# Constraining primordial curvature perturbations using dark matter
substructure

Shin’ichiro Ando  GRAPPA Institute, Institute of Physics, University of
Amsterdam, 1098 XH Amsterdam, The Netherlands  Kavli Institute for the Physics
and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583,
Japan  Nagisa Hiroshima  Department of Physics, University of Toyama, 3190
Gofuku, Toyama 930-8555, Japan  RIKEN iTHEMS, Wako, Saitama 351-0198, Japan
Koji Ishiwata  Institute of Theoretical Physics, Kanazawa University, Kakuma,
Kanazawa, 920-1192, Japan

###### Abstract

We investigate the primordial curvature perturbation by the observation of
dark matter substructure. Assuming a bump in the spectrum of the curvature
perturbation in the wavenumber of $k>1~{}{\rm Mpc}^{-1}$, we track the
evolution of the host halo and subhalos in a semi-analytic way. Taking into
account possible uncertainties in the evaluation of the tidal stripping effect
on the subhalo growth, we find a new robust bound on the curvature
perturbation with a bump from the number of observed dwarf spheroidal galaxies
in our Galaxy and the observations of the stellar stream. The upper limit on
the amplitude of the bump is $\order{10^{-7}}$ for $k\sim 10^{3}~{}{\rm
Mpc}^{-1}$. Furthermore we find the boost factor, which is crucial for the
indirect detection of dark matter signals, is up to $\order{10^{4}}$ due to
the bump that is allowed in the current observational bounds.

††preprint: RIKEN-iTHEMS-Report-22††preprint: UT-HET-138††preprint:
KANAZAWA-22-03

Introduction: The observation of the cosmic microwave background (CMB)
radiation strongly supports inflation at the early stage of the universe. The
CMB observation constrains the amplitude $A_{s}$ and the spectral index
$n_{s}$ of the scalar perturbation as $A_{s}=(2.099\pm 0.029)\times 10^{-9}$
and $n_{s}=0.9649\pm 0.0042$ at the pivot scale $k_{*}=0.05~{}{\rm Mpc}^{-1}$
[1]. At a smaller scale, on the other hand, the constraint on the curvature
perturbation is relaxed. For instance, the amplitude for the wavenumber of
$k>\order{1}~{}{\rm Mpc}^{-1}$ is constrained by $\mu$\- and $y$-type
distortion in the CMB observation [2, 3], the overproduction of the primordial
black holes (PBHs) [4], density profile of ultracompact minihalos [5, 6], and
the free-free emission in the Planck foreground analysis [7]. Despite the
constraints, the scalar amplitude in the small scale can be much larger than
$\order{10^{-9}}$. In this letter, we point out that the curvature
perturbation in such a small scale gives impact on the evolution of the
hierarchical structures of galaxies, which is traced by dark matter halos of
the Universe.

Dark matter plays a crucial role in the structure formation; the quantum
fluctuation produced by inflation seeds the density fluctuation, which grows
in the gravitational potential of dark matter. Therefore the imprint of the
small scale perturbation during the inflation is expected to remain in the
current structure of dark matter halos. Subhalos, which reside in larger-scale
halos, are especially promising objects to reveal the nature of dark matter;
dwarf spheroidal galaxies (dSphs) can form inside subhalos and they have been
found and observed intensively these days in the prospects to detect dark
matter annihilation signals [8, 9, 10].

Figure 1: Excluded region on the primordial curvature perturbation. The tidal
model (a) is adopted. Upper regions separated by lines are excluded.
“Satellite counts” (orange, dashed) and “Stellar stream ” (red, dashed)
corresponds to the limits by the observed number of dSphs and the observation
of the stellar stream, respectively. See Eqs. (9) and (10). As a reference,
the constraint due to $\mu$-distortion is shown as “$\mu$-distortion”, which
is given in Ref. [4]. Shaded region on the left is disfavored from the
Lyman-$\alpha$ observations [11].

In this paper we study the cosmological consequences of the primordial
curvature perturbation in the small scale. Assuming an additional bump in the
curvature perturbation, we investigate the subhalo evolution by extending the
SASHIMI package, theoretically-motivated model for the tidal stripping process
calibrated by the $N$-body simulation [12,
13].111https://github.com/shinichiroando/sashimi-c We give a new conservative
and robust bound on the curvature perturbation by using the observed number of
the dSphs in the Galactic halo [14, 15] and the observations of the stellar
stream [16, 17]. Our main result is shown in Fig. 1. Additionally we give the
predictions for the annihilation boost factor, which will be useful for future
study to search for the nature of dark matter.

