# On the role of dissipative effects in the quantum gravitational onset of
warm Starobinsky inflation in a closed universe

Meysam Motaharfar<EMAIL_ADDRESS>Parampreet Singh<EMAIL_ADDRESS>Department
of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803,
USA

###### Abstract

A problematic feature of low energy scale inflationary models, such as
Starobinsky inflation, in a spatially closed universe is the occurrence of a
recollapse and a big crunch singularity before inflation can even set in. In a
recent work it was shown that this problem can be successfully resolved in
loop quantum cosmology for a large class of initial conditions due to a non-
singular cyclic evolution and a hysteresis-like phenomena. However, for
certain highly unfavorable initial conditions the onset of inflation was still
difficult to obtain. In this work, we explore the role of dissipative particle
production, which is typical in warm inflation scenario, in the above setting.
We find that entropy production sourced by such dissipative effects makes
hysteresis-like phenomena stronger. As a result, the onset of inflation is
quick in general including for highly unfavorable initial conditions where it
fails or is significantly delayed in the absence of dissipative effects. We
phenomenologically consider three warm inflation scenarios with distinct forms
of dissipation coefficient, and from dynamical solutions and phase space
portraits find that the phase space of favorable initial conditions turns out
to be much larger than in cold inflation.

##  I Introduction

Inflation is a finite period of quasi-de Sitter accelerated expansion in the
early universe which elegantly predicts the minimal late-time curvature as
well as reproduces the adiabatic, nearly-Gaussian, and quasiscale-invariant
spectrum of primordial density fluctuations in accordance with the
observational cosmological data. An important issue in inflationary models is
that of right initial conditions for the inflaton to successfully yield
sufficient e-foldings to confirm with observations. This issue becomes more
relevant in the case of low energy inflationary models such as with
Starobinsky potential [1] which are favored by observations [2, 3]. Starting
from Planck regime, the potential energy is suppressed in low energy inflation
models and inflaton starts with kinetic energy domination [4, 5]. If the
universe is spatially closed then such a model can undergo a recollapse before
the onset of inflation and encounter the big crunch singularity. It has been
expected that a quantum theory of gravity may shed some insights on this
issue. Since the main problem in above scenario is the existence of a
recollapse followed by a big crunch singularity, if quantum gravity effects
can resolve the big crunch singularity and result in a non-singular cyclic
evolution then one can hope that in subsequent cycles conditions on dominance
of kinetic versus the potential energy alter in such a way that recollapse can
be avoided and inflation can begin.

Before we investigate the above problem in this manuscript, it is important to
make some remarks to set the right context of this study and discuss
alternative strategies to solve above problem. Our study is based on assuming
a positive spatial curvature of the universe. It has been noted earlier that
one requires a high degree of fine tuning to start inflation in low energy
models with a positive spatial curvature [6]. Thus, in a sense we take the
most difficult case to understand the initial conditions problem because if
the universe is spatially-flat or spatially open the recollapse caused by
intrinsic curvature is absent. In fact, the initial condition problem in such
cases, especially with a compact topology, becomes much easier to address [7,
8, 4]. Though it has been recently claimed that a primordial spatial curvature
may partially account for the observed anomaly in the temperature anisotropy
spectrum at low multipoles [9], and a small amount of late-time curvature
consistent with current observational data has the potential to explain the
current discrepancy between dataset probing early universe and those exploring
late-time universe properties [10, 11, 12], when Planck results are combined
with baryon acoustic oscillations data one finds that the current observations
are consistent with a spatially-flat universe [3, 13]. But the almost spatial
flatness of the universe in current epoch does not imply that it was spatially
flat in the pre-inflationary epoch. Thus it is worthwhile to study all the
cases of spatial curvature to understand the problem of initial conditions.
Let us also note that if the universe has a positive spatial curvature, the
problem of recollapse can be avoided in low energy inflationary models by
considering an additional field in a quadratic potential or a similar
potential which drives inflation in the beginning which is carried over by the
low energy inflation.111We thank the anonymous referee for pointing out this
possibility. In such a model, the additional field starts from initial
conditions which are potential dominated at Planck density such that the
problem of recollapse is completely avoided before the low energy inflation
onsets. This strategy is expected to work for any other low energy inflation
model with an additional scalar field sourced by a potential allowing the
first phase of inflation to start near Planck density. Another possibility is
to consider alternatives to low energy inflation models, such as a chaotic
inflationary model with additional cubic and quartic terms which turn out to
be consistent with the Planck data [4, 5].

While above strategies exist to alleviate the problem of initial conditions in
low energy inflation models, our objective in this study is to understand
whether quantum gravity effects when included can resolve this problem without
any additional fields which start inflation near Planck density. Since the big
crunch singularity caused by a recollapse in the pre-inflationary phase is a
roadblock to solve this problem, it is pertinent to incorporate quantum
gravity modifications which resolve the big crunch singularity to understand
the onset of inflation in low energy inflation models. This problem was
recently addressed using non-perturbative quantum gravitational effects in
loop quantum cosmology (LQC) [14]. It was shown that although a large class of
unfavorable initial conditions do not result in inflation in the classical
theory and lead to a big crunch singularity in a few Planck seconds, the
universe successfully goes through an inflationary phase after multiple non-
singular cycles of expansion and contraction due to quantum gravity effects.
The goal of the current work lies in the same direction with an aim to improve
and generalize these results to demonstrate that inclusion of dissipative
particle production in LQC results in a rather quick, and more robust onset of
inflation even for those extreme initial conditions where inflation does not
occur with above quantum gravity effects.

Let us recall that the non-perturbative loop quantum gravitational effects
resolve the big bang/big crunch singularities replacing them by a non-singular
bounce when energy density reaches Planckian values [15, 16, 17]. For the
spatially-closed model, singularity resolution results in multiple non-
singular cycles of expansion and contraction [18, 19, 20]. It is to be noted
that loop quantum gravity effects are only dominant near the classical
singularities and diminish quickly at smaller energy densities resulting in
classical dynamics at the macroscopic scales. In an effective spacetime
description of these quantum gravity effects, modified Friedmann equations can
be obtained which have been shown to capture the underlying quantum dynamics
to an excellent approximation [16, 18, 21, 22]. From these modified Friedmann
equations, one can show a generic resolution of all strong curvature
singularities in isotropic and anisotropic models in LQC [23] including in the
presence of spatial curvature [24]. Given that LQC robustly solves the problem
of singularities, it provides an excellent stage to address the problem of
resolution of onset of inflation in low energy inflationary models in presence
of a positive spatial curvature.

An interesting feature of cosmic expansion/contraction which leads to a novel
hysteresis like phenomena in non-singular cyclic evolution is the difference
in pressure during expansion and contraction stages [25]. This phenomena
occurs even in the absence of dissipative effects for suitable scalar field
potentials [26]. Hence, the universe posses an arrow of time due to an
asymmetric equation of state during expansion-contraction phase [27] rather
than entropy production due to viscous pressure as it was the case in Tolman’s
model [25]. Of course, the most challenging issue to build such models is to
overcome big bang/crunch singularities and to achieve a non-singular
evolution. This task was completed in LQC where hysteresis-like phenomena was
demonstrated for chaotic $\phi^{2}$ inflation [28], a result which was
recently generalized for Starobinsky inflation [14]. An interesting feature of
such hysteresis-like period is that although the universe may fail to inflate
at first, conditions improve in subsequent cycles for the onset of inflation
because the ratio of kinetic to potential energy decreases and subsequent
equation of state $\omega$, defined as the ratio of total pressure and energy
density, becomes less than $-1/3$. This causes a phase of accelerated
expansion and as a result the recollapse is avoided. This phenomena of
occurrence of non-singular cyclic evolution followed by inflation turns out to
be a feature of a large class of initial conditions for $\phi^{2}$ and
Starobinsky inflation models [14]. However, for the latter the onset of
inflation is found to require a much larger number of cycles in contrast to
the $\phi^{2}$ inflation and for certain highly unfavorable initial
conditions, inflation was not found to occur even after numerous non-singular
cycles of expansion and contraction [14]. The reason for this was tied to the
weak hysteresis for low energy inflation models.

