# Quantum Gravity in Species Regime

Gia Dvali  Arnold Sommerfeld Center, Ludwig-Maximilians-University, Munich,
Germany,  Max-Planck-Institute for Physics, Munich, Germany

###### Abstract

A large number of particle species allows to formulate quantum gravity in a
special double-scaling limit, the species limit. In this regime, quantum
gravitational amplitudes simplify substantially. An infinite set of
perturbative corrections, that usually blur the picture, vanishes, whereas the
collective and non-perturbative effects can be cleanly extracted. Such are the
effects that control physics of black holes and of de Sitter and their
entanglement curves. In string theory example, we show that the entropy of
open strings matches the Gibbons-Hawking entropy of a would-be de Sitter state
at the point of saturation of the species bound. This shows, from yet another
angle, why quantum gravity/string theory cannot tolerate a de Sitter vacuum.
Finally, we discuss various observational implications.

##  I Introduction and Summary

The main technical obstacle in understanding the microscopic physics of
objects such as a black hole or a de Sitter space, is a presumed complexity of
quantum gravitational amplitudes in non-perturbative regimes. It is a common
perception that processes associated with non-perturbative objects, at weak
coupling $\alpha$, are exponentially suppressed ${\rm e}^{-1/\alpha}$.

From here, one is tempted to makes two conclusions. First, non-perturbative
effects are completely dwarfed by a variety of contributions that are power-
law in $\alpha$. Secondly, at weak coupling, the non-perturbative objects, for
all practical purposes, can be treated semi-classically.

However, in several instances the situation is much more subtle. This is the
case for the states such as a black hole or a de Sitter Dvali:2011aa ;
Dvali:2012en ; Dvali:2013eja ; Dvali:2014gua ; Dvali:2017eba . The reason is
that both states belong to the category of the saturated systems, or saturons
for short Dvali:2020wqi . Such systems, despite being non-perturbative and
macroscopic, depart from semi-classicality rather fast.

For understanding the meaning of saturation, let us first note that non-
perturbative processes can be strongly enhanced by the following three
factors:

  * •

The occupation number of quanta, $N$;

  * •

The micro-state entropy, $S$;

  * •

The number of interacting particle species, $N_{\rm sp}$.

In fact, the enhancement can be so strong that a given process can be pushed
to saturating the unitarity bound. The phenomenon is not limited to gravity
and is a generic property of saturons.

###  I.1 Saturons

This term describes the systems that satisfy the following limiting relation,
imposed by unitarity, Dvali:2020wqi :

$N=\frac{1}{\alpha}=S\,.$ (1)

Here, $\alpha$ must be understood as the scale-dependent quantum coupling of
the theory, evaluated at the size of the system, $R$. As usual, the entropy
$S$ is defined as the log from the number of degenerate micro-states.

The saturation condition (1) encodes three relations. The primary idea
Dvali:2020wqi ; Dvali:2019jjw  is that, for an arbitrary system at weak
coupling, unitarity imposes the following upper bound on the entropy,

$S\leqslant\frac{1}{\alpha}\,.$ (2)

Correspondingly, a self-sustained system with $N$ constituent quanta, at its
maximal entropy capacity, satisfies the relation (1).

It was also argued in Dvali:2020wqi ; Dvali:2019jjw  that for a self-sustained
system, satisfying (1), the entropy $S$ is equal to the area measured in units
of a Goldstone decay constant, $f$. The presence of a Goldstone is generic,
due to a spontaneous breaking of Poincare symmetry by any saturon. Thus, the
saturation relation (1), also implies the area-form of the
entropy111Throughout the paper, the unimportant numerical coefficients shall
be set equal to one.,

$S={\rm Area}\cdot f^{2}=(Rf)^{2}\,.$ (3)

For definiteness, we wrote the expression in four space-time dimensions, but
the same area-law holds in arbitrary dimensions.

Notice, in case of a black hole, the Goldstone of a spontaneously broken
Poincare symmetry, is the graviton mode. Its decay constant is a Planck mass
$f=M_{P}$. This, was argued in Dvali:2020wqi ; Dvali:2019jjw , shows that the
area-form of the Bekenstein-Hawking entropy Bekenstein:1973ur ,

$S=(RM_{P})^{2}\,,$ (4)

is a particular example of the area-form of the entropy (3), originating from
the phenomenon of saturation. The same is true about Gibbons-Hawking entropy
Gibbons:1977mu  of de Sitter. This entropy is also given by (4), with $R$
understood as the de Sitter curvature radius.

That is, both, a black hole and a de Sitter Hubble patch, are saturons, as
they satisfy (1) and (3). However, the above correlations appear to be
universal and to go well beyond gravity Dvali:2020wqi ; Dvali:2019jjw . The
meaning of the expressions (1) and (3), shall become more transparent later.

The goal of the present paper is to focus on the role of the third item, the
number of particle species $N_{\rm sp}$. Throughout the paper, unless
otherwise stated, species will be assumed to be lighter that the relevant
energy scale in a given process. We shall primarily be interested in the
species of low spin, although most of the results will be applicable in the
presence of spin-$2$ species, as it is the case in Kaluza-Klein theories.

The idea is to use the magnifying power of particle species Dvali:2007hz , for
cleanly extracting certain quantum gravitational effects. We then apply this
method to gravitational saturons in form of black holes and de Sitter.

###  I.2 Species Limit

We shall define a double-scaling limit in which the Planck mass $M_{P}$ and
the number of species $N_{\rm sp}$ are taken infinite while the following
relation among them is maintained:

$M_{P}\rightarrow\infty,~{}N_{\rm
sp}\rightarrow\infty\,,~{}\frac{M_{P}}{\sqrt{N_{\rm sp}}}={\rm finite}\,.$ (5)

This regime has to be supplemented by a proper scaling of the graviton
occupation number $N$. We shall refer to (5), as the species limit.

The significance of the species limit (5), from quantum gravitational
perspective, is immediately captured by the fact that in this limit the half-
decay time of a black hole,

$t_{\rm half}=R^{3}\frac{M_{P}^{2}}{N_{\rm sp}}\,,$ (6)

is finite, despite the fact that the black hole mass is infinite. The
microscopic meaning of this phenomenon, shall be discussed in details.

In the species regime (5), the quantum gravitational amplitudes simplify
significantly. The secondary processes vanish while the non-perturbative
effects of interest stay finite. In other words, the species limit (5), allows
to distill a set of essential quantum gravitational effects. Some aspects of
this extraction were already studied in Dvali:2020wqi ; Dvali:2020etd , on
which we shall expand.

Next, we shall apply this treatment to saturons in gravity. As already said,
such are black holes and de Sitter. The fact that these states represent
saturons, follows from the $S$-matrix formulation of quantum gravity/string
theory. In this formulation, both objects must be considered as excited
composite states constructed on top of a valid $S$-matrix vacuum, such as
Minkowski. In other words, the $S$-matrix formulation leads to their
corpuscular resolution.

The corpuscular picture Dvali:2011aa ; Dvali:2012en ; Dvali:2013eja , reveals
that black holes as well as de Sitter satisfy the saturation relation (1). Due
to saturation, despite of their non-perturbative nature, both systems are
susceptible to quantum gravitational effects that alter their semi-classical
properties on rather short time-scales. The relevant time-scales are only
power-law (or even logarithmic Dvali:2013vxa ) in $1/\alpha$.

###  I.3 Inner Entanglement and Quantum Break-Time

A particularly important time-scale, delivered by the corpuscular picture, is
the quantum break-time $t_{Q}$. The concept was introduced in Dvali:2013vxa ,
within a prototype model for a black hole $N$-portrait Dvali:2011aa ;
Dvali:2012en . It was applied to de Sitter and to black holes in Dvali:2013eja
and in subsequent papers. The physical meaning of $t_{Q}$ is that, beyond it,
a total breakdown of the semi-classical approximation takes place. The major
mechanisms contributing into this departure are: 1) Inner entanglement
Dvali:2013eja ; Dvali:2014gua ; Dvali:2017eba ; and 2) Memory burden effect
Dvali:2018xpy ; Dvali:2018ytn ; Dvali:2020wft . Both effects shall be reviewed
in due course.

As discussed originally in Dvali:2013eja , the above effect is fundamental for
understanding physics behind the so-called Page’s curve Page:1993wv .
According to Page, the entanglement must reach the maximum after a black hole
emits about half of its mass. The semi-classical picture cannot explain this.
This picture is blind to an inner structure of the black hole. Because of
this, the semi-classical theory captures only one sort of entanglement: The
entanglement between a black hole and an outgoing radiation.

The corpuscular picture reveals the existence of another type of entanglement,
which we can refer to as the inner entanglement Dvali:2013eja . This effect
describes the entanglement among the internal degrees of freedom composing a
black hole. Together with the memory burden effect Dvali:2018xpy ;
Dvali:2018ytn ; Dvali:2020wft , the inner entanglement grows in time and
reaches the maximum, latest, by a half-decay. This is the origin of the
quantum break-time $t_{Q}$. These effects offer a new physical meaning for the
Page’s curve.

Note, by providing an internal quantum clock, the corpuscular picture predicts
the existence of the entanglement curve also for de Sitter Dvali:2013eja . It
shows that the entanglement must reach its maximum at the quantum breaking
point. In this sense, de Sitter is similar to a black hole. However, beyond
$t_{Q}$, the time evolutions of a black hole and of de Sitter exhibit drastic
differences. While a black hole can continue its existence beyond $t_{Q}$, for
de Sitter this is deadly. The incompatibility of a de Sitter vacuum with the
$S$-matrix formulation of quantum gravity, manifests itself though the
anomalous quantum break-time Dvali:2020etd .

In the present paper, we shall re-analyse the above dynamics, by taking into
account the effect of species. It has already been shown previously that, with
other parameters fixed, the species shorten $t_{Q}$. More precisely, for a
generic saturated system, $t_{Q}$ scales as Dvali:2017eba ; Dvali:2020wqi ;
Dvali:2020etd ,

$t_{Q}\,=\,\frac{R}{\alpha N_{\rm sp}}\,.$ (7)

Notice, for a black hole or a de Sitter, this time-scale is equal to (6).

In the present case, we shall study the species effect in various double-
scaling regimes, such as (5). This allows us to isolate the three important
time scales of: 1) Half-decay; 2) Inner entanglement; 3) Memory burden. All
three time-scales are bounded by $t_{Q}$ (7) (and by $t_{\rm half}$ (6)) and
remain finite in the species limit (5).

###  I.4 String Theory

Next, we use the species limit (5) for monitoring the resistance of string
theory against the deformation towards a de Sitter state. We take this limit
in an explicit string theoretic example in which $n$ $D-\bar{D}$-brane pairs
are plied up on top of each other. In the species limit, the string coupling
$g_{s}$ vanishes while the number $N_{\rm sp}=n^{2}$ of Chan-Paton factors is
taken infinite. At the same time, the product $ng_{s}$ is kept finite. As
already shown in Dvali:2020etd , this product determines the quantum break-
time $t_{Q}$ of a would-be de Sitter state. Using $ng_{s}$ as a control
parameter, we can change $t_{Q}$.

This example reveals some interesting correlations. The entropy of open string
modes, matches the Gibbons-Hawking entropy of de Sitter space, when the number
of Chan-Paton species saturates the unitarity bound (2). This bound, in the
present case reads as, $ng_{s}\lesssim 1$.

At the same point, the thermal corrections from Gibbons-Hawking excitations of
the open strings can flip the sign of the tachyon mass and stabilize it.
However, for the same values of the parameters, the de Sitter curvature
approaches the string scale. Simultaneously, the quantum break-time becomes of
order the string length. This is the way the string theory responds to its
deformation towards a would-be de Sitter state.

