# Warm inflection

Rafael Cerezo<EMAIL_ADDRESS>Departamento de Física Teórica y del Cosmos and
CAFPE
Universidad de Granada, Granada-18071, Spain  João G. Rosa<EMAIL_ADDRESS>Departamento de Física da Universidade de Aveiro and I3N
Campus de Santiago, 3810-183 Aveiro, Portugal

###### Abstract

While ubiquitous in supersymmetric and string theory models, inflationary
scenarios near an inflection point in the scalar potential generically require
a severe fine-tuning of a priori unrelated supersymmetry breaking effects. We
show that this can be significantly alleviated by the inclusion of dissipative
effects that damp the inflaton’s motion and produce a nearly-thermal radiation
bath. We focus on the case where the slow-rolling inflaton directly excites
heavy virtual modes that then decay into light degrees of freedom, although
our main qualitative results should apply in other regimes. Furthermore, our
analysis shows that the minimum amount of dissipation required to keep the
temperature of the radiation bath above the Hubble rate during inflation is
largely independent of the degree of flatness of the potential, although it
depends on the field value at the inflection point. We then discuss the
relevance of this result to warm inflation model building.

##  I Introduction

Inflation is an extremely successful paradigm in addressing the shortcomings
of standard Big Bang cosmology, providing an elegant solution to the horizon
and flatness problems by considering a period of accelerated expansion in the
early universe inflation . The simplest and most commonly considered scenario
is that of a scalar field, the inflaton $\phi$, that slowly rolls down its
potential, acting as an effective cosmological constant for a finite period.
This typically ends with the field rolling towards the minimum of its
potential and, while oscillating about it, reheating the universe through
perturbative and non-perturbative particle production. This scenario also
provides a seed for the observed temperature anisotropies in the Cosmic
Microwave Background and the Large Scale Structure of the universe, as the
accelerated expansion amplifies small quantum fluctuations of the scalar
field.

The slow-roll inflationary paradigm requires the scalar potential to have a
very flat region, where its motion is overdamped and the nearly constant
potential is the dominant source of energy density in the early universe.
While a plethora of phenomenological potentials with this property have been
constructed in the literature, the main challenge has been to embed the
inflationary dynamics within a more fundamental theory that reduces to the
Standard Model at low energies. This is important not only in establishing a
connection between inflation and low-energy particle phenomenology but also
due to the sensitivity of the inflationary dynamics to ultraviolet effects
close to the Planck scale.

This has motivated a search for inflaton candidates in supersymmetric (SUSY)
theories, in particular in the context of supergravity/string theory (see e.g.
Baumann:2009ni ), which provides the best-known candidate for a fundamental
theory of quantum gravity. These scenarios have the appealing feature of
naturally including several additional scalar fields, in particular the
superpartners of the Standard Model fermions and also the Higgs fields, as
well as a plethora of extra-dimensional moduli. Moreover, these models
generically exhibit a multitude of directions in field space along which the
scalar potential is completely flat in the supersymmetric limit and which are
uplifted by different SUSY breaking effects. For example, even the simplest
supersymmetric extension of the Standard Model, the MSSM, includes nearly 300
flat directions corresponding to gauge invariant combinations of the matter
and Higgs superfields Gherghetta:1995dv .

Flat directions can be lifted by a variety of effects, including soft terms
from SUSY breaking in a hidden/sequestered sector, renormalizable and non-
renormalizable terms in the superpotential, as well as non-perturbative
effects (e.g. gaugino condensation). In the context of string theory, these
effects are generically related to the geometry and topology of the
compactified extra-dimensions, which typically involves different fluxes
and/or D-brane configurations. All these different effects may a priori yield
both attractive and repulsive contributions to the scalar potential, which may
conspire to produce an inflection point or even a saddle point in the
potential.

The resulting flatness thus provides a very attractive setup for inflationary
dynamics, and several successful models have been constructed in the
literature. In the context of the MSSM flat directions, inflection points may
for example result from the interplay between repulsive soft trilinear
$A$-terms and (non-)renormalizable terms in the superpotential, providing not
only inflationary models consistent with observational data but also
interesting connections to low-energy phenomenology, such as neutrino masses,
natural dark matter candidates and the recent Higgs mass from LHC
Allahverdi:2006iq ; Allahverdi:2006cx ; Bueno Sanchez:2006xk ;
Allahverdi:2006we ; Allahverdi:2007vy ; Allahverdi:2007wt ; Lalak:2007rsa ;
Allahverdi:2010zp ; Enqvist:2010vd ; Allahverdi:2011su ; Chatterjee:2011qr ;
Boehm:2012rh . Several different flat directions in the MSSM field space have
been analyzed so far, including simple extensions such as additional singlet
fields leading to hybrid inflation models Hotchkiss:2011am , as well as taking
into account supergravity corrections Enqvist:2007tf ; Mazumdar:2011ih  and
possible embeddings within the string theory landscape Allahverdi:2007wh . In
the context of string theory, several new possibilities arise, as for example
the case of warped D-brane inflation Kachru:2003sx ; warped_brane_potentials ,
where the D-brane potential receives a broad array of contributions such as
Coulomb-like interactions in brane-antibrane pairs, couplings to the four-
dimensional scalar curvature and several different moduli stabilization
effects in the bulk of the compactification. A recent statistical analysis of
these contributions has shown that successful models typically occur near an
inflection point in the potential Agarwal:2011wm . Such features may also be
found in closed-string moduli dynamics, for example in the context of
racetrack models BlancoPillado:2004ns  and the so-called accidental inflation
scenarios Linde:2007jn ; BlancoPillado:2012cb .

