On corpuscular theory of inflation
Eur. Phys. J. C
On corpuscular theory of inflation
Lasha Berezhiani 0
0 Department of Physics, Princeton University , Jadwin Hall, Princeton, NJ 08540 , USA
In order to go beyond the meanfield approximation, commonly used in the inflationary computations, an identification of the quantum constituents of the inflationary background is made. In particular, the homogeneous scalar field configuration is represented as a BoseEinstein condensate of the offshell inflaton degrees of freedom, with mass significantly screened by the gravitational binding energy. The gravitational counterpart of the classical background is considered to be a degenerate state of the offshell longitudinal gravitons with the frequency of the order of the Hubble scale. As a result, the origin of the density perturbations in the slowroll regime is identified as an uncertainty in the position of the constituent inflatons. While in the regime of eternal inflation, the scattering of the constituent gravitons becomes the relevant source of the density perturbations. The gravitational waves, on the other hand, originate from the annihilation of the constituent longitudinal gravitons at all energy scales. This results in the quantum depletion of the classical background, leading to the upper bound on the number of efolds, after which the semiclassical description is expected to break down; this is estimated to be of the order of the entropy of the initial Hubble patch. The inflationary paradigm is a tantalizing idea of producing the entire observable universe from a small homogeneous causally connected patch as a result of the accelerated expansion [13]. The inflationary computations are usually done in the meanfield approximation, in which one considers the quantum fluctuations around a fixed classical background [48] (see also [9] and references therein). Although this approach has been proven to be very powerful, there may be some quantum effects that lie beyond the reach of this semiclassical

approximation. Especially if we would like to understand
the nature of slowroll eternal inflation [10–15], which is
governed by quantum dynamics.
Recently, a microscopic description of inflation has been
proposed in [16]. In this work, the classical background is
considered to be composed of quanta, just like any other
classical object. This approach is similar to the particle number
conserving formalism for Bose–Einstein condensates used in
condensed matter literature. Even though in the case of
inflation the particle number is not expected to be precisely
conserved, this formalism will safeguard us from an unphysical
particle production which may be an artifact of the meanfield
approximation.
The main idea of [16] was to think of the classical
backgrounds as built on top of the Minkowski space using certain
creation operators. The reason for choosing the Minkowski
space as a base state for the construction is twofold. Firstly,
it is a true vacuum state for the conventional models of
inflation.1 Secondly, there is no particle production on the
Minkowski space, therefore in the picture of [16] the particle
production during inflation should be understood as a result
of the dynamics of the constituent degrees of freedom, rather
than as the vacuum process.
In order to motivate the quantum picture further, let us
consider the history of our universe. In particular, let us begin by
a radiation dominated era. In cosmological computations we
treat radiation as a classical fluid with relativistic equation of
state. However, this is only an approximation and in reality
this fluid is composed of photons; which simply means that
if we start removing photons from a given region of the
universe, eventually we would be left with a Minkowski space.
This immediately points toward the quantum compositeness
of the classical metric itself; the microscopic modification to
the photon gas must lead to the microscopic modification to
the gravitational background.
1 For simplicity, we will be assuming the absence of the cosmological
constant throughout this work.
Going further back into the past, and assuming that the hot
big bang was preceded by inflation, we reach the reheating
era. At this time it is a common practice to treat the
oscillating homogeneous scalar field as a Bose–Einstein condensate
(BEC) of inflatons. After all, in order to finish in the quantum
state with finite density of quantum constituents, we need to
begin with such a state.
Therefore, it is natural to describe the scalar background
as some sort of BEC even past the reheating point. During
inflation the scalar field undergoes a slowroll instead of the
rapid oscillations, which suggests that the inflationary BEC
may need to differ from the reheating BEC.
We will indeed show that the inflationary background can
be described as a degenerate state of offshell inflatons with
significantly screened mass. The mass of the constituents is
in fact the only distinguishing factor between the two
condensates. Moreover, we will give a microscopic account of
the wellknown effects such as Gibbons–Hawking radiation
[17], as well as scalar [4] and tensor [18] modes of
inflationary perturbations.
