PHYSICAL REVIEW D 102, 115011 (2020)
Heavy light inflaton and dark matter production
† Fedor Bezrukov * and Abigail Keats Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
(Received 15 October 2020; accepted 16 November 2020; published 7 December 2020)
We study the minimal extension of the Standard Model by a scalar with quartic interaction serving as an inflaton. For the model where scale symmetry is broken only in the inflaton sector, the mass of the inflaton is constrained to be relatively low. Here, we analyze the previously omitted situation of the inflaton masses mχ ≳ 250 GeV. Then, we provide a window of inflaton masses with viable inflationary properties that evade direct observational constraints due to the small inflaton mixing with the Higgs sector. The addition of heavy neutral leptons with Majorana masses induced by the interaction with the inflaton allow for cold dark matter in the model with masses Oð1–10Þ MeV.
DOI: 10.1103/PhysRevD.102.115011
I. INTRODUCTION In the previous works [8,11–14] the analysis was focused The observable Universe is homogeneous and isotropic on the region of parameters with the inflaton mass below and almost completely flat. It is filled with matter and the Higgs boson mass. Here we focus on another case, radiation and has an almost invariant spectrum of primor- where the inflaton is heavier and the channel of its direct dial density perturbations. These seemingly finely tuned decay to a pair of Higgs bosons is open, which allows for a initial conditions can be explained by the presence of an more efficient reheating. Requiring reheating to be above inflationary epoch prior to the hot big bang [1–5]. Viable electroweak temperatures then provides the upper bound inflationary models should also provide a mechanism to on the inflaton mass. DM production is also studied for initiate a reheating period postinflation, during which the this parameter window. Here DM is produced from direct Standard Model (SM) particles are produced. A complete inflaton decays during the reheating process, leading to a realistic model should also have a dark matter (DM) nonthermal velocity distribution for DM. We provide both a candidate, and means to produce it, to give over 80% of numerical study of the DM production using Boltzmann equations and analytic results for limiting cases. the total matter energy density today [6,7]. The paper first outlines the scalar quartic inflationary This paper studies the extension of the SM by a scalar model with a nonminimal coupling, including cosmo- inflaton with quartic self-interaction, which was suggested logical constraints on the inflaton self-coupling given by in [8]. With the addition of a small nonminimal coupling to the scalar density perturbation amplitude and limits on gravity, this model provides inflationary predictions in the tensor-to-scalar ratio. We then describe the non- agreement with cosmic microwave background (CMB) thermal inflaton distribution resulting from turbulent observations [9–11]. We assume here, similar to in preheating, which provides the initial condition for the Refs. [8,11–14], that the scale symmetry in the scalar reheating study and DM production, to find the relevant sector is broken only by the symmetry breaking massive inflaton decay widths. The first part of our analysis term of the inflaton. Thus, the scalar provides symmetry analytically estimates the DM mass and average momen- breaking in the Higgs sector, as well as inflation. Notably, it tum at the end of reheating, calculated in the limit of can also give Majorana masses to uncharged heavy neutral reheating temperatures much less/greater than the infla- leptons (HNLs), which in turn can be used as DM particles ton mass [8,15]. Next, we solve the Boltzmann equations in the model [8,14]. numerically across the entire inflaton parameter space; here the inflaton mass is constrained for a given self- coupling, by kinematics of the decay and the electroweak *[email protected] symmetry breaking scale. The analytical and numerical † [email protected] results are comparable only at the extremities of the parameter space. Our final results conclude that heavy Published by the American Physical Society under the terms of inflaton decay in the early Universe produces MeV sterile the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to neutrinos; with an average momentum over temperature the author(s) and the published article’s title, journal citation, at the end of reheating of Oð1 − 10Þ, they are classified as and DOI. Funded by SCOAP3. cold DM candidates.
