Non-thermal leptogenesis and UV freeze-in of dark matter: impact of inflationary reheating

We study a minimal scenario to realize non-thermal leptogenesis and UV freeze-in of a Standard Model (SM) gauge singlet fermionic dark matter (DM) simultaneously, with inflaton field playing a non-trivial role in their yields. The renormalizable interactions are restricted to the SM fields, two right handed neutrinos (RHN) and inflaton coupling exclusively to the RHNs, while the DM couples to both the SM and the RHNs only via operators of dimension $d>4$. Considering two separate cases of $d=\{5,6\}$, we show that for $d=5$, inflaton decay into RHNs followed by their subsequent decay into SM particles lead to both reheating as well as DM production from the SM bath. This requires a cut-off scale as large as $\Lambda\sim 10^{17}~\rm GeV$ depending on the DM mass. On the other hand, for $d=6$, DM production happens directly from scattering of RHNs (for $\Lambda\gtrsim 10^{14}~\rm GeV$) that results in a very non-trivial evolution of the DM yield. In both these cases, it is possible to explain the observed baryon asymmetry through successful non-thermal leptogenesis via the decay of the RHNs, together with the PLANCK observed relic density of the DM via pure UV freeze-in mechanism. Taking into account both instantaneous as well as non-instantaneous reheating separately, we constrain the parameter space of this minimal scenario from relevant phenomenological requirements including sub-eV scale active neutrino masses and their mixing.

The observed matter-antimatter asymmetry of the universe has been a longstanding puzzle [1,2]. While approximately 5% of our present universe is dominated by visible matter, most of it is made up of baryons. Quantitatively, the asymmetry is quoted as a ratio of excess of baryons over antibaryons to photons. According to the latest data from Planck satellite it is given as [2] η B = n B − nB n γ = 6.1 × 10 −10 .
The estimate from the cosmic microwave background (CMB) measurements also matches very well with the ones obtained from big bang nucleosynthesis (BBN). Since the universe is expected to originate in a matter-antimatter symmetric manner and any pre-existing initial asymmetry is likely to be diluted due to accelerated expansion during inflation, there arises the requirement of a dynamical explanation for the observed asymmetry. Such a dynamical process requires to satisfy certain conditions, given by Sakharov [3] as (1)  can not be satisfied in required amount in the standard model (SM) of particle physics and considering an expanding Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. Outof-equilibrium decay of a heavy particle leading to the generation of baryon asymmetry of the Universe (BAU) has been a well-known mechanism for baryogenesis [4,5]. One interesting way to implement such a mechanism is leptogenesis [6], where a net leptonic asymmetry is generated first which gets converted into baryon asymmetry through (B + L)-violating EW sphaleron transitions [7]. For the lepton asymmetry to be converted into baryon asymmetry, it is important that the processes giving rise to the leptonic asymmetry freeze out before the onset of the sphaleron transitions to prevent wash-out of the asymmetry [8]. An interesting feature of this scenario is that the required lepton asymmetry can be generated through CP violating out-of-equilibrium decays of the same heavy fields that take part in the seesaw mechanism [9][10][11][12][13][14][15] that explains the origin of tiny neutrino masses [1], another observed phenomenon the SM fails to address. In such generic seesaw models for light neutrino masses, the scale of leptogenesis remains very high. For example, in Type-I seesaw framework, the requirement of producing the correct lepton asymmetry pushes the scale of right handed neutrinos to a very high scale M > 10 9 GeV, known as the Davidson-Ibarra bound [16] of high scale or vanilla leptogenesis 1 . If the temperature of the universe can reach a value equal or above the mass of right handed neutrinos, they can be thermally produced from the bath particles. However, the reheat temperature of the universe after inflation (T RH ) can, in principle, be as low as a few MeV, typical temperature around the BBN epoch. Therefore, a low reheat temperature will forbid the production of right handed neutrinos in thermal bath. However, they can still be produced non-thermally from the inflaton field, leading to the scenario of non-thermal leptogenesis [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].
Similar to matter-antimatter asymmetry, observations of a mysterious non-luminous, non-baryonic form of matter, known as dark matter (DM) in the universe has been another puzzle. Cosmology based experiments like Planck [2], through precise measurements of cosmic microwave background (CMB) anisotropies, suggest around 26% of the present universe's energy density to be made up of DM. In terms of density parameter Ω DM and h = Hubble Parameter/(100 km s −1 Mpc −1 ), the present DM abundance is conventionally reported as [2]: Ω DM h 2 = 0.120 ± 0.001 at 68% CL. Astrophysics related observations have also been suggesting the same for a much longer period [37][38][39]. While none of the SM particles can be a viable DM candidate, several beyond the standard model (BSM) proposals have been studied out of which the weakly interacting massive particle (WIMP) paradigm is the most popular one. WIMP paradigm relies upon weak Type-Interactions between the DM and SM sectors which not only play crucial role in generating its relic abundance from thermal freeze-out [40][41][42] but can also lead to observable DM-nucleon scattering rates at direct detection experiments. However, null results at such direct detection experiments like XENON1T [43] have led to growing interest in alternative DM formalisms in recent times.
One such popular alternative is the feebly interacting massive particle (FIMP) scenario where DM interaction with the SM bath is so weak that it never attains thermal equilibrium at any epoch in the early universe [44,45]. The initial abundance of such a particle is considered to be negligible and it can start populating (or freeze-in) the universe due to decay or annihilation of other particles in the thermal bath. If there exists renormalizable DM-SM coupling in FIMP scenario, the non-thermal criterion enforces the couplings to be extremely tiny via the following condition Γ H T M 0 < 1 [46], where Γ is the decay width.
