Dark Matter and Baryogenesis in the Fermi-Bounce Curvaton Mechanism*

Chinese Physics C Vol. 42, No. 6 (2018) 065101

Dark matter and baryogenesis in the Fermi-bounce curvaton mechanism *

Andrea Addazi1;1) Stephon Alexander2;2) Yi-Fu Cai(éÅ)3;3) Antonino Marcian`o1;4) 1 Department of Physics & Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China 2 Department of Physics, Brown University, Providence, RI, 02912, USA 3 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, China

Abstract: We elaborate on a toy model of matter bounce, in which the matter content is constituted by two fermion species endowed with four fermion interaction terms. We describe the curvaton mechanism that is thus generated, and then argue that one of the two fermionic species may realize baryogenesis, while the other (lighter) one is compatible with constraints on extra hot dark matter particles.

Keywords: dark matter, matter bounce, fermi bounce, baryogenesis PACS: 98.80.-k, 98.80.Cq, 11.30.Fs DOI: 10.1088/1674-1137/42/6/065101

1 Introduction For a detailed introduction to the generation of scale- invariant perturbations we refer to Ref. [1], while for a The matter bounce scenario is an alternative to infla- recent review on the status of matter bounce cosmologies tion that fulfills the same observational constraints, but we refer to Ref. [2]. carries definite novel predictions about CMB observables Similarly to inflation, simple realizations of the mat- to be measured in forthcoming experiments. In this re- ter bounce scenario have been developed that deploy gard the matter bounce scenario is distinguishable from scalar matter fields, whose potentials are chosen ad inflation. Scale-invariant perturbations are generated in hoc so as to reproduce a vanishing pressure during the a contracting cosmology, which is then thought to be con- matter-dominated phase of contraction of the universe nected to the current phase of expansion of the universe [3]. Differently from inflation, observations allow us to thanks to the emergence of a non-singular bounce in the rule out the matter bounce scenario with a single scalar dynamics. This is the theoretical peculiarity of matter field [4]. Indeed, single scalar field matter bounce mod- bounce models with respect to inflationary models, as els predict an exactly scale-invariant spectrum, while the the cosmological singularity is solved, and completeness actual observed spectrum has a slight red tilt with a spec- of geodesics is restored. tral index of n =0.968 0.006 (65%) [5], and a tensor-to- s § Cosmological perturbations are also dealt with pecu- scalar ratio r significantly larger than the value allowed liarly in each of these frameworks. In inflation the differ- by the observational bound r<0.12 (95%) [6]. ent dynamical evolutions of the causal horizon and Hub- Nonetheless, there are few instantiations of the mat- ble horizon are at the origin of the generation of scale- ter bounce scenario that predict a slight red tilt in the invariant Fourier modes that reenter the horizon. In the spectrum of scalar perturbations and fulfill the con- matter bounce scenario it is during the phase of matter- straints on the tensor-to-scalar ratio [7–9]. The mecha- dominated contraction that Fourier modes of the co- nisms that are usually considered for this purpose hinge moving curvature perturbation become scale-invariant. on the inclusion of additional matter fields [3, 10], on the

Received 15 January 2018, Revised 11 April 2018, Published online 11 May 2018 ∗ The work of AA was partially supported during this collaboration by the MIUR research grant Theoretical Astroparticle Physics PRIN 2012CPPYP7 and by SdC Progetto speciale Multiasse La Societ`a della Conoscenza in Abruzzo PO FSE Abruzzo 2007-2013. The work of YFC is supported in part by the Chinese National Youth Thousand Talents Program (KJ2030220006), by the USTC start-up funding (KY2030000049), by the NSFC (11421303, 11653002), and by the Fund for Fostering Talents in Basic Science of the NSFC (J1310021). AM wishes to acknowledge support by the Shanghai Municipality, through the grant No. KBH1512299, and by Fudan University, through the grant No. JJH1512105 1) E-mail: [email protected] 2) E-mail: stephon [email protected] 3) E-mail: [email protected] 4) E-mail: [email protected] ©2018 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

065101-1 Chinese Physics C Vol. 42, No. 6 (2018) 065101 choice of a matter field that has a small sound speed (so In Section 4 we review the curvaton mechanism for a as to enhance the amplitude of vacuum fluctuations) [8], Fermi bounce cosmology that accounts for two fermionic and finally on the suppression of the tensor-to-scalar ra- species. In Section 5 we deepen the phenomenological tio during the bounce, that is explained due to quantum consequences that can be derived for CMB observables, gravity effects [11]. and comment on the falsifiability of this scenario with re- Here we will follow a different theoretical perspective, spect to the introduction of dark matter. In Section 6 we closer to the intuition developed in particle physics. In- study the application of this curvaton model to leptoge- deed, we intend not to deploy exotic matter fields, or nesis, and comment on the phenomenological constraints matter fields that have not been observed yet in terres- that can be inferred from the data. Finally, in Section 7 trial experiments, and not to resort to quantum gravity we consider some outlooks and conclusions. effects, extending our framework up to the Planck scale. In a more conservative fashion we rather consider here 2 The matter bounce scenario matter fields that belong to the Standard Model (SM) of particle physics, and that correspond to the simplest and It is nowadays common knowledge that FLRW met- most conservative extensions of it, so as to encode dark rics suffer from singularities in all the curvature invari- matter in the picture we will develop. And following ants. It was already remarked by Hawking and Pen- the particle-physics intuition that a definite energy scale rose [20] that the initial singularity is unavoidable if will correspond to definite physical degrees of freedom, space-time is described by General Relativity and mat- we assume as in Ref. [9] that both the energy scale and ter undergoes null energy conditions (NEC). Many non- the matter content of the universe during its contracting singular bouncing cosmologies have since been developed phase are comparable to that of the present universe, in order to solve the Big Bang singularity issue, but at which brings us to consider the importance of dark mat- the cost of dismissing some of the assumptions behind ter during the pre-bounce matter phase contraction of the Hawking-Penrose theorem, most notably NEC. the Universe. Bouncing mechanisms can be implemented within We wish to remark that bouncing cosmologies involv- frameworks very different from one another. A complete ing dark matter (and dark energy) have recently received review, comprehensive of all the bouncing models devel- much attention in the literature, and that distinctive and oped hitherto, would be too long to give in this paper, falsifiable predictions on CMB observables have been de- turning far from our current purpose of focusing on a rived that will be tested in the near future [9, 12–18] (for model of bounce cosmology that accounts for dark mat- a recent review see also Ref. [19]). With respect to this ter and only involves fermionic matter fields. Nonethe- vast literature the gist of our proposal relies on the de- less, before focusing on fermionic matter bounce models ployment of fermionic matter fields. and their instantiations able to encode dark matter, we Specifically, we develop here a toy model in which wish to briefly survey the landscape offered within the both matter and dark matter are described by fermionic literature, and highlight some paradigmatic cases that fields, the dynamics of which are governed by the Dirac have received much attention. action on curved space time, and a four-fermion inter- The bouncing behavior of the universe at early time action term. The latter term is actually due to the res- can be reconstructed from high-energy theory corrections olution of the torsional components of the gravitational to the effective equation of motion of the gravitational connection with respect to fermionic bilinears, and must field. It is then worth mentioning that quantum the- be accounted for in the first order formalism. We then ories of gravity, as well as effective models inspired by implement a curvaton mechanism, in which the fermion the problem of quantum gravity, have driven many au- field with lighter mass is responsible for the generation thors to efforts in this sector. For this purpose, a charac- of almost scale-invariant curvature perturbation modes, terization of the bouncing mechanisms inspired by loop and the heavy mass field drives the dynamics of the back- quantum gravity and its cosmological applications — ground. We then argue that while the light fermion field loop quantum cosmology — has been outlined in detailed can be assumed to be a neutrino, the heavy fermion analyses [7, 21]. field can be related to the sterile neutrino, and hence On the other side, there exists a flourishing literature by decaying into the lighter neutrinos can accommodate that takes into account bouncing models from the point baryogenesis through leptogenesis. of view of string theory, for a complete review of which We start in Section 2 by differentiating our approach we refer to Refs. [22–24] as preliminary introductions. from the many others present within the literature. In The so called Hoˇrava-Lifshitz proposal can also achieve Section 3 we then review the instantiation of the mat- a bouncing phase for early time cosmology, as empha- ter bounce mechanism that deploys one fermionic field, sized in Ref. [25], while the contiguity of the bouncing which from now on we will call Fermi bounce cosmology. scenario of f(R) and Gauss-Bonnet theories can be read

