Metamorphosis of dwarf halo density profile under dark matter decays

Dark matter (DM) is a crucial ingredient of modern cosmology. It is needed to account for the anisotropy of the cosmic microwave background (CMB) [Wright et al., 1992, Spergel et al., 2003], large-scale structure formation and evolution [see, e.g., Springel et al., 2006], dynamics of galaxy clusters and groups [see, e.g., Zwicky, 1933], flattened rotation curves of late-type galaxies [see, e.g., Bosma, 1981], and mismatch between mass centers revealed by gravitational lensing and X-ray emission respectively [see, e.g., Markevitch et al., 2002]. While the DM mass density is about $6$ times higher than that of ordinary matter [Planck Collaboration et al., 2018a], its nature remains a mystery. Regardless of its composition, the microscopic properties of DM directly affect its cosmic mass distribution. Therefore, through a comparison between theoretical calculation and observation on DM distributions, properties of DM can be constrained. From 1970s, N-body simulation has become an indispensable theoretical tool for dealing with DM’s nonlinear clustering. In 1980s, N-body simulations were used to rule out a neutrino-dominated Universe [White et al., 1983] and establish the canonical cold dark matter (CDM) model [Peebles, 1982, Blumenthal et al., 1984, Frenk et al., 1985, Davis et al., 1985]. With the improving resolution of N-body simulations, the inner structure of CDM halos were first resolved in 1990s [Dubinski & Carlberg, 1991, Navarro et al., 1996b]. These high-resolution simulations revealed a central cusp in CDM halos, characterized by a power-law density profile $\rho(r)\propto r^{-\alpha}$ with $\alpha=1$, independent of the halo mass. The cuspy density profile results in a steeply rising rotation curve as $V_{\rm{cir}}\propto r^{1/2}$, which is incompatible with the observed solid-body rotation , $V_{\rm{cir}}\propto r$, of dark matter-dominated dwarf and low surface brightness (LSB) galaxies [Flores & Primack, 1994, Moore, 1994, Moore et al., 1999b]. This tension is referred to as the core-cusp problem [de Blok, 2010]. Together with other small-scale problems it challenges the standard CDM model [Bullock & Boylan-Kolchin, 2017].

Within the CDM framework, the resolution of the core-cusp problem relies on baryonic feedbacks. The idea is that the energy released from starburst radiation or supernovae explosions is indirectly transferred to DM particles via frequent fluctuations of potential well driven by repeated and fast gas outflows from the galactic centre [Navarro et al., 1996a, Pontzen & Governato, 2012]. The extra energy gained by DM particles expands their orbits and converts an initial cuspy density profile into a cored one. The degree of conversion depends on the total energy released from feedbacks measured by stellar mass $M_{\star}$, the total baryonic matter $M_{\rm{b}}$ opposing the expansion, and the energy transfer efficiency. However, the last one is found to be very sensitive to the star formation gas density threshold $n$, a numerical parameter commonly adopted in sub-grid models of galaxy formation [Benítez-Llambay et al., 2019, Dutton et al., 2019]. Cosmological simulations with a large $n$ report core formation in simulated dwarf galaxies [Governato et al., 2010, Tollet et al., 2016, Chan et al., 2015] while the dark matter cusp remains in simulations using a smaller $n$ [Sawala et al., 2016]. Till now, a consensus has not been reached about whether baryonic processes can solve the core-cusp problem, and it continues to cast doubt on the CDM model.

Apart from the cored profile, many dwarf and LSB galaxies also have similar shape of rotation curves, implying a self-similar dark matter structure [Kravtsov et al., 1998, Salucci & Burkert, 2000]. This conformity is unlikely to be due to chaotic and dramatic baryonic processes [Burkert, 1995, Navarro, 2019], but rather a clue to the nature of dark matter beyond CDM. In this work we study a decaying dark matter (DDM) model, which describes two-body decays of DM:

where $\psi^{\ast}$ stands for the unstable mother DM particle, $\psi$ is a massive stable daughter DM particle and $l$ is a light and relativistic particle. The DM has therefore multiple components. The dynamics of the two-body decays in this model are fully controlled by two parameters: the decay rate $\Gamma$ , or half-life $\tau^{\ast}=\ln{2}/\Gamma$, of mother particles and $\epsilon$ the energy of $l$ in unit of the mother particle’s rest mass. The recoil velocity $V_{k}$ of daughter particles in the center-of-mass frame of their mothers can be expressed as $V_{k}=\epsilon c$, where $c$ is the speed of light.