For the case of scattering, one has to replace Γ by the interaction rate n eq σv , n eq being the equilibrium number density of mother particle. These types of freeze-in scenarios are known as infra-red (IR)-freeze-in [45, where DM production is dominated by the lowest possible temperature at which it can occur i.e. T ∼ M 0 , since for T < M 0 , the number density of mother particle becomes Boltzmann suppressed. Another possibility to realize feeble DM-SM interaction is to consider higher dimensional operators (dimension d > 4) connecting DM and SM sectors. Unlike the IR freeze-in scenario, here the DM production is effective at high temperatures and relies upon the details of reheating after the end of inflation. Since DM-SM operators are of dimension five or above, typically, DM is produced via scattering specially at temperatures above the electroweak scale. This particular scenario is known as the ultra-violet (UV) freeze-in [44,[68][69][70][71][72][73][74][75][76][77].
Motivated by these, here we study a minimal scenario where both non-thermal leptogenesis and UV freeze-in of DM can be realized simultaneously. While two singlet right handed neutrinos (RHN) as in Type-I seesaw framework 2 take care of leptogenesis, a Z 2 odd SM gauge singlet fermion is introduced as DM which interact with the SM along with the RHNs only via operators of dimension d > 4, which is necessary for UV freeze-in. Considering the inflaton to be a singlet scalar, we assume its renormalizable coupling with the RHNs alone so that the inflaton decays into a pair of RHNs followed by the latter's out-of-equilibrium decay into the SM particles leading to reheating. We consider both dimension five and dimension six operators separately to study the UV freeze-in of DM while simultaneously solving the coupled Boltzmann equations for non-thermal leptogenesis and reheating. Depending upon inflaton coupling with the RHNs, we outline two cases: instantaneous reheating where the inflation energy density is converted into the heavy neutrinos instantaneously) as well as non-instantaneous (where the inflation energy density is converted into the heavy neutrinos over some finite time) reheating where corresponding evolution of DM and lepton asymmetry are determined with suitable choice of parameters and initial conditions. Without going into the details of the dynamics of inflation, in the present scenario we look into the post slow-roll era when the inflaton energy starts converting into the energy of the heavy neutrinos. See [78] for similar studies separating the period of inflation from reheating and references therein for specific models of inflation which allow this possibility. This separation also justifies the fact that we have a cut-off scale in the theory which is lower than the Planck scale while inflationary field excursions can, in general, be super-Planckian. We typically probe the T RH M ϕ scenario where M ϕ is the inflaton mass and T RH is the reheating temperature. This is in contrast to most of the cases where generally M ϕ T RH is assumed.
From the requirement of successful non-thermal leptogenesis leading to observed BAU, DM 2 Even though three RHNs are typically introduced in Type-I seesaw, two are sufficient to generate observed neutrino oscillation data with vanishing lightest neutrino mass.
relic also the correct neutrino mass, we finally constrain the key parameters namely, the DM mass, reheat temperature, cut-off scale Λ as well as the inflaton coupling with RHNs.
Also, for the dim.6 case we find the DM abundance shows a very non-trivial evolution as it is produced only from the heavy right handed neutrinos which is absent when it is produced from the radiation as in the case of dim.5.
The paper is organised as follows. In Sec. II, we briefly discuss our framework followed by discussion of non-thermal leptogenesis in Sec. III. In Sec. IV we elaborate the UV freeze-in set-up for calculating the DM yield following the details of relevant coupled Boltzmann equations in Sec. V. Sec. VI summarizes our results for both instantaneous and non-instantaneous scenarios. In Sec. VII we briefly discuss possible UV completion of DM-SM operators and finally in Sec. VIII we conclude.

II. THE FRAMEWORK
We extend the Standard Model effective field theory (SMEFT) with two right handed neutrinos N R which is commonly known as the νSMEFT [79][80][81]. Assuming only SM gauge symmetries, the renormalizable interaction is given by the SM-Lagrangian with a sterile Majorana neutrino contribution as This Lagrangian is then extended by adding the dimension five (dim.5) and dim.6 terms of the form where a basis of the operators O i upto and including dim.6 will be elaborated in subsection II A and subsection II B assuming CP conservation. We further consider a SM gauge singlet fermion χ odd under an imposed Z 2 symmetry that makes it absolutely stable over the age of the universe and hence a potential dark matter (DM) candidate. Since it is a gauge singlet fermion (like the right handed neutrinos) and Z 2 -odd, it does not have any tree-level interaction either with the SM fields or with the RHNs. One can then write generalised DM-SM non-renormalizable interactions as where SM is the operator in the visible sector of mass dimension d. The parameter Λ ij is a dimensionful scale and c ij is the dimensionless Wilson coefficient. If d + d > 4, then the interaction is associated with an effective non-renormalizable operator of the form presented in Eq. (4). All DM-SM interactions are encoded by higher-dimensional operators, with a cut-off scale Λ, which is the mass scale of the heavy fields integrated out to obtain the low-energy Lagrangian: where L SM(DM) is the renormalizable SM (DM) Lagrangian and L d>4 corresponds to the operators of dimension d > 4. In the renormalizable level the Lagrangian for the DM field has the form as the DM is electroweak singlet with zero hypercharge. Note that, the cut-off scale in Eq. (3) and Eq. (4) are the same implying the low energy theory containing the right handed neutrinos and the DM is generated by integrating out all the heavy fields at a single scale Λ.