2470-0010=2020=102(11)=115011(10) 115011-1 Published by the American Physical Society FEDOR BEZRUKOV and ABIGAIL KEATS PHYS. REV. D 102, 115011 (2020)
2 ¼ II. THE MODEL with respect to the gauge basis. Measurement of mh 125 GeV [6] constrains the SM Higgs boson self- A. Inflationary model coupling λ ≃ 0.1. The scalar potential of the model, following The quartic self-coupling term β is constrained by Refs. [8,12,13], with X being the inflaton field, and Φ ensuring the model is consistent with the CMB measure- the Higgs doublet, is ment of the primordial scalar density perturbation amplitude [16]. Additionally, to put the model within the 1 β α 2 0 13 VðX; ΦÞ¼− m2 X2 þ X4 þ λ Φ†Φ − X2 : ð1Þ tensor-to-scalar ratio limit, r< . [17], a nonminimal 2 X 4 λ coupling of the scalar field to gravity ξX2R=2 is required [9–11]. For a coupling in the range Oð10−5Þ ≤ ξ < 1, β has We assumed that the only source of scale symmetry the following limits [11,12]: violation is due to the negative mass term in the inflaton sector. Then, the negative quartic inflaton-Higgs coupling Oð10−13Þ ≤ β ≤ Oð10−9Þ: ð7Þ allows for the transfer of symmetry breaking into the SM sector.1 During the slow roll inflationary evolution, the We limit our analysis with ξ ≲ 1, as larger ξ would ξ field values converge to the attractor solution along the introduce a new energy scale MP= below the MP scale. gradient of the potential. At inflation the quadratic term pcanffiffiffi be neglected, and this attractor is given by the line B. Preheating and reheating 2Φ ¼ θ θ inf X in the field space, where the angle inf is At the end of inflation, the total energy density of the given by inflaton resides in the homogeneous oscillations of the rffiffiffiffiffiffiffiffiffiffiffiffiffiffi inflaton field. Preheating for heavy inflaton, where β > 8α, 2α þ β proceeds slightly differently than for the light inflaton case, θ ¼ ; ð2Þ inf λ due to the misalignment of the inflationary attractor (2) with the position of the vacuum (4) in the field space. Thus, in the limit of α; β ≪ λ. Slow roll inflation terminates even for the study of the background dynamics, some ∼ ð Þ when X O MP , after which, scale invariance of the inflationary energy is deposited in the oscillations of the model is broken in the inflaton sector, giving rise to the Higgs field on top of the vacuum. We can approximate vacuum expectation values (VEVs) of the SM Higgs the magnitude of this effect by evaluating the ratio of the boson and the scalar field [11–13]: inflaton to Higgs fields’ energy densities [for simplicity, rffiffiffiffiffiffi we take α ≪ β, so the Higgs field in the vacuum (4) is v λ negligible compared to its value along the inflationary hΦi¼pffiffiffi ; hXi¼v ð3Þ 2 2α attractor (2)]: ρ β 4 λ with v ¼ 246 GeV [6]. The angle of rotation of the vacuum X ∼ X ∼ ∼ ð108–1012Þ ð Þ pffiffiffi 4 O O : 8 with respect to the gauge basis ðh; χÞ¼ð 2Φ − v; X − hXiÞ ρΦ λΦ β is given by Therefore, this mechanism does not transfer noticeable pffiffiffi rffiffiffiffiffiffi 2hΦi 2α energy in a Higgslike direction during the early stages of θ ¼ ¼ ð Þ preheating. v h i λ : 4 X The preheating stage continues with energy transfer from oscillations of the zero mode into excitations of the Higgs Spontaneous symmetry breaking gives rise to the following – and inflaton fields, in a way similar to the light inflaton masses of the excitations on top of the vacuum [11 13]: case [12]. Parametric resonance then quickly takes effect rffiffiffiffiffiffi in exponentially exciting inflaton particles to occupy a pffiffiffiffiffi β highly nonthermal infrared distribution function. At the m ¼ 2λv; mχ ¼ m ; ð5Þ h h 2α same time, the Higgs particles promptly rescatter from their resonance bands due to their large self-coupling λ, thereby – which are found in a basis rotated by angle [11 13] preventing parametric enhancement from taking effect 2α [18,19]. Rescattering of inflatons from their resonance θ ¼ θ ð Þ bands becomes significant once roughly half of the energy m v 6 2α − β of the inflaton condensate has been transferred, after which the inflaton enters a phase of free turbulence. During this 1 The domain wall problem can be avoidedpffiffiffiffiffiffiffiffiffiffi with the addition of 3 3 2 a small cubic term μχ with μ ≲ α =λv, not altering the θm reduces to the angle given in Refs. [11–13] in the limiting reheating dynamics and the inflaton mass [12]. case of light inflaton, where 2α ≫ β.