2 Classical picture
For simplicity we will consider the model in which
inflation is driven by the single massive scalar field, without
selfinteractions, minimally coupled to gravity [19]
S =
d4x √−g
Mp2l R − 21 gμν ∂μϕ∂ν ϕ − 21 m2ϕ2 .
The classical dynamics of the homogeneous background
follows from the equation of motion for the scalar field and
the Friedmann equation,
Depending on the value of the slowroll parameter ≡
m2/H 2, there are two distinct regimes of the evolution. In
particular, it is easy to see that, for 1, the equations of
motion simplify in the absence of the initial kinetic energy
and reduce to
ϕ
˙
− 3H ϕ,
In this approximation, the scalar field is slowly rolling
down the potential. In fact, it does not change significantly
within the Hubble time, resulting in the nearly constant H
and consequently in the quaside Sitter universe.
For 1, on the other hand, the scalar field undergoes
damped oscillations with frequency m and hence behaves
like a degenerate gas of dust. In the presence of other light
species in the spectrum, the universe would reheat at this
point.
3 Quantum picture
Following [16], we would like to begin the discussion starting
from the reheating era. As we have already mentioned, for
1 the scalar field behaves like a dust. Therefore, it should
be described as a BEC of nearly onshell ϕquanta in k = 0
state. Indeed, during reheating it is common practice to treat
the inflaton background as a degenerate quantum state.
Because of this, it is natural to try to describe the scalar
background as some sort of BEC even during inflation. Since
the inflationary background is a homogeneous field
configuration, it should be viewed as a condensate of the finite
number of ϕquanta in k = 0 state. Other properties of this
condensate can be identified by matching to the known
semiclassical results. For instance, we need to obtain the correct
amplitude of density perturbations.
There are two potential sources of the density
perturbations: the uncertainty principle and the scattering of the
constituents.
The scalar constituents have vanishing wavenumber.
Therefore, their wavefunction is completely delocalized
throughout the entire universe; which means that the number
of ϕquanta within a given Hubble patch is not fixed. In fact,
it must undergo quantum fluctuations due to the uncertainty
principle. Ignoring interactions, one can show that the
number of quanta in a given region of the universe is given by the
Poisson distribution.2 This means that if the expected number
of scalar quanta in a given region, let us say the Hubble patch,
is Nϕ then the typical number fluctuation is δ Nϕ = Nϕ .
In order to estimate the amplitude of perturbations we need
to be more specific about the scalar constituents. During the
reheating era, inflaton is undergoing the damped oscillations.
Therefore, at this time the scalar background is a standard
BEC of ϕquanta with k = 0 and the energy ω = m. The
question is: what kind of condensate does the inflationary
stage correspond to?
Let us assume, for now, that during inflation the scalar
background can still be thought of as a collection of k = 0
and ω = m weakly interacting quanta. Then, in order to
source the curvature H , we need the following number of
quanta within the Hubble patch:
2 Strictly speaking, this statement depends on the quantum state of
inflatons. We will elaborate on this in the appendix.
The typical density perturbation due to the uncertainty
principle would be
Equating this with (3.2), we obtain
This is inconsistent with the standard result δϕ = H .
In order to identify the problem in the above computation,
it is useful to pause and think about the background we are
describing. During reheating the background is undergoing
oscillations with the frequency m. Hence, when thinking of
it as a coherent state, it is natural for the constituents to have
ω = m. When it comes to the inflationary stage, on the other
hand, the scalar field is undergoing a slowroll with the rate
much smaller than m
ϕ˙ = − H ϕ.
