Contents

CERN-TH-2018-065 IFUP-TH/2018

Dark Matter in the Standard Model?

[1cm] Christian Gross, Antonello Polosa,

Alessandro Strumia, Alfredo Urbano, Wei Xue

[7mm]


Dipartimento di Fisica dell’Università di Pisa and INFN, Sezione di Pisa, Italy

[1mm] Dipartimento di Fisica e INFN, Sapienza Università di Roma, I-00185, Roma, Italy

[1mm] Theoretical Physics Department, CERN, Geneva, Switzerland

[1mm] INFN, Sezione di Trieste, SISSA, via Bonomea 265, 34136 Trieste, Italy




Abstract


We critically reexamine two possible Dark Matter candidates within the Standard Model. First, we consider the hexa-quark. Its QCD binding energy could be large enough to make it (quasi) stable. We show that the cosmological Dark Matter abundance is reproduced thermally if its mass is . However, we also find that such mass is excluded by stability of Oxygen nuclei. Second, we consider the possibility that the instability in the Higgs potential leads to the formation of primordial black holes while avoiding vacuum decay during inflation. We show that the non-minimal Higgs coupling to gravity must be as small as allowed by quantum corrections, . Even so, one must assume that the Universe survived in independent regions to fluctuations that lead to vacuum decay with probability 1/2 each.

1 Introduction

In this work we critically re-examine two different intriguing possibilities that challenge the belief that the existence of Dark Matter (DM) implies new physics beyond the Standard Model (SM).

DM as the hexa-quark

The binding energy of the hexa-quark di-baryon is expected to be large, given that the presence of the strange quark allows it to be a scalar, isospin singlet [1], called or , and sometimes named exa-quark. A large binding energy might make light enough that it is stable or long lived. All possible decay modes of a free are kinematically forbidden if is lighter than about . Then could be a Dark Matter candidate within the Standard Model [2, 3, 4].

In section 2.1 we use the recent theoretical and experimental progress about tetra- and penta-quarks to infer the mass of the hexa-quark. In section 2.2 we present the first cosmological computation of the relic abundance, finding that the desired value is reproduced for . In section 2.3 we revisit the bound from nuclear stability ( production within nuclei) at the light of recent numerical computations of one key ingredient: the nuclear wave-function [5], finding that seems excluded.

DM as primordial black holes

Primordial Black Holes (PBH) are hypothetical relics which can originate from gravitational collapse of sufficiently large density fluctuations. The formation of PBHs is not predicted by standard inflationary cosmology: the primordial inhomogeneities observed on large cosmological scales are too small. PBH can arise in models with large inhomogeneities on small scales, . PBH as DM candidates are subject to various constraints. BH lighter than are excluded because of Hawking radiation. BH heavier than are safely excluded. In the intermediate region, a variety of bounds make the possibility that problematic but maybe not excluded — the issue is presently subject to an intense debate. According to [6] DM as PHB with mass are not excluded, as previously believed. And the HSC/Subaru microlensing constraint on PBH [7] is partially in the wave optics region. This can invalidate its bound below .

Many ad hoc models that can produce PBH as DM have been proposed. Recently [8] claimed that a mechanism of this type is present within the Standard Model given that, for present best-fit values of the measured SM parameters, the SM Higgs potential is unstable at  [9]. We here critically re-examine the viability of the proposed mechanism, which assumes that the Higgs, at some point during inflation, has a homogeneous vev mildly above the top of the barrier and starts rolling down. When inflation ends, reheating adds a large thermal mass to the effective Higgs potential, which, under certain conditions, brings the Higgs back to the origin,  [10]. If falling stops very close to the disaster, this process generates inhomogeneities which lead to the formation of primordial black holes. In section 3 we extend the computations of [8] adding a non-vanishing non-minimal coupling of the Higgs to gravity, which is unavoidably generated by quantum effects [11]. We find that must be as small as allowed by quantum effects. Under the assumptions made [8] we reproduce their results; however in section 3.6 we also find that such assumptions imply an extreme fine-tuning.

