SU(2)$_R$ and its Axion in Cosmology: A common Origin for Inflation, Cold Sterile Neutrinos, and Baryogenesis

We introduce an axion-inflation model embedded in the Left-Right symmetric extension of the SM in which $W_R$ is coupled to the axion. This model merges three milestones of modern cosmology, i.e., inflation, cold dark matter, and baryon asymmetry. Thus, it can naturally explain the observed coincidences among cosmological parameters, i.e., $\eta_{B}\approx P_{\zeta}$ and $\Omega_{DM} \simeq 5~\Omega_{B}$. The source of asymmetry is spontaneous CP violation in the physics of inflation, and the lightest right-handed neutrino is the cold dark matter candidate with mass $m_{N_1}\sim 1~GeV$. The introduced 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. It has two unknown fundamental scales, i.e. scale of inflation $\Lambda_{\rm inf}=\sqrt{HM_{Pl}}$ and left-right symmetry breaking $\Lambda_{F}$. Sufficient matter asymmetry demands $\Lambda_{\rm inf}\approx\Lambda_{F}$. The baryon asymmetry and dark matter today are remnants of a pure quantum effect (chiral anomaly) in inflation, which, thanks to flavor effects, are memorized by cosmic evolution.

The two pillars of the post-inflationary scenarios of leptogenesis are: i) CP asymmetric decay of massive Righthanded neutrinos (RHN) after reheating, and ii) washout processes to enhance the efficiency and eliminate the preexisting asymmetry to avoid theoretical uncertainties [1]. Besides, the lightest sterile neutrino may account for the Dark Matter (DM) [2]. The source of CP asymmetry is the CP-violating phases in the neutrino sector, unconstrained by the current data, which are assumed to be large enough. Moreover, low-scale leptogenesis mostly requires fine-tuning of parameters, e.g. highly degenerate RHN masses, e.g. m N3 m N2 [3]. Besides, once flavor effects are considered, it is difficult for a pre-existing asymmetry to be washed out by the RH neutrino decays [4] ( See also Fig. S3).
This letter introduces a new framework for simultaneous baryogenesis and dark matter production within General Relativity (GR), which avoids the above issues. The source of asymmetry is spontaneous CP violation by a W R gauge field coupled to the inflaton that produces leptons and baryons in inflation. In this scenario, baryon asymmetry and DM are remnants of the same effect in inflation. Thus it can naturally explain the observed coincidences among cosmological parameters, i.e., η B ≈ P ζ and Ω DM 5Ω B .
Early universe physics is a subject that seeks answers for fundamental questions linking the very high energy physics (immensely small scales) with the extremely large cosmological scales. The Standard Model (SM) of particle physics, highly successful in formulating fundamental particles at low energy scales, is greatly incomplete when it meets cosmology and astrophysics. The most glaring shortcomings of the SM are (I) the neutrino mass, (II) baryon asymmetry of the Universe (BAU), and (III) particle nature of dark matter. Considering cosmic inflation as the leading paradigm for early universe [5], we should, as well, add (IV) the particle nature of the inflaton field to this list. SM as a theory for particle physics also faces a number of issues, i.e., (i) Higgs vacuum stability problem, (ii) accidental B − L global symmetry, and (iii) ad hoc parity violation at the Electro-Weak scale (EW).
FIG. 1. SU (2)R-axion inflation: a natural common origin for inflation, fermionic dark matter, and matter asymmetry and its observational signatures.
Axion fields are well-motivated candidates for inflaton and are naturally coupled to gauge fields. As first discovered by the author, non-Abelian gauge fields may survive inflation and contribute to its physics while respecting the cosmological symmetries [6]. That introduced a new class of inflation models accompanied by SU (2) gauge fields with an immensely rich phenomenology. The minimal realization of this idea is an SU (2) gauge field in GR coupled to a generic axion inflaton [6][7][8], so-called SU (2)-axion inflation. For a review on gauge fields in inflation, see [9]. The novel features shared by these models are (1) Spontaneous P and CP violation, and satisfying all Sakharov conditions in inflation [10], (2) Particle production by the gauge field in inflation through Schwinger effect [11][12][13] and chiral anomaly [13], (3) Naturally warm inflation [14], (4) Prediction of chiral and non-Gaussian gravitational wave (GW) background [9,15,16] detectable by future CMB missions and laser interferometers [17]. Therefore, this inflation setup merges three open issues of the SM and cosmology, i.e., inflation, DM, and BAU. It gains an additional value due to its unique observable signature on GW background induced by GW-SU (2) field interactions. The connection of this SU (2) field with the SM, however, was still missing, and in the present work, we attempt to fill the gap.
The aim of this letter is to embed the SU (2)-axion inflation setting in gauge extensions of the SM and study its phenomenological and cosmological consequences. The most well-motivated Beyond the SM (BSM) theories are SUSY, GUT, and Left-Right Symmetric Models (LRSM) of the weak interactions. We restrict the current work to the most minimal realization, i.e., LRSM. Originally proposed to explain P violation in low energy processes [18], LRSM predicted massive neutrinos years before experiment. Among its additional appealing features are: natural B − L symmetry [19], entailed seesaw mechanisms [20], and solution to vacuum stability problem at high scales [21].
In this letter, we assume that the gauge field in the SU (2)-axion inflation models is W R in the LRSM. Comparing with the minimal LRSM, here we have an axion ϕ, which is coupled to the SU (2) R and drives cosmic inflation. We call this particle physics model for inflation SU (2) R -axion inflation. This model is a complete setup that can simultaneously provide plausible explanations for the phenomena (I-IV) and (i-iii) named earlier. A more detailed analysis is presented in a followup work [22]. The present paper can be a starting point for further, more involved analysis of the rich and multifaceted phenomenology of these gauge extensions of the SM in inflation physics.

