C, we used that background to computeHiggs fluctuations during inflation. Here, we will showthat such a background can be produced in the stochas-tic inflation formalism, and that such a scenario yieldsresults consistent to those of §III C.
In the stochastic picture, our background on the unsta-ble side of the potential is formed by Higgs fluctuationswhich crossed the horizon before N ∼ −60. At horizoncrossing, each such fluctuation has an amplitude
δh ' H
2π, (A1)
and a velocity
δh′ ' H
2π. (A2)
Our background was formed when a cumulative se-ries of such stochastic kicks pushed our Hubble patch tothe unstable side of the barrier. We can reabsorb theseearly stochastic kicks into a background field that is spa-tially homogeneous on our Hubble patch, while subse-quent stochastic kicks lead to spatial perturbations onsmaller scales which we addressed in §III.
When absorbed into an FLRW background in our Hub-ble volume, a large-scale stochastic kick to the Higgs im-parts a velocity
∆v =H
2π. (A3)
where v ≡ dh/dN . The background then obeys the equa-tion of motion (29) and when the potential derivativeterm is negligible the velocity decays according to
d ln v
dN= −3. (A4)
However, the velocity of a superhorizon mode in the ab-sence of classical roll can be deduced from the de Sittermodefunction (40), which yields
d ln δh′
dN= −2. (A5)
The difference between these decay rates is a breaking ofthe separate universe condition. If we reabsorb the su-perhorizon stochastic kicks into a new FLRW backgroundwe make a small error.
Nonetheless, the superhorizon velocity decay is expo-nential and the field velocity is rapidly dominated byclassical roll. Only kicks which occurred just prior toN ∼ −60 contribute to the velocity of our background.The cumulative stochastic kicks therefore impart to ourbackground an initial velocity
h′60 ∼H
2π, (A6)
where ∼ indicates a multiplicative factor of order unity.As described in §III B, kicks after N = −60 do not back-react on the background and we treat those as linear per-turbations. Therefore after h60 the initial velocity decaysuntil it becomes less important than the potential deriva-tive term and our background has reached the attractorsolution.
The numerical solution of the FLRW backgroundEq. (29) with the initial condition Eq. (A6) shows thatthe initial position of the field should be shifted by
∆h60 ∼ 0.04H (A7)
away from the instability in order to achieve the samehend as in the attractor initial velocity case. Therefore,the nonattractor initial velocity (A6) has little impacton the position of our background. This is because thenonattractor phase lasts just a small fraction of an e-fold,which can be seen as follows. At h60, the Higgs potentialslope is
− 1
3H2V,h|h60
∼ 1
3× H
2π, (A8)
which means that from (A4) the initial velocity for theFLRW background becomes less than the potential slope
18
in (ln 3)/3 ' 0.4 e-folds. If we had used the correct su-perhorizon velocity decay Eq. (A5), then we would havefound that the initial velocity decays in (ln 3)/2 ' 0.5e-folds. Therefore this error is not important for small-scale modes that crossed the horizon after this epoch ifwe assume that the background in our Hubble volume isestablished by stochastic kicks.
Appendix B: Superhorizon Curvature Evolution
In this appendix, we discuss how and why curvatureperturbations can evolve on superhorizon scales bothduring and after inflation in this scenario.
Local conservation of the stress-energy of a noninter-acting fluid I
∇µT Iµν = 0, (B1)
yields at the perturbation level a continuity equation anda Navier-Stokes equation. For our purposes, I here willeither denote the Higgs fluid or the total fluid, during orafter inflation.
The continuity equation on a surface of uniform energydensity of I leads to a conservation equation for the cur-vature perturbation ζI on that surface. On superhorizonscales, the conservation equation takes the simple form
ζI′ = − δpNA
I
ρI + pI, (B2)
where pI is the fluid pressure, ρI the energy density, andδpNAI is the nonadiabatic pressure of the fluid (see, e.g.,
Ref. [39] for notation). The nonadiabatic pressure is thepressure perturbation on a surface of uniform density ofI.
The nonadiabatic pressure on the right-hand side ofthe superhorizon conservation equation (B2) can be com-puted either by directly studying local variations in pres-sure on a uniform density surface, or by subtracting theadiabatic pressure through
δpNAI = δpI −
pIρIδρI (B3)
where δpI and δρI are the pressure and density pertur-bations of the fluid I in any gauge. The appearance ofnonadiabatic pressure is associated with having multipledegrees of freedom or clocks so that the local density nolonger uniquely defines the local pressure. For examplefor a scalar field, if the kinetic energy were not uniquelyspecified by the potential energy, i.e. the field position,then there is nonadiabatic stress. Likewise if I is a com-posite of two systems with differing equations of statethen ρI does not uniquely define pI . We shall see thatboth mechanisms are operative for the Higgs instabilitycalculation.