Therefore, it appears to be more natural to think of the inflaton
background as a condensate of the offshell ϕquanta in k = 0
state with
Since meff m we will need more degrees of freedom than
(3.1) in order to source the curvature H . In particular, we
have
Nϕ = meff H 3 =
Now, the typical density perturbation is given by
Nϕ meff H 3 = Mplm H 2,
which corresponds to the correct amplitude of the scalar
fluctuation on the horizon scale; equating this with (3.3) gives
us
is the number of them per Hubble patch. Therefore, we are
describing the interior of the horizon by the following number
eigenstate3:
N ω=H , Nϕω= mH2 , with N =
Nϕ =
We would like to stress that the constituents of inflation are
not the same as the asymptotic degrees of freedom. For the
scalar field, the latter would be created by the usual creation
operator ak†, which creates a ϕquantum with the dispersion
relation ωk2 = m2 + k2. If we were interested in
describing a state, in which degrees of freedom behaved as a weakly
interacting gas of these massive particles, then we could have
considered a state built using these asymptotic creation
operators. For instance, the scalar background during the
reheating era can be described as such a state. In the regime of the
slowroll inflation, on the other hand, this is not true
anymore, simply because the scalar field background does not
evolve appropriately. Instead, we need to define a new set
of creation operators bk†, which create quanta with the mass
gap ωk=0 = mH2 . The momentum dependence of the
dispersion relation is closely related to the dispersion relation
of the density perturbations computed using the meanfield
approximation. Moreover, in general we should not expect
their interaction vertices to be identical to the ones given by
(2.1). These points along with other related issues will be
discussed in detail in [20].
In the current work, we will simply rely on the universality
of the gravitational interactions and assume that the
interaction vertices are very similar to the ones of the asymptotic
theory. Next, we would like to estimate the interaction rate
between the constituents; since we should expect
interactions to excite particles out of the condensate state. One of
the relevant channels responsible for scalar excitations is
g + ϕ −→ g + ϕ.
This channel is expected to be significant because of the large
multiplicity of ϕquanta.
Keeping in mind the comments given above, the
interaction vertices responsible for this scattering process are
schematically given by
(∂ϕ)2,
where ϕ and h denote the offshell inflaton and graviton
degrees of freedom.
Another source of scalar perturbations is the scattering of
the constituents, which we will discuss shortly. According to
[16], the background geometry of inflation quantum
mechanically corresponds to the degenerate state of the offshell
longitudinal gravitons with ω = H and
3 This state ought to reproduce the classical background in largeN
limit, which means that it could have been chosen to be a coherent
state. However, for simplicity we will treat it as a number eigenstate.
The rate of the process (3.12) can be estimated as
N Nϕ = H,
where the expression in parentheses is the 2 → 2
scattering amplitude, H 2/meff represents the kinematic factor and
N Nϕ is the combinatoric factor.4 Therefore, the number of
excited quanta within Hubble time is
The corresponding density perturbation can be estimated as
Another interaction channel which could result in a
significant scalar amplitude is
g + g −→ ϕ + ϕ.
This corresponds to the annihilation of the constituent
gravitons into the pair of inflatons. The simple estimate shows
that the rate of this process is also of order H , therefore we
should expect to produce approximately few ϕquanta with
energy H through these channels. The estimate for the
corresponding magnitude of the density perturbation is given
by
Comparing this to (3.8), we find
H∗ ≡
In the low curvature regime H < H∗, the dominant source
of the density perturbations is the uncertainty principle. For
H > H∗, on the other hand, the Hamiltonian process (3.17)
begins to dominate.
For completeness, let us estimate the amplitude of the
scalar fluctuations for these two sources separately. Let us
begin with the uncertainty principle, for which we have (3.8).
In order to find the scalar amplitude, we need to relate the
density perturbation to δϕ. Notice that (3.3) is true as long as
the nonlinear terms in δϕ are negligible. In order to check the
validity of this approximation, we retain the largest nonlinear
correction. As a result, (3.3) becomes
4 I am thankful to JeanLuc Lehners for pointing out an error in the last
step of the original (3.14).
