Baryon-like Space Distribution of Dark Matter from Point of View of Explanation of Positron Anomaly

Abstract

In this work we test the possibility of accounting for the positron anomaly with annihilating dark matter particles without contradicting the gamma-ray constraints due to their unconventional space distribution. To achieve that, we consider two-component dark matter, whose major constituent is inert and forms the halo of the Galaxy, while the second, minor, component consists of annihilating particles that could form some different structure. This work is the next logical step after our previous “dark disk model” where an active DM component was considered to form a disk, allowing good suppression of accompanying gamma-radiation. Nowadays that model is not enough to avoid the contradiction, so we are testing a new, more complex one with a spiral spatial distribution like the one of baryons. We have previously tested two simplified toy models of ring-like density profiles and one simple spiral density profile that have shown good improvement compared to the disk case. In this work, we take things further and consider a more physically grounded density profile constructed on the base of a modern model of the baryon density of our Galaxy. Contrary to our expectations, this advanced model shows much worse agreement with the data than previous toy models.

1. Introduction

Cosmic rays (CRs) are an undoubtedly essential source of information on the nature of physical phenomena in the universe. Studying CR puzzles (observational data that cannot yet be explained) may help advance the study of other fundamental problems. Here we consider the positron anomaly (PA) in CRs [1,2,3]. There are attempts to solve it with the pulsars [4,5,6,7], dark matter [8], local CR overdensity (bubble) [9] or recent close SN [10]. All these hypotheses differ by the physical nature of the positron sources (and respectively by injection spectra) and their location in the Galaxy around us. None can be considered ultimate, and further research is necessary.

Here we consider the dark matter (DM) solution of the positron anomaly, developing our previous studies [11,12,13,14,15]. It is supposed that DM has (at least) two components: a dominant collisionless one, forming the dark halo, and a minor active component, which forms the small structures in the galactic disk and produces cosmic radiation.

This work continues our studies where the model of a dark disk was considered [11,12,13,16,17]. It allowed us initially to describe the PA, avoiding contradiction with the gamma-ray background (IGRB). But the next release of data on the IGRB took into account extra background gamma-ray sources (first of all, blazars’ radiation), which restored a big tension between the PA description and the (updated) data on the IGRB. Preliminary calculations [14,15] have shown that ring-like or spiral-like distributions of the DM sources of CRs allows the PA explanation and IGRB to be reconciled again. Here we extend this idea to, seemingly, a more natural CR source distribution—a baryon-like one.

2. Dark Matter Model with Baryon-like Spatial Distribution

2.1. Methodology

Our analysis algorithm for this part of the work mostly follows the one from our previous papers [13,14,15]. For the sake of clarity, it will be laid out in full detail here.

The algorithm consists of three main steps: generating the injection spectra for positrons and gammas, simulating the propagation using GALPROP v57 [18] and analyzing the results using Wolfram Mathematica v14.

In this work we consider leptophilic DM particles X with the mass $M_X$ able to annihilate into any of the leptonic channels:

• $X\overline{X}\to e^{+}{e}^{-}$;
• $X\overline{X}\to {\mu }^{+}{\mu }^{-}$;
• $X\overline{X}\to {\tau }^{+}{\tau }^{-}$.

We consider the possibility of gamma suppression due to the physics of DM interaction as a separate study [19,20,21] and focus here only on the effect caused by the DM spatial distribution. Therefore we do not specify here any other property of the DM particle. However, to calculate the spectra we need to model a specific process. We use the MC generator Pythia v6 [22] and its $Z\to l^{+}{l}^{-}$ decay with modified Z mass for each channel separately, which is only slightly model-dependent.

The branching ratios of these channels, alongside the mean cross-section $<\sigma v>$, are assumed to be the parameters of our model, and their values are obtained during the final analysis. The value of $M_X$, on the other hand, has to be chosen manually at the start and remain fixed during the whole procedure, so it is not considered as a parameter here.

