Black Holes and Gravitational Waves in Models of Minicharged Dark Matter

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Black holes and gravitational waves in models of minicharged dark matter

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Please note that terms and conditions apply. JCAP05(2016)054 hysics P le ic t a,c ar doi:10.1088/1475-7516/2016/05/054 strop A Paolo Pani a . Content from this work may be used 3 c osmology and and osmology Caio F.B. Macedo, C a,b 1604.07845 astrophysical black holes, dark matter theory, GR black holes, gravitational In viable models of minicharged dark matter, astrophysical black holes might rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License . ou An IOP and SISSA journal An IOP and CENTRA, Departamento de F´ısica,Instituto SuperiorUniversidade Técnico— de IST, Lisboa — UL,Perimeter Avenida Institute Rovisco for Pais Theoretical 1,31 Physics, 1049 Caroline Lisboa, Street Portugal North Waterloo,Dipartimento Ontario di N2L Fisica, 2Y5, “Sapienza” Canada Piazzale Universitàdi Roma A. and Moro Sezione 5, INFN 00185, Roma1, E-mail: Roma, Italy [email protected], [email protected], [email protected], [email protected] b c a Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI. Vitor Cardoso, Black holes and gravitationalmodels waves of in minicharged dark matter J and Valeria Ferrari Received April 29, 2016 Revised May 16, 2016 Accepted May 17, 2016 Published May 23, 2016 Abstract. be charged underNewman a solution hidden of Einstein-Maxwell U(1) theory. Thesedark symmetry objects and matter are unique are and probes ofa formally dark minicharged described binary photons. by black-holeblack the We coalescence hole’s show same by that Kerr- charge, aLIGOto the regardless provides constrain recent of various the gravitational-wave itsof detection observational dipolar nature. the bounds of emission quasinormal on The of modestoy the pre-merger (ordinary model set of inspiral and anthat a upper phase dark) point in limit can photons, charge on dynamical be whereas plungingobject the processes used the into are final the detection a excited black (hidden) to Reissner-Nordstromthe hole’s electromagnetic considerable black black charge. amplitude quasinormal hole hole, in By is modes wephotons the using nearly also of which gravitational-wave extremal. a show might spectrum the provide onlyin The final a when these coalescence possible scenarios. produces electromagnetic a counterpart burst to of black-hole low-frequency mergers Keywords: dark waves / sources ArXiv ePrint: JCAP05(2016)054 3 and M is assumed to be negligible em Q . Together with the celebrated BH no-hair 21 – 1 – 10 ≈ e = 1 units and unrationalized Gaussian units for the charge, unless ~ e/m = 1 c , since the electric BH charge 2 = G χM := J On the other hand, models of minicharged dark matter (DM) predict the existence of Through this work we use 2.1 Setup 2.2 Theoretical bounds on the charge-to-mass ratio of astrophysical3.1 BHs4 The3.2 inspiral of two Ringdown charged3.3 phase BHs and in bounds minicharged Excitation DM on3.4 of models7 the gravitational BH and charge Emission (hidden) of EM (hidden) modes EM in radiation dynamical in processes a binary BH merger 14 correspondence C.1 EM andC.2 metric perturbations Final17 equations C.3 Numerical procedure C.411 Supplemental results 21 25 28 27 29 1 otherwise stated. 1 Introduction Astrophysical black holesdischarge (BHs) effects are [1], consideredastrophysical electron-positron to plasmas. be pair These production electrically argumentsto-mass rely neutral2–4], [ — ratio due and one way of to charge or quantum neutralization the the other by electron, — on the huge charge- angular momentum theorems (cf. ref. [5]theory for — a vacuum review), astrophysicalmetric these [6], BHs arguments namely are imply the that described Kerr — by solution within [7]. a Einstein-Maxwell The special latter case is of characterized only the by Kerr-Newman its mass Contents 1 Introduction1 2 Charged BHs in minicharged DM models 3 new fermions which possesssymmetry a13]. [8– fractional electric Theircharge charge corresponding and or their charge are coupling charged isare to under naturally a the a much viable Maxwell hidden smaller sector U(1) cosmological candidate than observations is for and the suppressed. direct-detection cold experiments electron These [12, DM14–22]. minicharged and particles In their some other properties models have been constrained by several in astrophysical scenarios. 3 Gravitational-wave tests of charged BHs in minicharged DM models7 4 Discussion and final remarks A Quasinormal modes of Kerr-Newman BHs from the geodesic B Fisher matrix analysis C Technical details on the radial infall of a point charge into a charged BH 25 19 22 JCAP05(2016)054 6 2 3 − + 10 ϵ=0 2 h 4