For simplicity, we shall stick to scalar DM bilinears which tell that at the level of dim.5, the DM production takes place from the thermal bath through the gauge invariant interaction of the form [75] O DM-SM where H is the SM Higgs doublet. The only source for the heavy neutrinos, on the other hand, is the on-shell decay of the inflaton which is assumed to couple exclusively to them 3 .
This also implies that the kinematical condition M ϕ > 2M N always holds where M ϕ is the inflaton mass and M N is the mass of the heavy neutrino. In our case M N ≡ M 1 which is the lightest heavy neutrino that is responsible for generating the required lepton asymmetry.
As we shall argue, in the non-thermal leptogenesis set-up we consider hierarchical right handed neutrinos M 2 M 1 which, in turn, also implies 2M B. Dim.6 νSMEFT operators In dim.6 the tree-generated (TG) operators can be classified on the basis of number of Higgs fields associated: 3 Heavy right handed neutrinos can also be produced from the SM fields by virtue of their Yukawa couplings if the highest temperature of the SM bath exceeds the mass of heavy neutrinos. However, this is not the case in non-thermal leptogenesis pursued here.
To this end we have listed only TG operators which conserve baryon number. Note that as N R is a Majorana field, hence Γ µ ≡ γ µ γ 5 . All the SM left handed quark and lepton SU (2) L doublets are denoted by Q L , L respectively while the right handed singlet up-type quarks, down-type quarks and leptons are denoted respectively by u R , d R , e R . As per [82] there are 52 dim.7 independent operators containing sterile or right handed neutrinos. All The lowest dimensional operator that can be formed out of the DM and the sterile neutrino fields are dim.6 four fermion operators where Γ ∈ {1, γ µ γ 5 , γ 5 }. As one can understand, these operators are suppressed by 1/Λ 2 and hence in the presence of dim.5 DM-SM interaction the production of DM from the heavy neutrinos is negligible and can can safely be ignored. One can certainly write DM-SM interactions following Eq. (11) by replacing the right handed neutrino field with the SM fermion fields. But as we are considering only DM scalar bilinears, hence in a full SM basis such DM-SM operators shall not arise [75]. Thus, the sole source of DM production in dim.6 is via the 4-fermion interaction involving heavy neutrinos. With this, now we have all the necessary ingredients to address baryogenesis via (non-thermal) leptogenesis from the decay of the heavy neutrinos together with DM genesis via freeze-in in the early universe.

III. NON-THERMAL LEPTOGENESIS
In the standard vanilla leptogenesis [14] scenario the right-handed neutrinos are produced thermally by scattering processes in the thermal bath or are assumed to be initially in thermal equilibrium. The lower bound on the mass of the lightest right-handed neutrino, in turn, puts a lower bound on the reheat temperature: M 1 10 10 GeV [16,85]. Such a large reheat temperature can lead to gravitino overproduction in supersymmetric scenarios [86].
An elegant alternative to avoid such a catastrophe is to consider non-thermal production of the right handed neutrinos. In the present case we assume the inflaton ϕ decays exclusively to a pair of RHNs [32,87] via where y φ is the corresponding coupling strength 4 . The decay width for such a process: After the inflaton condensate has decayed away, the heavy neutrinos dominate the energy density of the universe. Subsequently, their out-of-equilibrium decay generates the required lepton asymmetry, and reheat the universe since their decay products, SM lepton and Higgs doublets, thermalise quickly. Thus, the decay of the heavy right handed neutrinos not only triggers leptogenesis but produce the thermal bath as well. This leads to interesting consequences contrary to the usual cases where the radiation is produced directly from inflaton 4 Here we consider the inflaton-Higgs coupling to be absent.
decay. Assuming that the energy stored in in the inflaton condensate is instantaneously transformed into radiation one finds which we shall use to parametrize the inflation-heavy neutrino coupling y ϕ in one of our scenarios, where the Hubble parameter at the end of reheating is given by [88] H (T RH ) = 8π 3 g 90 It is important to note here the instantaneous decay of inflaton energy density to radiation energy density is an assumption which may or may not always hold. As a result, we shall treat two cases separately: (i) instantaneous inflaton decay where Eq. (14) holds and the Yukawa coupling y ϕ is parametrized via the reheat temperature T RH and (ii) noninstantaneous inflaton decay where the Yukawa coupling y ϕ is a free parameter. In the latter case T RH is defined as the temperature where the inflaton energy density becomes equal to the radiation energy density and can be determined by solving the coupled Boltzmann equations as we shall elaborate [78]. In the later case, however, it is not possible to establish an analytical relation between y ϕ and T RH .
It is possible to define a reheating temperature for the reheating process initiated by the RHNs in the same spirit we define reheating temperature due to inflaton decay [32] T N RH = 90 (16) where Γ N is the decay width in the RHN rest frame. As advocated earlier, we are typically interested in the non-thermal leptogenesis scenario where we assume T RH < M 1 . For the choice of parameters which gives rise to the right DM abundance with observed baryon asymmetry (together with correct light neutrino mass), we find that Γ ϕ Γ N always holds as long as T RH < M 1 . This points out that the time span of the right handed neutrino dominated universe is extremely small, and hence the true reheat temperature of the universe is T RH due to inflaton decay and not T N RH . It is important to note here, as we are considering M ϕ 2M 1 , hence the RHNs can be considered to be non-relativistic (NR). Obviously this is an assumption that depends on the inflaton mass. If M ϕ M 1 the produced RHNs are relativistic and their energy density contributes to the radiation energy density. However that is not the case here because of the chosen mass hierarchy, and the heavy neutrinos are non-relativistic. Finally, note that once we have T RH < M i the lepton asymmetry washout processes can be neglected as the thermal bath does not have sufficient energy to produce the RHNs. We suppose that the RHN masses are hierarchical, with M 1 M 2 similar to vanilla leptogenesis but M 1 > T RH such that the lightest heavy neutrino N 1 with mass M 1 contributes dominantly to the asymmetry generation. This is precisely due to the fact that any source of lepton number violation at low scale washes out the asymmetries created due to similar violation at high energy scale. Additionally, in our case, we take M ϕ 2M 1 and hence inflaton decay into the lightest RHN is more favoured kinematically. Thus, if , then the inflaton only decays exclusively into two N 1 .

