# Emergent universe revisited through the CSL theory

Gabriel R. Bengochea<EMAIL_ADDRESS>Instituto de Astronomía y Física del
Espacio (IAFE), CONICET - Universidad de Buenos Aires, (1428) Buenos Aires,
Argentina  María Pía Piccirilli<EMAIL_ADDRESS>Grupo de Cosmología,
Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque S/N 1900 La Plata, Argentina.
CONICET, Godoy Cruz 2290, 1425 Ciudad Autónoma de Buenos Aires, Argentina.
Gabriel León<EMAIL_ADDRESS>Grupo de Cosmología, Facultad de
Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo
del Bosque S/N 1900 La Plata, Argentina.
CONICET, Godoy Cruz 2290, 1425 Ciudad Autónoma de Buenos Aires, Argentina.

###### Abstract

In this work we analyze how the spectrum of primordial scalar perturbations is
modified, within the emergent universe scenario, when a particular version of
the Continuous Spontaneous Localization (CSL) model is incorporated as the
generating mechanism of initial perturbations, providing also an explanation
to the quantum-to-classical transition of such perturbations. On the other
hand, a phase of super-inflation, prior to slow-roll inflation, is a
characteristic feature of the emergent universe hypothesis. In recent works,
it was shown that the super-inflation phase could generically induce a
suppression of the temperature anisotropies of the CMB at large angular
scales. We study here under what conditions the CSL maintains or modifies
these characteristics of the emergent universe and their compatibility with
the CMB observations.

Cosmology, Emergent Universe, Quantum Cosmology, Cosmic Microwave Background

##  I Introduction

The success of the standard $\Lambda$CDM cosmological model in explaining the
many accurate astronomical observations we have today (e.g. Ade et al.
(2016a); Aghanim et al. (2020a, b); Scolnic et al. (2018); Alam et al. (2021))
includes inflation, a phase of accelerated expansion during the very early
epoch of the universe Starobinsky (1979, 1980); Guth (1981); Mukhanov and
Chibisov (1981); Linde (1982, 1983); Albrecht and Steinhardt (1982); Bardeen
et al. (1983); Brandenberger (1984); Hawking (1982).

There are renowned merits attributed to the inflationary paradigm. In addition
to solving the horizon problem, predictions from the simplest slow-roll
inflationary model, such as a spatially flat geometry characterizing the
universe and a quantum origin of the spectrum of primordial perturbations
(i.e. a nearly scale-invariant power law), are some of them. Moreover, the
predictions are extremely consistent with recent observations from the cosmic
microwave background (CMB) radiation Akrami et al. (2020a). However, the
exploration of some alternatives seems to be interesting, in the light of
small features in the CMB power spectra that remain unexplained and also open
discussions in some recent works, which we mention below.

Some studies have drawn attention to the lack of large-angle correlations and
a weak power in the low-$\ell$ multipole moments of the angular power
spectrum, in the observed CMB temperature anisotropies with respect to that
predicted within the standard $\Lambda$CDM model. This was first mentioned in
the COBE results Hinshaw et al. (1996) and later confirmed in subsequent
generations of satellites Spergel et al. (2003, 2007); Larson et al. (2011);
Komatsu et al. (2011); Ade et al. (2014a, b, 2016b); Akrami et al. (2020b).
This feature on the largest angular scales was analyzed, with some
controversies included, by several authors (see for instance Copi et al.
(2007, 2009, 2010, 2015); Sarkar et al. (2011); Efstathiou et al. (2010)). As
is well known, the largest observable angular scales contain direct
information on primordial physical processes that occurred during the
inflationary era (or prior to it), and could only have undergone modifications
by the physics involved in the relatively recent past, e.g. through the late-
time integrated Sachs-Wolfe effect. In fact, several authors have shown that
some of the observed CMB anomalies could be explained in this way, e.g. Rakic
et al. (2006); Francis and Peacock (2010). However, there is still no verdict
on whether it is just a statistical fluke or if there really is something new
and interesting behind it. Today it constitutes one of the persistent large-
angle anomalies in the CMB data that makes up the list of current challenges
of the standard $\Lambda$CDM model Schwarz et al. (2016); Perivolaropoulos and
Skara (2021). In search of a convincing explanation, new theoretical ideas
have been considered from different approaches. The results for the
suppression of the low multipoles in the CMB spectrum can give us clues
towards new physics that determined the initial conditions for slow-roll
inflation or even tell us something about an earlier phase before slow-roll
inflation Sokolov (1993); Stevens et al. (1993); Rocha et al. (2004);
Campanelli et al. (2007); Bastero-Gil et al. (2003); Contaldi et al. (2003);
Bridle et al. (2003); Feng and Zhang (2003); Piao et al. (2004); Boyanovsky et
al. (2006); Cao et al. (2008); Destri et al. (2008, 2010); Handley et al.
(2014); Gessey-Jones and Handley (2021); Ramirez and Schwarz (2012); Ramirez
(2012); Biswas and Mazumdar (2014); Liu et al. (2013, 2014); Ashtekar et al.
(2020); Agullo et al. (2021a, b).

As mentioned above, the prediction that our current universe is spatially flat
is one of those that is often mentioned in relation to inflationary models,
and the data seem to indicate that indeed the observed universe is very close
to flat. However, it does not imply that the spatial sections are exactly
flat. Cosmological models in the context of non-flat cases have also been
explored with interesting results, see e.g. Tryon (1973); Vilenkin (1982);
Ellis (1987, 1988); Ellis et al. (1991); Linde (1995); Linde and Mezhlumian
(1995); White and Scott (1996); Linde (1998, 1999); Ellis et al. (2002); Linde
(2003); Lasenby and Doran (2005); Uzan et al. (2003); Zhang and Sun (2004);
Ratra (2017); Ooba et al. (2018a). Recently, in light of the analysis of some
observational data, the debate about what is the spatial curvature of the
universe has resurfaced, typically quantified via its fractional contribution
to the cosmic energy budget today parameterized by $\Omega_{K}$ Räsänen et al.
(2015); Park and Ratra (2019a); Ryan et al. (2018); Yu et al. (2018); Ooba et
al. (2018b); Park and Ratra (2019b); Ryan et al. (2019); Handley (2021);
Efstathiou and Gratton (2019); Riess et al. (2019); Di Valentino et al.
(2019); Di Valentino et al. (2021a); Efstathiou and Gratton (2020); Di
Valentino et al. (2021b); Benisty and Staicova (2021); Vagnozzi et al. (2021a,
b); Dhawan et al. (2021). This parameter has an important role in determining
the evolution of the universe and is closely related with the early universe
physics, because if the spatial curvature is positive, then the curvature term
will always dominate at early enough times in a slow-rolling inflationary
epoch. And indeed, some of these recent analyzes suggest that a small positive
spatial curvature could be present favoring, perhaps, the case of a closed
universe. It is also interesting to mention that the question of spatial
curvature participates in the so-called “$H_{0}$ tension” between sets of
early time probes and a number of late time observational data (see e.g. Verde
et al. (2019); Di Valentino et al. (2021c); Bolejko (2018); Collett et al.
(2019)). Additionally, spatial curvature also affects the search for the
nature of dark energy, an energy component to explain the current stage of
accelerated expansion and that would constitute around 70% of the energy
budget of the universe according to the $\Lambda$CDM model Clarkson et al.
(2007); Hlozek et al. (2008); Barenboim et al. (2010); Di Valentino et al.
(2021d, a). In particular, it is known that by assuming flatness, when in fact
$\Omega_{K}\neq 0$, would induce critically large errors in reconstructing the
dark energy equation of state, showing that including curvature as a free
parameter is imperative in any future analyses that attempts to determine
whether dark energy is a cosmological constant or is something more exotic.
And, on the other hand, closed universe models can generally relax the Hubble
tension between supernovae observations and the CMB. However, at this stage,
there is no conclusive evidence for a positive curvature, but this is at least
an attractive possibility supported by the data that warrants further
exploration.

An important feature that was highlighted in Biswas and Mazumdar (2014), and
of particular interest for the present work, is that a phase of super-
inflation (i.e. a period where the Hubble parameter increases with time) prior
to slow-roll inflation could be related to the suppression of power in the low
CMB multipoles. Furthermore, any mechanism that attempts to solve the
cosmological singularity problem, within a semiclassical spacetime
description, will naturally contain such a phase.

Starting with the very known pioneering works on singularity theorems by
Penrose and Hawking Penrose (1965); Hawking (1965); Hawking and Penrose
(1970), other authors have made contributions extending their cosmological
applications, including the case of inflationary models Hawking and Ellis
(2011); Wald (1984); Borde and Vilenkin (1994a, b); Borde (1994); Borde and
Vilenkin (1996, 1997); Guth (2001); Borde et al. (2003). To evade these
singularity theorems, some works have studied and developed alternatives in
the context of bouncing models, see e.g. Molina-Paris and Visser (1999);
Kanekar et al. (2001); Khoury et al. (2001); Steinhardt and Turok (2002);
Battefeld et al. (2004); Bojowald et al. (2004); Biswas et al. (2006); Peter
and Pinto-Neto (2002); Battefeld and Peter (2015); Lilley and Peter (2015);
Brandenberger and Peter (2017); Matsui et al. (2019); Barrau (2020). Another
alternative that manages to escape these theorems is the recently developed
framework of the _emergent universe_ Ellis and Maartens (2004). The idea of an
emergent universe is not new, it can be traced back to seminal works of
Einstein and Eddington Einstein (1917); Eddington (1930).

The emergent universe (EU) of Ellis and Maartens (2004); Ellis et al. (2004)
is one in which a spatially closed universe (based on General Relativity and
dominated by a scalar field minimally coupled to gravity) _emerges_ from an
initially past-eternal Einstein static state (with a finite initial size),
enters a phase of super-inflation and then evolves towards slow-roll inflation
to finally give rise to the standard hot-Big Bang. Because of how it is built,
there is neither a horizon problem nor singularity. This model has been
studied in recent years through different approaches and variants,
establishing its stability conditions, analyzing fine-tuning issues, its
viability both theoretically and observationally and making it clear which
questions are still open; see, for instance Mulryne et al. (2005); Gibbons
(1987); Barrow et al. (2003); Mukherjee et al. (2005, 2006); Banerjee et al.
(2007); Boehmer et al. (2007); Parisi et al. (2007); del Campo et al. (2007);
Debnath (2008); Banerjee et al. (2008); Beesham et al. (2009); del Campo et
al. (2009); Paul and Ghose (2010); Paul et al. (2010); Zhang et al. (2010);
Paul et al. (2011); Chattopadhyay and Debnath (2011); del Campo et al. (2011,
2010); Labrana (2012); Cai et al. (2012); Rudra (2012); Ghose et al. (2012);
Liu and Piao (2013); Aguirre and Kehayias (2013); Cai et al. (2014); Atazadeh
et al. (2014); Zhang et al. (2014); Bag et al. (2014); Huang et al. (2015);
Böhmer et al. (2015); Labraña (2015); Khodadi et al. (2016); Zhang et al.
(2016); Ríos et al. (2016); Khodadi et al. (2018); Martineau and Barrau
(2018); Labrana and Cossio (2019). Among the alternative scenarios of the
early universe Brandenberger (2010, 2020), a recent realization of an emergent
universe is the so-called string gas cosmology Brandenberger and Vafa (1989);
Nayeri et al. (2006); Brandenberger (2008, 2011). In this model the universe
begins in a long hot and almost static phase, dominated by a thermal gas of
closed fundamental strings. Recently, some conjectures such as Swampland Obied
et al. (2018) and Trans-Planckian Censorship (TCC) Bedroya and Vafa (2020)
have put very strong constraints on possible inflationary models Garg and
Krishnan (2019); Agrawal et al. (2018); Brandenberger and Martin (2013);
Bedroya et al. (2020). However, alternative cosmologies such as bouncing and
emergent models are consistent with these conjectures and trivially satisfied.
A good understanding of the emerging phase is still missing, but there are
some promising approaches Brandenberger (2021).

