# Model Independent Prediction of the Spectral Index of Primordial Quantum
Fluctuations

César Gómez  Raul Jimenez

###### Abstract

One of the most important achievements of inflationary cosmology is to predict
a departure from scale invariance of the power spectrum for cosmological
scalar perturbations. This tilt is understood as a consequence of a quasi de
Sitter classical equation of state describing the inflationary dark energy
dominated era. Here, following previous work, we find a departure of scale
invariance for the quantum Fisher information associated to de Sitter vacuum
for scalar quantum spectator modes. This gives rise to a purely quantum
cosmological tilt with a well defined dependence on energy scale. This quantum
tilt is imprinted, in a scale dependent energy uncertainty for the spectator
modes. The effective quasi de Sitter description of this model independent
energy uncertainty uniquely sets the effective quasi de Sitter parameters (
i.e., the small deviation from the cosmological constant equation of state) at
all energy scales. In particular, in the slow-roll regime characterized by an
almost constant $\epsilon$, the quantum Fisher –model independent– prediction
for the spectral index is $n_{s}=0.9672$. Moreover, the energy scale
dependence of the quantum cosmological tilt implies the existence of a
cosmological phase transition at energies higher than the CMB scale and of the
order of $1Mpc^{-1}$ where the tilt goes from red into blue. This strongly
suggest the existence of a pre-inflationary phase where the effective scalaron
contributes to the spectral index as normal relativistic matter and where the
corresponding growth of the power spectrum can result in dark matter in the
form of small mass primordial black holes. The source and features of the
quantum cosmological tilt leading to these predictions are determined by the
entanglement features of the de Sitter $\alpha-$ vacuum states.

##  1 Introduction and brief summary

In a series of recent papers [1, 2, 3, 4, 5] we have initiated a new approach
to inflationary cosmology based on the notion of quantum Fisher information.
It is fair to say that the idea has been, until now, mostly ignored by the
community. Very likely this lack of interest is due to two basic reasons.
First, the approach is based on notions as quantum Fisher information that are
not very familiar among theoreticians creating a discouraging potential
barrier for the reader. Secondly, we have probably been unable to convey in
more pristine terms the physics underlying the whole approach. In this
article, where we summarize some of the results and present new ones, we will
try to correct both deficiencies.

Let us start saying a few words about quantum Fisher information (see Ref. [6]
and references therein). The classical Fisher information, by contrast to the
most popular Shannon information, intends to quantify, in the case of
probability distributions depending on some statistical parameter, how much
information we can extract about the actual value of the parameter.
Classically this information is the width of the likelihood function when
estimating some random parameters: the smaller the Hessian of the likelihood
surface, the larger the value of the inverse of the Fisher matrix which is the
uncertainty on the estimated parameter. In that sense, Fisher information is
designed to account for the problem of estimating the value of a parameter
about which we only have indirect and statistical information. In the same way
that Shannon information has, as its quantum avatar, von Neumann entropy, in
the case of Fisher we can define quantum Fisher information for families of
quantum states (pure or mixed) depending on some parameter that by
construction is not directly measurable i.e. that is not associated with any
self adjoin physical observable. The first use of quantum Fisher information
is to provide the metric measuring the distinguishability distance between
states associated with different values of the external parameter. From a more
physical point of view the most basic example of quantum Fisher information is
obtained when we choose time as a external parameter. In this case the quantum
Fisher is simply the uncertainty of $G^{2}$ for $G$ the infinitesimal
generator of time translations i.e. quantum Fisher is just a way to measure
$\Delta(E^{2})$, where $E$ is the energy. In this case, the quantum estimation
problem relative to the time parameter becomes simply the Heisenberg time
energy uncertainty relation [7].

Why can this story about quantum estimation of time be relevant to understand
inflationary cosmology?

Inflationary cosmology [8, 9, 10, 11, 12, 13] is the present paradigm for the
cosmological description of the early Universe. The essence of inflationary
models is to find solutions to the Einstein equations representing an
expanding de Sitter phase (inflation). In addition, there is the need for a
natural graceful exit taking place after a finite amount of time. This time
needs to be long enough to account for the observable cosmic microwave
background (CMB) and large scale structure (LSS) spectrum. An initial
important step in that direction was taken by Starobinsky [8] who noticed,
while trying to solve the initial singularity problem, that Einstein’s
equations with an energy momentum tensor defined by the trace anomaly can give
rise to an initial unstable de Sitter phase ending naturally after a finite
amount of time. This initial model, where the physical parameters were
determined by the matter content involved in the vacuum polarization
responsible for the trace anomaly, was soon improved into a full fledged
inflationary model [14]. Indeed, the part of the trace anomaly that goes like
$\Box R$ can be reabsorbed by adding a counter term $R^{2}$ to the Lagrangian
with an arbitrary coefficient. This $R^{2}$ gravity has in its spectrum not
only the graviton but also a scalar field that was identified with the
original scalaron. However, the big secret of this model was only understood
with the development of the theory of quantum cosmological perturbations [15,
16, 17, 18, 19, 20, 21]. The main observation in Ref. [16] was that the
equation for the scalaron quantum fluctuations in the local de Sitter
background [15], already in the linear approximation, leads to a power
spectrum that is not scale invariant. The relevant parameter measuring the
lack of scale invariance, the tilt, being roughly given by
$\frac{M^{2}}{H^{2}}$ for $M$ parameterizing the scalaron mass. This non
vanishing tilt leads to a time dependent power spectrum for scalar
fluctuations and to a natural graceful exit whenever the amplitude of the
fluctuations reach the values defining the background metric [16]. In summary
the theory of quantum fluctuations immediately leads to the basic observation
that, provided a UV cutoff on the amplitude of the fluctuations, the existence
of tilt naturally leads to a graceful exit.

But what is the relation between all this well known stuff and quantum Fisher
information ?

Here comes our first basic observation. In the inflationary paradigm the
equation for the scalar quantum fluctuations describes time dependent quantum
oscillators with frequency $\omega^{2}(k,\eta)$ depending both on the comoving
momentum and conformal time $\eta$. Formally this frequency contains three
pieces. One is $k^{2}$ as it is the case for flat background. The second piece
goes like $\frac{-2}{\eta^{2}}$ and describes the modification due to exact de
Sitter. Finally, the last piece is the one that effectively accounts for the
scalaron mass and goes roughly as $-\frac{M^{2}}{H^{2}}\frac{1}{\eta^{2}}$
111Note that $M^{2}$ enters into the dispersion relation with negative sign..
The parameter $M$ that is model dependent and parametrized by the slow roll
coefficients accounts for the classical design of the quasi de Sitter
instability. In summary, the details of the quasi de Sitter slow roll are
imprinted in the extra piece of the frequency.

