# Enlightening the CSL model landscape in inflation

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.
Gabriel R. Bengochea<EMAIL_ADDRESS>Instituto de Astronomía y Física del
Espacio (IAFE), CONICET - Universidad de Buenos Aires, (1428) Buenos Aires,
Argentina

###### Abstract

We propose a novel realization for the natural extrapolation of the continuous
spontaneous localization (CSL) model, in order to account for the origin of
primordial inhomogeneities during inflation. This particular model is based on
three main elements: (i) the semiclassical gravity framework, (ii) a collapse-
generating operator associated to a relativistic invariant scalar of the
energy-momentum tensor, and (iii) an extension of the CSL parameter(s) as a
function of the spacetime curvature. Furthermore, employing standard
cosmological perturbation theory at linear order, and for a reasonable range
within the parameter space of the model, we obtain a nearly scale invariant
power spectrum consistent with recent observational CMB data. This opens a
vast landscape of different options for the application of the CSL model to
the cosmological context, and possibly sheds light on searches for a full
covariant version of the CSL theory.

Cosmology, Inflation, Quantum Cosmology

##  I Introduction

One of the most interesting challenges when trying to combine General
Relativity with Quantum theory to explain the very early epoch of the
universe, has to do with giving a satisfactory explanation of the quantum-to-
classical transition of perturbations originated during inflation. The
proposal that, at the very beginning, the universe went through an accelerated
phase, called inflation, is extremely successful since it allowed explaining
not only some well-known problems of the hot Big Bang model (e.g. the horizon
and flatness problems), but it has also offered a way to address the quantum
origin of large-scale structures that we see today in the sky, from primordial
cosmic seeds Starobinsky (1979, 1980); Guth (1981); Mukhanov and Chibisov
(1981); Linde (1982, 1983); Albrecht and Steinhardt (1982); Bardeen et al.
(1983); Brandenberger (1984); Hawking (1982). The inflationary paradigm
predictions for primordial power spectra are strikingly consistent with the
most recent accurate observations of cosmic microwave background (CMB)
radiation Aghanim et al. (2020a); Akrami et al. (2020); Aghanim et al.
(2020b).

Notice that, the aforementioned quantum to classical transition is closely
related to the so-called _measurement problem_ in Quantum physics Wigner
(1963); Omnes (1994); Albert (1994); Maudlin (1995); Becker (2018); Norsen
(2017); Durr and Lazarovici (2020); Okon (2014)111See Maudlin (1995), where
the author states the measurement problem in a formal and elegant way, and he
also describes what are the possible alternatives to address it.. Let us
mention why this problem is notoriously enhanced in the cosmological case Bell
(1981); Perez et al. (2006); Sudarsky (2011); Landau et al. (2013)222Non-
specialist readers interested in a introductory review on this subject can
find it in Bengochea (2020).. The central point here is that, according to
standard Quantum theory, the evolution of any quantum state is always unitary,
dictated by the Schrödinger equation, which does not break any initial
symmetry of the system or destroy quantum superpositions. If at the beginning
of inflation the spacetime is assumed to be spatially isotropic and
homogeneous, and the perturbations of matter fields (i.e. the inflaton field)
were in a quantum vacuum state also invariant under spatial rotations and
translations (i.e. the so-called Bunch-Davies vacuum); then one question
arises: how do we arrive at an anisotropic and inhomogeneous state (_with_
cosmic seeds of structures) from a quantum vacuum state that is perfectly
isotropic and homogeneous in space? The traditional early universe paradigm
seems to be incomplete in that sense333The standard explanation for such an
emergence is based on the study of the role of _quantum fluctuations_ of the
scalar field driving the accelerated expansion during the inflationary epoch.
However, that analysis lacks a complete description for the problem at hand
that is being discussed here Sudarsky (2011); Berjon et al. (2021).. And any
model that claims to provide a mechanism for the emergence of the seeds of
structure must be able to give a convincing answer to the mentioned scenario,
without appealing, of course, to the existence of observers or measuring
devices which cannot constitute fundamental elements in the very early
universe Hartle (1993). There are various proposals that try to explain the
quantum-to-classical transition from different approaches. For instance, by
invoking the decoherence framework Kiefer and Polarski (2009); Halliwell
(1989); Kiefer (2000); Polarski and Starobinsky (1996), it could provide a
partial understanding of the issue. However, it was shown that decoherence
does not fully addresses the problem Okon and Sudarsky (2016a); Adler (2003);
Schlosshauer (2004); Sudarsky (2011); Kent (1990); Stapp (2002). Other
authors, within the framework of the de Broglie-Bohm theory Bohm (1952), have
also explored applications to this cosmological puzzle, e.g. Valentini (2008,
2010); Pinto-Neto et al. (2012); Goldstein et al. (2015); Pinto-Neto and
Struyve (2018); Vitenti et al. (2019).

Known as _objective collapse theories_ , they constitute another approach that
seeks to address the aforementioned issue, since they were specifically
designed to solve the measurement problem in Quantum Mechanics Pearle (1976);
Ghirardi et al. (1986); Pearle (1989); Diosi (1987, 1989); Penrose (1996).
Reviews on these sort of theories can be found, for instance, in Bassi and
Ghirardi (2003); Bassi et al. (2013). The idea of invoking a self-induced
collapse in cosmology in order to generate the primordial perturbations and/or
accomplish the quantum-to-classical transition has been explored in great
detail since 2006 Perez et al. (2006), and it has led to numerous
investigations in recent years with varied proposals and results, e.g. Leon
and Sudarsky (2010); Sudarsky (2011); Diez-Tejedor and Sudarsky (2012); Diez-
Tejedor et al. (2012); Martin et al. (2012); Cañate et al. (2013); Das et al.
(2013); Bengochea et al. (2015); León and Sudarsky (2015); Markkanen et al.
(2015); Leon and Bengochea (2016); Alexander et al. (2016); León (2017);
Landau et al. (2012, 2013); Benetti et al. (2016); Bengochea and León (2017);
Drossel and Ellis (2018); 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);
Bengochea et al. (2021). In several of those works, a specific version based
on the continuous spontaneous localization (CSL) model Pearle (1976, 1989)
adapted to the situation of interest, was chosen to analyze this subject.

The CSL constitutes a particular proposal, based on a non-linear stochastic
modification of the standard Schrödinger equation, where spontaneous and
random collapses occur, resulting in an objective dynamical reduction of the
wave function. For the first time, in Ref. Martin et al. (2012) the CSL model
was applied to the inflationary universe and in Ref. Cañate et al. (2013) such
exploration was done within the semiclassical gravity framework. The debate on
the particular details involved in implementing the CSL model in the
cosmological context is still open, and contains an extensive landscape of
possibilities constituting an active research area at the moment Martin and
Vennin (2020); Bengochea et al. (2020a, b); Martin and Vennin (2021a); Gundhi
et al. (2021); Martin and Vennin (2021b). As was already pointed out in Okon
and Sudarsky (2014), objective collapse models have the appealing feature of
connecting plausible resolutions of other open problems in a single unified
picture; not only the measurement problem in Quantum Mechanics and the
emergence of the primordial inhomogeneities during inflation, but also the
problem of time in canonical quantum theories of gravitation, the black hole
information paradox Modak et al. (2015a, b); Bedingham et al. (2016) and other
applications to cosmology, for instance to account for the dark energy and the
late-time accelerated expansion of the universe Josset et al. (2017); Corral
et al. (2020); Linares Cedeño and Nucamendi (2021).

Recently, in Ref. Bengochea et al. (2020b), a wide variety of open
alternatives were discussed when applying the CSL dynamical collapse model to
the inflationary era. That exploration included the important aspects one must
face in order to address such a problem. These are: (i) the two different
approaches to deal with quantum field theory and gravitation, (ii) the
identification of the collapse-generating operator, and (iii) the general
nature and values of the CSL model parameters. In that same work, it was also
emphasized that all the choices connected with those issues have the potential
to dramatically alter the conclusions one can draw.

In this article, we will explore a particular CSL model in the inflationary
context, within the vast theoretical landscape available for the extrapolation
of the standard CSL (as constructed to deal with non-relativistic many
particle Quantum Mechanics) into the realms of relativistic quantum field
theory in curved spacetimes. To accomplish this task, we will follow some
options raised in Ref. Bengochea et al. (2020b). In this way, our model
possess three main elements which we mention below.

First, one must choose the setting within which one is going to combine
quantum field theory (QFT) with gravitation. While in some previous works, we
have explored the use of the CSL using the standard framework of quantization
(in which metric and matter fields are quantized simultaneously). Here, as in
other previous works, we will adopt the semiclassical gravity framework where
gravity is always classical while the matter fields are treated quantum
mechanically. This approach has received some known criticisms Eppley and
Hannah (1977); Page and Geilker (1981), but those arguments have been
discussed and refuted Mattingly (2005, 2006); Kent (2018); Tilloy and Diósi
(2016); Carlip (2008); Albers et al. (2008); Ford (2005, 2005); currently, the
theory continues to be of great interest. Moreover, it is not often emphasized
that most of the conclusions against are only valid in those contexts where
quantum mechanical evolution does not include any sort of measurement-related
or spontaneous collapse of the quantum state. We will assume such a framework
to be a valid approximation during the inflationary era, which is well after
the full quantum gravity regime has ended. This choice is due to the fact that
it appears favored from a theoretical and conceptual point of view, in
particular, when one wants to incorporate collapse models Diez-Tejedor and
Sudarsky (2012); Cañate et al. (2018); Juárez-Aubry et al. (2018, 2020);
Bengochea et al. (2020b). In fact, it was shown that, within semiclassical
gravity, the CSL model predicts generically a very small amplitude of the
tensor power spectrum León et al. (2015, 2017, 2018), which is very consistent
with the current non-detection of B-modes in the CMB data Akrami et al.
(2020).

Second, as mentioned in Ref. Bengochea et al. (2020b), there are a large
number of operators that are relativistically invariant and do reduce, in the
simple non-relativistic regimes encountered in laboratory situations, to the
required “mass density” or “energy density”. Some simple examples we might
consider are scalars constructed, for example, using the energy-momentum
tensor $T_{ab}$. In the present work, for the first time in this line of
research, we will explore the consequences of assuming a collapse generating
operator composed of a contraction of the energy-momentum tensor.
Specifically, we will choose the scalar $(T_{ab}T^{ab})^{1/2}$ as the collapse
operator in our proposal to extrapolate the CSL model into the cosmological
context.

Third, we will make a particular choice regarding the parameters of the CSL
model. The main idea is the following.

One of the central features of the CSL model, sometimes referred to as the
_amplification mechanism_ , is that the collapses must be rare for microscopic
systems, in order not to alter their quantum behavior as described by the
Schrödinger equation. But, on the other hand, their effects must increase when
several particles are hold together forming a macroscopic system. The
amplification mechanism can be characterized through the collapse rate
parameter $\lambda$ of the CSL model.

As was discussed at length in Bengochea et al. (2020b), such a parameter used
in the non-relativistic versions of the CSL model, let us call it
$\lambda_{0}$, might not be a fundamental constant after all. The original
motivation Pearle (1976); Ghirardi et al. (1986); Pearle (1989); Bassi and
Ghirardi (2003); Pearle and Squires (1994) for choosing the particular value
of $\lambda_{0}\approx 10^{-16}$ s-1 was based on some phenomena occurring at
laboratory scales, e.g. the number of atoms in a pointer apparatus, the matter
density of nuclear matter, etc. Therefore, it is not obvious at all that there
should be any comparison between the value of $\lambda_{0}$, informed by the
existence of hadrons, nuclei and the atomic densities of solids, with
parameters at an era when there were no nucleons, no atoms and certainly no
solids.

On the other hand, the second parameter of the non-relativistic CSL model,
$r_{c}$, is a length scale characterizing the level of de-localization
associated with the onset of the spontaneous collapse. For instance, for a
single particle, the effect of the collapse aspect of the modified dynamics is
negligible if the width of its wave function is smaller than $r_{c}$. But, if
it is larger than $r_{c}$, the collapse mechanism becomes very relevant. The
actual numerical value of $r_{c}$ comes from experimental constraints using a
non-relativistic version of the CSL model. Additionally, for that particular
CSL model, the parameter $r_{c}$ serves to characterize physical processes at
laboratory scales. For example, $r_{c}$ takes into account notions such as
atomic dimensions, mean spreads around the equilibrium positions of the
lattice points of a crystal, etc. (see Sec. 6.5 of Ref. Bassi and Ghirardi
(2003)). Thus, there are ample grounds to doubt any simple one-on-one
connection between the value of $\lambda_{0}$ and $r_{c}$, relevant for one
regime (involving laboratory scales), with parameters characterizing the model
in a completely different one, such as the inflationary universe.

Previous works Cañate et al. (2013); Leon and Bengochea (2016); León et al.
(2016); Piccirilli et al. (2019), where a particular version of the CSL model
was applied to inflation, were able to recover the correct shape and amplitude
of the CMB spectrum (in addition to being compatible with laboratory
constraints). Those results required making suitable assumptions regarding the
dependence of the model parameters on the Fourier mode wavelengths, which in
turn, might be viewed as tied to a more fundamental dependence on curvature.
However, in the present work, we will make explicit this dependence on the
curvature in the CSL parameters, and make no assumption involving a particular
decomposition used in the field’s modes (e.g. Fourier modes).