We see that string theory possesses the resources, in form of open string
modes, for accounting for Gibbons-Hawking entropy. However, the very same
degrees of freedom, speed-up the quantum breaking of the de Sitter “vacuum”.
In this way, the string theory exhibits a built-in mechanism restoring the
$S$-matrix consistency by abolishing de Sitter.

Finally, we discuss some observational imprints of quantum gravity coming from
inflation and from black holes. By consistency, the duration of inflation is
bounded from above by $t_{Q}$. Since species shorten $t_{Q}$, they strengthen
the quantum gravitational imprints from the inflationary phase, making them
potentially-observable, or possibly dominant Dvali:2020etd .

Witten has introduced the concept of meta-observables in de Sitter
Witten:2001kn . This concept is linked with the classical no-hair properties
of de Sitter. However, the finiteness of $t_{Q}$, effectively can promote them
into quantum-observables. Basically, we are saying that $t_{Q}$ endows de
Sitter with a quantum hair Dvali:2013eja ; Dvali:2018ytn . This hair is
suppressed by $1/t_{Q}$ and so are its observable imprints. Since the species
shorten $t_{Q}$ (see, (7)), these imprints can be significant, even for non-
far-future observations, if the number of species is high Dvali:2020etd . This
magnifying power, is demonstrated by the species limit (5). In this limit, the
de Sitter quantum hair is non-vanishing, even though the Planck mass is
infinite.

##  II Species and Scale of Quantum Gravity

The fundamental property of species is the enhancement of the quantum
gravitational effects. This phenomenon is linked with lowering the
gravitational cutoff, $M_{*}$, by species. Here, the cutoff $M_{*}$ is defined
as the scale at which the quantum gravitational interaction becomes strong. In
particular, a particle scattering process with momentum-transfer exceeding
$M_{*}$, cannot be described within the low energy QFT of gravity.
Correspondingly, no semi-classical treatment is possible for such systems. In
particular, this concerns the would-be cosmological scenarios with either
temperatures or curvatures exceeding the scale $M_{*}$.

The scale $M_{*}$ is highly sensitive to the number of species $N_{\rm sp}$.
This follows from the arguments that are independent of perturbation theory.
In particular, the black hole physics tells us Dvali:2007hz  that in theory
with $N_{\rm sp}$ particle species in $4$-dimensions, there is the following
upper bound on the scale $M_{*}$,

$M_{*}=\frac{M_{P}}{\sqrt{N_{\rm sp}}}\,.$ (8)

Correspondingly, in $d$ space-time dimensions the bound reads,

$M_{*}=\frac{M_{d}}{N_{\rm sp}^{\frac{1}{d-2}}}\,,$ (9)

where $M_{d}$ is a $d$-dimensional Planck mass. This bound is non-perturbative
and cannot be removed by re-summation. One can arrive to it in several
alternative ways. Correspondingly, the bound (8) can be given the several
different physical meanings.

We shall briefly recount two physical arguments of Dvali:2007hz . More non-
perturbative evidence supporting (8), shall emerge later. The first argument
is based on the black hole evaporation. Within the validity of semi-classical
gravity, a black hole of mass $M$ and radius $R=M/M_{P}^{2}$, emits a thermal
Hawking radiation of temperature $T=1/R=M_{P}^{2}/M$ Hawking:1974sw . Because
of its thermal nature and the universality of gravitational interaction,
Hawking radiation is democratic in all particle species. Due to this, the mass
of a black hole changes in time according to the following Stefan-Boltzmann
law,

$\dot{M}=-N_{\rm sp}T^{2}\,.$ (10)

Using the relation between the mass and the temperature, we can rewrite this
equation in the following form,

$\frac{\dot{T}}{T^{2}}=N_{\rm sp}\frac{T^{2}}{M_{P}^{2}}\,.$ (11)

This form is highly instructive. The left hand side of the equation represents
a measure of a departure from semi-classicality and thermality. Within the
validity of the semi-classical treatment, this quantity must be less than one.
This puts an universal upper bound on a temperature of a semi-classical black
hole,

$T_{*}=\frac{M_{P}}{\sqrt{N_{\rm sp}}}\,.$ (12)

This maximal temperature marks the gravitational cutoff of the low energy
theory (8). An extrapolation of the semi-classical regime above this scale,
would lead us into an obvious inconsistency. In particular, a black hole of
temperature $T\gg T_{*}$, would lose energy faster than its own inverse
temperature. Notice, it would radiate its entire mass within the time $\Delta
t<1/T\sim R$. This is absurd. Under no circumstances can such a black hole be
treated semi-classically.

Alternatively, we can arrive to the bound (8), by considering a maximal
possible Gibbons-Hawking temperature of a de Sitter like state in a theory
with $N_{\rm sp}$. Since the pioneering work by Gibbons and Hawking
Gibbons:1977mu , it is well-known that a de Sitter like Universe (such as an
inflationary Universe) is constantly creating particles with thermal-like
spectrum of temperature $T_{GH}=H$. Here $H$ is the Hubble parameter, which is
related with the de Sitter curvature (Hubble) radius as $H=1/R$. The value of
$H$, is determined by the energy density $V$ as, $H^{2}=V/M_{P}^{2}$. In
inflation, $V$ is dominated by the potential energy density of the inflaton
field.

Similarly to the Hawking radiation of a black hole, within the validity of
semi-classical gravity, the Gibbons-Hawking radiation is nearly-thermal. Its
energy density is given by,

$\rho_{GH}=N_{\rm sp}T_{GH}^{4}\,.$ (13)

It is easy to check that, for a temperature $T_{GH}\gg M_{*}$, this expression
would exceed the energy density $V$ of the de Sitter. Of course, this is
impossible.

For example, just like in the case of a black hole, a Hubble patch with
$H>M_{*}$, would convert its entire energy into the Gibbons-Hawking radiation
in less than one Hubble-time, $\Delta t=1/H$. This is again an absurd.

Thus, similarly to the temperature of a black hole (12), $T_{GH}$ is bounded
by the scale $M_{*}$ (8). Equivalently, the inflationary Hubble scale is
bounded as,

$H\leqslant\frac{M_{P}}{\sqrt{N_{\rm sp}}}=M_{*}\,.$ (14)

We shall adopt this as an absolute upper bound. Discussions of other aspects
of species bound in de Sitter can be found in Dvali:2008sy .

The bound (8) has also been derived Dvali:2008ec ; Dvali:2008jb ;
Brustein:2009ex ; Palti:2019pca  from the requirement that a black hole should
not exceed the maximal capacity of information storage and processing. In
particular, the species entropy should not exceed the standard Bekenstein-
Hawking entropy Bekenstein:1973ur  of a black hole.

In fact, the information storage capacity is also bounded by the unitarity of
the scattering process in which two gravitons, of center of mass energy
$E=N_{\rm sp}M_{*}$, produce $N_{\rm sp}$ species Dvali:2020wqi . The
unitarity of this process reinforces the bound (2), on the entropy of the
final state. This effectively translates as the bound (8). This connection
shall be discussed in more details below.

There also exist perturbative arguments supporting (8) (see, e.g.,
Dvali:2001gx ; Veneziano:2001ah ). We shall limit ourselves with non-
perturbative ones, as they cannot be avoided by the re-summation of
perturbation series.

##  III Species Regime of Quantum Gravity

We have seen that non-perturbative arguments unambiguously indicate that
quantum gravity is becoming strong at the scale $M_{*}$, given by (8). Already
this fact shows that species possess a magnifying power over quantum gravity.
Using this power, we would like to cleanly extract certain quantum
gravitational effects.

For this, we first define a basic quantum gravitational coupling $\alpha$. Let
us consider a tree-level scattering process involving gravitons with momentum-
transfer $q$. For example, an annihilation of a pair of gravitons, of a center
of mass energy $\sim q$, into a pair of some species. This process is governed
by an effective quantum coupling given by

$\alpha=\frac{q^{2}}{M_{P}^{2}}\,.$ (15)

This is what we shall call a basic quantum gravitational coupling. In general,
in a process with more complicated diagrammatic structure, the coupling
$\alpha$ in each vertex is determined by the momentum flow through it.

The momentum-transfer in a graviton vertex shall be restricted by the cutoff
$q\leqslant M_{*}$. Correspondingly, the coupling shall satisfy

$\alpha\leqslant\frac{M_{*}^{2}}{M_{P}^{2}}=\frac{1}{N_{\rm sp}}\,.$ (16)

In what follows, we shall work in the regime of a weak coupling $\alpha$ and
effectively take the limit,

$\alpha\rightarrow 0\,.$ (17)

Naively, one may expect that, in this limit, all quantum gravitational effects
vanish. However, this is not the case. In reality, (17) allows to make certain
phenomena more transparent. For achieving this, the scalings of other
quantities must be chosen properly. For the convenience of the description, we
shall introduce some useful parameters.

###  III.1 Species Limit

First, let us define a species coupling,

$\lambda\equiv N_{\rm sp}\alpha=\frac{q^{2}}{M_{*}^{2}}\,,$ (18)

where in the very last expression we used the relations (8) and (15).

For a reader familiar with the planar treatment of QCD with large number of
colors tHooft:1973alw , it must be clear that the species coupling (18)
represents a gravitational analog of ’t Hooft’s coupling. Due to universality
of the gravitational interaction, the role of “colors” in gravity is played by
the number of all particle species.

This said, there are fundamental differences. Unlike the gluons in QCD, which
carry color and anti-color indexes, no species label is carried by the
graviton. This suppresses a large number of quantum gravitational processes
analogs of which in gluodynamics would be non-zero.

We now take the double-scaling limit (5). In this regime, both $N_{\rm sp}$
and $M_{P}$ become infinite while the scale $M_{*}$ is kept finite. We shall
refer to this as the species limit. It is clear from (16) that in this regime
$\alpha$ vanishes while $\lambda$ remains finite. The behaviour of various
parameters in the species limit is summarized as,

$\displaystyle M_{P}\rightarrow\infty\,,~{}N_{\rm
sp}\rightarrow\infty\,,~{}\alpha\rightarrow 0\,,$ (19) $\displaystyle
M_{*}={\rm finite}\,,~{}\lambda={\rm finite}\,.$

In this regime, the quantum gravity simplifies substantially. All quantum
gravitational processes, in which each power of $\alpha$ is not accompanied by
$N_{\rm sp}$, vanish. An example is provided by Fig.1a.

Figure 1:  1.a) An example of a quantum gravitational amplitude that vanishes
in the double-scaling limit (19). The gravitons are denotes by black wavy
lines and the species by solid red ones. This amplitude scales as
$\alpha^{5}N_{\rm sp}=\lambda\alpha^{4}$ and is zero for (19). 1.b) A process
of graviton annihilation into $n$ species. For $n\sim N_{\rm sp}$ the cross
section is non-zero and for $\lambda\sim 1$ can even saturate unitarity due to
a maximal entropy of the final state. It therefore provides an example of a
quantum gravitational process that survives in the limit (19).

Let us consider the theory of Einstein gravity, minimally coupled to $N_{\rm
sp}$ particle species. We shall treat it as an effective QFT of a graviton
($h_{\mu\nu}$) and of other particle species, formulated on an $S$-matrix
vacuum of Minkowski. In such a theory, the linearized metric perturbations are
described by excitations of a canonically normalized graviton field around the
Minkowski space, $\delta g_{\mu\nu}=\eta_{\mu\nu}+\frac{h_{\mu\nu}}{M_{P}}$.
Likewise, the states of arbitrary complexity are viewed as excited states
constructed on top of the Minkowski vacuum. For definiteness, we assume that
the species are light and stable. We now take the limit (19).