SU(2) R -Axion Inflation Model
The minimal gauge group that implements the hypothesis of left-right symmetry is G ≡ SU (2) L × SU (2) R × U (1) B−L (suppressing color). The theory includes three gauge fields W L,R and A B−L associated with SU (2) L,R and U (1) B−L respectively. The fermionic content is the SM quarks and leptons extended by three RHNs as where ν iR are three RHNs interacting by SU (2) R . It is accompanied by an extended Higgs sector consists of a Higgs bi-doublet Φ, and SU (2) L,R triplets ∆ L,R . The Spontaneous Symmetry Breaking (SSB) structure of the LRSM is Below the scale Λ F , the first SSB happens which breaks the LR symmetry and gives a VEV to the SU (2) R triplet, i.e. ∆ R = 0. At this point, W ± R , Z R , and N i ≡ ν i + ν c i become massive. Next, when the temperature gets below EW scale, T < Λ W , the Higgs bi-doublet acquires a VEV, i.e. Φ = 0, and second SSB occurs which gives Dirac mass to the SM particles, active neutrinos included [20]. Now we add the inflaton, i.e. axion field, which is coupled to the W R . As a concrete example we consider where W Rµν is the strength tensor of For the sake of generality, we assume V (ϕ) is an arbitrary axion potential, flat enough to support the slow-roll inflation. For instance an axion monodromy inspired potential form [23]. This SU (2)-axion inflation model and its cosmic perturbations has been studied in [8]. The SU (2) R -axion inflation has two unknown fundamental scales, i.e., the scale of inflation Λ inf = √ M Pl H, and LR symmetry breaking Λ F . Moreover, W R may or may not have a VEV in inflation. Thus, we can distinguish four different types of scenarios.
Scenarios I and I v describe the case Λ inf > Λ F , while II and II v when Λ inf < Λ F . The v subscript denotes systems in which the SU (2) R acquires a VEV in inflation. In this work, we focus on scenarios I and II, W R = 0, and leave I v and II v cases for future works. The RH fermions are coupled to the W R field and its axion as whereλ is a constant, D µ is the spinor covariant derivative, and RH fermions are collectively shown as

