Also at ]Physics Department, Faculty of Science, Alexandria University,
Alexandria, Egypt

# Enhanced spectrum of primordial perturbations, galaxy formation and small
scale structure

Karim H. Seleim  [  Amr A. El-Zant<EMAIL_ADDRESS>Centre for
Theoretical Physics, The British University in Egypt, Sherouk City 11837,
Cairo, Egypt  A. M. Abdel-Moneim  Physics Department, Faculty of Science,
Alexandria University, Alexandria, Egypt

###### Abstract

The standard cosmological structure formation scenario is successful on large
scales. Several apparent problems affect it however at galactic scales, such
as the small scale problems at low redshift and more recent issues involving
early massive galaxy and black hole formation. As these arise where complex
baryonic physics becomes important, this is often assumed to be behind the
problems. But the same scales are also those where the primordial spectrum is
relatively unconstrained, and there are several ways in which it can be
modified. We focus on that arising from effects possibly associated with the
crossing of high energy cutoff scale by fluctuation modes during inflation.
Elementary arguments show that adiabatic evolution cannot modify the near
scale invariance, we thus discuss a simple model for the contrary extreme of
sudden transition. Numerical calculations and simple arguments suggest that
its predictions, for parameters con2sidered here, are more generic than may be
expected, with significant modifications requiring a rapid transition. We
examine the implications of such a scenario, in this simplest form of sudden
jump as well as gradual variants, on the matter power spectrum and halo mass
function in light of the limitations imposed by particle production. We show
the resulting enhancement and oscillation in the power spectrum on currently
nonlinear scales can potentially simultaneously alleviate both the apparent
problem of early structure formation and, somewhat counterintuitively,
problems at low redshift concerning the abundance of dwarf galaxies, including
those too big to fail. We discuss consequences that can observationally
constrain the scenario and its parameters, including an inflationary Hubble
scale $\lesssim 10^{-8}M_{\rm Pl}$, while touching on the possibility of
simultaneous modification of power on the largest scales.

††preprint: APS/123-QED

##  I Introduction

Structure can condense from small density perturbations in a nearly
homogeneous universe through gravitational instability. In the context of
contemporary cosmology the density perturbations are seeded by quantum
fluctuations in a primordial scalar field driving inflation, which later
decays as the universe reheats and the standard model particles (and putative
dark ones) arise (e.g., [1, 2]). The statistical properties of the primordial
perturbations thus leave their mark on the cosmic microwave background (CMB)
and large scale structure of the galaxy distribution. On these scales, on
which they can be inferred with precision, the properties of the perturbations
are consistent with a nearly scale invariant primordial spectrum essentially
determining their statistics [3].

At a phenomenological level, the simplest models of inflation do predict a
near scale invariant primordial power spectrum if the inflaton potential is
specified in such a way that the resulting Hubble parameter is nearly constant
over a sufficient number of e-folds. Nevertheless, this prediction is not
unique [4, 5]; indeed, little is known of the microscopic physics of
inflation, or the wider particle physics model it may be a part of, and the
coupling of the inflaton driving inflation to other fields may lead to changes
in the potential that can ruin the predictions of standard slow roll models
(.e.g., [6, 7, 8, 9] and [10, 11, 12] for reviews). Stages of singular or
rapid evolution of the potential or its derivatives, interrupting slow roll,
leave imprints on the primordial spectrum of fluctuations [13, 14, 15, 7, 16,
17, 18, 19, 20, 21, 22]. Such effects may lead to a variety of ’features’ and
changes that break the scale free spectrum, and may have observable
consequences associated with significant enhancement or suppression of power
on large scales [23, 24, 25, 26, 27, 28, 29], as well as on smaller (galactic
and sub-galactic) scales [30, 31, 32, 33, 34, 35, 36], including the formation
of primordial black holes [37, 38]. However the anomalous variation in the
inflaton potential needs to be localized in order for single field slow roll
inflation to proceed for a sufficient number of e-folds, and is also limited
on large scales by observations suggesting a nearly scale invariant primordial
power spectrum [39, 3].

In some models, such as DBI inflation, the breaking of scale invariance and
associated features in the power spectrum are best represented in terms of
sharp temporal variations in the sound speed of perturbations [40, 41, 42, 43,
44, 45, 46]. This is still accompanied by corresponding changes in the
equation of state of the inflaton and therefore, as in the cases above, by
anomalous background evolution. On the other hand, if inflation proceeds
modestly longer than the minimal number of e-folds needed to solve problems
such as the apparent causal connection in the CMB field (’horizon problem’),
primordial perturbations are expected to arise from well within a high energy
cutoff scale where new physics may transpire. This gives rise to what is often
termed as the trans-Planckian problem. The minimal length scale that it
entails can be introduced by modifying the commutation relations or
introducing nonlinear dispersion relations for the propagation of fluctuations
(early studies include [47, 48, 49, 50, 51, 52, 53], while a relatively recent
review can be found in [54]).

If the Hubble scale of inflation $H$ is smaller than the scale $k_{c}$, where
new physics may appear, the background evolution is unmodified by possible new
physics; modifications to the scale invariant spectrum may then arise solely
from anomalous evolution of the fluctuation modes, rather than of the
inflation potential. However, if some form of decoupling is assumed — e.g.
motivated by the fact that the power spectrum is evaluated at horizon exit
scales $H\ll k_{c}$ — then the corrections can be quite small; of order
$(H/k_{c})^{q}$, with $q\gtrsim 1$. In the context of local effective field
theory $q=2$ [55]; if the fluctuation modes are assumed to simply emerge from
the quantum foam at conformal time $\eta_{c}(k)$, depending on the comoving
mode wavenumber $k$ — instead of the usual Bunch Davies initial conditions
taken at $\eta\rightarrow-\infty$ — then $q=1$ [56, 57]; if the modes emerge
from the local adiabatic vacuum then $q=3$ [58]. A covariant small scale
cutoff, introduced by removing field configurations that are off shell by more
than a Planck scale leads to $q=1$ [59].

Much larger corrections are nevertheless possible in principle. This is the
case, in particular, if the emergence of the inflating modes from the high
energy cutoff scale is assumed to be preceded by non-adiabatic evolution
arising from a nonlinear dispersion relation at scales $>k_{c}$ [47, 48, 60,
61, 62]. Modes then do not emerge from the foam in their lowest energy states.
Excited states arise, providing anomalous initial conditions for further
evolution and leading to enhancement and oscillations in the power spectrum.
The most general parametrization of the effects of a non-adiabatic high energy
scale exit would therefore appear to include both phenomena [51, 63].

Though physics at these scales is largely unknown, it can in principle be
envisioned that the introduction of a cutoff scale in itself can modify the
effective dynamics of the fluctuations. Indeed, the introduction of a minimal
length scale and a large variety of phenomenological descriptions of ’quantum
spacetimes’, can be characterized by nonlinear dispersion relations [64, 65].
At the most intuitive level, a simple hydrodynamic analogy suggests such a
modification to the propagation of fluctuations [66, 67, 68]. This, much in
the same way that an effective macroscopic description of wave propagation
through a fluid or lattice may still be employed at wavelengths approaching
the interparticle distances, provided this is phenomenologicaly taken into
account through a modified dispersion relation. As, at scales smaller than the
interparticle distances waves cannot propagate at all, it is in the transition
between such a cutoff scale and the scale on which the standard effective
macroscopic description applies that a nonlinear dispersion relation may
describe the propagation of fluctuations. At this simplest intuitive level,
one may expect the dynamics of a sound wave, initially moving in a medium
where interparticle spacing is large and comparable with its wavelength, to
keep memory of the anomalous evolution, even after it crosses and propagates
into a medium with smaller interparticle spacing, where the effective theory
is perfectly valid and ’decoupling’ is guaranteed. The rough analogy here
would be with an inflaton fluctuation mode inflating from wavenumbers above
the high energy cutoff scale to ones below it, on its way to the horizon.
Applications of more sophisticated ’analogue’ models to the inflationary
scenario show that modifications of the dispersion relation can indeed lead to
significant changes of power spectrum of field correlations [69].

An important limitation on modifications of the power spectrum through the
inclusion of excited states relates to the fact that these are necessarily
associated with departures from a vacuum state. And too much excitation can
lead to departures significant enough to prohibit inflation from starting and
persisting in the first place [70, 71, 72, 73, 74]. However, as has been
pointed out, the limitations that arise thus may not be very constraining [75,
76, 77]. In the present investigation we wish to examine whether, within the
limits imposed by particle production, excitations of inflaton modes, stemming
from the presence of high energy cutoff scale, can lead to significant and
astrophysicaly interesting modifications of the primordial power spectrum.

At present, the non-observation of significant departures from scale
invariance on large (linear) scales, where the primordial spectrum can be
rather precisely inferred, seem to embody the main evidence against such
modifications. Indeed, observations on scales on which the density
perturbation is linear preclude even relatively small modifications [78, 79,
80, 81, 82]. Observations are however much less constraining at smaller
scales, where they are limited by the Silk damping of the CMB, and by
nonlinear structure formation erasing the possibility of directly mapping the
observed power spectrum to the primordial one. Precise CMB and large scale
structure inference is therefore limited to scales $\gtrsim 10~{}{\rm Mpc}$.
For a horizon scale of $\sim 10~{}{\rm Gpc}$ this spans three orders of
magnitudes. More model dependent constraints are available from the
Lyman-$\alpha$ forest down to wavenumbers roughly corresponding to comoving
spatial scales of order of Mpc. Beyond that, the spectrum is currently quite
weakly constrained [83, 84, 85]. On the other hand, the smallest structures
that form in the context of the standard cold dark matter scenario have earth
mass and roughly solar system size $\sim 10^{-4}{\rm pc}$. From such scales to
the smallest scales at which the linear power spectrum can be directly
recovered one counts 11 orders of magnitude — nine more than those separating
the nonlinear scale to the horizon.

It is not inconceivable that the scale invariance of the primordial power
spectrum does not hold in some parts of the aforementioned range. On the
contrary, despite the significant successes of the current model of structure
formation on large scales [86], through the past couple of decades a variety
of problems have arisen on galactic scales. There is a group of quite possibly
related longstanding issues connected to the central densities of dark matter
halos, and the abundance and dynamical properties of local dwarf galaxies [87,
88]; and, in apparent contradiction, more recent issuees related to an
apparent preponderance of massive old galaxies and supermassive black holes at
redshifts $3\lesssim z\lesssim 9$ that may pose a challenge the current
$\Lambda$CDM-based structure formation paradigm [89, 90, 91, 92, 93, 94, 95,
96, 97, 98, 99, 100, 101, 102].

Such problems appear in the highly nonlinear regime of structure formation;
where small density perturbations, born of primordial ones in the presumed
inflaton, have sufficiently grown under gravity to form gravitationaly bound
objects. Since it is also at such scales that complex baryonic physics becomes
important, it was natural to suppose that the main determinant lies in complex
baryonic physics of galaxy formation and evolution. For example, for the small
scale problems at $z=0$, processes involving energy input to dark matter halos
through dynamical friction with baryons [103, 104, 105, 106, 107, 108, 109,
110, 111] or through random potential fluctuations driven by starbursts or
active galactic nuclei (AGN) [112, 113, 114, 115, 116, 117, 118], were
invoked. (In addition to suggestions modifying the dark matter particle
physics models, as in warm dark matter [119, 120, 121, 122, 123], self
interacting dark matter [124, 125, 126, 127, 128, 129] and fuzzy dark matter
[130, 131, 132, 133, 134]). Similar attempts are ongoing in the case of the
the more recent early structure formation issue (some are discussed in Section
IV.3.2).

However, it is also precisely at the nonlinear scales, where baryonic physics
becomes important, that the primordial power spectrum is relatively
unconstrained. That modifications thereof can be relevant to small scale
problems associated with galaxy formation has long been realized [30], but not
as extensively investigated as the baryonic solutions discussed above. While
feedback from starbursts and AGN is now recognized as a central ingredient of
galaxy formation, independently of its possible role in alleviating the
aforementioned issues, and whereas massive baryonic clumps, proposed to
mediate dynamical friction coupling to dark matter, have since been observed
in forming galaxies (e.g. [135, 136]), it is also important to further examine
mechanisms that address galactic scale problems through modification of the
primordial spectrum. Deriving the consequences of such modifications is in
itself an interesting tool for understanding the processes from which they may
arise in an inflationary era.

In this study we investigate the effect on the power spectrum from field
excitations, stemming from non-adiabatic transition through a high energy
cutoff regime corresponding to currently nonlinear scales, and within the
limits imposed by particle production. We attempt to do this in generic terms,
starting from well defined initial conditions, with linear dispersion relation
(but with sound speed different from unity) and examining the effect of the
transition. As this solely affects the fluctuation modes, the equation of
state of the inflaton, and thus the background evolution, remains unmodified
(unlike in cases such as DBI inflation mentioned above), this helps isolate
the effect of excitations on the spectrum. We then look for associated effects
on the matter power spectrum and dark matter halo mass function.