A. Light neutrino mass and Casas-Ibarra parametrization
The extension of the SM particle spectrum with SM singlet RHNs allows us to write its Yukawa interaction with the SM lepton doublet and Higgs whereas at the same time, being a neutral SM singlet one cannot forbid its bare mass term which violates the lepton number (an essential ingredient for leptogenesis) by two units. The moment neutral component of the SM Higgs doublet acquires a vacuum expectation value (VEV) leading to the spontaneous breaking of the gauge symmetry, neutrinos which were massless in the SM obtains a Dirac mass which can be written as Here, v = 246 GeV denotes the VEV of the SM Higgs field. The presence of this Dirac mass together with the bare mass term of the heavy RHNs can explain the non-zero light neutrino masses through Type-I seesaw [9,11]. Here, the light neutrino masses can be expressed as, with M N being the mass of the heavy RHN. The mass eigenvalues and mixing are then obtained by diagonalising the light neutrino mass matrix as with m d ν = diag(m 1 , m 2 , m 3 ) consisting of mass eigenvalues and U being the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1] (the charged lepton mass matrix is considered to be diagonal). These discussions suggest the importance of the Yukawa coupling in generating the neutrino masses. The same Yukawa couplings also play a non-trivial role in determining the lepton asymmetry of the universe as they dictate the decay width of RHNs into leptons.
In order to obtain a complex structure of the Yukawa coupling which is essential from the leptogenesis point of view we use the well-known Casas-Ibarra (CI) parametrization [89].
Using the CI parametrization one can write Yukawa coupling as, where R is a complex orthogonal matrix R T R = I, which we choose as where z = a+ib is a complex angle. In order to obtain m d ν = diag(m 1 , m 2 , m 3 ), we consider the lightest neutrino mass eigenvalue to be vanishingly small with m 1 = 0 (m 1 < m 2 < m 3 ) for normal hierarchy of the neutrino masses. Then m d ν is calculable using the best fit values of solar and atmospheric mass alignedtings obtained from the latest neutrino oscillation data [1]. Now, the elements of Yukawa coupling matrix, y N for a specific value of z can be obtained for different choices of the heavy neutrino masses. The complex angle z can be chosen in a way that enhances the CP asymmetry while keeping the Yukawa couplings within perturbative limits. The CP asymmetry parameter is defined as Neglecting flavour effects we can write the final expression for the CP-asymmetry as [14,90] 1 3 16π which occurs due to the decay of the lightest right handed neutrino N 1 . Note that since we are working in two RHN set-up, hence no summation is involved on the right hand side of above equation. Although N 2 decay can also generate lepton asymmetry, in principle, we consider the asymmetry generated by N 2 decay or any pre-existing asymmetry to be negligible due to strong washout effects due to entropy injection. Finally, we would like to give an explicit structure for the complex Yukawa matrix y N for the set of parameters: 04 that provides the desired CP asymmetry satisfying light neutrino mass

IV. DARK MATTER PRODUCTION VIA UV FREEZE-IN
As the SM bath is getting produced from the decay of the lightest right handed neutrino, the DM can be produced from the thermal bath assuming effective interactions of the form χχH † H in dim.5. This essentially indicates that the DM is getting produced out-ofequilibrium via freeze-in from the thermal bath through a dim.5 interaction, while at dim.6 it can also be produced directly from RHNs following the interaction in Eq. (11). Since in the presence of dim.5 operators, higher dimensional interactions (of dimension d > 5) are naturally suppressed hence we will show the effect of dim.5 and dim.6 interactions on DM production individually. Now, the key requirement for freeze-in [44,45] DM production is to assume that DM was not present in the early universe. The DM is then produced via annihilation and/or decay of the particles in the thermal bath 5 . Due to the extremely feeble coupling of the DM with the visible sector particles the DM never really enters into thermal equilibrium. Such feeble interaction is ensured due to higher dimensional operators suppressed by a large cut-off scale, as mentioned earlier. Now, the Boltzmann equation (BEQ) for DM production via annihilation or 2 → 2 processes of the form a, b → 1, 2 is then given by [91]: 3 are Lorentz invariant phase space elements, and f i is the phase space density of particle i with corresponding number density being with g i is the internal degrees of freedom (DOF). Here we have made two crucial assumptions: the initial abundance for particle 1 is negligible such that f 1 ≈ 0 initially and we also neglect Pauli-blocking/stimulated emission effects, i.e. approximating 1 ± f i ≈ 1. The BEQ in Eq. (25) can be rewritten as an integral with respect to the centre of mass energy as [44,68] where The BEQ in terms of the yield or comoving number density Y χ = n 1 /s (s being the entropy density) can be written in the differential form as The total yield then turns out to be Since we are dealing with pure UV freeze-in case where the DM production is sensitive to high temperature, hence we will ignore masses of all the SM states (as well as the DM) associated with the processes contributing to DM abundance. This is legitimate as long as the reheat temperature and the cut-off scale is large compared to the masses involved.
In Eq. (29), we consider the maximum temperature of the universe T max to be the reheat temperature T RH . Now, the reheat temperature is very loosely bounded, as mentioned before.
