Adiabatic out-of-equilibrium solutions to the Boltzmann equation in warm inflation

Journal of High Energy Physics, Feb 2018

Mar Bastero-Gil, Arjun Berera, Rudnei O. Ramos, João G. Rosa

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Adiabatic out-of-equilibrium solutions to the Boltzmann equation in warm inflation

HJE Adiabatic out-of-equilibrium solutions to the Boltzmann equation in warm in ation Mar Bastero-Gil 0 1 2 4 5 Arjun Berera 0 1 2 3 5 Rudnei O. Ramos 0 1 2 5 Jo~ao G. Rosa 0 1 2 5 0 20550-013 Rio de Janeiro, RJ , Brazil 1 Edinburgh , EH9 3FD , United Kingdom 2 Granada-18071 , Spain 3 School of Physics and Astronomy, University of Edinburgh 4 Departamento de F sica Teorica y del Cosmos, Universidad de Granada 5 Campus de Santiago , 3810-183 Aveiro , Portugal We show that, in warm in ation, the nearly constant Hubble rate and temperature lead to an adiabatic evolution of the number density of particles interacting with the thermal bath, even if thermal equilibrium cannot be maintained. In this case, the number density is suppressed compared to the equilibrium value but the associated phasespace distribution retains approximately an equilibrium form, with a smaller amplitude and a slightly smaller e ective temperature. As an application, we explicitly construct a baryogenesis mechanism during warm in ation based on the out-of-equilibrium decay of particles in such an adiabatically evolving state. We show that this generically leads to small baryon isocurvature perturbations, within the bounds set by the Planck satellite. These are correlated with the main adiabatic curvature perturbations but exhibit a distinct spectral index, which may constitute a smoking gun for baryogenesis during warm in ation. Finally, we discuss the prospects for other applications of adiabatically evolving out-of-equilibrium states. Cosmology of Theories beyond the SM; Thermal Field Theory 1 Introduction 2 3 4 5 Adiabatic baryogenesis during warm in ation Generation of baryon isocurvature perturbations Summary and future prospects A Boltzmann equation for scattering processes if strong dissipation is attained at the end of the slow-roll evolution regime (see, e.g., refs. [8{11]). The greatest appeal of warm in ation lies perhaps in the fact that thermal in aton uctuations are directly sourced by dissipative processes, changing the form of the primordial spectrum of curvature perturbations and thus providing a unique observational window into the particle physics behind in ation [11{18]. In addition, we have recently shown that warm in ation can be consistently realized in a simple quantum eld theory framework requiring very few elds, the Warm Little In aton scenario [19], where the required atness of the in aton potential is not spoiled by thermal e ects (see also refs. [20{26] for earlier alternative models), paving the way for developing a complete particle physics description of in ation that can be fully tested with CMB and Large-Scale Structure (LSS) observations and possibly have implications for collider and particle physics data. Independently of the particle physics involved in sustaining a thermal bath during ination, a generic feature of warm in ation is the slow evolution of both the temperature and the Hubble parameter for the usually required 50{60 e-folds of expansion. Since both scattering and particle decay rates typically depend on the former, this implies that the ratio =H will generically evolve slowly during in ation. Consequently, as we explicitly show in this work for the rst time, particle species in the warm in ationary plasma can maintain distributions that are slowly evolving and out-of-equilibrium throughout in ation, whether or not they are directly involved in the dissipative dynamics. This may have an important impact not only on the in ationary dynamics and predictions themselves but also on the present abundance of di erent components. As an example of application of this novel observation, we show that this can lead to the production of a baryon asymmetry during in ation, a possibility that can be tested in the near future with CMB and LSS observations. This work is organized as follows. In section 2 we analyze the Boltzmann equation for a particle species interacting with a thermal bath for an adiabatic evolution of the ratio =H, focusing explicitly on the case of decays and inverse decays for concreteness. We then show that this leads to slowly varying out-of-equilibrium con gurations, obtaining the overall particle number density and its phase space distribution. We brie y discuss how similar results can be obtained for scattering processes. In section 3 we use these results to develop a generic baryogenesis (or leptogenesis) mechanism during in ation. Then, in section 4, we discuss how this generically leads to baryon isocurvature modes that give a small contribution to the primordial curvature perturbation spectrum. In section 5 we summarize our results and discuss their potential impact on other aspects of the in ationary and post-in ationary history. An appendix is included where some technical details are given. 