065101-2 Chinese Physics C Vol. 42, No. 6 (2018) 065101 out respectively from Refs. [26] and [27]. ECHSK action via the Cartan equation (see e.g. [38]), Nevertheless, the bouncing scenario does not neces- which is instead recovered by varying the ECHSK ac- sarily require (quantum) gravitational corrections to the tion with respect to the spin-connection ωIJ . One fi- energy density, but instead many studies deploy fields nally finds that the total action STot =SEH+SDirac+SInt that violate the null energy condition in order to achieve in which the Einstein-Hilbert action SEH and the Dirac the bounce. Among many examples that can be pointed action SDirac are cast in terms of metric compatible vari- out, we may cite the ghost condensate scenario [28], the ables, and an additional term SInt is present, which en- so-called Fermi bounce mechanism [30, 31], and the Lee- codes a four-fermion interaction potential. Specifically, Wick theory [32]. Because of their peculiarity of result- the Einstein-Hilbert action, in terms of the metric coming from known theories of particle physics, which have patible variables ω˜(e)IJ , reads been corroborated on the flat gravitational background 1 by means of high-energy terrestrial experiments, we will S = d4x e eµeν RIJ , EH 2κ | | I J µν focus in the next section on the Fermi bounce models. ZM while the Dirac action SDirac on curved space-time, once 3 One-field Fermi bounce cosmology the covariant derivative with respect to the torsionless connection has been denoted with µ, is recast as The action for the matter-gravity sector under ∇ 1 4 I µ scrutiny results from the sum of the gravitational SDirac = d x e ψγ eI ı µeψ mψψ +h.c.. 2 M | | ∇ − Einstein-Hilbert action, further endowed with a topolog- Z ³ ´ ical term a` la Holst, plus a non-minimal covariant Dirac The interaction four-fermion potene tial is action. Following previous literature [33], we refer to this 4 L M theory as the Einstein-Cartan-Holst-Sciama-Kibble the- SInt = ξκ d x e J J ηLM , − | | 5 5 ory (ECHSK). In the first order formalism, when gravity ZM is coupled to fermion fields, we must allow for a tor- in which the we have used the definition of the axial cur- L L sionful part of the spin-connection. Thus the ECHSK rent J5 = ψγ5γ ψ, and introduced a function ξ of the theory is necessarily a theory of gravity with torsionful real parameters α and γ, namely connection [34]. Nonetheless, a second order approach 3 γ2 2 1 is always possible [35], which adopts a torsionless (Levi- ξ:= 1+ . 16 γ2+1 αγ −α2 Civita) connection ω˜[e], and thus differs from the former µ ¶ first order treatment in that a four-fermion interaction Notice that the sign of ξ is crucial while discussing the term emerges. Notice however that the action written, physical applications in cosmology of the ECHSK action. no matter if cast in terms of a torsionful or Levi-Civita For instance, a positive value of ξ corresponds to a cos- connection, is invariant under diffeomorphisms and local mological Fermi liquid scenario in which the repulsive Lorentz transformations. potential is sustaining an accelerated phase of expan- From now on we will focus on the ECHSK theory, sion of the Universe [39]. Conversely, a negative value of which in the first order formalism reads ξ provides a violation of NEC, and hence determines a bounce in cosmological [30, 31] or astrophysical scenarios 1 4 µ ν IJ KL SHolst = d x e eI eJ P KLFµν (ω), — for instance in Ref. [38] it was shown that for a suit- 2κ | | ZM able choice of the parameter space region of the theory, in which black holes may never form. IJ IJ IL J It is worth commenting on the most common objec- Fµν (ω)=dω +ω ωL ∧ tion against this type of model, which concerns the even- IJ is the field-strength of ω , the Lorentz spin-connection, tual appearance of instabilities. For instance, it has been π κ=8 GN is the square of the reduced Planck length, and shown in Ref. [40] that some scalar field actions that vi- 1 olate NEC might hold ghost and/or tachyonic instabili- P IJ =δ[I δJ] ²IJ KL K L −2γ KL ties, which naturally suggests similar issues might arise in the Fermi bounce context. Nonetheless, despite the involves the Levi-Civita symbol ²IJKL and the Barbero– analysis of Ref. [40] being performed under very general Immirzi parameter γ. The Dirac action reads SDirac = 1 4 assumptions, it still relies heavily on the effective La- d x e Dirac, in which 2 | |L grangian being second or higher order in the space-time R 1 I µ ı derivatives, so the same conclusions cannot be easily ex- Dirac = ψγ eI 1 γ5 ı µψ mψψ +h.c., L 2 −α ∇ − tended to any generic action, nor to a fermionic action α R represenhting the³ non-minimal´ couplingi parameter. which is not quadratic in the canonical momenta. Fur- The∈ crucial observation [36, 37] is that the torsionful ther work is needed to show whether for the latter system part of the spin-connection can be integrated out of the the linearity in the canonical momenta prevents stability