There are diverse behaviours of the DDM model depending on the values of $V_{k}$ and $\tau^{\ast}$. For a very small $\tau^{\ast}$ such that almost all mother particles decay into daughter particles before any non-linear structures have formed, the DDM model is similar to the warm dark matter (WDM) model with a free-streaming length determined by the recoil velocity $V_{k}$ [Kaplinghat, 2005, Strigari et al., 2007]. For a large $V_{k}$ such that the decays convert a fair amount of energy from the matter component to the relativistic species, the expansion history of the whole Universe will be altered [see, e.g., Vattis et al., 2019]. For a broad range of $\tau^{\ast}$ while $V_{k}\lesssim 40$ km s^-1, the 1D Lyman-$\alpha$ forest power spectrum predicted by the DDM model is shown to be consistent with observation data [Wang et al., 2013], implying that the model behaves like CDM at large scale. Inside a parameter region outlined by $10.0\lesssim V_{k}\lesssim 40.0$ km s^-1 (or equivalently $3.0\times 10^{-5}\lesssim\epsilon\lesssim 1.3\times 10^{-4}$) and $0.1\lesssim\tau^{\ast}\lesssim 14$ Gyr, decays significantly heat up DM inside dwarf-sized halos, and are relevant for the small-scale problems of CDM [Abdelqader & Melia, 2008, Peter & Benson, 2010, Wang et al., 2014]. In light of this, we revisit the core-cusp problem and study the density profiles of dwarf halos in the DDM model using high-resolution cosmological N-body simuations, which is absent from previous studies.

This paper is structured as follows: we first review previous DDM algorithms for N-body simulations and then introduce our new DDM algorithm, test its performance and calibrate numerical parameters for high-resolution zoom-in simulations in Section 2. An overview of our highest-resolution simulation suite is given in Section 3. Section 4 details our mathematical modelling of the evolution of an isolated halo in the DDM cosmology. We present our main results in Section 5, followed by more extensive discussions about the DDM model in Section 6. We summarize in Section 7.

Our level-$12$ simulation suite comprises 10 cosmological zoom-in simulations: 1 CDM and 9 DDM realizations. All 10 simulations use the same initial condition at redshift $99$. The high-resolution volume in the initial condition is a cuboid with three edge-lengths being $0.09$, $0.09$ and $0.15$ $h^{-1}$ Mpc, centering at its parent simulation box. The mass of each high-resolution particle is $3.89\times 10^{2}$ $h^{-1}$ $M_{\odot}$. The gravitational softening length of high-resolution CDM particles is set to be $33.7$ $h^{-1}$ pc, frozen to a physical length about $50$ pc after $z=10$. All DDM realizations follow the same softening length assignment scheme such that the force resolution of our simulation suite is uniform. All our zoom-in halos are free of contamination by low-resolution particles within their virial radii. The $9$ DDM realizations differ from each other in the combination of $V_{k}$ and $\tau^{\ast}$. We have used $3$ values of $\tau^{\ast}$: $3.0$, $6.93$ and $14.0$ Gyr, with corresponding decayed fractions $0.959$, $0.748$ and $0.495$, respectively. For each $\tau^{\ast}$, we use $3$ different values of $V_{k}$: $20.0$, $30.0$ and $40.0$ km s^-1. These $9$ realizations constitute a rough sampling of the interesting region in the $\tau^{\ast}-V_{k}$ parameter space. Halo expansion has been observed in the 9 DDM zoom-in halos as they generally have smaller virial masses but lower concentrations compared to the corresponding CDM halo. The virial masses, virial radii and other global properties of all zoom-in halos are summarized in Table 1. We study the physics accounting for the DDM halo expansion in next section.