Throughout the paper, we adopt the Cosmological parameters based on the Planck
2018 results [1] (TT,TE,EE+lowE+lensing); the density parameter of dark matter
$\Omega_{\rm dm}h^{2}=0.1200$ and that of baryon $\Omega_{\rm b}h^{2}=0.02237$
with $h=0.6736$.

Figure 2: Primordial curvature perturbation ${\cal P}_{R}(k)$ (left), variance
$\sigma(M)$ of the power spectrum (middle), and average of host halo mass
evolution $M_{\rm host}(z)$ (right). $k_{b}=1.0\times 10^{2}~{}{\rm
Mpc}^{-1}h$ is taken for all and each line corresponds to $A=2.5\times
10^{-3}$, $1.6\times 10^{-4}$, $1.0\times 10^{-5}$, $6.3\times 10^{-7}$, and
$4.0\times 10^{-8}$. As a reference, the result without the bump is shown as
‘No bump’. $M_{\rm host}(0)=1.3\times 10^{12}M_{\odot}$ is taken for $M_{\rm
host}(z)$. See Appendix C for additional figures with different values of
$k_{b}$.

The primordial curvature perturbation: We consider a model in which the
primordial power spectrum has a bump in the small scale
$k\gtrsim\order{1}~{}{\rm Mpc}^{-1}$. In order to investigate the impact of
the curvature perturbation in the region, we consider an additional bump on
top of the nearly scale-invariant curvature perturbation that is consistent
with the CMB observation:

$\displaystyle{\cal P}_{R}={\cal P}_{R}^{(0)}+{\cal P}_{R}^{\rm bump}\,,$ (1)

where ${\cal P}_{R}^{(0)}(k)=A_{s}(k/k_{*})^{n_{s}-1}$ and

$\displaystyle{\cal P}_{R}^{\rm
bump}(k;k_{b})=\left\\{\begin{array}[]{ll}(A-{\cal
P}_{R}^{(0)}(k_{b}))\left(\frac{k}{k_{b}}\right)^{n_{b}}&k\leq k_{b}\\\
0&k>k_{b}\end{array}\right.\,.$ (4)

Here we have introduced three parameters, $A$, $k_{b}$, and $n_{b}$. In Ref.
[4] the steepest spectral index is $n_{b}=4$ in single-field inflation. On the
other hand, Ref. [18] claims that the spectral index can be as large as $8$
after encountering a dip in the amplitude and then the amplitude reaches to a
peak with the index less than 4. In our study we adopt $n_{b}=4$ and take $A$
and $k_{b}$ as free parameters. We plot several examples of the ${\cal P}_{R}$
in Fig. 2. The parameters are given in the figure caption.

Figure 3: Mass function $dN_{\rm sh}/dm$ of subhalo (left) and cumulative
maximum circular velocity function $N_{\rm sh}(>V_{\rm max})$ (right). We take
tidal model (a) and the other parameters are the same as Fig. 2. See Appendix
C for additional figures with different values of $k_{b}$.

From the curvature perturbation, the variance of the linear power spectrum in
the comoving scale $R$ is given by

$\displaystyle\sigma^{2}(M)=\int d\ln
k~{}\frac{k^{3}}{2\pi^{2}}P(k)W^{2}(kR)\,,$ (5)

where $P(k)$ is the power spectrum calculated from ${\cal P}_{R}$ and $W(kR)$
is the window function. We adopt the sharp-$k$ window, $W(kR)=\Theta(1-kR)$,
where $\Theta$ is the Heaviside step function. This is because for the power
spectrum that has a steep cutoff, it is shown in Ref. [19] that the sharp-$k$
window gives a good agreement with the simulation. The mass scale $M$ is given
as $M=(4\pi/3)(Rc)^{3}\rho_{m}$ where $\rho_{m}=\Omega_{m}\rho_{c}$
($\rho_{c}$ is the critical density) and a parameter $c=2.7$ is determined by
comparing with the simulation [19].

The result of $\sigma(M)$ is given in Fig. 2. It is seen that the bump with
$A\gtrsim 10^{-6}$ significantly affects the variance.222A similar variance is
obtained due to the formation of the PBHs [20]. $\sigma$ is enhanced below a
mass scale, which is, for instance, $(4\pi/3)(c/k_{b})^{3}\rho_{m}\sim
10^{7}\,M_{\odot}$ for $k_{b}=10^{2}\,{\rm Mpc}^{-1}h$. The scale gets smaller
as $k_{b}$ becomes larger. The variance becomes almost constant for $A\gtrsim
10^{-6}$. This is because the bump contributes dominantly in the integral
below the scale $k_{b}$.