To overcome the problem of onset of inflation in such cases we note that
dissipation is an indispensable part of any physical system interacting with
its environment, and there are two different dynamical realizations for
inflation: cold inflation and warm inflation [29], depending on whether non-
equilibrium dissipative particle production processes due to the couplings of
the inflaton field with other field degrees of freedom are negligible or not
during inflation. In fact, dissipative processes determine the way ultimately
the vacuum energy density, stored in the inflaton field, ends up converting
into radiation, thus allowing the universe to transit from the accelerating
phase to the radiation-dominated epoch. In the standard inflationary or cold
inflation scenarios, dissipative effects are typically ignored during the
inflationary phase if any pre-inflationary radiation energy density is
diluted. The universe then ends up in a supercooling phase requiring a
reheating mechanism [30], where the inflaton starts oscillating around the
minimum of its potential and progressively dissipates its energy into other
relativistic light degrees of freedom, to heat up the universe again as
required by the standard big bang cosmology. On the contrary, dissipative
effects may be strong enough during inflation where pre-inflationary radiation
energy density can be sustained during inflation and also become dominant at
the end of inflation whereby the universe smoothly enters into radiation
dominated epoch without a need for a separate reheating period [31]. Such
dissipative effects bring about much richer dynamics for inflation at both
background and perturbative levels (for reviews see for eg. [32]) introducing
warm inflation as a promising complimentary version of cold inflation by
addressing some of long-lasting problems related to (post-)inflationary
picture in cold inflation scenarios.

For a comparison with cold inflation it is useful to recall some of the
features of warm inflation. It is interesting to note that dissipative effect
appears as a supplementary friction term in background equations allowing
embedding of steeper potentials in warm inflation solving so-called
$\eta$-problem [33]. Also it leads to several different possibilities for
graceful exit depending on the form of potential, form of dissipation
coefficient, and whether the dynamics is in strong or weak dissipative regime
[34]. Moreover, dissipative effects also modify primordial spectrum of
curvature perturbations, resulting in a smaller energy scale of inflation, and
reconciling steeper potentials with observational data [35]. Such appealing
features of warm inflation allows it to simultaneously satisfy the so-called
swampland conjectures, provided warm inflation can occur in sufficiently
strong dissipative regime [36, 37, 38]. Although it is enormously challenging
to achieve a strong dissipative regime in warm inflation, two models were
successfully constructed to push warm inflation into strong dissipative regime
with inspiration from particle physics [39, 40]. Furthermore, the inflaton
itself can be a source and responsible for cosmic magnetic field generation
[41] and in combination with the intrinsic dissipative effects lead to a novel
dissipative baryogenesis scenario during inflation [42]. More recently, it was
shown that warm inflation enables a stable remnant of inflaton in the post-
inflationary epoch which can behave either like cold dark matter accounting
for all the dark matter in the universe [43] or like a quintessence at late
time generating the present phase of accelerated expansion [44] (see also [45,
46] for unifying all conventional ingredients of modern cosmology using
dissipative effects).

The goal of this manuscript is to investigate the dissipative particle
production effects222For brevity, we label these effects in the following as
dissipative effects. We note that the source of such dissipative effects is
particle production. on pre-inflationary dynamics of $k=1$ LQC and understand
their role on the hysteresis-like phenomena and the onset of inflation for
Starobinsky potential starting from highly unfavorable initial conditions. Our
goal will be to consider those cases which failed to lead to inflation in
absence of dissipative effects. In Sec. II, we give a brief review of the
effective dynamics of $k=1$ LQC and the warm inflation and discuss the way
dissipative effects are implemented in $k=1$ LQC. In Sec. III, we solve
dynamical system of equations in the presence of dissipative effects and show
that even small amount of dissipation enlarges the phase space of initial
conditions for which inflation occurs. We phenomenologically investigate these
solutions for three models of warm inflation: the warm little inflaton (linear
temperature dependent dissipation coefficient), variant of warm little
inflaton (inverse temperature dependent dissipation coefficient) and minimal
warm inflation (cubic temperature dependent dissipation coefficient).
Moreover, we also investigate some features of the qualitative dynamics using
phase-space portraits. These results show that in presence of dissipation the
hysteresis-like phenomena becomes much stronger and results in a quick onset
of inflation for even those initial conditions where inflation could not start
in absence of dissipation. We conclude the manuscript with a summary of our
results in Sec. IV.

##  II Effective Dynamics in $k=1$ LQC and Warm Inflation

In this section we first briefly review the effective dynamics of spatially
closed loop quantum cosmology in the holonomy quantization [18]. This is
followed by a discussion of the dynamical equations in the warm inflation
scenario and the way warm inflation can be implemented in the effective
spacetime description of $k=1$ model in LQC.

###  II.1 Effective dynamics of k=1 LQC

LQC is a canonical quantization based on Ashtekar-Barbero variables – the
connection $A^{i}_{a}$ and its conjugate triad $E^{i}_{a}$ which due to
homogeneity and isotropy, symmetry reduce to $c$ and $p$ for the $k=1$
Friedmann–Lemaître–Robertson–Walker (FLRW) model . In the improved dynamics or
the $\bar{\mu}$ scheme of LQC [16], it turns out that an equivalent set of
variables defined as $b=c|p|^{-1/2}$ and $v=|p|^{3/2}$ are more convenient to
obtain the quantum and effective description. Here $v$ denotes the physical
volume of the unit sphere spatial manifold and is related to the scale factor
of the universe as $v=2\pi^{2}a^{3}$. The phase space variables $b$ and $v$
satisfy $\\{b,v\\}=4\pi G\gamma$ where $\gamma$ denotes the Barbero-Immirzi
parameter whose value is generally taken to be $\gamma\approx 0.2375$ in LQC
following the calculations of black hole thermodynamics in loop quantum
gravity.

The effective Hamiltonian in the holonomy based quantization of the $k=1$
model in LQC for the lapse chosen as unity is given by

$\displaystyle\mathcal{H}_{\mathrm{eff}}=-\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}v\left[\sin^{2}(\lambda
b-D)-\sin^{2}D+(1+\gamma^{2})D^{2}\right]+\mathcal{H}_{\mathrm{matt}}\approx
0$ (1)

where $D=\left(\lambda(2\pi^{2})^{1/3}\right))/v^{1/3}$ and
$\lambda^{2}=4(\sqrt{3}\pi\gamma)\ell_{pl}^{2}$. Here we have ignored the
modifications from the inverse volume effects which turn out to be negligible
in comparison to the holonomy modifications [18].333Note that a non-singular
dynamics results solely from inverse volume modifications too in $k=1$ model
in LQC [47], which has been used to understand conditions for the onset of
inflation [48]. Before we examine the dynamics resulting from this Hamiltonian
let us note that there exists another quantization of the $k=1$ model in LQC
which is known as the connection based quantization [20]. Though there exist
some qualitative differences in the way singularity resolution occurs in this
prescription as compared to the holonomy based quantization [20, 49], the main
features of dynamics remain the same. Especially, the existence of hysteresis
which plays an important role in the onset of inflation is robust in both the
quantization prescriptions and the difference between the two approaches turn
to be small for inflationary dynamics [28]. For this reason we consider only
the effective dynamics for the holonomy quantization in this analysis.