Let us now come back to our discussion on quantum Fisher. Imagine yourself
doing quantum field theory for a scalar spectator in pure de Sitter. In the
$(k,\eta)$ hyperplane with $k$ representing the comoving momentum and $\eta$
the conformal time, we can define the hyperboloid ${\cal{H}}_{\Lambda}$ by the
condition $k|\eta|H=\Lambda$. Thus each hyperboloid is associated with a given
energy scale $\Lambda$. For each comoving momentum $k$ we define, using the
standard Bogolioubov transform, the creation-annihilation operators
$a_{k}(\Lambda)$ (for details see Ref. [4] and references therein) and the
corresponding $|\Lambda\rangle$ vacuum. In essence, this state represents the
vacuum for modes of comoving momentum $k$ and energy scale $\Lambda$. A
peculiar property of quantum field theory in de Sitter is that for a given
momentum $k$ the energy scale depends on the conformal time. This leads to
some difficulties at the time of defining effective field theory in Wilsonian
terms. We will not touch this issue in this article. The key ingredient of our
approach is the computation of the quantum Fisher information for the state
$|\Lambda\rangle$ defined relative to a Fock like basis. This Fock basis is
defined using the standard euclidean Chernikov-Tagirov [22]/Bunch-Davies [23]
vacuum. In this basis the $|\Lambda\rangle$ vacuum is defined, as usual,
relative to the corresponding Bogoloubov transformed oscillators. Physically,
this information is telling us the maximal information we can extract, about
the value of $\Lambda$ by performing measurements on the state
$|\Lambda\rangle$. Note that by the definition of de Sitter energy scale this
information is telling us about the quantum uncertainty in the conformal time
and consequently in the physical energy scale.

The main result we have derived in Ref. [4] is the renormalization group
equation for the Fisher information. In reduced notation, and denoting
$F(\Lambda)$ the quantum Fisher information modulo, the conformal piece (see
section 3 for details and definitions), the basic result is

$\Lambda\frac{\partial}{\partial\Lambda}F=\alpha_{F}F$ (1.1)

or equivalently $\frac{d\ln F}{d\ln\Lambda}=\alpha_{F}(\Lambda)$. This means
that quantum Fisher is not scale invariant and identifies the quantum tilt
$\alpha_{F}$ as well as its dependence on the energy scale. It is important to
stress that this anomalous scale transformation of quantum Fisher has as its
origin the specific form of the quantum phases introduced through the
Bogolioubov transform defining the state $|\Lambda\rangle$. It is well known
that the state $|\Lambda\rangle$ is a very entangled squeezed state (see [24]
see also [4] and references therein). The quantum Fisher as well as the shape
of the quantum tilt strongly depends on the entanglement features 222In order
to avoid some potential missunderstandings it is important to make some basic
distinctions. For the squeezed state that is strongly entangled we can define
von Neumann density matrix integrating over one of the partner modes defining
the squeezed state and to evaluate the corresponding entanglement entropy.
Moreover we can consider families of von Neumann density matrices depending on
de Sitter energy scale and to evaluate the relative entanglement entropy
between them and its relation with a Fisher information metric on the space of
mixed states. In this note we focus on the quantum Fisher of the pure state.

After these preliminaries, we are in a good position to establish our main
claim about how to establish a correspondence between the tilt of the spectral
index of scalar fluctuation in quasi de Sitter and the value of the quantum
Fisher when the estimated parameter is time. From Fisher information we can
derive the uncertainty of the energy square on the state $\Lambda$; let us
denote it $\Delta(E^{2})(k\eta)$ and we can compare it with the corresponding
energy uncertainty defined in the quasi de Sitter set up
$\delta(E^{2})(k,\eta)$, where this second energy depends on the slow roll
parameters. Both transform anomalously with respect to changes of energy
scale. In the case of Fisher $\Delta(E^{2})$ the tilt is given by
$\alpha_{F}$, while in the quasi de Sitter case is given by the cosmological
spectral index. To put both in correspondence let us identify both quantities.
This defines a self consistency constraint between the slow roll parameter
$2\delta=1-n_{s}$ and the value of the Fisher quantum tilt $\alpha_{F}$ that
uniquely fix the value of the slow roll parameter,

$\boxed{\frac{6\delta+2\delta^{2}}{\alpha_{F}^{-1}(4\delta)}=1},$ (1.2)

where $\alpha_{F}^{-1}$ is the inverse of $F$. Solving this simple equation
(which is derived in detail in the sections below) for $\delta$ for the Fisher
tilt $\alpha_{F}$ fixes the value of the cosmological spectral index $1-n_{s}$
in the slow roll to be

$\boxed{1-n_{s}=0.0328}$ (1.3)

This result that magically fits the one from Planck18 [25] $1-n_{s}=0.0348\pm
0.0042$, strongly indicates that the cosmological spectral index is fully
determined by identifying the quantum Fisher energy uncertainty with the
energy uncertainty we associate with quasi de Sitter slow roll, i.e. with the
quasi de Sitter piece of $\omega^{2}(k,\eta)$ simply defined as
$\omega^{2}-k^{2}-\frac{2}{\eta^{2}}$. Note that in order to get the value of
$1-n_{s}$ we don’t have any model dependent freedom. This number is fully
determined by the quantum Fisher information. The nice agreement with
experiment implies that in the CMB scales the slow roll approximation is
extremely good.

Before entering into the technical derivation of (1.2) let us summarize for
the benefit of the reader the main steps of our construction.

  1. 1.

First, we define the quasi de Sitter contribution to the oscillator energy
$\Delta E^{2}(k,\eta,\delta)$ as
$E^{2}(k,\eta,\delta)-E^{2}(k,\eta,\delta=0)$, where
$E^{2}(k,\eta,\delta)\equiv\frac{\omega^{2}(k,\eta,\delta)}{a^{2}(\delta)}$
with $a(\delta,\eta)=\frac{1}{H_{0}\eta}\frac{1}{(k_{0}\eta)^{\delta}}$. Here
$k_{0}$ is a pivot scale defined by the condition
$a(\delta,\eta_{0})=\frac{1}{H_{0}\eta_{0}}$ ( for more details see section 2
). Note that this quantity depends on $k,\eta,\delta$ and is zero for
$\delta=0$.

  2. 2.

Now, you put yourself in pure de Sitter and identify the quantum variance
$\delta(E^{2}(k,\eta))$ defined by the quantum Fisher information evaluated at
$(k,\eta)$. The derivation of this quantity is the main result of section 3.
This quantity only depends on pure de Sitter data. The key point that
substantiate our derivation is that this pure de Sitter quantity depends on
the quantum tilt $\alpha_{F}$ defined by the anomalous scale dependence of the
pure de Sitter quantum Fisher information ( see (1.1)).

  3. 3.

Finally we ask ourselves about what quasi de Sitter background can lead to
exactly the same value that the pure de Sitter quantum Fisher result. The
answer to this question is determined by the equation:

$\Delta E^{2}(k,\eta,\delta)=\delta(E^{2}(k,\eta))$ (1.4)

This identification defines a very non trivial constraint. First of all
$\Delta E^{2}(k,\eta,\delta)$ was defined in the slow roll parametrization of
quasi de Sitter. Secondly $\delta(E^{2}(k,\eta))$ was defined in pure de
Sitter. Both quantities define a correction to the oscillator frequency of the
same order in $\hbar$. Moreover both quantities include the effect of
anomalous scale invariance. In the case of $\Delta E^{2}(k,\eta,\delta)$ this
effect is parametrized by $\delta$ while in the case of
$\delta(E^{2}(k,\eta))$ this effect is encoded in $\alpha_{F}$. Now note that
the solution to equation (1.4) with data $\alpha_{F}$ selects a slow roll
parameter very non trivially. First of all the quantum anomalous scale
invariance selects a function $k\eta(\delta)$ through the relation, discussed
in section 3, $\alpha_{F}(k\eta)=4\delta$. This defines the matching between
the scale dependence set by the slow roll parameter $\delta$ with the one
defined, model independently, by the quantum Fisher tilt $\alpha_{F}$. The
important thing now is that in order to achieve (1.4) we need an extra
condition that combined with the matching for the scale dependence described
above leads to the main equation (1.2). Solving this equation allows you to
identify a unique value of $\delta$ and consequently of the spectral index.