To calculate the positron and gamma fluxes we use GALPROP v57 code [18,23,24] with our own modifications (to access the modifications contact the authors of this paper via the provided e-mail) to use custom injection spectra and DM density profiles in the form of text files containing necessary data. These modifications are reduced to reading a specified text file and filling out the internal GALPROP grids with its data, using some light interpolation when necessary. We use the MED model from [25] for diffusion and magnetic field parameters. At the end of the GALPROP run we obtain FITS files containing the spatial distribution of the positron fluxes across the Galaxy and gamma-rays in sky-map format using galactic coordinates. We then use Wolfram Mathematica to extract the positron fluxes near the solar system position from the FITS and save them as simple text files for later use, while for the gamma-rays we also calculate the mean primary and secondary fluxes coming from the IGRB region of galactic latitude $\left|b\right|>{20}^{\circ }$.

We then use Wolfram Mathematica to evaluate the proposed model using the following chi-square expression:

$${\chi }_{\mathrm{tot}}^2={\displaystyle \frac{1}{N_{\mathrm{d}.\mathrm{o}.\mathrm{f}}}}\left(\sum _{\mathrm{AMS}\mathrm{data}}{\displaystyle \frac{\mathrm{\Delta }{\mathrm{\Phi }}_e^2}{{\sigma }_e^2}}+\sum _{\mathrm{IGRB}\mathrm{data}}{\displaystyle \frac{\mathrm{\Delta }{\mathrm{\Phi }}_{\gamma }^2}{{\sigma }_{\gamma }^2}}{H}_{\theta }\left(\mathrm{\Delta }{\mathrm{\Phi }}_{\gamma }\right)\right),$$

(1)

where $N_{\mathrm{d}.\mathrm{o}.\mathrm{f}}$ is the number of degrees of freedom that equals the difference between the number of eligible datapoints and number of our model parameters (three in total), $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_{model}-{\mathrm{\Phi }}_{exp}$ denotes the difference between our model prediction and the experimental data and $\sigma$ the corresponding uncertainty, and index e stands for AMS-02 positron fraction data [26] and $\gamma$ for the Fermi-LAT IGRB [27]. In our analysis we use both AMS-02 and Fermi-LAT datapoints starting from the energy of 30 GeV. As we do not aim to explain IGRB spectra and only need to not exceed the data, we multiply the term in the second sum by the Heaviside theta-function $H_{\theta }$, which nullifies the contribution to chi-square of predictions below the data. For the same reason, in our $N_{\mathrm{d}.\mathrm{o}.\mathrm{f}}$ calculations we only take into account the gamma datapoints that the model predictions exceed. We use the electron and positron spectra from [28] to construct the background positron fraction, and take into account the solar modulations [29] to transform predicted fluxes near the solar system to observable near-Earth ones. As for the gamma-ray data, we take into account the contribution of unresolved sources [30].

2.2. Density Profile

Let us briefly address the topic of the feasibility of such an unconventional DM profile and the constraints it could face. First of all, the formation of such structures is a rather complex area of research, beyond the scope of this paper. However, there are papers on this topic showing the formation of the disk-like DM structures from a dynamics point of view [31,32,33,34,35]. Secondly, our model predictions do not directly depend on either the DM density or $<\sigma v>$, instead being proportional to the value of their product $<\sigma v>\times {\rho }^2$. Due to that, we can satisfy any constraints set on the DM density by changing the cross-section. In our previous “dark disk model” we normalized the density profile used for calculations at one such limit. Namely, the total DM density around the solar system, including both dark halo and disk contributions, should not exceed 1 GeV/cm³ [36,37]. In this work, we abandoned this step due to the complexity of the model involved. The only thing left is the possible constraints on the resulting $<\sigma v>$, which come from, mainly, CMB data [38]. This limitation is much more model-dependent, and there could be a physics-based way to circumvent it [39,40]. The subdominant nature of the considered active component of DM also plays in our favor in this case. By our crude estimations, our “dark disk model” optimal parameters led to a disk structure whose mass amounted to very small fraction of the total DM mass, which should make avoiding CMB constraints much easier.

