Journal of Cosmology and Astroparticle Physics       PAPER • OPEN ACCESS Constraining the general oscillatory inflaton potential with freeze-in dark matter and gravitational waves To cite this article: Jose A.R. Cembranos and Mindaugas Karčiauskas JCAP08(2024)051   View the article online for updates and enhancements. You may also like Intelligent mission planning of sky survey based on the NSGA-III algorithm Rongsheng Jia and Hongfei Wang - The Earth radiation balance as driver of the global hydrological cycle Martin Wild and Beate Liepert - Cosmic Concordance and Quintessence Limin Wang, R. R. Caldwell, J. P. Ostriker et al. - This content was downloaded from IP address 147.96.28.131 on 11/11/2024 at 19:45 https://doi.org/10.1088/1475-7516/2024/08/051 https://iopscience.iop.org/article/10.1088/1742-6596/2764/1/012092 https://iopscience.iop.org/article/10.1088/1742-6596/2764/1/012092 https://iopscience.iop.org/article/10.1088/1748-9326/5/2/025203 https://iopscience.iop.org/article/10.1088/1748-9326/5/2/025203 https://iopscience.iop.org/article/10.1086/308331 J C A P 0 8 ( 2 0 2 4 ) 0 5 1 ournal of Cosmology and Astroparticle Physics An IOP and SISSA journalJ Received: July 2, 2024 Accepted: August 13, 2024 Published: August 28, 2024 Constraining the general oscillatory inflaton potential with freeze-in dark matter and gravitational waves Jose A.R. Cembranosa and Mindaugas Karčiauskasb,c aDepartamento de Física Teórica and IPARCOS, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain bDepartamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain cCenter for Physical Sciences and Technology, Saulėtekio av. 3, 10257 Vilnius, Lithuania E-mail: cembra@ucm.es, mindaugas.karciauskas@ftmc.lt Abstract: The reheating phase after inflation is one of the least observationally constrained epochs in the evolution of the Universe. The forthcoming gravitational wave observatories will enable us to constrain at least some of the non-standard scenarios. For example, models where the radiation bath is produced by the perturbative inflaton decay that oscillates around a minimum of the potential of the form V ∝ ϕ2n, with n > 2. In such scenarios a part of the inflationary gravitational wave spectrum becomes blue tilted, making it observable, depending on the inflation energy scale and the reheating temperature. The degeneracy between the latter two parameters can be broken if dark matter in the Universe is produced via the freeze-in mechanism. The combination of the independent measurement of dark matter mass with gravitational wave observations makes it possible to constrain the reheating temperature and the energy scale at the end of inflation, at least within some parameter ranges. Keywords: dark matter theory, primordial gravitational waves (theory), inflation ArXiv ePrint: 2311.00378 © 2024 The Author(s). Published by IOP Publishing Ltd on behalf of Sissa Medialab. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. https://doi.org/10.1088/1475-7516/2024/08/051 mailto:cembra@ucm.es mailto:mindaugas.karciauskas@ftmc.lt https://doi.org/10.48550/arXiv.2311.00378 http://creativecommons.org/licenses/by/4.0/ http://creativecommons.org/licenses/by/4.0/ https://doi.org/10.1088/1475-7516/2024/08/051 J C A P 0 8 ( 2 0 2 4 ) 0 5 1 Contents 1 Introduction 1 2 The freeze-in DM production with an arbitrary inflaton potential 2 2.1 The setup 2 2.2 Evolution of the inflaton and radiation during reheating 4 2.3 Freeze-in production of dark matter 6 3 Gravitational waves 8 3.1 The spectrum 8 3.2 The bound from the Big Bang Nucleosynthesis 9 4 The parameter space 10 5 Summary and conclusions 15 1 Introduction Freeze-out is a popular mechanism to explain the production of Dark Matter (DM) [1–5]. According to this mechanism DM is in thermal equilibrium with the hot plasma of the early Universe. As the Universe expands and cools down, the DM interaction rate becomes smaller than the expansion rate. This makes DM particles decouple from the rest of plasma, leaving the comoving particle number density constant thereafter. The final DM abundance in this scenario is not very sensitive to the physics of the early Universe; at least not for models with a standard thermal history after the freeze-out. An alternative DM production mechanism is the so called “freeze-in” mechanism (see e.g. [6]). According to this paradigm, the interactions of DM are so weak that they are never in thermal equilibrium with the cosmic plasma. However, such interactions are strong enough that the scattering of the thermal bath particles can produce DM in sufficient abundance. After the production, DM particles travel unimpeded. In contrast to the freeze-out scenario, the DM abundance from the freeze-in scenarios is very sensitive to the details of the early Universe physics. This offers an opportunity: if Nature did select this scenario for DM production, by measuring DM properties will enable us to learn and constrain the physics of the early Universe. The potential of such studies is, for example, investigated in refs. [7, 8], where it is demonstrated how the measurement of DM particle mass could give us information about the maximum themperature and the reheating temperature after inflation. Any new method to constraint the epoch of reheating by observations is very welcome, as this remains to be a very poorly understood period due to the lack of direct observational signatures. The authors of ref. [7] assumed hypercharged minimal DM and a standard reheating scenario. According to such a scenario the inflaton oscillates around the minimum of the quadratic potential after inflation and perturbatively decays into relativistic particles [9]. The sensitivity of – 1 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 the freeze-in mechanism to the early Universe physics also means that any change in this standard reheating scenario could affect the properties of DM particles. For example, the effects of the non-standard expansion history after inflation is studied in refs. [10–14] and the authors of ref. Becker et al. [15] investigate other ways how freeze-in DM could be used to constrain the reheating epoch. In the past several years the detection of Gravitational Waves (GW) [16–18] opened a new window to investigate the Universe. More recently, the detection of the stochastic GW background by pulsar timing array collaborations NANOGrav, EPTA/InPTA, PPTA and CPTA has reached another big milestone in the GW astronomy [19–22]. Although the detected background signal is generated by astrophysical sources, it demonstrates the potential to study the physics of the early Universe too. There are numerous processes that could operate in the early Universe and produce a detectable background of GW [23, 24]. One of such processes is the non-standard expansion history after inflation [25–28]. Due to vacuum fluctuations, inflation leaves the Universe filled with the primordial GW background that is (quasi) scale invariant [9]. This spectrum is modified as the Universe undergoes the various phases of its evolution [29]. In particular, any epoch, where the expansion rate deviates from the radiation dominated one, tilts the spectrum in the range of subhorizon modes. In many models the tilt can be large enough that the inflationary GW signal falls within the sensitivity limits of future observatories (see for example [28, 30]). In this work we investigate how the combination of DM searches and GW observations could shed some light on inflation and the reheating epochs. In section 2 we compute the freeze-in production of DM particles during reheating. In contrast to the previous calculations we allow for the inflaton to oscillate around a non-quadratic potential. This modifies the expansion rate during reheating and the final density of DM. The non-standard expansion rate also affects the spectrum of the background GW. The computation of such a spectrum is summarised in section 3. In section 4 we find the relation between that spectrum and the DM properties. We also explore the allowed parameter range and determine regions which are accessible by future observations. 