Heavy Light Inflaton and Dark Matter Production

MAN/HEP/2020/010

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 (Dated: October 2020) We study the minimal extension of the SM 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 analysed the previously omitted situation of the inflaton masses mχ & 250 GeV. Therefore, we provide a window of inflaton masses with viable inflationary properties that evade direct observational constraints, due to their small 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 O(1 − 10) MeV.

I. INTRODUCTION limiting cases. The paper first outlines the scalar quartic inflation- The observable Universe is homogeneous and isotropic, ary model with a non-minimal coupling, including cos- and almost completely flat. It is filled with matter and mological constraints on the inflaton self-coupling given radiation and has an almost invariant spectrum of pri- by the scalar density perturbation amplitude and lim- mordial density perturbations. These seemingly finely its on the tensor-to-scalar ratio. We will then describe tuned initial conditions can be explained by the presence the non-thermal inflaton distribution resulting from tur- of an inflationary epoch prior to the Hot Big Bang [1–5]. bulent preheating, which provides the initial condition Viable inflationary models should also provide a mecha- for the reheating study and DM production, and find nism to initiate a reheating period post-inflation, during the relevant inflaton decay widths. The first part of our which the Standard Model (SM) particles are produced. analysis analytically estimates the DM mass and aver- A complete realistic model should also have a Dark Mat- age momentum at the end of reheating, calculated in the ter (DM) candidate and means to produce it, to give over limit of reheating temperatures much less/greater than 80% of the total matter energy density today [6,7]. the inflaton mass [8, 15]. Next, we solve the Boltzmann This paper studies the extension of the SM by a scalar equations numerically across the entire inflaton param- inflaton with quartic self-interaction, which was sug- eter space; here the inflaton mass is constrained, for a gested in [8]. With the addition of a small non-minimal given self-coupling, by kinematics of the decay and the coupling to gravity, this model provides inflationary pre- electroweak symmetry breaking scale. The analytical and dictions in agreement with the Cosmic Microwave Back- numerical results are comparable only at the extremities ground (CMB) observations [9–11]. We assume here, of the parameter space. Our final results conclude that similar to [8, 11–14], that the scale symmetry in the scalar heavy inflaton decay in the early universe produces MeV sector is broken only by the symmetry breaking massive sterile neutrinos; with an average momentum over tem- term of the inflaton. Thus, the scalar provides symmetry perature at the end of reheating of O(1 − 10), they are breaking in the Higgs sector, as well as inflation. Notably, classified as Cold DM candidates. it can also give Majorana masses to uncharged heavy neutral leptons (HNLs), which in turn can be used as DM II. THE MODEL particles in the model [8, 14]. In the previous works [8, 11–14] the analysis was fo- cused on the region of parameters with the inflaton mass A. Inflationary Model below the Higgs boson mass. Here we focus on another

arXiv:2010.06358v1 [hep-ph] 13 Oct 2020 case, where the inflaton is heavier and the channel of its The scalar potential of the model, following [8, 12, 13], direct decay to a pair of Higgs bosons is open, which al- with X being the inflaton field, and Φ the Higgs doublet, lows for a more efficient reheating. Requiring reheating is to be above electroweak temperatures then provides the 1 β α V (X, Φ) = − m2 X2 + X4 + λ(Φ†Φ − X2)2. (1) upper bound on the inflaton mass. DM production is also 2 X 4 λ studied for this parameter window. Here DM is produced We assumed that the only source of scale symmetry vio- from direct inflaton decays during the reheating process, lation is due to the negative mass term in the inflaton sec- leading to a non-thermal velocity distribution for DM. tor. Then, the negative quartic inflaton-Higgs coupling We provide both a numerical study of the DM produc- allows for the transfer of symmetry breaking into the SM tion using Boltzman equations and analytic results for sector1. During the slow roll inflationary evolution, the