The next point to consider is the fact that any model that claims to provide a
mechanism for the generation of the seeds of cosmic structure, must be able to
give a convincing answer to the following issue. In the early stages at the
beginning of the universe (and well after the Planck era has ended), the
spacetime is assumed to be spatially isotropic and homogeneous. In addition, a
standard assumption is that the perturbations of matter fields (e.g. the
inflaton field) were in a quantum vacuum state also perfectly symmetric (the
symmetry being spatial isotropy and homogeneity), usually described by the so-
called Bunch-Davies vacuum. Then, an important puzzle arises, namely to
explain the transition from a perfect symmetric state portraying the early
universe to the non-symmetric state that characterizes the current universe,
which cannot be attributed to quantum unitary evolution. The standard approach
to address that issue is based on the study of the role of _quantum
fluctuations_ during, for instance, the inflationary epoch. However, since the
evolution of any quantum state, according to standard quantum theory, is
always dictated by the Schrödinger equation (which does not break any initial
symmetry of the system or destroy quantum superpositions), the traditional
early universe paradigm is incomplete in that sense. In other words, the
solely existence of vacuum fluctuations is in no way sufficient to claim that
there are actual inhomogeneities of any kind present in the universe. The
aforementioned puzzle is usually referred in the literature as the quantum-to-
classical transition of the primordial perturbations. In fact, this subject is
closely related to what is known as _the measurement problem_ in quantum
physics Wigner (1963); Omnes (1994); Maudlin (1995); Becker (2018); Norsen
(2017); Durr and Lazarovici (2020); Albert (1994); Okon (2014), and is
notoriously exposed in the case of the quantum description of the primordial
universe Bell (1981); Perez et al. (2006); Sudarsky (2011); Landau et al.
(2013). This is because measuring devices and observers who decide when and
how to perform some kind of measurements cannot be fundamental notions in a
theory which seeks to describe the early universe where neither existed Hartle
(1993). Cosmologists have tried to account for the quantum-to-classical
transition by different types of arguments. A recent critique of different
efforts at explaining this subject can be found in Berjon et al. (2021)111The
reader interested in a pedagogical review on this subject, can find it in
Bengochea (2020)..

In order to classify the possible alternatives to address the issue described
above, in Maudlin (1995) the measurement problem was written in a elegant
manner showing that there are three statements that are mutually inconsistent.
In short: (A) the physical description provided by the quantum state is
complete, (B) quantum states always evolve according to the Schrödinger
equation, and (C) measurements always have definite results.

If statement (A) is denied, then the quantum state does not contain all the
information necessary for the description of a quantum system. In this way,
the addition of _hidden variables_ and the equations that determine their
evolution is required. The best known proposal for this case is the _de
Broglie-Bohm model_ Bohm (1952). Some applications to the cosmological case
can be found in Valentini (2010); Pinto-Neto et al. (2012); Goldstein et al.
(2015); Pinto-Neto and Struyve (2018); Vitenti et al. (2019).

Works based on decoherence Kiefer and Polarski (2009); Halliwell (1989);
Kiefer (2000); Polarski and Starobinsky (1996) led to a partial understanding
of the issue. Nevertheless, this argument by itself (i.e. without any extra
assumptions) cannot address the fact that a single (classical) outcome emerges
from the quantum theory. In other words, decoherence alone cannot solve the
quantum measurement problem Okon and Sudarsky (2016); Adler (2003);
Schlosshauer (2004). Other cosmologists seem to adopt the Everett “many-
worlds” interpretation of quantum mechanics Everett (1957) plus the
decoherence process when confronted with the quantum-to-classical transition
in the inflationary universe, e.g. Mukhanov (2005). Regarding this point, we
would like to refer the reader to Sudarsky (2011); Kent (1990); Stapp (2002)
where arguments against decoherence and the Everett interpretation are also
presented. This proposal and the Everettian interpretations are some of the
approaches that somehow discard statement (C).

The last remaining choice is to negate statement (B). This path leads to non-
standard quantum theories, i.e. theories where the collapse of the wave
function is self-induced by some novel mechanism. Known as _objective collapse
theories_ , from the mid-1970s several authors began to develop modifications
to the Schrödinger equation, with the aim to alter the evolution of the wave
function. In this way, the collapse of the wave function would occur without
any reference to external observers or devices present that should perform
some sort of measurements Pearle (1976); Ghirardi et al. (1986); Pearle
(1989); Diosi (1987, 1989); Penrose (1996). One virtue of collapse models is
that they have shown in recent years to have the attractive feature of
connecting plausible resolutions of other open problems in a single unified
picture Okon and Sudarsky (2014). Reviews on these sort of theories can be
found, for instance, in Bassi and Ghirardi (2003); Bassi et al. (2013).

In the present work, we will approach the emergent universe from the
perspective of the Continuous Spontaneous Localization (CSL) model Pearle
(1976, 1989), which will be incorporated into the situation at hand as a
mechanism to break the original symmetries of the quantum vacuum state of the
field driving the expansion of the early universe, and generating the
primordial cosmological perturbations. In this manner, the CSL model naturally
provides an explanation of the quantum-to-classical transition of such
perturbations. The incorporation of objective collapse schemes and theories in
the cosmological context has been studied since 2006 Perez et al. (2006). This
has led to numerous investigations in recent years with varied proposals,
particularly in the framework of semiclassical gravity (but also with
exploratory works in the framework of standard quantization), with very
encouraging results Leon and Sudarsky (2010); Sudarsky (2011); Diez-Tejedor
and Sudarsky (2012); Diez-Tejedor et al. (2012); Cañate et al. (2013);
Bengochea et al. (2015); León and Sudarsky (2015); Leon and Bengochea (2016);
León (2017); Landau et al. (2012, 2013); Benetti et al. (2016); Bengochea and
León (2017); Cañate et al. (2018); Juárez-Aubry et al. (2018); Piccirilli et
al. (2019); León et al. (2015); Mariani et al. (2016); León et al. (2017,
2018, 2016); Josset et al. (2017); Leon and Piccirilli (2020). Other authors
have investigated similar ideas and some of these works can be seen, for
instance, in Martin et al. (2012); Das et al. (2013); Markkanen et al. (2015);
Alexander et al. (2016); Drossel and Ellis (2018). The debate about the
particular details involving the implementation of the CSL theory into the
cosmological context is still open. In fact, there is an extensive landscape
of possibilities, which constitutes an active line of research at the moment
Martin and Vennin (2020); Bengochea et al. (2020a, b); Martin and Vennin
(2021a); Gundhi et al. (2021); Martin and Vennin (2021b); León and Bengochea
(2021).

In particular, a relevant aspect to examine is the following: when one decides
to combine quantum field theory (QFT) with gravitation, one must choose the
setting within which such a link is to be made. At the time of writing there
is no complete and finished program merging successfully both theories, so a
couple of options arise. In Ref. Bengochea et al. (2020b) some of these
approaches were analyzed, evaluating their pros and cons. There, it was argued
that the semiclassical gravity framework appears favored from a theoretical
and conceptual point of view when one wants to incorporate collapse models.
Therefore, our present analysis will be based on the semiclassical gravity
(SCG) framework, in which gravity is treated classically and the matter fields
are treated quantum mechanically Perez et al. (2006); Diez-Tejedor and
Sudarsky (2012); Cañate et al. (2013, 2018); Juárez-Aubry et al. (2018, 2020).
This approach accepts that gravity is quantum mechanical at the fundamental
level, but considers that the characterization of gravity in terms of the
metric is only meaningful when the spacetime can be considered classical.
Namely, we will be dealing with the description of an epoch well after the
full quantum gravity regime has ended (i.e. from when the emergent universe
begins to evolve), where the energy scales involved allow one to suppose valid
the consideration of the metric as classical and well described by
semiclassical gravity equations. Therefore, semiclassical gravity can be
treated as an effective description of quantum matter fields inhabiting a
classical spacetime. While this approach has received some criticisms Eppley
and Hannah (1977); Page and Geilker (1981), those arguments have been refuted
Mattingly (2005, 2006); Kent (2018); Tilloy and Diósi (2016); Carlip (2008);
Albers et al. (2008); Ford (2005, 2005). In the particular case of CSL theory,
its first implementation into the primordial universe, based on the SCG
framework, was done in Cañate et al. (2013), and some of us have continued to
explore its consequences. For example, in León et al. (2015, 2017, 2018) it
was shown that a strong suppression of primordial B-modes in the CMB is
predicted generically. In addition, within that same framework, observational
constraints were analyzed in Piccirilli et al. (2019), and in León (2017) it
was found that the condition for eternal inflation can be bypassed. In the
next section, some additional motivations for this choice in the present case
of analysis will become apparent.

To finish this Introduction, let us mention that in some previous works León
and Sudarsky (2015); Benetti et al. (2016); Bengochea and León (2017), where a
spontaneous collapse of the wave function was implemented during a phase of
slow-roll inflation, certain features on the low multipoles of the CMB were
analyzed as consequence of the collapse. On the other hand, recently in
Labraña (2015) the emergent universe model, originally put forward in Ellis
and Maartens (2004); Ellis et al. (2004), was analyzed and it was shown that
the super-inflation phase (a characteristic shared by all emergent universe
models) could be responsible for part of the anomaly in the low multipoles of
the CMB; in particular, for the observed lack of power at large angular
scales. Motivated by these results, and under the same assumptions of Labraña
(2015), here we calculate the primordial power spectrum of scalar
perturbations, but incorporating a particular version of the CSL model for the
situation at hand. We will analyze whether the super-inflation phase in the
framework of the emergent universe plus CSL continues (or not) to produce the
power suppression in the low multipoles. Lastly, we will study under what
conditions the CSL maintains or modifies such characteristics and their
compatibility with the CMB observations.

Our manuscript is divided as follows. We start in section II presenting the
theoretical framework of the emergent universe plus the CSL proposal; we also
obtain the predicted scalar power spectrum. Next, in section III we present
and discuss our results; there, we also perform a further exploration of the
corresponding parameter space. Finally, in section IV, we present our
conclusions. Regarding conventions and notation, we use a $(-,+,+,+)$
signature for the spacetime metric and units where $c=1=\hbar$.

##  II Emergent universe in the CSL framework

In this section, we present the implementation of the CSL model into the
emergent universe (EU) model.

###  II.1 Theoretical background

As we mentioned in the Introduction, many models of the emergent universe have
been studied in recent years. Since we will be closely following the results
of Ref. Labraña (2015), here we will do our analysis under the same
assumptions considered there, which in turn are based on Ellis and Maartens
(2004). In particular, we assume the action of General Relativity with a
scalar field $\phi$, which represents the dominant matter driving the universe
early expansion, minimally coupled to gravity, with canonical kinetic term.
The scalar potential employed (inspired by $R^{2}$-inflation) takes the form
$V(\phi)=(4\pi G)^{-1}(e^{C\phi}-1)^{2}$ as the one reconstructed in Ellis et
al. (2004) (using techniques developed in Ellis et al. (1991)). In obtaining
that potential, the evolution of the scalar factor $a(t)\simeq
a_{0}+A\>e^{H_{0}t}$ was taken into account; where, $a_{0}>0$ is the (initial)
radius of the Einstein static universe, $C$, $A$ are positive constants, and
$H_{0}$ is the Hubble parameter at the onset of slow-roll inflation222The type
of potential $V$ is as the one shown in Figs. 1 of Refs. Ellis et al. (2004);
Labraña (2015)..

In the EU model, after leaving its initial static state, the universe enters a
slow-rolling regime (at a few e-foldings after leaving its initial static
state) where the scale factor grows sufficiently quickly to mitigate
neglecting the curvature effects. This period of a de Sitter type of inflation
comes naturally to an end (as the scalar field starts oscillating around the
minimum of the potential), it is then followed by a re-heating phase, and
finally continues to the standard hot Big Bang expansion. In Ref. Ríos et al.
(2016), it was shown that the temporal evolution, given by Friedmann equation
along with the scalar field Klein-Gordon equation, leads the system towards an
attractor where $H$ tends to a constant and $\dot{\phi}^{2}\to 0$; this is,
the system evolves from an Einstein static state to a de Sitter type of
expansion.

The dynamics of this model is such that, prior to the traditional slow-roll
inflation, there is a phase of super-inflation where the Hubble parameter
increases with time, i.e. $\dot{H}>0$. The mechanism which generates this
superinflationary period depends on the particular model under consideration,
but it is a generic characteristic of the EU scenario. For example, in the
models of Refs. Ellis and Maartens (2004); Ellis et al. (2004), it is
considered a FLRW closed universe where the spatial curvature is responsible
for the superinflationary period. However, let us note that we could have
chosen another model from those mentioned in the Introduction, provided that
the evolution of the background (given by the mentioned scale factor $a(t)$)
produces the phases of super-inflation and slow-roll inflation, in which the
generation of curvature perturbations is analyzed here.

As discussed in depth in Ellis et al. (2004), even though the traditional
emergent universe is with positive spatial curvature, it is quickly negligible
in a few e-foldings and furthermore slow-roll inflation can always be made to
end for some negative value of $\phi$. On the other hand, it is also possible
to find the (finite) $N$ number of e-foldings for the total slow-roll phase
within the emergent universe. Analogously to Labraña (2015), we will now make
a first approach to the problem at hand and therefore we will neglect the
contributions of the space curvature to the primordial perturbation333See, for
instance, Appendix A of Ref. Labraña (2015) for details about this point..

We follow the standard procedure and separate the scalar field and the metric
into a homogeneous background plus small perturbations, i.e.
$g_{\mu\nu}=g_{\mu\nu}^{(0)}+\delta g_{\mu\nu}$ and
$\phi=\phi_{0}+\delta\phi$. We will now fix the gauge of the perturbations,
and work in the so called longitudinal (or Newtonian) gauge. In this gauge, at
first order in the scalar metric perturbations, and assuming no anisotropic
stress components, the corresponding line element is

$ds^{2}=a^{2}(\eta)[-(1+2\Psi)d\eta^{2}+(1-2\Psi)\delta_{i,j}dx^{i}dx^{j}].$
(1)

In these coordinates, the scale factor can be modeled by

$a(\eta)=\frac{a_{0}}{1-e^{a_{0}H_{0}\eta}}$ (2)

We define $\mathcal{H}\equiv a^{\prime}/a$; the prime over variables denotes
derivative with respect to conformal time $\eta$.