2 Adiabatic out-of-equilibrium dynamics Let us consider a particle X interacting with a thermal bath at temperature T in an expanding Universe.1 For concreteness, let us explicitly consider the case where X decays into particles Y1 and Y2 in the thermal bath, and assume that these maintain a near 1Throughout this work we consider a spatially at Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time in which the metric is given by ds2 = dt2 a(t)2dx2, where t is physical time, x are the comoving spatial coordinates and a(t) is the cosmological scale factor. { 2 { equilibrium distribution through other processes that do not directly involve X. Let us also, for simplicity, focus on the case where X is a boson with gX degrees of freedom and the decaying products Y1 and Y2 are either bosons or fermions (extending our results to decaying fermions is straightforward). The number density of X particles, nX = gX Z d3p (2 )3 fX (p) ; n_X + 3HnX = C ; has an evolution, in an expanding at FLRW universe, that is described by the Boltzmann equation [27, 28] HJEP02(18)63 where 1 f eq 1 f2eq = f1eqf eq f eq2 ; X f Xeq = e (EX 1 X) 1 Y1 + Y2 ! X and takes the form2 where the collision term C includes the e ects of decays X ! Y1 + Y2 and inverse decays where d i = gid3pi=2Ei(2 )3, M is the matrix element associated with both the decay and inverse decay processes (related by CPT invariance) and fi are the phase-space distribution functions for each particle. Note also that the plus (minus) sign in 1 fi in eq. (2.3) refers to bosons (fermions) and it is related to the usual Bose enhancement (Pauli-blocking) e ect. Since we assume Y1;2 are in equilibrium, we have i f eq = 1 e (Ei i) 1 ; i = 1; 2 where = 1=T in natural units and the plus (minus) sign is for a Fermi-Dirac (BoseEinstein) distribution. Taking into account conservation of energy, EX = E1 + E2, and assuming chemical equilibrium, X = 1 + 2 (an assumption that we may drop if chemical potentials can be neglected), then, it is easy to show that (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) C = = Z Z d3pX (2 )3 X (fX f Xeq) ; is the distribution of the X particles when they are in equilibrium. This then allows us to write the collision term in the form d X d 1d 2(2 )4 4(pX p1 p2)jMj 2 f1eqf eq 2 f eq X fX f eq X 2The inclusion of Landau damping e ects will not change our conclusions. { 3 { where the equilibrium decay width of the X boson is given by X = 1 2EX f Xeq Z of the decay width, a common procedure in the literature, by considering its thermal average [27]: The Boltzmann equation (2.2) can then be cast into the familiar form X = 1 Z neq X d3pX X f Xeq : n_X + 3HnX = X (nX neXq) : It is straightforward to obtain an analogous result for an arbitrary number of particles in the nal state. In a non-expanding Universe, the collision term will then naturally drive the number density of X particles towards its equilibrium value, while in an expanding Universe this only occurs for X H, which is the familiar rule of thumb in cosmology. Now, in a cosmological setting such as warm in ation, since H and T are slowly-varying and X will depend only on T and on the masses of the parent and daughter particles, we can take the ratio X =H to be slowly-evolving. For all cases where X X =H is slowly-evolving on the Hubble scale, i.e., _ X = X H 1 , we may take X to be a constant as a rst approximation and write the Boltzmann equation in terms of the number of e-folds of expansion, dNe = Hdt, as n0X + 3nX = X (nX neXq) : If the temperature is slowly-varying, we may take neXq to be constant as well, yielding the solution nX (Ne) = X 3 + X neq 1 X e (3+ X )Ne + nX (0)e (3+ X )Ne : Hence, X is driven exponentially fast to the solution nX ' 3 + X X neXq : Note that this is not exactly stationary due to the slow variation of both X and neXq, so we refer to this solution as adiabatic. This solution is attained in less than a e-fold if the initial number density is not too far from the quasi-stationary value, and even large discrepancies will be quickly washed away within a few e-folds. This agrees with the statement made above that nX is driven towards its equilibrium value for X 1, but reveals a novel feature that is absent in general cosmological settings | that even for X 1 a small but not necessarily negligible number density of X particles remains constant despite the fast expansion of the Universe. This is thus a new type of solution that only arises in cosmological { 4 { (2.8) (2.9) (2.10) (2.11) (2.12) settings where the temperature and Hubble rate are varying slowly, such as warm in ation. In an adiabatic approximation, we can include the slow variation of these parameters, namely X = X (Ne) and neXq = neXq(Ne) in the quasi-stationary solution in eq. (2.12). We may also take a step further and compute the phase-space distribution of Xparticles, which follows the momentum-dependent Boltzmann equation in a at FLRW universe, where the collision term is given by Hp C(p) = X (p) fX (p; t) = C(p) ; f Xeq(p) ; from the results obtained above. For illustrative purposes, we will consider two distinct cases where an X scalar boson decays into either fermion or scalar boson pairs, through Yukawa or scalar trilinear interactions, respectively. We neglect all chemical potentials for simplicity. The corresponding decay widths are given by [23] X (B) = X (F ) = 0 0 (B) mX (F ) mX !p !p " " 1 + 2 log 1 + 2 log T p T p 1 1 e !