arXiv:2011.07285v1 [astro-ph.HE] 14 Nov 2020 icst fsl-neatn D udwoedamping whose shear fluid weak ADM to conven- self-interacting leads A viscosity of molecular viscosity treatment. of different recipe a totally tional resemble a not condensate, require Bose-Einstein may gigantic which a they rather since but con- star, not star, fluid do We dark stars. Bosonic dark sider of frequency rotational the on oscil- the of damping weak. viscous sufficiently the are as lations far slowly as for even stars effective rotating be may momen- the 13] [12, Especially angular instability the r-mode emission. its wave limit removing gravitational to through by enough tum speed powerful rotation be may stellar stars, rotating instability secular for oscil- to leads stellar wave of gravitational with coupling lations where ro- 11], Chandrasekhar-Friedman- stellar [10, important stars, instability to Schutz relativistic and related For instability possible of tation. the appearance of is One effects investigated. fully intro- and matter prob- dark astrophysical [6–9]. cold and lems cosmological of some solve candidate to a duced self-interaction is made with are which (ADM) stars [5] matter dark dark These alternatives asymmetric discussed. be of to is them stars of neutron possibility to as and termed 4] are [3, considered.They stars dark also are matter dark from candidates[2]. hole black which observed flashes in thermonuclear absent exhibit X-ray is to accreting shown in is alternatives X-raybinaries of these compact One in However, be candidates may binaries. hole self-gravity. objects black exotic to their particles these alternative by that an matter is state made dark suggestions bound These the stan- a [1] the form with may interact) particles. not model do which dard (or model, weakly are standard very candidates dark the interact plausible of beyond most nature particles the Though elementary elusive, object still compact particles. is a matter matter is stars dark category of neutron such One made as holes. models black object and compact standard such ∗ [email protected] nti ae esuyteeeto -oeinstability r-mode of effect the study we paper this In been not have alternatives these on rotation of Effects made and stars neutron of mass typical with Objects against presented been have models alternative Some eateto at cec n srnm,Gaut School Graduate Astronomy, and Science Earth of Department rqec ftesasls hnhl fteKpeinfreque K Keplerian rotating the rapidly of or pulsars half spinning argue than fast is less to It stars alternatives the stars). of (dark frequency matter dark Fermionic interacting esuyrmd ntblt frttn opc bet com objects compact rotating of instability r-mode study We .INTRODUCTION I. oaa381 euok,Tko1380,Japan 153-8902, Tokyo Meguro-ku, 3-8-1, Komaba emoi aksassi slowly spin stars dark Fermionic Dtd oebr1,2020) 17, November (Dated: hnihruYoshida Shin’ichirou ev sa lentv oarpdyrttn erblack Kerr rotating not rapidly may a model to star a hole. alternative dark For an a lower. as case, becomes serve hole the frequency black Higher the mass stellar model, pulsars. lower the millisecond be of observed may mass frequency of mass balancing those typical the with than star, model neutron a a For smaller of limit. considerably re- Keplerian are The the stars than r-modes. of frequencies unstable spin of sulting growth the balances effect ula atr h akmte saself-interacting a is field. matter mediator vector zero-temperature dark a a The with Fermion is matter matter. baryonic is nuclear interaction The mutual whose matter negligible. dark and baryonic of esai n peial ymti,tesaeieal- spacetime ( the coordinate Schwarzschild symmetric, the spherically lows and static be pctm erci rte as, written is metric spacetime h teseeg esro h akmte switnas, written is matter dark the of tensor stress-energy The h emtie nt e.g., unit, geometrized The h nrydensity energy The emo hs qaino tt ES sta fdegen- of that mass is of Fermions (EOS) state erate of equation whose Fermion ds ξ ehr td o-oaigselreulbi composed equilibria stellar non-rotating study here We ic h tr ntepeetsuyaeasmdto assumed are study present the in stars the Since ( 2 x = ) ǫ = r lc holes. black err fAt n cecs h nvriyo Tokyo, of University The Sciences, and Arts of = htteisaiiylmtteangular the limit instability the that d g ∗ µν c.Ti a osri h tr as stars the constrain may This ncy. 8 m ¯ h π 1 dx 3 X 4 2 h µ " x oe faymti n self- and asymmetric of posed dx ξ T p ( I FORMULATION II. µν x .Dr trmodel star Dark B. ν + ) + 1 .Assumptions A. = ( = ǫ − n h pressure the and x 9 ǫ e π 2 2 m 2 + (2 3 ν X α dt x p ¯ h , X ) 2 c 2 u 1) + + m = 1 = µ m u e X 2 φ 2 ν 2 λ x − + dr 6 # ,θ ϕ θ, r, ln( pg G 2 ; + sasmdhere. assumed is , µν p x r r hto dark of that are . + 2 nwihthe which in ) sin p 2 + 1 θdϕ x 2 2 . ) i (3) (1) (2) , 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 . sumed. Finding faster pulsars may strongly constrain a dark star as an alternative to millisecond pulsars.