2 The freeze-in DM production with an arbitrary inflaton potential 2.1 The setup We consider a similar setup as discussed in ref. [6]. The model consists of three components: an inflaton field, radiation and the non-relativistic DM. After inflation the inflaton ϕ oscillates around the minimum of the potential and initially dominates the energy budget of the Universe. To simplify the discussion, we assume that the inflaton decays almost exclusively into light degrees of freedom (radiation) with the decay rate Γϕ. The thermalisation rate is taken to be fast enough so that we can assume these degrees of freedom to be in a local thermodynamic equilibrium. We can then relate the energy density of radiation to the thermodynamic temperature by ρr = π2g∗ (T ) 30 T 4 , (2.1) – 2 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 where g∗ (T ) is the number of effective relativistic degrees of freedom at a given temperature. The Universe is reheated with the temperature Treh when radiation takes over other contribu- tions to the energy budget. At that moment the Universe becomes radiation dominated. It is important to recognise that Treh is not the maximum temperature, as we will see later. The heavy and stable DM particles are non-relativistic throughout the whole of reheating process. Moreover, they are so weakly coupled to radiation that DM particles never come into thermal equilibrium. Nevertheless, the interaction is sufficiently strong that the thermal bath produces DM particles, which are then freely diluted by the expansion of the Universe. The Boltzman equations (in terms of energy densities) for such a setup can be written as [6] ρ̇ϕ + 3H (1 + wϕ) ρϕ = −Γϕρϕ , (2.2) ρ̇r + 4Hρr = Γϕρϕ , (2.3) ρ̇DM + 3HρDM = ⟨σ |v|⟩ MDM (ρeq DM)2 , (2.4) where ρϕ, ρr and ρDM are the energy densities of the inflaton, radiation and DM respectively. MDM is the mass of DM particles and ⟨σ |v|⟩ is the thermal average cross section for annihila- tion of DM. As DM is assumed to be far from equilibrium, only the (ρeq DM)2 term dominates the R.H.S. of eq. (2.4), which is the non-relativistic equilibrium energy density, given by ρeq DM (T ) = M 5/2 DMT 3/2 (2π)3/2 · e−MDM/T . (2.5) The inflaton decay rate Γϕ is assumed to be much lower than the Hubble parameter at the end of inflation, HI, Γϕ ≪ HI . (2.6) This leads to a long period of reheating, before the inflaton decays into radiation completely. Equations (2.2)–(2.4) differ from analogous equations in ref. [6] by that we do not assume the inflaton to be oscillating in a quadratic potential after inflation. We rather allow for a much flatter form of the potential at the minimum, such that V (ϕ) ∝ ϕ2n , (2.7) where n is a natural number. Moreover, since we are interested in observable stochastic background of GWs, we only consider values n > 2. These types of potentials can be encountered, for example, in α-attractor [31, 32] or monodromy [33] models of inflation. As it is well known (see [34–37]), an oscillating scalar field in such a potential leads to an effective equation of state of the form wϕ = n− 1 n+ 1 . (2.8) This value of wϕ is used in eq. (2.2) above. n = 2 corresponds to wϕ = 1/3; as n increases, wϕ approaches 1. However, wϕ = 1 describes a non-oscillatory inflaton potential [38], hence we cannot assume the standard perturbative reheating scenario and apply our calculations to – 3 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 estimate DM abundances. The wϕ = 1 case rather serves as an upper limit for the allowed range of wϕ values. With non-oscillatory potentials one has to employ other ways to reheat the universe. In summary, the range of wϕ values that will be used in this work is 1/3 < wϕ < 1. Inflaton oscillations in a non-quadratic potential also induce the time dependence of the decay rate Γϕ [14, 35, 39–42], which we parametrise by Γϕ = ΓIa ν , (2.9) where ΓI is the inflaton decay rate at the end of inflation and we normalised the scale factor to equal unity at that moment aI = 1. The scaling ν depends on the shape of the potential in eq. (2.7) as well as on the dominant inflaton decay channel. According to [35] if the inflaton decays primarily into fermions then ν = −3wϕ and ν = 3wϕ if it decays into scalar particles. One can device a more complicated behaviour of Γϕ (see e.g. [43]) but we consider only this generic case in the current work. 2.2 Evolution of the inflaton and radiation during reheating The condition in eq. (2.6) ensures that before the end of reheating the R.H.S. of eq. (2.2) is negligible. Here, we define the moment of reheating to be the moment when radiation energy density takes over that of the inflaton, i.e. reheating happens at areh when ρreh r = ρreh ϕ . The interval between inflation and this point we call the period of reheating. It is also a sufficiently good approximation to assume that the oscillating inflaton dominates the energy budget of the Universe before radiation takes over. In other words, we can approximate ρ ≃ ρϕ during reheating, where ρ is the total energy density. Solving eq. (2.2) then leads to ρϕ ≃ ρIa −3(1+wϕ) , (2.10) where a is the scale factor and ρI is the energy density at the end of inflation. Plugging this result into the Friedman equation 3m2 PlH 2 = ρ we can also compute the scaling of the Hubble parameter H ≃ HIa − 3 2 (1+wϕ) . (2.11) The evolution of the radiation energy density is governed by eq. (2.3). To find the solution of this equation during reheating, we can plug eqs. (2.10), (2.11) and (2.9) into eq. (2.3). At the lowest order in Γϕ/H ≪ 1 the integral is equal to ρr = ρI 4α ΓI HI ( a4α − 1 ) a−4 , (2.12) where α is defined as α ≡ 1 8 (5 − 3wϕ + 2ν) . (2.13) With the upper bound wϕ < 1, that were specified in the paragraph bellow eq. (2.8), and for −3wϕ ≤ ν ≤ 3wϕ, the range of possible α values is −1/2 < α < 1. Notice, that the above – 4 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 Figure 1. The evolution of the inflaton and radiation energy densities ρϕ and ρr respectively during reheating. The dotted lines represent approximate scalings derived in the text. solution is not valid if α = 0 (2ν = 3wϕ − 5). In that case ρr ∝ a−4 ln a. But since we are mainly interested in ν = ±3wϕ values, this case will not be analysed any further. Whichever the sign of α, the maximum value of ρr is achieved at amax = (1 − α)− 1 4α . (2.14) After this point the evolution of radiation does depend on the sign of α. In the case of α > 0, the first term in eq. (2.12) dominates. We call this case the “lasting inflaton decay case”, or the “lasting decay” briefly. In this case the production of radiation by the oscillating inflaton field continues to be significant all the way up until reheating is complete. Newly produced radiation continuously sources the radiation bath and makes its energy density decrease as ρr ∝ a− 3 2 (1+wϕ)+ν , i.e. slower than a−4. In the opposite regime α < 0, which we call the “brief decay” case, the radiation production is very inefficient. After reaching ρmax r , the energy density of the radiation bath cannot keep up with the expansion of the Universe and decays adiabatically, ρr ∝ a−4. Whichever the decay regime, the lasting or the brief one, we can compute the maximum temperature of the Universe. If we assume that the number of effective relativistic degrees of freedom g∗ (T ) in eq. (2.1) is constant during reheating, which we denote by g∗, the maximum value of ρr in eq. (2.12) can be related to the maximum temperature of radiation, which is T 4 max = ρI 15 2π2g∗ ΓI HI (1 − α) 1 α −1 . (2.15) To check if the above described scenario is correct, we also numerically integrate the exact system of equations (2.2)–(2.4). A few examples of those solutions are shown in figure 1 together with a few lines illustrating analytic solutions in eqs. (2.10) and (2.12). We can – 5 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 