Point Sources from Dissipative Dark Matter

TL;DR

Dissipative dark matter could form dense compact objects that produce point-like signals, potentially explaining gamma-ray excesses from the galactic center through novel indirect detection signatures.

Contribution

This paper introduces models of dissipative dark matter that form compact objects, providing a new mechanism for gamma-ray signals and linking dark matter properties to observable astrophysical phenomena.

Findings

Dissipative dark matter can form dense astrophysical objects.

Such objects can produce point-like gamma-ray signals.

Models can explain the galactic center gamma-ray excess.

Abstract

If a component of dark matter has dissipative interactions, it can cool to form compact astrophysical objects with higher density than that of conventional cold dark matter (sub)haloes. Dark matter annihilations might then appear as point sources, leading to novel morphology for indirect detection. We explore dissipative models where interaction with the Standard Model might provide visible signals, and show how such objects might give rise to the observed excess in gamma rays arising from the galactic center.

Figures

Tables

Table 1. Table 1: Particle content for the massless Z ′ superscript 𝑍 ′ Z^{\prime} portal model.

(22)

Equations

$$2\pi Gk\sum _{\mathrm{collisional}}\frac{{\Sigma }_i}{{\kappa }^2+k^2{c}_i^2-{\omega }^2}+2\pi Gk\sum _{\mathrm{collisionless}}\frac{{\Sigma }_i\mathcal{F}(\omega /\kappa ,k^2{\sigma }_i^2/{\kappa }^2)}{{\kappa }^2-{\omega }^2}$$

$$\mathcal{F}(s,\chi )$$

$$2\pi Gk\sum _{\mathrm{collisional}}\frac{{\Sigma }_i}{{\kappa }^2+k^2{c}_i^2}+2\pi Gk\sum _{\mathrm{collisionless}}\frac{{\Sigma }_i\mathcal{F}(0,k^2{\sigma }_i^2/{\kappa }^2)}{{\kappa }^2}$$

$$c_{XC}\sim \frac{{\alpha }_D}{\sqrt{40}}\sqrt{\frac{m_C}{m_X}}$$

$$k_{\pm }$$

$${\Sigma }_{XC}(r)$$

$$\mathcal{L}$$

$$f_{(XC)}$$

$$f$$

$$\mathcal{L}$$

$$V_{\mu }\to V_{\mu }-ϵ\frac{e}{g_D}{A}_{\mu }$$

$$\mathcal{L}$$

$$\frac{n_e⟨\sigma v⟩}{H}$$

$$ϵ\lesssim {\left(\frac{m_C}{{\alpha }^2{M}_{pl}}\right)}^{\frac{1}{2}}\sim 10^{-9}{\left(\frac{m_C}{1\mathrm{MeV}}\right)}^{\frac{1}{2}}$$

$$\mathcal{L}$$

$$ϵ\lesssim 1.8\times 10^{-6}{\left(\frac{m_X}{\text{GeV}}\right)}^{1/2}$$

$$⟨\sigma v{⟩}_{X\bar{X}\to \gamma Z^{\prime }/{\gamma }_D}$$

$$\frac{⟨\sigma v{⟩}_{X\bar{X}\to \gamma Z^{\prime }/{\gamma }_D}}{⟨\sigma v{⟩}_{X\bar{X}\to f\bar{f}}}$$

$$\mathcal{L}$$

$$r$$

$$\eta =Y_X-Y_{\bar{X}}=Y_C-Y_{\bar{C}}.$$

$$\frac{dr}{dx}$$

$$\lambda (x)$$

$$r_{eq}(x)$$

$$Y_{eq}(x)$$

$$⟨\sigma v{⟩}_{ann}$$

$$\frac{d{r}_{eq}(x_f)}{dx}$$

$$x_f$$

$$r_{\infty }$$

$$Y_{\infty }^{\eta =0}\simeq \frac{x_f}{\lambda (x_f)},$$

$$r_{\infty }$$

Full text

1 Introduction

A vast amount of astrophysical evidence exists for the presence of dark matter, but its particle physics nature remains entirely unknown. Various direct and indirect detection experiments that search for small non-gravitational interactions between the dark and Standard Model sectors hope to find this dark component of our energy density.

If the Standard Model (SM) is a guide, the dark sector may contain multiple species, with small components possessing stronger interactions, amongst itself and/or with the SM. Even if this component is subdominant – so that it does not impact gravitational measurements of dark matter so far – it might generate the first signals we discover.

One such class of models is where such a dark matter fraction has dissipative dynamics Fan:2013yva ; Fan:2013tia . In analogy with baryonic matter, this could lead to the formation of interesting astrophysical structures, without disrupting qualitative features of cold dark matter halo formation. If this component also interacts non-gravitationally with the Standard Model, the first signals of dark matter might have unexpected properties CyrRacine:2012fz ; McCullough:2013jma ; Fan:2013bea ; Fischler:2014jda ; 2014PhRvL.112p1301R ; Randall:2014kta ; Reece:2015lch ; Kramer:2016dew ; Kramer:2016dqu ; Agrawal:2017rvu .

In this paper, we consider the possibility that dark matter forms dense clouds that are sites of enhanced annihilation. These novel astrophysical structures – such as a dark disk or dense compact halo objects – will give rise to distinctive morphology for indirect detection. If small enough, these clouds would appear as point sources in indirect detection experiments. We explore models with both dissipative interactions and couplings to the SM that can give rise to visible signatures from dark compact objects, with the specific signal depending on the particular nature of this coupling. We will see that one of the most interesting aspects of such models is that the dissipative dark matter can be a negligible fraction of the total dark matter composition, yet still give rise to observable effects.