In the next section, after illustrating in simplest terms how the power
spectrum is essentially an adiabatic invariamt of the dynamics of inflaton
fluctuations, we present and discuss a simple model representing the other
extreme of a sudden transition (in an appendix, we show results that suggest
it is generic for a large range of parameters; in a second appendix we discuss
the situation when the assumption is relaxed). In Section III we discuss what
this model entails in more formal terms, evaluating the limits on power
spectrum modification in terms of particle production. In Section IV, we
study, within these limits, the possible modifications on the matter power
spectrum and halo mass function. We discuss possible astrophysical
consequences and constraints, before presenting our conclusions

##  II Adiabaticity, scale invariance and the sudden extreme

###  II.1 The evolution of fluctuations

The general quadratic action for inflationary perturbations with sound speed
$c_{s}$ can be expressed in terms of the Mukhanov-Sasaki (MS) variable $v$ as
[137, 138, 139]:

$S^{(2)}=\frac{1}{2}\int\mathrm{d}^{4}x\left(v^{\prime
2}-c_{\mathrm{s}}^{2}(\nabla v)^{2}+\frac{z^{\prime\prime}}{z}v^{2}\right),$
(1)

where $z=a\frac{\phi_{0}^{\prime}}{\mathcal{H}c_{\mathrm{s}}}$, $\phi_{0}$ is
the background inflaton field, $\mathcal{H}=a^{\prime}/a$, and the primes
denote derivative with respect to conformal time $\eta$. The evolution of each
Fourier mode $v_{k}(\eta)$ is governed by the MS equation

$v_{k}^{\prime\prime}+\left(c_{s}^{2}k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0.$
(2)

The MS variable is related to the curvature perturbations by $v=z\mathcal{R}$.
This is a quantity of fundamental interest, as it relates primordial quantum
fluctuations to the observables, such as CMB anisotropies; the power spectrum
of the large scale galaxy distribution; and, ultimately (more indirectly), the
formation of smaller scale structures, such as the dark matter halos hosting
galaxies. The dimensionless power spectrum of such perturbations is given by

$\Delta_{\mathcal{R}}^{2}(k)\equiv\frac{k^{3}}{2\pi^{2}}\left|\mathcal{R}_{k}\right|^{2},$
(3)

where the right-hand side is evaluated at the horizon ($c_{s}k=aH$); as in the
absence of isocurvature perturbations, the comoving curvature perturbations
$\mathcal{R}$ are conserved on super-horizon scales [140]. Scale-invariant
perturbations correspond to $\Delta_{\mathcal{R}}^{2}(k)={\rm const}$.
Departures from this can arise if $c_{s}$, or the inflationary Hubble scale
$H$, depend on time.

In the standard inflationary scenario, a massless field, and quasi de Sitter
evolution is assumed (and so $H$ is nearly constant throughout the
inflationary stage). The associated slow roll parameters, defined as

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}\quad\tilde{\eta}\equiv\frac{\dot{\epsilon}}{H\epsilon}\quad\text{
and }\quad\kappa\equiv\frac{\dot{c}_{s}}{Hc_{s}},$ (4)

are always much smaller than unity. Any departure from scale invariance is
small, and is usually quantified by the spectral tilt parameter $n_{s}$:

$n_{s}-1\equiv\frac{d\ln\Delta_{\mathcal{R}}^{2}}{d\ln
k}=-2\epsilon-\tilde{\eta}-\kappa.$ (5)

The slow roll parameters being small implies that $n_{s}\approx 1$.

To first order in the slow roll parameters, and assuming canonical kinetic
terms, one can also show that

$\frac{z^{\prime\prime}}{z}=\frac{1}{\eta^{2}}\left(2+3\epsilon+\frac{3}{2}\tilde{\eta}\right).$
(6)

In this case, to a good approximation, one can rewrite (2) as

$v_{k}^{\prime\prime}+\left(c_{s}^{2}k^{2}-\frac{2}{\eta^{2}}\right)v_{k}=0.$
(7)

If no non-standard dispersion relation is invoked then $c_{s}=1$, and the
standard scenario may be fully recovered.

###  II.2 Adiabaticity, adiabatic invariants and primordial power spectrum

####  II.2.1 General context

In the context of equation (2), setting $c_{s}=1$ can be interpreted as the
result of assuming a massless field with linear dispersion relation between
the physical frequency $\omega_{\rm phys}$ and the physical wavenumber $k_{\rm
phys}=\frac{k}{a}$, $\omega_{\text{phys }}=k_{\text{phys }}$. However, as
discussed in the introduction, this is not a necessity; a modification of the
equation of state of the inflaton (e.g. such as in DBI inflation), or
modification of the dispersion relation due to modes probing a high energy
cutoff scale, beyond which new physics may arise, can change the situation.

In the latter case, beyond a cutoff scale $k_{c}$, one can introduce the
relevant modification by replacing the square of the comoving wavenumber
$k^{2}$ in (2) with

$k^{2}\rightarrow k_{\mathrm{eff}}^{2}(k,\eta)\equiv
a^{2}(\eta)\omega_{\mathrm{phys}}^{2}\left[\frac{k}{a(\eta)}\right],$ (8)

the main requirement being that the new dispersion relation recovers the
linear one for scales $k\ll k_{\mathrm{c}}$ [141].This dispersion relation is
thus necessarily time dependent, as it must transit between two regimes. It
can be used to parametrize and reflect the effect of a varying sound speed in
equations (2) and (7), the latter applying when the background dynamics is
well approximated by standard slow roll. Indeed, in this context, Eq. (2) can
be rewritten as

$v_{k}^{\prime\prime}+\left[k_{\mathrm{eff}}^{2}(k,\eta)-\frac{z^{\prime\prime}}{z}\right]v_{k}=0.$
(9)

(A more rigorous derivation, based on a variational principle, can be found
here[142]).

How does the extra time dependence, that thus arises, affect the power
spectrum derived from the above equation? As noticed in several studies, mere
time dependence in itself is not sufficient to alter the nearly scale
invariant nature of the primordial spectrum of fluctuations. The adiabaticity
condition — that is, $|\frac{d\omega}{d\eta}|/\omega^{2}<1$ — must be
violated. A well known example where this condition is indeed violated invokes
the Corley-Jacobson dispersion relation

$k_{\mathrm{eff}}^{2}(k,\eta)=k^{2}-k^{2}\left|b_{m}\right|\left[\frac{k}{k_{c}a(\eta)}\right]^{2m}$
(10)

(where $b_{m}$ is a constant and $m$ an integer). The studies [141, 143, 48],
indeed indicated that a modification of the power spectrum, in the form of a
change in the spectral index and superimposed oscillations, was possible.
However, several criticisms were raised, including the possibility of complex
frequencies arising at early times, rendering the quantum field theory ill-
defined, and problems related to setting the initial conditions in non-
adiabatic regime. To circumvent such issues, a new dispersion relation [142]
was proposed, which exhibits linear behaviour in the small and large
wavenumbers, but has intermediate concave region where the adiabaticity is
violated locally.

Here we will be considering a simpler scenario, which assumes standard Bunch
Davies type initial conditions, with modified sound speed but still linear
dispersion relation. The effective sound speed transits to the standard
relation $\omega=k$ as the boundary around $k_{c}$ is crossed. In this simple
controlled context, we wish to estimate the rapidity and steepness of the
transition required in order to produce palpable change in the power spectrum.
It turns out that such a transition must be quite rapid.

####  II.2.2 The power spectrum as an adiabatic invariant

We now wish to show, in explicit simple terms, that the primordial power
spectrum is in fact an adiabatic invariant of the evolution of inflationary
perturbation, and thus cannot be snigficiantly modified by any changes in the
dispersion relation that keeps the dynamics of the perturbations sufficiently
adiabatic.

We will be interested in the case when the dispersion relation is modified due
to the fluctuation modes probing scales beyond a high energy cutoff, before
they inflate into lower energy scales on their way to horizon exit. Only the
effective speed of mode propagation is modified and slow roll inflation of a
massless field is assumed to hold in all stages. So, Eq. (7) holds to a good
approximation. However, because of the non-standard dispersion relation
assumed, $c_{s}$ in that equation will not be necessarily unity in all stages.
In fact its variation would incorporate changes parametrized by $k_{\rm eff}$
in Eq. (9) above.

In principle $c_{s}$ in (7) can be either larger or smaller than unity.
Perhaps a scenario in which modes do not propagate at all for $k_{\rm phys}\gg
k_{c}$, and then do so at increasing $c_{s}\rightarrow 1$ as they emerge from
the ’quantum foam’ at $k_{\rm phys}\sim k_{c}$, is appealing; it qualitatively
connects, for example, to waves propagating in a lattice, which are scattered
and dispersed to smaller speeds as one approaches the interparticle spacing,
before ceasing to propagate. However ’analogue’ models with superluminal
speeds $c_{s}>1$, beyond the cutoff scale, have also been proposed [68]. As we
discuss in Section III, for our purposes both situations lead to similar
results.

Equation (7) refers to a simple harmonic oscillator with variable frequency.
If $c_{s}$ is constant, the variation solely comes from the second term in the
bracket. To separate this effect from that connected to possible variation in
$c_{s}$ at a high energy cutoff transition, we exploit the fact that $k_{c}\gg
H$. This enables one neglect the second term in the brackets of equation (7),
at scales ($\sim k_{c}$) around the high energy cutoff transition; as, when
modes transit from beyond the cutoff scale $k_{c}$ to below it, the conformal
time $\eta_{c}=\frac{-1}{a_{c}H}=\frac{-k_{c}}{Hk}$. The term in the brackets
in the aforementioned equation is then
$c_{s}^{2}k^{2}\left(1-\frac{2}{c_{s}^{2}}\frac{H^{2}}{k_{c}^{2}}\right)$. The
second term inside this latter bracket is small compared to unity when

$c_{s}^{2}>2\left(\frac{H}{k_{c}}\right)^{2}.$ (11)

Since we already assume that $k_{c}\gg H$, this is always the case when
$c_{s}>1$. We will also assume that this condition is satisfied when
considering the case of $c_{s}<1$ 111Indeed, as discussed in Section IV.1.2,
in order to get modification of the matter power spectrum and halo mass
function at scales of the order of a comoving Mpc, which are of particular
interest, one needs $H/k_{c}\lesssim 10^{-4}$, much smaller than $c_{s}=0.01$,
which is the smallest we consider..

This leaves us with an equation of a harmonic oscillator with frequency
$\omega=c_{s}k$. The adiabatic invariant for a standard harmonic oscillator
with specific energy $E$ and frequency $\omega$ is $J=\frac{E}{\omega}$.
Taking the modulus of the amplitude and the velocity
$v_{k}^{\prime}\left(\eta\right)=i\omega_{k}v_{k}\left(\eta\right)$, the
energy of the oscillator is $E=\omega^{2}|v_{k}|^{2}$. Whatever the evolution
at scales above $k_{c}$, as long as it is adiabatic $J$ is conserved.
Moreover, at scales $<k_{c}$ one must recover the standard linear dispersion
relation, and so $\omega=k$. At such scales, relevant to eventual horizon
crossing, one then has

$J=k\left|v_{k}\right|^{2}.$ (12)

Comparing this with the standard slow roll inflationary power spectrum

$\Delta_{\mathcal{R}}^{2}(k)=\frac{kH^{2}}{2\pi^{2}}\left|v_{k}\right|^{2},$
(13)

one finds that they are equivalent up to a (nearly) constant factor
$\frac{H^{2}}{2\pi^{2}}$.

With sufficiently adiabatic evolution through the transition at $k_{c}$ no
significant change to the power spectrum can occur. Any appreciable effect
could then result solely from small variations in $H$, or from second term in
bracket of Eq. (7), which also turns out to be quite modest, as may be
expected given the quadratic correction at any physical scale $(k_{\rm
phys}/H)^{-2}$. For this implies again that this term is smaller than the
first until modes are quite close to existing the horizon. In Appendix A.1, we
show numerical calculations that corroborate this contention in the context of
the simple model described below.

###  II.3 A toy model of the sudden extreme

As we have seen, any adiabatic frequency change, due to nonstandard evolution
of modes beyond a high energy cutoff scale, will not alter the nearly scale
free form of the resulting power spectrum. We thus consider the opposite
extreme; that of a sudden change in the sound speed at $k_{c}$, while
employing the same approximation of neglecting the second term in the bracket
of Eq. (7). The procedure again separates changes in the power spectrum due to
variations in $c_{s}$ at around $k_{c}\gg H$ from any time dependence
connected to the second term in the above equation at much smaller physical
wavenumbers.