clearly see that after the initial burst of radiation production, the latter passively decays away as a−4 in the brief decay regime (α < 0). This is demonstrated in the upper row of the figure. Conversely, for α > 0 (the lower row of the figure) the radiation production continues to be significant until the moment of reheating. As was mentioned above, we define the moment of reheating, to be the instance when the radiation energy density starts dominating over the inflaton one. This can happen in both, brief or lasting, decay scenarios. In the former case, even if the production of radiation is negligible after amax, the radiation redshifts away slower than the inflaton (see the upper row of figure 1), eventually taking it over. In the lasting decay case several things can happen. First, as in the previous case, ρr might take over ρϕ at a = ard. But the evolution of ρr was computed assuming Γϕ/H ≪ 1, which scales as Γϕ/H ∝ aν+ 3 2 (1+wϕ). For the most of parameter values this ratio is growing with time. It is therefore possible that Γϕ/H becomes larger than 1 at a = adec < ard, before the domination of ρr in eq. (2.12). When the Γϕ < H bound is violated, the inflaton decays away in less than a Hubble time, giving its energy to radiation. Hence, in the lasting decay scenario we can define the moment of reheating to be areh = min [ard, adec]. In practice, however, it is sufficient to use the approximation areh ≃ ard. To see this, notice that ard/adec = (4α)1/(4α−1+3wϕ), which is never much larger than 1 for the interesting parameter values. In conclusion, for the rest of the paper the moment of reheating will be determined using eq. (2.12) and finding the moment ρr (areh) ≃ ρϕ (areh). With this in mind, we find the scale factor at reheating to be areh = ( 4 |α| HI ΓI ) 1 4ϑ−1+3wϕ , (2.16) where we introduced the parameter ϑ ≡ max {0, α} (2.17) to take into account both cases, α > 0 and α < 0. It is easy to compute the temperature of the Universe at areh, which is given by T 4 reh = ρI ( 30 π2g∗ )( ΓI 4 |α|HI ) 3(1+wϕ) 4ϑ+3wϕ−1 . (2.18) 2.3 Freeze-in production of dark matter Next, we integrate eq. (2.4), which specifies the evolution of DM. Before the epoch of matter-radiation equality DM is subdominant to other components of the Universe. This allows us to substitute H and T by the values computed in the previous section when we integrate eq. (2.4) next. Moreover, for concreteness, we assume that the DM annihilation is dominated by the s-wave channel, that is very common for renormalizable interactions. In such a case, the thermal average cross section is just a constant value. To do the integration it is first convenient to rewrite eq. (2.4) as a3ρreh DM = κ a∫ 1 e−g(a′)da′ , (2.19) – 6 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 where we used ρDM (aI = 1) = 0. The dimension-4 constant κ in the above expression depends on the DM cross section, and is given by κ = M2 DM ⟨σ |v|⟩ (λπ)3 · M 5 DM HI . (2.20) While the g (a) function is defined as g (a) ≡ −λ ( a4α − 1 4α )− 1 4 a+ 3 4 ln ( a4α − 1 4α ) + 1 2 (1 + 3wϕ) ln a , (2.21) where λ ≡ √ 2aα−1 maxxmax (2.22) and we introduced a dimensionless parameter x ≡ MDM T , (2.23) with xmax ≡ x (Tmax) and Tmax given in eq. (2.15). If the mass of DM particles is sufficiently smaller than the maximal temperature, MDM < Tmax, the integral in eq. (2.19) can be evaluated using the Laplace approximation. In that case we can take the freeze-in scale factor to be afi > 1. The approximate value of afi, which depends on the sign of α, is afi ≃ [ 6ϑ+ 1 + 3wϕ 2 √ 2λ |α| 1 4 (1 − θ) ] 1 1−ϑ . (2.24) At afi the comoving energy density of DM particles freezes in and remains constants thereafter. That value is given by a3ρDM = 16a4 fi π5/2 ·M2 DM ⟨σ |v|⟩ M 5 DM HI ( 1 − ϑ 6ϑ+ 1 + 3wϕ ) 7 2 ( a1−ϑ fi e ) 6ϑ+1+3wϕ 2(1−ϑ) a−3ϑ fi 1 − ϑ . (2.25) Utilising the constancy of a3ρDM we can estimate the present day DM abundance ΩDM Ωr = ρreh DM ρreh r [ g∗ (Ttd) g∗ ] 1 3 ( atd areh ) , (2.26) where ‘td’ denotes values evaluated today (remember that g∗ ≡ g∗ (Treh)). ΩDM and Ωr denote the current values of DM and radiation respectively. Plugging eq. (2.1) into the above expression we find ΩDM Ωr = 30 π2g∗ ρreh DM T 3 rehTtd . (2.27) Computing ρreh DM from eq. (2.25) we can write xreh ≡ MDM Treh =  g∗ 80 √ 3π ΩrmPl ΩDMTtd M2 DM ⟨σ |v|⟩ e− 6ϑ+1+3wϕ 2(1−ϑ) 1−ϑ · (6ϑ+1+3wϕ 4(1−ϑ) ) 7ϑ+2+3wϕ 2(1−ϑ)  2(1−ϑ) 10ϑ−1+3wϕ , (2.28) – 7 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 where Treh is given in eq. (2.18). The ΩrmPl/ΩDMTtd factor in this expression is fixed by observations. Taking Ttd = 2.725 K [44], which translates to Ωrh 2 ≃ 2.5 × 10−5, and ΩDMh 2 ≃ 0.12 [45], where h is the dimensionless Hubble parameter today defined as Htd = 100h km/s/Mpc, we can compute this factor to be ΩDM Ωr Ttd mPl ≃ 4.6 × 10−28 . (2.29) Eq. (2.28) is one of the main equations of this work. It gives the relation between DM particle mass and the reheating temperature. For later convenience we can also compute the xmax value, which is xmax = xreh a1−α max |α| 1 4 ( π2g∗ 30 M4 DM x4 rehρI ) 1−ϑ 3(1+wϕ) . (2.30) 3 Gravitational waves 3.1 The spectrum After inflation is over, the Universe is unavoidably filled with a background of primordial GWs that have an almost scale invariant spectrum [9] Ph = 8 m2 Pl ( Hk 2π )2 , (3.1) where Hk is the Hubble parameter during inflation at the moment when the k mode exits the horizon. For our purpose it is sufficient to assume that Hk ≃ const during inflation, which makes Ph in eq. (3.1) scale invariant. After inflation, when those modes reenter the horizon, they behave as radiation, in particular, they are redshifted according to ρhk (k < aH) ∝ a−4. On the contrary, the behaviour of the dominant scalar field energy density depends on the shape of the potential in eq. (2.7). For n > 2, or equivalently wϕ > 1 3 in eq. (2.8), the energy density of the oscillating inflaton dilutes faster than radiation. In those cases the relative contribution of GWs to the total energy budget of the Universe increases. The earlier the mode reenters the horizon before the end of reheating, the longer the relative energy density of that modes grows. This modifies the primordial spectrum of GWs by making the large frequency part blue tilted. More concretely, the GW spectrum today can be written as [27, 28] ΩGW (f) ≃ Ωrd GW ×  1 f < freh As ( f freh )−2 1−3wϕ 1+3wϕ f > freh , (3.2) where f denotes the present day frequency of GW modes in units of Hz and freh is the frequency of the mode that reenters the horizon at the end of reheating. Ωrd GW is the present day amplitude of GW modes that reenter the horizon after reheating, that is, during the radiation domination. Because the radiation dominated background scales with the – 8 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 same power as GWs at this epoch, the slope of the GW spectrum is not modified for such frequencies. Hence, Ωrd GW is given by Ωrd GW ≃ Gk Ωr 36π2 ρI m4 Pl . (3.3) In ref. [27] Gk is approximated by Gk ≃ 0.39, which we also adopt in this work. The wϕ dependent constant As is given by As = 1 π (1 + 3wϕ) 4 1+3wϕ Γ2 ( 5 + 3wϕ 2 (1 + 3wϕ) ) , (3.4) where Γ denotes the Gamma function. In eq. (3.2) we do not include the modes that reenter the horizon after matter-radiation equality. These modes are of a very low frequency and will play no role in our analysis. As can be seen from eq. (2.28), the DM abundance today fixes the ratio between the reheating temperature and DM particle mass. In its turn, knowing Treh, we can compute kreh ≡ arehHreh and therefore the corresponding frequency freh of that mode today [27] Treh = 9 × 10−12 (200 g∗ ) 1 4 ( freh Hz ) mPl , (3.5) where we used Gk ≃ 0.39 as stated above. We can invert this expression and write freh = 1011 xreh ( g∗ 200 ) 1 4 MDM mPl Hz , (3.6) where xreh is given in eq. (2.28). The above expression is another main result of this work, which relates the DM mass with the GW frequency corresponding to reheating. 