Chiral Anomaly in SU(2)${}_R$-Axion Inflation and the New Prediction for Particle Cosmology

Upon embedding the axion-inflation in the minimal left-right symmetric gauge extension of the SM with gauge group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$, [arXiv:2012.11516] proposed a new particle physics model for inflation. In this work, we present a more detailed analysis. As a compelling consequence, this setup provides a new mechanism for simultaneous baryogenesis and right-handed neutrino creation by the chiral anomaly of $W_R$ in inflation. The lightest right-handed neutrino is the dark matter candidate. This setup has two unknown fundamental scales, i.e., the scale of inflation and left-right symmetry breaking $SU(2)_R\times U(1)_{B-L}\rightarrow U(1)_{Y}$. Sufficient matter creation demands the left-right symmetry breaking scale happens shortly after the end of inflation. Interestingly, it prefers left-right symmetry breaking scales above $10^{10}~GeV$, which is in the range suggested by the non-supersymmetric SO(10) Grand Unified Theory with an intermediate left-right symmetry scale. Although $W_R$ gauge field generates equal amounts of right-handed baryons and leptons in inflation, i.e. $B-L=0$, in the Standard Model sub-sector $B-L_{SM}\neq 0$. A key aspect of this setup is that $SU(2)_R$ sphalerons are never in equilibrium, and the primordial $B-L_{SM}$ is conserved by the Standard Model interactions. This setup yields a deep connection between CP violation in physics of inflation and matter creation (visible and dark); hence it can naturally explain the observed coincidences among cosmological parameters, i.e., $\eta_{B}\simeq 0.3 P_{\zeta}$ and $\Omega_{DM}\simeq 5\Omega_{B}$. The $SU(2)_R$-axion inflation comes with a cosmological smoking gun; chiral, non-Gaussian, and blue-tilted gravitational wave background, which can be probed by future CMB missions and laser interferometer detectors.


Introduction
Modern cosmology has been remarkably successful in describing the Universe from a second after the Big Bang until today. However, its physics before that time is still much less certain. It profoundly involves particle theory beyond the Standard Model (BSM) to explain long-standing puzzles: I) the origin of the observed matter asymmetry, II) neutrino mass, III) nature of dark matter, and IV) cosmic inflation. Apart from the above problems, the standard model of particle physics (SM) faces some conceptual issues: i) ad hoc parity violation, ii) accidental B − L global symmetry, iii) vacuum instability, and iv) strong CP problem. Recently, [1] proposed a new setup for physics of inflation by embedding axion inflation in SU (2) R gauge extensions of the SM. For concreteness, as the most minimal realization of this idea 1 , the axion inflaton is coupled to SU (2) R gauge field in the minimal left-right symmetric model (LRSM) [2][3][4][5]. This new particle physics model for inflation gives rise to a new mechanism for simultaneous baryogenesis and Right-Handed Neutrino (RHN) creation through the chiral anomaly of SU (2) R , which provides the source of CP violation in inflation. This new mechanism does not rely on the largeness of the unconstrained CP-violating phases in the neutrino sector nor fine-tuned masses for the heaviest right-handed neutrinos. On the other hand, it makes a deep connection between inflation, baryon asymmetry, and DM relic density. Therefore, it can naturally explain the observed coincidences among cosmological parameters, i.e., η B 0.3P ζ and Ω DM 5Ω B . If the primordial relic density of the lightest RHN makes all the DM today, Ω DM specifies its mass as 1.7 GeV . Therefor its radiative decay may produce active neutrinos and gamma-rays with energy E γ = m N 1 /2 in the highly-dense DM regions. As a compelling consequence, this setup can simultaneously provide plausible explanations for the phenomena (I-IV) named earlier. In this work, we present a more detailed analytical and numerical analysis.
Originally proposed to explain P violation in low energy processes [2], LRSM predicted massive neutrinos years before experiment. Among its additional compelling consequences are: natural B−L symmetry [6], natural entailed seesaw mechanisms [7], solution to vacuum stability problem [8], and strong CP problem without an axion. The LRSM can solve the conceptual issues (i-iv) named earlier. In the minimal LRSM, the Electro-Weak (EW) gauge symmetry is extended to SU (2) L × SU (2) R × U (1) B−L [2][3][4][5]. As a result, it introduces a new fundamental scale, Λ F , during which the extended gauge symmetry is broken as SU (2) R × U (1) B−L → U (1) Y . Upon embedding axion-inflation in LRSM, we have two unknown energy scales, the scale of inflation and Λ F . Based on that we can classify the setup into two types; type-I with √ HM Pl > Λ F and type-II with √ HM Pl < Λ F . (See Fig.  1) Axion fields are abundant in string theory, and therefore very well-motivated candidates for the inflaton field [9][10][11]. Thanks to their natural shift symmetry, their effective potential is protected from dangerous quantum corrections, which guarantees the flatness of the potential. Besides their appealing theoretical stability, models of axion inflation are attractive phenomenologically and are naturally coupled to gauge fields. Non-Abelian gauge fields may contribute to the physics of inflation while respecting the cosmological symmetries [12,13]. The first inflationary model based on non-Abelian gauge fields has been introduced as Gauge-flation [12], which is an EFT of a larger model, i.e., Chromo-natural [14], after integrating out the axion [15]. 2 Inspired by the original models, several more inflationary models with the SU (2) fields have been proposed and studied, which share the same key features. In this work, we consider the minimal realization of SU (2)-axion inflation introduced and studied in [17]. For review see [18], Sec. 2 of [19] and references therein. Including SU (2) gauge fields in physics of inflation give rise to a rich phenomenology which we summarize in the following. The SU (2)-axion inflation [17] is a natural setting for warm inflation [20][21][22]. The Chern-Simons interaction drains kinetic energy from the axion and injects it into the radiation. This gauge field produces particles in inflation; charged Higgs via the Schwinger effect [23] and charged fermions by both Schwinger effect and chiral anomaly [1,24,25]. Another consequence of this Schwinger effect is sourced primordial gravitational waves [19]. All the Sakharov conditions [26] are satisfied in inflation [27,28], hence it provides a natural setting to explain the matter asymmetry (see Sec. 3.3). 3 As a cosmological smoking gun, it predicts chiral [18,31,32] and non-Gaussian [33] Gravitational Wave Background (GWB) which leads to parity odd CMB cross-spectra (see [34] and Sec. 6) [35].
The new setup proposed in [1] extended the field content of the minimal LRSM with an axion field which drives the cosmic inflation. It is assumed that the axion and SU (2) R gauge field are coupled by a Chern-Simons interaction, i.e., SU (2) R -axion inflation. 4 Here 2 In Chromo-natural, the form of the axion potential is assumed to be cosine, which requires a large value of λ to support slow-roll inflation [14]. However, the large coupling is hard to achieve in a controlled string compactification [16]. Therefore, we are interested in flat axion potentials and small values of λ, e.g. f 0.01 and λ 0.1 [17]. 3 Within the SM and through (global) gravitational anomaly, this setting naturally accompanies an inflationary leptogenesis [28][29][30]. 4 In principle we can couple the axion to both SU (2)R and SU (2)L gauge fields. However, any primordial left-handed baryons and leptons produced by SU (2)L (i.e. BSM = LSM) will be completely washed out by the SU (2)L sphaleon effects which are in thermal equilibrium between T reh and mW L . Therefore in the minimal realization of this idea we neglect this interaction.
both Parity and CP are spontaneously violated by the axion and its interaction with W R in the physics of inflation. Within this setup, SU (2) R gauge field is generated in inflation and creates right-handed chiral fermions coupled to it, i.e., SM baryons B, SM leptons L SM , and three Right-Handed Neutrinos (RHN) L N . In type-I scenario in which the first SSB happens after inflation, equal baryon and lepton numbers are created in inflation, i.e. B = L where L ≡ L SM + L N , yet B − L SM = 0. Shortly after inflation, the first SSB happens at Λ F , and eventually, the SU (2) R interactions freeze out at temperature T W R . The lightest of RHNs with feeble Yukawa couplings (our DM candidate) is decoupled at this point, while N 2,3 decay at T = m N 2,3 . Between reheating and EW scale, the spectator effects reshuffle the primordial densities. The summary of this new mechanism for simultaneous baryogenesis and RHN production is presented in Fig. 2. Ref. [1] was the first step to further, more involved analysis on the rich and multifaceted phenomenology of the gauge extensions of the SM in inflation physics. In the current work, we present a more detailed analytical and numerical analysis of the setup. L Ni (gray box) during cosmic evolution. Here B ≡ B SM and L ≡ L SM + L N . The CP violation by the chiral anomaly of W R in inflation simultaneously produces baryons, SM leptons, and RHNs in inflation. The lightest RHN (our DM candidate N 1 ) freezes out at T = T W R . Between reheating and electoweak scale, the spectator effects reshuffle the primordial densities while N 2,3 decay at T = m N2,3 . The net baryon density and dark matter today are the remnant of that quantum effect in inflation. Notice that T reh < m W R condition is essential to keep W R sphalerons out of equilibrium (see App D.1). The W L sphalerons, however, contribute to the spectator effects and washout B + L SM but conserve B − L SM .
The paper is structured as follows. In Sec. 2 we review the setup of SU (2) R -axion inflation embedded in LRSM. In Sec. 3, we work out the inflationary particle production. In Sec. 4, we study the post reheating evolution of the system. Next in Sec. 5, we work out the final baryon asymmetry and dark matter in the modern era. Sec. 6 presents a quick discussion on the setup's observational constraints and signatures. We finally conclude in Sec. 7. Technical details of the computations and the underlying mathematical tools are provided in App.s A-D.
Notations and conventions: In this work, we deal with 4 and 2-component spinors, which are acted upon by 4 × 4 and 2 × 2 matrices, respectively. The 4-spinors and 4 × 4 matrices are remained unchanged, while the 2-spinors and 2 × 2 matrices are written in boldface. The 2 × 2 identity matrix is represented as I 2 . L and R subscripts denote the left-and right-handed fermions. The lepton and baryon numbers are presented by L and B. The Hubble parameter in inflation is denoted by H and M Pl = (8πG) −1 is set to one, unless otherwise specified. We use the Einstein summation notation, i.e., repeated indices (one upper and one lower) are summed. The beginning of the Latin alphabets, i.e. a, b, c denote the SU (2) group indices. Greek letters starting from the middle of the alphabet, i.e., µ, ν, . . . are used for the space-time indices, whereas the starting ones, i.e., α, β, . . . , present the indices of the tangent space (non-coordinate) bases.