Inflection point inflation thus appears in a broad range of different setups,
being quite successful in terms of consistency with observational data, as
well as providing a natural embedding within ultraviolet completions of the
Standard Model and desirable links to low-energy phenomenology. However, these
models are far from generic and typically require a fine-tuning of the
different contributions to the scalar potential, making inflation rather
special within the vast landscape of different possibilities.

In this work, we revisit inflationary dynamics near an inflection point in the
potential taking into account the effects of dissipation in the inflaton’s
motion. Dissipation is a natural outcome of the interactions between the
inflaton field and other degrees of freedom, which are in fact required to
ensure a graceful exit into a radiation-dominated era, and is intrinsically
associated with particle production. If such dissipative effects are
sufficiently strong, particle production may actually balance the dilution
effect of the accelerated expansion, resulting in an inflationary state that
is far from the supercooled vacuum that is conventionally considered. In
particular, if the resulting particles have sufficiently strong interactions
between them, they can possibly reach a nearly-thermal state at a temperature
$T>H$, where $H$ is the inflationary Hubble rate, thus potentially changing
the dynamics of inflation. Although it encompasses a much more general class
of models where fluctuation-dissipation effects are significant during
inflation, the resulting paradigm is thus known as warm inflation, having been
originally proposed in wi ; Berera:1996nv  following on earlier work in
earlydissp .

Most scenarios considered in the literature so far in the context of quantum
field theory Berera:1998gx ; Yokoyama:1998ju ; Berera:1999ws ; BR1 ;
Berera:2002sp ; Hall:2004zr ; br05 ; Moss:2006gt ; Berera:2008ar ; Moss:2008yb
; Graham:2008vu ; BasteroGil:2009ec ; BasteroGil:2010pb ; BasteroGil:2012cm ,
as well as phenomenological models ph  consider the case where a nearly-
thermal bath of radiation is produced concurrently with accelerated expansion
from the adiabatic motion of the inflaton field, which may be analyzed within
the framework of linear response theory. In this regime, the non-local effects
of dissipation yield, to leading order, an additional friction term in the
inflaton’s equation of motion that helps overdamping its trajectory, thus
allowing for longer periods of slow-roll inflation and alleviating the need
for a very flat potential (see e.g. BasteroGil:2009ec ), which is particularly
important in the context of supergravity and string theory warm_string ;
warm_brane , where one typically finds a severe ‘eta-problem’. In this sense,
we expect the inclusion of dissipative effects to minimize the fine-tuning of
different terms in the scalar potential required for a sufficiently long
period of inflation near an inflection point. On the other hand, most studies
of warm inflation so far have focused on relatively steep potentials,
typically suffering from an eta-problem, where dissipative effects may play a
more prominent role, so that this will allow us to explore a new regime of
warm inflationary dynamics with a very flat potential.

Warm inflation has several other attractive features, such as providing an
alternative to the graceful exit problem, since radiation, although
necessarily sub-leading during inflation, may become the dominant component of
the energy density as soon as the conditions for slow-roll evolution break
down. The spectrum of primordial perturbations is also typically modified in
this context, as thermal fluctuations of the inflaton field overcome vacuum
fluctuations for $T>H$ wi ; Berera:1999ws ; Taylor:2000ze ; Hall:2003zp ;
Moss:2008yb . This generically suppresses the amount of primordial gravity
waves produced during inflation and induces a significant non-gaussian
component in the spectrum, which is generically expected to be within the
reach of the Planck mission Gupta:2002kn ; Chen:2007gd ; Moss:2007cv ;
Moss:2011qc . Furthermore, the inclusion of dissipative effects and the finite
temperature during inflation may address other outstanding problems in modern
cosmology, such as the generation of a baryon asymmetry BasteroGil:2011cx  and
the overproduction of gravitinos in supersymmetric models Taylor:2000jw ;
Sanchez:2010vj ; Bartrum:2012tg .

This work is organized as follows. In the next section, we review the
conventional dynamics of inflation near an inflection point in the potential,
taking as a working example that we will follow throughout our discussion a
renormalizable flat direction in a $U(1)_{\mathrm{B-L}}$ extension of the
MSSM. In section III we give a basic review of the generic features and
conditions for a successful model of warm inflation, applying these to a
potential with an inflection point in section IV. We then perform numerical
simulations of the evolution of the coupled inflaton-radiation system as a
function of the parameters characterizing the potential and the dissipation
coefficient, presenting our results in section V. Finally, in section VI we
summarize our main conclusions and discuss their impact on inflationary model
building, as well as prospects for future research in this topic.