115011-2 HEAVY LIGHT INFLATON AND DARK MATTER PRODUCTION PHYS. REV. D 102, 115011 (2020)
1 period the inflaton distribution evolves self-similarly β 2 mχ ≲ 1.25 GeV: ð12Þ toward thermalization, and so the following ansatz is 1.5 × 10−13 used [19,20]: Here we analyze the reheating process for heavy inflaton −q −p ðmχ > 2m Þ in detail. In the restored electroweak sym- fχðk; τÞ¼τ fχ 0ðkτ Þ; ð9Þ h ; metry regime, the dominant reheating mechanism is via the decay process χ → hh=gg, with decay width [12] τ ¼ t=t0 is a dimensionless timescale, where t0 is some arbitrarily late time, and k is conformal momenta. The β m4 Γ ¼ Γ þ Γ ¼ h ð Þ exponents derived numerically using lattice simulations are SM χ→hh χ→gg 3 : 13 4π mχ p ≈ 1=5 and q ≈ 3.5p [19,20]. The momentum distribution follows a power law at low momenta k−s with exponent The upper mass bound is provided by ensuring the s ¼ 3=2, which corresponds to a free turbulence period minimum reheating temperature is greater than the electro- dominated by three-particle scatterings; this is verified by weak symmetry breaking scale, T ¼ 160 GeV [21]; the the comparative analysis of lattice simulations with wave EW results of this are given later in the paper, where we evaluate kinetic theory, details of which are given in Refs. [19,20]. the bound numerically. Larger momenta are bounded by an ultraviolet cutoff [19,20], which we will model in the form of an exponential function parametrized by k0: 2. Dark matter production The neutrino minimal standard model ðνMSMÞ can − 7 −1 −3 −1 explain the origin of baryon asymmetry and DM [22] t 10 k t 5 2 k t 5 fχðk; tÞ¼ exp − : through the addition of three right-handed singlet neutrinos, t0 k0 t0 k0 t0 or HNLs, NI (I ¼ 1, 2, 3), to the SM. By extending νMSM ð Þ 10 with a NI − X Yukawa coupling [8,12]
1 f Our analysis proceeds in the perturbative reheating period, ¼ þ ð∂ Þ2 − I ¯ c þ LνMSMþX LνMSM 2 μX 2 NI NIX H:c: during the later stages of thermalization, from t ∼ t0.Once the Hubble expansion rate has decreased to the order of the þ VðX; ΦÞ; ð14Þ inflaton decay width, the inflaton can efficiently transfer its energy into the SM via the Higgs portal. where spontaneous symmetry breaking in the inflaton sector X → hXiþχ generates the sterile neutrino mass, 1. Reheating: Standard Model production m ¼hXif ; ð15Þ The inflaton mass governs the dominant mechanism for N I reheating. As a result, we analyze the reheating period and a coupling to the inflaton field. As in the scalar sector, separately for two regions of the inflaton parameter space: 2 we assume that the massive parameters enter only in the for light inflaton, with mχ < mh, and for heavy inflaton, 2 inflaton operators; i.e., we do not add bare Majorana masses with mχ > mh. for NI. The active-sterile neutrino coupling is strongly The mechanism for SM production from light inflaton constrained from above by the absence of x rays observed ð 2 Þ mχ < mh is dominated by the scattering process ν ≲ 10−15 from the radiative decay of sterile neutrinos (fIα ) χχ → hh=gg, which has been previously analyzed in ν ≪ [23,24]. We can therefore assume fIα fI, and neglect NI Refs. [12,13]. The light inflaton mass is bounded from production from active-sterile neutrino oscillations. In this below by ensuring that quantum corrections to the infla- limit, all sterile neutrinos are produced via the freeze-in ’ ton s quartic self-coupling are sufficiently small, so that mechanism from inflaton decay χ → N N¯ in the early α2 ≲ 0 1β I I slow roll inflation is not ruined ( . ) [13]: Universe. The corresponding decay width [8,13]