The first mechanism is affected by the observed baryon asymmetry, but does not depend on the unknown physics that generates the baryon asymmetry. The second possibility depends on inflation, but the mechanism only depends on the (unknown) value of the Hubble constant during inflation. In both cases the DM candidate is part of the SM. Conclusions are given in section 4.

2 DM as the hexa-quark

The hexa-quark is stable if all its possible decay modes are kinematically closed:

(1)

A stable is a possible DM candidate. A too light can make nuclei unstable. Scanning over all stable nuclei, we find that none of them gets destabilised by single emission if , with Li giving the potentially highest sensitivity to .

2.1 Mass of the hexa-quark from a di-quark model

We estimate the mass of the hexa-quark viewing it as a neutral scalar di-baryon constituted by three spin zero di-quarks

(2)

where are color indices. This is possible thanks to the strange quark, while spin zero di-quarks of the kind are forbidden by Fermi statistics because of antisymmetry in color and spin. We assume the effective Hamiltonian for the hexa-quark [12]

(3)

where the are effective couplings determined by the strong interactions at low energies, color factors, quark masses and wave-functions at the origin. The are the masses of the di-quarks in made of and constituent quarks [13]. is the spin of -th quark. Another important assumption, which is well motivated by studies on tetra-quarks [12], is that spin-spin interactions are essentially within di-quarks and zero outside, as if they were sufficiently separated in space.

Considering di-quark masses to be additive in the constituent quark masses, and taking and constituent quark masses from the baryons one finds

(4)

The chromomagnetic couplings could as well be derived in the constituent quark model using data on baryons

(5)

However it is known that to reproduce the masses of light scalar mesons, interpreted as tetraquarks,  [14], we need

(6)

Spin-spin couplings in tetra-quarks are found to be about a factor of four larger compared to the spin-spin couplings among the same pairs of quarks in the baryons, which make also di-quarks. It is difficult to assess if this would change within an hexa-quark. At any rate we can attempt a simple mass formula for

(7)

which in terms of light tetra-quark masses means . Using the determination of chromo-magnetic couplings from baryons we would obtain

(8)

whereas keeping the chromo-magnetic couplings needed to fit light tetra-quarks gives

(9)

if the same values for the chromo-magnetic couplings to fit light tetra-quark masses are taken (or 1.4 GeV using (6)). There is quite a lot of experimental information on tetra-quarks [12], whereas hexa-quarks, for the moment, are purely hypothetical objects. On purely qualitative grounds we might expect that the mass of could be closer to the heavier value being a di-baryon and not a di-meson (tetra-quark) like light scalar mesons.

In the absence of any other experimental information it is impossible to provide an estimate of the theoretical uncertainty on .

Lattice computations performed at unphysical values of quark masses find small values for the binding energy, about 13, 75, 20 MeV [15, 16, 17]. Extrapolations to physical quark masses suggest that does not have a sizeable binding energy, see e.g. [18]. Furthermore, the binding energy of the deuteron is small, indirectly disfavouring a very large binding energy for the (somehow similar) , which might too be a molecule-like state.111We thank M. Karliner, A. Francis, J. Green for discussions. Despite of this, we over-optimistically treat as a free parameter in the following.

We also notice that the particle could be much larger than what envisaged in [2, 3, 4] and that its coupling to photons, in the case of  fm size (see the considerations on diquark-diquark repulsion at small distances in [19]), could be relevant for momentum transfers as small as  MeV, compared to  GeV considered by Farrar.