Framework
The aim of this work is to embed the axion inflation in the gauge extensions of the SM when the inflaton is directly coupled to the BSM fields. Here we consider the model proposed in [1], i.e. SU(2) R -axion inflation, in which the axion-inflation is embedded in the minimal Left-Right Symmetric Model (LRSM). The minimal gauge group that implements the hypothesis of left-right symmetry is [2][3][4][5] where the color (SU (3) group) is suppressed. The subscripts L and R denotes left-and right-handed fields, while B and L represent baryon and lepton numbers respectively. The LRSM includes three gauge fields; W L,R are associated with SU (2) L,R and B µ corresponds to U (1) B−L . The fermionic content is consists of the SM quarks and leptons extended by three RHNs as where ν iR are three RHNs. The right-handed fermions interact with W R gauge field which is SU (2) R -valued, and is given as The extended Higgs sector of the model consists of a Higgs bi-doublet Φ, and SU (2) L,R triplets ∆ L,R . The Spontaneous Symmetry Breaking (SSB) structure is The first SSB occurs at T = Λ F which breaks the left-right symmetry and gives a VEV to the SU (2) R triplet, i.e. ∆ R = 0. That gives mass to W ± R , Z R , and provides Majorana masses for N i ≡ ν i + ν c i . Next, when the temperature gets below EW scale, T < Λ W , the second SSB happens, and the Higgs bi-doublet acquires a VEV, i.e., Φ = 0. It gives Dirac masses to the SM particles, active neutrinos included. In the minimal LRSM, the origin of the mass for the SM neutrino is a hybrid (I+II) seesaw mechanism [7]. For an overview on LRSM, see Appendix A and the references therein.

SU(2) R -Axion Inflation
Cosmic inflation is given by Friedmann-Lemaitre-Robertson-Walker (FRLW) metric in which the Hubble parameter is almost constant H(t) H, and the scale factor is a(t) e Ht . As for the inflaton field we consider an axion field ϕ which is coupled to the W R gauge field in the LRSM as [1] For the sake of generality, we assume V (ϕ) is an arbitrary axion potential, flat enough to support the slow-roll inflation. This SU(2)-axion inflation model and its cosmic perturbations, for a generic dark SU (2) field, has been introduced and studied in [17] (See also [28,29]). One of the most popular and well-motivated axion models of inflation to provide the flat potential is the axion monodromy. While the underlying periodicity of the theory continues to protect the inflaton potential from corrections, here the periodic field space of the axion is effectively unfolded due to the monodromy [11,36,37].
In this setup, we have two unknown high energy scales, i.e., the scale of inflation Λ inf = √ M Pl H, and LR symmetry breaking scale Λ F . Besides, the SU (2) R may or may not acquire a VEV. Therefore, we can distinguish four different types of scenarios, which are classified in Table 1. Scenario I and I v describe the case Λ inf > Λ F , while II and II v otherwise, i.e. Λ inf < Λ F . Moreover, the v subscript denotes systems in which the SU (2) R acquires a VEV in inflation. In scenarios II and II v , the W R is massive in inflation. 5 In this work we solely focus on types I and II and leave I v and II v for future work. Right-handed fermions in SU(2) R -axion inflation: The U (1) B−L and SU (2) L gauge fields and left-handed fermions in inflation have conformal symmetry, hence are negligible in physics of inflation. The W R gauge field, however, is coupled to the axion which breaks its conformal symmetry and sources it in inflation. The right-handed fermions are coupled to W R as (see Eq. (A.14)) 6 where D µ is the spinor covariant derivative. The generated W R gauge field, hence, produces right-handed quarks and leptons in inflation. Finally, the right-handed fermions can have an effective interaction with the axions as whereλ is a constant of the order of λ, and the right-handed current is There are two source terms for the fermions, i.e. the SU (2) R gauge field and its axion. However, the axion cannot generate Weyl fermions. The reason is that a Peccei-Quinn type U P Q (1) rotation of fermions as [40] removes the axion interaction and transforms the fermion mass matrix as [41] M → e Therefore, the axion only contributes to the generation of massive fermions in inflation.

Inflationary Particle Production
In this inflation model, W R gauge field is generated by the axion. The field equation of W Rµ in the massless case (type-I scenarios) is and in the massive case (type-I scenarios) W ± R and Z 0 R acquire m W R and m Z R respectively. Moreover, apart from the exponential expansion of the Universe, both axion and W R gauge field are active in inflation and produce right-handed quarks and leptons. The P and C are maximally broken by the chiral nature of the SU (2) R interaction, and CP is violated by the Chern-Simons interaction. Both right-handed baryon and lepton numbers are violated by the non-perturbative effects of the W R , i.e. chiral (Adler-Bell-Jackiw) anomaly [42,43]. The sterile neutrinos are massless (massive with mass m N i ) in scenario type-I (type-II). That gives the right-handed baryons and leptons the following anomalies where J µR B,L is the baryon and lepton number densities, and N R is the number of right-handed fermion generations. Note that the B and L violating interactions of the left-handed fermions remains negligible in inflation. The Chern-Simons term can be written as a total derivative where K µ is the Chern-Simons current, i.e.
In our setup N L = N R = 3, hence we neglect the effect of global gravitational anomaly. 7 The total baryon and lepton numbers are related to their corresponding quantities in SM as  The details of the discussion depend on whether the first SSB happens before or after inflation. Therefore, these two cases will be treated separately. Before going any further, let us fix the notations that will be used in this section. For later convenience, we define H as H ≡ aH, (3.8) which in terms of conformal time, i.e. dt = adτ , and during slow-roll is H − 1 τ . The rescaled physical momentum is defined as Finally, we define dimensionless parameters ξ andξ as During slow-roll ξ has slow-roll dynamics, and is related to the slow-roll parameter ≡ −Ḣ H 2 as As a result, ξ gradually increases with time.