2 1 −7 1 f mχ β 2 10 2 Γ ¼ I ð Þ mχ > 100 MeV: ð11Þ N 16 1.5 × 10−13 α 16π is much less than the Hubble expansion rate throughout The mass is bounded from above by requiring the minimum reheating, so sterile neutrinos remain decoupled from the reheating temperature to be greater than the sphaleron thermal bath. The lightest of the three sterile neutrinos N1 is freeze-out temperature ð∼150 GeVÞ to ensure efficient both massive and stable and so is an ideal feebly interacting sphaleron conversion of lepton to baryon asymmetry [13]: massive particle DM candidate. We will assume in our
115011-3 FEDOR BEZRUKOV and ABIGAIL KEATS PHYS. REV. D 102, 115011 (2020) analysis that N1 makes up the total DM energy density in the Additionally, we assume the SM thermalizes in- Universe, thereby constraining its abundance using [6] stantaneously on production to some temperature TSM, and therefore implements the detailed balance s0 Ω0 ¼ m Y ðtÞ ∼ 0 25: ð Þ condition. DM ρ N N . 17 c The above simplifications reduce the set of Boltzmann Eqs. (18)–(20) to the following [28,29]: ρ Here s0 is the entropy density of the Universe today, c is the critical density, and YN is the ratio of the sterile neutrino ∂fχðk; tÞ amχ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Γ ½ eqð Þ − ð Þ number density n , over entropy sðtÞ, evaluated once all DM SM fχ TSM fχ k; t ; N ∂t 2 2 has been produced. Constraining the coupling f1 from the ðamχÞ þ k abundance, we can then evaluate the mass of DM, using ð Þ Eq. (15), which has important consequences for structure 22 formation [25]. Z ∂ ð Þ m Γ a ∞ 0 fN kN;t ¼ χ N 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik ð 0 Þ 2 dk fχ k ;t : ∂t k k0 2 02 III. BOLTZMANN EQUATIONS FOR REHEATING N min ðamχÞ þ k AND DM PRODUCTION ð23Þ Three Boltzmann collision integral equations fully describe the dynamics of the system of particles during k and kN are the conformal inflatons and sterile neutrino reheating [26,27]: momenta, respectively. The lower bound on the integral is χ χ ˆ 2 Lfχ ¼ Cχ↔ þ Cχχ↔ þ C ¯ ð18Þ ð Þ hh=gg hh=gg χ↔NN; 0 amχ k ¼ k − : ð24Þ min N 4k ˆ ¼ N ð Þ N LfN Cχ↔NN;¯ 19 eq fχ ðk; tÞ is the Bose-Einstein distribution function of the ˆ ¼ SM þ SM þ SM ð Þ ð Þ LfSM Cχ↔hh=gg Chh=gg↔SMSM Ch=g↔SMSM: 20 inflaton thermalized at the SM temperature, TSM t : 1 1 The rate of particles x scattering in and out of their eq χ ð Þ¼ hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ð Þ f k; t 3 2 2 : 25 distribution functions fx is calculated using the collision ð2πÞ ðamχ Þ þk exp ð Þ − 1 integral. The standard definition for a general 2 − 2 aTSM t scattering process, aðpÞa0ðp0Þ ↔ bðqÞbðq0Þ, with ampli- 0 0 The differential equation for aðtÞ is found using the tude Maa →bb is Friedmann equation for the Hubble expansion rate, neglect- Z 3 0 3 3 0 1 0 0 ing the contribution of the sterile neutrinos to the total a ga d p gbd q gb d q C 0 0 ¼ 0 0 aa ↔bb 2 a a 3 b 3 b 3 energy density: Ep 2E 0 ð2πÞ 2Eqð2πÞ 2E 0 ð2πÞ p q sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 0 0 2 × ð2πÞ δ ðp þ p − q − q ÞjM 0→ 0 j aa bb _ð Þ ρχðtÞþρ ðtÞ ð Þ¼a t ¼ SM ð Þ 0 0 H t 2 ; 26 ½ ð Þ 0 ð Þ − ð Þ 0 ð Þ ð Þ ð Þ 3 × fb q; t fb q ;t fa p; t fa p ;t : 21 a t MP