065101-3 Chinese Physics C Vol. 42, No. 6 (2018) 065101 issues. For example, within the context of the Galileon a torsion-induced four-fermion interaction might yield a models, several examples in which the NEC can be con- non-singular bounce. Further developments include the sistently violated, without any instabilities, were found study of Ref. [52], deepening the role of a parity-violating in Refs. [41–43]. four-fermion self-interaction term within the framework Nonetheless, here we give a simple argument in fa- of a torsion-free theory. Production of scale-invariant vor of stability that deploys mean field approximation. scalar curvature perturbations has been finally investi- It is not difficult to show (for a detailed discussion, we gated in Ref. [31], for the case of one fermion species, refer the reader to Ref. [44]) that the four fermion in- and then extended in Ref. [30] to case of the curvaton teraction potential can be recast as a redefinition of the mechanism. mass. As a hint, we may consider to Fierz decompose the By considering a non-minimal coupling in the Dirac four-fermion potential, and then focus only on the lower action (see e.g. Refs [53–55]) and a topological term energy channel of the decomposition, which entails the for the torsionful components of the spin-connection densitized field χ = a3/2ψ, from which we can construct ωIJ (see for instance Ref. [37]), the inspection within bilinear perturbations, Refs. [30, 31] has considerably enlarged the parameter space of the fermionic theories previously examined in γI eµı m 2ξκ√ g χχ χ=0, I ∇µ− − − h i view of a bounce. This has allowed not only for a four- the brackets³ denoting the expectation válue of the back- fermion interaction which is regulated by the parame- e ground fermion bilinear in the mean field approxima- ters of the theory via the ξ function, but also for the tion. At every energy scale lower than the energy scale emergence of a scale-invariant power-spectrum. It is in- of the bounce, the effective mass will remain positive, or deed thanks to the presence of a torsion background that at most — as happens at the bounce — it will vanish. the topological term within the Holst action turns from This suggests there should not be any issue of instabili- a surface term into a contribution to the four-fermion ties that originate from the perturbed fields. interaction term. It is this latter term that entails an Any successful theory of the early universe must be almost scale-invariant power-spectrum of gravitational able to reproduce the observed nearly scale-invariant scalar perturbations. spectrum of adiabatic fluctuations in the CMBR. Scale Notice however that the four-fermion density modi- invariance has been investigated hitherto within the fies the Friedman equations to have a negative energy density that redshifts like a(t)6, thus the issues with framework of bouncing models with a contracting phase, ∼ such as ekpyrotic [45], string gas [46] and pre-Big Bang anisotropies are not yet solved in this scenario, when scenarios [47]. On the other hand, for a number of these we only take into account the tree-level contributions to models it has proven difficult to obtain adiabatic scale in- the energy density. Nonetheless, quantum corrections variant fluctuations in the contracting phase, mainly due to the effective action of fermion fields may provide an to issues in resolving the singularity or mode matching “ekpyrotic-like” contribution that redshifts faster than a(t)6, and is then able to wash out anisotropies when between contracting and expanding phases [47]. ∼ Nonetheless, Brandenberger and Finelli, and inde- the universe approaches the non-singular bounce [56]. pendently Wands [47, 48], have shown that a scale in- Finally, we should also mention an important issue, variant power spectrum can be generated in a matter- with relevant observational consequences for the obser- dominated contracting universe, proving the existence of vation of power spectra and cross-correlation functions some “duality” between the scale invariant power spec- of the CMBR. This has to deal with the semi-classical trum generated in the inflationary epoch and a contract- limit of fermion fields, and the appropriate way of deal- ing matter-dominated phase. During this latter phase, ing with objects that fulfil the Pauli exclusion princi- gauge invariant perturbations that cross the Hubble scale ple. Dirac fields indeed become physical observables turn out to be scale invariant because of the time be- that satisfy micro-causality only when they form bilin- havior of the scale factor — in the cosmological time, ears that belong to the Clifford algebra. We shall then al- a(t) ( t)2/3. It is also worth mentioning that at non- ways deal with these combinations of fields, while adopt- singular∼ −bounces, scale-invariant modes are matched to ing macroscopic states that represent coherent states in scale-invariant modes in the expanding phase. group theoretical meaning — coherent fermionic states A very seminal investigation on the role of fermion are SU(2) coherent states, known in condensed matter fields in cosmology was reported in Ref. [49], in which a as BCS states of superconductivity. We refer for this wide class of generic potentials of the Dirac field’s scalar discussion to the work developed in Ref. [57]. bilinear was considered, and a detailed scrutiny of dif- We close this section emphasizing that the advantage ferent cosmological scenarios was made available. The of the Fermi bounce mechanism mainly relies on the fact first analyses of the Fermi bounce mechanism then trace that it does not require the existence of any fundamental back to Refs. [50, 51], in which the authors realized that scalar field not observed through terrestrial experiments

065101-4 Chinese Physics C Vol. 42, No. 6 (2018) 065101 in order to drive the space-time background evolution. and finally the interaction terms of the theory read The fermionic field added to the gravitational action is sufficient to account both for the matter bounce scenario S = ξκ d4x e J L J M +2J L J M +J L J M η , (4) Int − | | ψ ψ ψ χ χ χ LM and the generation of nearly scale-invariant scalar per- ZM ¡ ¢ turbations. which only involve the axial vector currents Jψ and Jχ of the ψ and χ fermionic species, but in the three possible 4 Two-field curvaton mechanism combinations. For the two fermionic species the Dirac Lagrangians, In this section we review the curvaton mechanism provided with interactions, respectively read for Fermi bounce cosmologies, which was first studied in Tot 1 I µ Ref. [30]. We will deploy a similar strategy to the one ψ = ψγ eI ı µψ mψψψ +h.c. L 2 ∇ − outlined in Ref. [30] for the analysis of the curvature per- ξ³κJ L(J K +J K )η , ´ (5) turbations. Nonetheless, we are aware that, in order to − ψ ψe χ LK prove cosmological perturbations of fermionic fields to be and non-vanishing at the linear order, the procedure first de- Tot 1 I µ scribed in Ref. [57] must be implemented. We will then χ = χγ eI ı µχ mχχχ +h.c. L 2 ∇ − move consistently along the lines drawn in Ref. [57]. The ξ³κJ L(J K +J K )η , ´ (6) manipulation of the perturbed fermionic bilinear that we − χ χe ψ LK perform here will give at the end very similar results to with the energy-momentum tensors those in Ref. [30]. There are only a few differences that 1 concern the observable quantities for CMBR, namely the T ψ = ψγ eI ı ψ+h.c. g Tot , (7) µν 4 I (µ ∇ν) − µν Lψ scalar power spectrum and the tensor to scalar ratio parameter r, but these are not significant experimentally, and e and are completely due to the four-fermion interaction χ 1 I Tot Tµν = χγI e(µı ν)χ+h.c. gµν χ . (8) between the two fermionic species that we are taking into 4 ∇ − L account here. The background dynamicse of the fermionic bilinears Differently from the approach within Ref. [30], in must be solved along the lines of Ref. [57]. Nonethe- which four-fermion interaction terms were added for each less, the conclusions are here quite similar to what was of the fermionic species considered by simply following found in Ref. [30]. On the states of semiclassicality (re- a phenomenological recipe, the four-fermion terms we spectively) for the ψ-fermion field, namely the coherent focus on here follow directly from the ECHSK action. state αψ , and for the χ-fermion field, namely the coher- As a consequence, when two fermionic species are taken | i ent state αχ , we can easily recover (see e.g. Ref. [57]) into account, a novel four-fermion interaction term be- that on-shell,| i tween the two species arises. We will then show that n n once a mass hierarchy between the two species is con- ¯ ψ χ ψψ αψ = 3 , χχ¯ αχ = 3 , (9) sidered, the curvaton mechanism is again realized: the h i a h i a spacetime background evolution encodes a bounce, and in which the fermionic densities arise from the integra- a scale-invariant scalar power spectrum is generated. tion of the modes’ distributions of the coherent states, i.e. 4.1 The ECHSK action and background dynamics n = dµ(k) α (k) 2 , n = dµ(k) α (k) 2 , (10) ψ | ψ | χ | χ | We start directly from the action for gravity and Z Z dµ(k) denoting the appropriate relativistic measure on Dirac fermions in which torsion has been integrated out, the Fourier mode space. namely Using the Fierz identities, evaluating the product of S=SGR+Sψ+Sχ+SInt , (1) the fermionic bilinears on the coherent states, the first where again the Einstein-Hilbert action is written using Friedmann equation can be used, accounting for the con- IJ IJ tributions due to the two fermionic species, as follows: the mixed-indices Riemann tensor Rµν =Fµν [ω(e)], i.e. κ m n +m n κ2 (n +n )2 1 2 ψ ψ χ χ ψ χ 4 µ ν IJ H = 3 +ξ 6 , (11) SGR = d x e eI eJ Rµν , e (2) 3 a 3 a 2κ M | | Z in which the double product of fermionic densities in the the Dirac action Sψ on curved space-time is last term now accounts for the interaction between the