Let us first consider only the states of finite energies. That is, the total
energy is allowed to be arbitrarily large, as long as it is finite. Obviously,
in such states, the occupation numbers of finite energy (frequency) quanta are
also finite.

In such a case, the theory effectively reduces to a linear theory of a
massless graviton $h_{\mu\nu}$, coupled to species. The Lagrangian can be
written as,

${\mathcal{L}}=h^{\mu\nu}{\mathcal{E}}_{\mu\nu}\,+\,\frac{1}{M_{*}\sqrt{N_{\rm
sp}}}h_{\mu\nu}\sum_{j=1}^{N_{\rm sp}}T_{j}^{\mu\nu}+\,...\,,$ (20)

where ${\mathcal{E}}_{\mu\nu}$ is the linearized Einstein tensor and
$j=1,2,...,N_{\rm sp}$ is the species label. $T_{j}^{\mu\nu}$ are their energy
momentum tensors. All the self-couplings of the graviton vanish. The same is
true about its couplings to species that are higher than quadratic in
$h_{\mu\nu}$.

At the loop level, a relevant correction is a renormalization of the graviton
kinetic term by species. In the parameterization of (20), this correction is
of zeroth order in $1/N_{\rm sp}$ and, therefore, survives in the species
limit (19).

On the other hand, the corrections to non-linear couplings of gravitons
vanish. Correspondingly, the same applies to any transition amplitude between
finite numbers of initial and final gravitons,

$N_{in}\rightarrow N_{f}\,.$ (21)

Such amplitudes are zero to all orders within the validity of $1/N_{\rm
sp}$-expansion. An example of a four-loop correction to $2\rightarrow 2$
graviton scattering amplitude, is provided by Fig.1a. This correction scales
as $\alpha^{5}N_{\rm sp}=\lambda\alpha^{4}$ and is zero for (19).

Notice, for the suppression of higher order corrections, the relation (8) is
of fundamental importance. It is the ability of species, to lower the cutoff
relative to $M_{P}$, that enables us to consistently restrict the momentum-
transfer among the particles (both, virtual and real) by the scale $M_{*}$.
Correspondingly, this restricts the coupling $\alpha$ by $1/N_{\rm sp}$, as
indicated by (16).

We also wish to remark that starting with a minimal version of Einstein-
Hilbert action, is not essential. We could have instead started with a theory
in which infinite series of high order curvature invariants are preemptively
added to this action. An each additional derivative would be accompanied by
the factor $1/M_{*}$. After canonically normalizing the fields, an
$n$-graviton interaction vertex is suppressed by $1/M_{P}^{n-2}$. Due to this,
the expansion in terms of the curvature invariants, would effectively be
converted into a $1/\sqrt{N_{\rm sp}}$-expansion in terms of graviton
interactions. Consequently, all nonlinear self-interactions of graviton would
again vanish in the species limit (19).

The simplification of quantum gravity in the species regime (19), allows us to
extract non-perturbative processes. As already pointed out in Dvali:2020wqi ,
in this limit certain collective processes are magnified to the level of
saturating unitarity. One example considered there is the process,

${\rm graviton+graviton}~{}\rightarrow~{}N-{\rm particles}\,,$ (22)

in which two gravitons, of center of mass energy $E=Nq$, annihilate in $N$
quanta of various species 222Of course, since the final state quanta
gravitate, they must be properly dressed by soft gravitons. This standard IR-
dressing is independent of the entropy of species and is assumed to be done in
all processes of interest.. An each final particle, carries the energy $q$,
which is a $1/N$-fraction of the initial energy. A particular diagram
contributing to such a process is given by Fig.1b. The amplitude of this
process is proportional to $\alpha^{N/2}$ where the effective coupling
$\alpha$ is given by (15). Thus, it is of order $N/2$ in $\alpha$-expansion.

The saturation regime is established when the occupation number in the final
state is $N=1/\alpha$ while $N_{\rm sp}$ approaches the same value from below.
Starting from $\lambda\ll 1$ ($N_{\rm sp}\ll N$), we can scan the process by
gradually increasing $\lambda$, keeping other parameters fixed, until it
saturates unitarity.

Usually, such a process would be expected to exhibit an exponential
suppression, ${\rm e}^{-\frac{1}{\alpha}}$, characteristic of non-perturbative
amplitudes. And indeed, it would, if the number of species were small. But, in
the present case the story is different. For (19), the transition probability
is enhanced by the exponentially large number of possible final states, since
the quanta can come from various combinations of species.

This multiplicity can be quantified by the entropy $S$, which can be expressed
as a function of $\alpha$ and $\lambda$. Its generic form is,

$S=\frac{1}{\alpha}\ln(f(\lambda))\,,$ (23)

where $f(\lambda)$ is a function of $\lambda$. It satisfies $f(\lambda)={\rm
e}$ for a certain critical value $\lambda=\lambda_{*}$, typically order one.
The precise form of the function $f(\lambda)$ is representation-dependent and
is unimportant for the present discussion. For the intuitive usefulness, we
just say that its typical form is

$f(\lambda)=(1+1/\lambda)^{c_{1}\lambda}(1+\lambda)^{c_{2}}{\rm
e}^{-c_{3}}\,,$ (24)

where $c_{1},c_{2},c_{3}$ are positive numbers of order one. More explicit
forms can be found in Dvali:2020wqi .

Now, near the saturation point, the cross section (measured in units of $1/q$)
takes the form

$\sigma={\rm e}^{-\frac{1}{\alpha}+S}\,.$ (25)

Notice that, thanks to the limit (19), the saturation point is insensitive to
a non-exponential pre-factor which, therefore, can be set equal to one. The
error committed in this way is of order $\alpha$ and vanishes in the limit
(19).

From (25) it is clear that the saturation takes place for

$S=\frac{1}{\alpha}\,.$ (26)

As already said, this is reached for $\lambda=\lambda_{*}\sim 1$. It is clear
that at this point, the final state of species satisfies a saturon relation
(1). Notice, for $q=M_{*}$, the unitarity of the process bounds $N_{\rm sp}$
exactly by (8). This gives an alternative derivation of the species bound on
$M_{*}$ from saturation.

###  III.2 Collective Limit

We have seen that the species limit (19) allows to cleanly extract a set of
non-perturbative processes. This is possible, despite the fact that the
gravitational coupling $\alpha$ vanishes. In the considered process, the
suppression is compensated by the number of species and the corresponding
entropy of the state. We shall now turn to a different class of quantum
gravitational processes that survive in the species limit (19). These are the
transitions that are enhanced by the occupation number of gravitons, $N$.

In such cases, an interaction vertex of gravitons that vanishes in the limit
(19), can nevertheless generate a non-trivial transition. For example, a
$2\rightarrow 2$ transition amplitude, caused by a tree-level four-graviton
vertex, is suppressed by $\alpha$. Nevertheless, the transition matrix element
can be non-zero in the limit (19), provided the occupation number of quanta
scales as,

$N\propto\frac{1}{\alpha}\,.$ (27)

In order to account for such cases, it is convenient to introduce the notion
of the collective coupling, defined as,

$\lambda_{c}\equiv N\alpha\,.$ (28)

Originally, this concept was introduced in the context of the black hole
$N$-portrait.

It is important not to confuse the occupation number of gravitons $N$ in a
state, with the number of species $N_{\rm sp}$ in the theory. Correspondingly,
we should not confuse the collective coupling $\lambda_{c}$, with the species
coupling $\lambda$ defined in (18). The species coupling $\lambda$, is a
parameter of the theory, since it depends on $N_{\rm sp}$. In contrast, the
collective coupling $\lambda_{c}$, is a parameter of the state, which depends
on the occupation number $N$ in that state.

The concept of collective coupling, allows us to define the second double-
scaling limit,

$\displaystyle
M_{P}\rightarrow\infty\,,~{}N\rightarrow\infty\,,~{}\alpha\rightarrow 0\,,$
(29) $\displaystyle\lambda_{c}={\rm finite}\,,~{}N_{\rm sp}={\rm finite}\,.$

We shall refer to it as the collective limit. This limit requires some
clarification. Since we kept $N_{\rm sp}$ finite, the fixed hierarchy between
the cutoff $M_{*}$ and $M_{P}$ is maintained. In such a case, setting $\alpha$
to zero implies a restriction on $q$, for example, by considering a tree-level
process with a finite momentum transfer among gravitons. Such restrictions
need not be obeyed by the virtual quanta. As a result, the corrections from
UV-sensitive loops, translate as $1/N_{\rm sp}$-corrections. This is different
from the species regime (19), in which the analpogous corrections vanish.

Obviously, in the regime (29), the species coupling $\lambda$ vanishes
together with $\alpha$. Because of this, we may expect that quantum gravity
becomes trivial in IR Brustein:2009ex . However, this expectation does not
capture the collective effects that are enhanced by $N$.

Naturally, those processes that are controlled by the collective coupling
$\lambda_{c}$, are non-zero in the limit (29). For example, we choose an
initial state $\left|\,N\right>$, in which gravitons of certain wavelength $R$
are macroscopically occupied, so that $\lambda_{c}$ is non-zero. Let us
consider a process, in which two initial gravitons re-scatter into a pair of
final gravitons, as it is sketched in Fig.2b. Alternatively, the same pair can
annihilate into a pair of species, as described by Fig.2a. The matrix elements
of both transitions are proportional to $\lambda_{c}$. Correspondingly, they
are both non-vanishing in the collective limit (29). At the same time, all the
higher order corrections in $\alpha$, not accompanied by the corresponding
powers of $N$, vanish.

The processes of the sort of Fig.2, can have a wide range of physical
applications. In particular, they describe the processes of the gravitational
particle-creation by black holes and by de Sitter in the corpuscular theory of
Dvali:2011aa ; Dvali:2013eja .

The sketch in Fig.2b, also reveals why the self-binding potential of $N$
overlapping gravitons, survives in the collective limit (29). This diagram
describes the origin of the attractive potential, at the level of a virtual
one-graviton exchange among the real gravitons. The attractive potential
energy, experienced by each graviton from its $N$ neighbours, is $E_{\rm
pot}\sim\lambda_{c}/R$. For $\lambda_{c}\sim 1$, this potential can balance
the kinetic energy of a would-be free graviton ($E_{\rm kin}\sim 1/R$). At
this point, $N$ gravitons form a self-sustained bound-state. This bound-state
represents a corpuscular description of a black hole Dvali:2011aa ;
Dvali:2012en ; Dvali:2013eja .

###  III.3 Species Magnifier Limit

Finally, we shall discuss a regime, which is of special interest for our
applications. This regime was previously introduced in Dvali:2020wqi  and we
shall refer to it as the limit of species magnifier. It has the following
form,

$\displaystyle\alpha\rightarrow 0\,,~{}N\rightarrow\infty\,,~{}N_{\rm
sp}\rightarrow\infty\,,$ $\displaystyle M_{*}={\rm finite}\,,~{}\lambda={\rm
finite}\,,~{}\lambda_{c}={\rm finite}\,.$

Basically, the above limit describes the species regime (19), in the presence
of a state with infinite occupation number $N$ of gravitons, such that the
ratio $N/N_{\rm sp}$ is finite.