An important signal is the excess of gamma rays observed from towards the galactic center Goodenough:2009gk ; Hooper:2010mq ; Daylan:2014rsa ; Calore:2014xka ; TheFermi-LAT:2015kwa , with some statistical evidence supporting this excess as arising from point sources Lee:2015fea ; Bartels:2015aea ; Fermi-LAT:2017yoi , though this may be due to substructure in background that is not modeled in the range of diffuse background models considered (see also Horiuchi:2016zwu ). The more conventional assumption for sources that would produce the observed point-like spectrum could be a new population of millisecond pulsars (MSPs), with luminosity just below the Fermi point source detection threshold. Direct evidence for such a population has not yet been observed (see also Brandt:2015ula ; Hooper:2016rap ; Fermi-LAT:2017yoi ). If we are to definitely establish the origin, it is worthwhile to consider less conventional potential sources as well. An alternative admittedly more speculative possibility that is also worth investigating is that the excess is associated with dark point sources. In this paper we consider pointlike sources that arise from dissipative dynamics in the dark sector. In this scenario the dark objects cool analogous to the objects in the SM sector. In our case we only have a $U(1)$ interaction that leads to cooling, and we study the compact objects that would result from this interaction. An alternative is ultra-compact minihaloes (UCMHs) Berezinsky:2003vn ; Ricotti:2009bs ; Scott:2009tu ; Josan:2010vn ; Erickcek:2011us ; Bringmann:2011ut ; Berezinsky:2013fxa ; Clark:2015sha ; Aslanyan:2015hmi ; Clark:2016pgn and we discuss ways to distinguish them at the end.

This paper is organized as follows. We start in section 2 explaining how dissipative dark matter can give rise to compact objects and present a model-dependent estimate for their expected sizes. In section 3 we introduce a simple dissipative dark matter model and specific portal couplings to the Standard Model. We analyze the resulting phenomenology in section 4. Section 5 shows how such a model could account for the Galactic Center excess, and we conclude in section 6.

2 Compact objects from dissipative dark matter

If there is a component of dark matter that has dissipative dynamics, it can cool efficiently with the potential creation of compact objects. The sizes and distribution of such objects depends on nonlinear dynamics, and a careful prediction would require modeling and simulations. However, simple estimates demonstrate that compact objects, with size determined by fraction of dark matter in the charged dark component, are a viable possibility.

We assume the dissipative component leads to formation of an unstable disk which then fragments via Toomre instabilities to form compact objects. An alternative possibility is that compact objects form in smaller subhaloes and might survive as subhaloes merge to form our galaxy. This latter scenario could potentially lead to compact objects with a distribution different from the one we now consider but is worth independent consideration.

When formed from disk fragmentation, the typical size of the objects can be estimated using a simple stability analysis, which we perform below. For simplicity, we assume that the clumps are uniform so that we can characterize parameterize the distribution and mass functions with a single characteristic size and mass, $\{M,R\}$, dictated by the stability analysis. More generally, one could parameterize the number of objects of mass $M$ by the mass function, $dN/dM$ and assume a nonuniform density even within the objects.

We will consider a simple dissipative model with a weak-scale particle $X$ and a light particle $C$, both charged under a dark $U(1)$ with coupling strength $\alpha_{D}$. This model is discussed in further detail in section 3.

2.1 Dark Disk Fragmentation and the size of clouds

The stability condition for the dark disk in the presence of various gas and stellar components can be written using the dispersion relation for a density perturbation Romeo1992 ; Rafikov2001 ; binney2008galactic ; Randall:2014kta ; Shaviv:2016umn ,

[TABLE]

The gas and the dissipative component of dark matter are collisional, while the various population of stars are collisionless. The epicyclic frequency $\kappa=36\,\text{km}\,\text{s}^{-1}\,\text{kpc}^{-1}$, and $k$ denotes the wavenumber of the perturbation. $\Sigma_{i}$ is the column density of component $i$, and $c_{i}$($\sigma_{i}$) is the sound speed (velocity dispersion) of the collisional (collisionless) component. The function $\mathcal{F}$ is

[TABLE]

$I_{n}$ is the Bessel function of order $n$. The perturbation is unstable when $\omega^{2}<0$. Since the l.h.s in equation 1 is a monotonically increasing function of $\omega$ when $\omega<\kappa$, the instability criteria can be restated as the condition that the l.h.s is greater than 1 at $\omega=0$,

[TABLE]

The sound speed for the dark disk can be estimated by the temperature dissipative dark matter cools down to, which is roughly the temperature at which recombination occurs and rapid cooling ceases: $T\sim\frac{1}{40}m_{C}\alpha_{D}^{2}$,

[TABLE]

We show the stability condition in figure 1. Note that in the Milky Way, galaxy stars and gas are relevant to stability only on larger length scales. For our choice of parameters, the dark disk induces instability at much shorter length scales, where the contribution of stars and gas to the dispersion relation can be neglected. It therefore suffices here to to consider the stability of the dark disk by itself and find a simple expression for the wavenumber of unstable modes $k_{-}<k<k_{+}$,

[TABLE]

The fastest growing mode corresponds to $k={\pi G\Sigma_{XC}}/{c_{XC}^{2}}$.

We assume the dark disk surface density is

[TABLE]

where $r_{d}\simeq 3$ kpc is the assumed scale radius of the dark disk, and $\Sigma_{XC,\odot}$ is the local dark disk surface density which can be as large as $\sim 10M_{\odot}/\,\text{pc}^{2}$ Kramer:2016dqu . Therefore the wavenumber varies as a function of distance from the center since the density and hence the stability criterion vary.

The dark compact objects that will be formed then have mass approximately $M=\pi\Sigma_{XC}\mathcal{R}_{Toomre}^{2}$, where $\mathcal{R}_{Toomre}$ is the wavelength of the fastest growing mode. In figure 2 we show the size and mass of this mode as a function of the distance from the Galactic Center. We see that in this scenario we expect objects of mass $10^{4}M_{\odot}$, with a radius $\sim 10$ pc in the inner part of the galaxy. Of course, this is a very crude estimate of the initial size, and this object might undergo further collapse, tidal disruption, evaporation or accretion.