##  II Cold inflation near an inflection point

As discussed earlier, scalar potentials exhibiting an inflection point may
arise in a variety of models in supersymmetric theories and
supergravity/string theory models. For concreteness, we will consider
throughout most of our discussion a simple example introduced in
Allahverdi:2006iq ; Allahverdi:2007vy  and also considered in Hotchkiss:2011am
, consisting of a low-scale extension of the MSSM with an additional
$U(1)_{\mathrm{B-L}}$ symmetry and right-handed neutrino superfields. In
particular, we focus on the scalar potential induced for the $NH_{u}L$ flat
direction, parametrized by a scalar field $\phi$ that plays the role of the
inflaton and which, without loss of generality, we take to be real. This flat
direction is lifted by a renormalizable term in the superpotential and by
soft-SUSY breaking terms, yielding:

$V(\phi)=\frac{1}{2}m_{\phi}^{2}\phi^{2}+\frac{h^{2}}{12}\phi^{4}-\frac{Ah}{6\sqrt{3}}\phi^{3}~{}.$
(1)

For $A\simeq 4m_{\phi}$, this potential exhibits an approximate saddle point
for a field value $\phi_{0}\simeq\sqrt{3}m_{\phi}/h$, such that
$V^{\prime}(\phi_{0})\simeq V^{\prime\prime}(\phi_{0})\simeq 0$, which is thus
suitable for inflation. We may then define Hotchkiss:2011am :

$A=4m_{\phi}\sqrt{1-{\beta^{2}\over 4}}$ (2)

and expand the potential about the generic point of inflection, yielding for
$\beta\ll 1$, to lowest order:

$V(\phi)\simeq
V_{0}\left(1+3\beta^{2}\left({\phi-\phi_{0}\over\phi_{0}}\right)+4\left({\phi-\phi_{0}\over\phi_{0}}\right)^{3}\right)~{},$
(3)

where $V_{0}=V(\phi_{0})$. This clearly shows that, for $\beta=0$, $\phi_{0}$
is a saddle point in the potential, with $\beta$ determining the deviations
from this case, i.e. the fine-tuning of the parameters in the potential
required for a sufficiently flat inflationary potential. Note that for real
values of $\beta$, the potential exhibits an inflection point at $\phi_{0}$,
whereas for imaginary values of $\beta$ it develops a local minimum at
$\phi>\phi_{0}$, as illustrated in Fig. 1. This latter option could be suited
for inflation with the field trapped in the false vacuum and then tunneling
into the true minimum, as in the old inflationary picture. However, this does
not lead to a graceful exit into a radiation-dominated era, so we will not
consider this case in the remainder of our discussion.

Figure 1: Normalized scalar potential for different values of the fine-tuning
parameter $\beta$.

The inflationary dynamics, in the absence of dissipation, is determined by the
slow-roll parameters, which are in this case given by:

$\displaystyle\epsilon_{\phi}$ $\displaystyle=$ $\displaystyle{1\over
2}m_{P}^{2}\left({V^{\prime}(\phi)\over V(\phi)}\right)^{2}\simeq{1\over
2}\left({m_{P}\over\phi_{0}}\right)^{2}\left(3\beta^{2}+12\Delta_{\phi}^{2}\right)^{2}~{},$
$\displaystyle\eta_{\phi}$ $\displaystyle=$ $\displaystyle
m_{P}^{2}{V^{\prime\prime}(\phi)\over V(\phi)}\simeq
24\left({m_{P}\over\phi_{0}}\right)^{2}\Delta_{\phi}~{},$ (4)

where $m_{P}=2.4\times 10^{18}$ GeV is the reduced Planck mass,
$\Delta_{\phi}=(\phi-\phi_{0})/\phi_{0}$ and we have taken $V(\phi)\simeq
V_{0}$, which holds for $\Delta_{\phi},\beta\ll 1$. From these quantities we
may determine the amplitude and tilt of the resulting spectrum of density
perturbations, given by:

$\displaystyle\mathcal{P}_{R}$ $\displaystyle=$ $\displaystyle{1\over
24\pi^{2}}{V_{0}/m_{P}^{4}\over\epsilon_{\phi_{*}}}~{},$ $\displaystyle n_{s}$
$\displaystyle=$ $\displaystyle 1+2\eta_{\phi_{*}}-6\epsilon_{\phi_{*}}\simeq
1+48\left({m_{P}\over\phi_{0}}\right)^{2}\Delta_{\phi_{*}}~{},$ (5)

where $\phi_{*}$ denotes the value of the field when the relevant CMB scales
left the horizon about 40-60 e-folds before the end of inflation, and we have
used that $|\eta_{\phi_{*}}|\gg\epsilon_{\phi_{*}}$ for
$\Delta_{\phi},\beta\ll 1$. These two conditions can be used to determine the
constant term in the potential $V_{0}$ and $\phi_{*}$, leaving $\phi_{0}$ and
$\beta$ as the only undetermined parameters.

