#### 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.
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[1] A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys.
Lett. B 91 (1980) 99 [INSPIRE].
Not. Roy. Astron. Soc. 195 (1981) 467 [INSPIRE].
Problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
[2] K. Sato, First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon.
[3] A.H. Guth, The In ationary Universe: A Possible Solution to the Horizon and Flatness
[4] A. Albrecht and P.J. Steinhardt, Cosmology for Grand Uni ed Theories with Radiatively
Induced Symmetry Breaking, Phys. Rev. Lett. 48 (1982) 1220 [INSPIRE].
[5] A.D. Linde, A New In ationary Universe Scenario: A Possible Solution of the Horizon,
Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B 108
(1982) 389 [INSPIRE].
[6] A. Berera and L.-Z. Fang, Thermally induced density perturbations in the in ation era, Phys.
Rev. Lett. 74 (1995) 1912 [astro-ph/9501024] [INSPIRE].
[7] A. Berera, Warm in ation, Phys. Rev. Lett. 75 (1995) 3218 [astro-ph/9509049] [INSPIRE].
[8] A. Berera, I.G. Moss and R.O. Ramos, Warm In ation and its Microphysical Basis, Rept.
Prog. Phys. 72 (2009) 026901 [arXiv:0808.1855] [INSPIRE].
[9] M. Bastero-Gil and A. Berera, Warm in ation model building, Int. J. Mod. Phys. A 24
(2009) 2207 [arXiv:0902.0521] [INSPIRE].
[10] A. Berera, Interpolating the stage of exponential expansion in the early universe: A possible
alternative with no reheating, Phys. Rev. D 55 (1997) 3346 [hep-ph/9612239] [INSPIRE].
in ationary dynamics in quantum
[hep-ph/9904409] [INSPIRE].
(2007) 007 [astro-ph/0701302] [INSPIRE].
[arXiv:0905.3500] [INSPIRE].
HJEP02(18)63
[15] R.O. Ramos and L.A. da Silva, Power spectrum for in ation models with quantum and
thermal noises, JCAP 03 (2013) 032 [arXiv:1302.3544] [INSPIRE].
[arXiv:1303.3508] [INSPIRE].
in ation, JCAP 12 (2014) 008 [arXiv:1408.4391] [INSPIRE].
Lett. 117 (2016) 151301 [arXiv:1604.08838] [INSPIRE].
[20] A. Berera, M. Gleiser and R.O. Ramos, A First principles warm in ation model that solves
the cosmological horizon/ atness problems, Phys. Rev. Lett. 83 (1999) 264 [hep-ph/9809583]
[21] A. Berera and R.O. Ramos, Construction of a robust warm in ation mechanism, Phys. Lett.
[INSPIRE].
B 567 (2003) 294 [hep-ph/0210301] [INSPIRE].
hep-ph/0603266 [INSPIRE].
[22] I.G. Moss and C. Xiong, Dissipation coe cients for supersymmetric in atonary models,
[23] M. Bastero-Gil, A. Berera and R.O. Ramos, Dissipation coe cients from scalar and fermion
quantum
eld interactions, JCAP 09 (2011) 033 [arXiv:1008.1929] [INSPIRE].
[24] M. Bastero-Gil, A. Berera and J.G. Rosa, Warming up brane-antibrane in ation, Phys. Rev.
D 84 (2011) 103503 [arXiv:1103.5623] [INSPIRE].
[25] M. Bastero-Gil, A. Berera, R.O. Ramos and J.G. Rosa, General dissipation coe cient in
low-temperature warm in ation, JCAP 01 (2013) 016 [arXiv:1207.0445] [INSPIRE].
[26] R. Cerezo and J.G. Rosa, Warm In ection, JHEP 01 (2013) 024 [arXiv:1210.7975]
[INSPIRE].
[27] E.W. Kolb and M.S. Turner, The Early Universe, Front. Phys. 69 (1990) 1 [INSPIRE].
[28] D. Lyth and A. Liddle, The Primordial Density Perturbation: Cosmology, In ation and the
Origin of Structure, Cambridge University Press, Cambridge (2009).
[arXiv:1507.07834] [INSPIRE].
the relativistic Boltzmann equation in the Friedmann-Lema^tre-Robertson-Walker spacetime,
Phys. Rev. D 94 (2016) 125006 [arXiv:1607.05245] [INSPIRE].
Phys. Rev. D 30 (1984) 2212 [INSPIRE].
HJEP02(18)63
(1986) 45 [INSPIRE].
astro-ph/0601514 [INSPIRE].
Rev. D 20 (1979) 2484 [INSPIRE].
(1990) 158 [INSPIRE].
Rev. D 44 (1991) 970 [INSPIRE].
[36] D.V. Nanopoulos and S. Weinberg, Mechanisms for Cosmological Baryon Production, Phys.
712 (2012) 425 [arXiv:1110.3971] [INSPIRE].
In ationary Universe, Phys. Lett. B 216 (1989) 20 [INSPIRE].