As a matter of fact, as suggested by L. Diosi and R. Penrose in e.g. Diósi
(1984); Diosi (1987, 1989); Penrose (1996), it seems quite natural to think
that, at a fundamental level, the spontaneous collapse dynamics might be
intimately tied with gravitation. Therefore, accepting such a hypothesis, one
reasonable option for exploring the relation between the collapse of the wave
function and gravitation, is to replace the collapse parameter of the ordinary
CSL model by a function of some geometrical scalar associated with curvature
(see e.g. Okon and Sudarsky (2014)). In particular, in a series of works
devoted to the examination of the famous black hole information puzzle Modak
et al. (2015a, b), it was found that, during the black hole evaporation,
information is lost as a result of the modified evolution described by the CSL
model with the additional hypothesis that the collapse rate $\lambda$ is
enhanced by the curvature of the spacetime, through the Ricci scalar $R$.

In this paper, we will adopt the same parameterization of $\lambda$ as the
authors of Refs. Modak et al. (2015a, b), i.e. we will assume that $\lambda$
is a function of $R$ (a more detailed discussion about this point will be
given in Sect. III.3). This idea was first suggested in Pearle and Squires
(1994, 1996), and was also one of the options mentioned in Bengochea et al.
(2020b). We acknowledge that there are other possibilities for including
curvature scalars in the parameterization of $\lambda$. For instance, one
could consider the Kretschmann scalar
$R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$, or the scalar
$W_{\alpha\beta\mu\nu}W^{\alpha\beta\mu\nu}\>$, where $W_{\alpha\beta\mu\nu}$
denotes the Weyl tensor. In fact, as argued in Okon and Sudarsky (2014), the
latter choice seems to have the characteristics that could lead to an
association of low entropy with the early state of the universe and a large
entropy with its late-time state. That is, the adoption of an objective
collapse model with a curvature-dependent collapse parameter involving the
Weyl tensor, might result in dynamical-based explanation of the Weyl curvature
hypothesis (see Ref. Okon and Sudarsky (2016b) and references within for more
details). The point is that including an explicit dependence of some curvature
scalar in $\lambda$, appears to be a valid option if one accepts the premise
that the objective collapse is enhanced by the gravitational degrees of
freedom.

The dependence on curvature could result, of course, in an effective temporal
dependency of the model parameters. In fact, this kind of effective time
dependence (i.e. a time dependence that appears in a certain regime to be a
constant, but is in fact a coupling with some other dynamical variable) is one
we have already encountered in cosmology; e.g., the masses in the standard
model of particles depend on a vacuum expectation value in the Higgs sector,
and that field experienced dramatic changes between the inflationary epoch and
the present day. So an effective time dependence of the CSL parameters, in the
cosmological context, might very well be considered likely.

With respect to the second parameter of the non-relativistic CSL model
$r_{c}$, which is related to the localization of the wave function
corresponding to a system of (non-relativistic massive) particles, in this
article we will propose a novel approach to generalize such element when
analyzing the inflationary universe. In particular, we will assume that, when
taking into account the collapse of the wave function corresponding to the
inflaton, the quantum two-point correlation function of the field variables is
a straightforward way to extrapolate the notion of “localization” as given by
the ordinary CSL model (we will be more precise in Sec. III.4). Recall that,
in the latter, the level of localization is encoded in the quantum uncertainty
of the position operator.

Finally, let us mention that we will base our treatment on a model of
spontaneous collapses that is strictly speaking not covariant. We acknowledge
that a truly satisfactory proposal, to deal with the subject considered here,
should be based on a fully covariant theory. However in absence of such a
theory, we can move forward by making reasonable assumptions that could help
in finding a complete and workable theory. Fortunately, recent proposals for
special relativistic versions of these type of theories indicate that we
should be able to address this shortcoming in near future Tumulka (2006);
Bedingham (2011); Pearle (2015); Bedingham et al. (2016).

Our goal in this paper can be summarized as follows: based on the three main
elements mentioned above, and employing standard cosmological perturbation
theory at linear order, we will obtain the primordial scalar power spectrum.
Also, we will analyze the characteristics needed for the spectrum to be
consistent with both the laboratory constraints and those from the CMB
observations.

This manuscript is divided in 6 sections plus two Appendices, where in the
latter we have included most of the calculations. We start in section II with
a brief review of the standard CSL model by discussing the _mass proportional_
CSL model. Next, in section III we describe our proposed framework, in section
IV we find an equivalent scalar power spectrum, and in section V we analyze
the features of the obtained spectrum. Finally, in section VI, we present our
conclusions. Regarding conventions and notation, we use a $(-,+,+,+)$
signature for the spacetime metric and units where $c=1=\hbar$.

##  II Brief review of the CSL model

In this section, we provide a very brief review of the non-relativistic CSL
model, so there is no original work here. Specifically, we will focus on the
mass proportional CSL model. Those readers familiar with the subject can
safely skip to the next section. For a pedagogical review, including the main
technical aspects, we suggest Ref. Pearle (2012), and for a general overview
of collapse models see Bassi and Ghirardi (2003); Bassi et al. (2013).

The CSL model is a generalization of the spontaneous collapse model originally
proposed by Ghirardi, Rimini and Weber Ghirardi et al. (1986) to a system of
identical particles. In the CSL model the collapse of the wave function occurs
continuously in time, while in the GRW one the collapse happens discretely.
The mass proportional CSL model has been probed in experiments, where
relativistic effects can be ignored (see e.g. Ref. Gasbarri et al. (2021)). It
consists of two main equations. The first one is a modified version of the
Schrödinger equation, whose general solution is

$\displaystyle|\psi,t\rangle$ $\displaystyle=$ $\displaystyle{\hat{\cal
T}}e^{-i\int_{0}^{t}dt^{\prime}\hat{H}(t^{\prime})-\frac{1}{4\lambda_{0}}\int_{0}^{t}dt^{\prime}\int
d\textbf{x}^{\prime}[w(\textbf{x},t^{\prime})-2\lambda_{0}\hat{A}(\textbf{x})]^{2}}$
(1) $\displaystyle\times$ $\displaystyle|\psi,0\rangle$

Here, $\hat{\cal T}$ is the time-ordering operator and $w(\textbf{x},t)$
characterizes a stochastic process of white noise type. The probability for
the noise $w(\textbf{x},t)$ is given by the second main equation, which
represents the Probability Rule:

$P(w)dw\equiv\langle\psi,t|\psi,t\rangle\prod_{t_{i}=0}^{t-dt}\frac{dw(\textbf{x},t_{i})}{\sqrt{2\pi\lambda_{0}/dt}}$
(2)

The state vector norm evolves dynamically, i.e. it is not equal to 1. In
particular, Eq. (2) implies that state vectors with largest norm are the most
probable. The total probability is then $\int P(w)dw=1$, provided that
$\langle\psi,0|\psi,0\rangle=1$.

The parameter $\lambda_{0}$ is the collapse rate for a reference particle in a
spatially superposed state, and usually the reference particle is chosen as a
nucleon. On the other hand, the operator $\hat{A}(\textbf{x})$ is called the
collapse generating operator, and for the mass proportional CSL model it is
defined as

$\hat{A}(\textbf{x})=\sum_{i}\frac{m_{i}}{m_{0}}\frac{1}{(\pi
r_{c}^{2})^{3/4}}\int
d^{3}y\>e^{-|\textbf{x}-\textbf{y}|^{2}/2r_{c}^{2}}\hat{N}_{i}(\textbf{y})$
(3)

where
$\hat{N}_{i}(\textbf{y})=\hat{\chi}_{i}^{\dagger}(\textbf{y})\hat{\chi}_{i}(\textbf{y})$
is the particle number density operator, constructed from (non-relativistic)
creation an annihilation operators for a particle of type $i$ at y. In
addition, $m_{i}$ is the mass of the corresponding particle species and
$m_{0}$ is the mass of the reference particle, e.g. a nucleon. The second
parameter of the model is the smearing length $r_{c}$, a length scale
characterizing the level of de-localization associated with the onset of the
spontaneous collapse. In this way, $\hat{A}(\textbf{x})$ is considered as a
smeared mass density operator at x.

Equation (1) describes the complete evolution of an individual system prepared
in a given initial state. However, the presence of the noise $w(\textbf{x},t)$
clearly makes the evolution highly random. The only certainty is that, by
construction, the system will be driven towards one of the eigenstates of
$\hat{A}(\textbf{x})$. Therefore, it is useful to investigate what happens
with an ensemble of identically prepared systems. Because of the CSL dynamics,
the ensemble will be constituted of a set of different evolved state vectors,
each characterized by a different realization of $w(\textbf{x},t)$, and the
density matrix is the object that best serves to describe the ensemble’s
evolution.

The density matrix is defined from Eqs. (1) and (2) as

$\hat{\rho}(t)\equiv\int_{-\infty}^{\infty}P(w)dw\frac{|\psi,t\rangle\langle\psi,t|}{\langle\psi,t|\psi,t\rangle}$
(4)

From Eq. (4), it can be shown that the corresponding density matrix evolution
satisfies

$\frac{\partial\hat{\rho}(t)}{\partial
t}=-i[\hat{H},\hat{\rho}(t)]-\frac{\lambda_{0}}{2}\int
d^{3}x\>[\hat{A}(\textbf{x}),[\hat{A}(\textbf{x}),\hat{\rho}(t)]].$ (5)

As a consequence, the ensemble expectation value of any observable
corresponding to an operator $\hat{O}$ can be obtained as
$\overline{\langle\hat{O}\rangle}=$ Tr $\\{\hat{O}\hat{\rho}(t)\\}$.

Note that, from Eq. (1) [or equivalently (5)], one can also identify an
effective collapse rate for each particle species
$\lambda_{i}\equiv\lambda_{0}(m_{i}/m_{0})^{2}$, i.e. the strength of the
collapse scales with the mass of the particle. This leads to one of the most
important features of collapse models: the amplification mechanism. For
microscopic systems (for instance, a single electron) the collapse is very
weak, so superpositions can occur. For macroscopic objects (for example,
$10^{24}$ atoms) the collapse is very strong, implying a rapid and efficient
collapse of the wave function.

If the model is correct, accurate values of $\lambda_{0}$ and $r_{c}$ have to
be determined by empirical evidence. It should be noted that as the way the
model was constructed, relativistic effects have not been taken into account;
also, in ordinary matter, the collapse is mostly caused by nucleons. Until
experiments can provide precise values of the parameters, one can adopt some
estimated values based on reasonable arguments. For example, the choice
$r_{c}\simeq 10^{-5}$ cm, originally suggested by GRW, was based on the
assumption that $r_{c}$ should be large with respect to the atomic dimensions
and to the mean spreads around the equilibrium positions of the lattice points
of a crystal. Similar arguments, which involve human perception time and the
number of nucleons marking the quantum-classical threshold, lead to the choice
$\lambda_{0}\simeq 10^{-16}$ s-1 Pearle (1976); Ghirardi et al. (1986); Pearle
(1989); Bassi and Ghirardi (2003); Pearle and Squires (1994).

##  III The proposed framework

In this section, we present one of the possible extrapolations of the non-
relativistic CSL model, described in the previous section, to more general
situations in which relativistic effects cannot be ignored. In our view, the
proposed model is a natural extension of the original one, since it captures
the essence and main features of the mass proportional CSL model. Also, it can
be applied directly to the cosmological case; and in particular, to the
inflationary universe. To accomplish this, we will follow some options
discussed in Bengochea et al. (2020b).

The background metric will be given by a spatially flat FLRW spacetime. In
conformal comoving coordinates the metric components are
$g_{\mu\nu}=a^{2}(\eta)\eta_{\mu\nu}$, with $\eta_{\mu\nu}$ the Minkowski
metric with signature $(-,+,+,+)$. We also introduce the Hubble parameter $H$,
and then the comoving Hubble parameter is given as
$\mathcal{H}=a^{\prime}/a=aH$, with primes over functions denoting derivative
with respect to conformal time $\eta$. Inflation is characterized by a
shrinking comoving Hubble radius $d/d\eta[\mathcal{H}^{-1}]<0$, with $H\simeq$
constant. Also, if needed, we can fix the normalization of the scale factor
today, so $a_{0}=1$.

We proceed now to characterize the metric perturbations. In particular, we
choose to work in the longitudinal gauge where the line element associated to
the metric is

$ds^{2}=a^{2}(\eta)\left[-(1+2\Psi)d\eta^{2}+(1-2E)\delta_{ij}dx^{i}dx^{j}\right]$
(6)

with $E$ and $\Psi$ being scalar fields corresponding to scalar perturbations
at first order. In fact, by assuming that there is no anisotropic stress,
Einstein’s equations lead to $\Psi=E$. From now on, we will use this result
and refer to $\Psi$ as the Newtonian potential. Moreover, in this gauge,
$\Psi$ represents the curvature perturbation (i.e. the intrinsic spatial
curvature on hypersurfaces on constant conformal time for a flat universe).

