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arXiv:gr-qc/0512057v1 9 Dec 2005
Asymptotic behavior of the warm inflation scenario with viscous
pressure
Jos´ e P. Mimoso,1, ∗Ana Nunes,2, †and Diego Pav´ on3, ‡
1Department of Physics, Faculdade de Ciˆ encias da Universidade de Lisboa,
and Centro de F´ ısica Te´ orica e Computacional da Universidade de Lisboa,
Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal
2Department of Physics, Faculdade de Ciˆ encias da Universidade de Lisboa,
and Centro de F´ ısica Te´ orica e Computacional da Universidade de Lisboa,
Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal
3Departamento de F´ ısica, Universidad Aut´ onoma de Barcelona, Facultad de Ciencias
08193 Bellaterra (Barcelona) Spain
(Dated: February 7, 2008)
Abstract
We analyze the dynamics of models of warm inflation with general dissipative effects. We con-
sider phenomenological terms both for the inflaton decay rate and for viscous effects within matter.
We provide a classification of the asymptotic behavior of these models and show that the existence
of a late-time scaling regime depends not only on an asymptotic behavior of the scalar field poten-
tial, but also on an appropriate asymptotic behavior of the inflaton decay rate. There are scaling
solutions whenever the latter evolves to become proportional to the Hubble rate of expansion re-
gardless of the steepness of the scalar field exponential potential. We show from thermodynamic
arguments that the scaling regime is associated to a power-law dependence of the matter-radiation
temperature on the scale factor, which allows a mild variation of the temperature of the mat-
ter/radiation fluid. We also show that the late time contribution of the dissipative terms alleviates
the depletion of matter, and increases the duration of inflation.
PACS numbers: 98.80.Cq, 47.75+f
Keywords:
∗Electronic address: jpmimoso@cii.fc.ul.pt
†Electronic address: anunes@lmc.fc.ul.pt
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‡Electronic address: diego.pavon@uab.es
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I. INTRODUCTION
There is a widespread belief that our Universe, or at least a sufficiently large part of
it causally connected to us, experienced an early period of accelerated expansion, called
inflation. This happened before the primordial nucleosynthesis era could take place and
likely after the Planck period. Many inflationary scenarios have been proposed over the
years (see [1] and references therein). Most of them rely on the dynamics of a self-interacting
scalar field (the “inflaton”), whose potential overwhelms all other forms of energy during
the relevant period. They generally share the unsatisfactory feature of driving the Universe
to such a super-cooled state that it becomes necessary to introduce an ad hoc mechanism
-termed “reheating”- in order to raise the temperature of the universe to levels compatible
with primordial nucleosynthesis. Therefore this reheating phase appears as a subsequent,
separate stage mainly justified by the need to recover from the extreme effects of inflation
[2] during which a rather elaborate process of multifield parametric resonances followed by
particle production [3, 4, 5] takes place.
As an alternative some authors have looked for inflationary scenarios leading the universe
to a moderate temperature state at the end of the superluminal stage so that the reheating
phase could be dispensed with altogether. It was advocated that this can be accomplished
by coupling the inflaton to the matter fields in such a way that the decrease in the energy
density of the latter during inflation is somewhat compensated by the decay of the inflaton
into radiation and particles with mass. This would happen when the inflaton rolls down its
potential, but keeping the combined pressure of the inflaton and radiation negative enough
to have acceleration. This kind of scenario, known as “warm inflation” as the radiation
temperature never drops dramatically, was first proposed by Berera [6, 7]. It now rests on
solid grounds since it has been forcefully argued in a series of papers that indeed the inflaton
can decay during the slow-roll (see, e.g. [8, 9, 10] and references therein). Besides, this
scenario has other advantages, namely: (i) the slow-roll condition˙φ2≪ V (φ) can be fulfilled
for steeper potentials, (ii) the density perturbations generated by thermal fluctuations may
be larger than those of quantum origin [11, 12, 13], and (iii) it may provide a very useful
mechanism for baryogenesis [14].