Scenario Type-I
In scenario type-I, the W R and right-handed fermions are all massless in inflation. Here we first study the gauge field production by the axion. Next, we turn to the fermion production by the gauge field.
Massless SU(2) R gauge bosons: The linearized field equation of SU (2) R is and a constraint equation ∇ µ W µ R = 0. The (charged) gauge field has two degrees of freedoms and can be decomposed in terms of its two transverse modes as where a k,σ (b k,σ ) is the annihilation operator of the particle (anti-particle), and e ± (k) are ±1 helicity polarization vectors, defined as wherer = −k,θ andφ are the local orthogonal unit vectors in the directions of increasing r, θ, and φ. The polarization vectors satisfy in the following equations k.e ± (k) = 0 and k × e ± (k) = ∓ik e ± (k). (3.15) The function f σ (k) can be expanded in terms of the mode functions as 16) which are governed by the field equations below The field equation can be written as a Whittaker equation with parameters κ ± = ∓iξ and µ 2 = 1 4 . (3.18) Imposing the Bunch-Davies vacuum condition in the asymptotic past, we have the mode functions as [17] f a ± (τ, k) = e iκ ± π/2 (2π) Therefore one helicity state of the gauge field (plus/minus for positive/negative ξ) has a short period of tachyonic growth (See the left panel of Fig. 3). 8

Massless right-handed fermions:
In scenario type-I, all the fermions are massless in inflation. Therefore, the right-handed fermions are generated by the SU (2) R gauge field as 20) which implies that the produced right-handed fermions are given by the Chern-Simons current (3.5). After linearization, the Chern-Simons charge is given as For later convenience, we define K(ξ) as with κ ± = ∓iξ and µ = 1 2 . Using solution (3.19), we can write n CS as That gives the lepton and baryon number densities as The number density of N i is The right panel of Fig 3 shows K(ξ) vs ξ. It increases exponentially with the increase of ξ as (3.26)

Scenario Type-II
In scenario type-II, the gauge symmetry SU (2) R × U (1) B−L breaks to U (1) Y . It gives masses to the gauge boson, charged and neutral, i.e. 27) as well as at least two of the right-handed fermions. Here we first study the gauge field production by the axion. Next, we turn to the fermion production by the gauge field and the axion.
Massive SU(2) R Gauge Bosons: with a constraint equation The neutral component Z 0 R , satisfies the same equations with m W R replaced by m Z R . Since the gauge field is massive, in addition to the two transverse modes with polarization vectors e ± (k), there is another dynamical degree of freedom associated with k i W i R (k). For ease of notation, we define The gauge field in the massive case is given as where the polarization states are defined as Note that superscript ± denotes the charged of the field and subscript ± represents its helicity state. In addition to these dynamical fields, massive gauge field has W α R0 which is non-dynamical, specified by the constraint Eq. (3.29) Since it is only coupled to the longitudinal mode, it can be expanded as The field equation of the transverse modes with σ = 1, 2 (plus and minus helicity states) are given as The field equation of the longitudinal mode with σ = 3 is given as which is coupled to the W α R0 is given by the constraint Eq (3.33) as Like the massless case, the field equation of the transverse modes can be written as a Whitaker equation with parameters Imposing the Bunch-Davies vacuum condition in the asymptotic past, we have Since the longitudinal mode f α 3 and hence f α 0 are not coupled to the axion, they are strictly decaying and unimportant in inflation (see left panel of Fig. 4). Therefore, similar to the type-I case, the cosmological relevant modes in type-II scenarios are the transverse modes as well. Again f + polarization mode is generated by the axion which is shown in the right panel of Fig. 4.

Massive right-handed neutrinos:
In scenario type-II, the SM fermions are massless in inflation while at least two of the sterile neutrinos are massive. Therefore, we have The Chern-Simons charge given in Eq. (3.21) can be written as Therefore the baryon number density is given as The prefactor K(ξ, m α ) with respect to ξ and m W R is shown in the left and right panels of Fig. 5. It increases (decreases) with the increase of ξ (m α ) and for ξ > m α is  Due to the mass of the sterile neutrinos, the calculation of the lepton number is more involved and requires the mode functions. The leptonic field equations are where the bar emphasises that, unlike chiral anomaly, it is a classical effect. Using the point-splitting regularization, we computed ν i,R and D(ξ, m N i ) analytically in App. B. The exact form of D(ξ, m N i ) is presented in Eq. (B.12) and we show it in Fig. 6. Here we discuss its qualitative behavior which in the large and small mass limits is Then N i is directly proportional to (and an odd function of)ξ which is the (classical) source of particle production. Moreover, it increases with the mass of the sterile neutrinos, m N i , which is the cause of their chiral symmetry breaking in inflation. Therefore, the total number density of RHNs are given as The final total lepton number is

Baryon and Lepton Numbers
Here we summarize the main features of inflationary baryon and lepton generation.
• The transverse modes of W R are generated by the axion that subsequently sources right-handed baryons and leptons.
• All the Sakharov conditions required for BAU are satisfied in inflation: i) Out of thermal equilibrium condition holds during inflation, ii) C is violated by the chiral nature of the SU (2) R interaction, iii) B, L, and CP are violated by the non-perturbative effects of W R .
• n B and n L are the total baryon and lepton number densities respectively. n B and n L SM are the contributions of the SM fermions. The RHNs number density is n N = n L − n L SM .
• The B−L is conserved (violated) in scenario type-I (type-II ). However, B−L SM = L N is violated in both scenarios.
• In scenario type-I, the baryon and lepton numbers are both generated by the chiral ). It can be written as where α inf (ξ) is given as (see Fig. 7) • In scenario type-II, the baryon number is specified entirely by the chiral anomaly .44)). In the leptonic sector, however, the massive RHNs are also generated by the axion. Therefore, the total lepton number is n L n B +n N wheren N is the RHNs produced by the axion (Eq. (3.47)).
• The number density of the RHNs generated in inflation is where α inf for scenarios type-I (Eq. (3.25)) and II (Eq. (3.49)) are given as

Post Reheating Evolution
To study the post-inflationary evolution, we need to specify our parameter space. For the sake of concreteness, we restrict the current analysis by assuming the following conditions considered by the author in [1]: Condition C1) A hierarchical mass spectrum for the RH neutrinos (as implied by the neutrino oscillations) as where N 1 is much lighter than the EW scale with feeble Yukawa interactions and hence a DM candidate. (See Fig. 8) Condition C2) The W R field is never in thermal equilibrium with the thermal bath, i.e. T W R > T reh . Condition C3) The CP-violating phases in the neutrino sector, unconstrained by the current data, are not large enough to create the observed BAU. Reference [1] introduces a stronger version of the above conditions by imposing a more restrictive version of C2. Restricted condition C2) The postinflationary generation of RHNs via W R interactions is negligible compared to their preexisting relics. We discuss and quantify conditions C2 & restricted C2 in Sec. 4.1 and condition C3 in Sec. 4.2.