2.2 Cosmological relic density of the hexa-quark

We here compute the relic density of Dark Matter, studying if it can match the measured value , i.e.  [20]. A key ingredient of the computation is the baryon asymmetry. Its value measured in CMB and BBN is . The DM abundance is reproduced (using  GeV for definiteness) for

(10)

Thereby the baryon asymmetry before decoupling must be

(11)

One needs to evolve a network of Boltzmann equations for the main hadrons: , , , , , , , and . Strange baryons undergo weak decays with lifetimes , a few orders of magnitude faster than the Hubble time. This means that such baryons stay in thermal equilibrium. We thereby first compute the thermal equilibrium values taking into account the baryon asymmetry. Thermal equilibrium implies that the chemical potentials satisfy

(12)

Their overall values are determined imposing that the total baryon asymmetry equals

(13)

The equilibrium values can be analytically computed in Boltzmann approximation (which becomes exact in the non-relativistic limit)

(14)

where the () holds for (anti)particles. We then obtain the abundances in thermal equilibrium plotted in fig. 1, assuming . We see that the desired abundance is reproduced if the interactions that form/destroy decouple at . This temperature is so low that baryon anti-particles have negligible abundances, and computations can more simply be done neglecting anti-particles.222Let us consider, for example, the process where denotes other SM particles that do not carry the baryon asymmetry, such as pions. Thermal equilibrium of the above process implies (15) Inserting with gives (16) Namely, at large ; at low . A DM abundance comparable to the baryon abundance is only obtained if reactions that form decouple at the in eq. (17). Then, the desired decoupling temperature is simply estimated imposing , and decreases if is heavier:

(17)
Figure 1: Left: thermal equilibrium values of hadron abundances assuming and : the desired abundances are obtained if decoupling of interactions occurs at . We see that in this phase anti-particles are negligible. Right: thermal equilibrium values of .

To compute the decoupling temperature, we consider the three different kind of processes that can lead to formation of :

  1. Strong interactions of two heavier QCD hadrons that contain the needed two quarks. One example is , where denotes pions. These are doubly Boltzmann suppressed by at temperatures .

  2. Strong interactions of one heavier strange hadron and weak interactions that form the other (as ) from lighter hadrons. One example is . These are singly Boltzmann suppressed by and by .

  3. Double-weak interactions that form two quarks starting from lighter hadrons. One example is . These are doubly suppressed by .

At the abundance of strange hadrons is still large enough that QCD processes dominate over EW processes: interactions that form and destroy proceed dominantly through QCD collisions of strange hadrons:

(18)

where can be a or a , as preferred by approximate isospin conservation. The can be substituted by the .

Figure 2: Thermal hexa-quark abundance within the SM with the baryon asymmetry in eq. (11).

Defining and , the Boltzmann equation for the abundance is

(19)

where the superscript ‘eqb’ denotes thermal equilibrium at fixed baryon asymmetry and is summed over all baryons, but the dibaryon . A second equation for is not needed, given that baryon number is conserved: . Furthermore, is negligible, and is negligible around decoupling. The production rate is obtained after summing over all processes of eq. (18). In the non-relativistic limit the interaction rate gets approximated as

(20)

The opposite process is more conveniently written in terms of the breaking width defined by and given by

(21)

This gets Boltzmann suppressed at , when hyperons disappear from the thermal plasma. Assuming , the Boltzmann equation is approximatively solved by evaluated at the decoupling epoch where , which corresponds to . This leads to the estimated final abundance

(22)

The fact that is in thermal equilibrium down to a few tens of MeV means that whatever happens at higher temperatures gets washed out. Notice the unusual dependence on the cross section for formation: increasing it delays the decoupling, increasing the abundance.

Fig. 2 shows the numerical result for the relic abundance, computed inserting in the Boltzmann equation a -wave , varied around . The cosmological DM abundance is reproduced for , while a large gives a smaller relic abundance. Bound-state effects at BBN negligibly affect the result, and in particular do not allow to reproduce the DM abundance with a heavier .

We conclude this section with some sparse comments. Possible troubles with bounds from direct detection have been pointed out in [4, 21, 22]: a DM velocity somehow smaller than the expected one can avoid such bounds reducing the kinetic energy available for direct detection. Using a target made of anti-matter (possibly in the upper atmosphere) would give a sharp annihilation signal, although with small rates. The magnetic dipole interaction of does not allow to explain the recent 21 cm anomaly along the lines of [23] (an electric dipole would be needed). The interactions of DM with the baryon/photon fluid may alter the evolution of cosmological perturbations leaving an imprint on the matter power spectrum and the CMB. However, they are not strong enough to produce significant effects. The particle is electrically neutral and has spin zero, such that its coupling to photons is therefore suppressed by powers of the QCD scale [3]. So elastic scattering of with photons is not cosmologically relevant.

