Update on gravitational wave signals from post-inflationary phase transitions

In view of recent interest in high-frequency detectors, broad features of gravitational wave signals from phase transitions taking place soon after inflation are summarized. The influence of the matter domination era that follows the slow-roll stage is quantified in terms of two equilibration rates. Turning to the highest-frequency part of the spectrum, we show how it is constrained by the fact that the bubble distance scale must exceed the mean free path.


Introduction
A cosmological first-order phase transition may produce a gravitational wave signal [1][2][3].The signal is expected to be peaked, with the peak frequency proportional to the phase transition temperature, T * .In particular, phase transitions at the QCD scale, T * ∼ 100 MeV, could yield a signal for the high end of the pulsar timing array observation window (f 0 ∼ 10 −8 ...10 −6 Hz); those at the weak scale, with T * ∼ 100 GeV, would match the LISA frequency range (f 0 ∼ 10 −4 ...10 −1 Hz); whereas exotic phase transitions with T * ∼ 1...100 TeV could leave an imprint to be discovered by DECIGO (f 0 ∼ 10 −1 ...10 1 Hz) or the Einstein Telescope (f 0 ∼ 10 1 ...10 3 Hz) [4].Phase transitions at much higher scales, related perhaps to GUTs, might in turn be probed at the GHz...THz range that has become of recent interest [5].
The signal from high-scale phase transitions has been estimated in ref. [5].The purpose of the current study, which was influenced by recent model investigations [6,7], is to derive an upper bound for a phase-transition-induced gravitational wave energy density in the GHz...THz range.We establish its peak frequency, and elaborate on the effect of a matter domination era that follows slow-roll inflation (cf., e.g., refs.[8][9][10]), if the relevant equilibration rates are small compared with the Hubble rate.The main differences with respect to ref. [5] will be recapitulated in the conclusions (cf.sec.5).Before that, we review the main features of the background evolution in the reheating epoch (cf.sec.2), broad features of the gravitational wave signal originating from phase transitions (cf.sec.3), and the maximal strain that this physics can lead to (cf.sec.4).

Background evolution
We have in mind a usual single-field inflationary scenario.As the slow-roll stage ends, the inflaton starts oscillating and the temperature reaches a maximal value.There is normally a period in which inflaton oscillations dominate the overall energy density, before the energy density of a thermal plasma takes over.
We assume the matter content during this time to reside in three components: • inflaton, ϕ, with energy density e ϕ .
• "dark sector", defined as the one undergoing a phase transition.The low-temperature phase of the dark sector is assumed massive (i.e.non-relativistic), due to confinement or Higgs mechanism, so it does not exert pressure.Its energy density is denoted by e d .
• "visible sector", constituted of Standard Model like particles, which are effectively massless during the epoch under consideration.Its energy density is denoted by e v .
Adding sectors together, we denote e ij ≡ e i + e j , notably e dv ≡ e d + e v .The overall energy density is then e ≡ e ϕdv , with the corresponding Hubble rate given by In general we parametrize moments in the reheating epoch by the value of H. Motivated by Planck data [11], we also conservatively set H ≤ H max ≡ 10 −5 m pl .We model the dynamics of this coupled system by equations of the type ėi + 3H(e i + p i ) ≃ − j Γ ij (e j − e j,eq ) . (2.2) Summing over i, the Friedmann equation needs to be recovered, implying i Γ ij = 0.The equilibrium values, e j,eq , represent the fixed point that would be attained if the system had infinite time.In order to simplify the setup, we assume that the visible sector is effectively thermalized, e v ≈ e v,eq .Only two rates are assumed to have a large influence:1 • inflaton interaction rate, Υ ≡ Γ ϕϕ = −Γ dϕ .In general Υ would be a function of the dark sector temperature, but we assume that the part originating from vacuum decays dominates.A possible temperature dependence can subsequently be mimicked by interpolating between various constant values.
Assuming also an initial state with e ϕ ≫ e ϕ,eq and e d ≫ e d,eq , we are then faced with ) We define with the corresponding scale factors denoted by a + and a − , respectively.We now consider the solution of eqs.(2.3)-(2.5) in various regimes, making use of the labelling in fig. 1.
Domain (1).During the reheating period, the inflaton is oscillating faster than the Hubble rate, satisfying then φ + ∂ φV ≃ 0. After multiplication with φ, this can be integrated into φ2 + V ≃ constant.By the virial theorem, the two terms contribute equally on average, implying that p ϕ ≃ 0, e ϕ ≃ φ2 .The regime in which these equations apply corresponds to a The key for us is the solution of eq.(2.4) in the presence of e ϕ .With the given assumptions, the right-hand side of eq.(2.4) simplifies to +Υe ϕ .Replacing t through H as an integration variable, the evolution equation can be re-expressed as The general solution is where C is an integration constant.If w d > − 1 2 , the latter term dominates at late times.An important consequence of eq.(2.8) is that we can determine the phase transition moment, denoted by H * .If e d, * is the dark energy density at the phase transition, and omitting the subdominant term from the right-hand side, eq.(2.8) can be inverted into Here w * is the value of w d at the phase transition point.We argued above that p d is vanishingly small in the low-temperature phase, but we keep it in the expressions here, because then the same formulae can be re-used in domain (3), with the substitution () d → () dv .
Domain (2).Because of Υ ≫ H, ϕ has equilibrated in this domain, and plays little role.
Given that Γ ≪ H, the right-hand side of eq. ( 2.4) can be omitted, and we are faced with ėd + 3H(e d + p d ) ≃ 0 . (2.10) Thus we obtain Let us anticipate that in sec.3, when we consider gravitational waves generated in a phase transition, only temperatures below the phase transition matter.Therefore we can insert w d ≈ 0 in eq. ( 2.11) according to our previous assumption.