The careful reader can find this claim paradoxical in the following sense. The
slow roll piece of $\omega^{2}$ is fully determined by the form of the
classical slow roll potential that we can model at will. Nevertheless, the
quantum Fisher is in origin quantum and comes from the phases of the state
$|\Lambda\rangle$, consequently we have not any freedom to change its value.
The big claim we are making in this article is that the quantum Fisher effect
fully accounts for the observed value of the cosmological tilt.

Superficially the claimed quantumness of the tilt seems to enter into conflict
with the general picture of the classicalization in time of the fluctuations.
Indeed, the problem of how the quantum coherence of the de Sitter state
affects the correlators we normally use to compare with observations has been
extensively studied (see for instance Ref. [26, 27, 28, 29]). The basic point
is that the observational effects of the phases responsible for the quantum
coherence decrease exponentially with time as $e^{-{\cal N}}$ for ${\cal N}$
the number of e-foldings. This puts, in particular, severe constraints to
cosmological Bell like experiments [30]. Hence the obvious question is: Why
this classicalization of fluctuations is not in contradiction with the quantum
Fisher approach? How can we claim that we can extract the observable tilt from
quantum effects of the de Sitter state when the quantum coherence of these
effects is exponentially suppressed in the limit of deep IR scales
corresponding to $k\eta<<1$ ? The essence of the answer is simply the
following. Using quantum Fisher information we discover a quantum cosmological
tilt for all energy scales. In the deep IR i.e. for $k\eta<<1$ the suppressed
contribution of quantum phases will lead in the limit $k\eta=0$ to a vanishing
quantum Fisher and a vanishing tilt. This effective classicalization is
however not the end of the story. Indeed the quantum Fisher and its associated
tilt is not vanishing for small values of the energy scale. Now the
possibility that quantum Fisher effects account for the cosmological tilt is
determined by the solution to (1.2) that depends on the particular energy
dependence of the quantum tilt away from the very IR limit $k\eta=0$. It is
the non trivial dependence of the quantum tilt away from the deep IR limit
what sets the observable value of $1-n_{s}$. In more plain words: the quantum
Fisher in pure de Sitter accounts for the effect we simulate classically using
quasi de Sitter slow roll. This makes the spectral index a fundamental label
of our Universe. More precisely it can be interpreted as the basic label of a
de Sitter universality class.

In the quantum Fisher approach presented in this article, we reduce the study
of quantumness to the de Sitter ”vacuum” for the scalar spectator. Other
quantum effects due to the pure gravitational constituency of the metric
background have been recently considered in [31, 32, 33, 34]. These effects go
as $1/N_{GH}$ for $N_{GH}$ the de Sitter entropy and define quantum back
reaction corrections on the top of the ones we discuss here.

A bonus of the Fisher approach is that it gives us a tilt with a complete
dependence on the energy scale (see Fig. 1)333In this figure we represent the
quantum tilt in terms of the energy scale. The sensibility of this numerical
result on the number of created pairs is discussed in the appendix.. Several
things about this dependence are worth to be highlighted. First of all we see
an almost scale invariant region corresponding to the inflationary period. As
discussed in the appendix the extension of this region depends on an effective
IR cutoff. Secondly, we observe a change of phase from red tilted into blue
tilted at high momentum. The change in the tilt reflects an effective change
in the nature of the scalaron. Indeed the tilt can be parameterized in terms
of a running effective scalaron mass that changes sign in the blue tilted
phase. This blue tilted phase can naturally account for the dark matter, in
the form of primordial black holes, needed to amplify gravitationally the
primordial quantum fluctuations (see e.g. Ref. [35] for a recent review and
references therein).

Figure 1: The quantum cosmological tilt as a function of energy scale as
determined by the quantum Fisher. For values of $|k\eta|<0$ the end of
inflation is determined by the IR cutoff on the number of particles $n$ in the
sum (3.2). In this plot, for illustrative purposes, we set $n=10^{9}$. The
sensibility of the quantum cosmological tilt on this IR cutoff is discussed in
the appendix showing that only affects the effective duration of inflation but
not the predicted value for $(1-n_{s})$. The different shaded regions
correspond to different scales. The blue region is where the dark matter forms
and the spectral index is blue. A detailed explanation of each region is given
in the appendix.

##  2 Standard theory of quantum cosmological fluctuations

The fundamental ingredient to develop the theory of gauge invariant quantum
cosmological scalar fluctuations is the Chibisov-Mukhanov [15] equation 444For
the derivation see appendix in [15]

$\phi^{{}^{\prime\prime}}-\Delta\phi-\frac{z^{{}^{\prime\prime}}}{z}\phi=0$
(2.1)

with $z=a\sqrt{\epsilon}$ and $\epsilon$ the standard first slow roll
parameter. For the Fourier transform components we get

$\phi_{k}^{{}^{\prime\prime}}+(k^{2}-\frac{z^{{}^{\prime\prime}}}{z})\phi_{k}=0$
(2.2)

leading to an effective frequency
$\omega^{2}(k,\eta)=(k^{2}-\frac{z^{{}^{\prime\prime}}}{z})(k,\eta)$, where
$\eta$ is the conformal time. Using

$\frac{z^{{}^{\prime\prime}}}{z}=\frac{a^{{}^{\prime\prime}}}{a}+\frac{a^{{}^{\prime}}}{a}\frac{\epsilon^{{}^{\prime}}}{\epsilon}+\frac{1}{2}\frac{\epsilon^{{}^{\prime\prime}}}{\epsilon}-\frac{1}{4}\frac{(\epsilon^{{}^{\prime}})^{2}}{\epsilon^{2}}$
(2.3)

we can define the simplest slow roll approximation by the condition $\epsilon$
constant but different from zero and

$\frac{z^{{}^{\prime\prime}}}{z}=\frac{\beta(\beta+1)}{\eta^{2}}$ (2.4)

This corresponds to use a scale factor

$a=\frac{1}{H_{0}\eta}\frac{1}{(k_{0}\eta)^{\delta}}$ (2.5)

where we introduce, for dimensional reasons, a reference momentum $k_{0}$.
Independently of the value of $k_{0}$ we get

$\frac{a^{{}^{\prime\prime}}}{a}=\frac{\beta(\beta+1)}{\eta^{2}}$ (2.6)

with $\beta=-2-\delta$. The Hubble parameter defined as $H=\frac{\dot{a}}{a}$
is given by

$H(\eta)=(\delta+1)H_{0}(k_{0}\eta)^{\delta}$ (2.7)

leading to a constant value of the slow roll parameter
$\epsilon=-\frac{\dot{H}}{H^{2}}$ given by

$\epsilon=\frac{\delta}{\delta+1}$ (2.8)

Let us now consider equation (2.2) in this slow roll limit for a given
$\beta$. In the long wave limit $k\eta<<1$ the equation reduces to a Bessel
equation 555In the case of constant $\epsilon$ the exact solution is given in
terms of Hankel functions.. The solution is given by

$k^{3}|\phi_{k}(\eta)|^{2}\sim\frac{1}{\eta^{2}}(k\eta)^{-2\delta}$ (2.9)

Thus defining the power spectrum as $P(k,\eta)\equiv\frac{1}{\epsilon
a^{2}M_{P}^{2}}k^{3}|\phi_{k}|^{2}$ we get in this long wave length limit [16]
and using (2.5)

$\boxed{P(k,\eta)=\frac{H_{0}^{2}}{\epsilon
M_{P}^{2}}(\frac{k_{0}}{k})^{1-n_{s}}}$ (2.10)

with $(1-n_{s})=2\delta$. Note that for this definition of the power spectrum:

  * •

$k_{0}$ plays the role of pivot scale

  * •

We work in the slow roll approximation with $\epsilon$ constant

  * •

The tilt $1-n_{s}$ is computed in the limit $k\eta<<1$

  * •

$H_{0}$ is the initial value at time $\eta_{0}=\frac{1}{k_{0}}$. More
precisely $H_{0}=\frac{H(\eta_{0})}{(1+\delta)}$.