Using Hamilton’s equations, the equation of motion for volume turns out to be

$\displaystyle\dot{v}=\\{v,\mathcal{H}_{\mathrm{eff}}\\}=\frac{3}{\gamma\lambda}v\sin(\lambda\beta-D)\cos(\lambda\beta-D)$
(2)

which results in the following modified Friedmann equation

$\displaystyle H^{2}=\frac{\dot{v}^{2}}{9v^{2}}=\frac{8\pi
G}{3}(\rho-\rho_{\rm{min}})\left(1-\frac{\rho-\rho_{\rm{min}}}{\rho_{\max}^{\rm{flat}}}\right)~{}.$
(3)

Here $\rho_{\max}^{\rm{flat}}=3/(8\pi G\gamma^{2}\lambda^{2})$ denotes the
energy density at the bounce for the spatially-flat model in LQC and

$\displaystyle\rho_{\rm{min}}=\rho_{\max}^{\rm{flat}}\left[(1+\gamma^{2})D^{2}+\sin^{2}(D)\right]~{}$
(4)

denotes the minimum allowed energy density in the evolution. In the classical
universe this value coincides with the value of energy density at which a
classical recollapse occurs. The maximum of the energy density is given by

$\displaystyle\rho_{\rm{max}}=\rho_{\rm{min}}+\rho_{\max}^{\rm{flat}}~{}.$ (5)

Note that in the quantum regime depending on the initial conditions, a bounce
as well as a recollapse can occur at $\rho_{\rm{min}}$ as well as at
$\rho_{\rm{max}}$ [14].

The Hamilton’s equations for phase space variable conjugate to $v$ is given by

$\displaystyle\dot{b}=\\{b,\mathcal{H}_{\rm{eff}}\\}=-4\pi
G\gamma\left[\rho+P-\rho_{1}\right]$ (6)

with

$\displaystyle\rho_{1}=\frac{\rho_{\max}^{\rm{flat}}D}{3}\left[2(1+\gamma^{2})D-\sin(2\lambda\beta-D)-\sin(2D)\right]$
(7)

where $P$ denotes the pressure which equals
$P=-\partial\mathcal{H}_{\mathrm{matt}}/{\partial v}$. The dynamical equations
for the scalar field matter variables with a potential $V(\phi)$ are

$\displaystyle\dot{\phi}$
$\displaystyle=\\{\phi,\mathcal{H}_{\rm{eff}}\\}=\frac{p_{\phi}}{p^{3/2}}$ (8)
$\displaystyle\dot{p}_{\phi}$
$\displaystyle=\\{p_{\phi},\mathcal{H}_{\rm{eff}}\\}=-p^{3/2}V_{,\phi}~{}.$
(9)

Using above equations it is straightforward to show that Klein-Gordon equation
follows along with the standard conservation law for matter energy density.

Above dynamical equations encode non-perturbative quantum gravitational
effects which result in a non-singular bounce of the universe in the Planck
regime [18, 28, 14]. This results in non-singular cycles of expansion and
contraction if the matter does not violate strong energy condition i.e. has
equation of state $\omega=P/\rho$ greater than $-1/3$. For latter type of
matter content, the universe undergoes a recollapse at late times resulting in
a contraction and a big crunch singularity in the classical theory. This
singularity is avoided in LQC resulting in a bounce and another phase of
expansion and a possible recollapse if the equation of state $w>-1/3$ in the
expanding phase. If the recollapses occur at the macroscopic scales, the
difference in the volumes of two consecutive recollapses is found to be [28]

$\delta v^{1/3}_{\mathrm{rec}}=\frac{-\oint
Pdv}{(2\pi^{2})^{2/3}\rho_{\mathrm{max}}^{\rm{flat}}\gamma^{2}\lambda^{2}}.$
(10)

This implies that in each cycle of expansion and contraction the maximum
volume $v_{\mathrm{rec}}$ changes. This occurs because of the asymmetry of the
pressure during different phases of a given cycle which results in a
hysteresis-like phenomena [26, 28]. This hysteresis-like phenomena has been
shown to be responsible for alleviating problems with onset of inflation for
different potentials, especially low energy scale models [14]. Before we
examine this phenomena in presence of radiation production in warm
inflationary scenarios, we summarize the latter and obtain the relevant
equations in LQC.

###  II.2 Warm inflationary dynamics in LQC

The dynamical realization of warm inflation is different from the cold
inflation due to the presence of radiation as well as the possibility of
energy exchange between inflaton and radiation energy density. Hence, the
total energy density of the universe in warm inflation reads

$\displaystyle\rho=\rho_{\phi}+\rho_{r}$ (11)

where $\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)$ is scalar field energy
density with $V(\phi)$ being some potential function and $\rho_{r}$ is the
radiation energy density. The inflaton field $\phi$ and the radiation energy
density form a coupled system in warm inflationary dynamics due to dissipation
of energy out of the inflaton system and into radiation. The background
evolution equations are respectively given by [34]

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=-\Upsilon(\phi,T)\dot{\phi}$
(12) $\displaystyle\dot{\rho}_{r}+4H\rho_{r}=\Upsilon(\phi,T)\dot{\phi}^{2}.$
(13)

Here $\Upsilon(\phi,T)$ is the dissipation coefficient which can be a function
of both inflaton and temperature, depending on the specifics of the
microscopic physics behind the construction of a warm inflation model. For a
radiation or a bath of relativistic particles, the radiation energy density is
given by $\rho_{r}=\left(\pi^{2}g/30\right)T^{4}$, where $g$ is the effective
number of light degrees of freedom ($g$ is fixed according to the dissipation
regime and interactions form used in warm inflation). Such radiation
production results in entropy production where the entropy density $s$ is
related to radiation energy density by $Ts=(4/3)\rho_{r}$, i.e., is related to
temperature as $s=\left(2\pi^{2}g/45\right)T^{3}$, where we have considered a
thermalized radiation bath as is typically the case in warm inflationary
scenarios. Then, eq. (13) can be written in terms of entropy as follows [38]

$\displaystyle T\left(\dot{s}+3Hs\right)=\Upsilon\dot{\phi}^{2}.$ (14)

As we will see in next section, such entropy production significantly changes
the hysteresis-like phenomena. In fact, the term $3Hs$, which is positive in
expanding universe ($H>0$) and negative in contracting universe ($H<0$),
produces a larger difference in pressure during expansion/contraction stages
making hysteresis-like phase stronger in comparison with the case without
dissipative effects.

Let us note that the richer dynamics of warm inflation sharpened the interest
for finding explicit models aiming at overcoming two important issues found in
earlier particle physics realizations of warm inflation. First, the
requirement of large field multiplicities so as to be able to sustain a
nearly-thermal bath and, second, the difficulty to achieve strong dissipative
regimes ($\Upsilon\gg H$), due to the interplay between inflaton and radiation
fluctuations, leading to appearance of growing modes in the scalar curvature
power spectrum and that can render it inconsistent with the observations. The
former problem was first solved with an introduction of a new class of warm
inflation model building realization motivated from the ingredients used in
“Little Higgs” models of electroweak symmetry breaking where the inflaton is a
pseudo-Nambu Goldstone boson of a broken gauge symmetry and its potential is
protected against large radiative corrections by symmetry obeyed by the model
while still having enough interactions to allow thermalization of light
degrees of freedom. This results in enough dissipation even if the mediators
are very light with respect to ambient temperature. In such a model also known
as warm little inflaton, the dissipation coefficient is given by [50]

$\displaystyle\Upsilon_{\mathrm{lin}}=C_{\mathrm{lin}}T.$ (15)

We refer to the above $\Upsilon_{\mathrm{lin}}$ as the linear dissipation
coefficient. Although warm little inflaton was successful in producing
sustainable thermalized radiation bath utilizing just a few mediator fields,
it could not obtain strong dissipative regime, which allows steeper potentials
to be embedded in warm inflation by making energy scale of inflation smaller.
To this end, a concrete model of warm inflation, the so called minimal warm
inflation [39], was recently constructed in which the inflaton has axion-like
coupling to gauge fields. Since the inflaton is an axion, its shift symmetry
protects it from any perturbative backreactions and thus from acquiring a
large thermal mass. Hence, the thermal friction from this bath can easily be
stronger than Hubble friction even for small number of fields. The
corresponding axion friction coefficient turn out to be as

$\displaystyle\Upsilon_{\mathrm{cub}}=C_{\mathrm{cub}}T^{3}.$ (16)

Hereafter, we refer to the above $\Upsilon_{\mathrm{cub}}$ as the cubic
dissipation coefficient. In this regard, another model was also recently
proposed inspired from an idea used in warm little inflaton where the inflaton
is directly coupled to light scalar bosonic fields rather than fermionic
fields which is known as a variant of warm little inflaton [40]. Although the
exact form of dissipation coefficient is complex, the leading behaviour of
dissipation, when the effective mass is dominated by thermal part, varies as

$\displaystyle\Upsilon_{\mathrm{inv}}=C_{\mathrm{inv}}T^{-1}.$ (17)

Hereafter, we refer to the above $\Upsilon_{\mathrm{inv}}$ as the inverse
dissipation coefficient. We should note that $C_{\mathrm{inv}}\ll
C_{\mathrm{lin}}\ll C_{\mathrm{cub}}$ since it should be fixed in such a way
that the condition for sustainable thermal bath, i.e. $T>H$, is satisfied
during inflationary phase.