= 0 and - 3 p

= / 1 M Q h =

Q/M >

> / 10 M Q q 2 [cf. eq. (3.42)]. 3 0 cm in which astrophysical / Log[m/GeV] 2 and charge m

+ icag limit discharge -2 4 GeV 2 h . 0 q ∼ e

-4 xlddb aatcdynamics galactic by excluded = DM are the fractional hidden and electric ρ q

lsaabsorption plasma ). As a reference, an electron-like particle -6 0 , -5

and

-20 -10 -15

h Log ] ϵ [ h m, and charge – 2 – 6 m =0 h ϵ

- 3

= / 1 M Q

LUX

> / 10 M Q = 0 our model reduces to that of DM with dark radiation [23] and 2 the hatched region is excluded by the eﬀects of the magnetic ﬁelds of when 0

Log[m/GeV] icag limit discharge -2 = 1) is denoted by a black marker. In each panel, the red and blue areas below the Left panel: , Right panel: would be absorbed by hidden plasma of density -4 = 1 can exist [cf. eq. (2.23)]. The region above the black dashed line is excluded because

. The parameter space of a minicharged DM fermion with mass xlddb aaymgei field magnetic galaxy by excluded M 5 MeV . 0 Having established that charged astrophysical BHs can exist in theories of minicharged Although rather diﬀerent, both these DM models introduce fermions with a small (elec- Charged BHs are remarkably sensitive to the presence of even tiny hidden charges but 60 Q/M -6 0 ∼ -5 ∼