1 4 I µ two fermionic species. Sψ = d x e ψγ eI ı µψ mψψψ +h.c., (3) The scale factor of the metric is easily determined to 2 M | | ∇ − Z ³ ´ e 065101-5 Chinese Physics C Vol. 42, No. 6 (2018) 065101 be variable to

1 mχ δ χχ 2 3 ζ h i , (18) 3κ(mψnψ+mχnχ) ξκ(nψ+nχ) a= (t t )2 , ' mψ ψψ 4 − 0 −(m n +m n ) h i µ ψ ψ χ χ ¶ if it also holds that (12) and its value in t , when the bounce takes place, imme- mψnψ mχnχ . (19) 0 À diately follows, Therefore, as in Ref. [30], we proceed to write the 1 1 x 2 3 2 3 autocorrelation function for ζ(t, ) in the notation of ξκ(nψ+nχ) ξκ(nψ+nχ) a0 = . (13) Ref. [57] as −mψnψ+mχnχ ' − mψnψ µ ¶ µ ¶ 2 2 mχ δ χχχχ 4.2 Cosmological perturbations S ζ = h i , (20) P ≡P m2 4 ψψ 2 ψ h i Cosmological perturbations can be studied in the flat in which we are now assuming that: gauge, in which the curvature perturbation variable is 1) mψ >> mχ, which entails suppression at super- proportional to the perturbation of the energy density of horizon scale of the perturbations due to the ψ field — the system, wavenumbers of perturbations of ψ-bilinears are more δρ blue-shifted than perturbations of χ-bilinears; ζ= . (14) ρ+p 2) cross-correlation between perturbations of the ψ field and perturbations of the χ field are negligible, since Perturbations of the energy densities of the fermionic they are due to an interaction involving a graviton loop, species are linear in the perturbations of the fermionic the latter being suppressed by the fourth power of the bilinear. In Ref. [57] it has been shown that linear per- Planck mass Mp. turbations of the fermionic bilinear are non-vanishing. The perturbations to the χ field are then computed Thus also the curvature perturbation variable defined in resorting to the same kind of assumptions outlined in Eq. (14), linear by definition, is non-vanishing for fermion Refs. [30, 31], but following the procedure outlined in fields. Ref. [57]. The analysis we report below shows explicitly Note that within the formalism introduced in that at perturbative level the fluctuations that are re- Ref. [57], given a generic operator in the spinorial in- O covered are free of gradient and ghost instabilities, which ternal space, the n-th infinitesimal order expansion of often exist in other bouncing cosmologies. This refines the expectation value (on a quantum macroscopic coher- our previous heuristic argument given in Section 3. ent states) of the fermionic bilinear ψ ψ is defined by O First, we notice that away from the bounce the scale the expansion factor reads a(η) η2/η2, with ' 0 n 1/2 δ ( ψ ψ α ) n! α0 ψ ψ α0 , (15) η0 =[κ(mψnψ+mχnχ)]− , (21) ψ ψ ψ n h O i ≡ h | O | i O(δαψ ) ¯ which becomes, because of the requirement specified in ¯ in which the perturbation of the modes distribution¯ func- Eq. (19), 0 1/2 tion αψ αψ+δαψ+... has been considered. For simplicity η0 (κmψnψ)− . (22) of notation,' we will remove the subscript α , and de- ' ψ The dynamics of the perturbations of the χ-field bilinear, note perturbations of fermionic bilinears on the coherent which differently from Ref. [30] is now provided with a space simply as δ ψ ψ . four-fermion term interaction with the ψ field, can then If we now takehinOto accouni t the two fermionic species be found once the equations of motion for the field are with different values of the bare mass, we find that the solved: two main contributions to the variation of the energy I µ densities read γ e ı µ mχ 2ξκ χχ + ψψ χ=0, (23) I ∇ − − h i h i ³ ´ δρ=mχδ χχ +mψδ ψψ +..., (16) in which we haeve used the mean field approximation h i h i for the terms arising from the four-fermion interactions. having neglected contributions suppressed by ξκ. On Upon reshuffle of the equation of motion for the χ-spinor the other hand, similarly to Eq. (11), the denominator and densitization of its components we find of Eq. (14) becomes I µ 2 γ eI ı µ mχ 2ξκ√ g( χχ + ψψ ) χ=0. (24) mχnχ+mψnψ (nχ+nψ) ∇ − − − h i h i p+ρ= +2ξκ . (17) a3 a6 Deplo³ ying the background solution for the´ densities of e ee e e e Not surprisingly, at the zeroth order in ξκ a similar result the two fermionic species, this can be rewritten as to that in Ref. [30] is obtained. For values of mχ <

