The Higgs Boson can delay Reheating after Inflation
The Standard Model Higgs boson, which has previously been shown to develop an effective vacuum expectation value during inflation, can give rise to large particle masses during inflation and reheating, leading to temporary blocking of the reheating process and a lower reheat temperature after inflation. We study the effects on the multiple stages of reheating: resonant particle production (preheating) as well as perturbative decays from coherent oscillations of the inflaton field. Specifically, we study both the cases of the inflaton coupling to Standard Model fermions through Yukawa interactions as well as to Abelian gauge fields through a Chern-Simons term. We find that, in the case of perturbative inflaton decay to SM fermions, reheating can be delayed due to Higgs blocking and the reheat temperature can decrease by up to an order of magnitude. In the case of gauge-reheating, Higgs-generated masses of the gauge fields can suppress preheating even for large inflaton-gauge couplings. In extreme cases, preheating can be shut down completely and must be substituted by perturbative decay as the dominant reheating channel. Finally, we discuss the distribution of reheat temperatures in different Hubble patches, arising from the stochastic nature of the Higgs VEV during inflation and its implications for the generation of both adiabatic and isocurvature fluctuations.
Inflation was first proposed by Guth in Ref. Guth:1980zm () to solve several cosmological puzzles that plagued the old Big Bang Theory: an early period of accelerated expansion explains the homogeneity, isotropy, and flatness of the Universe, as well as the lack of relic monopoles. Subsequently, Linde Linde:1981mu (), as well as Albrecht and Steinhardt Albrecht:1982wi () suggested rolling scalar fields as a mechanism to drive the dynamics of inflation. In these models, quantum fluctuations of the inflaton field generate density fluctuations that leave imprints on the cosmic microwave background (CMB) and lead to the formation of the large scale structure in the Universe, i.e. the galaxies and clusters we observe today. After driving the exponential expansion of the Universe, inflation must also successfully reheat the Universe to ensure the transition to a radiation dominated state before Big Bang Nucleosynthesis (BBN). In models with rolling scalar fields, the transfer of energy from the inflaton to relativistic particles can be facilitated by couplings of the inflaton to Standard Model (SM) fields111There is also the possibility of the inflaton transferring its energy to one or more intermediary fields, which then couple to SM particles. We will not delve into such models here, due to their inherent ambiguity.. While the details of such couplings are highly model dependent, in general, the associated reheating can occur through the perturbative decay of the inflaton or through resonant particle production (preheating). Although the decay products of the inflaton are usually considered to be massless, recent work suggests that SM fields can acquire large masses due to the Higgs boson condensate which develops during inflation Enqvist:2013kaa (). In this work we examine the delay of the reheating process due to large SM masses during and after inflation and the possible implications of inhomogeneities arising from the stochastic motion of the Higgs field during inflation.
Enqvist et al. Enqvist:2013kaa () showed that during inflation, the SM Higgs boson can develop a mass and electroweak (EW) symmetry can be treated as effectively broken Kusenko:2014lra (). The effective Higgs mass arises due to quantum fluctuations of the Higgs field — similar to the quantum fluctuations of the inflaton field that generate the density fluctuations responsible for large scale structure Starobinsky:1994bd (). The expectation value of the Higgs amplitude over the entire inflating patch is vanishing due to the symmetric potential (where subscript indicates initial value at the onset of inflaton oscillations). However, the variance is non-zero and the typical Higgs amplitude (the effective Vacuum Expectation Value or VEV for short) in a random Hubble patch at the end of inflation is given by a root mean square value where is the Hubble scale at the end of inflation. The effective nonzero Higgs VEV during inflation then gives mass to all SM particles that couple to the Higgs. For reheating to occur, the inflaton must decay to other particles; yet this decay may be blocked if the inflaton mass is lower than the mass of decay products induced by the effective nonzero Higgs VEV . In short, in any model where the inflaton decays to SM particles which are coupled to the Higgs, reheating can be delayed until the Hubble-valued Higgs condensate dissolves. The same analysis as the one presented here holds in principle if the inflaton is coupled to a similarly Higgsed Dark sector, although the quantitative details can differ.
The lower reheating temperature that results due to the nonzero Higgs mass during inflation has various consequences. In the approximation that the Higgs condensate acquires a space-independent VEV, a constant suppression of the reheat temperature occurs across the observable Universe (however, see below for discussion of inhomogeneties). A global suppression of the reheat temperature may affect constraints on inflation models arising from the CMB which are often portrayed in the “” plane. Here, refers to the spectral index of the density perturbations produced by the model and is the tensor-to-scalar ratio, i.e. the ratio of the amplitude of gravitational waves to density perturbations. Predictions of specific inflation models may be visually compared to data from CMB experiments by plotting both the theoretical predictions and the data in the plane. However, the location of the predictions in the plane depends on the reheat temperature Cook:2015vqa (); Liddle:2003as (); Martin:2006rs (); Lorenz:2007ze (); Adshead:2010mc (); Easther:2011yq (); Dai:2014jja (); Martin:2014nya (); Drewes:2015coa (); Feng:2003nt (). The delayed reheating we find in this paper generally shifts predictions towards lower values of and larger values of . Since current constraints on inflationary models consider scenarios in which reheating occurs at an unspecified point between the end of inflation and BBN, our work does not aim to extend these bounds, but rather to shrink them by providing a more accurate description of (p)reheating, given a specific model.