The host halo and subhalo evolution: The enhancement on the variance of the
power spectrum due to the bump can affects the merger history of both the host
and subhalos. To see this we evaluate the halo evolution history based on the
extended Press-Schechter (EPS) formalism [21, 22]. See Appendix A and B for
details.

The host halo mass evolution $M_{\rm host}(z)$ is shown in Fig. 2, where the
average of 200 host halo realizations is given. We have checked that the
result without the bump agrees with the fitting formula given in Ref. [23] in
the low $z$ region, which is calibrated against the simulations in $z\lesssim
10$. It is seen that at the low $z$ region the host halo evolution coincides
with the one without the bump. However, the difference becomes significant in
high $z$ region, especially for a large $A$; we find a plateau for large
values of $A$, such as $A\gtrsim 10^{-6}$. In the conventional model, i.e.,
with no bump in the curvature perturbation, the halos with small mass scales
are formed in the past and they grow to much massive halos to the present.
This is true for the model with bump with a small amplitude, such as
$A\lesssim 10^{-8}$. When the amplitude is large, on the other hand, the halo
formation and growth happen almost at a certain redshift. For
$k_{b}=10^{2}\,{\rm Mpc}^{-1}h$, for instance, halos with mass $\sim
10^{7}\,M_{\odot}$ form at once. We see a mild dependence of the amplitude on
the mass scale; the mass scale is larger for the larger amplitude. This is
because $\sigma$ is altered in the larger mass-scale region as the amplitude
is larger. After the formation they merely grow for some period since $\sigma$
changes drastically for the small-mass scales. This period corresponds to the
plateau for $M_{\rm host}$. Eventually, the host mass rebegins to grow in
accordance with the case without the bump, which is a reasonable behavior
since $\sigma$ coincides with the one calculated without the bump.

The evolution of the subhalos is similar to that of the host halo, but it
suffers from the tidal stripping due to the gravitational potential of the
host halo after the accretion. To evaluate the subhalo evolution, we modify
SASHIMI package to implement the results of $\sigma(M)$ and the host halo
evolution $M_{\rm host}(z)$ with the existence of a bump in the primordial
curvature perturbation. For consistency, we adopt the concentration-mass
relation given by Ref. [24], where the fitting function is given in terms of
$\sigma(M)$. In the code the Navarro-Frenk-White (NFW) profile [25] with
truncation is assumed, which is characterized by typical mass density
$\rho_{\rm s}$, the scale radius $r_{\rm s}$ and the truncation radius
$r_{t}$. For the tidal process, we consider three models;

  * (a)

$g=0.86$ and $\zeta=0.07$ [26],

  * (b)

$z$-dependent $g(z)$ and $\zeta(z)$ [12],

  * (c)

No tidal stripping,

where $g$ and $\zeta$ are the parameters in the evolution of the subhalo mass
[26]

$\displaystyle\derivative{m}{t}=-g\frac{m}{\tau_{\rm
dyn}}\left(\frac{m}{M(z)}\right)^{\zeta}\,.$ (6)

Here $\tau_{\rm dyn}$ is the halo’s dynamical time. Since $g(z)$ and
$\zeta(z)$ are given in $z\leq 7$ in model (b), we take $g(z)=g(7)$ and
$\zeta(z)=\zeta(7)$ for $z>7$ in the current calculation. The model (c)
corresponds to the so-called un-evolved mass function. We note that the model
(c) might be more realistic than the others when we compute the boost factor.
This is because the tidal stripping effect may not change the inner structure
of the halo profile in the case of highly concentrated profile [27], which is
expected in the current case. The mass distribution function of subhalos at
the accretion is given by the EPS formalism as function of $m_{a}$, $z_{a}$
and the host halo mass $M_{0}$ at $z=z_{0}$. Using the number $d^{2}N_{{\rm
sh},a}$ of subhalos with mass $m_{a}$ that accrete at $z=z_{a}$, the subhalo
mass function after the tidal stripping is obtained by

$\displaystyle\frac{dN_{{\rm sh}}}{dm}$ $\displaystyle=\int d^{2}N_{{\rm
sh},a}\int dc_{{\rm vir},a}P^{c_{{\rm vir},a}}(m_{a},z_{a})\delta(m-m_{0})\,,$
(7)

where $P^{c_{{\rm vir},a}}(m_{a},z_{a})$ is the distribution function for
$c_{{\rm vir},a}$ that is computed from the one for the concentration-mass
relation. A subscript “0” stands for the values at $z=z_{0}$.