Particle Production in Inflation
Due to conformal symmetry, the gauge fields associated with SU (3) c × SU (2) L × U (1) B−L group, as well as all the left-handed fermions, are exponentially decaying in inflation. However, the W R associated with SU (2) R is coupled to the inflaton and sourced by it. Subsequently, the generated SU (2) R gauge field produces RH fermions. Note that the axion cannot create Weyl fermions [30] and they are merely produced by W R . The main particle physical consequences of this setup as the inflation physics are: i) P and C are maximally broken by the chiral nature of the SU (2) R interaction with the axion, ii) CP , B, and L are all violated by the non-perturbative effects of the W R , i.e. chiral (Adler-Bell-Jackiw) anomaly [24], iii) B − L is conserved (violated) in scenario type I (type II), while B − L SM is violated in both scenarios, and iv) out of thermal equilibrium condition holds during inflation. Thus, all the Sakharov conditions required for a BAU [25] are satisfied in inflation. The field equation where g R is the gauge coupling of W R . A massless gauge field (type I) with momentum k has two (transverse) polarization states specified by the polarization vectors e ± (k) where k.e ± (k) = 0. The massive gauge field (type II) has an extra (longitudinal) mode with polarization vector e 3 (k) = k/k and its zero component coupled to it given by the constraint equation. Interestingly, the longitudinal mode and the zero component are decoupled from the axion and decay in inflation. Thus we neglect them and refer the interested reader to [22] for the detailed calculations. Now we define the following slowly increasing parameters Imposing the Bunch-Davies vacuum, the transverse modes f a ± , associated with e ± (k) polarization states, are where τ is conformal time, W κ±,µ is the W -Whittaker function, κ ± = ∓iξ, and it has power-low behavior and softly increases with the increase of m W R (See Fig. S1). During slow-roll, ξ is an almost constant (gradually increasing) parameter. As a result, the axion slowly injects more and more energy into the gauge field sector, and inflation is warm.
The generated gauge boson field produces RH leptons and baryons in inflation. The anomaly of baryon and lepton currents are respectively as However, the B and L violating interactions of the lefthanded fermions remains negligible in inflation. The total lepton number is related to the SM one as where n Ni are the sterile neutrino lepton numbers. Using Eq. (8) in (10), we find the baryon and lepton numbers respectively as where K(ξ, m W R ) is the contribution of chiral anomaly (a pure quantum effect) and D(ξ, m Ni ) is the contribution of the mass term of RHNs (in type II scenarios). The explicit forms of these prefactors are presented in Eq.s (S2) and (S3), and their plots are shown in Fig. S2. The prefactor K(ξ, m W R ) increases (decreases) with the increase of ξ (m W R ) and for ξ > M W R as The prefactor D(ξ, m Ni ) is of order one, and symmetric wrtξ. Net B − L asymmetry (L ≡ L SM ) created by inflation is

Evolution after Reheating
The study of the post-inflationary evolution requires to specify our parameter space further. For the sake of concreteness, we restrict the current analysis by assuming the following conditions: 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, and with feeble Yukawa interactions, i.e., a DM candidate. C2) The CP-violating phases in the neutrino sector, unconstrained by the current data, are not enough to create the observed BAU. C3) The post-inflationary generation of RHNs with W R interactions via freeze-out and freeze-in is negligible compared to their pre-existing relics. N 3,2 decay to lighter particles, while N 1 freezes out soon after inflation. Due to its feeble Yukawa interactions it can account for the DM with a relic number density as Photon number density: Reheating starts at some point after the end of inflation. Here, we consider the phenomenological reheating model ρ reh = ε a inf a reh 4 ρ inf in which ε is the efficiency of the reheating process and relates ρ reh and the energy density at the end of inflation, ρ inf (See S3). Thus the photon number density today (g eff,0 = 43 11 ) is Note that, due to condition C3, entropy injection by the decay of heavier RHNs after reheating is negligible. Demands imposed by C3: Negligible freeze-out and freeze-in production of RHN by W R interactions requires where A = 1 2 g eff 10 2 1 2 is of the order one. C3 imposes an upper bound on the scale of inflation. For the details of the freeze-in production in this setup see [22].
Baryon to Photon Ratio: today we have It is directly related to the amplitude of the primordial curvature power spectrum P ζ (k) = P ζ (k 0 )( k k0 ) ns −1 as where is the slow-roll parameter, η B 6 × 10 −10 , and P ζ (k 0 ) ≈ 2 ×10 −9 [26]. The scale of inflation, then, is Combining Eq.s (20) and (23), we find the allowed parameter space. Interestingly, it demands Λ F ≈ Λ inf , i.e. the LR SSB should coincide with the geometrical transition that ends inflation. More precisely, we need HM Pl ≈ g −2 R m 2 W R . Moreover, the values of these parameters are within the natural range of parameters in GUT theories [22]. For instance with a ξ ∈ (2, 4) and ∆N ≡ ln( a reh a inf ) 2, we find 1 Cold Dark Matter Relic Density: The number density of N 1 neutrino today is n N1,0 2.8 n B,0 . If it makes all the DM today, its mass is where m P is the proton mass. Since the production mechanism is independent of the active-sterile mixing angles, N 1 can have a lifetime much larger than the age of the Universe. Nevertheless, via its loop-mediated radiative decay channel, it can decay to gamma-ray photons with energy E γ ≈ m N1 /2 [31]. Thus, it may provide observable effects to be probed by gamma-ray telescopes.