Another consequence of the lower reheat temperature is the possible effect on baryogenesis. Several models have been proposed that tie the origin of the matter/antimatter asymmetry of the Universe to inflation. In these models a lepton (helicity) asymmetry is produced at the end of inflation and is later converted to a baryon asymmetry through the electroweak sphaleron process Giudice:1999fb (); Adshead:2015jza (); Kusenko:2014lra (); Pearce:2015nga (); Yang:2015ida (); Alexander:2004us (); Caldwell:2017chz (). In these models, if the reheat temperature is too high, the lepton asymmetry generated during reheating could be suppressed by rapid lepton number violating interactions Yang:2015ida (); Adshead:2017znw (). A lower reheat temperature could avoid such processes.
However, the stochastic evolution of the Higgs field during inflation will inevitably lead to an inhomogeneous delay of the reheating process, hence the Universe will be comprised at the end of inflation by a collection of Hubble patches with different reheat temperatures. This can lead to both adiabatic and (in some leptogenesis models) baryon isocurvature perturbations. The study of fluctuations arising due to the stochasticity of the Higgs VEV opens up a new avenue for analyzing and constraining inflationary models or mapping the Higgs potential at high energies. We will refrain from a detailed discussion of these aspects until section V.
Reheating can occur in multiple stages, including preheating (resonant particle production), followed by perturbative decays (from coherent oscillations of the inflaton field)222Depending on the specific model, each of these two stages can be subdominant or even absent.. In this paper, we assume that the inflaton couples primarily to SM particles that develop masses when the Higgs field acquires a mass. Instead, if the inflaton were to decay to a massless gauge mode in the broken phase (the analog of the photon at lower temperatures) or to neutrinos, then the effective Higgs mass during inflation would not in any way affect reheating, since the Higgs does not provide a mass to these particles333Majorana neutrinos that acquire their mass due to the see-saw mechanism are also Higgsed Adshead:2015jza (). However, since the nature of neutrinos is uncertain at present, we do not try to provide any detailed treatment here.. We note that the direction of the Higgs VEV may or may not coincide with the current direction of Spontaneous Symmetry Breaking, hence the massless direction during inflation is not in general today’s photon. We will examine in detail the various types of reheating of the inflaton, given the nonzero Higgs VEV and resultant SM particle masses during inflation.
Perturbative Decay: After inflation the inflaton field can decay perturbatively into light particles forming a thermal bath, assuming the requisite coupling of the inflaton to SM fields. While often not as efficient at draining power from the inflaton condensate as resonant particle production, a strong enough interaction between the inflaton and the SM particles can ensure that the Universe be radiation dominated before BBN. We specifically calculate the inflaton decay to fermions, which develop a mass due to the effective nonzero Higgs VEV during inflation. For simplicity, we only consider the case of a Yukawa interaction of the inflaton with an arbitrary SM fermion of undetermined mass today (after the usual EW phase transition), such that we can calculate the perturbative decay width of the inflaton. We are thus able to determine, in general, the range of SM Yukawa couplings, for a given inflationary energy scale, for which the effects of the Higgs condensate on perturbative reheating are significant. Compared to the masses SM fields receive from the effective Higgs VEV which develops during inflation (with the VEV set by the inflation scale), the masses from the usual EW symmetry breaking are negligible, provided that the inflation scale is not too low. Thus the results of our calculations of the perturbative width should remain essentially the same for any SM final state particles (as long as the final state particles become massive due to the Higgs mechanism); the details of the phase space of the decay products would vary for different inflaton-SM interaction terms and particular final states but would not substantially change our final results.
Non-perturbative Resonant Particle Production: The first type of reheating to typically take place, and frequently the most effective one, is non-perturbative preheating, which leads to the resonant production of bosons. There are two possible contributions to preheating: tachyonic resonance (where the effective inflaton oscillation frequency-squared first crosses zero with ), followed by parametric resonance. To reduce complications, we will only consider a massive field during inflation, as a proxy for the massive electroweak gauge bosons during (and immediately after) inflation. To be more concrete, we study the case of natural inflation Freese:1990rb () for our investigation of non-perturbative decay. Here, the inflaton is an axion, designed to avoid the fine-tuning that plagues most other models of inflation. A shift symmetry provides the flat potential required for a successful inflation model Adams:1990pn ().
The coupling of an axion-like inflaton to a gauge boson has been shown to allow for complete preheating, in which the entirety of the inflaton’s energy density is transferred to the gauge fields within a single inflaton oscillation Adshead:2015pva (). In the particular case where the gauge field is identified as the hypercharge sector of the SM, this very efficient energy transfer can lead to the generation of large-scale magnetic fields with possible cosmological relevance for the explanation of the observed Blazar spectra Adshead:2016iae (); Neronov:1900zz (); Tavecchio:2010mk (); Dolag:2010ni (); Essey:2010nd (); Taylor:2011bn (); Takahashi:2013lba (); Finke:2013tyq (); Kachelriess:2012mc (). Although our calculations specifically assume a derivative coupling of the inflaton to gauge bosons which arises in models of natural inflation Adams:1992bn (); Gaillard:1995az (); Kawasaki:2000ws (); Banks:2003sx (); Hsu:2003cy (); Hsu:2004hi (); Freese:2004un (); Kim:2004rp (), the results of this work can be easily generalized to any model in which preheating occurs through resonant massive boson production. We also discuss perturbative decays of the inflaton field in this case of axion-like couplings when resonant particle production is completely blocked by the induced mass of the final state bosons.
