# Gravitational waves in a cyclic Universe: resilience through cycles and
vacuum state

Mariaveronica De Angelis<EMAIL_ADDRESS>Adam Smith
<EMAIL_ADDRESS>William Giarè<EMAIL_ADDRESS>Carsten van de
Bruck<EMAIL_ADDRESS>School of Mathematics and Statistics, The
University of Sheffield, Hounsfield Road, S3 7RH Sheffield, United Kingdom

###### Abstract

We present a generalised calculation for the spectrum of primordial tensor
perturbations in a cyclic Universe, making no assumptions about the vacuum
state of the theory and accounting for the contribution of tensor modes
produced in the dark energy phase of the previous cycle. We show that these
modes have minimal impact on the spectrum observed in the current cycle,
except for corrections on scales as large as the comoving Hubble radius today.
These corrections are due to sub-horizon modes produced towards the end of the
dark energy phase, persisting into the ekpyrotic phase of the next cycle as
additional quanta. In relation to the vacuum state, we argue that non-Bunch-
Davies quanta can easily overwhelm the energy density driving the dark energy
phase, potentially compromising the model. Therefore, avoiding backreaction
effects sets restrictive constraints on deviations away from the Bunch-Davies
vacuum during this phase, limiting the overall freedom to consider alternative
vacua in the cyclic Universe.

##  I Introduction

The most compelling observational evidence supporting cosmological inflation
Guth (1981); Linde (1982); Albrecht and Steinhardt (1982); Vilenkin (1983) as
the leading theory of the early Universe is currently provided by the Planck
satellite measurement of the spectral index of scalar perturbations,
$n_{s}=0.9649\pm 0.0042$ Akrami  _et al._ (2020). In the simplest single-field
slow-roll inflationary models, the spectrum of scalar modes is expected to be
almost but not exactly flat Mukhanov and Chibisov (1981); Bardeen  _et al._
(1983); Hawking (1982); Guth and Pi (1982), with deviations from flatness are
quantified in terms of how much $n_{s}$ deviates from 1 Lidsey  _et al._
(1997); Lyth and Riotto (1999); Baumann (2011); Martin  _et al._ (2014). As a
result, the Planck data seem to be in excellent agreement with the theoretical
predictions of inflationary models Akrami  _et al._ (2020); Aghanim  _et al._
(2020), ruling out a Harrison-Zeldovich scale-invariant spectrum Harrison
(1970); Zeldovich (1972); Peebles and Yu (1970) (corresponding to $n_{s}=1$)
at a statistical level exceeding $8.5$ standard deviations and lending weight
to the inflationary paradigm.

That being said, with no aim to downplay the significance of this result or
its interpretation, it is crucial to emphasise that, on its own, it does not
provide conclusive evidence for cosmological inflation. Even hinging on a
certain level of optimism and setting aside the uncertainty surrounding
constraints on $n_{s}$ from CMB experiments other than Planck111Over the
years, constraints on the spectral index have been released by a multitude of
Planck-independent CMB experiments such as WMAP Bennett  _et al._ (2013);
Hinshaw  _et al._ (2013), the Atacama Cosmology Telescope (ACT) Choi  _et al._
(2020); Aiola  _et al._ (2020), and the South Pole Telescope (SPT) Benson  _et
al._ (2014); Dutcher  _et al._ (2021). When considering these data at face
value, Planck is currently the only experiment excluding $n_{s}=1$ at a
statistical significance much larger than $3\sigma$. Conversely, ACT shows a
preference for $n_{s}=1$ Aiola  _et al._ (2020); Giarè  _et al._ (2023a).
Different combinations of these data overall support the result $n_{s}\neq 1$,
although sometimes they lead to discordant results in terms of the other
inflationary parameters or the preferred inflationary models Forconi  _et al._
(2021); Giarè  _et al._ (2023b). – or the potential implications arising from
the well-known tensions Bernal  _et al._ (2016); Verde  _et al._ (2019); Di
Valentino  _et al._ (2021a, b); Abdalla  _et al._ (2022) characterising the
recent debate222For studies suggesting potential implications of cosmological
tensions for inflation, see, e.g., Refs. Di Valentino  _et al._ (2018); Ye
_et al._ (2021, 2022); Jiang and Piao (2022); Jiang  _et al._ (2023a);
Takahashi and Yin (2022); Lin (2022); Hazra  _et al._ (2022); Braglia  _et
al._ (2022); Keeley  _et al._ (2020); Jiang  _et al._ (2023b) – alternative
theoretical mechanisms have been put forth, yielding an almost scale-invariant
spectrum of primordial density fluctuations without invoking inflation.

An illustrative example of such mechanisms is the cyclic Universe Steinhardt
and Turok (2002); Steinhardt  _et al._ (2002); Steinhardt and Turok (2003);
Khoury  _et al._ (2004); Turok and Seinhardt (2005); Khoury (2004); Lehners
(2008) that, in contrast to the conventional cosmological framework, suggests
a periodic history for the Cosmos. The model has been extensively studied and
discussed in relation to a broad range of topics, including quantum gravity,
modified gravity, gravitational waves and dark energy, see e.g., Refs. Boyle
_et al._ (2004); Ashtekar  _et al._ (2003); Bojowald  _et al._ (2004); Xiong
_et al._ (2009); Frampton (2007); Narlikar  _et al._ (2007); Xiong  _et al._
(2009); Baum and Frampton (2008); Biswas (2008); Cailleteau  _et al._ (2009);
Brandenberger (2009); El-Nabulsi (2011); Cai  _et al._ (2012); Cai and
Saridakis (2011); Nojiri  _et al._ (2012); Chang and Scherrer (2012); Ivanov
and Prodanov (2012); Saaidi  _et al._ (2012); Bars  _et al._ (2013); Tavakoli
and Fabris (2015); Oriti  _et al._ (2016); de Cesare  _et al._ (2016);
Pavlovic and Sossich (2017); Saridakis  _et al._ (2018); Das  _et al._ (2018);
Ijjas (2018); Ahmed and Alamri (2019); Li  _et al._ (2019); Scherrer (2019);
Ijjas and Steinhardt (2022); Gorkavyi (2022); Martín-Benito  _et al._ (2021);
Calcinari and Gielen (2023); Giovannetti and Montani (2022); Giovannetti  _et
al._ (2023) or Refs. Battefeld and Peter (2015); Brandenberger and Peter
(2017) for reviews. In broad terms, each cycle comprises a phase recasting the
standard Hot Big Bang theory (during which large-scale structures take shape),
followed by a phase of slow, accelerated expansion mirroring the present-day
observational evidence for a Dark Energy dominated dynamics. In the cyclic
Universe, this latter stage also serves to dilute inhomogeneities and flatten
the spatial geometry. Subsequently, a contraction phase ensues, generating
nearly scale-invariant density perturbations. Finally, the cycle concludes
with a big-crunch/big-bang transition, during which matter and radiation are
generated, setting the stage for the next cycle.

Notice that both inflation and the cyclic Universe provide physical mechanisms
to produce an almost scale-invariant spectrum of density perturbations Khoury
_et al._ (2002); Lehners  _et al._ (2007); Buchbinder  _et al._ (2007). In
addition, they can both explain observational facts such as the homogeneity in
the cosmic microwave background (CMB) radiation Lehners and Steinhardt (2013)
and the fact that the present-day spatial geometry of the Universe appears to
be flat, or at the very least nearly flat333For recent discussions surrounding
the spatial geometry of the Universe, see, e.g., Park and Ratra (2019);
Handley (2021); Di Valentino  _et al._ (2019); Efstathiou and Gratton (2020);
Di Valentino  _et al._ (2021c); Benisty and Staicova (2021); Vagnozzi  _et
al._ (2021a, b); Di Valentino  _et al._ (2021d); Yang  _et al._ (2021); Cao
_et al._ (2021); Dhawan  _et al._ (2021); Dinda (2022); Gonzalez  _et al._
(2021); Akarsu  _et al._ (2023); Cao and Ratra (2022); Glanville  _et al._
(2022); Bel  _et al._ (2022); Yang  _et al._ (2023); Stevens  _et al._ (2023);
Favale  _et al._ (2023). Therefore, at first glance, one might wonder how to
distinguish between the two models. Focusing solely on scalar modes, this is a
challenging knot to unravel Khoury  _et al._ (2003); Gratton  _et al._ (2004).
However, the two scenarios yield significantly distinct predictions for the
stochastic background of gravitational waves Boyle  _et al._ (2004). Similar
to scalar modes, inflation predicts a nearly scale-invariant (red-tilted)
spectrum of tensor modes Baumann (2011); Martin  _et al._ (2014); Caprini and
Figueroa (2018). Conversely, in the cyclic Universe, the tensor spectrum is
typically blue-tilted, and its amplitude is many orders of magnitude lower
than that predicted by inflation, remaining well below any observable
threshold achievable in the near future. Consequently, any measurement of
primordial gravitational waves (e.g., through the effects left in the CMB
B-mode polarisation at large angular scales) would offer conclusive evidence
for inflation, discounting the cyclic model.