Taken together, to consider the dissipative effects during both pre-
inflationary and inflationary phases all the way from the bounce until the end
of inflation, we phenomenologically implement dissipative effects into the
effective equations of spatially closed model LQC. The resulting dynamical
equations are,

$\displaystyle\dot{v}$ $\displaystyle=\frac{3}{\gamma\lambda}v\sin(\lambda
b-D)\cos(\lambda b-D)$ (18) $\displaystyle\dot{b}$ $\displaystyle=-4\pi
G\gamma\left[\frac{p_{\phi}^{2}}{v^{2}}+\frac{4}{3}\rho_{r}-\rho_{1}\right]$
(19) $\displaystyle\dot{\phi}$ $\displaystyle=-\frac{p_{\phi}}{v}$ (20)
$\displaystyle\dot{p}_{\phi}$
$\displaystyle=-v\,V_{,\phi}-\Upsilon(\phi,T)\,p_{\phi}$ (21)
$\displaystyle\dot{\rho}_{r}$
$\displaystyle=-\left[\frac{4}{\gamma\lambda}\sin(\lambda b-D)\cos(\lambda
b-D)\right]\rho_{r}+\frac{\Upsilon(\phi,T)\,p_{\phi}^{2}}{v^{2}}.$ (22)

In the next section, we will first discuss the way such dissipative effects,
or equivalently entropy production, change the dynamics of pre-inflationary
phase and also enlarge the phase-space of initial conditions which result in a
(warm) inflation. Then, we perform a qualitative analysis of the dynamical
equations to understand the attractor behavior of the solutions and gain
insights on the way dissipative effects help in the onset of inflation even
starting from highly unfavorable initial conditions.

##  III Dissipative effects on pre-inflationary dynamics of k=1 LQC

In this section we investigate the consequences of dissipative effects in LQC
to address the problem of onset of inflation for the Starobinsky potential. We
discussed in Sec. 2 the way non-singular cycles of expansion and contraction
result in a hysteresis-like phenomena which arises due to differences in
pressure during expansion and contraction stages of cosmic evolution. Due to
this difference in pressure, the work done during one cycle can be positive or
negative depending on the potential function. For sufficiently flat
potentials, the work can be positive resulting in increasing the size of the
universe in the successive cycles. Because of this even if the inflaton starts
with a kinetic energy dominated conditions and an equation of state close to
unity, the equation of state decreases in each cycle and eventually becomes
less than $-1/3$ which leads to an onset of inflation. Hence, if the universe
fails to inflate after the first cycle, it can do so after subsequent cycles
enlarging the phase-space of initial conditions which results in inflation
[28]. Recently it was shown that for Starobinsky potential universe can
inflate for a large part of initial conditions, however, it should go though
numerous cycles of expansion and contraction [14]. Further, for some of the
initial conditions the inflation does not commence even after a large number
of non-singular cycles. As we will see the dissipation or entropy production,
leads to larger differences in pressure during expansion-contraction phase
resulting in larger amplitude of the cycles. Therefore, we expect that entropy
production due to radiation particle production make the hysteresis phenomena
stronger leading into the universe with bigger size in successive cycles,
causing the universe to inflate after small number of cycles. In the following
we first obtain the background solutions demonstrating above phenomena which
is followed by discussion of phase space portraits in qualitative dynamics of
this model.

###  III.1 Dissipative effects for Starobinsky potential

Starobinsky inflation is a prominent example of low energy inflation models
favored by current observations. In classical cosmology, this model results
from adding $R^{2}$ term to action which translates to adding the following
potential in the Einstein frame

$\displaystyle
U(\phi)=\frac{3m^{2}}{32\pi}\left(1-e^{-\sqrt{\frac{16\pi}{3}}\phi(t)}\right)^{2}~{}.$
(23)

But in LQC, the above potential is not obtained from an $R^{2}$ term in the
action, since the covariant action in LQC does results in higher order
curvature terms but in a Palatini framework [51]. As in previous works in LQC,
we consider above potential as a phenomenological input in effective dynamics.

In Starobinsky model the inflation is supposed to start at energy scales far
lower than Planck scale and as a result the initial conditions in the Planck
regime are such that kinetic energy dominates the potential energy. If one
numerically solves the classical cosmological dynamics of above potential one
finds that the universe undergoes a recollapse before potential energy can
dominate and encounters a big crunch singularity [4, 5]. We would see that
this situation changes dramatically in the effective dynamics in LQC. Below we
numerically solve dynamical equations (18) for various initial conditions,
using explicit Runge-Kutta algorithm and stiff-switching method in Mathematica
with accuracy and precision goals set to eleven. The initial value of $b$ (the
conjugate to volume $v$) is fixed by the vanishing of the effective
Hamiltonian constraint. Moreover, we also set the initial value of $p_{\phi}$
using the condition for the bounce, i.e. $\rho=\rho_{\mathrm{max}}$.
Therefore, we are left with just three initial conditions on volume ($v_{0}$),
scalar field ($\phi_{0}$) and initial radiation energy density ($\rho_{r0}$).
We choose initial conditions such that the radiation energy density is sub-
dominated in comparison with both the kinetic energy density and potential
energy density of the inflaton field and the bounce happens with kinetic
dominated initial conditions ($\rho_{r0}\ll
U(\phi_{0})\ll\dot{\phi}_{0}^{2}/2$).

Figure 1: The evolution of volume for different values of cubic dissipation
coefficient. Initial conditions are chosen at the bounce with $v_{0}=5\times
10^{7}$, $\phi_{0}=-1$, $\rho_{r0}=10^{-12}$, and $g=17$. Figure 2: The
evolution of equation of state for cubic dissipation coefficient and same
initial conditions as in Fig. 1.