Meanwhile the left-hand side of the superhorizon con-servation equation (B2), ζ ′I can be computed by taking
a derivative of the curvature perturbation computed inthe δN approach (see §IV B for an overview of δN), orby taking a derivative of the curvature perturbation ob-tained by transformation from the spatially flat gauge.For the Higgs, the Fourier mode of the flat gauge fieldperturbation satisfies the Klein-Gordon equation(
d2
dη2+ 2
a
a
d
dη+ k2
)δhk + a2δV k,h
= (A− kB)h− 2a2AV,h. (B4)
The terms appearing on right-hand side of the Klein-Gordon equation (B4) involve the k-modes A of thelapse perturbation and B of the scalar shift perturba-tion, in the spatially flat gauge, which we neglected in(32). These terms take different forms during and afterinflation and we shall provide them shortly. We droptheir k superscripts for compactness.
These different approaches allow us to better un-derstand why the curvature evolves on superhorizonscales during inflation, at reheating, and after inflation,and they enable us to check our calculations for self-consistency, including that of various metric and fieldterms which will appear.
1. During Inflation
During inflation, the Higgs has energy density
ρh =1
2
h2
a2+ V, (B5)
and pressure
ph =1
2
h2
a2− V. (B6)
In the flat gauge, the Fourier mode of the Higgs densityperturbation is
δρkh =1
a2
(h ˙δhk − h2A
)+ V,hδh
k, (B7)
and the pressure perturbation is
δpkh =1
a2(h ˙δhk − h2A)− V,hδhk. (B8)
The metric terms in these equations and in the Klein-Gordon equation (B4) satisfy the momentum constraintand Hamiltonian (Poisson) constraint equations
a
aA =
1
2hδhk +
1
2φδφk,
a
akB =
1
2a2(δρkh + δρkφ) + 3
(a
a
)2
A. (B9)
The Klein-Gordon equation (B4) then becomes(d2
dη2+ 2
a
a
d
dη+ k2
)δhk + a2δV k,h
= δhk1
a2
d
dη
(a3
ah2
)+ δφk
1
a2
d
dη
(a3
ahφ
). (B10)
19
The metric terms contain contributions from the Higgsperturbations themselves as well as inflaton perturba-tions induced by the metric perturbations in the anal-ogous KG equation for the inflaton. Intrinsic inflatonfluctuations should not be included here since they areuncorrelated with the Higgs and computed separately.
We have verified numerically that including these met-ric terms on the right-hand side of the Klein-Gordonequation (B10), including accounting for the induced in-flaton perturbations, has no significant effect on the so-lution for δhk or δρkh, which justifies our treatment in§III C. This can be analytically seen as follows.
Given that the KG source to the field velocity is thepotential slope, we can estimate
h′ ∼ −c V,h3H2
, (B11)
where c = 1 for Higgs slow roll and an order unity fac-tor otherwise. Assuming that the fields in the back-ground evolve on the Hubble time or slower, the RHSof Eq. (B10) can then be estimated as(
d2
dη2+ 2
a
a
d
dη+ k2
)δhk + a2δV k,h
' δhk c2a6
a2V 2,h − δφkc
a3
aV,hφ. (B12)
The δhk metric term on the RHS can be comparedto the nominal δV k,h = δhkV,hh term on the left handside. Using the analytic form for the Higgs potential,V = λh4/4, with λ only logarithmically varying, showsthat the metric δhk term is suppressed relative to thenominal term as
c2V 2,h
3H2V,hh' c2
9
λh4
H2=
4c2
9
V
H2� 1 (B13)
where the last inequality follows because the Higgs is aspectator during inflation.
The second metric term, proportional to δφk, is fur-ther suppressed relative to the δhk metric term becauseδφk is sourced by the Higgs metric term itself. In par-ticular, solving the inflaton Klein-Gordon equation withthe Higgs metric source we find
δφk ∼ δhkh′φ′, (B14)
and therefore in the Higgs Klein-Gordon equation theinflaton metric term becomes suppressed relative to theHiggs one by a factor
φ′2 ∼ εH . 1. (B15)
Therefore all metric terms in the Klein-Gordon equa-tion (B10) for the Higgs can be neglected when the Higgsis a spectator. This justifies the approximated Klein-Gordon equation (32) and the subsequent analysis basedon it. As shown in §III C, the curvature perturbation on
uniform Higgs density (UHD) surfaces during inflationthen obeys (39), i.e.