We plot the mass function $dN_{\rm sh}/dm$ of the subhalo at $z_{0}=0$
computed using the tidal model (a) in Fig. 3. We found that the mass function
is affected significantly, depending on $A$ and $k_{b}$. As $k_{b}$ becomes
small, the mass function is altered in large subhalo mass, which is expected
from the behavior of $\sigma(M)$. Due to the bump, the number of the subhalo
of a mass scale tends to be enhanced. On the contrary, the mass function is
suppressed below that mass scale. This effect is significant for large $A$ and
small $k_{b}$. Such a drastic change leads to change the prediction of the
number of dSphs, which are formed in subhalos. Additionally, we found that the
result is almost independent of the tidal models and $z_{\rm max}$ if $z_{\rm
max}\geq 7$. Here $z_{\rm max}$ is the maximum redshift to track the subhalo
evolution. Therefore, we expect the observable consequences are determined by
the evolution in the low redshift regime and that they are not significantly
affected by the details of the tidal evolution models, which we confirm below.

Astrophysical observables and constraint: It is considered that subhalos which
satisfy certain conditions form galaxies inside. A quantity for the criterion
is the maximum circular velocity. Based on the conventional theory of galaxy
formation, for instance, the dSphs formation occurs for $V_{{\rm
max},a}>18~{}{\rm km/s}$, where $V_{{\rm max},a}$ is the maximum circular
velocity at the time of the accretion. Using this condition, we can predict
the number of the present dSphs in the Galaxy, which is one of the important
observables of the dSphs. However, this criterion is under debate. A recent
study suggests a different criterion of $V_{\rm max,a}>10.5~{}{\rm km/s}$
[28]. Therefore, the predicted number of dSphs can change, depending on the
choice of criteria. To avoid such uncertainties on the condition for formation
of dSph, we take a much conservative approach based on the observations.
Focusing on the present maximum circular velocity $V_{\rm max}$ that is
actually observed for dSphs, the number of dSphs in our Galaxy whose $V_{\rm
max}$ is over 4 km/s is given by [29, 30]:

$\displaystyle N_{\rm dSph}^{\rm low}(V_{\rm max}>4~{}{\rm km/s})=94\,.$ (8)

The condition $V_{\rm max}>4\ {\rm km/s}$ is determined by the minimum of the
observed velocity dispersion among dSphs [14, 15]. We apply this observational
bound directly to our calculation by imposing the following condition

$\displaystyle N_{\rm sh}(V_{\rm max}>4~{}{\rm km/s})\geq N_{\rm dSph}^{\rm
low}(V_{\rm max}>4~{}{\rm km/s})\,,$ (9)

and see whether the curvature perturbation with the bump contradicts the
observation.

Fig. 3 shows the cumulative maximum velocity function $N_{\rm sh}\,(>V_{\rm
max})$ of subhalos. It is seen that $N_{\rm sh}\,(>V_{\rm max})$ is enhanced
compared to the case with no bump at a large value of $V_{\rm max}$,
corresponding to a massive case, as $A$ becomes large and $k_{b}\lesssim
10^{2}\,{\rm Mpc}^{-1}h$. In the exchange for the enhancement at large $V_{\rm
max}$ region, it is suppressed in small $V_{\rm max}$ region. For instance, it
is below the observed value for $k_{b}=10^{2}\,{\rm Mpc}^{-1}h$ and $A\gtrsim
2.5\times 10^{-3}$. Therefore, the bump in the primordial curvature
perturbation in such a parameter space is excluded. On the other hand, for
$k_{b}\gtrsim 10^{2}\,{\rm Mpc}^{-1}h$, the cumulative maximum circular
velocity function is almost unchanged. This reflects the fact that the bump
with a large $k_{b}$ affects less massive halos than those responsible for
dSphs.

Making a comprehensive analysis on the bump model, we compute the cumulative
number of the subhalos $N_{\rm sh}\,(V_{\rm max}>4~{}{\rm km/s})$ on
$(A,\,k_{b})$ plane. We found that it is smaller than the observed value in
the region $A\gtrsim 10^{-6}$ and $k_{b}\lesssim 10^{2}\,{\rm Mpc}^{-1}h$ so
that the region is excluded. The result is shown in Fig. 1. As expected from
the results of the mass function, we confirmed that the number of subhalos
satisfying the condition of the maximum circular velocity and the resultant
exclusion region do not depend on the tidal stripping models. Therefore, it is
concluded that the bound is conservative and robust.