The appearance in Ref. (25) of the density numbers for far from the bounce, which ensures initial states for both the fermionic species is peculiar of the theory un- fermionic perturbations to be close to the vacuum fluc- der scrutiny here, derived directly from the ECHSK ac- tuations in Minkowski space. Eventual deviations from tion, and represents a main difference with respect to scale invariance of the perturbations (before exiting the the analysis in Ref. [30], especially for what concerns Hubble radius), which may arise along with the uni- CMBR phenomenology — the other main phenomeno- verse’s contraction because of the presence of the imagi- logical difference will be spelled out in the next section, nary part in Eq. (31), can be switched off by proper fine and concerns baryogenesis. tuning, requiring the imaginary part to be at most of the Following the recipe implemented in Refs. [30, 31], same order as the real one. we can find solutions for the spinorial components of the Finally, we can find solutions that interpolate among Dirac equation (25) in terms of the spinorial functions two different limits: 1) When the gradient term is dominant, namely at ˜ 1 k k f h = [u˜L,h( ,η)+u˜R,h( ,η)], sub-Hubble scales with kη 1, we impose the initial § √2 condition for the Fourier| mo| Àdes of the fermionic bilin- 1 g˜ h = [v˜L,h(k,η)+v˜R,h(k,η)]. (26) ear, which are the observable quantity, resorting to a § √2 Wentzel-Kramers-Brillouin (WKB) approximation. This These are recovered by rescaling densitized spinors up to yields u˜ = a3/2u and v˜ = a3/2v. Densitized spinors are in turn ˜ 4 mχ ikη expressed in terms of their chiral and helical components f h e− . (32) § ' 2k r u˜ (k,η) u˜(t,k)= u˜ (t,k)= L,h ξ , (27) This initial condition exactly coincides with the vacuum h k h h h Ã u˜R,h( ,η) ! fluctuations. Second, we study the asymptotic solution X X k to the perturbation equation in the limit of kη 1, v˜R,h( ,η) | | ¿ v˜(t,k)= v˜h(t,k)= ξh , (28) i.e. at super-Hubble scales. To apply the relation v˜ (k,η) 2 2 h h Ã L,h ! a(η) η /η and Eq. (22), one can write down the effec- X X ' 0 in which we have introduced the helicity 2-eigenspinor, tive mass term as kˆ written in terms of the unit vector , which is γ 2ξ(nχ+nψ)mχ 2 with γ= . (33) 1 h(kˆ ıkˆ ) −η − nψmψ ξ = x− y , kˆ σξ =hξ , (29) h ˆ ˆ · h h 2) When the (other) asymptotic solution of the equa- 2(1 hkˆ ) Ã ıkx hky ! − z − tion of motion has a leading term of the form q σ 2(1+γ √1+4γ) standing for the Pauli matrices. − ˜ 3 √1+4γ ˜ f h c(k)η − , (34) We can recast Eq. (25) in terms of the fh functions: § ' ˜ 2 ˜ c(k) being a k dependent coefficient that must be deter- f 00h+ω (k,η)f h =0, (30) § § mined by matching the above two asymptotic solutions in which we have introduced an effective frequency, de- (32) and (34) at the moment of Hubble crossing. fined by The asymptotic solution at super-Hubble scales is

2 2 2 2 mχ a0 therefore determined to be 0 ω (k,η)=k +mχa +ımχa +2ξκ(nχ+nψ) ı 3 . a a 1+γ √1+4γ − mχ − µ ¶ ˜ 4 3 √1+4γ f h (kη) − , (35) (31) § ' 2k r In this framework χ is a curvaton, and does not con- having now normalized the expectation value of a tribute to drive the dynamics of spacetime background. fermionic bilinear in the asymptotic future.

So the condition mχ mψ holds naturally, and the sec- After the contraction phase, the universe would even- ond term of the effectiv¿e frequency can then be neglected. tually enter the non-singular bouncing phase by avoiding For the third and the last term, which are imaginary, the spacetime singularity due to the help of the back- we proceed to smooth them out, by taking the time- ground fermionic condensate. During this phase, the evo- averaged evolution at super-Hubble scales. Thanks to lution of the Hubble parameter H can be approximated this consideration, the effective frequency will turn out as a linear function of the cosmic time t [3, 58]. Accord- to depend mainly on the gradient term k2 and the effec- ingly, the scale factor behaves roughly as a exp(t2). In ∼ tive mass term 2ξκ(nχ+nψ)mχ/a, in which the effect of this case, one can read that the evolution of the non- the new interaction between the two fermionic species is singular bounce can occur when the scale factor reaches evident. the minimal value. By inserting back into the pertur- Notice also that the imaginary part of the effective bation equation (30), the latter can be solved both nu- frequency can be suppressed in the contracting phase merically and semi-analytically. The procedure is similar

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to the analyses performed in Ref. [32] (see also Ref. [4] rived in Eqs. [5, 6], which sets ns =0.968 0.006, γ takes for the observational constraint on the growth of pri- values in the range § mordial fluctuations during the bounce). We can learn γ=2.048 0.009. (40) that, for a fast bounce with the energy scale much lower § than the Planck scale, the perturbations passing through Consistently, for this range of values the power spectrum the non-singular bouncing phase would not be much af- appears to be red-tilted. fected by the background evolution. In the present study, for simplicity we assume that the perturbations were al- 5 CMBR phenomenology and dark mat- most conserved during the non-singular bounce and can ter be smoothly inherited from the contracting phase to the expanding phase. In this section, we show how hot dark matter con- 4.3 Power spectrum of scalar gravitational per- straints can be satisfied in a heuristic but successful fash- turbations ion within the two-fermion-species toy model we have discussed so far. But before tackling hot dark matter We now consider all the necessary ingredients to cal- constraints we first focus on the phenomenological con- culate the power spectrum of the primordial scalar gravi- sequences of the toy model for CMBR observables. tational perturbations. Indeed, if we substitute Eq. (35) into Eq. (20), we find that the power spectrum of the 5.1 Constraints for the mass from CMBR primordial curvature perturbations is expressed by the We start by commenting on the production of pri- relation mordial gravitational waves, and hence on the testable

3 2 2 4(1+γ √1+4γ) mχ δαχ k − implications for the scalar to tensor ratio. We notice = | | (kη) 3 √1+4γ , (36) S 2 2 π2 − that the dynamics of primordial gravitational waves are P mψnψ 4 uniquely determined by the spacetime background dy- in which we have used Eq. (9). Exact scale invariance of namics, and that their evolution is decoupled from other the power spectrum corresponds to the value γ =2 (i.e. perturbation modes at linear order. Thus the deriva- ξ= nψmψ/[(nχ+nψ)mχ]). In this case, during the mat- tion of the tensor perturbations power spectrum follows − 2 ter contracting phase, the amplitude scales as η− and Ref. [3], and allows us to find the power spectrum generated in the fermion curvaton 2 mechanism is 1 E π T = 2 2H 2 , where ϑ=8 (2q 3)(1 3q). (41) 3 P ϑ aE Mp − − m δnχ 1 = χ , (37) S 2 2 π2 2 2 P mψnψ 4 a η The coefficient q is typically required to be less than unity, and is determined by the detailed procedure of in which we have assumed δn = δα 2 to retain only a χ | χ| the phase transition, since it represents a background mild, and phenomenologically negligible, dependence on parameter associated with the contracting phase. k. If we assume that the universe is evolving through the Evaluating Eq. (37) at the end of the matter con- bounce, and neglect for the moment its fermionic contracting phase tE , when the scale factor equals the value traction phase, we can estimate the maximal amplitude aE , enable us to recast it as of the Hubble rate to be of the same order of magnitude 3 2 mψ m δnχ H as . So at the fermionic matter-bounce phase the am- = χ E , (38) √ξ S 2 2 π2 plitude of the Hubble parameter cannot be bigger than P mψnψ 16 mψ √ξ , which allows us to approximate it with its maxi- in which we have used η = 2/ = 2/(a H ). In mψ E E E E mal value H , and find the corresponding power Eq. (38) we used the notation Hand a respectively E √ξ E E spectrum | |' for the values of the comoving HubbleH parameter and of 2 the scale factor at the end of matter contracting phase, 1 mψ T 2 2 . (42) just right before the time tE at which the phase transi- P ' ϑ ξ Mp tion takes place and perturbations, before reentering the | | Finally, the tensor to scalar ratio, which is by definition Hubble horizon, become constant. r / , is easily recovered to be Slight deviations of γ from 2 in Eq. (36) entail the ≡PT PS derivation of phenomenologically allowed relations for 16π2 m2 n2 r= ψ ψ , (43) the spectral index, i.e. 2 3 2 ϑ mχ δnχMp dln S 2 in which the condition of scale invariance γ 2 has been nS 1 P (γ 2). (39) − ≡ dlnk '−3 − assumed. This result is the same as was already' derived Consistently with the estimate of the spectral index de- in Ref. [30].