If we choose for $k_{0}$ the CMB scale $k_{\rm CMB}$ then equation (2.10) sets
the power spectrum for $H_{0}$ the value of Hubble at $\eta=\frac{1}{k_{\rm
CMB}}$ i.e. at horizon exit. This is the standard prescription where you use
the Hubble at horizon exit time of CMB modes and approximate the tilt using
constant $\epsilon$ and the solution to Bessel in the long wave limit. To
improve this approximation you need to consider the more general case with
$\epsilon$ depending on time and to solve exactly in that regime equation
(2.2).

###  2.1 Energy and scale invariance

The physical energy of a mode of comoving momentum $k$ at conformal time
$\eta$ is given, in the slow roll regime, by

$E^{2}(k,\eta)=\frac{1}{a^{2}}(k^{2}-\frac{\beta(\beta+1)}{\eta^{2}})$ (2.11)

Let us define variation of the energy relative to the pure de Sitter case.
This is given by

$\boxed{\Delta
E^{2}(k,\eta)=(3\delta+\delta^{2})H_{0}^{2}(k_{0}\eta)^{2\delta}}$ (2.12)

In the limit of $\delta<<1$ we simply get

$\Delta E^{2}(k,\eta)\sim 3\delta H(\eta)^{2}$ (2.13)

The lack of scale invariance can be nicely expressed in the form of a
renormalization group equation, namely

$\boxed{k_{0}\frac{\partial}{\partial k_{0}}\Delta E^{2}=2\delta\Delta E^{2}}$
(2.14)

It is an interesting exercise to evaluate $\Delta E^{2}$ for the Starobinsky
model [8] in terms of the scalaron mass $M$ as defined in [16]. In this case
$2\delta=\frac{2M^{2}}{3H^{2}}$ and we get

$\Delta E^{2}=M^{2}(k_{0}\eta)^{2\delta}$ (2.15)

In other words, the scalaron mass $M^{2}$ is simply $\Delta E^{2}$ evaluated
at the reference time $\eta_{0}=\frac{1}{k_{0}}$. Moreover the former
expression leads to a running scalaron mass

$M^{2}(\eta)=M^{2}(k_{0}\eta)^{2\delta}$ (2.16)

Thus defining $\eta_{0}$ as initial time the bound on the time life of quasi
de Sitter can be derived setting the final time $\eta_{f}$ by the bound
$M^{2}(\eta_{f})=M_{P}^{2}$ that leads to

$\delta t=\frac{1}{H}\frac{3H^{2}}{2M^{2}}\ln(\frac{M_{P}^{2}}{M^{2}})$ (2.17)

that is the estimation of quasi de Sitter time in Ref. [16].

As a comment note that in the pure de Sitter limit i.e. $M=0$ we get $\delta
t=\infty$ and $\Delta E^{2}=0$. This is in essence a manifestation of energy
time uncertainty relation, namely for eternal de Sitter $\Delta E^{2}$ is
zero.

In summary we observe the natural set of connections:

Lack of conformality $\Leftrightarrow$ Tilt $\Leftrightarrow$ graceful exit
$\Leftrightarrow$ energy ”uncertainty” $\Delta E^{2}\neq 0$

###  2.2 Connection to quantum breaking

In the approach developed in Ref. [31, 32, 33, 34] the life time of de Sitter
based on quantum breaking was estimated to be

$\delta t\sim\frac{1}{H}\frac{M_{P}^{2}}{H^{2}}$ (2.18)

It is instructive to compare this time with the bound (2.17). Formally the
quantum breaking corresponds to a scalaron mass

$\boxed{M=\frac{M_{P}}{N_{GH}}}$ (2.19)

with $N_{GH}\equiv\frac{M_{P}^{2}}{H^{2}}$ the de Sitter Gibbons-Hawking
entropy. The simplest interpretation is that pure quantum breaking effects in
de Sitter lead naturally to an effective scalaron mass of order $1/N_{GH}$.
This also highlights that eternal de Sitter corresponds to the $N_{GH}=\infty$
limit. This quantum scalaron mass has its origin in the quantum constituency
of de Sitter and leads to a tilt of order $1/N_{GH}$ 666In order to avoid any
confusion note that although a finite value of de Sitter entropy was
originally obtained using an eternal de Sitter metric the constituency
representation of de Sitter where we look for a coherent state – defined
relative to Minkowski vacuum– with the expectation value of the metric
reproducing classical de Sitter geometry has an occupation number of the order
$N_{GH}$.Hence constituency effects are always order $\frac{1}{N_{GH}}$.

As already discussed in the introduction, this $1/N_{GH}$ tilt is not the one
we discover using the quantum Fisher for the scalar spectator in de Sitter. In
the latter case the Fisher quantum tilt extracts the anomalous scale
transformations relative to changes of energy scale for the dS vacuum for the
spectator modes. We will use the next section to derive the quantum Fisher
tilt and present its connection with the cosmological tilt for quantum scalar
fluctuations.

##  3 The quasi de Sitter-Fisher correspondence

###  3.1 de Sitter kinematics and Fisher information

Observables in inflationary cosmology are functions of the comoving momentum
$k$ and the conformal time $\eta$. Conformal time is defined in the interval
$[-\infty,0]$ covering the full physical time interval $[-\infty,+\infty]$. It
is convenient to foliate the $k,\eta$ hyperplane into hyperboloids defined by
the condition $k\eta=C$, for $C$ arbitrary constant (see Fig. 2). In pure de
Sitter we can define the energy scale $\Lambda=-k\eta H$ so these hyperboloids
define equal energy hypersurfaces.

Figure 2: The plane $k,\eta$ foliated in hyperboloids representing the same
energy scale. The horizon exit hyperboloid is $k\eta=1$.

For a scalar spectator we can define for each couple $(k,\Lambda)$ a dS
”vacuum” state as

$|k,\Lambda\rangle=\sum_{n}c(\Lambda,n)e^{in\Phi(\Lambda)}|n_{k},n_{-k}\rangle$
(3.1)

where we sum over all integers and

$c(\Lambda,n)=C\tanh({r(\Lambda)})^{n}$ (3.2)

with $C$ a normalization constant,

$r(\Lambda)=-\sinh^{-1}(\frac{H}{2\Lambda})$ (3.3)

and

$\Phi(\Lambda)=-\frac{\pi}{4}-\frac{1}{2}\tan^{-1}(\frac{H}{2\Lambda}).$ (3.4)

Using the definition of $\Lambda$ this state represent the dS vacuum for mode
$k$ at conformal time $\eta=\frac{\Lambda}{kH}$. Thinking of $\Lambda$ as
statistical parameter the dependence of the state on $\Lambda$ simply reflects
the conformal time dependence for the mode $k$.