Another observable effect appears in the stellar stream, where gaps are caused
by a passage of subhalos in the Galaxy. Too large or too small number of
subhalos may conflict with the observation of the stellar stream. We make use
of the results by Ref. [16], which analyzes GD-1 stream [17] using data from
Gaia [31, 32] and Pan-STARRS survey [33]. We adopt the most conservative limit
on the number of subhalos whose mass is in $10^{5}M_{\odot}$ –
$10^{9}M_{\odot}$, which is given by

$\displaystyle N_{\rm sh}/N_{\rm sh,CDM}<2.7~{}(95\%~{}{\rm CL})\,,$ (10)

where $N_{\rm sh,CDM}$ corresponds to one without the bump. Consequently we
found that the amplitude in $k_{b}=\order{10^{2}\,\mathchar
45\relax\,10^{4}}\,{\rm Mpc}^{-1}h$ is constrained; the most stringent upper
limit is $10^{-7}$, which is shown in Fig. 1.

The limits from the observations of satellite number and stellar stream for
tidal models (b) and (c) are given in Appendix C. We found that the bounds are
almost unchanged by choice of the tidal models. Therefore, the limits shown in
Fig. 1 is conservative and robust.

Finally we discuss the annihilation boost factor due to the substructure in
the host halo. The enhancement of the subhalo clustering leads to a large
enhancement of the pair-annihilation signals of dark matter. We define the
boost factor as $B\equiv J_{{\rm sh}}^{\rm tot}/J_{\rm h}$, where

$\displaystyle J_{\rm h}=\int d^{3}x\rho_{\rm h}^{2}\,,~{}~{}J_{\rm sh}^{\rm
tot}$ $\displaystyle=\int d\rho_{\rm s}dr_{\rm s}dr_{t}\frac{d^{3}N_{\rm
sh}}{d\rho_{\rm s}dr_{\rm s}dr_{t}}\int d^{3}x\rho_{\rm sh}^{2}\,,$ (11)

assuming the NFW profile $\rho_{\rm h}$ and $\rho_{\rm sh}$ for both the host
and subhalos, respectively.

Note that the boost factor depends on the minimum halo mass [24]. In our
analysis we take minimum halo mass as $10^{-6}M_{\odot}$, assuming neutralino-
like dark matter [34]. We found that the boost factor is significantly
enhanced by the amplitude of the bump in the regions which are not excluded
from current observations. (See Appendix LABEL:qpp:figs for details.) It
becomes as large as $\order{10^{4}}$ for $A\sim\order{10^{-2}}$ and
$k_{b}\sim\order{10^{5}}$ Mpc${}^{-1}h$. Additionally, we observe a mild
dependence on $k_{b}$, i.e, the boost factor gets larger for larger $k_{b}$.

Careful readers may think that the resultant boost factor in the model (c)
should be enhanced compared to the model (a) or (b) since the tidal stripping
process reduces the subhalo mass. Although the subhalos lose their masses due
to the tidal process, the inner structure of subhalo is hardly affected. This
is because the concentration parameter is much larger than $\order{1}$ [24].
The concentration-mass relation at high redshifts is still under debate (e.g.,
see Ref. [35]). Hence the model of the concentration-mass relation would be
the main source of the uncertainty in the estimation of the boost factor. If
the issue is settled, then the clustering of subhalos would be another
important observable to constrain the unconventional curvature perturbation in
the future experiment.

Conclusion: In this work, we propose a new scheme for investigating the
primordial curvature perturbation of the small scale by using the observation
of the dark matter substructure. Assuming an additional bump in the primordial
curvature perturbation, an enhancement on the variance of the linear power
spectrum appears. We track the evolution of both the host halo and subhalos
from the power spectrum, based on the EPS formalism and the semi-analytic
calculation. In the evolution of the subhalo, we take into account uncertainty
in the evaluation of tidal stripping effect by comparing three types of
models. We have found that the extra bump in the curvature perturbation
significantly affects the evolution of the host and subhalos, depending on the
amplitude and the wavenumber scale of the bump. On the other hand, it turns
out that the mass function of the subhalos merely depends on the tidal models.
This fact enables us to compute the number of dSphs that is free from the
uncertainty in the tidal models. We focus on the the cumulative number of
subhalos with maximum circular velocity over $4~{}{\rm km/s}$, which is
observed directly. Imposing the most conservative bound from observations of
the dSphs, we have found that the bump with amplitude $A\gtrsim 10^{-6}$ in
$k\lesssim 10^{2}~{}{\rm Mpc}^{-1}h$ is excluded. Another consequence that has
a direct connection with the number of subhalo is the stellar stream. Adopting
the most conservative limit from the observation, the amplitude of the bump in
the region $k=\order{10^{2}\,\mathchar 45\relax\,10^{4}}\,{\rm Mpc}^{-1}h$ is
constrained as $A\gtrsim 10^{-7}$ at most. We also obtain an indication for
the boost factor, which is crucially important for detecting dark matter
annihilation signals. The boost factor of $\order{10^{4}}$ can be expected in
the parameter region allowed by the existing observations. In future, the
predictions in this work could be tested by various probes of small-scale
halos, such as gravitational lensing observations [36, 37, 38, 39, 40] or
pulsar timing array experiments [41, 42, 43, 44, 45, 46]. We leave it for
future work.