Correspondingly, while this regime maintains benefits of the species limit
(19), it enables to expose non-trivial collective phenomena. For example, a
four-graviton interaction vertex can trigger a non-trivial transition from an
initial $N$-graviton state via a re-scattering of two constituents. The matrix
element of this process is controlled by $\lambda_{c}$. At the same time, the
loop corrections to the same vertex, such as Fig.1a, vanish.

In order to capture more essence of the limit (III.3), let us compare the
outcomes of the process of Fig.2a in the regimes (29) and (III.3). As already
mentioned, this process describes the scattering of two gravitons into an
arbitrary pair of species. The rate of this process is enhanced by $N_{\rm
sp}$ due to the number of available final states. It scales as,

$\Gamma\sim q\lambda_{c}^{2}N_{\rm sp}\,,$ (31)

where $q=1/R$ is a characteristic energy of the initial gravitons. The inverse
of this rate, gives the time-scale during which the number of initial
gravitons is reduced roughly by two. Of course, there are other processes that
contribute but this one suffices for making the point.

As a reference unit of time, we can adopt $R=1/q$. During this time, the
relative reduction of the graviton number is,

$\frac{\Delta N}{N}\sim\frac{N_{\rm sp}}{N}\,\lambda_{c}^{2}\,.$ (32)

This quantity represents a measure of the quantum gravitational back reaction
on the state.

Correspondingly, the time-scale $t_{\rm half}$, during which the initial state
would loose roughly half of its constituents, is given by,

$t_{\rm half}\sim\frac{R}{\lambda_{c}^{2}}\frac{N}{N_{\rm sp}}\,.$ (33)

We shall now compare the behaviour of the above three quantities
(31)-(32)-(33), in the two different regimes of (29) and (III.3).

We start with the collective limit (29). In this regime, the emission rate
(31) is finite. However, the quantum gravitational back reaction (32) is zero.
Correspondingly, the half-decay time (III.3) is infinite. The physical reason
behind this behaviour is no secret. In the collective limit (29), the initial
graviton occupation number $N$, is infinite. Such a state behaves as a
graviton reservoir of infinite capacity. A finite depletion rate, is not
capable of destroying such a system within any finite time. In other words,
the fraction of gravitons depleted within any finite interval of time, is of
measure-zero. This explains why the back reaction (32) is vanishing.

In the species magnifier regime (III.3), the situation changes drastically.
The infinite number of species $N_{\rm sp}$, opens up an infinite number of
the decay channels. Correspondingly, the rate (31) blows up. However, the
problem is regular. The back-reaction (32) is finite and so is the half-decay
time (33) of the graviton condensate.

The important aspect of the magnifier limit is that it captures the quantum
gravitational back-reaction, despite the fact that $\alpha=0$. At the same
time, the secondary effects, that are higher order in $\alpha$, vanish.

To summarize, working in the regime (III.3) provides us with a double
advantage. First, it allows to cleanly distill some collective and non-
perturbative quantum gravitational phenomena. Secondly, it shows that, for
large $N_{\rm sp}$, the quantum gravity can leave significant imprints at
distances much larger than the Planck length. The species literally act as a
“magnifying glass” for quantum gravity.

In what follows, we shall apply the species regime for extracting certain
quantum gravitational effects in de Sitter and in black holes and for
understanding their observational implications.

##  IV Corpuscular Theory

###  IV.1 Motivation

As we have seen, the magnifying effect of species can be deduced already
within the semi-classical treatment. The relations such as (8), do not require
the knowledge of an internal corpuscular structure of either a black hole or a
de Sitter. The semi-classical picture is of course incomplete. Although this
picture allows to derive the species bound (8), it cannot grasp the
fundamental physics behind it. For example, it is not clear, how a back-
reaction from particle-creation, affects a black hole or a de Sitter.

In order to answer the next layer of questions, we must work within a more
fundamental framework, in which the corpuscular structure is visible. Once we
have such a picture, we shall supplement it with a large number of light
particle species. This shall enable us to work in the regime (III.3) and to
use species as a magnifying tool.

The corpuscular picture we shall work with, is the quantum $N$-portrait
Dvali:2011aa ; Dvali:2012en ; Dvali:2013eja . In this picture, a black hole or
a de Sitter Hubble patch, is described as a composite state of soft gravitons
of mean occupation number $N$. This state satisfies the saturation relation
(1).

A particularly strong motivation for the corpuscular picture, is provided by
the $S$-matrix formulation of quantum gravity. This formulation demands the
existence of a valid $S$-matrix vacuum. All other states, must be described as
excited states, constructed on top of this vacuum.

The de Sitter cannot serve as a valid $S$-matrix vacuum for quantum gravity.
The absence of a globally-defined time, is part of the problem. However, the
issue is more subtle and is linked with the very nature of quantum
gravity/string theory Dvali:2020etd . The point is that, in general, a valid
vacuum must not recoil and absorb information in a scattering process. A de
Sitter can satisfy this requirement, but exclusively in the following double
scaling limit,

$M_{P}\rightarrow\infty,~{}R={\rm finite}\,.$ (34)

In this limit, the quantum graviton coupling (15), vanishes. Hence, whenever
de Sitter becomes a rigid vacuum, gravitons (closed strings) decouple. In
other words, de Sitter can be promoted into a “vacuum” only at the expense of
trivializing the graviton (closed string) $S$-matrix. Notice, at the same
time, non-gravitational interactions can remain non-trivial for (34). In
particular, for interactions that are asymptotically-free, an approximate
$S$-matrix treatment becomes better and better at short distances and can be
consistently formulated in the limit (34). 333In this sense, there is no a
priory inconsistency in an approximate $S$-matrix treatment on a temporary
cosmological backgrounds (for various examples, see, e.g., Gorbenko:2019rza ,
Pajer:2020wnj ).

The important message we take from this analysis, is that the $S$-matrix
formulation excludes a de Sitter vacuum for any finite value of $M_{P}$. The
only remaining option is to treat de Sitter as an excited (coherent) state
constructed on top of a valid $S$-matrix vacuum such as Minkowski Dvali:2011aa
; Dvali:2013eja ; Dvali:2014gua ; Dvali:2017eba . Various other aspects of
this idea have been discussed Kuhnel:2014gja ; Kuhnel:2015yka ;
Casadio:2015xva ; Berezhiani:2016grw , including string context Brahma:2020htg
.

The $S$-matrix justification for the corpuscular structure of a black hole
Dvali:2011aa , is more straightforward. There is a little doubt that a black
hole of size $R$, can be produced in an $S$-matrix process of center of mass
energy $E\sim M_{P}^{2}R$ tHooft:1987vrq ; Amati:1987wq ; Gross:1987kza . The
new insight, brought by the corpuscular $N$-portrait Dvali:2011aa  is the
identification of the relevant $S$-matrix process in form of $2\rightarrow N$
graviton scattering, where the occupation number of the final gravitons is
$N\sim(RM_{P})^{2}$ and their wavelengths are $\sim R$. It has been observed
Dvali:2014ila ; Addazi:2016ksu  that such a $2\rightarrow N$ graviton
scattering process, saturates unitarity precisely when the entropy of the
final state matches the Bekenstein-Hawking entropy of a black hole of size
$R$. It is therefore natural to describe a black hole as a “loose” bound-state
of $N$ such quanta. This justifies the idea of the black hole $N$-portrait,
from the $S$-matrix perspective.

###  IV.2 Two Types of Constituents

The corpuscular pictures for a black hole and for de Sitter exhibit close
similarities. Therefore, we shall discuss them in a common language. This
language also allows to grasp the fundamental differences between the two
systems. These differences, become important at the later stages of their
time-evolutions.

In both cases, the relevant gravitational radius shall be denoted by $R$. In
classical theory, this scale describes a Hubble radius ($R=1/H$) for de Sitter
and a Schwarzschild radius for a black hole.

The constituents are divided in two main categories: The master modes and the
memory modes.

The master modes, to be symbolically denoted by creation/annihilation
operators $\hat{a}^{\dagger}$/$\hat{a}$, are the main contributors into the
energy of the system (a black hole or a Hubble patch). The diversity of the
master modes is small, but they come in high occupation numbers. Because of
this, they contribute predominantly into the energy but very little into the
information capacity of the system.

The memory modes, to be denoted by $\hat{b}^{\dagger}$/$\hat{b}$, play the
opposite role. They are responsible for the information storage capacity of
the system. This capacity is measured by the micro-state entropy $S$. This
stands for Bekenstein-Hawking entropy for a black hole and Gibbons-Hawking
entropy for de Sitter. The memory modes are initially (nearly) gapless. Due to
this, they contribute negligibly into the energy. However, unlike the master
modes, the memory modes come in a very large diversity of “flavors”. This is
the reason for why they account for the entropy of the system.

We shall briefly review each category.

####  IV.2.1 Master Modes

The characteristic frequencies (${\mathcal{E}}_{\rm master}$) and wavelengths
($l_{master}$) of the master modes are both set by the scale $R$,

${\mathcal{E}}_{\rm master}\sim\frac{1}{l_{\rm master}}\sim R\,.$ (35)

That is, the dispersion relations of the master modes are not too far from the
dispersion relations of the free gravitons, propagating on top of the
Minkowski vacuum.

This is explained by the self-sustainability condition that was already
discussed previously. The attractive potential, experienced by each master
graviton, is comparable to its kinetic energy. Therefore, the collective
gravitational interaction, while sufficiently strong for holding the master
modes together, only mildly modifies their dispersion relations. The essence
of this collective interaction was already discussed and illustrated by a
sketch in Fig.2b.

The master gravitons are the main contributors into the initial energy of the
system. An each quantum contributes roughly $1/R$. The total energy is thus
proportional to their occupation number,

$E=N{\mathcal{E}}_{\rm master}=\frac{N}{R}\,.$ (36)

Given the standard relation,

$E=RM_{P}^{2}\,,$ (37)

between the energy (mass) and a size of a black hole (or a Hubble patch), we
get,

$N=(M_{P}R)^{2}\,.$ (38)

The basic quantum gravitational coupling of the master gravitons is given by
(15) evaluated for $q\sim 1/R$,

$\alpha=\frac{1}{(RM_{P})^{2}}\,.$ (39)

This given two important relations. First, for both systems the above
expression is equal to the inverse of the entropy $S$ (Bekenstein-Hawking
entropy for a black hole and Gibbons-Hawking entropy for de Sitter). Both
entropies are equal to (4).

Secondly, the number of constituents $N$ is equal to the inverse of their
coupling (and equivalently, to the entropy). Thus, both, a black hole and a de
Sitter, satisfy the saturation relation (1). Taking into account the explicit
form of the entropy (4), the saturation relation reads as,

$N=\frac{1}{\alpha}=S=(RM_{P})^{2}\,.$ (40)

As pointed out in Dvali:2020wqi ; Dvali:2019jjw  and also explained earlier in
the present paper, the area form of the entropy (4) represents a particular
case of the generic property of saturons. The entropy of a saturon is always
given by its area measured in units of the decay constant of the Goldstone
boson of spontaneously broken Poincare symmetry. The emergence of such a
Goldstone boson is generic, because any saturon breaks Poincare symmetry
spontaneously. In case of a black hole or a de Sitter, the Goldstone boson in
question is the graviton, but the effect is universal and goes beyond gravity
(See, Dvali:2020wqi ; Dvali:2019jjw  for explicit examples.)

The concept of spontaneous breaking of Poincare symmetry by the saturated
$N$-graviton state, is best defined in large-$N$ limit (29). The order
parameter of this breaking is $\sqrt{N}/R$, which, by Goldstone theorem,
determines the decay constant of a canonically normalized Goldstone mode. As
it is clear from (38), this quantity is nothing but the Planck mass, $M_{P}$,
which represents the graviton decay constant. This fully matches the fact that
the Goldstone boson of Poincare symmetry, originates from the graviton. The
above explains why $M_{P}$ enters in the expression (4) for the area-form of
the entropy.