A light would affect neutron stars, as they are expected to contain particles, made stable by the large Fermi surface energy of neutrons. Then, would give a loss of pressure, possibly incompatible with the observed existence of neutron stars with mass  [1]. However, we cannot exclude on this basis, because production of hyperons poses a similar puzzle. as DM could interact with cosmic ray giving and photon and other signals [24] and would be geometrically captured in the sun, possibly affecting helioseismology.333We thank M. Pospelov for suggesting these ideas.

In the next section we discuss the main problem which seems to exclude as DM.

Figure 3: Sample diagram that dominates nucleon decay into inside nuclei. The initial state can also be or .

2.3 Super-Kamiokande bound on nuclear stability

Two nucleons inside a nucleus can make a double weak decay into , emitting , or  [2]. This is best probed by Super-Kamiokande (SK), which contains Oxygen nuclei. No dedicated search for (where can be one or two and can be , depending on the charge of ) has been performed,444SK searched for di-nucleon decays into pions [25] and leptons [26] and obtained bounds on the lifetime around years. However these bounds are not directly applicable to where the invisible takes away most of the energy reducing the energy of the visible pions and charged leptons, in contrast to what is assumed in [25, 26]. but a very conservative limit

(23)

is obtained by requiring the rate of such transitions to be smaller than the rate of triggered background events in SK, which is about  Hz [27]. A more careful analysis would likely improve this bound by three orders of magnitude [2].

Figure 4: Wave functions as function of the relative distance between two nucleons in . The darker (ligther) green curve shows , obtained from [5], using the AV18 (AV18+UIX) potential, the darker (ligther) red curve shows using the AV18 (AV18+UIX) potential, the blue curve is the MS wave function used in [2], the dashed black curve is the BBG wave function with hard core radius . Left: Wave functions over the entire range. Right: Zoom-in to the region relevant for our calculation.

The amplitude for the formation of is reasonably dominated by the sample diagram in fig. 3: doubly-weak production of two virtual strange baryons (e.g. through and ; at quark level and ), followed by the strong process :

(24)

The predicted life-time is then obtained as [2]555A numerical factor of 1440 due to spin and flavor effects has already been factored out from here and in the following. Note also that the threshold GeV neglects the small difference in binding energy between and .

(25)

where the smaller value holds if is so light that the decay can proceed through real or emission, while the longer life-time if obtained if instead only lighter or can be emitted.

The key factor is the dimension-less matrix element for the transition inside a nucleus, that we now discuss. Following [2], we assume that the initial state wave function can be factorized into wave functions of the two baryons and a relative wave function for the separation between the center of mass of the ’s. The matrix element is given by the wave-function overlap

(26)

Here, are center-of-mass coordinates which parametrise the relative positions of the quarks within each . Using the Isgur-Karl (IK) model [28] the wave functions for the quarks inside the and inside the are approximated by

(27)
(28)

where and are the radii of the nucleons respectively of .666One should be aware that the IK model has serious shortcomings. One issue is that it is a non-relativistic model — an assumption which is problematic in particular for small . Another problem is that the value of that gives a good fit to the lowest lying and baryons —  fm — is smaller than the charge radius of the proton:  fm. Therefore we consider both  fm and  fm, as done in [2]. Performing all integrals except the final integral over gives

(29)

As shown in fig. 5 below (and as discussed in [2]), if is not much smaller than , the overlap integral is not very much suppressed and is tens of orders of magnitude below the experimental limit, and is clearly excluded. This conclusion is independent of the form of .

