# Implications of a holographic density of states on inflation

Satish Ramakrishna<EMAIL_ADDRESS>Department of Physics &
Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road
Piscataway, NJ 08854-8019

(31-Jan-2022)

###### Abstract

There is theoretical evidence that the number of degrees of freedom in quantum
fields decreases as one studies them at extremely short distances. This
emerges from the study of entropy of black holes, as well as from holographic
theories in AdS geometries. Presumably a theory of quantum gravity will
provide an explicit description of how the number of degrees of freedom thin
out as one studies high energy scales.

We do not have a comprehensive theory of how such a thinning of degrees of
freedom would occur. It is likely that there might be residual (and
measurable) effects at larger length scales, though this might be significant
only near the Planck scale. There are very few instances in Nature where one
might be able to see effects of this thinning. One promising venue is in the
phenomenon of inflation, produced in the simplest models through a scalar
inflaton field in a potential with a flat (“slow-roll”) part as well as a
potential well. During inflation, fluctuations of small length scales are
stretched to large scales and then exit the Hubble horizon. We compute the
effect of such a thinning of degrees of freedom upon the running of the
spectral index of quantum fluctuations of the inflaton and deduce that this
will lead to a small positive power of wave-vector (opposite to the usual
$\sim-\epsilon$, i.e., negative power correction). Some comments are then made
about the impact on observations (or non-observations) of such fluctuations
Martin .

##  I Introduction

The AdS/CFT correspondence Dawid ; Susskind ; Maldacena  as well as various
hints we have from the study of entropy of quantum systems Bekenstein ;
Srednicki  appears to drive towards a conclusion that space-time is
holographic. A related idea, that of thinning of the density of states
available for fluctuations in a quantum field, has been explored, for instance
by Banks Banks1  \- though this idea has been explored in various parts of the
literature in different contexts Weinberg ; Reuter1 ; Percacci ; Reuter2 . In
concrete terms, this would mean that the number of degrees of freedom in a
quantum field, for instance, would decrease, i.e., thin out, from the expected
$(3+1)$-dimensional to lower-dimensional at short distances. This thinning
would be expected to be continuous and should be a result of a quantum theory
of space-time and gravity. It would be manifested as the measure in Fourier
space modes $d^{3}k\>d\omega$ changed to $d^{3-q}k\>d\omega$ when computing
loop diagrams in quantum-field-theory. Specifically, one would expect $q$
itself to be a function of $k$ \- an example of such a behavior might be as in
Fig. 1, where $G$ would be a scale where the density of states drops by one
power of momentum above the scale $G$ \- we have plotted the function
$q=\frac{4-D}{2}$ where $D$ is the number of space-time dimensions. However,
one would expect $q$ to differ from $0$ in some range around $G$, which might
conceivably extend a few orders of magnitude below the scale $G$. An example
is shown in Fig. 1 in a Log-Linear Plot.

Figure 1: Approximation for q(k/G)

We see this method as an approximation technique, much along the lines of
dimensional regularization. The key consequence of semi-classical quantum
gravity, for instance the Bekenstein Bekenstein  formula is that there are
fewer degrees of freedom in a given patch of spacetime than would be estimated
in quantum field theory. That can be modelled by a modification of the measure
used to perform integrals in quantum field theory. The theory of inflation
naturally involves very short-distance physics – we make no claim to
constructing a quantum theory of gravity, merely a simulation of one, by
modeling the reduction in degrees of freedom of the quantum field.

A key point is that, like dimensional regularization, we apply this
approximation method in momentum-space integrals. The corresponding logic is
not intuitive at all in real-space. Hence, we write the action (see Equation
(1)) in real-space, then re-write it in momentum-space, followed by applying
the condition that the degrees of freedom are a function of momentum.

The various theories of inflation Riotto  usually incorporate scalar fields
(one or many) that have quantum fluctuations at all scales during inflation,
while their corresponding length scales are under the then Hubble horizon.
These fluctuations are then frozen, as these modes exit the Hubble horizon
(see Fig. 2) and they are then seen when they re-enter the horizon billions of
years later. These quantum fluctuations occur at all length scales and are
expanded by a factor $\frac{a(t_{0})}{a(t_{I})}$, where $a(t_{0})$ is the
current scale factor, while $a(t_{I})$ is the scale factor at the start of
inflation. This number is $\sim 10^{40}$ at least (the uncertainty is in the
exponential) and hence very short wavelength fluctuations (Planck scale) are
massively expanded in size (by this factor), then, they re-enter the horizon.
These fluctuations can be either scalar or tensor in nature and a rather
simple prediction from the simplest theories of inflation is that the scalar
modes (as a function of momentum) are very close to scale invariant. A
spectral index, defined from $P_{\chi}\sim k^{n_{s}-1}$ (the power spectrum
Riotto ) is used to characterize the deviation from a scale-invariant
($n_{s}=1$) prediction. From experimental observationsWang ,
$n_{s}-1\approx-0.04$.