We note that the spatial distribution of the objects would require more detailed simulations. Clumps might be dominantly clustered in the inner galaxy, spread through the disk, or distributed throughout the dark matter halo. We will assume that the dark matter also forms a disk + bulge system, with the disk subject to fragmentation. By analogy with the bulge, this would lead to a population of dark compact objects at the center of the galaxy in a roughly spherical distribution. In principle compact objects can form throughout the bulk as well, though the answer would depend on whether such objects fall to the center and shock heating repeats itself. For now we assume compact objects are concentrated in the center and the disk, and ask what the consequences for indirect detection might be. We will focus on the objects in the inner galaxy, and not consider other potential visible signals from the plane of the disk or the halo which would require detailed simulations.

3 Dissipative Dark Matter Models

Dark matter with dissipative dynamics was considered in Ref. Fan:2013yva ; Fan:2013tia . In this section we expand the discussion to include the possibility that in addition to dissipative self-interactions there are also interactions with the Standard Model. In this case we can have dissipative interactions that lead to the formation of compact objects and these in turn can potentially give rise to visible signatures.

3.1 A simple dissipative dark matter model

The model consists of a heavy dark proton $X$ and a lighter dark electron $C$, both charged under a dark U(1) gauge group, as in Fan:2013yva ; Fan:2013tia ; Cline:2012is .

[TABLE]

A sufficiently large relic population of a light enough $C$ would lead to efficient cooling in dark matter haloes. Being light, $C$s would annihilate efficiently, and survive only if produced through non-thermal processes – such as a matter-antimatter asymmetry in the population. We consider models in which there is both a symmetric relic abundance of $X$ and $\bar{X}$ particles, as well as an asymmetric population of $C$ and $X$. Such a symmetric population would occur naturally for dark matter masses comparable to the weak scale, just as for the more usual WIMPs. By charge neutrality $n_{C}-n_{X}+n_{\bar{X}}=0$. We define the fraction of $XC$ dark matter in the asymmetric component by

[TABLE]

We also define the fraction of dark matter density in the charged system,

[TABLE]

with $\Omega_{X}$ the energy total density in $X,\bar{X}$ and $C$. The rest of the dark matter is assumed to be made up of a distinct cold dark matter (CDM) component.

To generate visible indirect signatures, the model must also include “portal” interactions with the SM. A simple option is a kinetic mixing portal, in which the dark photon kinetically mixes with the SM photon. However, constraints on millicharged particles coupled with constraints on the survival of the compact objects necessitate an additional dark gauge $Z^{\prime}$, which can be either massless or massive– resulting in different signatures that we outline below.

3.2 Portal Models

We first pursue the simplest possibility for interactions between the dark and Standard Model sectors and see why that is not so promising for current indirect detection measurements. Dark matter can interact with the SM through kinetic mixing of the dark photon with the SM photon Okun:1982xi ; Holdom:1985ag ,

[TABLE]

When the dark photon mass is zero, it is convenient to redefine the dark photon

[TABLE]

After rescaling $V$ to obtain a canonical kinetic term, this yields the Lagrangian (to leading order in $\epsilon$)

[TABLE]

There is no direct coupling of $V$ to matter in this basis, but the dark matter particles pick up a millicharge $e\epsilon$ under the SM photon Holdom:1985ag .

Models with millicharged particles (MCPs) have rich phenomenology, and the millicharge has many constraints depending on the mass of the particle Davidson:1991si ; Davidson:2000hf ; Dubovsky:2003yn ; McDermott:2010pa ; Wilkinson:2013kia ; Dolgov:2013una ; Brust:2013xpv ; Dvorkin:2013cea ; Vogel:2013raa ; Vinyoles:2015khy ; Munoz:2015bca ; Ali-Haimoud:2015pwa ; Kadota:2016tqq ; Kamada:2016qjo . We summarize the constraints in figure 3.

For very light particles, $m_{C}<10\,\text{keV}$, star cooling constraints restrict $\epsilon<10^{-14}$ Vogel:2013raa ; Vinyoles:2015khy . In the range of $C$ masses where cooling occurs efficiently, $m_{C}\sim$ MeV, very stringent bounds on $\epsilon$ are needed to prevent dark photons from contributing $N_{\mathrm{eff}}\sim 1$ to the cosmic microwave background (CMB) and during Big Bang nucleosynthesis (BBN) Brust:2013xpv ; Vogel:2013raa . The bound relies on thermal coupling and corresponds to when the sectors thermally decouple. The dark sector is coupled to the photon bath through the process $e^{+}e^{-}\to C\bar{C}$. The rate for this process relative to Hubble is,

[TABLE]

Thus, the condition that the dark sector not get into thermal contact with the SM at temperature $T=m_{C}$,

[TABLE]

This is an extremely severe bound and forces $\epsilon$ to be too small to produce observable indirect detection signals. Interesting signals arise only in the presence of additional interactions, as we now propose.

3.2.1 A massless $Z^{\prime}$ portal

The kinetic mixing of the dark photon with the SM photon leads to a millicharge for the light particle $C$, which is also severely constrained. However, an additional massless gauge boson, $Z^{\prime}$, which couples to $X$ but not to $C$, can have a moderate kinetic mixing with the hypercharge gauge boson and produce observable signals at direct and indirect detection experiments, or at colliders.