Already at this level we can define the quantum Fisher information as

$F(k,\Lambda)=k^{2}(\sum_{n}c(\Lambda,n)^{2}(\frac{\partial\Phi(\Lambda,n)}{\partial\eta})^{2}-(\sum_{n}c(\Lambda,n)^{2}(\frac{\partial\Phi(\Lambda,n)}{\partial\eta}))^{2})$
(3.5)

which is, note the overall factor $k^{2}$, the quantum Fisher information
defined relative to the conformal time $\eta$. For pure de Sitter we get

$F(k,\Lambda)=\frac{1}{\eta^{2}(k\eta)^{6}}(k\eta)^{\alpha_{F}(k\eta)}$ (3.6)

where $\alpha_{F}(k\eta)$ is the quantum Fisher tilt that has a very precise
dependence on $(k\eta)$ (see Fig. 1). In order to unravel the physical meaning
of this quantum tilt we need to understand how this quantum Fisher tilt enters
into the quantum uncertainty of physical energy. 777See the appendix for a
detailed description of the energy dependence of $\alpha_{F}(k\eta)$.

First of all, note that for each value of the energy scale $\Lambda$ we can
define a family of quantum states corresponding to use the different values of
$k,\eta$ on the hyperboloid defined by $\Lambda$. In reality, we have two
identical hyperboloids: one for the mode $k$ and another for the mode $-k$
entering into the definition of the state (3.1). Thus after taking into
account the integration factor over the two hyperboloids we get

$F=\frac{1}{\eta^{2}}(k\eta)^{\alpha_{F}(k\eta)}$ (3.7)

This leads to a quantum uncertainty on $k$ at energy scale $\Lambda$ given by
the square root of the quantum Fisher function, namely

$\delta(k)=\frac{1}{\eta}(k\eta)^{\alpha_{F}/2}$ (3.8)

Now we can define, at first order, $\delta E_{F}^{2}$ at energy scale
$\Lambda$ as

$\boxed{\delta
E_{F}^{2}(k\eta)=\frac{k\delta(k)}{2a^{2}}=(\frac{k\eta}{2})H^{2}(k\eta)^{\alpha_{F}/2}}$
(3.9)

We should interpret this quantity as the quantum energy square uncertainty for
spectator modes living on the hyperboloid defined by $\Lambda$.

###  3.2 The correspondence

In order to define the quasi de Sitter-Fisher correspondence we need to
compare the quasi de Sitter energy uncertainty given in (2.12) and Fisher
quantum uncertainty given in (3.9). Formally:

$\boxed{\delta E_{F}^{2}(k\eta)\Leftrightarrow\Delta E^{2}(k,\eta)}$ (3.10)

where $\Delta E^{2}(k,\eta)$ is the quasi de Sitter expression. To define the
exact correspondence in the slow roll limit we choose $H$ used in the Fisher
construction to be $H_{0}$ and we put ourselves in the energy scale implicitly
defined by $\hat{(k\eta)}$ where

$\alpha_{F}(\hat{(k\eta)})=4\delta$ (3.11)

with $\delta$ defined in (2.5). In these conditions we get the slow roll-
Fisher correspondence at the scale defined by (3.11)

$\boxed{\Delta E^{2}=\delta
E_{F}^{2}\frac{6\delta+2\delta^{2}}{\hat{(k\eta)}}(\frac{k_{0}}{k})^{2\delta}}$
(3.12)

This equation has a very nice meaning. Indeed, it gives us the renormalization
group running of the physical energy uncertainty in terms of the pivot scale
and uses (model independent) quantum Fisher information to set the value at a
pivot scale defined by (3.11).

###  3.3 Fisher derivation of spectral index

Recall that in the slow roll approximation we have $2\delta=1-n_{s}$. Let us
now derive this spectral index from imposing the correspondence $\Delta
E^{2}=\delta E_{F}^{2}$. From (3.12) this implies, for $k=k_{0}$,

$\frac{6\delta+2\delta^{2}}{\hat{(k\eta)}}=1$ (3.13)

Thus, given as data the quantum Fisher tilt function $\alpha_{F}(k\eta)$ the
spectral index in slow roll is determined by the equation

$\boxed{\frac{6\delta+2\delta^{2}}{\alpha_{F}^{-1}(4\delta)}=1}$ (3.14)

Note that this equation is setting the value of the slow roll spectral index
by imposing that:

The quasi de Sitter energy uncertainty is equal to the quantum Fisher energy
uncertainty.

Note that the only data entering into this equation is the quantum Fisher tilt
$\alpha_{F}(k\eta)$ that is completely model independent. Since we are
interested in extracting $(1-n_{s})$ at slow roll we impose as a self
consistency condition the equality in this regime between the quasi de Sitter
energy uncertainty and the quantum Fisher energy uncertainty. This a priori
leaves the value of the slow roll parameter $\delta$ completely determined by
$\alpha_{F}$. It is easy to solve the self consistency equation (3.14).
Surprisingly enough the result is

$\boxed{1-n_{s}=0.0328}$ (3.15)

i.e. $n_{s}=0.9672$. This apparently magic result can induce the skeptic
reader to suspect that we are cheating and that we are already introducing the
information we want to extract. It is easy to convince yourself that the value
of $1-n_{s}$ could be very different from the correct result for a different
value of the quantum Fisher tilt $\alpha_{F}(k\eta)$. The key point is that
this quantum tilt is completely independent of any quasi de Sitter modeling.

Coming back to our former discussion on quantum breaking, we should add to
$1-n_{s}$ the quantum constituency contribution [31, 32, 33] $1/N_{GH}$ that
is in general, for realistic values of $H$, negligible.

###  3.4 Master equation

For generic scale dependence of $\epsilon$ we have for
$\frac{z^{{}^{\prime\prime}}}{z}$ the complicated function of the first and
second derivative of $\epsilon$ given in (2.3). The exact tilt in this case
will require to solve equation (2.2) exactly. The quasi de Sitter-Fisher
correspondence described above leads to an implicit master equation relating
the function of $\epsilon$ defined by the exact value of
$\frac{z^{{}^{\prime\prime}}}{z}$ and the corresponding exact tilt that we
will represent as $(1-n_{s})(\epsilon)$. Let us define888Note that
$A(\epsilon)=2H^{2}a^{2}\eta^{2}(1-\frac{\epsilon}{2}-\frac{3}{4}\frac{d\ln\epsilon}{dN}+\frac{1}{4}\epsilon\frac{d\ln\epsilon}{dN}+\frac{1}{8}(\frac{d\ln\epsilon}{dN})^{2}+\frac{1}{4}\frac{d^{2}\ln\epsilon}{d^{2}N})-2$
with $dN=-aHd\eta$. In what follows we will assume that $\epsilon$ is an
arbitrary power law function of $k\eta$.