The metric degrees of freedom will remain classical because of the
semiclassical gravity approach, these include the background and the
perturbation $\Psi$. In the matter fields sector, the background scalar field
$\phi_{0}$ will be treated also in a classical fashion; however, the perturbed
part $\delta\phi$ will be subjected to quantization. Taking into account that
the CSL theory modifies the Schrödinger equation, it will be convenient to
carry out the quantization in the Schrödinger picture. Therefore, we focus on
finding the total Hamiltonian of the system.

We introduce the (re-scaled) field variable $y=a\delta\phi$. Expanding the
action of the system (i.e. a single scalar field minimally coupled to gravity)
up to second order in the perturbations, one can find the action associated to
$y$. In this way, the second order action is $S^{(2)}=\int
d^{4}x\mathcal{L}^{(2)}$, where

$\mathcal{L}^{(2)}_{y}=\frac{1}{2}\bigg{[}y^{\prime 2}-(\nabla
y)^{2}+\frac{a^{\prime\prime}}{a}y^{2}\bigg{]}.$ (3)

We define the canonical momentum
$p(\textbf{x},\eta)\equiv\partial\mathcal{L}^{(2)}_{y}/\partial y=y^{\prime}$,
in this way, the Hamiltonian density is given by

$\mathcal{H}^{(2)}_{y}=\frac{p^{2}}{2}+\frac{(\nabla
y)^{2}}{2}-\frac{y^{2}}{2}\frac{a^{\prime\prime}}{a}.$ (4)

We now promote the fields $y$ and $p$ to quantum operators satisfying the
following equal time commutator relation

$[\hat{y}(\textbf{x},\eta),\hat{p}(\textbf{y},\eta)]=i\delta(\textbf{x}-\textbf{y}).$
(5)

Our next step is to decompose the field and the conjugated momentum in Fourier
modes. This is justified by the fact that we work with a linear theory and,
hence, all the modes evolve independently. In Fourier space, the total
Hamiltonian corresponding to Eq. (4) takes the form

$\hat{H}=\int_{\mathbb{R}^{3+}}d^{3}k\quad\left[\hat{p}^{*}_{\textbf{k}}\hat{p}_{\textbf{k}}+\hat{y}^{*}_{\textbf{k}}\hat{y}_{\textbf{k}}\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)\right].$
(6)

Furthermore, it will be convenient to work with real variables. In this way,
we separate the canonical variables into their real and imaginary parts, i.e.

$\hat{y}_{\textbf{k}}\equiv\frac{1}{\sqrt{2}}(\hat{y}_{\textbf{k}}^{\text{R}}+i\hat{y}_{\textbf{k}}^{\text{I}}),\qquad\hat{p}_{\textbf{k}}\equiv\frac{1}{\sqrt{2}}(\hat{p}_{\textbf{k}}^{\text{R}}+i\hat{p}_{\textbf{k}}^{\text{I}}).$
(7)

The quantum commutator in Eq. (5), implies

$[\hat{y}_{\textbf{k}}^{s},\hat{p}_{\textbf{q}}^{s^{\prime}}]=i\delta(\textbf{k}-\textbf{q})\delta_{ss^{\prime}}$
(8)

where $s=$R,I and $\delta_{ss^{\prime}}$ is Kronecker’s delta. Using this
separation the Hamiltonian becomes
$\hat{H}=\int_{\mathbb{R}^{3+}}d^{3}k(\hat{H}^{\text{R}}_{\textbf{k}}+\hat{H}^{\text{I}}_{\textbf{k}})$,
with the following definitions

$\hat{H}^{R,I}_{\mathbf{k}}\equiv\frac{(\hat{p}_{\mathbf{k}}^{R,I})^{2}}{2}+\frac{(\hat{y}_{\mathbf{k}}^{R,I})^{2}}{2}\left(k^{2}-\frac{a^{\prime\prime}}{a}\right).$
(9)

In order to apply the CSL model into the EU scenario, we will follow the
approach first introduced in Cañate et al. (2013) for the inflationary regime.
There, it was found that with an appropriate selection of the field collapse
operators and using the corresponding CSL evolution law, it is possible to
attain a “collapse” in the relevant operators corresponding to the Fourier
components of the field. Furthermore, we will assume linearity in the collapse
generating operator, therefore, the reduction mechanism will act on each mode
of the field independently, i.e. there will be no mode mixing because of the
CSL process.

In view of the above, the evolution of the state vector characterizing each
mode of the quantum field as given by the CSL theory is:

$\displaystyle|\Phi_{\textbf{k}}^{\textrm{R,I}},\eta\rangle$ $\displaystyle=$
$\displaystyle\hat{T}\exp\bigg{\\{}-\int_{\tau}^{\eta}d\eta^{\prime}\bigg{[}i\hat{H}_{\textbf{k}}^{\textrm{R,I}}$
(10) $\displaystyle+$
$\displaystyle\frac{1}{4\lambda_{k}}(\mathcal{W}_{\textbf{k}}^{\text{R,I}}(\eta)-2\lambda_{k}\hat{y}_{\textbf{k}}^{\textrm{R,I}})^{2}\bigg{]}\bigg{\\}}|\Phi_{\textbf{k}}^{\textrm{R,I}},\tau\rangle$

where $\hat{T}$ is the time-ordering operator, and $\tau$ denotes the
conformal time at the beginning of the EU regime. Note that the stochastic
field
$\mathcal{W}_{\textbf{k}}=\mathcal{W}_{\textbf{k}}^{\text{R}}+i\mathcal{W}_{\textbf{k}}^{\text{I}}$
depends on k and the conformal time. In other words, it is reasonable to
introduce a stochastic function for each independent degree of freedom given
that we are applying the CSL collapse dynamics to each mode of the field.
Consequently, the stochastic field $\mathcal{W}_{\textbf{k}}(\eta)$ might be
regarded as a Fourier transform on a stochastic spacetime field
$\mathcal{W}(\textbf{x},\eta)$. The probability for the stochastic field is
given by the second main CSL equation: the Probability Rule, i.e.

$P(\mathcal{W}_{\textbf{k}}^{\text{R,I}})d\mathcal{W}_{\textbf{k}}^{\text{R,I}}=\langle\Phi_{\textbf{k}}^{\text{R,I}},\eta|\Phi_{\textbf{k}}^{\text{R,I}},\eta\rangle\prod_{\eta^{\prime}=\tau}^{\eta-d\eta}\frac{d\mathcal{W}_{\textbf{k}}(\eta^{\prime})^{\text{R,I}}}{\sqrt{2\pi\lambda_{k}/d\eta}}.$
(11)

As can be seen from the CSL evolution equation (10), we have chosen the field
variable $\hat{y}_{\textbf{k}}^{\text{R,I}}$ as the collapse generating
operator. Technically, this means that the CSL process will drive the initial
state vector towards an eigenstate of $\hat{y}_{\textbf{k}}^{\text{R,I}}$. The
motivation for this choice is based on the fact that when one implements the
SCG formulation into cosmological perturbation theory, one obtains, at linear
order in Fourier space, the following relation:

$\Psi_{\textbf{k}}+\mathcal{H}^{-1}\Psi_{\textbf{k}}^{\prime}=\sqrt{\frac{\mu}{2}}\frac{\langle\hat{y}_{\textbf{k}}\rangle}{aM_{P}}$
(12)

where $\mu\equiv\phi_{0}^{{}^{\prime}2}/(2\mathcal{H}^{2}M_{P}^{2})$. We
observe in the above expression that the quantum expectation value
$\langle\hat{y}_{\textbf{k}}\rangle$ acts as a source for the curvature
perturbation. This might be interpreted as indicating that the collapse is
tied with some aspect of the quantum matter that “gravitates” (i.e. that would
characterize the interaction between gravitation and matter degrees of
freedom). Moreover, this view is, in principle, consistent with the proposals
by R. Penrose and L. Diosi suggesting that gravity might play a fundamental
role in the so called collapse of the wave function Diosi (1987); Penrose
(1996).

We denote by $\Phi[y,\eta]$ the wave functional characterizing the quantum
state of the field. In Fourier space, the wave functional can be factorized
into mode components
$\Phi[y_{\textbf{k}},\eta]=\Pi_{\textbf{k}}\Phi_{\textbf{k}}^{\textrm{R}}[y_{\textbf{k}}^{\textrm{R}},\eta]\times\Phi_{\textbf{k}}^{\textrm{I}}[y_{\textbf{k}}^{\textrm{I}},\eta]$.

It is known that the ground state of the Hamiltonian (9), characterized by a
wave functional $\Phi^{\text{R,I}}_{0}[y^{\text{R,I}},\eta]$, is a Gaussian.
Also, the Hamiltonian (9) and the CSL evolution equation (10) are quadratic in
both $\hat{y}_{\textbf{k}}^{\textrm{R,I}}$ and
$\hat{p}_{\textbf{k}}^{\textrm{R,I}}$; consequently, the wave functional at
any time can be written in the form:

$\Phi^{\text{R,I}}[y_{\textbf{k}}^{\text{R,I}},\eta]=\exp[-A_{k}(\eta)(y_{\textbf{k}}^{\text{R,I}})^{2}+B_{k}^{\text{R,I}}(\eta)y_{\textbf{k}}^{\text{R,I}}+C_{k}^{\text{R,I}}(\eta)].$
(13)

The initial state of the field $|\Phi_{\textbf{k}}^{\textrm{R,I}},\tau\rangle$
will be the chosen as the standard Bunch-Davies (BD) vacuum. The corresponding
wave functional for the BD vacuum is characterized by the initial conditions

$A_{k}(\tau)=\frac{k}{2},\qquad B_{k}^{\text{R,I}}(\tau)=0,\qquad
C_{k}^{\text{R,I}}(\tau)=0.$ (14)

###  II.2 Power spectrum

After having introduced the theoretical basis of our model, here we focus on
deriving a prediction for the primordial spectrum, which is the observational
quantity of interest. The standard expression of the primordial spectrum,
associated to the curvature perturbation, is normally expressed in the so-
called comoving gauge, while our main equations were obtained in the
longitudinal gauge. For a single scalar field, the relation between the
curvature perturbation in the comoving gauge $\mathcal{R}$ and in the
longitudinal gauge $\Psi$ is given by Mukhanov and Chibisov (1981); Mukhanov
(2005)

$\mathcal{R}=\Psi\left(1+\frac{1}{\mu}\right)+\frac{\mathcal{H}^{-1}}{\mu}\Psi^{\prime}.$
(15)

Let us note that, in cosmic time $t$ coordinates, $\mu$ can be expressed as
$\mu=\dot{\phi}_{0}^{2}/(2M_{P}^{2}H^{2})$. Given that the system has an
attractor point such that $H\to$ constant and $\dot{\phi}_{0}^{2}$ tends to an
infinitesimal small number, then it follows that $\mu\to 0$. Therefore, in
Fourier space, Eq. (15) implies

$\mathcal{R}_{\textbf{k}}\simeq\frac{1}{\mu}\left(\Psi_{\textbf{k}}+\mathcal{H}^{-1}\Psi_{\textbf{k}}^{\prime}\right)=\frac{\langle\hat{y}_{\textbf{k}}\rangle}{aM_{P}\sqrt{2\mu}}$
(16)

where in the last equality we have used our main equation (12).

The scalar power spectrum associated to $\mathcal{R}_{\textbf{k}}$ is defined
as

$\overline{\mathcal{R}_{\textbf{k}}\mathcal{R}^{*}_{\textbf{q}}}\equiv\frac{2\pi^{2}}{k^{3}}\mathcal{P}_{s}(k)\delta(\textbf{k}-\textbf{q})$
(17)

where $\mathcal{P}_{s}(k)$ is the dimensionless power spectrum. The bar
appearing in (17) denotes an ensemble average over possible realizations of
the stochastic field $\mathcal{R}_{\textbf{k}}$. In the CSL model, each
realization will be associated to a particular realization of the stochastic
process characterizing the collapse. We can use approximation (16) to compute
the spectrum associated to $\mathcal{R}_{\textbf{k}}$, i.e.