Typically, the lower bound on T RH comes from the measurement of light element abundance during BBN, which requires T RH 1.8 MeV [92]. The upper bound, on the other hand, may emerge from (i) cosmological gravitino problem [93,94] in the context of supersymmetric framework that demands T RH 10 10 GeV to prohibit thermal gravitino overproduction and (ii) simple inflationary scenarios that require at most T RH ∼ 10 16 GeV [95,96] for a successful inflation. In view of this, the reheat temperature of the universe can thus be regarded as a free parameter.

V. THE COUPLED BOLTZMANN EQUATIONS
Let us recapitulate our scenario so far. For successful leptogenesis, we rely upon the out-of-equilibrium decay of the heavy neutrino N 1 that is being produced from the decay of inflaton. The out-of-equilibrium decay eventually produces the observed baryon asymmetry via non-thermal leptogenesis. Since inflaton does not decay into radiation directly, right handed neutrino decay into the SM particles is also responsible for reheating the universe.
The DM, on the other hand, is produced from the SM bath via dim.5 DM-SM interaction through the standard UV freeze-in mechanism. In dim.6 the DM can be produced both from the SM bath and from the right handed neutrinos via 4-fermion interaction. Similar to the previous work [75], we consider only scalar bilinears for DM and hence DM-SM operators do not appear at dim.6 level. Fig. 1 shows a cartoon depicting the sources of different components of the universe in our scenario based on which the coupled BEQ needs to be formulated. Note that, since there is no direct inflaton-DM coupling, hence the DM can not be produced directly from the inflaton decay. A loop-induced radiative decay of inflaton to a pair of DM is possible in dim.6 due to the presence of N N χχ operator. Such a process will be suppressed by a factor of ∼ y 2 ϕ /16π 2 Λ 4 and can be neglected as Λ ∼ O (10 14 ) GeV and y ϕ ∼ O (10 −7 ), required to satisfy relic criteria of DM in TeV mass range or below. At dim.5 level no such radiative inflaton decay can arise since we are considering the Higgs-inflaton coupling to be absent. 6 With this, the set of coupled BEQ readṡ 6 DM production via 2 → 2 annihilation of the inflaton mediated via gravitons are studied in [97][98][99][100].
where is the CP-asymmetry generated by the lightest RHN decay (as in Eq. (23)) and Γ ID is the inverse decay rate and the DM number density evolution follows from Sec. IV.
Here onward we are going to ignore the inverse decay rate. Since the RHNs are assumed to be non-relativistic hence n N = ρ N /m N where in our case m N ≡ M 1 . Note that in the BEQ governing the DM abundance we have the equilibrium number density n 2 eq which is applicable for dim.5 as the DM is produced only from the thermal bath. In case of dim.6 (where we ignore the dim.5 operators), as we will see later, one has to replace the equilibrium number density with the heavy neutrino number density as the RHNs become the source of DM production via Eq. (11). Next we are going to make appropriate variable transformation in order to solve the set of coupled equations.

A. Variable transformation & initial conditions
It is convenient to make suitable variable transformation while solving Eq. (30). We make the following set of transformations by scaling the energy and number densities with the scale factor a [22,32] E ϕ = ρ ϕ a 3 , We also define .
Instead of the temperature T we use a dimensionless quantity z = M 1 /T as a variable and define it as a function of R mentioned in Eq. (20) In terms of theses rescaled variables Eq. (30) can be written as In the above set of equations all the primes correspond to derivative with respect to y = a/a I . In obtaining the initial energy density for the inflaton we use the condition Γ ϕ = H (a I ) that leads to Excepting for the inflaton, which has an initial energy density, all other quantities are being produced from different sources. Hence we choose the initial values of the variables appearing in Eq. (34) as: The observed baryon asymmetry depends on the final B − L abundance which can be derived from the redefined N B−L via where we have recasted n γ = g ζ(3)T 3 /π 2 in terms of ρ γ = (π 2 /30) g T 4 and also assumed which is allowed since the heavy neutrinos are non-relativistic 7 . The predicted baryon to photon number ratio has to be compared with the value η B measured at the recombination epoch. It is related to N B−L by [14,32] where a sph = 28/79 is the fraction of B − L asymmetry converted into a baryon asymmetry by sphaleron processes and f = N rec γ /N γ = 2387/86 is the dilution factor calculated assuming standard photon production from the onset of leptogenesis till recombination [14].
The DM abundance can be calculated via where we have used s ≈ 1.8g s n γ [42] to calculate the co-moving entropy density and y ∞ corresponds to the asymptotic value of y = a/a I .

VI. RESULTS AND DISCUSSIONS
In this section we would like to discuss and summarize our findings. We mainly focus on dim.5 DM-SM interaction where the DM is produced via UV freeze-in from interaction of the form in Eq. (8). Later we shall briefly discuss the effect of dim.6 N R -DM operators 7 If the produced heavy neutrinos were relativistic then T = 30(ρ R +ρ N ) . on DM yield. As clarified in the beginning, we investigate the instantaneous and noninstantaneous inflaton decay cases separately and point out the differences thereof. We begin with the instantaneous inflaton decay scenario and illustrate how the resulting parameter space can give rise to the observed DM relic abundance, together with the correct baryon asymmetry. Then we will move on to the more general non-instantaneous decay case and make a comparative analysis between the two.

A. Instantaneous inflaton decay
In the instantaneous inflaton decay approximation Eq. (14) holds and hence the inflatonheavy neutrino coupling is automatically determined once T RH is chosen. Keeping this in mind, let us first look into the dependence of the yield of {E N , E ϕ , R} on the heavy neutrino mass M 1 and reheat temperature T RH as these two parameters mainly control the above three quantities while the effective cut-off scale becomes important for the DM relic abundance.