3.2 The bound from the Big Bang Nucleosynthesis The epoch of the Big Bang Nucleosynthesis (BBN) can be used to put some of the tightest bounds on new physics at that period. Any modification of the standard scenario results in different ratios of light element abundances in the Universe, which are tightly constrained by observations [46]. In particular, gravitons are relativistic species. If their contribution to the overall energy density during BBN is too large, the thermal history of the Universe changes and the very successful standard BBN predictions are modified. This limits the amount of GWs at BBN. The bound becomes especially severe for strongly blue tilted spectra [47, 48]. To compute the total energy density of GWs we can integrate the spectral energy distribution over all frequencies ΩBBN GW ≡ fmax∫ fBBN ΩGW (f) df f , (3.7) where ΩGW (f) is the GW density parameter today. fBBN corresponds to the frequency of modes that reenter the horizon at BBN. The largest frequency fmax is taken for kmax modes that exit the horizon at the end of inflation. To leave BBN predictions intact, the upper limit – 9 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 ΩBBN GW h2 < 1.12 × 10−6 must be satisfied [24]. The function ΩGW (f) is provided in eq. (3.2). As can be seen, in the region f > freh the spectrum is blue tilted for wϕ > 1/3. Thus, in practice, the integral in eq. (3.7) is dominated by the largest frequency modes. Using this fact, we can write the bound in eq. (3.7) as [27] ΩGW (fmax)h2 < ΩBBN GW h2 = 2.24 × 10−6 × 3wϕ − 1 3wϕ + 1 . (3.8) The value of fmax can be computed in terms of freh. By construction we have fmax/freh = kmax/kreh = HI/arehHreh, where the Hubble parameter scales as in eq. (2.11) during reheating. Using the Friedman equation ρI = 3m2 PlH 2 I and eq. (2.1) we find fmax freh = ( 4 × 1010 ρ 1/4 I mPl Hz freh ) 2(1+3wϕ) 3(1+wϕ) , (3.9) where we also employed eq. (3.5). Plugging this value into the bound in eq. (3.8), we finally obtain the upper limit of the inflation energy scale that is consistent with BBN bound ρ 1/4 I mPl < ( 36π2 GkAs ΩBBN GW Ωr ) 3(1+wϕ) 8(1+3wϕ) ( 2 × 10−11 freh Hz ) 3wϕ−1 2(1+3wϕ) . (3.10) 4 The parameter space To find models that are compatible with observations we scan over the range of MDM and ρI parameters for given values of wϕ and ν. We also identify the regions of this parameter space that could be probed by the future GW observatories. Bellow we explain how the upper and lower limits of MDM and ρI are computed. First, to compute eq. (2.25) we assumed the lower bound of MDM to be MDM > Treh. But to ensure that the universe is radiation dominated at BBN we must also take Treh > TBBN ≃ 1 MeV. Conveniently, the bound MDM > Treh > TBBN can be written as xreh > 1 ; MDM > xrehTBBN . (4.1) The second inequality provides us with the lower bound for MDM values. As we will see bellow, in some cases this allows for very small MDM, which, one might worry, must be excluded by DM detection experiments. However, that depends on the DM cross section. To remain generic, we do not take into account such experiments in our analysis. To set the upper bound, we note that to compute eq. (2.25), we also assumed MDM < Tmax. For larger MDM values DM production is exponentially suppressed. Using the value of xmax in eq. (2.30) we can write this bound as MDM < xreh ( 30ρI π2g∗ ) 1 4  |α| 1 4 xreha 1−α max  3(1+wϕ) 4(1−ϑ) . (4.2) – 10 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 As can be seen from the above inequality, the largest allowed DM particle mass depends on the energy scale of inflation ρI. There are two possible upper limits for ρI. First, ρI is limited by the observational constraints on the GWs at the pivot scale k∗ = 0.05 Mpc−1, that are measured from CMB observations [49]. This constraint is usually expressed in terms of the tensor-to-scalar ratio r ≡ Ah As , (4.3) where Ah and As are the amplitudes of the tensor and scalar spectra at k∗ respectively. Ah is defined as Ah = Ph (k∗), where Ph is given in eq. (3.1) and As is determined by the Planck normalisation ln ( 1010As ) = 3.043 ± 0.014 [45]. Using eq. (3.1) we can write the inflation energy scale ρ∗ as ρ∗ = 3 2rπ 2Asm 4 Pl . (4.4) At the end of inflation ρI < ρ∗. Hence, plugging in r < 0.036 [49] results in ρ 1/4 I < 5.8 × 10−3mPl . (4.5) The above constraint might not be the tightest one. In certain parameter regions, the requirement that GW do not spoil the predictions of BBN, as it is expressed in eq. (3.10), might provide more stringent constraint. Hence, the maximum value of ρI will be determined by the tighter bound of the two. Plugging the upper bound of ρI into eq. (4.2), we can find the maximum value of allowed DM particle mass. If the constraint in eq. (4.5) is tighter than the one in eq. (3.10), the upper bound on MDM is MDM mPl < 5.8 × 10−3xreh ( 30 π2g∗ ) 1 4  |α| 1 4 xreha 1−α max  3(1+wϕ) 4(1−ϑ) . (4.6) In the opposite case, plugging the upper value of eq. (3.10) into (4.2), we find MDM mPl < 0.81 − 5+9wϕ 3(1+wϕ)xreh ( 1080 g∗GkAs ΩBBN GW Ωr ) 1 4  |α| 1 4 xreha 1−α max  2(1+3wϕ) 4(1−ϑ) . (4.7) In the range of MDM values that satisfy the lower limit in eq. (4.1) and the upper limits in eq. (4.6) and (4.7), the lower value of ρI can be computed by inverting eq. (4.2). That is ρ 1 4 I > MDM xreh ( π2g∗ 30 ) 1 4 ( xreha 1−α max |α| 1 4 ) 3(1+wϕ) 4(1−ϑ) (4.8) To explore the parameter space of this model we adopt the following values g∗ = 200 , (4.9) M2 DM ⟨σ |v|⟩ = 10−2 . (4.10) – 11 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 100 102 104 106 108 MDM, GeV 10 11 10 9 10 7 10 5 10 3 1/ 4 I /m Pl r=0.036 limit = +3w ; w 0.33 0.5 0.6 0.67 0.69 Figure 2. The parameter range with allowed values of MDM and ρI. The dashed blue horizontal line represents the upper limit on ρI given in eq. (4.5). The actual upper limit of ρI is constrained either by this bound or the BBN bound in eq. (3.10). The lower limit of ρI is determined by the condition in eq. (4.8). The lower limit of MDM is given in eq. (4.1) and the upper one in eq. (4.6) or (4.7), whichever is the tighter one. As for the values of ν we consider two possibilities, as mentioned in subsection 2.1. If the dominant inflaton decay branching ratio is determined by the ϕ → ψψ̄ process, where ψ is a spin-1 2 particles, then ν = −3wϕ [35]. However, in this case we do not find a parameter region which would result in the right abundance of DM. In the other case, when the dominant decay channel is ϕ → χ2 and/or the scattering ϕ2 → χ2, the decay rate scaling is given by ν = +3wϕ. This scenario can be successfully realised in a limited range of wϕ values, which is demonstrated in figure 2. We are also interested in the possibility to probe such models with GW observations. To this goal we compute GW spectra for all the allowed range of MDM, ρI and wϕ values and inspect if they cross the sensitivity curves of several GW observatories such as aLIGO, aVirgo, the Kamioka Gravitational-Wave Detector (KAGRA) [50–53], NANOGrav [54–56], PPTA [57, 58], IPTA [59–62], SKA [63–65], LISA [66, 67], BBO [68–70], DECIGO [71– 73], CE [74, 75], ET [76–79]. The power-law-integrated sensitivity curves [80] of these observatories are taken from [81]. We demonstrate GW spectra in figure 3. In this figure the blue shaded regions denote the range of spectra ΩGW (f)h2 for a fixed value of wϕ and MDM, while ρI varies within the above discussed limits. In these plots one can also see sensitivity curves of GW observatories. The lighter shaded part of the blue region denotes the range of inflation energy scale ρI that results in observable GW. In the first plot of figure 3 we show the spectrum for n = 2. As expected, it is flat. The third plot serves to illustrate the parameter values for which the maximum value of ρI is limited not by eq. (4.5) but by the BBN bound in eq. (3.10). The spectra in figure 3 are calculated for fixed values of g∗ and M2 DM ⟨σ |v|⟩ given in eqs. (4.9) and (4.10). However, the results are very insensitive to the exact values of