$A(\epsilon)\equiv\eta^{2}\frac{z^{{}^{\prime\prime}}}{z}-2$ (3.16)

The master equation defining the quasi de Sitter-Fisher correspondence is
formally given in terms of the exact Fisher tilt $\alpha_{F}$ by

$\boxed{\alpha_{F}^{-1}(2(1-n_{s})(\epsilon))=A(\epsilon)}$ (3.17)

In the former paragraph we have considered this equation in the slow roll
limit with $\epsilon$ constant but non vanishing. In this case
$A(\epsilon)=6\delta+2\delta^{2}$ and $2(1-n_{s})(\epsilon)=4\delta$ 999Recall
that this is the exact Bessel parametrization in the limit of $\epsilon$
constant.. In this limit the master equation becomes equation 3.14 with the
advertised solution $1-n_{s}=0.0328$.We can describe in words the meaning of
the master equation as follows:

$\alpha_{F}$ (quasi de Sitter energy difference) = spectral index

i.e. the quantum cosmological Fisher tilt maps the deviation from de Sitter of
the spectator dispersion relation, at a given scale, into the spectral index
at that scale.

However the master equation allows us to extract information about the running
of the exact spectral index beyond the slow roll regime. The physics
implications of such a running dictated by the quasi de Sitter-Fisher
correspondence will be discussed in next section.

##  4 The non perturbative information of the quantum Fisher tilt: glimpses of
a theory of dark matter

In standard inflation the spectral index is computed using the solutions to
equation (2.2) in the approximation where the slow roll parameters are almost
constant. Deviations from this approximation require a model dependent
specification of the slow roll parameters at different energy scales. The key
bonus of the Fisher approach is that it provides a well defined dependence of
the quantum tilt $\alpha_{F}(k\eta)$ on energy scale. In this section we will
try to extract some glimps on how a full fledged cosmology can fit the shape
of the quantum tilt.

###  4.1 The critical hyperboloid: the effective scalaron mass

From the explicit expression of $\alpha_{F}(k\eta)$ discussed in Ref. [2] we
discover a critical hyperboloid defined by

$-(k\eta)_{cr}=1.87$ (4.1)

where the quantum tilt goes from red into blue. For a fixed value of conformal
time $\eta$ we can define $k_{exit}\sim\frac{1}{\eta}$ and the critical
$k_{cr}$ defined by $k_{cr}\eta=1.87$. This momentum is much larger than the
horizon exit momentum. Thus this larger momentum will exit the horizon at a
later conformal time.

What is the meaning of this cosmological phase transition?

The simplest way to address this question is using our master equation (3.17).
Formally this equation is telling us that $\alpha_{F}$ at the effective
scalaron mass is giving us the value at that scale of the spectral index
$(1-n_{s})$. In a very rough approximation. we can define the effective
scalaron mass $M_{\rm eff}(k\eta)$ at energy scale $k\eta$ in terms of
$\alpha_{F}$ as follows

$\alpha_{F}(k\eta)/2=3\sqrt{1+\frac{4M^{2}_{\rm eff}(k\eta)}{9H^{2}}}-3$ (4.2)

Once we use this effective scalaron fit of the quantum tilt we inmediately
observe that when crossing the critical hyperboloid we transit from the
scalaron describing the inflationary period corresponding to the sign $+$
inside the square root to a pre-inflationary period where the sign changes
into $-$ (see also Ref. [5]). This means that for these large energies, larger
than the critical energy determined by the critical hyperboloid, the effective
scalaron mass square changes sign. This is extremely interesting since it
indicates that the quantum effects imprinted in the quantum tilt are defining
a phase transition to a phase where the Universe is contracting in time until
reaching the critical time where the Universe enters into the standard
inflationary phase. In this pre-inflationary phase the effective scalaron mass
is entering in the dispersion relation as a normal matter mass effect.

In simpler words; the blue tilt regime is dominated by a form of matter that
is responsible for the change of sign in (4.2). An interesting interpretation
for cosmology of our derived value for the quantum tilt is to use the Hubble
defined by (2.7), which contains the information on $\alpha_{F}$. The result
is depicted in Fig. 3. This figure indicates a pre-inflationary contracting
phase and an end of inflation depending on the IR cutoff on the contributing
number of pairs (see appendix). A pre-inflationary contracting phase is very
much reminiscent of bouncing cosmological models [36]. Here we reduce
ourselves to suggest this simple interpretation of the quantum cosmological
tilt. It is well known that bouncing solutions enter into conflict with the
null energy condition NEC [37]. In our approach the change red to blue of the
tilt is a pure quantum effect and we don’t have an effective way to describe
this change of phase semiclassically, thus we don’t have anything to say on
the interesting problem of violations of the NEC.

Figure 3: The Hubble parameter as a function of scale for $k_{0}=1$. As in
previous figures we have adopted $n=10^{9}$ for illustration purposes to set
the end of inflation. Note that at very large and small scales, the Hubble
parameter approaches zero. For the largest scales, the ones entering the
horizon today, the expansion resembles that of a quintessence field after the
FLRW phase.

###  4.2 Is the pre-inflationary high energy phase the source of dark matter?

Although the primordial quantum fluctuations play the natural role of seeds of
the large scale structures, the gravitational amplification of these seeds
requires the presence of dark matter that interacts extremely weakly with the
electro-magnetic force. In other words, we need (cold) matter but cannot be
visible to do the job of galaxy formation properly. A very nice and natural
suggestion on the nature of this matter are the so called primordial black
holes. The advantage of this candidate is that they have a pure gravitational
origin if we manage to create quantum fluctuations intense enough to create
small black holes in a very concrete range of masses, bounded by Hawking
evaporation, to allow them to survive longer than Hubble time. To have such
strong primordial fluctuations we need as a necessary condition to have a
region of large momentum where a blue tilt dominates. But a blue tilt seems to
be natural if we somehow change the sign of the effective scalaron mass i.e.
if we convert, in the corresponding high energy region the scalaron in a
normal form of matter contributing to the cosmological tilt with the opposite
sign creating a blue tilted phase. There are some models, for instance, the
curvaton axion model that achieve this goal. In any case the important message
is that the primordial black hole solution to the dark matter problem requires
already a form of matter changing at large energy scales the tilt from red to
blue.

What we want to suggest, and we have worked our in detail in Ref. [5] is that
the blue tilted phase is the natural quantum source of the dark matter in the
sense that i) contributes to the tilt as ordinary matter and ii) triggers the
formation of primordial black holes that can appear to us as the relic of such
pre-inflationary phase.

###  4.3 Two phenomenological comments

We conclude this section pointing out two potentially observable consequences
of the quantum cosmological tilt. The first, is that the blue tilt for value
of $k\eta>1$ could yield the dark matter in the form of primordial black
holes. This can be tested in the near future with upcoming experiments.

A feature that can be already tested with current observations is the behavior
of the quantum cosmological tilt at scales ($\sim 10$ Mpc) of about $100$
times smaller than the current horizon. In this region, the small bump gives
us a red tilt of order ($1-n_{s}\sim 0.1-0.3$) deviating significantly from
scale invariance in the sense of generating less power on these scales. These
scales can be tested observationally independently of the CMB.

The level of the power spectrum is usually represented by the combination of
the root mean square of the fluctuations in a comoving top-hat filter of 8 Mpc
and the current matter density ($\Omega_{m}$) as
$S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}$). The fluctuations can be measured by
using the CMB determined value and extrapolating it to low redshift adopting
the $\Lambda$CDM model. Alternatively, they can be measured directly from the
low redshift universe by computing, for example, the correlation function of
the weak lensing shear field. Extrapolating the $\Lambda$CDM model parameters,
as measured by the Planck18 mission at the CMB, yields a value of
$S_{8}=0.83\pm 0.013$ [25] while measurements of the weak lensing correlation
function by the KiDS team [38] at redshifts of $z\sim 1$ yield to
$S_{8}=0.759\pm 0.022$. There is a discrepancy of about $3\sigma$ between the
two values, with the present Universe being less clustered than the early one.