After equating this expression with (3.8) we find that the
nonlinear term becomes important for H > H∗, resulting in
uncertainty principle =
Repeating the computation for the Hamiltonian process
(3.17), we arrive at
scattering =
To summarize, we find that the scalar amplitude δϕ is of the
order of the curvature scale H , irrespective of the energy
density. As for the origin of the perturbation, the uncertainty
principle is the dominant source of the density perturbations
for H < H∗, while for H > H∗ the scattering of the
constituents takes over.
Interestingly, the crossover curvature scale H∗ is also the
scale above which the universe enters the selfreproductive
regime. In other words, at this scale the amplitude of quantum
fluctuations begins to exceed the classical displacement of the
scalar field within the Hubble time.
Also, it should be noted that (3.14) is finite in the →
0 limit; unlike the result of [16], where the corresponding
scattering rate was found to diverge in this limit. This means
that, in our picture, we can view the de Sitter space as a
limiting case of inflation.
Having identified the source of the density perturbations,
let us discuss the production of other particles; those not
being the constituents of the background. For instance, the
gravitational waves, or any other spectator field, have
vanishing occupation number on the background state. This means
that the only possible source for their production can be the
annihilation of the constituent degrees of freedom into these
states. According to [16], the dominant channel for the
production of the gravitational waves is
g + g −→ g + g,
with the rate of this process being given by
= H.
It is easy to show that this production rate corresponds to
the correct powerspectrum for the tensor modes
In the next section, we show that the process responsible for
the production of the gravitational waves also gives rise to a
quantum clock.
Let us conclude this section by pointing out a caveat. In
light of the previous discussion, that the dispersion relations
for the constituents can be identified by studying the
collective excitations around the classical background, there can be
only one dynamical scalar. In other words, both scalar field
and metric backgrounds can be thought of as a collection of
the offshell particles with gap set by the frequency of the
classical background; however, only one species of the
constituents will have different klevels excitable. In particular, if
we choose a spatially flat gauge, in which the scalar
perturbation of the gauge field is fixed to zero, then we have chosen
a formalism with gauge constituents frozen in the ground
state. The aftermath of this stipulation would be the
prohibition of the process (3.12) as a potential source of the density
perturbations, the details will be discussed in [20]. However,
by no means should the number of constituent gravitons be
considered as fixed; e.g. they could annihilate, resulting in
the depletion of the condensate.
4 Depletion of the condensate
As we have discussed in the previous section, the production
of the gravitational waves is the result of the annihilation of
the constituent longitudinal gravitons. Therefore, this process
must drain the graviton condensate, just like the Schwinger
pair production results in the discharge of the background
electric field. This depletion can easily be quantified using
the scattering rate of the constituents
Here, the first term on the right hand side is a classical
increase of the number of constituent gravitons as a result of
the slowroll, while the second term represents the quantum
depletion. This relation simply tells us that quantum
mechanically the condensate loses approximately one constituent
graviton in every Hubble time. This immediately suggests
that the semiclassical description is valid as long as the first
term dominates over the second one in (4.1). In other words
we have the following consistency bound:
As one might have expected, this coincides with the threshold
of selfreproduction for the m2ϕ2model. This inequality can
be rewritten as the upper bound on number of efolds
The physical meaning of these bounds is very simple. If we
begin inflation with < 1/N , then the entire condensate will
be depleted before the end of inflation.5
We would like to emphasize that the depletion channel,
identified in the current work as the relevant one, is
independent of the occupation number of the constituent inflatons.
Therefore, it should not be surprising that the obtained bound
on the lifetime of the inflationary background is identical to
the quantum breaktime of the de Sitter space found in [21].
In this section we have considered two sources for
changing number of constituent gravitons: classical slowroll and
the depletion through the scattering of the constituents.
However, there is a third source not included in (4.1), namely the
density perturbations. In particular, the graviton condensate
is a dressing field for the scalar field configuration, dictated
by the quantum constraint equations and correspondingly by
the consistency with the underling diffeomorphism
invariance. Hence, scalar field fluctuations, which originate from
the uncertainty in the position of the constituent inflatons,
will necessarily result in the fluctuation of the number of
constituent gravitons.