Despite this fact being acknowledged as a strength for inflation and perhaps a
limitation in predictive capacity for the cyclic model, it is worth
emphasising a few caveats surrounding this conclusion. Firstly, despite the
best efforts, the detection of primordial tensor modes remains elusive at
present Ade  _et al._ (2021), making it impossible to discriminate between the
two scenarios. Therefore, the cyclic Universe remains an alternative worth
considering. Secondly, the inflationary predictions concerning the amplitude
and tilt of the tensor spectrum depend significantly on the specific model.
While well-known consistency relations among inflationary parameters can be
derived within single-field slow-roll inflation minimally coupled to gravity
Baumann (2011); Martin  _et al._ (2014), these relations can be violated by
various physical mechanisms. A long yet not exhaustive list of possibilities
include considering modified gravity Kobayashi  _et al._ (2010); Kawasaki  _et
al._ (2013); Nozari and Shafizadeh (2016); Giarè and Renzi (2020), multi-field
inflation Kaiser and Sfakianakis (2014); Price  _et al._ (2015); Achucarro
_et al._ (2011); De Angelis and van de Bruck (2023); Giarè  _et al._ (2023c),
additional (spectator) rolling axion fields Mukohyama  _et al._ (2014); Namba
_et al._ (2016); Peloso  _et al._ (2016); Özsoy (2021), couplings to axion-
gauge or spin-2 fields Dimastrogiovanni  _et al._ (2017); Iacconi  _et al._
(2020), breaking spatial and/or temporal diffeomorphism invariance Endlich
_et al._ (2013); Cannone  _et al._ (2015); Graef and Brandenberger (2015);
Ricciardone and Tasinato (2017), higher curvature corrections to the effective
gravitational action Baumann  _et al._ (2016); Giarè  _et al._ (2021), higher
order operators in effective field theory Capurri  _et al._ (2020); Giarè  _et
al._ (2023d), violations of the null energy condition Cai (2023); Ye  _et al._
(2024), alternative vacuum state/initial conditions Ashoorioon  _et al._
(2014a, b); Choudhury (2023), sound speed resonances Cai  _et al._ (2021),
inflation in an Universe filled with an elastic medium Gruzinov (2004), and
possible effects/models inspired by quantum gravity Ashoorioon  _et al._
(2005); Brandenberger  _et al._ (2007, 2014); Baumgart  _et al._ (2022). Many
of these more elaborated scenarios yield completely different predictions,
often resulting in a blue-tilted spectrum and possibly leaving signatures in
different cosmological and astrophysical observables Stewart and Brandenberger
(2008); Cai  _et al._ (2015); Wang and Xue (2014); Kuroyanagi  _et al._ (2015,
2021); Giarè and Melchiorri (2021); Vagnozzi (2021, 2023); Jiang  _et al._
(2023c); Oikonomou  _et al._ (2024). Furthermore, models with an arbitrarily
small tensor amplitude can always be constructed (see, e.g., Ref Stein and
Kinney (2023)), making it virtually impossible to rule out inflation based
solely on a lack of detection of primordial gravitational waves. This is a
critique frequently raised against inflationary cosmology as it questions its
actual predictive capability.

As concerns the cyclic Universe, since any difference with respect to
inflation in terms of predictions is likely to be confined to the spectrum of
tensor modes, it becomes interesting to test whether similar caveats apply or
if the model demonstrates greater resilience.

In light of this, we review the production of primordial gravitational waves
in a cyclic Universe, identifying (and eventually clarifying) some conceptual
aspects related to its concrete predictivity. Specifically, prevailing
calculations in the existing literature conventionally establish initial
conditions for primordial scalar and tensor modes during the ekpyrotic
contracting phase Boyle  _et al._ (2004). While for scalar perturbations the
implications of setting the initial conditions in different phases have been
examined Erickson  _et al._ (2007), the calculation of the tensor spectrum has
always been performed starting in the ekpyrotic phase, assuming a Bunch-Davies
(BD) vacuum state and neglecting any potential contributions arising from
tensor modes originating during the dark energy phase of the previous cycle.
This leads us to question whether they exert any influence on the spectrum
observed in the current cycle. Taking a broader perspective, one may wonder
whether the predictions concerning tensor modes remain resilient throughout
the diverse cycles of the model itself. Yet another aspect that is imperative
to clarify is to what extent the predictions depend on the choice of the
vacuum state, addressing the crucial question of what freedom exists in the
cyclic Universe regarding the choice of the vacuum state and whether
substantial alterations can arise in the tensor spectrum by assuming different
vacua, akin to what is found in inflationary cosmology.

To address these points, we present a general model for the evolution of
gravitational waves produced in a cyclic Universe, making no assumptions about
the initial vacuum state and starting the calculation from the dark energy
phase of the previous cycle. We find that the additional tensor models
originated in the previous cycle have minimal impact on the tensor spectrum
observed in the current cycle, except for corrections on scales as large as
the comoving Hubble radius today that are due to sub-horizon modes produced
towards the end of the previous dark energy phase. Most importantly, we find
that non-BD quanta in the dark energy phase can easily overwhelm the energy
density associated with the modulus field, potentially spoiling the model.
Avoiding these backreaction effects sets restrictive constraints on deviations
away from the BD vacuum during the dark energy phase, thereby limiting the
overall freedom to consider alternative vacua in the cyclic Universe.

The paper is organised as follows. In section II, we introduce the cyclic
Universe model and review its background dynamics. In section III, we consider
the evolution of gravitational waves in such a Universe, starting from the
previous cycle’s dark energy phase and deriving the evolution in full
generality. In section IV, we discuss the implications for the model’s
predictions, deriving constraints on the choice of the vacuum state and
analysing the contribution coming from tensor modes originated in the previous
cycle. Finally, in section V, we derive our main conclusions.

##  II Cyclic model and Background Dynamics

We consider a simple scalar field setup in which the dynamics of the cyclic
model in the Einstein frame are well described by the 4$D$ effective
Lagrangian Erickson  _et al._ (2007)

$\mathcal{L}=\sqrt{-g}\left(\frac{M_{\rm
Pl}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right),$
(1)

where $g$ is the determinant of the metric $g_{\mu\nu}$, $R$ is the Ricci
scalar and we adopt units where $c=1$. The scalar field $\phi$ is a modulus
field, driving the dark energy dominated phase and the subsequent ekpyrotic
and contracting kinetic phases, which we discuss below. Assuming a spatially
flat FRLW background, the scalar field satisfies the usual equation of motion

$\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0,$ (2)

where dots denote derivatives with respect to cosmological time $t$. On the
other hand, ignoring any coupling between the scalar field and the other
standard model species and neglecting any additional contributions from the
latter to the total Universe energy density, the evolution of the scale factor
is governed by the Friedmann equation that, in terms of the Hubble parameter
$H=\dot{a}/a$, reads

$H^{2}=\frac{1}{3M_{\rm
Pl}^{2}}\biggl{(}\frac{1}{2}\dot{\phi}^{2}+V(\phi)\biggl{)}.$ (3)

In what follows, to efficiently describe the dynamics of the cyclic Universe
we focus on a phenomenological potential of the form

$V=V_{0}\left(1-e^{-c\phi/M_{\rm Pl}}\right)Y(\phi),$ (4)

where $V_{0}$ is of the same order of the vacuum energy observed in today’s
Universe, $c$ is a positive constant value and $Y(\phi)$ is a step function.
Notice that our choice concerning the specific potential employed in the work
is, in part, motivated by the fact that the exponential form is convenient for
analysis, and in part from the fact that the same potential has been widely
adopted in similar studies, allowing a direct comparison between our findings
and other results documented in the existing literature. However, it is
important to emphasise that cyclic models can emerge from a broad spectrum of
different potentials that should ultimately emerge from the higher-dimensional
theory. Without loss of generality, the only constraint comes from requiring
an acceptable spectrum of scalar perturbations that implies considering a
steep, strongly negatively curved region across observational ranges of the
field to reproduce.