The change in sound speed across the transition is equivalent to a sudden
change in frequency. To illustrate such a situation in simplest terms, we
consider the effect of such a change on a simple harmonic oscillator, with
initial amplitude $A$, frequency $\omega_{\rm in}$ and phase $\phi$. Its
evolution is given by

$\displaystyle X_{in}(t)=A\cos\left(\omega_{\rm in}t+\phi\right)$ (14)
$\displaystyle X^{\prime}_{\rm in}(t)=-A~{}\omega_{\rm
in}\sin\left(\omega_{\rm in}t+\phi\right),$ (15)

with

$\displaystyle A=\sqrt{X^{2}_{\rm in}(t)+\frac{V^{2}_{in}(t)}{\omega^{2}_{\rm
in}}}$ (16) $\displaystyle\phi=\arccos{\left(\frac{X_{\rm
in}(t)}{A}\right)}-\omega_{\rm in}t.$ (17)

Suppose that at some moment $t=t_{s}$ the spring constant is suddenly altered,
and the corresponding frequency of the oscillator changes to $\omega_{\rm
out}$. Then, for $t\geq t_{s}$, rCl X_out(t)&= B cos(ω_out(t - t_s))
+Csin(ω_out(t - t_s))
X^′_out(t)=-Bω_outsin(ω_out(t - t_s))
+Cω_outcos(ω_out(t - t_s)). Matching the initial and final states at $t_{s}$
one obtains

$\displaystyle B=X_{\rm in}(t_{s})$ (18) $\displaystyle C=\frac{\dot{X}_{\rm
in}(t_{s})}{\omega_{\rm out}}.$ (19)

So the evolution after the jump can be expressed in terms of the initial state
at the jump as rCl X_out(t)&=X_in(t_s)cos(ω_out(t - t_s))
+˙Xin(ts)ωoutsin(ω_out(t-t_s)), with the amplitude and phase changing after
the jump.

We now apply this toy model to attempt to mimic the evolution of fluctuations
due to a sudden change in frequency (or again, effectively sound speed) of
propagation of inflaton fluctuations. In our approximation the evolution is
effectively governed by two independent harmonic oscillators, due to the
complexity of the mode functions in (7). Thus, for the real part,

$\displaystyle X_{{\rm in}_{r}}(\eta)=A_{r}\cos({\omega_{\rm
in}\eta+\phi_{r}})$ (20) $\displaystyle\dot{X}_{{\rm
in}_{r}}(\eta)=-A_{r}\omega_{\rm in}\sin({\omega_{\rm in}\eta+\phi_{r}}),$
(21)

and for the imaginary part we have

$\displaystyle X_{{\rm in}_{i}}(\eta)=A_{i}\cos({\omega_{\rm
in}\eta+\phi_{i}})$ (22) $\displaystyle X^{\prime}_{{\rm
in}_{i}}(\eta)=-A_{i}\omega_{\rm in}\sin({\omega_{\rm in}\eta+\phi_{i}}).$
(23)

The sudden step will here correspond to conformal time $\eta_{c}$, when an
inflating mode crosses a physical the wavenumber $k_{c}$, where ’new physics’
may arise. Applying the step condition as previously, for the real part we
find rCl X_out_r = X_in_r (η_c) cos[ω_out(η-η_c)]
\+ X’inrωout sin[ω_out(η-η_c)]. Similarly, for the imaginary part rCl X_out_i
= X_in_i (η_c) cos[ω_out(η-η_c)]
\+ X’inrωout sin[ω_out(η-η_c)]. The complete solution then is

$v_{k}(\eta)=X_{{\rm out}_{r}}(\eta)+iX_{{\rm out}_{i}}(\eta).$ (24)

This can be evaluated for each $k$, with $\omega_{k_{\rm out}}=k$, given
$\omega_{k_{\rm in}}=c_{s}k$. As mentioned above (and checked in Appendix A.1)
usage of Eq. (24), in order to evaluate the effect on the primordial power
spectrum of a sudden step in $c_{s}$ and $\omega$ at $k_{c}$, returns a good
approximation. The results of Appendix A.2 also suggest that the sudden jump
scenario itself turns out to be much more generic to any appreciable change in
the power spectrum than may seem a priory. We now discuss how the power
spectrum is evaluated and the modifications to the standard near scale
invariant form that arise.

####  II.3.1 The power spectrum of primordial fluctuations

A mode corresponding to comoving wavenumber $k$ crosses the high energy cutoff
scale at $\eta_{c}=-\dfrac{k_{c}}{Hk}$. At $\eta\ll\eta_{c}$ we assume Bunch
Davies type initial conditions but with $\omega_{k}=c_{s}k$, with $c_{s}\neq
1$. Thus, before the transition in sound speed (and frequency),

$v_{k}(\eta)\rightarrow\frac{1}{\sqrt{\omega_{k_{\rm in}}}}e^{i\omega_{k_{\rm
in}}\eta}.$ (25)

For the initial amplitudes one then has

$\displaystyle A_{r}=A_{i}=\frac{1}{\sqrt{\omega_{k_{\rm
in}}}},\quad\phi_{r}=0,\quad\phi_{i}=\frac{\pi}{2}.$ (26)

Modes with physical wavenumbers larger than $k_{c}$ at the start of inflation
undergo a frequency change such that $\omega_{\rm in}/\omega_{\rm out}=c_{s}$
(where $c_{s}$ refers to the value, different from unity, before the
crossing). Modes with smaller wavenumbers do not cross the high energy cutoff
scale and their frequency remains unmodified (we discuss how the transition
scale is connected to current comoving scales in Section IV).

All modes eventually cross the horizon. Using equations (24) and (3), one can
evaluate the power spectrum in the context of our simplified model when $H$ is
given. This is done at horizon crossing when $\eta=\eta_{H}$. Alternatively,
one can also use equation (7) to calculate the power spectrum numerically at
superhorizon scales, as done in the Appendix for purpose of comparison and
evaluating the relevance of the model.

In de Sitter inflation $H$ is exactly constant, and all modes are assumed to
exit the horizon at time $\eta_{H}=-1/k$. Since again a standard dispersion
relation must reign beyond the high energy cutoff scale $k_{c}$, one expects
$\omega_{\rm out}=k$. All modes then leave the horizon at the same phase and
oscillations implied by equations (II.3) and (II.3) do not appear in the power
spectrum; only enhancement is found at scales undergoing the jumps.
Numerically, equation (7) can be used to obtain similar results (cf. Appendix
A.1).

The Hubble parameter in more realistic models of inflation must vary slowly
with time. The variations imply that modes do not leave the horizon at the
same phase, and oscillations as well as enhancement appear in the primordial
power spectrum. As a simple generic example, we will adopt power-law
inflation[145, 146] where (in proper time), $a(t)\approx t^{p}$, with $p>1$.
This corresponds to an inflation potential of the form

$V(\phi)=V_{0}\exp\left[-\frac{1}{M_{\rm
Pl}}\sqrt{\frac{2}{p}}(\phi-\phi_{i})\right],$ (27)

where $M_{\rm Pl}$ is the reduced Planck mass. with slow-roll parameters given
by

$\epsilon_{v}=\frac{M^{2}_{\rm
Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}=\frac{1}{p},\qquad\tilde{\eta_{v}}=M^{2}_{\rm
Pl}\frac{V_{\phi\phi}}{V}=\frac{2}{p},$ (28)

where $V_{\phi}=\frac{dV}{d\phi}$. The scale factor and the Hubble parameter
become

$a(\eta)=\left(\frac{\eta}{\eta_{\rm in}}\right)^{\frac{p}{1-p}},\qquad
H(\eta)=-\frac{p}{p-1}\left(\frac{\eta}{\eta_{\rm
in}}\right)^{\frac{p}{p-1}}\frac{1}{\eta},$ (29)

with $\eta_{\rm in}=\frac{t_{\rm in}}{1-p}$. With these forms for the
evolution of the scale factor and Hubble parameter one can again use Eq. (24)
in conjunction with (3) to evaluate the power spectrum, or numerically
integrate Eq. (2), which now takes the following form

$v^{\prime\prime}_{k}(\eta)+\left[c_{s}^{2}k^{2}-\frac{2p^{2}-p}{(1-p)^{2}}\frac{1}{\eta^{2}}\right]v_{k}(\eta)=0.$
(30)

We will generally use $H=10^{-4}M_{\rm Pl}$, $p=55$. and
$\eta_{i}=-10^{4}M^{-1}_{\rm Pl}$, in order to get the correct normalization
and tilt of the power spectrum on linear scales using power law inflation. The
results, when rescaled accordingly, are valid for other values of $H$, since
the relative enhancement of the power spectrum depends only on the in and out
frequency ratio of oscillations. The connection between the cutoff scale
$k_{c}$ and the corresponding comoving scale depends on $H/k_{c}$ rather than
the absolute values (Section IV.1.2). As shown in Appendix A.1, the step
enhancement in the power spectrum is accompanied by oscillations in this more
generic (as opposed to de Sitter) case.

##  III Broken invariance and particle production

###  III.1 General solution in terms of Bogoliubov coefficients

We now consider what the simplified sudden step model actually implies in
terms of quantum fluctuations in an inflaton. For this purpose we translate it
to the language Bogoliubov expansion and coefficients. In this context, the
high energy cutoff transition will be seen to lead to excitations of the field
and particle production. The excitations will take place as a result of
transitions between well defined time independent in and out states. They
invariably lead to enhancement in the power spectrum, generically accompanied
by oscillations.

###  III.2 Generic enhancement in power spectrum

Using (7), and again invoking the approximation of neglecting the
$\frac{-2}{\eta^{2}}$ term due to $k_{c}\gg H$, we get the mode function
differential equation of a massless free scalar field in Minkowski spacetime,
with $\omega_{k}=k$. Assuming the field to be initially in the vacuum state
$|0\rangle$, the amplitude of the vacuum fluctuations (the square root of the
power spectrum) are given in terms of the vacuum mode function corresponding
to the Bunch Davies initial conditions (25). Then, non-adiabatic evolution
(whether sudden or not), can transform this initial vacuum state $|0\rangle$
to one with excitations, with respect to the old annihilation operator
$\hat{a}_{\mathrm{k}}^{-}$.

To find the effect of such excitations on the power spectrum after the
transition is complete, one can proceed as follows. First by writing the mode
expansion of the field operator in terms of the annihilation operator
$\hat{b}_{\mathrm{k}}^{-}$ of $|0\rangle$ and its complex conjugate

$\hat{\chi}=\frac{1}{\sqrt{2}}\int\left(e^{i\mathbf{k}\cdot\mathbf{x}}\mu_{k}^{*}\hat{b}_{\mathbf{k}}^{-}+e^{-i\mathbf{k}\cdot\mathbf{x}}\mu_{k}\hat{b}_{\mathbf{k}}^{+}\right)\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}},$
(31)

and then computing the two point correlation function in the state $|0\rangle$
using this operator. One can then define the amplitude of the quantum
fluctuations in terms of the new mode function $\mu_{k}(\eta)$ as

$\Delta_{\mu}(\eta)=\frac{1}{2\pi}k^{3/2}\left|\mu_{k}(\eta)\right|.$ (32)

This new normalized mode function is a linear combination of the old one and
its complex conjugate. Using Bogoliubov coefficients, it can be written as

$\mu_{k}(\eta)=\alpha_{k}v_{k}(\eta)+\beta_{k}v_{k}^{*}(\eta).$ (33)

Thus we have

$\Delta_{\mu}(\eta)=\frac{1}{2\pi}\frac{k^{3/2}}{\sqrt{\omega_{k}}}\left[\left|\alpha_{k}\right|^{2}+\left|\beta_{k}\right|^{2}+2\operatorname{Re}\left(\alpha_{k}\beta_{k}^{*}e^{2i\omega_{k}\eta}\right)\right]^{1/2}.$
(34)

The second coefficient $\beta$ refers to excitations away from the vacuum
state; as we will see below it directly counts particle production. The ratio
of the primordial power spectrum after and before the sudden change can be
expressed as

$\frac{\Delta_{\mu}^{2}(\eta)}{\Delta_{v}^{2}(\eta)}=1+2\left|\beta_{k}\right|^{2}+2\operatorname{Re}\left(\alpha_{k}\beta_{k}^{*}e^{2i\omega_{k}\eta}\right).$
(35)

Averaging over a period larger the periodic time of the system (or in generic
inflation models, over horizon exists of the different modes with different
$H$), eliminates the oscillating term. The main result is that excitations
away from the vacuum state lead to typically larger RMS fluctuations and power
spectrum.

An important point to note here is that the generic enhancement in the power
spectrum will occur whether the ’jump’ in frequency is upward — that is
whether $\omega_{\rm in}<\omega_{\rm out}$ — or the downward, with
$\omega_{\rm in}>\omega_{\rm out}$. Or, assuming a dispersion relation
$w=c_{s}k$ to govern the propagation of fluctuations before and after the
jump, the power spectrum will be enhanced whether $c_{s}$ is larger before the
jump, or whether it is larger afterwards. This is seen explicitly below.

###  III.3 Relations between the coefficients and the frequencies

The above does not necessarily assume instantaneous transition jumps between
the well defined in to out states, just that a time dependent transition
occurred. In our simplified model we have two regions connected by a sudden
jump, which enables one to calculate the Boguliubov coefficients explicitly in
terms of the in and out frequencies.