To be more specific, we have argued in the previous section
that the typical density perturbation can be estimated as (3.8).
This results in the following change in the Hubble scale:
which results in the following change in the number of
gravitons within the Hubble patch:
The depletion Eq. (4.1) is valid only if δ N 1. It is
straightforward to show that this is indeed so in the selfreproductive
regime 1/N .
5 Summary
We have argued that the homogeneous classical scalar field
of the inflationary background corresponds to the Bose–
Einstein condensate of the offshell inflaton degrees of
freedom with the mass gap of the order of m2/H . Furthermore,
we have shown that the physics behind the origin of the
density perturbations depends on the curvature scale. In the
slowroll regime, the perturbations originate from the quantum
5 We would like to point out that, in the presence of a large number of
light particles (with mass less than the Hubble scale), the consistency
bound (4.2) would read
This would, obviously, lower the upper bound on number of efolds
down to N /nspecies.
uncertainty in the position of the constituent inflatons.
However, in the regime of eternal inflation, they originate from
the annihilation of the constituent gravitons. The latter
process is also responsible for the production of the primordial
gravitational waves.
The annihilation of the constituent gravitons causes the
gradual depletion of the graviton condensate, giving rise to
the upper bound on the lifetime of the quaside Sitter space.
Namely, as was first suggested in [21], the Hubble patch of the
de Sitter space with the curvature scale H has N = Mp2l/H 2
number of longitudinal gravitons in the zero momentum
state. Thus, if the condensate loses approximately one
graviton in the Hubble time it will take N efolds to deplete the
entire reservoir. In the current work, we have shown that the
situation of the slowroll inflation is very similar to the case
of the de Sitter space. If the Hubble patch with N
longitudinal gravitons had started deep in the selfreproductive regime,
then after spending of order N efolds in this regime it would
have ceased to be described by a semiclassical geometry.
Acknowledgements I am especially grateful to Gia Dvali for
illuminating discussions and comments. I would also like to thank Justin
Khoury, Valery Rubakov and Herman Verlinde for useful discussions,
and JeanLuc Lehners for useful comments on the manuscript. This
work is supported by US Department of Energy Grant DESC0007968.
Open Access This article is distributed under the terms of the Creative
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Considering a large box of size V and total number of
particles N , it is easy to show that the probability distribution for
the number of particles n inside the smaller volume v V
is given by the Poisson distribution with
Strictly speaking, this is true for the distinguishable
particles. For the indistinguishable ones, on the other hand,
quantum fluctuations tend to be much more pronounced. In fact,
the result is sensitive to quantum state the particles are in. If
the state is a direct product of oneparticle states or a coherent
one, then one gets the same result as for the distinguishable
ones. However, if the system is believed to be in equilibrium
and hence all physically different configurations are assigned
the same probability, then the situation changes drastically.
To discuss the latter case in detail, let us consider the
abovementioned large box to have been split into r ≡ V /v
number of equal size cells. If we label the states of the system
in terms of the number of particles in each cell, then every
possible state can be written as
(n1, n2, . . . , nr ),
with ni = N . Moreover, in the case of the uniformly
distributed noninteracting particles, all these states must have
equal probabilities. The total number of such states is
(r + N − 1)! . (AIV)
(r − 1)!N !
Now, we would like to find a probability of having n particles
in one of the cells, e.g. n1 = n. This can be done by dividing
the number of distinguishable states with n1 = n by the total
number of distinguishable states
(r + N − n − 2)! (r − 1)!N ! .
P(n1 = n) = (r − 2)!(N − n)! × (r + N − 1)!
As a check for the proper normalization we have
n=N
n=0
n =
P(n1 = n) = 1.
n=N
n=0
We can also find the expectation value of n, as
Which is an intuitive result, considering the uniform
distribution. The amplitude of the number fluctuation can be
computed in a similar way
= n2 − n 2 =
n2 P(n1 = n) − n 2. (AVIII)
n=N
n=0
After a little bit of algebra, we obtain
which implies an order one number fluctuation.
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