Reheating neutron stars with the annihilation of self-interacting dark matter

Compact stellar objects such as neutron stars (NS) are ideal places for capturing dark matter (DM) particles. We study the effect of self-interacting DM (SIDM) captured by nearby NS that can reheat it to an appreciated surface temperature through absorbing the energy released due to DM annihilation. When DM-nucleon cross section $\sigma_{\chi n}$ is small enough, DM self-interaction will take over the capture process and make the number of captured DM particles increased as well as the DM annihilation rate. The corresponding NS surface temperature resulted from DM self-interaction is about hundreds of Kelvin and is potentially detectable by the future infrared telescopes. Such observations could act as the complementary probe on DM properties to the current DM direct searches.


I. INTRODUCTION
Dark matter (DM) composes one-fourth of the Universe, however, its essence is still elusive. Many terrestrial detectors are built to reveal the particle nature of DM either from measuring the coupling strength between DM and the Standard Model (SM) particles [1][2][3][4][5][6][7] or the indirect signal from DM annihilation in the space [8][9][10][11][12][13]. But the definitive evidence is yet to come.
A compact stellar object such as neutron star (NS) is a perfect place to capture DM particles even when the DM-nucleon cross section σ χn is way smaller than the current direct search limits. Investigations on DM in compact stellar objects are studied recently in Refs. [14][15][16][17][18][19][20][21][22][23]. Due to the strong gravitational field, DM evaporation mass for NS is less than 10 keV [19]. Therefore, NS is sensitive to a broad spectrum of DM mass from 10 keV to PeV, sometimes it can be even extended to higher mass region. Unlike the Sun, it loses its sensitivity to DM when m χ 5 GeV as a consequence of evaporation [24]. In the later discussion, we will focus on the Weakly Interacting Massive Particle (WIMP) scenario with mass from MeV to 10 TeV.
An old NS has the age greater than billions of years, its temperature might drop out a few Kelvins after processing several cooling mechanism by emitting photons and neutrinos [25,26]. However, if the residing DM particles in NS can annihilate to SM particles, it could inject energy and heat up the host star [14,15]. In addition, recent literature also suggests that the halo DM particles constantly bombard NS would deposit their kinetic energy to the star. This is called dark kinetic heating [27]. These two contributions might prevent NS from inevitable cooling and maintain its surface temperature T surface at [28] T surface = 2480 K σ eff χn σ c where σ eff χn is the effective DM-nucleon cross section and σ c the critical DM-nucleon cross section. Their mathematical expressions will be given in the later sections. In principle, NS surface temperature T surface is different from its interior temperature T but correlated. The relation connecting T and T surface will be given later.
However, current direct searches have put more stringent limits on σ χn to test. If it is small enough, DM self-capture will eventually take over. In this region, DM self-interaction will re-enhance the captured DM particles and such effect is in the same way as having a relatively large σ χn . It also increases the DM annihilation rate so does the energy injection.
Our study shows the reheating mechanism is induced by the thermal effect. DM cannot thermalize with NS if σ χn is too weak. In such case, DM temperature T χ will decouple from NS interior temperature T in the early stage of evolution. This decoupled temperature T dec χ is in general higher than the current NS interior temperature. A higher T dec χ indicates larger thermal area and has better chance to capture other halo DM particles. Therefore, NS will experience a reheating process when σ χn is small enough. Such process is possible to maintain T surface up to hundreds of Kelvins.
An old, isolated NS nearby the Solar System emits infrared is a very good candidate to pin down such effect coming from DM self-interaction. The corresponding blackbody peak wavelength is potentially detectable in the future telescopes, e.g. the James Webb Space Telescope (JWST) [50], the Thirty Meter Telescope (TMT) [51] and the European Extremely Large Telescope (E-ELT). In the following context, we consider a nearby NS with age t NS 2 × 10 9 years, mass M = 1.44M where M ≈ 1.9 × 10 33 g is the Solar mass and radius R = 10.6 km. It also has the halo density ρ 0 = 0.3 GeV cm −3 , the DM velocity dispersionv = 270 km s −1 and the NS velocity relative to the Galactic Center (GC) v N = 220 km s −1 . For discussion convenience, we will use natural unit c = = k B = 1 and G = M −2 P l in this paper. This paper is structured as follows: In Sec. II and Sec. III, we briefly review the formalism of DM captured by NS and the cooling and heating mechanism respectively. In Sec. IV, numerical results are presented as well as the discussion on the reheating effect. We will summarize our work in Sec. V.

