# Dirac-Born-Infeld warm inflation realization in the strong dissipation
regime

Meysam Motaharfar<EMAIL_ADDRESS>Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA  Rudnei O.
Ramos<EMAIL_ADDRESS>Departamento de Fisica Teorica, Universidade do Estado
do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil

###### Abstract

We consider warm inflation with a Dirac-Born-Infeld (DBI) kinetic term in
which both the nonequilibrium dissipative particle production and the sound
speed parameter slow the motion of the inflaton field. We find that a low
sound speed parameter removes, or at least strongly suppresses, the growing
function appearing in the scalar of curvature power spectrum of warm
inflation, which appears due to the temperature dependence in the dissipation
coefficient. As a consequence of that, a low sound speed helps to push warm
inflation into the strong dissipation regime, which is an attractive regime
from a model building and phenomenological perspective. In turn, the strong
dissipation regime of warm inflation softens the microscopic theoretical
constraints on cold DBI inflation. The present findings, along with the recent
results from swampland criteria, give a strong hint that warm inflation may
consistently be embedded into string theory.

##  I Introduction

Warm inflation (WI) Berera:1995wh ; Berera:1995ie ; Berera:1998px  is an
alternative dynamical realization for conventional cold inflation (CI)
Guth:1980zm ; Sato:1981ds ; Albrecht:1982wi ; Linde:1981mu ; Linde:1983gd
during which the inflaton field dissipates its vacuum energy into an ambient
radiation bath, thus eliminating the necessity of a postinflationary reheating
process Berera:1996fm  and leading into different possibilities for a graceful
exit mechanism Das:2020lut . Such a nonequilibrium dissipative particle
production process acts to slow the motion of the inflaton field, allowing the
embedding of steeper potentials in the WI context and helping to solve, for
example, the so-called $\eta$-problem Berera:1999ws ; BasteroGil:2009ec ;
Bastero-Gil:2019gao . In WI the thermal fluctuations play the dominant role in
the formation of the seeds for the large scale structure (LSS) by producing a
quasi scale-invariant spectrum of primordial curvature perturbations. At the
same time, the shape of the produced spectrum depends on the field content of
the model, leading to signatures able to make it distinguishable from the one
produced in CI Bastero-Gil:2014raa ; Moss:2011qc ; Moss:2007cv ; Bastero-
Gil:2014jsa ; Ramos:2013nsa ; BasteroGil:2011xd ; Graham:2009bf ; Hall:2003zp
; Bastero-Gil:2014oga . Furthermore, the rich dynamics of WI allows it to
address/alleviate some of long-lasting problems related to the
(post-)inflationary phase in the CI scenarios Bartrum:2012tg ; Sanchez:2010vj
; Motaharfar:2021gwi ; Bastero-Gil:2016mrl ; BasteroGil:2011cx ; Rosa:2019jci
; Dimopoulos:2019gpz ; Levy:2020zfo ; Rosa:2018iff ; Lima:2019yyv ; Sa:2020fvn
; Berghaus:2019cls  (see also Refs. Dymnikova:1998ps ; Dymnikova:2001ga ;
Dymnikova:2001jy  for related work).

Despite its tremendous success, earlier particle physics realizations of WI
were confronted with two important difficulties. The first one was that
achieving a thermal radiation bath during inflation can result in potentially
large thermal corrections to the inflaton’s potential, thus, hindering the
slow-roll dynamics. Earlier WI model building constructions circumventing this
problem made use of models with large field multiplicities Moss:2006gt ;
BasteroGil:2010pb ; BasteroGil:2012cm ; Bartrum:2013fia , making them
technically unappealing. This issue was later resolved with the introduction
of a new class of WI model building realization able to sustain a nearly-
thermal bath, yet with a small number of field species Bastero-Gil:2016qru .
These type of models were dubbed “warm little inflaton (WLI)” models. The
second difficulty was that the backreaction of the thermal radiation bath on
the inflaton power spectrum due to a temperature dependent dissipation
coefficient leads to the appearance of growing/decreasing modes in the scalar
of curvature power spectrum Graham:2009bf . As a consequence of this,
consistency with the observations could only be achieved in weak dissipation
regimes of WI, thus preventing WI from going into a strong dissipation regime.
However, the strong dissipation regime of WI is particularly appealing from
both a theoretical and effective field theory point of view. In particular,
the strong dissipation regime of WI can naturally result in a smaller energy
scale inflation with sub-Planckian field excursions. These are features that
have been attracting considerable attention, especially more recently, e.g.,
based on the swampland program aiming at finding effective field theories
which can consistently be embedded in quantum gravity theories and the role
played by WI in achieving this goal Das:2020xmh ; Brandenberger:2020oav ;
Goswami:2019ehb ; Berera:2019zdd ; Kamali:2019xnt ; Das:2019hto ; Das:2018rpg
; Motaharfar:2018zyb . This issue was also partially addressed recently by
building two distinct explicit WI models, namely the “variant of warm little
inflaton” (VWLI) Bastero-Gil:2019gao  and the “minimal warm inflation” (MWI)
Berghaus:2019whh ; Laine:2021ego , utilizing different field contents in each
construction. In the VWLI, the strong dissipative regime is obtained by
interpolating between decreasing and growing modes, while in the MWI it is
achievable only for some particular form of potentials, while
growing/decreasing modes for the power spectrum can still exist in both cases.

As far as well-motivated theoretical field theory constructions for inflation
are concerned, there has been considerable progress towards this goal in
recent years, with the construction of several concrete string inflation
models for instance (for reviews, see, e.g., Refs. Kallosh:2007ig ;
Baumann:2009ni  and references therein), with particular attention given to
brane inflation scenarios111There are two realizations of brane inflation,
namely ultraviolet (UV) and infrared (IR) models, depending on whether the
brane is moving towards or away from the tip of the throat. In this paper, we
consider an UV model (see Refs. Chen:2005ad ; Chen:2004gc  for IR models)..
Dirac-Born-Infeld (DBI) inflation is one such string theoretic motivated model
in which the inflaton is interpreted as a modulus parameter of a D-brane
propagating in a warped throat region of an approximate Calabi-Yau flux
compactification Silverstein:2003hf ; Alishahiha:2004eh . Hence, the effective
action for DBI inflation contains a special form of the DBI kinetic term. The
DBI kinetic term introduces some novel speed limits on the inflaton field
velocity and helps in keeping it near the top of potential, even when it is
too steep, thus resulting in the slow-roll phase through a low sound speed,
instead of from dynamical friction due to expansion. A low sound speed,
smaller than unity, allows the inflaton field to have a sub-Planckian
evolution, even for steep potentials, thus circumventing the $\eta$-problem
Easson:2009kk . Moreover, fluctuations also propagate with a low sound speed
parameter, resulting in both a smaller tensor-to-scalar ratio and a
significant non-Gaussianity, which can potentially be distinguishable from
other scenarios Silverstein:2003hf ; Alishahiha:2004eh .