During inflation, the matter degrees of freedom are represented by a single
scalar field $\phi(\textbf{x},\eta)$, called the inflaton. Furthermore, we
separate $\phi$ in its homogeneous part $\phi_{0}(\eta)$ plus a small
inhomogeneous perturbation $|\delta\phi(\textbf{x},\eta)|\ll|\phi_{0}|$. The
field $\phi_{0}$ will be treated in a classical way, while the perturbation
$\delta\phi$ can be described as a QFT in a flat FLRW background metric. The
condition for an inflationary phase requires that the potential energy $V$ of
the scalar field dominates over the kinetic term $\phi^{\prime 2}/2a^{2}$.
Furthermore, we will focus on slow roll inflation, where the slow roll
parameter is defined as $\epsilon\equiv
1-\mathcal{H}^{\prime}/\mathcal{H}^{2}$. During slow roll inflation
$\epsilon\ll 1$, and the inflationary phase ends when $\epsilon\simeq 1$.

###  III.1 Semiclassical Gravity

Given the fact that we have not yet at our disposal a complete and
satisfactory theory of quantum gravity, we will employ the semiclassical
gravity (SCG) formulation. This framework describes the way in which the
gravitational and matter degrees of freedom are related to each other. We
consider the SCG approach as an effective setting rather than something akin
to a fundamental theory. The SCG framework is characterized by Einstein
semiclassical equations

$R_{ab}-\frac{R}{2}g_{ab}=8\pi G\langle\hat{T}_{ab}\rangle$ (7)

where $R_{ab}$ and $R$ are the Ricci tensor and scalar respectively; also we
are neglecting the contribution of a cosmological constant. The expectation
value of the energy-momentum tensor $\langle\hat{T}_{ab}\rangle$,
characterizing the matter fields, acts as a source of the spacetime geometry.
In this way the description of gravity in terms of the metric is always taken
as classical.

The inflationary epoch is presumed to develop at energy scales smaller than
the Planck mass, therefore, one expects that the SCG approach can be
considered as a suitable framework. For our case of interest, the use of SCG
possess two main conceptual advantages. First, the spacetime, and thus the
metric, is always classical. There is no issue with the “quantum-to-classical
transition” in the characterization of the spacetime. We will not need to
justify the transition from “metric operators” (e.g. $\hat{\Psi}$) to
classical metric variables (such as $\Psi$). The fact that the spacetime
remains classical is of great importance when including the CSL model. This is
because the “localization” (or collapse) of the wave function is regarded as a
physical process occurring in time. Consequently, it is preferred to have a
setting that admits consideration of full spacetime notions Diez-Tejedor and
Sudarsky (2012); Cañate et al. (2018). The second advantage is that it allows
to present a transparent picture of how the primordial perturbations are born
from the wave function collapse: the initial state of the universe (i.e. the
one resulting after a few e-folds since the beginning of inflation) is
described by the completely symmetric Bunch-Davies (BD) vacuum and the equally
symmetric FLRW spacetime; the symmetry being spatial homogeneity and isotropy.
Then, after the CSL mechanism has ended, i.e. when the wave function has been
effectively localized/collpased, the state emanating from the collapse needs
not to share the symmetries of the initial state. After the collapse, the
gravitational degrees of freedom are assumed to be accurately described by Eq.
(7). Thus, in this post-collapsed state, in which the symmetries of the
original state have been lost, the expectation value
$\langle\hat{T}_{ab}\rangle$ yields a geometry that generically will no longer
be homogeneous and isotropic.

###  III.2 Main equations of the extended CSL model

We now introduce the two main equations that characterize our version of the
CSL model. The first is the evolution equation adapted to the cosmological
context:

$\displaystyle|\Phi,\eta\rangle$ $\displaystyle=$
$\displaystyle\hat{T}\exp\bigg{\\{}\int_{\tau}^{\eta}d\eta\int
d^{3}x\sqrt{|g|}\bigg{[}-i\hat{\mathcal{H}}(\textbf{x},\eta)$ (8)
$\displaystyle-$
$\displaystyle\frac{1}{4\lambda}\bigg{(}W(\textbf{x},\eta)-2\lambda\hat{C}(\textbf{x},\eta)\bigg{)}^{2}\bigg{]}\bigg{\\}}$
$\displaystyle\times$ $\displaystyle|\Phi,\tau\rangle$

where $\hat{T}$ is the time-ordering operator, $d\eta d^{3}x\sqrt{|g|}$ is the
4-volume associated to the background metric $g_{\mu\nu}$, and
$\tau\to-\infty$ is the conformal time at the beginning of inflation. The
operator $\hat{\mathcal{H}}$ corresponds to the Hamiltonian density of the
system. In addition, the CSL parameter $\lambda$ and the collapse generating
operator $\hat{C}$ appear here. We will be more specific about these objects
later on. The classical scalar field $W(\textbf{x},\eta)$ is of white noise
type, whose probability is given by the second equation which constitutes the
Probability Rule:

$P(W)dW=\langle\Phi,\eta|\Phi,\eta\rangle\prod_{\eta^{\prime}=\tau}^{\eta-d\eta}\frac{|g|^{1/4}dW(\textbf{x},\eta^{\prime})}{\sqrt{2\pi\lambda/d\eta}}.$
(9)

Note that we have chosen the time coordinate to be the conformal time $\eta$,
which implies $|g|^{1/2}=a^{4}$. The justification of such an election is
based on the widely known result indicating that, in these coordinates, the
“free” Hamiltonian of a scalar field in a flat FLRW spacetime is
mathematically equivalent to the one of a scalar field in Minkowski spacetime
with time dependent mass. Furthermore, from Eqs. (8) and (9), and assuming
that the initial states are normalized $\langle\Phi,\tau|\Phi,\tau\rangle=1$,
one obtains $\int P(W)dW=1$ as expected.

The density matrix operator, which is an useful expression describing the
ensemble’s evolution, is constructed from the state vectors and their
associated probabilities, here given by Eqs. (8) and (9):

$\hat{\rho}(\eta)\equiv\int_{-\infty}^{\infty}P(W)dW\frac{|\Phi,\eta\rangle\langle\Phi,\eta|}{\langle\Phi,\eta|\Phi,\eta\rangle}.$
(10)

According to Eq. (10), the corresponding density matrix evolution is

$\frac{\partial\hat{\rho}(\eta)}{\partial\eta}=-i[\hat{H},\hat{\rho}(\eta)]-\frac{\lambda
a^{4}}{2}\int
d^{3}x\>[\hat{C}(\textbf{x},\eta),[\hat{C}(\textbf{x},\eta),\hat{\rho}(\eta)]]$
(11)

where $\hat{H}$ is the total “free” Hamiltonian444Here, we have abused the
notation, so keep in mind that $\hat{H}$ is not related to the Hubble
parameter., i.e. $\hat{H}\equiv\int d^{3}x\sqrt{|g|}\hat{\mathcal{H}}$. As a
first step, let us ignore the CSL term in the evolution equations and
concentrate on finding the “free” Hamiltonian of the theory during inflation.
Given that we are working within the semiclassical gravity framework, only the
matter degrees of freedom will be quantized. Consequently, the field variable
operator can be taken as $y(\textbf{x},\eta)\equiv
a(\eta){\delta\phi}(\textbf{x},\eta)$. Expanding the action of the standard
inflationary model (i.e. a single scalar field–the inflaton–with canonical
kinetic term and minimally coupled to gravity) up to second order in the
perturbations, one can find the action associated to the matter perturbations.
Thus, the second order action is $S^{(2)}=\int d^{4}x\mathcal{L}^{(2)}_{y}$,
where

$\displaystyle\mathcal{L}^{(2)}_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}y^{\prime 2}-(\nabla
y)^{2}-y^{2}a^{2}V_{,\phi\phi}+\frac{a^{\prime\prime}}{a}y^{2}\bigg{]}$ (12)
$\displaystyle+$ $\displaystyle
a[4\phi_{0}^{\prime}\Psi^{\prime}y-2a^{2}V_{,\phi}\Psi y]$

and $V_{,\phi}$ indicates partial derivative with respect to $\phi$.

The previous Lagrangian can be used to obtain the Hamiltonian density involved
in Eq. (8). We define the canonical momentum
$p(\textbf{x},\eta)\equiv\partial\mathcal{L}^{(2)}_{y}/\partial y=y^{\prime}$,
in this way, the Hamiltonian $\mathcal{H}_{y}\equiv
py^{\prime}-\mathcal{L}^{(2)}_{y}$ is

$\displaystyle\mathcal{H}^{(2)}_{y}$ $\displaystyle=$
$\displaystyle\frac{p^{2}}{2}+\frac{(\nabla
y)^{2}}{2}+\frac{y^{2}}{2}\bigg{(}a^{2}V_{,\phi\phi}-\frac{a^{\prime\prime}}{a}\bigg{)}$
(13) $\displaystyle-$ $\displaystyle
a^{2}HM_{P}\sqrt{\epsilon}(4\Psi^{\prime}+6aH\Psi)y$

where we have used $\phi_{0}^{\prime}=aHM_{P}\sqrt{2\epsilon}$ and the slow
roll approximation $-V_{,\phi}\simeq 3H\phi_{0}^{\prime}/a$.

At this point we promote $y$ and $p$ to quantum variables, with equal time
commutator

$[\hat{y}(\textbf{x},\eta),\hat{p}(\textbf{y},\eta)]=i\delta(\textbf{x}-\textbf{y})$
(14)

Also, recall that we are working within the SCG framework, where the metric
variables are always classical and only the matter fields are quantized. In
particular, in our approach, because of the CSL mechanism together with SCG,
the metric would be changing from a homogeneous and isotropic spacetime to
another one with actual inhomogeneities/anisotropies. The latter in fact would
become manifested in the metric perturbations such as $\Psi$. As a
consequence, the terms involving $\Psi$ in Hamiltonian (13), could be treated
as reflecting the effect on $\phi$ of the metric response to the
“fluctuations” of the field itself $\delta\phi$. In other words, the
$\Psi$-terms can be considered as a back-reaction on the evolution of the
field. However, in Leon and Piccirilli (2020) was shown that these terms are
of second order in the slow-roll parameters. Therefore, the $\Psi$-terms in
the Hamiltonian (13) can be ignored at the leading order considered in this
work. Thus, the total “free” Hamiltonian during inflation, turns out to be
[see the first term in the r.h.s. of Eq. (11)]

$\displaystyle\hat{H}=\int d^{3}x\sqrt{|g|}\hat{\mathcal{H}}(\textbf{x},\eta)$
$\displaystyle\int
d^{3}x\left[\frac{\hat{p}^{2}}{2}+\frac{(\nabla\hat{y})^{2}}{2}+\frac{\hat{y}^{2}}{2}\bigg{(}a^{2}V_{,\phi\phi}-\frac{a^{\prime\prime}}{a}\bigg{)}\right].$

###  III.3 The collapse rate parameter

Let us turn our attention to the CSL terms in the evolution equation, Eq. (8)
[or equivalently Eq. (11)]. As mentioned above, the parameter $\lambda$ is the
collapse rate. Notice that, in the original GRW collapse model (as well as in
the mass proportional CSL model), there is an effective collapse rate that
depends on the mass of the particle involved. This feature leads to the
amplification mechanism, which basically states that as more massive particles
are involved, their collapse rates become strengthened. On the other hand, R.
Penrose and L. Diosi have, for a longtime, advocated that the collapse of the
wave function might be a dynamical process, in which the underlying mechanism
might be related to gravitational interaction Diósi (1984); Diosi (1987,
1989); Penrose (1996). Driven by these ideas, and as extensively discussed in
Ref. Bengochea et al. (2020b), we think it is natural to consider that the
collapse rate should incorporate some aspects tied to the spacetime curvature
for systems where this factor is important. A possible extrapolation of
$\lambda$ to more general regimes is the one that assumes an explicit
dependence of $\lambda$ upon the spacetime curvature. That is, in laboratory
experiments where one can safely presume a flat spacetime, the value of the
CSL parameter is approximately $\lambda_{0}$. While in a regime where the
spacetime curvature is strong, such a value would completely change.

As we have mentioned in the Introduction (as well as in Ref. Bengochea et al.
(2020b)), there are several ways to include a spacetime curvature dependence
in the collapse rate parameter. For example, if one were restricted only to
include the geometrical features of the spacetime, a reasonable
parameterization would be $\lambda(W^{2})$, where $W^{2}\>\equiv
W_{\alpha\beta\mu\nu}W^{\alpha\beta\mu\nu}$ with $W_{\alpha\beta\mu\nu}$ the
Weyl tensor. However, in the case of a FLRW spacetime $W=0$, hence one would
need to include the first order perturbations of the metric so $W\neq 0$. The
corresponding analysis, although valid, would require second-order
perturbation theory [see e.g. Eq. (11)] which is beyond the scope of the
present work. On the other hand, if it is assumed that the relevant curvature
scalar, present in the parameterization of $\lambda$, is that due to the
presence of a local matter distribution, then a dependence of scalars
constructed using the Riemann tensor seems to be the reasonable option
(because of Einstein’s field equations). Once again, there are many choices
for expressing $\lambda$ as a function of a curvature scalar constructed using
$R^{\alpha}_{\>\beta\mu\nu}$, e.g. $R$, $R_{\mu\nu}R^{\mu\nu}$,
$R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$, etc.555In Appendix A of Ref.
Bengochea et al. (2020b) we introduced a toy model in which the CSL parameter
$\lambda$ depends on the Kretschmann curvature scalar
$R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$ (the case for $\lambda(W^{2})$
was also considered). This model, however, did not involve a FLRW spacetime;
instead, we analyzed the spacetime corresponding to a static spherical object
of certain radius and a homogeneous mass-density distribution.