The halo expansion in the DDM model is driven by two primary physical processes. The first one is the decay itself. On average, the kinetic energy of newly born daughter particles are greater than those of their mothers. This extra kinetic energy drives the orbits of daughters outwards, hence expanding the whole halo. We call this the Step-$1$ expansion. A consequence of this expansion is the weakening of the gravitational potential, triggering the Step-$2$ expansion: the bulk particles’ orbits expand outwards to rebalance the weakened gravity with the inertial force seen in the orbits’ rotating frames. Cen [2001] considers a special case of the two-step expansion: $V_{k}\gg v_{e}$ and $\tau>t_{\mathrm{dyn}}$, where $v_{e}$ and $t_{\mathrm{dyn}}$ are the halo’s typical escape velocity and dynamical time, respectively. The large $V_{k}$ unbinds all daughter particles during the Step-$1$ expansion while the slow decay simplifies the Step-$2$ evolution to an adiabatic expansion. Starting from an NFW density profile, the resulting density profile turns out to remain an NFW shape, but with a smaller concentration and a lower normalization density (see also the relavant discussion in Peter [2010]). Sánchez-Salcedo [2003] considers the situation where most of daughter particles are bound to the halo ($V_{k}<v_{e}$) and the halo expands adiabatically ($\tau>t_{\mathrm{dyn}}$). Through simple semi-analytic calculations, he argued that the cored profile is a natural result of the two-step expansion. In this section, we present a general formalism to implement the two-step expansion.

Our model assumes that a dark matter halo forms at a high redshift when only a small fraction of mother particles have decayed. Initially the halo is CDM-like with an NFW density profile. Then decays proceed and the density profile evolves. The decay of a mother particle is a random process and does not have a preferential direction. On average, each newborn daughter particle acquires an additional amount of kinetic energy from the mass deficit of its mother particle:

where $E_{k,\mathrm{mom}}$ is the kinetic energy of the decayed mother particle and $\langle E_{k,\mathrm{dau}}\rangle$ is the expected kinetic energy of its daughter particle with mass $m_{\rm{dau}}$. Similarly, the mean angular momentum of a daughter particle $\langle\boldsymbol{L_{\rm{dau}}}\rangle$ is the same as that of its mother, $\boldsymbol{L_{\rm{mom}}}$,

Based on equation (5) and equation (6), we build our simplified model using Schwarzschild’s orbit-based method [Schwarzschild, 1979] (see Chanamé [2010] for a short overview), which represents a collisionless system by a large library of particle orbits. Physical quantities, such as mass distribution, are then derived through constructing superpositions of these orbits.

Given a gravitational potential $\Phi(r)$, a particle’s orbit is a function of its total energy $E$ and angular momentum $\boldsymbol{L}$. The joint distribution of $E$ and $\boldsymbol{L}$ gives the whole library of particle orbits. To make the model as simple as possible, we assume all mother particles take circular orbits around the halo center, with random orientations of orbital planes. The daughter particles take up rosette orbits.

Now we consider the enclosed mass profile $M(R)$ as a superposition of orbits. Mathematically, all orbits in the library form a set. We name it as Olib. For each element $x$ in Olib, a weighting factor $g_{x}(R)$ is assigned such that $M(R)$ is the summation of all particle masses weighted by $g_{x}(R)$ over the set Olib:

where $m_{x}$ is the mass of the particle moving in the orbit $x$. The weighting factor $g_{x}(R)$ can be constructed as the fraction of time the particle spends inside the sphere $r=R$:

where $\Delta t_{x}(R)$ is the duration that a particle travels inside the sphere $r=R$ within an orbital period $T_{x}$. The weighting factors for circular orbits $g_{\rm{cir}}(R,r)$ are step functions since a circle with radius $r$ is either totally inside or totally outside the sphere $R$:

Similarly for a rosette orbit, if its perihelion $r_{\rm{min}}$ (aphelion $r_{\rm{max}}$) is larger (smaller) than $R$, then its weighting factor $g_{\rm{ros}}(R,r_{\rm{min}},r_{\rm{max}})$ is 0 (1). If $r_{\rm{min}}<R<r_{\rm{max}}$, the weighting factor is calculated as follows:

where $V_{\rm{eff}}(r)$ is the effective potential.

Consider the Step-$1$ expansion during a small time interval $\Delta t$ such that the gravitational potential $\Phi(r)$ remains static. Consider a pair of mother and daughter particles: a circular orbit with radius $r_{0}$ before decay and the corresponding rosette orbit after decay. According to equation (6), they share the same specific angular momentum $l_{0}$, thus the same effective potential:

The value of $l_{0}$ can be obtained by considering the circular motion at radius $r_{0}$, where $V_{\rm{eff}}$ reaches its minimum. As for the rosette orbit, its perihelion and aphelion are the two roots of the following equation from equation (5):

They both depend on the radius $r_{0}$ of its mother particle’s orbit. If equation (12) has only one root, daughter particle is considered to be unbound and makes no contribution to the mass profile $M$($R$).