Although DBI inflation models were greatly successful, it was realized that
the microscopic bound on the maximal field variation due to compactification
is able to put strong constraints on the throat volume Baumann:2006cd  and
bulk volume Chen:2006hs . These issues, together with the Lyth bound, lead to
a model independent upper bound on the tensor-to-scalar ratio which turns out
to be inconsistent with the stringent observational lower bound on
gravitational waves that are produced by the inflationary dynamics
Lidsey:2007gq . Moreover, a viable reheating process, which typically involves
brane/antibrane annihilation (see, e.g., Refs. Barnaby:2004gg ; Frey:2005jk ;
Chialva:2005zy ; Firouzjahi:2005qs  for earlier studies on reheating in brane
inflation), is highly constrained due to overproduction of long-lived Kaluza-
Klein (KK) modes Kofman:2005yz  (note, however, that subsequently Refs.
Chen:2006ni ; Berndsen:2008my ; Dufaux:2008br ; Frey:2009qb  argued that
warped KK modes can instead serve as dark matter candidates).

Reference Cai:2010wt  showed that the cold DBI inflation model can be
reconciled with the observations, removing the aforementioned inconsistencies,
provided that the strong dissipation regime of Dirac-Born-Infeld warm
inflation (DBIWI) is achieved. This is also because the WI scenario violates
the Lyth bound. However, the work done in Ref. Cai:2010wt  made use of a
phenomenological dissipation coefficient, assuming it to be independent of the
temperature222The stringent observational lower bound on tensor-to-scalar
ratio found in Ref. Lidsey:2007gq  can be relaxed in other DBI setups without
dissipation, for instance, in multi-brane Huston:2008ku  and multi-field
Langlois:2009ej  models.. However, almost all explicit particle physics
realizations of WI result in an explicit temperature dependent dissipation
coefficient Berera:2008ar ; BasteroGil:2010pb ; BasteroGil:2012cm ; Bastero-
Gil:2016qru ; Bastero-Gil:2019gao ; Berghaus:2019whh . Besides this, pushing
WI into the strong dissipation regime is challenging due, again, to the
aforementioned problem related to the appearance of a growing/decreasing
function in the power spectrum, which tends to lead to results inconsistent
with the observations, e.g., for the spectral tilt Benetti:2016jhf . Although
there have been some previous papers considering DBIWI realizations Cai:2010wt
; Zhang:2014dja ; Rasouli:2018kvy , none of them made an investigation of the
role of the sound speed on the backreaction of the radiation perturbations on
the inflaton ones. Thus, none of the earlier references on DBIWI have studied
how the growing/decreasing function in WI gets affected by the sound speed,
which requires a detailed study of the perturbations in DBIWI. Taken all
together, the purpose of this paper is to cover this issue and to properly
understand the primordial perturbations in DBIWI and the consequences that it
brings to model realizations, as far as the observations constraints are taken
into account. Our results show that DBIWI is able to go into the strong
dissipation regime for well-motivated field theory realizations of WI that
result in explicit temperature dependent dissipation coefficients. Our results
also shed new light on the potential use of DBI type of models in the context
of WI.

The outline of the remainder of this paper is as follows. In Sec. II, we
present the dynamics of DBIWI realization and investigate the behavior of the
dynamical parameters in the model. Then, in Sec. III, we study the
perturbation equations for the DBIWI and present the backreaction of the
thermal radiation bath on the inflaton perturbations and scalar of curvature
power spectrum. In Sec. IV we discuss the results obtained for the DBIWI and
demonstrate the effect of a low sound speed on the spectrum of DBIWI. Finally,
in Sec. V, we give a summary of the results and conclude by discussing the
implications of the DBIWI realization from a model building perspective.
Throughout this paper, we work with the natural units, in which Planck’s
constant, the speed of light and Boltzmann’s constant are set to $1$ and we
also work with the reduced Planck mass, $M_{\rm Pl}\equiv 1/\sqrt{8\pi
G}\simeq 2.4\times 10^{18}$GeV.

##  II DBIWI background dynamics

The dynamical realization of WI is different from the CI one due to the
presence of radiation and energy exchange between the inflaton field and the
radiation energy density. Hence, the total energy density of the Universe in
WI contains both the inflaton field and a primordial radiation energy density,
i.e., $\rho=\rho_{\phi}+\rho_{r}$, where $\rho_{\phi}$ and $\rho_{r}$ are the
inflaton field and the radiation energy densities, respectively. Even when
$\rho_{r}$ is subdominated at the beginning and throughout inflation, i.e.,
$\rho_{r}\ll\rho_{\phi}$, the underlying dissipation effects generating this
radiation energy density are still able to modify the inflationary dynamics
and the perturbations in a nontrivial way in WI. The inflaton field and
radiation energy density form a coupled system in the WI dynamics due to
dissipation of energy out of the inflaton system and into radiation. In the
spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric, the
background evolution equations are, respectively, given by

$\displaystyle\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=-\Upsilon\left(\rho_{\phi}+p_{\phi}\right),$
(1)
$\displaystyle\dot{\rho}_{r}+3H(\rho_{r}+p_{r})=\Upsilon\left(\rho_{\phi}+p_{\phi}\right),$
(2)

where $p_{\phi}$ and $p_{r}$ are the inflaton and radiation pressures,
respectively, and $\Upsilon$ is the dissipation coefficient, which in general
can be a function of both the inflaton field $\phi$ and temperature $T$ of the
produced radiation bath.

In the DBIWI realization, the inflaton is a modulus parameter of a D-brane
moving in a warped throat region which dissipates its vacuum energy into
radiation through Eqs. (1) and (2), while its evolution is effectively
governed by the following Lagrangian density,

$\displaystyle\mathcal{L}_{DBI}=f^{-1}(\phi)\left[1-\sqrt{1-2f(\phi)X}\right]-V(\phi),$
(3)

where $X=-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi$, with $V(\phi)$
being the potential function for the inflaton and $f(\phi)$ is the redefined
warp factor. The form for the warp factor function can be phenomenologicaly
deformed depending on the desired model construction. For the well-studied
anti-de Sitter throat Silverstein:2003hf ; Alishahiha:2004eh  the warp factor
function is given by $f(\phi)={f_{0}}/{\phi^{4}}$ with $f_{0}$ being a
positive constant. For definiteness, this is the form for the warp factor
function that we will be assuming in this work.

Varying the action with respect to the metric, the energy density
$\rho_{\phi}$ and the pressure $p_{\phi}$ of the DBI field are given,
respectively, by

$\displaystyle\rho_{\phi}=\frac{\dot{\phi}^{2}}{c_{s}(1+c_{s})}+V(\phi),$ (4)
$\displaystyle p_{\phi}=\frac{\dot{\phi}^{2}}{1+c_{s}}-V(\phi),$ (5)

where $c_{s}\equiv\sqrt{1-2Xf(\phi)}=\gamma^{-1}$ is the sound speed, which is
smaller than unity as a consequence of the nontrivial kinetic structure of the
DBI Lagrangian, with $\gamma$ similarly being the Lorentz factor in five
dimensions. Inserting Eq. (4) in Eqs. (1) and (2), the dynamical equation for
the inflaton field in the DBIWI is found to be given by

$\displaystyle\ddot{\phi}+3c_{s}^{2}H(1+Q)\dot{\phi}+c_{s}^{3}V^{\prime}+\frac{f^{\prime}}{2f^{2}}\left(1-3c_{s}^{2}+2c_{s}^{3}\right)=0,$
(6)

while for the radiation energy density is

$\displaystyle\dot{\rho}_{r}+4H\rho_{r}=c_{s}^{-1}\Upsilon{\dot{\phi}^{2}},$
(7)

where $Q=\Upsilon/(3H)$ gives a measure for the strength of the dissipative
processes in WI. Equation (6) for the inflaton background evolution can also
be obtained from the following covariant equation (when taking $\phi$ as an
homogeneous field),