We ask the natural question - if the density of states of a scalar field (in
the simplest theories of inflation) is indeed constrained for high-
momentum/short-wavelength modes, can we see this in the fluctuation in a
contribution to the spectral index? If so, what is the form of this
contribution and does the result exclude some modifications to the density of
states.

The question is complicated by the fact that we have a variety of theories of
inflation and so a number of possible contributions to the spectral index. In
that case, it is possible that the density of states formulation we choose,
constrains (and is constrained by), the kinds of theories of inflation that
would be consistent with it.

Figure 2: Momentum modes entering and exiting the Hubble horizon

###  I.1 The plan of the calculation

We carry out the calculation for inflation caused by a single scalar field. We
write down the action and then compute the equation obeyed by the Fourier
modes in $D$-dimensions. We then proceed to solve this equation for modes
whose wavelengths are much smaller than and also much larger than the Hubble
horizon. The full solution is then obtained by matching the two limiting cases
- this allows us to write down the full solution. We then use this to compute
the power spectrum for fluctuations for the various modes and thus the
spectral index, i.e., the $k$-dependence of the power spectrum.

This calculation is a prelude to the full computation of the power spectrum of
scalar perturbations to the metric from such a scalar field. This calculation
is performed in a convenient gauge (the flat gauge) and the solution is used
to compute the power spectrum of the fluctuations. We thus obtain the spectral
index for these scalar modes. The scalar modes are directly observed in the
angular dependence of the cosmic-microwave-background (CMB) radiation and the
spectral index can be calculated from the observed data. This spectral index
is compared to what we would expect.

In each case, we let the parameter $q=\frac{4-D}{2}$ be slightly different
from $0$ and judge the impact upon the spectral index. This can be directly
compared to observation and useful conclusions may be drawn about excluding
some proposals for the density of high momentum states in quantum-field-
theory.

###  I.2 Comparison to previous analyses

Previous work Brandenberger1 ; Danielsson ; Cheung  has been done on similar
aspects in this area (study of the spectral index). In particular, Martin &
Brandenberger1 (hereafter referred to as M&B) study non-linear dispersion
relations, i.e., where the action studied in the next section is quite non-
linear for large momentum. In such a situation, due to this large-$k$
behavior, the ground state does not admit of plane wave solutions - in
particular, (M&B) refer to three regions of the solutions, viz.,

  1. 1.

Region I: $k>>\frac{1}{l_{Pl}}$ where $l_{Pl}$ is the Planck length

  2. 2.

Region II: $\frac{1}{l_{Pl}}>k>H$ where $H$ is the Hubble constant (inverse
Hubble radius)

  3. 3.

Region III: $k<<H$ which is well past the point where the modes exit the
Hubble horizon

As it turns out, all relevant parameters and scaling behavior (i.e., the
spectral index) are found from matching between regions. In Region III, we
find, in all generality, with the modification to the measure discussed in the
above, corrections to the spectral index proportional to $q$ (with $q(k)$
measured at the relevant scale, see Section III). This is purely a consequence
of the matching condition and the nature of the solution to the problem, so it
appears as a correction to all the cases that are considered in the references
mentioned above, even though the ground states for large-$k$ are very
different in all the non-linear cases. Concretely, in all the cases in (M&B),
the spectral index will have the $q$-related correction discussed below.

The studies of Danielsson Danielsson  concern corrections to Equation (10) in
the following section due to adiabatic evolution from a standard vacuum at
some particular conformal time. While there is no non-linearity here, the same
matching conditions lead to corrections exactly as below in the spectral
index.