The dynamics of inflation is governed by the slow-roll equation:

$3H\dot{\phi}\simeq-V^{\prime}(\phi)~{},$ (6)

with $H^{2}\simeq V(\phi)/3m_{P}^{2}$. Inflation ends in this case when the
slow-roll condition $|\eta_{\phi}|<1$ is violated, such that
$\Delta_{\phi_{e}}\simeq-(\phi_{0}/m_{P})^{2}/24$. This allows us to compute
the total number of e-folds of inflation from horizon-crossing, which is then
given by:

$N_{e}=\int_{t_{*}}^{t_{e}}Hdt\simeq-\int_{\phi_{*}}^{\phi_{e}}{3H^{2}\over
V^{\prime}(\phi)}d\phi\simeq{\arctan(1/2\xi)+\arctan\left((n_{s}-1)/4\xi\right)\over\xi}~{},$
(7)

where $\xi=6\beta(m_{P}/\phi_{0})^{2}$. Thus, in the limit $\beta\rightarrow
0$ for any given $\phi_{0}$, this yields a maximum of $N_{e}\simeq 119$
e-folds of inflation for $n_{s}=0.967$ Komatsu:2010fb , which exceeds the
observationally required range. We can invert this to determine the value of
$\beta$ required for 40-60 e-folds of inflation, yielding:

$\beta\simeq(3.1-5.2)\times 10^{-3}\left({\phi_{0}\over m_{P}}\right)^{2}~{},$
(8)

with smaller values of $\beta$ yielding longer periods of inflation, since the
resulting potential is flatter. This illustrates the generic fine-tuning
problem of inflection point models, and in this particular case the soft
inflaton mass and the trilinear term in Eqs. (1) and (2) have to compensate
each other to at least one part in $10^{6}$ for a successful model with sub-
planckian inflaton values, as can be seen by inserting Eq. (8) in Eq. (2).

##  III Warm inflation dynamics

In warm inflation, the interactions of the inflaton field with other degrees
of freedom can, as discussed above, significantly affect its dynamics and the
standard inflationary observables, such as the amplitude and the tilt of the
power spectrum, its non-gaussianity and the spectrum of primordial
gravitational waves. The time non-local component of these interactions leads
to a transfer of energy from the inflaton field to other degrees of freedom,
yielding a dissipative process which is in general described by non-
equilibrium dynamics. However, in the slow-roll regime, the motion of inflaton
may be sufficiently slow compared to the overall relaxation time of the system
to allow for an adiabatic description of the evolution, giving to leading
order an additional friction term in the scalar equation of motion:

$\ddot{\phi}+(3H+\Upsilon)\dot{\phi}+V^{\prime}(\phi)=0.$ (9)

The dissipative coefficient can be computed microscopically given a particular
particle physics realization of inflation Berera:2008ar  and may in general
depend on the value of the inflaton field and the properties of the multi-
particle state produced by dissipation. Successful realizations of warm
inflation have been constructed in a regime where the fields coupled to the
inflaton acquire large masses during inflation, proportional to the inflaton
value, and then decay into light degrees of freedom, which is generically
known as the two-stage mechanism Berera:2002sp . For sufficiently strong
interactions the produced particles may then thermalize in less than a Hubble
time, producing a quasi-thermal bath of radiation at a temperature $T$
BasteroGil:2012cm . If the intermediate fields are sufficiently heavy compared
to the temperature of the radiation, thermal corrections to the inflaton mass
are Boltzmann-suppressed, thus preserving the tree-level flatness of the
inflaton potential, in particular in the case of supersymmetric flat
directions. Such large masses are naturally obtained for large values of the
inflaton field, hence avoiding the problems of early realizations of warm
inflation in a high-temperature regime Berera:1998gx ; Yokoyama:1998ju .

Supersymmetry also provides a natural framework for keeping quantum and
thermal corrections to the scalar potential under control Hall:2004zr , even
though it is broken by finite temperature and energy density effects during
inflation. A generic superpotential realizing the two-stage interactions is
given by BasteroGil:2009ec ; Moss:2008yb :

$W=W(\Phi)+g\Phi X^{2}+hXY^{2}~{},$ (10)

where the scalar component of $\Phi$ is the inflaton field, its expectation
value giving large masses to the bosonic and fermionic components of the
intermediate superfield(s) $X$. They decay in turn into the components of the
superfield $Y$, which remain light and form the radiation fluid. For $T\ll
m_{X}$ and a broad range of couplings and field multiplicities, the dominant
contribution to the dissipative coefficient corresponds to virtual excitations
of the intermediate scalar fields decaying into the bosonic $Y$ components and
has the form Moss:2008yb ; BasteroGil:2010pb ; BasteroGil:2012cm :

$\Upsilon\approx C_{\phi}\frac{T^{3}}{\phi^{2}}~{},$ (11)

where $C_{\phi}$ is a constant that depends on the coupling $h$ and the field
multiplicities in the $X$ and $Y$ sectors and which, for the purposes of our
discussion, we will take as a free parameter of the model. Note that
renormalizable superpotentials of the form in Eq. (10) are ubiquitous in
supersymmetric models, such as for example the NMSSM, where the additional
singlet could play the role of the inflaton and dissipate its energy into
(s)quarks and (s)leptons through the Higgs portal, e.g. $W=g\Phi
H_{u}H_{d}+hQH_{u}U+\ldots$ Notice, however, that a much larger number of
fields is required with the dissipation coefficient in Eq. (11) to achieve a
sufficient number of e-folds of inflation than in the MSSM. More generically,
one expects fields coupled to supersymmetric flat directions to acquire large
masses during inflation (see e.g. Allahverdi:2007zz ), while fields that do
not couple directly to flat directions should remain light. Such a
superpotential also arises in D-brane constructions, where dissipative effects
have been shown to play an important role in overcoming the associated eta-
problem warm_brane .

