#### 4.1. Models Based on Structure Formation and Tidal Evolution

In an analytical approach in

Section 2, the subhalo luminosity

${L}_{\mathrm{sh}}$ is characterized with the mass of the subhalo and the redshift of interest; see, for example, Equation (

10). The mass and redshift, however, are not the only quantities that fully characterize the subhalo properties. Indeed, they depend on the accretion history and mass loss after they fall onto their host halo, that is, two subhalos that have the identical mass could have formed with different masses and accreted at different redshifts, evolved down to

$z=0$ reaching the same mass. Bartels and Ando [

46] and Hiroshima et al. [

50] developed an analytical prescription to take these effects into account, which we follow in this section.

A subhalo is characterized with its mass and redshift when it accreted onto its host, (

${m}_{a}$,

${z}_{a}$). The concentration parameter

${c}_{a}$ is drawn from the log-normal distribution with mean

${\overline{c}}_{a}({m}_{a},{z}_{a})$ [

64] and

${\sigma}_{logc}=0.13$ [

65]. Since the subhalo was a

field halo when it just accreted, one can use the relations in

Section 2.2 to obtain

${r}_{s,a}$ and

${\rho}_{s,a}$ for the NFW profile.

After the accretion, the subhalos evolve by losing their mass through tidal forces. The mass-loss rate is typically characterized by a dynamical timascale at the redshift

z,

as follows [

90]:

where

$H\left(z\right)={H}_{0}{[{\mathsf{\Omega}}_{m}{(1+z)}^{3}+{\mathsf{\Omega}}_{\mathsf{\Lambda}}]}^{1/2}$,

$m\left(z\right)$ and

$M\left(z\right)$ are the subhalo and host-halo masses at

z, respectively. Following Jiang and van den Bosch [

90], Hiroshima et al. [

50] adopted simple Monte Carlo simulations to estimate

$\dot{m}$ based on the assumption that the subhalo loses all the masses beyond its tidal radius in one complete orbit at its peri-center passage. While Jiang and van den Bosch [

90] found

$A=0.81$ and

$\zeta =0.04$, Hiroshima et al. [

50] extended the mass and redshift ranges of applicability and found that these parameters are weakly dependent on both

M and

z:

One can solve Equation (

26) to obtain the subhalo mass at a redshift of interest

z,

$m\left(z\right)$, with a boundary condition of

$m\left({z}_{a}\right)={m}_{a}$. For the evolution of the host,

$M\left(z\right)$, Hiroshima et al. [

50] adopted a fitting formula given by Correa et al. [

147].

The subhalo density profile after accretion is also well described with the NFW profile with a sharp truncation at

${r}_{t}$:

This is indeed a good approximation found in the simulations [

37]. In addition to

${r}_{t}$, Peñarrubia et al. [

129] found that the internal structure changes. If the inner profile is

$\propto {r}^{-1}$ just like NFW, the maximum circular velocty

${V}_{\mathrm{max}}$ and its corresponding radius

${r}_{\mathrm{max}}$ evolve as

respectively. After computing

${V}_{\mathrm{max}}$ and

${r}_{\mathrm{max}}$ at

z, one can convert them to

${\rho}_{s}$ and

${r}_{s}$ through

which are valid for the NFW profile. Finally by solving the condition

the truncation radius

${r}_{t}$ is obtained. Hiroshima et al. [

50] omitted subhalos with

${r}_{t}<0.77{r}_{s}$ from the subsequent calculations assuming that they were tidally disrupted [

148]. This criterion, however, might be a numerical artifact [

130]. Either case, Hiroshima et al. [

50] checked that whether one implements this condition or not did not have impact on the results of, for example, subhalo mass functions.

Thus, given

$({m}_{a},{z}_{a},{c}_{a})$, one can obtain all the subhalo parameters after the evolution,

$(m,{r}_{s},{\rho}_{s},{r}_{t})$, in a deterministic manner. The differential number of subhalos accreted onto a host with a mass

${m}_{a}$ and at redshift

${z}_{a}$,

${d}^{2}{N}_{\mathrm{sh}}/\left(d{m}_{a}d{z}_{a}\right)$, is given by the excursion set or the extended Press-Schechter formalism [

55]. Especially Yang et al. [

89] obtained analytical formulation for the distribution that provides good fit to the numerical simulation data over a large range of

$m/M$ and

z. Hiroshima et al. [

50] adopted their model III.

The subhalo mass function is obtained as

where

$P\left({c}_{a}\right|{m}_{a},{z}_{a})$ is the probability distrbution for

${c}_{a}$ given

${m}_{a}$ and

${z}_{a}$, for which Hiroshima et al. [

50] adopted the log-normal distribution with the mean

${\overline{c}}_{a}({m}_{a},{z}_{a})$ [

64] and the standard deviation

${\sigma}_{log{c}_{a}}=0.13$ [

65]. We show the subhalo mass functions obtained with Equation (

35) for various values of

M and

z in

Figure 4, where comparison is made with simulation results of similar host halos. Halos and subhalos formed in these simulations were identified with ROCKSTAR phase space halo finder [

149]. The bound mass is used as the subhalo mass, which nearly corresponds to the tidal mass [

48]. For all these halos, one can see remarkable agreement between the analytic model and the corresponding simulation results in resolved regimes. Successfully reproducing behaviors at resolved regimes, this analytic model is able to make reliable predictions of the subhalo mass functions below resolutions of the numerical simulations, without relying on extrapolating a single power-law functions, from which most of the previous studies in the literature had to suffer. The subhalo mass fraction is then obtained as

and is shown in

Figure 4 (bottom right) for various values of redshifts. The subhalo mass fraction is found to increase as a function of

M and

z. At higher redshifts, since there is shorter time for the subhalos to experience tidal mass loss,

${f}_{\mathrm{sh}}$ is larger. Again, a good agreement in

${f}_{\mathrm{sh}}$ is found between the analytic model and the simulation results by Giocoli et al. [

86].