$\overline{\mathcal{R}_{\textbf{k}}\mathcal{R}_{\textbf{q}}^{*}}=\frac{\mathcal{H}^{2}}{a^{2}\phi_{0}^{\prime
2}}\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}.$
(18)

From definition (17) and Eq. (18), we can identify an equivalent scalar power
spectrum as:

$\mathcal{P}_{s}(k)\delta(\textbf{k}-\textbf{q})=\frac{k^{3}\mathcal{H}^{2}}{2\pi^{2}a^{2}\phi_{0}^{\prime
2}}\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}.$
(19)

In terms of the Real and Imaginary parts of $\hat{y}_{\textbf{k}}$, the
ensemble average in (19) is

$\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}=(\overline{\langle\hat{y}_{\textbf{k}}^{\textrm{R}}\rangle^{2}}+\overline{\langle\hat{y}_{\textbf{k}}^{\textrm{I}}\rangle^{2}})\delta(\textbf{k}-\textbf{q}).$
(20)

Furthermore,
$\overline{\langle\hat{y}_{\textbf{k}}^{\textrm{R}}\rangle^{2}}=\overline{\langle\hat{y}_{\textbf{k}}^{\textrm{I}}\rangle^{2}}$,
thus we will omit the indexes R,I from now on.

Using the main equations of the CSL model, Eqs. (10) and (11) one obtains:

$\overline{\langle\hat{y}_{\textbf{k}}\rangle^{2}}=\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}-\frac{1}{\textrm{Re}[A_{k}(\eta)]}.$
(21)

Substituting Eqs. (20) and (21) into Eq. (19), we find that the power spectrum
can be expressed as:

$\mathcal{P}_{s}(k)=\frac{k^{3}\mathcal{H}^{2}}{2\pi^{2}a^{2}\phi_{0}^{\prime
2}}\left(\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}-\frac{1}{4\textrm{Re}[A_{k}(\eta)]}\right).$
(22)

We observe that the prediction for the power spectrum depends on the terms
$\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}$ and
$(\textrm{Re}[A_{k}(\eta)])^{-1}$, which can be obtained from the CSL
equations.

The quantity $(\textrm{Re}[A_{k}(\eta)])^{-1}$ represents the variance of the
field variable, which in turn is related to the width of the wave functional
(13). The evolution equation for this quantity can be found by taking the time
derivative of (10), and applying the resulting operator to the wave functional
(13). Then, regrouping terms of order $y^{2}$, $y^{1}$ and $y^{0}$, the
evolution equations corresponding to these terms become decoupled. In
particular, the evolution equation associated to $y^{2}$ only contains
$A_{k}(\eta)$, so it decouples from the other variables $B_{k}(\eta)$ and
$C_{k}(\eta)$. The evolution equation is then

$A_{k}^{\prime}=-2iA_{k}^{2}+\frac{i}{2}\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)+\lambda_{k}.$
(23)

Performing the change of variable $A_{k}\equiv f^{\prime}/(2if)$, Eq. (23) can
be expressed as

$f^{\prime\prime}+\left(q^{2}-\frac{a^{\prime\prime}}{a}\right)f=0$ (24)

where:

$q^{2}\equiv k^{2}\left(1-2i\frac{\lambda_{k}}{k^{2}}\right).$ (25)

The solution to Eq. (24), which satisfies the BD initial condition
corresponding to $A_{k}(\tau)=k/2$, is

$f=\frac{e^{-iq\eta}}{\sqrt{2k}(1-e^{a_{0}H_{0}\eta})}\>\>_{2}F_{1}(q_{-},q_{+},b;e^{a_{0}H_{0}\eta})$
(26)

where $\>{}_{2}F_{1}$ is the hypergeometric function, and

$q_{\pm}\equiv-1-\frac{iq}{a_{0}H_{0}}\pm\left[1-\left(\frac{q}{a_{0}H_{0}}\right)^{2}\right]^{1/2}$
(27) $b\equiv 1-\frac{2iq}{a_{0}H_{0}}.$ (28)

With solution $f$, one can return to the original variable $A_{k}$ and obtain
the sought quantity $(\textrm{Re}[A_{k}(\eta)])^{-1}$. As a matter of fact,
one has

$\textrm{Re}[A_{k}(\eta)]=\frac{W}{|f|^{2}4i}$ (29)

where $W\equiv f^{\prime}f^{*}-f^{\prime*}f$ is the corresponding Wronskian.
Note that if $\lambda_{k}=0$, then $W=i$ for all $\eta$, and $q=k$.

The other important term in the power spectrum is
$\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}$. In order to find this
quantity, it will be useful to define the following objects:
$Q\equiv\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}$,
$R\equiv\overline{\langle\hat{p}_{\textbf{k}}^{2}\rangle}$ and
$S\equiv\overline{\langle\hat{p}_{\textbf{k}}\hat{y}_{\textbf{k}}+\hat{y}_{\textbf{k}}\hat{p}_{\textbf{k}}\rangle}$.
The evolution equations for $Q,R$ and $S$ obtained from the CSL equations are:

$Q^{\prime}=S,\enskip R^{\prime}=-w_{k}(\eta)S+\lambda_{k},\enskip
S^{\prime}=2R-2Qw_{k}(\eta)$ (30)

with $w_{k}(\eta)\equiv k^{2}-a^{\prime\prime}/a$. Therefore, we have a linear
system of coupled differential equations, whose general solution is a
particular solution to the system plus a solution to the homogeneous equation
(with $\lambda_{k}=0$). In this way, the solution can be written as:

$Q(\eta)=C_{1}y_{1}^{2}+C_{2}y_{2}^{2}+C_{3}y_{1}y_{2}+Q_{p}$ (31)

where the constants $C_{1},C_{2}$ and $C_{3}$ are found by imposing the
initial conditions corresponding to the Bunch-Davies vacuum state:
$Q(\tau)=1/2k,R(\tau)=k/2$, $S(\tau)=0$. The functions $y_{1}$ and $y_{2}$ are
two linearly independent solutions of $y^{\prime\prime}=-w_{k}y$, and the
function $Q_{p}$ is a particular solution of

$Q_{p}^{\prime\prime\prime}+4w_{k}Q_{p}^{\prime}+2w_{k}^{\prime}Q_{p}=2\lambda_{k}.$
(32)

The exact solutions $y_{1}$ and $y_{2}$ are

$y_{1}(\eta)=\frac{e^{-ik\eta}}{\sqrt{2k}(1-e^{a_{0}H_{0}\eta})}\>\>_{2}F_{1}(k_{-},k_{+},b;e^{a_{0}H_{0}\eta})$
(33)

and $y_{2}=y_{1}^{*}$, also $k_{\pm}$ and $b$ are defined in the same manner
as in (27) (28) but replacing $q\to k$.

On the other hand, the exact solution of Eq. (32) is difficult to find, but we
can find approximate solutions in the regimes of interest. In particular, we
are interested in the static regime, which corresponds also to the regime
where the BD initial conditions are imposed. The other regime involved is the
de Sitter phase, where the power spectrum is evaluated for the purpose of
comparing it with the standard prediction. Therefore, in the static regime
$w_{k}\simeq k^{2}$ while in the de Sitter phase $w_{k}\simeq
k^{2}-2/\eta^{2}$. It is remarkable that, in these two regimes, $Q_{p}$ can be
approximated by the same solution, i.e.

$Q_{p}(\eta)\simeq\frac{\lambda_{k}\eta}{2k^{2}}.$ (34)

Thus, the constants obtained from imposing the initial conditions are,

$C_{1}=\frac{-i\lambda_{k}}{4k^{2}}e^{2ik\tau},\qquad C_{2}=C_{1}^{*},\qquad
C_{3}=1-\frac{\lambda_{k}\tau}{k}.$ (35)

Now that we have all the elements needed for obtaining the power spectrum
(22), it is straightforward to check first that if $\lambda_{k}=0$ then
$\mathcal{P}_{s}=0$, because $Q(\eta)=(4\textrm{Re}[A_{k}(\eta)])^{-1}$
exactly in that case. This result is also consistent with our view in which,
if there is no collapse, then the metric perturbations are zero, i.e. there
are no inhomogeneities in the spacetime.

On the other hand, considering the modes in the super-Hubble limit ($-k\eta\to
0$), the power spectrum (22) can be approximated by

$\mathcal{P}_{s}(k)=A_{s}\chi^{2}|F(\chi)|^{2}C(k)$ (36)

where

$A_{s}\equiv\frac{H_{0}^{4}}{4\pi^{2}\dot{\phi}_{0}^{2}}$ (37)
$\chi\equiv\frac{k}{a_{0}H_{0}}$ (38)
$F(\chi)\equiv\frac{2\Gamma(1-2i\chi)}{\Gamma(2-i\chi-\sqrt{1-\chi^{2}})\Gamma(2-i\chi+\sqrt{1-\chi^{2}})}$
(39) $C(k)\simeq 1+\frac{\lambda_{k}|\tau|}{k}+\frac{\lambda_{k}}{k^{2}}\sin
2\delta$ (40)
$\delta\equiv\arctan\left(\frac{\text{Im}F}{\text{Re}F}\right)-\chi
a_{0}|\tau|.$ (41)

Note that in the definition of the amplitude $A_{s}$, we have used that the
quantity $\mathcal{H}^{2}/\phi_{0}^{\prime 2}=H^{2}/\dot{\phi}_{0}^{2}$ tends
to a constant, given the existence of the aforementioned attractor point for
the dynamical background variables in the limit $-k\eta\to 0$.

Thus, we have found the main prediction of this section, namely the primordial
scalar power spectrum originated by the CSL mechanism within the EU model.

##  III Results

In this section, we shall proceed to examine the observational effects of
implementing the CSL model in the EU scenario. It will be useful to set as a
reference model the one described in Labraña (2015), we will refer to it as
the original model.

Furthermore, as also argued in Labraña (2015), we can generalize the power
spectrum obtained in order to include the small scale dependence normally
associated with the scalar spectral index $n_{s}$. Thus, for the present
section we will use the following expression for the primordial power spectrum
(PPS):

$\mathcal{P}_{s}(k)=A_{s}\chi^{2}|F(\chi)|^{2}C(k)\left(\frac{k}{k_{P}}\right)^{n_{s}-1}$
(42)

where $k_{P}$ is a pivot scale, which is traditionally set as $k_{P}=0.05$
Mpc-1

The next step in our analysis is to introduce a parameterization of
$\lambda_{k}$, in order to explore the possible observational features of our
model. We propose the linear parameterization in $k$ given by

$\lambda_{k}=\lambda_{0}\,(k+B)$ (43)

with $\lambda_{0}$ acting as a proportionality constant plus a parameter
$B\geq 0$. In fact, if $B=0$, we recover a very similar expression for the PPS
as the one obtained in the original model. In this way, $B$ quantifies small
deviations from the original model reflecting the inclusion of the CSL model.
A very similar parameterization was also implemented in a recent work
involving the CSL proposal during inflation Leon and Piccirilli (2020). Also,
we will fix the value of the proportionality constant as
$\lambda_{0}=10^{-14}$ s-1. This choice is motivated by the fact that such a
value is within the range allowed by laboratory experiments testing non-
relativistic versions of the CSL model Piscicchia et al. (2017), where
$\lambda_{0}$ corresponds to the CSL parameter for these kind of models. In
the units used in the present paper, the former choice is equivalent to
$\lambda_{0}=1.029\,{\rm Mpc^{-1}}$. Given that $k$ has also units of ${\rm
Mpc^{-1}}$, the constant $B$ has units of ${\rm Mpc^{-1}}$ too. For ease of
notation, from now on we will neglect the units of these quantities with the
understanding that the corresponding units have been well established.

At this point, we would like to discuss the following issue. Numerical
calculations set a restriction for implementing the exact formula (42). That
is, as long as the value of $k$ increases, the Gamma functions become
exponentially small beyond the capability of machine representation. This
demands a cutoff value $k_{max}$, which in principle we set it to $0.015$, to
be consistent with the value chosen in the original model Labraña (2015). On
the other hand, we can also approximate expression (42), obtaining

$\mathcal{P}_{s}(k)\simeq
A_{s}\frac{\chi^{2}}{(1+\chi)^{2}}\frac{\lambda_{k}\tau}{k}\left(\frac{k}{k_{P}}\right)^{n_{s}-1}.$
(44)

The advantage of this approximation is that no numerical restrictions are
imposed, hence it can be used in the numerical calculations up to the end of
the observable window $k=1$.

Consequently, we have two options for performing the analysis. The first one
is to use the exact expression for the PPS (42) up to $k_{max}$, and then
perform an analytic continuation such that it approaches smoothly to the
standard expression corresponding to the canonical model, i.e. to the PPS of
the standard cosmological model. The second option we can consider is to use
the approximate formula (44) for the whole $k$ range. Both alternatives can be
seen in Fig. 1, which shows the PPS for different values of the $B$ parameter
comparing both the exact and approximate expressions. After an exhaustive
exploration, we find that the values $10^{-3}$ and $10^{-4}$ are
representative to show the behavior of the power spectrum curve. For each
case, both calculations, exact (solid line) and approximate (dashed line), are
shown.