Having that said, the potential (4) serves multiple purposes, including
describing dark energy responsible for cosmic acceleration observed today.
More importantly, it plays a crucial role in transitioning the Universe from
accelerated expansion to contraction. This is achieved by rolling from
positive to negative values of the potential until reaching a time where
$H^{2}=0$ and consequently triggering a phase characterised by an equation of
state $\omega\gg 1$. For instance, by solving Eq. (3) it can be shown that
when the negative potential dominates $V\simeq-V_{0}e^{-c\phi}$ (i.e.
ekpyrotic phase), the scale factor behaves as Erickson  _et al._ (2007)

$a(t)\sim(-t)^{\tilde{\alpha}},$ (5)

where $t$ has negative values and $\tilde{\alpha}=2/c^{2}$. At this point it
is also convenient to introduce the conformal time $d\tau=dt/a(t),$ which we
will frequently use later. In terms of the conformal time, the scale factor
during the ekpyrotic phase evolves as:

$a(\tau)\sim\biggl{[}(-1)^{\tilde{\alpha}}(\tau-\tilde{\alpha}\tau)^{\tilde{\alpha}/(1-\tilde{\alpha})}\biggl{]}_{\tau_{i}}^{\tau_{f}},$
(6)

underscoring that the Universe is gradually contracting while the scalar field
slowly descends along its sharply decreasing negative potential, to produce an
acceptable spectrum of cosmological scalar perturbations.

In the literature, the ekpyrotic phase is typically assumed as the starting
point of the cycle where initial conditions of primordial perturbations are
imposed, and the calculations of the relative scalar and tensor spectra begin.
However, in this study, we want to extend the model to include the
contribution of tensor perturbations produced during the dark energy phase of
the previous cycle to investigate whether they could have any impact on the
spectrum we observe in the current cycle and eventually clarify why (not). To
do so, the overall strategy will be to start the calculation in the dark
energy phase of the previous cycle (making no assumptions on the vacuum state)
and evolve the system through four regimes. For this reason, before dealing
with the explicit calculation of the tensor spectrum, given that in our case
we consider one more phase than in previous studies, it is useful to dedicate
the following two subsections to reviewing the background dynamics of the
model. In the same spirit of the discussion outlined so far, we start from the
dark energy phase of the previous cycle and evolve the scale factor, ensuring
its continuity across the boundaries of each phase. Additionally, we derive
constraints on the model’s parameters based on minimum theoretical
requirements, such as the continuity of $H(t)$ and the consistency of the
theory across cycles.

###  II.1 Evolution and continuity of the scale factor across stages

####  II.1.1 Dark Energy phase

We start from the dark energy phase, in which the expansion rate $H$ is
roughly constant and the scale factor $a(t)$ behaves as

$a(t)=a(t_{\rm{tr}})e^{H\left(t-t_{\rm{tr}}\right)}\;\;\;\;t<t_{\rm{tr}},$ (7)

with $t_{tr}$ transition time between dark energy and ekpyrotic phase. The
equation above can be translated in terms of conformal time as

$a(\tau)=\frac{1}{H(B-\tau)},$ (8)

where, for the continuity across $\tau=\tau_{\rm{tr}}$, $B$ is fixed to444The
value of $B$ is achieved by considering $H_{0},a(\tau_{\rm{r}})$ and
$\tau_{\rm{r}}$ given in the next sections.

$B=\frac{1}{a(\tau_{\rm{tr}})H}+\tau_{\rm{tr}}.$ (9)

Additionally, by means of Eq. (6), we can infer

$\frac{a(\tau_{\rm{tr}})}{a(\tau_{\rm{end}})}=\left(\frac{\tau_{\rm{tr}}-\tau_{\rm{ek}}}{\tau_{\rm{end}}-\tau_{\rm{ek}}}\right)^{\alpha},$
(10)

where $\alpha\equiv\tilde{\alpha}/(1-\tilde{\alpha})$ and
$\tau_{\rm{ek}}\equiv(1-2\alpha)\tau_{\rm{end}}$ is the conformal time
corresponding to when the potential diverges to minus infinity.

####  II.1.2 Ekpyrotic phase

As a next step, we transition to the ekpyrotic phase. In this phase the
potential becomes negative and the Einstein frame expansion forces the scale
factor to contract

$\frac{a(\tau)}{a(\tau_{end})}=\left(\frac{\tau-\tau_{ek}}{\tau_{end}-\tau_{ek}}\right)^{\alpha}\;\;\;\;\;\tau_{tr}<\tau<\tau_{end}.$
(11)

We note again that, being $\alpha\ll 1$, the contraction is very slow.

####  II.1.3 Contracting kinetic phase

Once $\tau>\tau_{\rm{end}}$ we enter the region where the effects of the
potential are negligible, namely $\phi<\phi_{\rm{end}}$. During this period –
known as contracting kinetic phase – we have:

$\frac{a(\tau)}{a(\tau_{\rm{r}})}=\left(\frac{-\tau}{(1+\chi)\tau_{\rm{r}}}\right)^{\frac{1}{2}}\;\;\;\;\;\tau_{\rm{end}}<\tau<0,$
(12)

where $\chi$ is a small positive constant that measures the amount of
radiation created at the bounce ($\tau=0$).

####  II.1.4 Expanding kinetic phase

Finally, for the last phase of the cycle, the so-called expanding kinetic
phase, we get

$a(\tau)=\left(\frac{\tau}{\tau_{\rm{r}}}\right)^{\frac{1}{2}},\;\;\;\;\;0<\tau<\tau_{\rm{r}},$
(13)

Notice that, for convenience, we work in a coordinate frame where the scale
factor is set to $a(\tau_{\rm{r}})=1$ for the time $t_{\rm{r}}$ corresponding
to the beginning of the radiation-dominated era. The corresponding conformal
time $\tau_{\rm{r}}=(2H_{\rm{r}})^{-1}$ is constrained by the radiation
temperature $T_{\rm{r}}$, being $H_{\rm{r}}\propto
T_{\rm{r}}^{2}/M_{\rm{Pl}}$. Following Ref. Khoury  _et al._ (2004), we choose
a quite conservative value $T_{\rm{r}}\sim 10^{7}$ GeV, akin to that obtained
in the more familiar standard cosmology at the end of the reheating phase
following inflation. This choice also ensures that we can safely recover
predictions of primordial Big Bang Nucleosynthesis (BBN).