To simplify the study of the dynamics of warm inflation, previous works treated the par-
ticles created in the decay of the inflaton purely as radiation, thereby ignoring the existence
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of particles with mass in the decay fluid. Here, we will go a step beyond by taking into
account the presence of of particles with mass as part of the decay products, and give a
hydrodynamical description of the mixture of massless and non-massless particles by an
overall fluid with equation of state p = (γ −1)ρ, where the adiabatic index γ is bounded by
1 ≤ γ ≤ 2.
On very general grounds, this fluid is expected to have a negative dissipative pressure,
Π, that somewhat quantifies the departure of the fluid from thermodynamical equilibrium,
which we will consider to be small but still significant. This viscous pressure arises quite
naturally via two different mechanisms, namely: (i) the inter-particle interactions [15], and
(ii) the decay of particles within the matter fluid [16].
A well known example of mechanism (i) of prime cosmological interest is the radiative
fluid, a mixture of massless and non-massless particles, as it plays an essential role in the
description of the matter-radiation decoupling in the standard cosmological model [17, 18,
19].
A sizeable viscous pressure also arises spontaneously in mixtures of different particles
species, or of the same species but with different energies -a typical instance in laboratory
physics is the Maxwell-Boltzmann gas [20]. One may think of Π as the internal “friction”
that sets in as a consequence of the diverse cooling rates in the expanding mixture, something
to be expected in the matter fluid originated by the decay of the inflaton.
As for mechanism (ii), it is well known that the decay of particles within a fluid can be
formally described by a bulk dissipative pressure Π. This is only natural because the decay
is an entropy–producing scalar phenomenon associated with the spontaneous enlargement
of the phase space (we use the word “scalar” in the sense of irreversible thermodynamics)
and the bulk viscous pressure is also a scalar entropy–producing agent. There is an ample
body of literature on the cosmological applications of this analogy -see e.g. [16], [21]. In
the case of warm inflation, it is natural to expect that, at least, some species of particles
directly produced by the decay of the inflaton will, in turn, decay into other, lighter species.
In this connection, it has been proposed that the inflaton may first decay into a heavy boson
χ which subsequently decays in two light fermions ψd [22]. This is an obvious source of
entropy, and therefore it can be modelled by a dissipative bulk pressure Π.
Our purpose in this paper is to generalize the usual warm inflationary scenario by in-
troducing the novel elements mentioned above, namely the decay of the scalar field into a
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fluid of adiabatic index γ rather than just radiation, and especially the dissipative pressure
of this fluid, irrespective of the underlying mechanism. We will not dwell on the difficult
question of the quantum, non–equilibrium thermodynamical problem underlying warm in-
flation [8, 23, 24, 25, 26, 27, 28], but rather take a phenomenological approach similar to
that considered in several works [29, 30, 31, 32, 33] (which can be traced back to the early
studies of inflation [34]). Instead of adopting a model building viewpoint and looking for
the implications of specific assumptions, we aim at identifying typical features of models
that yield interesting asymptotic behavior. We resort to a qualitative analysis of the corre-
sponding autonomous system of differential equations using the approach developed in [35]
that allows the consideration of arbitrary scalar field potentials. We will characterize the
implications of allowing for various forms of the rate of decay of the scalar field, as well
as for various forms for the dissipative pressure. We consider, for instance, models with
scalar field potentials displaying an asymptotic exponential behavior. These arise naturally
in generalized theories of gravity emerging in the low-energy limits of unification proposals
such as super-gravity theories or string theories [36]. On the one hand, after the dimen-
sional reduction to an effective 4-dimensional space-time and the subsequent representation
of the theories in the so-called Einstein frame typical polynomial potentials become expo-
nential [37, 38]. On the other hand, the theories are then characterised by the existence of a
scalar field that couples to all non-radiation fields, with the coupling depending, in general,
on the scalar field. The simplest example of these features can be found in the so-called
non-minimal coupling theories. We provide a classification of the relevant global dynamical
features of the cosmological model associated with those possible choices. A limited account
of some of the results of the present work was reported in [39].