The outline of this work is as follows. In section II, we review the dynamics of the Higgs condensate during and after inflation. In section III, we investigate the effects of SM particle masses on the perturbative decay of the inflaton. In section IV, we consider resonant production of massive gauge bosons during preheating. We summarize our conclusions and consider further applications of this work in section V.
Ii Higgs Condensate
We consider the Higgs doublet and its potential in the form
where GeV and is the quartic self-coupling, which is taken to be positive during inflation. During inflation, the SM Higgs boson with minimal coupling to gravity is a light spectator field. Though initially rolling classically, the Higgs field soon reaches a regime dominated by quantum fluctuations. To set the initial conditions for the Higgs field at the end of inflation (and the start of the inflaton oscillations leading to the reheating or preheating stages) we use the fact that the superhorizon modes of the Higgs field follow a random walk during the final stages of inflation, with the probability distribution function (PDF) given by Starobinsky:1994bd ()
where and are the Hubble rate and the Higgs quartic self-coupling evaluated at the end of inflation, is the Gamma function and . Throughout the paper, the subscript refers to the initial time of onset of inflaton oscillations (referred to as the end of inflation). It is straightforward to check that the above PDF is properly normalized as . The Higgs field is thus distributed in different Hubble patches according to the above PDF, so that we can take the Higgs VEV in each Hubble patch to be constant and have a magnitude with a probability given by Eq. (3). The dispersion of the PDF is
yielding an effective VEV
which gives masses to the Higgs and other SM particles. We take to be the initial value of the Higgs field at the end of inflation, . We note that Enqvist:2014tta ().
Once the inflaton begins to oscillate, the Universe is matter dominated and the Hubble constant drops as . The dynamical evolution of the Higgs field may be described by the equation of motion for a scalar field in a quartic potential,
and we define the effective Higgs mass squared as
Since the self-interaction term in Eq. (6) is negligible immediately after the the end of inflation, the amplitude of the Higgs field remains “frozen” at the value it had at the end of inflation , until it starts oscillating. Using Eq. (5), we obtain the effective Higgs mass squared from the end of inflation until the onset of Higgs oscillations,
Oscillations of the Higgs field begin once the Hubble constant has dropped to the value , that is when the mass term overcomes the friction term in Eq. (6), at which point the value of the Higgs field is
where the subscript “osc” indicates the time when the Higgs oscillations begin and we have assumed there is very little running of between the end of inflation and the onset of Higgs oscillations. The ratio between the Hubble constants at these two times is 444This equation differs from Eq. (2.6) in Ref. Enqvist:2013kaa (), which was derived using the assumption that even for , an assumption that does not apply during this time period.
For , the Higgs field begins to oscillate about five Hubble times after the end of inflation Enqvist:2013kaa () (as we show in Fig. 1 below, this corresponds to a few -folds). As we will see below, the time approximately coincides with the time at which the transition between the two contributions to inflaton reheating occurs. Preheating (both tachyonic and parametric resonance) takes place before while perturbative decay begins after . As a consequence, the Higgs field varies slowly during preheating and can be taken to be constant, while a detailed treatment of its evolution is relevant for perturbative inflaton decays due to the oscillatory behavior of the Higgs field after .
As discussed in Ref. Enqvist:2013kaa (), the energy density in the Higgs field is eventually dissipated through resonant production of W bosons via the dominant decay mode . The dynamics of the Higgs field doublet coupled to the gauge field is described by the Lagrangian term
where the covariant derivative of the Higgs to the gauge field is , is a coupling constant, is a set of generators for the gauge group, and is the field strength, defined through . Eq. (6) for the dynamical evolution of the Higgs field can be obtained from this Lagrangian by neglecting the gauge couplings of the Higgs, a reasonable approximation throughout most of the reheating process prior to the final dissipation of the Higgs into gauge bosons. As we will show in the case of perturbative inflaton decay, for , the Higgs field redshifts significantly as before backreaction effects become important. The decay time of the Higgs condensate can then be reasonably approximated by the time at which the backreaction term in the equation of motion becomes comparable to the co-moving amplitude of the Higgs field. With this approximation in mind, we write the equation of motion Enqvist:2013kaa () for the gauge field 555We ignore the non-Abelian self-interactions of the gauge fields, which may change the Higgs condensate decay time somewhat, but should not drastically affect our overall results. ,
where is the Fourier transform of the transverse component of the gauge field in Eq. (11) and , while the time-dependent frequency of the mode with wavenumber is
Note that the sum of the last two terms in the expression for vanishes in a matter-dominated background. The corresponding occupation number is given by Greene:1997fu ()
from which we calculate the effective Higgs mass term induced by W-bosons using an approximate expression for the expectation value ,
Assuming that the dominant decay mode of the Higgs condensate is non-perturbative W-boson production, the condensate decays approximately when has grown to reach the value
This condition is an approximation for the conservation of energy density stored in the Higgs condensate. Since the dominant decay channel for the Higgs condensate is resonant W-boson production, then, if the condition in Eq. (16) is met, the energy density implied by the effective mass of the Higgs field, , will have been depleted. As we show in the next section, the details of the condensate decay are only relevant in very specific cases of perturbative reheating.