-15 -20 -10

= 0 of the three dimensional parameter space ( Log ] ϵ [ m charged BHs can exist (here andcharges in of the following the darkthe fermion, region respectively). excluded by Interestingly, direct-detection such experiments region and does by not cosmological observations. overlap with Figure 1 (see main text for details). The left and right panels respectively show the planes and in this region extremal[cf. BHs eq. would (2.20)]. discharge by plasma accretion within less than the Hubble time two threshold lines denote the regions where charged BHs with a charge-to-mass ratio ( galaxy clusters [21] andthe it region is excluded the mostconstraints). by stringent the observational direct-detection constraintthe experiment on region the LUX above model [24], the (we solidIn cf. also black the ref. line show is region21] [ excluded aboveM for by the soft-scattering details effects dark and on red the other dot-dashed galaxy dynamics line [23]. hidden photons emitted during the ringdown of a tric and/or dark) charge.vent It the is stringent therefore constraints natural thatIn to are this expect in work that place we minichargedBHs will for DM are show charged can that astrophysically BHs circum- this allowed in in is Einstein-Maxwell the indeed theory. presence the of case minicharged andare DM. that otherwise even extremal insensitive Kerr-Newman equivalence to principle the of details generaldiscuss of relativity. BHs charged their We under shall interaction. athe take fractional same advantage electric This footing. of charge is this Figure1 orsummarizes universality under aspace the and of a consequence a main hidden minicharged results dark of fermion of interaction with the on mass section2, showing the parameter dark fermions do notthrough possess the (fractional) exchange electric of chargeinteraction dark but with photons, interact no the among coupling each latter to other being Standard-Model only the particles mediators [23]. of a long-range gauge JCAP05(2016)054 , . 2 h h e Q (2.5) (2.6) (2.1) (2.3) (2.2) (2.4) . The + 2 e , 2 em 2 h Q µ q := A 2 h µ h = e of the photon fields Q θ πej and hidden charge 0. are decoupled (cf. refs. [10, + 4 , ] µ → ) e µν h ν θ B B µ h j j is the gauge coupling of the h , + and h ) e h ) ν πe , em ν j µν . This kinetic coupling is constrainted j ) µ are the electromagnetic (EM) and F θ [ , ( ν h + 4 µ ) µ ( h B j µ j cos h h are the field strengths of the ordinary µν e A T + µ πe , πeA µ 8 em B + 8 and ν h ν em ν j j − ∂ − h ( em πej µν µ em 2 defines the kinetic mixing angle 2 j − T πe πe B ( θ F ν 4 4 + 4 π – 3 – = 0), the most general stationary solution [5] of µν B µν − − g g µ µν µ h 1 4 and to dark photons with coupling ∂ j 4 1 = 8 = = = tan B 2 h − − = e µν := µν µν µν 2 α ν α / ν B F G B µ µν em B 1 4 F µ µ j B is the electron charge, and ∇ µα ∇ µα − B F e 2 µν and F := := µ µν A h µν em F µν ν T ∂ T 4 1 for the standard and hidden currents follow directly from eqs. (2.2) are free. − ν − h h j ν ν π A R ∇ µ 16 ∂ and g := − µν √ = 0 = F = In the absence of currents ( The field equations arising from the Lagrangian (2.1) read ν em In the model studied in refs. [10, 18], L j 2 ν 18] for details).by arguments In related this toformation case, Big (cf., the Bang e.g., ref. nucleosynthesis, effectiveas [26]). cosmic electron fundamental. microwave For charge This simplicity background, corresponds is we and neglect to large-scale such the structure coupling model here studied and in consider refs. the [10, Lagrangian18] (2.1) when before the diagonalization leading to the Lagrangian (2.1) in which the above field equations is the Kerr-Newman [6] metric with total charge DM, we proceed to studythe their coalescence gravitational-wave (GW) of signatures inby a3. section the binary LIGO/Virgo We consider BH Collaborationdifferent using system the phases similar GW of to interferometers aLIGOin the GW150914, [25], the the coalescence and binary show event can and that the recently be of excitation detected the used of final theunequal-mass to ringdown BH BHs, constrain modes produced in the and a inwith U(1) the perturbative the previous, charge total restricted approach, post-merger results and radiated of phase.calculations. in show energy the Numerical that Finally, during Relativity, BHs they as we the well are explore collision as in of with good simplified flat-space agreement 2 Charged BHs in2.1 minicharged DM models Setup We consider the following classical Lagrangian [10] for the standard Maxwell field and∇ for the dark photon, respectively. The continuity equations and (2.3). Note that a hidden electron carries both electric charge where where we defined the effective stress-energy tensors photon and of the dark photon, respectively, parameters the hidden number currents, hidden sector. Theto model ordinary photons (2.1) with describes coupling a theory in which a charged fermion is coupled JCAP05(2016)054 . ). 2 , = 0 h (2.7) (2.8) (2.9) 0 + (2.11) (2.12) (2.13) (2.10) = 0, a Q 2 h provides ν em j q ν h e j and throw in := q em . In this case the h Q = 0, i.e. when DM Q 4 h = . em Q em , eQ 2 . 2 2 em r r r qQ + 22 Q On the other hand, the dark coupling h − = and electric charge 3 Q , := 10 h e e M 2 2 h tr × B = 5 , Mm h . For the electron to be absorbed by the BH, e e e tr ≈ 1 + 2 ≤ em as defined in the previous section. m e – 4 – r , 0 e Q + eF to ordinary photons. As such, they are insensitive m em h Q e 2 0 h 2 2 r + Q r r Q Q = ≤ = eQ h tr e em tr , and the force felt by a hidden electron reduces to = = B em 2 h Q h M = e eF Q tr tr and mass / h F 2 B e F := := h = 0 and em 1 + F . When the BH is spinning the presence of a charge induces a h F h q Q h Q h Q e e = which is indeed typically neglected. + Q 0 h e Q 0 the model (2.1) simply describes a dark fermion coupled to ordinary photons with coupling = → h . In other words, em . µ em 2 Q j e Observational bounds on minicharged DM typically constrain the coupling to Standard- We are now considering the standard scenario in which the electric charge is produced by an ordinary When plays a crucial role in models of dark radiation [23]. Gravitational tests do not require 4 3 2 e h 2 current then the (classical) electric force must satisfy where we have defined the effective charge of a dark fermion in this model, e ordinary photons as mediatorsIn and are particular, indeed the sensitive effects todoes we the not are entire couple going parameter to space to Standard-Model ( discuss particles, are as2.2 present in also dark when radiation Theoretical models bounds [23]. There on are several the mechanisms charge-to-mass thatOne ratio conspire is to of purely limit astrophysical kinematical. thea BHs electric Take low-energy charge a electron of BH of astrophysical with BHs. charge mass which can be written in terms of the dimensionless charge-to-mass ratio of the BH as to the coupling Model particles, especially the coupling where magnetic field along the angular directions [6]. Note that the hidden current where the (EM andhidden) hidden) Maxwell BH equations charges in are static defined equilibrium, from the solution of the (standard and both electric chargeelectron and feel hidden respectively charge the to force the BH. Thus, a standard electron and a dark If the BH acquirescondition charge that only is through enforced theand by accretion the several EM of considerations and as a hidden discussed hidden charge in current areBH section (i.e., proportional charge),2.2 if to reads then each other, JCAP05(2016)054 (2.16) (2.17) (2.14) (2.15) (in geometric units), 21 − 10 . ∼ yr , of elementary charges (each with /M e q . e q N em , to zero. Thus, in the rest of this work Q