The thermalized radiation fluid has an energy density

$\rho_{r}\simeq\frac{\pi^{2}}{30}g_{*}T^{4}~{},$ (12)

where $g_{*}$ is the effective number of light degrees of freedom, and is
sourced by the dissipative motion of the inflaton field, yielding

$\dot{\rho}_{r}+3H(\rho_{r}+p_{r})=\Upsilon\dot{\phi}^{2},$ (13)

where $p_{r}$ is the pressure associated with the radiation fluid. In warm
inflation, this fluid is not redshifted away during inflation, due to the
additional dissipative source term wi ; wi2 . The radiation energy density
needs, however, to be subdominant to achieve a period of accelerated
expansion, i.e. $\rho_{r}\ll\rho_{\phi}$, where
$\rho_{\phi}=\dot{\phi}^{2}/2+V(\phi)$. However, the associated temperature
may be larger than the expansion rate, $T>H$, which allows one to neglect the
effects of expansion in computing the dissipation coefficient and modifies the
evolution of inflaton fluctuations. Otherwise, when $T<H$, dissipative effects
can be neglected and the standard cold inflation scenario is recovered.

Slow-roll inflation, whether cold or warm, requires an overdamped evolution of
the inflaton field. In warm inflation this can be achieved due to the friction
term $\Upsilon$ in addition to Hubble damping. Once the field $\phi$ is in the
slow-roll regime, the evolution of the radiation fluid is also generically
damped, and the equations of motion reduce to

$\displaystyle 3H(1+Q)\dot{\phi}$ $\displaystyle\simeq-V^{\prime}(\phi)\,,$
(14) $\displaystyle 4\rho_{R}$ $\displaystyle\simeq 3Q\dot{\phi}^{2}\,,$ (15)

where we have introduced the dissipative ratio $Q=\Upsilon/(3H)$, which,
depending on the particular model, may increase or decrease during inflation.
The slow-roll conditions are now given by $\epsilon_{\phi},\eta_{\phi}\ll
1+Q$, alleviating the need for a very flat potential, and additionally one
requires that Moss:2008yb :

$\displaystyle\beta_{\Upsilon}$ $\displaystyle=$ $\displaystyle
m_{P}^{2}\left(\frac{\Upsilon_{\phi}V_{\phi}}{\Upsilon V}\right)\ll 1+Q\,,$
(16) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{TV_{T\phi}}{V_{\phi}}<1\,,$ (17)

where $\beta_{\Upsilon}$ measures the variation of the dissipation coefficient
with respect to the inflaton field and the last condition ensures that thermal
corrections to the potential are small, which is the case in the regime where
Eq. (11) is valid.

##  IV Warm inflation near an inflection point

As previously discussed, the additional friction term in Eq. (9) alleviates
the flatness of the potential required in order to achieve a sufficient amount
of inflation. In the context of inflection point inflation, we have seen that
the $\beta$ parameter determines the shape of the potential in the vicinity of
the inflection point, measuring the fine-tuning of the underlying parameters.
Therefore, we expect that a warm realization of these models can naturally
reduce the amount of fine-tuning required.

We will use the Eq. (1) as a working example of a potential with an inflection
point to analyze the generic dynamics of warm inflation in this context,
although this does not correspond to a concrete realization of warm inflation
in the MSSM. Writting Eq. (1) in the form of Eq. (3), the dynamics is
described by six independent parameters, in particular the value of the field
at the inflection point $\phi_{0}$, the corresponding height of the potential
$V_{0}$, the fine-tuning parameter $\beta$, the value of the field at horizon-
crossing $\phi_{*}$, the dissipative constant $C_{\phi}$ and the effective
number of light degrees of freedom $g_{*}$. We can use the WMAP 7-year results
giving a power spectrum with an amplitude $\mathcal{P}_{\mathcal{R}}=(2.43\pm
0.11)\times 10^{-9}$ and a spectral index $n_{s}=0.967\pm 0.014$
Komatsu:2010fb  to determine $V_{0}$ and $\phi_{*}$. As mentioned above, for
$T>H$ the dominant contribution to the spectrum of primordial perturbations
are the thermal fluctuations in the coupled inflaton-radiation system, and the
resulting amplitude of the power spectrum in the slow-roll regime is given by
wi ; Berera:1999ws ; Hall:2003zp :

$\mathcal{P}^{1/2}_{\mathcal{R}}\simeq\left(\frac{H}{2\pi}\right)\left(\frac{3H^{2}}{V^{\prime}(\phi)}\right)(1+Q)^{5/4}\left(\frac{T}{H}\right)^{1/2}~{},$
(18)

where all quantities are implicitly evaluated at horizon-crossing. In order to
solve this equation, it is useful to write it in a more convenient way. Using
the slow-roll equations (14) and (15), one obtains