After $\Delta t$, the mass increment of daughter particles inside the sphere $r=R$ can be calculated by considering the contributions from the daughter particles which are born during this time interval:

where $\Delta f$ is the fraction of mother particles decayed during $\Delta t$:

$\rho_{\rm{mom}}(r,t_{i})$ is the density profile of mother particles at time $t_{i}$. The mass profile of mother particles declines uniformly by a factor of $\Delta f$:

Then after the Step-$1$ expansion, the total mass profile $M(R)$ changes by an amount of

We proceed to consider the Step-$2$ expansion in the time interval $(t_{i},t_{i}+\Delta t)$. For the sphere $r=R$ at $t_{i}$, the adiabatic expansion reads:

from which $R_{f}(t_{i+1})$, the radius at $t_{i+1}=t_{i}+\Delta t$, can be derived. Initially the total mass inside $R$ is $M(R,t_{i})$. After $\Delta t$, the total mass inside $R_{f}(t_{i+1})$ is $M(R,t_{i})+\Delta M(R,\Delta t,t_{i})$. Therefore the mass profile $M$($R,t$) evolves a bit.

Through the two-step expansion we have evolved the whole system from $t_{i}$ to $t_{i+1}$. We loop the procedures listed above and evolve the whole system from the initial time $t_{0}$ to the final time $t_{f}$.

The simplified semi-analytic model presented in Section 4 is implemented with a well-tested numerical code called SemiCore. Starting from the best-fit NFW profile of the level-$12$ CDM halo ($M_{\rm{vir}}=5.17\times 10^{9}$ $h^{-1}$ $M_{\odot}$ and $c_{\mathrm{vir}}=21.6$), we run SemiCore with the combinations of $V_{k}$ and $\tau^{\ast}$ listed in Table 1, with the initial and final times being the same as those of cosmological N-body simulations. In Figure 3, we show the present ($z=0$) average density profile ratios, $\bar{\rho}_{\rm{ddm}}(r)/\bar{\rho}_{\rm{cdm}}(r)$, between the DDM-CDM halo pairs which evolve from the same primeval local overdensity field. We compare the results of the SemiCore model with those from the level-$12$ N-body simulations. For the SemiCore model, $\bar{\rho}_{\rm{cdm}}(r)$ is calculated from the input NFW profile.

The simulation data reveals that dark matter decays reduce the mass profile throughout the dwarf halos. The global reduction amplitude increases as $V_{k}$ increases or $\tau^{\ast}$ decreases. For a given pair of decay parameters, the difference between DDM and CDM halos become more pronounced as $r$ approaches to the halo center. These trends are well reproduced by the SemiCore model. Furthermore, the average density ratios $\bar{\rho}_{\rm{ddm}}/\bar{\rho}_{\rm{cdm}}$ predicted by SemiCore agree with those from sophisticated N-body simulations to better than $40\%$, for all resolved radii and for all combinations of $V_{k}$ and $\tau^{\ast}$ considered in this study. The triumph of SemiCore confirms the two-step expansion scenario for explaining the halo expansion induced by dark matter decays.

Figure 3 also shows that SemiCore systematically overpredicts the mass reduction in the inner region ($r\lesssim 0.1R_{\rm{vir}}$). It may be related to the simplified assumption of circular orbits for all mother particles. The same effect was observed in modelling adiabatic contraction of dark matter by circular orbits, which leads to an enhancement of the central density relative to the results of high resolution simulations [Gnedin et al., 2004]. In the outer region ($r\gtrsim 0.8R_{\rm{vir}}$), N-body simulations show that the ratios $\bar{\rho}_{\rm{ddm}}/\bar{\rho}_{\rm{cdm}}$ continue to grow while the SemiCore model predicts a flattened or slightly declining shape. Notice that the decay of a mother particle in a rosette orbit generally produces daughter particles that reach larger $r$, compared to those from a circular mother orbit. Also, there is mass accretion from the environment in N-body simulations, which is absent in the SemiCore model. Both factors can be responsible for the discrepancy seen in the outer halo region.