+=T !# e ! =T 1 + e !+=T !# 1 + e ! =T where we have neglected the masses of the decay products, with ! !p = qp2 + m2X , p = jpj. The decay widths at zero-temperature and zero-momentum = (!p are, in the two cases, given by 0 (B) = g B2 M 2 32 mX ; 0 (F ) = F mX ; g 8 2 where gB;F are dimensionless couplings and M is the mass scale of the trilinear scalar coupling. We have solved the Boltzmann equation (2.13) numerically in both cases for di erent values of the X boson mass and decay width, taking fX (p; 0) = 0 and imposing fX (p; t) ! 0 in the limit p ! 1 (in practice at a su ciently large momentum value). We illustrate the resulting time evolution of the phase-space distribution for the bosonic and fermionic cases in gure 1(a) and gure 1(b) respectively. In both cases the decay is of a relativistic X boson with mass set at the value mX = 0:001T and with 0 (B;F )=H = 0:5, which corresponds to (B) = 0:0245 ( X(F ) = 10 4 in the case of decay into fermions). It X is clear from the results shown in gure 1 that in both cases the distribution reaches a stationary con guration after only a couple of Hubble times, a result that we obtained generically for di erent values of the mass and decay width, for both fermionic and bosonic decay. Naturally, for larger values of X the stationary con guration is attained faster. equation (2.13), yielding a rst-order inhomogeneous di erential equation for f Xstat(p). Following standard methods, we may formally write the stationary solution in the integral form f Xstat(p) = f X(h)(p) Z 1 dp0 X (p0) f Xeq(p0) p p0 H f X(h)(p0) ; (2.17) { 5 { (2.13) (2.14) (2.15) p)=2, (2.16) 0.030 0.025 T 0.010 0.005 0.000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 HJEP02(18)63 4 (b) Decay into fermions. a function of the momentum p, for decay into bosons (a) and for decay into fermions (b). The (bosons) and X(F ) = 10 4 (fermions). parameters used are mX = 0:001T , and 0 (B;F )=H = 0:5, which corresponds to (B) = 0:0245 X X where f (h) = eR dp0 X (p0)=Hp0 is the homogeneous solution, which takes the form f (h) = p X if one replaces the decay width by its thermal average eq. (2.8), as explained above. X For practical purposes, however, this integral form is not very useful, since integrals involving the Bose-Einstein distribution do not have, in general, a simple analytical form and have to be computed numerically. In gure 2, we show the obtained stationary distributions for bosonic and fermionic decays for the same values of the X mass and average decay width considered in gure 1. For comparison, we also show in gure 2 the distribution obtained when replacing X (p) by X . We can see that for fermionic decay this yields a good approximation to the full solution, while for bosonic decay there are more prominent di erences. In particular, the latter distribution is peaked at lower momentum values than the one obtained using X , which is related to the Bose enhancement of the decay at low-momentum values p . mX . For mX T there is thus a substantial variation of the bosonic decay width with { 6 { Γ(XB)(p) Γ(B) X Γ(XF)(p) Γ(F) X 0 1 2 4 5 6 3 (a) Decay into bosons. HJEP02(18)63 0.005 0.000 3 0.030 0.020 T 0.010 0.005 0.000 (b) Decay into fermions. times (when the distributions have already reached the stationary state), for decay into bosons (a) and for decay into fermions (b). The parameters used are mX = 0:001T and corresponds to (B) = 0:0245 (bosons) and X (F ) = 10 4 (fermions). Solid lines yield the solution X when considering the full momentum dependent decay widths, while dashed lines correspond to the (B;F )=H = 0:5, which 0 solution obtained using the constant thermally averaged decay widths. momentum in the relevant range p . T , while for fermionic decay it is a good approximation to use the thermally averaged decay width in place of the full momentum-dependent expression. Note that the larger the mass of the X boson the closer the distributions are to the one obtained using X , since thermal corrections to the decay width become less important in this regime for p . T . mX . All stationary distributions are nevertheless reasonably well tted by an expression of the form f Xstat(p) = A X (p; Tstat) 1 3H + X (p; Tstat) epp2+m2X =Tstat 1 : (2.18) We show in gure 3 the results for the t amplitude A and e ective temperature Tstat for mX = 0:001T and di erent values of the thermally averaged decay width, for both the bosonic and fermionic decays. { 7 { distribution with mX = 0:001T . Dashed lines are for the case of decay into bosons, while solid lines are for the case of decay into fermions. As we can see in gure 3, for relativistic X bosons the coe cient A O(1) for fermionic decays and also bosonic decays, unless the decay width is much smaller than the Hubble parameter, while the e ective temperature Tstat . T . The larger variation of the t parameters for bosonic decay is again due to the above mentioned Bose enhancement e ect, a variation that becomes smaller for larger values of the X mass. With the results shown in gure 3, we also note that, generically, the e ective temperature Tstat approaches (asymptotically) the thermal bath temperature T from below and likewise for the overall amplitude in front of