Gravitational Waves from Particle Decays during Reheating

Gravitational waves have become an irreplaceable tool for exploring the post-inflationary universe. Their cosmological and astrophysical origins have been attracting numerous attention. In this Letter, we point out a novel source of ultra-high frequency gravitational waves: the decay of particles produced during the reheating era. We highlight the decay of the Higgs boson as a representative case, showing how it yields a testable gravitational wave spectrum by future observations.

In this Letter, we explore a novel source of GWs that may provide insights into the dynamics of a nonthermal particle during reheating.As testable scenarios, we consider decays of inflaton and the Higgs boson in the SM and its extension after inflation, resulting in GWs at frequencies higher than those aimed by LISA [51], DE-CIGO [52], BBO [53], but accessible at other proposed experiments searching for high-frequency GWs [54][55][56][57][58][59].
The Letter will proceed as follows.After clarifying the inflationary sector in Sec.II and evaluating the GW spectrum produced from the inflaton decay based on a Starobinsky-type model [60] as an example in Sec.III, we apply a similar method to the case of the decay of the Higgs boson produced by the inflaton decay during reheating in the SM in Sec.IV and its extension in Sec.V. Finally, we summarize the results in Sec.VI and give discussions and conclusions in Sec.VII.

II. INFLATIONARY SECTOR
As a viable example of inflation, we consider a Starobinsky-type model [60], whose potential for the inflaton ϕ is given by where M P ≃ 2.4 × 10 18 GeV is the reduced Planck mass, and m ϕ is the inflaton mass, which is determined by the amplitude of the curvature power spectrum A S as where N * is the number of e-folds when the pivot scale k * = 0.05 Mpc −1 exits the horizon during inflation.From the Planck data [11], the amplitude of the scalar perturbation is given by A S = 2.099×10 −9 .By taking N * = 50 for definiteness, we obtain m ϕ ≃ 3 × 10 13 GeV.
After inflation is over at the scale factor a = a e , the inflaton starts oscillating about the potential minimum where approximately V (ϕ) ≃ m 2 ϕ ϕ 2 /2.The inflaton energy density, ρ ϕ , follows the Boltzmann equation, where H = ȧ/a is the Hubble parameter, and the energy transfer rate Γ (E) ϕ is defined by [61,62] Γ with n being a label of the n-th Fourier mode of the inflaton oscillation.The decay amplitude of the n-th mode is denoted by M n with the transferred energy E n = nm ϕ .The Lorentz-invariant phase-space element dLIPS should be taken according to the final state kinematics.We suppose that the inflaton predominantly decays into a pair of the SM fermions (f ), whose interaction is given by L ϕ,int = y ϕ ϕf f. ( We note that the energy transfer rate in the present case coincides with the normal decay width of the inflaton, Graviton bremsstrahlung with the initial state being either the inflaton or the Higgs boson. The reheating completes when ρ ϕ (a reh ) = ρ rad (a reh ), where ρ rad represents the radiation energy density.The reheating temperature is estimated as [61] T reh ≃ 7.7 × 10 4 GeV × 427/4 g reh 1/4 y ϕ 10 −10 , (7) where g reh = g * (T reh ) = 427/4 if all the SM particles are in the thermal bath at T = T reh .

III. GW SPECTRUM FROM INFLATON DECAY
Associated with the decay into fermions, a single graviton is emitted via the bremsstrahlung process [34][35][36].The relevant diagrams for the three-body decay are shown in Fig. 1.The total decay width can then be divided into two pieces, where Γ (0) ϕ ≃ Γ ϕ→f f does not involve gravitons in the final state, while Γ ϕ has a single graviton in addition to the fermions.It is convenient to further decompose Γ (1) ϕ into two pieces [35], Γ Γ (1) n being the corresponding amplitude, where E GW is the energy of graviton.
The evolution of the graviton energy density, ρ GW , follows the Boltzmann equation, from which we obtain We note that dΓ ϕ→GW /dE GW is defined as the differential decay width of the inflaton, where dLIPS 3 /dE GW indicates the integration for the three-body phase space, without integrating for E GW .
The gravitational wave spectrum can be obtained by with ρ cr,0 h −2 = 8.0992 × 10 −47 GeV 4 being the present critical density.Based on the above argument, we can readily evaluate the GW spectrum associated with the inflaton decay into fermions.The graviton h µν is defined by g µν = η µν + 2h µν /M P with η µν = diag(+1, −1, −1, −1), and couples to the energy-momentum tensor as For a spin-1/2 particle we have where We have here neglected the terms proportional to η µν as it vanishes due to the traceless condition of the graviton.After summing over all the spin and the polarization states, we obtain the differential energy-transfer rate as where ) under the approximation of m f ≪ m ϕ .Thus, we arrive at for E GW ≪ m ϕ , where ρ ϕ (a e ) ≡ ρ e ≃ 0.175m 2 ϕ M 2 P in the Starobinsky model [63], and H(a e ) ≡ H e = ρ e /3/M P ≃ 7.2 × 10 12 GeV × (ρ e /0.175m 2 ϕ M 2 P ) 1/2 .Using E GW = 2πf (a 0 /a reh ) with f and a 0 being the frequency of the GWs in the present universe and the today's scale factor, respectively, we obtain with the peak frequency being a reh a 0 ≃ 3.7 × 10 18 y ϕ 10 −10 −1

Hz. (21)
Our estimation is consistent with the results in Ref. [35].

