Cosmological constraints on the velocity-dependent baryon-dark matter coupling

TL;DR

This paper constrains baryon-dark matter interaction cross sections using Planck CMB and SDSS Lyman-alpha data, revealing mass-dependent bounds for dark matter below 1 MeV, contrasting previous mass-independent claims.

Contribution

It introduces a mass-dependent analysis of baryon-dark matter coupling constraints for dark matter masses below 1 MeV, incorporating free-streaming effects similar to warm dark matter.

Findings

Mass-dependent bounds on baryon-dark matter cross section for m_χ down to 5 keV.

Constraints show mass dependence, unlike previous mass-independent results.

Provides new limits on dark matter interactions in the sub-MeV mass range.

Abstract

We present the cosmological constraints on the cross section of baryon-dark matter interactions for the dark matter mass below the MeV scale from the Planck CMB (cosmic microwave background) and SDSS (Sloan Digital Sky Survey) Lyman-$\alpha$ forest data. To explore the dark matter mass $m_{\chi}\lesssim 1$ MeV for which the dark matter's free-streaming effect can suppress the observable small scale density fluctuations, in addition to the acoustic oscillation damping in existence of the baryon-dark matter coupling, we apply the approximated treatment of dark matter free-streaming analogous to that of the conventional warm dark matter. We also demonstrate the mass dependence of the baryon-dark matter cross section bounds (for the dark matter mass down to $m_{\chi} \sim 5~{\rm keV}$), in contrast to the dark matter mass independence of the cross section constraints for the light dark matter below the MeV scale claimed in the previous literature.

Tables

Table 1. Table 1: 95 % percent 95 95\% confidence limits for the upper bounds on the baryon-DM cross section constant σ 0 subscript 𝜎 0 \sigma_{0} in units of cm 2 superscript cm 2 \rm cm^{2} , where only the CMB data was used in the analysis.

	CMB ($95\%$ C.L.)
$n$	$m_{\chi }=5$ keV	$m_{\chi }=7$ keV	$m_{\chi }=10$ keV
$-4$	$1.6\times {10}^{-41}$	$1.6\times {10}^{-41}$	$1.6\times {10}^{-41}$
$-2$	$2.0\times {10}^{-33}$	$2.0\times {10}^{-33}$	$2.0\times {10}^{-33}$
$-1$	$1.1\times {10}^{-29}$	$1.1\times {10}^{-29}$	$1.1\times {10}^{-29}$
$0$	$3.4\times {10}^{-26}$	$3.4\times {10}^{-26}$	$3.7\times {10}^{-26}$
$+2$	$7.1\times {10}^{-24}$	$1.0\times {10}^{-23}$	$1.6\times {10}^{-23}$

Table 2. Table 2: 95 % percent 95 95\% confidence limits for the upper bounds on the σ 0 subscript 𝜎 0 \sigma_{0} in units of cm 2 superscript cm 2 \rm cm^{2} , where the CMB and Ly − α 𝛼 -\alpha data were used.

	CMB + Lyman-$\alpha$ ($95\%$ C.L.)
$n$	$m_{\chi }=5$ keV	$m_{\chi }=7$ keV	$m_{\chi }=10$ keV
$-4$	$1.8\times {10}^{-41}$	$1.9\times {10}^{-41}$	$1.9\times {10}^{-41}$
$-2$	$1.8\times {10}^{-34}$	$1.9\times {10}^{-34}$	$2.2\times {10}^{-34}$
$-1$	$2.9\times {10}^{-31}$	$3.6\times {10}^{-31}$	$3.6\times {10}^{-31}$
$0$	$3.2\times {10}^{-29}$	$4.8\times {10}^{-29}$	$6.7\times {10}^{-29}$
$+2$	$4.0\times {10}^{-31}$	$1.2\times {10}^{-29}$	$3.9\times {10}^{-28}$