eq. (2.18). This then means that the stationary solution f stat gets suppressed at large momenta relative to the equilibrium one f eq. This result is similar X to the one obtained recently for an exact solution of the Boltzmann equation in a FLRW background [ 29, 30 ], though it di ers fundamentally from the solution found here. In particular, the result of refs. [ 29, 30 ] applies to a massless gas of particles with MaxwellBoltzmann distribution in a radiation dominated epoch and it only approaches a stationary and decay into fermions (dotted line). The solid line is the result of the solution given by eq. (2.12). distribution at asymptotically large times, while the present one applies generically to a relativistic distribution in an in ationary regime and reaches a stationary state in only a few e-folds. in gure 4. Finally, we can obtain explicitly the number density from our results. We have integrated the numerically obtained distributions over momenta and compared the results with the adiabatic solution for the number density in eq. (2.12). These results are shown We conclude that eq. (2.12) is a good approximation to the numerical value of the so that adiabatic evolution regime. number density, particularly for fermionic decays, while again we observe some discrepancies in the case of bosonic decays for intermediate values of X =H. These only occur for relativistic X bosons and eq. (2.12) becomes a better approximation as mX increases. We note that, in any case, the observed discrepancies correspond at most to O(1) factors, X )neXq is generically a good approximation to the number density in the Our generic analysis would not be complete without considering scattering processes, which may also contribute to the collision term in the Boltzmann equation. Although, for the same type of interactions, scattering processes are typically suppressed compared to decays since they involve larger powers of the couplings in the perturbative regime, there may be cosmological settings where they correspond to di erent types of interactions and may dominate the collision term. The analysis of scattering processes is somewhat more involved than for decays, since in general one cannot easily express the collision term in terms of the number density, as we detail in appendix A. This is nevertheless possible in the limit of small occupation numbers, fi 1, i.e., in the absence of Bose enhancement or Fermi degeneracy [27]. In this regime, the Boltzmann equation can be written as n_X + 3HnX ' 2 h vi nX X neq 2 ; (2.19) where h vi is the thermally averaged cross section times velocity de ned in appendix A. In the adiabatic limit, where both the latter and H vary slowly, we obtain the adiabatic { 9 { (2.20) (3.1) (3.2) (3.3) (X ! B) = (X ! B) = (1 + ) X ; (1 + ) X ; 1 2 1 2 (X ! Y ) = (X ! Y ) = 1 2 1 2 (1 (1 ) X ; ) X ; where 6 = yield the amount of C/CP violation. Note that the total decay widths of the X particle and of its anti-particle are equal, as required by CPT invariance, (X ! B) + (X ! Y ) = (X ! B) + (X ! Y ) : Both X and X will then obey the same Boltzmann equation (2.10) when the particles in the B; Y nal states are in equilibrium (which we assume to be maintained by other interactions), and evolve towards the adiabatic solution (2.12), which will be the same for both nX and nX . The full adiabatic solution (2.11) may be di erent for both nX and nX if the initial values are distinct, but any discrepancies are quickly erased by expansion. where X(s) = h vineXq=H is the ratio between the thermally averaged scattering rate and the Hubble parameter. Although this may seem more complicated than the adiabatic solution for decays, we note that the suppression factor with respect to the equilibrium number density becomes X(s)=3 for small values of X(s) and also tends to 1 in the limit X 1, as in eq. (2.12). The cases where the collision term is dominated by decays or by scattering processes thus yield quite similar adiabatic solutions for the number density, at least for small occupation numbers. Adiabatic baryogenesis during warm in ation The fact that we have obtained new solutions to the Boltzmann equation that are inherently out-of-equilibrium immediately suggests an application: the production of a baryon asymmetry during in ation. Let us then consider a generic model where the X particles (bosons or fermions) decay violating B (or L or B L) and C/CP. In particular, let us consider a simple case with two possible decay channels (as discussed e.g. in ref. [31]), X ! B ; X ! Y ; where the nal state B carries a baryon number b > 0 and Y has no baryonic charge. In practical applications these will typically correspond to 2-body decays, although the number of particles in the nal state can be arbitrary. B-violation then occurs because there is no consistent assignment of a baryonic charge to the X particle. We also have the conjugate decays X ! B and X ! Y with opposite baryonic charges, and C/CP violation implies that the partial decay widths satisfy the relations solution for the number density nX ' 3 2 X(s) r Now, the baryon number density, i.e., the di erence between the number density of baryons and that of anti-baryons, evolves according to the Boltzmann equation given by n0B + 3nB = b 2 h X (1 + )(nX neXq) 2 X (1 + )(nX neq)i : X Let us consider the case where nX (0) = nX (0), such that n0B + 3nB = b X 2 (nX neXq) ; where = corresponds to the amount of CP violation. Using the solution given by eq. (2.11), this yields the adiabatic solution for the baryon number density, nB(Ne) ' 2 b 1 e (3+ X )Ne X 3 + X neXq + nX (0)e 3Ne e X Ne 1 ; (3.6) (3.4) (3.5) (3.7) HJEP02(18)63 which approaches (exponentially fast) the quasi-stationary solution, nB ' 2 b X 3 + X Hence, we see that a constant baryon asymmetry is produced during warm in ation (or in fact during any analogous period of quasi-adiabatic temperature and Hubble rate evolution), for any value of X = X =H. Interestingly, nB ! b limit for which the X particles are in equilibrium. This may seem to contradict Sakharov's conditions for baryogenesis [ 32 ], but it is simply associated with the fact that we are taking decays and inverse decays as the main processes responsible for driving the X particles towards equilibrium. In this case, to get closer to equilibrium, we need to increase X , which also increases the rate of production of baryon number, in such a way that we obtain a nite baryon number density for arbitrarily large X . Note that, in the large X limit, from neXq=2 for X ! 1, the the solution given by eq. (2.11), we have that nX source term on the right-hand-side of eq. (3.5) tends to a nite value 3b neq X ! 3neXq= X . Thus, the baryon neXq=2 in the limit X ! +1. However, note that in any physical setting X is not at the limiting value but rather is nite, which means the X particles are always out-of-equilibrium. Nevertheless this analysis demonstrates that this parameter can be arbitrarily large and still produce a signi cant baryon asymmetry. The baryon asymmetry produced during warm in ation can set the nal cosmological asymmetry if the source term in the baryon number density equation becomes suppressed after the slow-roll regime and throughout the subsequent the cosmic history. This is, of course, model-dependent, but we can envisage scenarios where some other processes, such as e.g. scatterings, are more suppressed than decays during warm in ation, but become the dominant processes once the slow-roll period is over and radiation becomes the dominant component. In this case X will be driven towards equilibrium after in ation more quickly than through decays and inverse decays and the baryon source will quickly shut down. If it remains in equilibrium until it is su ciently non-relativistic, there should be no signi cant sources of baryon number at late times that could substantially modify the asymmetry 3Potentially electroweak sphalerons [ 33 ] may convert a lepton asymmetry into a baryon asymmetry in a produced during in ation.3 leptogenesis scenario [34]. The smallness of the observed cosmological baryon-to-entropy ratio nB=s can have di erent sources in this scenario: (i) CP-violation may be small, 1, (ii) the X during warm in ation, neXq=s particles may be far from equilibrium, X nB=s 10 10 (see, e.g., ref. [35] for BBN constraints on this ratio). 1. Any combination of these may thus easily explain why 1 or (iii) X particles may be non-relativistic Let us consider in more detail the particular case where the X particles are relativistic during in ation, mX T , as well as fully out-of-equilibrium, X 1. In this case, the baryon-to-entropy ratio is given by nB s ' where gX is the number of degrees of freedom in X, to which each bosonic (fermionic) degree of freedom contributes by a factor 1 (3/4), and g is the e ective total number of relativistic degrees of freedom in the thermal bath. The smallness of the observed baryon-to-entropy ratio may in this case be due to a small amount of CP violation in and/or large deviations from thermal equilibrium, X 1, during in ation, or a combination of both these factors, with an additional suppression by gX =g . This will of course be a model-dependent issue that is not pursued further here. We note that the high temperatures typically attained during warm in ation (see, e.g., refs. [17, 19]) suggest possible implementations of this mechanism within grand uni ed theories [36]. 4 Generation of baryon isocurvature perturbations The interesting aspect that we would like to discuss in more detail is the fact that nB=s depends on the ratio X =H, which, albeit nearly constant, exhibits a small variation during in ation and, moreover, is associated with the in ationary dynamics. As such, this ratio will necessarily acquire uctuations on superhorizon scales due to uctuations in the in aton eld. This implies that the latter will induce both adiabatic curvature uctuations and baryon isocurvature uctuations, the latter corresponding to relative uctuations in the baryon and photon uids, which would be absent if the baryon asymmetry were not generated during in ation. This is similar to the warm baryogenesis scenario proposed in ref. [37], where the baryon asymmetry is directly sourced by the dissipative processes that sustain the thermal bath in warm in ation, but the spectrum of isocurvature modes may be di erent. This yields the interesting possibility of looking for baryon isocurvature modes with CMB and LSS observations to assess whether the observed baryon