## acknowledgment

We thank T. Ishiyama, T. Sekiguchi and K. Yang for valuable discussion. This
work is supported by JSPS KAKENHI Grants No. JP19K23446 and JP22K14035 (N.
H.), MEXT KAKENHI Grants No. JP20H05852 (N. H.), JP20H05850, and JP20H05861
(S. A.). The work of KI is supported by JSPS KAKENHI Grant No. JP18H05542,
JP20H01894, and JSPS Core-to-Core Program Grant No. JPJSCCA20200002. This work
was supported by computational resources provided by iTHEMS.

##  Appendix A The extended Press-Schechter formalism

The collapse of the overdensity to form the halos is characterized by two
quantities:

$\displaystyle\delta(z)=\delta_{c}/D(z)\,,~{}~{}~{}\sigma(M)\,,$ (12)

where $z$ is the redshift, $\delta_{c}\simeq 1.686$ is the critical
overdensity, and $D(z)$ is the linear growth factor defined by

$\displaystyle D(z)=D_{\rm
norm}H(z)\int_{z}^{\infty}dz^{\prime}\,\frac{1+z^{\prime}}{H^{3}(z^{\prime})}\,.$
(13)

Here $D_{\rm norm}$ is determined to satisfy $D(0)=1$ and $H(z)$ is the Hubble
parameter. Since $D(z)$ is monotonically decreasing function of $z$,
$\delta(z)$ monotonically decreases as $z$ gets small. On the other hand,
$\sigma(M)$ is cumulative as $M$ becomes small. Namely, the parameters $z$ and
$M$ can be translated into $\delta(z)$ and $\sigma(M)$, respectively. Based on
the EPS theory, the evolution of the halo is described by the following
probability distribution function (PDF) [47, 48, 49],

$f(S_{2},\delta_{2}|S_{1},\delta_{1})=\frac{1}{\sqrt{2\pi}}\frac{\delta_{2}-\delta_{1}}{\left(S_{2}-S_{1}\right)^{3/2}}\exp\left[-\frac{\left(\delta_{2}-\delta_{1}\right)^{2}}{2\left(S_{2}-S_{1}\right)}\right]\,,$
(14)

where $S_{i}\equiv\sigma^{2}(M_{i})$ and $\delta_{i}\equiv\delta(z_{i})$ and
$z_{1}<z_{2}$. Namely, $f(S_{2},\delta_{2}|S_{1},\delta_{1})$ is the PDF that
a halo with a mass of $M_{2}$ is a progenitor at $z=z_{2}$ of a halo with a
mass of $M_{1}$ at $z=z_{1}$. In the canonical $\Lambda$CDM case, halos with
small mass are created in the past and evolve by accretions and mergers to the
present. With the additional bump in the primordial curvature perturbation,
however, this picture changes.

In the calculation, we construct the evolution history of the host halo by
applying the inverse function method to the above distribution function (see
details for Ref. [50]). We start our calculation of the host halo evolution
from $z=0$ and track its merger history up to $z=300$, taking 4000 points of a
constant interval in the $\ln(1+z)$ space. Considering the Milky-Way like host
halo at $z=0$, we take the host halo mass as $M_{\rm host}(z=0)=1.3\times
10^{12}M_{\odot}$ [51].