##  II Calculation of spectral index for a scalar field in $D$-dimensions

We start with the action for a scalar field in $D$-dimensional space-time
($D-1$-space dimensions and $1$ time dimension). We work with the metric
signature $(+---)$ and set $\hbar=c=1$. We work with an FLRW-metric in
conformal time (written as $\tau$). The dimensionless scale factor is the
function $a(\tau)$. The Hubble parameter $H$ is, as usual $\frac{\dot{a}}{a}$
where $\dot{a}$ is the time derivative of the scale factor. The action
integral is

$\displaystyle{\cal S}[\chi]=\frac{1}{2}\int
d^{D-1}x\>d\tau\>\sqrt{-det\>g}\>\>g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi$
(1)

Here

$\displaystyle
det\>g=a^{2D}\>\>\>,\>\>\>g^{\mu\nu}\partial_{\mu}\chi\>\partial_{\nu}\chi=\frac{1}{a^{2}}\bigg{(}(\chi^{{}^{\prime}})^{2}-(\nabla\chi)^{2}\bigg{)}\>\>\>,\>\>\>f^{{}^{\prime}}\equiv\frac{\partial
f}{\partial\tau}$ (2)

Next, we define $q=\frac{4-D}{2}$ and we set
$\chi=\frac{\sigma}{a^{\frac{D-2}{2}}}=\frac{\sigma}{a^{1-q}}$ and rewrite the
action as

$\displaystyle{\cal S}[\sigma]=\frac{1}{2}\int
d^{D-1}x\>d\tau\>\bigg{(}(\sigma^{{}^{\prime}})^{2}+\sigma^{2}\bigg{(}\frac{a^{{}^{\prime}}}{a}\bigg{)}^{2}\bigg{(}\frac{D-2}{2}\bigg{)}^{2}-(\nabla\sigma)^{2}-\frac{D-2}{2}(\sigma^{2})^{{}^{\prime}}\bigg{(}\frac{a^{{}^{\prime}}}{a}\bigg{)}\bigg{)}$
(3)

Integrating by parts allows us to write this action as

$\displaystyle{\cal S}[\sigma]=\frac{1}{2}\int
d^{D-1}x\>d\tau\>\bigg{[}(\sigma^{{}^{\prime}})^{2}+\sigma^{2}\bigg{(}\frac{a^{{}^{\prime}}}{a}\bigg{)}^{2}\bigg{(}\frac{D-2}{2}\bigg{)}\bigg{(}\frac{D-4}{2}\bigg{)}-(\nabla\sigma)^{2}+\frac{D-2}{2}\sigma^{2}\frac{a^{{}^{\prime\prime}}}{a}\bigg{]}$
(4)

Now, we write the action integral in momentum space and then set
$q=\frac{4-D}{2}$ to be a function of momentum $k$. We can then write down the
equation for the Fourier modesComment  of $\sigma$ as

$\displaystyle\sigma_{k}^{{}^{\prime\prime}}+\bigg{(}k^{2}-(1-q(k)\>)\frac{a^{{}^{\prime\prime}}}{a}+q(k)\>(1-q(k)\>)\bigg{(}\frac{a^{{}^{\prime}}}{a}\bigg{)}^{2}\bigg{)}\sigma_{k}=0$
(5)

This equation can be exactly solved, though a rather more intuitive way to do
so is to realize that $a(\tau)=-\frac{1}{H\tau}$ for de-Sitter evolution and
$\frac{a^{{}^{\prime\prime}}}{a}\sim\bigg{(}\frac{a^{{}^{\prime}}}{a}\bigg{)}^{2}\sim\frac{1}{\tau^{2}}$,
so that the regimes $|k\tau|=\frac{k}{aH}$ is either much larger or smaller
than 1 are interesting, with a matching condition at $k=aH$.

In the limit $k>>aH$, which corresponds to sub-Hubble-horizon length scale
fluctuations of $\sigma$, the equation (in the Bunch-Davies Riotto ; Davies
vacuum with the usual normalization) can be solved to give

$\displaystyle\sigma_{k}(\tau)=\frac{1}{\sqrt{2k}}e^{-ik\tau}$ (6)

Note that, in line with the analysis in M&BBrandenberger1 , the Unruh
dispersion relation Unruh  and a different choice of vacuum, determine that
the relevant $\sigma_{k}\sim\frac{1}{k}$ in our notation. That affects the
result they get (and for this analysis too), as described below.

In the opposite limit, $k<<aH$, which corresponds to super-Horizon-length
scale fluctuations, there is a growing solution to the equation

$\displaystyle\sigma_{k}^{{}^{\prime\prime}}=\bigg{(}(1-q)\frac{a^{{}^{\prime\prime}}}{a}-q(1-q)\bigg{(}\frac{a^{{}^{\prime}}}{a}\bigg{)}^{2}\bigg{)}\sigma_{k}$
(7)

The solution can be checked to be $\sigma_{k}\sim a^{1-q}$. This amounts to a
correction to the usual “freeze-out”, where the fluctuations increase in sync
with expansion - note that $\chi_{k}=\frac{\sigma_{k}}{a^{1-q}}$, so the
$\chi_{k}$ modes are indeed frozen out. As (a correction to the) usual, the
solution can be written as $\sigma_{k}=B(k)a^{1-q}$ and we obtain $B(k)$
through matching the solutions at $k=aH$. This yields

$\displaystyle B(k)=\frac{H^{1-q}}{\sqrt{2k^{3}}}k^{q}$ (8)

Here, $q$, which is a function of the momentum $k$, is then set to be the
value at $k=aH$, so we are sensitive to the dimensional effects at the scale
of the Hubble horizon.