As an example, we consider a model with 3 $U(1)$ gauge symmetries – hypercharge $U(1)_{Y}$, the dark photon $U(1)_{D}$ and the $Z^{\prime}$ portal $U(1)_{Z^{\prime}}$. The SM particles are not charged under the $U(1)_{D}$ or the $U(1)_{Z^{\prime}}$, and the dark sector particles are not charged under the SM gauge group. We assume that there is no kinetic mixing between the $U(1)_{D}$ and either $U(1)$, only between $U(1)_{Z^{\prime}}$ and $U(1)_{Y}$,

[TABLE]

We summarize the particle content of the model in table 1. The presence of only vectorlike fermions beyond the SM ensures that there are no anomalies. If we want to have a cosmological population of $XC$ atoms (which have a net charge under the $U(1)_{Z^{\prime}}$), we need a corresponding population of $Y$ particles to ensure that the universe is charge neutral under $U(1)_{Z^{\prime}}$. We note that for this particular choice of charges, there is no kinetic mixing induced by $X,Y,C$ loops between the $U(1)_{D}$ and the $U(1)_{Z^{\prime}}$. For simplicity we assume below that the $Y$ particle has roughly the same mass as $X$, $m_{Y}\sim m_{X}$.

The dominant constraints on this scenario come from cosmological and astrophysical observations (for proposals to cover this parameter space in collider experiments see Haas:2014dda ; Gabrielli:2016rhy ). There are bounds on this scenario from CMB and Lyman-$\alpha$ measurements McDermott:2010pa ; Dvorkin:2013cea ; Dolgov:2013una . The bound on kinetic mixing assuming $X$ makes up all of dark matter is

[TABLE]

in order for $X$s to be kinetically decoupled from baryons during recombination. For larger values of the mixing, $X$ and $Y$ can only make up small fraction, of the total dark matter Dolgov:2013una , $\Omega_{X}<0.001$ (where $\Omega_{X}$ is the energy density of the free $X,Y$ particles). There are also potentially strong constraints ($\epsilon\lesssim 10^{-14}(m_{X}/\,\text{GeV})$) from the measurement of shapes of cluster haloes, which would be affected due to the cluster magnetic force on the MCP dark matter Kadota:2016tqq . However, this constraint is likely to be dramatically weakened if the MCP makes up only a fraction of the dark matter density.

A promising signal in this model arises from the annihilation of dark matter to SM photons through $X$ millicharge Aisati:2014nda . The annihilation cross section for $X\bar{X}\to\gamma Z^{\prime},\gamma\,\gamma_{D}$ is,

[TABLE]

There is also continuum gamma-ray emission originating from $X\bar{X}$ annihilations to SM fermions through an $s$-channel photon. However, for comparable mass and couplings, as is expected in a dissipative scenario, the relative rate of the line emission and continuum emission are comparable:

[TABLE]

The FERMI sensitivity for the line signal Ackermann:2015lka is about 3 orders of magnitude stronger than the continuum emission sensitivity. This implies that for $\alpha_{Z^{\prime}}+\alpha_{D}>10^{-5}$, the line constitutes a more promising avenue for detection of this model.

A search for a line spectrum from a point source will be interesting to carry out since it has the cleanest spectrum and morphology in indirect searches. A spectral analysis of unassociated point sources will be relatively simple, but a stacked analysis or a statistical analysis along the lines of Lee:2014mza ; Lee:2015fea would likely have a higher sensitivity. Note that even in presence of a small mixing as in equation 24, it can be possible to get an observable signal from point sources.

3.2.2 A massive $Z^{\prime}$ portal

We now consider a massive $Z^{\prime}$, which in some respects most closely mimics the gauge structure in the Standard Model. In this case the dominant signal is continuum photons. As above, $C$ does not couple to $Z^{\prime}$. In this case, we do not require the presence of the additional fermion $Y$. This model is similar in some respects to that considered in Izaguirre:2015eya .

We can write the leading interactions of the $Z^{\prime}$ as,

[TABLE]

We work in the basis where only the visible matter coupling to the $Z^{\prime}$ boson is through kinetic mixing, and the dark matter does not pick up any charge under the usual photon. This interaction can produce signals at direct or indirect detection experiments, as well as collider signals. Our focus is the indirect detection signal of an excess in gamma rays from the Galactic Center. In that respect, the annihilation to $Z^{\prime}$ pairs mimics the annihilation to the $b$-quarks in the models of Ref. Daylan:2014rsa ; Berlin:2014tja ; Agrawal:2014una ; Izaguirre:2014vva , which is seen to give a good fit.

We show the collider limits on the massive $Z^{\prime}$ portal in figure 4 (also see Alexander:2016aln for a detailed review of constraints and future prospects, and for further references). A number of future experiments/analyses aim to cover new parameter space for this model Echenard:2014lma ; Battaglieri:2014hga ; Balewski:2013oza ; Abrahamyan:2011gv ; Essig:2010xa ; Bjorken:2009mm ; Anelli:2015pba . In the next section we study in detail the phenomenology of dissipative dark matter with the massive $Z^{\prime}$ portal. This model can potentially explain the excess gamma ray photons observed from the galactic center, which we return to in section 5.

4 Phenomenology

4.1 Relic abundance

A crucial ingredient for dissipative dynamics is the presence of the light $C$ particles. This requires an asymmetry in $C$, since the symmetric population annihilates efficiently. On the other hand, the indirect detection signal from $X\bar{X}$ annihilation requires a symmetric component to survive as well. Thus, we are led to consider a freezeout where both a symmetric as well as an asymmetric component survives.

An alternative where the halo also consisted of charged dark matter was considered in Agrawal:2016quu ; Agrawal:2017rvu . Here we restrict to the possibility that only a fraction of the total dark matter is in $X,\bar{X},C$ particles.