$\displaystyle c_{s}^{-1}\Box\phi-
c_{s}^{-3}f(\phi)(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\phi\nabla^{\nu}\phi)-V^{\prime}$
$\displaystyle-\frac{f^{\prime}(\phi)}{2f^{2}(\phi)}\left(c_{s}^{-3}-3c_{s}^{-1}+2\right)=-c_{s}^{-1}\Upsilon
u^{\mu}\partial_{\mu}\phi.$ (8)

In the slow-roll regime, $\ddot{\phi}\ll H\dot{\phi}$ and $\dot{\rho}_{r}\ll
4H\rho_{r}$, i.e., the inflaton is slowly varying and radiation is produced in
a quasistationary way. Both slow-roll conditions hold as long as

$\epsilon_{s}\equiv\frac{d\ln c_{s}}{dN}=\frac{\dot{c}_{s}}{Hc_{s}}\ll 1,$ (9)

with $\epsilon_{s}$ quantifying the variation of the sound speed, along with
the usual condition that the standard slow-roll parameters in WI are small.
Then, under the slow-roll approximation, the background equations reduce to

$\displaystyle\dot{\phi}\simeq-\frac{c_{s}V^{\prime}}{3H(1+Q)},\ \
\rho_{r}\simeq\frac{3Q\dot{\phi}^{2}}{4c_{s}},\ \
3H^{2}\simeq\frac{V(\phi)}{M_{\rm Pl}}.$ (10)

To study the model quantitatively, we need to fix the functionality of the
potential and of the dissipation coefficient. In this paper, we consider a
monomial power-law potential for the inflaton,

$V(\phi)=\frac{V_{0}}{2n}\left(\frac{\phi}{M_{\rm Pl}}\right)^{2n},$ (11)

where $V_{0}$ is the amplitude of the potential and $n$ is a real number.
Moreover, the dissipation coefficient can be very well parameterized by333See
also Refs. Zhang:2009ge ; Visinelli:2016rhn  for some earlier studies also
considering this functional form for the dissipation coefficient in WI.

$\Upsilon(\phi,T)=C_{\Upsilon}\,T^{c}\phi^{p}M^{1-p-c},$ (12)

where $C_{\Upsilon}$ is a constant, $T$ is the temperature, $M$ is some
appropriate mass scale and both quantities can be associated with the
specifics of the microscopic model parameters leading to Eq. (12) (see, e.g.,
Refs. Gleiser:1993ea ; Berera:1998gx ; Berera:2008ar ; BasteroGil:2012cm ;
Bastero-Gil:2016qru ; Laine:2021ego  for some explicit examples of quantum
field theory model derivations for dissipation coefficients used in WI).

The scalar of the curvature power spectrum in WI Graham:2009bf ;
BasteroGil:2011xd ; Bastero-Gil:2014jsa ; Ramos:2013nsa  has strong dependence
on the dissipation ratio $Q$ and on the temperature over the Hubble parameter
ratio, $T/H$. One also notes that strictly speaking $T/H$ can also be
considered as a function of $Q$, once we use the CMB amplitude value to
constrain the scalar power spectrum. In this case, $Q$ can be considered the
relevant quantity parametrizing the WI dynamics. It is then useful to look at
the dynamical behavior of these quantities during inflation. In particular,
the analysis of their behavior is important to determine regimes where the
spectral tilt of the scalar of curvature power spectrum increases (leading
potentially to a blue-tilted spectrum) or decreases (and that can lead to a
red-tilted spectrum). Such an analysis is particularly relevant in modeling WI
models and constraining them with the observations Benetti:2016jhf .
Furthermore, since $Q$ appears explicitly in the slow-roll parameters of WI,
the behavior of $Q$ during WI is important for determining regimes where the
dynamics are consistent and can have a graceful exit Das:2020lut . In the case
of DBIWI, the slow-roll coefficients also depend on the sound speed $c_{s}$,
hence, for completeness, we also study its behavior here. During slow-roll we
find that the dissipation ratio $Q$, the temperature over Hubble parameter
$T/H$, and the sound speed $c_{s}$ in DBIWI evolve with the number of e-folds
$N$, respectively, as

$\displaystyle\frac{d\ln Q}{dN}$ $\displaystyle=$
$\displaystyle\frac{c_{s}\left\\{\left[4+c(1+c_{s}^{2})\right]\epsilon_{V}-c(1+c_{s}^{2})\eta_{V}-(2c(1-c_{s}^{2})+4p)\kappa_{V}\right\\}}{4-c+4Q+cc_{s}^{2}Q},$
(13) $\displaystyle\frac{d\ln(T/H)}{dN}$ $\displaystyle=$
$\displaystyle\frac{c_{s}\left\\{\left(6+c_{s}^{2}+5Q+c(-1+c_{s}^{2}Q)\right)\epsilon_{V}-(1+c_{s}^{2})(1+Q)\eta_{V}+\left[p(c_{s}^{2}Q-1)-2(1+Q)(1-c_{s}^{2})\right]\kappa_{V}\right\\}}{(1+Q)(4-c+4Q+cc_{s}^{2}Q)},$
$\displaystyle\frac{d\ln c_{s}}{dN}$ $\displaystyle=$
$\displaystyle\frac{c_{s}(1-c_{s}^{2})\left\\{(-4+c+cQ)\epsilon_{V}-(-4+c)(1+Q)\eta_{V}-2\left[4-c+(4+c+2p)Q\right]\kappa_{V}\right\\}}{(1+Q)(4-c+4Q+cc_{s}^{2}Q)},$
(15)

where $\epsilon_{V}=M_{\rm Pl}^{2}(V^{\prime}/V)^{2}/2$ and $\eta_{V}=M_{\rm
Pl}^{2}V^{\prime\prime}/V$ are the usual slow-roll inflaton potential
parameters and $\kappa_{V}=M_{\rm Pl}^{2}V^{\prime}/(\phi\,V)$.

We note from Eqs. (13)-(15) that the denominator of those expressions is
always positive. This is because the power $c$ in the temperature dependence
of the dissipation coefficient satisfies $-4<c<4$ (see, e.g., Refs.
Moss:2008yb ; delCampo:2010by ; BasteroGil:2012zr ). On the other hand, the
sign of the numerator in the above expressions will depend on both the form of
the dissipation coefficient and on the inflaton potential exponent $n$. As
examples of representative cases of WI models, we can consider two cases of
dissipation coefficient that are well motivated microscopically: (a)
$c=1,\,p=0$, i.e., a linear in the temperature dependence for the dissipation
coefficient, $\Upsilon\propto T$, that was first derived in Ref. Bastero-
Gil:2016qru  in the case of the WLI; and (b) $c=3,\,p=0$, i.e., a cubic power
in the temperature dependence for the dissipation coefficient,
$\Upsilon\propto T^{3}$, and derived recently in Refs. Berghaus:2019whh ;
Laine:2021ego  in the case of the MWI. From the Eqs. (13)-(15) we find that in
case (a) both $Q$ and $T/H$ are growing functions with the number of e-folds
for $n>1/3$ for all $c_{s}$, while $c_{s}$ initially decreases for $n<3$ when
$Q\ll 1$, and for $Q>1$ it will increase for $n>13/7$ and decreases otherwise.
In case (b) $Q$ and $c_{s}$ are decreasing functions with the number of
e-folds whenever $n<3$ and $c_{s}^{2}<1/3$ or $n>3$ and $c_{s}^{2}>1/3$ and
increasing functions with $N$ otherwise, while $T/H$ is in general a growing
function with $N$. Both cases (a) and (b) then reinforce the WI condition
$T/H>1$ throughout the inflationary dynamics.