One question we address is whether non-trivial scaling solutions [33, 35, 40, 41] (hereafter
simply termed scaling solutions) exist, i.e., solutions where the ratio of energies involving
the matter fluid and scalar field keep a constant ratio. Another class of solutions refered in
the literature as having a scaling asymptotic behavior are those for which both the energy
density of the scalar field and that of the matter fluid decay with different power laws of the
scale factor of the universe [40, 45, 46]. In this latter case one of the components eventually
dominates and thus the ratio of their energy densities becomes evanescent, in clear contrast
to the case of the non-trivial scaling solutions. We shall term these solutions as trivial scaling
solutions to contrast them with the previous ones sometimes dubbed tracker solutions. The
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trivial case arises in association with scalar field potentials of a power-law type and, as we
shall see, they occur when the scalar field decays have the the same type of time-dependences
as those required by the (non-trivial, tracking) scaling solutions.
One of the reasons why non-trivial scaling solutions are important is that they provide an
asymptotic stationary regime for the energy transfer between the scalar field and radiation.
This stationary (sometimes termed “quasi-static”) regime is an assumption in the standard
treatment of warm inflation [11] to evaluate the temperature of matter in the final stages.
On the other hand, introducing this class of solutions in the kinetic analysis of interacting
fluids [42, 43] leads to an alternative to the usual Γ ≫ 3H case, generalizing the example of
Ref. [44] where temperature of the matter (radiation) bath is nearly constant.
We show that this class of scaling behavior depends not only on the asymptotic form of
the inflaton [35], but also on having an appropriate time-dependent rate for the scalar field
decay. The additional consideration of bulk viscosity, besides being a natural ingredient in
models with one or more matter components as well as in models with inter-particle decays,
facilitates the Universe to have a late time de Sitter expansion.
An outline of this work is as follows. Section II studies the model underlying the original
idea of the warm inflation proposal, namely the model in which the inflaton field decays into
matter during inflation thus avoiding the need for the post-inflationary reheating. This decay
is characterized by a rate Γ which we shall initially assume to be a constant. Our results
though will argue in favour of a varying Γ and we shall thus consider the case where Γ ∝ H.
This yields late time scaling solutions whenever the scalar field potentials asymptotes to an
exponential behavior. This happens regardless of the slope of the potential. Subsequently,
section IIII, analyses more realistic models where a bulk viscous pressure term Π is also
present in the equation of state of matter. We first envisage the usual form Π = −3ζH for
that pressure and, subsequently, analyse a general model with both a varying rate of decay
and a general form for the bulk viscosity Π = −3ζ ραHβ, where 2α + β = 2 on dimensional
grounds. Finally, section IV provides a discussion of our results.
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II.THE DYNAMICS OF WARM INFLATION
A. Warm inflation with constant Γ
We consider a spatially flat Friedmann-Robertson-Walker universe filled with a self-
interacting scalar field and a perfect fluid consisting of a mixture of matter and radiation,
such that the former decays into the latter at some constant rate Γ. For the time being
we ignore the dissipative pressure. We also neglect radiative corrections to the inflaton
potential [12, 24]. The corresponding system of equations reads
3H2= ρ +
˙φ2
2+ V (φ),
2(˙φ2+ γρ),
(1)
˙H = −1
¨φ = −(3H + Γ)˙φ − V′(φ),
(2)
(3)
where here and throughout we use units in which 8πG = c = 1. The first two are Einstein’s
equations, the third describes the decay of the inflaton. From these, it follows the energy
balance for the matter fluid,
˙ ρ = −3γ H ρ + Γ˙φ2.(4)
As usual H ≡ ˙ a/a denotes the Hubble factor.
To cast the corresponding autonomous system of four differential equations it is expedient
to introduce the set of normalized variables
x2=
˙φ2
6H2
V (φ)
3H2
Γ
3H,
(5)
y2=
(6)
r =
(7)
along with the new time variable N = lna. Thus we get
x′= x (Q − 3(1 + r)) − W(φ)y2,
y′= (Q + W(φ)x)y ,
(8)
(9)
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r′= rQ , (10)
φ′=
√6x , (11)
where a prime means derivative with respect to N, and the definitions
W(φ) =
?
3
2
?∂φV
V
?
(12)
and
Q =3
2
?
2x2+ γ (1 − x2− y2)
?
, (13)
as well as ρ/(3H2) = 1 − x2− y2were used. Equation (11) was first considered in [35],
and is crucial for the consideration of general potentials V (φ) besides the particular case of
the exponential potential. The function Q defined by Eq. (13) is related to the deceleration
parameter q = −¨ aa/˙ a2by Q = 1 + q.