Equations

$$\sigma (v)={\sigma }_0{v}^n,$$

$${\dot{T}}_{\chi }=$$

$${\dot{T}}_{\mathrm{b}}=$$

$$+\frac{2{\mu }_{\mathrm{b}}}{m_{\mathrm{e}}}{R}_{\gamma }(T_{\gamma }-T_{\mathrm{b}}),$$

$$R_{\chi }=\frac{{\rho }_{\mathrm{b}}{\sigma }_0{c}_n}{m_{\chi }+m_{\mathrm{H}}}\frac{{\mathcal{F}}_{\mathrm{He}}}{1+z}{\left(\frac{T_{\mathrm{b}}}{m_{\mathrm{H}}}+\frac{T_{\chi }}{m_{\chi }}+\frac{V_{\mathrm{RMS}}^2}{3}\right)}^{\frac{n+1}{2}},$$

$$V_{\mathrm{RMS}}^2=\{\begin{array}{ll}10^{-8}& z>10^3\\ 10^{-8}{\left(\frac{1+z}{10^3}\right)}^2& z\le 10^3.\end{array}$$

$$T_{\chi }=\{\begin{array}{ll}{T}_{\mathrm{b}},& R_{\chi }\frac{m_{\chi }}{m_{\chi }+m_{\mathrm{b}}}>\frac{H}{1+z}\\ T_{\mathrm{dec}}{\left(\frac{1+z}{1+z_{\mathrm{dec}}}\right)}^2,& R_{\chi }\frac{m_{\chi }}{m_{\chi }+m_{\mathrm{b}}}<\frac{H}{1+z}.\end{array}$$

$${\dot{\delta }}_{\mathrm{b}}=$$

$${\dot{\theta }}_{\chi }=$$

$${\dot{\theta }}_{\mathrm{b}}=$$

$$+R_{\gamma }({\theta }_{\gamma }-{\theta }_{\mathrm{b}})+k^2\psi ,$$

$$c_{\mathrm{b}}^2$$

$$c_{\chi }^2$$

$$\mathcal{T}(k)={\left[1+(\alpha k{)}^{2\nu }\right]}^{-5/\nu },$$

$$\alpha =0.24{\left(\frac{m_{\chi }/T_{\chi }}{1\mathrm{keV}/T_{\nu }}\right)}^{-0.83}{\left(\frac{{\Omega }_{\chi }{h}^2}{0.12}\right)}^{-0.16}\mathrm{Mpc},$$

$$\frac{T_{\chi }}{T_{\nu }}={\left(\frac{10.75}{g_{\ast }(T_{\mathrm{dec}})}\right)}^{1/3},$$

$$P(k)={\mathcal{T}}^2(k)P_{\ast }(k).$$

$$100\Theta \in (0.5,10),{\Omega }_{\mathrm{b}}{h}^2\in (0.005,0.04),$$

$${\Omega }_{\chi }{h}^2\in (0.01,0.5),{\tau }_{\mathrm{reio}}\in (0.005,0.5),$$

$$\ln (10^{10}{A}_{\mathrm{s}})\in (0.5,10),n_{\mathrm{s}}\in (0.5,1.5).$$

Full text

PACS numbers

95.35.+d ,98.80.Es

††preprint: APS/123-QED

I Introduction

While there is a growing support for the existence of dark matter (DM) from the astrophysical observables, its precise nature such as its mass and how it interacts with the visible matter remains unknown. The current studies favor the cold dark matter (CDM) paradigm Planck in which DM particles are “cold” (non-relativistic) and can interact with other particles only through the gravitational force. While CDM can explain well the large-scale structure observations, there are still unsettled challenges in the CDM paradigm on small scales Kauffmann:1993gv ; Flores:1994gz ; Moore:1994yx ; Moore:1999nt ; Klypin:1999uc ; Read:2005zv ; BoylanKolchin:2011de ; Wetzel:2016wro ; Kim:2017iwr and there is hence still a room to seek the hint on the new physics beyond the CDM paradigm. We in this paper study the possibility for the DM non-gravitationally scattering with the baryons and its consequences on the cosmological observables.

Many theoretical works suggest the DM particle models which have non-gravitational interactions with baryons, such as the DM interacting through its dipole moment (dipole DM) and the millicharged DM Sigurdson:2004zp ; dav ; turnoff ; essig2 ; eee ; cora ; kenjisn ; celine ; boesh ; rocky ; beam ; bur ; pos ; dubo ; Melchiorri:2007sq ; mas ; bank ; ess ; semi ; light ; barg2 ; heo3 ; ilidio ; gary ; kst ; gk which have received the revived interests in view of the recent global 21 cm signal measurement by the EDGES Bowman:2018yin ; Tashiro:2014tsa ; Munoz:2015bca ; Barkana:2018lgd ; Berlin:2018sjs ; Kovetz:2018zan .