To check the assumption that the DM spatial distribution follows closely the one of baryons, and because of that allows us to explain the positron anomaly without contradiction of the gamma-ray observations, a realistic model of the Milky Way structure is needed. We chose the model described in the work [41], which is dedicated to estimating systematic errors for one of the techniques of distance measurement taking into account a realistic galactic potential. While the model in question, according to the authors, is not the most recent or accurate one and is tuned to the specific task of their paper, it still provides a comprehensive model of the Milky Way and it conveniently has all of its constituent parts described in one place.

In our work most of them are used, and only the potential of supermassive black holes, contribution of the DM halo and contribution of the baryon disk, symmetrical in regard to the galactic center, are discarded. The latter is due to our previous results showing a preference towards DM disk absence [15]. Therefore, the approximation for baryon density is constructed as follows:

$${\rho }_{\mathrm{baryon}}={\rho }_{centre}+{\rho }_{\mathrm{gas}}+{\rho }_{\mathrm{spiral}},$$

(2)

$${\rho }_{centre}={\rho }_{\mathrm{NSC}}+{\rho }_{\mathrm{NSD}}+{\rho }_{\mathrm{bar}1}+{\rho }_{\mathrm{bar}2}+{\rho }_{\mathrm{bar}3}.$$

(3)

Here ${\rho }_{\mathrm{NSC}}$ and ${\rho }_{\mathrm{NSD}}$ describe the star cluster and star disk in the most central region; ${\rho }_{\mathrm{bar}}$ with the indexes from 1 to 3 are the contributions to the galactic bar, necessary to account for its complex shape, forming the central structure of the profile ${\rho }_{centre}$; ${\rho }_{\mathrm{gas}}$ represents the $H_{\mathrm{I}}$ and $H_2$ disks; and ${\rho }_{\mathrm{spiral}}$ corresponds to the spiral structure of the Galaxy. The first two of its constituents are defined below:

$${\rho }_{\mathrm{NSC}}=\left(3-\gamma \right){\displaystyle \frac{M_{\mathrm{NSC}}}{4\pi q_{\mathrm{c}}}}{\displaystyle \frac{a_0}{a_c^{\gamma }{\left(a_c+a_0\right)}^{4-\gamma }}},$$

(4)

$$a_c=\sqrt{r^2+{\left({\displaystyle \frac{z}{q_c}}\right)}^2},$$

(5)

$${\rho }_{\mathrm{NSD}}={\rho }_{1}exp\left[-{\left({\displaystyle \frac{a_d}{R_1}}\right)}^{n_1}\right]+{\rho }_{2}exp\left[-{\left({\displaystyle \frac{a_d}{R_2}}\right)}^{n_2}\right],$$

(6)

$$a_d=\sqrt{r^2+{\left({\displaystyle \frac{z}{q_d}}\right)}^2},$$

(7)

where r and z are the coordinates in the galactic cylindrical system of coordinates, and the parameter values are listed in

Table 1. Values of the parameters for ${\rho }_{\mathrm{NSC}}$ and ${\rho }_{\mathrm{NSD}}$.

NSC			NSD
Parameter	Value	Units	Parameter	Value	Units
$q_c$	0.73	–	$q_d$	0.37	–
$\gamma$	0.71	–	$n_1$	0.72	–
$a_0$	$5.9\times {10}^{-3}$	kpc	$n_2$	0.79	–
$M_{\mathrm{NSC}}$	$6.1\times {10}^7$	$M_{\odot }$	$R_1$	$5.06\times {10}^{-3}$	kpc
–			$R_2$	$24.6\times {10}^{-3}$	kpc
–			${\rho }_1$	$2.00583\times {10}^{12}$	$M_{\odot }$${\mathrm{kpc}}^{-3}$
–			${\rho }_2$	$1.53\times {10}^{12}$	$M_{\odot }$${\mathrm{kpc}}^{-3}$

.