The special case where r = 0, naturally, corresponds to the absence of interaction between
the scalar field and the perfect fluid, and it is an invariant manifold of the dynamical
system (8)-(11). It is appropriate to refer here its major features in order to better appreciate
the implications of the decay of the scalar field (see Table I).
We distinguish the fixed points of the system into those occurring for finite values of φ
and those associated with the asymptotic limit, φ → ∞. In the former case, i.e., for finite
φ = φ∗, the fixed points always require the vanishing of the kinetic energy of the scalar field
(x = 0). They are located at the origin (x = y = 0), and at (x = 0,y = 1), on the frontier
of the phase space domain x2+ y2= 1, which is an invariant manifold. For x = 0, y = 0,
the potential must have a vanishing critical point at φ∗, a case that cannot be dealt with
the variables in use, but it is well-know that if φ∗is a minimum at the origin, then it is
a stable point and the scale factor evolves as a(t) ∝ t2/(3γ)[46, 50]. The fixed points on
x2+ y2= 1 are given by x = 0, y = 1 and require that W = 0. This means that they can
only occur in association with extrema of the potential. Their stability is defined by the
sign of W′(φ∗), where φ∗is the value of φ where V′(φ) (and hence W) vanishes. When V
has a non-vanishing minimum and, hence W′> 0, the critical point is a stable node. When
V has a maximum and, hence W′< 0, we have a saddle point (an unstable fixed point).
These fixed points correspond to the de Sitter exponential behavior and are accompanied
by the depletion of the matter component (ρ = 0).
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To study the critical points that occur at φ → ∞ (which we shall label φ∞), we carry
out the regularization produced by the change of variable ψ = 1/φ. Then Eq. (11) becomes
ψ′= −√6xψ2, (14)
and the critical points correspond to either to x = 0, as previously seen, or ψ = 0. The
φ∞critical points depend on the asymptotic behavior of V (φ) [35]. If V (φ) exhibits some
non-vanishing asymptotic value we have again x = 0, y = 1 corresponding to a cosmological
constant and, hence, to a de Sitter late-time behavior. If V (φ) asymptotes towards the
exponential potential, say V ∝ e−λφ, with λ constant, there are several possible fixed values
dependent on the ratio between λ2and γ (see, for instance, [48] for details). There are
unstable fixed points on the invariant manifolds bounding the phase-space domain for all
possible choices of both W = −
x = 0 and y = 0, which is a saddle and corresponds to a(t) ∝ t2/(3γ), (ii) two solutions
dominated by the scalar field kinetic energy at x = ±1 and y = 0 which are either unstable
nodes or saddles, and correspond to the stiff behavior a(t) ∝ t1/3, φ∞(t) ∼ lntK0, where
K0is an arbitrary constant defining the scalar field initial velocity. There is another fixed
?
3/2λ and γ, namely: (i) a matter dominated solution at
point on the x2+ y2= 1 boundary representing a scalar field dominated solution, when
W2< 9. This fixed point is stable when W2< 9γ/2, and unstable otherwise (saddle). Thus
for W2> 9γ/2 (i.e., λ2> 3γ), there is a stable fixed point in the interior of the phase space
domain. This latter point corresponds to scaling behavior between the matter and scalar
field energy-densities [38, 40, 41, 47, 48, 49]. This attractor solution is characterized by
a(t) ∝ t2/3γand φ − φ0= lnt±2/λ.
There are also trivial scaling solutions for which ρφ∝ a−nand ρ ∝ a−m, where n > m
are positive constants, when [46]
V (φ) = A2
?
1 −n
m
?2?6 − n
2n
??φ
A
?̟
(15)
where
̟ =
2n
n − m.