x Ep is the energy of particle x with physical momentum p, _ð Þ¼da where a t dt, MP is the reduced Planck mass, and the and gx is the number of effective degrees of freedom of energy densities of the inflaton and SM are particle x. Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Equations (18)–(20) are simplified using the following: 4π ∞ χ gχ 2 2 2 Γ ¼ ð106–108ÞΓ ρχðtÞ¼ dkk k þðamχÞ fχðk; tÞ; ð27Þ (i) SM O N,soCχ↔ ¯ , is neglected. 4 Γ ≳ ð107ÞΓ NN a k¼0 (ii) SM O χχ→hh=gg,soCχχ↔hh=gg, is neglected. (iii) Sterile neutrino density remains low during reheat- π2g Γ ρ ðtÞ¼ SM T4 ðtÞ: ð Þ ing, as N is much less than the Hubble expansion SM 30 SM 28 rate throughout the time of production; therefore, the N backward reaction CNN¯ →χ is neglected. The number of degrees of freedom of the inflaton is gχ ¼ 1. (iv) Number conserving and violating interactions We take the number of effective degrees of freedom of the ¼ 100 between W =Z bosons are at rates of at least SM as constant throughout reheating at gSM . This is Oð1011Þ times greater than the Hubble expansion an acceptable approximation even for lighter inflatons, ð20 80Þ rate during reheating. As a result, kinetic and which produce sterile neutrinos at 115011-4 HEAVY LIGHT INFLATON AND DARK MATTER PRODUCTION PHYS. REV. D 102, 115011 (2020) The differential equation for TSM is derived from the the sterile neutrino distribution. The sterile neutrino number covariant conservation of the energy-momentum tensor density is given by [8] Z Z 30 ∞ 4π ∞ dTSM 2 gN 2 ¼ −HT − dkk mχΓ n ðtÞ¼ dk k f ðk ;tÞ SM πð Þ3 SM N 3 N N N N dt gSM aTSM 0 a k¼0 eq 3 ½ ð Þ − ð Þ ð Þ 3Γ M0ζð5ÞT ðtÞ × fχ TSM fχ k; t ; 29 ¼ N SM ð Þ 2 ; 30 2πmχ and replaces the Boltzmann Eq. (20). Equations (22), (26), and (29) form a closed set of where the number of degrees of freedom of the sterile 3 differential equations. The solution for fχðk; tÞ will then be ¼ 2 ≈ pMffiffiffiffiffiffiP neutrino gN , and M0 g . Having evaluated the f ðk ;tÞ SM used in Eq. (23) to solve for N N . entropy of the Universe at the end of reheating, IV. ANALYTICAL TREATMENT 4 ρ ðtÞ 2π2 sðtÞ¼ SM ¼ g T3 ðtÞ; ð Þ 3 ð Þ 45 SM SM 31 The DM mass and momentum heavily depends on the TSM t inflaton distribution function at the time of production. At the extremities of the inflaton parameter space, where the the relative DM abundance (17) is calculated using h i Γ reheating temperature is much less/greater than the inflaton Eqs. (3), (15), and (16) for X , mN, and N, respectively, β mass, the DM is produced via two different mechanisms. to obtain the following power law relations between mχ, , f1 m Here we define the reheating temperature Teq as the , and N: temperature when the energy densities of the inflaton f3 and the SM are equal. In these two regions we can use Ω0 ∝ p1ffiffiffi ; ð32Þ analytical approximations of the inflaton distribution func- DM β tions to investigate the dependence of the DM properties on mχ the inflaton parameters. In the parameter space between, ∼ ð9 94 10−6Þ ð Þ mN . × 1 MeV: 33 the DM is produced via both mechanisms, so a more careful β3 numerical analysis is required. The analytical approximation of the average momentum A. Relativistic inflaton particles ðTeq ≫ mχ Þ over temperature at the end of reheating is [8] Light inflatons with larger self-couplings have ≫ hp i Teq mχ, as shown in Fig. 1. In this limit, the inflaton N ¼ 2.45: ð34Þ has thermalized with the SM prior to the production of T ∼ mχ sterile neutrinos, at TSM 2 . The thermal inflaton distri- bution function at the SM temperature, given by Eq. (25),is B. Nonrelativistic inflaton particles ðTeq ≪ mχ Þ used in Eq. (23) to obtain the analytical approximation of Heavier inflatons with smaller self-couplings have ≪ Teq mχ, as shown in Fig. 1. In this limit, sterile neutrinos are produced prior to thermalization from a highly infrared inflaton distribution, at the same time as SM production: −3 2 −ð k þΓ Þ −Γ t k SMt