III. RESULTS For a reference value, choosing µ ∼ 1.6 and κ ∼ 2.0 we have a ratio of rotational frequency to Keplerian limit A. Timescale of r-mode instability and critical ∼ 0.2 and the time it takes for a star to spin down from angular frequency maximal rotation to this value is estimated (by using Eq.(13)) to be 2 × 105(yrs).

We compute critical rotation parameter ωc andτ ˜GW as functions of µ and κ for fixed gravitational mass M.

As the Chandrasekhar’s limit of degenerate star is in- 3.0 versely proportional to square of mass of the constituent 1.8 2.5 0.6 1.2 1.5 0.0 0.9

Fermion, we have smaller maximum mass for larger µ -0.6 2.0 when κ is fixed. Therefore as is seen in Fig.1 the right κ

0.3 corners of the parameter space is excluded. Eq.(10) also 1.5

-0.3 limit the possible region in the parameters space consis- 1.0 tent with observations (dashed and dotted lines in Figs.1, 3.0 2, and 3), though this may not be a very strong constraint due to precision of indirect observations. We simply plot 2.5

-0.70 -0.60 -0.90 them for references. 2.0 -1.10 κ

3.0 1.5

2.4 1.0 2.5 1.2 1.5 1.8 2.1 0.0 0.6 -0.80 0.5 -1.00 1.0 1.5 2.0 2.5 3.0 2.0

κ 0.9 0.3 1.5

1.0 FIG. 2. Same as Fig.1 except M = 2.0M⊙. 3.0

2.5

-0.60 -0.50 -0.40 -0.80 2.0 -1.00 κ For the more massive stars as in Fig.2 and Fig.3 the 1.5 trend is more enhanced. In Fig.2 the mass is 2.0M⊙

1.0 which is close to the theoretical maximum of a neutron -0.70 0.5 -0.90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 star. ωc become smaller and so does the timescale factor τ˜GW since the effect of relativistic gravity is stronger. In Fig.3 we have M = 10M⊙ that corresponds to a typi- FIG. 1. Timescale factorτ ˜ of r-mode instability and the cal stellar mass black hole alternative. For this case the GW instability timescale is less than 1yr and the critical rota- factor ωc of critical angular frequency for M = 1.5M⊙ mod- els. Dark shaded region in the bottom right corner is the ex- tional frequency is less than 10% of Keplerian one. The 2 cluded region where the maximum mass of a star is less than models have 1 − 3 × 10 km in radius. Therefor the corre- 1.5M⊙. Dashed and dotted curves corresponds to the edge sponding dimensionless Kerr parameter is approximately 2 of the allowed parameter region by the current observations a =2/5(Ω/ΩK) GM/c R, which amounts to 1-2 times (see Eq.(10)). The region between two curves satisfies the Ω/ΩK. It is therefore at most 0.2 for the M = 10M⊙ τ p constraint. Upper panel: Contours of log10 ˜GW (yrs). Lower models. This may be another argument against an at- ω panel: Contours of log10 c. tempt to identify a dark star with a black hole candidate (BHC), since some of the observed stellar-mass BHCs In the lower panel of Fig.1 the contour plot of criti- in X-ray binaries have the Kerr parameter close to the cal factor ωc is presented. Gravitational mass M is fixed maximum allowed value of unity [19]. 4

1.0 IV. SUMMARY 0.8 -3.6

-2.8 -2.6 0.6 -3.4 -3.2 -3.0 In this paper the effect of r-mode instability in rotating κ

0.4 dark stars composed of asymmetric and self-interacting dark Fermions, which are alternatives of conventional 0.2 neutron stars and black holes, are investigated. The r- 1.0 mode instability spins down a star through gravitational 0.8 wave emission and viscosity of the dark Fermion fluid

0.6 -1.75 tends to damp the instability. We consider the critical -1.70 -1.65 -1.60 -1.55 κ angular frequency for which both mechanisms balances, 0.4 which marks the final spin frequency of the star. By ap- 0.2 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 plying the conventional model of molecular viscosity to dark Fermions, we obtain weak shear viscosity. The crit- ical angular frequency is shown to be less than half of the break-up limit of the star. The higher the mass, the lower FIG. 3. Same as Fig.1 except M = 10M⊙. the frequency becomes. For typical mass of millisecond pulsars (1.5M⊙) to massive neutron stars (2M⊙) the rel- ative critical frequency to the Keplerian one is less than 20%, which may constrain the alternative models in the presence of the fastest spinning pulsars. For a stellar- mass black hole case (10M⊙), it may be less than 10%, which may be a strong constraint to the existence of the alternative model to nearly extreme-Kerr black holes in X-ray binaries. The timescale of the r-mode instability is also computed and it is shown to be of astrophysical importance.

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