these parameters, as they are suppressed by the power 2 (1 − ϑ) / (10ϑ− 1 + 3wϕ) in eq. (2.28). To demonstrate this fact, we provide figure 4, where several spectra are shown with various values of wϕ, g∗ and M2 DM ⟨σ |v|⟩ (and ν = 3wϕ). – 12 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 10 10 10 6 10 2 102 106 f [Hz] 10 51 10 44 10 37 10 30 10 23 10 16 10 9 h2 GW BBO CEDECIGO ET HLVK IPTA LISANANOGrav SKA BBN bound w = 0.33; MMD = 0.1 GeV; = 3w 1/4/mPl 5.78x10 3 4.20x10 3 1.31x10 12 10 9 10 4 101 106 1011 f [Hz] 10 42 10 36 10 30 10 24 10 18 10 12 10 6 h2 GW BBO CEDECIGO ET HLVK IPTA LISANANOGrav SKA BBN bound w = 0.5; MMD = 0.1 GeV; = 3w 1/4/mPl 5.78x10 3 4.66x10 4 3.41x10 10 10 9 10 4 101 106 1011 f [Hz] 10 23 10 20 10 17 10 14 10 11 10 8 10 5 h2 GW BBO CEDECIGO ET HLVK IPTA LISANANOGrav SKA BBN bound w = 0.67; MMD = 0.1 GeV; = 3w 1/4/mPl 6.76x10 4 1.16x10 4 6.38x10 5 Figure 3. A few examples of GWs spectra for several freeze-in models of DM with three values of wϕ and three DM particle masses. Each spectrum corresponds to one curve. The lowest (green) curve denotes a model with the minimal inflation energy scale that is allowed by eq. (4.8). The upper (blue) curve corresponds to the spectrum of the maximal allowed inflationary scale. In the first two subplots this corresponds to the bound in eq. (4.5). In the last subplot the maximal value of ρI is given by the inequality in eq. (3.10). The shaded region denotes the range of spectra that are allowed by the aforementioned bounds, the lighter part denoting the potentially observable range. In all of these models we take g∗ = 200, MDM ⟨σ |v|⟩ = 10−2. We also show the sensitivity curves of the present and future planned GW detectors. A full scan of the MDM and ρI values is shown in figure 5. In this figure MDM varies from the value given in eq. (4.1) to the value given in either eq. (4.6) or eq. (4.7), whichever is smaller. The lower limit of ρI is determined by eq. (4.8) and the upper limit by the tighter constraint of the one given in eq. (3.10) or (4.5). We perform this scan for several values of wϕ = 0.5, 0.6, 0.667 and 0.69. The first three correspond to n = 3, 4, and 5 in eq. (2.7) respectively. Models with any larger value of n are excluded. wϕ = 0.69 is plotted just for illustrative purposes, it does not correspond to any integer n. In this plot we also display regions that will be accessible by future GW observatories. They are denoted by coloured curves. Regions above a given curve falls within the sensitivity curve of the experiment. The colour coding is the same as in figure 3. We also display the predicted sensitivity of the CMB Stage-4 project [82]. As can be seen from the figure, within some parameter range – 13 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 10 13 10 10 10 7 10 4 10 1 102 105 108 1011 f [Hz] 10 29 10 25 10 21 10 17 10 13 10 9 10 5 h2 GW BBO CEDECIGO ET HLVK IPTA LISA NANOGrav SKA g * 10 200 4000 w 0.4 0.69 10 13 10 10 10 7 10 4 10 1 102 105 108 1011 f [Hz] 10 29 10 25 10 21 10 17 10 13 10 9 10 5 BBO CEDECIGO ET HLVK IPTA LISA NANOGrav SKA M2 DM |v| 1e-12 1e-02 1e+06 w 0.4 0.69 Figure 4. The sensitivity of the GW spectrum to the effective number of relativistic degrees of freedom during reheating, g∗, and the thermal average cross section for annihilation of DM, ⟨σ |v|⟩. For all models we use MDM = 104 GeV and ρ 1/4 I = 10−3mPl. 10 1 100 101 102 103 104 105 106 10 9 10 7 10 5 10 3 1/ 4 I /m Pl w = 0.5 (n = 3); = +3w ; 10 1 100 101 102 103 10 7 10 6 10 5 10 4 10 3 10 2 w = 0.6 (n = 4); = +3w ; 10 1 100 MDM, GeV 10 4 10 3 1/ 4 I /m Pl w = 0.667 (n = 5); = +3w ; 4 × 10 2 5 × 10 2 6 × 10 2 MDM, GeV 10 3 w = 0.69; = +3w ; Figure 5. Parameter ranges (blue regions) for ρI and MDM for various values of wϕ during reheating. The red dashed line corresponds to the ρI bound in eq. (4.5) and the grey dotted line corresponds to the sensitivity limit of CMB Stage-4 project. The colour coded curves mark the regions that will be de- tectable by future GW observatories. Such regions are above the curves, where the colours are the same as of sensitivity curves in figure 3. Notice, that there is no allowed parameter range for n = 6 or large. The lower right plot is just for illustrative purposes, it does not correspond to any integer value of n. the detection of GWs can establish constraints on the inflation energy scale that are tighter by several orders of magnitude, depending on the value of MDM. This is true at least for slow-roll inflation, where the energy scale of inflation does not change much. This is due to the fact that ρI in figure 5 corresponds to the energy density at the end of inflation, while CMB polarisation measurements, such as CMB Stage-4, constrain the epoch that corresponds to tens of e-folds before that. – 14 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 5 Summary and conclusions The properties of DM particles, that are required to reproduce the present day abundance, depend on the thermal history of the Universe. This is especially true if DM is produced via the so called freeze-in mechanism. We can use this sensitivity to constrain the earliest epochs of the evolution of the Universe. In this work we study the freeze-in DM production mechanism and how it is affected by the non-standard expansion history during reheating. In order to reproduce the observed present day abundance, DM particle mass is related to the reheating temperature as shown in eq. (2.28). The relation, among other parameters, also depends on the effective equation of state, wϕ, during reheating. To fix that value another set of observations is needed, which is provided by GW observatories. Observations of GWs give us a new tool to study the Universe and the very early stages of its evolution. Quantum vacuum fluctuations during inflation leave the Universe filled with a stochastic GW background. The primordial spectrum of this background is (quasi-)flat. The ensuing evolution modifies a part of that spectrum. Any period where the expansion rate of the Universe deviates from the radiation dominated one, tilts the spectrum within the range of subhorizon modes. In particular, the expansion rate that is faster than the radiation dominated one, makes the corresponding part of the spectrum blue tilted. Depending on the magnitude of the deviation and the scale of inflation, the tilt can be sufficiently large to make such GW signal directly observable. In such cases, we can use eq. (2.28) to constrain the inflation energy scale, the reheating temperature and the expansion rate during the reheating. In figure 3 we show how the spectrum of GWs is related to the main three parameters of the model: DM particle mass MDM, the expansion rate during reheating and inflation energy scale. In principle, the spectrum depends on the number of relativistic degrees of freedom during reheating, g∗, and the thermal average cross section for annihilation of DM ⟨σ |v|⟩. However, as we show in figure 4, this dependence is very strongly suppressed. Finally, we show the allowed parameter space of such models for several expansion rates (parametrised by wϕ) in figure 5. We can see that in the case of large enough inflation energy scale and 3 ≤ n ≤ 5 (1/2 ≤ wϕ ≤ 2/3), such models produce GWs that are detectable by future observatories. We can also look at this result differently: if DM are produced by the freeze-in process, the measurement of the MDM and the GW spectrum will allow us to constrain the reheating temperature and inflation energy scale. For some values of MDM the latter constraint can more than an order of magnitude tighter than the predicted bound from the future CMB Stage-4 project. Acknowledgments M.K. is partially supported by the María Zambrano grant, provided by the Ministry of Univer- sities from the Next Generation funds of the European Union. This work is also partially sup- ported by the MICINN (Spain) projects PID2019-107394GB-I00/AEI/10.13039/501100011033 and PID2022-139841NB-I00 (AEI/FEDER, UE), COST (European Cooperation in Science and Technology) Actions CA21106 and CA21136. – 15 – J C A P 0 8 ( 2 0 2 4 ) 0 5 1 References [1] B.W. Lee and S. Weinberg, Cosmological Lower Bound on Heavy Neutrino Masses, Phys. Rev. Lett. 39 (1977) 165 [INSPIRE]. [2] R.J. Scherrer and M.S. Turner, On the Relic, Cosmic Abundance of Stable Weakly Interacting Massive Particles, Phys. Rev. D 33 (1986) 1585 [Erratum ibid. 34 (1986) 3263] [INSPIRE]. [3] M. Srednicki, R. Watkins and K.A. Olive, Calculations of Relic Densities in the Early Universe, Nucl. Phys. B 310 (1988) 693 [INSPIRE]. [4] P. Gondolo and G. Gelmini, Cosmic abundances of stable particles: Improved analysis, Nucl. Phys. B 360 (1991) 145 [INSPIRE]. [5] E.W. Kolb, The Early Universe, Taylor and Francis (2019) [DOI:10.1201/9780429492860] [INSPIRE]. [6] D.J.H. Chung, E.W. Kolb and A. Riotto, Production of massive particles during reheating, Phys. Rev. D 60 (1999) 063504 [hep-ph/9809453] [INSPIRE]. [7] B. Feldstein, M. Ibe and T.T. Yanagida, Hypercharged Dark Matter and Direct Detection as a Probe of Reheating, Phys. Rev. Lett. 112 (2014) 101301 [arXiv:1310.7495] [INSPIRE]. [8] P.N. Bhattiprolu, G. Elor, R. McGehee and A. Pierce, Freezing-in hadrophilic dark matter at low reheating temperatures, JHEP 01 (2023) 128 [arXiv:2210.15653] [INSPIRE]. [9] D.H. Lyth and A.R. Liddle, The primordial density perturbation: Cosmology, inflation and the origin of structure, Cambridge University Press (2009) [INSPIRE]. [10] F. D’Eramo, N. Fernandez and S. Profumo, Dark Matter Freeze-in Production in Fast-Expanding Universes, JCAP 02 (2018) 046 [arXiv:1712.07453] [INSPIRE]. [11] N. Bernal, J. Rubio and H. Veermäe, Boosting Ultraviolet Freeze-in in NO Models, JCAP 06 (2020) 047 [arXiv:2004.13706] [INSPIRE]. [12] A. Ghoshal, L. Heurtier and A. Paul, Signatures of non-thermal dark matter with kination and early matter domination. Gravitational waves versus laboratory searches, JHEP 12 (2022) 105 [arXiv:2208.01670] [INSPIRE]. [13] B. Barman et al., Gravity as a portal to reheating, leptogenesis and dark matter, JHEP 12 (2022) 072 [arXiv:2210.05716] [INSPIRE]. [14] B. Barman, A. Ghoshal, B. Grzadkowski and A. Socha, Measuring inflaton couplings via primordial gravitational waves, JHEP 07 (2023) 231 [arXiv:2305.00027] [INSPIRE]. [15] M. Becker et al., Confronting dark matter freeze-in during reheating with constraints from inflation, JCAP 01 (2024) 053 [arXiv:2306.17238] [INSPIRE]. [16] LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE]. [17] LIGO Scientific and Virgo collaborations, GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116 (2016) 241103 [arXiv:1606.04855] [INSPIRE]. [18] LIGO Scientific and Virgo collaborations, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 (2017) 161101 [arXiv:1710.05832] [INSPIRE]. [19] NANOGrav collaboration, The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys. J. Lett. 951 (2023) L8 [arXiv:2306.16213] [INSPIRE]. – 16 – https://doi.org/10.1103/PhysRevLett.39.165 https://doi.org/10.1103/PhysRevLett.39.165 https://inspirehep.net/literature/119522 https://doi.org/10.1103/PhysRevD.33.1585 https://inspirehep.net/literature/17641 https://doi.org/10.1016/0550-3213(88)90099-5 https://inspirehep.net/literature/261944 https://doi.org/10.1016/0550-3213(91)90438-4 https://doi.org/10.1016/0550-3213(91)90438-4 https://inspirehep.net/literature/304505 https://doi.org/10.1201/9780429492860 https://inspirehep.net/literature/299778 https://doi.org/10.1103/PhysRevD.60.063504 https://doi.org/10.1103/PhysRevD.60.063504 https://doi.org/10.48550/arXiv.hep-ph/9809453 https://inspirehep.net/literature/476526 https://doi.org/10.1103/PhysRevLett.112.101301 https://doi.org/10.48550/arXiv.1310.7495 https://inspirehep.net/literature/1262294 https://doi.org/10.1007/JHEP01(2023)128 https://doi.org/10.48550/arXiv.2210.15653 https://inspirehep.net/literature/2172223 https://inspirehep.net/literature/853992 https://doi.org/10.1088/1475-7516/2018/02/046 https://doi.org/10.48550/arXiv.1712.07453 https://inspirehep.net/literature/1644325 https://doi.org/10.1088/1475-7516/2020/06/047 https://doi.org/10.1088/1475-7516/2020/06/047 https://doi.org/10.48550/arXiv.2004.13706 https://inspirehep.net/literature/1793271 https://doi.org/10.1007/JHEP12(2022)105 https://doi.org/10.48550/arXiv.2208.01670 https://inspirehep.net/literature/2131794 https://doi.org/10.1007/JHEP12(2022)072 https://doi.org/10.1007/JHEP12(2022)072 https://doi.org/10.48550/arXiv.2210.05716 https://inspirehep.net/literature/2164553 https://doi.org/10.1007/JHEP07(2023)231 https://doi.org/10.48550/arXiv.2305.00027 https://inspirehep.net/literature/2655647 https://doi.org/10.1088/1475-7516/2024/01/053 https://doi.org/10.48550/arXiv.2306.17238 https://inspirehep.net/literature/2673377 https://doi.org/10.1103/PhysRevLett.116.061102 https://doi.org/10.48550/arXiv.1602.03837 https://inspirehep.net/literature/1421100 https://doi.org/10.1103/PhysRevLett.116.241103 https://doi.org/10.48550/arXiv.1606.04855 https://inspirehep.net/literature/1469395 https://doi.org/10.1103/PhysRevLett.119.161101 https://doi.org/10.48550/arXiv.1710.05832 https://inspirehep.net/literature/1630824 https://doi.org/10.3847/2041-8213/acdac6 https://doi.org/10.48550/arXiv.2306.16213 https://inspirehep.net/literature/2672619 J C A P 0 8 ( 2 0 2 4 ) 0 5 1 [20] H. Xu et al., Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res. Astron. Astrophys. 23 (2023) 075024 [arXiv:2306.16216] [INSPIRE]. [21] EPTA and InPTA: collaborations, The second data release from the European Pulsar Timing Array — III. Search for gravitational wave signals, Astron. Astrophys. 678 (2023) A50 [arXiv:2306.16214] [INSPIRE]. [22] D.J. Reardon et al., Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array, Astrophys. J. Lett. 951 (2023) L6 [arXiv:2306.16215] [INSPIRE]. [23] M.C. Guzzetti, N. Bartolo, M. Liguori and S. Matarrese, Gravitational waves from inflation, Riv. Nuovo Cim. 39 (2016) 399 [arXiv:1605.01615] [INSPIRE]. [24] C. Caprini and D.G. Figueroa, Cosmological Backgrounds of Gravitational Waves, Class. Quant. Grav. 35 (2018) 163001 [arXiv:1801.04268] [INSPIRE]. [25] M. Giovannini, Gravitational waves constraints on postinflationary phases stiffer than radiation, Phys. Rev. D 58 (1998) 083504 [hep-ph/9806329] [INSPIRE]. [26] M. Giovannini, Production and detection of relic gravitons in quintessential inflationary models, Phys. Rev. D 60 (1999) 123511 [astro-ph/9903004] [INSPIRE]. [27] D.G. Figueroa and E.H. Tanin, Ability of LIGO and LISA to probe the equation of state of the early Universe, JCAP 08 (2019) 011 [arXiv:1905.11960] [INSPIRE]. [28] M.R. Haque, D. Maity, T. Paul and L. Sriramkumar, Decoding the phases of early and late time reheating through imprints on primordial gravitational waves, Phys. Rev. D 104 (2021) 063513 [arXiv:2105.09242] [INSPIRE]. [29] K. Saikawa and S. Shirai, Primordial gravitational waves, precisely: The role of thermodynamics in the Standard Model, JCAP 05 (2018) 035 [arXiv:1803.01038] [INSPIRE]. [30] K. Dimopoulos, Observable Primordial Gravitational Waves from Cosmic Inflation, in the proceedings of the 40th Conference on Recent Developments in High Energy Physics and Cosmology, Ioannina, Greece (2023) [arXiv:2308.00777] [INSPIRE]. [31] Y. Ueno and K. Yamamoto, Constraints on α-attractor inflation and reheating, Phys. Rev. D 93 (2016) 083524 [arXiv:1602.07427] [INSPIRE]. [32] A. Di Marco, P. Cabella and N. Vittorio, Constraining the general reheating phase in the α-attractor inflationary cosmology, Phys. Rev. D 95 (2017) 103502 [arXiv:1705.04622] [INSPIRE]. [33] L. McAllister, E. Silverstein, A. Westphal and T. Wrase, The Powers of Monodromy, JHEP 09 (2014) 123 [arXiv:1405.3652] [INSPIRE]. [34] M.S. Turner, Coherent Scalar Field Oscillations in an Expanding Universe, Phys. Rev. D 28 (1983) 1243 [INSPIRE]. [35] Y. Shtanov, J.H. Traschen and R.H. Brandenberger, Universe reheating after inflation, Phys. Rev. D 51 (1995) 5438 [hep-ph/9407247] [INSPIRE]. [36] K.D. Lozanov and M.A. Amin, Equation