1 ∗ [email protected] The domain wall problem can be avoided with the addition of 3 p † [email protected] a small cubic term, µχ with µ . α3/λv, not alterating the 2

field values converge to the attractor solution along the B. Preheating and Reheating gradient of the potential. At inflation the quadratic term √can be neglected, and this attractor is given by the line 2Φ = θinfX in the field space, where the angle θinf is At the end of inflation, the total energy density of the given by: inflaton resides in the homogeneous oscillations of the inflaton field. Preheating for heavy inflaton, where β > 8α, r 2α + β proceeds slightly differently than for the light inflaton θinf = , (2) λ case, due to the misalignment of the inflationary attractor (2) with the position of the vacuum (4) in the field in the limit of α, β λ. Slow roll inflation terminates space. Thus, even for the study of the background dy- when X ∼ O(MP), after which scale invariance of the namics, some inflationary energy is deposited in the os- model is broken in the inflaton sector, giving rise to the cillations of the Higgs field on top of the vacuum. We can vacuum expectation values (VEVs) of the SM Higgs bo- approximate the magnitude of this effect by evaluating son and the scalar field [11–13]: the ratio of the inflaton to Higgs fields’ energy densities (for simplicity we take α β, so the Higgs field in the r v λ vacuum (4) is negligible compared to its value along the hΦi = √ , hXi = v ; (3) inflationary attractor (2)): 2 2α with v = 246 GeV [6]. The angle of rotation of the vac- √ 4 ! uum with respect to the gauge basis (h, χ) = ( 2Φ − ρX βX λ 8 12 ∼ 4 ∼ O ∼ O(10 − 10 ). (8) v, X − hXi) is given by ρΦ λΦ β √ r 2hΦi 2α θv = = . (4) hXi λ Therefore, this mechanism does not transfer noticeable energy into the Higgs-like direction during the early Spontaneous symmetry breaking gives rise to the follow- stages of preheating. ing masses of the excitations on top of the vacuum [11– 13]: The preheating stage continues with energy transfer from oscillations of the zero mode into excitations of the r √ β Higgs and inflaton fields, in a way similar to the light in- mh = 2λv, mχ = mh , (5) flaton case [12]. Parametric resonance then quickly takes α 2 effect in exponentially exciting inflaton particles to oc- which are found in a basis rotated by angle [11–13] cupy a highly non-thermal infra-red distribution function. At the same time, the Higgs particles promptly 2α re-scatter from their resonance bands due to their large θm = θv (6) 2α − β self-coupling, λ, thereby preventing parametric enhance- ment from taking effect [18, 19]. Rescattering of infla- 2 with respect to the gauge basis . Measurement of mh = tons from their resonance bands becomes significant once 125 GeV [6] constrains the SM Higgs boson self coupling roughly half of the energy of the inflaton condensate has λ ' 0.1. been transferred, after which the inflaton enters a phase The quartic self-coupling term, β, is constrained by of free-turbulence. During this period the inflaton distri- ensuring the model is consistent with the CMB measure- bution evolves self-similarly towards thermalisation, and ment of the primordial scalar density perturbation am- so the following ansatz is used [19, 20]: plitude [16]. Additionally, to put the model within the r < . tensor-to-scalar ratio limit, 0 13 [17], a non-minimal −q −p coupling of the scalar field to gravity, ξX2R/2, is required fχ(k, τ) = τ fχ,0(kτ ); (9) −5 [9–11]. For a coupling in the range O(10 ) 6 ξ < 1, β has the following limits [11, 12]: τ = t/t0 is a dimensionless time scale, where t0 is some −13 −9 k O(10 ) 6 β 6 O(10 ). (7) arbitrarily late time, and is conformal momenta. The exponents, derived numerically using lattice simulations, We limit our analysis with ξ 1, as larger ξ would in- are: p ≈ 1/5 and q ≈ 3.5p [19, 20]. The momentum . −s troduce a new energy scale MP /ξ below the MP scale. distribution follows a power law at low momenta, k , with exponent s = 3/2, which corresponds to a free turbulence period dominated by three-particle scatterings; this is verified by the comparative analysis of lattice sim- reheating dynamics and the inflaton mass [12]. ulations with wave kinetic theory, details of which are 2 θm reduces to the angle given in [11–13] in the limiting case of given in [19, 20]. Larger momenta is bounded by an ultra- light inflaton, where 2α β. violet cut-off [19, 20], which we will model in the form of 3 an exponential function, parameterised by k0: 2. Dark matter production

− 3 − 7 − 1 ! 2 ν t 10 k t 5 Neutrino Minimal Standard Model ( MSM) can ex- fχ(k, t) = × plain the origin of baryon asymmetry and DM [22] t k t 0 0 0 through the addition of three right-handed singlet neutri- " − 1 # N I , , k t 5 nos, or HNLs, I ( = 1 2 3), to the SM. By extending exp − . (10) νMSM with a NI − X Yukawa coupling [8, 12]: k0 t0