Thermal Evolution
Reheating starts at some point after the end of inflation and ends at the formation of a dominant thermal bath with temperature T reh . Here, we consider the phenomenological reheating model below in which ρ inf and ρ reh are the energy density at the end of inflation and reheating respectively. Moreover, δ reh is the efficiency of the reheating process given as (See App. C.1) where w X is the effective equation of state in the intermediate period between the end of inflation and the formation of the thermal bath. The radiation energy density is given by where g eff is the effective number of relativistic degrees of freedom. For SM particles at the time of reheating we have g eff = 427/4. The reheating temperature is The photon number density at the time of reheating is where ζ(x) is the Riemann zeta function and ζ(3) 1.2. The photon number density today is related to n γ,reh as where S is the entropy injection factor. It captures the increase of entropy by the out of thermal equilibrium decay of heavy RHNs, and is worked out in App. C.1. We found that the entropy injection is negligible in our setup, i.e. 9 S 1.
Condition C2: The W R gauge interaction has essential effects on thermal properties of our setup. After the 1st SSB, they keep sterile neutrinos in thermal equilibrium by scattering with the SM fermions. The temperature of the freeze-out of W R gauge field can be estimated as where g * is the number of relativistic degrees of freedom at T W R and m W R is the mass of W ± R . The particles which are only coupled through the W R interactions with the thermal bath, e.g. N 1 , gets decoupled at this point. The thermal evolution after inflation depends on whether sterile neutrinos are in thermal equilibrium initially or not. If in thermal equilibrium, W R interactions generate thermal abundances of RHNs, i.e., freeze-out production. The focus of this work, however, is the region in the parameter space in which W R interactions are never in thermal equilibrium, i.e. T reh < T W R which demands (See Fig. 9) .
(4.10) reh < reh > Figure 9. The freeze-out temperature of W R interactions in terms of m W R . The (pink) shaded area shows regions with T reh < T W R (condition C2) and below the solid line, the secondary (postinflationary) abundance of RHNs are not generated by freeze-out but instead by the freeze-in mechanism. In the (blue) shaded region we have T reh > m W R , so W R sphalerons are never in thermal equilibrium below the dashed line.
The above condition guarantees that N i does not have thermal abundances by freezeout mechanism. 10 However, as we will see shortly, W R scatterings may still create a post inflationary abundance of RHNs via freeze-in mechanism. Fig. 9 presents T W R vs m W R and the (pink) shaded area shows the region with T reh < T W R which is the focus of the 10 In case that T W R < T reh , Fermi-type theory of WR field keeps sterile neutrinos in thermal equilibrium even at temperatures lower than m W R . The freeze-out relic abundance of Ni is current work. The (blue) shaded region marks where T reh > m W R . Therefore, in the region of our interest the SU (2) R sphalerons are never in thermal equilibrium to cause any B + L violating interaction (see Eq. (D.5)). That is in contrast to the SU (2) L sphalerons which are in thermal equilibrium in the wide temperature interval of m W L < T < 10 12 GeV . Another constraint on Hubble parameter in inflation comes from the current upper bound on the tensor to scalar ratio, r 0.05 < 0.07 at 95% confidence [45], which implies H 10 −5 M Pl .

Restricted Condition C2:
At reheating, the pre-existing RHNs, generated in inflation, is Eq. (3.54) One of the consequences of condition C2 is that the RHN does not have a thermal abundance, i.e. no freeze-out production. However, post-inflationary W R scatterings produce RHNs via freeze-in as [46] n s N i ≈ 10 13 × g eff 10 2 T reh M Pl The superscripts p in Eq. (4.11) and s in Eq. (4.12) denote contributions of pre-existing (inflationary) and secondary (freeze-in) production respectively. One can restricted condition C2 such that this secondary RHN production is subleading comparing to the pre-existing one. Using Eq. (4.5), we can quantify restricted condition C2 as once inequality in Eq. (4.10) holds. Type-I Scenarios: In this case, the W R gauge fields and N i s are massless in inflation. On the other hand, condition C2 demands m W R T reh which implies the first SSB must happen shortly after the end of inflation. The shaded areas in Fig. 10 show the parameter space in which restricted C2 is satisfied for a given m W R . As we see, most of the parameter space is accessible in case of w X = 1 while the case with w X = 0 requires larger values of m W R and ∆N and/or ξ. The ratio of the pre-existing N 1 to its freeze-in production in type-I can be analytically approximated as (4.14) Type-II Scenarios: In this case the SU (2) R gauge fields and N i s are massive in inflation. However, N 1 which is the dark matter candidate is very light compared to H. Therefore, the pre-existing N 1 in Eq.  Fig. 11 shows the parameter space corresponding to each ξ in which restricted C2 is satisfied. Comparing with the type-I scenario, the C2 is satisfied in a smaller part of the parameter space and only for w X = 1 case.  Figure 11. The parameter space (∆N, m W R ) for different values of ξ that satisfies condition restricted C2 in type-II scenarios with w X = 1. The accessible region for w X = 0 case is in the region with ∆N > 8 and it is not shown here.
To summarize, condition restricted C2 prefers type-I scenarios and w X = 1. In particular, it holds in a wide part of the parameter space when Λ F Λ inf , i.e., the first SSB coincides with the end of inflation. Interestingly, it relates left-right gauge symmetry breaking to a geometrical phase transition in cosmology, i.e., the end of exponential expansion of the Universe. Moreover, as Fig.s. 10-11 show, it demands m W R > 10 10 GeV which is the scale suggested by the non-supersymmetric SO(10) GUT model with an intermediate left-right symmetry scale [47][48][49].

Spectator Effects, RHN decay, and Matter Asymmetry
Throughout the Early Universe, particles experience a whole cascade of interactions that eventually equilibrate in the Early Universe. Many of them can potentially redistribute the initial asymmetries to the spectator degrees of freedom. These processes do not participate directly in the generation or washout of the asymmetries (hence the name spectator). Still, they have important effects in finial B and L by imposing certain relations between different species. In addition to the spectator effects, the CP asymmetric decay of N 2,3 produces SM leptons and simultaneously partially washes out the pre-existing lepton asymmetries. In this section, we consider washout effects, lepton flavor effects, and sphaleron processes. For the ease of notation, we denote the SM leptons as L, i.e.

Spectator Effects
The SU (2) L sphalerons (SU (2) R sphalerons) transmit the asymmetry from left-handed (right-handed) leptons to left-handed (right-handed) quarks and vice versa. The W L gauge field is inactive and unimportant in inflation. Later on, however, they attain a thermal equilibrium, and together with W R , they can have significant impacts on the final B and L asymmetries. The B + L violating processes due to W L,R sphalerons shuffle the initial baryons and leptons coupled to them. In App. D.1 we showed that W R sphalerons are never in thermal equilibrium in our setup (see also Fig. 9). Hence they can not give rise to B + L violating processes. After the W R and sterile neutrinos' freezeout, the SM particles remain in thermal equilibrium up to the electroweak scale. Quarks, SM leptons, and Higgs bosons interact via gauge and Yukawa interactions as well as non-perturbative sphaleron processes. All the SM gauge interactions and W L sphaleron processes are in equilibrium in the temperature range of 100 GeV T 10 12 GeV . The thermal equilibrium of Yukawa interactions is flavor-dependent. Nevertheless, all of them are in equilibrium at T < 85 T eV [50]. Using the sphaleron effects and hypercharge constraint, we find that B, L, and B − L are related as where c sph = 28 79 is the sphaleron conversion factor.