1 p 1 p ν em M q m m M j m m m − − 2 2 q 11 e q 18 m s. The above results show that — in Einstein- − − 7 – 5 – ∼ 10 − , corresponding to a discharge time scale 10 39 1 × 10 − ∼ 10 star 5 M × yr Q ∼ , the total number ∼ 8

N M ∼ M M plasma ) ), respectively. For standard Maxwell theory, the charge- =

M 2 is typically very small. In addition, BHs can be neutralized , q discharge em 2 τ Q /M discharge m M/M τ ). ( em 8 , e − Q /M ) that it needs to accrete from the surrounding plasma to be neutralized m 10

m ) and ( = × M 1 ( 2 . m , q 19 1 − m = 2 10 and being the proton mass. This plasma mass is easily available under the form of and mass e ≈ 100 C. It is reasonable to expect that, if they collapse to a BH, this small charge Edd p q ˙ = m ≈ M Another possibility to form charged BHs is if the latter were born through the gravita- The above discussion imposes an extremely stringent upper bound on the EM charge The above discussion also shows that the bounds on the charge-to-mass ratio of astro- /M q em tional collapse of (charged)effects, stars. as Self-gravitating shown stars by are(electrons) Eddington, globally are Rosseland charged easily and due kicked others toare out [28–30]: pressure stuck of in the in star the afor by interior. nutshell, positive pressure lighter The and effects charges calculation, negative whereas charges, which heavy yields proceeds ions the by (protons) following assuming result thermal for equilibrium the charge of a star [28–30], and that — if such BH ever acquires a charge — it discharges very quickly. Maxwell theory — it is difficult to charge a BH past For where the stellar material is assumedand to be mass composed mainly ( of two species of ions with charge to-mass ratio inwith the star iswill of remain the hidden orderQ behind of the eq. horizon, (2.13). giving Thus, rise stars to are a typically (very charged weakly) charged BH with of astrophysical BHs. To fulfillordinary this charges and, bound, therefore, we we assume set that the astrophysical current BHs do not accrete physical BHs become much less stringent in minicharged DM models. For a hidden electron This corresponds to a plasma mass of we can safely assume that the force felt by a putative hidden electron is given by eq. (2.11). These numbers change if the particlecharge-to-mass is thrown ratio at large velocities, butby show that surrounding the maximum plasma.separation can If occur and theFor the an eletrical BH extremal force charge BH can overwhelms with be the neutralized gravitational by particles force, of charge opposite charge. with interstellar matter withinneeded a to small accrete regionassume surrounding an that the amount plasma BH ofaccretion accretion [27]. plasma disk. occurs such at In Torate, that estimate the the an most the same extremal conservative time rate scenario BH of mass is gas accretion accretion discharged, occurs from let at an us the ordinary Eddington is [27] charge JCAP05(2016)054 Q/M (2.23) (2.21) (2.20) (2.22) (2.18) (2.19) might be a . Likewise, a , q yr plasma and e q M 1 in our units, where the m , p ≤ m p m m Q/M . e q m . ) parameter space. In the region 2

or M e M e m, q Q M m 10 , m 2 3 v q

M ) plane in which BHs with a certain , 3 M M m q πρ m . m 5 GeV cm 4 m, q − / ∼ ≤ ∼ ρ – 6 – 10 18 of astrophysical BHs. This constraint can be also m Q − qE M ) eV. In the rest of this work we shall focus on this ∼ depending on the parameters 4 GeV 10 2 Bondi . 5 is the relative velocity between DM and the BH. Using ˙ /M 0 , Schwinger discharge is important unless × q M 2

