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Fermionic Dark Stars Spin Slowly

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Fermionic Dark Stars Spin Slowly Fermionic dark stars spin slowly Shin’ichirou Yoshida∗ Department of Earth Science and Astronomy, Graduate School of Arts and Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan (Dated: November 17, 2020) We study r-mode instability of rotating compact objects composed of asymmetric and self- interacting Fermionic dark matter (dark stars). It is argued that the instability limit the angular frequency of the stars less than half of the Keplerian frequency. This may constrain the stars as alternatives to fast spinning pulsars or rapidly rotating Kerr black holes. I. INTRODUCTION effect balances the growth of unstable r-modes. The re- sulting spin frequencies of stars are considerably smaller Some alternative models have been presented against than the Keplerian limit. For a model with typical mass such standard compact object models as neutron stars of a neutron star, the balancing frequency may be lower and black holes. One such category is a compact object than those of observed millisecond pulsars. Higher the made of dark matter particles. Though nature of dark mass of the model, the frequency becomes lower. For a matter is still elusive, the most plausible candidates are stellar mass black hole case, a dark star model may not elementary particles beyond the standard model, which serve as an alternative to a rapidly rotating Kerr black interact very weakly (or do not interact) with the stan- hole. dard model particles. [1] These dark matter particles may form a bound state by their self-gravity. One of the suggestions made is that these exotic objects may be II. FORMULATION an alternative to black hole candidates in compact X-ray binaries. However, these alternatives in accreting X-ray A. Assumptions binaries is shown to exhibit thermonuclear flashes which is absent in observed black hole candidates[2]. We here study non-rotating stellar equilibria composed Objects with typical mass of neutron stars and made of baryonic and dark matter whose mutual interaction is from dark matter are also considered.They are termed as negligible. The baryonic matter is a zero-temperature dark stars [3, 4] and possibility of them to be alternatives nuclear matter. The dark matter is a self-interacting to neutron stars is discussed. These dark stars are made Fermion with a vector mediator field. of asymmetric dark matter (ADM) with self-interaction [5] which is a candidate of cold dark matter and intro- duced to solve some cosmological and astrophysical prob- B. Dark star model lems [6–9]. Effects of rotation on these alternatives have not been fully investigated. One of the possible and important Since the stars in the present study are assumed to effects is appearance of instability related to stellar ro- be static and spherically symmetric, the spacetime al- tation. For relativistic stars, Chandrasekhar-Friedman- lows the Schwarzschild coordinate (r, θ, ϕ) in which the Schutz instability [10, 11], where coupling of stellar oscil- spacetime metric is written as, lations with gravitational wave leads to secular instability 2 µ ν 2ν 2 2λ 2 2 2 2 for rotating stars, may be powerful enough to limit the ds = gµν dx dx = −e dt +e dr +r sin θdϕ . (1) stellar rotation speed by removing its angular momen- The geometrized unit, e.g., c =1= G, is assumed here. arXiv:2011.07285v1 [astro-ph.HE] 14 Nov 2020 tum through gravitational wave emission. Especially the r-mode instability [12, 13] may be effective even for slowly The stress-energy tensor of the dark matter is written as, rotating stars as far as the viscous damping of the oscil- µν µ ν µν lations are sufficiently weak. T = (ǫ + p)u u + pg . (2) In this paper we study the effect of r-mode instability on the rotational frequency of dark stars. We do not con- The energy density ǫ and the pressure p are that of dark sider Bosonic dark star, since they may not resemble a Fermion whose equation of state (EOS) is that of degen- fluid star, but rather a gigantic Bose-Einstein condensate, erate Fermions of mass mX , which require a totally different treatment. A conven- 4 2 tional recipe of molecular viscosity leads to weak shear mX 2 αX mX 6 ǫ = 3 ξ(x)+ 3 2 x ; viscosity of