(16)
Coming back to the model that includes the interaction and thus letting r be non-
vanishing, we immediately see from Eq. (10) that, along the r-direction, all the points are
singular points if and only if Q = 0. For finite values of φ, as x = 0 at the fixed points, this
requires once more y2= 1 so that the singular points are associated with ρ = 0, i.e., with the
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depletion of the matter component. Moreover, as in the r = 0 case, these singular points are
extrema of the potential V (φ). They correspond to a de Sitter behavior (a(t) ∝ e
φ = φ0 constant) and are either stable or unstable depending on the extremum being a
√
V (φ0)/3)t,
minimum (W′> 0) or a maximum (W′< 0). In fact, at the singular points corresponding
to extrema of the potential V (φ), the eigenvalues found in the linear stability analysis are
µy = −3γ ,
µx,φ = −3(1 + r)
(17)
2
1 ±
?
?
?
?1 −4√6W′(φ0)
9(1 + r)2
.(18)
On the other hand, we no longer have fixed points at x = 0, y = 0 (unless γ = 0 which
corresponds to the perfect fluid being a cosmological constant). This happens because the
system then evolves along the r-axis towards r → ∞, a behavior that can only be prevented
by the existence of a positive minimum of the potential V (φ). At φ → ∞ the system does
not exhibit scaling solutions anymore. The only fixed points allowed in this asymptotic
limit are those associated with a non-vanishing, asymptotically flat potential, which thus
corresponds to the de Sitter exponential behavior.
Accelerated expansion corresponds to the region of the phase space where Q < 1, so that
3γy2− 3(2 − γ)x2> 3γ − 2 . (19)
This condition does not carry any dependence either on r or φ. Thus we may restrict our
discussion to a (x,y) projection of the phase space. The condition (19) defines for 1 < γ < 2
the region between the upper branch of the hyperbolae 3γy2− 3(2 − γ)x2= (3γ − 2) and
the boundary x2+ y2= 1 of the phase space domain (see Figure 1). The asymptotes of the
hyperbolae are y = ±
becomes progressively smaller. In fact the region shrinks vertically towards the x = 0, y = 1
?
(2 − γ)/γ, and we see that, as γ increases, the inflationary region
point and it reduces to it in the limit case of γ = 2.
In Ref. [11] the end of inflation is given by the condition ρφ ≃ ργ and this event is
associated with the beginning of the matter (radiation) domination. As it becomes apparent
from the above discussion, the condition for the end of inflation, Q = 1, is more general and
does not strictly require matter domination. Taylor and Berera’s condition [11] corresponds
to the end of slow-roll inflation (i.e.,˙φ2≃ 0 ≃ x) and is extended, in the present study, to
general γ-fluids as
ρm≃
2
3γ − 2ρφ. (20)
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The independence on r of the size of inflationary region should not though be understood
as the interaction having no effect on inflation. From Eqs (17) and (18) we see that the
eigenvalues of the linearized system at the fixed points carry a dependence on r which is
such that it renders the minima of the potential more stable and the maxima less unstable
(as if the potential became shallower). Thus the transfer of energy from the scalar field to the
perfect fluid favors inflation in that the system spends a longer time in the neighborhood
of the extrema of the potential. This is exactly what is meant to happen in the warm
inflation scenario where it is assumed that slow roll holds and argued that r allows for
steeper potentials than those required in its absence. As discussed in [11], it is a simple
matter to see that the slow-roll condition on˙φ
˙φ = −
V′
3H(1 + r)≃ −V′
3rH,
(21)
is easier to satisfy if the scalar field decays, that is, if r > 1 and much easier if r ≫ 1.
The fact that r increases indefinitely in the present model is a consequence of its definition,
and merely translates the fact that, unless the system is trapped at a non-vanishing minimum
of V (φ), H decreases towards zero. Since this is a direct result of assuming a constant Γ, we
consider in the next section a more appropriate model where Γ decreases as the Universe’s
expansion proceeds.
B.Warm inflation with Γ ∝ H
We assume that Γφ = 3Γ∗H where Γ∗ is a dimensionless, positive constant. As H is
expected to be a non-increasing function of time in an expanding universe, this is a simple
choice for the time dependence of Γφsuch that the decays have a stage of maximum intensity
(when inflation occurs) followed by a progressive attenuation until it vanishes altogether.