fχðk; tÞ ∼ e SM fχðk; 0Þ¼ e k0 : ð35Þ k0 The sterile neutrino number density is obtained by solving the equation 3 ∂nNðtÞa 3 ¼ 2Γ nχðtÞa ; ð36Þ ∂t N where the inflaton number density is FIG. 1. Reheating temperature T defined when the energy Z eq 4π ∞ pffiffiffi 3 densities of the SM and inflaton are equal, against the inflaton gχ 2 3 k0 −Γ ð Þ¼ ð Þ¼ð2 πÞ2 SMt nχ t 3 dkk fχ k; t e : mass. The dotted lines give the lower inflaton mass bound at a 0 a 2 ¼ 250 mh GeV and the lower reheating temperature bound at ¼ 160 ð37Þ the electroweak symmetry breaking scale, TEW GeV [21]. 115011-5 FEDOR BEZRUKOV and ABIGAIL KEATS PHYS. REV. D 102, 115011 (2020) This leads to reheating temperature, which is required to exceed the ¼ 160 electroweak symmetry breaking scale, TEW GeV pffiffiffi 3 3 ΓN k0 [21]. For the inflaton that entered thermal equilibrium, we ð Þ¼2ð2 πÞ2 ð Þ nN t Γ : 38 SM a can roughly estimate sffiffiffiffiffiffi Solving for the entropy in terms of k0: pffiffiffiffiffiffiffiffi β ∝ Γ ∝ ð Þ Teq SM 3: 44 ∂ ð Þ 3 mχ nSM t a 3 ¼ 2Γ nχðtÞa ; ∂t SM pffiffiffi 4 4 3 3 2π n ðtÞ 4π ð2 πÞ2 k0 However, the proper equilibrium temperature for inflatons sðtÞ¼ SM ¼ ; 45ζð3Þ 45ζð3Þ a starting from a nonthermal distribution is found by solving the Boltzmann equations numerically. The mass bounds for 45ζð3Þ Γ −12 −9 Y ¼ N : ð39Þ inflatons with couplings ð10 ≤ β ≤ 10 Þ are defined in N 4π4 Γ −13 SM Fig. 1; inflatons with coupling β ¼ 10 are not included here, as T 2 determines when the DM leptons are produced relative to 3 −3 mN∼ð1.05 × 10 Þmχ MeV: ð41Þ the thermalization of the inflaton distribution with the SM, and therefore the properties of the DM. This is demon- ∼ Given that sterile neutrinos are produced at TSM Teq, and strated in Fig. 2, which shows the time of sterile neutrino presuming all the sterile neutrinos are created from inflaton and SM production from 260 GeV inflaton as a function of particles at rest, their average momentum at the end of the SM temperature. The temperature when the inflaton reheating is distribution thermalizes is indicated by the vertical dot- ted lines. h i mχ pN ∼ ð Þ Inflaton particles of mass 260 GeV and coupling 2 : 42 β ¼ 10−9 ∼ 2400 T Teq have a reheating temperature of Teq GeV. V. NUMERICAL RESULTS AND DISCUSSION We assume the free turbulent evolution of the inflatons driven by three particle scatterings up to some moment t ¼ t0, leading to the distribution function (10). Starting from a universe filled with only inflaton particles at t0, the initial Hubble expansion rate is approximated by sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ρχðt0Þ mχk0 Hðt0Þ¼ ∼ : ð Þ 3 2 3 2 ð Þ3 43 MP MPa t0 ð Þ ≳ Γ The parameter k0 is chosen so that H t0 SM; i.e., we choose the moment slightly before the inflaton decays. When the Hubble expansion rate has decreased to FIG. 2. The plot gives numerical results from 260 GeV inflatons, ð Þ ∼ Γ with self-coupling ð10−12 ≤ β ≤ 10−9Þ, represented by different H t SM, the Universe will start to reheat. First we need to define the parameter space of the heavy colors given in the legend. The dashed lines are the relative energy −13 densities of inflaton/SM, ρχ=ρSM, and the full lines are the Oð10 Þ ≤ β ≤ 3 inflaton with self-couplings in the range normalized conformal number densities of sterile neutrino n a ð10−9Þ N O . The inflaton mass is bounded from below by the plotted against the SM temperature, T . The vertical dotted lines χ → 2 SM kinematics of the decay, hh=gg, requiring mχ > mh. give the SM temperature at which the inflaton distribution The inflaton mass is bounded from above by the minimum thermalizes. 