v Q/M 2 , eq.) (2.13 becomes Mm 3 M m ( < ≤ s Q/M 10 e q / by their typical local values. Thus, the discharge process is − Q ∼ M v 10 v given by eq. (2.15). If E 220 km and and mass m q yr obtained from eq. (2.20) in the ( ρ plasma 4 . 10 0 M 10 ∼ × 4 . = 1 is the local DM density and 1, i.e. for any discharge τ ρ To summarize, in minicharged DM models eq. (2.18) provides an intrinsic constraint on Furthermore, BHs with a large charge also have a large electric field close to the hori- We recall that the charge of a nonspinning charged BH is bounded by 5 discharge considerable fraction of thein BH this scenario. mass Furthermore, soaround DM that being compact a almost objects collisionless charged sofluids, it BH does that would not dark be form plasma difficult accretion is disks to accreted discharge at the Bondi rate for collisionless inequality is saturated in the extremal case. BH with hidden chargeaccreting surrounded a by mass a plasma of hidden electrons can be neutralized by For electric fields of order which is much longerwe than have eq. normalized (2.16)much since slower the than for DM ordinary densityτ plasma. is The low. black dashed In line the in above figure1 expression shows the threshold It is clear that theMm upper bound (2.22) is much less stringent than the bound (2.18) whenever the maximum charge-to-mass ratio regime, i.e. we dogravitational not radius consider of ultralight the DM BH. whose Compton wavelength is larger than the turned around to determine the region in the ( zon. Such electricmechanism field [31]. is prone This to effectCompton produce becomes wavelength is spontaneous relevant of when pair the the production order work via of done the the by Schwinger rest the mass electric of field the on lightest a particle [1], with effective charge which can be less stringent than unity where below this line, the timelonger scale than to the discharge Hubble an extremal time, BH which through we accretion assume would as be a much very conservative limit. can exist, namely the above equation and eq. (2.15), we estimate the discharge time JCAP05(2016)054 6 = and 1 (3.1) q h e , from the G √ , respectively. i = 0. We stress G h m i e √ λ / 1 q := i ≤ q . Interestingly, existing 2 m Q/M and , r ). G is the position vector of the mass 2 2 i √ 3 q λ r 1 r / 1 q 1 λ q An (electric or hidden) charge in circu- ∓ − . ≤ r c 7 1 are allowed, even when 1 = 2), 2 (1 i m G ∼ m and 3 1 r G := – 7 – = 0, a regime which cannot be ruled out by EM Gm = 1 ( eff Q/M i ± G = → i r ¨ G i , with masses 2 2 m + 2 2 h, q e = 2 is the relative position of the bodies. If we define q 1 r and − 2 1 2 r + := 1 Let us assume circular orbits for simplicity. The GW-driven coalescence of a compact binary can be characterized by three phases– [34 2 h, r For clarity in this section we reinsert factors of Orbits are circularized under GW-radiation reaction, so it is natural to expect that the last stages in the , 6 7 i q lar motion emits dipolardominates radiation over the governed by quadrupolarrole Larmor’s GW in formula. flux the at The early large dipolar inspiral. distances, energy so Because flux it both might particles play an are important charged under the standard Maxwell equation above it is clearof that two the uncharged problem particles can where be mapped into a standard Keplerian motion life of a compact binary are quasi-circular [39]. To leading order, by using eq. (2.11), the motion is governed by the equation where the upper (lower) sign refers to 36]: the inspiral, thearations merger and and the is ringdown. wellto approximated The inspiral by the corresponds post-Newtonian highly to theory; nonlinear largecurately the orbital through stage merger sep- numerical right phase simulations before corresponds ofphase and the corresponds full after Einstein’s to merger, equations;equilibrium and the finally, solution the can relaxation ringdown of only the ofory be field [36–38]. the described equations, highly-deformed ac- In anddegrees end-product the of can approximation, to rest be in of a describedsignal. order this by stationary, to paper BH estimate perturbation the we effect will theof focus a3.1 on hidden each BH of charge The in these inspiral theWe phases, of GW model with two charged various the BHs initial in minicharged inspiral DM of models the BH binary by considering two point charges, This region isconstraints shown on in minicharged DM1 figure extremal modelsfor BHs do with not two charge-to-mass rule differentthat ratio out EM-based values charged constraints BHs. of on minicharged Onthat DM the charged models contrary, are BHs even insensitive canobservations. to the exist Finally, coupling the also colored when threshold regions line shown for in figure1 discharge plasma inlay discharge well this [cf. below scenario. eq. the black)], (2.20 dashed so that3 charged BHs should Gravitational-wave not tests easily ofThe charged BHs recent in GW minichargedcess detection DM to models of theregime, a strong-field/highly-dynamical precision binary regime GW of BH measurementstal can the physics coalescence be gravitational [32, by usedGW interaction.].33 aLIGO observations to can develop [25] It In be BH-basedminicharged used has is this tests DM to therefore given scenarios. constrain of a natural us fundamen- putative to ac- hidden investigate charge of whether astrophysical present BHs and in upcoming e m JCAP05(2016)054 | (3.8) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) /dt 3 dip / 2 dE − | = 0, provided i . πf , 2 3 M 3 c eff eff 1 . ζ G e G 0 2 . Note that if the charge- √ 2 , 2 2 2 2 ζ m . By assuming an adiabatic 5 /M ,