###  II.2 Parameter constraints

####  II.2.1 Continuity of the Hubble parameter

In order to constrain the length of the ekpyrotic contracting phase, we
require that the Hubble parameter returns to its original value after every
cycle. Following Ref. Erickson  _et al._ (2007), this implies that

$H_{\rm{end}}/H_{\rm{0}}\approx\sqrt{\frac{-V_{\rm{end}}}{V_{0}}}$ (14)

where $V_{\rm{end}}$ is the depth of the potential well and $V_{0}$ is the
height of the potential plateau. The spectral range of perturbations produced
when the field rolls from $V\approx 0$ to $V\approx-V_{\rm{end}}$, satisfies

$\frac{k_{\rm{max}}}{k_{\rm{min}}}\approx\sqrt{\frac{-V_{\rm{end}}}{V_{0}}},$
(15)

and it needs to span at least $N=60$ e-folds for the ekpyrotic phase to
produce a scale-invariant spectrum over a broad enough range of scales for us
to observe today Akrami  _et al._ (2020); Aghanim  _et al._ (2020). Hence, the
transition time between dark energy and ekpyrotic phase is constrained to

$\frac{H_{\rm{tr}}}{H_{\rm{end}}}=2\,\alpha\,\tau_{\rm{end}}\,\frac{a_{\rm{end}}\,a_{\rm{tr}}}{{(\tau_{\rm{tr}}-\tau_{\rm{ek}})}}<e^{-60}.$
(16)

Moreover, as $a(\tau)\approx\rm{const}$ during the ekpyrotic contracting phase
Erickson  _et al._ (2007), from Eq. (16) it follows that

$\left|\tau_{\rm{tr}}-\tau_{\rm{ek}}\right|>2\,\alpha\,\tau_{\rm{end}}\,a_{\rm{end}}^{2}\,e^{60}.$
(17)

####  II.2.2 Cycling constraint

To place constraints on the duration of the kinetic evolution phases, we
require that they must last enough time for the scalar field to have started
at the potential minimum (i.e. $\phi=\phi_{\rm{end}}$), moved off to the
Bounce (i.e. $\phi\to-\infty$), and returned all the way back past
$\phi_{\rm{end}}$ and made it up to the potential plateau to begin a
radiation-dominated Universe.

Barring some brief $\omega\gg 1$ period (which divides the expanding kinetic
phase into two parts) as the field moves back up past $\phi_{\rm{end}}$ to the
potential plateau, from Eq. (2) and Eq. (4) we find

$\phi-\phi_{end}=c_{1}\ln\left(\frac{t}{t_{end}}\right),$ (18)

where the factor $c_{1}^{2}=2/3$ comes from Eq. (3) during kinetic domination,
$t_{\rm{end}}$ is the time taken to reach $\phi_{\rm{end}}$ starting at
$\phi\to-\infty$.

Notice that in the region where $\omega\gg 1$, contributions from $Y(\phi)$
becomes relevant and $V\approx V_{0}\left(1-e^{-c\phi}\right)$. Therefore, in
this case, the time $t_{\rm{r}}$ required to climb the potential well and
reach the plateau at $V\approx V_{0}$ can be bounded to

$\frac{t_{\rm{r}}}{t_{\rm{end}}}>\left(\frac{V_{\rm{end}}}{V_{0}}\right)^{\sqrt{\frac{3}{2c^{2}}}}.$
(19)

Since the time taken for the field to cross the negative region of the
potential before radiation domination begins is given by

$\frac{t_{\rm{r}}}{t_{\rm{end}}}\approx\frac{\sqrt{V_{\rm{end}}}}{H_{\rm{r}}},$
(20)

from Eq. (19) we can infer an upper limit on the Hubble parameter at $t_{r}$
which reads

$H_{\rm{r}}\lesssim\frac{\sqrt{V_{\rm{end}}}}{M_{\rm{Pl}}}\left(\frac{V_{0}}{V_{\rm{end}}}\right)^{\frac{3}{2c^{2}}}.$
(21)

This upper limit constrains the ratio between $\tau_{r}$ and $\tau_{\rm end}$
to

$\Gamma=\left|\frac{\tau_{\rm{r}}}{\tau_{\rm{end}}}\right|\gtrsim\left(\frac{V_{\rm{end}}}{V_{0}}\right)^{\sqrt{\frac{2}{3c^{2}}}}\simeq
10^{8},$ (22)

where we used that the vacuum energy density $\rho_{\Lambda}=V_{0}\sim
10^{-120}M_{\rm{Pl}}^{4}$ and $V_{\rm{end}}\sim 10^{-20}M_{\rm{Pl}}^{4}$.
Following Ref. Boyle  _et al._ (2004), throughout this paper, we always
consider the dimensionless parameter $\Gamma\sim 10^{8}$.

In conclusion, the final constraints we derive for the cyclic model at the
background level (and that are important to bear in mind for the following
discussion on tensor perturbations) are:

$\displaystyle\left|\frac{\tau_{\rm{r}}}{\tau_{\rm{end}}}\right|$
$\displaystyle\gtrsim 10^{8},$ $\displaystyle(\tau_{\rm{tr}}-\tau_{\rm{ek}})$
$\displaystyle\gtrsim 10^{9},$
$\displaystyle\tau_{\rm{r}}=\frac{1}{2H_{\rm{r}}}.$ (23)

##  III General primordial tensor spectrum

Considering a spatially flat FLRW metric, in the synchronous gauge the
perturbed line element reads:

$ds^{2}=a^{2}(\tau)\left[d\tau^{2}-\left(\delta_{ij}+h_{ij}\right)dx^{i}dx^{j}\right].$
(24)

Tensor modes (i.e., metric perturbations) are described in terms of the
transverse and traceless part of the symmetric 3×3 matrix $h_{ij}$. To
characterise the contribution of each wavenumber $k$ to
$h_{ij}(t,\mathbf{x})$, we consider a Fourier representation
$\tilde{h}_{ij}(t,\mathbf{k})$. Moving to the Fourier space, focusing on one
particular polarisation state, and assuming isotropy, the gravitational wave
field $h_{k}$ satisfies the following equation:

$h_{k}^{\prime\prime}+2\frac{a^{\prime}}{a}h_{k}^{\prime}+k^{2}h_{k}=0,$ (25)

where $(..)^{\prime}$ denotes the derivative with respect to conformal time
$\tau$. However, it is more convenient to use a new variable
$f_{k}(\tau)\equiv a(\tau)h_{k}(\tau)$ satisfying

$f_{k}^{\prime\prime}+\left(k^{2}+\frac{a^{\prime\prime}}{a}\right)f_{k}=0.$
(26)

After redefining $f_{k}=i\sqrt{\tau}\,u_{k}$, Eq. (26) assumes the more
familiar form of a Bessel equation and, for each phase of the cyclic model,
the general solution involves a linear combination of the Hankel functions of
the first and second kind, that we denote as $H^{(1,2)}$.

In this section, we present a generalised calculation for the primordial
tensor spectrum in the cyclic Universe. In subsection III.1, we derive the
general solution of Eq. (26) making no assumptions about the vacuum state of
the theory and considering the contribution of tensor modes produced in the
dark energy phase of the previous cycle rather than starting directly from the
ekpyrotic phase of the present cycle. In subsection III.2, we require internal
consistency in the evolution of the tensor mode amplitudes throughout the four
phases of the model, ensuring that both $h_{k}(\tau)$ and
$h_{k}^{\prime}(\tau)$ remain continuous functions and matching the general
solutions across the different phases. Finally, in subsection III.3, we derive
the expression of the primordial tensor spectrum and briefly discuss its
strain today.

###  III.1 General Solutions in the different Phases

####  III.1.1 General solution in the Dark Energy phase

During the dark energy phase, $a(\tau)$ is given by Eq. (8), and the general
solution of Eq. (26) reads

$f_{k}(\eta)=\sqrt{-k\eta}\left(D_{1}(k)H^{(1)}_{\frac{3}{2}}(-k\eta)+D_{2}(k)H^{(2)}_{\frac{3}{2}}(-k\eta)\right)\;\;\;\tau<\tau_{\rm{tr}},$
(27)

where $H_{n}^{(1)}$ and $H_{n}^{(2)}$ denote the Hankel functions of first and
second kind respectively, $\eta=\tau-B$, where $B$ is given by Eq. (9) and,
for each wave-number $k$, $D_{1,2}(k)$ are arbitrary constants.