115011-6 HEAVY LIGHT INFLATON AND DARK MATTER PRODUCTION PHYS. REV. D 102, 115011 (2020) As shown by the red line in Fig. 2, sterile neutrinos are produced from remnant thermalized inflaton particles post- SM production. This is most efficient when the Universe ∼ 2 has cooled to TSM mχ= ; at lower temperatures, produc- tion is inefficient as the inflaton occupation number is highly Boltzmann suppressed. However, nonrelativistic inflatons most efficiently produce sterile neutrinos at the ∼ same time as the SM, at TSM Teq; at this time, the inflaton distribution is nonthermal and the occupation number is at its largest. The orange line in Fig. 2 demonstrates the nonthermal production of sterile neutrinos from inflaton particles of mass 260 GeV, coupling β ¼ 10−12, and a ∼ 240 reheating temperature of Teq GeV. The intermediate couplings β ¼ 10−10=10−11 generate sterile neutrinos by both mechanisms that govern the highly relativistic/non- relativistic inflaton regions. The blue and green lines in Fig. 2 show the increasing efficiency of sterile neutrino β production at Teq with decreasing . Analytical approximations of the sterile neutrino mass as a function of inflaton mass, that lead to the proper DM abundance, show a positive correlation for relativistic inflaton particles (33), and a negative correlation for nonrelativistic inflaton particles (41). Plotting the sterile neutrino mass against the inflaton mass, shown in Fig. 3 (top), allows us to clearly identify which production mechanism dominates in different regions of the inflaton parameter space. The inflaton mass which produces the FIG. 3. Top: sterile neutrino mass against inflaton mass. The maximum sterile neutrino mass is analytically approxi- solid lines are the numerical results and the dashed lines are the analytical approximations. The analytical results for thermal mated, using Eqs. (33) and (41): inflaton given by Eq. (33) have a dependence on β so are color 4 1 coded accordingly. The analytical result for nonrelativistic mχ ∼ ð5.1 × 10 Þβ5 GeV: ð45Þ inflaton given by (41) has no dependence on β,soitisgiven by the black dashed line. Bottom: average sterile neutrino The left/right of the peak corresponds to the inflaton momentum over temperature at the end of reheating hpNi=T parameter space where the thermal/nonthermal production against inflaton mass. Different colors represent inflatons with mechanism dominates. In agreement with Fig. 3 (top), different self-coupling β. Eq. (45) demonstrates that as β increases, the peak sterile neutrino mass moves to larger values of the inflaton mass, and Eq. (33) states that relativistic inflatons with smaller β approximation for thermal production (34). While produce heavier sterile neutrinos. mχ 115011-7 FEDOR BEZRUKOV and ABIGAIL KEATS PHYS. REV. D 102, 115011 (2020) decays needs MeV neutral leptons, which are cold DM to Oð1–10Þ MeV, which puts our results well within candidates with the requirements for structure formation given by the Lyman-alpha data. 1 h i ð Þ 3 h i pN ¼ gSM Tν pN ∼ 0 5–11 ð Þ Let us turn to the limitations of our analysis. First, our ð Þ . ; 46 Tν gSM T T results depend on the assumption that the initial inflaton distribution is governed by turbulence driven by three- ð Þ¼10 75 where gSM Tν . . Our model is therefore well particle scatterings, as suggested in Refs. [19,20].We within the constraints from the Lyman-alpha data, which assessed the level of influence of these assumptions by puts an upper bound on the DM free streaming parameter, comparing the