$Q_{*}(1+Q_{*})^{13/2}\simeq\mathcal{P_{R}}\left(\frac{C_{\phi}}{3}\right)\left(\frac{\pi
C_{\phi}}{2C_{R}}\right)^{2}(2\epsilon_{\phi_{*}})^{3}\left(\frac{m_{P}}{\phi_{*}}\right)^{6}~{},$
(19)

where $C_{R}=g_{*}\pi^{2}/30$. Eq. (19) and the expression for the spectral
index BasteroGil:2009ec

$(1+Q_{*})(1+7Q_{*})(n_{s}-1)+(2+9Q_{*})\epsilon_{\phi_{*}}+3Q_{*}\eta_{\phi_{*}}+(1+9Q_{*})\beta_{\Upsilon_{*}}\simeq
0$ (20)

form a coupled system of equations for $Q_{*}$ and $\phi_{*}$ that needs to be
solved numerically for given values of $\phi_{0}$, $\beta$, $C_{\phi}$ and
$g_{*}$. Once the system is solved, we can obtain the value of $V_{0}$ using
Eq. (18):

$V_{0}\simeq\left(\frac{C_{R}}{C_{\phi}}\right)\frac{144\pi^{2}\mathcal{P_{R}}\phi^{2}_{*}m^{2}_{P}}{\sqrt{1+Q_{*}}\left(1+3\beta^{2}\left({\phi_{*}-\phi_{0}\over\phi_{0}}\right)+4\left({\phi_{*}-\phi_{0}\over\phi_{0}}\right)^{3}\right)}~{}.$
(21)

The system has in general three possible solutions satisfying the
observational constraints, and we have consistently chosen the one that
maximizes the difference $\phi_{*}-\phi_{0}$, since as we discuss below this
minimizes the amount of dissipation required for a sufficiently long period of
inflation. To simplify the numerical procedure, one can use the approximate
solutions in the strong and weak dissipative regimes, $Q_{*}\ll 1$ and
$Q_{*}\gg 1$, respectively, where the equations decouple, to find the initial
guess required to calculate numerically the full solution to the coupled
system of equations. In the intermediate regime, $Q_{*}\sim 1$, it is
sufficient to use an initial guess in this range.

##  V Numerical results

Having determined $V_{0}$ and $\phi_{*}$ from the observational constraints,
we may now study the evolution of the coupled inflaton-radiation system as a
function of the remaining parameters, $C_{\phi}$, $\beta$, $\phi_{0}$ and
$g_{*}$. For concreteness, we first fix the number of relativistic degrees of
freedom $g_{*}=100$, corresponding to the order of magnitude of the number of
MSSM scalar fields, although we study the effect of varying this parameter at
the end of this section. Our main goal is to determine which is the lowest
value of $C_{\phi}$ required for a sufficiently long period of inflation as a
function of the fine-tuning parameter $\beta$ and for different values of
$\phi_{0}$. The number of e-folds of warm inflation can be computed by
including the effects of dissipation in Eq. (7):

$N_{e}\simeq-\int_{\phi_{*}}^{\phi_{e}}{\frac{3H^{2}(1+Q)}{V^{\prime}(\phi)}d\phi}.$
(22)

However, due to the $T$\- and $\phi$-dependent dissipative ratio $Q$, this
integral cannot be solved analytically as in the cold inflation case. Besides,
the value of the field at the end of inflation cannot be calculated a priori.
Hence, the equations of motion for both the inflaton and the radiation fluid
have to be integrated numerically. In most areas of the parameter space, the
inflaton field is always in the slow-roll regime and therefore we may
integrate Eq. (14). However, in some regions of the parameter space the
radiation fluid is not slow-rolling, in which case we integrate the full
equation (13). The consistency of our analysis is determined by three main
conditions:

  * •

$\epsilon_{H}=-\frac{\dot{H}}{H}<1$, which is required for accelerated
expansion and generalizes the slow-roll conditions;

  * •

$m_{X}\gg T$, for which the form of the dissipative coefficient in Eq. (11) is
valid.

  * •

$T>H$, which allows one to neglect the effects of expansion in computing the
dissipation coefficient and generically defines the regime where inflation is
warm.

These conditions need to hold for $40-60$ e-folds of inflation in order to
solve the horizon and flatness problems, and in Fig. 2 we show the regions in
the plane $C_{\phi}-\beta$ where this is obtained for different values of the
inflection point $\phi_{0}$.

Figure 2: Values of $C_{\phi}$ and $\beta$ required to obtain
$N_{e}\in[40,60]$ for $g_{*}=100$ and different values of $\phi_{0}/m_{P}$.

As one can easily see in this figure, lower values of $\phi_{0}$ require more
dissipation in order to obtain the same number of e-folds, which is related to
the associated increase in the slow-roll parameters in Eq. (II), due to a
steeper shape of the potential. In addition, we find two distinct regions of
parameter space in Fig. 2, corresponding to small and large values of the
fine-tuning parameter $\beta$. The separation between these regions depends on
the value of $\phi_{0}$, with the small-$\beta$ region moving to lower values
of $\beta$ for smaller $\phi_{0}$.