##  III Perturbations for DBIWI

Let us describe the scalar perturbation equations in DBIWI. We start with the
fully perturbed FLRW metric,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-(1+2\alpha)dt^{2}-2a\partial_{i}\beta dx^{i}dt$ (16)
$\displaystyle+$ $\displaystyle
a^{2}[\delta_{ij}(1+2\varphi)+2\partial_{i}\partial_{j}\gamma]dx^{i}dx^{j},$

where $\alpha$, $\beta$, $\gamma$ and $\varphi$ are the spacetime-dependent
metric perturbation variables. In WI the evolution of field fluctuations
$\delta\phi$, which is effectively described by a stochastic evolution, is
determined by a Langevin-like equation Graham:2009bf ; Bastero-Gil:2014jsa .
In the case of DBIWI, perturbing the covariant equation (II) around the
background, i.e., $\Phi({\bf x},t)=\phi(t)+\delta\phi({\bf x},t)$, the
corresponding Langevin equation for $\delta\phi$ reads

$\displaystyle\delta\ddot{\phi}$ $\displaystyle+3H\
\left(1+Q-\epsilon_{s}\right)\delta\dot{\phi}+H^{2}\left\\{z^{2}+\frac{3\dot{f}}{2Hf}\left[(1-c_{s}^{2})(1+Q)-\epsilon_{s}\right]-\frac{1-3c_{s}^{2}+2c_{s}^{3}}{H^{2}}\left(\frac{{f^{\prime}}^{2}}{f^{3}}-\frac{f^{\prime\prime}}{2f^{2}}\right)+\frac{c_{s}^{3}V^{\prime\prime}}{H^{2}}\right\\}\delta\phi$
$\displaystyle+c_{s}^{2}\delta\Upsilon\dot{\phi}=c_{s}^{3}\left(\xi_{q}+\xi_{\Upsilon}\right)+\left[(3c_{s}^{-2}-1)\ddot{\phi}+3H(1+Q)\dot{\phi}\right]\alpha+(\dot{\alpha}+c_{s}^{2}\kappa)\dot{\phi}-\frac{3f^{\prime}}{2f^{2}}(1-c_{s}^{2})(1-c_{s}^{-2})\alpha,$
(17)

which is also supplemented by the first-order perturbation equations for the
radiation energy density, $\delta\rho_{r}$, and for the radiation momentum
perturbation, $\Psi_{r}$, given, respectively, by

$\displaystyle\delta\dot{\rho}_{r}$
$\displaystyle+4H\delta\rho_{r}=\frac{k^{2}}{a^{2}}\Psi_{r}+\dot{\rho}_{r}\alpha+\frac{4}{3}\rho_{r}\kappa+\delta
Q_{r},$ (18)

and

$\displaystyle\dot{\Psi}_{r}$
$\displaystyle+3H\Psi_{r}=-\frac{\delta\rho_{r}}{3}-\frac{4}{3}\rho_{r}\alpha+J_{r}.$
(19)

In the Eqs. (17), (18) and (19), we have that

$\displaystyle\delta\Upsilon$ $\displaystyle=$ $\displaystyle
3HQ\left(\frac{c\delta{\rho_{r}}}{4\rho_{r}}+\frac{p\delta\phi}{\phi}\right),$
(20) $\displaystyle\delta{Q}_{r}$ $\displaystyle=$ $\displaystyle
c_{s}^{-1}\delta\Upsilon\dot{\phi}^{2}+3H^{2}Qc_{s}^{-3}(1-c_{s}^{2})\frac{\dot{f}}{2Hf}\dot{\phi}\delta\phi$
(21) $\displaystyle+$ $\displaystyle
3c_{s}^{-3}(1+c_{s}^{2})HQ\dot{\phi}\delta\dot{\phi}$ $\displaystyle-$
$\displaystyle 3c_{s}^{-3}(1+c_{s}^{2})HQ\dot{\phi}^{2}\alpha,$ $\displaystyle
J_{r}$ $\displaystyle=$ $\displaystyle-
c_{s}^{-1}\Upsilon\dot{\phi}\delta\phi,$ (22)

where $z=c_{s}k/(aH)$, $\kappa=3(H\alpha-\dot{\varphi})+k^{2}\chi/a^{2}$ and
$\chi=a(\beta+a\dot{\gamma})$. Moreover, in Eq. (17)
$\xi_{q,\Upsilon}\equiv\xi_{q,\Upsilon}({\bf k},t)$ are stochastic Gaussian
sources related to quantum and thermal fluctuations with appropriate
amplitudes chosen such as to match the analytical derivation for the scalar of
curvature power spectrum in WI Ramos:2013nsa . This leads in particular that
$\xi_{\Upsilon}$ and $\xi_{q}$ both having zero mean and satisfying two-point
correlation functions given, respectively, by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\langle\xi_{\Upsilon}({\bf
k},t)\xi_{\Upsilon}({\bf k}^{\prime},t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\frac{2\Upsilon
T}{a^{3}}\delta(t-t^{\prime})(2\pi)^{3}\delta({\bf k}+{\bf k}^{\prime}),$ (23)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\langle\xi_{q}({\bf
k},t)\xi_{q}({\bf k}^{\prime},t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\frac{H^{2}(9+12\pi Q)^{1/2}(1+2n_{*})}{\pi a^{3}}$ (24)
$\displaystyle\times$ $\displaystyle\delta(t-t^{\prime})(2\pi)^{3}\delta({\bf
k}+{\bf k}^{\prime}),$

where $n_{*}$ represents the statistical distribution state for the inflaton
quanta at the Hubble radius Ramos:2013nsa  (see also the Appendix B of Ref.
Das:2020xmh  for details).

The set of perturbation equations (17), (18) and (19) are gauge-ready
equations. They can be used with any appropriate gauge choice or also be
worked out in terms of gauge-invariant quantities Kodama:1985bj ; Hwang:1991aj
. For instance, they can be taken in the Newtonian-slicing (or zero-shear)
gauge $\chi=0$, with the relevant metric equations becoming

$\displaystyle\kappa=\frac{3}{2M_{\rm
Pl}^{2}}(c_{s}^{-1}\dot{\phi}\delta\phi-\Psi_{r})\,,$ (25)
$\displaystyle\alpha=-\varphi\,,$ (26)
$\displaystyle\dot{\varphi}=-H\varphi-\frac{1}{3}\kappa\,.$ (27)

It can also be easily checked that Eq. (17) reduces to the standard Langevin
equation in WI Graham:2009bf ; Bastero-Gil:2014jsa  for $c_{s}=1$, i.e.,
$\epsilon_{s}=0$. As it is clear from Eqs. (17) and (18), the term
$\delta\Upsilon$ is responsible for coupling the inflaton perturbations with
those of the radiation whenever $c\neq 0$ and which is known to lead to a
growing mode for the resulting power spectrum Graham:2009bf .