Discussion
This letter introduced the SU (2) R -axion inflation model embedded in the Left-Right symmetric extension of the SM. It is a new framework for simultaneous baryogenesis and darkgenesis. One of the most well-studied leptogenesis scenarios with new gauge interactions is the LRSMs [27,28]. Let us explore the differences between previous studies and the current proposal. The scenarios so far discussed in the literature rely on unconstrained CP-violating phases in the neutrino sector. (a) a relic abundance of RHNs is generated after reheating by W R interactions via freeze-out or freeze-in mechanisms. (b) the asymmetric decay of RHNs, then, creates matter asymmetry. As an alternative mechanism, the current proposal set parameters such that phenomena (a)-(b) named earlier are negligible. The source of asymmetry is spontaneous CP violation in the physics of inflation, and the lightest right-handed neutrino is the cold dark matter candidate. Relic abundances of SM leptons, baryons, and RHNs are generated by the chiral anomaly of W R in inflation. Sufficient asymmetry does not require finetuned masses for the heaviest right-handed neutrinos, but it demands Λ inf ≈ Λ F . Therefore this new framework relates the scale of SU (2) R ×U (1) B−L breaking with the end of inflation and prefers scales above 10 10 GeV . Interestingly, it is in the range suggested by the nonsupersymmetric SO(10) GUT with an intermediate leftright symmetry scale. Due to the common origin of inflation, cold dark matter, and baryon asymmetry it can naturally explain the observed coincidences among cosmological parameters, i.e., η B ≈ P ζ and Ω DM 5 Ω B with m N1 ∼ 1 GeV .
Acknowledgments: The author would like to thank Eiichiro Komatsu for insightful discussions and valuable input during previous collaborations on which part of the present work is based. She thanks Marco Drewes Supplemental Materials: SU(2) R and its Axion in Cosmology: A common Origin for Inflation, Cold Sterile Neutrino, and Baryogenesis

Azadeh Maleknejad Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
Here we merely discuss some essential calculations. A more detailed analysis is presented in [22].

S1. PARTICLE PRODUCTION IN INFLATION
The prefactor T (ξ, m W R ) in Eq. (9) is whereτ = k aH and κ ± = ∓iξ and Fig. S1. The prefactor K(ξ, m W R ) in Eq.s (13) and (14) is the following momentum integral with the same µ and κ σ values as Eq. (S1). K(ξ, m W R ) is shown in the left panel of Fig. S2. Moreover, the prefactor D(ξ, m Ni ) in Eq.s (14) is the contribution of axion in the production of massive sterile neutrinos (in type II scenarios) in inflation. More precisely, apart from W R , the massive fermions can get generated by the background axion field as well. However, the axion can not create chiral (massless) fermions. The reason is that a Peccei-Quinn type U P Q (1) [29] rotation of fermions as Ψ → e − iλ f ϕ Ψ, removes the fermion-axion interaction and simply transforms the fermion mass term as [30] m Ψ → e 2iλ f ϕ m Ψ . At the leading order, this effect can be captured as [22] wheren Ni is the the number density of massive RHNs generated by the background and the bar emphasises that, unlike chiral anomaly, it is a classical effect. This calculation is more involved and is done analytically by the author in [22]. Here we show D(ξ, m Ni ) in the right panel of Fig. S2. For |ξ| > 1, it has the following asymptotic forms

S2. SPECTATOR EFFECTS IN LEPTOGENESIS ERA
Throughout the Early Universe, particles experience a whole cascade of interactions that eventually equilibrium in the Early Universe. Many of them can potentially redistribute the initial asymmetries to the spectator degrees of freedom. The spectator effects in this scenario are studied in [22]. Here we present a summary of these effects, i.e., sphaleron processes and lepton flavor effects.