Since the baryon-DM interactions can give the significant effects on the fluid dynamics of primordial plasma and structure formations, the observations of cosmic microwave background (CMB) anisotropies and large-scale structure can tightly constrain the baryon-DM couplings Loeb:2005pm ; Bertschinger:2006nq ; Dvorkin:2013cea . We in this paper study the constraints on such a DM model by using the Planck CMB data  Planck and the Lyman-$\alpha$ forest data from the Sloan Digital Sky Survey (SDSS) SDSS . Although there already exist recent works on this subject Xu:2018efh ; Boddy:2018wzy ; Slatyer:2018aqg , our study extends the previous literature by covering the DM mass range below the MeV scale and finds the unique features peculiar to the small mass such as the dependence of the baryon-DM cross section bounds on the DM mass.

The remainder of the paper is organized as follows: Sec. II discusses the temperature and perturbation evolutions of the DM and baryon in existence of the baryon-DM coupling. Sec. III describes our likelihood analysis for constraining the cosmological parameters including the baryon-DM coupling cross section, followed by the presentation of the results in subsections along with the comparison to the previous works. Finally, the summary of this work is given in Sec. IV.

II Cosmological evolution with baryon-dark matter coupling

The baryon-DM coupling gives an impact on the thermal history and the structure formation of the Universe. To evaluate such effects, we consider the velocity-dependent baryon-DM scattering cross section parameterized as Dvorkin:2013cea ; zurekyu ; raby ; plasma

[TABLE]

where $v$ denotes the baryon-DM relative velocity. While $n$ depends on the type of the coupling, we present our discussions for several representative values of $n$ without specifying the concrete particle theory models ArkaniHamed:2008qn ; Buckley:2009in ; Sigurdson:2004zp ; Melchiorri:2007sq to keep our discussions as general as possible so that our cosmological constraints can be applicable to a wide range of DM scenarios. We set $c=\hbar=k_{\rm B}=1$ in our discussions.

II.1 Thermal evolution

The evolution equations for the DM and baryon temperatures, $T_{\chi}$ and $T_{\rm b}$ read, in existence with the baryon-DM coupling,

[TABLE]

where the dot denotes a derivative with respect to the conformal time, $\rho_{\chi}$ is the DM energy density, $\rho_{\rm b}$ is the baryon energy density, and $m_{\chi}$, $m_{\rm H}$ and $m_{\rm e}$ are the DM, hydrogen and electron masses respectively. $\mu_{\rm b}=m_{\rm H}(n_{\rm H}+4n_{\rm He})/(n_{\rm H}+n_{\rm He}+n_{\rm e})$ is the mean molecular weight for baryons with the number density for hydrogen, helium and free electrons, $n_{\rm H},~{}n_{\rm He},~{}n_{\rm e}$, and $R_{\gamma}=4n_{\rm e}\sigma_{\rm T}\rho_{\gamma}/{3\rho_{\rm b}}/(1+z)$ is the Thomson scattering rate with the Thomson cross section $\sigma_{\rm T}$. In Eq. (2), $R_{\chi}$ represents the baryon-DM scattering rate. To calculate $R_{\chi}$, it is required to evaluate the relative velocity $v$ which is related not only to the thermal velocity dispersions of both baryons and DM but also to the relative peculiar velocity between them. Following Refs. Dvorkin:2013cea ; Xu:2018efh , $R_{\chi}$ is given by

[TABLE]

where $V^{2}_{\rm RMS}$ is the rms peculiar velocity and we adopt the evolution of $V^{2}_{\rm RMS}$ in the standard cosmology Xu:2018efh ,

[TABLE]

In Eq. (4), $c_{n}$ is the $n$-dependent constant and ${\mathcal{F}}_{\rm He}$ represents a ratio of the cross section for helium to hydrogen to take account of the difference in DM-hydrogen and DM-helium scatterings Dvorkin:2013cea . In our calculation, following previous works Dvorkin:2013cea ; Xu:2018efh , both $c_{n}$ and ${\mathcal{F}}_{\rm He}$ are set to $c_{n}\approx\{0.27,0.53,1,2.1,13\}$ with $n=\{-4,-2,-1,0,2\}$ and ${\mathcal{F}}_{\rm He}=0.76$ for simplicity.