ζkh′
ζkh= −2εH , (B16)
on superhorizon scales at leading order in the Hubbleslow-roll parameter εH . This superhorizon evolution isimportant for the estimation (44) of the relative ampli-tude of ∆2
ζhon CMB and PBH scales.
We can alternately derive the superhorizon evolu-tion (B16) by using the nonadiabatic pressure rela-tion (B3) or by using the δN formalism (59). Let us firstfocus on the nonadiabatic pressure. On a UHD surface,the Higgs energy density fluctuation vanishes
δ(ρh)UHD ≡ 0 =1
2δ(H2h′2)UHD + V,hδ(h)UHD, (B17)
where we have again assumed metric perturbations arenegligible. The nonadiabatic pressure (B3) is then
δpNAh ≡ δ(ph)UHD =
1
2δ(H2h′2)UHD − V,hδ(h)UHD
= δ(H2h′2)UHD. (B18)
Plugging this nonadiabatic pressure into the super-horizon conservation equation for ζh (B2) we have
ζ ′h = −δ(H2h′2)UHD
H2h′2. (B19)
When the Higgs is slowly rolling, we can use (B11) withc = 1, i.e.
h′ ' − V,h3H2
. (B20)
and on uniform Higgs field slicing, the Higgs energy den-sity varies only if H itself varies. Conversely, ignoringsuch higher order corrections in δ lnH (see below), wesee that on UHD surfaces V,h is uniform and
ζ ′h = 2δ(lnH)UHD. (B21)
Even though the Higgs is a spectator field, H varies onthe UHD slice because of shifts in the number of e-foldsinduced by the gauge transformation from spatially flatslicing δN = ζh. We therefore can obtain the desiredresult
ζ ′hζh
= 2δ lnH
δN= −2εH . (B22)
Using the same logic, we can check our assumption thatuniform Higgs field (UHF) and UHD curvatures coincideto leading order. Since
δ(ρ)UHF =1
2δ(H2h′2)UHF = −εH
3ρ′h(ζh)UHF (B23)
20
and the gauge transformation between UHF and UHDinvolves an e-fold shift of δ(ρh)UHF/ρ
′h, we obtain
(ζh)UHD ' (ζh)UHF
(1 +
εH3
). (B24)
We therefore use UHF and UHD interchangeably duringinflation.
Finally, notice that these explanations make use ofδN as computed from gauge transformations. We caninstead compute it directly in the δN formalism usingthe position-dependent number of e-folds Nh(hi; h) to aUHF slice through the UHF equivalent of Eq. (59),
ζh =∂Nh(hi; h)
∂hiδhi. (B25)
To find Nh(hi; h), we can exploit that the local Higgsevolves along the attractor according to Eq. (B20), andbecause the Higgs is a spectator the Hubble rate has nodependence on the Higgs. Working to linear order in thenumber of e-folds N − Ni, we expand the denominator3H2 and we have
h′(N,~x) ' − V,h|h(~x)
1
3H2i (1− 2εH(N −Ni))
, (B26)
where Hi ≡ H(Ni). For analytic purposes, we approx-imate the potential as V (h) = 1
4λh4, with λ only loga-
rithmically dependent on h, solving this equation withh(Ni, ~x) = hi to find
N −Ni =1
2εH
(1− e
−3H2
i εHλ
(1
h2−
1
h2i
)). (B27)
This is the number of e-folds that takes to get from somehi at time Ni to a field value h, and thus we have com-puted
N −Ni = Nh(hi; h). (B28)
Taking a partial derivative with respect to hi to get thelinear ζh, we thus have
ζh = δhi ×3H2
h3iλe−
3H2i εHλ
(1
h2−
1
h2i
). (B29)
To obtain ζ ′h we take a derivative with respect to the finalsurface h and find,
ζh′
ζh= h′
∂ ln ζh∂h
= −2εH , (B30)
at leading order in εH . h′ here is evaluated on the back-ground. This gives a third way of understanding thesuperhorizon evolution (B16).
We therefore have a consistent picture where the be-havior of the flat gauge perturbations, the nonadiabaticpressure, and the δN formalism all consistently show thatζh evolves and decays outside the horizon as H evolvesduring inflation because a uniform Higgs slice is not auniform Hubble slice.
2. At Reheating
At reheating, ζh is instantaneously boosted andchanges sign. This is due to an instantaneous source inthe conservation equation for the Higgs stress-energy dueto the direct interaction between radiation and the Higgs,rather than a nonadiabatic pressure. The sign changein ζh = −δρh/ρ′h at reheating occurs because a posi-tive δh fluctuation corresponds to a negative δρh fluctu-ation before uplift but a positive one after (see Eq. (63)).Note that in a realistic inflation model where H variessmoothly, ζ would be continuous at reheating.