##  Appendix B Subhalo evolution

To begin with, we collect important quantities for the subhalo properties. We
assume that the halos follow the Navarro-Frenk-White (NFW) profile [25] with
truncation, which is characterized by typical mass density $\rho_{\rm s}$, the
scale radius $r_{\rm s}$ and the truncation radius $r_{t}$ as

$\displaystyle\rho_{\rm NFW}(r)=\left\\{\matrixquantity{\frac{\rho_{\rm
s}}{(r/r_{\rm s})(1+r/r_{\rm s})^{2}}&~{}~{}~{}~{}r\leq r_{t}\\\
0&~{}~{}~{}~{}r>r_{t}}\right.\,.$ (15)

Given a mass parameter $m_{200}$, a concentration parameter $c_{200}$ is
evaluated in the simulations at the redshift $z$. Since $\sigma(M)$ is
significantly altered by the bump compared to the conventional case, we adopt
the concentration-mass relation given by Ref. [24], where the fitting function
is given in terms of $\sigma(M)$.333See also Refs. [52, 53, 54] and Appendix B
of Ref. [12]. On the other hand, $m_{200}$ relates to $r_{200}$ via
$m_{200}=(4\pi/3)200\rho_{c}(z)r_{200}^{3}$, where $\rho_{c}(z)$ is the
critical density at the redshift $z$. Using the relation
$c_{200}=r_{200}/r_{\rm s}$, the scale radius $r_{\rm s}$ is determined. Then,
$\rho_{\rm s}$ is obtained as

$\displaystyle m$ $\displaystyle=\int d^{3}x\,\rho=4\pi\rho_{\rm s}r_{\rm
s}^{3}f(c)\,,$ (16)

where $f(c)=\ln(1+c)-c/(1+c)$ and $m=m_{200}$ and $c=c_{200}$ are taken. We
consider the $(r_{\rm s},\rho_{\rm s})$ as the values at the accretion of a
subhalo onto a host halo, which are denoted as $(r_{{\rm s},a},\rho_{{\rm
s},a})$. In order to discuss the tidal stripping process of subhalo after the
accretion, it is appropriate to use the virial mass $m_{{\rm vir},a}$ instead
of $m_{200}$. $m_{{\rm vir},a}$ is obtained by using Eq. (16) where $m=m_{{\rm
vir},a}$ and $c=c_{{\rm vir},a}=r_{{\rm vir},a}/r_{{\rm s},a}$, and by

$\displaystyle m_{{\rm vir},a}$
$\displaystyle=\frac{4\pi}{3}\Delta_{c}(z_{a})\rho_{c}(z_{a})r_{{\rm
vir},a}^{3}\,.$ (17)

Here $\Delta_{c}(z)$ is given in Ref. [55]. To sum up, we get the parameters
$\rho_{{\rm s},a}$, $r_{{\rm s},a}$ and $m_{{\rm vir},a}$ at the accretion.

After accretion, the subhalo loses its mass due to the tidal stripping. Given
a value of the subhalo mass $m_{0}$ at the redshift $z_{0}$ after tidal
stripping, $(\rho_{{\rm s},a},r_{{\rm s},a})$ are translated into $(\rho_{{\rm
s},0},r_{{\rm s},0})$ at the redshift $z_{0}$ using the relation among
$V_{{\rm max},0}/V_{{\rm max},a}$, $r_{{\rm max},0}/r_{{\rm max},a}$, and
$m_{0}/m_{{\rm vir},a}$ [56]. Here $V_{\rm max}$ and $r_{\rm max}$ are the
maximum circular velocity and the radius which relate to the NFW profile
parameters as

$\displaystyle V_{\rm max}=\sqrt{\frac{4\pi G\rho_{\rm s}}{4.625}}r_{\rm
s}\,,~{}~{}~{}r_{\rm max}=2.163r_{\rm s}\,,$ (18)

where $G$ is the Newtonian constant. Using Eq. (16) with $m=m_{0}$, $r_{\rm
s}=r_{s,0}$, $\rho_{\rm s}=\rho_{{\rm s},0}$, and $c_{t,0}=r_{t,0}/r_{{\rm
s},0}$, we obtain the truncation radius $r_{t,0}$ at $z=z_{0}$. To summarize,
given a subhalo mass $m_{{\rm vir},a}$ at the accretion and $m_{0}$ after
tidal stripping, we obtain subhalo properties, such as $\rho_{\rm s}$, $r_{\rm
s}$, $r_{t}$, and $V_{\rm max}$, which are important to discuss the observable
consequences.