The galactic bar contributions are defined as follows:

$${\rho }_{\mathrm{bar}1}={\rho }_{b1}·\mathrm{sech}\left(a_{b1}^m\right)\left(1+{\alpha }_b\left(exp\left[-a_{+}^{n_{b1}}\right]+exp\left[-a_{-}^{n_{b1}}\right]\right)\right)exp\left[-{\displaystyle \frac{r^2+z^2}{R_b^2}}\right],$$

(8)

$$a_{b1}={\left({\left({\left({\displaystyle \frac{\left|x\right|}{x_{b1}}}\right)}^{c_{\perp 1}}+{\left({\displaystyle \frac{\left|y\right|}{y_{b1}}}\right)}^{c_{\perp 1}}\right)}^{{\displaystyle \frac{c_{\parallel }}{c_{\perp 1}}}}+{\left({\displaystyle \frac{\left|z\right|}{z_{b1}}}\right)}^{c_{\parallel }}\right)}^{{\displaystyle \frac{1}{c_{\parallel }}}}$$

(9)

$$a_{+}={\left({\left({\displaystyle \frac{x+c·z}{x_{cb}}}\right)}^2+{\left({\displaystyle \frac{y}{y_{cb}}}\right)}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

(10)

$$a_{-}={\left({\left({\displaystyle \frac{x-c·z}{x_{cb}}}\right)}^2+{\left({\displaystyle \frac{y}{y_{cb}}}\right)}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

(11)

$${\rho }_{\mathrm{bar}2}={\rho }_{b2}exp\left(-a_{b2}^{n_{b2}}\right)\mathrm{sech}{\left({\displaystyle \frac{z}{z_{b2}}}\right)}^{2}exp\left[-{\left({\displaystyle \frac{r}{R_{o2}}}\right)}^{n_{o2}}-{\left({\displaystyle \frac{R_{i2}}{r}}\right)}^{n_{i2}}\right],$$

(12)

$${\rho }_{\mathrm{bar}3}={\rho }_{b3}exp\left(-a_{b3}^{n_{b3}}\right)\mathrm{sech}{\left({\displaystyle \frac{z}{z_{b3}}}\right)}^{2}exp\left[-{\left({\displaystyle \frac{r}{R_{o3}}}\right)}^{n_{o3}}-{\left({\displaystyle \frac{R_{i3}}{r}}\right)}^{n_{i3}}\right],$$

(13)

$$a_{b2}={\left({\left({\displaystyle \frac{\left|x\right|}{x_{b2}}}\right)}^{c_{\perp 2}}+{\left({\displaystyle \frac{\left|y\right|}{y_{b2}}}\right)}^{c_{\perp 2}}\right)}^{{\displaystyle \frac{1}{c_{\perp 2}}}},$$

(14)

$$a_{b3}={\left({\left({\displaystyle \frac{\left|x\right|}{x_{b3}}}\right)}^{c_{\perp 3}}+{\left({\displaystyle \frac{\left|y\right|}{y_{b3}}}\right)}^{c_{\perp 3}}\right)}^{{\displaystyle \frac{1}{c_{\perp 3}}}},$$

(15)

where x, y, z are galactic Cartesian coordinates and the parameters are listed in

Table 2. Values for the parameters in bar density components.

${\mathit{\rho }}_{\mathbf{bar}1}$			${\mathit{\rho }}_{\mathbf{bar}2}$			${\mathit{\rho }}_{\mathbf{bar}3}$
Par	Val	Unit	Par	Val	Unit	Par	Val	Unit
$c_{\perp 1}$	2.232	–	$c_{\perp 2}$	0.97	–	$c_{\perp 3}$	1.879	–
$n_{b1}$	1.94	–	$n_{b2}$	3.051	–	$n_{b3}$	0.98	–
${\rho }_{b1}$	$3.16$ × ${10}^9$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^3}}$	${\rho }_{b2}$	$0.5\times {10}^9$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^3}}$	${\rho }_{b3}$	$1.743\times {10}^{13}$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^3}}$
$x_{b1}$	0.49	kpc	$x_{b2}$	5.364	kpc	$x_{b3}$	0.478	kpc
$y_{b1}$	0.392	kpc	$y_{b2}$	0.959	kpc	$y_{b3}$	0.267 1	kpc
$z_{b1}$	0.229	kpc	$z_{b2}$	0.611	kpc	$z_{b3}$	0.252	kpc
$x_{cb}$	0.751	kpc	$R_{i2}$	0.558	kpc	$R_{i3}$	7.607	kpc
$y_{cb}$	0.469	kpc	$R_{o2}$	3.19	kpc	$R_{o3}$	2.204	kpc
$R_b$	4.37	kpc	$n_{i2}$	3.196	–	$n_{i3}$	1.63	–
${\alpha }_b$	0.626	–	$n_{o2}$	16.731	–	$n_{o3}$	−27.291	–
c	1.342	–	–			–
$c_{\parallel }$	1.991	–	–			–
m	0.873	–	–			–