A detailed calculation of the relic abundance in a very similar model was presented in Agrawal:2017rvu (see also Feng:2008mu ; Graesser:2011wi ; Baldes:2017gzw ), so here we just recall the important results. The key differences from the standard relic abundance calculation are the presence of an asymmetric component and a relative temperature between the dark and SM sectors, $\xi=T_{D}/T$. If the interactions between $X$ and the SM are small, the dominant annihilation channels for $X$ will be in the dark sector, $\gamma_{D}\gamma_{D}/Z^{\prime}Z^{\prime}/\gamma_{D}Z^{\prime},C\bar{C}$. We define the ratio of the population of $\bar{X}$ and $X$ as $r$,

[TABLE]

such that the value of the ratio today, $r_{\infty}$, characterizes how much of the symmetric and asymmetric components survive. We define the asymmetry,

[TABLE]

$\eta$ is a constant that is set at early times, and we treat it as a free input parameter.

The Boltzmann equation in terms of $r$ and $x=m_{X}/T$ is,

[TABLE]

We have defined $\lambda$ and $r_{eq}$,

[TABLE]

The values for $g_{*}^{1/2},h_{eff}$ are tabulated in Gondolo:2004sc , and $g_{X}=2$ is the number of degrees of freedom of $X,\bar{X}$. $\lambda(x)$ is roughly constant, since the annihilation cross section is

[TABLE]

to leading order in $m_{Z^{\prime}}/m_{X}$ (for the couplings we are interested in, the Sommerfeld enhancement during thermal freezeout is negligible).

We use an approximation for $r_{\infty}$ by integrating the Boltzmann equation from the freeze-out temperature $x_{f}$ to today neglecting $r_{eq}$, and with the boundary condition $r(x_{f})=r_{eq}(x_{f})$. The value $x_{f}$ can be found by solving the following implicit equation,

[TABLE]

with the approximate solution,

[TABLE]

The value of $r_{\infty}$ obtained through this procedure is a good approximation to the numerical solution.

[TABLE]

Note that $\lambda$ is directly related to the relic abundance in the limit $\eta\to 0$ (i.e. in the absence of an asymmetry),

[TABLE]

so that,

[TABLE]

For the parameter space of interest to us, $r_{eq}(x_{f})\simeq 1$. The dark matter density today is

[TABLE]

where $\Omega_{B},\eta_{B}$ are the baryonic energy density and asymmetry respectively, and $x_{0}$ is the value of $x$ today.

In figure 5 we show the region of $\alpha_{\mathrm{eff}}$–$\eta$ parameter space where a significant symmetric dark matter relic abundance survives in the presence of an asymmetry. The blue curves are contours of constant dark matter relic density $\Omega_{X}$, and the orange lines denote the value of $f_{(XC)}$, defined in equation 8. The shaded region is excluded in this model as too much dark matter would survive.

We see that there are two limiting behaviors of the solutions. When $\eta\ll Y_{\infty}^{\eta=0}$, the result is essentially the symmetric freezeout value. Conversely, for $\eta>Y_{\infty}^{\eta=0}$, the relic abundance is mainly set by the asymmetry. For appreciable amounts of both symmetric and asymmetric components, we choose parameters in the turnaround region, $\eta\sim Y_{\infty}^{\eta=0}$.

4.2 Cooling

We now consider how dark matter haloes might cool through dissipation to form compact objects with enhanced density. The cooling calculation and requisite parameter space follows closely the one outlined in Fan:2013yva , with the modification due to additional $\bar{X}$ particles Agrawal:2017rvu .

The dissipative component of dark matter behaves in analogy to baryonic matter during galaxy formation. Clumps of $X,C,\bar{X}$ accrete onto the CDM halo, and are subsequently shock heated to its virial temperature. If this temperature is larger than the binding energy of the $XC$ bound state, then the charged $X$ and $C$ can radiate. The lighter $C$ dominates the cooling and leads to collapse into a region with lower velocity than that in the dark matter halo and can in principle form a disk. The cooling time of dissipative dark matter must be smaller than the age of the universe (more precisely the dynamical time of the virialized halo) for disks or compact structures to form.

The subsequent behavior of the dark matter gas depends on the nature of the collapse and other details of the model. For baryonic matter this collapse leads to sites of star formation. In principle high density compact objects can result from this collapse in the dark sector as well, though in the absence of additional forces one would not expect nuclear burning or feedback.

We assume the initial condition of a shock heated gas inside a CDM halo of mass $M_{gal}\simeq 10^{12}M_{\odot}$ and virial radius $R_{vir}\simeq 110\,\text{kpc}$. We estimate the cooling timescale for the scenario where the $XC$ dark matter initially falls into the CDM halo with a Navarro-Frenk-White (NFW) profile with a scale radius $R_{s}=20\,\text{kpc}$. The virial temperature for this case is

[TABLE]

where $\mu=m_{X}/(1+f_{(XC)})$ is the average mass of a particle in the dark matter gas, and $M_{s}\simeq 2\times 10^{11}M_{\odot}$ is total halo mass inside the scale radius $R_{s}$. The density of $X$,$\bar{X}$ and $C$ particles inside the scale radius is

[TABLE]

where we have expressed the density of $C$ particles in terms of the asymmetry $f_{(XC)}$ defined in equation 8.

We assume Bremsstrahlung and Compton scattering off the dark CMB photons that can lead to cooling Fan:2013yva (for a detailed analysis of cooling in the dark sector see Rosenberg:2017qia ). The time scale for bremsstrahlung is,

[TABLE]

The Compton cooling off dark CMB occurs on a timescale

[TABLE]

The time scale for equilibration between $X$ and $C$ is

[TABLE]

If the $C$ particles are in kinetic equilibrium with $X$ and $\bar{X}$, i.e. the equilibration timescale is shorter than the cooling time scale, the cooled compact objects will be composed of all of $X,\bar{X},C$ particles. On the other hand, if $t_{eq}>t_{cool}$, then the $C$ particles cool but cannot cool $X$ and $\bar{X}$ particles through equilibrium processes.