In the small-$\beta$ region, the potential is extremely flat and intuitively
one would expect less friction to be required for a given period of
accelerated expansion. However, Fig. 2 clearly shows that the required value
of $C_{\phi}$ becomes constant for low values of $\beta$, which suggests
taking a closer look at the physical mechanism behind dissipation. Since it is
the motion of the inflaton field that produces light particles in a quasi-
thermal bath, the amount of radiation produced depends on how fast the
inflaton is rolling, as can be explicitly seen in Eq. (13). If the potential
is too flat, the inflaton will roll too slowly, which suppresses the amount of
radiation produced and consequently decreases the temperature of the thermal
bath. In fact, it is the condition $T>H$ that determines the end of warm
inflation in this region of parameter space, as one can see in Fig. 3, where
we plot the evolution of the relevant quantities in this regime. This also
explains why the initial condition farther away from the inflection point
yields the lowest value of $C_{\phi}$, since an initially steeper potential
can more easily produce a radiation bath with $T>H$.

(a) $(\phi-\phi_{0})/\phi_{0}$

(b) $\epsilon_{H}$ and $\rho_{r}/\rho_{\phi}$

(c) $T/H$

Figure 3: Evolution with the number of e-folds of $(\phi-\phi_{0})/\phi_{0}$,
$\epsilon_{H}$, $\rho_{r}/\rho_{\phi}$ and $T/H$ for $\phi/m_{P}=10^{-2}$,
$g_{*}=100$ and $\beta=10^{-7}$ when inflation lasts for $40$ e-folds.

In Fig. 3(a), one can see that the inflaton starts above the inflection point
and ends close to the latter, with the temperature dropping below the Hubble
rate after 40 e-folds of inflation. Notice, however, that inflation does not
necessarily end at this point, since $\epsilon_{H}<1$ and decreasing, but our
analysis is no longer consistent at this stage since de Sitter effects may
modify the dissipation coefficient. It may, in fact, be possible for an
additional period of cold inflation to follow, thus decreasing the amount of
dissipation required to achieve the desired number of e-folds, although a
detailed analysis of this possibility is beyond the scope of this work.
Finally, in Fig. 3 we see that $\epsilon_{H}$ follows closely the evolution of
the radiation energy density, which in this case is becoming more and more
sub-leading compared to the inflaton field, thus requiring an additional
reheating stage.

In the large-$\beta$ region the potential is steeper, therefore the production
of radiation is enhanced and $T>H$ is no longer the dominant constraint. In
fact, in this regime radiation tends to be overproduced and dominate the
energy density, thus allowing for a graceful exit from inflation, as shown in
Fig. 4 where we plot the evolution with the number of e-folds of the relevant
quantities in the large-$\beta$ region.

(a) $(\phi-\phi_{0})/\phi_{0}$

(b) $\epsilon_{H}$ and $\rho_{r}/\rho_{\phi}$

(c) $T/H$

Figure 4: Evolution with the number of e-folds of $(\phi-\phi_{0})/\phi_{0}$,
$\epsilon_{H}$, $\rho_{r}/\rho_{\phi}$ and $T/H$ for $\phi/m_{P}=10^{-2}$,
$g_{*}=100$ and $\beta=10^{-2}$ when inflation lasts for $40$ e-folds.

In Fig. 4(a), one can see that the inflaton field starts away from the
inflection point, remains close to it for a few e-folds but that, due to the
slope of the potential, inflation ends beyond the point of inflection, in
contrast with the small-$\beta$ behavior. Notice that $\rho_{r}/\rho_{\phi}$
decreases sharply when the field slows down close to the inflection point, in
agreement with the discussion above, but then increases as the field moves to
lower values and eventually ends inflation with a smooth exit into a
radiation-dominated era. In Fig. 4(c) it is also clear that $T>H$ for the
whole duration of inflation.

Although we have not plotted the condition $m_{X}\gg T$ in Figs. 3 and 4, we
have checked that it is satisfied in all the parameter space shown, for
couplings $g\sim 1$. On the other hand, we may consider more general
potentials, associated with different SUSY breaking effects, yielding a
different value for the numerical coefficient of the slow-roll parameter
$\eta_{\phi}$ in Eq. (II). We then find that, for lower values of this
coefficient, the condition $m_{X}\gg T$ is more stringent than $T>H$. However,
the amount of dissipation required does not change significantly even for an
order of magnitude change in this coefficient, so we do not explore this
possibility in more detail.

Finally, we analyze the effect of the number of relativistic degrees of
freedom on the amount of dissipation required for successful inflation. In
Fig. 5 we show the $C_{\phi}-\beta$ region where 40-60 e-folds of inflation
are obtained with different values of $g_{*}$.

Figure 5: Values of $C_{\phi}$ and $\beta$ required to obtain
$N_{e}\in[40,60]$ for $\phi_{0}/m_{P}=1$ and $g_{*}=1,10^{4}$.

In Fig. 5, it can be observed that the required value of $C_{\phi}$ decreases
for smaller $g_{*}$. In order to understand this behavior, we compute the
explicit dependence of the dissipation coefficient on $g_{*}$ by substituting
Eq. (12) into Eq. (11):

$\Upsilon=\frac{30^{3/4}C_{\phi}}{\pi^{3/2}g_{*}^{3/4}}\frac{\rho^{3/4}_{r}}{\phi^{2}}.$
(23)

Hence, the relevant quantity is an effective dissipation constant:

$\tilde{C}_{\phi}=\frac{C_{\phi}}{g^{3/4}_{*}}$ (24)

that remains constant in Fig. 5 for the different values of $g_{*}$, which is
also the case for smaller (sub-planckian) values of the inflection point.