The relation (40) indicates that both systems (black hole and de Sitter)
represent saturons. This is defining for their physical properties.

####  IV.2.2 Memory Modes

Unlike the master modes, the memory modes have very short wavelengths (of
order the cutoff) but almost zero frequencies. Namely, the energy gap for
exciting a memory mode is,

${\mathcal{E}}_{\rm memory}\sim\frac{1}{NR}\ll\frac{1}{l_{\rm memory}}\,.$
(41)

Obviously, the dispersion relations of the memory modes are very different
from those of free gravitons. A free graviton, of some wavelength $l$,
propagating on Minkowski vacuum, would have a frequency, $\sim 1/l$. This is
due to the Poincare symmetry of the Minkowski background.

However, the dispersion relations of the memory modes, are affected by the
presence of the master modes. As already discussed, the state of $N$ master
modes, breaks the Poincare symmetry spontaneously. As a result of this
breaking, the dispersion relations of the would-be “hard” memory modes are
strongly affected. Their wavelengths stay short, but their frequencies are
redshifted to almost zero. In fact, the memory modes become exactly gapless in
the collective limit (29). At finite $N$, their gaps scale as (41).

The variety of the “flavors” of the memory modes is of order $1/\alpha$. This
is determined by the number of different momentum oscillators that become
gapless at the saturation point. Due to this, the memory modes are the main
contributors into the entropy of the system. These modes are gapless because
of the critical occupation number of the master modes. That is, the energy
gaps of the memory modes depend on the occupation number of the master mode.
The gaps collapse to (almost) zero when the occupation number of the master
mode reaches a critical value.

This phenomenon, schematically, can be described by the following Hamiltonian
(see, e.g., Dvali:2017nis ; Dvali:2018xpy ),

$\hat{H}=\sum_{b}\omega_{b}\left(1-\alpha\,\hat{a}^{\dagger}\hat{a}\right)^{2}\hat{b}^{\dagger}\hat{b}\,.$
(42)

For simplicity of illustration, we took a single master mode. The summation is
over various memory modes. $\omega_{b}$ are their would-be frequencies in the
Minkowski vacuum, which typically are very high (of order the cutoff or so).

The reader should not be alarmed by the oversimplified appearance of the above
Hamiltonian. In particular, by its number-conserving form. These details are
unimportant and have been neglected for making the essence of the phenomenon
maximally transparent. Thanks to the power of saturation, the (42) captures
this essence surprisingly well. In fact, near the saturation point (40), the
dynamics of a generic system effectively reduced to the above, up to a proper
Bogoliubov transformation.

In a state with the occupation number of the master mode
$\langle\hat{a}^{\dagger}\hat{a}\rangle\equiv N$, the effective energy gaps of
the memory modes are given by

${\mathcal{E}}_{\rm memory}=\omega_{b}\left(1-\alpha N\right)^{2}\,.$ (43)

It is clear that the memory modes become effectively gapless over a state in
which $N$ is critical,

$N=\frac{1}{\alpha}\,.$ (44)

Since the memory modes are gapless, an exponentially large number of
degenerate micro-states emerges. These states differ by the occupation numbers
of the memory modes. This is the reason behind the maximal entropy of the
saturated system.

Thus, we can say that in a saturated state the master mode assists
Dvali:2018tqi  the memory modes in becoming gapless. Away from saturation
(40), the memory modes are very costly in energy, but at the saturation point,
they cost nothing. Thanks to this phenomenon, the memory modes can store a
large amount of information at almost zero energy expense.

Thus, a black hole or a de Sitter state is described by the occupation number
of the master mode and the ones of the memory modes. This state can be denoted
as,

$\left|\,N\right>=\left|\,N\right>_{a}\left|\,n_{1},n_{2},...\right>_{b}\,,$
(45)

where $\left|\,N\right>_{a}$ is a state of a master mode, with occupation
number $N$, and $\left|\,n_{1},n_{2},...\right>_{b}$ is a state of memory
modes, with various occupation numbers. An each sequence, represents a memory
pattern. The number of distinct patterns is exponentially large,

$n_{st}\sim{\rm e}^{1/\alpha}\,.$ (46)

As long as the system is saturated (40), the patterns are degenerate in energy
(up to $1/N$ corrections). They form a set of micro-states that are
classically indistinguishable. The corresponding micro-state entropy is,

$S=\ln(n_{st})=\frac{1}{\alpha}=(RM_{P})^{2}\,.$ (47)

This is the origin of Bekenstein-Hawking and Gibbons-Hawking entropies in the
corpuscular picture.

As said, the above presentation captures the essence of the story. We needed
to display the necessary ingredients of the phenomenon before highlighting the
magnifying role of the particle species in it.

###  IV.3 Saturation

The saturation relation (1) can be written as,

${\rm Number}=\frac{1}{\rm Coupling}={\rm Entropy}\,.$ (48)

The saturons (the systems satisfying this relation) exhibit certain universal
phenomena. For the present discussion, the most important is the effect of
quantum breaking and the internal mechanisms leading to it.

The main engine of the time-evolution of a saturated system, is the re-
scattering of the master modes. This re-scattering leads to the processes that
are observable on the time-scales $\sim R$. For example, such is a process of
particle-emission, which leads to the decay of a saturon.

Since the memory modes have very low frequencies (41), the time-scale of their
evolution would be,

$t_{\rm memory}\sim NR\,.$ (49)

In addition, the memory modes are affected by the re-scatterings of the master
modes, over a similar time-scale.

There exist three important effects contributing into the time-evolution of
the saturon state:

  * •

The reduction of the occupation number of the master mode. This is the flip
side of the particle emission.

  * •

The inner (or self) entanglement Dvali:2013eja ; Dvali:2014gua ; Dvali:2017eba
: The constituents of a black hole (or a de Sitter) become internally
entangled among each other.

  * •

The memory burden effect Dvali:2018xpy ; Dvali:2018ytn ; Dvali:2020wft : With
the decrease of the occupation number of the master mode, the system departs
from saturation. Correspondingly, the gaps of the memory modes grow. This
generates a quantum back-reaction force.

All three effects lead to a complete breakdown of the semi-classical picture.
This phenomenon is referred to as the quantum breaking. The corresponding
time-scale is called a quantum break-time $t_{Q}$. For a de Sitter and a black
hole, $t_{Q}$ was derived in Dvali:2013eja ; Dvali:2014gua ; Dvali:2017eba .
For a generic saturated system (without classical instability) Dvali:2017eba ;
Dvali:2020wqi  it is given by (7), which can be presented as,

$t_{Q}\sim\frac{t_{\rm cl}}{\lambda}\,.$ (50)

Here, $t_{\rm cl}$ is the classical time-scale set by the frequencies of the
constituents, $t_{\rm cl}\sim R$. $\lambda\equiv\alpha N_{\rm sp}$ is the
analog of species coupling (18) (’t Hooft coupling), for the generic saturated
system.

The various aspects of the above effects have already been studied in
saturated gravitational systems, such as black holes and de Sitter. In the
present paper we are interested in the magnifying role of species in these
processes. For completeness, in parallel, we shall briefly reproduce the
essence of the phenomena.

###  IV.4 Quantum Clock: Re-scattering and Depletion

Initially, the main engine of the time-evolution is the re-scattering of the
master modes. The memory modes, since at this stage they are essentially
gapless, back react very little and can be ignored. However, back reaction
becomes very important at later times.

The initial re-scattering of the master modes, takes place regardless of the
number of species in the theory. However, species speed it up. One outcome of
this re-scattering, is the emission of free particles. The typical process is
depicted in Fig.2a. For a black hole and a de Sitter, the produced particles
describe Hawking and Gibbons-Hawking quanta, respectively.

For describing the process, we can use the formulas (31), (32), (33) obtained
for a generic $N$-graviton state. The new specifics is that, at initial times,
the parameters are related by the saturation relation (40).

The rate of particle-creation is,

$\Gamma=\frac{1}{R}(N\alpha)^{2}N_{\rm sp}=\frac{1}{R}N\lambda_{c}\lambda\,.$
(51)

As expected, at initial times, the rate of particle-production, reproduces the
semi-classical Hawking and Gibbons-Hawking rates. Note Dvali:2013eja , a
particular channel, describing a creation of a graviton, corresponds to the
standard tensor perturbation mode of inflation Starobinsky:1979ty .

However, the corpuscular picture allows us to go beyond the semi-classical
approximation. Using the microscopic description, it is instructive to
consider the physical meaning of particle-creation from different
perspectives.

Figure 2:  2.a) Decay of the multi-graviton state into species (red line)
illustrates the enhancement of the rate of the process due to high occupation
number of the initial state. The suppression factor $\alpha^{2}$ is
compensated by multiplicity factor $N^{2}$. 2.b) Illustrates the finiteness of
the collective effect, experienced by each graviton due to its interactions
with the neighbours.

First, using the saturation relation (40), valid for initial times, we can
write the rate (51) in terms of the species coupling,

$\Gamma_{in}=\frac{1}{R}N_{\rm sp}=\frac{1}{R}\frac{\lambda}{\alpha}\,.$ (52)

This form, clearly shows the magnifying power of species. As already discussed
earlier for a generic $N$-state, in the double-scaling limit (III.3), the rate
(51) blows up. However, this blow-up does not lead to any sort of
inconsistency. The saturation relation (40) guarantees that, in the same
limit, the mass of a black hole (or of a de Sitter Hubble patch), (37),
becomes infinite.

As a result, the back-reaction over time $R$ (32), is finite. Due to
saturation (40), in the present case, the back-reaction (32) takes the form,

$\frac{\Delta N}{N}=\frac{N_{\rm sp}}{N}\,.$ (53)

Correspondingly, the time of half-decay,

$t_{\rm half}=\frac{R}{\lambda}=\frac{R^{3}M_{P}^{2}}{N_{\rm
sp}}=R^{3}M_{*}^{2}\,,$ (54)

is also finite.

The corpuscular picture makes it very clear that the particles are not
produced “for free”. Due to back-reaction, the coherent state depletes and
looses coherence. This is true for black holes, as well as, for de Sitter.

The rate of the departure from coherence can be estimated by using the fact
that the number of quanta in a coherent superposition diminishes with roughly
the same rate (51) as the particles are emitted. That is,

$\dot{N}=-\Gamma\,.$ (55)

For estimating the back-reaction, in the leading order, the coupling $\alpha$
can be taken constant. Then, at initial times, the rate of particle-production
changes as (as everywhere, unimportant numerical coefficients are set equal to
one),

$\displaystyle\frac{\dot{\Gamma}}{\Gamma}$ $\displaystyle=$
$\displaystyle-\frac{1}{R}N\alpha^{2}N_{\rm
sp}=-\frac{1}{R}\lambda_{c}\lambda\,.$ (56)

Now, approximating the quantities on the right hand side by their initial
saturated values (40), we get,

$\Gamma(t)=\Gamma_{in}{\rm e}^{-\frac{t}{t_{\rm half}}}\,,$ (57)

where $t_{\rm half}$ is given by (54).

The expressions (51) and (57) are instructive due to several reasons. First,
for de Sitter, they reinforce the upper bound (14) on $H$. We see from (51)
that the violation of this bound would lead to an unacceptably high rate of
depletion. With such a gigantic decay rate, the de Sitter state would cease to
exist in less than one Hubble time. In full agreement with this, (57) shows
that the production rate would drop in less than one Hubble time.