of State and Duration to Radiation Domination after Inflation, Phys. Rev. Lett. 119 (2017) 061301 [arXiv:1608.01213] [INSPIRE]. [37] J.A.R. Cembranos, A.L. Maroto and S.J. Núñez Jareño, Cosmological perturbations in coherent oscillating scalar field models, JHEP 03 (2016) 013 [arXiv:1509.08819] [INSPIRE]. [38] G.N. Felder, L. Kofman and A.D. Linde, Inflation and preheating in NO models, Phys. Rev. D 60 (1999) 103505 [hep-ph/9903350] [INSPIRE]. – 17 – https://doi.org/10.1088/1674-4527/acdfa5 https://doi.org/10.48550/arXiv.2306.16216 https://inspirehep.net/literature/2672606 https://doi.org/10.1051/0004-6361/202346844 https://doi.org/10.48550/arXiv.2306.16214 https://inspirehep.net/literature/2672722 https://doi.org/10.3847/2041-8213/acdd02 https://doi.org/10.48550/arXiv.2306.16215 https://inspirehep.net/literature/2672611 https://doi.org/10.1393/ncr/i2016-10127-1 https://doi.org/10.1393/ncr/i2016-10127-1 https://doi.org/10.48550/arXiv.1605.01615 https://inspirehep.net/literature/1455861 https://doi.org/10.1088/1361-6382/aac608 https://doi.org/10.1088/1361-6382/aac608 https://doi.org/10.48550/arXiv.1801.04268 https://inspirehep.net/literature/1647939 https://doi.org/10.1103/PhysRevD.58.083504 https://doi.org/10.48550/arXiv.hep-ph/9806329 https://inspirehep.net/literature/471600 https://doi.org/10.1103/PhysRevD.60.123511 https://doi.org/10.48550/arXiv.astro-ph/9903004 https://inspirehep.net/literature/496010 https://doi.org/10.1088/1475-7516/2019/08/011 https://doi.org/10.48550/arXiv.1905.11960 https://inspirehep.net/literature/1737294 https://doi.org/10.1103/PhysRevD.104.063513 https://doi.org/10.48550/arXiv.2105.09242 https://inspirehep.net/literature/1864194 https://doi.org/10.1088/1475-7516/2018/05/035 https://doi.org/10.48550/arXiv.1803.01038 https://inspirehep.net/literature/1658596 https://doi.org/10.48550/arXiv.2308.00777 https://inspirehep.net/literature/2684590 https://doi.org/10.1103/PhysRevD.93.083524 https://doi.org/10.1103/PhysRevD.93.083524 https://doi.org/10.48550/arXiv.1602.07427 https://inspirehep.net/literature/1423269 https://doi.org/10.1103/PhysRevD.95.103502 https://doi.org/10.48550/arXiv.1705.04622 https://inspirehep.net/literature/1599380 https://doi.org/10.1007/JHEP09(2014)123 https://doi.org/10.1007/JHEP09(2014)123 https://doi.org/10.48550/arXiv.1405.3652 https://inspirehep.net/literature/1296294 https://doi.org/10.1103/PhysRevD.28.1243 https://doi.org/10.1103/PhysRevD.28.1243 https://inspirehep.net/literature/190271 https://doi.org/10.1103/PhysRevD.51.5438 https://doi.org/10.1103/PhysRevD.51.5438 https://doi.org/10.48550/arXiv.hep-ph/9407247 https://inspirehep.net/literature/374623 https://doi.org/10.1103/PhysRevLett.119.061301 https://doi.org/10.48550/arXiv.1608.01213 https://inspirehep.net/literature/1479144 https://doi.org/10.1007/JHEP03(2016)013 https://doi.org/10.48550/arXiv.1509.08819 https://inspirehep.net/literature/1395226 https://doi.org/10.1103/PhysRevD.60.103505 https://doi.org/10.1103/PhysRevD.60.103505 https://doi.org/10.48550/arXiv.hep-ph/9903350 https://inspirehep.net/literature/496766 J C A P 0 8 ( 2 0 2 4 ) 0 5 1 [39] M.A.G. Garcia, K. Kaneta, Y. Mambrini and K.A. Olive, Reheating and Post-inflationary Production of Dark Matter, Phys. Rev. D 101 (2020) 123507 [arXiv:2004.08404] [INSPIRE]. [40] A. Ahmed, B. Grzadkowski and A. Socha, Implications of time-dependent inflaton decay on reheating and dark matter production, Phys. Lett. B 831 (2022) 137201 [arXiv:2111.06065] [INSPIRE]. [41] B. Barman, N. Bernal, Y. Xu and Ó. Zapata, Ultraviolet freeze-in with a time-dependent inflaton decay, JCAP 07 (2022) 019 [arXiv:2202.12906] [INSPIRE]. [42] A. Banerjee and D. Chowdhury, Fingerprints of freeze-in dark matter in an early matter-dominated era, SciPost Phys. 13 (2022) 022 [arXiv:2204.03670] [INSPIRE]. [43] R.T. Co, E. Gonzalez and K. Harigaya, Increasing Temperature toward the Completion of Reheating, JCAP 11 (2020) 038 [arXiv:2007.04328] [INSPIRE]. [44] D.J. Fixsen, The Temperature of the Cosmic Microwave Background, Astrophys. J. 707 (2009) 916 [arXiv:0911.1955] [INSPIRE]. [45] Planck collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [Erratum ibid. 652 (2021) C4] [arXiv:1807.06209] [INSPIRE]. [46] Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022) 083C01 [INSPIRE]. [47] P.G. Ferreira and M. Joyce, Cosmology with a primordial scaling field, Phys. Rev. D 58 (1998) 023503 [astro-ph/9711102] [INSPIRE]. [48] H. Tashiro, T. Chiba and M. Sasaki, Reheating after quintessential inflation and gravitational waves, Class. Quant. Grav. 21 (2004) 1761 [gr-qc/0307068] [INSPIRE]. [49] BICEP and Keck collaborations, Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett. 127 (2021) 151301 [arXiv:2110.00483] [INSPIRE]. [50] KAGRA collaboration, Detector configuration of KAGRA: The Japanese cryogenic gravitational-wave detector, Class. Quant. Grav. 29 (2012) 124007 [arXiv:1111.7185] [INSPIRE]. [51] KAGRA collaboration, Interferometer design of the KAGRA gravitational wave detector, Phys. Rev. D 88 (2013) 043007 [arXiv:1306.6747] [INSPIRE]. [52] KAGRA collaboration, KAGRA: 2.5 Generation Interferometric Gravitational Wave Detector, Nature Astron. 3 (2019) 35 [arXiv:1811.08079] [INSPIRE]. [53] KAGRA collaboration, First cryogenic test operation of underground km-scale gravitational-wave observatory KAGRA, Class. Quant. Grav. 36 (2019) 165008 [arXiv:1901.03569] [INSPIRE]. [54] M.A. McLaughlin, The North American Nanohertz Observatory for Gravitational Waves, Class. Quant. Grav. 30 (2013) 224008 [arXiv:1310.0758] [INSPIRE]. [55] NANOGRAV collaboration, The NANOGrav 11-year Data Set: Pulsar-timing Constraints On The Stochastic Gravitational-wave Background, Astrophys. J. 859 (2018) 47 [arXiv:1801.02617] [INSPIRE]. [56] A. Brazier et al., The NANOGrav Program for Gravitational Waves and Fundamental Physics, arXiv:1908.05356 [INSPIRE]. [57] R.N. Manchester et al., The Parkes Pulsar Timing Array Project, Publ. Astron. Soc. Austral. 30 (2013) 17 [arXiv:1210.6130] [INSPIRE]. – 18 – https://doi.org/10.1103/PhysRevD.101.123507 https://doi.org/10.48550/arXiv.2004.08404 https://inspirehep.net/literature/1791818 https://doi.org/10.1016/j.physletb.2022.137201 https://doi.org/10.48550/arXiv.2111.06065 https://inspirehep.net/literature/1966379 https://doi.org/10.1088/1475-7516/2022/07/019 https://doi.org/10.48550/arXiv.2202.12906 https://inspirehep.net/literature/2039411 https://doi.org/10.21468/SciPostPhys.13.2.022 https://doi.org/10.48550/arXiv.2204.03670 https://inspirehep.net/literature/2064804 https://doi.org/10.1088/1475-7516/2020/11/038 https://doi.org/10.48550/arXiv.2007.04328 https://inspirehep.net/literature/1806017 https://doi.org/10.1088/0004-637X/707/2/916 https://doi.org/10.1088/0004-637X/707/2/916 https://doi.org/10.48550/arXiv.0911.1955 https://inspirehep.net/literature/836577 https://doi.org/10.1051/0004-6361/201833910 https://doi.org/10.1051/0004-6361/201833910 https://doi.org/10.48550/arXiv.1807.06209 https://inspirehep.net/literature/1682902 https://doi.org/10.1093/ptep/ptac097 https://inspirehep.net/literature/2106994 https://doi.org/10.1103/PhysRevD.58.023503 https://doi.org/10.1103/PhysRevD.58.023503 https://doi.org/10.48550/arXiv.astro-ph/9711102 https://inspirehep.net/literature/450928 https://doi.org/10.1088/0264-9381/21/7/004 https://doi.org/10.48550/arXiv.gr-qc/0307068 https://inspirehep.net/literature/623457 https://doi.org/10.1103/PhysRevLett.127.151301 https://doi.org/10.1103/PhysRevLett.127.151301 https://doi.org/10.48550/arXiv.2110.00483 https://inspirehep.net/literature/1938051 https://doi.org/10.1088/0264-9381/29/12/124007 https://doi.org/10.48550/arXiv.1111.7185 https://inspirehep.net/literature/1079406 https://doi.org/10.1103/PhysRevD.88.043007 https://doi.org/10.1103/PhysRevD.88.043007 https://doi.org/10.48550/arXiv.1306.6747 https://inspirehep.net/literature/1240490 https://doi.org/10.1038/s41550-018-0658-y https://doi.org/10.48550/arXiv.1811.08079 https://inspirehep.net/literature/1704344 https://doi.org/10.1088/1361-6382/ab28a9 