The virial mode $k_{v}$ approach to Structure Formation with Warm Dark Matter

The standard $\Lambda$CDM model has been a very successful model to describe our Universe and is consistent with cosmological observational evidence such as the cosmic microwave background (CMB) anisotropies [4], galaxy redshift surveys [5], type Ia Supernovae [6] reach to the conclusion that the content of the Universe is composed of 69% dark energy driving the accelerated expansion of the Universe, 31% matter whose clustering feature influence the large scale structure formation, corresponding to 27% Dark Matter (DM) and the remaining 4% is baryonic matter.

The nature of dark matter has received a great deal of attention in the last decade, due in part of the missing satellites in the universe, and the amount of structure at different scales and redshifts puts strong constraints on the nature of dark matter. In recent times the large surveys such as SDSS-IV [7] and in the near future, DESI [8] in the near future will have an important impact in determining the properties of dark energy (DE) and dark matter (DM). Despite the efforts in both particle physics and cosmology, the nature and composition of the DM are still unknown. Candidates for DM can be classified according to its velocity dispersion, $v_{nr}$, when the DM particle become non-relativistic (given by the scale factor $a_{\rm nr}$). For thermal relics one can relate the mass of the WDM particle to $a_{\rm nr}$. In this case DM is cold for mass larger than $m_{\rm cdm}\sim\mathcal{O}$(MeV). This kind of DM particles stop being relativistic and start clustering object at early times. DM with a mass around $m_{\rm wdm}\sim\mathcal{O}$(keV) are known to be warm, WDM, whose main attribute is that its dispersion velocity wipes out some density concentrations of matter and, therefore, induce a cut-off scale into the mass halo function [9]. The amount of energy density today $\Omega_{\rm dmo}$ along with the scale factor when the WDM particle becomes non-relativistic $a_{nr}$ are crucial for determining the properties of the large scale structure of the universe. For instance, the dispersion velocity of DM particles wipes out density concentrations of matter and, therefore, induce a cut-off scale in the mass halo function [9]. Most DM candidates have a smooth evolution of the DM velocity, however, phase transitions in the underlying particle physics model for dark matter shows that an abrupt transition to a non-relativistic limit is plausible [10, 11].

The large velocity dispersion of DM at early stages on the evolution of the Universe tends to suppress gravitational clustering at small scales and conciliates with what it is observed. Cosmological N-body simulations of the $\Lambda$CDM model predict the number of satellite galaxies in Milky Way-like galaxies is smaller than the expected, the so-called missing satellite problem [12, 13, 14] and high concentrations of DM in the innermost regions of galaxies (cusp-core problem [15, 16]). Baryon physics also pursued the solutions to this problem by integrating star formation and halo evolution in the galaxy, however, the discussion is still in progress [17, 18, 19]. The details of the suppression at small scales depend on the DM particle nature, which takes us to connect the DM models and astrophysical observations. It can be determined by the parametrization of the transfer function $T_{X}(a,k)=(P_{X}/P_{cdm})^{1/2}$ in terms of the power spectrum of $X$ DM particles and $\Lambda$CDM model, where $T_{X}$ is scale and time-dependent. Is has been conventional to compare the $X$ and CDM models at the epoch when the amplitude of the fluctuations of model $X$ is half the size of $\Lambda$CDM , i.e. $T_{X}(a,k)^{2}=1/2$. Different ansatzes have been proposed to parametrize the transfer function $T_{X}(a,k)$ and the parameters involved must be numerically fitted using numerical Boltzmann codes [2, 3].

Here, we will present a new approach to structure formation where the virial radius places a dominant role in structure formation. This new approach is physically motivated and is consistent with previous works in WDM structure formation and it allows for an understanding of the suppression of small scale structure in terms of the virial mass and radius. We introduce a new analytical transfer function, physically motivated, that reproduces the clustering of large classes non-thermal DM models and preserving the connection with the physics and nature of the DM.

We present and compare the standard and our virial transfer function in section 2.1 and section 3 , and the free streaming scale in Section 4, the dark matter analysis and finally the conclusion in section 5.

The clustering properties of Dark Matter have a direct impact on the number of halo as a function of mass and redshift and can be contrasted with several observational large scale structure experiment as [7]. The velocity dispersion of DM particles plays a crucial roll in structure formation. While DM particles are still relativistic, primordial density fluctuations are suppressed due to the velocity dispersion of DM particles and inhibit the formation of structure below the free streaming scale $\lambda_{fs}$ [20]. Here, we will present two approaches to extract cosmological clustering properties of warm dark matter (WDM) in terms of Transfer Function $T(a,k)$ defined as the quotient of the linear matter perturbations between WDM and CDM. We refer to the standard approach the work of M. Viel et al. [2, 21] presented in section 2.1 while our virial approach is given in section 3.

Throughout this paper, we adopt Planck 2018 cosmological parameters [4] in a flat Universe with $\omega_{dmo}=0.12$, and $\omega_{bo}=0.02237$ as the CDM matter and baryonic with $\omega_{i}=\Omega_{i}h^{2}$ (with $i=dm,b$) and $h=0.6736$ the Hubble constant in units of 100 km s^-1Mpc^-1, $z_{\rm reio}=7.67$ the reionization redshift , $n_{s}=0.965$ the tilt of the primordial power spectrum and $\ln(10^{10}A_{s})=3.044$ with $A_{s}$ the amplitude of primordial fluctuations.