####  III.1.2 General solution in the ekpyrotic contraction

In the stage of the ekpyrotic contraction, $a(\tau)$ is given by Eq. (11) and
the general solution reads

$f_{k}(\tau)=\sqrt{y}\left(A_{1}(k)H^{(1)}_{n}(y)+A_{2}(k)H^{(2)}_{n}(y)\right)\;\;\;\tau_{\rm{tr}}<\tau<\tau_{\rm{end}},$
(28)

where $y\equiv-k(\tau-\tau_{\rm{ek}})$.

####  III.1.3 General solution in the kinetic contracting phase

Moving to the kinetic contracting phase and making use of Eq. (12), we achieve

$f_{k}(\tau)=\sqrt{-k\tau}\left(B_{1}(k)H^{(1)}_{0}(-k\tau)+B_{2}(k)H^{(2)}_{0}(-k\tau)\right)\;\;\;\tau_{\rm{end}}<\tau<0,$
(29)

####  III.1.4 General solution in the kinetic expanding phase

Considering Eq. (13) in the kinetic expanding phase the genral solution is

$f_{k}(\tau)=\sqrt{k\tau}\left(C_{1}(k)H^{(1)}_{0}(k\tau)+C_{2}(k)H^{(2)}_{0}(k\tau)\right)\;\;\;0<\tau<\tau_{\rm{r}}.$
(30)

###  III.2 Matching phases

In what follows we obtain expressions for the coefficients
$D_{1,2}(k),A_{1,2}(k),B_{1,2}(k)$ and $C_{1,2}(k)$ by matching $h_{k}(\tau)$
and $h_{k}(\tau)^{\prime}$ at the boundaries of each phase.

####  III.2.1 Dark energy - ekpyrotic

At the boundary between dark energy and ekpyrotic stage, we require continuity
of $f_{k}(\tau)$ namely

$\sqrt{x_{\rm{tr}}}\left(D_{1}(k)H^{(1)}_{\frac{3}{2}}(x_{\rm{tr}})+D_{2}(k)H^{(2)}_{\frac{3}{2}}(x_{\rm{tr}})\right)=\sqrt{y_{\rm{tr}}}\left(A_{1}(k)H^{(1)}_{n}(y_{\rm{tr}})+A_{2}(k)H^{(2)}_{n}(y_{\rm{tr}})\right),$
(31)

where $x_{\rm{tr}}=-k\eta(\tau_{\rm{tr}})$, and continuity of
$f^{\prime}(\tau)$ which we write in the matrix form
$\underline{\underline{\textbf{y}}}_{1}\textbf{A}=\underline{\underline{\textbf{x}}}_{1}\textbf{D}$
where

$\displaystyle\underline{\underline{\textbf{y}}}_{1}$
$\displaystyle=\left(\begin{matrix}\sqrt{y_{\rm{tr}}}H^{(1)}_{n}(y_{\rm{tr}})&\sqrt{y_{\rm{tr}}}H^{(2)}_{n}(y_{\rm{tr}})\\\\[5.0pt]
\sqrt{y_{\rm{tr}}}H^{(1)}_{n-1}(y_{\rm{tr}})-\frac{(n-\frac{1}{2})}{\sqrt{y_{\rm{tr}}}}H^{(1)}_{n}(y_{\rm{tr}})&\qquad\sqrt{y_{\rm{tr}}}H^{(2)}_{n-1}(y_{\rm{tr}})-\frac{(n-\frac{1}{2})}{\sqrt{y_{\rm{tr}}}}H^{(2)}_{n}(y_{\rm{tr}})\end{matrix}\right),$
$\displaystyle\underline{\underline{\textbf{x}}}_{1}$
$\displaystyle=\left(\begin{matrix}\sqrt{x_{\rm{tr}}}H^{(1)}_{\frac{3}{2}}(x_{\rm{tr}})&\sqrt{x_{\rm{tr}}}H^{(2)}_{\frac{3}{2}}(x_{\rm{tr}})\\\\[5.0pt]
\sqrt{x_{\rm{tr}}}H^{(1)}_{\frac{1}{2}}(x_{\rm{tr}})-\frac{1}{\sqrt{x_{\rm{tr}}}}H^{(1)}_{\frac{3}{2}}(x_{\rm{tr}})&\qquad\sqrt{x_{\rm{tr}}}H^{(2)}_{\frac{1}{2}}(x_{\rm{tr}})-\frac{1}{\sqrt{x_{\rm{tr}}}}H^{(2)}_{\frac{3}{2}}(x_{\rm{tr}})\end{matrix}\right),$
(32)

and

$\textbf{A}=\left(\begin{matrix}A_{1}\\\
A_{2}\end{matrix}\right),\;\;\;\;\;\;\textbf{D}=\left(\begin{matrix}D_{1}\\\
D_{2}\end{matrix}\right),\;\;\;\;\;\;\text{with}\;\;x_{\rm{tr}}\equiv-k\tau_{\rm{tr}}.$
(33)

####  III.2.2 Ekpyrotic \- kinetic contraction

Similarly here, at $\tau=\tau_{\rm{end}}$ namely the end of the ekpyrotic
phase, we require
$\underline{\underline{\textbf{y}}}_{2}\textbf{B}=\underline{\underline{\textbf{x}}}_{2}\textbf{A}$
where

$\displaystyle\underline{\underline{\textbf{y}}}_{2}$
$\displaystyle=\left(\begin{matrix}H^{(1)}_{0}(x_{\rm{e}})&H^{(2)}_{0}(x_{\rm{e}})\\\\[5.0pt]
\sqrt{x_{\rm{e}}}H^{(1)}_{-1}(x_{\rm{e}})+\frac{H^{(1)}_{0}(x_{\rm{e}})}{2\sqrt{x_{\rm{e}}}}&\qquad\sqrt{x_{\rm{e}}}H^{(2)}_{-1}(x_{\rm{e}})+\frac{H^{(2)}_{0}(x_{\rm{e}})}{2\sqrt{x_{\rm{e}}}}\end{matrix}\right),$
$\displaystyle\underline{\underline{\textbf{x}}}_{2}$
$\displaystyle=\left(\begin{matrix}\sqrt{2\alpha}H^{(1)}_{n}(2\alpha
x_{\rm{e}})&\sqrt{2\alpha}H^{(2)}_{n}(2\alpha x_{\rm{e}})\\\\[5.0pt]
\sqrt{2\alpha x_{\rm{e}}}H^{(1)}_{n-1}(2\alpha
x_{\rm{e}})-\frac{(n-\frac{1}{2})}{\sqrt{2\alpha
x_{\rm{e}}}}H^{(1)}_{n}(2\alpha x_{\rm{e}})&\qquad\sqrt{2\alpha
x_{\rm{e}}}H^{(2)}_{n-1}(2\alpha
x_{\rm{e}})-\frac{(n-\frac{1}{2})}{\sqrt{2\alpha
x_{\rm{e}}}}H^{(2)}_{n}(2\alpha x_{\rm{e}})\end{matrix}\right),$ (34)

and

$\textbf{B}=\left(\begin{matrix}B_{1}\\\
B_{2}\end{matrix}\right),\;\;\;\;\;\;\text{with}\;\;x_{\rm{e}}\equiv-k\tau_{\rm{end}}.$
(35)

####  III.2.3 Kinetic contraction- kinetic expansion

The matching for these two final stages arises at $\tau=0$, which is trivial
as we have

$C_{1,2}=-\sqrt{1+\chi}B_{2,1}.$ (36)

###  III.3 Strain spectrum today ($\tau=\tau_{0}$)

The quantity in terms of which we assess the production of primordial
gravitational waves in the cyclic Universe is the dimensionless strain
spectrum

$\Delta h=k^{\frac{3}{2}}\frac{\left|h_{k}(\tau)\right|}{\pi}.$ (37)

It is convenient to evaluate the dimensionless strain spectrum in the
radiation-dominated epoch. Using the general solutions we derived and matched
across the different phases of the model, it reads:

$\Delta h(k,\tau_{r})=\frac{k^{2}\sqrt{2\,\tau_{\rm{r}}}}{a(\tau_{\rm{r}})\pi
M_{\rm{Pl}}}\left|C_{1}(k)H_{0}^{(1)}(x_{\rm{r}})+C_{2}(k)H_{0}^{(2)}(x_{\rm{r}})\right|,$
(38)

where $x_{\rm{r}}\equiv k\,\tau_{\rm{r}}$. The strain tensor spectrum today
$\Delta h(k,\tau_{0})$ can be easily related to the spectrum in the radiation
dominated epoch by means of the transfer function formalism as:

$\Delta h(k,\tau_{0})\equiv\mathcal{T}(k)\Delta h(k,\tau_{\rm{r}}),$ (39)

where $\mathcal{T}(k)$ is the transfer function, responsible for propagating
the spectrum forwards to today’s observed spectrum. Following Ref. Boyle  _et
al._ (2004), we use a transfer function of the form

$\mathcal{T}(k)\approx\left(\frac{k_{0}}{k}\right)^{2}\left(1+\frac{k}{k_{\rm{eq}}}+\frac{k^{2}}{k_{\rm{eq}}k_{\rm{r}}}\right),$
(40)

where $k_{\rm{r}}=a_{\rm{r}}H_{\rm{r}}\approx T_{r}^{2}/M_{\rm{Pl}}$ is the
wave number of modes crossing the horizon at the start of radiation
domination.