A. DM evolution equation
When the halo DM particles scatter with NS and lose significant amount of energies, they will be gravitationally bounded in the star. The DM number evolution can be characterized by the differential equation where N χ is the DM number, C c the capture rate due to neutrons in NS, C s the DM selfcapture rate and C a the DM annihilation rate. A general solution to N χ is given by where τ = 1/ C c C a + C 2 s /4 is the equilibrium timescale. In the case of t τ , dN χ /dt = 0.
It reaches DM number equilibrium in NS. Hence we have Thus, R signifies how crucial that the DM self-capture is in the DM evolution in NS.
Additionally, we can obtain two solutions to N χ when dN χ /dt = 0 by examining Eq. (3), That means, either a dominant C c or a dominant C s , can accumulate the same amount of DM particles in NS. In the later discussion, we will show that the similar degeneracy effect will cause the same NS surface temperatures.

B. Rates of DM capture and annihilation
The capture rate due to DM scattering with target neutrons in NS is given by [19] where ρ 0 is the DM density,v the DM velocity dispersion, N n = M/m n the total number of target neutrons in NS, and M and R are the mass and radius of NS respectively. The suppression factor ξ = δp/p F is due to the neutron degeneracy effect. The momentum transfer in each scattering is δp √ 2m r v esc where m r = m χ m n /(m χ + m n ) the reduced mass and v esc 1.8 × 10 5 km s −1 . Since the DM-nucleon cross section cannot exceed the geometric limit that is given by N n σ c = πR 2 where σ c 2 × 10 −45 cm 2 is the DM-nucleon critical cross section in NS. Thus, σ eff χn ≡ min(σ χn , σ c ). The last factor Unless m χ 10 TeV, the term in the parentheses is roughly unity.
Another way of capture is due to the halo DM particle scatters with the trapped DM particle. This is DM self-capture and is given by [20] where v esc (R) is the escape velocity at the surface of NS. For a rather conservative calculations, we take φ χ = 1 [20].
In addition, DM self-capture suffers from the geometric limit as well. The critical cross section for DM self-interaction is defined by N χ σ c χχ = πr 2 th where [19] is the thermal radius of DM and T χ is the DM temperature. If DM has higher T χ , it results in larger thermal area πr 2 th . Hence we have σ eff χχ = min(σ χχ , σ c χχ ). When the geometric limit for DM self-capture attains, we have C s N χ ∝ σ c χχ N χ = πr 2 th . The effect of C s saturates at this point and depends only on the size of r th regardless of the value of σ χχ .
In the end, having more and more DM particles accumulated in NS, the chance of DM annihilation becomes appreciated. Thus, and the total annihilation rate is In the later discussion, we will use thermal relic annihilation cross section σv = 3 × 10 −26 cm 3 s −1 unless otherwise specified.

III. NEUTRON STAR COOLING AND ENERGY INJECTION DUE TO DM AN-NIHILATION
After the birth of NS, it undergoes the cooling mechanism due to neutrino and photon emissions [25,26]. Nonetheless, if the residing DM particles in NS can annihilate, they will inject extra energies to heat the host star up. The NS interior temperature T can be described by the following differential equation where ν,γ,χ are the emissivities due to neutrino, photon emissions, and DM respectively.
They are given by [14,25] where n 4 × 10 38 cm −3 is the NS baryon number density and n 0 0.17 fm −3 the baryon density for the nuclear matter [14]. In this calculation, it is n/n 0 2.3. The rate of heat loss due to photon emission is given by L γ = 4πR 2 σ SB T 4 surface where σ SB is the Stefan-Boltzmann constant. We can further connect the NS surface temperature T surface to the NS interior temperature T by the relation [26,52,53] T surface = 0.87 × 10 6 K g s 10 14 cm s −2 where g s = GM/R 2 = 1.85 × 10 14 cm s −2 is the surface gravity. Hence we have the effective photon emissivity [14] On the other hand, NS heating comes from the contributions of DM annihilation and dark kinetic heating. They are given by for annihilation and for dark kinetic heating. The factor f χ characterizes the energy absorption efficiency which runs from 0 to 1 and E s = m χ (γ − 1) the DM kinetic energy deposited in NS. It is given by γ 1.35 [28]. Therefore, we have for DM.
The last quantity c V is the heat capacity of NS. It is expressed as [14] c where index i runs over n, p, e and the corresponding Fermi momenta are For calculation convenience, we have expressed all these quantities in terms of natural unit.