https://doi.org/10.48550/arXiv.1901.03569 https://inspirehep.net/literature/1713407 https://doi.org/10.1088/0264-9381/30/22/224008 https://doi.org/10.1088/0264-9381/30/22/224008 https://doi.org/10.48550/arXiv.1310.0758 https://inspirehep.net/literature/1256458 https://doi.org/10.3847/1538-4357/aabd3b https://doi.org/10.48550/arXiv.1801.02617 https://inspirehep.net/literature/1646662 https://doi.org/10.48550/arXiv.1908.05356 https://inspirehep.net/literature/1749725 https://doi.org/10.1017/pasa.2012.017 https://doi.org/10.1017/pasa.2012.017 https://doi.org/10.48550/arXiv.1210.6130 https://inspirehep.net/literature/1192875 J C A P 0 8 ( 2 0 2 4 ) 0 5 1 [58] R.M. Shannon et al., Gravitational waves from binary supermassive black holes missing in pulsar observations, Science 349 (2015) 1522 [arXiv:1509.07320] [INSPIRE]. [59] G. Hobbs et al., The international pulsar timing array project: using pulsars as a gravitational wave detector, Class. Quant. Grav. 27 (2010) 084013 [arXiv:0911.5206] [INSPIRE]. [60] R.N. Manchester, The International Pulsar Timing Array, Class. Quant. Grav. 30 (2013) 224010 [arXiv:1309.7392] [INSPIRE]. [61] J.P.W. Verbiest et al., The International Pulsar Timing Array: First Data Release, Mon. Not. Roy. Astron. Soc. 458 (2016) 1267 [arXiv:1602.03640] [INSPIRE]. [62] J.S. Hazboun, C.M.F. Mingarelli and K. Lee, The Second International Pulsar Timing Array Mock Data Challenge, arXiv:1810.10527 [INSPIRE]. [63] C.L. Carilli and S. Rawlings, Science with the Square Kilometer Array: Motivation, key science projects, standards and assumptions, New Astron. Rev. 48 (2004) 979 [astro-ph/0409274] [INSPIRE]. [64] G. Janssen et al., Gravitational wave astronomy with the SKA, PoS AASKA14 (2015) 037 [arXiv:1501.00127] [INSPIRE]. [65] A. Weltman et al., Fundamental physics with the Square Kilometre Array, Publ. Astron. Soc. Austral. 37 (2020) e002 [arXiv:1810.02680] [INSPIRE]. [66] LISA collaboration, Laser Interferometer Space Antenna, arXiv:1702.00786 [INSPIRE]. [67] J. Baker et al., The Laser Interferometer Space Antenna: Unveiling the Millihertz Gravitational Wave Sky, arXiv:1907.06482 [INSPIRE]. [68] J. Crowder and N.J. Cornish, Beyond LISA: Exploring future gravitational wave missions, Phys. Rev. D 72 (2005) 083005 [gr-qc/0506015] [INSPIRE]. [69] V. Corbin and N.J. Cornish, Detecting the cosmic gravitational wave background with the big bang observer, Class. Quant. Grav. 23 (2006) 2435 [gr-qc/0512039] [INSPIRE]. [70] G.M. Harry et al., Laser interferometry for the big bang observer, Class. Quant. Grav. 23 (2006) 4887 [Erratum ibid. 23 (2006) 7361] [INSPIRE]. [71] N. Seto, S. Kawamura and T. Nakamura, Possibility of direct measurement of the acceleration of the universe using 0.1-Hz band laser interferometer gravitational wave antenna in space, Phys. Rev. Lett. 87 (2001) 221103 [astro-ph/0108011] [INSPIRE]. [72] S. Kawamura et al., The Japanese space gravitational wave antenna DECIGO, Class. Quant. Grav. 23 (2006) S125 [INSPIRE]. [73] K. Yagi and N. Seto, Detector configuration of DECIGO/BBO and identification of cosmological neutron-star binaries, Phys. Rev. D 83 (2011) 044011 [Erratum ibid. 95 (2017) 109901] [arXiv:1101.3940] [INSPIRE]. [74] LIGO Scientific collaboration, Exploring the Sensitivity of Next Generation Gravitational Wave Detectors, Class. Quant. Grav. 34 (2017) 044001 [arXiv:1607.08697] [INSPIRE]. [75] D. Reitze et al., Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51 (2019) 035 [arXiv:1907.04833] [INSPIRE]. [76] M. Punturo et al., The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27 (2010) 194002 [INSPIRE]. [77] S. Hild et al., Sensitivity Studies for Third-Generation Gravitational Wave Observatories, Class. Quant. Grav. 28 (2011) 094013 [arXiv:1012.0908] [INSPIRE]. – 19 – https://doi.org/10.1126/science.aab1910 https://doi.org/10.48550/arXiv.1509.07320 https://inspirehep.net/literature/1394654 https://doi.org/10.1088/0264-9381/27/8/084013 https://doi.org/10.48550/arXiv.0911.5206 https://inspirehep.net/literature/838240 https://doi.org/10.1088/0264-9381/30/22/224010 https://doi.org/10.48550/arXiv.1309.7392 https://inspirehep.net/literature/1256215 https://doi.org/10.1093/mnras/stw347 https://doi.org/10.1093/mnras/stw347 https://doi.org/10.48550/arXiv.1602.03640 https://inspirehep.net/literature/1421203 https://doi.org/10.48550/arXiv.1810.10527 https://inspirehep.net/literature/1700193 https://doi.org/10.1016/j.newar.2004.09.001 https://doi.org/10.48550/arXiv.astro-ph/0409274 https://inspirehep.net/literature/658983 https://doi.org/10.22323/1.215.0037 https://doi.org/10.48550/arXiv.1501.00127 https://inspirehep.net/literature/1336039 https://doi.org/10.1017/pasa.2019.42 https://doi.org/10.1017/pasa.2019.42 https://doi.org/10.48550/arXiv.1810.02680 https://inspirehep.net/literature/1697132 https://doi.org/10.48550/arXiv.1702.00786 https://inspirehep.net/literature/1512187 https://doi.org/10.48550/arXiv.1907.06482 https://inspirehep.net/literature/1744110 https://doi.org/10.1103/PhysRevD.72.083005 https://doi.org/10.1103/PhysRevD.72.083005 https://doi.org/10.48550/arXiv.gr-qc/0506015 https://inspirehep.net/literature/684100 https://doi.org/10.1088/0264-9381/23/7/014 https://doi.org/10.48550/arXiv.gr-qc/0512039 https://inspirehep.net/literature/699718 https://doi.org/10.1088/0264-9381/23/15/008 https://doi.org/10.1088/0264-9381/23/15/008 https://inspirehep.net/literature/724937 https://doi.org/10.1103/PhysRevLett.87.221103 https://doi.org/10.1103/PhysRevLett.87.221103 https://doi.org/10.48550/arXiv.astro-ph/0108011 https://inspirehep.net/literature/566200 https://doi.org/10.1088/0264-9381/23/8/S17 https://doi.org/10.1088/0264-9381/23/8/S17 https://inspirehep.net/literature/715756 https://doi.org/10.1103/PhysRevD.83.044011 https://doi.org/10.48550/arXiv.1101.3940 https://inspirehep.net/literature/884842 https://doi.org/10.1088/1361-6382/aa51f4 https://doi.org/10.48550/arXiv.1607.08697 https://inspirehep.net/literature/1478569 https://doi.org/10.48550/arXiv.1907.04833 https://inspirehep.net/literature/1743201 https://doi.org/10.1088/0264-9381/27/19/194002 https://inspirehep.net/literature/874889 https://doi.org/10.1088/0264-9381/28/9/094013 https://doi.org/10.1088/0264-9381/28/9/094013 https://doi.org/10.48550/arXiv.1012.0908 https://inspirehep.net/literature/879607 J C A P 0 8 ( 2 0 2 4 ) 0 5 1 [78] B. Sathyaprakash et al., Scientific Objectives of Einstein Telescope, Class. Quant. Grav. 29 (2012) 124013 [Erratum ibid. 30 (2013) 079501] [arXiv:1206.0331] [INSPIRE]. [79] M. Maggiore et al., Science Case for the Einstein Telescope, JCAP 03 (2020) 050 [arXiv:1912.02622] [INSPIRE]. [80] E. Thrane and J.D. Romano, Sensitivity curves for searches for gravitational-wave backgrounds, Phys. Rev. D 88 (2013) 124032 [arXiv:1310.5300] [INSPIRE]. [81] K. Schmitz, New Sensitivity Curves for Gravitational-Wave Signals from Cosmological Phase Transitions, JHEP 01 (2021) 097 [arXiv:2002.04615] [INSPIRE]. [82] K. Abazajian et al., CMB-S4 Science Case, Reference Design, and Project Plan, arXiv:1907.04473 [INSPIRE]. – 20 – https://doi.org/10.1088/0264-9381/29/12/124013 https://doi.org/10.1088/0264-9381/29/12/124013 https://doi.org/10.48550/arXiv.1206.0331 https://inspirehep.net/literature/1116935 https://doi.org/10.1088/1475-7516/2020/03/050 https://doi.org/10.48550/arXiv.1912.02622 https://inspirehep.net/literature/1768678 https://doi.org/10.1103/PhysRevD.88.124032 https://doi.org/10.48550/arXiv.1310.5300 https://inspirehep.net/literature/1261378 https://doi.org/10.1007/JHEP01(2021)097 https://doi.org/10.48550/arXiv.2002.04615 https://inspirehep.net/literature/1779852 https://doi.org/10.48550/arXiv.1907.04473 https://inspirehep.net/literature/1743173 Introduction The freeze-in DM production with an arbitrary inflaton potential The setup Evolution of the inflaton and radiation during reheating Freeze-in production of dark matter Gravitational waves The spectrum The bound from the Big Bang Nucleosynthesis The parameter space Summary and conclusions