Now r = Γ∗ is a constant parameter and the dynamical system reduces to the three
equations
x′= x [Q − 3(1 + r)] − W(φ)y2,
y′= [Q + W(φ)x]y ,
√6x ,
(22)
(23)
φ′=
(24)
where Q is still given by Eq. (13) We see that these equations are analogous to those of
the r = 0 case of the previous section with a different coefficient on the linear term in x
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of Eq. (22). Thus the basic qualitative dynamical features remain the same as those found
for that model (see Table II). The decay of the scalar field though introduces two major
consequences worthing to be emphasized.
Besides the fact that the origin x = 0, y = 0 is again a fixed point associated with the
vanishing of the scalar field’s energy and, hence, corresponds to the matter domination, the
interaction given by a non-vanishing r has the relevant effect (already found in the constant
Γ model) that the stability of the minima is reinforced and that the maxima become less
unstable. Moreover, the scalar field decay prevents the existence of the fixed points at
x = ±1, y = 0, that would correspond to a behavior completely dominated by the scalar
field’s kinetic energy (and which was, therefore, associated with a stiff behavior in the r = 0
case).
The other major effect of the interaction arises when we look for fixed points with x2+y2<
1 at φ → ∞. Now, we find that there are always attracting scaling solutions for potentials
that have an asymptotic exponential behavior, that is, for potentials for which W → const
when φ → ∞ [35]. Moreover, this happens independently of the steepness of the late time
exponential behavior which is a remarkable effect of the present model for the transfer of
energy from the scalar field to the matter.
Indeed the latter solutions are given by the roots of the system of equations
(u − 1)(u −a
b) − ru = 0 ,(25)
and
cos2θ =
λ2
6(1 + r)2ξ2,(26)
where u = ξ2and θ are polar coordinates, x = ξ cosθ and y = ξ sinθ, and where we have
defined
a =
γ
2
(27)
b =
λ2
6(1 + r)2, (28)
as well as W∞= −
r is such that equations (25) and (26) always have one non-vanishing root within the range of
?
3/2λ. It is a simple matter to conclude that the effect of a non-vanishing
allowed values for ξ and for cosθ, and hence there are scaling solutions regardless of the ratio
between λ2and 3γ. Furthermore linear stability analysis shows that the scaling solutions
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are stable. It is important nevertheless to remark that although scaling solutions emerge
for any ratio of λ2/γ, the way that the γeffindex associated with the effective equation of
state inducing the power-law scaling behavior is shifted from the corresponding γ value of
the scaling solutions in the absence of decays depend on λ2being larger or smaller than γ.
Assuming the potential to be asymptotically given by V ∝ e−λφ, the latter solutions are
a(t) ∝ tA, φ − φ0= lnt±2/λ, where A is given in implicit form by
?
3γ
3γA −
2
??
A −2
λ2(1 + r)
?
−4r
λ2= 0 . (29)
Notice that we can define γeff= 2/(3A). A linear expansion about r = 0 in the neighborhood
of the scaling solution (for λ2?= 3γ) yields
A =
2
3γ
1 +
2
λ2
?
2
3γ−
2
λ2
?r
, (30)
when λ2?= 3γ. So the decays have the effect of increasing (resp. decreasing) the scale factor
rate of expansion with regard to the r = 0 case if λ2> 3γ (resp. λ2< 3γ). In particular
we can see that the scaling behavior can be inflationary, for cases where this would not
happen in the absence of decays. For instance, taking γ = 4/3 and λ2> 4, the condition
for the scaling solution to be inflationary is 1 + r > λ2/4 > 1. Thus, in this model, the
solutions yield endless power-law inflation even for a modest scalar field decay, provided that
the asymptotic behavior of the potential is steep enough, i.e., λ2> 3γ (> 4 in the present
example).
Naturally, besides the scaling solutions, there can also be fixed points corresponding to
de Sitter behavior x = 0, y = 1, whenever the scalar field potential exhibits an asymptotic,
non-vanishing constant value. However, when the potential is asymptotically exponential,
that there are no fixed points on the boundary x2+y2= 1 at φ∞in contrast to what happens
in the r = 0 case.
From a thermodynamical viewpoint, the above scaling solutions are particularly interest-
ing. In a universe with two components, it can be shown [42, 43] that the temperature of
each of the components satisfies the equation
˙Ti
Ti
= −3˙ a
a
?