The value for $1-n_{s}$ as inferred from the $S_{8}$ measurement of KiDS is
shown as the blue dot in Fig. 4 with corresponding uncertainties. This lack of
power seems to fit well with the predictions of the quantum Fisher tilt. We
also show the spline model independent reconstruction of $1-n_{s}$ by Ref.
[39] (their Fig. 4), where we show the 95% confidence contours. This is a
reconstruction for larger scales and agrees well with our the model
prediction. In addition, the gray shaded area shows the value of $1-n_{s}$
assuming its value at all scales is that of the CMB.

Figure 4: The black line shows the tilt in the power spectrum of primordial
fluctuations as a function of scale computed from the quantum Fisher. Note
that for scales a factor ten smaller than the current horizon, $1-n_{s}\sim
0.03$. Overplotted as orange and blue lines are the best constraints, at the
95% level, on the reconstruction of $1-n_{s}$ from Planck data and large scale
structure from Ref. [39] allowing the tilt to be a free function modeled by
splines; this allows for the most freedom in the reconstruction. The blue dot
and associated uncertainty bars shows the effective value of $1-n_{s}$ as
derived from the local universe by the KidS-1000 weak lensing survey [38].
This is a low redshift value and independent of the CMB.

## Appendix: IR sensibility of the quantum cosmological tilt

Figure 5: The effect of the number of pair production $n$ on the cosmological
tilt $1-n_{s}$. The red, blue, orange and black curves are for
$n=1,10,1000,100,000$ respectively. The nearly scale invariant tilt is
produced by the increase in $n$. See text for more details on how to set $n$
and end inflation. Figure 6: The value of the spectral index $1-n_{s}$ as a
function of pair production $n$. For $n>1000$ it converges to $1-n_{s}=0.0328$
and remains at that value.

The quantum Fisher information for the state $|\Lambda\rangle$ has been
computed in the basis of entangled pairs dictated by the de Sitter Bogolioubov
transformation. Numerically, we have performed the sum over the number of
pairs until a certain cutoff value $N$. In this section we discuss how the
shape of the quantum cosmological tilt $\alpha_{F}$ depends on this cutoff.
First of all, note that in the high energy regime of large values of $k\eta$,
the contribution of states with large number of pairs is very suppressed, i.e.
the quantum states in that region are coherent and not squeezed. Contrary, in
the IR regime, with small $k\eta$ and large squeezing parameter, states with
arbitrarily large value of soft entangled pairs contribute. In this sense, the
UV regime of the quantum cosmological tilt should not be very dependent on the
value of the cutoff $N$ while the IR regime should be very sensible to this
cutoff. This phenomena is explicitly shown in Fig. 5 where we observe two
interconnected facts. On one side, we observe that the blue tilted UV phase is
not changing when we increase the value of $N$. On the other side, we also
observe that the almost scale invariant region becomes larger when we increase
the value of $N$. Physically this effect of $N$ is teaching us two main
lessons. First, that the almost scale invariant region we associate with the
inflationary period appears as a consequence of the contribution of large
numbers of entangled pairs. Until what value of the scale $k\eta$ extends this
region is a consequence of the state normalization. The normalization factor
depends on the squeezing parameter as $\frac{1}{cosh(r(k\eta))}$ with
$r(k\eta)=\log(\sqrt{1+\frac{1}{4(k\eta)^{2}}}+\frac{1}{2k\eta})$. Thus in the
deep IR region this normalization factor should depend on the number $N$ of
pairs as $\frac{1}{\sqrt{N}}$. This sets the smaller value of $k\eta$ for the
cutoff $N$ as

$N\sim\frac{1}{(k\eta)^{2}}$ (4.3)

In other words, for a given value $N$ the computed quantum cosmological tilt
is reliable only until this IR scale. A priori we do not have any obvious
physical criteria to set this cutoff on $N$. What we observe is that for a
given large value of $N$ the extension in time of the almost scale invariant
inflationary period can be estimated using (4.3) to be

${\cal{N}}\sim\frac{1}{2}\ln N$ (4.4)

with ${\cal{N}}$ the number of e-foldings during the inflationary period. In
other words, the end of inflation is set by the IR cutoff on $N$. In the
numerical analysis we have performed we get some hints on how the system with
an IR cutoff $N$ behaves when we formally push $(k\eta)$ to super horizon
scales much smaller than (4.3). In this regime, where we push $k\eta$ beyond
the bound (4.3) the system tendency, triggered by the normalization condition,
is to leave inflation creating a large red tilt formally implying an epsilon
larger than one. This is an appealing feature that is however out of the
effective control of our computation so it should be taken with a grain of
salt.

Finally, let us comment on the prediction of $(1-n_{s})$ in CMB scales. This
prediction was done solving the master equation and obviously depends on the
form of $\alpha_{F}$ so we could wonder if this prediction is sensible to the
IR cutoff $N$. In Fig. 6 we show that the value of $1-n_{s}$ converges for
$n>1000$ and remains at a value $1-n_{s}=0.0328$ for any value of $n>1000$. In
a certain sense the spectral index is a property of the effective universality
class defined by taking the limit $N=\infty$ in the IR cutoff. In summary this
value of the spectral index is very insensitive to the IR cutoff $N$.

So summarizing:

  * •

The sum over large number of entangled pairs is crucial to have an almost
scale invariant inflationary region.

  * •

The UV blue tilted preinflationary region is a robust prediction that is not
affected by the IR cutoff on $N$.

  * •

The predicted value of the spectral index using the master equation is equally
robust and not affected by the IR cutoff.

  * •

The IR cutoff $N$ sets formally the duration of the inflationary period.

As a very speculative final comment we could try to use some entropy argument
to set a bound on the value of $N$. If we interpret $\ln N$ as an entropy
measure we could bound $N$ by requiring $\ln N$ to be smaller than $N_{GH}$
the Gibbons Hawking de Sitter entropy. We will investigate quantitatively
further this superficial comment in the future.

## Acknowledgments

The work of CG was supported by grants SEV-2016-0597, FPA2015-65480-P and
PGC2018-095976-B-C21. The work of RJ is supported by MINECO grant
PGC2018-098866-B-I00 FEDER, UE. RJ acknowledges “Center of Excellence Maria de
Maeztu 2020-2023” award to the ICCUB (CEX2019- 000918-M).