The tight coupling can hold roughly until the redshift $z_{\rm dec}$ satisfying $R_{\chi}{m_{\chi}}/({m_{\chi}+m_{\rm b}})=H_{\rm dec}/(1+z_{\rm dec})$ where the subscript $\rm dec$ represents the value at $z_{\rm dec}$. $(1+z)R_{\chi}/H$ can increase for a bigger $z$ if $n>-3$, and the DM temperature can tightly couple with the baryon temperature at a high redshift for $n>-3$ Dvorkin:2013cea . We hence adopt the sudden decoupling approximation in treating the thermal evolution, for $n>-3$, until $z=10^{4}$ for the case with $z_{\rm dec}>10^{4}$ in order to reduce the computational cost111This ”sudden decoupling approximation” induces a negligible difference in our calculations for the cross section bounds compared with the exact ones Xu:2018efh .,

[TABLE]

For $z<10^{4}$, the temperature evolution is obtained from Eqs. (2) and (II.1) numerically with the initial condition Eq. (6).

On the other hand, when $z_{\rm dec}<10^{4}$, we take the tight thermal coupling $T_{\rm b}=T_{\chi}$ before $z=10^{4}$ and we thereafter numerically solve Eqs. (2) and (II.1).

For $n=-4$, since the coupling between baryon and DM components is weaker at a higher redshift, the DM temperature evolves adiabatically, $T_{\chi}\propto(1+z)^{2}$, after becoming non-relativistic and is much lower than the CMB and baryon temperatures. We therefore solve Eqs. (2) and (II.1) from $z=10^{4}$ with the initial condition $T_{\chi}=0$.

For an illustration purpose, we show the evolutions of photon, baryon and DM temperatures with $n=-4$ for different DM masses in Fig. 1. The low DM temperature can be heated up by the higher baryon temperature due to the baryon-DM couplings. We can see in this figure that, after the baryon temperature decoupled from the CMB temperature, the baryon temperature is dragged toward the DM temperature and, resultantly, the DM and baryon temperatures can couple at low redshifts.

II.2 Density fluctuation evolution

We now consider the evolution of the density fluctuations in existence of the baryon-DM coupling. We work in the Newtonian gauge in the following. For a given Fourier mode $k$, the density fluctuations, $\delta_{\rm b}$ and $\delta_{\chi}$, and the divergences of the fluid velocities222Note that $\theta=ikv$, here $v$ is the fluid velocity., $\theta_{\rm b}$ and $\theta_{\chi}$, for baryon and DM components evolve as

[TABLE]

where $\psi$ and $\phi$ are the Newtonian gravitational potential and the spatial metric perturbation, respectively. $c_{\rm b}$ and $c_{\chi}$ represent the sound speeds of each fluid

[TABLE]

For the initial condition of the fluctuations, we adopt the standard adiabatic initial condition in which the initial density fluctuations and the velocity divergence match the baryon ones.

We focus on the DM mass below the MeV scale. In such a mass range, we have to take into account a relativistic free-streaming effect which erases small scale density fluctuations in the early universe, as in the analysis for the warm dark matter (WDM) model.

The public Boltzmann code, CLASS, provides us the treatment to solve the Boltzmann equations for the WDM model, and we modified the CLASS in order to calculate Eqs. (7)–(9). This method is however computationally demanding in performing the Markov chain Monte Carlo (MCMC) analysis to obtain the observational constraint on the baryon-DM coupling parameters. To facilitate our analysis (without losing much precision as demonstrated in the following section), we adopt the following WDM approximation to take into account the free-streaming effect due to the small DM mass.

II.3 The matter power spectrum suppression

The matter power spectrum suppression due to the acoustic damping can affect the Ly-$\alpha$ observations if the DM couples to the baryon around $z\sim 10^{6}$. The mode relevant for the Ly-$\alpha$ observations ($k\sim 1\ h$/Mpc) enters the horizon around $z\sim 10^{6}$ (corresponding to the temperature of order keV), and the DM hence can be treated as non-relativistic around $z\sim 10^{6}$ of our interest if the mass is above keV.