1 −Γi
3H
?∂pi
∂ρi
+
ni˙ si
∂ρi/∂Ti
, (31)
where i = 1 or 2, nidenotes the number density of particles of the i-species, Γitheir rate
of decay, and Ti the temperature of this component. In the important case of particle
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production with constant entropy per particle, ˙ si= 0, we also have (ρ1+ p1)Γ1= −(ρ2+
p2)Γ2. Thus, taking the first component to be the matter/radiation fluid and the second to
be the inflaton scalar field, we have
Γφ=(ρ + p)
˙φ2
Γm/r∝ Γm/r. (32)
As (ρ + p)/˙φ2= γ(1 − x2− y2)/2x2is a constant in the scaling solutions, Γφ= 3rH, with
r a constant, implies Γm/r= 3σH, where σ is another constant that depends both on r and
on the location of the scaling solution. This yields a temperature of the matter/radiation
component evolving as a power-law T ∝ a−3(γ−1)(1−σ). Thus for σ close to 1, the temperature
of the matter/radiation remains quasi-static, whereas for σ > 1 (resp. σ < 1), it increases
(resp. decreases). Notice also that for σ ≃ 0, we recover the temperature law for perfect
fluids without dissipative effects. Provided we guarantee enough inflation, r need not be
very large (contrary to what is usually assumed to facilitate slow-rolling).Indeed, the
temperature of the radiation at the end of inflation is
Tend= Tbeginninge−N(1−σ), (33)
where N is the number of e-foldings. Thus, a value of sigma lower but sufficiently close to
1 has the potential to avoid a serious decrease of the temperature of the universe. As
σ =
3x2
∗
2(1 − x2
∗− y2
∗)r ,(34)
at the scaling solutions, we see that r need not be very large to ensure that σ ∼ 1.
Trivial scaling solutions generalizing those given Eq. (15) in the r = 0 case, arise in these
models for scalar field potentials of the form
V (φ) = A2
?
1 −n
m
?2?6(1 + r) − n
2n
? ?φ
A
?̟
, (35)
where ̟ is still given by Eq. (16) and A = Ar=0/√1 + r. For the potential to be positive
one also requires 0 < m < n < 6(1 + r). The only difference with respect to the r = 0
case lies in the dependence on r of the constant factor multitplying φ̟, and translates the
fact that there is now a different distribution of the scalar field energy density between its
kinetic and potential parts. Indeed, we have that
V (φ) =
?6(1 + r)
n
− 1
?
˙φ2
(36)
14
Page 15
which shows that the extra damping of the kinetic energy part when r ?= 0, as expected.
As in the r = 0 case, the possible emergence of the trivial scaling behavior in association
with monomial potentials (that might or not be part of double wells) happens when the
matter fluid is already dominating and is thus of a lesser importance in the context of warm
inflation.
III.WARM INFLATION WITH BULK VISCOSITY
One of the main purposes of the present work is to assess the implications for warm
inflation of the presence of a viscous pressure, Π, in the matter component, so that the total
fluid pressure is p = (γ − 1)ρ+ Π. We may assume the expression Π = −3ζH which, albeit
some causality caveats, is the simplest one may think of and has been widely considered in
the literature [51, 52, 53]. If the mixture of massive particles and radiation is taken as a
radiative fluid, it is admissible to adopt ζ ∝ ργτ, where τ denotes the relaxation time of the
dissipative process. For the hydrodynamic approach to apply the condition tcolH < 1 should
be fulfilled. Since τ ∝ tcol(a reasonable assumption) and the most obvious time parameter
in this description is H−1, one concludes that Π ≃ −β ργ, with 0 < β < 1.
This modifies the field equations (2) and (4) which now read
˙H = −
˙φ2+ γρ + Π
2
?
(37)
˙ ρ = −3γ +Π
ρ
?
H ρ + Γ˙φ2, (38)
while equations (1) and (3) remain in place and thus the dynamical system is now
x′= x [Q − 3(1 + r)] − W(φ)y2,
y′= [Q + W(φ)x]y ,
??Π′
Π
r′= rQ
√6x ,
(39)
(40)
χ′= χ
?
+ 2Q
?
, (41)
(42)
φ′=
(43)
where
χ = Π/(3H2),(44)
15
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