If one labels the initial vacuum as $\left|0_{\mathrm{in}}\right\rangle$, and
the final vacuum $\left|0_{\mathrm{out}}\right\rangle$. Before the jump, and
assumes the scalar field is in the initial vacuum state, the mode function is

$v_{k}^{(\mathrm{in})}(\eta)=\frac{1}{\sqrt{\omega_{k_{\mathrm{in}}}}}e^{i\omega_{k_{\mathrm{in}}}\eta},$
(36)

for $\eta<\eta_{c}$. Before the jump, the frequency $\omega_{\rm in}=c_{s}k$
with $c_{s}\neq 1$. In order to connect with standard inflationary scenario,
the frequency after the jump is $\omega_{k_{\rm out}}=k$. The final frequency
$\omega_{k_{\rm out}}$ is therefore necessarily different from the initial one
$\omega_{k_{\rm in}}$. This causes excitations in the field, which modify the
power spectrum.

After the jump, the mode function $v_{k}^{(\mathrm{in})}(\eta)$ evolves into
the superposition of $v_{k}^{(\text{out})}(\eta)$ and its complex conjugate:

$v_{k}^{(\mathrm{in})}(\eta)=\frac{1}{\sqrt{\omega_{k_{\mathrm{out}}}}}\left[\alpha_{k}^{*}e^{i\omega_{k_{\rm
out}}\left(\eta-\eta_{c}\right)}-\beta_{k}e^{-i\omega_{k_{\mathrm{out}}}\left(\eta-\eta_{c}\right)}\right].$
(37)

The Bogoliubov coefficients $\alpha_{k},\beta_{k}$ are determined by the
requirement that the solution and its first derivative must be continuous at
the jump, that is at $\eta=\eta_{c}$. The result is

$\alpha_{k}=\frac{e^{-i\omega_{k_{\mathrm{in}}}\eta_{c}}}{2}\left(\sqrt{\frac{\omega_{k_{\mathrm{in}}}}{\omega_{k_{\mathrm{out}}}}}+\sqrt{\frac{\omega_{k_{\mathrm{out}}}}{\omega_{k_{\mathrm{in}}}}}\right)$
(38)
$\beta_{k}=\frac{e^{i\omega_{k_{\mathrm{in}}}\eta_{c}}}{2}\left(\sqrt{\frac{\omega_{k_{\mathrm{in}}}}{\omega_{k_{\mathrm{out}}}}}-\sqrt{\frac{\omega_{k_{\mathrm{out}}}}{\omega_{k_{\mathrm{in}}}}}\right)$
(39)

This explicitly shows that the time dependence introduced by assuming a
sufficiently rapid transition from in to out states can lead to significant
excitation in the inflaton field for large enough frequency ratio. As is
clear, the absolute values of $\beta_{k}$ and $\alpha_{k}$ derived above do
not depend on whether $\omega_{\rm in}>\omega_{\rm out}$ or the reverse. This
again shows that generic enhancement in the power spectrum is expected,
independent of the direction of the jump.

###  III.4 Limits from particle production

As we have seen, excitations of the inflaton generically lead to enhanced
power spectrum. In section IV below, we will suggest that these may have
important consequences at galactic scales, at both high and low redshifts,
pertaining to such apparent problems as the dearth of dwarf galaxies, ’too big
to fail’ and early galaxy formation, while maintining a standard spectrum at
scales where it is highly constrained. But how much excitations of the field
can one have without ruining the inflationary scenario itself? Indeed, the
exponential expansion during inflation hinges on a dark energy equation of
state, too much excitation and particle production can turn it instead into a
radiation field, with deceleration replacing the exponential expansion.

The radiation energy density associated with the relativistic particles, which
can be assumed to be produced through excitations of the field, may be
expressed as [76]

$\langle\rho\rangle=\int_{k_{\rm phys}=H}^{k_{\rm phys}=k_{c}}d^{3}{\bf
k}_{\rm phys}\omega_{\rm phys}(k_{\rm phys})n_{k_{\rm phys}},$ (40)

where $k_{\rm phys}$ and $\omega_{\rm phys}$ are the physical wavenumbers and
frequencies. The occupation number of excited states can be expressed in terms
of the second Bogoliubov coefficient as $n(k)=|\beta_{k}|^{2}$. In the
relevant integration range the relation between the wavenumbers and
frequencies is linear, and the integral is dominated by larger values of
$k_{\rm phys}$. In this case, $\langle\rho\rangle\approx\beta^{2}k_{c}^{4}$,
where $\beta$ corresponds to $\beta_{k}$ at larger values of $k$ dominating
the integral.

In the context of the sudden step scenario $\beta_{k}$ is some non-zero
constant for modes affected by the jump (and zero otherwise), and the above
estimate is rigorously justified. In order for inflation to start and proceed
then, $\beta^{2}k_{c}^{4}$ must be smaller than the energy density scale of
inflation $H^{2}M_{\rm Pl}^{2}$. This leads to the condition

$|\beta|<\frac{M_{\rm Pl}H}{k_{c}^{2}}.$ (41)

If $k_{c}=M_{\rm Pl}$ this is small for $H\ll k_{c}$. However much smaller
cutoff scales may in principle be allowed (and claimed all the way down to the
TeV scale e.g. [147, 148, 149]; also [150] for review). For the largest field
inflation allowed by recent data, with $H\lesssim 3\times 10^{-5}M_{\rm Pl}$
and relatively conventional high scale $k_{c}\gtrsim 10^{-3}M_{\rm Pl}$, one
finds $|\beta|\lesssim 30$ as an upper limit. In general, one only needs
$k_{c}/M_{\rm Pl}\approx H/k_{c}$ to get a Bugoliubov coefficient of order
one. The backreaction condition above may thus in principle allow for large
modifications that could be tested and constrained observationally, even in
the nonlinear regime of structure formation.

As long as (41) is satisfied inflation can start and proceed, but in order to
obtain a near invariant spectrum on large scales, the time derivative of the
backreaction energy must also be small as the large scale modes exit the
horizon. The limits of integration in Eq. (40) is an upper limit on
backreaction energy, which assumes that the whole interval between $k_{c}$ and
$H$ is filled with excited states corresponding to modes that have already
crossed $k_{c}$. In this case the time derivative
$\frac{d\langle\rho\rangle}{dt}\sim\beta^{2}H^{3}\dot{H}$ is much smaller in
absolute value than the change in energy density of the inflaton $\sim M_{\rm
Pl}^{2}H\dot{H}$ for values of $\beta^{2}$ of interest. However, at earlier
times, as modes are crossing $k_{c}$ and filling up the interval down to $H$,
the integration interval is variable, the time derivative of the backreaction
energy $d\frac{\langle\rho\rangle}{dt}\sim\beta^{2}k^{3}_{\rm
phys}(k_{c})\dot{k}_{\rm phys}(k_{c})\approx\beta^{2}Hk^{4}_{\rm phys}(k_{c})$
can be much larger (here $k_{\rm phys}(k_{c})$ refers to the physical
wavenumber of the first scale that crosses $k_{c}$; it decreases as the mode
inflates towards the horizon, when $k_{\rm phys}(k_{c})=H$). This leads to the
constraint $\beta^{2}\lesssim\epsilon\frac{H^{2}M_{\rm
Pl}^{2}}{k_{c}^{4}}\left(\frac{k_{c}}{k_{\rm phys}}\right)^{4}$, where
$\epsilon=-\dot{H}/H^{2}$. A more detailed treatment gives a similar
constraint, $\beta^{2}\lesssim 2(6\pi)^{2}\epsilon\frac{H^{2}M_{\rm
Pl}^{2}}{k_{c}^{4}}\left(\frac{k_{c}}{k_{\rm phys}}\right)^{4}$ 222cf. Ref
[75]. Their more rigorous formulation using effective field theory invokes an
’earliest time’, which is defined as the time the smallest CMB scale leaves
the cutoff scale. The earliest time here would correspond to that when the
first scales cross the high energy cutoff. In both situations the origin of
the time variation of the backreaction lies in the same change in number of
excited states in the interval between $k_{c}$ and $H$..

The effect of the changing energy density as the excited states are filling up
the interval between $k_{c}$ and $H$ can be quite complicated, as it would
require evaluation of the modified evolution, taking into account the
rescaling of the energy density (which itself can act as vacuum energy [76]).
Here we just point out that, simply assuming the usual relation
$\epsilon=\frac{H^{2}}{8\pi^{2}P_{0}}$ to hold when the effect is small
enough, leads to the condition

$|\beta|\lesssim\left(\frac{2\times 10^{-9}}{P_{0}}\right)^{1/2}\times
6.7\times 10^{4}\left(\frac{H}{k_{\rm phys}(k_{c})}\right)^{2},$ (42)

with $P_{0}\approx 2\times 10^{-9}$ the standard characteristic value of the
standard primordial power spectrum of scalar fluctuations, This rough estimate
suggests that $|\beta|$ can be of order one, without affecting the power
spectrum on larger scales exiting the horizon, if these scales exit when the
spatial physical scale that first crosses the high energy threshold has
inflated enough to be about $0.004$ times the size of the horizon. We further
discuss the possible interpretation of this constraint in Section IV.1.2.

As the ratio of the power spectrum modified by excitations to the vacuum power
spectrum scales as $1+2\beta^{2}$, considerable modifications may be allowed
in principle, if $|\beta|$ is of order one or larger. In the following we
consider possible consequences of, and constraints on, such modification on
currently nonlinear scales, where existing observational constraints are
relatively weak and apparent problems with galaxy formation at low and high
redshift arise.

##  IV Matter power spectrum and halo mass function

In this section we examine some possible astrophysical implication of the
sudden change of frequency at a high energy cutoff. For this purpose we
compute the linear matter power spectrum and the dark matter halo mass
function. The modified halo mass function will be of interest, particularly in
terms of its possible observable consequences on the galaxy stellar mass
function. For the actual calculations we assume a $\lambda$CDM universe with
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, $h=0.7$, and RMS dispersion in the
density field at $8~{}h^{-1}{\rm Mpc}$ at $z=0$, $\sigma_{8}=0.8$.

###  IV.1 The matter power spectrum

Figure 1: Dimensionless matter power spectra at $z=0$. The perturbed cases
correspond to sudden jumps of a factor of 100 in mode frequency (and therefore
sound speed) due to shift in dispersion relation at the high energy cutoff
scale (chosen to correspond to a comoving wavenumber $k=1h/{\rm Mpc}$ as
discussed in the text). Spectra are shown for a de Sitter background (left)
and corresponding power law inflation model as a generic example (cf. Section
II.3.1). The oscillations in the latter case are absent in the de Sitter one
due to all modes leaving the horizon at the same phase. The frequency ratio
corresponds to a Boguliubov coefficient $|\beta|=4.95$ (Eq. 39). Note that,
for power law inflation, there is net enhancement despite the strong
oscillations, which appear symmetric around the uperturbed spectrum on the
logscale. This will result in similar mass dispersions in the de Sitter and
power law models, where the smoothing also leads to gradual enhancement
despite the sudden jump at the cutoff scale (Fig. 2).

####  IV.1.1 Evaluation procedure

The power spectrum of perturbation in the CDM is evaluated from

$P(k,a)=\frac{4}{9}\frac{k^{4}P_{i}(k)}{\Omega_{m}^{2}H_{0}^{4}}T^{2}(k)D^{2}(a),$
(43)

where $P_{i}(k)\equiv\Delta_{\mathcal{R}}^{2}(k)$ is the primordial power
spectrum, $D(a)$ the linear growth factor, and $H_{0}$ is the present value of
the Hubble parameter.

As we will be primarily interested in generic consequences, rather than
detailed comparison with data, for this purpose we genrerally use the BBKS
fitting form [152]

$T\left(x\equiv\frac{k}{k_{eq}}\right)=\frac{\ln~{}(1+0.171x)}{0.171x}\left[F(x)\right]^{-1/4},$
(44)

with $k_{\rm eq}=0.073~{}\Omega_{m}h^{2}{\rm Mpc}^{-1}$ and

$F(x)=1+0.284x+(1.18x)^{2}+(0.399x)^{3}+(0.490x)^{4}.$ (45)

For the growth factor, we use[153]

$D(z)=\frac{D^{+}(z)}{D^{+}(z=0)},$ (46)

where

$D^{+}(z)=\frac{5\Omega_{m}}{2}\frac{H(z)}{H_{0}}\int_{z}^{\infty}\frac{\left(1+z^{\prime}\right)dz^{\prime}}{\left[H\left(z^{\prime}\right)/H_{0}\right]^{3}},$
(47)

with

$H(z)=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+\left(1-\Omega_{m}\right)}.$ (48)

We have verified our results against full solution of the perturbation
equations using the public code CLASS (class-code.net), and show results using
this code in examining the predictions of the enhanced primordial spectrum
scenario in the more general case when the assumption of sudden jump is
relaxed (Appendix B and Fig. 8).