Flavor Effects:
Due to a CP-violating source, by the end of inflation we have a lepton quantum state |l inf as where C inf α are specified by physics of inflation. The composition of this primordial initial leptons and their CP conjugated anti-leptons are different. Moreover, the CP violating decays of the heavy sterile neutrinos can modify these initial states. The leptons produced in N i decays can be described in terms of quantum states denoted as |l i that can be decomposed in SM flavor space as where C iα are coefficients given by the leptonic Yukawa matrix. Note that |l i s do not form an orthonormal bases, i.e. in general l i |l j =i = 0. The processes has two stages of decay and wash-out, one for each of N 3 and N 2 . Given the mass hierarchy considered in (17), at second stage with T < 10 12 GeV , the τ -lepton Yukawa interactions are thermalized. Hence the evolution can distinguish between τ flavor and τ ⊥ = e + µ. That breaks the initial coherency between components parallel and orthogonal to τ which demands separate Boltzmann equations for each. The decay of N i washes out the pre-existing asymmetry in the direction of heavy neutrino lepton flavor associated with its decay, i.e., |l i . while leaves its perpendicular components, i.e. |l i ⊥ , unchanged. The geometry of this evolution in the SM flavor space is illustrated in Fig. S3. After the second stage of RNH decays with T < m N2 , the remnant asymmetry which remains unchanged by both washout effects is |l inf 3 ⊥ 2 ⊥ . This flavor-vector consists of two incoherent components in τ and τ ⊥ directions. Interestingly, eliminating the effect of this pre-existing asymmetry requires tightly fine-tuned relations between the flavored decay rates, hence on leptonic Yukawa couplings, as well as the flavor-space direction of the pre-existing lepton 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 and |l inf is in the plane of |l 2 − |l 3 . Given that |l i and |l inf are specified by different physical parameters, these fine-tuning assumptions are extremely unnatural. In other words, the pre-existing asymmetry is memorized by cosmic evolution. In most of the parameter space, the remnant asymmetry, n p,f , is significant, i.e.
n p,f where n p,i is the primordial value of asymmetry. Note that total elimination of the pre-existing asymmetry requires highly fine-tuned 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 |l2 and |l3 , or ii) |l2 and |l3 are perpendicular to each other which |l inf is in the plane of |l2 − |l3 .

Sphaleron Effects:
Finally the SU (2) L,R sphalerons reshuffle the asymmetry of left-/right-handed leptons and quarks. In our setup the right-handed sphalerons are never in thermal equilibrium. Using the weak 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. The combination of Eq.s (S7)-(S9) relates the final asymmetry to its primordial value as (S11)

S3. PHENOMENOLOGICAL MODEL OF REHEATING
Reheating starts at some point, a reh , after the end of inflation, a inf . Yet, the precise physics of reheating is not well understood. Here to quantify our analysis, we use a phenomenological model for reheating. 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 them (See Fig. S4). In that case, the energy density of reheating is related to ρ inf as ρ reh ≈ a inf a reh 3(1+w X ) ρ inf . (S12) We can write the above relation in the following phenomenological form where ε is the efficiency of the reheating process given as ε ≈ a inf a reh 3w X −1 .
(S14) From the combination of Eq.s (20) and (23), we find where A = O(1), and ∆N is the number of e-folds between end of inflation to reheating, i.e.
a inf a reh = e −∆N . (S16) The relation (S15) provides the parameter space in which the freeze-out and freeze-in production of RHNs after inflation is negligible (condition C3), while the remnant of the primordial asymmetry has the right baryon to photon ratio.