In the following we first solve the dynamical equations for the case of
minimal warm inflationary model i.e. with a cubic dissipation coefficient. In
this case inflaton has an axionic coupling to a non-abelian gauge theory and
the sphaleron transitions between gauge vacua, existing at sufficiently high
temperatures. And, if the corresponding non-Abelian gauge theory has gauge
group $SU(3)$, there are 8 gauge bosons, each of which contribute two
relativistic degrees of freedom. Including the inflaton itself, there are in
total 17 relativistic degrees of freedom. So we set the number of relativistic
degree of freedom $g=17$ [37]. In Figs. 1 and 2, we plot the evolution of
volume and equation of state for three different values of $C_{\mathrm{cub}}$
in LQC. For the initial conditions $v_{0}=5\times 10^{7}$, $\phi_{0}=-1$,
$\rho_{r0}=10^{-12}$ the dynamical evolution is non-singular for all the
considered values of $C_{\mathrm{cub}}$. The initial conditions are chosen
such that in absence of dissipation, inflation does not start after various
cycles of non-singular evolution. As can be seen, in case of
$C_{\mathrm{cub}}=0$ (dissipationless universe), the universe does not enter
an inflationary phase even after many cycles of non-singular evolution. This
is because the hysteresis-like phenomena is not large enough to set scalar
field at the flat part of potential function in subsequent cycles. However,
dissipative effects make hysteresis phase stronger (decreasing the number of
cycles and increasing its amplitudes) whereby the universe begins inflationary
phase after small number of cycles. This is evident in the dynamical evolution
for $C_{{\mathrm{cub}}}=0.4$ and $C_{\mathrm{cub}}=1000$. We see from the
former case that even a small non-zero value of $C_{\mathrm{cub}}$, resulting
in small dissipative effects, has substantial effects on the hysteresis-like
phenomena and the onset of inflation. We find from the volume and equation of
state plot that a phase of inflation starts after a few non-singular cycles
when volume grows exponentially and equation of state becomes less than
$-1/3$. But such a small value of dissipation coefficient can not sustain the
thermal bath during inflation and one needs a larger value of
$C_{\mathrm{cub}}$. As one increases the value of $C_{\mathrm{cub}}$, the
dissipative effects make hysteresis phase very strong and inflationary phase
starts just after just one bounce. This is shown in Fig. 1 for the case of
$C_{\mathrm{cub}}=1000$. Here we should note that the curves for
$C_{\mathrm{cub}}=0.4$ and $C_{\mathrm{cub}}=1000$ start from the same initial
volume but because of the use of logarithmic scale in the plot, the figure
does not show the same value of volume for both the curves. We note that the
evolution of equation of state in Fig. 2 shows that for $C_{\mathrm{cub}}=0$,
the equation of state oscillates between 1 and 0.5 for the entire range of
evolution, however for non-vanishing dissipation coefficients it decreases
quickly below $w=-1/3$ and becomes $w\approx-1$ indicating an onset of slow-
roll inflation. As one can see, the equation of state becomes $-1$ much
earlier for larger dissipation coefficient.

Figure 3: The evolution of volume for different values of linear dissipation
coefficient. Initial conditions are chosen at the bounce with $v_{0}=10^{7}$,
$\phi_{0}=-1.5$, $\rho_{r0}=10^{-11}$, and $g=12.5$. Figure 4: The evolution
of equation of state for linear dissipation coefficient and same initial
conditions as in Fig. 3.

We now discuss the case of warm little inflaton in LQC. In Figs. 3 and 4, we
plot evolution of volume and equation of state for linear dissipation
coefficient and three different values of $C_{\mathrm{lin}}$ for initial
conditions $v_{0}=10^{7}$, $\phi_{0}=-1.5$, and $\rho_{r0}=10^{-11}$. We also
fix $g=12.5$ using analysis in [50]. We consider non-zero values of
$C_{\mathrm{lin}}$ as 0.00004 and 0.00008 which are typical values for warm
inflation to happen in spatially-flat spacetime. As before, the chosen initial
conditions correspond to the unfavorable ones where inflation does not start
in LQC even after various cycles of non-singular evolution when dissipation is
absent. This can be seen from the curve corresponding to $C_{\mathrm{lin}}=0$,
where universe oscillates in non-singular evolution but there is no onset of
inflation since equation of state never becomes less than $-1/3$. However,
when we add dissipative effects, the hysteresis becomes stronger and we see
that the universe experiences an inflationary phase after a small number of
cycles. We see that the equation of state becomes less than $-1/3$ after a few
cycles for $C_{\mathrm{lin}}=0.00004$ and $C_{\mathrm{lin}}=0.00008$. As we
increase the value of $C_{\mathrm{lin}}$, the number of cycles prior to onset
of inflation decrease and the amplitude of the cycles become larger.

Figure 5: The evolution of volume for different values of inverse dissipation
coefficient. Initial conditions are chosen at the bounce with $v_{0}=2.5\times
10^{6}$, $\phi_{0}=-2$, $\rho_{r0}=10^{-9}$, and $g=12.5$. Figure 6: The
evolution of equation of state for inverse dissipation coefficient and same
initial conditions as in Fig. 5.

Finally, we consider the variant of warm little inflaton with inverse
dissipation coefficient which is shown in Figs. 5 and 6. As in previous cases,
initial conditions are chosen such that there is no inflationary phase even
after many cycles in absence of dissipative effects. But choosing a non-zero
dissipation coefficient, even if small, leads to a striking difference in
dynamics and results in a stronger phenomena of hysteresis. In these figures
we choose $C_{\mathrm{inv}}=3\times 10^{-11}$ and $6\times 10^{-11}$ and fix
$g=12.5$. The chosen values of $C_{\mathrm{inv}}$ are smaller than two other
cases as we discussed previously. We find that as we increase the value of
$C_{\mathrm{inv}}$, the number of cycles decrease and the amplitude of the
cycles become larger and the universe enter into inflationary phase sooner.

To summarize the results so far, we have found the dissipative effects
resulting in radiation production, make the hysteresis phenomena stronger and
setting the condition for inflation to happen sooner for dissipation
coefficients which have cubic, linear and inverse relationship to temperature.
Though we discussed a sample of initial conditions, our results are robust to
changes in initial conditions. To gain some insights on the qualitative
dynamics and robustness of results we study the phase-space portraits in the
following.

###  III.2 Qualitative dynamics in phase-space portrait

It is useful to understand the phase space portraits for qualitative dynamics
by introducing following variables,

$\displaystyle X(t)$ $\displaystyle=\chi_{0}\left(1-e^{-\sqrt{\frac{16\pi
G}{3}}\phi(t)}\right)$ (24) $\displaystyle Y(t)$
$\displaystyle=\frac{p_{\phi}(t)}{v(t)\sqrt{2\rho_{\mathrm{max}}}}~{}~{}Z(t)=\sqrt{\frac{\rho_{r}(t)}{\rho_{\mathrm{max}}}}$
(25)

where $\chi_{0}=m\sqrt{\frac{3}{32\pi G\rho_{\mathrm{max}}}}$ and
$\rho_{\mathrm{max}}$ denotes the maximum energy density (5) determined by the
initial conditions. Our goal will be to find the inflationary attractors for
different choices of dissipation coefficients and initial conditions. These
have been studied earlier for cold inflation in detail in LQC [52]. The
inflationary attractor lies at $(X=0,Y=0)$ which corresponds to the reheating
phase in cold inflation and the beginning of radiation epoch in warm
inflation.

Figure 7: Projection of 3 dimensional phase space portrait on $Z=0$ plane for
cubic dissipation coefficient with $m=0.62$, $C_{\mathrm{cub}}=7.5$,
$v_{0}=35$, $\rho_{r0}=10^{-3}$ and seven distinct initial conditions for
$\phi_{0}$.

Figure 8: Evolution of energy components as well as equation of state for
cubic dissipation coefficient with $m=0.62$, $C_{\mathrm{cub}}=7.5$,
$v_{0}=35$, $\rho_{r0}=10^{-3}$ and $\phi_{0}=0.5$ (gray curve in Fig. 7).

Starting with the cubic dissipation in Fig. 7, the left plot shows the
projection of entire phase space region on plane $Z=0$ and the right plot
zooms in on the attractor near the origin. The vertical dashed black line
corresponds to $X=\chi_{0}$ and all real solutions lie to the left of this
line. For a better visualization of the qualitative features it is useful to
consider a large value of inflaton mass which is here chosen to be $m=0.62$.
The solid black circular curve corresponds to the energy density of the first
bounce at $t=0$ where the initial conditions are set. The left plot in Fig. 7
shows curves corresponding to seven distinct initial conditions for
$\phi_{0}$, while other parameters are fixed, for which the universe undergoes
inflation in presence of dissipative effects. For the same initial conditions
in absence of dissipative effects the universe goes through many cycles
without onset of an inflationary phase. It should be noted that we chose
$C_{\mathrm{cub}}$ large enough to see the end of inflation and also post-
inflationary phase. So in most of the cases, the hysteresis-like phenomena go
away and universe experience an inflationary phase after just one bounce.