3. After Reheating
After reheating, the background Higgs Klein-Gordonequation (29), and the definition
ρh =1
2
h2
a2+ V T (B31)
implies
ρ′h = −3h2
a2− V T,TT, (B32)
since the thermal bath redshifts as T ∝ a−1. This lastterm in the right-hand side of (B32) was omitted inRefs. [11, 12]. We can define an effective pressure forthe Higgs as
ph,eff =1
2
h2
a2− V T + V T,T
T
3, (B33)
such that the continuity equation
ρ′h + 3(ρh + ph,eff) = 0, (B34)
is satisfied.Likewise the Higgs perturbations carry terms associ-
ated with the redshifting of the radiation bath. The flatgauge energy density perturbation is
δρkh =1
a2
(hδhk − h2A
)+ V T,h δh
k + V T,T δTk, (B35)
and the flat gauge effective pressure perturbation is
δpkh,eff =1
a2(hδhk − h2A)− V T,h δhk −
2
3V T,T δT
k
+1
3
(V T,TTTδT
k + V T,hTTδhk), (B36)
where again the new terms involve the temperaturederivatives of V T . The lapse perturbation A is related tothe total momentum density by the Einstein constraintequation
A =1
2
aH
k
ρtot + ptot
H2(vtot −B), (B37)
21
where the total momentum density,
(ρtot + ptot)(vtot −B) =k
a2hδhk +
4
3ρr(vr −B), (B38)
satisfies momentum conservation[d
dN+ 4
](ρtot + ptot)(vtot −B)
=k
aH
[δpktot + (ρtot + ptot)A
], (B39)
with δpktot = δpkh,eff + δpkr and
δpkr =δρkr3
=4
3ζkr ρr. (B40)
The shift B is then determined from the lapse using thetrace-free Einstein equation
B′ + 2B = − k
aHA, (B41)
from which we can also see that the expansion shear−(k/aH)B is negligible for k/aH � 1 which is requiredfor the δN construction of curvature fluctuations to hold[27, 40]. Therefore the initial value for A satisfies theHamiltonian constraint
A = − 1
6
δρktot
H2+
1
3
k
aHB ' −1
6
δρktot
H2, (B42)
for k/aH � 1.We then solve the linearized Klein-Gordon equation
(B4) with (71) and the total momentum and metric equa-tions above.
Note that this construction assumes any momentumexchange between the Higgs and radiation fields impliedby the Klein-Gordon equation cancel to conserve the to-tal momentum. While this exchange itself may not befully accounted for by the additional effective potentialterms, above the horizon all momentum terms are negli-gible in their impact on energy density fluctuations and
δρ′h + 3(δρh + δph,eff) ' 0 (B43)
to leading order in k/aH. Similarly, though we do notexplicitly solve the analogous radiation continuity equa-tion since we assume ζkr is constant, we have validatedthat assumption by checking that its momentum sourceis negligible above the horizon. Moreover, we have alsochecked numerically that the lapse and shift perturba-tions have no significant impact on the evolution of δhk
or on the final ζ for superhorizon modes. This calculationvalidates the assumptions in the main text.
After solving the Higgs background and perturbationequations, as shown in §IV C, we find that the curvatureperturbation ζ evolves outside the horizon after inflation,as shown in Figs. 6–9. This occurs due to a nonadiabaticpressure on a surface of uniform total energy density ρtot,
δpNA,ktot = δpktot −
ptot
ρtotδρktot, (B44)
where all quantities are sums of the Higgs and radiationcomponents.
In §IV C, we saw that ζ oscillates within each Higgscycle. The total density perturbation has predominantlycanceling components while the adiabatic sound speedptot/ρtot oscillates only fractionally around 1/3. Thenonadiabatic pressure is then dominated by the pressureperturbations on the flat slice, with an oscillatory con-tribution from the Higgs, explaining the oscillation in ζ.The oscillations in ζ increase in amplitude over time, be-cause of the small anharmonic terms in the potential.Note that curvature oscillations should appear in otherrelated contexts, e.g. the curvaton scenario [41], thoughthey are usually averaged over.
Likewise on the cycle average, at first order the radi-ation pressure perturbation cancels the Higgs pressureperturbation and ζ is constant. However, there is a sec-ular drift to ζ induced by a small non-canceling pieceto the cycle-averaged total pressure perturbation. Thispiece decays as (ρtot + ptot), inducing a constant secu-lar contribution to ζ ′ and therefore the small linear driftin ζ which we discussed in §IV C. This is the usual en-tropy fluctuation mechanism to convert isocurvature tocurvature fluctuations through a change in the equationof state of the components [41, 42], applied here to theHiggs field.
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