LνMSM+X = LνMSM + Our analysis proceeds in the perturbative reheating 1 2 fI c period, during the later stages of thermalisation, from (∂µX) − N¯I NI X + h.c. + V (X, Φ), (14) t ∼ t0. Once the Hubble expansion rate has decreased 2 2 to the order of the inflaton decay width, the inflaton can spontaneous symmetry breaking in the inflaton sector, efficiently transfer its energy into the SM via the Higgs X → hXi + χ, generates the sterile neutrino mass, portal. mN = hXifI , (15)

1. Reheating: Standard Model production and a coupling to the inflaton field. As in the scalar sector, we assume that the massive parameters enter only The inflaton mass governs the dominant mechanism in the inflaton operators, i.e. we do not add bare Ma- for reheating; as a result, we analyse the reheating pe- jorana masses for NI . The active-sterile neutrino cou- riod separately for two regions of the inflaton parameter pling is strongly constrained from above by the absence m < m of X-rays observed from the radiative decay of sterile neu- space: for light inflaton, with χ 2 h; and for heavy ν −15 m > m trinos (fIα . 10 )[23, 24]; we can therefore assume inflaton, with χ 2 h. ν The mechanism for SM production from light infla- fIα fI , and neglect NI production from active-sterile neutrino oscillations. In this limit, all sterile neutrinos ton (mχ < 2mh) is dominated by the scattering process χχ → hh/gg, which has has been previously analysed in are produced via the freeze-in mechanism from inflaton ¯ [12, 13]. The light inflaton mass is bounded from below decay, χ → NI NI , in the early Universe. The corre- by ensuring quantum corrections to the inflaton’s quar- sponding decay width [8, 13], tic self-coupling are sufficiently small, so that slow-roll 2 2 fI mχ inflation isn’t ruined (α . 0.1β)[13]: ΓN = , (16) 16π 1 1 ! 2 ! 2 β 10−7 is much less than the Hubble expansion rate throughout mχ > 100 MeV. (11) reheating, so sterile neutrinos remain decoupled from the 1.5 × 10−13 α thermal bath. The lightest of the three sterile neutrinos, N1, is both massive and stable, and so is an ideal Feebly The mass is bounded from above by requiring the min- Interacting Massive Particle (FIMP) DM candidate. We imum reheating temperature to be greater than the will assume in our analysis that N1 makes up the total sphaleron freeze-out temperature (∼ 150 GeV), to ensure DM energy density in the Universe, thereby constraining efficient sphaleron conversion of lepton to baryon asym- its abundance using [6]: metry [13]: 0 s0 1 Ω = mN YN (t) ∼ 0.25. (17) ! 2 DM β ρc mχ . 1.25 −13 GeV. (12) 1.5 × 10 Here s0 is the entropy density of the Universe today; ρc is the critical density; and YN is the ratio of the sterile Here we analyse the reheating process for heavy infla- neutrino number density nN , over entropy, s(t), evalu- ton (mχ > 2mh) in detail. In the restored electroweak ated once all DM has been produced. Constraining the symmetry regime, the dominant reheating mechanism is coupling f1 from the abundance, we can then evaluate via the decay process χ → hh/gg, with decay width [12]: the mass of DM, using (15), which has important conse- quences for structure formation [25]. 4 β mh ΓSM = Γχ→hh + Γχ→gg = 3 . (13) 4π mχ III. BOLTZMANN EQUATIONS FOR The upper mass bound is provided by ensuring the min- REHEATING AND DM PRODUCTION imum reheating temperature is greater than the electroweak symmetry breaking scale, TEW = 160 GeV [21]; Three Boltzmann collision integral equations fully de- the results of this are given later in the paper, where we scribe the dynamics of the system of particles during re- evaluate the bound numerically. heating [26, 27]: 4

Lfˆ Cχ C Cχ χ = χ↔hh/gg + χχ↔hh/gg + χ↔NN¯ (18) ˆ N LfN = Cχ↔NN¯ (19) ˆ SM SM SM LfSM = Cχ↔hh/gg + Chh/gg↔SMSM + Ch/g↔SMSM (20)