In Refs. Modak et al. (2015a, b), motivated by finding a resolution to the
black hole information loss paradox based on the CSL mechanism, the authors
assumed that $\lambda$ is determined by the Ricci scalar $R$ as

$\lambda=\lambda_{0}\left[1+\left(\frac{R}{\mu}\right)^{\alpha}\right],$ (16)

with $\lambda_{0}$ the value of the CSL parameter corresponding to the non-
relativistic CSL models, $\alpha>0$ is a free parameter and $\mu$ corresponds
to an appropriate physical scale with dimensions of length-2. We point out
that, in Refs. Modak et al. (2015a, b), the parameter $\alpha$ was constrained
to $\alpha\geq 1$. In flat spacetime (or in laboratory experiments perfomed in
regions of the spacetime where the curvature can be safely neglected) Eq. (16)
reduces to $\lambda_{0}$, i.e. the standard (non-relativistic) CSL parameter.
The advantage of Eq. (16), is that it shows explicitly how the collapse rate
is now sensitive to the local curvature. In other words, the local curvature
scalar $R$ serves to adjust the dominance of the non-linear, stochastic term
over the linear term in Eq. (8) (or Eq. (11) equivalently). Note the former
term is the one responsible for breaking the linear superposition of various
basis vectors. Of course, at the present moment, one cannot verify or falsify
the hypothesis that led to Eq. (16). However, in the future, it is quite
possible that direct observations of gravitational effects in the quantum
regime could be achieved at laboratory scales, which, in turn, will provide
strong empirical evidence in favor or against of the proposal that resulted in
Eq. (16).

Motivated by the aforementioned discussion, we will assume that the collapse
rate parameter $\lambda$, used in the CSL model considered in the present
paper, is also given by Eq. (16) identically. In order to provide a reasonable
value of $\mu$, we note first that the CSL model, shown in Eq. (8), evolves
the state of the quantum field with respect to the conformal time $\eta$.
Therefore, we are explicitly considering a particular foliation of the
spacetime, in which the states evolve from one space-like surface of constant
$\eta$ to another one. Additionally, in our proposal, the rate of collapse is
enhanced by the curvature of the spacetime, according to Eq. (16). Hence, in
principle, the collapse rate depends on the time parameter defined by the
foliation $\lambda=\lambda(\eta)$.

Assuming that the matter content in the universe can be described by the
energy-momentum tensor of a perfect fluid
$T_{\mu\nu}=(\rho+P)u_{\mu}u_{\nu}+Pg_{\mu\nu}$, where $\rho$, $P$ correspond
to the energy density and pressure in the rest frame of the fluid
respectively, and $u^{\mu}$ represents its 4-velocity (relative to the
observer). Then, from Einstein equations (in trace form) $R=-T/M_{P}^{2}$,
together with equation of state $P/\rho=\omega\simeq$ constant, one obtains

$R=\frac{\rho}{M_{P}^{2}}(1-3\omega).$ (17)

At this point, we can discuss the specific physical scale $\mu$ employed in
our case of interest. During slow roll inflation,
$\omega_{\text{inf}}\simeq-1$ and
$\rho_{\text{inf}}=3M_{P}^{2}H_{\text{inf}}^{2}\simeq$ constant, so $R\simeq
12H_{\text{inf}}^{2}$. Let us introduce a spatial physical length of size
$\ell_{\text{phys}}=a|\tau|$. This scale666We note that $|\tau|$ is, to a very
good degree of approximation, the size of the comoving particle horizon at any
time $\eta$ after the end of inflation. The particle horizon at time $\eta$
can be envisaged as the intersection of the past light cone of an observer at
point $p$ with the spacelike surface at $\eta=\tau$. Causal influences must
originate inside this region. Only comoving particles whose worldlines
intersect the past light cone of $p$ can send a signal to an observer at $p$.
Therefore, the physical particle horizon $a|\tau|$ is a good candidate for a
physical length to be compared with $R$ during inflation. Conversely, other
natural choices do not enhance the term $R/\mu$ during inflation, e.g.
assuming $\mu=\ell_{P}^{-2}$, where $\ell_{P}$ is Planck’s length, implies
$R/\mu\ll 1$ because inflation involves energy scales less than Planck scale.
So, the term $R/\mu$ does not affect the collapse rate, i.e.
$\lambda\simeq\lambda_{0}$. Evidently, a similar situation occurs if one
chooses $\mu=H_{\text{inf}}^{2}$, which is essentially $R$ during inflation,
hence $R/\mu\simeq 1$, so $\lambda\simeq\lambda_{0}$. That is, once again the
curvature term does not dominate in Eq. (16). induces the particular choice
$\mu=\ell_{\text{phys}}^{-2}$. In this case, we can estimate $|\tau|\simeq
1/(a_{\tau}H_{\text{inf}})$, and by taking into account the number of e-folds
$N$ since the beginning of inflation $a=e^{N}a_{\tau}$, we conclude that
$R/\mu\simeq e^{2N}\gg 1$. Therefore, the choice $\mu=\ell_{\text{phys}}^{-2}$
ensures that, during the full inflationary regime, the collapse rate $\lambda$
in Eq. (16), is enhanced by the term $R/\mu$. In particular, one has

$\lambda(\eta)\simeq\lambda_{0}e^{2N(\eta)\alpha}$ (18)

and from now on we will use this expression for the collapse rate when
analyzing the CSL model during inflation.

It is also relevant to analyze the behavior of $\lambda$ in the subsequent
cosmic stages after inflation. For pure radiation $\omega_{\text{rad}}=1/3$,
one obtains that $R^{\text{rad}}$ vanishes exactly. For pressure-less matter
(dust) $\omega\simeq 0$, the Ricci scalar is
$R^{\text{matt}}\simeq\rho^{\text{matt}}/M_{P}^{2}$. After inflation ends, we
can approximate the matter content in the universe by a two-component mixture
of radiation and pressure-less matter. The Ricci scalar is then
$R=\sum_{i}\rho_{i}(1-3\omega_{i})/M_{P}^{2}=R^{\text{matt}}$, where the index
$i$ labels each component of the matter content, $i=$ $\\{$rad, matt$\\}$.
Therefore, after the inflationary regime, the evolution777The total energy
density $\rho\equiv\rho_{\text{matt}}+\rho_{\text{rad}}$ evolves as
$\rho=\frac{\rho_{\text{eq}}}{2}\left[\left(\frac{a_{\text{eq}}}{a}\right)^{3}+\left(\frac{a_{\text{eq}}}{a}\right)^{4}\right].$
of the quantity $R/\mu$ is given as

$\frac{R}{\mu}=\frac{3\rho_{\text{eq}}a_{\text{eq}}^{2}}{2\rho_{\text{inf}}a_{\tau}^{2}}\frac{a_{\text{eq}}}{a}$
(19)

where we used the choice $\mu=\ell_{\text{phys}}^{-2}$.

The quantity $a_{\text{eq}}$ denotes the scale factor evaluated at the matter-
radiation equality epoch. In particular, one has
$a_{\text{eq}}=e^{N_{\text{eq}}}a_{\tau}$, where $N_{\text{eq}}$ is the total
number of e-folds from the beginning of inflation up to $a_{\text{eq}}$.
Moreover, assuming an inflationary energy scale of
$\rho_{\text{inf}}^{1/4}\simeq 10^{15}$ GeV and that the energy scale at the
radiation-matter equality epoch is $\simeq 1$ eV, we have
$\rho_{\text{eq}}\simeq 10^{-96}\rho_{\text{inf}}$. Also, if we consider that
inflation lasts approximately $60$ e-folds and that $a_{\text{eq}}\simeq
10^{-4}a_{0}$, then $N_{\text{eq}}\simeq 115$. Taking into account all these
estimates, Eq. (19) yields

$\frac{R}{\mu}\simeq\frac{a_{0}}{a}.$ (20)

The above equation implies that at the onset of the matter-radiation equality
era $R^{\text{eq}}/\mu\simeq 10^{4}$, and then it decays888 In the radiation
dominated epoch, $a<a_{\text{eq}}$, the quantity $R/\mu$ also decays as $a$.
However, taking e.g. the primordial nucleosynthesis epoch as the beginning of
the radiation dominated epoch, one obtains $R/\mu\simeq 10^{9}$, so from Eq.
(16) we find that the CSL parameter is constrained between
$\lambda_{0}10^{9\alpha}\geq\lambda(\eta)\geq\lambda_{0}10^{4\alpha}$.
Assuming the allowed experimental range $10^{-13}$s-1 $\geq\lambda_{0}\geq
10^{-19}$ s-1 and $\alpha\leq 2$, we find that the parameter $\lambda(\eta)$
is consistent with the bound $\lambda_{\text{RD}}\leq 10^{5}$s-1, which was
obtained in the radiation dominated epoch. That value was found in Ref. Lochan
et al. (2012) using the mass proportional non-relativistic CSL model and
spectral distortions in the CMB.  as $a^{-1}$. Specifically, assuming a matter
dominated regime, the quantity $R/\mu$ evaluated today is approximately of
order $10^{0}$. Consequently, in the matter dominated epoch,
$a>a_{\text{eq}}$, from expression (16), we find that the CSL parameter
$\lambda$ will be constrained between

$10^{4\alpha}\lambda_{0}\geq\lambda(\eta)\geq\lambda_{0}.$ (21)

The experimental constraints on $\lambda_{0}$ are between Gasbarri et al.
(2021)

$10^{-10}\textrm{s}^{-1}\geq\lambda_{0}\geq 10^{-19}\textrm{s}^{-1}.$

Thus, for $\alpha\leq 2$ the variation of $\lambda$ within the interval (21)
due to the matter dominated cosmological epoch, can fit the allowed
experimental range. In particular, if we consider values close to the lowest
empirical bounds, e.g. $\lambda_{0}=10^{-19}$s-1.

The previous analysis serves to illustrate that, as a result of the choice
$\mu=\ell_{\text{phys}}^{-2}$, the term $R/\mu$ becomes very dominant in Eq.
(16) during inflation. However, after the end of inflation $R/\mu$ decays as
$a^{-1}$. In particular, in the matter dominated epoch, the CSL rate parameter
is essentially the same as the non-relativistic version of the CSL model. This
is a desirable feature to be included in our model because all results and
observational bounds, regarding the CSL model, obtained from contemplating
this cosmic stage, remain intact Lochan et al. (2012); Corral et al. (2020);
Linares Cedeño and Nucamendi (2021).

###  III.4 The collapse generating operator

The next important element for the CSL cosmological model is the collapse
generating operator. As we have argued in the Introduction, we are interested
in a collapse operator that is obtained from covariant objects that reduce, in
the simple non-relativistic regimes encountered in laboratory situations, to
the “mass density” or the “energy density”. Specifically, we would like to
recover something that can be interpreted as the mass density operator used in
the mass proportional CSL model [see Eq. (3)]. One possible covariant object,
as mentioned in Bengochea et al. (2020b), is the scalar
$\Gamma\equiv(T_{\mu\nu}T^{\mu\nu})^{1/2}$. Moreover, given that our main
focus is to implement the CSL model during inflation, and obtain the
primordial spectrum, we will be working within the framework of cosmological
perturbation theory at linear order. As a consequence, our choice for the
collapse operator $C$ will be motivated by the first order perturbations
associated to the scalar $\Gamma$. In particular, the collapse operator will
be derived from:

$C(\textbf{x},\eta)\equiv\frac{1}{M_{P}}\int
d^{3}y\sqrt{|h|}s(\textbf{x},\textbf{y};\eta)\delta\Gamma(\textbf{y},\eta)$
(22)

where, $d^{3}y\sqrt{|h|}$ is the 3-volume associated to the spatial metric
$h_{ij}$ , the integration is over one of the hypersurfaces of constant time
$\eta$, and

$\delta\Gamma=\frac{1}{2(\bar{T}_{\mu\nu}\bar{T}^{\mu\nu})^{1/2}}(\bar{T}^{\mu\nu}\delta
T_{\mu\nu}+\bar{T}_{\mu\nu}\delta T^{\mu\nu})$ (23)

with $\bar{T}_{\mu\nu}$ denoting the background part. The function
$s(\textbf{x},\textbf{y};\eta)$ is a smearing function defined over the
spatial hypersurfaces of constant $\eta$ with dimensions of length-3/2 (as it
occurs in the non-relativistic CSL model). The factor $M_{P}^{-1}$ in Eq.
(22), fixes the correct dimensions of $C$ and can also be considered to play a
similar role as the reference mass usually introduced in non-relativistic CSL
models (in which the mass of a nucleon is commonly used).

With the same spirit as the original CSL model, where the smearing function
helps to localize a macroscopic body in position space, here we adopt the view
that $s(\textbf{x},\textbf{y};\eta)$, induces a “localization” of the fields
associated to $\delta\Gamma$. In other words, at first sight
$s(\textbf{x},\textbf{y};\eta)$ should depend on the “quantum uncertainties”
of the field variables rather than the uncertainties of the position operator
as in the non-relativistic CSL model.