Lepton Flavor Effects
One potentially very significant aspect of (post inflationary) leptogenesis is the flavor effect. The flavor-dependent washout and L violating interactions can significantly change the value, and even sign of the final baryon asymmetry [51][52][53]. By the end of inflation, and due to our flavor blind CP violating source, we have a lepton quantum state |l inf as The decays of the heavy sterile neutrinos modify these initial states. More precisely, the CP asymmetric decay of N i produces leptons as 11 and simultaneously washes out the pre-existing (inflationary) leptons in this direction, i.e.
However, the pre-existing leptons normal to |l i direction, i.e.
elude the washout. As discussed earlier, we assume that N 1 has feeble Yukawa interactions with the SM and hence a DM candidate (condition C1). Therefore, only N 2 and N 3 contribute to the seesaw mechanism as well as decays and washouts. As a result, the component |l inf 3 ⊥ 2 ⊥ which is normal to both |l 3 and |l 2 remains as the remnant of the initial asymmetry. For the mass spectrum in Eq. (4.1), the corresponding Boltzmann equations and details are presented in App. D.2 and here we report the final results. The geometry of this process in the SM flavor basis is schematically shown in Fig. 12. Condition C3: The SM lepton asymmetry after decay of N 2 at T = M 2 10 9 GeV is where n p,f B−L is the remnant of the primordial asymmetry n p,i B−L , and n N B−L is the lepton number produced by the CP asymmetric decay of N 2 as where ε 2 is the CP asymmetry and κ 2 is the associated efficiency factor. Interestingly, when flavour effects are considered, it is very difficult for the pre-existing asymmetry to be washed out by the RH neutrinos [54,55]. The value of n N B−L depends on the leptonic Yukawa matrix and the unconstrained CP violating phases in the neutrino sector. In this work, we assume that the amount of this asymmetry is not sufficient to account for the observed matter asymmetry, i.e. condition C3: (4.24) Condition C3 is the opposite limit of what is assumed in leptogenesis scenarios [56].
Finally the remnant of the primordial asymmetry is given as below in terms of the initial B − L where C is a parameter less than one ( see Eq. (D.35) and Fig. 15). For most of the parameter space we have Eliminating the effect of this pre-existing asymmetry is very hard and requires tightly finetuned relations between leptonic Yukawa couplings and the physics of inflation which is discussed in App D.2.

Modern Era Baryon Asymmetry and Dark Matter
In this section we work out the baryon to photon ratio and dark matter density today.
Here we only consider type-I scenarios in which the 1st SSB happens after inflation. The remnants of the inflationary baryon and SM lepton asymmetries after the decay of the heavy RHNs and the getting redistributed by the spectator effects are respectively as The final N 1 number density is where n s N 1 is the secondary (freeze-in) production of N 1 given in Eq. (4.12). As is assumed in [1], if the restricted version of condition C2 in Eq. (4.13) holds, we have n N 1 (a) 2.8 n B (a). (5.4) The particle production mechanism throughout cosmic evolution is then summarized in Fig. 2.

Baryon to Photon Ratio
To the best of our knowledge, the cosmos is highly matter-dominated. The baryonantibaryon asymmetry can be quantified by the baryon to photon ratio at the present time as [57] η in which a 0 superscript denotes the present time value. Our setup predicts the baryon to photon ratio as where g eff = 427/4. One can write η 0 B in terms of the curvature power spectrum as in which is the slow-roll parameter, and β is

Right-handed Neutrino as Cold Dark Matter
As discussed in Sec. 4, we assume that the lightest RHN, N 1 , has feeble Yukawa couplings, hence decouples after freezeout of W R interactions at T W R with a relic density given as where Ω B is the baryon density parameter, m p is the proton mass. If N 1 makes all of the DM that we observer today, i.e. Ω N 1 5Ω B , it specifies the mass of N 1 in terms of the proton mass as ) .

(5.12)
Condition C2 implies that 0 ≤ n s N 1 n p N 1 < 10 6 which specifies the mass of N 1 in the wide range of a few keV to a few GeV . That mass range is associated to different DM spectra from warm DM to cold DM. On the other hand if following [1] we consider restricted condition C2 which grantees that the N 1 relic density is primordial, we have which makes a specific prediction for the mass of N 1 as m N 1 1.8 m p = 1.7 GeV. (5.14) That leads to a cold DM spectrum that is consistent with structure formation. Next, we study the stability of N 1 as a DM particle.

Decay of N 1
Given that W R is very heavy and freezes out early (see Sec. 4.1), the dominant decay channel of N 1 is N 1 → 3ν with the total decay width [58,59] where G F is the Fermi constant, α = e, ν, τ and θ α,1 are the mixing angles of left-handed neutrinos with N 1 . Demanding that the lifetime of this process, t N 1 , is larger than the age of the Universe, i.e. t U ≈ 4.4 × 10 17 s, we arrive at where θ 2 1 ≡ α θ 2 α,1 . Demanding that N 1 is stable over the lifetime of the universe gives In this framework, the generation mechanism of N 1 is independent of its Yukawa mixing with active neutrinos, and θ 1 can be any number that satisfies the above upper bound. The next leading decay channel is the loop-mediated radiative decay of N 1 to active neutrinos and a gamma-ray photon with energy E γ ≈ M N 1 /2 as [58] Although the radiative decay has a branching ratio of order 2%, it can provide upper bounds from not observing gamma-ray photons with energy E γ . The current strongest gamma-ray bounds in the GeV scale are on mass range 10-100 GeV [60] which is much heavier than our DM.

Quick on Observational Constraints and Signatures
In this section, we briefly discuss the cosmological, astrophysical, and collider constraints and signatures of our setup. The current work is based on embedding the minimal SU (2)axion inflation model [17] in minimal LRSM [2][3][4]. The cosmic perturbations of the minimal SU (2)-axion inflation in the presence of the gauge field VEV has been studied and compared with Planck data in [17]. In the current work, however, we sorely focus on scenarios with vanishing VEV. Hence it enjoys a wider accessible parameter space. As a cosmological smoking gun, all SU (2)-axion inflation models predict chiral [18,31,32] and non-Gaussian [33] gravitational wave background which leads to parity odd CMB cross-spectra [35]. This chiral GW Background (GWB) is blue tilted and can also be detected by future laser interferometer detectors. The parity odd features can be used as an observational marker to distinguish it from the standard GWB produced by the vacuum fluctuations [61][62][63]. This signal has been extensively studied in the literature. For an exhaustive discussion on the measurement of this effect, see [34].
Direct production or virtual contributions at astrophysical and collider processes put several constraints on the charged and neutral SU (2) R gauge boson mass and mixing parameters. The K L − K S kaon mass difference measurement [64] places a lower bound on the mass of W R as m W R > 1.6 T eV and the mixing angle between Z R and Z L is constrained to be less than 10 −4 . The possible low-energy W R has been the target of several LHC collaborations which puts the current bound as m W R > 3 T eV [65]. For an exhaustive discussion of the phenomenological implications and constraints of LRSM, see [66]. Our current setup with high scale SU (2) R SSB, i.e., m W R > 10 10 GeV , satisfies the above lower bounds. The most distinctive astrophysical signal of our DM candidate with GeV mass is the gamma-ray line at E = m N 1 /2 produced in the one-loop decay N 1 → γν. Gamma-ray lines have been probed by the Fermi-LAT [60], H.E.S.S. [67], and MAGIC telescopes [68]. However, the strongest current bounds are on DM masses above 10 GeV [60] which is heavier than our DM. We leave the further study of the observable signatures of this setup for future work.