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15 3 Cosmological observations constrain the power spec- mψ =10 GeV and mχ =10− eV. For these parameters, 9 7 trum of scalar perturbations to be S 2.2 10− [59], the r-parameter rapidly converges to r 10− in a time P ' × 12 31 ' and set an upper bound on the detection of primordial t 10 tPl 10− s, which is close to the reheating time gravitational waves, allowing for the tensor-to-scalar ra- scale' in inflationary' scenarios. tio values within the range r <0.12 (95%) [6]. If we ne- 5.2 Including dark matter glect, as a working assumption, the null hypothesis for r and estimate it with its higher bound r 0.12, we can ∼ So far we have developed a fermion curvaton mech- constrain up to two parameters of the theory. Therefore, anism consistent with the latest cosmological data. The deploying the experimental value and bound respectively gist of this framework is in the realization of a see-saw for and r, we use Eqs. (38) and (43) to derive con- PS mechanism that has phenomenological consequences in straints on the masses of the fermion fields involved. cosmology. Assuming that the fluctuation of the abun- The first constraint we can derive is on the mass of dance of the light χ fermions is related to the abundance the heavy species, which is the same as in Ref. [30], i.e. of the heavy ψ fermions by 2 11 2 mψ .10− ξ Mp . (44) 2 nψ 7 | | − δnχ 3 10 , (46) Differently from Ref. [30], if we now assume the to- ' mχ tal mass hierarchy (19) we find that large values of we can concretely realize a see-saw mechanism in this ξ =(nψmψ)/[(nχ+nψ)mχ] are favored, once we look at |values| of γ that allow for a nearly scale-invariant power framework. The two fermionic species accounted for spectrum (γ 2). The constraint (44) now links the should correspond to: mass of the hea' vy species to the GUT scale, if we make 1) a regular neutrino, which would correspond in this 4 framework to the light χ-fermion species, the mass of a proper choice of ξ 10 . 3 The second constrain' t we can derive on the masses which would be then mχ <10− eV, fulfilling in this way of the two fermionic species arises from combining the constraints from Big Bang nucleosynthesis and being Eqs. (38) and (43) into consistent with all the experimental data [60, 61]; 2) a sterile neutrino, which would drive the pri- 2 2 mψ nψ 2 mordial spacetime background dynamics, namely the ψ 3 2 O(10 ) . (45) mχ δnχMp ∼ species. Its mass would be at the GUT scale for a choice This value is actually slightly different from the one re- of the ratio that appears in Eq. (46). Nonetheless, in this ported in Ref. [30], and can be easily achieved within model even smaller values than the GUT scale would be this scenario, while linking the mass of the light species consistent for the mass of the background species ψ. to the mass of the heavy one. We end this section with a comment on the perturbation theory in fermion field cosmologies. For this type of cosmology, the stress energy tensor behaves as a per- fect fluid only at the background level. But if we consider perturbations, anisotropic components may appear in the stress-energy tensor. These latter would not affect our conclusions on the curvature perturbations generated in the primordial era, but may affect the propagation of primordial gravitational waves, and thus the estimate of the scalar to tensor ratio. This is a very interesting topic which deserves further investigation, especially in light of the considerations in Ref. [57].

6 Baryogenesis in the curvaton Fermi bounce scheme

Fig. 1. (color online) The power spectrum of scalar In this section, we discuss a scenario in which baryo- (blue line) and tensor (red line) perturbations is 3 15 genesis is realized from the decay of the ultra-massive displayed for parameter scales ξ =10 , mψ =10 3 fermions, described by the ψ-species, into the SM par- GeV and mχ = 10− eV. After a time scale t > 5×1012 (Planck time units), their ratio, i.e. ticles, corresponding to the χ-species. Thus ψ is now 7 the r-parameter, converges to r'10− . deﬁned as a singlet of the SM gauge group. Conse- quently, this ultra massive ﬁeld is not protected by the In Fig. 1, we display the numerical results of power electroweak symmetry, and for the Georgi’s missing sin- spectra of scalar and tensor perturbations, for ξ = 103, glet mechanism, it should have a mass much higher than

065101-9 Chinese Physics C Vol. 42, No. 6 (2018) 065101 the SM vacuum expectation value (VEV) scale [62]. We while the annihilation cross section σ(ψψ χχ) is as- will not only show that the correct baryon asymmetry is sumed to be subdominant, for the momen→t. Once the reproduced, but we also set severe limits on the bounce temperature of the universe drops below the critical value ¯ scale. In Fig. 1, we obtain a number density asymmetry T = mψ, the heavy neutrinos cannot follow the rapid nB L which is compatible with the baryogenesis consis- variation of the equilibrium distribution. The deviation | − | 10 tency condition nB /s 10− . from thermal equilibrium is related to too high a number ∼ density of heavy fermions, compared to the equilibrium 6.1 Minimal coupling with SM leptons density. However, the heavy fermion decay, and a lep- The ψ-species can be then identified with the right- ton asymmetry, could be generated owing to the pres- handed (RH) neutrino, considering a type I see-saw ence of CP -violating transitions. CP -violating transi- mechanism for the left-handed (LH) neutrino mass. A tions involve the quantum interference between the tree- Fukugita-Yanagida leptogenesis mechanism [63, 64] can level amplitude and the one-loop diagrams (vertex and be then realized, once all the Sakharov conditions [65] self-energy contributions). The resulting CP violation are satisfied. parameters can be conveniently defined as A minimal Lagrangian for this instantiation of the Γ (ψ lφ) Γ (ψ ¯lφ¯) see-saw mechanism reads ²Da → − → . (48) ≡ Γ (ψ lφ)+Γ (ψ ¯lφ¯) α T 1 T 1 → → =yψl φ +m ψ C− ψ+m χ C− χ L α ψ χ At tree-level, this parameter is equal to zero. However, ξ µ the CP asymmetry appears at the leading order of per- + 2 (ψγ5γµψ)(χγ γ5χ)+h.c., 2 MP l turbation theory y (1-loop), entailing the two contribu- 2 1 in which we have reinserted MP l = κ− . We empha- tions size that within our model the four-fermion interaction 2 M 1 Im(y†y)ik term is recovered from integrating out (non-dynamical) ²a = , (3level)+(self energy), (49) −8π (y†y) torsion in the ECHSK theory. At the same time, the bouncing evolution of the background itself, as specified where i,k are ψ-generation indices, 2 in Section. 4.1, is due to the four-fermion interaction, 1 m Im(y†y) ²V = f ψi ik , (50) when negative values of ξ are taken into account. a π −8 mψi (y†y) The decay channels of the heavy fermion to the SM µ ¶ and particles are 1+x ψ lφ, ψ ¯lφ¯, f(x)=√x 1 (1+x)ln (51) → → − x in which l are lepton fields, φ denotes the Higgs, and y is ½ µ ¶¾ These results hold for m >> Γ – for small mass dif- the Yukawa matrix of the ψ and l generations. In partic- ψ | | ular, we assume that the number of ψ-generations will be ferences, we may expect an enhancement of the mixing more than one. Such a Lagrangian mediates L-violating contribution. channels with several different L-number assignments for 6.2 The Boltzmann equations ψ. For instance, if L:ψ=0, the first Yukawa term violates the L-number of ∆L = 1. However, such an assignment The generation of a baryon/lepton asymmetry can for the lepton number of ψ is problematic, since ψ could be treated with Boltzmann equations. While accounting also couple to quarks, destabilizing the proton. A strong for these latter, we have to consider the main processes, fine-tuning for the coupling of ψ to the quarks should be which are the decays and inverse decays of the heavy then invoked. The most elegant choice is to consider a fermions, and the lepton number conserving ∆L=0 and L-preserving Yukawa term, while the L-number is vio- violating ∆L=2 processes. These processes read, respec- lated by the Majorana mass term. This choice coincides tively, with L:ψ =1, as for RH neutrinos. On the other hand, dn ψ +3Hn = γ(ψ lφ)+γ(lφ ψ) the CP violation is encoded in the complex phases of dt ψ − → → the Yukawa coupling y, which is a matrix in the space of γ(ψ ¯lφ¯) γ(¯lφ¯ ψ), (52) leptonic generations of ψ. − → − → Below we will then consider the case of a negligi- dn l +3Hn = γ(ψ lφ) γ(lφ ψ) ble four-fermion interaction between the extra massive dt l → − → fermion and the light one. This imposes a bound on the +γ(¯lφ¯ lφ) γ(lφ ¯lφ¯), (53) 2 suppression scale ξMP−l . → − →