###  III.1 The scalar of curvature power spectrum

Given the perturbation equations, Eqs. (17)-(19), the scalar power spectrum is
determined from the comoving curvature perturbation ${\cal R}$,

$\Delta_{\cal R}(k)=\frac{k^{3}}{2\pi^{2}}\langle|{\cal R}|^{2}\rangle\,,$
(28)

where “$\langle\cdots\rangle$” means averaging over different realizations of
the noise terms in Eq. (17) (see, for instance Refs. Graham:2009bf ;
BasteroGil:2011xd ; Bastero-Gil:2014jsa  for details of the numerical
procedure). Given an appropriate gauge, ${\cal R}$ is composed of
contributions from the inflaton momentum perturbations
$\Psi_{\phi}=-c_{s}^{-1}\dot{\phi}\delta\phi$ and from the radiation momentum
perturbations $\Psi_{r}$,

$\displaystyle{\cal R}=\sum_{i=\phi,r}\frac{\rho_{i}+p_{i}}{\rho+p}{\cal
R}_{i}\;,$ (29) $\displaystyle{\cal
R}_{i}=-\varphi-\frac{H}{\rho_{i}+p_{i}}\Psi_{i}\,,$ (30)

with $p=p_{\phi}+p_{r}$, $\rho_{\phi}+p_{\phi}=c_{s}^{-1}\dot{\phi}^{2}$ and
$\rho_{r}+p_{r}\simeq 4\rho_{r}/3=c_{s}^{-1}Q\dot{\phi}^{2}$.

An explicit analytic expression for the scalar of curvature power spectrum can
be obtained when neglecting the coupling between inflaton and radiation
perturbations (i.e., taking $c=0$ in Eq. (20)) and it was determined in Ref.
Ramos:2013nsa . The result in this case is well approximated by Ramos:2013nsa

$\Delta_{{\cal
R}}\simeq\left(\frac{H_{*}^{2}}{2\pi\dot{\phi}_{*}}\right)^{2}\left(1+2{n_{BE}}+\frac{2\sqrt{3}\pi
Q_{*}}{\sqrt{3+4\pi Q_{*}}}{T_{*}\over H_{*}}\right),$ (31)

where $n_{BE}$ is the Bose-Einstein distribution and the effect of sound speed
parameter is encoded in the inflaton field velocity through Eq. (10). One
should note that in DBIWI, fluctuations are frozen at the sound horizon, where
${c_{s}}_{*}k_{*}=a_{*}H_{*}$, rather than at the Hubble radius, where
$k_{*}=a_{*}H_{*}$. In general we can also replace $n_{BE}$ in Eq. (31) by
$n_{*}$, representing the statistical distribution state of the inflaton at
the sound horizon, which might not be necessarily that of thermal equilibrium.
The form given by Eq. (31) is typically the result used in most of the recent
literature in WI when the growing function is absent.

We want to first understand the effect of the coupling between inflaton and
radiation perturbations and how a sound speed $c_{s}<1$ might change the
results for the power spectrum in DBIWI. To possibly do something analytical
and to obtain a feeling for these effects, we can start by considering some
simplifying assumptions. Since we are interested in the strong dissipation
regime $Q>1$, we can start by neglecting the contribution of quantum
fluctuations to the power spectrum. For $Q>1$, the contribution from the
thermal stochastic fluctuations $\xi_{\Upsilon}$ in Eq. (17) dominates over
that from the quantum fluctuations $\xi_{q}$. Moreover, we can also
analytically solve Eq. (17) explicitly by neglecting the slow-roll order
corrections (and likewise the metric contributions, which also give slow-order
corrections only). Under these simplifying assumptions, the authors in Ref.
Graham:2009bf  have explicitly shown that the power spectrum grows with
$Q_{*}$ like $\Delta_{\mathcal{R}}\propto\Delta_{{\cal R},c=0}Q_{*}^{3c}$,
with

$\Delta_{{\cal
R},c=0}\simeq\frac{\sqrt{3}}{4\pi^{\frac{3}{2}}}\frac{H^{3}_{*}T_{*}}{\dot{\phi}^{2}_{*}}Q_{*}^{\frac{1}{2}},$
(32)

when taking $Q_{*}\gg 1$ in Eq. (31). By utilizing the same techniques
developed in Ref. Graham:2009bf , but now adapted to the perturbation
equations in DBIWI, we arrive at the result, when $Q_{*}\gg 1$, that
$\Delta_{\mathcal{R}}$ is now given by

$\displaystyle\Delta_{\mathcal{R}}\simeq\Delta_{{\cal
R},c=0}\left(\frac{Q_{*}}{Q_{c}}\right)^{3cc_{s*}^{2}},$ (33)

where $Q_{c}$ is a function of both $c$ and $c_{s}$. The result given by Eq.
(33) explicitly shows that $c_{s}<1$ can compensate for the result of a
growing scalar spectrum amplitude the larger is $Q$ and whenever $c>0$ (or,
likewise, a decreasing amplitude when $c<0$). The approximations assumed in
Ref. Graham:2009bf  do allow us to solve for the inflaton perturbations and,
hence, they help to find an analytical expression for $\Delta_{\mathcal{R}}$,
it has been shown in Ref. BasteroGil:2011xd  that these approximations tend to
overestimate the effect of the coupling between the inflaton and radiation
perturbations. By accounting for the full expressions, i.e., not dropping all
slow-roll order terms in the equations, the dependence of
$\Delta_{\mathcal{R}}$ on $Q_{*}$ tends to be more suppressed, with a smaller
power in $Q_{*}$. Hence, the results from the (numerical) solution for the
full perturbation equations shows that $\Delta_{\mathcal{R}}\propto
Q_{*}^{\beta}$, with $\beta<3c$, when $c_{s}=1$. This result is also observed
in our analysis shown in the next section, showing that $\beta<3cc_{s*}^{2}$
also holds here in the DBIWI. However, the effect is still large enough to
make the spectrum to depart considerably from the result given by Eq. (31). By
explicitly numerically solving the set of perturbations equations, as in Ref.
BasteroGil:2011xd , the result of the coupling between inflaton and radiation
perturbations can be expressed as an overall correction to Eq. (31), which
modifies it to

$\Delta_{{\cal
R}}=\left(\frac{H_{*}^{2}}{2\pi\dot{\phi}_{*}}\right)^{2}\left(1+2n_{BE}+\frac{2\sqrt{3}\pi
Q_{*}}{\sqrt{3+4\pi Q_{*}}}{T_{*}\over H_{*}}\right)G(Q_{*}),$ (34)

where $G(Q_{*})$ accounts for the effect of the coupling of the inflaton and
radiation fluctuations. Some explicit forms for $G(Q_{*})$ have been given in
the literature, depending on the form of the dissipation coefficient in WI
BasteroGil:2011xd ; Bastero-Gil:2014jsa ; Benetti:2016jhf ; Bastero-
Gil:2016qru ; Motaharfar:2018zyb ; Kamali:2019xnt ; Lima:2019yyv ; Das:2020xmh
. In the next section we will give results for $G(Q_{*})$ in the DBIWI case
for some of the most representative dissipation coefficient forms used in
recent literature and we will also explicitly see how a $c_{s}<1$ effectively
suppresses the effects of a growing $Q_{*}$ in the amplitude of the scalar
power spectrum. But before going into that analysis, let us also derive some
useful results concerning the spectral tilt in DBIWI.