Conclusions
Recently [1] proposed a new particle physics model for inflation, based on embedding axioninflation in gauge extensions of the SM. To unify cosmic inflation and BSM, it utilized the minimal Left-Right Symmetric Model (LRSM) [2][3][4] with gauge group SU (2) L × SU (2) R × U (1) B−L . As the name implies, the model includes W R gauge bosons and three righthanded neutrinos (RHN). As the inflaton field, an axion is added to the field content of LRSM, which is directly coupled to the SU (2) R gauge field. In this work, we presented the analytical and numerical details of this setup.
LRSM in cosmology introduces a new fundamental cosmic scale, i.e., feeble scale Λ F , where the extended gauge symmetry breaks down to the SM one, i.e., SU (2) R ×U (1) B−L → U (1) Y . At feeble scale, the W ± R and Z 0 R become massive, and the RHNs acquire Majorana masses [7]. Later, at the electroweak scale, the second spontaneous symmetry breaking happens, which gives mass to the SM particles. At this point, the left-handed neutrinos acquire mass by seesaw mechanism [7] (for cosmic evolution see Fig. 1). Based on the scale of inflation Λ inf = √ HM Pl , feeble scale Λ F , and the (possible) SU (2) R field's VEV in inflation, one can separate four different types of scenarios (see table 1). Following [1], we solely focused on scenarios with vanishing SU (2) R VEV, i.e. type- The SU (2) R gauge field is produced by inflaton while other gauge fields, i.e., SU (3), SU (2) L and U (1) B−L , are diluted by the exponential expansion. The chiral anomaly of W R breaks CP in physics of inflation and gives rise to simultaneous baryogenesis, leptogenesis, and RHN creation in inflation (see Eq.s (3.24) & (3.25) for type-I and Eq.s (3.44) & (3.50) for type-II scenarios). Even in type-I scenarios in which B − L is a gauge symmetry in inflation, we have B − L SM = 0. For cosmic evolution after inflation, we future specified our parameters and imposed the three conditions which are used in [1]. Condition C1 considered a hierarchical mass spectrum for RHNs with feeble Yukawa interactions for N 1 such that it is a DM candidate (Eq. (4.1)). Condition C2 demands that W R is never in thermal equilibrium with the thermal bath. Consequently, it implies; 1) the SU (2) R sphalerons were never in equilibrium as well (Eq. (4.10) and Fig. 9), and 2) there is no secondary freeze-out production of RHNs. However, the post-inflationary scatterings of W R can generate RHNs via freeze-in mechanism (Eq. (4.12)). Following [1], one can also consider a restricted version of condition C2 which demands that this secondary RHN production is subleading comparing to the pre-existing one (Eq. (4.13)). Finally, condition C3 assumed that the unconstrained CP-violating phases in the neutrino sector are not strong enough to make a sizable contribution to the matter asymmetry (Eq. (4.24)). C3 is the opposite limit of what is assumed in leptogenesis scenarios.
The lightest RHN gets decoupled after the freeze-out of W R field at T W R (Eq. (4.9)). The heavier RHNs decay after temperature gets below their masses, and the spectator effects reshuffle the primordial baryon and SM lepton numbers. The final baryon to photon ratio and DM relic density are presented in Eq.s (5.6) and (5.11) respectively. This setup can explain η B and Ω DM in a wide range of its parameter space (see Fig. 13). If N 1 makes all the DM relic density, then its mass is in the range of keV − GeV . In case that restricted C2 condition holds, the mass is predicted to be m N 1 ≈ 1.7 GeV , i.e. a cold DM spectra consistent with structure formation (Eq. (5.14)). In that case, baryogenesis and DM today are the remnants of a pure quantum effect (chiral anomaly of W R ) in inflation. Consequently, it can naturally explain the observed coincidences among cosmological parameters, i.e., η B = 0.3P ζ and Ω DM = 5Ω B . Besides, this model is a complete setup that can simultaneously provide plausible explanations for the phenomena (I-IV) named in the introduction. The summary of this new mechanism is illustrated in Fig. 2.
It is noteworthy to mention that we can couple the axion to both SU (2) R and SU (2) L gauge fields. However, the inflationary production of left-handed baryons and leptons by SU (2) L (i.e. B SM = L SM ) will be completely washed out by the SU (2) L sphaleon effects which are in thermal equilibrium between T reh and m W L . Since SU (2) L -axion interaction leaves no fermionic remnants today, it is neglected in the minimal realization of this idea proposed in [1].
In this setup, P and CP are broken by the VEV of the axion and its interaction with the gauge field. It provides a deep connection between inflation, matter asymmetry, and DM relic density. This alternative mechanism, therefore, does not rely on the largeness of the unconstrained CP-violating phases in the neutrino sector nor fine-tuned masses for the heaviest right-handed neutrinos. Interestingly, sufficient matter creation relates the feeble scale to a geometrical phase transition in cosmology, i.e., the end of exponential expansion of the Universe. Moreover, it demands m W R > 10 10 GeV (see Fig.s 10-11) which is the scale suggested by the non-supersymmetric SO(10) GUT model with an intermediate left-right symmetry scale [47][48][49]. The above relations between the energy scales may be hints of a fundamental connection that we leave for future work. As yet another added benefit, this setup comes with a cosmological smoking gun; chiral, non-Gaussian, and blue-tilted gravitational wave background, which can be probed by future CMB missions and laser interferometer detectors. For an exhaustive discussion on the measurement of this effect, see [34].

Acknowledgments
The author especially thanks Eiichiro Komatsu for insightful discussions and valuable input during previous collaborations. She also likes to thank G. Giudice, J. Kopp, and M. Shaposhnikov for valuable discussions.

A Overview of Minimal Left-Right Symmetric Theories
Here we review the aspects of minimal left-right symmetric models (LRSM) [2][3][4][5] that we need in this paper.

Field and Matter Content:
The model's field content is presented in Table 2, and in the following, we explain the gauge field, extended Higgs, and fermionic sectors, respectively. The baryon and lepton numbers are denoted by B and L, respectively. Moreover, L and R subscribes represent left-and right-handed fields.
Gauge Group of the minimal left right symmetric interaction (suppressing color) is where (W R , g R ) and (W L , g L ) are the SU (2) R and SU (2) L gauge fields respectively where τ a denotes the Pauli matrices which acts on the SU (2)-color indices. The W L,R act on left-and right-handed fields respectively. The strength tensor of W L,R are given as The electric charge Q is defined as where Y is the hypercharge.
Scalar Sector involves three Higgs fields, i.e. a Higgs bi-doublet to produce the Dirac masses, and two triplet Higgs to create Majorana masses for the neutrinos. The

Scalars
Higgs SU (2) bi-doublet Higgs SU (2) triplets and the SU (2) R,L triplets with B − L = 2 are given as The gauge-covariant derivatives of Φ and ∆ L,R are given as The theory of the Higgs sector is given as where the Higgs potential V Higgs (Φ, ∆ R , ∆ L ) is the most general renormalizable, gauge and parity invariant potential for Φ and ∆ L,R [7,69]. The Higgs mass spectrum and the scale of each spontaneous symmetry breaking are given by minimizing the Higgs potential. Here we are interested in the cosmological consequences of such potential. For an exhaustive discussion we refer the interested reader to [8,70,71].
Fermionic Sector consists of three generations of quarks and leptons as where ν iR are three RHNs interacting via the SU (2) R and U (1) B−L . Given that we assume neutrinos are Majorana, and define two Majorana fields associated to the left-and righthanded neutrinos as where the c superscript denotes the charge conjugated field. For simplicity, we present the left-and right-handed fermions collectively as which are specified by the Lagrangian where the spinor gauge-covariant derivatives are where ω µ is the spin connection. 12 For the cosmological background, we have σ µ ω µ = 3 2 HI 2 and where σ i are the Pauli matrices which carries spatial index. 13 The fermions pick up their mass by the Yukawa interactions where l c Ri = C l * Ri is the charged conjugated l Ri , and The spin connection is defined as ωµ ≡ i 2 ω αβ µ Σ αβ where Σ αβ = i 4 [γα, γ β ] and ω αβ µ ≡ e αν ∇µe β ν . 13 Note that σµ is the curved space form of the flat space σα = (I2, σi) as σµ = e α µ σα where e α µ are the tetrads.