asymmetry was or not produced during in ation, but also whether its generation was directly linked with dissipative dynamics. Note that other baryogenesis models [38{43], such as A eck-Dine baryogenesis, may also lead to baryon isocurvature modes in the primordial spectrum, but these are in this case uncorrelated with the main adiabatic curvature component, since they are associated with distinct elds. The degree of correlation between isocurvature and adiabatic curvature modes may thus be used to test the warm in ation paradigm itself and isolate a particular mechanism for baryogenesis. Let us then analyze in more detail the spectrum of baryon isocurvature perturbations produced in the present model. Since nB=s will always depend on X in the adiabatic parameters these become ; j j where is the dissipation coe cient. In the slow-roll regime, valid when the slow-roll 1 + Q, where Q = =3H, = (MP2 =2)(V 0=V )2 and = MP2 V 00=V , Combining both equations for R = CRT 4, where CR = ( 2=30)g , we obtain after some algebra, that _ ' V 0( ) 3H(1 + Q) ; R ' 4 3 Q _2 : T H 4 3 ' 2 R C 1 MP H 2 Q (1 + Q)2 ; dynamics under consideration, baryon isocurvature modes will always be generated, independently of the value of X or whether X particles are relativistic during in ation, but let us focus, for concreteness, on the weakly-coupled high-temperature case leading to eq. (3.8). At high temperature, we typically have that Baryon isocurvature modes are characterized by the quantity [44] X / T , such that nB=s / T =H. SB = B B 3 4 R = R (nB=s) nB=s = (T =H) T =H evaluated when the relevant CMB scales become superhorizon during in ation. The subscript `R' used in eq. (4.1) and in the quantities below refers to radiation. We thus need to determine how the uctuations in the ratio T =H are related to in aton uctuations. The dynamics of warm in ation is dictated by the coupled in aton and radiation equations, which for the homogeneous background components are given by  + (3H + ) _ + V 0( ) = 0 ; _R + 4H R = _2 ; where one can see that a warm thermal bath, i.e., T & H, can easily be attained for H MP in the slow-roll regime even if the dissipative ratio Q is not very large. Considering perturbations in the above equation, we obtain after a straightforward calculation, SB = (T =H) T =H 1 ' 4 6 2 1 + Q + 1 Q Q0 1 + Q Q R ; where R ' (H= _) is the gauge-invariant comoving curvature perturbation (written in the = 0 gauge [28]) and primes denote derivatives with respect to the number of efolds of in ation, dNe = Hdt. The dynamical quantity Q0=Q depends on the form of the dissipation coe cient , but it is in general a linear combination of the slow-roll parameters divided by a linear polynomial in Q (see, e.g., refs. [9, 17, 19]). This implies that typically SB=R O(ns 1) O(Ne 1). The Warm Little In aton (WLI) scenario of ref. [19], where the in aton interacts with relativistic fermion elds and / T , constitutes the simplest and most appealing particle physics realization of warm in ation. In this case, we have Q0=Q = (6 2 )=(3 + 5Q). (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) For thermal in aton uctuations and weak dissipation at horizon-crossing, Q have for the scalar spectral index ns Although the exact relation between SB and R is model-dependent, we expect these quantities to be in general proportional with a proportionality constant of this magnitude as argued above. We note that the e ects of baryon and cold dark matter isocurvature modes (CDI) on the CMB spectrum are indistinguishable, although, e.g., the trispectrum may in principle distinguish between them [45]. As such, the e ective contribution to the cold dark matter isocurvature spectrum from the baryon modes above is given by The Planck analysis of CDI modes in the CMB spectrum [46] uses the variable PCDI = B c 2 SB R ISO = PCDI P R + PCDI 2 ; P : R and for the WLI model this gives ISO 10 5 when ns ' 0:96{0:97, which, as argued above, should also give the generic magnitude of the e ect. This is still well below the state-of-theart constraints set by the Planck collaboration, yielding ISO . 10 2 for generic CDI models and using di erent data sets [46]. Particular models of CDI modes can be constrained by an additional order of magnitude, but the predictions of our baryogenesis mechanism are still compatible with the Planck results and may be tested in the future with increased precision measurements. The proportionality between SB and R shows that these quantities are naturally correlated, since both adiabatic and isocurvature modes are generated by thermal in aton uctuations. Their correlation is, however, scale-dependent, since the baryon isocurvature spectral index di ers from the adiabatic spectral index, 1 nI ' d ln PCDI = dNe d ln P dNe R 2n0s 1 ns ' (1 ns) 1 2n0s (1 ns)2 : where 2 = MP4 V 000V 0=V 2. If we consider a quartic chaotic in aton potential, V ( ) = 4 which yields predictions for both ns and the tensor-to-scalar ratio r in excellent agreement with the Planck results within the WLI scenario for warm in ation [19], one nds n0s = (1 ns)2=2, such that (1 nI ) ' 2(1 ns), yielding nI 0:92{0:94 for ns = 0:96{0:97. This implies, in particular, that the Planck constraints for CDI modes fully correlated with the main