4.3 Direct detection

In principle, the kinetic mixing of the $Z^{\prime}$ with the photon in the portal models can give rise direct detection, which in principle could be a significant constraint. The direct detection constraints for millicharged dark matter (i.e. through the massless $Z^{\prime}$ portal) are very strong, with the WIMP-nucleon cross section estimated by

[TABLE]

where $q$ is the typical momentum transfer and $Z$ and $A$ are the atomic and mass number of the target. However, in much of the parameter space, the millicharged dark matter is evacuated from the disk by shock waves generated during supernova explosions, and is shielded from (re)entering the disk by the large-scale magnetic field of the galaxy Chuzhoy:2008zy ; McDermott:2010pa . This mechanism is efficient for

[TABLE]

which is a very large range of $\epsilon$ for $m_{X}\sim 30\,\text{GeV}$.

The constraints for the massive $Z^{\prime}$ portal depend on the mass of the $Z^{\prime}$. For our case, where $m_{Z^{\prime}}\gg q$, the direct detection cross section is,

[TABLE]

In fact this constraint is further relaxed in our setup. The $X,\bar{X}$ component of dark matter makes up only a fraction of the total dark matter. Most of $X$ and $\bar{X}$ are in collapsed objects, so the direct detection signal exists only if these objects overlap with our solar system. Finally, the velocity dispersion in the collapsed object is much lower than the velocity dispersion for the CDM halo, leading to typical recoil energies lower than the experimental threshold for most direct detection experiments. We conclude that direct detection does not put strong model-independent constraints on the model we are investigating.

4.4 Constraints on compact objects

We now estimate the size of compact objects in our models and evaluate existing constraints. We show these constraints in figure 8.

• •

**Total number

**Given a certain mass of the objects and the fraction of dark matter in the $XC$ component, we can derive a bound on the number of such objects in our halo,

[TABLE]

in the entire galaxy. If this bound is not saturated the rest of dissipative dark matter might be in more diffuse clouds.

• •

Longevity

The symmetric component in dense objects can annihilate, putting a constraint on the size of a dark photon gauge coupling for a given value of $M$ and $R$. This constraint is relevant for the existence of compact objects even if there is no $\gamma$-ray signal.

[TABLE]

where we have ignored corrections of $\mathcal{O}(m_{Z^{\prime}}/m_{X})$, and ${S}_{ann}$ is the Sommerfeld enhancement,

[TABLE]

• •

**Further cooling

**

If the virial velocity for a compact object is higher than the binding energy for the $XC$ system, then there may be a non-negligible ionized $C$ component, which can lead to further cooling and collapse. For a virially supported object, we can impose the following conservative bound,

[TABLE]

We impose this constraint for our analysis, but in principle such objects can also be stabilized by other mechanisms. We are also ignoring other cooling precesses such as atomic and molecular cooling which could cool the compact object further.

• •

**Tidal Stripping

**The tidal radius for a compact object of mass $M$ at a distance $r$ from the galactic center is,

[TABLE]

Objects larger than this radius will be tidally disrupted. If the density profile of the compact object is constant, then objects of all size with the given density would not remain. But if the density towards the inner parts the compact object is higher, then tidal disruption can leave behind a dense core of the original object.

In the inner galaxy, the mass enclosed is dominated by baryons in the form of the bulge and the stellar disk. We use the parametrization of the bulge and disk from Binney:1996sv ; McGaugh:2008nc ,

[TABLE]

with $b^{2}=x^{2}+(y/\eta)^{2}+(z/\zeta)^{2}$, $\rho_{B,0}=3.5\times 10^{12}M_{\odot},b_{m}=1.9\,\text{kpc},b_{0}=0.1\,\text{kpc},\eta=0.5$ and $\zeta=0.6$.

The stellar disk is modeled as

[TABLE]

with $\Sigma_{0}=562M_{\odot}/\,\text{pc}^{2}$ and scale radius $r_{d}=3\,\text{kpc}$. As a benchmark, we impose the constraint that the compact object is smaller than the tidal radius at $\sim$ 0.3 kpc from the galactic center,

[TABLE]

5 The Galactic Center Excess

A number of analyses have found that the FERMI telescope has observed an excess of gamma ray emission from the center of the galaxy Goodenough:2009gk ; Hooper:2010mq ; Daylan:2014rsa ; Calore:2014xka ; TheFermi-LAT:2015kwa . There are also hints for excess gamma-ray emission from Andromeda Ackermann:2017nya and from the dwarf galaxy Reticulum II  Geringer-Sameth:2015lua ; Bonnivard:2015tta ; Fermi-LAT:2016uux . The origin of the GC excess is intensely debated, but one exciting possibility is that it originates from dark matter annihilations Goodenough:2009gk ; Hooper:2010mq ; Daylan:2014rsa ; Alves:2014yha ; Berlin:2014tja ; Agrawal:2014una ; Agrawal:2014aoa ; Izaguirre:2014vva ; Ipek:2014gua ; Boehm:2014bia ; Abdullah:2014lla ; Martin:2014sxa ; Bell:2014xta ; Freytsis:2014sua ; Agrawal:2014oha ; Buckley:2014fba ; Elor:2015tva ; Kaplinghat:2015gha ; Karwin:2016tsw .

However, Refs. Lee:2015fea ; Bartels:2015aea showed that there are some hints for a statistical preference for point-source emission in the GCE (but this conclusion might be sensitive to structure in the diffuse background that is not modeled by the range of background models considered, see also Horiuchi:2016zwu ). The morphology of the excess if pointlike would be in tension with the smooth distribution expected for a CDM annihilation Clark:2016mbb ; Fermi-LAT:2017yoi . Most physicists would then attribute the observed point-like spectrum to a new population of Millisecond Pulsars (MSPs) Brandt:2015ula ; Fermi-LAT:2017yoi . We investigate the alternative possibility that these point sources could arise from dark matter annihilation in compact objects.