In the other (Unruh vacuum) case studied by M&B Brandenberger1 , $B(k)\sim
k^{q-2}$.

Hence, the solution for the original $\chi_{k}$ mode (in Fourier space) is

$\displaystyle\chi_{k}=\frac{H^{1-q}}{\sqrt{2k^{3}}}k^{q}$ (9)

This reduces to the usual solution if $q=0$.

In the reference mentioned in Section IIA Brandenberger1 , various other vacua
with non-linear dispersion relations are studied. They lead to different
matching conditions, however, they all are corrected by the same power of
$q(k=aH)$, since that derives from the small-$k$ solution, which is known to
be in the regime of linear dispersion anyway and Equation (7) applies.

We can now compute the power spectrum of the scalar field. It would be

$\displaystyle{\cal P}_{\chi}(k)=\frac{k^{3}}{2\pi^{2}}|\chi_{k}|^{2}$
$\displaystyle=\bigg{(}\frac{H^{1-q}}{2\pi}\bigg{)}^{2}k^{2q}$ (10)

And, in parallel, for the Unruh vacuum case, ${\cal P}_{\chi}(k)\sim
k^{2q-1}$.

We could generalize the above discussion for a slight variation of the Hubble
parameter for different modes Riotto  and obtain the spectral index

$\displaystyle n_{\chi}-1=2q-2(1-q)\epsilon$ (11)

This is one of two central results of this paper. There is a modification to
the spectral index due to this effect, for non-zero $q$. For zero $q$, the
result reduces to the usual result for a massless scalar field. As described
earlier, $q=q(k=aH)$.

In parallel with the above analysis, which studies the behavior with the
Bunch-Davies vacuum, the Unruh vacuum result would be

$\displaystyle n_{\chi}-1=2q-1-2(1-q)\epsilon$ (12)

##  III Calculation of the spectral index for scalar perturbations with non-
zero $q$

In this case, we simply follow Riotto  and study the comoving (scalar)
curvature perturbation $\zeta$. The theory of perturbations of the scalar
field in a background metric is well developed - the general metric we work
with is explained thoroughly in various introductory papers and texts
Brandenberger2 ; Riotto ; Baumann

$\displaystyle(ds)^{2}=(1+2\Phi)d\tau^{2}-2B_{i}d\tau
dx^{i}-((1-2\psi)\delta_{ij}+(\partial_{i}\partial_{j}-\frac{1}{D-1}\delta_{ij}\partial^{2}E)dx^{i}dx^{j}$
(13)

In the above, $\Phi$ is called the lapse function (or the gravitational
potential), $\psi$ , $E$, $B_{i}$ are all as usually defined to obtain scalar
metric perturbations. The metric has some gauge freedom, since physics is
insensitive to general coordinate transformations (diffeomorphisms) Baumann .
Hence, one usually works in a particular gauge.

We work in the flat gauge, with zero lapse function and work to lowest order
in fluctuations, so we neglect the graviational potential fluctuation. The $D$
related terms do not change the result obtained vs. the usual computation
since we are considering only scalar perturbations and further, work in a
gauge that drops even them (the computation for the tensor terms is
substantially more involved). In this gauge, with fluctuations in the scalar
field $\delta\chi$,

$\displaystyle\zeta=H\frac{\delta\chi}{\dot{\chi}}$

The variance is computed (as demonstrated, for instance, in Riotto Riotto ) to
be

$\displaystyle<\zeta^{2}>=H^{2}\frac{(\frac{H}{2\pi})^{2}}{(\dot{\chi})^{2}}$
(14)

and obtain the power spectrum (using $M_{P}^{-2}=8\pi G_{N}$)

$\displaystyle
P_{\zeta}=\frac{1}{2M_{P}^{2}\epsilon}\bigg{(}\frac{H}{2\pi}\bigg{)}^{2(1-q)}k^{2q}$
(15)

from analyzing the fluctuations of the scalar field, with the Bunch-Davies
vacuum, as in Section III.