Broad characteristics of the gravitational wave signal
Considering a phase transition taking place in one of the domains mentioned above, we wish to work out its gravitational wave signature.Here we are concerned with an upper bound, omitting the complicated hydrodynamics by which it gets formed (cf., e.g., ref. [4]).The assumption is that immediately after the transition, gravitational waves carry the energy density e gw, * .The first task is to estimate which energy fraction this corresponds to today.
We focus on gravitational waves whose wavelength was within the horizon at the time of their formation. 2Their energy density scales with expansion like radiation.The relation to the critical energy density today, when the scale factor is a 0 , is where e crit is the current critical energy density and the notation e ± , a ± corresponds to that introduced around eq. (2.6).We have expressed the result as a product of three factors, which can be estimated as follows, considering the domains in fig. 1.
Domain (1).For the first factor in eq. ( For the second factor in eq.where we parametrized the thermal energy and entropy densities as e dv = g e π 2 T 4 /30, s dv = g s 2π 2 T3 /45.For the numerical value, we have employed e γ,0 /e crit = 2.473× 10 −5 /h 2 , where h is the reduced Hubble rate, as well as g s,0 ≃ 3.92, which is related to though not studied as much as the parameter N eff that captures the energy density of the universe after neutrino decoupling (cf.ref. Role of mean free path.If microscopic information about the hydrodynamics is inserted, we can be more precise about the factor e gw, * /e dv, * in eqs.(3.5), (3.7) and (3.8).In particular, since gravitational waves are tensor excitations, their production rate is proportional to a bubble length scale breaking translational invariance, which we denote in the following by ℓ B .In fact, a quadrupole moment requires a quadratic dependence on ℓ B , but this could be partly compensated for by a long duration of a process (cf., e.g., ref. [15]).Furthermore, if ℓ B goes towards zero, the production rate does not vanish, but is then taken over by that from thermal fluctuations [16].In the so-called hydrodynamic regime, the fluctuation rate is proportional to the shear viscosity, which in turn is proportional to the mean free path, ℓ free .We treat the fluctuation contribution as a separate source (cf.sec.4).To get a conservative upper bound for the phase transition contribution, we set e gw, * /e dv, * → (ℓ B /ℓ H )θ(ℓ B − ℓ free ), where ℓ H ≡ H −1 * is the Hubble radius.The value of ℓ B /ℓ H is strongly model dependent, so we vary it in the range 1...10 −6 , indicating the variation as an error band.
We still need to estimate the numerical value of ℓ free .If α < ∼ 1 is a characteristic coupling, then the mean free path is ∼ 1/(α 2 T ), however we would not like to make assumptions about the magnitude of the coupling.Therefore we set The most conservative estimate is obtained when ℓ free is smallest, or T * is highest.This is the case when the dark sector energy density saturates H * , i.e. when Υ > H * .The results obtained after determining ℓ free with this recipe are shown in fig.2(left). 3eak frequency.The ratio ℓ B /ℓ H plays an important role also for the peak frequency of the gravitational wave spectrum.As we push towards high temperatures, with correspondingly small values of ℓ H , we must make sure that we do not underestimate ℓ B , i.e. we must maintain ℓ B > ℓ free .The peak gravitational wave frequency today, f 0 , is expressed as Starting with domain (1) in fig. 1, the first factor can be approximated as in eq.(3.3), yielding (H − /H * ) 2/3 , whereas the second factor can be expressed like in radiation domination, where T 0 = 3.57 × 10 11 Hz.Putting the factors together yields (3.12)As indicated, the same formula is obtained in domains ( 2) and (3).The standard formula for a radiation-dominated epoch is recovered by omitting the first factor in eq.(3.10), and by replacing the point between the matter and radiation dominated epochs by the phase transition point, whereby The results of eqs.(3.12) and (3.13) are plotted in fig.2(right).