However if were a few times smaller than — a possibility which seems unlikely due to diquark repulsions (see e.g. [19]) but cannot firmly be excluded — then is extremely sensitive to the probability of the overlap of two nucleons inside the oxygen core at very small distances (less than, say, 0.5 fm). The wave function of nucleon pairs at such small distances has not been probed experimentally. In fact, at such small distances nucleons are not the appropriate degrees of freedom.777Data indicate that about 20% of the nucleons form pairs so close (about 1 fm) that the local density reaches the nucleon density (about 2.5 times larger than the nuclear density) and thus that the quark structure of nucleons starts becoming relevant already at 1 fm [29]. Thus, for a very small one can only make an educated guess of , since the form of is uncertain. Nevertheless, we will show in the following that for a reasonable form of a stable is excluded even if it were very small.

Figure 5: The dimension-less squared matrix element for nuclear decay into , , as a function of the radius in units of the nucleon radius , using different nuclear wave functions. The color coding, defined in fig. 4, refers to the nuclear wave functions used. The thinner (thicker) curves assume (). The Super Kamiokande bound would be evaded for for (for ).

Numerical computations of the ground-state wave-functions of nuclei, including have been performed e.g. in [5]. The quantity that determines is the two-nucleon point density , defined in eq. (58) of [5]. We obtain by interpolating the data given in [5] and adding the constraint , which is a conservative assumption for our purposes since would lead to a larger matrix element. There are 28 neutron-neutron pairs and 64 proton-neutron pairs in so one has and . We therefore define the wavefunctions

(30)

These wave functions are plotted in fig. 4, together with the Miller-Spencer (MS) and the Brueckner-Bethe-Goldstone (BBG) wave function used in [2]. The BBG wave functions assume a hard repulsive core between nucleons such that vanishes at . We take for illustration. This is not realistic but allows to see what kind of nuclear wave function would sufficiently suppress the rate of -formation in nuclei, if is small enough. The resulting is plotted in fig. 5, again compared to that obtained using the Miller-Spencer and BBG wave functions.

The resulting matrix elements from the MS wave function qualitatively agree to what is obtained using the wave functions extracted from [5]. By contrast, the matrix element using the BBG wave function with hard core radius is very much suppressed, especially if is small. The reason is that, according to the assumption of a hard core repulsive potential, the nucleons can’t get close enough to form the small state . Since we do not consider a which vanishes for  fm realistic, we conclude that a stable is excluded.

Weaker bounds on production are obtained considering baryons containing ’s.

Figure 6: Left: RGE running of (solid red lines) obtained by changing the top mass in its interval defined by . For comparison, we show the running of at 3-loops (dashed blue lines) without including the Coleman-Weinberg corrections. Right: RGE running of a small .

3 DM as black holes triggered by Higgs fluctuations

We here present the technical computations relative to the mechanism anticipated in the Introduction. The SM potential is summarized in section 3.1. In section 3.2 we outline the mechanism that generates black holes. Section 3.3 studies the generation of Higgs inhomogeneities. Post-inflationary dynamics is studied in section 3.4. Formation of black holes is considered in section 3.5. The viability of a critical assumption is discussed in section 3.6.

3.1 The Higgs effective potential

The effective potential of the canonically normalised Higgs field during inflation with Hubble constant is

(31)

at . Here is the effective quartic coupling computed including quantum corrections. The second mass term in can be generated by various different sources [8]. We consider the minimal source: a Higgs coupling to gravity, , with Ricci scalar during inflation. Finally, during inflation the effective potential in eq. (31) is augmented by the vacuum energy associated to the inflaton sector, , where GeV is the reduced Planck mass.

We implement the RG-improvement of the effective potential at NNLO precision: running the SM parameters at 3-loops and including -loop quantum corrections to the effective potential. We consider fixed values of and GeV, and we vary the main uncertain parameter, the top mass, in the interval  [32]. In fig. 6a we show the resulting as function of .

The non-minimal coupling to gravity receives SM quantum corrections encoded in its RGE, which induce even starting from at some energy scale. The RGE running of small values of is shown in fig. 6b. As mentioned before, a non-zero can be considered as a proxy for an effective mass term during inflation. The latter, for instance, can be generated by a quartic interaction between the Higgs and the inflaton field or by the inflaton decay into SM particles during inflation. For this reasons, it makes sense to include as a free parameter in the analysis of the Higgs dynamics during inflation, at most with the theoretical bias that its size could be loop-suppressed.