results of the three-particle scattering function or equivalently, velocity v<10−3 at temperature ∼1 eV to a four-particle scattering function [with the power law − [25,35]. The velocity of the DM in our model is distribution k s with s ¼ 5=3 instead of Eq. (10)]inthe nonanalytical region of the parameter space. We found a 1 ð1 Þ 3 h i relatively weak dependence on the initial distribution func- h i ¼ 10−6 MeV gSM eV pN v T¼1 eV ð Þ tion, with up to a (10–20)% difference between the results. mN gSM T T Second, we ignored the details of symmetry restoration in ∼ ð10−8–10−6Þ ð Þ O ; 47 the electroweak sector after preheating, which would require ð1 Þ ∼ 3 91 a full thermal field theory treatment. Thus, our results for where gSM eV . [30]. Sterile neutrinos with the inflaton masses approaching the kinematic limit of decay highest velocity are produced from 7600 GeV inflatons into two Higgs bosons may be modified by exact study. with coupling β ¼ 10−9, and the lowest velocity from −12 A potentially interesting region of inflaton masses could inflatons with coupling β ¼ 10 . ≃ be when mχ mh, when the mixing angle (6) becomes large. However, we expect that reheating in this range is VI. CONCLUSION still inefficient. Although this would significantly enhance We studied a singlet scalar model with a quartic self- the inflaton decay rate via inflaton-Higgs mixing, such interaction and a coupling to the Higgs sector. With the processes cannot contribute to reheating in restored electro- addition of a nonminimal coupling of the scalar field to weak symmetry, and χχ → hh is inefficient for inflatons gravity, this model can successfully produce inflations in this mass range. Nonetheless, we cannot rule out the within CMB bounds of the tensor-to-scalar ratio and the possibility of significant SM production here as a result of amplitude of primordial scalar perturbations, for self- the misalignment of the inflationary attractor with the coupling in the range β ¼ Oð10−13–10−9Þ. With scale vacuum, without a careful study of the preheating period. invariance only broken in the scalar sector, the inflaton- Overall, the model provides a viable inflationary mecha- Higgs coupling gives rise to symmetry breaking in the nism and DM generation, while evading all current exper- Higgs sector and provides the mechanism to initiate imental constraints, due to extremely low mixing of the reheating. In the parameter space of heavy inflaton particles inflaton with the Higgs sector, and strongly sterile leptons ðmχ > 2mhÞ, the mixing angle with the Higgs sector is at cold DM velocity. very small, thus evading direct experimental constraints. A future study that could lead to potentially interesting Our analysis restricts the heavy inflaton mass range to detectable signatures would require extensions of the basic ð250 115011-8 HEAVY LIGHT INFLATON AND DARK MATTER PRODUCTION PHYS. REV. D 102, 115011 (2020) Future work on this model may also include studying the significantly larger, as new detection channels in particle effects of adding the renormalizable trilinear inflaton-Higgs colliders, such as χ → qq¯, may become accessible. coupling, χH†H. A small trilinear coupling is necessary to avoid a domain wall problem; however, a sizeable coupling ACKNOWLEDGMENTS may significantly enhance χ → hh in the heavy inflaton parameter space, thus reheating the Universe more effi- The authors of the paper are grateful to D. Gorbunov for ciently and extending the upper mass bound of the inflaton. valuable discussions. 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