In addition to such acoustic oscillation damping effects, the DM free-streaming effects can also suppress the matter power spectrum. The DM can free-stream once it decouples from the baryons, and the matter power spectrum due to such free-streaming effects can potentially appear at the small scales observable by the Ly-$\alpha$ measurements if the DM is relativistic at the decoupling epoch. If the DM is non-relativistic at the decoupling, on the other hand, the free-streaming length would be too small to be observable by the Ly-$\alpha$. In our computation, we regard the free-streaming effect is negligible when the decoupling temperature become $T_{\rm dec}<0.1\ m_{\chi}$.

To illustrate the DM decoupling epoch and the epoch when DM becomes non-relativistic, we plot the ratio of two time scales, $R_{\chi}$ and $H/(1+z)$, in Fig. 2 333The sudden decoupling approximation is reflected on the slight kink in the scattering rate evolution.. The orange vertical line represents the redshift, $z_{\rm rel}$, before which the DM with mass $m_{\chi}=10\ {\rm keV}$ is relativistic. The yellow vertical line is the redshift $z_{\rm in}$ around which the largest observable wavenumber $k=1~{}\rm Mpc^{-1}$ enters the horizon.

If the ratio of two scales $(1+z)R_{\chi}/H$ is much larger than unity, the baryon-DM coupling is effective.

Let us now outline our treatment of the free streaming effects.

First, for $n\geq-3$, we solve Eqs. (7)–(9) numerically without the free-streaming effect and obtain the power spectrum $P_{*}(k)$ in Eq. (15). Then, similar to the manner in the warm DM analysis Bode:2000gq , we postprocess the free-streaming effect on the power spectrum by using the transfer function ${\cal T}(k)$.

[TABLE]

with the fitting parameters, $\alpha$ and $\nu$. We adopt the fitting parameters suggested in Ref. Viel:2005qj ;

[TABLE]

and $\nu=1.12$. $\Omega_{\chi}h^{2}$ is the current value of the DM density parameter multiplied by the square of the Hubble constant $H_{0}$ (here, $H_{0}=100h\ {\rm km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$). The temperatureratio between the DM and the neutrino components is written as Viel:2005qj

[TABLE]

where $g_{*}(T_{\rm dec})$ denotes degrees of freedom at temperature $T_{\rm dec}$ when the DM decoupling occurs. We compute $g_{*}(T_{\rm dec})$ by using Table A1 in Ref. Husdal:2016haj .

In short, our final matter power spectrum with considering the free-streaming effect of the light DM mass is given by (compared with the standard CDM matter power spectrum $P_{*}$ without the free streaming effects)

[TABLE]

Fig. 3 shows the effects of free streaming and compare the power spectra between the one evaluated from Eq. (15) and the one computed with the full numerical treatment by the CLASS. We can see the discrepancy is less than 1 % for the parameter range of our interest, and our using the fitted transfer function instead of using the computationally demanding full numerical solution suffices for our purpose of demonstrating the possible bounds on the baryon-DM cross sections.

We note that such a free-streaming effect approximation discussed above however does not work for $n<-3$. As shown in Fig. 2, the baryon-DM coupling becomes weaker as the redshift increases. In this case, the DM particles are not thermally coupled with the baryon in the early epoch. One expects that the baryon-DM interaction cross section can be more tightly constrained by the CMB than by the Ly-$\alpha$ for $n=-4$ because the baryon-DM interactions become stronger (enhanced by the quartic power of the velocity) at the epoch $z\sim 10^{3}$ relevant for the CMB rather than $z\sim 10^{6}$ relevant for Ly-$\alpha$. Hence, even though the treatment of the non-thermally produced DM is model-dependent, we simply consider the parameter ranges where the dispersion of our DM is too small to be relevant for the Lyman-$\alpha$ observations444Even though we do not specify a concrete model for the purpose of general discussions, the neglection of the free streaming effects for $n=-4$ can be applicable, for instance, when the cold (thermally decoupled) parent particle with the mass $M$ decays into the lighter DM at $T=T_{\rm dec}$ for $M\ll T_{\rm dec}$. The dispersion scales as the momentum over the mass bond1980 ; colo1995 ; Bode:2000gq ; Viel:2005qj , and such a scenario can lead to a much smaller DM velocity dispersion scale than the conventional warm DM scenario at least by a factor $(M/T_{\rm dec})(m_{\rm WDM}/m_{\chi})$. See, for instance, Refs. boya2008 ; Bae:2017dpt for the calculation of the phase space density and the resultant velocity dispersion for the non-thermally produced DM..