We propose to determine the transfer function $T_{v}(k_{v})$ as a function of the viral mode $k_{v}$, defined in terms of the free streaming mode with $k_{v}=2\pi/\lambda_{v}$ with $r_{v}=\lambda_{v}/2$ half the radius of as structure with a size of the free streaming scale $\lambda_{fs}$. We refer to he transfer function $T_{v}(k_{v})$ as the virial approach. The free streaming scale $\lambda_{fs}$ and mass $M_{fs}$ are

with $r_{fs}=\lambda_{fs}/2$ and $\rho_{o}$ present matter content of the universe. The virial mode, radius and viral mass are given by

The mass contained in the virialized structure is conserved and the virial mass is $M_{v}=M_{fs}$ and since $r_{v}$ is half the size of $r_{fs}$ the density of the virial sphere is $\rho_{v}=8\rho_{0}$. Of course, a halo density profile such as NFW [25, 26] for example, would be more appropriate to determine the energy density inside this structure but this lies beyond the scope of this work. We take the transfer function for the virialized mode $k_{v}$ as a two-parameter linear transfer function $T_{v}(k_{v})$ as a function of the viral mode $k_{v}$, similar as the Standard Transfer,

with $\beta_{v}$ and $\gamma_{v}$ constant parameters. We determine the values of these parameters in the next section (3), where we find that $\beta_{v}$ and $\gamma_{v}$ are correlated leaving $T_{v}(k)$ with only one free parameter.

The thermal velocities of the dark matter particles have a direct influence on structure formation. While DM particles are still relativistic, primordial density fluctuations are suppressed on scales of order the Hubble horizon at that time. This is called the free-streaming scale and depends on the time when a massive particle becomes non-relativistic $t_{nr}$. It is defined by

The corresponding free-streaming scale mode $k_{fs}$ and a mass $M_{fs}$ contained in sphere of radius $\lambda_{fs}/2$ are given by

and in terms of the viral mode

with $\rho_{v}=8\rho_{mo}$. For mass-scales $M\lesssim M_{v}$ the free-streaming of particles erases all peaks in the initial density field therefore the number of structures below this mass scale should be significantly reduced in numbers. We show this behavior in Fig. 4, where we compare CDM and WDM mass functions. However, it is the virialized scale that accounts for the mass contained within the structure to be formed.

We studied the clustering properties of WDM particles and we present the viral approach, where the transfer function depends on the viral wave number $k_{v}=4\pi/\lambda_{fs}$, and we compare it to the standard approach. The velocity dispersion of WDM at early stages on the evolution of the universe suppress gravitational clustering and may conciliates observations of the number of small scales galaxies. The details of the suppression depend on the properties of the DM particles, however a key ingredient is the time when these particles become non-relativist given by the scale factor $a_{nr}$.

We have presented here a new approach to structure formation where the virial radius $r_{v}$ places a dominant role. The viral mode $k_{v}$ is defined in terms of the free streaming scale $\lambda_{fs}$ and depends thus directly on $a_{nr}$. This new approach is physically motivated, is consistent with previous works in WDM structure formation, and allows for an understanding the suppression of small scale structure in terms of the virial mass and radius, $M_{v}$ and $r_{v}$ respectively. The transfer function $T(a,k)$ is determined in terms of the virial mode $k_{v}$ which has a clear interpretation in terms of the free streaming scale and is easily calculated, contrasting with the $\alpha$ parameter of eq.(2.4) [2, 3] which requires a numerical fit.

From Table (1) we see that our virial transfer function performs better than the standard transfer function for WDM with masses in the range (1-10) keV. The standard transfer function has up to two constant parameters ($\beta,\gamma$) which can be reduced to only one $\mu$ with $\beta=2\mu,\;\gamma=-5/\mu$ with $\mu=1.12$ supplemented by the $\alpha$ containing the relevant clustering parameters of the WDM model in eq.(2.4) and has been fitted to give the correct value of $k_{1/2}$ for different WDM models. Since $\alpha$ has been numerically fitted we consider it as a free parameter. Therefore the standard transfer function has up to three parameters and can be reduced to two parameters by taking the fitted constraint $\gamma\beta=-10$. On the other hand our viral Transfer function parametrization in eq.(3.7) has only two parameters $\beta,\gamma$. As in the Standard Transfer function we can reduce a parameter by taking $\gamma_{v}\beta_{v}$ constant. Doing so, we found $\beta_{v}=2\nu,\,\gamma_{v}=-9/\nu$ with $\nu=1.09$, and $\gamma_{v}\,\beta_{v}=-18$. Taking $\beta_{v}$ and $\gamma_{v}$ independent we obtained $\beta_{v}=2.05\pm 0.04$, $\gamma_{v}=-8.94\pm 0.10$ with $\gamma_{v}\,\beta_{v}=-18.32$ at the central values. Clearly the one and two parameter Transfer Function in the viral approach are consistent. We obtained for a WDM thermal particle that becomes non-relativistic at $a_{nr}=2.77\times 10^{-8}$, corresponding to a 3 keV mass, a viral radius $r_{v}=26.4$M_pc with wave number $k_{v}=21.64$ in the viral approach and a $k_{1/2}=21.86$ in the standard approach, corresponding to a structure with a contained mass of $\mathcal{O}(10^{7})\;{\rm M}_{\odot}$.

We have shown that our virial parametrization of the transfer function is physically motivated and has an excellent agreement with numerical results from Boltzmann CLASS codes shown in Table (1) and Fig.(2) and Fig.(1) and outperforming slightly the standard transfer function [2, 20] given in eq.(2.3) for masses in the range 1-10 keV. The parameter $\alpha$ in the Standard Transfer function must be fitted numerically using Boltzmann code, while in our approach the physical parameter $k_{v}=2k_{fs}$ is determined in terms of the free streaming mode $\lambda_{fs}$ an easily computed quantity.

AM acknowledges partial support from Project IN103518 PAPIIT-UNAM, PASPA DGAPA-UNAM and CONACyT. JM acknowledges Catedras-CONACYT financial support and MCTP/UNACH as the hosting institution of the Catedras program.