The rest of the calculation proceeds rather straightforwardly and we get

$\displaystyle
n_{\zeta}-1=2q-2\epsilon(1-q)-2\epsilon+2\delta=2q+2\eta-6\epsilon+2q\epsilon$
(16)

This is a second central result of this paper. There is a correction to the
spectral index $n_{\zeta}$, the $2q$ extra in the sum. Again, in the limit
$q\rightarrow 0$, we recover the usual result. This correction would appear
for large-$k$, which represent the modes that would have been smaller than the
Planck length before and during inflation.

As defined, $q=\frac{4-D}{2}$. It is bigger than $0$ ($0.5\>to\>1.5$) for
modes that are much shorter in wavelength than the Planck scale. As mentioned
earlier, the various holographic models as well as density of state
modifications apply for modes with large wave-vector that would have been
close to (or at the) the Planck scale during inflation.

##  IV Numerical Estimates, Discussion & Conclusions

Starting with the Planck length $\sim 10^{-35}m$, if we expect to have $60$
e-folds during inflation, followed by expansion post-inflation by a factor of
$10^{30}$, we would see the length expanded by a factor of $10^{56}$. This
would lead to fluctuations with a wavelength of $10^{21}$ meters, this
corresponds to a length of $10^{6}$ light-years. This would correspond to
$l\sim 10^{3}-10^{4}$. This is at the boundary of resolution of the angular
distribution seen with Planck’s data, but the above results would lead to (on
the usual angular distribution plot) an increasing power spectrum ($\sim
k^{2q-2(1-q)\epsilon}$). There are other effects, of course, that are
complicated to disentangle from this one (Silk damping, etc. Silk ), so limits
could possibly be placed on $q$ as a consequence. Interestingly, the larger
error bars at these ranges of $l$ are consistent with rising CMB spectrum
amplitude - more accurate measurements are awaited.

One can ponder the result of the above calculations in a few ways.

It is clear from observation Wang  that the spectral index for scalar
fluctuations is approximately scale invariant, i.e., $n_{\zeta}\approx 0.96$.

First, the variety of possible values for $q$ and where it starts to become
relevant are $q=0.5$ if space were two-dimensional at length scales near the
Planck length, to $q=1.5$ in the Banks-Draper ansatz Banks1 . These are large
deviations and would not consistent with observation; therefore, they should
be considered excluded - if we trust the inflation models that need a small
$\epsilon$, we would perforce need to exclude such large density of states
modifications. Interestingly, if we see a suitably complex ground state for
large-$k$, as in the cases studied by M&BBrandenberger1 , we could well allow
a larger value of $q$ to compensate for their $n_{\zeta}=0$.

Second, that we can actually have reasonably large values of $q$ (say
$q=0.5$), but allow either larger values for $\epsilon$ in a suitably
constructed theory of inflation, or a more complex ground state Brandenberger1
. In the first case, since (with the definition
$\epsilon=3\frac{\frac{(\dot{\Phi})^{2}}{2}}{\frac{(\dot{\Phi})^{2}}{2}+V(\Phi)}$),
the value $\epsilon=1$ corresponds to the the equation of state parameter
$w=\frac{P}{\rho}=-\frac{1}{3}$, so one can allow for values of epsilon almost
all the way to $1$ to still maintain the conditions, i.e., $\ddot{a}>0$, that
inflation produces and still solve the flatness and horizon problems. The
second possibility is explored in Section III.

There is a third interesting possibility that this allows us to explore the
situation with negative $q$ through comparison to experiment. In the simplest
theories of inflation, deviations of $q$ away from $0$ appear to be excluded.

A fourth possibility, if one believes that $q$ is indeed non-zero, is that it
puts limits on how large the number of e-foldings can be - if $N$ were larger
than around 120, then we would definitely see the effect of a non-zero $q$ in
the spectral index for simple models of inflation.

In conclusion, we have extended the theory of quantum perturbations due to
inflation to account for a varying density of states and have obtained
potentially experimentally observable corrections to the running spectral
index for short wavelengths. If we trust the simplest models for inflation,
the result excludes a reduction of the density of states as well as of
dimensionality. It is conceivable that alternative inflation models that can
work with larger values of $\epsilon$ would be consistent with a reduced
density of states as well as observations.

##  V Acknowledgments

Discussions with Professor Scott Thomas at the NHETC Rutgers are gratefully
acknowledged, as also the hospitality of the Rutgers Physics Department. The
perceptive comments of two anonymous reviewers is gratefully acknowledged,
especially the pointer to some of the trans-Planckian work.

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