IV. GW SPECTRUM FROM THE SM HIGGS BOSON DECAY
We now turn to a discussion of the GWs generated by the Higgs boson decay after the end of inflation.We suppose that the Higgs doublet Φ is produced from the inflaton decay through the interaction term, where µ is a dimensionful coupling.By denoting χ as a real scalar degree of freedom, the decay rate of ϕ → χχ is given by Thus, the branching fraction is obtained as where a factor of 4 counts the number of real scalar degrees of freedom in Φ.In the following, we take Br χ as a free parameter.We note that the coupling (22) induces a temporal contribution to the squared mass of the Higgs boson given by µ 2ρ ϕ /m 2 ϕ .In the parameter space of our interest, the temporal mass contribution is sufficiently smaller than m ϕ , otherwise, the nonperturbative effect becomes important [62].
The produced Higgs boson decays into gauge bosons and fermions.To illustrate how the Higgs boson decay generates GWs, for the sake of brevity, we assume that the Higgs boson predominantly decays into a pair of fermions through the Yukawa interactions.By writing the relevant Yukawa coupling as y f , the decay width of χ into a pair of fermions is given by where m χ is the Higgs boson mass during reheating.Both m χ and y f are to be renormalized values at the relevant scale of the decay at early times, while we take them as free parameters based on the agnostic viewpoint, as explained below.
For simplicity, we consider the case where the squared Higgs boson mass is positive so that the electroweak (EW) symmetry is not broken spontaneously during reheating; namely, the masses of the four real scalar degrees of freedom in Φ are m χ .The following argument, however, does not depend on whether the Higgs field acquires a vacuum expectation value or not, as long as the nonthermal condition is satisfied, as described in the following paragraph.Although the Higgs boson mass today is m χ ≃ 125 GeV, it can be much larger during reheating because of the unbroken EW symmetry.
When the Higgs mass during reheating is in excess of the temperature, the produced Higgs boson maintains a nonthermal spectrum [64][65][66], which is given by where p denotes the absolute value of the Higgs momentum, and θ(x) is the Heaviside step function.The number density of the Higgs doublet, n χ (t), follows the Boltzmann equation given by where the inflaton number density is obtained by solving Eq. ( 3), yielding n ϕ = (ρ e /m ϕ )(a/a e ) −3 for a ≪ a reh .Integrating Eq. ( 27) from a e to a, we obtain Note that when the decay rate of χ becomes nonnegligible, n χ starts decreasing as a −3 , resulting in a suppression in the GW production.The time scale of when this happens can be estimated from Br In the following, we require a d > a reh , corresponding to Γ ϕ m ϕ ≳ Γ χ m χ .In a similar manner to the inflaton decay, a single graviton is emitted via the bremsstrahlung process in the Higgs boson decay, as shown in Fig. 1.By taking the nonthermal Higgs distribution into account, the Boltzmann equation for ρ GW is written as where dΓ χ→GW /dE GW is the differential decay rate of the Higgs boson at rest, and for m χ ≪ m ϕ .The differential decay rate is given by where the kinetic factor F (x) is the same as the inflaton decay width.Integrating Eq. ( 30) from a = a e to a = a reh , we obtain For a reh ≫ a e and E GW ≪ m χ , we arrive at with the peak frequency given by Hz. (35) It is important to notice that in Eq. ( 33), dρ GW (a reh )/dE GW is independent from a reh /a e at leading order for a reh ≫ a e , while the same quantity in the case of the inflaton decay becomes dρ GW (a reh )/dE GW ∝ (a reh /a e ) −3/2 as can be seen in Eq. ( 19); i.e., a larger suppression than the case of the Higgs boson decay.The difference comes from the fact that, in the case of the Higgs boson decay, the inflaton keeps producing the Higgs boson during reheating, enhancing the GW production from the Higgs boson decay, and hence the dilution effect for ρ GW is mitigated.Such an enhancement is absent in the case of inflaton decay as the inflaton is the primary source of producing GWs.
On the other hand, by noticing that the maximal GW amplitude is obtained when a d ≃ a reh , corresponding to Γ ϕ m ϕ ≃ Γ χ m χ .In this case, we obtain h 2 Ω GW (f peak ) ≃ 1.2 × 10 −19 Br χ .This result implies that the ratio of the energy densities of radiation and GW is crucial, suggesting the following new venue for new physics to be tested by GWs.