Analytic approximation

We will show precise numerical results for the SM case. However the discussion is clarified by introducing a simple approximation that encodes the main features of the SM effective potential in eq. (31):

(32)

where is the position of the maximum of the potential with no extra mass term, . The parameters and depend on the low-energy SM parameters such as the top mass: they can be computed by matching the numerical value of the Higgs effective potential at the gauge-invariant position of the maximum, . The result is shown in the right panel of fig. 7.

Results will be better understood when presented in terms of the dimensionless parameters , , and , where is the temperature, as they directly control the dynamics that we are going to study. The parameter controls the flatness of the potential beyond the potential barrier at , with smaller corresponding to a flatter potential. The non-minimal coupling controls the effective Higgs mass during inflation. Finally will set the reheating temperature in eq. (35) and thus the position and size of the thermal barrier.

The position of the potential barrier — defined by the field value where the effective potential has its maximum — strongly depends on the value of the top mass, on the non-minimal coupling to gravity, and, after inflation, on the temperature of the thermal bath which provides and extra mass term. For , the maximum of the Higgs potential gets shifted from to

(33)

where is the product-log function defined by . The condition

(34)

must be satisfied otherwise the effective mass is too negative and it erases the potential barrier, thus leading to a classical instability.

The thermal potential

After the end of inflation, the Higgs effective potential receives large thermal corrections from the SM bath at generic temperature . The initial temperature of the thermal bath is fixed by the dynamics of reheating after inflation. We assume instantaneous reheating, as this is most efficient for rescuing the falling Higgs field. The reheating temperature is then given by

(35)

where is the number of SM degrees of freedom. After reheating the Universe becomes radiation-dominated, the Ricci scalar vanishes, and so the contribution to the effective potential from the non-minimal Higgs coupling to gravity.

The effective Higgs potential at finite temperature is obtained adding an extra thermal contribution which can be approximated as an effective thermal mass for the Higgs field, (see e.g. [10])

(36)

At we can neglect the exponential suppression in the thermal mass, and the maximum of the effective potential in eq. (36) is given by

(37)
Figure 7: Left: Number of -folds at the beginning of the Higgs fall that gives the maximal rescued by the reheating temperature. This is computed as function of the Hubble constant during inflation, for 3 different values of the uncertain parameter that approximates the Higgs potential. Continuous (dashed) curves correspond to (). Right: SM values of and of the position of the top of the SM potential as function of the top mass.

3.2 Outline of the mechanism

During inflation, the Higgs field is subject to quantum fluctuations. Depending on the value of , these quantum fluctuations could lead the Higgs beyond the barrier, and make it roll towards Planckian values. If is high enough and is not too far, thermal corrections can “rescue” the Higgs, bringing it back to the origin [10]. The mechanism relies on a tuning such that the following situation occurs [8]:

  1. At -folds before the end of inflation, the Higgs background value is brought by quantum fluctuation to some . This configuration must be spatially homogeneous on an inflating local patch large enough to encompass our observable Universe today. We consider the de Sitter metric in flat slicing coordinates, . We will discuss later how precisely this assumption must be satisfied, and its plausibility.

  2. When the classical evolution prevails over the quantum corrections, the Higgs field, starting from the initial position , begins to slow roll down the negative potential. This condition reads

    (38)

    where is a constant of order 1, fixed to in [8]. We will explore what happens choosing or . From this starting point on, the classical evolution of the background Higgs value is described by

    (39)

    where the subscript indicates that this is a classical motion. Dots indicate derivatives with respect to time .

  3. At the end of inflation, , the Higgs is rescued by thermal effects. This happens if the value of the Higgs field at the end of inflation is smaller than the position of the thermal potential barrier at reheating, . A significant amount of PBH arises only if this condition is barely satisfied in all Universe. This is why the homogeneity assumption in is needed.