####  IV.1.2 Choice of jump scale

CMB and large scale structure observations place quite tight constraints on
the amplitude of the primordial power spectrum on scales $\sim 10{\rm Mpc}$ or
larger, We now show how smaller scales can be affected by a modified power
spectrum, while larger scales remain unaffected, if inflation proceeds for
approximately the number of e-folds needed to solve the horizon problem.

Observable inflation takes place after the comoving spatial scale
$k^{-1}(H_{0})\sim H_{0}^{-1}$ exits the horizon; it is characterized by the
minimum number of e-folds needed to solve the horizon problem:
$N[k(H_{0})]=\ln\left[a_{\mathrm{end}}/a_{k(H_{0})}\right]$, where the
subscripts denote the end of inflation and the epoch of horizon exit of the
scale $k^{-1}(H_{0})$. The condition can be written as

$\frac{a_{k(H_{0})}}{a_{0}}=\frac{H_{0}}{H},$ (49)

where, $a_{0}$ is the current scale factor. $H$ is the Hubble parameter at the
horizon exit of the scale $k(H_{0})^{-1}$ during inflation.

Given a comoving scale $k$, one may ask when it was equal to a given physical
scale $k_{c}$ during inflation. This gives the following condition

$\frac{k}{a_{0}H_{0}}=\frac{a_{c}k_{c}}{a_{k(H_{0})}H}.$ (50)

As an example, we set $k(k_{c})=1~{}{\rm Mpc}^{-1}$, $H=10^{-4}{M_{\rm Pl}}$,
and $k_{c}={M_{\rm Pl}}$. We then find that $a_{c}\sim a_{k(H_{0})}$; that is,
at the time the current horizon scale $H_{0}^{-1}$ exits the horizon during
inflation, the comoving spatial scale $\sim 1~{}\mathrm{Mpc}$ is of the order
of the Planck length.

This general picture is reproduced even if the cutoff scale $k_{c}$ is not the
Planck scale. All one needs is $H/k_{c}\approx 10^{-4}$. If inflation proceeds
for a number of e-folds larger than the number $N_{\rm min}=N[k(H_{0})]$
required to solve the horizon problem, then the ’jump scale’ can still
correspond to $k(k_{c}))=1{\rm Mpc^{-1}}$ if $H/k_{c}<10^{-4}$. In general,
the number of e-folds allowed, with $k(k_{c})$ corresponding to the smallest
comoving spatial scale affected by the high energy cutoff transition, is

$N=N_{\rm
min}+\ln\left[\left(\frac{k_{c}}{H}\right)\left(\frac{k(H_{0})}{k(k_{c})}\right)\right],$
(51)

The scale $k(H_{0})\approx 10^{-4}~{}{\rm Mpc}^{-1}$ is fixed by the present
size of the horizon, while $k(k_{c})=1~{}{\rm Mpc^{-1}}$ happens to roughly
correspond to the largest scale on which significant modification of the power
spectrum would not affect its inference from galaxy cluster counts and lensing
surveys (but, depending on the exact scale, not necessarily Lyman-$\alpha$
bounds, as discussed in Section IV.3.3). Larger values of $k(k_{c})$ are in
principle possible, and in this case the power spectrum can be modified on
smaller scales, affecting smaller nonlinear structures. However, if one takes
into account our crude estimate of the time variation of the backreaction,
this may be constrained. For, as mentioned in relation to Eq. (42), to
maintain $|\beta|$ of order 1, one may need $H/k_{\rm phys}(k_{c})\gtrsim
0.004$. If $k(k_{c})\approx 1~{}{\rm Mpc^{-1}}$, the comoving scale exiting
the horizon when this is satisfied is $\approx 0.004~{}{\rm Mpc^{-1}}$. Larger
scales, with smaller wavenumbers, can be affected if one insists on
$|\beta|\gtrsim 1$. In the context of the simplest scenario with constant
$\beta_{k}$ beyond the cutoff regime, the power spectrum may be modified on
such scales. This may be allowed on comoving scales $k<0.004~{}{\rm
Mpc}^{-1}$, and may even be relevant to supposed anomalies of the CMB on large
scales, but not on smaller spatial scales, where modifications are tightly
constrained. That changes in the power spectrum on the largest scales may be
connected with backreaction associated with initial evolution has already been
noted (e.g. [154]), and may be of interest in the present context, but its
proper examination is beyond our present scope.

Here we will be mainly interested in the enhancement of the power spectrum on
large nonlinear scales, corresponding to $k\approx{\rm Mpc}^{-1}$, because of
the particularly interesting consequences for galaxy formation we discuss in
Section IV.3. Fig. 1 shows the resulting dimensionless matter power spectrum
for a jump corresponding to ratio of sound speeds (or in and out frequencies)
of $100$ on such scales. The ratio is associated with a Boguliubov coefficient
$|\beta|$ of about 5. The large value is chosen as to clearly delineate
phenomena associated with significant excitation on nonlinear structure
formation. This fixes our basic fiducial model. We will examine, in addition,
the effect of smaller enhancements and comoving spatial cutoff scales (Section
IV.3.3), as well as the effect of relaxing the sudden jump assumption
(Appendix B; Section IV.3.3).

Figure 2: RMS mass fluctuations corresponding to power spectra shown in Fig.
1. Note that the strong oscillations in the power spectrum, in the case of
power law inflation, have little effect here, as they are smoothed over and
integrated out as the dispersions are extracted from the power spectra.
Despite the sharp jump in linear power spectra, the change in the RMS mass
fluctuations beyond the cutoff scale is also gradual.

###  IV.2 Halo mass function

####  IV.2.1 Evaluation procedure

On nonlinear scales, modifications of the primordial power spectrum are
primarily encoded in the mass function of self gravitating dark matter
objects, the halos hosting galaxies. We evaluate this function using the
Press-Schecter formalism, which estimates the number of dark matter halos per
unit mass and comoving volume, given the linear matter power spectrum via a
spherical collapse model [155, 156]. This is given by

$\frac{dn}{dm}=\frac{\rho_{0}}{M^{2}}f(\sigma)\mid\frac{d\ln\sigma}{d\ln
M}\mid$ (52)

where $\rho_{0}$ is the mean matter density at $z=0$ and $f(\sigma)$ is given
by

$f(\sigma)=\sqrt{\frac{2}{\pi}}\nu\exp\left(\frac{-\nu^{2}}{2}\right),$ (53)

where $\nu=\delta_{c}/\sigma$, with $\delta_{c}=1.686$ the critical
overdensity for spherical collapse and $\sigma$ the RMS variance of mass
fluctuations within a sphere of radius $R$ and containing mass
$M=\vartheta_{f}\rho_{0}R^{3}$, where $\vartheta_{f}$ a constant that depends
on the filter function $W$. For Gaussian filter it is
$\vartheta_{f}=(2\pi)^{3/2}$. The filter function is characterized by its size
$R$ or mass $M$. In the case of Gaussian filter we use here, the relation
between them is

$M=4.37\times
10^{12}\Omega_{m}h^{-1}\left(\frac{R}{h^{-1}\mathrm{Mpc}}\right)^{3}M_{\odot}.$
(54)

As our primary aim is to illustrate generic consequences of enhanced small
scale power spectrum, we generally kept to the aforementioned simplest form of
the Press-Schecter formalism. However we have also verified the insensitivity
of our results to that choice by comparing with an ellipsoidal collapse
fitting function, which provides better fits to mass functions of halos
identified in cosmological simulations [157, 158, 159, 160],

$f(\sigma)=A\sqrt{\frac{2a_{s}}{\pi}}\left[1+\left(\frac{\nu^{2}}{a_{s}}\right)^{p_{s}}\right]\nu\exp\left(-\frac{a_{s}\nu^{2}}{2}\right),$
(55)

where we set $p_{s}=0.3$, $A=0.3222$ and $a_{s}=0.707$. Results using that
form are shown in Appendix B, where we examine implication of an enhanced
small scale spectrum when the assumption of sudden jump is relaxed, and also
in Fig. 8.

The mass variance is calculated through the integral,

$\sigma^{2}(R)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}k^{2}P(k)W^{2}(kR)dk,$ (56)

where $P(k)$ is the linear power spectrum and $W(kR)$ is the Fourier transform
of the Gaussian filter function

$W(kR)=\exp\left(-\frac{(kR)^{2}}{2}\right).$ (57)

Fig. 2 shows the thus calculated dispersion for de Sitter and power low
inflation models. As can be seen the strong oscillations in the power spectrum
of the latter case are smoothed and integrated over, and the results are quite
similar in the two cases. Also, despite the sharp jump in the corresponding
power spectra, the change in the RMS mass fluctuations in the nonlinear regime
beyond the cutoff scale is gradual.

Figure 3: Multiplicity function, describing the fraction of mass in dark halos
of mass $M$, for the power spectra shown in Fig. 1 and dispersions of Fig. 2.
As may already be expected from the latter figure, the results are similar in
de Sitter and power law inflation (labelled DS and PL respectively), due to
the smoothing and integration over the oscillations as the mass function is
derived.

####  IV.2.2 Mass function at redshift zero

Fig. 3 shows the resulting Press-Schecter halo multiplicity function, which
estimates the fraction of mass in halos of mass $M$, corresponding to the
unperturbed and perturbed (with jump) matter power spectra shown in Fig. 1. As
may be expected given the mass dispersions shown in Fig. 2, the results are
virtually identical in case of de Sitter and power law inflation, despite the
strong oscillations in the spectrum in the latter case. Perhaps also expected
is the enhancement at larger masses embodied in the bump encompassing a scale
around a few $10^{12}{M_{\odot}}$, when the power spectrum is boosted. More
conterintutively, there is a dearth of small halos when the power spectrum is
perturbed. This is due to those smaller halos more rapidly merging into larger
ones, as we discuss further below (next subsection).

The bump at higher masses is not in itself directly observable; as it can be
accounted for by changing the galaxy-halo occupation numbers. Indeed it
roughly corresponds to the highest mass to light ratio inferred when fitting
galaxies to halos in context of the standard model. Nevertheless, the
compatibility of such enhancement with data can be tested through a
combination of abundance matching and dynamical modelling. In fact, for
galaxies with stellar mass above $5\times 10^{10}M_{\odot}$, abundance
matching with standard power spectrum seems to overpredict the observed
stellar masses for a given dynamical mass [161]. As the galaxy number density
(determined by the stellar mass function), is a decreasing function of mass,
the discrepancy may in principle be accommodated in our current context as
follow: by increasing the abundance of halos with larger dynamical masses
(with the bump at higher masses), the galaxy population associated with such
halos will then be one with correspondingly larger number density, and hence
smaller masses. This is essentially the same effect that may help alleviate
the apparent problems with early massive galaxy formation as we discuss in
Section IV.3.2. Thus the solution to such potential problems does not only
appear consistent with the distribution of dynamical masses at low redshift
but may even resolve certain problems there; at both high and low mass scales.

####  IV.2.3 Enhancement at high $z$ and large $M$, and suppression at the
opposite ends

Figure 4:  Same as in Fig. 3 but at the indicated redshifts (using power law
inflation model). Solid lines show the results with standard (nearly) scale
invariant primordial spectrum, which are compared to those obtained when the
power spectrum is boosted at smaller scales, as a result of an imposed step
(corresponding to a ratio of hundred fold) in mode frequency and propagation
speed (cf. Fig. 1).

.

For the same parameters used above, Fig. 4 shows the multiplicity function at
selected redshifts. Four trends are clear. First, the enhancement in the
primordial power spectrum leads to enhancement in the number of halos at
intermediate masses, culminating approximately at the mass scale corresponding
to the length scale to where the jump in the power spectrum is placed. The
second is that the effect is larger, and is apparent for a larger range in
masses, at higher redshifts. Indeed, Fig. 5 shows a maximum enhancement of
almost four orders of magnitude at $z=8$, compared to only a factor of a few
at $z=0$. This is because high mass halos are rarer at higher redshifts and
thus the relative increase due to the enhancement resulting from the
discontinuity in the power spectrum is larger. It also results in the rate of
change in the mass function with redshift, at fixed mass, being smaller in the
case of enhanced spectrum than in the unperturbed case.

Finally, there is the somewhat counterintuitive effect, mentioned at the
conclusion of the previous subsection, of significant suppression of the
multiplicity function contribution of halos at smaller masses, below a few
$10^{11}{M_{\odot}}$. This interesting result may be understood by recalling
that enhancement in the power spectrum at comoving scales associated with
masses of $\gtrsim 10^{12}{\rm M_{\odot}}$, implies that all smaller scales
are also enhanced. And enhancement at smaller (length and mass) scales in turn
implies that smaller halos form at higher redshift and that by the redshifts
considered here they have already been typically subsumed in larger ones; that
is, the typical mass scale, for a given fluctuation level at a given redshift,
shifts up. This leads to a relative decrease in the number of halos with
masses $\lesssim 10^{9}{M_{\odot}}$. As opposed to the case of the enhancement
of the multiplicity of relatively high mass halos, the de-enhancement is here
relatively larger at smaller $z$, as the lower mass halos are now those that
are rarer at such redshifts.