In Figure 3, the average density ratios from N-body simulations display a common shape: rising inner and outer regions connected by an extended plateau. A positive slope of the $\bar{\rho}_{\rm{ddm}}/\bar{\rho}_{\rm{cdm}}$ curve implies the flattening of DDM density profile compared to that of its CDM countpart. We calculate the rotation curve $V_{\rm{cir}}(r)$ scaled by $V_{0.3}$, the circular velocity at radius $R_{0.3}$ where $\mathrm{d}\ln{V_{\rm{cir}}}/\mathrm{d}\ln{r}=0.3$ [Hayashi & Navarro, 2006]. In Figure 4, we plot the scaled rotation curves for all level-$12$ DDM halos, and their values of $R_{0.3}$ and $V_{0.3}$ are listed in Table 1. The scaled rotation curves based on the cuspy NFW profile and cored Burkert profile [Burkert, 1995] are also plotted together for comparison. The DDM curves spread between the two theoretical curves in two groups. One is made up of $4$ DDM halos: $\rm{V}20\rm{T}14$, $\rm{V}30\rm{T}14$, $\rm{V}40\rm{T}14$ and $\rm{V}20\rm{T}7$. Their scaled rotation curves are closer to the NFW curve than the Burkert one. The remaining $5$ halos form the second group which features a significant deviation from the NFW curve and is much closer to the Burkert curve. For $\rm{V}20\rm{T}14$, a member of the pro-NFW group, and $\rm{V}40\rm{T}3$, a member of the pro-Burkert group, we further plot their scaled rotation curves of mother and daughter components in Figure 4. Surprisingly, the scaled daughter (mother) rotation curve of $\rm{V}40\rm{T}3$ ($\rm{V}20\rm{T}14$) follows the Burkert (NFW) shape. Since $\rm{V}40\rm{T}3$ has the strongest decay effect while $\rm{V}20\rm{T}14$ has the weakest, the simulation results show that the halo expansion due to dark matter decays can flatten the halo density profile and transform it from a cuspy shape to the cored shape, depending on the combination of $V_{k}$ and $\tau^{\ast}$ for a given dwarf halo mass.

The flattening of the central density profile of dwarf halos is needed to resolve the core-cusp problem of CDM. We show DDM and CDM’s rotation curves together with observational data in Figure 5. Halo $\rm{V}20\rm{T}3$ is a representative for the flattened DDM rotation curves. Two dwarf galaxies UGC07866 and UGCA444 are selected from the Spitzer Photometry $\&$ Accurate Rotation Curves (SPARC) database [Lelli et al., 2016] for comparison as they have roughly the same mass as the halo $\rm{V}20\rm{T}3$ [Li et al., 2020]. In order to cover the possible mass range where the two SPARC galaxies reside, we use SemiCore to calculate the rotation curves of two DDM halos, with current virial masses of $4.12\times 10^{9}$ $h^{-1}$ $M_{\odot}$ and $2.0\times 10^{10}$ $h^{-1}$ $M_{\odot}$, respectively. The two SemiCore runs both use $V_{k}=20.0$ km s^-1 and $\tau^{\ast}=3.0$ $\mathrm{Gyr}$. Their initial halos are set to follow the CDM $c-M$ relation [Diemer & Joyce, 2018]. For the CDM rotation curve, we assume an NFW profile and a fixed virial mass $4.22\times 10^{9}$ $h^{-1}$ $M_{\odot}$, the same as the halo $\rm{V}20\rm{T}3$. In the left panel of Figure 5, the DM contributed rotation curves of dwarf galaxy UGC07866 and UGCA444 are shown with observational uncertainties. The simulated or modelled DDM and CDM’s rotation curves have been scaled by a factor of $\sqrt{1-f_{b}}$, where $f_{b}$ is the cosmic baryon fraction of global matter density. It shows that the DDM rotation curves naturally follow the observational data while the CDM ones give good fit to data only with a small concentration, which is unusual for halos formed in CDM N-body simulations [Dutton & Macciò, 2014]. The data points at small radii ($R\lesssim 0.7$ $\rm{kpc}$) favour DDM curves as the NFW curves systematically overpredict the rotation velocities there. In the right panel of Figure 5, we plot the corresponding scaled rotation curves. With considerably large uncertainties, the observational data distribute between the NFW and Burkert curves at small radii, in good agreement with the DDM scaled rotation curves for dwarf galaxies. Our results agree with Sánchez-Salcedo [2003] in that the core-cusp problem can be solved in the DDM model with a recoil velocity $V_{k}$ smaller than the typical escape velocities of dwarf halos, provided that the decay half-life $\tau^{\ast}\lesssim 7.0$ $\mathrm{Gyr}$.