As mentioned before, we are interested in pure UV freeze-in production of the DM. This entails to only four-point contact interaction for the DM production via HH → χχ process where H is the SM-like Higgs doublet. However, one has to take into account of the fact that before electroweak symmetry breaking we have where the massless Goldstone bosons (GB) φ ±,0 are the propagating DOFs. One can also note that in dim.5 although we have N N → HH process via the operator in Eq. (8) which also contributes to the thermal bath. But the out-of-equilibrium N → H decay will always dominate over pair annihilation of N with the latter being Λ suppressed. Hence, heavy neutrino decay is considered to be the dominant source of entropy production. and DM abundance. In all these plots we ensure that the measured light neutrino mass is obtained (by employing Casas-Ibarra parametrization). At dim.5 the thermally averaged 2 → 2 cross section for DM production is given by considering a pure UV freeze-in scenario where we ignore masses of the DM and the SM fields with respect to the cut-off scale and the reheat temperature 8 . With this assumptions incorporated here we see the following features: • In the top left panel of Fig. 2 we show the variation of E N with y = a/a I for four different choices of the heavy neutrino mass M 1 = {10 9 , 10 10 , 10 11 , 10 12 } GeV in red, green, blue and black respectively for a fixed reheat temperature of 10 8 GeV. Here we see for a larger M 1 the plateau region is wider. This is because for heavier RHN, corresponding Dirac Yukawa coupling is comparatively large, hence its decay starts competing with the production from a much earlier epoch leading to a wider plateau region. Note that all these curves start falling sharply around the same y = a/a I because of fixed reheat temperature that keeps the inflaton-heavy neutrino decay unchanged as Γ ϕ ∝ T 2 RH . As a consequence, when the inflaton decay is completed, the heavy neutrino production stops and E N falls. For a low mass N 1 the yield is larger simply because a smaller M 1 gives rise to a comparatively smaller decay width for the RHN. As a result, the decay rate starts competing with the production at a much later epoch leading to a larger N 1 yield (and a smaller plateau).
• The effect of reheat temperature on the heavy neutrino yield is shown in the top right panel of Fig. 2. For a fixed M 1 the heavy neutrino decay width Γ N is fixed. Now, a larger T RH results in larger Γ ϕ . This implies that the inflaton decay ϕ → N N completes earlier for a large T RH . Thus, for large reheat temperature we see the curves start falling at a lower y showing a smaller plateau since the production of RHN becomes comparable with its decay over a shorter period of time. For large reheat temperature, inflaton decay width is larger and hence its decay into RHNs gets completed at an earlier epoch. For smaller T RH , on the other hand, the plateau is also wider as the inflaton decay takes place over a longer epoch. Therefore, in this case, the RHN decay can remain comparable to its production over a longer epoch. For large T RH the heavy neutrino yield is also larger as E N ∝ Γ ϕ .
• In the middle left panel of Fig. 2 we show the behaviour of radiation with y for a fixed reheat temperature. Here we see for larger M 1 the radiation yield is large as R ∝ Γ N ∝ M 1 . However, as the reheat temperature is fixed and so is the inflaton decay width, hence the heavy neutrino production stops as soon as the inflaton decay is completed. This results in saturation of the radiation yield at y 10 9 . Also, as the entire energy contained in the inflaton field (whose mass is fixed) gets transferred into radiation eventually, all the curves corresponding to different values of M 1 merge for higher values of y. For a particular M 1 = 10 12 GeV we see R increases initially upto y ∼ 10 4 during which the rate of production of the RHN dominates over its decay. Then the radiation yield rises linearly though at a rate slower than initial, as RHN starts to decay till y ∼ 10 10 where the decay rate of the RHN starts overcoming its rate of production. Finally, at y 10 10 the yield saturates when practically the RHN production has stopped as the inflaton decay is over and the RHN decay into the radiation is also completed. It is possible to make an one-to-one correspondence with the top left panel for the same choice of M 1 . In the middle right panel we show the effect of varying reheat temperature on the radiation yield. As for large T RH the inflaton decays faster to the RHN. This results in an earlier drop in the heavy neutrino yield. As a result, for a fixed M 1 , the radiation yield from the RHN decay is also smaller. The fact that the heavy neutrino yield drops earlier for larger T RH , is exactly reflected in the blue curve where we see the radiation starts saturating earlier than the other two. Hence the asymptotic yield of radiation for small T RH (and hence smaller Γ ϕ ) wins over those due to comparatively larger T RH (and hence larger Γ ϕ ).
• Finally in the bottom panel of Fig. 2  Type-I seesaw realization, they can decay quickly leading to quick depletion in their number density once their production from inflaton decay stops. One can also notice that all the curves except for the black one (which corresponds to the inflaton) has a 'knee' around y ∼ 10 4 . This occurs due to the entropy injection from the decay of the heavy neutrino when the RHN decay starts becoming comparable to its production. In all these plots the The effect of maximum temperature during reheating has been taken into account as the maximum temperature T max corresponds to y max 1.4 (see Appendix. A). Production of the right amount of N B−L will lead to the observed baryon asymmetry η B following Eq. (39).
The evolution of the B − L asymmetry for dim.5 interaction is shown in the top left panel of Here we show the relic density allowed points in T RH −Λ plane where the colouring is with η B .
We see correct baryon asymmetry is obtained for a very narrow range of T RH ∼ 10 10 GeV.