¹ This value is taken from [42], as [41] seems to contain a misprint.

.

The last contributions to the density profile are defined as follows:

$${\rho }_{\mathrm{gas}}={\displaystyle \frac{{\mathrm{\Sigma }}_{g1}}{4z_{g1}}}exp\left(-{\displaystyle \frac{R_{m1}}{r}}-{\displaystyle \frac{r}{R_{g1}}}\right)\mathrm{sech}{\left({\displaystyle \frac{z}{2z_{g1}}}\right)}^2+{\displaystyle \frac{{\mathrm{\Sigma }}_{g2}}{4z_{g2}}}exp\left(-{\displaystyle \frac{R_{m2}}{r}}-{\displaystyle \frac{r}{R_{g2}}}\right)\mathrm{sech}{\left({\displaystyle \frac{z}{2z_{g2}}}\right)}^2,$$

(16)

$${\rho }_{\mathrm{spiral}}={\rho }_{\mathrm{disk}}·{\alpha }_{sp}·{\displaystyle \frac{r^2}{R_{sp}}}·S,$$

(17)

$${\rho }_{\mathrm{disk}}={\displaystyle \frac{{\mathrm{\Sigma }}_{d1}}{2z_{d1}}}exp\left(-{\displaystyle \frac{r}{R_{d1}}}-{\displaystyle \frac{R_{dcut}}{r}}-{\displaystyle \frac{\left|z\right|}{z_{d1}}}\right)+{\displaystyle \frac{{\mathrm{\Sigma }}_{d2}}{2z_{d2}}}exp\left(-{\displaystyle \frac{r}{R_{d2}}}-{\displaystyle \frac{R_{dcut}}{r}}-{\displaystyle \frac{\left|z\right|}{z_{d2}}}\right),$$

(18)

$$S=\sum _{k=1}^2\left(exp\left[-{\displaystyle \frac{r^2}{{\sigma }_{sp}^2}}\left(1-f_k\right)\right]-exp\left[-{\displaystyle \frac{r^2}{{\sigma }_{sp}^2}}\right]·I_0\left[-{\displaystyle \frac{r^2}{{\sigma }_{sp}^2}}\right]\right),$$

(19)

$$f_k=cos\left(2\left(\phi +{\gamma }_k\right)-{\displaystyle \frac{2}{tan\left[i_{sp}\right]}}ln\left[{\displaystyle \frac{r}{R_a}}\right]\right),$$

(20)

where $I_0$ is the modified Bessel function of the first kind and zeroth order and the values of the parameters are listed in

Table 3. Values for the parameters in disk and spiral density components.