The value of the quartic Higgs coupling at the initial time when the inflaton first starts oscillating is unknown. However, in the case where there is no new physics between the EW scale and , we may estimate the value in the following way. In the absence of new physics, the stability of the Higgs vacuum all the way up to the Hubble scale at the end of inflation, , requires values of the top mass, Higgs mass and strong coupling constant to vary up to from the central measured values Enqvist:2014bua (). Here, we assume that the value of the Hubble scale during inflation ranges within . With these assumptions, the running of the Higgs quartic self-coupling yields at the inflationary scale Degrassi:2012ry (). We will use this value of as our canonical value when estimating numbers throughout the text, but stress that this value is uncertain and heavily depends on the value of the SM parameters as well as inputs from additional new physics beyond the SM.
We note that, for much smaller values of the Hubble scale during inflation, the delayed reheating effects considered in this paper become unimportant. The effective Higgs VEV in Eq. (5) becomes smaller both because of the smaller value of and also because runs to higher values. The resulting fermion masses due to the Higgs VEV are then also smaller and no longer important.
Iii Perturbative Inflaton Decay
iii.1 Neglecting backreaction
If we assume the inflaton has begun to oscillate as a massive scalar field by the end of inflation, under the influence of a quadratic potential , the set of equations which describes the perturbative reheating is
where is the energy density of the inflaton and is the energy density of radiation resulting from the decay of the inflaton with a decay rate . For simplicity, we assume the inflaton has a Yukawa coupling to SM fermions, yielding
where is the decay width in the massless fermion limit and
is the effective fermion mass induced by the Higgs condensate with Yukawa coupling . In Eq. (20), we have included the -function (step-function) to model the phase-space blocking due to large effective fermion masses. As mentioned in section I, the results of our calculations of the perturbative width should remain essentially the same for any SM final particles (as long as they become massive due to the Higgs); the details of the phase space of the decay products would vary for different inflaton-SM interactions or choice of final states but would not substantially change our basic results.
The decay of the inflaton field is controlled by the decay rate in Eq. (20) and it is thus negligible (blocked) when , i.e. when
Fig. 1 shows the value of the Higgs field squared amplitude in units of (black solid line), as a function of the number of -folds after the inflaton starts oscillating, for the choice and . We also plot the quantity for (green dotted line), (red dot-dashed line), and (blue dashed line). For a given value of , whenever falls below the value given by the blocking condition in Eq. (22), the inflaton field decays in a series of burst events. This picture is valid whenever the period of the oscillations in the Higgs field is much smaller than the characteristic Hubble timescale . In the opposite limit , the blocking in Eq. (22) still applies provided that the Higgs field is replaced with its root-mean-squared value , shown with the black dashed line in Fig. 1666We wish to distinguish the initial condition average (that was defined in terms of the stochastic superhorizon probability distribution) from the root-mean-square which is obtained by where .. In this second scenario, the decay does not occur in bursts since the oscillations in are suppressed. Unfortunately the true oscillations of the Higgs field are likely to be in an intermediate regime where the period is comparable to the Hubble rate, , in which case the usual calculation of a perturbative decay width is not appropriate since the mass of the final states is varying too slowly to yield a root-mean-squared value and too quickly to resolve the times when Eq. (22) is satisfied. Thus, we present our results assuming the two limiting cases in which we are confident of the perturbative width calculation, and we leave a more realistic calculation to future work. The two cases split when the Higgs starts oscillating777Before , non-perturbative particle production described in the next section can be much more efficient in reheating than any perturbative particle production described in this section. (i.e. at .
If the Higgs condensate decays too soon after it starts oscillating, then inflaton decay is never blocked, and the effects of this work are not present. Thus we are interested in the case in which the timescale for Higgs decay is longer than the timescale for reheating, i.e., , where is the Hubble scale when the approximate condition for the decay of the Higgs condensate, is satisfied. In the other limit, with , we expect the Higgs condensate to decay away before most of the energy is transferred from the inflaton to the thermal bath, and the blocking effects described here do not significantly alter the reheating process. Note that, while depends on the precise values of and at the inflationary scale and on the arbitrary direction of the EW symmetry breaking arising from the effective VEV, the decay of the Higgs condensate is largely independent of the perturbative reheating process for . Furthermore, the particular details which determine the exact value of are only relevant when . In this subsection we consider only cases in which the inflaton decays completely before the Higgs experiences backreaction from the gauge bosons, i.e. . Then Eq. (6) suffices to study the effects of effective fermion masses on perturbative reheating.
Initial conditions for the inflaton are set by the Hubble scale as , where we use the Planck mass GeV. For radiation, initial conditions follow from Eq. (18) as , where when the inflaton decay is blocked by particle masses, while when the blocking does not occur. Eq. (6) determines the dynamics of the time-dependent mass, defined by Eq. (21), for the fermion byproducts of perturbative inflaton decay.
In Fig. 2, we show the evolution of and determined by simultaneously solving Eqs. (17)-(19), where we have taken , , , and . The top panel corresponds to a constant value of , while the other two panels correspond to the given in Eq. (20) for the two limiting cases of (middle panel) and (bottom panel). In all panels, the blue (solid) curve shows , and the red (dashed) curve shows . In the middle panel, the radiation energy density has a step-function behavior, corresponding to the periods when and , allowing the inflaton field to decay into bursts of SM particles. For the choice of in Fig. 2 there is only one burst; for higher values of there would be many such bursts. Such behavior is not present in the bottom panel, since the decay occurs as the condition in Eq. (22) is violated and the monotonic falls below the value . Defining the reheat time through , we find for a constant decay rate (top panel), for the decay rate in Eq. (20) and (middle panel), and for the decay rate in Eq. (20) and (bottom panel). Inclusion of phase space blocking in Eq. (20) delays reheating.