self-interacting ADM fluid whose damping ¯h " 9π ¯h mφ # 1 ξ(x)= x 1+ x2(2x2 + 1) − ln(x + 1+ x2) , 8π2 ∗ [email protected] h p p i(3) 2 and C. r-mode instability m4 2 α m2 X X X 6 We adopt the simple estimation of r-mode instability p = 3 χ(x)+ 3 2 x ; ¯h " 9π ¯h mφ # in [16]. A stellar model is assumed to be spherically sym- metric and the lowest-order post-Newtonian approxima- 1 2 2 χ(x)= x 1+ x2 x − 1 + ln(x + 1+ x2) , tion is adopted to evaluate gravitational radiation from 8π2 3 r-mode oscillations. p p (4) The gravitational wave timescale τGW of r-mode oscil- lation is estimated by the ratio of luminosity to oscillation Here x = pX /mX is dimensionless Fermi momentum of dark Fermion. The self-interaction of dark Fermions are energy as, mediated by a boson with the rest mass mφ, whose cou- 2ℓ 2ℓ+2 −1 32πG 2ℓ+2 (ℓ − 1) ℓ +2 pling constant is αX (see [14] and [15]). In this paper we τ = − Ω GW c2ℓ+3 K [(2ℓ + 1)!!]2 ℓ +1 focus our interest on the repulsive self-interaction of dark Fermions, thus α > 0. R 2ℓ+2 X 2ℓ+2 Ω Assuming the cross section of the self-interaction to be × ρr dr , (13) ΩK [5], Z0 where ρ is mass density, ℓ is the order of spherical har- 2 2 −4 αX mX mφ σ =5 × 10−23(cm2) , monics, Ω is the rotational angular frequency, and ΩK is 10−2 10GeV 10MeV the Keplerian frequency of the given mass M and R of a (5) star [16]. As for the mass density we simply use the mass and the constraints of cross section to be [3], energy density ǫ divided by c2, which is the solution of σ TOV equation. ≤ ≤ 2 −1 0.1 1 (cm g ), (6) As for viscosity of dark Fermion fluid we adopt the mX simple estimate of molecular viscosity. Microscopic ori- we have a constraint. gin of shear viscosity is the transport of momentum by 2 4 2 dark matter particles. The shear viscous coefficient η is 1 2 −1 αX mX mX written as (e.g., see [17]) (cm g ) ≤ −2 3 10 100mφ 1GeV −1 −1 1 10 2 −1 σX kB TX σX ≤ (cm g ). (7) η = a = ac θ 2 (14) 3 m m c2 m X X s X X As is easily seen, we have two independent parame- where a is a factor of order of unity and kB is Boltz- ters for dark Fermion EOS. We choose the following two mann constant. T is the ’temperature’ of dark Fermions parameters, µ and κ X characterizes random motions of dark Fermions. θX is 2 mX the normalized temperature parameter k T /m c . µ ≡ , (8) B X X 1GeV From the assumption of our analysis on degeneracy of dark Fermion, we expect θ ≪ 1. Wep also note that and X σX /mX and θX are degenerate as parameters. We here 2 re-normalize θX so that the combination aσX /mX is re- mX αX κ ≡ −2 . (9) duced to unity. 100mφ 10 The damping timescale τV of r-mode is [16] Then the constraint (7) is rewritten as −1 R R −1 − 2ℓ 2ℓ+2 1 −3 2 10 τV = c(ℓ 1)(2ℓ + 1) ηr dr ρr dr . ≤ µ κ ≤ . (10) 0 0 ! 3 3 Z Z (15) With the EOS, Tolman-Oppenheimer-Volkoff equa- The critical rotational frequency at which growth of tions, r-mode instability and viscous damping of the mode are in balance is determined by, dν m +4πpr3 = , (11) dr r(r − 2m) τGW + τV =0, (16) r ′2 ′ whose solution Ω/ΩK is the stellar rotational frequency where m(r) = 0 ǫ4πr dr and the hydrostatic balance is, at which both effect balances. A larger value of Ω/ΩK R than ωc means r-mode instability dominates and makes dp (ǫ + p)(m +4πpr3) the star spin down. The equation is concisely written as = − , (12) − 2 +2 dr r(r 2m) ℓ −1 1 1 Ω τ˜V 2 2ℓ+2 2 = −1 θX ≡ ωc θX (17) are numerically solved to obtain equilibrium stars. ΩK τ˜ GW 3 where as M = 1.5M⊙. The region between dashed and dotted 2ℓ+2 curves corresponds to the one satisfying Eq.(10). The fac- 32πG (ℓ − 1)2ℓ ℓ +2 R τ˜−1 = Ω2ℓ+2 ρr2ℓ+2dr tor is O[0.1]. Therefore the critical angular frequency of GW c2ℓ+3 K [(2ℓ + 1)!!]2 ℓ +1 Z0 the dark star is at most O[0.1], if θX < 1. Above this ro- (18) tational frequency, r-mode instability quickly makes the and star spin down. We conclude that the rotational fre- −1 R R quency of dark stars with this mass may be smaller than −1 2ℓ 2ℓ+2 τ˜V = c(ℓ − 1)(2ℓ + 1) r dr ρr dr . the half of Keplerian frequency. It is interesting that Z0 Z0 ! the fastest known pulsar PSR J1748-2446ad (716Hz) [18] (19) may be spinning marginally faster than the limit here, Notice that r-mode timescale is written as when canonical mass 1.5M⊙ and radius 12km are as- 2ℓ+2 (ΩK /Ω) τ˜GW .
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