${\mathit{\rho }}_{\mathbf{spiral}}$			${\mathit{\rho }}_{\mathbf{disk}}$			${\mathit{\rho }}_{\mathbf{gas}}$
Par	Val	Unit	Par	Val	Unit	Par	Val	Unit
${\alpha }_{sp}$	0.36	–	${\mathrm{\Sigma }}_{d1}$	$1.3719\times {10}^9$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^2}}$	${\mathrm{\Sigma }}_{g1}$	$53.1\times {10}^6$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^2}}$
$R_{sp}$	8.179	kpc	${\mathrm{\Sigma }}_{d2}$	$9.2391\times {10}^8$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^2}}$	${\mathrm{\Sigma }}_{g2}$	$2.18\times {10}^9$	${\displaystyle \frac{M_{⨀}}{{\mathrm{kpc}}^2}}$
$R_a$	9.64	kpc	$R_{d1}$	2	kpc	$R_{g1}$	7	kpc
${\sigma }_{sp}$	5	kpc	$R_{d2}$	2.8	kpc	$R_{g2}$	1.5	kpc
$i_{sp}$	12.5	deg	$z_{d1}$	0.3	kpc	$z_{g1}$	0.085	kpc
${\gamma }_1$	139.5	deg	$z_{d2}$	0.9	kpc	$z_{g2}$	0.045	kpc
${\gamma }_2$	69.75	deg	$R_{dcut}$	2.4	kpc	$R_{m1}$	4	kpc
–			–			$R_{m2}$	12	kpc

.

As we have excluded the disk and dark matter contributions from the original density profile and because the minimums of ${\rho }_{\mathrm{spiral}}$ from Equation (17) go down to negative values, we made an extra step in the calculations to ensure our total density remains positive, nullifying it otherwise.

The section of the resulting density profile corresponding to the galactic plane $z=0$ is shown in Figure 1a. However, this profile still must undergo a couple of small transformations. First of all, it is much easier to use GeV·cm⁻³ units for our work. And finally, the angle between the bar major axis and Sun–galactic center line must be 28°. As the default solar system position in GALPROP coordinates is $\{8.5,0,0\}$, this density profile should be rotated. This is achieved by constructing the density profile tabulated data file in the following format:

$$xyz{\rho }_{\mathrm{spiral}\left(x^{\prime },y^{\prime },z\right)}$$

where

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=M\left(\theta \right)·\left[\begin{array}{c}x\\ y\end{array}\right]$$

(21)

and $M\left(\theta \right)$ is a 2D rotation matrix with $\theta ={28}^{\circ }$. The same section for the finalized density profile is shown in Figure 1b.

2.3. Results

The first simulation was conducted for the case of a 550 GeV initial DM particle. This is roughly in the same mass range of the best-fit scenarios for our previous “toy” models of a dark disk and spiral [14,15]. The obtained results are shown in Figure 2.

Unlike our previous models, this one overproduces the positrons in the 10 GeV range to such an extent that the high-energy part of the spectra becomes virtually negligible. The situation remains unchanged with the increase in the initial DM particle mass, even up to 100 TeV (see Figure 3 for reference).

This overproduction seems to arise due to very high density in the galactic center region compared to other parts of the structure. To test this assumption and also the feasibility of trying to suppress the low-energy part of the spectrum via tuning the properties of the interstellar medium, we tested a modified density profile with the central part cut out of it:

$${{\rho }_{DM}}^{\prime }={\rho }_{\mathrm{gas}}+{\rho }_{\mathrm{spiral}}.$$

(22)

The results for such a profile and a 500 GeV DM particle are shown in Figure 4.

While exclusion of the galactic center region did indeed solve the overproduction problem and allowed a much better description of the positron fraction, the resulting value of ${\chi }^2$ increased to 20, mostly due to overproduction of gamma-rays. It seems that the baryon density profile “as is” is too thick in the z-axis and therefore does not allow this gamma-ray tension to be resolved.

3. Conclusions

This work is devoted to the search for a solution of the positron anomaly problem. This work is continuation of our previous search for a space distribution of DM that could serve as its source. Previously, ring-like and spiral-like DM density profiles (of subdominant components) were considered, and they demonstrated a positive trend for PA explanation. Here we considered a seemingly natural development of this idea and used a baryon-like distribution of DM sources of CRs. The result was found to be negative. The central regions of the Galaxy in this case greatly overproduce lower-energy positrons near the solar system, making the explanation of high-energy excess impossible. Their exclusion, on the other hand, allows reproduction of the positron fraction data but fails to suppress the gamma-radiation due to the “thickness” of the remaining components of the density profile.