Maximal gravitational strain from post-inflationary phase transitions
We have seen in sec.3 that a phase transition during a matter-dominated epoch, present if min{Υ, Γ} < H * , leads to a suppressed gravitational wave signal (cf.eqs.(3.5) and (3.7)), and reduces the peak frequency of a given transition (cf.eq.(3.12)). 4In the present section we wish to find the maximal possible signal, at the highest possible peak frequency, and therefore consider the regime min{Υ, Γ} > H * .In this case the transition takes place during radiation domination.We represent the results in the same form as in fig. 2 of ref. [5], underlining then also the differences with respect to this standard reference.
For quantifying the gravitational wave signal, it has become standard to employ the experimentally meaningful strain, rather than the theoretically preferred energy density.However, there are a number of possibilities for its definition.In terms of eq.(3.1), the current gravitational energy density can be expressed as e gw,0 = m 2 pl i,j ḣt ij ḣt ij /(32π), where h t ij is a metric perturbation in the tensor channel, and ... can be interpreted as a time average.The GeV, corresponding to H max ≤ 10 −5 m pl .The last part consists of two known domains, namely an IR tail from hydrodynamic fluctuations at small f 0 [16] and a peak part from elementary particle scatterings at large f 0 [17]; we have connected these with a straight line.The amplitude of the peak part increases in BSM scenarios (cf.refs.[18,19] for concrete examples).
critical energy density is defined as e crit ≡ 3m 2 pl H 2 0 /(8π), where H 0 is the current Hubble rate.We go to frequency space (∂ t → ω 0 = 2πf 0 ), and replace the sum over polarizations by a numerical factor (4, for two polarization states and an additional factor for the symmetry in i ↔ j), whereby h t ij → h t .In the absence of spectral information, we may effectively assign all the energy density to the peak frequency, f 0,peak .Then the strain can be defined as Alternatively, if spectral information is available, we can parametrize a differential spectrum, we see that h  (4.4) We then interpret the upper bound ∆N eff ≤ 0.2 [20] as a bound on h 2 t (f 0,peak ) like in eq.(4.1).The second comparison concerns the irreducible background that originates from thermal fluctuations.This contribution is strongly dependent on the maximal temperature reached in the early universe.Evaluating the result by setting T to the temperature corresponding to H max , T max ≃ 9 × 10 15 GeV, the dotted curve in fig. 3 is obtained, this time as an actual spectrum in accordance with eq.(4.2). 6 Like for phase transitions (cf.fig.2), a period of matter domination, present if min{Υ, Γ} ≪ H max , would reduce the signal.

Conclusions
Our main findings can be summarized as follows.For phase transitions taking place soon after inflation, the gravitational wave signal is suppressed by a matter domination epoch, in an analytically quantifiable manner (cf.eqs.(3.5) and (3.7)), unless all equilibration rates are larger than the Hubble rate at the time of the transition.In the latter case (which could be realized for instance in models leading to a "strong regime" of warm inflation, or if the phase transition takes place after matter domination has ended), the phase transition signal can in principle saturate the N eff bound, at f 0 < 20 GHz (cf.fig.3).However, whether this actually happens depends on model-dependent characteristics of the phase transition.At f 0 > 100 GHz, in contrast, the phase transition signal must merge with the irreducible background from thermal fluctuations (cf.fig.3).The reason is that at distances less than the mean free path, "bubbles" are nothing but regular thermal fluctuations.We note that the latter physics seems to be missing from the considerations leading to fig. 2 of ref. [5]. 7

a 4 *e + a 4 +e + a 4 +e − a 4 − 3 . ( 3 . 3 )
For the third factor, we express a via s dv a 3 dv ≃ constant, resulting in

Figure 3 :
Figure3: A combination of the two panels in fig.2with min{Υ, Γ} ≫ H * , expressed in terms of the strain h t from eq. (4.1), compared with the ∆N eff bound from eq. (4.4), as well as the thermal fluctuation result with T max ≤ 9 × 10 15 GeV, corresponding to H max ≤ 10 −5 m pl .The last part consists of two known domains, namely an IR tail from hydrodynamic fluctuations at small f 0[16] and a peak part from elementary particle scatterings at large f 0[17]; we have connected these with a straight line.The amplitude of the peak part increases in BSM scenarios (cf.refs.[18,19] for concrete examples).
3.1), we parametrize the energy density released into gravitational radiation by its relation to the dark sector energy density at that time.Making use of eq.(2.9), this then yields * )H * .(3.2)

5
Adopting eq.(4.1), the results of fig.2are replotted in fig.3.We compare the phase transition result with two other considerations.The first is the parameter N eff , characterizing the energy density carried by additional relativistic species at the time of primordial nucleosynthesis.If we write the gravitational energy density at that time as e gw,bbn ≡ ∆N eff (7/8)(4/11) 4/3 e γ,bbn , and redshift until today, then