In Fig. 3, we show the increase in the number of -folds (relative to the case of no Higgs blocking) by which reheating after inflation is delayed as a function of the parameters and . The left column shows results obtained by solving the set of Eqs. (17)-(19) (relevant when we do not take backreaction into account); the right column includes the effects of backreaction onto the Higgs condensate and will be discussed in the next subsection. Here we have fixed and . Specifically, we show the difference in the number of -folds for the field-dependent in Eq. (20) with respect to the results for a constant . The top panel is for the case of and the bottom panel for . Relative to a fixed , an earlier inflaton decay leads to a more significant delay in the reheating time. Furthermore, for fixed and , the delay in the reheating time is more significant for a larger because the time intervals at which the phase space for perturbative decay is open, , become shorter. The delay in reheating, , obtained in the two scenarios typically differ by a factor of order two. For either of the two scenarios, the largest delay in reheating is expected to be for and , which are the largest values of and we considered.
iii.2 Including backreaction
So far, we have neglected the backreaction of the production of the gauge bosons on the Higgs field dynamics. This approximation is expressed in Eq. (6), where the excitation of the gauge bosons does not affect the evolution of the Higgs field. Depending on the couplings and , the inclusion of backreaction could change the conclusions drawn so far. In this section we will choose value of high enough to demonstrate the effect of backreaction, but note that for a SM value evaluated at inflationary scales backreaction effects may be insignificant.
We sketch the effect due to the inclusion of gauge boson production by introducing the effective mass of the gauge boson , as defined in Eq. (15). When equals the effective mass of the Higgs field, , we assume that the Higgs field decays and the decay rate of the inflaton is no longer blocked, reaching the value . For this, we consider a set of Boltzmann equation analogous to what we have previously discussed in Eqs. (17)-(18),
where the decay rate is expressed in terms of , see Eq. (20), and as
The expression for above states that the Higgs blocking only affects the dynamics during the time period when backreaction can be neglected, . We solve Eq. (12) in the illustrative case , i.e., , where is the resonance parameter which arises from writing the equation of motion of the W-boson as a Mathieu equation, see for example Ref. Greene:1997fu (). Although the value of the gauge coupling is somewhat higher than what one would expect in the SM by a factor 4-5, we choose such a high value in order to illustrate the effects of backreaction on the Higgs blocking, in the case of the perturbative inflaton decay to fermions. However, even for a smaller value of , the effects of backreaction could still be present for values of which are outside of the perturbative regime888As shown in the lower panels of Fig. 3, backreaction only becomes important for . This is because, as shown in Fig. 1, larger induced fermion masses will lead to more efficient phase space blocking and larger . Thus, only for sufficiently large is the blocking in effect long enough for the decay time of the inflaton to become similar the decay time of the Higgs. If we had calculated the decay of the Higgs with , then the Higgs would have decayed later, requiring an inflaton Yukawa coupling larger than allowed by perturbative unitarity, , in order for the effects of backreaction to be important.. In principle, the effective gauge boson mass is obtained by computing the expression in Eq. (15), which involves an integration over the gauge boson momenta . Here, we simplify such computation by considering that, for , all modes are excited up to the cutoff scale Greene:1997fu ()
This approximation is valid when the solutions to Eq. (12) lie in a resonant band, as it is for our choice of , and allows us to write the occupation number as
where is the occupation number when assuming . Using this expression for to evaluate the integral in Eq. (15) , we obtain the induced mass of the W boson as
where in the last expression we have used Eq. (26) and the relation , which is valid after the Higgs has started oscillating. We have checked that the result in Eq. (28), obtained in the approximation, is consistent with the numerical integration in Eq. (15). Backreaction becomes important once , or when the occupation number is999This result differs from Eq. (4.10) in Ref. Enqvist:2013kaa (), where the authors obtain a time-dependent occupation number. This difference is related to the inconsistency pointed out in Footnote 4.
corresponding to -folds after the inflaton field has started to oscillate. Including backreaction in the Higgs dynamics modifies the results obtained in the previous section, which can be seen when comparing the left panels of Fig. 3 to the right panels in Fig. 3, where there is a cutoff introduced in the delay of reheating, . Backreaction affects the region , where the delay of reheating becomes less pronounced.
iii.3 Effects on the reheat temperature
In previous subsections, we have calculated the delay in reheating in terms of independently of the precise value of . However, in a realistic model of inflation we want to achieve the correct density perturbations and translate delays in reheating into predictions for the reheat temperature. To be illustrative, we will consider a quadratic shape for the form of the inflaton potential near its minimum and an inflaton mass GeV. A quadratic potential with this mass throughout inflation for all values of would produce the correct density perturbations; yet it is ruled out since it yields a value of , placing it outside the current experimental bounds. However, it is possible to construct models where the inflaton potential is non-quadratic during the period when density fluctuations and tensor modes are produced (60-50 -folds before the end of inflation), and yet is quadratic near the bottom (e.g. some variants of axion monodromy McAllister:2008hb ()). Henceforth to be concrete we will assume a quadratic potential near the minimum and mass GeV.
When the inflaton field decays, the Universe transitions to a radiation dominated state with the Hubble rate
where is the number of the relativistic degrees of freedom at temperature . We compute the reheating temperature by first evaluating the Hubble rate when and then using Eq. (32). Clearly, a prolonged inflaton oscillation (matter domination) stage affects the reheating temperature . In general, we expect that lowers when the Higgs blocking effect is present, because the coherent inflaton oscillation period lasts longer for the same value of .