The annihilation boost factor is then

which is to be compared with Equation (

10) that was derived with a simpler (and unrealistic) discussion. The superscript (0) represents the quantity

in the absense of sub-subhalos and beyond. The subhalo luminosity,

${L}_{\mathrm{sh}}^{\left(0\right)}({m}_{a},{z}_{a},{c}_{a}|M,z)$, is proportional to the volume integral of density squared

${\rho}_{\mathrm{sh}}^{2}\left(r\right)$ out to the truncation radius,

where

${\rho}_{s,\mathrm{sh}}$,

${r}_{s,\mathrm{sh}}$ and

${r}_{t,\mathrm{sh}}$ are functions of

$({m}_{a},{z}_{a},{c}_{a})$ as well as

M and

z.

Then, the effect of sub

${}^{n}$-subhalos (for

$n\ge 1$) can be estimated iteratively. At

nth iteraction, when a subhalo accreted onto its host at

${z}_{a}$ with

${m}_{a}$, it is assigned a sub-subhalo boost factor

${B}_{\mathrm{sh}}^{(n-1)}({m}_{a},{z}_{a})$. After the accretion, the outer region of the subhalo is stripped away by the tidal force and thus all the sub-subhalos within this stripped region will disappear, reducing the sub-subhalo boost accordingly. Hiroshima et al. [

50] assumed that the sub-subhalos were distributed within the subhalo following

${n}_{\mathrm{ssh}}\left(r\right)\propto {[1+{(r/{r}_{s})}^{2}]}^{-3/2}$. The luminosity due to sub-subhalos within a radius

r is therefore proportional to their enclosed number

and it gets suppressed by a factor of

${N}_{\mathrm{ssh}}(<{r}_{t}|{r}_{s})/{N}_{\mathrm{ssh}}(<{r}_{\mathrm{vir}}|{r}_{s,a})$ due to the tidal stripping.

4 The luminosity due to the smooth component also decreases as

${L}_{\mathrm{sh}}^{\left(0\right)}(<{r}_{t}|{\rho}_{s},{r}_{s})/{L}_{\mathrm{sh}}^{\left(0\right)}(<{r}_{\mathrm{vir}}|{\rho}_{s,a},{r}_{s,a})$, where

Thus the sub-subhalo boost after the

nth iteration,

${B}_{\mathrm{ssh}}^{\left(n\right)}$ is obtained by

Similarly, the sub-subhalo mass fraction

${f}_{\mathrm{ssh}}$ is obtained by

where

${f}_{\mathrm{sh}}({m}_{a},{z}_{a})$ is obtained with Equations (

36) and

${m}_{\mathrm{sm}}(<r|{\rho}_{s},{r}_{s})\propto {\rho}_{s}{r}_{s}^{3}f(r/{r}_{s})$ is the enclosed mass within

r of the smooth component of the subhalo. The subhalo boost factor after

nth iteration is obtained with Equation (

37) by replacing

${L}_{\mathrm{sh}}^{\left(0\right)}$ with

$[1-{f}_{\mathrm{ssh}}^{2}+{B}_{\mathrm{ssh}}^{\left(n\right)}]{L}_{\mathrm{sh}}^{\left(0\right)}$ [see discussions below Equation (

15)]:

The host luminosity in the absence of the subhalos

${L}_{\mathrm{host},0}(M,z)$ is defined by marginalizing over the concentration parameter

${c}_{\mathrm{vir}}$:

with the log-normal distribution

$P\left({c}_{\mathrm{vir}}\right|M,z)$.

Figure 5 shows the subhalo boost factors

${B}_{\mathrm{sh}}$ as a function of the host mass

M at various redshifts

z (top left). The boost factors are on the order of unity, while it can be as larger as ∼5 for cluster-size halos. It is also noted that they are larger at higher redshifts, because the subhalos have less time to be disrupted. The top right panel of

Figure 5 shows the effect of sub

${}^{n}$-subhalos, which is saturated after the second iteration. The contribution to the boost factors due to sub-subhalos and beyond is ≲10% for the hosts with

${M}_{\mathrm{host}}\ge {10}^{13}{M}_{\odot}$. The bottom left panel of

Figure 5 shows the luminosity ratio

${L}_{\mathrm{total}}/{L}_{\mathrm{host},0}=1-{f}_{\mathrm{sh}}^{2}+{B}_{\mathrm{sh}}$ (Equation (

15)) as a function of the host masses for various values of the redshifts. The bottom right panel of

Figure 5 shows comparison with the results of the other work [

41,

44,

48]. We note that the analytic models do not rely on the subhalo mass function prepared separately, as the models can provide them in a self-consistent manner. The resulting boost factors are, however, found to be more modest than the previous results. This is mainly because the subhalo mass function adopted in the literature is larger than the predictions of the analytic models. However, they might be larger because of halo-to-halo variance. See discrepancy between predictions of the subhalo mass function for the

$1.8\times {10}^{12}{M}_{\odot}$ halo by Hiroshima et al. [

50] and the result of Springel et al. [

37] shown in the top left panel of

Figure 4.

Finally, for convenience of the reader who might be interested in using the results without going into details of the formalism, we provide fitting functions for both the subhalo mass functions and the annihilation boost factors. They are summarized in

Appendix A.