The rate of particles x scattering in and out of their distribution functions, fx, is calculated using the collision integral; 0 0 0 the standard deﬁnition for a general 2 − 2 scattering process, a(p)a (p ) ↔ b(q)b(q ), with amplitude Maa0→bb0 is

Z 3 0 3 3 0 a 1 ga0 d p gb d q gb0 d q C 0 0 aa ↔bb = a a0 3 b 3 b0 3 (21) 2Ep 2Ep0 (2π) 2Eq (2π) 2Eq0 (2π) 4 4 0 0 2 0 0 (2π) δ (p + p − q − q )|Maa0→bb0 | fb(q, t)fb0 (q , t) − fa(p, t)fa0 (p , t) .

x eq Ep is the energy of particle x with physical momemtum fχ (k, t) is the Bose-Einstein distribution function of the p, and gx is the number of effective degrees of freedom of inflaton thermalised at the SM temperature, TSM(t): particle x. 1 1 f eq k, t √ . Equations (18),(19) and (20) are simplified using the χ ( ) = 3 2 2 (25) (2π) h (amχ) +k i exp − 1 following: aTSM(t) O 6 − 8 Cχ • ΓSM = (10 10 )ΓN , so χ↔NN¯ is neglected. The differential equation for a(t) is found using the 7 Friedmann equation for the Hubble expansion rate, ne- • ΓSM O(10 )Γ , so C , is ne- & χχ→hh/gg χχ↔hh/gg glecting the contribution of the sterile neutrinos to the glected. total energy density: • Sterile neutrino density remains low during reheat- s a˙(t) ρχ(t) + ρSM(t) ing, as ΓN is much less than the Hubble expansion H(t) = = 2 ; (26) rate throughout the time of production; therefore a(t) 3MP N the backward reaction CNN¯→χ is neglected. da where a˙(t) = dt , MP is the reduced Planck mass and the • Number conserving and violating interactions be- energy densities of the inflaton and SM are: tween W ±/Z bosons are at rates of at least O(1011) Z ∞ q 4πgχ 2 2 2 ρχ(t) = dk k k + (amχ) fχ(k, t), (27) times greater than the Hubble expansion rate dur- a4 ing reheating. As a result, kinetic and chemi- k=0 π2g cal equilibrium are reached very quickly, and so ρ t SM T 4 t . SM SM SM( ) = SM( ) (28) Chh/gg↔SMSM and Ch/g↔SMSM are neglected. Ad- 30 ditionally, we assume the SM thermalises instanta- The number of degrees of freedom of the inflaton is gχ = neously on production to some temperature TSM, 1. We take the number of effective degrees of freedom and therefore implement the detailed balance con- of the SM as constant throughout reheating, at gSM = dition. 100. This is an acceptable approximation even for lighter inflaton, which produce sterile neutrinos at (20 < TSM < The above simplifications reduces the set of Boltzmann 80) GeV; with gSM = 86.25 [30], the corresponding error equations (18, 19, 20) to the following [28, 29]: on the inflaton-sterile neutrino coupling, f1, is still less ∂fχ(k, t) amχ h eq i than 5%. = ΓSM f (TSM) − fχ(k, t) , p 2 2 χ The differential equation for TSM is derived from the ∂t (amχ) + k (22) covariant conservation of the energy-momentum tensor: Z ∞ 0 ∂fN (kN , t) mχΓN a 0 k 0 dTSM = dk fχ(k , t). = −HTSM− 2 p 2 02 dt ∂t kN k0 (amχ) + k min Z ∞ h i (23) 30 2 eq 3 dk k mχΓSM fχ (TSM) − fχ(k, t) , gSMπ(aTSM) k k 0 and N are the conformal inflaton and sterile neutrino (29) momenta respectively. The lower bound on the integral is and replaces the Boltzmann equation (20). 2 (22), (26) and (29) form a closed set of differential amχ 0 equations; the solution for fχ(k, t) will then be used in kmin = kN − . (24) 4kN (23) to solve for fN (kN , t). 5