The decay width of the heavy neutrino ψ at tree level dn¯l +3Hn¯ = γ(ψ ¯lφ¯) γ(¯lφ¯ ψ) is dt l → − → ¯¯ 1 ΓD =Γ (ψ φl)+Γ (ψ φl)= y†ymψ , (47) +γ(lφ ¯lφ¯) γ(¯lφ¯ lφ), (54) → → 8π → − →

065101-10 Chinese Physics C Vol. 42, No. 6 (2018) 065101 with the reaction densities expressed as follows, the equilibrium distribution. On the other hand, reso- nances may be treated as on-shell particles, falling out 2 γ(ψ lφ)= dΦ123fN (p1) (ψ lφ) , (55) of thermal equilibrium. Because of CPT invariance, for → |M → | Z thermal equilibrium distributions we must have and d(n n¯) 2 l l eq γ(lφ ¯lφ¯)= dΦ f (p )f (p ) 0(ψ lφ) , (56) − +3H(nl n¯l)=∆γ =0. (61) → 1234 l 1 φ 2 |M → | dt − Z where H is the Hubble parameter, dΦ1,..,n denotes the The resonance contributions (the decay and inverse phase space integration over particles in the initial and decay) final states, eq eq 3 3 d p d p ∆γres = 2²γ (ψ lφ), 1 n π 4 4 − → dΦ1,..,n = 3 ... 3 (2 ) δ (p1+... pn) (57) (2π) 2E1 (2π) 2En − and the weights have a compensating term from 2 2 contribution processes → d3p fi(p)=exp( βEi(p)), ni(p)=gi 3 fi(p) (58) − (2π) eq eq ∆γ2 2 = 2 dΦ1234fl (p1)fφ (p2) Z → represent the Boltzmann distribution (for simplicity we Z 2 2 ( 0(lφ ¯lφ¯) 0(¯lφ¯ lφ) ), (62) use a Boltzmann distribution rather than Bose-Einstein × |M → | −|M → | or Fermi-Dirac distributions, neglecting the distribution ¯¯ functions in the final state, which is a good approxima- Such a lφ lφ compensation is a consequence of CPT symmetry/unitarit↔ y. From unitarity, one can find tion for small number densities) and the number density of particle i=N,l,φ at temperature T = 1 , respectively, β ( (lφ X) 2 (X lφ) 2)=0, and and 0 denote the scattering matrix elements |M → | −|M → | M M X of the indicated processes at T =0 (the prime indicates X that in 2 2 scatterings the contribution of the internal where X denotes all possible generic channels. In the → resonance state has been subtracted). case of weak coupling y, the channel is restricted to 2- The ratio of number density and entropy density particle states to the leading order of perturbation ex- 2 (namely YX =nX /s) remains constant for an expanding pansion. Therefore, at the leading order y , one obtains universe in thermal equilibrium. The heavy fermions are weakly coupled to the thermal bath, so that they freeze eq eq eq ∆2 2 = 2 dΦ1234fl (p1)fφ (p2) out of the thermal equilibrium at T¯ =m . This implies → ψ Z that the decay rate is too slow to follow the rapidly de- (lφ ψ) 2 (ψ ¯lφ¯) 2 creasing equilibrium distribution fN exp( βmψ). As a × −|M → | |M → | ∼ − ³ π consequence, the system will evolve toward an excess of + (¯lφ¯ ψ) 2 (ψ lφ) 2 δ(s m2 ) eq ψ |M → | |M → | mψΓ − the number density nN >nN . eq eq ´ The Boltzmann equations are classical dynamical = 2²γ (ψ lφ)= ∆γ . (63) → − res equations. However, they are endowed with S-matrix elements, in turn containing quantum mechanical inter- The incorporation of off-shell effects requires a formalism ferences of different amplitudes. The S-matrices are con- that goes beyond the Boltzmann equations (Kadanoff- tained in collisions terms in the Boltzmann equations. Baym equations [66]). However, the corrections to Boltz- The scattering matrix elements are evaluated at zero mann’s model are expected to be negligible. temperature. However, the quantum mechanical inter- The numerical integration of the Boltzmann equa- ferences must be affected by interactions with the ther- tions for reasonable parameters is displayed in Fig. 2. mal bath. For 2 2 scatterings, one finds In particular, the number density of the ψ-particle de- → creases in cosmological time, generating a small lepton 2 2 2 (lφ ¯lφ¯) = 0(lφ ¯lφ¯) + (lφ ¯lφ¯) , (59) |M → | |M → | |Mres → | number asymmetry. where the resonance contribution reads 6.3 Baryon number asymmetry from sphalerons 2 (lφ ¯lφ¯) (lφ ψ) (ψ ¯lφ¯)∗ = (lφ ψ) . M → ∼M → M → |M → | (60) Let us assign a chemical potential µ to each SM par-

The particles involved in the process may be treated as ticle: quarks, Higgs, and leptons. In the SM – with Nf massless: their distribution functions will coincide with generations and one Higgs doublet – we must assign1)

1) In addition to the Higgs doublet, the two left-handed doublets qi and li and the three right-handed singlets ui, di and ei of each generation will each be treated with independent chemical potential.