m / 2 4 2 1 2 2 2 − /R . This simple estimates suggest M η R m M 2 , 2 m m 3 1 4 can be written as eff 2 1 5 c , 60 m 42 # m 2 eff R m 1 G e 3 2 2 m 3 1 / G − c ∼ √ GW m 2 = m dt 2 eff 2 4 1 1 2 1 1 η eff dE G R " m 3Hz m . 3 m G e 2 3 πf 2 c 0 2 / , 3 eff − 5 and h, 0 if the two BHs have similar charge-to-mass m 5 2 e , as + − G M 2 2 3 h = 2 dip 2 m ∼ 5 c M dt eff dt ζ , we can obtain a differential equation for the − m Ω f dE R dE ζ – 8 – 4 3 E 4 eff G − e + 1 eff 1 + /dt G 1 1 e , M e 3 1 h, 2 := m G 5 2 c 2 m m eff η / dip em h, √ 2 dt 5 G 2 = c dip − dE 1 32 dE . 4 3 4 3 5 dt ζ η m 0 M = dE − ∼ ∼ ∼ − 3 η 128 dip h 9 dt /dt eff 55 M em 2 + dt e dE GW dt G c dE 1 dt eff c + 1, the condition GW dE . Note that the latter is nonvanishing even when h, √ Φ G dE ζ 1 dE − m ). This equation can be solved analytically in the limit 756 ζ c 480 t − 3715 ( R = πft := + ˙ = R 2 E ζ R ) = 2 f are nonzero. On the other hand, ( , i.e. when the dipolar loss is a small correction compared to the quadrupolar GW | 2 + is the radius of the orbit, h, Ψ e /dt R To quantify the effect of the dipolar energy loss more precisely, we can compute the The inspiral phase of GW150914 was compatible with the prediction of post-Newtonian or GW 1 h, dE where where orbital radius GW phase associated with suchbinding effect energy through of a simple the quasi-Newtonian two-body evolution [43]. system is The flux. In thisphase case, of a the quadrupolar standardphase GWs procedure of emitted the [43] by “+” allows the GW us system polarization through to reads an compute adiabatic the evolution. amplitude The and the field and under thedissipation hidden [40] field, with the differentNamely, dipolar couplings, flux in consists addition of to two the copies standard of GW Larmor’s energy energy flux41]. [ to-mass ratio of the twoin objects this is case) the the same,write the leading the corresponding term total dipole would dipolar term be flux, is zero quadrupolar and like (only the GW flux. It is convenient to From the above equationonly it on is the clear combination thate the dipolar emission in the inspiral phase depends where Ω is thethat orbital the frequency dipolar and correction we might be assumed small when the binary enters the aLIGO band. approximation, | theory [25, ].42 Thisbe implies small. that any In putative the dipolar limit contribution to the energy flux must ratios. JCAP05(2016)054 := /G (3.9) eff M G [25] and The ratio