In the inflationary case, one can find the explicit dependence on the field
variables for the operator $\delta\hat{\Gamma}$. Using Eq. (23), and taking
into account the canonical quantum variables $\hat{y},\hat{p}$, while
neglecting the terms proportional to $\Psi$ [see discussion after Eq. (14)],
one has

$\delta\hat{\Gamma}=HM_{P}\sqrt{2\epsilon}\left(\frac{\hat{p}}{a^{2}}+5H\frac{\hat{y}}{a}\right)$
(24)

In the non-relativistic CSL model, the smearing function introduces the second
parameter of the model, namely the smearing length $r_{c}$. The parameter
$r_{c}$ is a length scale characterizing the localization of the wave function
through the collapse mechanism. For a single particle, if the width of its
wave function is much less than $r_{c}$, the effect of the collapse becomes
negligible. On the contrary, if the width is larger than $r_{c}$ the collapse
effects become significant. Therefore, as we have argued, the quantum
uncertainties of the position operator, associated to the width of the wave
function in position space, play an important role in defining the smearing
function and in characterizing the parameter $r_{c}$.

In our view, it is not at all obvious that there is a simple one-on-one
connection between the values of $r_{c}$, characterizing laboratory
experiments effectively described by non-relativistic Quantum Mechanics, and
the parameters of a CSL-like model describing a complete different regime such
as the inflationary universe Bengochea et al. (2020a, b). As a matter of fact,
in this regime certain notions are lost, like the position operator of a
particle; but, more fundamental ones emerge, e.g. the quantum fields.

In order to move forward, we will adopt as fundamental the relation between
“quantum uncertainties” and the smearing function. Intuitively, we will think
the smearing function $s$ as strongly dependent on the “quantum uncertainties”
of the system under consideration. For example, if the system can be
effectively described by non-relativistic Quantum Mechanics, then “quantum
uncertainties” in the position operator should serve to define the size of the
smearing region, and thus set the value of $r_{c}$. On the other hand, if the
system we want to analyze is, e.g. the radiation emitted by Black Holes, then
$s$ should depend on the “quantum uncertainties” of the matter fields and,
quite possibly, on the background spacetime curvature. However, it is a well-
known fact that, strictly speaking, the uncertainty of a quantum field is not
well-defined generically because vacuum expectation values like
$\langle\hat{\varphi}^{2}(x)\rangle_{\text{vac.}}$ diverge. In order to take
the next step, we can focus on the two-point correlation function(s). In fact,
until a full covariant CSL theory is formulated, we can only make progress by
considering particular smearing functions with the desired characteristics.

Returning to our case of interest, the operator $\delta\hat{\Gamma}$ in Eq.
(24) depends linearly on the field variables $\hat{p}/a^{2}$ and $\hat{y}/a$.
Without loss of generality, we will analyze the 2-point function of
$\hat{y}/a$, and the corresponding analysis of $\hat{p}/a^{2}$ will proceed in
an equivalent manner. Let us recall that $\hat{y}/a=\hat{\delta}\phi$, and
concentrate on very small proper wavelengths (high-frequencies)
$\lambda_{\text{phys}}\ll H^{-1}$. In the vacuum state, characterized by the
standard Bunch-Davies vacuum Birrell and Davies (1984), the 2-point function
of $\hat{y}/a$ is

$\langle\hat{\delta}\phi(\textbf{x}_{1},\eta)\hat{\delta}\phi(\textbf{x}_{2},\eta)\rangle_{\text{BD}}\propto\frac{1}{a^{2}r^{2}}$
(25)

where $r\equiv|\textbf{x}_{1}-\textbf{x}_{2}|$. On the opposite regime,
$\lambda_{\text{phys}}\gg H^{-1}$, the 2-point function in the BD vacuum
becomes

$\langle\hat{\delta}\phi(\textbf{x}_{1},\eta)\hat{\delta}\phi(\textbf{x}_{2},\eta)\rangle_{\text{BD}}\propto
H^{2}$ (26)

Therefore, as inflation takes place, the 2-point function decreases because
the physical distance ($ar$) increases. But, as $ar$ becomes larger than
$H^{-1}$, the 2-point function remains basically a constant of order $H^{2}$.
Thus, we can make use of $H$ as characterizing the size of the smearing region
for the field variable $\hat{y}/a$ whose “uncertainty” the CSL model will
attempt to decrease. In view of the previous discussion and the analysis of
the 2-point function, we propose the following smearing function:

$s(\textbf{x},\textbf{y};\eta)\equiv\frac{H^{3/2}}{(2\pi)^{3/2}}e^{-\frac{1}{2}H^{2}a(\eta)^{2}|\textbf{x}-\textbf{y}|^{2}}$
(27)

Here, $H$ acts akin to the $r_{c}$ parameter of the non-relativistic CSL
model. To further illustrate this point, we can write the explicit form of the
collapse operator obtained from Eq. (22) by gathering all its elements. Using
the smearing function in Eq. (27), the collapse operator is then

$\hat{C}(\textbf{x},\eta)=\frac{a^{3}H^{3/2}}{(2\pi)^{3/2}M_{P}}\int
d^{3}y\>e^{-\frac{1}{2}a^{2}H^{2}r^{2}}\delta\hat{\Gamma}(\textbf{y},\eta)$
(28)

Switching to Fourier space, and using Eq. (24), the collapse operator can be
expressed as:

$\hat{C}_{\textbf{k}}(\eta)=\left(\frac{2\epsilon}{H}\right)^{1/2}e^{-\frac{k^{2}}{2a^{2}H^{2}}}\left(\frac{\hat{p}_{\textbf{k}}}{a^{2}}+5H\frac{\hat{y}_{\textbf{k}}}{a}\right)$
(29)

We observe that, for modes such that $k\gg aH$, the operator
$\hat{C}_{\textbf{k}}(\eta)$ becomes exponentially suppressed. That is, the
effect of the CSL term on the evolution of the quantum state is negligible. On
the contrary, for modes such that $k\ll aH$, the effect of the collapse
(induced by the CSL evolution), grows stronger. This is the reason why in our
proposed model $H$ acts in a similar manner as the $r_{c}$ parameter.
Furthermore, this is consistent with the fact that super-Hubble modes ($k\ll
aH$) contribute the most to the observed spectrum. Namely, we expect the
collapse affects the most to the state characterizing the modes which have the
strongest presence in the primordial inhomogeneities, and which later become
imprinted in the CMB. For a more precise analysis regarding this subject, we
invite the reader to check B.

After presenting our version of the CSL model, and how it is adapted to
inflation, in the next section, we will focus on obtaining one of the main
predictions of the inflationary paradigm: the primordial power spectrum.
However, before ending this section, let us discuss how the modified
Schrödinger equation, corresponding to the mass proportional CSL model
employed in ordinary laboratory circumstances, can be recovered from our
proposal.

The generalized collapse operator, $C(\textbf{x},\eta)$, as given by Eq. (22),
involves three distinctive elements: (i) the perturbation associated to the
quantity $\Gamma$, (ii) the smearing function $s$ and (iii) the reference mass
$M_{P}$. Regarding (i), given the definition
$\Gamma\equiv(T_{\mu\nu}T^{\mu\nu})^{1/2}$, it is clear that in the non-
relativistic limit, $\Gamma\to\rho$ where $\rho$ denotes the mass density
function.999This case is analogous to that encountered in General Relativity
when one is interested in recovering the Poisson equation for the Newtonian
potential from the Einstein field equations, $\nabla^{2}\Phi=4\pi G\rho$ . The
matter density source $\rho$ comes from the energy-momentum tensor In
particular, for several non-relativistic massive particles
$\rho(\textbf{x})=\sum_{i}m_{i}N_{i}(\textbf{x})$, where $N_{i}(\textbf{x})$
is the particle number density for a particle of type $i$ at x and $m_{i}$ is
the mass of the corresponding particle species. With respect to element (ii),
from the arguments presented in this section [recall discussion after Eq.
(24)], in the regime where non-relativistic Quantum Mechanics can be used, the
smearing function takes the form $s\to(\pi
r_{c}^{2})^{-3/4}\>e^{-|\textbf{x}-\textbf{y}|^{2}/2r_{c}^{2}}\>$. And as
regards to (iii), the reference mass suitable at laboratory scales corresponds
to the mass of a nucleon $m_{0}$ instead of the reduced Planck mass $M_{P}$.
Furthermore, since ordinary experiments are performed in regions of the
spacetime where the local curvature can be ignored, from Eq. (16), we have
that $\lambda(R)\to\lambda_{0}$. In view of the previous arguments, in the
non-relativistic quantum regime, the CSL term proposed in Eq. (8), which
modifies the Schrödinger equation, reduces to

$\displaystyle\int_{\tau}^{\eta}d\eta\int
d^{3}x\sqrt{|g|}\frac{1}{4\lambda(R)}\bigg{(}W(\textbf{x},\eta)-2\lambda(R)\hat{C}(\textbf{x},\eta)\bigg{)}^{2}$
(30) $\displaystyle\to$ $\displaystyle\int_{t_{i}}^{t}dt\int
d^{3}x\frac{1}{4\lambda_{0}}\bigg{(}w(\textbf{x},t)-2\lambda_{0}\hat{A}(\textbf{x},t)\bigg{)}^{2},$

where $\hat{A}(\textbf{x},t)$ is given in Eq. (3). Note that we have
deliberately made a distinction between $w$ and $W$ due to the fact that there
is very little knowledge about the physical origin of this term, so, in
principle we cannot assure that the noise term is the same for all physical
systems and for all scales (however its statistical features are the same).

##  IV The Newtonian potential and an equivalent power spectrum

In this section we present the derivation of an important equation, relating
the primordial curvature perturbation and the proposed CSL model of the
previous section. Also, we provide an expression for the primordial power
spectrum, which is equivalent but not exactly equal to the standard one.

The perturbed Einstein equations (EE) at linear order in the longitudinal
gauge can be combined into Mukhanov et al. (1992); Mukhanov (2005):

$\nabla^{2}u=z\left(\frac{v}{z}\right)^{\prime},\qquad
v=\theta\left(\frac{u}{\theta}\right)^{\prime}$ (31)

with the following definitions:

$v\equiv
a\left(\delta\phi+\frac{\phi_{0}^{\prime}}{\mathcal{H}}\Psi\right),\qquad
u\equiv\frac{2M_{P}^{2}a\Psi}{\phi_{0}^{\prime}}$ (32a)
$z\equiv\frac{a\phi_{0}^{\prime}}{\mathcal{H}},\qquad\theta\equiv\frac{1}{z}.$
(32b)

By using the definition of the slow roll parameter $\epsilon$ and
$z=aM_{P}\sqrt{2\epsilon}$, we can combine equations (31) to obtain

$\Psi+\mathcal{H}^{-1}\Psi^{\prime}=\sqrt{\frac{\epsilon}{2}}\frac{\langle\hat{y}\rangle}{aM_{P}}$
(33)

where we have made use of the SCG approach. Also, Eq. (33) is exact, i.e no
approximations were made. This equation relates the quantum expectation value
of the matter degrees of freedom, obtained from the CSL collapse mechanism,
and the primordial curvature perturbation represented by the Newtonian
potential, which is always classical. In particular, in the vacuum state
$\langle\hat{y}\rangle_{\text{vac}}=0$, meaning that no curvature
perturbations are present. It is only after the collapse has taken place that
$\langle\hat{y}\rangle\neq 0$, and the primordial perturbation is “born.”

We introduce a well-known quantity, defined generically as

$\mathcal{R}\equiv\Psi+\left(\frac{2\rho}{3}\right)\left(\frac{\mathcal{H}^{-1}\Psi^{\prime}+\Psi}{\rho+P}\right)$
(34)

where $\rho$ and $P$ are associated to the type of matter driving the
expansion of the universe. A main feature of the quantity $\mathcal{R}$ is
that, for adiabatic perturbations, it is conserved for super-Hubble scales,
irrespective of the cosmological epoch one is considering Mukhanov et al.
(1992).

From the components of the energy-momentum tensor of a single scalar field, we
have $\rho+P=\phi_{0}^{\prime
2}/a^{2}=M_{P}^{2}\mathcal{H}^{2}2\epsilon/a^{2}$, and because of Friedmann’s
equation $\mathcal{H}^{2}=a^{2}\rho/3M_{P}^{2}$, one obtains

$\mathcal{R}=\Psi\left(1+\frac{1}{\epsilon}\right)+\frac{\mathcal{H}^{-1}}{\epsilon}\Psi^{\prime}$
(35)

The above equation is exact, but during slow roll inflation $\epsilon\ll 1$;
consequently, at the lowest order in $\epsilon$, we can approximate the latter
expression as

$\mathcal{R}\simeq\frac{1}{\epsilon}\left(\Psi+\mathcal{H}^{-1}\Psi^{\prime}\right)=\frac{\langle\hat{y}\rangle}{aM_{P}\sqrt{2\epsilon}}$
(36)

where in the last equality we have used our main equation (33).