## References

  * [1] C. Gomez and R. Jimenez, “Cosmology from Quantum Information,” Phys. Rev. D 102 (2020) no.6, 063511 doi:10.1103/PhysRevD.102.063511 [arXiv:2002.04294 [hep-th]].
  * [2] C. Gomez and R. Jimenez, “Dark matter from primordial quantum information,” JCAP 10 (2020), 004 doi:10.1088/1475-7516/2020/10/004 [arXiv:2003.08402 [astro-ph.CO]].
  * [3] C. Gomez and R. Jimenez, “How Gaussian can the Sky be? Primordial Non-Gaussianity from Quantum Information,” JCAP 07 (2020), 047 doi:10.1088/1475-7516/2020/07/047 [arXiv:2005.09506 [astro-ph.CO]].
  * [4] C. Gomez and R. Jimenez, “The Quantum Origin of Quasi de Sitter: a Model Independent Quantum Cosmological Tilt,” [arXiv:2012.04003 [hep-th]].
  * [5] C. Gomez and R. Jimenez, “The quantum cosmological tilt and the origin of dark matter,” [arXiv:2012.07883 [hep-th]].
  * [6] M. G. A. Paris, Quantum estimation for quantum technology - International Journal of Quantum Information, 2009 -
  * [7] J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65(1990) 1697-1700
  * [8] A. A. Starobinsky, Phys. Lett. 91B (1980) 99 [Adv. Ser. Astrophys. Cosmol. 3 (1987) 130]. doi:10.1016/0370-2693(80)90670-X
  * [9] K. Sato, “First Order Phase Transition of a Vacuum and Expansion of the Universe,” Mon. Not. Roy. Astron. Soc. 195 (1981), 467-479 NORDITA-80-29.
  * [10] A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23 (1981) 347 [Adv. Ser. Astrophys. Cosmol. 3 (1987) 139]. doi:10.1103/PhysRevD.23.347
  * [11] A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Adv. Ser. Astrophys. Cosmol. 3 (1987), 149-153 doi:10.1016/0370-2693(82)91219-9
  * [12] A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48 (1982) 1220 [Adv. Ser. Astrophys. Cosmol. 3 (1987) 158]. doi:10.1103/PhysRevLett.48.1220
  * [13] A. D. Linde, “Chaotic Inflation,” Phys. Lett. B 129 (1983), 177-181 doi:10.1016/0370-2693(83)90837-7
  * [14] Kofman, L. A. and Linde, Andrei D. and Starobinsky, Alexei A. ”Inflationary Universe Generated by the Combined Action of a Scalar Field and Gravitational Vacuum Polarization”, Phys. Lett. B 157 (1985) 361-367
  * [15] G. V. Chibisov and V. F. Mukhanov, “Galaxy formation and phonons,” Mon. Not. Roy. Astron. Soc. 200 (1982), 535-550
  * [16] V. F. Mukhanov and G. V. Chibisov, “Quantum Fluctuations and a Nonsingular Universe,” JETP Lett. 33 (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549].
  * [17] S. W. Hawking, “The Development of Irregularities in a Single Bubble Inflationary Universe,” Phys. Lett. 115B (1982) 295.
  * [18] A. A. Starobinsky, “Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations,” Phys. Lett. B 117 (1982), 175-178 doi:10.1016/0370-2693(82)90541-X
  * [19] A. H. Guth and S. Y. Pi, “Fluctuations in the New Inflationary Universe,” Phys. Rev. Lett. 49 (1982), 1110-1113
  * [20] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, “Spontaneous Creation of Almost Scale - Free Density Perturbations in an Inflationary Universe,” Phys. Rev. D 28 (1983), 679 doi:10.1103/PhysRevD.28.679
  * [21] V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger,“Theory of Cosmological Perturbations” Physics Reports 215, 203 (1992)
  * [22] N. A. Chernikov and E. A. Tagirov, Ann. Inst. H. Poincare Phys. Theor. A 9 (1968), 109
  * [23] T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 360 (1978), 117-134 doi:10.1098/rspa.1978.0060
  * [24] A. Albrecht, P. Ferreira, M. Joyce and T. Prokopec, “In-flation and squeezed quantum states,” Phys. Rev. D50,4807-4820 (1994) [arXiv:astro-ph/9303001 [astro-ph]]
  * [25] N. Aghanim et al. [Planck], “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020), A6 doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * [26] R. H. Brandenberger, V. F. Mukhanov and T. Prokopec,“Entropy of a classical stochastic field and cosmologicalperturbations,” Phys. Rev. Lett.69, 3606 (1992) [astro-ph/9206005].
  * [27] R. H. Brandenberger, T. Prokopec and V. F. Mukhanov,“The Entropy of the gravitational field,” Phys. Rev. D48, 2443 (1993) [gr-qc/9208009].
  * [28] D. Campo and R. Parentani, “Decoherence and entropyof primordial fluctuations. I: Formalism and interpreta-tion,” Phys. Rev. D78, 065044 (2008) [arXiv:0805.0548[hep-th]];D. Campo and R. Parentani, “Decoherence and entropyof primordial fluctuations II. The entropy budget,” Phys.Rev. D78, 065045 (2008) [arXiv:0805.0424 [hep-th]]
  * [29] J. Martin and V. Vennin, “Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?,” Phys. Rev. D 93 (2016) no.2, 023505 doi:10.1103/PhysRevD.93.023505
  * [30] D. Green and R. A. Porto, “Signals of a Quantum Universe,” Phys. Rev. Lett. 124 (2020) no.25, 251302 doi:10.1103/PhysRevLett.124.251302 [arXiv:2001.09149 [hep-th]].
  * [31] G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation, JCAP1401,023 (2014) [arXiv:1312.4795 [hep-th]].
  * [32] G. Dvali and C. Gomez, “Quantum Exclusion of Positive Cosmological Constant?,” Annalen Phys.528, 68-73 (2016) doi:10.1002/andp.201500216 [arXiv:1412.8077[hep-th]].
  * [33] G. Dvali, C. Gomez and S. Zell, “Quantum Break-Timeof de Sitter,” JCAP1706, 028 (2017) [arXiv:1701.08776[hep-th]];G. Dvali, C. Gomez and S. Zell, “Quantum BreakingBound on de Sitter and Swampland,” Fortsch. Phys.67,no. 1-2, 1800094 (2019) [arXiv:1810.11002 [hep-th]]
  * [34] G. Dvali, “$S$-Matrix and Anomaly of de Sitter,” Symmetry 13 (2020) no.1, 3 doi:10.3390/sym13010003 [arXiv:2012.02133 [hep-th]].
  * [35] B. Carr and F. Kuhnel, “Primordial Black Holes as Dark Matter: Recent Developments,” Ann. Rev. Nucl. Part. Sci. 70 (2020), 355-394 doi:10.1146/annurev-nucl-050520-125911 [arXiv:2006.02838 [astro-ph.CO]].
  * [36] A. Ijjas and P. J. Steinhardt, “Bouncing Cosmology made simple,” Class. Quant. Grav. 35 (2018) no.13, 135004 doi:10.1088/1361-6382/aac482 [arXiv:1803.01961 [astro-ph.CO]].
  * [37] V. A. Rubakov, “The Null Energy Condition and its violation,” Usp. Fiz. Nauk 184 (2014) no.2, 137-152 doi:10.3367/UFNe.0184.201402b.0137 [arXiv:1401.4024 [hep-th]].
  * [38] M. Asgari et al. [KiDS], “KiDS-1000 Cosmology: Cosmic shear constraints and comparison between two point statistics,” Astron. Astrophys. 645 (2021), A104 doi:10.1051/0004-6361/202039070 [arXiv:2007.15633 [astro-ph.CO]].
  * [39] A. Ravenni, L. Verde and A. J. Cuesta, “Red, Straight, no bends: primordial power spectrum reconstruction from CMB and large-scale structure,” JCAP 08 (2016), 028 doi:10.1088/1475-7516/2016/08/028 [arXiv:1605.06637 [astro-ph.CO]].