So far, we have obtained the results by considering the initial value of the Higgs field to be the “central value” in Eq. (5). However, for a given value of , the initial value of the Higgs field is distributed according to the PDF in Eq. (3), so in some Hubble patches the value of can differ greatly from what is expected by its central value. For each value of , we obtain a different reheat temperature , hence the reheat temperature of each Hubble patch of our observable Universe will depend on the PDF . In particular, the probability of finding the reheat temperature in a Hubble patch between and is
where the PDF for the reheat temperature is given by
We solve the set of the Boltzmann Eqs. (17)-(19) for given and , obtaining the one-to-one relation without the effects of backreaction included. We also solve Eqs. (23)-(24) and Eq. (19) when backreaction is included. Since we also consider the evolution of the Higgs with either or , we obtain a PDF of the reheat temperature for each of the four cases considered before. Since, for cases with , the numerical function has oscillatory features, we have fit to a polynomial in order to obtain a smooth derivative for the calculation of in Eq. (34). Although we expect the oscillations of to manifest as features in , we note that these features would be less apparent in a more realistic treatment of the Higgs field, with , and leave a calculation of the associated effects on for future work.
We plot results in Fig. 4, where we show the PDF defined in Eq. (34) for (top row panels) and (bottom row panels), as well as when backreaction is neglected (left column panels) and when its effect is included (right column panels). For each panel, the different lines represent different values of (red solid line), (blue dotted line), and (green dot-dashed line). For comparison, the reheat temperature without any blocking effects is obtained from the Hubble equation, given by Eq. (32) as , or
Setting and GeV, we find GeV when , GeV for , and GeV for .
As stated previously, to be concrete, we have taken a quadratic potential near the minimum and mass GeV. We note that, for values of the inflaton mass within several orders of magnitude of this number and a fixed ratio of , our results for the relative delay in reheating remain roughly the same: the ratio of changes by factors up to (compared to the results in Fig. 4).
The results in Fig. 4 are consistent with the features of Fig. 3. Naively, for a fixed delay in the reheating, an increase in the value of leads to higher . In contrast, as seen in Fig. 3, increases with higher for a fixed value of . Both these effects are present when considering the respective PDFs for different in the top-left panel of Fig. 4, when and without considering the effects of backreaction. When increasing , the most likely values of are higher. At the same time, the overall delay in reheating from (as shown in the legend of Fig. 4) to the most likely values of also increases.
From the the bottom-left panel of Fig. 4 (no backreaction but ), we again see the scenario where generically leads to an even more delayed reheating, as the most likely values of decrease for all relative to the previous case with . Also, the largest decreases in the most likely values of , relative to the case with , occur for larger values of .
As shown in the right panels of Fig. 4, backreaction inhibits the blocking effect and restores the standard unblocked reheating picture, thus it generally predicts higher reheat temperatures. However, since backreaction only acts when with our choice of parameters, the PDF for the red solid curve (corresponding to ) in the top-right panel predicts that higher reheat temperatures are more likely than in the top-left panel. The blue and green lines remain identical because backreaction does not effect reheating with these choices of .
The main result of this section is the following: In the case of perturbative inflaton decay, the Higgs blocking considered in this paper can lead to a reheat temperature that is suppressed by roughly an order of magnitude compared to the standard case without blocking (the curves in Fig. 4 can peak at a lower values than the unblocked numbers in the legend).
Iv Gauge Preheating
iv.1 Resonant gauge boson production
If, instead of perturbative reheating through a Yukawa coupling of the inflaton to SM fermions, we consider Chern-Simons couplings of the inflaton to SM gauge bosons, resonant gauge boson production can reheat the Universe almost instantaneously after the end of inflation for sufficiently large values of the inflaton-gauge couplings Adshead:2015pva (); Adshead:2016iae (). Although previous analyses of gauge preheating have considered effectively massless gauge bosons, we analyze the effects of a large gauge boson mass on the structure of the resonance. We assume an axion-like derivative coupling of the inflaton to an Abelian gauge boson with an effective mass given by the coupling of the gauge boson to the SM Higgs101010Breaking of the Electroweak symmetry during inflation would leave one of the resulting gauge bosons massless. In a general effective field theory setting, the inflaton would couple to both the massive and the massless gauge bosons alike. If one considers a definite UV completion of the SM, the couplings of the inflaton to the separate gauge bosons would depend on the charges of the fermions under the SM and the Peccei-Quinn-like symmetry which provides for the shift symmetry of the axion Freese:1990rb (); Adams:1990pn (). Such an analysis is beyond the scope of the current paper. We instead consider only an effectively massive gauge field, as an indicative case of a more complicated and model-dependent process.. Chern-Simons couplings to gauge fields are generic for example in models of a pseudo scalar inflaton as in Natural Inflation Freese:1990rb (); Adams:1992bn (), since they respect the underlying shift symmetry. The relevant terms in the potential of the corresponding effective Lagrangian are given by
where and . Note that the electroweak symmetry breaking due to the large Higgs condensate, which develops when the Higgs is a spectator field during inflation, yields the mass term for the gauge boson , where is the Abelian gauge coupling111111A detailed derivation of the equations of motion is given in Appendix A.. While the dynamics of the Higgs field are still technically determined by Eq. (6), the Higgs field does not begin oscillating until well after gauge preheating, which only lasts for a few -folds after the end of inflation. We thus take the gauge boson mass to be constant during the last few -folds of inflation and immediately thereafter and depend only on the Hubble scale at the end of inflation.