Similarly Dvali:2012uq , for a black hole this reinforces the upper bound (12)
on Hawking temperature. A hotter black hole cannot exist. Such a black hole,
would emit all its constituents within the time less than its size. This would
make no sense.

To summarize, species produce the two inseparable effects. First, they enhance
the rate of particle-production. Secondly, as a result of back-reaction,
species speed up an intrinsic quantum clock. This clock measures the depletion
of $N$-graviton state which describes a black hole or a de Sitter. The flip
side of this process is the generation and growth of entanglement which we
shall now discuss.

###  IV.5 Entanglement and Page’s Curve

The corpuscular picture reveals a new phenomenon, which we can call a self-
entanglement or an inner entanglement Dvali:2013eja . This takes place, both
for de Sitter and for a black hole. In both cases, the growth of inner
entanglement happens at the same rate as the particle emission. Together with
the depletion, the self-entanglement contributes into the departure from
classicality. It reaches the maximum at quantum break-time Dvali:2013eja ;
Dvali:2014gua ; Dvali:2017eba ,

$t_{Q}=\frac{R^{3}M_{P}^{2}}{N_{\rm sp}}=\frac{R}{\lambda}=R^{3}M_{*}^{2}\,.$
(58)

It is easy to notice that this is a particular case of (50), applied to a
gravitational saturon. The time-scale (58) signals a complete break-down of
the semi-classical picture. The quantum break-time has been verified in
various contexts Dvali:2017ruz ; Kovtun:2020ndc ; Kovtun:2020kcl ;
Berezhiani:2020pbv ; Blumenhagen:2020doa .

From (54) it is clear that $t_{Q}$ matches the time of half-decay, but brings
a new physical meaning. The effect is intrinsically corpuscular in nature and
cannot be observed in semi-classical theory. This is why, unless additional
assumptions are invoked, the behaviour of entanglement in semi-classical
treatment is related with certain puzzling features of Page curve. According
to Page Page:1993wv , the entanglement must reach a maximal value after
$t_{\rm half}$. This is exactly what corpuscular theory predicts. The semi-
classical picture cannot explain this.

In semi-classical treatment, the sole engine for the Page’s curve, is the
entanglement between the black hole and an outgoing Hawking radiation. This
creates a false impression that the entanglement grows indefinitely throughout
the black hole existence. However, this picture is incomplete and requires a
corpuscular resolution which incorporates the inner entanglement and memory
burden effects.

As already shown in Dvali:2013eja , the corpuscular theory immediately
accommodates Page’s central point. The entanglement of a saturated
$N$-particle state cannot grow indefinitely. It reaches the maximum after the
quantum break-time $t_{Q}$. The fact that $t_{Q}$ comes out equal to Page’s
time, can be taken as consistency check of the picture.

The corpuscular approach shows that the analog of Page’s time exists also in
de Sitter Dvali:2013eja . For both systems, the re-scattering of the
constituent gravitons (which also leads to particle emission), generates an
internal entanglement among the remaining constituents Dvali:2013eja .
Simultaneously, it creates the memory burden effect Dvali:2018xpy  in de
Sitter Dvali:2018ytn  and in black holes Dvali:2020wft .

###  IV.6 Inner entanglement

After the emission of a single particle, the state (45) evolves into a
following superposition,

$\left|\,N\right>\rightarrow\sum_{j}^{N_{\rm
sp}}c_{j}\left|\,N^{\prime},j\right>\times\left|\,j\right>\,,$ (59)

where $c_{j}$ are constants. Here, $j=1,2,...,N_{\rm sp}$ is the species label
that characterizes various states of emitted quanta, which are denoted by
$\left|\,j\right>$. Since we are not in a semi-classical picture, the term
“emitted quanta” requires a specification. It describes quanta with dispersion
relations that are close to quantum particles propagating in a background de
Sitter (or Schwarzschild) metric.

Notice, in de Sitter, this is a fine definition at initial times. Later,
towards the quantum break time, the classical de Sitter geometry looses the
meaning. By then, the distinction between the emitted and retained quanta
becomes blurry. However, this is not an issue, since after $t_{Q}$ we are
already in a quantum-broken phase. Beyond this point, the de Sitter cannot
exist as a valid state. A graceful exit must take place before the time
$t_{Q}$ elapses.

In contrast, the black holes can exist beyond $t_{Q}$. Also, for them, the
notion of the emitted quantum is well defined at all times.

Of course, at the initial times, the emission is democratic in species, since
they are produced via graviton processes. The states
$\left|\,N^{\prime},j\right>$ denote what remains from a black hole or a
Hubble patch after the emission. The expression (59) shows that these states
are entangled with outgoing quanta through the label $\left|\,j\right>$.

The crucial point is that each $\left|\,N^{\prime},j\right>$ is also self-
entangled. That is, even if initial $\left|\,N\right>$ was a tensor product
state of various master and memory modes,
$\left|\,N_{a_{1}}\right>\left|\,N_{a_{2}}\right>...\left|\,n_{b_{1}}\right>\left|\,n_{b_{2}}\right>$,
in the state $\left|\,N^{\prime},j\right>$ they become entangled. In other
words, the state evolves into a superposition,

$\left|\,N^{\prime}\right>=\sum_{a,b}c_{a_{1},a_{2},..b_{1},b_{2},...}\left|\,N_{a_{1}}\right>\left|\,N_{a_{2}}\right>...\left|\,n_{b_{1}}\right>\left|\,n_{b_{2}}\right>\,,$
(60)

where $c_{a_{1},a_{2},..b_{1},b_{2},...}$ are coefficients. We have
deliberately ignored the label of the emitted species $j$, in order to focus
on the internal entanglement. In addition, of course, the state is entangled
with respect to the species index $j$.

Let the number of linearly-independent states participating in the
superposition (60) be $\tilde{n}_{\rm st}$. We shall measure the level of the
entanglement by the log of this number:

$S_{\rm ent}\equiv\ln(\tilde{n}_{\rm st})$ (61)

Obviously, $\tilde{n}_{\rm st}$ grows with each act of re-scattering and so
does $S_{\rm ent}$. The maximum is reached when the inner entanglement
saturates the capacity of $N$-particle state. Obviously, $\tilde{n}_{\rm st}$
cannot exceed the total number of available micro-states, $n_{\rm st}$. Thus,
at its maximum, the quantity $S_{\rm ent}$ equals to the micro-state entropy
of the system (47).

The shortest time for reaching the maximal entanglement,

$S_{\rm ent}=\frac{1}{\alpha}\,,$ (62)

from an initial unentangled state, is bounded by unitarity. In an $N$-particle
saturated system, with coupling constant $\alpha=1/N$ and frequencies $1/R$,
this time is given by:

$t_{\rm ent}=\frac{R}{\lambda}=R\frac{N}{N_{\rm sp}}\,,$ (63)

where $\lambda$ is the species coupling (18).

Intuitively, the above can be understood by noticing that (63) is the minimal
time required for an order-one fraction of the master modes to re-scatter.
$t_{\rm ent}$ is also a shortest time, set by unitarity, required for a start
of decoding the information stored in a generic saturated system. This will
become much clearer after considering the memory burden effect.

As we see, for de Sitter and black holes, the time (63) is equal to quantum
break time $t_{Q}$. It also matches the Page time. We thus arrive to yet
another physical meaning of this time-scale.

The phenomenon of inner entanglement, originates from the multi-particle
nature of the state $\left|\,N\right>$. This state hides complexity, which is
not visible at initial times. In order to resolve it, one needs to be
sensitive to $1/N$ effects, which are extremely suppressed for macroscopic
values of $N$. However, the complexity steadily grows in time. The re-
scattering of constituents (which also leads to the emission of species $j$),
simultaneously generates the entanglement among them. At the early times, the
self-entanglement is small. However, it grows at the same rate (51) that
governs the emission. The number of species can dramatically accelerate this
growth.

###  IV.7 Memory burden effect

In the course of depletion, the gaps of the memory modes grow. This happens,
because the occupation number of the master mode, $N$, diminishes.
Correspondingly, the system moves away from the saturation point (40).
Consequently, the mechanism behind the gaplessness of the memory modes is
abolished.

This effect is clearly illustrated by the equation (43). The energy gap of a
memory mode, is non-zero away from criticality, $N<1/\alpha$. When the master
mode depletes, $N$ decreases and the memory gaps grow. Correspondingly, the
energy cost of the memory pattern (45) becomes higher and higher.

For example, after reducing the occupation number of the master mode to half
of (40), the gap (43) would grow from zero to $\omega_{b}/4$. Correspondingly,
the energy cost of the memory pattern (42) would become,

$E_{\rm
pattern}=\left<n_{1},n_{2},..\,\right|\hat{H}\left|\,n_{1},n_{2},..\right>=\frac{1}{4}\sum_{b}\omega_{b}n_{b}\,.$
(64)

This results into a back-reaction force, called the “memory burden”
Dvali:2018xpy ; Dvali:2018ytn ; Dvali:2020wft . This force resists to any
departure of the system from criticality. In particular, it slows down the
decay process.

The memory burden exerts the maximum force, latest by the time the system
looses about half of the quanta of its master mode. This happens after the
time of half decay $t_{\rm half}$ (54). As we already saw, this is also equal
to the time $t_{\rm ent}$, required for developing the inner entanglement.
Notice, since by $t_{\rm half}$ the memory modes acquire non-negligible
frequencies, the information pattern stored in them starts to become readable.
Beyond this point, the system cannot be described within any semi-classical
approximation, even qualitatively. The corresponding time-scale has a meaning
of the quantum break-time $t_{Q}$ (58).

Some observational implications of the memory burden effect for black holes
has been studied in Dvali:2020wft . In cosmology, the memory burden can leave
potentially-observable imprints from the inflationary epoch Dvali:2018ytn ;
Prihadi:2020pqi . These imprints, as well as other quantum gravitational
effects, are magnified by the number of species $N_{\rm sp}$ Dvali:2020etd .
The theories in which this number is large, give better observational
prospects for such effects.

###  IV.8 Physical meanings of $t_{Q}$

As we have seen, the three equal time scales,

$t_{Q}=t_{\rm half}=t_{\rm ent}=\frac{R}{\lambda}\,,$ (65)

represent different manifestations of one and the same corpuscular quantum
clock. It is important that this time-scale stays finite in the species
magnifier limit (III.3) of quantum gravity. This allows us to cleanly distill
the quantum gravitational effects leading to it. These effects are behind
different physical meanings of the quantum clock. We summarize them below.

  * •

A time of half-decay of a saturated system.

  * •

A quantum break-time: Time of full departure from the semi-classical regime.

  * •

A time during which the memory burden reaches its maximum.

  * •

A time after which an observer can start decoding the quantum information
(memory pattern) stored in a saturated system.

  * •

A time during which a saturated system can reach the maximal inner
entanglement.

###  IV.9 Comparing the Curves

As already explained, together with the memory burden effect, the inner
entanglement is the main mechanism behind the Page’s curve. Both effects reach
the maxima by the time $t_{Q}$. Until this point, the resulting Page’s curves
for de Sitter and black holes are very similar. The fundamental differences
start beyond this point.

Unlike a black hole, de Sitter cannot exist as a consistent state after
$t_{Q}$. The reason is the conflict between the quantum state and the
classical source in form of a cosmological constant. Because of this conflict,
the system must gracefully exit from the de Sitter phase, latest by the time

$t_{E}\leqslant t_{Q}\,.$ (66)

Therefore, the continuation of the Page’s curve beyond $t_{Q}$, depends on the
mechanism of the graceful exit and the physics beyond it. For example, the
exit can be provided by a time-dependent scalar field (inflaton).