###  IV.3 Interpretation and possible consequences

As, in the current model of structure formation, galaxies form in seeds
provided by the potential wells of dark matter halos, the significant
modifications to the halo mass function are expected to leave imprints on the
associated galaxy stellar mass function. Pertinent questions here thus include
whether those modifications have consequences for problems arising at small
scales within the current standard scenario of structure formation, as
outlined in the introduction; or, in contrast, whether such modifications can
constrained also on nonlinear scales.

####  IV.3.1 Small scale problems at low redshift and the dearth of dwarf
galaxies

One straightforward consequence of the suppression of halo multiplicity at
small scales pertains to the longstanding issue of the dearth of dwarf
galaxies in the standard scenario: a galaxy like the Milky Way is expected, in
the context of the $\Lambda$CDM with a standard primordial power spectrum, to
have hundreds of satellites that are not observed, and some of the predicted
hosting halos are too ’large’ to have ’failed’ to form galaxies. These are
aspects of the so-called small scale problems of the standard scenario has
given rise to various explanations, e.g in terms of baryonic physics, warm
dark matter, fuzzy dark matter, as well as direct suppression of the small
scale power spectrum.

Our somewhat counter-intuitive result, on the other hand, is that an
enhancement of power on small scales can also lead to a suppression in the
number of small halos (as these ‘overmerge’ into larger entities). This
suppression at $z=0$ at the scales where issues such as the dearth of small
galaxies ($M_{h}\lesssim 10^{9}M_{\odot}$) and too big to fail
($10^{9}\lesssim M_{h}\lesssim 10^{11}M_{\odot}$) problems appear, can
therefore be of relevance to apparent small scale crises arising in the
context the standard model of structure formation. The order of magnitude
suppression at smaller masses is directly relevant to resolving the apparent
discrepancy between the number of observed small satellite galaxies and large
number of small halos found in cold dark matter simulations. The suppression
on the larger mass scales, on the other hand, may help alleviate the too big
to fail issue; when this is posed as an abundance matching problem, whereby
the abundance of simulated halos is too large at the masses inferred from the
dynamics of observed galaxies in the range $10^{9}\lesssim
M_{h}/M\odot\lesssim 10^{11}$ (e.g., ref. [162]).

####  IV.3.2 The excess of early massive galaxies

The enhancement at higher mass scales may, on the other hand, have
consequences for the more recently raised issues associated with early galaxy
formation. These are extensions of longstanding phenomena related to what is
referred to as ’downsizing’ (e.g. [163]), required to account for
preponderance of early massive galaxies; a phenomenon that does not appear
entirely natural in a hierarchical structure formation scenario, where the
smaller halos embodying the potential wells hosting the galaxies form first.

The problem of early galaxy formation has been termed ‘impossibly early’ in
the context of the standard $\Lambda$CDM scenario of structure formation [93].
In that work, the authors attempt to infer the halo mass function at high-$z$,
primarily from stellar mass functions derived using photometric spectral
energy distribution templates and ultraviolet luminosity functions. The halo
mass is then inferred by assuming a stellar to halo mass of
$M_{*}/M_{h}=1/70$. If this local value of $M_{*}/M_{h}$ is used, then Fig. 1
of the aforementioned work suggests that the number density of massive
galaxies can greatly exceed that of the halos they should inhabit for
$z\gtrsim 4$ in the standard $\Lambda$CDM structure formation scenario. The
discrepancy becomes more severe as one moves up in redshift and mass, reaching
four orders of magnitude or more.

The above would seem to rule out the standard scenario of structure formation
in the context of $\Lambda$CDM cosmology. However, a couple of caveats have
been pointed out. First, regarding the assumption that $M_{*}/M_{h}$ does not
vary with redshift. For, as can also be seen from Fig. 1 of [93], instead of
moving the points inferred from the observed stellar number densities down
orders of magnitude to fit the corresponding halo number densities, one can
move the points horizontally to the left by an order of magnitude. This
fitting procedure in effect invokes a $z$ (and $M_{*}$) dependent
$M_{*}/M_{h}$, to replace the fiducial local value of $M_{*}/M_{h}=1/70$
assumed by the authors. The procedure, requiring $M_{*}/M_{h}\sim 1/7$, is
still in principle consistent with a universal baryon fraction of $1/6.3$,
associated with the standard cosmological scenario, but only just [98] 333Note
that in [98] the more conservative cumulative stellar mass function, which
averages over the increasing $\frac{M_{*}}{M_{h}}(M_{*})$, is employed. The
data for $M_{*}=10^{8}{M_{\odot}}$ to $10^{10}{M_{\odot}}$ are consistent with
$\Lambda$CDM. Tension still arises for the data point shown for $M_{*}\geq
10^{11.7}{M_{\odot}}$ at $z\approx 5.5$ (their Fig. 2 top-left). The results
in Fig. 18 of [184] show data consistent with $M_{*}/M_{h}=1/6.3$, or even
larger, already at $z\sim 5$, for $M_{*}\gtrsim 10^{11}{M_{\odot}}$. In
general, it seems that galaxies can be accommodated into halos with
$M_{*}/M_{h}\ll 1/6.3$ for $z\gtrsim 5$ if one keeps to $M_{*}\lesssim
10^{10.5}{M_{\odot}}$ (e.g., [185], and Fig. 9 of [186]). On the other hand,
the most extreme cases in the data presented by Steinhardt et. al. [93] (where
$M_{*}/M_{h}\approx 1/7$ seems required), are just barely consistent with
standard $\Lambda$CDM..

Another caveat that has been pointed out concerns the extraction of $M_{*}$
and associated number densities from the ultraviolet luminosity function at
high $z$, which some of the data points of [93] relies on [165]. However, a
multi-wavelength analysis of a sample of massive galaxies at $z>3$ also leads
to a cumulative mass function that can be consistent (within estimated errors)
with $M_{*}/M_{h}$ approaching the universal baryon fraction at $z\sim 5.5$
and $M_{*}>10^{11}{\rm M_{\odot}}$ [166]. That work also shows (Fig. 14) that
the number densities of massive galaxies are very difficult to reproduce in
hydrodynamic numerical simulations — with significant underestimate for $z>3$
— which may be expected, as their reproduction would seem to require that all
available baryons reside inside galaxies, and their near total conversion to
stars over a short time ($\sim{\rm Gyr}$). This would have as consequence the
presence of a significantly ‘quenched’, quiescent population of massive
galaxies already at high redshift. The presence of such a population, which is
indeed observed, poses significant challenges. Synthesizing the stellar
populations of one such object, observed at $z=3.717$, for example, seems to
again require prior evolution involving a $M_{*}/M_{h}$ reaching the universal
baryon fraction [167] (see also [168]). There now appears to be a substantial
population of such galaxies, observed at increasing redshift [169, 170, 171,
172, 173, 174, 175], and not easily reproduced by either hydrodynamic
simulations [172, 173] or semi analytical models [170, 174].

Although questions as to the ultimate severity of these problems will only be
settled with the next generation surveys (e.g. with the JWST), the situation
warrants pointing out that they can in principle be alleviated by invoking
small scale enhancement of the primordial power spectrum examined here.

Figure 5:  Ratios of mass functions with modified power spectrum to those
resulting from unmodified power spectrum, at the same redshift. The results
correspond to the ratios of dashed to solid lines in Fig. 4 and Fig. 3.

Figure 6: Ratios of the mass functions at different redshifts, for the
standard case (left) and that with modified power spectrum (right). Note the
slower evolution (reflected in the smaller ratios) at higher $z$ for most of
the mass range in the modified case.

Fig. 5 shows that significant enhancements can be achieved at mass scales
$10^{12}{M_{\odot}}\lesssim M\lesssim 10^{13}{M_{\odot}}$, with a peak at a
scale corresponding to highest dark matter to stellar mass ratio in standard
modelling, at which the enhancements reach even the ‘impossibly’ large levels
claimed in [93]. Perhaps no less important is the slower evolution of the mass
function for $z\gtrsim 4$, observed in Fig. 6, which is more consistent with
the redshift evolution of the inferred stellar mass densities in [93] than the
much faster evolution in the standard case (the slow evolution of the stellar
mass function for $4\lesssim z\lesssim 7$ was also observed for example by
Song et. al. [176]). This would seem to waive the apparent requirement of a
$M_{*}/M_{h}$ that is high dependent on redshift in order to fit the data.

With better statistics, and firmer grip on observational systematics, it
should be possible to distinguish between scenarios involving enhancement in
the primordial power spectrum, such as the one presented here and
reconciliation with data through improvement of the baryonic model; by
invoking further ‘downsizing’ physics input, in terms of feedback, quenching
and other ‘subgrid’ physics (assuming the data remain consistent with the
strict upper bounds placed in the context of $\Lambda$CDM [98]). As the
baryonic models become better constrained, there may be particular
consequences that could also constrain (or confirm) the sort of scenario
discussed here. We now discuss some of these.

####  IV.3.3 Other observables, constraints and variation on basic model

Figure 7: Same as in Fig. 5 but with step frequency ratios of 10 instead of
100, leading about to about an order of magnitude less modification in the
power spectrum, with Bogulibov coefficient $|\beta|=1.42$ instead of $4.95$
(left); and for power spectrum comoving modification scale $k=3h/{\rm Mpc}$,
instead of $k=1h/{\rm Mpc}$ (right).

In the context of the enhanced spectrum scenario presented here, the
clustering of halos, on mass scales and redshifts where numbers are predicted
to be significantly enhanced, may be measurably different from the standard
case. This is because the biasing with respect to the matter distribution
would be expected to be different (since they would correspondent to less rare
density peaks). Combined clustering and abundance matching analysis in the
context of a ‘halo model’ (e.g., [177]), particularly at higher redshifts
[178], could thus in principle test, and place constraints on, scenarios
involving primordial power spectrum enhancements. The galaxy-matter
correlation function, entering into calculations of galaxy-galaxy weak lensing
signals, should also be different in the present scenario from the standard
case. The difference should again be especially significant at higher
redshifts, where the abundance of high mass halos is strongly increased,
making for a relatively clumpy matter distribution. Tests are also possible at
low redshift. particularly as regards to the peak in abundance of Milky Way
sized halos, which seems consistent with observations (Section IV.2.2). Some
observations on the other hand suggest that the standard model itself may
overpredict the halo mass function at scales $10^{13}M_{\odot}\lesssim
M\lesssim 10^{14}M_{\odot}$ [179]; further enhancements of the mass function
at such scales may be thus constrained.

Another observable that can potentially place immediate constraints on the
scenario discussed here is the Lyman-$\alpha$ forest. Here, detailed
comparison with data involves complex simulations that depend on assumptions
regarding the state of the intergalactic medium, which become less robust at
nonlinear scales [180, 134]. In the nonlinear regime the modifications in the
power spectrum are primarily imprinted in the RMS dispersion and the halo mass
function, where the complex pattern of enhancement and suppression at
different scales and redshifts would contribute to the mass fluctuations
probed by one dimensional Lyman-$\alpha$ spectra. The large enhancement at
higher masses and redshifts may also affect the thermal history of the
intergalactic medium. It may therefore be worth investigating if and how such
changes affect the standard constraints regarding the power spectrum. Pending
such investigation, as the modifications to the linear power spectrum and mass
dispersion considered above are large and fall within the region relevant to
Lyman-$\alpha$ observations, it apt to probe what is to be expected if more
modest modifications are made.

In Fig. 7 (left panel) we show the relative change in the mass function for
(about an order of magnitude) smaller perturbation in the power spectrum, as
well as on scales deeper in the nonlinear regime. As can be seen, in the
former case, significant enhancement in the mass function can still occur at
the right scale at higher redshift (where halos are exceedingly rare in the
standard scenario), so as to alleviate the apparent early galaxy formation
problem. The reduction in number of small halos, relevant to the dearth of
small galaxies and too big to fail problems at low $z$, is smaller however.

When the modification in the power spectrum is placed on a smaller spatial
scales, deeper in the nonlinear regime (Fig. 7, right panel), the decrease in
number of small halos at $z=0$ is again significant and relevant to the dearth
of dwarf galaxies, but does not cover all the mass range relevant to the too
big to fail problem. As may be expected, the enhancement at high redshifts
happens at a smaller mass scale (they are also smaller because halos in the
standard scenario are already more abundant at such scales). Enhancement at
such scales is not directly applicable to the problem of high $M_{*}/M_{h}$ at
$M_{*}\gtrsim 10^{10.5}M_{\odot}$ and $z\gtrsim 4$, as discussed in previous
subsection. It may nevertheless be relevant at higher redshift, as the
progenitors of massive quiescent galaxies were assembled (especially if the
star formation rate density does not steeply decline beyond $z=8$, as
suggested by some authors; c.f. ref. [168], particularly the discussion
Section 7.3 and references therein). As one further increases the comoving
wavenumber associated with the high energy cutoff $k_{c}$, this general trend
persists. The suppression on small scales at $z=0$ are found to correspond to
masses of small halos that are overabundant in CDM up to comoving cutoff
$k\sim 9~{}{\rm Mpc^{-1}}$, which essentially avoids Lyman-$\alpha$ bounds. At
larger modification wavenumbers, however, one finds enhancement rather than
suppression at mass scales $\lesssim 10^{9}M_{\odot}$, relevant to the dearth
of dwarf galaxies issue. Although enhancing the power spectrum at such smaller
spatial scales would not appear to alleviate any of issues regarding galaxy
formation discussed here, it could still have consequences for early black
hole formation and the epoch of ionization (see also [92]).