8 M 30 1. In particular, ≈ < = 0, and eq. (3.8) | i λ ζ | , obtained from eq. (3.9) for ⊙ . ⊙ ⊙ 1 G measured 6= 0. Because to leading 1 =0.7 m . 2 =0 =0.7 100 M i = 2 2 69M 46M 135M λ =λ =λ =-λ 1 1 1 λ λ λ eff , is the GW frequency and G 2 f are nonzero, we obtain two types of 80 λ 1 for fixed h,i λ e 1 , measured − i m ) ⊙ 1 60 M . The black curve corresponds to the uncharged i (M 1 = 0, then λ = 0, provided = m 5 ζ / – 9 – h,i 3 e 40 − , a rescaling of Newton’s constant is degenerate with = Mη i M = 0 we recover the standard result, namely a minimum 20 := eff i as a function of of a binary system as a function of λ G 2 M 0 the total mass of the system can be significantly smaller than in M m < + 0 2 1 0 λ 50

200 150 100

m 1

. However, signiﬁcant changes occur if 0

⊙

) ( . When M M λ

i and for diﬀerent values of = λ M

M M 69 are the time and phase at coalescence, 30 ≈ c ≈ 0 the effective Newton’s constant is larger and the real total mass of the system min < M 2 and Φ is the chirp mass. When . Total mass λ c measured 1 5 t / λ However, when at least one of the parameters The other correction appearing in eq. (3.8) is the second term inside the square brackets 3 M − For scalar tensor theories the dipole contribution is proportional to the sensitivities, which vanish identi- 8 Mη the uncharged case. where cally for BHs [33]. yields the standard leading-orderpost-Newtonian term result, in to the which second we line added of the thecorrections. above first The equation. first next-to-leading one order isresult. a rescaling of Newton’s This constant, which affects correction also the is Newtonian present even if case [25]. Note that when fixed order the GW phasethe depends on measurement of theobtained by chirp neglecting mass. charge effectsrelative Extracting to would the yield the latter a real result from chirp that mass the is of Newtonian rescaled the GW by system, a phase namely factor Figure2 shows the total mass Figure 2 for different values of total mass can be significantly smallershows than that in neglecting the charge effectsBH uncharged might case. masses. systematically This lead property to is overestimate the intriguing measured sincein it eq. (3.8).the The leading-order latter correction is for larger neutron-star at small binaries frequencies, in as scalar-tensor expected, theories. and in fact reassembles when JCAP05(2016)054

M (3.11) (3.10) (3.12) 30 Hz, = 30 2 f > m = 1 m , ] , N 3 / δ 2 − − , 1000 3. . 3 [1 / 0 01. On the other hand, these 4 . min 0 3 − ≈ 500 / 5 ζ πf − max ζ < πf f M . 3 c − eff 2 M 3 3 c G / eff 5 1 . G ζ − min 0 f ! 100 f[Hz] 03 times smaller than the first post-Newtonian 3 2 3 to simplify the final expression. In the small- . 01% when / 3 / ζ . 0 5 / 5 min max 02 η 5 30 Hz, we obtain . f – 10 – f 0 ∼ − min 50 5 ∼ f / ≈ 4 1 + η M min N

3 δ 18 f ζ=0.1 ζ=0.01 ζ=0.3 c [cf. eq. (3.6)] as a function of the GW frequency. Note that 2 eﬀ ζ 743 + 924 ζ G 5 max / f − 2 η 3 / := 8 10 1 -4 -5 -6 -2 -3 25 1 π Ψ 168 0.1