###  III.2 The spectral tilt $n_{s}$ in DBIWI

From Eq. (34) and also using Eqs. (13)-(15), we can explicitly find
expressions for the spectral tilt $n_{s}$,

$n_{s}-1=\frac{d\ln\Delta_{{\cal R}}(k)}{d\ln k}\simeq\frac{d\ln\Delta_{{\cal
R}}(k)}{dN},$ (35)

which in the weak and strong dissipation regimes of DBIWI are given,
respectively, by

$\displaystyle\left(n_{s}-1\right)\Bigr{|}_{Q_{*}\ll 1}$ $\displaystyle=$
$\displaystyle\frac{c_{s*}}{4-c}\left\\{-\left[10+7c_{s*}^{2}-c(3+2c_{s*}^{2})\right]\epsilon_{V}\right.$
(36) $\displaystyle+$
$\displaystyle\left.\left[-1+(7-2c)c_{s*}^{2}\right]\eta_{V}\right.$
$\displaystyle+$
$\displaystyle\left.\left[(14-4c)(1-c_{s*}^{2})-p\right]\kappa_{V}\right\\},$

and

$\displaystyle\left(n_{s}-1\right)\Bigr{|}_{Q_{*}\gg 1}$
$\displaystyle=-\frac{c_{s*}}{2(4+cc_{s*}^{2})Q_{*}}\left\\{\left[18-c(1-c_{s*}^{2})\right]\epsilon_{V}\right.$
(37)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!+\left.\left[2+c+(-14+5c)c_{s*}^{2}\right]\eta_{V}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!+\left.2\left[c(1-c_{s*}^{2})+2(-7+p)+7(2+p)c_{s*}^{2}\right]\kappa_{V}\right\\}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!+\frac{c_{s*}}{4+cc_{s*}^{2}}\left\\{\left(4+c+cc_{s*}^{2}\right)\epsilon_{V}-c(1+c_{s*}^{2})\eta_{V}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!-\left.\left[2c(1-c_{s*}^{2})+4p\right]\kappa_{V}\right\\}\frac{G^{\prime}(Q_{*})}{G(Q_{*})}.$

Note that in the case of the primordial inflaton potential of the form Eq.
(11), we have in Eqs. (36) and (37) that

$\displaystyle\epsilon_{V}$ $\displaystyle=$ $\displaystyle\frac{2n^{2}M_{\rm
Pl}^{2}}{\phi_{*}^{2}},$ (38) $\displaystyle\eta_{V}$ $\displaystyle=$
$\displaystyle\frac{2n(2n-1)M_{\rm Pl}^{2}}{\phi_{*}^{2}},$ (39)
$\displaystyle\kappa_{V}$ $\displaystyle=$ $\displaystyle\frac{2nM_{\rm
Pl}^{2}}{\phi_{*}^{2}}.$ (40)

In deriving the Eqs. (36) and (37) for $n_{s}$ we have also considered that in
WI that $T_{*}/H_{*}\gg 1$ (in particular, $T_{*}/H_{*}\gg 1$ can be
explicitly verified in the examples to be considered in the next section).
Finally, in the regime $T_{*}/H_{*}\ll 1$ and $Q_{*}\ll 1$ we recover the cold
inflation standard result, $n_{s}-1=-6\epsilon_{V}+2\eta_{V}$. We can note
from Eq. (37) that when $c_{s*}\ll 1$, in the absence of the growing mode,
$G^{\prime}(Q)\to 0$, we have that $n_{s}-1\propto-1/Q$ and the spectral tilt
will tend to be red-tilted for large $Q$. However, in the presence of the
growing mode, the last term in Eq. (37) dominates at large $Q$ and it will
tend to drive the spectrum to be blue-tilted (for $G^{\prime}(Q)>0$). To
better illustrate the latter case, let us consider a few relevant WI
dissipation terms that have commonly been used in the literature, namely, (a)
the linear in the temperature dissipation coefficient Bastero-Gil:2016qru ,
with $c=1,\,p=0$; (b) the cubic in the temperature dissipation coefficient,
with $c=3,\,p=0$ (e.g., from Ref. Berghaus:2019whh ) and (c) the dissipation
coefficient with $c=3,\,p=-2$ (see, e.g., Refs. BasteroGil:2012cm ;
Bartrum:2013fia ). For definiteness, we will also consider the example of a
primordial potential with a quartic inflaton potential ($n=2$), which has been
the primordial potential mostly considered with these dissipation coefficients
in WI. We will also assume that the growing mode function is well described by
a polynomial function in the dissipation ratio $Q$, with the leading term for
$Q\gg 1$ of the form $G(Q)\propto Q^{\beta}$. Let us see the results for each
of these three cases separately.

####  III.2.1 Case (a)

For case (a), with $c=1,\,p=0$, we obtain for Eq. (37) that

$\left(n_{s}-1\right)\Bigr{|}_{Q_{*}\gg
1}\simeq\frac{-2c_{s*}\left[(17+c_{s*}^{2})-2\beta(5+c_{s*}^{2})\right]}{(4+c_{s*}^{2})Q_{*}(\phi_{*}/M_{\rm
Pl})^{2}}.$ (41)

Hence, for $c_{s*}\ll 1$ we have that the spectral tilt will turn blue, i.e.,
$n_{s}-1>0$, if $\beta>1.7$, while for $c_{s*}=1$, the spectral tilt is blue
for $\beta>1.5$. We recall that for this case Bastero-Gil:2016qru  we have
that $\beta\simeq 2.315$ (for $c_{s*}=1$), thus, for large $Q$ this
dissipation coefficient with a quartic inflaton potential will always disagree
with the observations. But from Eq. (33), we expect that $\beta$ will decrease
proportional to $c_{s}^{2}$. Thus, we need at least $c_{s*}^{2}\lesssim
1.7/2.315\sim 0.7$ for the model to be able to sustain a large dissipation
DBIWI regime and to have a red-tilted scalar of curvature power spectrum,
consistent with the observations. This expectation will be confirmed by our
numerical results shown in Sec. IV.

####  III.2.2 Case (b)

For case (b), with $c=3,\,p=0$, we obtain for Eq. (37) that

$\left(n_{s}-1\right)\Bigr{|}_{Q_{*}\gg
1}\simeq\frac{-2c_{s*}\left[(23+31c_{s*}^{2})-2\beta(-1+3c_{s*}^{2})\right]}{(4+3c_{s*}^{2})Q_{*}(\phi_{*}/M_{\rm
Pl})^{2}}.$ (42)

Hence, for $c_{s*}\ll 1$ we have that the spectral tilt, different from the
previous case, will always be red, since in this case
$n_{s}-1\propto-(23+2\beta)$. When $c_{s*}=1$, we see from Eq. (42) that
$n_{s}-1\propto-(54-4\beta)$ and the spectral tilt is, thus blue for
$\beta>13.5$. But in this case Benetti:2016jhf  we have that $\beta\simeq
4.33$ (for $c_{s*}=1$), hence, for large $Q$ this dissipation coefficient with
a quartic inflaton potential can quite robustly sustain a red-tilted spectrum.
In fact, the effect of having a $c_{s*}<1$ here can potentially make the
spectrum too red tilted to be consistent with the observations, since a value
of $c_{s*}<1$ would make the exponent $\beta$ smaller and lead to a redder
spectrum. This is confirmed by our numerical results shown in Sec. IV. Since
this type of dissipation coefficient already leads to quite satisfactory
results when compared to the observations Berghaus:2019whh , even in the
strong dissipation regime of WI, it does not benefit from a DBIWI
construction, at least in the context of monomial chaotic inflaton potentials.