Figure 8:  Same as in Figures 5 and 7 but with gradually modified power
spectrum, rather than sudden jump. The spectrum is modified using Eqs. (63)
and (64) with $S=200$, $b=2$ and $k_{c}=3h~{}{\rm Mpc}^{-1}$ comoving. The
results correspond to ratios of dashed and solid lines in Fig. 13 (taking the
dashed line at $b=2$ for the left hand panel). The associated modifications to
the power spectra are those shown in Fig. 12

On the other hand, relaxing the assumption of a sudden jump transition leads
to suppression and enhancement of halo abundances on larger mass scales, This
allows for retaining the advantages of a sudden cutoff at comoving wavenumber
$1h/{\rm Mpc}$, while keeping the mass function at $10^{13}M_{\odot}$ at $z=0$
unchanged, and modifying the power spectrum much more modestly at $1h/{\rm
Mpc}$ (cf. Appendix B, Fig. 12). Fig. 8 shows the relative change in the mass
function for such a gradual transition in the power spectrum around a
characteristic wavenumber of $3h/{\rm Mpc}$ comoving. As can be seen,
significant suppression at $z=0$ is again recovered at mass scales of order
$10^{11}M_{\odot}$, relevant to both the dearth of dwarf galaxies and the too
big to fail problems. Enhancement at higher $z$ also occurs at scales
significantly larger than the corresponding sudden jump case (shown on right
panel of Fig. 7), which renders the enhancement more directly relevant to
early massive galaxy formation issues. The mass function is unmodified at
scales $10^{13}M_{\odot}$ at $z=0$.

Thus, potential resolution of all or some of the galactic and sub-galactic
scale issues through modification of the power spectrum, rather than (or in
addition to) baryonic physics input, may in principle be tested and
constrained through distinctive predictions. This is true in general and is
not confined to our particular simple model of a sudden jump; such tests will
thus become more relevant if the small scale issues connected to the standard
structure formation scenario are confirmed to persist with incoming
observations. In the context of the present scenario, such observations can
potentially probe imprints (or lack thereof) of high energy cutoff physics on
the relevant astrophysical scales, and place constraints on the duration of
inflation, as the ratio of the Hubble scale of inflation to the high energy
cutoff scale and the number of inflationary e-folds fix the scale at which the
matter power spectrum and halo mass function is modified (cf. Section IV.1.2).
For the minimal number of e-folds required to solve the horizon problem for
example, $H/k_{c}\approx 10^{-4}$ is required to address the galactic scale
issues discussed. Significant power spectrum modification also require
$k_{c}\lesssim H/k_{c}$ (Section III.4). These tests can be stringent; as,
given this scale, and the level of excitation determined by the Boguliubov
$\beta_{k}$, the predictions of the simplest scenario of sudden jump through a
high energy transition scale, are unique in terms of the expected effect on
the power spectrum.

##  V Conclusion

Slow roll inflation predicts a nearly scale invariant spectrum of primordial
fluctuations, which is borne out by precise observations of the cosmic
microwave background and large scale structure in the universe. Nevertheless,
that prediction is not unique, a variety of effects invoking discontinuous or
phased evolution between slow rolls, for example, can lead to anomalous
’features’ in the spectrum. Excited states arising from modes crossing a high
energy cutoff scale can also lead to significant modifications to the scale
free spectrum. Although these are essentially ruled out at the scales where
the aforementioned observations are effective, the primordial spectrum is
relatively unconstrained on smaller, currently nonlinear scales, where the
matter distribution has collapsed into bound self gravitating objects, washing
out the primordial signature by largely encoding it in the halo mass function.

On the contrary, at such scales — which span many more octaves of observable
structure than the three that are probed in the linear regime — a variety of
issues arise in the context of the standard model of structure formation; such
as the ’small scale problems’ at low redshift and the apparent problems
involving early galaxy and supermassive black hole formation at higher $z$,
which can be seen as extension of longstanding phenomena requiring
’downsizing’ in galaxy formation. As these issues arise precisely at the
scales where complex baryonic physics comes to play a central role in the
standard scenario of structure formation, it was natural that extensive
investigation of solutions in these terms have been pursued. However, as these
are also the scales where the primordial spectrum of fluctuations is
relatively weakly constrained, this aspect, with its effects and consequences,
may also warrant further investigation.

Here we considered the effects of excited states arising from the transiting
of fluctuation modes through a high energy cutoff scale. As the power spectrum
of primordial fluctuations is effectively an adiabatic invariant of inflaton
dynamics (Section II.2.2), adiabatic evolution necessarily leaves the nearly
scale free spectrum intact. We next considered a simple model of the opposite
extreme; of a sudden jump across the transition. The initial conditions for
the fluctuations before the jump are well defined, taking the Bunch Davies
form, but with propagation speed $c_{s}\neq 1$. An intuitive, simple
’analogue’ model approximated by such a transition corresponds to the case of
a gas or lattice where sound waves do not propagate at all below the
interparticle distance, then propagate at an anomalous speed in an effective
macroscopic approximation, before finally propagating with the standard sound
speed and dispersion relation as the wavelength become progressively larger
than the interparticle distance.

In this context, the primordial spectrum is invariably enhanced rather than
suppressed (whether the initial $c_{s}>1$ or is $<1$), for all scales
undergoing the transition through the high energy cutoff (Section III). This
is accompanied by strong, tightly spaced oscillations in the power spectrum of
generic (as opposed to pure de Sitter) models of inflation, where modes exit
the horizon at different phases. Numerical calculations suggest that
sufficiently non-adiabatic evolution, leading to significant modification of
the power spectrum implies an effectively sudden transition for all $c_{s}>1$
and for $0.01\lesssim c_{s}<1$ (Appendix A.2). The simple model of sudden
jump, and its predictions, are in this range thus generic. We also considered
the possibility of a more gradual transition when the aforementioned
conditions are not satisfied (Appendix B).

Given the excitation level induced in the inflaton field, and the current
comoving scale corresponding to the jump across the high energy cutoff scale
during inflation, the predictions of the simple sudden jump models are
essentially unique (in terms of its effect on the matter power spectrum, mass
variance and the dark matter halo mass function). The level of excitation can
be quantified through a Bogoliubov coefficient $\beta_{k}\neq 0$ for scales
that undergo the jump, and is easily evaluated in terms of the in and out
frequency ratio (or equivalently $c_{s}$ ratio; Section III.3). If assumed to
be within a few orders of magnitude of the Planck scale, the jump scale
corresponds to currently nonlinear scales if inflation proceeds for
approximately the number of e-folds necessary to solve the horizon problem. In
general, the comoving jump scale corresponds to currently nonlinear scales for
minimal inflation if $H/k_{c}\sim 10^{-4}$, with smaller ratios allowing for
larger e-folds (Section IV.1.2). In this context, the nonlinear scales can be
modified, while leaving the standard spectrum intact on linear ones.

Backreaction bounds on $|\beta|$ must be imposed, as ’over-excitation’ of the
inflaton would result in radiation domination rather than inflation; these may
however still allow for major enhancements of the power spectrum $\sim
1+2\beta^{2}$ (and oscillations in the generic inflation case). As we discuss
in Section III.4, this would be generally the case if
$k_{c}\lesssim\left(H/k_{c}\right)M_{\rm Pl}$. Such enhancements can have
observable consequences, confirming or constraining the effect of excitations
on structure formation on nonlinear scales. In order to impose modifications
on such scales in particular, and still keep the excitations from overwhelming
the inflaton vacuum state, one thus requires $H/k_{c}\lesssim 10^{-4}$ and
$k_{c}\lesssim\left(H/k_{c}\right)M_{\rm Pl}$. This implies $k_{c}\lesssim
10^{-4}M_{\rm Pl}$ and $H\lesssim 10^{-8}M_{\rm Pl}$. A rough estimate of the
derivative of the backreaction suggests possible modification of the power
spectrum on the largest scales, and may place tight constraints on the
comoving scale at which enhancement of the small scale power spectrum can
occur (to about a comoving Mpc; Section III.4). That modification on the
largest scales can accompany the changes on small, nonlinear ones, is an
interesting possibility that may be worth studying in detail.

To probe for possible characteristic signatures of the modifications on
nonlinear scales, we evaluate (in Section IV) the dark halo multiplicity
function, quantifying the fraction of mass in halos of mass $M$. In our
fiducial example, the peak, resulting from power spectrum enhancement, is
chosen to correspond to a few times $10^{12}{M_{\odot}}$. This is the mass
scale where the highest mass to light ratio is inferred when associating
galaxies with halos in the context of halo models derived within the standard
scenario. It is also the scale where issues related to the apparent
preponderance of early massive galaxies, particularly quiescent ones, appear
(Section IV.3.2). For relatively small enhancements at small redshifts $z$,
the enhancement at larger $z\sim 8$ is dramatic, as such massive halos are
very rare at these redshifts in the standard scenario. The change in the
number densities of massive galaxy-hosting halos with redshift is also much
smaller than in the standard case. Combined, these effects may alleviate the
apparent ’impossibly’ early galaxy formation problem, even in the most extreme
form claimed.

Perhaps more surprisingly, an enhancement of the spectrum at these
intermediate nonlinear scales leads to suppression of small halos at low $z$,
thus potentially alleviating longstanding issues related to the dearth of
small galaxies, including those ’too big to fail’, in the standard structure
formation scenario. This is due to the enhanced spectrum leading to
overmerging of small mass objects at high $z$, so as to lead to a suppression
of such objects at low $z$.

The halo mass function, in itself, cannot place strong constraints on
enhancements of the primordial power spectrum on currently nonlinear scales,
as one can vary the galaxy halo occupation number to match the data (in the
standard scenario, the early galaxy formation issues at high $M$ and $z$
arises because this seems to sometimes require very large stellar mass
fraction, which the enhanced halo mass function here may resolve; Section
IV.3.2 ). However, combined abundance matching and dynamical analysis at
low-$z$ can. Halo abundance enhancement at the scales considered here appears
consistent with such analyses; it may in fact alleviate the apparent
overprediction of stellar masses for given dynamical mass for galaxies with
stellar mass $\gtrsim 5\times 10^{10}M_{\odot}$ ([161];Section IV.2.2).

Major modifications in the spectrum of primordial fluctuation are eventually
encoded in more minor modifications to the nonlinear matter power spectrum, as
these enter primarily through the modified halo mass function rather than the
statistics of the spatial distribution. Nevertheless, the scenario of an
enhanced primordial power spectrum at scales corresponding to currently
nonlinear ones, can also be tested through its signature on halo biasing. The
fact that more massive halos would be less rare may be expected to
particularly impact such observables as galaxy-mass correlations and leasing
signals (especially at higher redshift where the effects of enhancement at
higher mass scales are more prominent). To address both aforementioned issues
— of massive high-$z$ massive galaxies and small local ones — simultaneously
in the most severe forms claimed, through sudden transitions, also entails
significant modifications at scales probed by Lyman-$\alpha$ observations (the
required modifications are more modest, or at scales that may be less
constrained, if only partial resolution of both issues is sought or if the
assumption of sudden transition is relaxed; Section IV.3.3).

Thus, observations, coupled with modelling and simulations with modified
spectrum, may place constraints on scenarios invoking enhanced power on
currently nonlinear scales, distinguishing them from baryonic solutions to the
same problems. In the context of the analytical sudden step model primarily
considered here, this includes constraints on $H/k_{c}$, $k_{c}$, $|\beta|$,
and the number of inflationary e-folds, as discussed above. Given the field
excitation levels (i.e.$|\beta|$) and the comoving scale of the high energy
transition, the consequences for the matter spectrum and halo mass function
are essentially unique. Variants that could also be tested include those
involving phased or discontinuous stages of inflation with relatively
localized peaks in the primordial spectrum. This will become perhaps more
pressing if next generation surveys (e.g. employing the JWST) confirm problems
related to early galaxy formation. On smaller (sub-galactic) scales still,
primordial power spectrum enhancement may be relevant to early supermassive
black hole formation, and the formation of the first dark matter objects, and
may be tested through such effects as CMB spectral distortions.

###### Acknowledgements.

We wish to thank Adel Awad, Alexey Golovnev, Sergio E. Joras, Giovanni
Marozzi, Jerome Martin and Joe Silk for useful discussions, and the referee
for constructive comments and suggestions. This project was supported
financially by the Science and Technology Development Fund (STDF), Egypt.
Grant No. 25859.