Symmetry Breaking Structure, New Fundamental Scale, and Mass:
Once the neutral component of ∆ R acquires a VEV as both of the B−L and left-right symmetries are spontaneously broken. That introduces a new fundamental scale, i.e. Λ F = κ R , which is much higher than the EW scale, Λ W 246 GeV . The 1st SSB breaks the gauge symmetry down to the SM electroweak symmetry as All non-Standard Model heavy particle masses are related to the VEV of ∆ R . The charged and neutral SU (2) R gauge bosons pick up the following masses where g Y is given as (A.23) The right-handed neutrinos get Majorana mass terms as which leads to the Majorana mass matrix M Rij = κ R Y R ij . Finally, when the temperature drops below the EW phase transition, i.e. T = Λ W , the 2nd SSB happens and the neutral components of the bi-doublet receive its VEVs as That breaks the gauge symmetry to U (1) em , i.e.
which provides Dirac masses for the SM particles, SM neutrinos included. After 2nd SSB, therefore, all the SM massive particles pick a Dirac mass similar to SM. The interaction between ∆ R and Φ with ∆ L in Higgs potential imposes a VEV for the latter once the former fields acquired their VEVs. The VEV of ∆ L is of the order of O( Φ 2 κ R ) Φ [7]. The value of the κ 1,2 is related to the EW scale κ as (A. 27) In the limit of our interest, κ R κ 1 , κ 2 , κ L in which the left and right charged and neutral gauge bosons are decoupled. Thus, we can consider W ± L,R and Z L,R as physical states. Here for simplicity we also assume κ 1 κ 2 and κ 2 κ.
We summarize the symmetry-breaking structure of the setup in Table 3. Below we will discuss its consequences on active neutrinos.

Symmetry Group
After 1st SSB After 2nd SSB Massive Particles W ± R , Z R and N i SM and ν i (Seesaw type-I & II) Neutrino masses; Natural Seesaw Mechanism: The first SSB provides Majorana masses for the RHNs and the second SSB gives Dirac masses to neutrinos as well as an induced Majorana mass to left-handed neutrinos. The neutrino mass matrix as where the Majorana mass matrices M R,L are .29) and the Dirac mass matrix is Given the fact that m D M R , we can diagonalize the mass matrix and find the masses of the active neutrinos as Note that κ L is a (small) induced VEV and the contribution of both the first term (seesaw type-I) and the second term (seesaw type-II) are of the same order. Thus, in minimal LRSM, the neutrino mass is a hybrid seesaw type-I, and II [7].

Experimental Constraints on Parameters:
Various Experimental limits can be placed on the mass scales and mixing parameters of the LRSM. First, considering the charged lepton Yukawa couplings as a guide to the neutrino ones suggests 10 −10 y 2 Y 1 which implies a successful seesaw requires 10 T eV κ R 10 15 GeV. (A.32) Next, regardless of the details of the SSB, there is a theoretical lower bound on the ratio of g R to g L [70] g R g L ≥ tan θ w 0.55.
That gives m Z R ≈ 1.7m W R . Finally, there are several constraints for the right-handed charged and neutral gauge boson mass and mixing parameters. These arise due to their direct production or virtual contributions at colliders or astrophysical processes. The K L − K S kaon mass difference measurement [64] places a lower bound on the mass of W R as m W R > 1.6 T eV and the mixing angle between Z R and Z L is constrained to be less than 10 −4 . The possible low-energy W R has been the target of several LHC collaborations which puts the current bound as m W R > 3 T eV [65]. For an exhaustive discussion of the phenomenological implications and constraints of LRSM, see [66].

Gauge coupling evolution
There is a significant difference between a high scale inflation and electroweak scale. Thus the running of the gauge couplings might be sizable. In the one-loop approximation, the RGE for the SU (N c ) gauge coupling with N f Weyl or Majorana fermions in the fundamental representation and N s Higgs fields in the R s representation is given as where k is the momentum, µ is a given scale associated with our renormalization and b i is The gauge coupling of SU (2) R at the scale of inflation is Using the theoretical lower bound on g R in Eq. (A.33), we arrive at

B Massive Sterile Neutrino production in Inflation
This appendix presents the analytical calculations of massive RHN production by the axion in inflation. In this work, we restrict ourselves to the cases with W R = 0. 14 From Eq.
(3.46), we find the linearized field equation of ν jR as As a Majorana fermion, N j ≡ ν jR + ν c jR can be decomposed as where c s jk and c s † jk are the annihilation and creation operators of the RHNs as and E ± k are the ± 1 2 helicity polarization states These helicity 2-spinors are the eigenstates of the helicity operator and satisfy E −s k = −isσ 2 E s * k . The Majorana condition then requires The pair of coupled first order differential equations for ν iR and ν c iR coming from Eq. (B.1) can be decoupled into two second order differential equations for the mode functions X ± jk (τ ). Upon field redefinitionX we have where κ s and µ j are Setting the initial conditions with Bunch-Davies vacuum, the solutions arẽ Notice that the − 1 2 helicity mode of the right-handed neutrinos is proportional to their mass. Thus, as we expected, the massless ν jR has only the + 1 2 helicity state. Working out the mode functions of the massive sterile neutrinos, we are ready to compute its contribution to the lepton number in Eq. (3.47) as Using point splitting regularization, we analytically calculated the above momentum integral in [24]. 15 We here use the final result which is Re e 2|µ j |π − e −2sκ I π sinh(2|µ j |π) in which κ I ≡ 2ξ and ψ (0) (z) ≡ dΓ(z) z is the digamma function.

C Phenomenological Model of Reheating
Reheating starts at some point after the end of inflation and ends at a reh with the formation of a dominant thermal bath with temperature T reh . Yet, the precise physics of reheating is not well understood. Depends on the details of the post-inflation physics, there may be an intermediate phase X with the average equation of state w X , which connects inflation to the final thermal bath (See Fig. 14). To quantify our analysis and capture these ambiguities, in this appendix, we introduce a phenomenological model for reheating. Next, we compute the entropy injection by the decay of RHNs in our setup. 15 The details of the calculation and point splitting regularization that is used in computing the integral (B.11) can be found in Appendix D of [24]. Notice that κ+ and κ− parameters in the current work are denoted as κ+ andκ− in [24]. The dimensionless parameter ξA in the latter is associated with the VEV of the SU (2) gauge field, which is set to zero in the current work. In that case, the energy density at the end of reheating is related to ρ inf as The parameter δ reh is the efficiency of the reheating process which models our ignorance about physics of reheating in terms of two parameters; w X and ∆N given as which is the number of e-folds between end of inflation until the formation of the thermal bath. The ratio The factor e − 3 2 (3w X +1)∆N for these two preheating scenarios are presented in the right panel of Fig. 14.

C.1 Entropy Injection
The out of thermal equilibrium decay of heavy RHNs, i.e. N 2,3 , at T = m N 2,3 injects entropy to the hot plasma and increase the energy of radiation as where S is the entropy injection factor given as 10) where n N i (a reh ) is the total number density of N i , i.e.
From Eq. (4.12), the freeze-in part of the density is while the contribution of the primordial density in Eq. (4.11) gives Given that H M Pl < 10 −5 GeV and the mass of the heaviest RHN is around 10 12 GeV , Eq.s (C.12) implies that contribution of n s N i (freeze-in mechanism) to the entropy injection is negligible in our setup. Therefore, we have 14) In case that the entropy injection is sizable in our setup, the baryon to photon ratio is where A(ξ, ∆N ) is The quantity A(ξ, ∆N ) is negligible in the region of our interest (see Fig. 7). Therefore, our setup has negligible entropy injection

D Spectator Effects
This Appendix is devoted to the spectator effects on matter asymmetry. First, we work out the temperature windows in which each of the W L,R sphalerons are in thermal equilibrium and hence can violate the left-/right-handed B+L. Next, we discuss the lepton flavor effects in our setup.