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On the other hand, SU(3) QCD instanton-mediated processes must generate eﬀective interactions between LH and RH quarks, which leads to the constraint

(2µ µ µ )=0. (66) qi− ui− di i X A third condition (valid in all temperature ranges) arises from the requirement that the total hypercharge of the plasma must vanish. So, from Eq. (64) and the SM hy- percharges, we obtain 2 (µqi+2µui µdi µli µei+ µφ)=0. (67) − − − Nf i X Finally, from the Yukawa couplings, one ﬁnds, in turn Fig. 2. (color online) The evolution of the number densities as functions of z = mψ/T are displayed µqi µφ µdj =0, µqi+µφ µuj =0, µli µφ µej =0. (68) (log10 n,log10 z) scale. In dashed black lines, we − − − − − show the evolution of the number density of mas- These relations are valid if and only if the system is in sive fermions ψ starting from a thermal initial thermal equilibrium1), at temperatures 100GeV < T < density nψ(z << 1) ' 3/4. In thick black lines, 1012 GeV. we plot the evolution of the number density of 2 Let us deﬁne the baryon number density nB =gBT /6 massive fermions ψ starting from a thermal ini- 2 and the lepton number densities nli = LigT /6. So, we tial density nψ(z<<1)'0. In the red dashed and express baryon and lepton numbers B and Li in terms thick red lines the B−L number density |nB L| − of the chemical potentials, i.e. is pictured from initial thermal and zero nψ respectively. The characteristic parameters cho- 10 6 B= (2µqi+µui+µdi), (69) sen in this plot are mψ ' 10 GeV, ² ' 10− , 3 6 8 i mχ ≡mν1 '10− eV, ξ610 −10 . X

5Nf+1 chemical potentials. The asymmetry in the parti- Li =2µli+µei, L= Li . i cle and antiparticle number densities, for βµi <<1, reads X 3 gT 3 Solving this simple system of algebric equations, with ni n¯i = βµi+ [(βµi) ] (for fermions), (64) − 6 O respect to µl, one obtains and 2 4Nf 14Nf +9Nf 3 B= µl, L= µl , (70) gT 3 n n¯ = 2βµ + [(βµ ) ] (for bosons), − 3 6Nf +3 i− i 6 i O i having considered the thermal bath as a non-interacting They yield the important connections between the B, gas of massless particles. B-L and L asymmetries: However, the plasma of the early Universe is very B=a(B L); L=(a 1)(B L), (71) different from a weakly coupled relativistic gas. This is − − − because of unscreened non-Abelian gauge interactions, where a=(8Nf +4)/(22Nf +13). which have very important nonperturbative effects to 6.4 Four-fermion interaction between the two take into account. species Leptons, quarks and Higgs bosons will interact via perturbative operators - Yukawa and gauge couplings. We wish now to comment on the possible relevance of However, they will also interact via the non-perturbative the torsion-mediated four-fermion interactions, namely sphaleron processes. These processes lead to a set of con- ξ straints between the SM chemical potentials. The effec- µ OΓ = 2 ψγµγ5ψ χγ γ5χ. (72) tive interactions induced by sphalerons (SU(2) electro- MP l weak instantons) must imply If χ is weakly interacting with the SM particles, ψψ χχ represent entropy-leaking collisions that could affect→the (3µqi+µui µdi)=0. (65) 1 − leptogenesis scenario if m ξ− M . In particular, if i ψ P l X ∼

1) Yukawa interactions are in equilibrium only within a more restricted temperature window depending on the strength/magnitude of the Yukawa couplings. In the following discussions, we will ignore these technical complications. In fact, only a small eﬀect on our discussion of leptogenesis is expected from taking them into account. 065101-12 Chinese Physics C Vol. 42, No. 6 (2018) 065101 the fermion ψ has a lepton number assignment L=0, the leptogenesis, and thus to explain baryogenesis. Speciﬁ-

OΓ operator violates the lepton number as ∆L=2. For cally, we have focused on a toy model for the curvaton example, the initial number density of nψ can be affected mechanism, which is an instantiation of a Fermi bounce by the annihilation process cosmology that singles out as a dark matter candidate ξs a sterile neutrino-like field, hence deriving phenomeno- σ(ψψ χχ) , → ∼ 8πM 2 logical consequences of these assumptions. It is quite P l remarkable that this scenario comes out to be falsifiable, 1) 1 2 estimated at s < ξ− MP l. This process becomes out- since it predicts a non-vanishing value of r. of-equilibrium for s T 2 6 4m2 , which means that χχ ' ψ A peculiarity of the model we have shown here is collisions cannot reproduce two heavy ψ particles. that the results we have derived are not confined to an In the scenario we deepened here, we imposed the hi- effective analysis in which the dynamics of dark matter is 1/2 erarchy mψ < ξ− MP l, i.e. the bounce scale must be only addressed from a hydrodynamical perspective. Con- higher than the heavy fermion mass. On the other hand, versely, the Fermi matter bounce scenario described here a successful leptogenesis requires a fermion mass scale of encodes a microphysical description in terms of fermionic m 109 GeV or similar. This imposes an indirect bound ψ ∼ fields, which provides a natural way to overcome short- on the bounce scale. In particular, from numerical cal- comings that usually arise in cosmology because of the culations, we checked that the safety bound for a good use of auxiliary scalar fields, with consequent issues of 1/2 leptogenesis is ξ− MP l > (10 15)mψ, assuming natu- arbitrariness, as happens in inflation. − 1 ral initial conditions n (z < 10− ) 0,3/4 respectively. ψ ' It might indeed seem surprising at first sight that From these values, the same plots displayed in Fig. 2 are matter fields belonging to the SM and to its simplest obtained. extension reproduce the desired background and pertur- On the other hand, the case in which mψ > (10 bation features. But it should be not surprising that 1/2 − 15)ξ− MP l cannot correspond to a successful leptoge- in this framework, retrieved from particle physics, it is nesis, without an unnatural initial superabundance of possible to tackle questions that concern the nature of 1/2 heavy fermions. In particular, the case of m ξ− M ψ ' P l dark matter and the origin of baryogenesis. Indeed the is undesirable, first of all because the unitarity and the model we have deepened here turns out to be compati- calculability of z <1 in Fig. 2 cannot be controlled, and ble with hot dark matter constraints. Several issues con- second because the annihilation process will dominate tinue to be unexplored, and we cannot deny that this over all other channels around z 1, provoking a too fast ' line of research will require more detailed investigations number density decay of ψ. in the future. Nonetheless we wish to mention that the Fermi bounce scenario might entail, as distinctive phe- 7 Discussion nomenological predictions, able to falsify this paradigm among the others in the literature, the appearance of In this paper we have shown that the matter bounce non-vanishing cross-correlation functions between polar- scenario, as an alternative framework to inflation, allows ization modes, coming from the parity-violating elements us to encode hot dark matter in a fairly natural way of the Clifford algebra bilinears [57]. However, more work when fermion matter fields are taken into account. We is also required in this direction. have further shown that the model is able to generate

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