To compute condition we fix the initial value of the classical motion such that eq. (38) is satisfied with ; next, we maximise the obtained solving eq. (39) by tuning the amount of inflation where the fall happens, as parameterized by . The left panel of fig. 7 shows the initial value obtained following this procedure as a function of in units of . Smaller values of (i.e. smaller values of ) imply a flattening of the potential, and the classical dynamics during inflation is slower. The right side of the curves is limited by the classicality condition in eq. (38). A shifts the position of the potential barrier towards the limiting value in eq. (37) — which does not depend on — above which the rescue mechanism due to thermal effects is no-longer effective: its net effect is to reduce the number of -folds during which classical motion can happen (for fixed ).

We anticipate here the feature of PBH formation which implies the restriction on the parameter space mentioned at point : Higgs fall must start at least -folds before the end of inflation. The collapse of the mass inside the horizon -folds before inflation end forms a PBH with mass (see also section 3.5)

(40)

PBH must be heavy enough to avoid Hawking evaporation. The lifetime of a PBH with mass due to Hawking radiation at Bekenstein-Hawking temperature is

(41)

where at g. BH heavier than g are cosmologically stable, and BH heavier than g are allowed by bounds on Hawking radiation as a (significant fraction of) DM. Since , imposing g implies a conservative lower limit on :

(42)

3.3 Higgs fluctuations during inflation

We now consider the evolution of Higgs perturbations during inflation. Expanding in Fourier space with comoving wavenumber ,888The comoving wavenumber is time independent, and it is related to the physical momentum via , which decreases as the space expands. the equation for the mode takes the form

(43)

where we neglected metric fluctuations. In terms of the number of -folds and of the Mukhanov-Sasaki variable it becomes

(44)

It is convenient to consider the evolution of the perturbation making reference to a specific moment before the end of inflation: at the initial value defined in section 3.2. We recall that in our convention at the end of inflation. Eq. (44) becomes

(45)

In this form, the Mukhanov-Sasaki equation is particularly illustrative. Consider the evolution of the perturbation for a mode of interest that we fix compared to the reference value at . In particular, we consider the case of a mode that is sub-horizon at the beginning of the classical evolution, that is . From eq. (45), we see that in the subsequent evolution with the exponential suppression will turn the mode from sub-horizon to super-horizon.

Figure 8: Left: Sample evolution of the classical Higgs background (, red solid line) and of a perturbation with (dashed lines). Right: Higgs curvature perturbation during inflation. We compare the full numerical result with the analytical approximation (last term in eq. (52), solid horizontal black line). The vertical dashed gray line marks the instant of horizon exit. We use the analytical approximation in eq. (32) with , (which corresponds to GeV) and .

We are now in the position to solve eq. (45). To this end, we need boundary conditions for and its time derivative. We use the Bunch-Davies conditions at for modes that are sub-horizon at the beginning of the classical evolution, , and we treat the real and imaginary part of separately since they behave like two independent harmonic oscillators for each comoving wavenumber . At generic -fold time , the perturbation is related to the Mukhanov-Sasaki variable by

(46)

We show in the left panel of fig. 8 our results for the time evolution of the classical background and the perturbation (both real and imaginary part) during the last -folds of inflation. As a benchmark value, we consider an initial sub-horizon mode with . After few -folds of inflation such mode exits the horizon: oscillations stop, and from this point on, further evolution is driven by the time derivative of the classical background. This is a trivial consequence of the equations of motion on super-horizon scales. Differentiating eq. (39) with respect to the cosmic time shows that and satisfy the same equation on super-horizon scales, and, therefore, they must be proportional, for  [8]. The proportionality function can be obtained by a matching procedure. Deep inside the horizon, in the limit , the Mukhanov-Sasaki variable reproduces the preferred vacuum of an harmonic oscillator in flat Minkowski space, and we have, after introducing the conformal time as , . Roughly matching the absolute value of the solutions at horizon crossing we determine the absolute value of as

(47)

where we indicate with the time of horizon exit for the mode — the time at which (equivalently, ). The number of -fold at horizon exit is given by

(48)

Primordial curvature perturbations

The primordial curvature perturbation