IV. ANALYTICAL TREATMENT B. Non-relativistic inﬂaton particles (Teq mχ)

The DM mass and momentum heavily depends on the Heavier inflaton with smaller self-couplings has Teq inflaton distribution function at the time of production. mχ, as shown in Figure 1. In this limit, sterile neutri- At the extremities of the inflaton parameter space, where nos are produced prior to thermalisation from a highly the reheating temperature is much less/greater than the infra-red inflaton distribution, at the same time as SM inflaton mass, the DM is produced via two different mech- production: anisms; here we define the reheating temperature, Teq, as 3 k − 2 k −ΓSMt − +ΓSMt the temperature when the energy densities of the infla- fχ(k, t) ∼ e fχ(k, 0) = e k0 . (35) ton and the SM are equal. In these two regions we can k0 use analytical approximations of the inflaton distribution The sterile neutrino number density is obtained by solv- functions to investigate the dependence of the DM prop- ing the equation erties on the inflaton parameters. In the parameter space between, the DM is produced via both mechanisms, so a 3 ∂nN (t)a 3 more careful numerical analysis is required. = 2ΓN nχ(t)a , (36) ∂t where the inflaton number density is

A. Relativistic inflaton particles (Teq mχ) Z ∞ 3 4πgχ √ 3 k0 2 2 −ΓSMt nχ(t) = 3 dk k fχ(k, t) = (2 π) e . a 0 a Light inflaton with larger self-couplings have Teq (37) mχ, as shown in Figure1. In this limit, the inflaton This leads to has thermalised with the SM prior to the production of mχ !3 sterile neutrinos, at TSM ∼ 2 . The thermal inflaton dis- √ 3 ΓN k0 tribution function at the SM temperature, given by (25), nN (t) = 2(2 π) 2 . (38) ΓSM a is used in (23) to obtain the analytical approximation of the sterile neutrino distribution. The sterile neutrino Solving for the entropy in terms of k0: number density is given by [8]: 3 ∂nSM (t)a 3 Z ∞ = 2ΓSMnχ(t)a , (39) 4πgN n t dk k2 f k , t ∂t N ( ) = 3 N N N ( N ) (30) 3 a k=0 4 4 √ 3 ! π n t π π 2 k 3 2 SM ( ) 4 (2 ) 0 3ΓN M0ζ(5)T (t) s(t) = = , SM 45ζ(3) 45ζ(3) a = 2 ; 2πmχ 45ζ(3) ΓN YN = 4 . where the number of degrees of freedom of the sterile 4π ΓSM 3MP neutrino gN = 2, and M0 ≈ √ . Having evaluated the gSM The relative DM abundance, (17), is calculated using (3), entropy of the Universe at the end of reheating, (15) and (16) for hXi, mN and ΓN , to obtain the following power-law relations between mχ, β, f1 and mN : 2 4 ρSM(t) 2π 3 s(t) = = gSMTSM(t), (31) 3 5 3 TSM(t) 45 0 ΓN fI mχ ΩDM ∝ mN ∝ 3 , (40) ΓSM β 2 the relative DM abundance (17), is calculated using (3), − 2 m ∼ . × 3 m 3 . (15) and (16) for hXi, mN and ΓN respectively, to obtain N (1 05 10 ) χ MeV (41) the following power-law relations between mχ, β, f1 and Given that sterile neutrinos are produced at TSM ∼ Teq, mN : and presuming all the sterile neutrinos are created from f 3 inflaton particles at rest, their average momentum at the Ω0 ∝√1 , (32) end of reheating is: DM β −6 mχ hpN i mχ mN ∼ (9.94×10 ) 1 MeV. (33) ∼ . (42) β 3 T 2Teq

The analytical approximation of the average momentum over temperature at the end of reheating is [8]: V. NUMERICAL RESULTS AND DISCUSSION

hpN i We assume the free turbulent evolution of the inﬂatons = 2.45. (34) T driven by three particle scatterings up to some moment 6

1.0 3000 β=10-9 2500 β=10-10 0.8 ρχ/SM n a3 β=10-11 N 2000 thermal β=10-12 β=10-9