The typical matter power spectra and angular power spectra of the CMB temperature fluctuations are shown in Figs. 4 and 5 where the baryon-DM cross section is set to the upper bound values allowed by the 95% C.L. limits. We now discuss how these bounds on the cross sections are obtained.

III constraint

Our final aim is to provide the constraint on the baryon-DM coupling for the DM mass below 1 MeV scale. After outlining our analysis method in Sec. III.1, we show the constraint from the CMB alone in Sec. III.2 which should be compared with the limits by adding Ly-$\alpha$ data in Sec. III.3.

III.1 MCMC analysis

To compute the temperature and polarization fluctuations in the CMB and the matter power spectrum, we numerically solve the Boltzmann equations including the baryon-DM coupling by modifying the publicly available numerical code, CLASS CLASS2 as described in the previous section.

Our analysis uses the CMB angular power spectrum data (TT + lowP + lensing) from the Planck  Planck and the Lyman-$\alpha$ flux power spectrum from SDSS SDSS which gives the matter power spectrum at redshift $z=3$ around $k\simeq 1\ h\,{\rm Mpc^{-1}}$. In this paper, we adopt the MCMC method with Monte Python Audren developed in the CLASS code.

For the MCMC analysis, we have seven free parameters. The free parameter for the baryon-DM coupling is only $\sigma_{0}$, while we fix the DM mass $m_{\chi}$ and the spectral index for the velocity dependence, $n$. For the other six free parameters, we set the priors for the standard cosmological parameters as

[TABLE]

where $\Theta$ is the angular size of the sound horizon at recombination, $\Omega_{\rm b}h^{2}$ and $\Omega_{\chi}h^{2}$ are the current values of the baryonic and DM density parameters multiplied by the square of the Hubble constant $H_{0}$ (here, $H_{0}=100h\ {\rm km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$), $\tau_{\rm reio}$ is the reionization optical depth and $A_{\rm s}$ and $n_{\rm s}$ are the amplitude and spectral index of the primordial power spectrum.

III.2 The bounds from the CMB data alone

Let us show the results of the parameter constraints. First we present the limits from the MCMC analyses by using the CMB data alone. Such analysis using only the CMB data have been performed in the previous works Dvorkin:2013cea ; Xu:2018efh ; Boddy:2018wzy , but our study extends the previous works by covering the DM mass lighter ($\lesssim 10$keV) than that discussed in the the previous literature and found the unique features peculiar to such small DM masses.

Table 1 lists the results of the constraint on the cross section of the baryon-DM coupling $\sigma_{0}$ from the MCMC analyses by using the CMB data alone. We show the marginalized posterior distribution of $\sigma_{0}$ in Fig. 6. In the previous works, Ref. Xu:2018efh has investigated the constraint with the mass range $m_{\chi}\geq 10\ {\rm MeV}$ for $-4\leq n\leq 2$ cases by using the CMB+Ly$\alpha$ data and Ref. Boddy:2018wzy has studied the baryon-DM coupling with the mass range $m_{\chi}\geq 10\ {\rm keV}$ for $n=-4,-2$ cases by using the CMB data alone. Their works show that, when $m_{\chi}\ll 1\ {\rm GeV}$, the constraint on $\sigma_{0}$ becomes independent of the DM mass. As shown in Table 1 and Fig. 6, our constraint is consistent with them and we confirmed that there is no mass-dependence in the constraints even for the mass range, $\leq 10\ {\rm keV}$, except for the $n=+2$ case which gives a unique feature for a small DM mass.

This interesting mass dependence for $n=2$ which has not been found in the previous works shows up because of the strong $n$-dependence of the coupling rate. The smaller the DM mass is, the larger the $T_{\chi}/m_{\chi}$ term becomes in Eq. (4). A consequent enhancement of the baryon-DM interaction can result in the tighter bounds on $\sigma_{0}$ as the DM mass decreases. This characteristic feature becomes more apparent for a bigger $n$ because of the bigger $(n+1)/2$ power dependence of $R_{\chi}$ and a bigger value of $T_{\chi}$ in Eq. (4), and it is most prominent for $n=2$ compared with the other smaller values of $n$. For instance, for $n=-4$, the DM decoupled sufficiently early such that $T_{\chi}$ is too small to see the appreciable dependence of $\sigma_{0}$ on $m_{\chi}$.