As T RH determines the inflaton decay width, hence a large T RH causes a larger inflaton decay width resulting in larger N 1 abundance (as one can also notice in the top left panel of n B s − 28 79 which was derived in the context of inflaton field and leptogenesis without the presence of DM in the same framework. The branching ratio of the inflaton decay into RHNs in Eq. (43) namely, B = 1 since the inflaton decays only into the RHNs. In Fig. 5 we have plotted the baryon asymmetry η B as a function of the reheating temperature for a fixed RHN and inflaton mass. We find, the asymmetry obtained from the exact numerical solution, (shown by the red curve) is slightly less than the one obtained from approximate analytical solution can be explained from the definition of reheat temperature in instantaneous inflaton decay approximation in Eq. (14) and the coupled BEQ in (34). A larger T RH , will result in a larger decay width of the inflaton (as per Eq. (14)), leading to larger abundance of N 1 , as we can see from the top right panel of Fig. 2. With larger N 1 abundance we will have a larger lepton asymmetry, which, in turn, shall give rise to an increase in the baryon asymmetry.
where σv N N →χχ is the pair annihilation cross-section of the heavy neutrinos to DM.
One should note here the presence of E 2 N /M 2 1 term which accounts for the DM production from the heavy neutrino. As the heavy neutrinos are always out-of-equilibrium, hence we can write On the other hand, several operators connecting the heavy neutrinos with the SM fields appear at dim.6 as mentioned in Sec. II. However, the dominant source of leptogenesis can still be considered to be due to N 1 → H decay as 2 → 2 scattering processes involving N 1 in the initial or the final states are suppressed for M 1 > T RH . With this, we can now solve the set of BEQs in Eq. (44). The evolution of yield for dim.6 interactions with y = a/a I is shown in Fig. 6 for a fixed cut-off scale, reheat temperature, inflaton mass and lightest right handed neutrino mass. For a DM mass of 10 GeV, the DM yield (shown by the blue curve) produces the right relic abundance. Also, an observed baryon asymmetry is obtained corresponding toÑ B−L (y → ∞). We see, the yield of the different components, excepting for the DM, follow the pattern of dim.5 interaction as we have seen in Fig. 3. The DM yield flattens and becomes constant at a very early time y ∼ 10 5 . This happens at the same time where the decay of the RHN becomes comparable to its production as we can see comparing the red curve with the blue one. Since the DM yield depends on the pair annihilation of the heavy neutrinos, hence the moment the heavy neutrino decay starts becoming comparable to its production, the DM abundance freezes in and becomes constant. In Fig. 7  The B − L asymmetry also saturates at around the same epoch. This happens because at dim.6 the heavy neutrino becomes the common origin of B − L asymmetry as well as DM genesis and hence the B − L and DM yield build up in a similar fashion. Since at dim.6 the DM abundance via UV freeze-in approximately becomes Ω χ h 2 ∝ m χ T 3 RH /Λ 4 hence compared to the dim.5 case, for a much lower Λ observed DM abundance can be achieved for a given DM mass and reheat temperature. We find that for a DM mass of 10 GeV, in order to satisfy the observed relic abundance, one can have Λ = 10 13 GeV for T RH = 10 9 GeV which is at least four orders of magnitude smaller than the dim.5 case for a DM mass of the same order. This also establishes our earlier claim that dim.6 interactions become ineffective in the presence of dim.5 operators.

B. Non-instantaneous inflaton decay
In case where the instantaneous inflaton decay approximation breaks down, one can no more make the assumption Γ ϕ = H to parametrize the inflaton-heavy neutrino Yukawa coupling in terms of the reheat temperature T RH . In this case the inflaton-heavy neutrino coupling becomes a free parameter. The reheat temperature, on the other hand, can be obtained by defining the temperature at which E Φ (y RH ) = R (y RH ) where y RH defines the corresponding scale factor ratio 10 . Similar definition of reheating temperature for such noninstantaneous inflaton decay scenario have also been used in earlier works, for example [78].
In fact, we have checked, by taking larger R/E Φ ratio at y = y RH , there is only a slight change in T RH that can be neglected. Thus, we stick to the conventional definition already used in earlier works. We first make an estimation of the inflaton-heavy neutrino Yukawa coupling y ϕ and establish a relation between y ϕ and T RH for dim.5 when the instantaneous decay is no longer an approximation. In the top right panel of Fig. 8 we show the dependence of the Yukawa coupling y ϕ on the reheat temperature during instantaneous decay (in black) and non-instantaneous decay (red dashed) of the inflaton field. In the non-instantaneous decay limit the coupling y ϕ becomes a free parameter, independent of the reheating temperature unlike the instantaneous decay limit where T RH determines y ϕ . Thus, under the non-instantaneous decay approximation one can choose a much lower y ϕ with respect to the instantaneous case (corresponding to a given T RH ) as evident from the top right panel of the instantaneous one as large T RH becomes equivalent to large y ϕ . One has to, however, be careful about the fact that for a given M ϕ and M 1 arbitrarily large Yukawa coupling (even if within the perturbative limit) needs to be avoided as that might correspond to a reheat temperature T RH > M 1 contrary to the requirement for non-thermal leptogenesis.
It is also important to note that changing Λ keeping everything else fixed does not change y RH . Therefore, the top right panel plot remains unaltered irrespective of the choices of the cut-off scale. We now show the yields of the different components as before with y = a/a I in the bottom panel. The nature of the yields remain the same as one can see by comapring with Fig. 3. For y ϕ = 10 −5 and Λ = 10 17 GeV we can obtain observed DM abundance for a DM of mass m χ = 1 GeV and baryon asymmetry of the correct order in the limit y → ∞.