3 0.6

a -10

N β=10

1500 / n -11 [ GeV ] β=10 rel eq -12 ρ T 0.4 β=10 1000

500 0.2

TEW

2mh 1000 2000 3000 4000 5000 6000 7000 8000 0.0 5000 2000 500 mχ 100 50 20 10 mχ[GeV] TSM [GeV] FIG. 1. Reheating temperature, Teq, defined when the energy densities of the SM and inflaton are equal, against the inflaton FIG. 2. The plot gives numerical results from 260 GeV infla- −12 −9 mass. The dotted lines give the lower inflaton mass bound, at ton, with self-coupling (10 6 β 6 10 ), represented by 2mh = 250 GeV, and the lower reheating temperature bound different colours given in the legend. The dashed lines are the at the electroweak symmetry breaking scale, TEW = 160 GeV relative energy densities of inflaton/SM, ρχ/ρSM, and the full [21]. lines are the normalised conformal number densities of sterile 3 neutrino, nN a , plotted against the SM temperature, TSM. The vertical dotted lines give the SM temperature at which the inflaton distribution thermalises. t = 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 this article, so the precise results at mχ ∼ 2mh should s s be treated with caution. ρ t m k3 χ( 0) χ 0 The reheating temperature relative to the inflaton H(t0) = 2 ∼ 2 3 ; (43) 3MP 3MPa(t0) mass determines when the DM leptons are produced relative to the thermalisation of the inflaton distribution with the parameter k0 is chosen so that H(t0) & ΓSM, i.e. we the SM, and therefore the properties of the DM. This is choose the moment slightly before the inflaton decays. demonstrated in Figure2, which shows the time of sterile When the Hubble expansion rate has decreased to H(t) ∼ neutrino and SM production from 260 GeV inflaton, as a ΓSM, the Universe will start to reheat. function of the SM temperature. The temperature when First we need to define the parameter space of the inflaton distribution thermalises is indicated by the the heavy inflaton with self-couplings in the range vertical dotted lines. −13 −9 O(10 ) 6 β 6 O(10 ). The inflaton mass is bounded Inflaton particles of mass 260 GeV and coupling β = χ → hh/gg −9 from below by the kinematics of the decay, , 10 have a reheating temperature of Teq ∼ 2400 GeV. requiring mχ > 2mh. The inflaton mass is bounded from As shown by the red line in Figure2, sterile neutrinos above by the minimum reheating temperature, which is are produced from remnant thermalised inflaton parti- required to exceed the electroweak symmetry breaking cles post SM production. This is most efficient when the T scale, EW = 160 GeV [21]. For inflaton that entered Universe has cooled to TSM ∼ mχ/2; at lower temper- thermal equilibrium, we can roughly estimate atures, production is inefficient as the inflaton occupa- s tion number is highly Boltzmann suppressed. However, p β non-relativistic inflaton most efficiently produce sterile Teq ∝ ΓSM ∝ 3 . (44) mχ neutrinos at the same time as the SM, at TSM ∼ Teq; at this time, the inflaton distribution is non-thermal However, the proper equilibrium temperature for infla- and the occupation number is at its largest. The or- ton starting from a non-thermal distribution is found by ange line in Figure2 demonstrates the non-thermal pro- solving the Boltzmann equations numerically. The mass duction of sterile neutrinos from inflaton particles of −12 −9 −12 bounds for inflaton with couplings (10 6 β 6 10 ) mass 260 GeV, coupling β = 10 , and a reheating −13 are defined in Figure1; inflaton with coupling β = 10 temperature of Teq ∼ 240 GeV. The intermediate cou- −10 −11 is not included here, as Teq < TEW for mχ > 2mh. plings, β = 10 /10 , generate sterile neutrinos by The dotted lines in Figure1 give the area of parame- both mechanisms that govern the highly relativistic/non- ter space where we would expect the thermal mass of the relativistic inflaton regions. The blue and green lines in Higgs boson to suppress the inflaton decay width. Pre- Figure 2 show the increasing efficiency of sterile neutrinos cise analysis of this region requires full thermal quantum production at Teq with decreasing β. treatment of the evolution, which is beyond the scope of Analytical approximations of the sterile neutrino mass 7 as a function of inflaton mass, that lead to the proper DM abundance, show a positive correlation for relativis- -9 tic inflaton particles (33), and a negative correlation for 12.5 β=10 β=10-10 non-relativistic inflaton particles (41). Plotting the ster- 10.0 ile neutrino mass against the inflaton mass, shown in β=10-11 Figure3 (top), allows us to clearly identify which pro- 7.5 β=10-12 duction mechanism dominates in different regions of the [ MeV ] inflaton parameter space. The inflaton mass which pro- N 5.0 m duces the maximum sterile neutrino mass is analytically approximated, using (33) and (41), by:

4 1 m ∼ . × β 5 χ (5 1 10 ) GeV; (45) 2.5 the left/right of the peak corresponds to the inflaton pa- 500 1000 2000 5000 m rameter space where the thermal/non-thermal produc- χ[GeV] tion mechanism dominates. In agreement with Figure3 (top), (45) demonstrates that as β increases, the peak sterile neutrino mass moves to larger values of the inflaton mass; and (33) states that relativistic inflaton with 20 β=10-9 smaller β produce heavier sterile neutrinos. β=10-10 The numerical results in Figure3 (top) are given by the -11 10 β=10 solid lines, and the analytical results by the dashed lines. β=10-12

The relativistic approximations (33), have a dependence > N on β, and are coloured accordingly; the non-relativistic T 5 approximation (41), has no dependence on β, so is given < p by the black line. The analytical and numerical results match to good accuracy for the lightest and heaviest in- 2 flaton particles with self-coupling β = 10−9. The inflaton particles with smaller β do converge towards the 1 analytical approximations, however numerical analysis is 500 1000 2000 5000 necessary for arbitrary values of the parameters. mχ[GeV] Figure3 (bottom) plots the average sterile neutrino momentum over temperature at the end of reheating, hpN i/T , across the inflaton parameter space. The lightest inflaton particles, with coupling β = 10−9, have FIG. 3. (top) Sterile neutrino mass against inflaton mass. hpN i/T ∼ 2.4, which is in agreement with our analyt- The solid lines are the numerical results and the dashed lines ical approximation for thermal production (34). Whilst are the analytical approximations. The analytical results for thermal inflaton, given by (33), have a dependence on β so mχ < Teq, increasing the inflaton mass increases the effi- are colour-coded accordingly. The analytical result for non- ciency of sterile neutrino production at Teq, thereby de- relativistic inflaton, given by (41), has no dependence on β, creasing hpN i/T until a minimum is reached at Teq = mχ, so is given by the black dashed line. (bottom) Average sterile hp i/T ∼ hp i/T corresponding to N 1. N rapidly increases neutrino momentum over temperature at the end of reheating, once mχ > Teq, as hpi and Teq are increasing and de- hpN i/T , against inflaton mass. Different colours represent creasing functions of mχ, respectively. Analytical results inflaton with different self-coupling, β. are consistent with our numerical results for heavy non- relativistic inflaton, given by (42); for example, the analytical estimate for 7600 GeV inflaton is hpN i/T ∼ 24. where gSM(Tν ) = 10.75. Our model is therefore well Other models for sterile neutrino DM production in- within the constraints from the Lyman-alpha data, which clude light scalar decay [8], the resonant or non-resonant puts an upper bound on the DM free streaming param- production from active neutrinos [31, 32] and thermal eter, or equivalently velocity v < 10−3 at temperature production with further entropy dilution from the dark ∼ 1 eV [25, 35]. The velocity of the DM in our model is sector [33, 34]. These models produce keV sterile neu- 1 3 trinos; with an average momentum over temperature at −6 MeV gSM(1 eV) hpN i hviT =1 eV = 10 active neutrino decoupling of hpN i/Tν = O(1), they are mN gSM(T ) T warm DM candidates. By comparison, production in ∼ O(10−8 − 10−6), (47) heavy inflaton decays needs MeV neutral leptons, which are Cold DM candidates with where gSM(1 eV) ∼ 3.91 [30]. Sterile neutrinos with the 1 3 highest velocity are produced from 7600 GeV inflaton hpN i gSM(Tν ) hpN i −9 = ∼ 0.5 − 11; (46) with coupling β = 10 , and the lowest velocity from Tν gSM(T ) T inflaton with coupling β = 10−12. 8