Another important aspect of the quantity $\mathcal{R}$ is that, in the
comoving gauge, it represents the curvature perturbation. In fact, the
primordial power spectrum usually shown in the literature is associated to
$\mathcal{R}$. The scalar power spectrum (associated to the curvature
perturbation in the comoving gauge and in Fourier space) is defined as

$\overline{\mathcal{R}_{\textbf{k}}\mathcal{R}^{*}_{\textbf{q}}}\equiv\frac{2\pi^{2}}{k^{3}}\mathcal{P}_{s}(k)\delta(\textbf{k}-\textbf{q})$
(37)

where $\mathcal{P}_{s}(k)$ is the dimensionless power spectrum. The bar
appearing in (37) denotes an ensemble average over possible realizations of
the stochastic field $\mathcal{R}_{\textbf{k}}$. In the CSL inflationary
model, each realization will be associated to a particular realization of the
stochastic process characterizing the collapse mechanism, which in turn is
related to a specific realization of the noise $W$.

On the other hand, our main equation (33), was obtained in the longitudinal
gauge. Fortunately, Eq. (34) relates $\Psi$, $\Psi^{\prime}$ and $\mathcal{R}$
exactly. That is, we can compute the curvature perturbation in the
longitudinal gauge (where the SCG framework together with the CSL collapse
mechanism generate the primordial inhomogeneities as given in Eq. (33)), and
then, we can switch to the comoving gauge in order to compare the primordial
spectrum obtained in our model with the standard one. In particular, we can
use approximation (36) to compute the scalar power spectrum, associated to
$\mathcal{R}_{\textbf{k}}$, that results from our main equation (33). This is,

$\overline{\mathcal{R}_{\textbf{k}}\mathcal{R}_{\textbf{q}}^{*}}=\frac{1}{2M_{P}^{2}a^{2}\epsilon}\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}$
(38)

Finally, from definition (37) and Eq. (38), we can identify an equivalent
scalar power spectrum as:

$\mathcal{P}_{s}(k)\delta(\textbf{k}-\textbf{q})=\frac{k^{3}}{4\pi^{2}M_{P}^{2}a^{2}\epsilon}\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}$
(39)

##  V Analysis of the scalar power spectrum

The next step is to use the CSL model described in section III, to compute the
expectation values in Eq. (39). Furthermore, one needs to evaluate
$\mathcal{P}_{s}(k)$ in the super-Hubble regime $-k\eta\to 0$ to be able to
compare the corresponding theoretical prediction with the observational data.
The calculations are long but straightforward; all the technical details
employed are presented in A.

At the lowest order in the slow roll parameter $\epsilon$, the primordial
power spectrum obtained is

$\displaystyle\mathcal{P}_{s}(k)$ $\displaystyle=$
$\displaystyle\frac{H^{2}}{8\pi^{2}M_{P}^{2}\epsilon}\bigg{\\{}1-\frac{2\sigma_{k}}{9}\bigg{[}21+29\gamma_{\varepsilon}-58N$
(40) $\displaystyle+$ $\displaystyle
58\ln(-k\tau)\bigg{]}-F(\sigma_{k},-k\eta)|_{-k\eta\to 0}\bigg{\\}}$

where $N$ is the total number of e-folds since the beginning of inflation, and
$\gamma_{\varepsilon}$ is the Euler-Mascheroni constant. We also introduce the
coefficient

$\sigma_{k}\equiv k^{3}|\tau|^{3}\frac{\lambda_{0}\epsilon}{H}$ (41)

and the function $F(\sigma_{k},-k\eta)|_{-k\eta\to 0}$ is defined in (99) [see
also Eq. (98)].

One of the main assumptions used to obtain the primordial spectrum (40) was to
fix the free parameter $\alpha=3/2$, see Eq. (18). This is consistent with the
previous analysis, in which for $\alpha\leq 2$ and the matter dominated
cosmological epoch, the parameter $\lambda$ can fit the experimental bounds
coming from the non-relativistic CSL model. We have analyzed other values of
the free parameter, for example $\alpha=1,2$ , and we found that these values
do not yield the correct shape of the spectrum, i.e. the one favored by
observational data.

We proceed to examine the shape and amplitude of the predicted power spectrum
(40). As shown in A, in the super-Hubble regime, the function $F$ appearing in
the expression for the power spectrum behaves as

$F(\sigma_{k},-k\eta)|_{-k\eta\to 0}\simeq(-k\eta)^{6}\to 0$ (42)

which means that $F(\sigma_{k},-k\eta)|_{-k\eta\to 0}\ll 1$. The latter result
implies that, in Eq. (40), the last term of $\mathcal{P}_{s}(k)$ can be
neglected.

Additionally, we note that it is possible to express $F(\sigma_{k},-k\eta)$ as

$F(\sigma_{k},-k\eta)=\frac{\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}-\overline{\langle\hat{y}_{\textbf{k}}\rangle^{2}}}{\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}_{\text{BD}}}.$
(43)

Hence, the fact that $F(\sigma_{k},-k\eta)\ll 1$, when $-k\eta\to 0$, means
that the width of the wave functional associated to $y_{\textbf{k}}$, which in
the BD vacuum state is a Gaussian centered at zero, is decreasing during
inflation due to the CSL mechanism. In other words, the CSL process is
effectively localizing (or “collapsing”) the initial wave function.

On the contrary, if we turn off the collapse mechanism (i.e. $\lambda=0$),
then $\sigma_{k}=0$ and $F(0,-k\eta)=1$ [see also Eq. (98)]. Thus, in the
absence of the CSL collapse term, $\mathcal{P}_{s}(k)=0$. This is an expected
result because if there is no collapse, then the quantum state of the field
remains homogeneous/isotropic, and, as a consequence, there are no
inhomogeneities/anisotropies in the spacetime either.

The expression for $\mathcal{P}_{s}(k)$ was obtained working at the lowest
order in the slow roll parameter $\epsilon$. Henceforth, at this order, the
predicted spectrum must be nearly scale invariant for it to be consistent with
the observational data. At the next leading order in $\epsilon$, one could
find the standard scale dependence reflected in the so called scalar spectral
index. It is thus clear that any significant departure from a perfect scale
invariant spectrum in Eq. (40) would mean a fatal failure for the CSL model
proposed in this work. We have argued that the last term in Eq. (40) can be
neglected if $\lambda\neq 0$. Therefore, the other term that might induce a
strong departure from scale invariance is the one with the coefficient
$\sigma_{k}$. Let us focus on this term.

The observed amplitude of the temperature aniso-tropies in the CMB constrains
the value of $H$ during inflation, and this value is approximately $H\simeq
10^{-5}M_{P}\epsilon^{1/2}$. Moreover, we can choose a particular value of
$\lambda_{0}$ consistent with laboratory experiments, for instance
$\lambda_{0}\simeq 10^{-17}$s-1, or in Planck units $\lambda_{0}\simeq
10^{-61}M_{P}$. From the definition of $\sigma_{k}$, Eq. (41), it then follows
the estimate

$\sigma_{k}\simeq(k|\tau|)^{3}\epsilon^{1/2}10^{-56}$ (44)

Considering a typical energy scale for inflation $V^{1/4}\simeq 10^{-3}M_{P}$,
implies $\epsilon\simeq 10^{-3}$. Assuming also a total duration for inflation
of $N=65$, yields $|\tau|\simeq 10^{5}$ Mpc. In addition, by taking into
account $k$ in the range of observational interest Akrami et al. (2020), one
has $k\in[10^{-4},10^{-1}]$ Mpc-1. Putting together all these values, we find
that the coefficient is within the following estimated range:

$\sigma_{k}\in[10^{-55},10^{-46}]$ (45)

Consequently, $\sigma_{k}\ll 1$ for the range of $k$ that contributes the most
to the observed spectrum. We point out that other values of $\lambda_{0}$,
within the window allowed by laboratory experiments, also lead to a strong
suppression of the coefficient $\sigma_{k}$.

The former analysis indicates that the primordial power spectrum (40) is scale
invariant to a very good degree of approximation. Thus, we have found that, at
the lowest order in the slow roll parameter, the predicted spectrum is
essentially the same as the traditional one, including its amplitude, i.e.

$\mathcal{P}_{s}(k)\simeq\frac{H^{2}}{8\pi^{2}M_{P}^{2}\epsilon}$ (46)

##  VI Conclusions

The adequate implementation of the CSL model during inflation has been
critically analyzed in recent works Martin and Vennin (2020); Bengochea et al.
(2020b, a); Martin and Vennin (2021a); Gundhi et al. (2021); Martin and Vennin
(2021b). In particular, in Ref. Bengochea et al. (2020b) the most important
elements to be considered were discussed, according to our point of view.
These are: (i) a suitable framework for treating the quantum degrees of
freedom and gravity, which we think is provided by the semiclassical gravity
approximation, (ii) a collapse-generating operator that can be constructed
from covariant objects and, at the same time, can be interpreted as the “mass
density” operator usually encountered in the laboratory applications of the
CSL model, and (iii) a generalization of the CSL parameters that includes a
dependence on the spacetime curvature. In this work, we have provided a
concrete realization based on these three hypotheses. We have shown that the
proposed model yields a primordial scalar power spectrum with the
characteristics required to be consistent with observational CMB data, i.e. we
have found a nearly scale invariant power spectrum. Moreover, there is no
conflict between our proposal and empirical constraints, found in laboratory
experiments, on the parameters of the non-relativistic version of the CSL
model.

On the other hand, given that our model is not fully covariant, some potential
problems might arise. However, we have taken a first step in exploring the
theoretical landscape corresponding to the extrapolation of the CSL model
suitable to more complex systems, in which gravity and quantum fields play an
important role. Evidently, open issues remain to be analyzed, and we hope to
contribute in addressing some of them in the future ahead.

###### Acknowledgements.

The authors thank the anonymous referees for their comments and suggestions.
G.L. is supported by CONICET (Argentina), and also acknowledges support from
the following project grants: Universidad Nacional de La Plata I+D G175 and
PIP 112-2020-0100729CO of CONICET (Argentina). G.R.B. is supported by CONICET
(Argentina) and he acknowledges support from grant: PIP 112-2017-0100220CO of
CONICET (Argentina). We thank Philip Pearle for encouraging this study. We are
also especially grateful to Daniel Sudarsky for his constructive comments and
suggestions.

##  Appendix A Computation of the primordial spectrum

The standard primordial power spectrum is expressed in Fourier space. So, we
start by defining the Fourier transform of any scalar field as:

$f(\textbf{x},\eta)=\frac{1}{(2\pi)^{3/2}}\int_{\mathbb{R}^{3}}d^{3}k\>f_{\textbf{k}}(\eta)e^{i\textbf{k}\cdot\textbf{x}}$
(47)

In Fourier space, Eq. (III.2) takes the form

$\displaystyle\int_{\mathbb{R}^{3+}}d^{3}k\quad
a^{4}\hat{\mathcal{H}}_{\textbf{k}}=$
$\displaystyle\int_{\mathbb{R}^{3+}}d^{3}k\quad\left[\hat{p}^{*}_{\textbf{k}}\hat{p}_{\textbf{k}}+\hat{y}^{*}_{\textbf{k}}\hat{y}_{\textbf{k}}\left(k^{2}-\frac{a^{\prime\prime}}{a}+a^{2}V_{,\phi\phi}\right)\right].$

The quantization procedure will be carried out in the Schrödinger picture, so
it is more convenient to work with real variables, which later can be
associated to Hermitian operators. In this way, we separate the canonical
variables in their real and imaginary parts. In Fourier space, this is given
as

$\hat{y}_{\textbf{k}}\equiv\frac{1}{\sqrt{2}}(\hat{y}_{\textbf{k}}^{\text{R}}+i\hat{y}_{\textbf{k}}^{\text{I}}),\qquad\hat{p}_{\textbf{k}}\equiv\frac{1}{\sqrt{2}}(\hat{p}_{\textbf{k}}^{\text{R}}+i\hat{p}_{\textbf{k}}^{\text{I}}).$
(49)

The quantum commutator in Eq. (14) implies

$[\hat{y}_{\textbf{k}}^{s},\hat{p}_{\textbf{q}}^{s^{\prime}}]=i\delta(\textbf{k}-\textbf{q})\delta_{ss^{\prime}}$
(50)

where $s=$R,I and $\delta_{ss^{\prime}}$ is Kronecker’s delta.