adiabatic component with nI = ns do not apply in the present scenario. The correction due to the running of the adiabatic spectral index can be signi cant, since in most models n0 s O((1 dissipation at horizon-crossing, Q 1, ns)2). For instance, for thermal in aton uctuations and weak n0s ' 3 2 Note that for larger values of X , the baryon-to-entropy ratio becomes less dependent on the latter, thus suppressing the associated baryon isocurvature modes. Also, when X is non-relativistic, nB=s exhibits an additional dependence on the ratio mX =T that must be taken into account. This may potentially enhance SB, depending on the particular model of warm in ation considered, although relative uctuations in the ratio above are also typically proportional to combinations of slow-roll parameters, and hence necessarily O(ns 1). 5 Summary and future prospects In this work we have shown that particle number densities during a period of warm in ation can follow out-of-equilibrium adiabatic solutions to the Boltzmann equation and which are suppressed relative to the equilibrium value by a factor is the ratio between the thermally averaged decay rate of the particle species X of interest and the in ationary Hubble rate, obtaining a similar result for the case where scattering processes yield the dominant interactions. We have also shown numerically that the corresponding phase space-distributions tend to a stationary con guration with a modi ed equilibrium distribution, given by eq. (2.18), with essentially the above amplitude suppression (up to some distortion due to the momentum-dependence of the decay width) and a slightly smaller e ective temperature. Such adiabatic solutions are achieved after a small number of e-folds that naturally decreases when X increases. This shows that particles can remain out-of-equilibrium throughout warm in ation, with small but not necessarily negligible number densities. To illustrate the impact of this result, we have shown that the observed cosmological baryon asymmetry could be produced by the out-of-equilibrium decay of a generic X particle interacting with the in ationary thermal bath, violating baryon number and C/CP. The smallness of the resulting baryon asymmetry can in this case be a consequence of the small value of X in combination with a small amount of CP violation, and also of the Boltzmann suppression in the case of non-relativistic particles. An interesting feature of this generic scenario is the generation of superhorizon baryon isocurvature modes, correlated with the main adiabatic curvature perturbations. The spectrum of such modes is model-dependent but we have shown that generically the predicted amplitude is below the current constraints on (e ective) cold dark matter isocurvature perturbations by the Planck collaboration. Evidence for such modes could, in the future, constitute a smoking gun for the production of a baryon asymmetry during in ation and, in fact, for a warm in ation scenario, and the detailed properties of the spectrum may help to distinguish the present scenario from the warm baryogenesis mechanism, where a baryon asymmetry is produced directly by the dissipative e ects that sustain the thermal bath during in ation. As for the warm baryogenesis mechanism, the present scenario may also be generalized to an asymmetry in other particles carrying di erent charges, possibly producing an asymmetry in the dark matter sector. The resulting CDI power spectrum has an amplitude larger than the baryonic modes by a factor ( c= B)2 30 [47], which may thus be more easily probed. Our results may have a signi cant impact in other aspects of the in ationary dynamics. For instance, it is often assumed that the particles interacting directly with the in aton and dissipating its energy are in equilibrium with the overall thermal bath. This is required for consistently using equilibrium phase-space distribution functions to compute the associated dissipation coe cients (see, e.g., refs. [23, 25]). Typically this requires such particles to decay faster than the Hubble rate, posing constraints on the coupling constants and particle masses considered that could be relaxed if out-of-equilibrium distribution functions are known. Our results can thus potentially be employed to this e ect, a possibility that we plan to investigate in detail in future work. Another important aspect of warm in ation where the results obtained in this work should be of relevance is the primordial spectrum of curvature perturbations, since the latter depends on the phase space distribution of in aton uctuations. In particular, for weak dissipation at horizon-crossing, predictions for ns and r di er signi cantly for the limiting cases where in aton uctuations are in a vacuum or in a thermal state (i.e., in equilibrium with the overall thermal bath) [17, 19]. Although for strong dissipation this issue becomes less relevant, since dissipation becomes the dominant source of in aton uctuations, agreement with observations in most scenarios considered so far typically favours the Q . 