VI. CONCLUSION A potentially interesting region of inflaton masses could be when mχ ' mh, when the mixing angle (6) becomes large. However, we expect that reheating in We studied a singlet scalar model with a quartic self- this range is still inefficient. Although this would signifi- interaction and a coupling to the Higgs sector. With cantly enhance the inflaton decay rate via inflaton-Higgs the addition of a non-minimal coupling of the scalar field mixing, such processes can not contribute to reheating in to gravity, this model can successfully produce inflation restored electroweak symmetry, and χχ → hh is ineffi- within CMB bounds of the tensor-to-scalar ratio and the cient for inflaton in this mass range. Nonetheless, we can amplitude of primordial scalar perturbations, for self- not rule out the possibility of significant SM production −13 −9 coupling in the range β = O(10 − 10 ). With scale here as a result of the misalignment of the inflationary invariance only broken in the scalar sector, the inflaton- attractor with the vacuum, without a careful study of the Higgs coupling gives rise to symmetry breaking in the preheating period. Higgs sector and provides the mechanism to initiate re- Overall, the model provides a viable inflationary mech- heating. In the parameter space of heavy inflaton parti- anism and DM generation, while evading all current ex- cles (mχ > 2mh), the mixing angle with the Higgs sec- perimental constraints, due to extremely low mixing of tor is very small, thus evading direct experimental con- the inflaton with the Higgs sector, and strongly sterile straints. Our analysis restricts the heavy inflaton mass leptons at CDM velocity. range to (250 < mχ . 7600) GeV, by ensuring efficient A future study that could lead to potentially inter- reheating of the Universe above the electroweak scale, via esting detectable signatures would require extensions of inflaton decay into two Higgs bosons. the basic model studied here. In particular, it is possible to modify the potential so that the DM is warmer A mechanism for freeze-in DM production is realised and therefore more visible. Mass terms for the Higgs in our model through the addition of a Yukawa coupling doublet and sterile neutrino were removed from the La- of the inflaton to sterile neutrinos, within the framework grangian so scale invariance is only broken in the infla- of νMSM. We assume DM is made up entirely of the ton sector, however these terms can be used to tune the lightest sterile neutrino and is produced via inflaton de- sterile neutrino mass so the DM is lighter and there- cay. For inflaton with mχ Teq, DM is produced once fore warmer. For example, the Majorana sterile neutrino the inflaton has thermalised and so the model param- mass term, − MI N¯ cN, can be tuned to have the oppo- eters can be deduced analytically. For heavy inflaton 2 site sign and have an equal magnitude to that acquired with mχ Teq, DM is produced simultaneously with & from the Yukawa coupling, thereby giving a smaller ef- the SM from a highly non-thermal infra-red inflaton dis- fective sterile neutrino mass. Alternatively, the inclu- tribution, and so it is necessary to solve the Boltzmann sion of the symmetry breaking Higgs doublet mass term, equations numerically. In the heavy inflaton parameter +µ2H†H, would change the VEV of the inflaton field, space the DM is strongly non-thermal and cold, with and thus change the contribution to the sterile neutrino hpN i/T ∼ O(1 − 10) at the end of reheating. Using the mass from the Yukawa term. Producing lighter and known abundance of DM in the Universe, the Yukawa therefore warmer DM would allow the model to be con- coupling constrains the DM mass to O(1−10)MeV, which strained from the observation of the smallest DM struc- puts our results well within the requirements for struc- tures formed in the Universe. ture formation given by the Lyman-alpha data. Future work on this model may also include studying Let us turn to the limitations of our analysis. First, the effects of adding the renormalisable trilinear inflaton- † our results depend on the assumption that the initial in- Higgs coupling, χH H. A small trilinear coupling is flaton distribution is governed by turbulence driven by 3- necessary to avoid a domain wall problem, however a particle scatterings, as suggested in [19, 20]. We assessed sizeable coupling may significantly enhance χ → hh in the level of influence of these assumptions by compar- the heavy inflaton parameter space, thus reheating the ing the results of the three-particle scattering function Universe more efficiently and extending the upper mass to a 4-particle scattering function (with the power law bound of the inflaton. Additionally there are experimen- −s distribution k with s = 5/3 instead of (10)) in the tal motivations if θm is significantly larger, as new detec- non-analytical region of the parameter space. We found tion channels in particle colliders, such as χ → qq¯, may a relatively weak dependence on the initial distribution become accessible. function, with up to a (10 − 20)% difference between the results. Secondly, we ignored the details of symmetry restoration in the electroweak sector after preheating, ACKNOWLEDGMENTS which would require a full thermal field theory treatment. Thus our results for inflaton masses approaching The authors of the paper are grateful to D. Gorbunov the kinematic limit of decay into two Higgs bosons may for valuable discussions. The work is supported in part be modified by exact study. by STFC research grant ST/P000800/1. 9

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