In general, a Chern-Simons coupling to gauge bosons can induce a coupling of the inflaton to the axial vector current,
Even though such couplings can source interesting phenomenology Adshead:2015kza (); Adshead:2015jza (), Pauli’s exclusion principle does not allow for sufficient transfer of energy from the inflaton to fermions during preheating. We will thus neglect these processes here.
where is the Fourier transform of the transverse component of the gauge field and . Note that, since the last two terms in the definition of are subdominant after inflation, we will neglect them in all analytic estimates although we still include them in the numerical calculations. The mass term is equivalent to the second term on the right hand side of Eq. (13). Using Eq. (5), we may write so that we may use the quantity as an effective coupling constant. We can use the WKB approximation to solve Eq. (38) in regions where the effective frequency is slowly varying, .
Preheating begins with a period of tachyonic instability, with . At the end of this period, the coefficient of the solution which describes the amplitude of modes is given by Adshead:2015pva ()
where the time period during which is taken to be 121212If one roughly thinks of Eq. (38) as a harmonic oscillator equation, one can think of as the regime of exponential growth rather than oscillation. Since it is not exactly an harmonic oscillator, we use the terminology “effective frequency.”. Subsequent to the first tachyonic burst, the effective frequency acquires an oscillatory component, due to the fact that the inflaton condensate is oscillating around the minimum of its potential. This can lead to a period of parametric resonance and further particle production for later times . For the massless gauge boson case, the parametric resonance was discussed for example in Ref. Adshead:2015pva ().
In Ref. Adshead:2016iae () it was shown that gauge preheating dynamics can be divided into three cases, depending on the size of the axion-gauge coupling and the resulting effects:131313The values of the couplings mentioned in this section refer to a model with a quadratic inflaton potential. For shallower potentials, like axion monodromy, the values of the couplings in each regime are increased, but the characteristics of these regions persist. A numerical comparison between the cases of quadratic and axion monodromy potential was performed in Ref. Adshead:2015pva ().
Small coupling : The analysis can be done entirely using linear equations of motion, neglecting backreaction effects and mode-mode coupling. The resulting fraction of the energy density that is transferred non-perturbatively into the gauge fields is a few percent or less.
Intermediate coupling : The linear equations of motion can be used and provide very accurate results, until the time when the energy density of the gauge fields becomes comparable to the energy density of the background inflaton field. In this case preheating can be complete and lasts typically a few inflaton oscillations and a few -folds.
Large coupling : In this case the entirety of the energy density of the inflaton is transferred onto the gauge fields within one axion oscillation, driven by an initial period of tachyonic amplification, henceforth denoted as the first tachyonic burst. Proper study of this case requires the use of lattice simulations, which is beyond the scope of the present work.
We begin by closely examining the case of , which was considered as the typical example of the “intermediate coupling case” in Ref. Adshead:2016iae (). We use a linear no-backreaction analysis, an approach that has previously been shown to agree very well with the full lattice results during the initial stages of preheating. Indeed, we show that this analysis also works equally well for the subsequent evolution when we introduce a non-zero mass due to the Higgs mechanism.
Fig. 5 shows the square of the effective frequency as a function of the number of -folds relative to the end of inflation (the end of inflation is here taken to be at ), for . We show results for different values of the ratio ranging from zero to one: as described before, this ratio may be thought of as an effective coupling constant. The upper panel shows the first tachyonic burst, which takes place during the last few -folds of inflation; the lower panel shows the parametric resonance right after inflation ends (right after for the first time). In the upper panel, the behavior of during the first tachyonic burst is qualitatively independent of the value of : the negative effective frequency-squared differs by only about for typical values of the gauge boson mass. However, behaves differently for different values of the ratio during the parametric resonance after the first inflaton zero-crossing, as shown in the lower panel of Fig. 5. In fact, for larger values of this ratio, the oscillation of the effective mass-squared acquires a constant positive offset, leading to a severe suppression of the parametric resonance.
The behavior just described can be explained in the language of the Mathieu equation , where we consider the effective frequency given by Eq. (38) and neglecting the last two terms arising from the expansion,
The expression for the effective frequency in Eq. (40) allows for a very simple estimation of the effect of the Higgs mass during the initial tachyonic amplification phase. Considering the mode that exits the horizon at the end of inflation , and using a quadratic inflaton potential for the illustrative purpose, we obtain and at the end of inflation. Inserting these expressions into Eq. (40) and using the condition , which is in the intermediate coupling regime, we get the simple estimate
where generically . This is a useful estimate of the effect of the gauge field mass on the tachyonic amplification. However, we must also consider the point where the effective frequency-squared acquires its largest (negative) value. By using the slow-roll expression for , it is easy to compute
Setting , we see that the effect of the gauge field mass at this point is less than for and rises to over for . Hence it is reasonable to expect that massive gauge fields exhibit similar early tachyonic behavior to their massless counterparts, especially for masses that do not significantly exceed the Hubble scale. The estimations given in Eq. (41) and (42) agree with the plot shown in Fig. 5.
Fig. 6 shows the time evolution of the gauge field energy density relative to the inflaton energy density for several values of the coupling and the gauge field mass. We see that the behavior expected from the evolution of the effective mass squared is verified by the numerical calculation: the initial tachyonic burst shows little deviation for massive and massless gauge bosons, while the subsequent parametric resonance is severely suppressed or completely shut off for sufficiently large mass values.