So far, the whole calculation is done by setting $a(\tau_{\rm{r}})=1$.
However, for ease of comparison with data and the other results documented in
the literature, it is convenient to return to the usual coordinate system
where $a_{0}=1$. This can be easily done by means of the well-known inverse
scaling between the scale factor and the CMB temperature (i.e. $a\propto
1/T$). This relation fixes the ratio $a(\tau_{0})/a(\tau_{\rm{r}})\propto
T_{\rm{r}}/T_{0}\sim 10^{20}$. As a result, we can re-scale the comoving
frequency accordingly, which now takes the more familiar value
$\hat{k}_{\rm{r}}=k_{\rm{r}}/a(\tau_{0})=10^{-1}\,\text{Hz}$. From now on, we
will name the re-scaled comoving frequency $\hat{k}_{\rm{r}}$ as $k_{\rm{r}}$.
Additionally, always following from the inverse proportionality between the
scale factor and the CMB temperature, we expect a spectral range of modes
entering between the start of radiation domination and today of the order of

$\frac{k_{0}}{k_{\rm{r}}}\propto\frac{T_{0}}{T_{\rm{r}}}\sim 6.6\times
10^{-20}.$ (41)

Taking in mind that during matter domination $H\sim a^{-3/2}$, we get

$\frac{k_{\rm{eq}}}{k_{0}}\approx\sqrt{1+z_{\rm{eq}}}\sim 10^{2},$ (42)

where $z_{\rm{eq}}$ is the redshift at the equivalence.

We can also place a bound on the wave number of modes on the horizon at the
previous dark energy-ekpyrotic transition, $k_{\rm{tr}}$, which will be useful
later. The wavelength of such modes can be estimated using the duration of the
ekpyrotic phase, which we require to last at least 60 e-folds to effectively
homogenise and flatten the Universe for the subsequent cycle. However, a much
stronger constraint comes from Eq. (15), as the number of e-folds during the
ekpyrotic phase is given by
$N\sim\ln(H_{\rm{end}}/H_{0})\approx\ln(\sqrt{-V_{\rm{end}}/V_{0}})$. Taking
our previous constraints on $V_{0}$ at today’s dark energy density and
$V_{\rm{end}}$ at the GUT scale, we arrive at $N\sim 115$. This forces our
horizon to shrink by a factor of $\sim 10^{50}$ during the dark ekpyrotic
phase, and hence we would expect

$\frac{k_{\rm{end}}}{k_{\rm{tr}}}\approx 10^{50}.$ (43)

##  IV Resilience through cycles and the vacuum state

The calculation of the spectrum of gravitational waves introduced in the
previous section, in addition to considering the contribution of tensor modes
produced in the dark energy phase of the previous cycle, is entirely general
regarding the vacuum state and applies to any particular choice. This allows
us to test several conceptual aspects of the theory (partially highlighted in
the introduction), such as its resilience throughout cycles and the
implications of the choice of the vacuum states. In this section, we examine
both these issues in more detail. In subsection IV.1, we focus on the choice
of the vacuum state of the theory, deriving novel constraints on the latter
based on internal consistency through cycles and epochs. In subsection IV.2,
we quantify the extent to which including the contribution from tensor modes
originating during the dark energy phase of the previous cycle changes the
spectrum of tensor perturbations observed in the current cycle.

###  IV.1 Gravitational waves from a non-Bunch Davies vacuum

As often speculated both in quantum field theory and effective field theory,
the choice of the vacuum state represents an important topic of discussion and
significance as it plays a crucial role in determining the properties of the
theory and the physical predictions it makes. On the one hand, in quantum
field theory, the vacuum state is the lowest-energy state of a quantum field,
which usually corresponds to the state with no physical particles. On the
other hand, effective field theories often deal with specific energy scales or
regimes of a more fundamental theory (such as in our effective representation
of the cyclic Universe), and the choice of the vacuum state may involve
integrating out high-energy degrees of freedom and focusing on the low-energy
behavior. This makes the choice of the vacuum state far from trivial when
considering the possibility of new physics at sufficiently high energies.

The selection of the vacuum state has been the subject of intense study and
attention in the context of inflationary models where considering more exotic
(though physically motivated) vacuum states other than BD) can lead to
markedly different predictions for the spectrum of scalar and tensor modes,
see e.g. Ashoorioon  _et al._ (2014a, b); Choudhury (2023) and references
therein. Just to mention one example among many, even sticking to the
framework of single-field inflation minimally coupled to gravity, considering
an exotic vacuum can result in the violation of the usual slow-roll
consistency relations allowing for a blue-tilted spectrum of gravitational
waves, with an enhanced amplitude on small scales that can eventually produce
observable signatures visible by CMB and Gravitational Waves experiments.

However, the same issue has not been investigated in the cyclic Universe.
Previous analyses of the spectrum of the gravitational wave assume a BD vacuum
for perturbations produced in the ekpyrotic phase, enforcing the solution to
converge to that of a plane wave in flat space at sufficiently short distances
Boyle  _et al._ (2004). As argued in Ref. Steinhardt  _et al._ (2002), this
choice mostly relies on the classical treatment of the dynamics of the
evolving modulus field at arbitrarily small length scales. However, just like
in inflation, the choice may no longer be trivial if the dynamics of the
modulus field is influenced by new physics at some characteristic energy scale
$M$. As a result, one may wonder whether in the cyclic Universe, the choice of
the vacuum state is somehow affected by the same level of ”arbitrariness” as
inflation, or if additional constraints can be derived based on the strong
interconnection between the different phases and/or from the need to maintain
consistency through cycles. A different, yet related, question is whether,
also in the cyclic Universe, this potential degree of arbitrariness impacts
the predictions for the tensor spectrum or if they remain robust under the
choice of the vacuum state.

Here, we take a first step forward in the discussion and, considering a non-BD
vacuum state in the dark energy phase of the previous cycle, we argue that
this initial state imposes a more fundamental constraint on the cyclic model,
primarily due to the backreaction effects that a non-BD state could have on
the background evolution of the modulus field.