[38] M.S. Turner, A.G. Cohen and D.B. Kaplan, Isocurvature Baryon Number Fluctuations in an
[39] J. Yokoyama and Y. Suto, Baryon isocurvature scenario in in ationary cosmology: A particle
physics model and its astrophysical implications, Astrophys. J. 379 (1991) 427 [INSPIRE].
[40] S. Mollerach, On the Primordial Origin of Isocurvature Perturbations, Phys. Lett. B 242
[41] M. Sasaki and J. Yokoyama, Initial condition for the minimal isocurvature scenario, Phys.
[42] J. Yokoyama, Formation of baryon number uctuation in supersymmetric in ationary
cosmology, Astropart. Phys. 2 (1994) 291 [INSPIRE].
[43] K. Koyama and J. Soda, Baryon isocurvature perturbation in the A eck-Dine baryogenesis,
Phys. Rev. Lett. 82 (1999) 2632 [astro-ph/9810006] [INSPIRE].
[44] D.H. Lyth, C. Ungarelli and D. Wands, The Primordial density perturbation in the curvaton
scenario, Phys. Rev. D 67 (2003) 023503 [astro-ph/0208055] [INSPIRE].
[45] D. Grin, D. Hanson, G.P. Holder, O. Dore and M. Kamionkowski, Baryons do trace dark
matter 380,000 years after the big bang: Search for compensated isocurvature perturbations
with WMAP 9-year data, Phys. Rev. D 89 (2014) 023006 [arXiv:1306.4319] [INSPIRE].
[46] Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XX. Constraints on in ation,
Astron. Astrophys. 594 (2016) A20 [arXiv:1502.02114] [INSPIRE].
[47] Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological
parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE].
In ation with CMB data, arXiv:1710.10008 [INSPIRE].
scalar
[11] A. Berera , Warm in ation at arbitrary adiabaticity: A Model, an existence proof for eld theory , Nucl. Phys. B 585 ( 2000 ) 666 [12] L.M.H. Hall , I.G. Moss and A. Berera , Scalar perturbation spectra from warm in ation , Phys. Rev. D 69 ( 2004 ) 083525 [ astro -ph/0305015] [INSPIRE].
[13] I.G. Moss and C. Xiong , Non-Gaussianity in uctuations from warm in ation , JCAP 04 [14] C. Graham and I.G. Moss , Density uctuations from warm in ation , JCAP 07 ( 2009 ) 013 [16] S. Bartrum , A. Berera and J.G. Rosa , Warming up for Planck , JCAP 06 ( 2013 ) 025 [17] S. Bartrum , M. Bastero-Gil , A. Berera , R. Cerezo , R.O. Ramos and J.G. Rosa, The importance of being warm (during in ation) , Phys. Lett. B 732 ( 2014 ) 116 [18] M. Bastero-Gil , A. Berera , I.G. Moss and R.O. Ramos , Theory of non-Gaussianity in warm [19] M. Bastero-Gil , A. Berera , R.O. Ramos and J.G. Rosa , Warm Little In aton, Phys. Rev.
[29] D. Bazow , G.S. Denicol , U. Heinz , M. Martinez and J. Noronha , Analytic solution of the Boltzmann equation in an expanding system , Phys. Rev. Lett . 116 ( 2016 ) 022301 [30] D. Bazow , G.S. Denicol , U. Heinz , M. Martinez and J. Noronha , Nonlinear dynamics from [31] J.M. Cline , Baryogenesis, hep-ph/0609145 [INSPIRE].
[32] A.D. Sakharov , Violation of CP Invariance, c Asymmetry and Baryon Asymmetry of the Universe, Pisma Zh . Eksp. Teor. Fiz . 5 ( 1967 ) 32 [INSPIRE].
[33] F.R. Klinkhamer and N.S. Manton , A Saddle Point Solution in the Weinberg-Salam Theory , [34] M. Fukugita and T. Yanagida , Baryogenesis Without Grand Uni cation , Phys. Lett. B 174 [35] B. Fields and S. Sarkar , Big-Bang nucleosynthesis (2006 Particle Data Group mini-review) , [37] M. Bastero-Gil , A. Berera , R.O. Ramos and J.G. Rosa , Warm baryogenesis, Phys. Lett. B [48] M. Benetti and R.O. Ramos , Warm in ation dissipative e ects: predictions and constraints from the Planck data , Phys. Rev. D 95 ( 2017 ) 023517 [arXiv: 1610 .08758] [INSPIRE].
[49] M. Bastero-Gil , S. Bhattacharya , K. Dutta and M.R. Gangopadhyay , Constraining Warm [50] R. Arya , A. Dasgupta , G. Goswami, J. Prasad and R. Rangarajan , Revisiting CMB constraints on Warm In ation , arXiv: 1710 .11109 [INSPIRE].
[51] S. Bartrum , A. Berera and J.G. Rosa , Fluctuation-dissipation dynamics of cosmological elds , Phys. Rev. D 91 ( 2015 ) 083540 [arXiv: 1412 .5489] [INSPIRE].