As a way to gauge the efficiency of the tachyonic and parametric resonance, we calculate the number of -folds after inflation for which the energy density of the gauge fields is equal to the energy density of the background inflaton, . The calculation is performed neglecting backreaction, hence in our approximation denotes the total energy density of the Universe. While this is not a proof of complete preheating, it is a very strong indication of it. We stop the calculation at three -folds after the end of inflation, since all interesting dynamics for the vast majority of cases is contained within that period.
Fig. 7 shows the number of -folds after the end of inflation at which the energy density of the gauge fields first equals the inflaton energy density as a function of the inflaton-gauge coupling and the mass of the gauge field normalized by the Hubble scale at the end of inflation. We can again distinguish the intermediate and large coupling regimes. For intermediate couplings , when the first tachyonic burst is not strong enough and parametric resonance is needed to preheat the Universe, increasing the gauge field mass makes the parametric resonance less effective and smoothly shuts off preheating, before the inflaton can transfer all its energy into the gauge fields.
In the large coupling regime (for massless gauge bosons) the first tachyonic burst is efficient enough to allow for the majority or all the inflaton energy to be transferred into gauge fields. In this case, introducing a small gauge field mass does not significantly affect the early tachyonic burst. It would however suppress any subsequent parametric resonance. Once the gauge field mass is increased beyond the point when it does affect the early tachyonic burst, the subsequent parametric resonance is so far suppressed, that it is completely ineffective.
Hence, the contour plot of Fig. 7 clearly separates between the intermediate coupling regime , where an increase in the gauge field mass smoothly leads to a prolonged preheating stage until it is completely shut off, and the large coupling regime , where the gauge field mass either does not appreciably affect the preheating process, or it completely shuts it off.
It is instructive to take a few vertical cuts through the contour plot of Fig. 7 and closely look at how the duration of preheating changes as a function of the gauge field mass . We plot as a function of the gauge mass in the upper panel of Fig. 8 for four different values . The plot shows a clear difference in the behavior of the function between the intermediate and the large coupling regimes.
For intermediate couplings , red and blue lines in the upper panel of Fig. 8, an adequate fit to the results is a function of the form . Nevertheless, the fit does not capture the steps featured in the functions, which are a consequence of our prescription for defining the end of preheating as the time when . The lower panel of Fig. 8 shows the behavior of the energy density in the gauge bosons as a function of for the same value of the parameter as in the upper panel, while the black line shows the evolution of the inflaton energy density . Since and are both oscillating functions, the time when the preheating condition is satisfied jumps as a function of . This behavior is analogous to what we discussed in the middle panel of Fig. 2.
For larger couplings, the function has more pronounced features. For , green line in the upper panel of Fig. 8, the duration of preheating is almost constant for , at which point successful preheating abruptly ceases. To explain this behavior, we refer to Fig. 5 and to the lower panel of Fig. 8. By increasing the gauge field mass enough to suppress the initial tachyonic burst, we have made the parametric resonance completely absent, hence there is no gradual increase of , but rather an abrupt transition from complete to unsuccessful preheating. The case (brown line) shows even more features. For the Universe preheats completely at . At there is a discontinuity of . The reason is clearly shown by the brown curve of the lower panel of Fig. 8. The initial tachyonic burst becomes marginally inadequate to preheat the Universe at , but the subsequent parametric resonance is still effective. The second feature arises at , where we see from the orange curve of the lower panel of Fig. 8 that evolves almost parallel to for . The marginal cases would require a full lattice simulation, in order to determine whether the Universe completely preheats. While the boundaries of Fig. 7 will in general shift, the qualitative features are expected to survive, even if the system is properly simulated.
Before concluding this section, we want to examine the approximation . While this is a very good approximation after inflation, due to the delayed decay of the Higgs condensate, the exact dynamics of the Higgs VEV during inflation is dependent on its exact potential and must take into account the running of the Higgs self-coupling and any new physics that contributes to this running between the TeV and the inflationary energy scale. We performed a sample of our calculations by allowing the gauge field mass to vary during inflation and the results showed no significant change. On the qualitative level, the distinction between gradual and sharp shut-off of preheating remains and the quantitative changes on the exact preheating time were smaller than the inherent uncertainties of using a linearized analysis to probe non-linear physics.
iv.2 Effects on the reheat temperature
For models of inflation with coupling through higher dimensional operators as in Eq. (36), in many cases (e.g. cosine natural inflation) the requirement for sufficient inflation requires . In that regime, the potential during reheating can roughly be approximated as a simple quadratic with GeV in order to achieve the correct density perturbations. As discussed in section III.3, more realistic models of inflation which are allowed by Planck data, such as axion monodromy, reduce to a quadratic potential close to to their minimum. As has been shown in Adshead:2015pva (), the preheating behavior of quadratic and axion monodromy models are equally efficient, for properly chosen values of parameters.
Then, for a number of -folds after the end of inflation,
where . A simple analytical calculation of the energy density at the end of inflation would give , hence Eq. (43) is off by 35% at , while being very accurate for . The resulting reheat temperature is then
where is the Stefan-Boltzmann constant. Reading off the value for which we obtain a successful reheating from Fig. 7, we can compute the resulting reheat temperature.
To illustrate our results, we present numbers for the case . Without Higgs blocking the number of reheating -folds is and, using Eq. (44), the reheat temperature is