We expand on details of above dynamics in Fig. 8, where the evolution of the
potential energy density $V(\phi)$, kinetic energy density $\dot{\phi}^{2}/2$
and the radiation energy density $\rho_{r}$ are shown for gray curve in Fig.
7. We can see from the figure that the universe starts from a bounce in the
kinetic dominated regime (initial condition with equation of state
$\omega\approx 1$), with $\dot{\phi}^{2}/2\gg U(\phi)\gg\rho_{r}$, and the
kinetic energy very soon dilutes away since it behaves as $a^{-6}$. In the
subsequent evolution we find that radiation energy density becomes important
in comparison to kinetic and potential energies. Such radiation dominated
regime before the inflationary phase has also been reported for warm inflation
in spatially-flat LQC [53] (see also [54] for review on warm inflation in
spatially-flat LQC). After some cycles, when the hysteresis-like phenomena
causes an onset of inflation, the radiation energy density becomes subdominant
in evolution. Note that contrary to cold inflation, radiation is concurrently
produced during inflationary phase and it may reach an equality with potential
energy density at the end of inflation if dissipative effects are strong
enough to sustain the thermal bath, and the universe smoothly enters into
radiation dominated epoch without subsequent (p)reheating phase. However, as
it is clear in Fig. 8, the universe does not enters into radiation dominated
epoch at the end of inflation. This is because the considered value of
$C_{\mathrm{cub}}$, chosen due to computational constraints, is not large
enough and hence the inflationary phase is a cold-type and a reheating
mechanism is a must. Hence, although small dissipation coefficient could not
sustain thermal bath during inflation leading to warm-type inflation but it
has substantial effect on pre-inflationary dynamics and the onset of inflation
which is evident from the existence of a $\omega\approx-1$ phase from the plot
of equation of state. We further note that if dissipation coefficient is
chosen large enough the warm inflation starts quickly and the radiation energy
density reaches similar values as the potential energy density at the end of
inflation.

Figure 9: Projection of 3 dimensional phase space portrait on $Z=0$ plane for
linear dissipation coefficient with $m=0.62$, $C_{\mathrm{lin}}=0.8$,
$v_{0}=35$, $\rho_{r0}=10^{-3}$ and seven distinct initial conditions for
$\phi_{0}$.

Figure 10: Evolution of energy components as well as equation of state for
linear dissipation coefficient with $m=0.62$, $C_{\mathrm{lin}}=0.8$,
$v_{0}=35$, $\rho_{r0}=10^{-3}$ and $\phi_{0}=1.55$ (gray curve in Fig. 9).

Fig. 9 illustrates the projection of the entire phase-space portrait on $Z=0$
plane for linear dissipation coefficient and $m=0.62$. The left plot shows
that seven distinct initial conditions (different values of $\phi_{0}$ while
other parameters are fixed) starting from the first bounce all exhibit an
attractor behaviour and come to the center of circle (fixed point). In Fig.
10, the evolution of energy density components shows that inflation ends in
radiation dominated epoch due to dissipative effects. Although such
dissipative effects are large enough to sustain thermal bath during inflation,
they are not large enough to suppress kinetic energy during inflation and
terminate the universe in a radiation dominated epoch. Hence, there is very
short kinetic dominated regime before the universe transits into radiation
epoch which is typical in warm inflation scenario when the dissipation effect
is small (such kinetic dominated regime after warm inflation has also been
seen in [45]). However, such a kinetic dominated regime does not have any
adverse implications since it is very short. Moreover, since in this model
such kinetic dominated regime occurs around the minimum of potential, we see
spiral behavior which is typical in cold inflation due to reheating phase.
However, this oscillatory phase plays no role in making the universe hot and
the universe enters into a radiation dominated regime due to radiation
production during inflation and not a reheating phase. We also find that there
is a radiation dominated regime before inflation phase as it was seen in case
of cubic dissipation coefficient.

Figure 11: Projection of 3 dimensional phase space portrait on $Z=0$ plane for
inverse dissipation coefficient with $m=0.79$, $C_{\mathrm{inv}}=0.2$,
$v_{0}=30$, $\rho_{r0}=10^{-3}$ and six distinct initial conditions for
$\phi_{0}$.

Figure 12: Evolution of energy components as well as equation of state for
inverse dissipation coefficient with $m=0.79$, $C_{\mathrm{inv}}=0.2$,
$v_{0}=30$, $\rho_{r0}=10^{-3}$ and $\phi_{0}=2.2$ (brown curve in Fig. 11).

In Fig. 11 we illustrate the projection of the entire phase-space portrait on
$Z=0$ plane for inverse dissipation coefficient and $m=0.79$. We find that in
this case although there is an attractor behaviour and all initial conditions
starting from the first bounce come to the center of the phase space plot,
there is no spiral behaviour. This is because the universe enters into
radiation dominated regime due to strong dissipative effects. There is no
oscillatory behaviour which is typical of cold inflation or what is found for
previous cases of dissipation coefficient. In particular, the kinetic energy
always remain sub-dominated during inflation due to strong dissipative effects
and inflation ends when radiation becomes equal with potential energy.
Furthermore, we find that there is also a radiation dominated regime before
inflationary phase as it was seen for cubic and linear dissipation
coefficients.

##  IV Summary

Onset of inflation in low energy inflationary models in classical theory is
challenging in spatially closed models because of the recollapse of the
universe before inflation can set in. Indeed one requies a high degree of fine
tuning of initial conditions for the onset of inflation in such a case [6]. On
the other hand if the spatial curvature is zero or negative this problem is
non-existent and onset of inflation becomes highly probable [7, 8, 4]. The
recollapse in spatially closed model causes a big crunch singularity and a
closed universe ends in a big crunch singularity in a few Planck seconds
before the beginning of an inflationary phase. This is a long standing problem
whose resolution becomes important given that current observational data
favors low energy inflation models and a slight positive spatial curvature of
the universe. There are various ways to overcome this issue in classical
cosmology. Apart from the case of considering models with spatially flat and
spatially open universe, one can consider a low energy inflation model coupled
with an additional scalar field which drives an early phase of inflation near
Planck density which is taken over later by low energy inflation. Further one
can introduce additional higher order terms in quadratic potential to fit with
Planck data [4, 5]. But if one aims to understand this issue for a single
field set up in low energy inflation models, two issues need to be addressed
simultaneously. The first is a successful and a generic resolution of
singularities, and the second a mechanism to create favorable conditions for
inflation to begin. The challenges underlying the first problem are well known
and require insights from non-perturbative quantum gravity. The latter problem
is also non-trivial given that inflation in low energy inflation models begins
at very small energy scales compared to Planck scale due to which initial
conditions in the Planck regime are kinetic energy dominated which lead to a
recollapse of the universe. The above problem was recently analyzed in LQC
[14] where non-perturbative quantum gravity effects are known to result in a
generic resolution of all strong curvature singularities [23]. In particular,
the big bang/big crunch singularities are resolved and replaced by a non-
singular bounce [15, 16, 18]. It was found that for Starobinsky inflation
potential, the universe in LQC cycles through various periods of expansion and
contraction resulting in an onset of inflation even when inflaton starts from
kinetic energy dominated initial conditions which result in a big crunch in a
few Planck seconds. At the heart of this resolution likes a hysteresis-like
phenomena which changes the ratio of kinetic and potential energy in
subsequent cycles in such a way that the equation of state even when starting
from $\omega\approx 1$ becomes less than $-1/3$. The universe then enters a
phase of accelerated expansion, a recollapse is avoided and inflation sets in.