The mass distribution function ${\cal F}$ of subhalos at the accretion is
given by the EPS formalism as function of $m_{a}$, $z_{a}$ and the host halo
mass $M_{0}$ at $z=z_{0}$. Then the number of subhalos with mass $m_{a}$ that
accrete at $z=z_{a}$ is given by

$\displaystyle d^{2}N_{{\rm sh},a}={\cal
F}(s_{a},\delta_{a}|S_{0},\delta_{0})d\ln m_{a}dz_{a}\,,$ (19)

where $s_{a}=\sigma^{2}(m_{a})$, $\delta_{a}=\delta(z_{a})$,
$S_{0}=\sigma^{2}(M_{0})$, and $\delta_{0}=\delta(0)$. In the present
calculation, $M_{0}=M_{\rm host}(z=0)=1.3\times 10^{12}M_{\odot}$ is taken.
Combining all the discussion above, the subhalo mass function after the tidal
stripping is obtained by

$\displaystyle\frac{dN_{{\rm sh}}}{dm}=\int d^{2}N_{{\rm sh},a}\int dc_{{\rm
vir},a}P^{c_{{\rm vir},a}}(m_{a},z_{a})\delta(m-m_{0})\,,$ (20)

where $P^{c_{{\rm vir},a}}(m_{a},z_{a})$ is the distribution function for
$c_{{\rm vir},a}$ that is computed from the one for $c_{200}$-mass relation
discussed above Eq. (16).

##  Appendix C Additional figures

We give additional figures with various values of $k_{b}$ for the primordial
curvature perturbation ${\cal P}_{R}(k)$ (Fig. 4), the variance $\sigma(M)$ of
the power spectrum (Fig. 5), and the average of the host halo mass evolution
$M_{\rm host}(z)$ (Fig. 6). The subhalo mass function $dN_{\rm sh}/dm$ and the
cumulative maximum circular velocity function $N_{\rm sh}(>V_{\rm max})$ in
the tidal model (a), (b), and (c) are shown in Figs. 7 and 8, respectively.
Fig. 9 gives the color maps of the cumulative number of subhalos the maximum
circular velocity satisfying $V_{\rm max}>4$ km/s, the number of dSphs whose
mass is in $10^{5}M_{\odot}$ – $10^{9}M_{\odot}$ normalized by the one without
the bump, and the boost factor for the tidal model (a), (b), and (c). The mild
dependence of the boost factor on $k_{b}$, which is mentioned in the main
text. is seen for all the tidal models. This behavior can be qualitatively
understood from the results of the subhalo mass function. In the plot of
$dN_{\rm sh}/dm$, the bump is shifted to the smaller scale as $k_{b}$ becomes
large, meanwhile the intensity of $m^{2}dN_{\rm sh}/dm$ stays in the same
order. This means that many subhalos with lower masses are formed for bigger
$k_{b}$, which leads to the enhancement of the boost factor.

Figure 4: Primordial curvature perturbation ${\cal P}_{R}$ as function of
wavenumber. $k_{b}=1.0\times 10^{2}~{}{\rm Mpc}^{-1}h$ (left), $1.0\times
10^{4}~{}{\rm Mpc}^{-1}h$ (center), and $1.0\times 10^{6}~{}{\rm Mpc}^{-1}h$
(right). Each line corresponds to $A=2.5\times 10^{-3}$, $1.6\times 10^{-4}$,
$1.0\times 10^{-5}$, $6.3\times 10^{-7}$, and $4.0\times 10^{-8}$ from top to
bottom. As a reference, ${\cal P}_{R}^{(0)}$ is shown as ‘No bump’.  Figure 5:
Variance $\sigma$ of the power spectrum as functions of the mass scale $M$.
The parameters are the same as Fig. 4. As a reference, $\sigma(M)$ without the
bump is shown as ‘No bump’, which is calculated by using CAMB [57].  Figure 6:
Average of host halo mass evolution as functions of the redshift $z$. We take
$M_{\rm host}(z=0)=1.3\times 10^{12}M_{\odot}$ and the other parameters are
the same as Fig. 5. The result without the bump is also shown as ‘No bump’.

Figure 7: Mass function of subhalo. Tidal model corresponds to (a) Jiang & van
den Bosch [26], (b) Hiroshima et al. [12], (c) No tidal stripping, from top to
bottom. The parameters are the same as Fig. 5.

Figure 8: Cumulative maximum circular velocity function. Tidal models, the
parameters, and the ordering of panels are the same as Fig. 7.

Figure 9: (Color map of the cumulative number of subhalos with the maximum
circular velocity satisfying $V_{\rm max}>4$ km/s (left) , $N_{\rm sh}/N_{\rm
sh,CDM}$ (middle), and boost factor (right) on $(A,\,k_{b})$ plane. For both
panels the tidal models are (a) Jiang & van den Bosch [26], (b) Hiroshima et
al. [12], and (c) No tidal stripping, from the top to the bottom. As a
reference, the constraint due to $\mu$-distortion is shown as
“$\mu$-distortion”, which is given in Ref. [4]. Shaded region on the left is
disfavored from the Lyman-$\alpha$ observations [11].

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