A Cosmological Signature of the Standard Model Higgs Vacuum Instability: Primordial Black Holes as Dark Matter

For the current central values of the Higgs and top masses, the Standard Model Higgs potential develops an instability at a scale of the order of $10^{11}$ GeV. We show that a cosmological signature of such instability could be dark matter in the form of primordial black holes seeded by Higgs fluctuations during inflation. The existence of dark matter might not require physics beyond the Standard Model.

Introduction.It has been known since a long time that the Standard Model (SM) Higgs potential develops an instability at large field values [1,2].For the current central values of the Higgs and top masses, the quartic coupling λ in the Higgs potential becomes negative for Higgs field values of the order of 10 11 GeV,1 making our electroweak vacuum not the one of minimum energy.While some take this as motivation for the presence of new physics to change this feature, this is not necessarily a drawback of the SM.Indeed, our current vacuum is by far stable against both quantum tunneling in flat spacetime and thermal fluctuations in the early universe [2,4].
The situation is different when considering the SM during inflation [5].Assuming that the inflationary stage starts with a vanishing vacuum expectation value of the SM Higgs, if the effective mass of the Higgs is smaller than the Hubble rate H during inflation, quantum mechanical excitations of the Higgs will push the Higgs away from its initial value.The classical value (the long wavelength mode) of the Higgs field randomly walks receiving kicks of the order of ±(H/2π) each Hubble time and can surmount the potential barrier and fall deep into the unstable side of the potential [5][6][7].At the end of inflation, patches which have experienced this phenomenon will be anti-de Sitter regions and they are lethal for our universe as they grow at the speed of light [8].From this result, one can derive upper bounds on the Hubble constant during inflation, which depend on the reheating temperature and on the Higgs coupling to the scalar curvature or to the inflaton [8,9].
The fact that the upper bound on the Hubble rate during inflation depends on the reheating temperature T RH is intuitive: for sufficiently large values of T RH , patches in which the Higgs field probes the unstable part of the potential can be recovered thanks to the thermal effects after inflation.Indeed, the mass squared of the Higgs field receives a positive correction proportional to T 2 in such a way that in those would-be dangerous regions the Higgs field can roll back down to the origin and be safe.
As illustrated above, the physical implications of living in a metastable electroweak vacuum are fascinating and have far-reaching consequences for cosmology.This has triggered much activity in a field that involves inflationary dynamics, the physics of preheating, the interplay between Higgs properties and observables of cosmological interest, etc.In spite of this richness, a word of warning is in order: the energy scale of this physics is very high and we have no smoking-gun signature (comparable to proton decay for GUTs) that the electroweak vacuum metastability is actually realized in nature (with the exception of the vacuum decay itself!).
In view of this, one reasonable question to ask is how can we probe, even if indirectly, the SM Higgs vacuum instability.In this short note we will argue that there might be a cosmological signature of the SM vacuum instability: the very presence of dark matter in our universe.We will argue that the origin of dark matter does not need physics beyond the SM: dark matter may be associated to primordial black holes seeded by the perturbations of the Higgs itself generated during the last stages of inflation.
The picture we envisage is the following.During inflation there are patches where the Higgs has been pushed by quantum fluctuations beyond the potential barrier and is classically rolling down the slope away from it.Higgs fluctuations do not contribute significantly to the total curvature perturbation ζ which is ultimately responsible for the anisotropies in the Cosmic Microwave Background (CMB) and in the Large-Scale Structure (LSS) on observable scales.Higgs perturbations instead grow to relatively large values in the last e-folds of inflation, which are irrelevant for observations in the CMB and in the LSS.When inflation ends and reheating tales place, these regions are rescued by thermal effects and the Higgs rolls down to the origin of its potential.At later times, the Higgs perturbations reenter inside the Hubble radius and, if they are large enough, they provide high peaks in the matter power spectrum which give rise to Primordial Black Holes (PBH). 2 We show that these PBHs can provide the dark matter we see in the universe today.
In this sense, and within a more anthropic attitude, one could say that the electroweak SM instability is beneficial to our own existence as dark matter is necessary to form structures.In the absence of other dark matter candidates, the SM would be able to provide the right dark matter abundance.As discussed below, although the parameter choices needed for PBH formation might seem finetuned, they would be motivated anthropically.In particular, this mechanism offers an anthropic explanation of why the electroweak vacuum is metastable (but near-critical, very close to being stable). 3he dynamics during inflation.From now on we concentrate on an inflating local patch which is sufficiently large to encompass our observable universe today.We will be agnostic about the details of the model of inflation and the origin of the curvature perturbation responsible for the CMB anisotropies and LSS on large scales, which we call ζ st .This ζ st might be caused by a single degree of freedom [12] or by another mechanism such as the curvaton [13].Also, we will take a constant Hubble rate during inflation and suppose that it ends going through a period of reheating characterised by a reheating temperature T RH .Of course, one can repeat our calculations within a preferred model of inflation.