To prove this point, following Refs. Aravind  _et al._ (2013); Ashoorioon  _et
al._ (2014a); Holman and Tolley (2008), we allow the second Bogoliubov
coefficient $\beta_{k}$ to be non-zero. Notice that, since the Bogoliubov
coefficient is related to $D_{2}$ by

$\beta_{k}=2D_{2}\sqrt{(k/\pi)},$ (44)

and parameterizes deviations from the BD vacuum. As the effective theory
becomes invalid at energy scales beyond those of new physics, we ensure that
no modes are excited past this point, therefore requiring $\beta_{k}\to 0$ for
$k>Ma(\tau_{c})$. In addition, we choose $\beta_{k}$ of the form

$\beta_{k}\sim\beta_{0}e^{-\frac{k^{2}}{M^{2}a\left(\tau_{c}\right)^{2}}},$
(45)

where the energy scale of the effective theory is $M\sim 10^{-4}M_{\rm{Pl}}$
and $\beta_{0}$ is a proportionality factor. The effects produced by the
backreaction in the dark energy-dominated phase set stringent limits on how
large the non-vanishing $\beta_{k}$ can be. In particular, requiring that the
energy density of the non-BD quanta does not overwhelm the energy density
associated with the modulus field – hence spoiling the model – implies

$\rho_{non-
BD}\sim\frac{1}{a_{i}^{4}}\int{\frac{d^{3}k}{(2\pi)^{3}}}\left|\beta_{0}\right|^{2}k\sim\left(\frac{a(\tau_{c})}{a_{i}(\tau)}\right)^{4}\left|\beta_{0}\right|^{2}M^{4}\ll
M_{\rm Pl}^{2}H^{2},$ (46)

where $a_{i}$ refers to the value of the scale factor when we set our initial
vacuum conditions, and we have used $M_{\rm PL}^{2}H^{2}$ as the energy
density associated with the background evolution. We take the cutoff time
$\tau_{c}$, beyond which we cannot be certain of the modulus field dynamics to
be the start of radiation domination regime in the previous cycle
$\tau_{r(pc)}$. As long as $(a(\tau_{r(pc)})/a(\tau))^{4}\ll 1$, Eq. (46) sets
an upper bound for the parameter $\beta_{0}$:

$\beta_{0}<\frac{HM_{\rm Pl}}{M^{2}}\sim 10^{-53}$ (47)

where we used that, during the dark energy phase, the Hubble parameter is
approximately constant $H\simeq H_{0}=100\,h\simeq 2.1h\times 10^{-42}$ GeV.
Notice that this bound is independent of the cutoff time $\tau_{c}$ when
starting the calculation in the dark energy phase. This is because of the
three expansion phases preceding this epoch and the large net expansion of the
scale factor from cycle to cycle, ensuring that
$(a(\tau_{c})/a(\tau_{i}))^{4}\ll 1$ for any $\tau_{c}$.

We emphasise once again that avoiding problems with backreaction during the
dark energy-dominated phase leads to an extremely restrictive constraint on
deviations away from the BD vacuum. To gain a rough idea of how restrictive
our bound is, we can compare constraints on the same cutoff scale obtained in
inflationary cosmology. In that case, as highlighted in Ref. Holman and Tolley
(2008), the restriction reads $\beta_{0}<10^{-6}$; i.e., about 47 orders of
magnitude weaker than our bound. The reason for such a large difference lies
in the fact that, although inflation and dark energy share several common
aspects in terms of background dynamics, these two phases span energy scales
that differ by over 100 orders of magnitude in characteristic energy density.
The energy scale of inflation is much higher, making it way more challenging
for perturbations to overwhelm the dynamics and allowing larger freedom for
deviations away from BD.

To further appreciate the strength of our constraint, it is also worth briefly
discussing what happens when setting the vacuum state in a different phase of
the model rather than in the dark energy-dominated one. In particular, we
focus on the ekpyrotic phase, which, as highlighted multiple times in this
work, is the phase where initial conditions are typically fixed, and the
calculation of the spectrum of tensor perturbations is initiated. In this
phase, the Hubble parameter drastically grows, see Eq. (15), while the scale
factor remains approximately constant. As a consequence, the constraint on
$\beta_{0}$ is relaxed up to $\beta_{0}<10^{45}$ for $M\sim 10^{-4}M_{\rm
Pl}$, allowing us complete freedom to choose a wide range of vacuum states in
the ekpyrotic phase without compromising the model during this stage. That
being said, it is necessary to ensure that significant deviations away from a
BD vacuum during the ekpyrotic phase do not lead to other issues during the
subsequent evolutionary states of the cycle. However, given that the evolution
of the dark energy phase is governed by the background dynamics of the lowest
energy scale and considering the cyclic nature of the model and its resilience
through cycles (meaning that starting from the dark energy phase of the
previous cycle does not produce differences in observable quantities as we
prove in the next subsection), we argue that selecting the vacuum during the
dark energy phase is the most restrictive and conservative choice to
circumvent this issue entirely. This makes relying on the BD vacuum a safer
assumption from a model-building perspective.

###  IV.2 Gravitational waves produced from different phases

Our general calculation for the strain spectrum of gravitational waves enables
us to investigate how the predictions change when incorporating the
contribution from tensor modes originating in the dark energy phase of the
previous cycle. We can then test the robustness of these predictions by
comparing our results with those already documented in the literature, derived
from starting in the ekpyrotic contracting phase.

In this section, we compare the predictions for the strain spectrum of
gravitational waves obtained in the following two cases:

  * (a)

Using our general calculation and starting in the dark energy phase of the
previous cycle. In this case we impose BD vacuum conditions on the
coefficients of the dark energy stage $D_{1}$ and $D_{2}$ given by:

$\displaystyle D_{1}$ $\displaystyle=\frac{1}{2}\sqrt{\frac{\pi}{k}},$
$\displaystyle D_{2}$ $\displaystyle=0.$ (48)

  * (b)

Starting the calculation in the ekpyrotic phase (disregarding the matching at
$\tau=\tau_{tr}$) and imposing BD initial conditions as done in Res. Boyle
_et al._ (2004)

$\displaystyle A_{1}$ $\displaystyle=\frac{1}{2}\sqrt{\frac{\pi}{k}},$
$\displaystyle A_{2}$ $\displaystyle=0.$ (49)

Notice that, although our calculation is fully generic concerning the choice
of the vacuum state, as we proved in the previous subsection, deviations away
from the BD vacuum state in the dark energy phase are strongly constrained,
providing us with a valid physical reason to impose this vacuum state in the
dark energy phase for the case (a). Instead, for the case (b), we impose the
BD initial conditions in the ekpyrotic phase to work in the same framework as
Ref. Boyle  _et al._ (2004) and allow direct comparison.

Figure 1: Strain spectrum plotted against physical frequency $k$, of
gravitational waves produced starting the cycle with the dark energy phase in
blue, and with the ekpyrotic phase in orange. BD initial conditions are
assumed for both spectra. $k_{0},k_{eq},k_{r}$ and $k_{end}$ are the comoving
frequencies on the horizon today, at matter-radiation equivalence, at the
start of radiation domination, and at the ekpyrotic-kinetic transition
respectively. Figure 2: Illustrative plot of the comoving Hubble horizon,
$1/aH$, throughout the ekpyrotic, kinetic contracting, kinetic expanding,
radiation domination, and finally matter domination phases in magenta, orange,
dark blue, green and cyan respectively. Key modes on the horizon are
illustrated as horizontal dashed lines, along with a label of their comoving
wavenumber, and some less important modes are labelled.

After matching the relevant phases, the strain spectra of tensor modes
predicted in the two cases can be derived using Eq. (38) and Eq. (40) and are
shown in Figure 1 in blue for the case (a) and in orange for the case (b). As
evident from the figure, a difference of up to an order of magnitude in the
strain $\Delta h$ is observed for modes $k_{0}$ at the present-day horizon.

This difference can be understood by considering the evolution of modes
produced during the dark energy phase of the previous cycle. To further
clarify this point, we refer to Figure 2 where we show the evolution of the
comoving Hubble horizon, $1/aH$, throughout the different phases of the model.
From the figure we note that the horizon at the end of the dark energy phase
(i.e. $k_{\rm tr}^{-1}$) is several orders of magnitude greater than the
present-day horizon (i.e. $k_{0}^{-1}$) ensuring that none of these super
horizon modes can have re-entered by today or in any subsequent cycle. Modes
produced during this dark energy phase that exit the horizon become frozen and
subsequently experience further stretching throughout this epoch.