We highlight the parameters of our model that can reproduce this point source signal, specifically the mass $m_{X}$ and the coupling $\alpha_{Z^{\prime}}$, and the size of objects, $\{M,R\}$. We choose the simplified limit where all objects have similar masses and sizes, and have a uniform density. It would be straightforward to vary these assumptions to include any other mass function and a realistic density profile and distribution that might be motivated by N-body simulations. If there is a very sizeable spread in the mass function, there will be a large number of brighter compact objects, which should have been resolved as point sources by Fermi. We assume the spatial distribution of the compact objects in the inner galaxy to be spherically symmetric with an NFW-squared radial profile (more precisely $r^{-2.5}$) in the inner galaxy. This is motivated by the observed morphology of the galactic center excess.

The mass $m_{X}$ can be fixed by finding the best-fit spectrum to the excess. The point source analysis in Lee:2015fea uses a single wide energy bin, which is unsuitable for a spectral analysis. Instead we find the best-fit spectrum following the analysis in Calore:2014xka ; Agrawal:2014oha . We then consider the annihilation rate and the size and mass of compact objects required to reproduce the observed point-source excess.

5.1 Spectrum for the Galactic Center excess

For the spectral fit in this section, we will follow the analysis of Calore:2014xka . The region of interest (ROI) in this analysis extended to a $\pm 20^{\circ}$ square around the galactic center, with the inner $2^{\circ}$ latitude masked out. We further choose an NFW profile ($\rho_{\odot}=0.4\,\text{GeV}/\,\text{cm}^{3},R_{s}=20\,\text{kpc},\gamma=1.20$), which translates to $J=2.0\times 10^{23}\,\text{GeV}^{2}/\,\text{cm}^{5}$.

The flux from DM annihilation with DM mass $m_{X}$ is (for Dirac fermions)

[TABLE]

where the flux factor $J$ is the line-of-sight integral of the dark matter density, and $dN/dE$ is the spectrum of photons per annihilation. Given the $J$-factor and the spectrum per annihilation for different masses, we can in principle find the best-fit $m_{X}$ and $\langle\sigma v\rangle$. We will use only the $m_{X}$ value found this way, and our best-fit $\langle\sigma v\rangle$ will instead be determined by fitting to the point source flux in the next section.

When $\alpha_{D}>\alpha_{Z^{\prime}}$, the dominant annihilation mode that produces photons is $X\bar{X}\to\gamma_{D}Z^{\prime}$. The $Z^{\prime}$ then decays to SM fermions through the $\epsilon$ mixing which produces a continuum photon spectrum. The spectrum can be calculated by boosting the photons in the rest frame of the $Z^{\prime}$ to the galactic frame. The spectrum of photons produced in such cascade decays was also considered in a model-independent analysis in Elor:2015bho .

The spectrum of photons in the rest frame of $Z^{\prime}$ is

[TABLE]

where the sum is over the spectrum from each of the decay modes of the $Z^{\prime}$ through the kinetic mixing Curtin:2014cca , and $\frac{dN_{f}(E)}{dE}$ is the spectrum of photons from decay $Z^{\prime}\to f$. The spectrum from various standard model final states can be simulated using PYTHIA Sjostrand:2006za ; Sjostrand:2007gs , and has been tabulated in PPPC 4 DM ID Cirelli:2010xx ; Ciafaloni:2010ti .

The spectrum of photons per annihilation, in the galactic frame, is obtained by convolving the spectrum in the rest frame of $Z^{\prime}$ with a unit normalized “box” Agrawal:2014oha

[TABLE]

and

[TABLE]

In figure 7 we show the $\chi^{2}$-contours for the photon spectrum fit to the observed excess, as a function of the dark matter mass and the cross section (given the astrophysical $J$-factor) and the spectrum of the best-fit point. The best fit dark matter mass for our model, $m_{X}\simeq 25\,\text{GeV}$ is seen to provide a good fit to the observed spectrum of the excess.

5.2 The Point-like GeV Excess from Compact Objects

in Dissipative Dark Matter

According to  Lee:2015fea (see also Bartels:2015aea ; Fermi-LAT:2017yoi ), the signal appears to originate from point-like sources that are distributed with an NFW-squared profile. The region of interest for this analysis was taken to be inner $30^{\circ}$ with $|b|<2^{\circ}$ masked. The pixel size for the analyses is $0.5^{\circ}$ to a side, which translates to about $75\,\text{pc}$ at the center of the galaxy. The point spread function varies between $0.05^{\circ}$ to $0.2^{\circ}$ for the energy range $1$–$10\,\text{GeV}$. Therefore, any objects smaller than about $10-100\,\text{pc}$ would give rise to non-Poissonian photon statistics, appearing as point sources. We saw in section 2.1 that for our chosen parameters we expect our compact objects to be $\mathcal{O}$(10–100) pc in size, and hence to appear as point sources.

The analysis predicts a certain number of objects in the inner galaxy to account for the total flux absorbed by the point source template. In the inner $10^{\circ}$ of the galaxy with $|b|\geq 2^{\circ}$, $86^{+32}_{-25}$ point sources can explain about half of the excess from the galactic center when the Fermi 3FGL point sources are masked. For the unmasked analysis in Lee:2015fea the corresponding number is $132^{+31}_{-25}$.

We note that the flux and the number of objects is roughly fixed independently in addition to being constrained by the total amount of flux. If we had a much larger number of objects (with a corresponding smaller flux per object), they would have photon-per-pixel statistics closer to a smooth morphology predicted by standard CDM substructure. Much fewer objects would need to be brighter to account for the total flux, and hence would be above the threshold for detection as point sources by Fermi.