We suppose that the Hubble rate during inflation is large enough to have allowed the SM Higgs to randomly walk above the barrier of its potential and therefore to probe the potentially dangerous unstable region.As a representative value we take H 10 12 GeV.
Despite the Higgs negative potential energy, this region keeps inflating as long as the total vacuum energy during inflation is larger, that is, for where h c is the Higgs classical value and m P = 2.4 × 10 18 GeV is the reduced Planck mass.The equation of motion of the classical value of the SM Higgs is where, as usual, dots represent time derivatives and tildes field derivatives.For the sake of simplicity, from now on we will approximate the potential as with λ > 0 running logarithmically with the field scale.
During inflation, λ should in fact be evaluated at a scale µ given by µ 2 h 2 c + H 2 [7].A typical value (for h c 10 12 GeV) is λ 10 −2 .In order to make any prediction deterministic and not subject to probability arguments, we are interested in the regime in which the dynamics of the zero mode of the Higgs is dominated by the classical motion rather than by the randomness of the fluctuations.We require therefore that in a Hubble time, ∆t = 1/H, the classical displacement of the Higgs is larger (in absolute value) than the quantum jumps This implies that, inside the inflating region, h c must be bounded from below We define the initial time at which the Higgs starts its classical evolution by t * .In this estimate we have assumed that the motion of the Higgs is friction dominated, that is This is true as long as If so, the Higgs is slowly moving for a sufficient number of e-folds.Eqs. ( 6) and ( 8) provide the allowed range for the motion of the Higgs and we will assume from now on that the patch under consideration is characterized by such values.Under these circumstances, the evolution of the classical value of the Higgs is where we have introduced the number of e-folds till the end of inflation N and denoted by h e the value of the classical Higgs field at the end of inflation.Meanwhile, Higgs fluctuations are generated.Perturbing around the slowly-rolling classical value of the Higgs field and accounting for the metric perturbations as well, one finds that the Fourier transform of the perturbations of the Higgs field satisfy the equation of motion (in the flat gauge) where a is the scale factor and the last term accounts for the backreaction of the metric perturbations.Driven by the Higgs background evolution in the last e-folds of inflation, the Higgs perturbations grow significantly after leaving the Hubble radius.The reason is the following.Having numerically checked that the last term in Eq. ( 10) is negligible, the Higgs perturbations and ḣc solve the same equation on scales larger than the Hubble radius k aH, as can be seen by taking the time derivative of Eq. ( 2).Therefore the two quantities must be proportional to each other during the evolution and on super-Hubble scales Matching at Hubble crossing k = aH this super-Hubble solution for δh k with its standard wave counterpart on sub-Hubble scales implies that where t k is the instant of time when the mode with wavelength 1/k leaves the Hubble radius.The growth of δh k is therefore dictated by the growth of ḣc .These are the Higgs perturbations which will be responsible for the formation of PBHs.In fact, we should deal with the comoving curvature perturbation ζ which is gauge-invariant and reads (still in the flat gauge) where ζ h is the Higgs perturbation and we assume ζ st to be conserved during inflation on super-Hubble scales and, for simplicity, that there is no energy transfer with the Higgs fluctuations.Notice that in the curvaton model, for instance, ζ st could be even zero on large scales during inflation.Using Eqs. ( 2) and ( 10) (again with the negligible last term dropped), one then obtains Since ρh = ḣc ( ḧc +V (h c )) = −3H ḣ2 c , one can easily show (and we have checked it numerically) that during inflation and on super-Hubble scales ζ h reaches the plateau This is the quantity which gives the largest contribution to ζ in the last few e-folds before the end of inflation.