On the other hand, sub-horizon modes produced in the same phase oscillate with
decaying amplitude $h\propto a^{-1}$. This feature can be shown by solving Eq.
(26) in the dark energy background dynamics described by Eq. (7). In view of
that, we expect sub-horizon modes produced deep within the dark energy phase
(as well as in any previous phase of the previous cycle) to decay away to
negligible amplitudes when compared to modes produced at the end of the same
phase or during the subsequent ekpyrotic phase. Instead, sub-horizon modes
produced near the end of the dark energy phase can survive into the ekpyrotic
phase of the next cycle, acting as extra quanta in the vacuum initial
conditions for the subsequent ekpyrotic phase. This contribution is encoded in
the coefficient $A_{2}$ in Eq. (28) and can lead to observable effects on
scales corresponding to the long wavelength portion of the strain spectrum.
Referring back to Figure 1, we can appreciate how the differences produced by
amplitude oscillations expected from an under-damped simple harmonic
oscillator solution affect only the range of frequencies between $k_{0}$ and
$k_{\rm eq}$ before becoming frozen in during the ekpyrotic phase.

Moving forward, as outlined in Ref. Erickson  _et al._ (2007), the subsequent
phases will provide a red-tilt to the scale-invariant dark energy spectrum.
Notice that for an ekpyrotic phase lasting $N\sim 115$ e-folds, the magnitude
of the coefficient $A_{2}$ becomes $\mathcal{O}(10^{10})$ between $k_{0}$ and
$k_{\rm eq}$ then rapidly decays to order $\mathcal{O}(10^{-15})$ for modes
around $k_{\rm r}$. This explains the discrepancy between imposing BD initial
conditions in the dark energy phase given by Eq. (48), and in the ekpyrotic
phase, given by Eq. (49).

We conclude with a last important remark: as argued in subsection IV.1, to
preserve the background evolution of the modulus field, it is important to
have a BD-like vacuum state during the dark energy phase. This requirement
implies that perturbations existing in the current cycle must decay to
negligible levels in amplitude compared to the energy density present in the
BD vacuum, preventing them from acting as additional quanta on top of the
vacuum. We emphasise that this condition is satisfied by the evolution of
tensor perturbations after the bounce, particularly during the subsequent
radiation and matter-dominated phases. In particular, the transfer function,
Eq. (40), ensures that the amplitudes of short-wavelength modes re-entering
the horizon first (that are potentially the most problematic), are decreased
by a factor $\left(k_{0}/k\right)^{2}$. Consequently, the amplitude of the
shortest wavelength modes in the observable spectrum ($k\sim k_{\rm end}$), is
suppressed by more than $20$ orders of magnitude during the radiation and
matter-dominated phases. As a result, the evolution through these phases
guarantees the decaying amplitude of all modes in every cycle preceding each
dark energy phase and restoring the vacuum to a BD state in the latter. This
shields the dark energy phase from possible backreaction effects, underscoring
the resilience of the model and demonstrating once again that, from a
theoretical standpoint, fixing initial conditions in this phase is much safer
from a model-building perspective.

##  V Conclusion

In this study, we investigate the production of gravitational waves in a
cyclic Universe, focusing on certain conceptual aspects of the theory, such as
the resilience of observable predictions against the phase of the cycle in
which initial conditions are set and the choice of the vacuum state.

In most of the analyses carried out in the literature, the ekpyrotic phase is
typically assumed as the starting point of the cycle where initial conditions
of primordial perturbations are imposed, and the calculations of the relative
spectra begin. While for scalar perturbations the implications of setting the
initial conditions in different phases of the theory have been examined in a
few studies surrounding this topic, to the best of our knowledge, the
calculation of the tensor spectrum has always been performed starting in the
ekpyrotic phase, assuming a BD vacuum state and neglecting any potential
contributions arising from tensor modes originating during the dark energy
phase of the previous cycle.

In light of this, a few (we believe) interesting questions remained somewhat
pending. For instance, one might wonder whether setting initial conditions in
the dark energy phase of the previous cycle and considering the potential
contribution of tensor modes generated in this phase could lead to any
observable differences in the gravitational wave strain spectrum observed in
the current cycle and eventually clarify why (not). Most importantly, in
effective field theory descriptions of the cyclic Universe, the choice of the
vacuum state may involve integrating out high-energy degrees of freedom,
focusing on low-energy behaviors of the modulus field. This makes the choice
far from trivial when considering the possibility of new physics acting at
sufficiently high characteristic energy scales. Consequently, one might wonder
about the implications of assuming a BD vacuum state in the ekpyrotic phase
and, more broadly, what freedom exists in the cyclic Universe regarding the
choice of the vacuum state and how such freedom affects the predictions for
the spectrum of gravitational waves.

Fuelled by these questions, we consider a cyclic Universe described by the
effective 4D Lagrangian (1) with a potential given by Eq. (4). After reviewing
the background dynamics of the model (section II), we focus on the production
and evolution of tensor modes. In section III, we analytically solve the
equation of motion of the gravitational wave field through all the different
phases of the cycle by starting from the dark energy phase of the previous
cycle and making no assumptions about the vacuum state. The results presented
in this section generalise the predictions for the tensor spectrum to include
the contribution of tensor perturbations produced during the dark energy phase
of the previous cycle and apply to any choice of the vacuum state of the
theory. Therefore, they extend the treatment presented so far in the
literature, allowing us to gain important insights into the issues posed
earlier.

As argued in section IV, our findings reveal a significant resilience of the
cyclic Universe model concerning predictions for the spectrum of primordial
gravitational waves, with the most relevant results reading as follows.

  * •

Initial Conditions: To quantify the impact of the contribution arising from
tensor modes produced in the previous dark energy phase of the cycle, in
subsection IV.2 we compared the spectra obtained by starting from the
ekpyrotic contracting phase (in orange in Figure 1) and the dark energy phase
of the previous cycle (in blue in Figure 1), assuming in both cases a BD
vacuum. The two spectra are essentially identical except for a difference up
to an order of magnitude in the strain $\Delta h$ for modes on the scale
$k_{0}$ at the present-day horizon. The reason is that the horizon at the end
of the dark energy phase is several orders of magnitude greater than the
present-day horizon (see also Figure 2), implying that none of these super-
horizon modes can have re-entered by today or in any subsequent cycle. On the
other hand, sub-horizon modes produced in the same phase oscillate with
decaying amplitude $h\propto a^{-1}$. Therefore, sub-horizon modes produced
deep within the dark energy phase (as well as in any previous phase of the
previous cycle) decay away to negligible amplitudes, while sub-horizon modes
produced near the end of the dark energy phase can survive into the ekpyrotic
phase of the next cycle, being responsible for the small differences in the
strain spectrum observed around $k_{0}$.

  * •

Vacuum state: Although including tensor modes originating from the previous
dark energy phase does not lead to significant differences in the spectrum of
primordial perturbations, we argue that starting the calculation in this phase
imposes a more fundamental constraint on the choice of the vacuum state of the
model. To highlight this aspect, in subsection IV.1 we consider a non-BD
vacuum state in the dark energy phase parametrized in terms of the second
Bogoliubov coefficient given by Eq. (45). We show that requiring the energy
density of the non-BD quanta not to overwhelm the energy density associated
with the modulus field – hence, potentially spoiling the model – implies an
extremely restrictive constraint on deviations away from the BD vacuum. On the
other hand, for the same non-BD vacuum state, the constraint is completely
lost in the ekpyrotic phase where we are left with complete freedom to choose
a wide range of vacuum states without compromising the model during this stage
(but possibly spoiling the subsequent evolution). As a result, selecting the
vacuum during the dark energy phase (where we are basically forced to BD)
seems to be the most restrictive and conservative choice, as well as a safer
assumption from a model-building perspective. Given the cyclic nature of the
model and the resilience of its predictions against the phase where we set
initial conditions (as proven in the previous point), this strongly reduces
our freedom to consider exotic vacuum states in the cyclic Universe.

###### Acknowledgements.

CvdB is supported (in part) by the Lancaster–Manchester–Sheffield Consortium
for Fundamental Physics under STFC grant: ST/X000621/1. This article is based
upon work from COST Action CA21136 Addressing observational tensions in
cosmology with systematics and fundamental physics (CosmoVerse) supported by
COST (European Cooperation in Science and Technology).

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