We next turn to the normalization of our signal, which we fit to Lee:2015fea . The normalization depends on the $J$-factor, on the total annihilation cross section $\langle\sigma v\rangle$ and on the number of photons per annihilation.

[TABLE]

The total annihilation rate $X\bar{X}\to Z^{\prime}\gamma_{D}$ is

[TABLE]

to leading order in $m_{Z^{\prime}}/m_{X}$. The velocity of dark matter in the compact object ($c_{XC}$) is given in equation 4.

In this analysis, a single energy bin from 1.893–11.943 GeV was used. For $m_{X}=25\,\text{GeV}$, $m_{Z^{\prime}}=10\,\text{GeV}$, this corresponds to,

[TABLE]

The $J$-factor associated with the point sources is given by,

[TABLE]

where $\bar{J}$ is the $J$-factor for each point source if they are all identical. In the analysis of Lee:2015fea , the flux from each source was estimated to be $\bar{\Phi}=1.4\pm 0.3\times 10^{-10}$ photons/cm2/s. The $J$-factor for each point source is

[TABLE]

where $f_{(XC)}$ is the fraction of the dissipative component in bound states. The flux from each point source is,

[TABLE]

where the numerical prefactor above is the consistent with the typical flux from a point source in the analysis of Lee:2015fea ; Lee:2014mza . The value of $\langle\sigma v\rangle$ is enhanced by the Sommerfeld effect, and therefore can be much larger today than the canonical thermal freezeout value of $2.2\times 10^{-26}\,\text{cm}^{3}/\,\text{s}$ without affecting the relic abundance calculation.

In figure 8 we see the parameter space where the dark matter annihilations can explain the GCE. We also show constraints on the parameter space where the compact objects can undergo tidal disruption, annihilate away within the age of the universe, or are unstable to further cooling. We also show benchmark points from our instability analysis in section 2 for a few choices of the local dark disk density.

As noted above, we need an $\mathcal{O}(100)$ compact objects of mass $10^{4}M_{\odot}$ each to account for the total flux of the GCE. For a CDM component with an NFW profile the total mass of dark matter in the inner galaxy region is $\sim 10^{9}M_{\odot}$. Therefore, $XC$ dark matter can be as little as $10^{-3}$ of the total dark matter in the inner galaxy while still giving us interesting signals. Intriguingly, the benchmark values from our stability analysis, $\Sigma_{XC,\odot}\sim$ 1–10 $M_{\odot}/\,\text{pc}^{2}$, translate into a total dark disk mass of $10^{9}$–$10^{10}M_{\odot}$, or $10^{-3}$–$10^{-2}$ fraction of the total galactic dark matter.

6 Conclusion

Dissipative dark matter models can give rise to clumps of dark matter with enhanced density. Even if dissipative dark matter makes up a tiny fraction of the total dark matter density, these clumps can be the dominant site of annihilation, leading to novel indirect detection signatures.

We have considered a simple example of such a dissipative dark matter model and estimated the size and mass of collapsed objects we might expect as a function of the model parameters. We have shown that these objects could be small enough to appear as point sources in Fermi-LAT. The spectrum depends on the portal connecting the dark sector with the Standard Model.

With an additional massless dark photon, dark matter particles can annihilate to photon and a dark photon, which appear as a line spectrum in Fermi-LAT. A line from point sources would be a smoking gun signal of dissipative dark matter models. If the portal is a massive vector, the annihilation then leads to continuum photons, with a spectrum of that resembles the spectrum of photons from the $b\bar{b}$ final state. This can potentially provide an explanation of the point-source origin of the galactic center excess as arising from dark matter annihilation in these objects.

The point source morphology and continuum spectrum of gamma rays are characteristic signals of millisecond pulsars but as we have shown might possibly occur in certain classes of dark matter models. Our analysis gives us a target which we can try and distinguish from MSPs.

The authors of Calore:2015bsx consider observations of pulsars in other wavelengths and conclude that pulsar surveys in the radio frequencies will potentially be able to detect millisecond pulsars in the bulge. The absence of any signal in other wavelengths will strength to the hypothesis of dark matter compact objects as the origin of this signal.

The angular size of dark matter clouds in the inner galaxy can be comparable to the resolution of Fermi. If there are larger clouds, or clouds closer to us, Fermi-LAT might be able to resolve the structure, definitively discriminating it from pulsars.

UCMHs that form from large fluctuations in the power spectrum have also been considered as potential dark point sources Berezinsky:2003vn ; Ricotti:2009bs ; Scott:2009tu ; Josan:2010vn ; Erickcek:2011us ; Bringmann:2011ut ; Berezinsky:2013fxa ; Clark:2015sha ; Aslanyan:2015hmi ; Clark:2016pgn . For the UCMH, in addition to annihilation of dark matter within the UCMH, there will be annihilations of dark matter in the UCMH and in halo, leading to a different radial dependence of the excess. There will also be annihilations in the halo with a diffuse morphology. These features will help distinguish UCMHs from dissipative compact objects. Further, due to dissipation the compact objects considered in this paper will be concentrated towards the center of the halo (possibly along a plane), whereas we expect UCMHs to be isotropically distributed throughout the halo.

Very little is known about dark matter, and even less about components which might be a small fraction. However, it is precisely these small but interesting components – not unlike baryons – which might provide the most spectacular signals from dark matter.

Acknowledgements.

We thank Mariangela Lisanti for very useful comments on an earlier version of this manuscript. We are grateful to Francis-Yan Cyr-Racine, Matt Reece and Jakub Scholtz for discussions and comments on the manuscript. This work is supported by NSF grants PHY-0855591 and PHY-1216270.