Moreover, we will include in each Figure a plot called canonical model, which
is used as a second reference (in addition to the original model). As its name
suggests, the canonical model corresponds to the standard $\Lambda$CDM
cosmological model, with the cosmological parameters determined by latest data
from Planck collaboration Aghanim et al. (2020b). In particular, we focus on
the angular spectrum corresponding to the temperature and E–mode polarization
auto-correlation and cross correlation functions. These data leads to the
following set of parameters at the $68\%$ confidence level:
$\Omega_{b}h^{2}=0.02236$, $\Omega_{c}h^{2}=0.1202$, $H_{\rm
today}=67.27\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $A_{s}=2.101\,\times\,10^{-9}$,
$n_{s}=0.9649$, $\tau_{d}=0.0544$ (called the optical depth parameter), and
considering no running of the scalar spectral index.

Figure 1: Primordial power spectrum for the EU model including the CSL
proposal. Here we set $a_{0}H_{0}=2\times 10^{-4}$ from Labraña (2015) and
consider different values of the $B$ parameter (units in $\rm Mpc^{-1}$).
Solid line represents the exact formula, while dashed line corresponds to the
approximate expression. The canonical model is also plotted as a reference.
The PPS which results from the CSL model tends to the canonical model for some
scale between $0.01<k<0.1$. The green lines correspond to $B=0$, which is
essentially the original model.

The canonical model plot will be useful not only as a guide to quickly spot
the novel features introduced by our model, but also to confirm the fact that,
for given a value of $a_{0}H_{0}$, a certain scale $k_{max}$ can be found
where the PPS meets naturally the standard $\Lambda$CDM model. This was the
spirit in which the value of $k_{max}=0.015$ was adopted, as mentioned above.

The next step is to analyze the effects of varying the model parameters in the
angular power spectrum. To accomplish this, we perform the corresponding
modifications in the Code for Anisotropies in the Microwave Background (CAMB)
software Lewis et al. (2000). Figure 2 depicts the resulting angular spectrum
for different values of $B$.

In order to have a better visualization on the impact our model may have on
observational signatures, we show the best–fit $\Lambda$CDM prediction for the
angular spectrum and the corresponding Planck data444These data points
correspond to R3.01 baseline Planck TT, TE, EE+lowE+lensing for multipoles
between $2<l<2508$. Considering lensing effects in our calculations did not
show any difference in the result. Therefore, we consider that these data are
adequate for making comparisons with our model. together with their error bars
(shown in blue). The effect of varying $B$ is mainly seen in the low
multipoles ($l<50$). If $B$ tends to zero, the CSL spectrum approaches to the
one of the original model, and at the same time, it separates from the
canonical model at the bottom of the graph. On the contrary, if $B$ is
increased, the CSL spectrum splits from the canonical model at the top part of
the plot corresponding to the lowest multipoles.

The previous analysis indicates that our model has the potential to exhibit
different features in the low–$l$ range, approaching the $\Lambda$CDM angular
spectrum from upwards or below, and having the original model as the lower
limit. In other words, including the CSL mechanism in the EU model could
result not only in a suppression of the temperature anisotropies of the CMB at
large angular scales, but also in an excess.

Figure 2: Angular power spectrum of the temperature anisotropy auto-
correlation function. The canonical model is represented in orange. The solid
and dashed lines correspond to the exact and approximated expressions for the
PPS obtained from the CSL model, respectively. Here we considered
$a_{0}H_{0}=2\times 10^{-4}$, and three values for $B=0,10^{-4},10^{-3}$. The
plot depicts the behaviour of our model’s predictions, progressively
separating from the original model ($B=0$) and approaching to the canonical
model either from above or below. The effect is mainly observed at the lowest
multipoles, while larger multipoles are unaffected. Planck’s data and error
bars are shown in order to provide some intuition of the effects the model
might bring in when fitting observational data.

###  III.1 Further parameter space exploration

From the previous discussion, it is clear that the implementation of the CSL
model into the EU scenario could be in good agreement with observational data.
However, it is also important to check whether one can, in principle, truly
distinguish the predictions between the canonical model and the one proposed
in this work. Recall that we have introduced a new parameter $B$, while the
combination $a_{0}H_{0}$ comes from the original model. Therefore, we are
interested in testing the robustness of $B$ and $a_{0}H_{0}$ in the predicted
angular spectrum. In other words, we will vary $B$ and $a_{0}H_{0}$ enough to
see how much our predicted spectrum deviates from the standard one.

At first glance, one could argue that any well fitted value of $B$ would make
the model consistent with observations, because any new features introduced in
the spectrum could remain masked under the so called cosmic variance. However,
exploring a wide range of the $B$ parameter space, shows that, even though
highest multipoles are unaffected, the first acoustic peak is half missed for
increasing values of $B$, see Fig. 3. This suggests that $B$ has an upper
limit and its value can be constrained with observational data; therefore, the
model has predictability.

Figure 3: Exploring a wide range of $B$ values clearly shows that not any
value would fit observational data (even when our model mostly affects the
lowest multipoles, which are masked by the cosmic variance). Different $B$
values of order $10^{-3}$ are tested, fixing $a_{0}H_{0}=2\times 10^{-4}$. The
solid line represents the prediction from the CSL model (exact formula). The
dashed black line depicts the prediction from the canonical model. Increasing
the value of $B$ shows a progressive departure from the standard prediction
and the data. This implies an excess in the CMB angular power spectrum
anisotropies.

Up to this point, we have worked with the same fixed value of $a_{0}H_{0}$ as
the one considered in the original model Labraña (2015). Henceforth, it is
interesting to explore the possibility to regard the combination $a_{0}H_{0}$
as a free parameter. In Fig. 4, we vary $a_{0}H_{0}$ along four orders of
magnitude. Increasing $a_{0}H_{0}$, implies a strong suppression in the
angular power spectrum. This fact indicates that not any value of $a_{0}H_{0}$
would make the model consistent with the data. Furthermore, it also indicates
that the suppression observed in the original model could be in part explained
by a particular combination of the cosmological parameters that has been
chosen there.

Figure 4:  Four different orders of magnitude corresponding to $a_{0}H_{0}$
are explored, with fixed $B=10^{-4}$ (which has shown good compatibility with
data). The highest values show a strong suppression in the angular power
spectrum not compatible with observational data. This indicates that
$a_{0}H_{0}$ plays the role of a free parameter and could be estimated with
statistical analysis. Dashed black line represents the canonical model.
Planck’s data are shown in blue points along with their error bars.

The final issue we want to address is: given a particular value of
$a_{0}H_{0}$ that is not compatible with observational data, could it be
compensated in some way by varying the $B$ parameter introduced by the CSL
model? Figure 5 encompasses this question by plotting different values of $B$
for a fixed $a_{0}H_{0}=2\times 10^{-2}$ (which has been previously seen not
compatible with observational data). Even though the variation of $B$
introduces new features and the model predictions might approach to Planck’s
data at one end, other sectors of the data are highly missed. Consequently, we
can safely state that the $B$ parameter cannot compensate the suppression
induced by a non-favoured value of $a_{0}H_{0}$.

Figure 5: Different values for the $B$ parameter are explored considering
$a_{0}H_{0}=2\times 10^{-2}$ which has proven no good fit with data. The plot
shows that $B$ is a free parameter which can be constrained and is well
restricted by the data, but does not compensate the suppression given by the
choice of $a_{0}H_{0}$. For higher $B$ values, not only the lowest multipoles
are missed, but also the first acoustic peak fails to be matched. In fact,
this constitutes a promising result for accurate parameter estimation with
observational data under statistical analysis. The canonical model is shown in
point–dashed black line for reference.

##  IV Conclusions

In recent years, the _emergent universe_ (EU) has been studied from different
perspectives as a viable cosmological model, which could not only avoid the
horizon problem and the initial singularity of the standard approach, but
could also account for some anomalies in the observations of the CMB at large
angular scales.

Motivated by the recent results of Labraña (2015), where it was shown that a
phase of super-inflation prior to that of standard slow-roll inflation
(originated in the context of the emergent universe presented in Ellis and
Maartens (2004); Ellis et al. (2004)) could explain the power suppression
observed in the low multipoles of the CMB, we decided to explore in this work
the same emergent universe but within the framework of a particular objective
collapse theory, known as Continuous Spontaneous Localization (CSL). The CSL
theory has been studied with encouraging results in cosmological applications
for several years, as a mechanism to break the original symmetries of the
quantum vacuum state of the fields, generating the primordial cosmological
perturbations and giving an explanation to the quantum-to-classical transition
of such perturbations.

We obtained a prediction for the primordial power spectrum. The computed
spectrum can be consistent with CMB observational data, if a particular
parameterization for the collapse rate $\lambda_{k}$ is assumed. The
parametrization we considered was also proposed and analyzed in previous works
but in the framework of standard inflation. Such a parameterization introduces
an extra free parameter $B$, in addition to the parameter $a_{0}H_{0}$ of the
emergent model studied in Labraña (2015). We also found that the predictions
and results of such a work can be recovered, when $B=0$ is chosen.

From the analysis in Sec. III, we have found that implementing the CSL
collapse proposal to the emergent universe scenario introduces extra
modifications at the lowest multipoles. Specifically, through the CSL-
parameter $B$, the angular spectrum in the low multipoles sector, exhibits a
suppression or an increment. This is a different feature from what is
generically produced in models with a super-inflation phase which only
decrease the curve spectrum at large angular scales. On the other hand, in the
model proposed in this work, whether there is an excess or suppression, will
be determined by a combination of the parameters derived from the original
model and the novel one introduced by the collapse rate parameterization. This
fact enables our model to introduce some new features in the angular spectrum,
particularly in the sector of interest where the so-called low–$l$ anomaly is
located.

On the other hand, increasing the value of the combination $a_{0}H_{0}$
produces a suppression in the lowest multipoles. We have also seen that the
variation of the $B$ parameter produces opposite (and similiar) effects to
$a_{0}H_{0}$. Nevertheless, a suppression of the angular spectrum given by
$a_{0}H_{0}$ cannot be compensated with an increase of $B$ .

One important aspect of our model is that the primordial power spectrum
obtained smoothly approaches to the one from the standard $\Lambda\rm CDM$
model within the observational range of interest. This characteristic
constitutes a natural modification of the (temperature) angular spectrum,
affecting mainly the lowest multipoles without altering the highest ones
(which are already constrained by observations to a high degree of accuracy).

Exploration of the free parameters shows that not any value will make the
proposed model consistent with observational data. This suggests that there
are good opportunities of effectively constraining the parameter space with
the full machinery of statistical analysis based on Monte Carlo–Markov chains
for cosmology. Such a scenario gives good predictability to the CSL collapse
proposal in the emergent universe model. In fact, we expect that the extra
free parameter $B$ should take a small value (between $10^{-3}$ and $10^{-4}$)
but not centered at $B=0$, which clearly distinguishes our proposal from the
one explored in Labraña (2015).

###### Acknowledgements.

G.R.B. is supported by CONICET (Argentina) and he acknowledges support from
grant PIP 112-2017-0100220CO of CONICET (Argentina). G.L. and M.P.P. are
supported by CONICET (Argentina), they also acknowledge support from the
following project grants: Universidad Nacional de La Plata I+D G175 and PIP
112-2020-0100729CO of CONICET (Argentina). M.P.P. thanks Susana Landau for
useful discussions regarding the impact of non–standard cosmological models on
CMB angular power spectrum. The authors acknowledge the use of the
supercluster MIZTLI of UNAM through project LANCAD-UNAM-DGTIC-132 and thank
the people of DGTIC-UNAM for technical and computational support.