1 regime [17, 19, 48{50]. The dissipative dynamics itself is not su cient to determine the state of in aton uctuations, since other processes in the thermal bath can be responsible for a substantial creation and annihilation of in aton particles. Since the in aton's direct interactions with other particles cannot typically be very strong, it is unlikely that full thermal equilibrium of in aton particles is achieved in general. Nevertheless, production of in aton particles may play a substantial role. The adiabatic solutions obtained in this work could then be used to infer the in aton phase-space distribution at horizon-crossing in di erent models, eliminating the uncertainty in observational predictions. Adiabatic solutions to the Boltzmann equation require both the ratio X =H and the equilibrium distribution neXq to vary slowly compared to the expansion rate, thus requiring both H and T to remain nearly constant. This naturally makes warm in ation the type of dynamics to which such solutions can be applied. We may envisage, however, other cosmological scenarios where, in addition to the early in ationary period in which the observable CMB scales became superhorizon, there are other (shorter) periods of in ation where particle production can sustain the temperature of the cosmic thermal bath. This may be, for instance, the case of second order or crossover cosmological phase transitions, where a scalar eld rolls to a new minimum once the temperature drops below a critical value. It has been shown in ref. [ 51 ] that dissipative friction can make the eld's vacuum energy dominate over the radiation energy density, and in fact prevent the latter from redshifting due to expansion. This may e.g. dilute unwanted thermal relics produced during or after the rst period of warm in ation. Our solutions may, thus, also describe the evolution of the number density of di erent particle species during such periods, which may have a signi cant impact on their present abundances. One can consequently also envisage baryogenesis scenarios along the lines proposed above during these shorter in ationary periods, although these may not easily be tested if the associated isocurvature perturbations are generated at too small scales. We should not exclude other applications of our solutions where both particle production and an expanding environment are involved and have analogous adiabatic conditions as the ones we considered here, e.g., possibly in the quark-gluon plasma formation and subsequent hadronization process under study with heavy-ion collisions experiments. In summary, the novel adiabatic solutions to the Boltzmann equation found in this work can have a signi cant impact in cosmology and shed a new light on several of its presently open questions. Acknowledgments HJEP02(18)63 A.B. is supported by STFC. M.B.-G. is partially supported by \Junta de Andaluc a" (FQM101) and MINECO (Grant No. FIS1016-78198-P). R.O.R. is partially supported by Conselho Nacional de Desenvolvimento Cient co e Tecnologico | CNPq (Grant No. 303377/ 2013-5) and Fundac~ao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro | FAPERJ (Grant No. E-26/201.424/2014). J.G.R. is supported by the FCT Investigator Grant No. IF/01597/2015 and partially by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 and by the CIDMA Project No. UID/MAT/04106/2013. A Boltzmann equation for scattering processes Let us consider, for concreteness, the case where X bosons can annihilate into Y bosons in the thermal bath, XX $ Y Y , although our analysis can be easily generalized to di erent types of particles and processes such as XY $ XY . Labeling the X particles as (1; 2) and the Y particles as (3; 4), the collision term in the Boltzmann equation for the X number density, eq. (2.2) is given by C = Z d 1d 2d 3d 4(2 )4 4 (p1 + p2 p3 p4) jMj 2 [f1f2(1 + f3eq)(1 + f4eq) f3eqf4eq(1 + f1 + f2)] where M is the scattering matrix element, and we take the Y particles as part of the thermal bath. Using conservation of energy and the form of the equilibrium distributions, we can show that: C = fjeq)5 v; such that we may write the term in square brackets in eq. (A.1), after some algebra, as 1 + f3eq + f4eq = f3eqf4eq 1 + f1eq + f eq f1eqf2eq 2 ; 2 f3f4 f1eqf2eq 4f1f2 f1eqf2eq + X i6=j=1;2 fifieq(fj 3 fjeq)5 ; which clearly vanishes in equilibrium. The collision term can then be written in the form (A.1) (A.2) (A.3) (A.4) where the cross section times velocity factor is given by v = (f1eqf2eq) 1 Z 4E1E2 d 3d 4jMj2(2 )4 4 (p1 + p2 p3 p4) f3eqf4eq: (A.5) The cubic terms in the phase-space distribution functions fifieq(fj fjeq) prevent writing the collision term in terms of the number density in a simple form, but may be discarded when fi 1, in which case fi ' e Ei=T (discarding chemical potentials for simplicity). In this case we can write the collision term as where we approximated the energy dependent cross section times velocity of the scattering process by its thermal average [27], as done for the case of decays, C ' h vi(n2X neXq 2) ; h vi = 1 neq 2 X This then yields eq. (2.19) for the Boltzmann equation when scattering processes dominate the collision term. Open Access. 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Mar Bastero-Gil, Arjun Berera, Rudnei O. Ramos, João G. Rosa. Adiabatic out-of-equilibrium solutions to the Boltzmann equation in warm inflation, Journal of High Energy Physics, 2018, 63, DOI: 10.1007/JHEP02(2018)063