III.3 The bounds from CMB + Lyman-$\alpha$

We next show the constraint from the CMB + Lyman-$\alpha$ data, which is summarized in Table 2. The marginalized posterior distribution of $\sigma_{0}$ is shown in Fig. 7.

As the DM mass becomes sufficiently small ($m_{\chi}\lesssim 1$ MeV), the suppression of the power spectrum due to the free-streaming can become important on the small scales depending on $n$ values as discussed in the previous section for some parameter range of our interest. Therefore, the additional suppression due to the baryon-DM coupling can be tightly constrained, except for the $n=-4$ case. We find that the combination of CMB + Lyman-$\alpha$ can provide the stronger constraint than that from the CMB alone with $n>-3$. This is reasonable because $R_{\chi}/aH$ becomes bigger for a bigger redshift for $n>-3$, which gives the stronger suppression due to the baryon-DM coupling in the redshift $z\sim 10^{6}$ when the modes relevant for the Lyman-$\alpha$ ($k\simeq 1\ h\,{\rm Mpc^{-1}}$) enter the horizon. For $n=-4$, on the other hand, $R_{\chi}/aH$ increases for a smaller redshift and the baryon-DM momentum transfer rate becomes sufficiently large at the CMB epoch ($z\sim 10^{3}$), and we expect to get the tight constraint from the CMB rather than from the Lyman-$\alpha$. The comparison of Fig. 6 with Fig. 7 for $n=-4$ indeed shows no appreciable difference in the best-fit values for $\sigma_{0}$ and the inclusion of the Lyman-$\alpha$ data does not give us a significant improvement in constraining $\sigma_{0}$. For the $n=-4$ case, the marginalized posterior distribution in Fig. 7 shows that the MCMC analysis prefers a somewhat larger value of the cross section $\sigma_{0}$ than that preferred by the analysis including only the CMB data. We point out that this is due to the degeneracy between $n_{\rm s}$ and $\sigma_{0}$. The analysis tries to fit the deficit of the CMB power spectrum on large scales by a larger $n_{\rm s}$ which can suppress the power below the pivot scale $k\leq 0.05\ h$/Mpc. Such a larger $n_{\rm s}$ gives the excess on small scales ($k\geq 0.05\ h$/Mpc), and, to cancel this excess, a large $\sigma_{0}$ (which can suppress the small scale power without affecting the large scale) is preferred.

Fig. 8 illustrates our discussions in this section and summarizes the results of the constraints on the $\sigma_{0}-m_{\chi}$ plane along with results from the previous works.

IV Summary

We have investigated the cosmological constraints on the cross section of a baryon-DM coupling which is parameterized as the power of the relative velocity between DM and baryons. This parameterization allows us to generally treat many DM models which possess the couplings with baryons. We have performed the MCMC analysis by using the angular power spectrum of the CMB from the Planck and the small-scale matter power spectrum obtained from the Lyman-$\alpha$ forest data by the SDSS. Our obtained result includes the new constraint for a smaller DM mass below the MeV scale and a wider range of the power law index, $-4\leq n\leq 2$, compared with the previous works.

Even though we explored the DM mass range only down to $m_{\chi}\sim 5$ keV because of the uncertainty due to the treatment of the collision term for the relativistic species, we already could start seeing interesting characteristic features such as the potentially strong mass dependence of the momentum transfer rate and hence that of the cross section bounds so that the further exploration of the even lighter DM mass range than studied here would be warranted. Our result should offer a complementary study to other studies including other astrophysical constrains and DM direct detection bounds Sigurdson:2004zp ; Xu:2018efh . For instance, the Lyman-$\alpha$ forest observation by the Dark Energy Spectroscopic Instrument (DESI) DESI could be of great interest to get the stronger constraint on the baryon-DM coupling, which is left for our future work.

Acknowledgements.

We thank J. Silk for the useful discussions. This work was supported by Grants-in-Aid for Scientific Research from JSPS (Nos. 16J05446 (J.O.), 15K17646 (H.T.) and 17H01110 (H.T.)) and the IBS under the project code IBS-R018-D1.