## References

  * Ade et al. (2016a) P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A13 (2016a), eprint 1502.01589.
  * Aghanim et al. (2020a) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A1 (2020a), eprint 1807.06205.
  * Aghanim et al. (2020b) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020b), eprint 1807.06209.
  * Scolnic et al. (2018) D. M. Scolnic et al., Astrophys. J. 859, 101 (2018), eprint 1710.00845.
  * Alam et al. (2021) S. Alam et al. (eBOSS), Phys. Rev. D 103, 083533 (2021), eprint 2007.08991.
  * Starobinsky (1979) A. A. Starobinsky, JETP Lett. 30, 682 (1979).
  * Starobinsky (1980) A. A. Starobinsky, Phys. Lett. B91, 99 (1980).
  * Guth (1981) A. H. Guth, Phys. Rev. D23, 347 (1981).
  * Mukhanov and Chibisov (1981) V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532 (1981), [Pisma Zh. Eksp. Teor. Fiz.33,549(1981)].
  * Linde (1982) A. D. Linde, Phys. Lett. B108, 389 (1982).
  * Linde (1983) A. D. Linde, Phys. Lett. B 129, 177 (1983).
  * Albrecht and Steinhardt (1982) A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
  * Bardeen et al. (1983) J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D28, 679 (1983).
  * Brandenberger (1984) R. H. Brandenberger, Nuclear Physics B 245, 328 (1984).
  * Hawking (1982) S. W. Hawking, Phys. Lett. 115B, 295 (1982).
  * Akrami et al. (2020a) Y. Akrami et al. (Planck), Astron. Astrophys. 641, A10 (2020a), eprint 1807.06211.
  * Hinshaw et al. (1996) G. Hinshaw, A. J. Banday, C. L. Bennett, K. M. Gorski, A. Kogut, C. H. Lineweaver, G. F. Smoot, and E. L. Wright, Astrophys. J. 464, L25 (1996), eprint astro-ph/9601061.
  * Spergel et al. (2003) D. N. Spergel et al. (WMAP), Astrophys. J. Suppl. 148, 175 (2003), eprint astro-ph/0302209.
  * Spergel et al. (2007) D. N. Spergel et al. (WMAP), Astrophys. J. Suppl. 170, 377 (2007), eprint astro-ph/0603449.
  * Larson et al. (2011) D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu, M. R. Nolta, C. L. Bennett, B. Gold, M. Halpern, R. S. Hill, N. Jarosik, et al., Astrophys. J. Suppl. 192, 16 (2011), eprint 1001.4635.
  * Komatsu et al. (2011) E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page, et al., Astrophys. J. Suppl. 192, 18 (2011), eprint 1001.4538.
  * Ade et al. (2014a) P. A. R. Ade et al. (Planck), Astron. Astrophys. 571, A1 (2014a), eprint 1303.5062.
  * Ade et al. (2014b) P. A. R. Ade et al. (Planck), Astron. Astrophys. 571, A15 (2014b), eprint 1303.5075.
  * Ade et al. (2016b) P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A16 (2016b), eprint 1506.07135.
  * Akrami et al. (2020b) Y. Akrami et al. (Planck), Astron. Astrophys. 641, A7 (2020b), eprint 1906.02552.
  * Copi et al. (2007) C. Copi, D. Huterer, D. Schwarz, and G. Starkman, Phys. Rev. D75, 023507 (2007), eprint astro-ph/0605135.
  * Copi et al. (2009) C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, Mon. Not. Roy. Astron. Soc. 399, 295 (2009), eprint 0808.3767.
  * Copi et al. (2010) C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, Adv. Astron. 2010, 847541 (2010), eprint 1004.5602.
  * Copi et al. (2015) C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, Mon. Not. Roy. Astron. Soc. 451, 2978 (2015), eprint 1310.3831.
  * Sarkar et al. (2011) D. Sarkar, D. Huterer, C. J. Copi, G. D. Starkman, and D. J. Schwarz, Astropart. Phys. 34, 591 (2011), eprint 1004.3784.
  * Efstathiou et al. (2010) G. Efstathiou, Y.-Z. Ma, and D. Hanson, Mon. Not. Roy. Astron. Soc. 407, 2530 (2010), eprint 0911.5399.
  * Rakic et al. (2006) A. Rakic, S. Rasanen, and D. J. Schwarz, Mon. Not. Roy. Astron. Soc. 369, L27 (2006), eprint astro-ph/0601445.
  * Francis and Peacock (2010) C. L. Francis and J. A. Peacock, Mon. Not. Roy. Astron. Soc. 406, 14 (2010), eprint 0909.2495.
  * Schwarz et al. (2016) D. J. Schwarz, C. J. Copi, D. Huterer, and G. D. Starkman, Class. Quant. Grav. 33, 184001 (2016), eprint 1510.07929.
  * Perivolaropoulos and Skara (2021) L. Perivolaropoulos and F. Skara (2021), eprint 2105.05208.
  * Sokolov (1993) I. Y. Sokolov, Soviet Journal of Experimental and Theoretical Physics Letters 57, 617 (1993).
  * Stevens et al. (1993) D. Stevens, D. Scott, and J. Silk, Phys. Rev. Lett. 71, 20 (1993).
  * Rocha et al. (2004) G. Rocha, L. Cayón, R. Bowen, A. Canavezes, J. Silk, A. J. Banday, and K. M. Górski, Monthly Notices of the Royal Astronomical Society 351, 769 (2004).
  * Campanelli et al. (2007) L. Campanelli, P. Cea, and L. Tedesco, Phys. Rev. D 76, 063007 (2007).
  * Bastero-Gil et al. (2003) M. Bastero-Gil, K. Freese, and L. Mersini-Houghton, Phys. Rev. D 68, 123514 (2003), eprint hep-ph/0306289.
  * Contaldi et al. (2003) C. R. Contaldi, M. Peloso, L. Kofman, and A. D. Linde, JCAP 07, 002 (2003), eprint astro-ph/0303636.
  * Bridle et al. (2003) S. L. Bridle, A. M. Lewis, J. Weller, and G. Efstathiou, Monthly Notices of the Royal Astronomical Society 342, L72 (2003).
  * Feng and Zhang (2003) B. Feng and X. Zhang, Phys. Lett. B 570, 145 (2003), eprint astro-ph/0305020.
  * Piao et al. (2004) Y.-S. Piao, B. Feng, and X. Zhang, Phys. Rev. D 69, 103520 (2004).
  * Boyanovsky et al. (2006) D. Boyanovsky, H. J. de Vega, and N. G. Sanchez, Phys. Rev. D 74, 123007 (2006), eprint astro-ph/0607487.
  * Cao et al. (2008) F. J. Cao, H. J. de Vega, and N. G. Sanchez, Phys. Rev. D 78, 083508 (2008), eprint 0809.0623.
  * Destri et al. (2008) C. Destri, H. J. de Vega, and N. G. Sanchez, Phys. Rev. D 78, 023013 (2008), eprint 0804.2387.
  * Destri et al. (2010) C. Destri, H. J. de Vega, and N. G. Sanchez, Phys. Rev. D 81, 063520 (2010), eprint 0912.2994.
  * Handley et al. (2014) W. J. Handley, S. D. Brechet, A. N. Lasenby, and M. P. Hobson, Phys. Rev. D 89, 063505 (2014), eprint 1401.2253.
  * Gessey-Jones and Handley (2021) T. Gessey-Jones and W. J. Handley, Phys. Rev. D 104, 063532 (2021), eprint 2104.03016.
  * Ramirez and Schwarz (2012) E. Ramirez and D. J. Schwarz, Phys. Rev. D 85, 103516 (2012), eprint 1111.7131.
  * Ramirez (2012) E. Ramirez, Phys. Rev. D 85, 103517 (2012).
  * Biswas and Mazumdar (2014) T. Biswas and A. Mazumdar, Class. Quant. Grav. 31, 025019 (2014), eprint 1304.3648.
  * Liu et al. (2013) Z.-G. Liu, Z.-K. Guo, and Y.-S. Piao, Phys. Rev. D 88, 063539 (2013).
  * Liu et al. (2014) Z.-G. Liu, Z.-K. Guo, and Y.-S. Piao, Eur. Phys. J. C 74, 3006 (2014), eprint 1311.1599.
  * Ashtekar et al. (2020) A. Ashtekar, B. Gupt, D. Jeong, and V. Sreenath, Phys. Rev. Lett. 125, 051302 (2020), eprint 2001.11689.
  * Agullo et al. (2021a) I. Agullo, D. Kranas, and V. Sreenath, Gen. Rel. Grav. 53, 17 (2021a), eprint 2005.01796.
  * Agullo et al. (2021b) I. Agullo, D. Kranas, and V. Sreenath, Class. Quant. Grav. 38, 065010 (2021b), eprint 2006.09605.
  * Tryon (1973) E. P. Tryon, Nature (London) 246, 396 (1973).
  * Vilenkin (1982) A. Vilenkin, Physics Letters B 117, 25 (1982).
  * Ellis (1987) G. F. R. Ellis, Astrophys. J. 314, 1 (1987).
  * Ellis (1988) G. F. R. Ellis, Classical and Quantum Gravity 5, 891 (1988).
  * Ellis et al. (1991) G. F. R. Ellis, D. H. Lyth, and M. B. Mijić, Physics Letters B 271, 52 (1991).
  * Linde (1995) A. D. Linde, Phys. Lett. B 351, 99 (1995), eprint hep-th/9503097.
  * Linde and Mezhlumian (1995) A. D. Linde and A. Mezhlumian, Phys. Rev. D 52, 6789 (1995), eprint astro-ph/9506017.
  * White and Scott (1996) M. J. White and D. Scott, Astrophys. J. 459, 415 (1996), eprint astro-ph/9508157.
  * Linde (1998) A. D. Linde, Phys. Rev. D 58, 083514 (1998), eprint gr-qc/9802038.
  * Linde (1999) A. D. Linde, Phys. Rev. D 59, 023503 (1999), eprint hep-ph/9807493.
  * Ellis et al. (2002) G. F. R. Ellis, W. R. Stoeger, S. J., P. McEwan, and P. Dunsby, Gen. Rel. Grav. 34, 1445 (2002), eprint gr-qc/0109023.
  * Linde (2003) A. D. Linde, JCAP 05, 002 (2003), eprint astro-ph/0303245.
  * Lasenby and Doran (2005) A. Lasenby and C. Doran, Phys. Rev. D 71, 063502 (2005), eprint astro-ph/0307311.
  * Uzan et al. (2003) J.-P. Uzan, U. Kirchner, and G. F. R. Ellis, Mon. Not. Roy. Astron. Soc. 344, L65 (2003), eprint astro-ph/0302597.
  * Zhang and Sun (2004) D.-H. Zhang and C.-Y. Sun, Chinese Physics Letters 21, 1865 (2004).
  * Ratra (2017) B. Ratra, Phys. Rev. D 96, 103534 (2017), eprint 1707.03439.
  * Ooba et al. (2018a) J. Ooba, B. Ratra, and N. Sugiyama, Astrophys. J. 864, 80 (2018a), eprint 1707.03452.
  * Räsänen et al. (2015) S. Räsänen, K. Bolejko, and A. Finoguenov, Phys. Rev. Lett. 115, 101301 (2015), eprint 1412.4976.
  * Park and Ratra (2019a) C.-G. Park and B. Ratra, Astrophys. J. 882, 158 (2019a), eprint 1801.00213.
  * Ryan et al. (2018) J. Ryan, S. Doshi, and B. Ratra, Mon. Not. Roy. Astron. Soc. 480, 759 (2018), eprint 1805.06408.
  * Yu et al. (2018) H. Yu, B. Ratra, and F.-Y. Wang, Astrophys. J. 856, 3 (2018), eprint 1711.03437.
  * Ooba et al. (2018b) J. Ooba, B. Ratra, and N. Sugiyama, Astrophys. J. 864, 80 (2018b), eprint 1707.03452.
  * Park and Ratra (2019b) C.-G. Park and B. Ratra, Astrophys. Space Sci. 364, 134 (2019b), eprint 1809.03598.
  * Ryan et al. (2019) J. Ryan, Y. Chen, and B. Ratra, Mon. Not. Roy. Astron. Soc. 488, 3844 (2019), eprint 1902.03196.
  * Handley (2021) W. Handley, Phys. Rev. D 103, L041301 (2021), eprint 1908.09139.
  * Efstathiou and Gratton (2019) G. Efstathiou and S. Gratton (2019), eprint 1910.00483.
  * Riess et al. (2019) A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. 876, 85 (2019), eprint 1903.07603.
  * Di Valentino et al. (2019) E. Di Valentino, A. Melchiorri, and J. Silk, Nature Astron. 4, 196 (2019), eprint 1911.02087.
  * Di Valentino et al. (2021a) E. Di Valentino, A. Melchiorri, and J. Silk, Astrophys. J. Lett. 908, L9 (2021a), eprint 2003.04935.
  * Efstathiou and Gratton (2020) G. Efstathiou and S. Gratton, Mon. Not. Roy. Astron. Soc. 496, L91 (2020), eprint 2002.06892.
  * Di Valentino et al. (2021b) E. Di Valentino et al., Astropart. Phys. 131, 102607 (2021b), eprint 2008.11286.