D.1 W L,R Sphalerons
The SU (2) L,R sphaleron transitions start getting in thermal equilibrium once where Γ L,R sph is the transition rate per unit time per unit volume so dimensional estimate gives Γ L,R sph ∼ (α L,R T ) 4 where α L,R = g 2 L,R 4π . Using lattice simulations the transition rate for the SU (2) L weak sphalerons has been found in [72] as where χ ≈ 18 and the extra α L factor is due to specific plasma effects [73]. The W R,L switch off after their corresponding scale of SSB. Therefore, the SU (2) L weak lepton and baryon violating processes are in thermal equilibrium in the wide temperature interval 100 GeV < T L sph < 10 12 GeV.
As a rough estimate, we assume that the same relation holds for the SU (2) R sphalerons, i.e.
Thus, W R sphalerons are in thermal equilibrium in the following interval Given that in our setup T reh < T W R < m W R , the W R sphalerons are never in equilibrium to cause any B + L violating interaction.

D.2 Lepton Flavor Effects
One potentially very significant aspect of leptogenesis is the flavor effects. The flavordependent washout and L violating interactions can significantly change the value, and even sign of the final baryon asymmetry [51][52][53]. By the end of inflation, we have a lepton (anti-lepton) quantum state |l inf (|l inf ) as where C inf α andC inf α are specified by the physics of inflation as The composition of this primordial initial leptons and their CP conjugated anit-leptons are different. The CP violating decays of the heavy sterile neutrinos can modify these initial states. At very high temperatures T 10 12 GeV , however, the interactions are still flavor blind and we can describe leptons as a coherent superposition of charged leptons as where C iα are coefficients given by the Yukawa matrix which in terms of the active neutrino mass matrix we have . The flavored decay parameters of N i to l α are defined as , (D. 9) and K i = α K iα . The Yukawa couplings of neutrinos contain several CP-violating phases which remain unconstrained by the current data. Therefore, the decay of sterile neutrinos can be a CP asymmetric process quantified as where ε i = α ε iα is the CP-asymmetry.
In the light of the current neutrino oscillations data, the RH neutrino mass spectrum turns out to be typically highly hierarchical. For the sake of concreteness, in this work, we consider m N 3 10 12 GeV m N 2 10 9 GeV m N 1 , (D. 11) where m N 1 is assumed to be lower than the EW scale. Furthermore, we assume that the lightest sterile neutrino has feeble Yukawa interactions with the SM and hence a DM candidate, i.e.
(D.12) Therefore, only the two heavy sterile neutrinos, N 2 and N 3 contribute to the seesaw mechanism as well as decays and washouts. Moreover, due to the hierarchical neutrino mass spectrum, the decays and washout of N 2 and N 3 occur in separate stages with no overlaps. As a result, the decay processes can be studied by the following semi-classical Boltzmann equations for η X ≡ n X nγ (η eq X = n eq X nγ ) The decay terms and the related washout terms are given as (D.17) At very high temperatures T 10 12 GeV , the interactions are flavor blind and we can describe leptons as a coherent superposition of charged leptons. At temperatures 10 9 GeV < T < 10 12 GeV , τ lepton-Higgs interactions are fast and destroy the coherence of the lepton states produced by N i decay. Therefore the Boltzmann Eq.s (D.14) and (D.15) are effectively described by two incoherent SM flavors, i.e., τ and τ ⊥ = e + µ. The SM lepton asymmetry after decay of N 2 at T = M 2 10 9 GeV is n δ (z 2 ) = n p,f δ (z 2 ) + n N δ (z 2 ), (D. 18) where n p,f δ (z 2 ) is the contribution of the primordial asymmetry n p,i δ , as n p,f δ (z 2 ) = C n p,i δ + e − z 2 1 dW i (z ) dz dz n p,i δ 3 ⊥ , (D. 19) and n N δ (z 2 ) is the lepton number produced by the CP asymmetric decay of N 2 , i.e.
n N δ (z 2 ) ≈ ε 2 κ 2 (z 2 ), (D. 20) in which κ 2 (z 2 ) is the efficiency factor of the CP asymmetric decay. In this work we are interested in the limit n N δ (z 2 ) n p,f δ 1 (condition C3). (D.21) As a result, the SM lepton asymmetry after the washout effects is n p,f δ = C n p,i δ .
(D. 22) For our N i mass spectrum given in Eq. (D.11), the decay process consists of two separate stages, which we will study in the following to find the desired C.
First Stage -decay of N 3 (m N 3 10 12 GeV ): The decay of N 3 washes out the pre-existing asymmetry in the direction of heavy neutrino lepton flavor |l 3 while leaves the component normal to it unchanged. The pre-existing asymmetry can be decomposed as n p δ = n p δ 3 + n p δ 3 ⊥ , (D. 23) where n p δ 3 (n p δ 3 ⊥ ) is the asymmetry parallel (perpendicular) to |l 3 . The above superposition sum is due to the linearity of the Boltzmann equations. The residual values of the primordial asymmetries are where A 0 3 is the tree-level probability of the primordial asymmetry to be in the direction of |l 3 .
Second Stage -decay of N 2 (10 12 GeV m N 2 10 9 GeV ): This stage of our post inflationary evolution can be effectively described by two SM flavors, i.e., (τ, τ ⊥ = e + µ) and two relevant flavors of the sterile neutrinos (N 2 , N 3 ). At temperatures 10 9 GeV < T < 10 12 GeV , τ lepton-Higgs interactions are fast and destroy the coherence of the lepton states produced by N i decay. Thus we need to consider separate Boltzmann equations for the components parallel and orthogonal to τ , i.e. n p δ 3 = n p δ 3τ + n p δ 3τ ⊥ , (D. 25) in which n p δ 3τ = A 0 3τ n p δ 3 and n p δ 3τ ⊥ = (1 − A 0 3τ )n p δ 3 , (D. 26) where the probabilities A 0 iτ (i = 1, 2, 3) are given in terms of the flavored decay parameters as From that we can define n p δτ ≡ n p δ 3τ + n p δ 3 ⊥ τ and n p δ τ ⊥ ≡ n p δ 3τ ⊥ + n p δ 3 ⊥ τ ⊥ . At temperatures T ∼ m N 2 , the N 2 wash-out processes act on the flavored asymmetries. The final residual asymmetries in the end of its decay process is n p,f δ τ 2 = A 0 2τ e − 3π 8 K 2 n p δτ and n p,f δ τ 2 ⊥ = (1 − A 0 2τ )n p δτ .
(D.31) Similar relations hold for τ ⊥ = e + µ. Fig. 12 shows the geometrical structure of the flavor effects in the flavor space. The final residual asymmetry in the SM lepton frame is n p,f δτ = A 0 2τ e − 3π 8 K 2 n p δτ + (1 − A 0 2τ )n p δτ , (D.32) (D.33) Considering the most conservative assumption that the decaying terms experience strong washout effects and are negligible, the final remnant of the primordial (inflationary) asymmetry is n p,f δ = n p,f δτ + n p,f δ τ ⊥ = C n p,i δ , (D. 34) where C is 36) vs A 0 2τ and A 0 3τ where the dark shaded area denotes regions with A f < 0.1. As we see, in most of its parameter space, A f is close to one with an average value as

Fig. 15 presents
Given that our inflationary primordial asymmetry is flavor blind, it is a plausible assumption to consider A 0 3 = 1 3 . For typical values of flavored decay rates, the remnant of the primordial asymmetry is significant which is related to the inflationary asymmetry as Interestingly, eliminating the effect of this pre-existing asymmetry requires tightly finetuned relations between the flavored decay rates, hence on leptonic Yukawa couplings, and the flavor-space direction of the inflationary asymmetry. More precisely, one needs either i) |l inf coincides with one of |l 2 and |l 3 , or ii) |l 2 and |l 3 are perpendicular to each other which |l inf is in the plane of |l 2 − |l 3 . As a result, the relation presented in Eq. (D.38) is a good estimate for most of the possible flavor parameter space.