####  III.2.3 Case (c)

It is useful to compare case (b) with the results for case (c), when
$c=3,\,p=-2$, in which case the dissipation coefficient depends on the
temperature but also on the inflaton amplitude. In this case, we obtain for
Eq. (37) that

$\left(n_{s}-1\right)\Bigr{|}_{Q_{*}\gg
1}\simeq\frac{-2c_{s*}\left[3(5+c_{s*}^{2})-2\beta(7+3c_{s*}^{2})\right]}{(4+3c_{s*}^{2})Q_{*}(\phi_{*}/M_{\rm
Pl})^{2}}.$ (43)

Here, we have that for $c_{s*}\ll 1$ the spectral tilt is such that it will
turn blue when $\beta>15/14\simeq 1.07$, while for $c_{s*}=1$, the spectral
tilt is blue already for $\beta>0.9$. In this case we still have $\beta\simeq
4.33$ (for $c_{s*}=1$), thus, for large $Q$ the spectrum will always be quite
blue tilted. This agrees with the previous results for this type of WI model
Bartrum:2013fia ; Benetti:2016jhf , which have shown that this dissipation
coefficient, within the mononial chaotic inflaton potential models, can only
be consistent with the observational data444From the Planck Collaboration
Akrami:2018odb , the result for the spectral tilt is $n_{s}=0.9658\pm 0.0040$
(95$\%$ CL, Planck TT,TE,EE+lowE+lensing+BK15+BAO+running) at pivot scale
$k_{*}=0.05\,{\rm Mpc}^{-1}$.  when $Q_{*}\ll 1$, i.e., in the weak
dissipation regime of WI. In this regime of $Q_{*}\ll 1$, the growing mode is
not an issue, since $G(Q_{*})\to 1$. As in case (a) with a linear temperature
dissipation coefficient, this model can benefit from a DBIWI realization by
exploring the effect of a small $c_{s}$ value, which will help to suppress the
growing mode by decreasing $\beta$. However, this is expected to only happen
for a smaller value for $c_{s}$ than the one shown for the case (a),
$c_{s*}^{2}\lesssim 0.2$. This case also illustrates well the importance in
deriving the full dependence of the dissipation coefficient in both the
temperature and in the inflaton field amplitude. Despite cases (b) and (c)
have exactly the same temperature dependence, the fact that the dissipation
coefficient in (c) has an explicit dependence on $\phi$, implies that an
agreement with the observational value for the tilt of the scalar power
spectrum can only be achieved in the weak dissipation regime of WI (for the
standard case of $c_{s*}=1$), while in (b) we can robustly support the strong
dissipation regime of WI, even with the presence of the growing mode ()which
in that case is benign).

###  III.3 Non-Gaussianities in DBIWI

Before ending this section. Let us briefly comment on the expected non-
Gaussianities in DBIWI. In the DBIWI realization, non-Gaussianities may be
generated by both nonequilibrium dissipative effects and by a low sound speed.
It was shown in Ref. Moss:2011qc ; Bastero-Gil:2014raa  that non-Gaussianities
produced by temperature dependent dissipation coefficients are larger than
temperature independent dissipation coefficients due to the backreaction of
radiation bath and coupling of the inflaton to the radiation field
perturbations. However, we qualitatively expect that for low sound speeds,
$c_{s*}<1$, a suppressing effect, as seen for the scalar of curvature power
spectrum, will also work in the case of the nonlinear parameter $f_{NL}$.
Moreover, the non-Gaussianity parameter for WI in the strong dissipation
regime is roughly less than $10$, i.e., $|f_{NL}^{Warm}|\leq 10$, for
temperature dependent dissipation coefficients Moss:2011qc ; Bastero-
Gil:2014raa . Furthermore, the non-Gaussianity produced by cold DBI inflation
is inversely proportional to the square of the sound speed, i.e.,
$f_{NL}\simeq\frac{35}{108}(c_{s}^{-2}-1)$. Hence, the corresponding non-
Gaussianity parameter in cold DBI inflation is $f_{NL}\simeq 3$ for
$c_{s*}=0.3$, which is not very large. Therefore, depending on how these two
non-Gaussianity sources contribute to the non-Gaussianity parameter (where
they can either counterbalance or enforce each other), we may obtain large or
small non-Gaussianities in DBIWI Cai:2010wt ; Zhang:2014kwa . Nonetheless, a
comprehensive analysis is needed to find how large the non-Gaussianity will be
in the DBIWI realization.

##  IV Results

To demonstrate the effect of $c_{s}$ on the power spectrum in the DBIWI
realization, more precisely on the function $G(Q_{*})$, we have considered the
quartic monomial inflaton potential ($n=2$) and focused our studies for the
two representative more recent cases of WI dissipation coefficients, namely,
$\Upsilon=C_{\Upsilon}T$ and $\Upsilon=C_{\Upsilon}T^{3}/M^{2}$. Results for
other cases of monomial inflaton potential are found to be very similar to the
quartic one, so we refrain from showing those other cases with expected
similar results here. The growing function $G(Q_{*})$ is determined by the
numerical evaluation of the perturbation equations and obtaining the scalar of
curvature power spectrum (28), with $G(Q_{*})$ defined as

$G(Q_{*})=\frac{\Delta_{{\cal R},c\neq 0}}{\Delta_{{\cal R},c=0}},$ (44)

where $\Delta_{{\cal R},c\neq 0}$ is the solution for the scalar of curvature
power spectrum with the explicit coupling of the inflaton perturbations with
the radiation ones in Eq. (17) and $\Delta_{{\cal R},c=0}$ is the solution
when this coupling is explicitly dropped from the equation. We should note
that during the numerical analysis, although we dropped the metric
perturbations, which is justified in an appropriate gauge and in the strong
dissipation regime, we take into account the effect of first order slow-roll
parameters, i.e. $\epsilon_{V}$, $\eta_{V}$, $\kappa_{V}$, etc., in Eq. (17)
to obtain the precise behavior for the growing function. The results for the
linear and cubic dependencies in temperature for the dissipation coefficient
are shown in Fig. 1. The results shown in Fig. 1 indicate that for
$c_{s}\lesssim 0.1$ the growing function $G(Q_{*})$ can indeed be taken as
$G(Q_{*})\approx 1$.

(a) Linear in $T$ dissipation coefficient (b) Cubic in $T$ dissipation
coefficient

Figure 1: The growing function $G(Q_{*})$ as defined by Eq. (44).

As shown in the previous section, we may obtain a red-tilted spectral index in
the strong dissipation regime even if $c_{s*}$ has intermediate values, not
necessarily for values of $c_{s*}\ll 1$. This is because as $c_{s*}<1$, it can
already suffice to suppress the growing mode enough to allow for a red-tilted
spectrum, even for large $Q$ values. Thus small, but still reasonable values
for $c_{s*}$ can work to allow the spectrum in DBIWI to be red-tilted in the
strong dissipation regime. Hence, it is useful to obtain the exact
functionality of $G(Q_{*})$ for future analysis. In this regard, we present
the following fitting functions to the curves shown in Fig. 1,

$\displaystyle G(Q_{*})$
$\displaystyle=1+A_{c}Q^{\alpha}_{*}+B_{c}Q^{\beta}_{*},$ (45)

where the coefficients $\alpha$, $\beta$, $A_{c}$ and $B_{c}$ are given in the
Table 1 for the two representative values of $c$ (i.e., for a linear and a
cubic in $T$ dissipation coefficients, as obtained in the LWI and MWI models,
respectively) and also for three representative values for $c_{s}$ (at the
effective Hubble radius crossing).