Although hysteresis-like phenomena can enlarge the phase-space of initial
conditions for plateau-like potential, with Starobinsky potential as the most
known one, the inflationary phase either occurs after many cycles or does not
happen due to flatness of potential for some unfavorable initial conditions
[14]. A pertinent question is whether there exists a mechanism which can
result in onset of inflation in LQC even for such unfavorable initial
conditions. The goal of the paper was to successfully answer this question.
Motivated by Tolman’s model in which the hysteresis-like phase happens due to
entropy production sourced by viscous pressure and also warm inflationary
dynamics, we considered spatially closed LQC in which the scalar field
concurrently dissipates its kinetic energy into radiation field starting from
the first bounce. Hence, there are two contributions for work done in each
cycle; one from an asymmetric equation of state of scalar field during
expansion-contraction phase and the other from entropy production due to
dissipative effects. Due to dissipative effects one expects the phenomena of
hysteresis to become much stronger and inflation to set in far more easily.

We worked in the setting of effective spacetime description in LQC and
obtained Hamilton’s equations with non-perturbative quantum gravity
corrections in presence of dissipative effects for $k=1$ model using holonomy
quantization. The Hamilton’s equations were numerically solved for Starobinsky
potential and three different dissipation coefficients inspired from quantum
field theory all the way from the first bounce until the end of inflation.
These were with cubic, linear and inverse temperature dependence. We found
that even a small value of dissipation makes the hysteresis-like phenomena
strong. The effect is such that inflation sets in not only a few cycles but
also for those initial conditions which are extremely unfavorable for
inflation to begin even in LQC without radiation production. Moreover, we find
that as we make the dissipative effects large enough the hysteresis-like
phenomena goes away and the universe inevitably enters into inflationary phase
after just one bounce. To gain insights on the qualitative dynamics of the
universe from the bounce until the end of inflation we studied the phase-space
portraits for different initial conditions and all three dissipation
coefficients. In all three cases, we found that all initial conditions
experience an attractor dynamics showing that the universe goes through
inflationary phase. In other words, the universe starting from the first
bounce with stiff-like initial conditions dilutes its kinetic energy due to
both Hubble friction and dissipative effects and transfers it to radiation
field, as opposed to dissipationless universe, whereby after some cycles the
radiation energy density becomes the dominant energy component. However, such
radiation dominated epoch continues only for a very short period since
radiation energy density decays as $a^{-4}$. Then the potential energy becomes
dominant resulting in inflationary phase. Moreover, if the dissipative effects
are large enough, the inflationary phase will be of warm-type whereby the
universe smoothly enters into radiation dominated epoch without the need for
separate reheating epoch. Our analysis shows that with presence of dissipative
effects, non-singular quantum gravitational dynamics results in an onset of
inflation for low energy inflationary models even from highly unfavorable
initial conditions. In comparison to the cases where dissipation is absent, we
find hat inflation is set in much quicker due to stronger hysteresis-like
phenomena. Since our results establish phenomenological viability of low
energy inflationary models in spatially-closed universes at the level of
background dynamics, it will be interesting to investigate the model at
perturbative level to confront its predictions with observational data.

## Acknowledgements

This work is supported by the NSF grant PHY-1454832 and PHY-2110207. Authors
thank Bao-Fei Li for discussions, and the anonymous referee for valueable
comments.

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# The search of higher multipole radiation in gravitational waves from compact
binary coalescences by a minimally-modelled pipeline

O. Halim1 G. Vedovato2 E. Milotti3,1 G. A. Prodi4,5 S. Bini6,5 M. Drago7,8 V.
Gayathri9 C. Lazzaro10,2 D. Lopez11 A. Miani6,5 B. O’Brien9 F. Salemi6,5 M.
Szczepanczyk9 S. Tiwari11 A. Virtuoso3,1 S. Klimenko9 1INFN Sezione di
Trieste, I-34127 Trieste, Italy
2INFN, Sezione di Padova, I-35131 Padova, Italy
3Università di Trieste, Dipartimento di Fisica, I-34127 Trieste, Italy
4Università di Trento, Dipartimento di Matematica, I-38123 Povo, Trento, Italy
5INFN, TIFPA, I-38123 Povo, Trento, Italy
6Università di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy
7Università di Roma La Sapienza, I-00185 Roma, Italy
8INFN, Sezione di Roma, I-00185 Roma, Italy
9University of Florida, Gainesville, FL 32611, USA
10Università di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova,
Italy
11Physik-Institut, University of Zurich, Winterthurerstrasse 190, 8057 Zurich,
Switzerland<EMAIL_ADDRESS>
###### Abstract

The coherent WaveBurst (cWB) pipeline implements a minimally-modelled search
to find a coherent response in the network of gravitational wave detectors of
the LIGO-Virgo Collaboration in the time-frequency domain. In this manuscript,
we provide a timely introduction to an extension of the cWB analysis to detect
spectral features beyond the main quadrupolar emission of gravitational waves
during the inspiral phase of compact binary coalescences; more detailed
discussion will be provided in a forthcoming paper [1]. The search is
performed by defining specific regions in the time-frequency map to extract
the energy of harmonics of main quadrupole mode in the inspiral phase. This
method has already been used in the GW190814 discovery paper (Astrophys. J.
Lett. 896 L44). Here we show the procedure to detect the (3, 3) multipole in
GW190814 within the cWB framework.

Keywords: gravitational waves, analysis, multipoles, compact binary
coalescences

##  1 Introduction

Asymmetric binary blackhole systems are predicted to emit gravitational waves
(GWs) with higher modes (HMs) in addition to the quadrupole [2]. The analyses
of recent compact binary coalescence (CBC) signals by the LIGO-Virgo
Collaboration demonstrated the existence of HMs [2, 3]. The more sophisticated
waveform models are required to describe binary systems including HMs, which
are used for matched filter analysis pipelines. However, minimally-modelled
burst algorithms, such as cWB can also detect this effect [4, 5].

The cWB works at the level of time-frequency analysis. In terms of HM search
from CBCs, it looks for coherent excess power in chirp-like regions
corresponding to different HMs. This HM search strategy involving cWB, in some
sense, is similar to an alternative method in [6], which can be used to
compare cWB reconstructions with the estimates from Bayesian inference method
[7, 8].

This proceeding is organised as follows. Sec. 2 will explain the procedure.
Sec. 3 shows the implementation on GW190814. In the end, we conclude
everything in Sec. 4.

##  2 The procedure: waveform residual energy

In GW analysis with cWB pipeline [4, 5], waveform reconstructions are done
thanks to the the discrete Wilson–Daubechies–Meyer (WDM) transform [9]. The
pipeline first decompose the GW signals with this WDM in order to produce
time-frequency maps of the signals. The time-frequency pixels are selected by
retaining only a fixed fraction of them choosing those above a specified
excess network energy. Moreover, cWB estimates the coherence among GW
detectors by the maximization of the constrained likelihood [4]. These
coherent wavelet pixels provide a GW waveform reconstruction, as a point
estimate in the time domain.

In this proceeding, we summarise our recent paper [1] that has been submitted
to CQG, in which we focus on the procedure to specifically detect the HMs. We
use two compatible waveform models, in which the only difference is whether
there is HM content or not. All the waveforms are whitened by cWB pipeline
[7]. We define “on-source” and “off-source” data, where on-source means the
data contain the GW signal and off-source do not. These off-source data
provide independent noise instances. In this case, we can assess the effect of
noise with no assumption of the noise statistics except stationarity.

Our idea is to compare the cWB reconstructions with the model waveforms (with
or without HMs) from the Bayesian inference methods [10, 11]. Thus, we define
waveform residual energy, $E_{\mathrm{res}}$:

$E_{\mathrm{res}}=\sum_{k=1}^{\mathrm{det}}\;\sum_{i\in\langle\mathrm{pixels}\rangle}(w_{k}^{\mathrm{cWB}}[i]-w_{k}^{\mathrm{model}}[i])^{2},$
(1)

where $w_{k}^{\mathrm{cWB}}[i]$ and $w_{k}^{\mathrm{model}}[i]$ are the WDM
transforms of the cWB reconstruction and of a waveform model. Here, $k$ is the
detector index and $i$ is the WDM pixel index in the time-frequency map (more
details are in c.f Sec. III.A of [7]).