##  VI Summary and future prospects

In this work we have analyzed the dynamics of inflation close to an inflection
point in the scalar potential taking into account dissipative effects
resulting from the coupling of the inflaton field to other degrees of freedom.
We have focused on supersymmetric models, where the plethora of available flat
directions may be lifted by competing SUSY-breaking effects, producing
inflection and even saddle points in the potential, although at the expense of
fine-tuning a priori unrelated parameters. Moreover, supersymmetry provides a
natural framework for warm inflation, helping to protect the flatness of the
potential against both thermal and radiative corrections, in particular in the
regime where the fields coupled to the inflaton acquire large masses and their
effects are Boltzmann-suppressed, nevertheless allowing for the dissipative
production of virtual excitations that may decay into light fields in a quasi-
thermal bath. For concreteness, we have focused on the $NH_{u}L$ flat
direction in a low scale extension of the MSSM, although the resulting scalar
potential has a sufficiently generic form and our main results should apply to
other realizations of inflection point inflation.

Our numerical simulations of the dissipative dynamics of inflation in this
model have lead us to two main conclusions. Firstly, if dissipative effects
are sufficiently strong, a sufficiently long period of inflation may occur
independently of the fine-tuning of the parameters in the potential, which was
expected since the additional friction alleviates the need for a very flat
potential. Secondly, and more surprisingly, the required amount of dissipation
does not decrease arbitrarily for flatter potentials, given that if the scalar
potential is too flat and the inflaton evolves too slowly, it becomes more
difficult to sustain a radiation bath with a temperature above the Hubble
rate, which is required for consistency of our analysis. This results in a
field-dependent critical value of the fine-tuning parameter $\beta$ below
which the required dissipation parameter $C_{\phi}$ becomes constant. Above
this value, the potential is sufficiently steep to ensure that $T>H$
throughout inflation, with steeper potentials requiring larger values of the
dissipation parameter.

The value of $C_{\phi}$ depends on the coupling between the intermediate
fields and the light degrees of freedom, as well as on the multiplicities of
both heavy and light fields. The minimum value of $C_{\phi}\gtrsim 10^{6}$
obtained for $g_{*}=100$ is of the same order as that obtained for other forms
of the inflaton potential, such as monomial or hybrid models BasteroGil:2009ec
, which implies large couplings and field multiplicities, so one may ask
whether there is any gain from the model building perspective in trading a
large fine-tuning in the parameters of the potential for large couplings and a
large number of fields. On one hand, fine-tuning makes inflation less generic,
since it isolates a small region of the available parameter space, whereas
inflation should provide an explanation for the otherwise finely-tuned
conditions in the early universe. On the other hand, a large number of degrees
of freedom during inflation points towards more complicated beyond the
Standard Model scenarios, e.g. with fields in large representations, which may
be realized in generic GUT constructions or D-brane models warm_brane . As
discussed earlier in this work, strong dissipative effects may have other
interesting consequences in the dynamics of the early universe and, moreover,
we have seen that the minimum value of $C_{\phi}$ may be substantially reduced
in models with a smaller number of relativistic degrees of freedom, where the
temperature of the radiation bath is consequently larger, pointing towards
constructions with several fields coupled directly to the inflaton but with
few distinct decay channels.

One should point out that we have focused on a particular form of the
dissipation coefficient, corresponding to the decay of virtual excitations of
the heavy fields coupled to the rolling inflaton. In BasteroGil:2012cm , it
was pointed out that this is not the only possibility, as the excitation of
real modes becomes the dominant contribution to dissipative effects for
$m_{X}/T\lesssim 10$, yielding a much larger dissipation coefficient than
virtual excitations for $\mathcal{O}(1)$ couplings and field multiplicities,
nevertheless suppressing quantum and thermal corrections to the scalar
potential. While the dynamics of warm inflation in this regime remains
unexplored, it suggests a much more promising avenue from the model building
point of view, and we expect our main qualitative results to apply in this
case as well, since they depend more on the presence of significant
dissipative effects than on the specific form of the dissipation coefficient.

Whereas we intend to explore this regime in more detail in the future, our
results show that dissipative effects are an interesting and well-motivated
alternative to fine-tuning in inflationary models and we hope to motivate
further investigation of this topic in other inflationary scenarios.

###### Acknowledgements.

We would like to thank Mar Bastero-Gil, Arjun Berera and Anupam Mazumdar for
their useful comments and suggestions. R.C. is partially supported by MICINN
(FIS2010-17395) and “Junta de Andalucía” (FQM101) and would like to
acknowledge the hospitality of the Particle Theory Group at the University of
Edinburgh during the completion of this work. The work of J.G.R. was partially
supported by the STFC Particle Theory Group grant at the University of
Edinburgh (United Kingdom), the FCT grant PTDC/FIS/116625/2010 (Portugal) and
the Marie Curie action NRHEP 295189- FP7-PEOPLE-2011-IRSES.

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