##  Appendix A Comparison of simplified model with numerical solution, and the
effect of relaxing the sudden jump condition

In this appendix we test the approximation of the simplified model of Section
II.3, introduced to evaluate the effect of non-adiabatic transition at a high
energy cutoff scale $k_{c}$ on the primordial power spectrum. There are two
approximations that were invoked; the sudden step and the neglect the term
proportional to $-\frac{1}{\eta^{2}}$ in the Mukhanov-Sasaki equations (2) and
(7). We start by examining the latter, then we discuss the former.

###  A.1 Model versus numerical solution of Mukhanov-Sasaki equation with step

Figure 9: Comparison of primordial power spectrum obtained from simplified
analytical model with full numerical calculation, for sudden jumps
corresponding to ratios of the in and out sound speed, or equivalently
frequencies, of $10$ (left) and $100$ (right). This is done for a de Sitter
background with $H/k_{c}=10^{-4}$ and $k$ is shown in units of $k_{c}$.

Figure 10: Same as in Fig. 9, but for power law inflation model (discussed in
Section II.3.1).

We evolve the dynamics of fluctuation modes numerically, using the MS equation
(2) for de Sitter and power law inflationary backgrounds, while replacing the
term $k^{2}$ with

$k^{2}\rightarrow k_{\mathrm{eff}}^{2}(k,\eta)\equiv
a^{2}(\eta)\omega_{\mathrm{phys}}^{2}\left[\frac{k}{a(\eta)}\right].$ (58)

where

$\omega_{\mathrm{phys}}^{2}\left[\frac{k}{a(\eta)}\right]=\left(\frac{k}{a(\eta)}+\frac{\delta
k}{a(\eta)}\mathcal{H}\left[\frac{k}{k_{c}a(\eta)}-1\right]\right)^{2},$ (59)

and $\mathcal{H}$ is the Heaviside step function. The parameter $\delta$
quantifies the size of the step. such that the sound speed past the step given
by $c_{s}=1+\delta$; it can be positive or negative, corresponding to an
upward and downward jump in sound speed respectively. As discussed in Section
III, in the context of the sudden step scenario they are equivalent in terms
of the effect on the power spectrum.

The primordial spectra are evaluated, for both the analytical model and
numerical calculations, as described in Section II.3.1. The results are shown
and compared in figures 9 and 10, for the case of de Sitter and power law
inflation respectively. As noted in Section II.3.1, in the de Sitter case all
modes exit the horizon at exactly the same phase. And any initial shift in
phase, due to change in effective frequency related to the second term in
bracket of the MS equation, leads to corresponding constant difference in the
final power spectrum. This leads to a difference between the numerical and
analytical results, where the aforementioned term is neglected. Nevertheless,
the relative error in the ratio of the perturbed to unperturbed power spectrum
is still of order $25\%$ when the step frequency ratio is $10$. It is an order
of magnitude lower still when the change of frequency is hundred folds.

In the case of power law inflation the Hubble scale $H$ is not exactly
constant. The modes exit the horizon at different phases, and this leads to
the oscillations, which accompany the enhancement in Fig. 10. The
corresponding error is then primarily in phases, with the maxima and minima of
the oscillations practically equal in the simplified model and the numerical
calculations. The change in phase is generally unimportant for calculating
quantities with observable consequences; such as the mass fluctuations at a
given spatial or mass scale, and halo mass multiplicity function. For these
depend on integrals of the matter power spectrum (as discussed in Section IV).
The simple analytical model — with its simple interpretation in terms of well
defined in and out states; Section III — thus turns out to be a good
approximation.

Figure 11: The primordial power spectrum evaluated at different levels of
violations of the adiabaticty condition (60), when transition across the high
energy cutoff scale is interpolated using a logistic function (Eq. 61), and
numerically integrated. The numbers in the legend keys refer to the order of
magnitude above the critical value of $\gamma$ required to violate the
adiabaticity condition (Eq. 62). Left panel: interpolation between high energy
sound speed $c_{s}=100$ and standard regime ($c_{s}=1$). Right panel:
interpolation between high energy sound speed $c_{s}=0.01$ and standard
regime. The results are shown for de Sitter inflation, and wavenumbers on the
horizontal axis are expressed in terms of the high energy cutoff scale
$k_{c}$, with $k_{c}/H=300$.

###  A.2 Non-adiabaticity versus sudden jump condition

Oscillations can, in general, be considered adiabatic if $\omega(\eta)$
changes only slightly over a characteristic time $\Delta\eta$ of order of one
oscillation period. If the frequency $\omega$ changes from a value
$\omega_{1}$ to another value $\omega_{2}$, on a characteristic timescale
$\Delta\eta$, the change may thus be adiabatic if

$|\omega_{1}-\omega_{2}|<\omega^{2}\Delta\eta.$ (60)

In the context of our sudden step model, $\omega_{1}$ and $\omega_{2}$ will
correspond to $\omega_{\rm in}$ and $\omega_{\rm out}$, respectively. We take
the ’typical’ $\omega$ on the right hand side to correspond to the minimal
frequency; supposing that the dynamics may be affected non-adibaticaly if the
change in frequency is larger than this. To examine to what extent that model
may describe a more general situation, where change may be more gradual, we
use the logistic function to parametrize the transition:

$k_{\rm eff}\equiv\omega(\eta)=k+\frac{\delta
k}{1+\exp\left[-\gamma\left(\frac{\eta}{\eta_{c}}-1\right)\right]}.$ (61)

Here the parameter $\gamma$ describes the stiffness of the transition, this
being steep and steplike for $\gamma\gg 1$, and $\delta$ (which may be
positive or negative) the scale of the step in the transition. Thus, in the
high energy regime limit, the sound speed $c_{s}=1+\delta$, while $c_{s}=1$
when the transition to the standard low-energy physics regime is complete. In
these terms, the characteristic time over which $\omega$ changes between its
initial and final value is $\frac{\eta_{c}}{\gamma}$. The adiabaticity
violation condition can then be written as

$\gamma>\frac{{\rm Min}(c_{s}^{2})}{|c_{s}-1|}\left(\frac{k_{c}}{H}\right),$
(62)

where we have used $\eta_{c}=\frac{-k_{c}}{Hk}$, and $c_{s}\neq 1$ corresponds
to the high energy limit sound speed. Since, as we have seen in Section
II.2.2, the power spectrum is essentially an adiabatic invariant of the
dynamics of inflationary perturbations, it is necessary to satisfy this
condition in order to modify the standard power spectrum.

Two cases are of particular interest in the context of the present study:
$c_{s}\gg 1$, so that ${\rm Min}(c_{s})=1$ and $c_{c}\ll 1$, when ${\rm
Min}(c_{s})=c_{s}$. For sound speeds considered here, the adiaticity condition
(62) is violated at smallest possible when the sound speed is minimal, that is
$c_{s}=0.01$. Still, even in this case, $\gamma$ is of order one or larger if
$k_{c}/H\geq 10^{4}$, as required to keep the significant changes in power
spectrum in the nonlinear regime of structure formation (Section IV.1.2). For
our parameters, the transition is thus necessarily stiff.

Figure 12: Primordial (left) and matter (right) power spectra, modified at
small scales using Eqs. (63) and (64), showing progressively steeper
transitions with increasing $b$. The characteristic high energy transition
scale is taken as $k_{c}=3h{\rm Mpc}^{-1}$ comoving, and $S=200$.

We now show that the transition is stiff even form much smaller $k_{c}/H=300$,
which is a minimal value, in the sense that with smaller values the effect of
the second term in brackets of Eq. (7) becomes important at $\eta\sim\eta_{c}$
for $c_{s}=0.01$ (see discussion relating to inequality 11). For it turns out
that the adiabaticty condition needs to be quite strongly violated for
sufficient change in the dynamics significantly affects the power spectrum
(that significant changes occurs well beyond the adiabaticity breaking
condition is common in dynamical systems [181]). This can be seen from Fig.
11, where we show that large changes only occur when $\gamma$ is orders of
magnitudes above the value estimated from (62). This is the case for both the
$c_{s}=100$ and $c_{s}=0.01$, with the former being stiffer still as expected
from (62) 444Note also, in the latter case of $c_{s}=0.01$, the small bump on
the right, visible in the low $\gamma$ spectra beyond the cutoff scale. These
are due to non-adibaticity at the crossing of the high energy transition at
$\eta_{c}$. They are much less prominent for smaller $k_{c}/H\geq 10^{4}$
values, used in the rest of this work. They nevertheless represent another
potentially interesting effect associated with non-adiabatic evolution. The
transition is stiffer still for smaller values of $c_{s}>1$ and larger
$c_{s}<1$. Thus, for sound speed ratios considered here, our simple model of a
sudden jump, and accompanying signature of a sudden break in the power
spectrum, appears much more generic than may seem a priory.

##  Appendix B Modified mass function from non-sudden spectrum enhancements

Figure 13: Left panel: the halo multiplicity functions at $z=0$ for the
spectra shown in Fig. 12. Right panel: the multiplicity functions at shown
redshifts for the case $b=2$.

As we have seen in the previous appendix, the sudden jump transition in the
power spectrum at smaller scales is a good approximation for the parameters
primarily considered in this paper, namely for initial $c_{s}\geq 0.01$; for
such values, the sufficient violation of the adiabaticity condition, required
for significant modification of the power spectrum, practically implies a
sudden transition. Nevertheless, as is already apparent in Fig. 11, for
$c_{s}=0,01$ (right hand panel), the sudden jump approximation becomes less
accurate as a predictor of significant change at smaller $c_{s}$. If one
envisages a transition starting at significantly smaller sound speed still, it
may then take place more gradually while still imparting a palpable effect on
the power spectrum.

We consider potentially observable consequences of this effect by examining a
series of progressively steeper transitions. We do this by modifying the
primordial power spectrum in a parametric manner, such that

$\Delta_{\mathcal{R}}^{2}(k)=\Delta_{\mathcal{R}{\rm
St}}^{2}(k)\left[(S-1)~{}G\left(\frac{k}{k_{c}}\right)+1\right],$ (63)

where $\Delta_{\mathcal{R}{\rm St}}^{2}(k)$ is the standard, nearly scale
invariant, power spectrum of scalar perturbations. The transition function $G$
tends to unity as $k\gg k_{c}$ and vanishes as $k/k_{c}\rightarrow 0$, and
$S\geq 1$ is the enhancement factor (it determines the ratio of the asymptotic
values of the power spectrum at small and large scales). We have tried several
forms for $G$, and the resulting trends were verified to be generic. Here we
show results for the following form (also used in [183] for the purpose of
suppression of the spectrum rather than enhancement):

$G(x)=\frac{1}{2}\left[\tanh~{}(b~{}\log
x)+1\right]=\frac{1}{2}\left[\frac{x^{2b}-1}{x^{2b}+1}+1\right],$ (64)

where $b>0$ determines the steepness of the transition around $x=k/k_{c}$. In
the following we will take $k_{c}$ to correspond to a comoving scale of
$3h~{}{\rm Mpc}^{-1}$. As equations (34 )and (39) show $S$ to be about $50$
for a sudden step scenario with initial $c_{s}=0.01$, the requirement that the
initial $c_{s}<0.01$ implies $S>50$ (otherwise, in line with the
aforementioned considerations, a gradual transition may not have a significant
effect on the power spectrum). In what follows we take $S=200$.

Fig. 12 shows the primordial power spectrum, as well as the dimensionless
matter power spectrum calculated using publicly available CLASS code (class-
code.net), for several values of $b$. The corresponding multiplicity
functions, calculated using (52) and (55), are shown in Fig. 13, where the
left hand panel displays results at $z=0$. For large $b$, those results are
similar to the sudden jump case. For $b=1$ the effect is smeared out, with
increase at halo masses $\gtrsim 10^{14}M_{\odot}$, which would increase
tension with cluster counts, which is already present in thee standard model.
The suppression at small mass scales is enhanced by the gradual transition.

Of particular interest is the intermediate, $b=2$, case. The enhancement takes
place at larger masses than the corresponding case with sudden jump (at
$3h{\rm Mpc}^{-1}$). The suppression at $z=0$ takes place at larger masses as
well. This allows for largely retaining the advantages of the sharp cutoff at
$1h{\rm Mpc}^{-1}$ — in terms of simultaneously alleviating both the dearth of
dearth of dwarf galaxy and too big to fail problems at $z=0$, as well as
accounting for early galaxy formation at higher redshifts — while avoiding any
enhancement at scales of order $\gtrsim 10^{13}M_{\odot}$ at $z=0$, and
relatively mildly perturbing the matter power spectrum at $1h{\rm Mpc}^{-1}$
(Fig. 12; see also Section IV.3.3). The enhancement of the mass function at
$z=0$ may also be relevant for explaining the overestimation of abundance
matching within the standard model of stellar masses of massive galaxies
([161]; Section IV.2.2).

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