Dynamics after inflation: reheating.At the end of inflation, the vacuum energy which has driven inflation gets converted into thermal relativistic degrees of freedom, a process commonly dubbed reheating.For simplicity, we suppose that this conversion is instantaneous, in such a way that the reheating temperature is obtained by energy conservation and taking the number of relativistic degrees of freedom to be about 10 2 .For our representative value of H = 10 12 GeV, we obtain T RH 10 15 GeV.Due to the thermal effects, the Higgs potential receives thermal corrections such that the potential is quickly augmented by the term [8] (a fit that is accurate for h 10 T in the region of interest and includes the effect of ring resummation).If the maximum temperature is larger than the value of the Higgs h e at the end of inflation, or more precisely if the corresponding patch is thermally rescued and the initial value of the Higgs immediately after the end of inflation coincides with h e .The classical value of the Higgs field starts oscillating around the origin, see Fig. 1.The Higgs fluctuations oscillate as well with the average value remaining constant and the amplitude slowly increasing for a fraction of e-folds.At the same time, the curvature perturbation, with power spectrum given in Fig. 2, gets the largest contribution from the Higgs fluctuations.After inflation, the long wavelength Higgs perturbations decay after several oscillations into radiation curvature perturbation which, being radiation now the only component, will stay constant on super-Hubble scales.We have taken the Higgs damping rate to be γ h ∼ 3g 2 T 2 /(256πm T ) ∼ 10 −3 T [14] (where g is the SU(2) L coupling constant).This value has been derived by noticing that for a thermal Higgs mass m T 0.34 T , the one-loop absorption and direct decay channels for quarks and gauge bosons are forbidden, and the damping occurs through the two-loop diagrams involving gauge bosons.Therefore, we have evaluated the value of the curvature perturbation after a fraction of e-fold.FIG.1: Evolution of H, T , hc, δh k during the last e-folds of inflation, for k = 50 a(t * )H where t * is defined to be the time when hc starts its classical evolution.The region of hc beyond the top of the potential barrier is shaded green.Generation of Primordial Black Holes.After the end of inflation, the Hubble radius grows and the perturbations generated during the last e-folds of inflation are the first to reenter the horizon.If they are large enough, they will collapse to form PBHs almost immediately after horizon reentry, see for instance Ref. [15] and references therein.
There is a critical value ∆ c for the density contrast (during the radiation era) above which a given region collapses to a PBH.This value is typically ∆ c ∼ 0.45 [16].As a result, in order to obtain a significant number of PBHs, the power spectrum on small scales must be sizeable.The mass of a PBH at formation and corresponding to the density fluctuation leaving the Hubble radius N e-folds before the end of We first define the variance of the density contrast where W (k, R) is a Gaussian window function smoothing out the density contrast on the comoving horizon length R ∼ 1/aH.The mass fraction β(M ) of the universe which ends up into PBHs at the time of formation t M is The total contribution of PBHs at radiation-matter equality is obtained by integrating the corresponding fraction β(M, t eq ) = a(t eq )/a(t M )β(M ) [18] at the time of equivalence where M ev (t eq ) 10 −21 M is the lower mass which has survived evaporation at equality and M (t eq ) is the horizon mass at equality (which for our purposes can be taken equal to infinity).Fig. 3 shows the resulting mass spectrum of PBHs at their formation time.The position of the peak in the PBH mass spectrum is set by the mode k * that exits the Hubble radius during inflation when the Higgs zero mode starts its classical evolution.The width of the peak in Fig. 3 determines the width of the first peak in Fig. 2.This implies that the PBH mass spectrum peaks between a maximum mass M * (corresponding to the mode k * ) and M * /e 2 ∼ 0.1 M * , corresponding to the mode leaving the Hubble radius when its oscillating amplitude was at the minimum.
To be on the safe side we ask that the interesting range of PBH masses is large enough to avoid the bounds from evaporating PBHs by the present time.This requires the dynamics to last about 17 e-folds before the Higgs hits the pole in Eq. (9).Interestingly this can be achieved in the SM for realistic values of the Higgs and top masses and α s : In our numerical example we use M h = 125.09GeV, M t = 172 GeV, and α s = 0.1184.
From the time of equality to now, the PBH mass distribution will slide to larger masses due to merging.While the final word can only be said through N-body simulations, one can expect merging to shift the spectrum to higher masses even by orders of magnitude [20,21] and to spread the spectrum, but maintaining the abundance.Accretion, on the other hand, increases both the masses and the abundance of PBHs as dark matter.On the other side, both merging and accretion help to render the PBHs Conclusions.If the scenario we have presented were in fact realized in nature, we can highlight three points as the most relevant.First, the SM would be capable of explaining dark matter by itself (supplemented by a period of inflation that is well motivated by other reasons).This has a double side: the SM provides a dark matter candidate in the form of PBHs and also provides the mechanism necessary to create the PBH seeds during inflation via the quantum fluctuations of the Higgs field in the unstable part of the Higgs potential.Both aspects (dark matter candidate and PBH generation mechanism) go against the common lore that physics beyond the SM is needed.In fact, if this scenario were correct, the Higgs field would not only be responsible for the masses of elementary particles but also for the dark matter content of our universe.Second, the PBH generation mechanism gives an anthropic handle on Higgs near-criticality which would be explained as needed to get sufficient dark matter so that large enough structures can grow in the universe.Finally, the PBHs responsible for dark matter would represent a conspicuous cosmological signature of the actual existence of an unstable range in the Higgs potential at large field values.
[19] 3:The spectrum of PBHs at formation generated by the mechanism we discuss (in red), superimposed with the experimental constraints on monocromatic spectra of PBH (from Ref.[19]and references therein): in yellow, the observations of extra-galactic γ-ray background; in blue, femto-, micro-and milli-lensing observations from Fermi, Eros, Kepler, Subaru HSC; in green, dynamical constraints from White Dwarves and Ultra-Faint Dwarf galaxies; in orange, constraints from the CMB.