The variation of n B−L /s with y = a/a I for the non-instantaneous scenario is shown in the top left panel of Fig. 9 where we see a larger M 1 leads to a larger asymmetry for a given y ϕ following the earlier trend in Fig. 4. On the top right panel of Fig. 9 we show how the DM yield n DM /s evolves with y = a/a I for the same choice of M ϕ and y ϕ as in the top left panel. Here we see the same trend as in Fig. 4 viz. a larger Λ leads to smaller final DM abundance for a fixed y ϕ . This is again same as in the top right panel of Fig. 4 where T RH plays the same role as y ϕ . Thus, for a given DM mass, a larger Λ requires a smaller y ϕ to produce the observed relic abundance. In the bottom left panel of Fig. 9 we show the the region of parameter space where Planck observed relic density is satisfied for DM mass ranging m χ : {1 − 100} GeV. The inflaton-heavy neutrino Yukawa y ϕ , which is now a free parameter, is varied in the range y ϕ : {10 −8 −10 −4 }. Here we again see, similar to the bottom left panel of Fig. 4, a larger DM mass needs comparatively larger Λ to satisfy the observed DM abundance. Since in the case of non-instantaneous inflaton decay, as discussed earlier, a larger y ϕ indicates a larger reheat temperature, hence for a fixed Λ increasing y ϕ results in DM over-abundance. Thus, a large y ϕ at a fixed Λ requires a smaller DM mass to produce observed DM abundance. Again, since Γ ϕ ∝ y 2 ϕ hence increasing y ϕ increases N 1 abundance (as E N ∼ Γ ϕ ) which in turn results in excess in the baryon asymmetry. This can exactly be compared with the effect of increasing T RH in case of instantaneous decay scenario. The same as the instantaneous decay case and hence we do not elaborate them further.

VII. POSSIBLE UV COMPLETION
We have realized the possibility of non-thermal leptogenesis and UV freeze-in for a singlet fermionic DM with the inflaton field playing a non-trivial role. Since we are considering hierarchical right handed neutrinos, the scale of leptogenesis is high M 1 > 10 9 GeV. As we have found, to realize the freeze-in scenario, the cut-off scale needs to be Λ 10 15 [105], while the authors of [106,107] considered clockwork origin of FIMP couplings.

VIII. CONCLUSION
In this work we have addressed two long-standing problems in particle cosmology, namely the matter-antimatter asymmetry and the dark matter relic abundance in a minimal setup while keeping the DM analysis model independent. In order to explain both we have consid- interaction also arises due to 4-fermion operators that appear only in dimension 6 or higher.
We take up only scalar DM bilinears for simplicity and thus in dim.6 no SM gauge invariant operator appears connecting the DM with the SM field. At dim.5 level, however, DM-SM interactions can arise involving SM Higgs. Since we focus on non-thermal DM as well as non-thermal leptogenesis, we consider a singlet scalar field mimicking the inflaton field after slow-roll whose decay into RHNs play a crucial role in DM production, leptogenesis as well as overall reheating of the universe.
We consider baryogenesis via non-thermal leptogenesis where the heavy neutrino masses are larger than the reheat temperature of the universe for explaining the observed baryon asymmetry. The inflaton exclusively decays into a pair of lightest of the heavy neutrinos N 1 via a renormalizable interaction. The heavy neutrino N 1 thus produced further decays into SM bath producing a net B − L asymmetry that is converted into a baryon asymmetry by sphaleron processes. The decay of the inflaton field into N 1 pair can be either instantaneous and non-instantaneous where in the former case the inflaton-RHN coupling is parametrized by the reheat temperature while in the latter the coupling can be considered to be a free parameter. We analyse both the cases separately and show it is possible to obtain the observed baryon asymmetry depending on the choice of reheat temperature (for instantaneous decay case) or the inflation-RHN Yukawa coupling (for non-instantaneous decay case) together with the heavy neutrino mass.
The higher dimensional operators mediating DM-SM and DM-RHN interactions motivate us to consider UV freeze-in as the mechanism for DM genesis. At dim.5 level, the DM can get produced from the SM bath itself which is again generated from the out-of-equilibrium In all such cases we make sure that the inflaton mass M ϕ > 2M 1 and M 1 < T RH in compliance with our previous assumptions. As the DM in the case of dim.6 interaction is produced from the heavy neutrino annihilation here we see a very intriguing dynamics in the DM yield. The DM yield shows a peak at early times and then gets diluted with the subsequent entropy injection as the heavy neutrino starts decaying. This behaviour is absent in dim.5 as the DM there gets produced only from the radiation and hence the two, namely the radiation and the DM energy density evolve in the similar way. While we perform a complete numerical analysis by solving the coupled Boltzmann equations, for the sake of completeness, we have also compared our numerical estimation of baryon asymmetry with the one obtained using approximate analytical formula under the assumption of instantaneous RHN decay. Comparing the analytical and numerical results in one of the cases, we found that the overall baryon asymmetry behaves similarly with the reheat temperature under both approaches, although the numerical analysis lead to slightly larger values.
Finally, in presence of heavy neutrinos, inflaton decay alone can provide a common ori-gin of observed baryon asymmetry and DM abundance in the present universe along with successful reheating. While the role of inflaton field in non-thermal leptogenesis and DM production has been studied separately in earlier works, we have presented an analysis combining both and indicating the interesting interplay of inflaton, right handed neutrino and DM leading to the desired phenomenology including light neutrino masses and mixing. This where H I is determined via as E ϕ I = ρ ϕ I a 3 I . Now, at y = 1, R = 0 which gives T = 30 π 2 g (T ) (A6)