  * Benisty and Staicova (2021) D. Benisty and D. Staicova, Astron. Astrophys. 647, A38 (2021), eprint 2009.10701.
  * Vagnozzi et al. (2021a) S. Vagnozzi, E. Di Valentino, S. Gariazzo, A. Melchiorri, O. Mena, and J. Silk, Phys. Dark Univ. 33, 100851 (2021a), eprint 2010.02230.
  * Vagnozzi et al. (2021b) S. Vagnozzi, A. Loeb, and M. Moresco, Astrophys. J. 908, 84 (2021b), eprint 2011.11645.
  * Dhawan et al. (2021) S. Dhawan, J. Alsing, and S. Vagnozzi, Mon. Not. Roy. Astron. Soc. 506, L1 (2021), eprint 2104.02485.
  * Verde et al. (2019) L. Verde, T. Treu, and A. G. Riess, Nature Astron. 3, 891 (2019), eprint 1907.10625.
  * Di Valentino et al. (2021c) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021c), eprint 2103.01183.
  * Bolejko (2018) K. Bolejko, Phys. Rev. D 97, 103529 (2018), eprint 1712.02967.
  * Collett et al. (2019) T. Collett, F. Montanari, and S. Rasanen, Phys. Rev. Lett. 123, 231101 (2019), eprint 1905.09781.
  * Clarkson et al. (2007) C. Clarkson, M. Cortes, and B. A. Bassett, JCAP 08, 011 (2007), eprint astro-ph/0702670.
  * Hlozek et al. (2008) R. Hlozek, M. Cortes, C. Clarkson, and B. Bassett, Gen. Rel. Grav. 40, 285 (2008), eprint 0801.3847.
  * Barenboim et al. (2010) G. Barenboim, E. F. Martínez, O. Mena, and L. Verde, JCAP 03, 008 (2010), eprint 0910.0252.
  * Di Valentino et al. (2021d) E. Di Valentino, A. Melchiorri, O. Mena, S. Pan, and W. Yang, Mon. Not. Roy. Astron. Soc. 502, L23 (2021d), eprint 2011.00283.
  * Penrose (1965) R. Penrose, Phys. Rev. Lett. 14, 57 (1965).
  * Hawking (1965) S. W. Hawking, Phys. Rev. Lett. 15, 689 (1965).
  * Hawking and Penrose (1970) S. W. Hawking and R. Penrose, Proceedings of the Royal Society of London Series A 314, 529 (1970).
  * Hawking and Ellis (2011) S. W. Hawking and G. F. R. Ellis, _The Large Scale Structure of Space-Time_ , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011).
  * Wald (1984) R. M. Wald, _General Relativity_ (Chicago Univ. Pr., Chicago, USA, 1984).
  * Borde and Vilenkin (1994a) A. Borde and A. Vilenkin, Phys. Rev. Lett. 72, 3305 (1994a), eprint gr-qc/9312022.
  * Borde and Vilenkin (1994b) A. Borde and A. Vilenkin, in _8th Nishinomiya-Yukawa Memorial Symposium: Relativistic Cosmology_ (1994b), eprint gr-qc/9403004.
  * Borde (1994) A. Borde, Phys. Rev. D 50, 3692 (1994), eprint gr-qc/9403049.
  * Borde and Vilenkin (1996) A. Borde and A. Vilenkin, Int. J. Mod. Phys. D 5, 813 (1996), eprint gr-qc/9612036.
  * Borde and Vilenkin (1997) A. Borde and A. Vilenkin, Phys. Rev. D 56, 717 (1997), eprint gr-qc/9702019.
  * Guth (2001) A. H. Guth, Annals N. Y. Acad. Sci. 950, 66 (2001), eprint astro-ph/0101507.
  * Borde et al. (2003) A. Borde, A. H. Guth, and A. Vilenkin, Phys. Rev. Lett. 90, 151301 (2003), eprint gr-qc/0110012.
  * Molina-Paris and Visser (1999) C. Molina-Paris and M. Visser, Phys. Lett. B 455, 90 (1999), eprint gr-qc/9810023.
  * Kanekar et al. (2001) N. Kanekar, V. Sahni, and Y. Shtanov, Phys. Rev. D 63, 083520 (2001), eprint astro-ph/0101448.
  * Khoury et al. (2001) J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D 64, 123522 (2001), eprint hep-th/0103239.
  * Steinhardt and Turok (2002) P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 126003 (2002), eprint hep-th/0111098.
  * Battefeld et al. (2004) T. J. Battefeld, S. P. Patil, and R. Brandenberger, Phys. Rev. D 70, 066006 (2004), eprint hep-th/0401010.
  * Bojowald et al. (2004) M. Bojowald, R. Maartens, and P. Singh, Phys. Rev. D 70, 083517 (2004), eprint hep-th/0407115.
  * Biswas et al. (2006) T. Biswas, A. Mazumdar, and W. Siegel, JCAP 03, 009 (2006), eprint hep-th/0508194.
  * Peter and Pinto-Neto (2002) P. Peter and N. Pinto-Neto, Phys. Rev. D 66, 063509 (2002), eprint hep-th/0203013.
  * Battefeld and Peter (2015) D. Battefeld and P. Peter, Phys. Rept. 571, 1 (2015), eprint 1406.2790.
  * Lilley and Peter (2015) M. Lilley and P. Peter, Comptes Rendus Physique 16, 1038 (2015), eprint 1503.06578.
  * Brandenberger and Peter (2017) R. Brandenberger and P. Peter, Found. Phys. 47, 797 (2017), eprint 1603.05834.
  * Matsui et al. (2019) H. Matsui, F. Takahashi, and T. Terada, Phys. Lett. B 795, 152 (2019), eprint 1904.12312.
  * Barrau (2020) A. Barrau, Eur. Phys. J. C 80, 579 (2020), eprint 2005.04693.
  * Ellis and Maartens (2004) G. F. R. Ellis and R. Maartens, Class. Quant. Grav. 21, 223 (2004), eprint gr-qc/0211082.
  * Einstein (1917) A. Einstein, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin pp. 142–152 (1917).
  * Eddington (1930) A. S. Eddington, Mon. Not. Roy. Astron. Soc. 90, 668 (1930).
  * Ellis et al. (2004) G. F. R. Ellis, J. Murugan, and C. G. Tsagas, Class. Quant. Grav. 21, 233 (2004), eprint gr-qc/0307112.
  * Mulryne et al. (2005) D. J. Mulryne, R. Tavakol, J. E. Lidsey, and G. F. R. Ellis, Phys. Rev. D 71, 123512 (2005), eprint astro-ph/0502589.
  * Gibbons (1987) G. W. Gibbons, Nuclear Physics B 292, 784 (1987).
  * Barrow et al. (2003) J. D. Barrow, G. F. R. Ellis, R. Maartens, and C. G. Tsagas, Class. Quant. Grav. 20, L155 (2003), eprint gr-qc/0302094.
  * Mukherjee et al. (2005) S. Mukherjee, B. C. Paul, S. D. Maharaj, and A. Beesham (2005), eprint gr-qc/0505103.
  * Mukherjee et al. (2006) S. Mukherjee, B. C. Paul, N. K. Dadhich, S. D. Maharaj, and A. Beesham, Class. Quant. Grav. 23, 6927 (2006), eprint gr-qc/0605134.
  * Banerjee et al. (2007) A. Banerjee, T. Bandyopadhyay, and S. Chakraborty, Grav. Cosmol. 13, 290 (2007), eprint 0705.3933.
  * Boehmer et al. (2007) C. G. Boehmer, L. Hollenstein, and F. S. N. Lobo, Phys. Rev. D 76, 084005 (2007), eprint 0706.1663.
  * Parisi et al. (2007) L. Parisi, M. Bruni, R. Maartens, and K. Vandersloot, Class. Quant. Grav. 24, 6243 (2007), eprint 0706.4431.
  * del Campo et al. (2007) S. del Campo, R. Herrera, and P. Labrana, JCAP 11, 030 (2007), eprint 0711.1559.
  * Debnath (2008) U. Debnath, Class. Quant. Grav. 25, 205019 (2008), eprint 0808.2379.
  * Banerjee et al. (2008) A. Banerjee, T. Bandyopadhyay, and S. Chakraborty, Gen. Rel. Grav. 40, 1603 (2008), eprint 0711.4188.
  * Beesham et al. (2009) A. Beesham, S. V. Chervon, and S. D. Maharaj, Class. Quant. Grav. 26, 075017 (2009), eprint 0904.0773.
  * del Campo et al. (2009) S. del Campo, R. Herrera, and P. Labrana, JCAP 07, 006 (2009), eprint 0905.0614.
  * Paul and Ghose (2010) B. C. Paul and S. Ghose, Gen. Rel. Grav. 42, 795 (2010), eprint 0809.4131.
  * Paul et al. (2010) B. C. Paul, P. Thakur, and S. Ghose, Mon. Not. Roy. Astron. Soc. 407, 415 (2010), eprint 1004.4256.
  * Zhang et al. (2010) K. Zhang, P. Wu, and H. W. Yu, Phys. Lett. B 690, 229 (2010), eprint 1005.4201.
  * Paul et al. (2011) B. C. Paul, S. Ghose, and P. Thakur, Mon. Not. R. Astron. Soc. 413, 686 (2011), eprint 1101.1360.
  * Chattopadhyay and Debnath (2011) S. Chattopadhyay and U. Debnath, Int. J. Mod. Phys. D 20, 1135 (2011), eprint 1105.1091.
  * del Campo et al. (2011) S. del Campo, E. I. Guendelman, A. B. Kaganovich, R. Herrera, and P. Labrana, Phys. Lett. B 699, 211 (2011), eprint 1105.0651.
  * del Campo et al. (2010) S. del Campo, E. I. Guendelman, R. Herrera, and P. Labrana, JCAP 06, 026 (2010), eprint 1006.5734.
  * Labrana (2012) P. Labrana, Phys. Rev. D 86, 083524 (2012), eprint 1111.5360.
  * Cai et al. (2012) Y.-F. Cai, M. Li, and X. Zhang, Phys. Lett. B 718, 248 (2012), eprint 1209.3437.
  * Rudra (2012) P. Rudra, Mod. Phys. Lett. A 27, 1250189 (2012), eprint 1211.2047.
  * Ghose et al. (2012) S. Ghose, P. Thakur, and B. C. Paul, Mon. Not. R. Astron. Soc. 421, 20 (2012), eprint 1105.3303.
  * Liu and Piao (2013) Z.-G. Liu and Y.-S. Piao, Phys. Lett. B 718, 734 (2013), eprint 1207.2568.
  * Aguirre and Kehayias (2013) A. Aguirre and J. Kehayias, Phys. Rev. D 88, 103504 (2013), eprint 1306.3232.
  * Cai et al. (2014) Y.-F. Cai, Y. Wan, and X. Zhang, Phys. Lett. B 731, 217 (2014), eprint 1312.0740.
  * Atazadeh et al. (2014) K. Atazadeh, Y. Heydarzade, and F. Darabi, Phys. Lett. B 732, 223 (2014), eprint 1401.7638.
  * Zhang et al. (2014) K. Zhang, P. Wu, and H. Yu, JCAP 01, 048 (2014), eprint 1311.4051.
  * Bag et al. (2014) S. Bag, V. Sahni, Y. Shtanov, and S. Unnikrishnan, JCAP 07, 034 (2014), eprint 1403.4243.
  * Huang et al. (2015) Q. Huang, P. Wu, and H. Yu, Phys. Rev. D 91, 103502 (2015), eprint 1504.05284.
  * Böhmer et al. (2015) C. G. Böhmer, N. Tamanini, and M. Wright, Phys. Rev. D 92, 124067 (2015), eprint 1510.01477.
  * Labraña (2015) P. Labraña, Phys. Rev. D 91, 083534 (2015), eprint 1312.6877.
  * Khodadi et al. (2016) M. Khodadi, Y. Heydarzade, F. Darabi, and E. N. Saridakis, Phys. Rev. D 93, 124019 (2016), eprint 1512.08674.
  * Zhang et al. (2016) K. Zhang, P. Wu, H. Yu, and L.-W. Luo, Physics Letters B 758, 37 (2016).
  * Ríos et al. (2016) C. Ríos, P. Labraña, and A. Cid, Journal of Physics: Conference Series 720, 012008 (2016).
  * Khodadi et al. (2018) M. Khodadi, K. Nozari, and E. N. Saridakis, Class. Quant. Grav. 35, 015010 (2018), eprint 1612.09254.
  * Martineau and Barrau (2018) K. Martineau and A. Barrau, Universe 4, 149 (2018), eprint 1812.05522.
  * Labrana and Cossio (2019) P. Labrana and H. Cossio, Eur. Phys. J. C 79, 303 (2019), eprint 1808.09291.
  * Brandenberger (2010) R. H. Brandenberger, PoS ICFI2010, 001 (2010), eprint 1103.2271.
  * Brandenberger (2020) R. H. Brandenberger, _Beyond Standard Inflationary Cosmology_ (2020), eprint 1809.04926.
  * Brandenberger and Vafa (1989) R. Brandenberger and C. Vafa, Nuclear Physics B 316, 391 (1989).
  * Nayeri et al. (2006) A. Nayeri, R. H. Brandenberger, and C. Vafa, Phys. Rev. Lett. 97, 021302 (2006), eprint hep-th/0511140.
  * Brandenberger (2008) R. H. Brandenberger (2008), eprint 0808.0746.
  * Brandenberger (2011) R. H. Brandenberger, Class. Quant. Grav. 28, 204005 (2011), eprint 1105.3247.
  * Obied et al. (2018) G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa (2018), eprint 1806.08362.
  * Bedroya and Vafa (2020) A. Bedroya and C. Vafa, JHEP 09, 123 (2020), eprint 1909.11063.