The equation (66) has profound cosmological implications. In particular, it
puts an upper bound on the duration of inflation. This excludes any sort of
meta-stability. Since $t_{Q}$ is an upper bound on duration of any given
Hubble patch, it is impossible to satisfy in a sensible meta-stable state.

No such conflict arises for a black hole, which can continue its existence
beyond $t_{Q}$. Of course, after this time, due to maximal inner entanglement
and memory burden, a black hole internally is no longer describable by any
classical field configuration. The semi-classical approximation breaks down
fully. In particular, the process of a decay, can no longer be described as
Hawking’s thermal evaporation.

Since the decay is not self-similar, we cannot even claim that the black hole
shrinks in size and heats up. In fact, there are indications Dvali:2020wft
that after $t_{Q}$ the decay of a black hole may slow down due to the “memory
burden” effect. However, regardless of the slow-down, the entanglement, after
reaching the maximum, can only decrease.

Let us now turn to the role of species in Page’s curve. This role is obvious.
Since species shorten $t_{Q}$, they shorten (and sharpen) the entanglement
curves for a de Sitter and for a black hole.

Figure 3: A schematic description of Page’s like curve for de Sitter for
different numbers of species. The blue curve corresponds to a higher number
$N_{\rm sp}$ than the red one. The corresponding quantum break-times $t_{Q}$
are denoted by the same colors as the curves. Unlike a black hole case, the
curve cannot be extended beyond the quantum break-time $t_{Q}$. This time
formally equals to a would-be Page’s time but in de Sitter has a fundamentally
different effect. The growth is terminated by the graceful exit that takes
place at $t_{E}\leqslant t_{Q}$.

##  V String Theory

We now wish to discuss some implementations of the species magnifier limit
(III.3) in string theory. We shall mostly follow Dvali:2020etd  in which the
needed expression for string theoretic quantum break-time has been derived. A
complementary derivation of quantum break-time (58) in string theory context,
was also given in Blumenhagen:2020doa .

The regime (III.3) shows transparently, how the string theory responds to a
deformation towards a de Sitter type state. For producing such a deformation,
we shall use the standard uplifting of the energy via the tension of
(anti)$D$-branes Dvali:1998pa .

For definiteness, we shall work in type IIB in ten dimensions, with string
coupling $g_{s}$ and the string scale $M_{s}$. As in Dvali:2020etd , we shall
consider the background with number of coincident $D_{9}-\bar{D}_{9}$-brane
pairs. The world-volume (open string) physics of this system is rather well-
understood (see, Srednicki:1998mq  and subsequent papers). Our task is to
highlight how the system responds to “de Sitterization”. One response is a
finite quantum break-time, which was already derived in Dvali:2020etd . Here,
we wish to show that this time-scale carries a string theoretic information
about the Gibbons-Hawking entropy.

In string construction, the role of $M_{*}$ is played by $M_{s}$. The roles of
the low energy species are played by the zero modes of the open strings. With
$n$ $D_{9}$-branes and $n$ anti-$D_{9}$-branes piled-up on top of each other,
there exist $N_{\rm sp}\sim n^{2}$ light Chan-Paton species. They come from
the zero modes of the open strings and transform under the $U(n)\times U(n)$
gauge symmetry.

Of course, such a configuration is unstable since $D$-branes and
anti-$D$-branes can annihilate each other. This instability is manifested by
an open string tachyon. The $D-\bar{D}$ annihilation can be described as
tachyon condensation Sen:1999mg . During this condensation the gauge symmetry
is partially Higgsed and the Chan-Paton species become massive. In the
original “brane inflation” scenario Dvali:1998pa , the tachyon condensation
was used as the graceful exit mechanism from the inflationary phase.

In the present discussion, the main interest is the un-condenced phase, in
which tachyon is placed on top of the “hill”. Classically, such a phase can
produce a de Sitter like metric with the curvature radius,

$R=\frac{1}{\sqrt{ng_{s}}M_{s}}\,.$ (67)

Viewing this as a coherent state of gravitons, we can estimate the number $N$
of the constituent master modes. Since each constituent brings roughly the
energy $1/R$, we can determine $N$ by matching $N/R$ with the energy of the
Hubble patch. This gives,

$N=\frac{1}{g_{s}^{2}(ng_{s})^{4}}\,.$ (68)

It is easy to see that this quantity equals to a Gibbons-Hawking entropy of a
would-be ten dimensional de Sitter with the radius (67):

$S_{GH}=(RM_{10})^{8}=\frac{1}{g_{s}^{2}(ng_{s})^{4}}\,,$ (69)

where $M_{10}$ denotes the ten-dimensional Planck mass, related to $M_{s}$ and
$g_{s}$ through,

$M_{10}^{8}=\frac{M_{s}^{8}}{g_{s}^{2}}\,.$ (70)

On the other hand, the ten-dimensional quantum coupling of gravitons of
wavelength $R$ (67), is given by,

$\alpha=\frac{1}{(RM_{10})^{8}}=g_{s}^{2}(ng_{s})^{4}\,.$ (71)

The equations (68), (69) and (71) confirm that a (would-be) de Sitter obtained
in this construction is a saturated state (40).

At the same time, the species coupling (18) is equal to

$\lambda=\alpha n^{2}=(ng_{s})^{6}\,.$ (72)

The corresponding quantum break-time, according to (58), is given by

$t_{Q}=\frac{1}{M_{s}}\frac{1}{(ng_{s})^{13/2}}=\frac{R}{(ng_{s})^{6}}=\frac{R}{\lambda}\,.$
(73)

This expression was already derived in Dvali:2020etd .

We shall now take the species magnifier limit (III.3). In the present case,
this limit reads:

$\displaystyle n\rightarrow\infty\,,~{}g_{s}\rightarrow
0\,,~{}\lambda=(ng_{s})^{6}={\rm finite}\,,$ (74) $\displaystyle M_{s}={\rm
finite}\,,~{}R={\rm finite}\,.$

In this limit, all stringy processes that are not accompanied by proper powers
of $N$ or $n$, vanish. Notice, the quantum break-time (73) remains finite.

For $ng_{s}\ll 1$, the quantum break-time (73) is much longer than the
$D$-brane instability time due to tachyon, which is of order $1/M_{s}$. Thus,
unless additional stabilizing measures are taken, the tachyon condenses prior
to quantum breaking. That is, the tachyon condensation provides a graceful
exit that saves the would-be de Sitter from quantum breaking.

What shall happen if we try to stabilize the tachyon? This can be attempted in
the following way. Notice that the tachyon coupling to the Chan-Paton species
generates an additional contribution to the mass-square of the tachyon field.
This is due to their production via Gibbons-Hawking mechanism. The effect can
be approximated as coming from a thermal bath of temperature $T_{GH}=1/R$.
This generates an effective thermal mass for the tachyon,

$\Delta m^{2}_{tachyon}=\frac{(ng)^{2}}{R^{2}}=(ng_{s})^{3}M_{s}^{2}\,.$ (75)

This contribution is positive and could potentially balance the zero-
temperature negative mass-square of the tachyon, provided

$(ng_{s})\sim 1\,.$ (76)

In this case, the stack of branes could be stabilized.

This stabilization can be viewed as a restoration of $U(n)\times U(n)$
symmetry at high temperature. Such a stabilization of $D$-branes of top of
each other by thermal effects has been considered previously (see second
reference in Dvali:1998pa ). The difference in the present case is that the
stabilization would be “self-imposed” in the sense that it would come from the
Gibbons-Hawking temperature created by the brane configuration. That is,
branes create de Sitter, which creates a thermal density of open string modes.
These thermal corrections, in turn, stabilize the tachyon. This sounds too
good to be true. And indeed, there is a caveat.

Notice Dvali:2010vm , (76) represents the point for which the number of Chan-
Paton species saturates the ten-dimensional version of the species bound (8),

$M_{*}=\frac{M_{10}}{N_{\rm sp}^{1/8}}\,.$ (77)

Taking into account that $M_{*}=M_{s}$ and using the relation (70), we
translate (77) as the limiting relation (76) between $n$ and $g_{s}$.

Thinking of the relation (76) as the limit, we notice a rather interesting
tendency. This is about entropy of a would-be de Sitter state. Due to the
existence of the light Chan-Paton species, we have a non-gravitational
contribution to the micro-state entropy. This contribution, roughly, scales as
their number 444 More precisely, around the saturation point (see below) the
entropy scales as $S_{CP}\simeq\ln\begin{pmatrix}n^{2}+\frac{1}{g_{s}^{2}}\\\
n^{2}\end{pmatrix}$.,

$S_{CP}=n^{2}\,.$ (78)

We observe that, in the limit (76), the entropy of the open string Chan-Paton
species (78), matches the Gibbons-Hawking entropy (69) of a would-be de Sitter
state:

$S_{CP}=S_{GH}=\frac{1}{g_{s}^{2}}\,.$ (79)

However, at this point, the two critical effects take place in UV and IR
theories.

First, the curvature becomes of order the string scale. This is the warning
sign from the UV theory. We expect that, beyond this threshold, any notion of
the de Sitter geometry is lost. This expectation is fully supported by the
knowledge Bowick:1992qu  that the Hagedorn effects Hagedorn:1965st  set in
above the string temperatures. The fundamental stringy effects, such as Atick-
Witten phase transition Atick:1988si , must be taken into account. The
discussions of aspects of thermal phase can be found in Alvarez:1986sj ,
Dienes:2012dc , Blumenhagen:2020doa  (and references therein). Whatever the
theory is in this phase, it is not de Sitter.

It is remarkable, how the low energy theory accounts for this breakdown.
First, this is signalled by the fact that the quantum break-time (73) becomes
of order the Hubble radius, which is also given by the string length,
$t_{Q}\sim 1/M_{s}$.

The second indication from the low energy theory is the behaviour of the
scattering amplitudes. Namely, exactly at this point, the processes of
graviton-graviton scattering into Chan-Paton species, saturate unitarity bound
(40). The example is given by a process in which two gravitons scatter into
many Chan-Paton quanta with total occupation number $N=1/\alpha$.

This process falls in the category of generic $2\rightarrow N$ processes of
the type (22). As already discussed in section III.1, they saturate unitarity
when the entropy of species reaches a critical value (40) Dvali:2020wqi . Near
the saturation point the cross-section is given by (25). In the present case,
$\alpha$ is given by (71), whereas the entropy $S$ is given by (78).
Therefore, around saturation, the cross section can be written as,

$\sigma={\rm
e}^{-n^{2}\left(\frac{1}{(ng_{s})^{6}}-\frac{S_{CP}}{n^{2}}\right)}={\rm
e}^{-n^{2}\left(\frac{1}{\lambda}-\frac{S_{CP}}{n^{2}}\right)}\,.$ (80)

Taking into account (78), the unitarity is saturated for $\lambda\sim 1$. Or
equivalently, at the saturation of unitarity by the cross section (80), the
open string Chan-Paton entropy matches the Gibbons-Hawking one (79).

Through the above phenomenon, the theory is telling us something profound. In
the light of open-closed correspondence, which is intrinsic to string theory,
we would expect that the open string sector would carry information about the
Gibbons-Hawking entropy of de Sitter vacuum, if such were to exist. And,
indeed, we see that the theory does contain such a resource, through the
entropy of the open string modes (78). However, this entropy matches the would
be Gibbons-Hawking entropy, exactly at the point where unitarity is saturated.
Simultaneously, the curvature (67) becomes stringy and so does the quantum
break-time (73). As a result, the chance of getting a stable de Sitter state
is abolished. This fully matches the conclusion from the previous arguments
Dvali:2020etd  that de Sitter vacuum is incompatible with non-trivial closed
string $S$-matrix.