IV. SIDM IMPLICATION FOR NEUTRON STAR TEMPERATURE AND DM-NUCLEON CROSS SECTION
Before presenting the numerical results, we briefly introduce the setups of our calculations. In the presence of DM self-interaction, it might come to a time stamp t s that DM self-interaction cross section reaches its geometric limit σ c χχ = πr 2 th /N χ . If our numerical procedure detects that the input σ χχ is larger than σ c χχ , it will automatically return σ c χχ to avoid overestimating the effect of DM self-interaction. Since N χ still changes after t > t s unless reaching N χ,eq , σ c χχ will adjust itself accordingly as well. Moreover, σ c χχ depends on the DM temperature T χ due to its dependence on r th . To thermalize with NS, it costs a timescale [19] t th ≈ m 2 χ m n p F 6 √ 2n n σ χn m 3 where n n is the neutron number density of the NS. 1 Rigorous discussions on the topic of thermal exchange can be found in Refs. [19,20,54].
During NS cooling, when the NS interior temperature T drops too low, DM might not have enough time to thermalize with T within t NS . Hence, in each time stamp t of our 1 To determine t th one needs to solve the DM energy loss rate: dE/dt = −ξn n σ χn vδE. The detail discussion is beyond the scope of this work. It can be found in Refs. [19,20,54] and references therein. numerical calculation, we constantly check if t th < t NS holds. It returns T χ (t) = T (t) for true. On the contrary, if our program finds t th > t NS at time stamp t dec , we have T χ decoupled from the evolution of T . After this moment, T χ will always freeze at T dec the decoupled temperature. 2 In addition, if σ χn is too small to have DM particles became thermal equilibrium with NS at the beginning t = t 0 , we let T χ = T 0 where T 0 is the initial temperature at t = t 0 for both NS and DM that we put manually as the initial condition for numerical calculation.

A. Numerical results
Our numerical results for the evolutions of NS interior temperature T and the DM temperature T χ are shown in Fig. 1. The initial condition is T 0 = 10 9 K at t 0 = 100 years. The temperature evolution is insensitive to the initial condition. After the first few decades, the effect of T 0 becomes negligible. This agrees with Ref. [14]. In the case of σ χn = 10 −55 cm 2 , roughly before the first one million years, DM kept in thermal equilibrium with NS. However, when the NS temperature dropped out T 10 8 K, the 1 TeV DM mass cannot thermalize with the NS within t NS . Thus, T χ decoupled from T and lives on its own. For lighter 2 Assuming no dark radiation is emitted afterward, thus DM will not suffer from cooling mechanism. We have taken f χ = 1.
DM particle, it would be easier to thermalize with NS due to more violent thermal motion v ∝ T χ /m χ . Exception happens when m χ 1 GeV, its σ χn is subject to a suppression ξ = δp/p F due to neutron degeneracy effect. If DM can maintain at higher temperature, its thermal radius r th will be larger. Although the DM self-interaction cross section already saturated at σ c χχ in the very early period, with larger r th , σ c χχ ∝ πr 2 th results in larger impact to the DM self-capture rate. The situation is more interesting for σ χn = 10 −58 cm 2 . As a consequence of smaller σ χn , all T χ decoupled from T far earlier and hotter than those with larger σ χn .
Before continuing the discussion, let's take a look of Eq. (3). Since DM self-capture usually saturates at this stage with T χ = T dec χ , essentially we have C s N χ ∝ σ c χχ mχ N χ = πr 2 th /m χ . Therefore, dN χ /dt = 0 generally implies 3 where the energy injection does not concern with σv . When dT /dt = 0, we can simply equal Eq. (16) and Eq. (19). By incorporating Eq. (15), the scaling relation for T surface in the C s dominant region is expressed as where T dec χ is estimated as for m χ > 1 GeV and for m χ 1 GeV. The minimum function in Eq. (24) indicates that T dec χ cannot exceed the initial temperature T 0 . It means that the reheating process is not unlimited. When σ χn is too weak to make t th < t NS at t = t 0 even having T χ = T 0 , the corresponding T dec χ will always freeze at T 0 right after its capture as indicating previously. This corresponds to the largest thermal area πr 2 th , so does the maximum DM self-capture and energy injection. Therefore, it induces the highest T max surface by SIDM. Our scaling relation is valid for T 0 = 10 9 K which should cover most of the situation.
In Fig. 2, T surface versus σ χn is shown with different m χ . The temperature is calculated at t = t NS . For σ χn > 10 −55 cm 2 , it is in C c dominant region. Our numerical calculation agrees with Eq. (1) very well (gray dashed lines). However, for smaller σ χn 10 −55 cm 2 , the self-interaction starts taking over. In this region, T dec χ ∝ 1/σ χn thus T surface ∝ σ −1/4 χn . Until T dec χ = T 0 , the plateaus to each m χ in Fig. 2 represent the associated maximum NS surface temperature T max surface . As a remark, in this paper we set the initial NS temperature T 0 = 10 9 K at t 0 = 100 years.
However, the exact value of T 0 is uncertain and highly depends on the nuclear structure of NS in the very early stage of evolution. Although such effect does not directly affect NS cooling in the later stage, it determines T max surface . If T 0 is higher, then in general DM would have a larger T dec χ when σ χn is too small to thermalize with NS since the beginning t = t 0 .
Thus, NS will have a higher T max surface than the result we present here. To serve as an illustrative purpose, our setup is rather conservative.
In Fig. 3, a broad scan of T surface over m χ − σ χn plane are displayed. It is obvious that NS is warmer in the presence of SIDM than without SIDM. Higher temperature region is essentially induced by the larger DM self-capture as discussed earlier. Warmer region for m χ 1 GeV is explained by the suppression on σ χn due to neutron degeneracy effect.
Likewise, the thermalization would be more difficult because the thermal motion is less vital for m χ > 1 GeV. Either way leads to a hotter T dec χ , thus enhances the DM self-capture and T surface consequently.