In Figure 6, we show the positions of the $9$ zoom-in dwarf halos in the DDM parameter space. They reside inside a region, shown in gold, where dark matter decays have prominent effects on the number density and inner structure of dwarf galaxies [Peter & Benson, 2010]. The $5$ zoom-in halos with a much more flattened central density are indicated by blue squares and the remaining $4$ by blue triangles. For $V_{k}=20.0$ km s^-1 and $\tau^{\ast}=6.93$ $\mathrm{Gyr}$, Wang et al. [2014] have run a zoom-in simulation on a Milky-Way sized host halo with the PMK10 algorithm. They found that the circular velocity profiles of the $15$ most massive subhalos pass through most of the data points from the $9$ classical Milky-Way dSphs, and therefore the too-big-to-fail problem [Boylan-Kolchin et al., 2011] is potentially resolved. The brown strip is the parameter region, shown by Abdelqader & Melia [2008] using a semi-analytic model that incorporates dark matter decays in the hierarchical formation history of dark matter halos, that can account for the deficit of dwarf galaxies in our local group, a puzzle closely related to the missing-satellites problem [Klypin et al., 1999, Moore et al., 1999a]. It can be seen that several different CDM problems can be solved by a common parameter subspace in the DDM model.

In Figure 7, we show that $R_{0.3}$ and $V_{0.3}$ have a power-law relation for DDM zoom-in dwarf halos:

where the power index $\beta$ increases as $\tau^{\ast}$ decreases. Given the values of $R_{0.3}/R_{\rm{vir}}$ and $V_{0.3}$, both being observable quantities, a relation between $V_{k}$ and $\tau^{\ast}$ can be derived from relation (18), implying a possible degeneracy in the DDM parameter space as far as the halo’s inner structure is concerned. As $V_{k}$ and $\tau^{\ast}$ are constants in the DDM model, from relation (18) we also find

Since dwarf halos have a narrow virial mass spectrum and equation (19) depends only weakly on $M_{\rm{vir}}$, an approximate universal density profile is expected. The observed value of $\beta$ can then be used to measure $\tau^{\ast}$. We will study this possibility in a future work.

In this work we improved the DDM N-body algorithm by combining the advantages of the PMK10 and CCT15 algorithms. The new algorithm outperforms PMK10 in accuracy while demands much less computing resources than CCT15. Same as CCT15, the new algorithm samples dark matter decays only at certain times and evolves the whole N-body system in a collisionless way at other times. This feature enables the algorithm to be implemented in a plugin module called DDMPLUGIN, which can be used with any CDM N-body code.

We carried out high-resolution cosmological N-body simulations to study the density profiles of dwarf halos with the new DDM algorithm. Good numerical convergence was achieved, and we succeeded to resolve the halo structure robustly down to about $700$ pc. Compared to CDM counterparts, DDM dwarf halos have lower mass concentration and shallower density profile at the inner region. Adopting the orbit-superposition method for mass profile construction, we developed a simplified semi-analytic model for the DDM halo mass profile, which features rosette orbits for daughter particles and incorporates effects of dark matter decays and adiabatic expansion. Although simple, the model predicts DDM halo mass profiles that agree semi-quantitatively with resolved simulation profiles. It therefore illustrates clearly the physics mechanisms involved in the transformation from cusp to core density profiles.

We also calculated the scaled rotation curves for DDM simulation halos and compared them with $2$ dwarf galaxies from the SPARC database. The shape of the DDM rotation curve is shallower than that based on the NFW profile but steeper than the Burkert’s. The two SPARC dwarf galaxies favour the DDM shape of the rotation curve. Furthermore, we show that there is an approximate universal power-law relation between $V_{0.3}/V_{\rm{vir}}$ and $R_{0.3}/R_{\rm{vir}}$ for dwarf halos, which can be used to extract DDM parameters from observation data. Together with previous studies, this work supports the DDM cosmology, which keeps the success of CDM at large scale and reconciles the differences between observations and predictions from N-body simulations at small scale.