Notice, none of the above obstructions appear when accounting for Bekenstein-
Hawking entropy of an extremal supersymmetric black hole by string degrees of
freedom, as it is known from Strominger-Vafa construction Strominger:1996sh .
This difference, is also connected with the fact that de Sitter cannot
preserve any supersymmetry and, unlike an extremal black hole, exhibits a
finite quantum break-time.

Similarly, no quantum breaking is evident to take place (at least in
supersymmetric limit) by an anti de Sitter, physics of which is accounted by
CFT Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj . We shall leave open the
question of possible connection between this physics and the ideas of
saturation and corpuscular structure. Some preliminary thoughts can be found
in Dvali:2011aa ; Dvali:2013eja .

##  VI Some Observational Implications

In standard slow-roll picture, the inflaton plays a double role. First, it
provides a classical clock that enables inflation to end. Secondly, the
quantum fluctuations of this clock provide density perturbations. The quantum
corpuscular effects, enhanced by species, interfere with both mechanisms of
the inflaton clock.

###  VI.1 Enhancement of Gibbons-Hawking Perturbations

First, the density fluctuations from Gibbons-Hawking production of species, is
enhanced by their number,

$\delta_{\rm sp}^{2}=\frac{H^{2}}{M_{P}^{2}}N_{\rm sp}\,.$ (81)

No such enhancement is exhibited by inflaton fluctuations,

$\delta_{\rm inf}^{2}=\frac{H^{2}}{\epsilon M_{P}^{2}}\,,$ (82)

where $\epsilon\equiv(V^{\prime}M_{P}/V)^{2}$ is the standard inflationary
slow-roll parameter.

Using (8), it is instructive to rewrite the above expressions as functions of
$M_{*}$,

$\delta_{\rm sp}^{2}=\frac{H^{2}}{M_{*}^{2}}\,,~{}\delta_{\rm
inf}^{2}=\frac{1}{\epsilon N_{\rm sp}}\frac{H^{2}}{M_{*}^{2}}\,.$ (83)

We see that in the limit (III.3), which implies $\frac{1}{\epsilon N_{\rm
sp}}\rightarrow\infty$, the iflaton perturbations vanish.

We wish to comment that, in case of a large number of light spin-2 species,
the enhancement could potentially be experienced by the tensor mode
Starobinsky:1979ty .

###  VI.2 Quantum Clock

As we have seen, the species speed up a different clock in de Sitter. This is
an internal quantum clock originating from the corpuscular effects such as
depletion, self-entanglement and memory burden. This quantum clock advances
regardless of the classical one of a slowly-rolling inflaton field. If the
number $N_{\rm sp}$ is above a certain critical value, the quantum clock
elapses faster than the classical one. This puts an upper bound on $N_{\rm
sp}$.

In general, if the number of species is large, the effect of the quantum clock
cannot be ignored. In particular, the tilt in the spectral index will include
contributions from two different sources: The standard classical tilt, due to
change of $H$, and the quantum one which we shall now quantify.

The quantum clock gives a departure from a (would-be) scale-invariant spectrum
of perturbations. This departure is unrelated to a classical slow-roll clock
and is due to depletion of the de Sitter constituents. The corresponding tilt
in the spectral index can be estimated as,

$(n_{s}-1)_{\rm Q}=-\frac{N_{sp}H^{2}}{M_{P}^{2}}\,.$ (84)

This expression for $N_{\rm sp}=1$ was derived in Dvali:2013eja . The effect
is enhanced by the number of species Dvali:2020etd .

We shall try to illuminate this phenomenon from various angles. For this, we
represent (84) in three equivalent ways,

$\displaystyle(n_{s}-1)_{\rm Q}$ $\displaystyle=$ $\displaystyle-1/(t_{Q}H)$
$\displaystyle=$ $\displaystyle-\frac{H^{2}}{M_{*}^{2}}$ $\displaystyle=$
$\displaystyle\lambda\,.$

An each form, highlights a different physical meaning of the same phenomenon.
We shall review them separately.

The first equality in (VI.2) shows that the tilt is the ratio of Hubble-time
$1/H$ to quantum break-time $t_{Q}$. This makes sense, since, according to
(56), after time $t_{Q}$ the particle-creation rate drops by order one. The
effect of the number of species on the tilt is via shortening $t_{Q}$, as it
is clear from (58).

The second equality in (VI.2), tells us that the tilt is the square of the
ratio of Hubble $H$ to gravity cutoff $M_{*}$. According to (8), the job of
the species is to lower $M_{*}$. Correspondingly, they increase the tilt.

Finally, the last equality in (VI.2) tells us that the tilt is equal to the
species coupling $\lambda$. Again, the role of the species in increasing tilt
is transparent through their contribution into $\lambda$, as described by
(18).

Note, the information pattern carried by de Sitter is expected to contribute
into the tilt through the memory burden effect, Dvali:2018xpy ; Dvali:2018ytn
; Dvali:2020wft ; Prihadi:2020pqi . Some alternative descriptions can be found
in Gomez:2021yhd .

###  VI.3 Phenomenological Bounds

The large numbers of particle species are included in many extension of the
Standard Model. The extreme case is provided by the scenario Dvali:2007hz
which incorporates

$N_{\rm sp}\sim 10^{32}\,.$ (86)

This number is motivated by the Hierarchy Problem, an outstanding puzzles of
the Standard Model. The essence of it is that the mass term of the Higgs
scalar is quadratically sensitive to the UV-cutoff of the theory. Ordinarily,
the role of the cutoff would be played by $M_{P}$. Hence, having the Higgs
mass around $M_{P}$, would raise no concern. However, the Higgs is much
lighter than $M_{P}$. This is the source of the puzzle.

The idea of Dvali:2007hz  is to use the fact that species lower the cutoff
(8). In a theory with $N_{\rm sp}$ given by (86), the cutoff $M_{*}$ is pushed
down to its current experimental bound of few TeV. In this case, the Hierarchy
Problem gets effectively nullified.

Ordinarily, the inflationary cosmology is assumed to be associated with very
high energy scales. In this light, a theory with low cutoff raises concerns
because the Hubble parameter has to be small. The perception is that such a
theory has a difficulty of generating density perturbations. Again, this
concern is coming from an implicit assumption that density perturbations
originate from the fluctuations of a single slow-rolling scalar field.
However, the same physics that pushed the Hubble down, also provides an
alternative mechanism of enhancing the quantum gravitational particle-creation
by the number of species. This needs to be taken into account in inflationary
model building.

Since (86) represents an absolute phenomenological upper bound on the number
of particle species, it is useful to give some examples of enhancement due to
this number. For instance, for (86), the observable quantum gravitational
imprints can be generated by inflation with the Hubble parameter in a range of
nuclear energies, or so.

As another example, a primordial black hole of $10^{-3}$ earth mass, would
currently approach its quantum break-time and be at the peak of its Page’s
curve. Despite being macroscopic, such a black hole would represent a fully
quantum object, with a maximal inner entanglement. Also a memory burden effect
would be maximal. A scattering of radiation at such a black hole, would in
principle trigger an induced radiation coming from the memory modes. Similar
considerations apply to lighter black holes.

The observational prospects, of course, depend on the number density of such
black holes. As the preliminary studies indicate Dvali:2020wft , the memory
burden effect slows their decay after $t_{Q}$. As suggested, this effect can
potentially enlarge the allowed mass window of primordial black holes as dark
matter Hawking:1971ei ; Carr:1974nx ; Carr:2016drx  towards the lighter end.

##  VII Outlook: Background Matters

In the present paper we have discussed quantum gravity in species limit (5).
This allows to cleanly isolate certain collective and non-perturbative effects
that otherwise would be shadowed by irrelevant corrections. The example is a
quantum break-time (7), exhibited by saturated systems such as de Sitter and
black holes Dvali:2013eja ; Dvali:2014gua ; Dvali:2017eba ; Dvali:2020etd .
The species limit (5) allows to keep $t_{Q}$ finite, despite the fact that the
perturbative gravity decouples.

Working in the species limit, allows to visualize the time-scales that control
the inner entanglement and the back-reaction from the information pattern
(memory burden). These are the corpuscular mechanisms that lead the system to
quantum breaking.

We extracted the quantum break-time in string theoretic example in which the
species are counted by the Chan-Paton factors. This example shows how
profoundly the string theory resists to formation of a would-be de Sitter
vacuum. The synchronized warning signs are given by low and high energy
theories.

The quantum break-time, computed in low energy theory of zero-modes, becomes
of order the string length when the high energy theory enters the Hagedorn
phase. At the same time, the entropy of the open string modes matches the
Gibbons-Hawking entropy. However, this matching takes place at the saturation
point of the ten-dimensional species bound (77). This is accompanied by the
saturation of the unitarity bound (2) by the entropy of the low energy
species.

The above observations, reinforce the earlier conclusions Dvali:2013eja ;
Dvali:2014gua ; Dvali:2017eba ; Dvali:2020etd  that the finite quantum break-
time is the manifestation of inconsistency of the de Sitter state. The same
message is carried by more recent conjectures Palti:2019pca .

Next, we wish to put the concept of the exclusion of de Sitter by the
$S$-matrix, in a broader perspective of background dependence of quantum
gravity. The first point is to take into account the double-scaling argument
Dvali:2020etd , which shows that “vacuumization” of de Sitter is linked with
trivializing the graviton (closed string) $S$-matrix. Ordinarily, we assume
that an approximate $S$-matrix treatment should be valid at short distances.
This is true, as long as the short-distance limit commutes with the
formulation of the theory.

That is, it is OK to perform an approximate computation, after the theory has
been formulated properly. However, there is no a priory justification for
doing the same, within a theory that is approximately formulated. The last
concept not always makes sense. A good example of such an inconsistency, would
be a gauge theory that is approximately anomalous. In fact, the connection
between quantum break-time of de Sitter and anomaly was already made in
Dvali:2020etd , by drawing a parallel with anomalous $1/N$-effects of Witten-
Veneziano mechanism Witten:1979vv .

The second lesson is that, not every background of classical General
Relativity, must find its vacuum realization within quantum gravity/string
theory. Quantum gravity is more fundamental and more restrictive. It is
therefore natural that only certain backgrounds of the classical theory can be
given a tittle of a consistent vacuum. Many backgrounds, such as de Sitter,
that cannot fit in the vacuum category, must be viewed as excited states
constructed on top of a valid $S$-matrix vacuum. They will be subject to
severe constraints such as the quantum break-time Dvali:2013eja .

The non-acceptance of all possible classical backgrounds as vacua, must be
taken as an advantage of the theory. Needless to say, by eliminating de Sitter
vacua, the quantum gravity/string theory nullifies an outstanding cosmological
puzzle.

This suggests that the requirement of a background-independent formulation of
quantum gravity may not necessarily be the only viable strategy. Rather, an
impossibility of formulation of a theory on a given background, can be viewed
as a selection tool that increases the restrictive power of the theory.

Acknowledgements

We thank Goran Senjanović for valuable discussions and comments. We thank
Alexey Starobinsky for conversation on possible enhancement of the tensor
mode. This work was supported in part by the Humboldt Foundation under
Humboldt Professorship Award, by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy - EXC-2111 -
390814868, and Germany’s Excellence Strategy under Excellence Cluster Origins.

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