Table 1: The coefficients $\alpha$, $\beta$, $A_{c}$ and $B_{c}$ in growing function $G(Q_{*})$ with different values for $c$ and $c_{s}$. $c$ | $c_{s*}$ | $\alpha$ | $\beta$ | $A_{c}$ | $B_{c}$
---|---|---|---|---|---
|  1 |  1.364 |  2.315 |  0.335 |  0.0185
1 | 0.6 | 0.694 | 1.114 | 0.311 | 0.187
| 0.3 | 0.395 | 0.448 | 0.205 | 0.127
| 1 | 1.946 | 4.330 | 4.981 | 0.127
3 | 0.6 | 1.975 | 2.684 | 0.475 | 0.083
| 0.3 | 0.815 | 0.939 | 0.478 | 0.368

When these results are applied to the cubic in the temperature form for the
dissipation coefficient ($c=3,\,p=0$) and as already anticipated from the
discussion given for case (b) in Sec. III.2.2, for values of $c_{s}<1$, we get
that $n_{s}\simeq 0.95$ and even smaller values555More specifically, for
$c_{s*}=1$, we get for the model with $n=2,\,c=3,\,p=0$ that $n_{s}\simeq
0.97$ within the whole range of $Q_{*}\sim 1-100$, while for $c_{s*}\sim 0.7$
we already have that $n_{s}\simeq 0.95$ for this same range of $Q_{*}$ values.
The optimum range for $c_{s*}$ for this specific model is found to be
$0.9\lesssim c_{s*}\lesssim 1$, for which the spectral tilt can be kept within
the $95\%$ CL value $n_{s}=0.9658\pm 0.0040$, obtained from the Planck
TT,TE,EE+lowE+lensing+BK15+BAO+running data. for $n_{s}$ the larger is $Q_{*}$
and the smaller is $c_{s*}$, showing that the spectrum is already outside the
two-sigma range in the red-tilted side of the observational values for
$n_{s}$. This indicates that for this form of a cubic dissipation coefficient,
the model is not expected to benefit from a DBIWI realization, as already
discussed before. Let us then next focus on the linear dissipation coefficient
WI model of Ref. Bastero-Gil:2016qru . Some explicit examples of results
obtained in this case are given in Table 2 for the quartic inflaton potential
and with a linear in $T$ dissipation coefficient. As it is clear, for
$Q_{*}>10$, the tensor-to-scalar ratio is smaller than $10^{-7}$, hence, the
model is able to resolve the aforementioned inconsistency in the cold DBI
inflation Lidsey:2007gq  and discussed in the Introduction section.

Table 2: Numerical values of the parameters and the relevant cosmological quantities obtained for the case of a quartic inflaton potential ($n=2$) with a linear in $T$ dissipation coefficient ($c=1$, $p=0$) and for $c_{s*}=0.1$, when $G(Q_{*})=1$, and for $c_{s*}=0.3$, when $G(Q_{*})$ is given by Eq. (45). $c_{s*}$ | $Q_{*}$ | $n_{s}$ | $r$ | $N_{*}$ | $|\Delta\phi|/M_{\rm Pl}$ | $C_{\Upsilon}$ |  $T_{\rm end}$ (GeV) |  $V_{0}$ (GeV)4 | $\epsilon_{V_{*}}$ | $\eta_{V_{*}}$ | $V_{*}^{1/4}/M_{\rm Pl}$ | $f_{0}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| 1.03 | 0.9651 | $1.13\times 10^{-5}$ | 62.1 | 3.36 | 0.0067 | $2.67\times 10^{13}$ | $3.94\times 10^{59}$ | 0.68 | 1.02 | $1.56\times 10^{-5}$ | $3.30\times 10^{17}$
0.1 | 10.25 | 0.9628 | $3.00\times 10^{-7}$ | 61.0 | 1.42 | 0.023 | $1.08\times 10^{13}$ | $3.27\times 10^{59}$ | 3.80 | 5.71 | $6.31\times 10^{-6}$ | $2.14\times 10^{18}$
| 102.51 | 0.9625 | $3.88\times 10^{-9}$ | 60.0 | 0.46 | 0.076 | $3.68\times 10^{12}$ | $3.69\times 10^{59}$ | 35.55 | 53.33 | $2.16\times 10^{-6}$ | $1.71\times 10^{19}$
| 1.03 | 0.9672 | $3.33\times 10^{-5}$ | 62.4 | 5.85 | 0.0088 | $3.27\times 10^{13}$ | $1.27\times 10^{59}$ | 0.23 | 0.34 | $1.91\times 10^{-5}$ | $3.15\times 10^{17}$
0.3 | 10.30 | 0.9667 | $5.70\times 10^{-7}$ | 61.5 | 2.44 | 0.027 | $1.09\times 10^{13}$ | $7.24\times 10^{58}$ | $1.30$ | 1.95 | $6.39\times 10^{-6}$ | $2.91\times 10^{18}$
| 103.16 | 0.9684 | $3.49\times 10^{-9}$ | 60.3 | 0.79 | 0.074 | $3.08\times 10^{12}$ | $3.40\times 10^{58}$ | 12.31 | 18.47 | $1.80\times 10^{-6}$ | $4.73\times 10^{19}$

##  V Conclusions

We studied the effects of a low sound speed on the dynamics of perturbations
equations of WI inspired by string motivated models that include relativistic
D-brane motion. We numerically solved the coupled inflaton and radiation field
perturbation equations for the first time in the case of a noncanonical
kinetic term in DBIWI. We found that a low sound speed is able to suppress the
growing function that always appears in the scalar power spectrum of WI,
whenever the dissipation coefficient exhibits an explicit dependence with the
temperature of the radiation bath. As a consequence of this suppression effect
seen in DBIWI, the restrictions for constructing a WI realization in the
strong dissipation regime $Q\gg 1$ are considerably relaxed. We have also
complemented these results with the ones derived from the analytical
expressions for the spectral tilt. Based on our results, as the low sound
speed will push WI models into the strong dissipation regime, the severe
inconsistency problems seen in DBI cold inflation, e.g., due to an upper bound
on the tensor-to-scalar ratio arising from compactification constraints and a
lower bound from observations, can all be resolved due to large dissipation.
We have explicitly shown that among the most common dissipation coefficients
that have been derived from well-motivated particle physics realizations and
applied to the WI context, the dissipation coefficient with a linear
dependence on the temperature is the case that can mostly benefit from a DBIWI
realization. Hence, these type of models are able to soften all theoretical
and observational constraints, while predicting very small tensor-to-scalar
ratio and potentially significant non-Gaussianity, making them falsifiable in
near future.

From a model-building perspective, the present work, along with the previous
results on swampland conjectures in WI Kamali:2019xnt , gives a strong hint
that WI may consistently be embedded in string theory utilizing the physics of
the brane. Besides, phenomenologically realizing WI in the strong dissipation
regime is a significant step towards achieving such goal. Therefore, the last
step is building an explicit model describing how D-brane may be able to
dissipate its energy into radiation field. This and other implications of our
results are certainly worthwhile to explore further in future studies.

## Acknowledgments

R.O.R. is partially supported by research grants from Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), Grant No. 302545/2017-4, and
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro
(FAPERJ), Grant No. E-26/202.892/2017.

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