  * Garg and Krishnan (2019) S. K. Garg and C. Krishnan, JHEP 11, 075 (2019), eprint 1807.05193.
  * Agrawal et al. (2018) P. Agrawal, G. Obied, P. J. Steinhardt, and C. Vafa, Phys. Lett. B 784, 271 (2018), eprint 1806.09718.
  * Brandenberger and Martin (2013) R. H. Brandenberger and J. Martin, Class. Quant. Grav. 30, 113001 (2013), eprint 1211.6753.
  * Bedroya et al. (2020) A. Bedroya, R. Brandenberger, M. Loverde, and C. Vafa, Phys. Rev. D 101, 103502 (2020), eprint 1909.11106.
  * Brandenberger (2021) R. Brandenberger (2021), eprint 2102.09641.
  * Wigner (1963) E. P. Wigner, American Journal of Physics 31, 6 (1963).
  * Omnes (1994) R. Omnes, _The Interpretation of Quantum Mechanics_ (Princeton University Press, 1994).
  * Maudlin (1995) T. Maudlin, Topoi 14, 7 (1995).
  * Becker (2018) A. Becker, _What is Real? The Unfinished Quest for the Meaning of Quantum Physics_ (Basic Books, New York, 2018).
  * Norsen (2017) T. Norsen, _Foundations of Quantum Mechanics_ (Springer International Publishing AG, 2017).
  * Durr and Lazarovici (2020) D. Durr and D. Lazarovici, _Understanding Quantum Mechanics_ (Springer International Publishing AG, 2020).
  * Albert (1994) D. Z. Albert, _Quantum Mechanics and Experience_ , (Harvard University Press) (1994).
  * Okon (2014) E. Okon, Rev. Mex. Fis. E 60, 130 (2014).
  * Bell (1981) J. S. Bell, _Quantum Mechanics for cosmologists_ , Quantum Gravity 2 (eds. Isham, C., Penrose, R. and Sciama, D., Oxford University Press, 1981).
  * Perez et al. (2006) A. Perez, H. Sahlmann, and D. Sudarsky, Class. Quant. Grav. 23, 2317 (2006), eprint gr-qc/0508100.
  * Sudarsky (2011) D. Sudarsky, Int. J. Mod. Phys. D 20, 509 (2011), eprint 0906.0315.
  * Landau et al. (2013) S. Landau, G. León, and D. Sudarsky, Phys. Rev. D 88, 023526 (2013), eprint 1107.3054.
  * Hartle (1993) J. B. Hartle, arXiv e-prints gr-qc/9304006 (1993), eprint gr-qc/9304006.
  * Berjon et al. (2021) J. Berjon, E. Okon, and D. Sudarsky, Phys. Rev. D 103, 043521 (2021), eprint 2009.09999.
  * Bengochea (2020) G. R. Bengochea, Rev. Mex. Fis. E 17, 263 (2020), eprint 2007.03428.
  * Bohm (1952) D. Bohm, Phys. Rev. 85, 166 (1952).
  * Valentini (2010) A. Valentini, Phys. Rev. D 82, 063513 (2010), eprint 0805.0163.
  * Pinto-Neto et al. (2012) N. Pinto-Neto, G. Santos, and W. Struyve, Phys. Rev. D 85, 083506 (2012), eprint 1110.1339.
  * Goldstein et al. (2015) S. Goldstein, W. Struyve, and R. Tumulka, arXiv e-prints arXiv:1508.01017 (2015), eprint 1508.01017.
  * Pinto-Neto and Struyve (2018) N. Pinto-Neto and W. Struyve, arXiv e-prints arXiv:1801.03353 (2018), eprint 1801.03353.
  * Vitenti et al. (2019) S. D. P. Vitenti, P. Peter, and A. Valentini, Phys. Rev. D 100, 043506 (2019), eprint 1901.08885.
  * Kiefer and Polarski (2009) C. Kiefer and D. Polarski, Adv. Sci. Lett. 2, 164 (2009), eprint 0810.0087.
  * Halliwell (1989) J. J. Halliwell, Phys. Rev. D39, 2912 (1989).
  * Kiefer (2000) C. Kiefer, Nucl. Phys. Proc. Suppl. 88, 255 (2000), eprint astro-ph/0006252.
  * Polarski and Starobinsky (1996) D. Polarski and A. A. Starobinsky, Class. Quant. Grav. 13, 377 (1996), eprint gr-qc/9504030.
  * Okon and Sudarsky (2016) E. Okon and D. Sudarsky, Found. Phys. 46, 852 (2016), eprint 1512.05298.
  * Adler (2003) S. L. Adler, Studies in History and Philosophy of Modern Physics 34, 135 (2003), eprint quant-ph/0112095.
  * Schlosshauer (2004) M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2004), eprint quant-ph/0312059.
  * Everett (1957) H. Everett, Rev. Mod. Phys. 29, 454 (1957).
  * Mukhanov (2005) V. Mukhanov, _Physical Foundations of Cosmology_ (New York: Cambridge University Press, 2005).
  * Kent (1990) A. Kent, International Journal of Modern Physics A 5, 1745 (1990), eprint gr-qc/9703089.
  * Stapp (2002) H. P. Stapp, Canadian Journal of Physics 80, 1043 (2002), eprint quant-ph/0110148.
  * Pearle (1976) P. Pearle, Phys. Rev. D 13, 857 (1976).
  * Ghirardi et al. (1986) G. Ghirardi, A. Rimini, and T. Weber, Phys.Rev. D34, 470 (1986).
  * Pearle (1989) P. M. Pearle, Phys.Rev. A39, 2277 (1989).
  * Diosi (1987) L. Diosi, Phys.Lett. A120, 377 (1987).
  * Diosi (1989) L. Diosi, Phys.Rev. A40, 1165 (1989).
  * Penrose (1996) R. Penrose, Gen.Rel.Grav. 28, 581 (1996).
  * Okon and Sudarsky (2014) E. Okon and D. Sudarsky, Found. Phys. 44, 114 (2014), eprint 1309.1730.
  * Bassi and Ghirardi (2003) A. Bassi and G. Ghirardi, Phys. Rept. 379, 257 (2003), eprint quant-ph/0302164.
  * Bassi et al. (2013) A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Reviews of Modern Physics 85, 471 (2013), eprint 1204.4325.
  * Leon and Sudarsky (2010) G. Leon and D. Sudarsky, Class. Quant. Grav. 27, 225017 (2010), eprint 1003.5950.
  * Diez-Tejedor and Sudarsky (2012) A. Diez-Tejedor and D. Sudarsky, JCAP 07, 045 (2012), eprint 1108.4928.
  * Diez-Tejedor et al. (2012) A. Diez-Tejedor, G. León, and D. Sudarsky, General Relativity and Gravitation 44, 2965 (2012), eprint 1106.1176.
  * Cañate et al. (2013) P. Cañate, P. Pearle, and D. Sudarsky, Phys.Rev. D 87, 104024 (2013), eprint 1211.3463.
  * Bengochea et al. (2015) G. R. Bengochea, P. Cañate, and D. Sudarsky, Phys. Lett. B743, 484 (2015), eprint 1410.4212.
  * León and Sudarsky (2015) G. León and D. Sudarsky, JCAP 06, 020 (2015), eprint 1503.01417.
  * Leon and Bengochea (2016) G. Leon and G. R. Bengochea, Eur. Phys. J. C76, 29 (2016), eprint 1502.04907.
  * León (2017) G. León, European Physical Journal C 77, 705 (2017), eprint 1705.03958.
  * Landau et al. (2012) S. J. Landau, C. G. Scóccola, and D. Sudarsky, Physical Review D 85, 123001 (2012), eprint 1112.1830.
  * Benetti et al. (2016) M. Benetti, S. J. Landau, and J. S. Alcaniz, JCAP 12, 035 (2016), eprint 1610.03091.
  * Bengochea and León (2017) G. R. Bengochea and G. León, Phys. Lett. B 774, 338 (2017), eprint 1708.07527.
  * Cañate et al. (2018) P. Cañate, E. Ramirez, and D. Sudarsky, JCAP 08, 043 (2018), eprint 1802.02238.
  * Juárez-Aubry et al. (2018) B. A. Juárez-Aubry, B. S. Kay, and D. Sudarsky, Phys. Rev. D 97, 025010 (2018), eprint 1708.09371.
  * Piccirilli et al. (2019) M. P. Piccirilli, G. León, S. J. Landau, M. Benetti, and D. Sudarsky, International Journal of Modern Physics D 28, 1950041 (2019), eprint 1709.06237.
  * León et al. (2015) G. León, L. Kraiselburd, and S. J. Landau, Phys. Rev. D 92, 083516 (2015), eprint 1509.08399.
  * Mariani et al. (2016) M. Mariani, G. R. Bengochea, and G. León, Physics Letters B 752, 344 (2016), eprint 1412.6471.
  * León et al. (2017) G. León, A. Majhi, E. Okon, and D. Sudarsky, Phys. Rev. D 96, 101301 (2017), eprint 1607.03523.
  * León et al. (2018) G. León, A. Majhi, E. Okon, and D. Sudarsky, Phys. Rev. D 98, 023512 (2018), eprint 1712.02435.
  * León et al. (2016) G. León, G. R. Bengochea, and S. J. Landau, European Physical Journal C 76, 407 (2016), eprint 1605.03632.
  * Josset et al. (2017) T. Josset, A. Perez, and D. Sudarsky, Phys. Rev. Lett. 118, 021102 (2017), eprint 1604.04183.
  * Leon and Piccirilli (2020) G. Leon and M. P. Piccirilli, Phys. Rev. D 102, 043515 (2020), eprint 2006.03092.
  * Martin et al. (2012) J. Martin, V. Vennin, and P. Peter, Phys. Rev. D 86, 103524 (2012), eprint 1207.2086.
  * Das et al. (2013) S. Das, K. Lochan, S. Sahu, and T. P. Singh, Phys. Rev. D 88, 085020 (2013), eprint 1304.5094.
  * Markkanen et al. (2015) T. Markkanen, S. Rasanen, and P. Wahlman, Phys. Rev. D 91, 084064 (2015), eprint 1407.4691.
  * Alexander et al. (2016) S. Alexander, D. Jyoti, and J. Magueijo, Phys. Rev. D 94, 043502 (2016), eprint 1602.01216.
  * Drossel and Ellis (2018) B. Drossel and G. Ellis, New Journal of Physics 20, 113025 (2018), eprint 1807.08171.
  * Martin and Vennin (2020) J. Martin and V. Vennin, Phys. Rev. Lett. 124, 080402 (2020), eprint 1906.04405.
  * Bengochea et al. (2020a) G. R. Bengochea, G. Leon, P. Pearle, and D. Sudarsky (2020a), eprint 2006.05313.
  * Bengochea et al. (2020b) G. R. Bengochea, G. León, P. Pearle, and D. Sudarsky, Eur. Phys. J. C 80, 1021 (2020b), eprint 2008.05285.
  * Martin and Vennin (2021a) J. Martin and V. Vennin, Eur. Phys. J. C 81, 64 (2021a), eprint 2010.04067.
  * Gundhi et al. (2021) A. Gundhi, J. L. Gaona-Reyes, M. Carlesso, and A. Bassi, Phys. Rev. Lett. 127, 091302 (2021), eprint 2102.07688.
  * Martin and Vennin (2021b) J. Martin and V. Vennin, Eur. Phys. J. C 81, 516 (2021b), eprint 2103.01697.
  * León and Bengochea (2021) G. León and G. R. Bengochea, Eur. Phys. J. C 81, 1055 (2021), eprint 2107.05470.
  * Juárez-Aubry et al. (2020) B. A. Juárez-Aubry, T. Miramontes, and D. Sudarsky, J. Math. Phys. 61, 032301 (2020), eprint 1907.09960.
  * Eppley and Hannah (1977) K. Eppley and E. Hannah, Foundations of Physics 7, 51 (1977).
  * Page and Geilker (1981) D. N. Page and C. D. Geilker, Phys. Rev. Lett. 47, 979 (1981).
  * Mattingly (2005) J. Mattingly, _Is Quantum Gravity Necessary?_ (2005), vol. 11, pp. 327–338.
  * Mattingly (2006) J. Mattingly, Phys. Rev. D73, 064025 (2006), eprint gr-qc/0601127.
  * Kent (2018) A. Kent, Class. Quant. Grav. 35, 245008 (2018), eprint 1807.08708.
  * Tilloy and Diósi (2016) A. Tilloy and L. Diósi, Phys. Rev. D93, 024026 (2016), eprint 1509.08705.
  * Carlip (2008) S. Carlip, Class. Quant. Grav. 25, 154010 (2008), eprint 0803.3456.
  * Albers et al. (2008) M. Albers, C. Kiefer, and M. Reginatto, Phys. Rev. D78, 064051 (2008), eprint 0802.1978.
  * Ford (2005) L. H. Ford, 100 Years Of Relativity : space-time structure: Einstein and beyond pp. 293–310 (2005), eprint gr-qc/0504096.
  * Ford (2005) L. H. Ford, _Spacetime in Semiclassical Gravity_ (2005), pp. 293–310.
  * Piscicchia et al. (2017) K. Piscicchia, A. Bassi, C. Curceanu, R. Grande, S. Donadi, B. Hiesmayr, and A. Pichler, Entropy 19, 319 (2017), eprint 1710.01973.
  * Lewis et al. (2000) A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000), eprint astro-ph/9911177.