We test the consistency of cWB reconstruction comparing with the Bayesian
estimation by calculating the residual energy $E_{\mathrm{res}}^{\mathrm{(on-
source)}}$ in the on source reconstruction versus the maximum likelihood
(MaxL) sample waveform from the Bayesian inference without HMs111From the
Bayesian parameter estimate, we use the MaxL sample as the best estimate
instead of maximum a posteriori sample (MAP), which would be better motivated
in a Bayesian perspective, because of the flat priors on specific regions in
the parameter space. The MAP sample provides no real advantage when it is
within the interior of these regions, and in fact it may produce worse results
when it rails against their boundaries.. The significance is evaluated by the
empirical distribution of off-source injections from random samples of the
posterior distribution into a wide set of off-source, equally spaced
intervals.

The injected signals in off-source data are analysed and reconstructed by cWB
and again, compared with their whitened version without HMs by
$E_{\mathrm{res}}^{\mathrm{(off-source)}}$. Thus, the residual energies give
us an empirical distribution either $E_{\mathrm{res}}^{\mathrm{(on-source)}}$
or $E_{\mathrm{res}}^{\mathrm{(off-source)}}$. From here, the p-value can be
calculated to test the hypothesis whether the injected waveform is in
agreement with its cWB reconstruction.

The instantaneous frequency of the generic $(\ell,m)$ multipole222We use
$(\ell,m)$ as a shorthand for both $(\ell,m)$ and $(\ell,-m)$. emitted by
spinning, non-precessing black hole binaries, can be approximated by a scaling
from the dominant $(2,2)$ multipole [12, 13]

$f_{\ell,m}(t)\approx\frac{m}{2}f_{22}(t).$ (2)

Therefore, we look for the presence of significant residual energy along
“slices” of the time-frequency map; the slice is defined as
$f(t)=(\alpha\pm\delta\alpha)f_{22}(t)$, where $\alpha$ is a non-negative real
parameter [2, 6] and $\delta\alpha$ determines the strip width, between a
minimum and a maximum time. More details of the resolution $\delta\alpha$ can
be seen in c.f. Fig. 4 of [1].

We calculate the significance of the residual energy by Monte Carlo
simulations: random waveform samples from the Bayesian inference are injected
into the off-source data both with and without HMs. For each case, we compute
the residual energy. Thus, two empirical distributions are constructed:
without HMs as the null hypothesis and with HMs as the alternative hypothesis.
Our strategy can be seen step-by-step in the scheme of Fig 1.

Figure 1:  Scheme of our strategy using Monte Carlo program to produce the
null and alternative empirical distributions. Notice that the two different
cWB reconstructions are both compared with the waveform without HMs (solid
blue arrows). The Monte Carlo loop is repeated for $\sim$2000 off-source
injections.

##  3 GW190814

We highlight several results related to GW190814. For more details, the reader
is advised to see [1]. First of all, let us discuss the left panel of Fig. 2
for GW190814 analysis by cWB. This plot shows the $E_{\mathrm{res}}$ for all
the pixels selected in the on-source. Here the reference waveform comes from
the MaxL estimate by the SEOBNRv4_ROM model [14], implemented in LALSuite [15]
(LALSim version 1.10.0.1), without HMs (we use the same data as in GW190814
paper [3]). The time-frequency shape has the chirp-like pattern found in CBCs,
but somewhat wider, which corresponds to a deviation of $E_{\mathrm{res}}$
from the null hypothesis, associated to the $(3,3)$ multipole.

Figure 2:  Left: time-frequency map of $E_{\mathrm{res}}$ for the GW190814
with respect to the MaxL SEOBNRv4_ROM waveform, mapped into the LIGO
Livingston detector by the cWB with resolution $dt=1/32$ s and $df=16$ Hz.
Right: the time-frequency map of the time-frequency slice with $\alpha=1.5$
(meaning the $(3,3)$ HM). These pixels are used to evaluate the total
$E_{\mathrm{res}}(\alpha;\delta\alpha,\Delta t,\delta t,df)$ in the time-
frequency slice. The red vertical line shows the merger time444GPS time:
1249852257.0154 s. from the MaxL SEOBNRv4_ROM waveform. The dotted blue
vertical lines are the considered time-frequency slice,
$[t_{\mathrm{merger}}-0.5$ s$,t_{\mathrm{merger}}-0.03$ s$]$. The black dotted
curves are the limits of the time-frequency slice
$[\alpha-0.1,\alpha+0.1]\times f_{22}(t)$ with $\alpha=1.5$. Figure 3:
Empirical $E_{\mathrm{res}}$ distributions of the GW190814 HMs for
$\alpha=1.5$, $\delta\alpha=0.1$, $\Delta t=0.5$ s, and $\delta t=0.03$ s. Red
vertical line: on-source value. Purple histogram: $E_{\mathrm{res}}$
distribution for the null hypothesis. Green histogram: $E_{\mathrm{res}}$
distribution for the model with HMs, from SEOBNRv4HM_ROM injections in off-
source. The GW190814 on-source $e_{\mathrm{res}}$ with respect to the MaxL
with HMs is an outlier of the null model, SEOBNRv4_ROM with the p-value =
0.0068, but it is compatible with SEOBNRv4HM_ROM with HMs (p-value = 0.17).

The waveform for the HMs is SEOBNRv4HM_ROM including (2,1), (3,3), (4,4), and
(5,5) along with the dominant (2,2) multipole [16, 15, 17], in which its HMs
can be turned off and simply gives us SEOBNRv4_ROM. Fig. 3 gives an example of
the empirical distributions of GW190814 for the slice defined555The choices of
these parameters are discussed in [1]. by $\alpha=1.5$, $\delta\alpha=0.1$,
$\Delta t=0.5$ s, and $\delta t=0.03$ s. We see that the on-source
$E_{\mathrm{res}}$ with respect to the MaxL with HMs is an outlier of the null
hypothesis with the p-value = 0.0068, but it is compatible with SEOBNRv4HM_ROM
(the waveform including HMs) with p-value = 0.17.

Moreover, we also study other modes. Fig. 4 gives us the p-value for the null
hypothesis for several $\alpha$ values [3]. The p-value drops at $\alpha=1.5$
(corresponding to $(3,3)$ mode), while the p-values are larger for other
$\alpha$s that the null hypothesis cannot be rejected. Additionally, cWB does
not detect significant $E_{\mathrm{res}}$ at $\alpha\sim 1$, implying that the
dominant $(2,2)$ multipole (quadrupole) of the SEOBNRv4HM_ROM is consistent
with the SEOBNRv4_ROM.

Figure 4:  Plot of p-value vs. $\alpha$ for the null hypothesis of GW190814.
The p-value drops at $\alpha=1.5$ mode, corresponding to the $(3,3)$
multipole. We notice that the quadrupole fluctuation ($\alpha=1$) is
consistent with the null hypothesis. This figure is a deeper version (wider
$\alpha$-range and twice of the datapoints) of the lower panel in c.f. Fig. 7
in [3].

##  4 Conclusions

We have given a timely introduction of our method regarding the detection of
GW multipoles; the details will be provided in a forthcoming paper [1]. In
particular, we highlight the GW190814 analysis. The paper [1] is actually the
extension of our previous work in [7]. More details of the Receiver Operating
Characteristic curves regarding the choices of $\delta\alpha$ as well as the
analysis on GW190412 can be found in that forthcoming paper [1].

cWB makes use of data, software and/or web tools obtained from the
Gravitational Wave Open Science Center [18], a service of LIGO Laboratory, the
LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by
the U.S. National Science Foundation. Virgo is funded by the French Centre
National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale
della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by
Polish and Hungarian institutes. The authors are grateful for computational
resources provided by the LIGO Laboratory and supported by National Science
Foundation Grants PHY-0757058 and PHY-0823459.

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