V. GW SPECTRUM FROM THE B-L HIGGS BOSON DECAY
As an example of new physics models, we consider the gauged B − L model where U(1) B−L is gauged by adding three right-handed neutrinos.An additional Higgs field S, charged +2 under U(1) B−L , is introduced to spontaneously break U(1) B−L by acquiring a non-zero VEV, ⟨S⟩ = v S .In the broken phase where S = v S + s/ √ 2, the real part s obtains a mass m s , while the right-handed neutrinos and the U(1) B−L gauge boson acquire masses m N and m Z ′ , respectively.For simplicity, we focus on the case where the spectrum of these new particles are m s ≲ m ϕ ≪ m Z ′ , m N and where the inflaton decays only into a pair of s.
By integrating out N and Z ′ , s obtains an effective coupling to the Riemann tensor [67], with β α3 = 31/11520π 2 .Thus, the decay width of s into a pair of gravitons is The differential decay width becomes On the other hand, s may also decay into four SM fermions through off-shell which is the leading decay channel for the given mass spectrum.Taking the massless limit for the final state fermions, we obtain the decay width as where we have used m Z ′ ≫ m s .In a similar manner to the SM Higgs case, we estimate the GW spectrum as where a eq is the matter-radiation equality time, namely, a eq /a 0 ≃ 1/3388 [68].At the peak frequency we obtain

VI. RESULTS
In Fig. 2 we show our results, where we have taken y ϕ = 10 −5 for the inflaton decay (black dot-dashed) and v S = 0.3M P for the B − L Higgs decay (red dashed).As can be seen from Eqs. ( 20) and ( 21), the peak value of h 2 Ω GW ≃ 5.2 × 10 −18 from the inflaton decay does not depend on y ϕ , while the place of the peak varies with y ϕ .The black shaded region is excluded by the LIGO observing run 3 [69], and the other shaded regions are future projections of LISA and BBO, whose sensitivity curves are taken from Ref. [70].The sensitivity curves of CE and DECIGO are taken from Refs.[71] and [52], respectively.Shown in the brown line indicates the parameter space that can be explored by proposed experiments using resonant cavities [55,58].The constraint from the BBN, h 2 Ω GW < 1.3 × 10 −6 [72], is depicted by the dotted line.
For the GWs from the inflaton decay, if we require perturbativity as y ϕ ≲ 10, the lowest value of f peak ≃ 3.7 × 10 7 Hz is suggested, which is beyond the reach of the future interferometers.The resonant cavities, on the other hand, would have a chance to test such a scenario.The GWs from the B − L Higgs boson decay can reach the BBN bound when m s and v S are sufficiently large.On the other hand, for BBN to happen, we need T reh > 4 MeV [73], or equivalently Γ s→rad ≳ 10 −23 GeV(T reh /4 MeV) 2 .From Eq. ( 39), we obtain (m s /10 13 GeV) 7 (0.3M P /v S ) 6 ≳ (T reh /4 MeV) 2 , by which the peak value of Ω GW is limited as h 2 Ω GW (f peak )/7.9 × 10 −8 ≲ (m s /10 13 GeV) 14/3 (T reh /4 MeV) −4/3 .Note that for future projections including COrE and Euclid, we expect that h 2 Ω GW ≳ 7.6 × 10 −8 can be tested [74].

VII. DISCUSSIONS AND CONCLUSIONS
We briefly mention possible models that fit our argument.As we have seen, to maximize the GWs from the Higgs decay, m χ needs to be large enough, i.e., a large hierarchy between the Higgs boson mass during reheating and the EW scale.This type of hierarchy requires finetuning in a context of, for instance, the radiative breaking of EW symmetry in supersymmetric models [75][76][77][78].One of such models is discussed in, e.g., Ref. [79], where the Higgs boson mass at the EW scale is obtained by the radiative EW symmetry breaking together with cancellations among mass parameters of the order of the grand unification scale.
Finally, we emphasize that the GW spectrum discussed in this Letter is not limited to the decay of the Higgs boson.As one such example, we have considered the B − L breaking Higgs decay, in which the produced GWs may be tested in future observations.It is also applicable to heavy particles produced during reheating, including those that emerged from supersymmetry and extended Higgs sectors.The resultant GWs are notably characterized by ultra-high frequencies, which have recently gained increased attention, spurring numerous discussions and proposals for forthcoming experiments (see, e.g., Refs.[80,81]).Taking particle decay during reheating as a new source of GWs will open up avenues for extensive future research.
0 s n 5 z n e + c 7 4 4 U 8 m o N p 7 3 Z 8 d x 7 + z e v b d 3 v / X g 4 a P H T 9 r 7 T 8 + 0 y FIG. 2: Summary of the GW spectrum from the decays of the inflaton and the B − L Higgs boson.