B. Minimum NS temperature and maximum reheating effect
Here we present our final result in Fig. 4. In this figure, the purple thick line corresponds to the minimum NS surface temperature T min surface in the presence of SIDM. It associates to the valleys in Fig. 2 to each m χ . On top of this line, the σ χn is strong enough to make the capture dominated by C c effect. Thus, T surface scales as Eq. (1) very well. Below this line, C s effect dominates the capture. The NS temperature experiences a reheating due to the increasing capture rate by SIDM.
It is easily noticed that there would be two lines showing the same T surface . For instance, green lines indicate T surface = 100 K. Though one is due to C c (above the purple line) and the other is due to C s (below the purple line), they are in general indistinguishable by The purple thick line corresponds to the coldest T min surface in the presence of SIDM. Below this bound, NS can be reheated due to the increasing DM self-capture efficiency. This reheating region acts like it has a relatively larger σ χn . However, the reheating process saturates at T max surface when σ χn is too small to make t th < t NS since t = t 0 . Thus, r th is in its maximum so does the DM self-capture. The T max surface to each m χ is displayed in color gradient. The limits on σ χn from different direct searches such as DARWIN [3], LUX [4] and XENON1T [7] are shown in the plot as well.
measuring NS surface temperature solely. The color gradient indicates the maximum NS surface temperature T max surface to each DM mass. DM with m χ 60 MeV has the highest T max surface 700 K.

V. SUMMARY
In this work, we found that when DM self-capture dominates the evolution of DM in NS, T surface will increase by this effect. T surface will be in the same way as DM has a relatively large σ χn . In addition, there exists a minimum NS temperature T min surface ∼ O(10 K) before C s dominates the capture process. However, DM self-interaction cross section reaches its geometric limit σ c χχ very quickly and becomes irrelevant to σ χχ . The value of σ χχ determines how fast the DM self-capture attains the geometric limit but not the ultimate fate of DM in NS. Thus, to constrain σ χχ from NS temperature may not be possible. Nonetheless, if NS temperature smaller than T min surface is observed, it could be inferred that either non-existence of DM self-interaction, inefficient energy absorption f χ or even no DM annihilation. On the other hand, T surface has a maximum value T max surface when DM cannot thermalize with NS since the beginning t = t 0 . Such features due to SIDM presented in this paper are natural consequences when we consider T χ does have its own pace in the evolution. Additionally, a scaling relation for T surface in the C s dominant region is shown as well.
In a certain parameter space, both C c and C s will generate T surface which are essentially degenerate. The corresponding blackbody peak wavelength is infrared and could be detected by JWST, TMT and E-ELT in the future. If T surface predicted by such parameter space is observed, its interpretation should be made carefully.