The quantum “free” Hamiltonian term corresponding to (A) is then

$\int_{\mathbb{R}^{3+}}d^{3}k\>a^{4}\hat{\mathcal{H}}_{\textbf{k}}=\int_{\mathbb{R}^{3+}}d^{3}k(\hat{H}^{\text{R}}_{\textbf{k}}+\hat{H}^{\text{I}}_{\textbf{k}})$
(51)

with the following definitions

$\hat{H}^{R,I}_{\mathbf{k}}\equiv\frac{(\hat{p}_{\mathbf{k}}^{R,I})^{2}}{2}+\frac{(\hat{y}_{\mathbf{k}}^{R,I})^{2}}{2}\left(k^{2}-\frac{a^{\prime\prime}}{a}+a^{2}V_{,\phi\phi}\right).$
(52)

In the Schrödinger approach, the quantum state of the system is described by a
wave functional, $\Phi[y(\textbf{x},\eta)]$. In Fourier space (and since the
theory is still free in the sense that it does not contain terms with power
higher than two in the Lagrangian), the wave functional can also be factorized
into mode components as

$\Phi[y(\textbf{x},\eta)]=\prod_{\textbf{k}}\Phi_{\textbf{k}}(y_{\textbf{k}}^{\text{R}},y_{\textbf{k}}^{\text{I}})=\prod_{\textbf{k}}\Phi_{\textbf{k}}^{\text{R}}(y_{\textbf{k}}^{\text{R}})\Phi_{\textbf{k}}^{\text{I}}(y_{\textbf{k}}^{\text{I}}).$
(53)

In the field representation, the operators would take the form:

$\hat{y}_{\textbf{k}}^{\text{R,I}}\Phi_{\textbf{k}}^{\text{R,I}}=y_{\textbf{k}}^{\text{R,I}}\Phi_{\textbf{k}}^{\text{R,I}},\qquad\hat{p}_{\textbf{k}}^{\text{R,I}}\Phi_{\textbf{k}}^{\text{R,I}}=-i\frac{\partial\Phi_{\textbf{k}}^{\text{R,I}}}{\partial
y_{\textbf{k}}^{\text{R,I}}}.$ (54)

The decomposition shown in Eq. (53) and the linear evolution equation (8)
imply that the quantum state of each mode evolves independently (see also
Martin and Vennin (2021b)). Thus, in Fourier space, the CSL evolution equation
of the quantum state, corresponding to the wave functional(s) of each mode
$\Phi_{\textbf{k}}^{\text{\text{R,I}}}(y_{\textbf{k}}^{\text{\text{R,I}}})$,
is

$\displaystyle|\Phi_{\textbf{k}}^{\textrm{R,I}},\eta\rangle$ $\displaystyle=$
$\displaystyle\hat{T}\exp\bigg{\\{}\int_{\tau}^{\eta}d\eta\bigg{[}-i\hat{H}^{\text{R,I}}_{\textbf{k}}$
(55) $\displaystyle-$
$\displaystyle\frac{a^{4}}{4\lambda}\bigg{(}W_{\textbf{k}}(\eta)^{\text{R,I}}-2\lambda\hat{C}_{\textbf{k}}^{\text{R,I}}(\eta)\bigg{)}^{2}\bigg{]}\bigg{\\}}$
$\displaystyle\times$
$\displaystyle|\Phi_{\textbf{k}}^{\text{R,I}},\tau\rangle$

where the collapse generating operator (in Fourier space) is given as

$\hat{C}_{\mathbf{k}}^{\text{R,I}}=e^{-\frac{k^{2}}{2a^{2}H^{2}}}\left(\frac{2\epsilon}{H}\right)^{1/2}\left(\frac{\hat{p}_{\mathbf{k}}^{\text{R,I}}}{a^{2}}+5H\frac{\hat{y}_{\mathbf{k}}^{\text{R,I}}}{a}\right)$
(56)

and the Probability Rule is

$P(W_{\textbf{k}}^{\text{R,I}})dW_{\textbf{k}}^{\text{R,I}}=\langle\Phi_{\textbf{k}}^{\text{R,I}},\eta|\Phi_{\textbf{k}}^{\text{R,I}},\eta\rangle\prod_{\eta^{\prime}=\tau}^{\eta-d\eta}\frac{a^{2}dW_{\textbf{k}}(\eta^{\prime})^{\text{R,I}}}{\sqrt{2\pi\lambda/d\eta}}$
(57)

We will adopt the standard assumption regarding the vacuum state. That is, at
an early conformal time $\tau$, the modes are in their adiabatic ground state,
which is a Gaussian centered at zero with certain spread. This ground state is
commonly referred to as the Bunch-Davies (BD) vacuum. Taking into account that
the initial quantum state is Gaussian, and the CSL evolution equation (55) is
quadratic in the canonical variables
$\hat{y}_{\textbf{k}}^{\text{R,I}},\hat{p}_{\textbf{k}}^{\text{R,I}}$, the
form of the state vector in the field basis at any time is

$\Phi^{R,I}(\eta,y_{\textbf{k}}^{R,I})=\exp[-A_{k}(\eta)(y_{\textbf{k}}^{R,I})^{2}+B_{k}(\eta)y_{\textbf{k}}^{R,I}+C_{k}(\eta)].$
(58)

Therefore, the wave functional evolves according to the CSL equation (55),
with initial conditions given by

$A_{k}(\tau)=\frac{k}{2},\qquad B_{k}(\tau)=C_{k}(\tau)=0.$ (59)

Those initial conditions correspond to the BD vacuum, which is perfectly
homogeneous and isotropic in the sense of a vacuum state in quantum theory.

Considering that our main goal is to obtain the primordial power spectrum, see
Eq. (39), we focus our efforts in computing the quantity
$\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}$.
The expectation values, of course, will be evaluated at the evolved state
provided by (55). In terms of the real and imaginary parts we have

$\overline{\langle\hat{y}_{\textbf{k}}\rangle\langle\hat{y}_{\textbf{q}}\rangle^{*}}=\left(\overline{\langle\hat{y}_{\textbf{k}}^{\text{R}}\rangle^{2}}+\overline{\langle\hat{y}_{\textbf{k}}^{\text{I}}\rangle^{2}}\right)\delta(\textbf{k}-\textbf{q}).$
(60)

Also, from (60), it is clear that we are interested in computing the
quantities $\overline{\langle\hat{y}_{\textbf{k}}^{\text{R,I}}\rangle^{2}}$.
In fact, the calculation of the real and imaginary part are exactly the same,
so we will only focus on one of them. Additionally, we will simplify the
notation by omitting the indexes R,I from now on.

Using the Gaussian wave function (58), and the two main CSL equations for the
mode’s quantum state [Eqs. (55) and (57)], we obtain

$\overline{\langle\hat{y}_{\textbf{k}}\rangle^{2}}=\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}-\frac{1}{4\textrm{Re}[A_{k}(\eta)]}.$
(61)

###  A.1 First term of (61)

We now turn our attention to the first term on the right hand side of (61),
i.e $\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}$. The two main CSL
equations for the quantum state of each mode, Eqs. (55) and (57), can be used
to obtain the equation of evolution for the density matrix. This yields one
equation per Fourier mode, which can be written as

$\frac{\partial\hat{\rho}_{\textbf{k}}(\eta)}{\partial\eta}=-i[\hat{H}_{\textbf{k}},\hat{\rho}_{\textbf{k}}(\eta)]-\frac{\lambda
a^{4}}{2}[\hat{C}_{\textbf{k}}(\eta),[\hat{C}_{\textbf{k}}(\eta),\hat{\rho}_{\textbf{k}}(\eta)]]$
(62)

The latter equation can also be obtained form the equation of evolution of the
density matrix (11) in Fourier space. Thus, it serves as a consistency check.

The importance of Eq. (62) is that one can use it to derive the equation of
evolution for the stochastic mean of the quantum expectation value of any
operator $\hat{O}_{\textbf{k}}$. This is,

$\frac{\partial}{\partial\eta}\overline{\langle\hat{O}_{\textbf{k}}\rangle}=-i\overline{\langle[\hat{O}_{\textbf{k}},\hat{H}_{\textbf{k}}]\rangle}-\frac{a^{4}\lambda}{2}\overline{\langle[\hat{C}_{\textbf{k}},[\hat{C}_{\textbf{k}},\hat{O}_{\textbf{k}}]]\rangle}$
(63)

We take into account that $\lambda$ during inflation is given by Eq. (18).
This is,

$\lambda(\eta)=\lambda_{0}\left(\frac{a(\eta)}{a_{\tau}}\right)^{2\alpha}$
(64)

Moreover, by choosing $\alpha=3/2$ and estimating $a_{\tau}\simeq
1/(H|\tau|)$, we obtain

$\lambda(\eta)=\lambda_{0}H^{3}|\tau|^{3}a(\eta)^{3}$ (65)

We define the quantities
$Q\equiv\overline{\langle\hat{y}_{\textbf{k}}^{2}\rangle}$,
$R\equiv\overline{\langle\hat{p}_{\textbf{k}}^{2}\rangle}$ and
$T\equiv\overline{\langle\hat{p}_{\textbf{k}}\hat{y}_{\textbf{k}}+\hat{y}_{\textbf{k}}\hat{p}_{\textbf{k}}\rangle}$,
notation warning: do not confuse this $R$ with the Ricci scalar. The evolution
equations of $Q$, $R$ and $T$ obtained from (63) are:

$Q^{\prime}=T+a^{3}\lambda_{1}e^{-\frac{k^{2}}{a^{2}H^{2}}}$ (66a)
$R^{\prime}=-w_{k}(\eta)T+a^{5}\lambda_{1}25H^{2}e^{-\frac{k^{2}}{a^{2}H^{2}}}$
(66b) $T^{\prime}=2R-2Qw_{k}(\eta)$ (66c)

where

$w_{k}(\eta)\equiv k^{2}-\frac{a^{\prime\prime}}{a}+a^{2}V_{,\phi\phi}$ (67)

and

$\lambda_{1}\equiv\lambda_{0}2\epsilon H^{2}|\tau|^{3}$ (68)

We combine Eqs. (66) to obtain a single differential equation for $Q$,

$Q^{\prime\prime\prime}+4w_{k}(\eta)+2Qw_{k}^{\prime}(\eta)=S(\eta)$ (69)

where the source function $S$ is

$\displaystyle S$ $\displaystyle\equiv$ $\displaystyle
2\lambda_{1}e^{-\frac{k^{2}}{a^{2}H^{2}}}\bigg{[}a^{3}\bigg{(}5k^{2}+w_{k}\bigg{)}$
(70) $\displaystyle+$ $\displaystyle
31a^{5}H^{2}+\frac{2ak^{4}}{H^{2}}\bigg{]}$

In Ref. Martin and Vennin (2020) it is shown that Eq. (69) can be solved by
introducing the Green function of the free theory,

$G(\eta,\tilde{\eta})=\frac{1}{W}\left[y_{k}^{*}(\tilde{\eta})y_{k}(\eta)-y_{k}(\tilde{\eta})y_{k}^{*}(\eta)\right]\Theta(\eta-\tilde{\eta})$
(71)

where $y_{k}(\eta)$ is a solution of

$y^{\prime\prime}_{k}(\eta)+\left[k^{2}-w_{k}(\eta)\right]y_{k}(\eta)=0$ (72)

and $W=y_{k}^{\prime}y_{k}^{*}-y_{k}^{*^{\prime}}y_{k}$ is the Wronskian,
which for Eq. (72) is a constant. The solution to Eq. (69) is then

$Q_{k}(\eta)=|y_{k}(\eta)|^{2}+\frac{1}{2}\int_{\tau}^{\eta}d\tilde{\eta}\>S(\tilde{\eta})G(\eta,\tilde{\eta})^{2}$
(73)

where $\tau\to-\infty$ is the conformal time at the beginning of inflation.

Considering the approximated form of the scale factor during inflation
$a(\eta)\simeq-1/H\eta$, with $H\simeq$ constant; one has $w_{k}(\eta)\simeq
k^{2}-2/\eta^{2}$. The solution of (72) is then

$y_{k}(\eta)=\frac{1}{\sqrt{2k}}\left(1+\frac{i}{k\eta}\right)e^{ik\eta}$ (74)

Consequently, the Green function (71) is

$\displaystyle G(\eta,\tilde{\eta})$ $\displaystyle=$
$\displaystyle\frac{1}{k^{3}\eta\tilde{\eta}}\bigg{(}(1+k^{2}\eta\tilde{\eta})\sin[k(\eta-\tilde{\eta})]$
(75) $\displaystyle-$ $\displaystyle
k(\eta-\tilde{\eta})\cos[(\eta-\tilde{\eta})]\bigg{)}\Theta(\eta-\tilde{\eta})$

Moreover, in the super-Hubble limit, we have

$G(\eta,\tilde{\eta})|_{(-k\eta,-k\tilde{\eta})\to(0,0)}\simeq\frac{\eta^{3}-\tilde{\eta}^{3}}{3\eta\tilde{\eta}}\Theta(\eta-\tilde{\eta})$
(76)

On the other hand, the amplitude of the source function $S(\tilde{\eta})$ is
exponentially suppressed in the regime $k\tau\to-\infty$, while it reaches its
maximum in the opposite regime $-k\tilde{\eta}\to 0$. Therefore, when
computing the integral in (73), we can consider the approximated expression
(76); i.e. we use the form of $G(\eta,\tilde{\eta})$ in the super-Hubble
limit.

Furthermore, we are interested in computing the power spectrum that is used to
compare the theoretical predictions with the observational data. Specifically,
we have to evaluate $Q$ in the regime of observational interest, i.e. in the
super-Hubble limit. After taking this limit, at the leading order we finally
obtain

$\displaystyle Q_{k}(\eta)|_{-k\eta\to 0}$ $\displaystyle=$
$\displaystyle\frac{1}{2k^{3}\eta^{2}}\bigg{\\{}1-\frac{2\sigma_{k}}{9}\bigg{[}21+29\gamma_{\varepsilon}-58N$
(77) $\displaystyle+$ $\displaystyle 58\ln(k|\tau|)\bigg{]}\bigg{\\}}$

where $N$ is the total number of e-foldings from the beginning of inflation,
$\gamma_{\varepsilon}$ is the Euler-Mascheroni constant and the coefficient
$\sigma_{k}$ is defined in (41).

###  A.2 Second term of (61)

Continuing with the second half of the calculations, we focus now on the
second term on the right hand side of (61); i.e. the term
$[4\text{Re}(A_{k})]^{-1}$. We take the time derivative of (55), obtaining

$\frac{\partial}{\partial\eta}|\Phi_{\textbf{k}}^{\textrm{R,I}},\eta\rangle=-i\hat{H}_{\textbf{k}}^{\textrm{R,I}}+\hat{H}_{\textbf{k}\>\textrm{CSL}}^{\textrm{R,I}}|\Phi_{\textbf{k}}^{\textrm{R,I}},\eta\rangle$
(78)

with

$\displaystyle\hat{H}_{\textbf{k}\>\textrm{CSL}}^{\textrm{R,I}}$
$\displaystyle\equiv$
$\displaystyle-\frac{a^{4}(W^{\text{R,I}}(\textbf{k},\eta))^{2}}{4\lambda}+a^{4}{W}^{\text{R,I}}(\textbf{k},\eta)\hat{C}_{\textbf{k}}^{\textrm{R,I}}$
(79) $\displaystyle-$ $\displaystyle
a^{4}(\hat{C}_{\textbf{k}}^{\textrm{R,I}})^{2}\lambda$

We apply the CSL evolution operator, as characterized by Eq. (78), to the wave
function (58) and regroup terms of order $y^{2}$, $y^{1}$ and $y^{0}$; the
evolution equations corresponding to these terms are thus decoupled.
Fortunately, the evolution equation corresponding to $y^{2}$ only contains
$A_{k}(\eta)$, which is the function we are interested in. The evolution
equation is then

$\displaystyle A_{k}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{iw_{k}}{2}-2iA_{k}^{2}+e^{-\frac{k^{2}}{a^{2}H^{2}}}\lambda_{1}\bigg{(}-4A_{k}^{2}a^{3}$
(80) $\displaystyle+$ $\displaystyle 25H^{2}a^{5}+i20HA_{k}a^{4}\bigg{)}$

where $w_{k}$ and $\lambda_{1}$ are defined in Eqs. (67) and (68)
respectively.

Performing the following change of variable

$A_{k}=\frac{1}{N}\left(\frac{f_{k}^{\prime}}{f_{k}}-g\right)$ (81)

with

$N\equiv 2(i+2\lambda_{1}a^{3}e^{-\frac{k^{2}}{a^{2}H^{2}}})$ (82)

and

$g\equiv-\frac{1}{2}\left(\frac{N^{\prime}}{N}+i20H\lambda_{1}a^{4}e^{-\frac{k^{2}}{a^{2}H^{2}}}\right)$
(83)

equation (80) can be recast in a more useful way:

$f_{k}^{\prime\prime}+P(\eta)f_{k}=0$ (84)

where

$\displaystyle P(\eta)$ $\displaystyle\equiv$
$\displaystyle-\frac{iw_{k}N}{2}-\left(g^{\prime}-g\frac{N^{\prime}}{N}\right)+g^{2}$
(85) $\displaystyle-$ $\displaystyle
25H^{2}\lambda_{1}a^{5}Ne^{-\frac{k^{2}}{a^{2}H^{2}}}$ $\displaystyle+$
$\displaystyle i20H\lambda_{1}a^{4}e^{-\frac{k^{2}}{a^{2}H^{2}}}g.$

We note that if $\lambda_{1}=0$ (i.e. no collapse) $f_{k}$ satisfies the usual
equation $f_{k}^{\prime\prime}=-w_{k}f_{k}$. The exact solution of Eq. (84) is
difficult to find, but it can be done perturbatively in $\lambda_{1}$.
Therefore, at leading order in $\lambda_{1}$, the function $P(\eta)$ is
explicitly given as

$P(\eta)=w_{k}+\Sigma(\eta)\lambda_{1}+\mathcal{O}(\lambda_{1}^{2})$ (86)

where

$\displaystyle\Sigma(\eta)$ $\displaystyle\equiv$
$\displaystyle-\frac{2ia}{H^{2}}e^{-\frac{k^{2}}{a^{2}H^{2}}}\bigg{(}11a^{4}H^{4}$
(87) $\displaystyle-$ $\displaystyle
5a^{2}H^{2}k^{2}+2k^{4}+a^{2}H^{2}w_{k}\bigg{)}.$

The perturbed solution to Eq. (84) can be written as:

$f_{k}=f^{(0)}_{k}+\lambda_{1}f^{(1)}_{k}+\mathcal{O}(\lambda_{1}^{2})$ (88)

The function $f^{(0)}_{k}$ is the solution of the mode equation for
$\lambda_{1}=0$. Specifically, we can choose the normalized mode function in
the Bunch-Davies vacuum, which at leading order in slow roll is given by

$f^{(0)}_{k}=\frac{1}{\sqrt{2k}}\left(1+\frac{i}{k\eta}\right)e^{ik\eta}$ (89)

Inserting expansions (86), (88) in (84), one finds that the function
$f^{(1)}_{k}$ satisfies

$f^{(1)^{\prime\prime}}_{k}+w_{k}f^{(1)}_{k}=-\Sigma(\eta)f^{(0)}_{k}$ (90)

In order to solve Eq. (90), we will work at the leading order in slow roll,
this is, $a(\eta)\simeq-1/H\eta$, which implies $w_{k}=k^{2}-2/\eta^{2}$.
Therefore, the function $\Sigma(\eta)$ is explicitly

$\Sigma(\eta)=\frac{2ie^{-k^{2}\eta^{2}}}{\eta^{5}H^{3}}\left(9-4k^{2}\eta^{2}+2\eta^{4}k^{4}\right)$
(91)

The solution to Eq. (90) is

$f^{(1)}_{k}(\eta)=-\int_{\tau}^{\eta}d\tilde{\eta}\>\Sigma(\tilde{\eta})f^{(0)}_{k}(\eta)G(\eta,\tilde{\eta})$
(92)

where $G(\eta,\tilde{\eta})$ is the Green function introduced previously, Eq.
(75). As before, the amplitude of the source term is exponentially suppressed
in the regime $k\tau\to-\infty$, so the mayor contribution to the integral in
Eq. (92) comes from the super-Hubble limit. This means that we can use the
approximated expressions for $f^{(0)}_{k}(\eta)$ and $G(\eta,\tilde{\eta})$,
Eq.(76) in such a limit. The result is

$\displaystyle f^{(1)}(x)$ $\displaystyle=$
$\displaystyle\frac{k^{5/2}}{H^{3}6\sqrt{2}}\frac{e^{-x^{2}}}{x^{4}}\bigg{[}6-33x^{2}-11x^{4}$
$\displaystyle+$ $\displaystyle
11e^{x^{2}}x^{3}\bigg{(}4\sqrt{\pi}(1-\text{Erf}[x])-x^{3}\text{Ei}[-x^{2}]\bigg{)}\bigg{]},$

where we have used the definition $x\equiv-k\eta$. The function Ei$(z)$ is the
exponential integral function defined as

$\text{Ei}(z)\equiv-\int_{-z}^{\infty}dt\>\frac{e^{-t}}{t}$ (94)

and Erf$(z)$ is the Gaussian error function

$\text{Erf}(z)\equiv\frac{2}{\sqrt{\pi}}\int_{0}^{z}dt\>e^{-t^{2}}$ (95)

Returning to the original variable $A_{k}$, Eq. (81), with the perturbed
solution in terms of Eqs. (89) and (A.2), we obtain

$\displaystyle A_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2}k\left(1-\frac{1}{x(x+i)}\right)+\frac{\lambda_{1}k^{4}e^{-x(i+x)}}{12H^{3}x^{4}(x+i)^{2}}$
(96) $\displaystyle\times$
$\displaystyle\bigg{\\{}-18-6e^{ix}(1+x)(-5i-5x+2x^{3})$ $\displaystyle+$
$\displaystyle x\left[18i+x(15-21ix+11x^{3}(3i+x))\right]+11e^{x^{2}}x^{5}$
$\displaystyle\times$
$\displaystyle\left(-4\sqrt{\pi}\text{Erf}[x]+x(-3+x(3i+x))\text{Ei}[-x^{2}]\right)\biggr{\\}}$
$\displaystyle+$ $\displaystyle\mathcal{O}(\lambda_{1}^{2})$

From the observational point of view, we are interested in taking the limit
$x\to 0$. Retaining the dominant terms in the perturbed solutions, we find
that the real part of $A_{k}$ can be approximated by

$\text{Re}(A_{k})|_{x\to
0}\simeq\frac{kx^{2}}{2}\left(1+8\frac{\lambda_{1}k^{3}}{H^{3}x^{6}}\right)$
(97)

In fact, we rewrite the previous expression using the coefficient $\sigma_{k}$
defined in Eq. (41), which yields

$\text{Re}(A_{k})|_{x\to
0}\simeq\frac{kx^{2}}{2}\left(1+16\frac{\sigma_{k}}{x^{6}}\right)$ (98)

We also introduce the function

$F(\sigma_{k},x)|_{x\to 0}\equiv\frac{2kx^{2}}{4\text{Re}(A_{k})|_{x\to
0}}.\>$ (99)

As we can see from Eq. (98), in the limit $x\to 0$, the function
$F(\sigma_{k},x)|_{x\to 0}$ goes to zero as $x^{6}$, i.e.

$|F(\sigma_{k},x)|_{x\to 0}\ll 1$

On the other hand, if there is no collapse, this is, if $\lambda=0$, then
$\sigma_{k}=0$ and $F(0,x)|_{x\to 0}=1$

It is now straightforward to write the explicit expression for the equivalent
power spectrum. Substituting Eqs. (77) and (99) in (61), and making use of Eq.
(39), we obtain the primordial power spectrum as shown in Eq. (40).

##  Appendix B Dynamical evolution of subhorizon (and superhorizon) modes

In this Appendix, we provide an analysis regarding the evolution of the
subhorizon modes as dictated by the CSL evolution equation. For ease of the
calculations, we will rely on the evolution equation for the density matrix
Eq. (62), the analysis of course would be equivalent if the CSL master
equation (55) was used instead. Therefore, from Eq. (62), we observe that the
term affecting the dynamical evolution of any mode, due to the CSL process, is

$\hat{D}^{\text{CSL}}_{\textbf{k}}\equiv\frac{\lambda(a)a^{4}}{2}[\hat{C}_{\textbf{k}}(a),[\hat{C}_{\textbf{k}}(a),\hat{\rho}_{\textbf{k}}]].$
(100)

Additionally, the collapse rate parameter $\lambda$ as given in Eq. (18) can
be recast as

$\lambda(a)=\lambda_{0}\left(\frac{a}{a_{\tau}}\right)^{3},$ (101)

where in the above equation we have used $\alpha=3/2$ (we have chosen this
value due to the results of A) and the definition of $N$, i.e. the number of
e-folds since the beginning of inflation.

Substituting Eq. (101) and the expression for the collapse operator
$\hat{C}_{\textbf{k}}$, as given in Eq. (29), we can rewrite
$\hat{D}^{\text{CSL}}_{\textbf{k}}$ as

$\displaystyle\hat{D}^{\text{CSL}}_{\textbf{k}}$ $\displaystyle=$
$\displaystyle\lambda_{0}\frac{\epsilon}{H}\left(\frac{a}{a_{\tau}}\right)^{3}e^{-k^{2}/a^{2}H^{2}}\bigg{\\{}[\hat{p}_{\textbf{k}},[\hat{p}_{\textbf{k}},\hat{\rho}_{\textbf{k}}]]$
(102) $\displaystyle+$ $\displaystyle
5Ha\bigg{(}[\hat{y}_{\textbf{k}},[\hat{p}_{\textbf{k}},\hat{\rho}_{\textbf{k}}]]+[\hat{p}_{\textbf{k}},[\hat{y}_{\textbf{k}},\hat{\rho}_{\textbf{k}}]]\bigg{)}$
$\displaystyle+$ $\displaystyle
25H^{2}a^{2}[\hat{y}_{\textbf{k}},[\hat{y}_{\textbf{k}},\hat{\rho}_{\textbf{k}}]]\bigg{\\}}.$

We proceed to analyze the last term in the previous equation, i.e. the one
with the highest power of $a$. The analysis corresponding to the other terms
is equivalent. Thus, the dynamical evolution for the last term is dictated by
the function

$\Delta_{k}(a)\equiv a^{5}e^{-k^{2}/a^{2}H^{2}}.$ (103)

We note that $\Delta_{k}(a)<1$ if and only if $\kappa^{2}>5\ln a$, where
$\kappa\equiv k/aH$. Assuming that inflation lasts approximately 65 e-folds,
then $a\sim e^{65}$. As a consequence, $\Delta_{k}(a)<1$ if and only if
$\kappa>\mathcal{O}(10)$. In particular, for a subhorizon mode $\kappa\gg 1$,
one has that $\Delta_{k}(a)\ll 1$. This demonstrates that, for subhorizon
modes, the additional dynamical evolution introduced by the CSL mechanism is
exponentially suppressed.

On the other hand, an analogous analysis for superhorizon modes $\kappa\ll 1$,
indicates that $\Delta_{k}(a)\gg 1$, i.e. the superhorizon modes are the most
affected by the CSL evolution term.

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