r 10 10 10 10 10

Ψ r | | 32 1 and by less than 0 100 Hz and = . 0 ∼ = N between the hidden-charge-induced GW phase and the first post-Netwonian δ ≈ N ζ Ψ max 1, but they can be as large as 25% when r . f 0 . In our case we obtain ˙ f f . df ζ . Ratio max min f f R 3 saturates the bound presented in eq. (3.13). Another relevant quantity is the number of cycles spent in the detector bandwidth, . 0 := ≈ term when N ζ between the second term and the third term inside the square brackets in eq. (3.8), namely and different values of the coupling and we have also expanded for charge limit, for is shown in figure3 as a function of the GW frequency for a typical inspiral. For where Figure 3 correction [cf. eq. (3.10)] for the quasicircular inspiral of two charged masses with the charge-induced corrections are at least Therefore, dipolar effects changefew the percent number when of cycles relative to the Newtonian case by a corrections become important at smallerspace-based frequencies interferometers and such might as produce eLISA detectable effects [44]. for JCAP05(2016)054 2 χM (3.15) := ) triplet, defined in J of the final 220 9 B , τ = 2 ringdown 330 , spin m , ω M = 220 l in eq. (3.8) would still ω eff 0 and the dipolar emission , G ) t ≈ ζ lmn ω sin( lmn t/τ − e ) 0) gravitational modes [34, 35, 51]. Because the r , ( – 11 – 3 , lm eLISA-aLIGO (projected) (3.14) is an overtone index. Usually, the modes excited to A n ∼ 4 − lmn X 3 aLIGO (3.13) are angular indices describing how radiation is distributed 0) and (3 . ) , 0 10 ), and 2 . In our case these bounds translate into l m , 9 t, r . . . The analysis of ref. [45] shows that GW150914 sets the upper − 2 | | Ψ( ζ ζ ζ | ≤ | | 10 and 5 24 m l | defined in eq.). (3.6 Note that the bound derived from the aLIGO × = 3 , whereas a putative eLISA detection of a GW150914-like event with an ζ 2 are the (2 . B − | 10 10 B | × of the final BH. 2 Q . | = 2 Maxwell modes which might be excited to considerable amplitude. We neglect this possibility B | m To complete this program we must first know the QNMs of Kerr-Newman BHs, which For simplicity, here we focus on the small-charge case. The Finally, we stress that if the two BHs have similar charge-to-mass ratios (which might Very recently, ref. [45] performed a detailed analysis to derive GW-based constraints on = As discussed in section 3.3, when the BHs are highly charged the final ringdown might also depend on Very recently, ref. [47] showed that compact objects without an event horizon but with a light ring might l 9 10 for example, allows us to invert the problem and to determine the mass has been an opensolved in problem a series for of papers morenumerically. [52–56] than and the 50 QNMs of years. a Kerr-Newman Fortunately, BH this can be problem now computed was recently final state only depends on three parameters,and the charge knowledge of the ( frequencies of Kerr-Newman BHs in the small-charge limit are well approximated by the here for simplicity. the display a ringdown signalis similar due to to BHs the even differentany when boundary role their for conditions QNM charged that BHs. occur spectrumwithout for For is distinction. this a completely reason horizonless different. in ultracompact the Such object rest effect and of does the not paper play we will refer to QNMs or to ringdown modes larger amplitudes called quasinormal modes (QNMs), whichBH are the [37, characteristic oscillation4850].– modes on Here, the final BH’s sky ( be the case if their formation mechanisms are similar), then optimal detector configuration or aas combined stringent eLISA-aLIGO as detection set a projected bound for the combination detection is roughly consistent with our simplified analysis. generic dipolar emissions in compact-binaryto inspirals map (see eq. also ref. (3.5) [46]). into It this is generic straightforward parametrization. In our case the parameter refs. [45, ]46 reads bound is suppressed. In this case the zeroth-order corrections due to be present [cf.quadrupolar figure2]. term that will In also addition, modify the the3.2 standard first Newtonian quadrupole nonvanishing Ringdown formula. radiative phaseOnce effect and the bounds would two on BHs berelax merge, the a they to BH form charge its aradiation. final single The deformed stationary final, charged (Kerr-Newman “ringdown” andof stage [6]) spinning exponentially of BH damped state this which sinusoids, by process will is emission well-described of by the GWs, superposition EM and dark JCAP05(2016)054 330 KN KN 330 220 ω ω δω (3.18) (3.16) (3.19) (3.20) (3.21) (3.17) = = , 2 1 ω ω , , 2716 5% in the region . . 2 2 1 / / ) 1 1 χ and the damping 07585 0) mode. Because . , − #) #) 0 , 3 , frequency KN 2 330 , frequency , 2 2 2 1 (1 2 / 1 ω 1 Q Q A A 3 2 3 1 − , associated with frequency 3 Q Q #) are the vibration frequency 2 2 2 1 i KN χ 220 τ 2 2 1 + 4 1 + 4 ω A A σ 220 Q 3 2 2 1 τ ω ω Q 09443 . and 2 2 , + + i 1 + 4 A and ω + 0 σ 2 1 2 2 2 2 220 ω Q Q 505 χ 3 1 3 2 . ω , 0 + ) Q Q i 2 1 2 2 τ χ i 2 3506 = 3) with amplitude 2 1 1 + 4 1 + 4 . A A = 2) with amplitude ω 02314 − 0 , and . Q m 3 1 1 2 m = (1 ω ω – 12 – Q 220 indices), the Fisher matrix is simply an addition i = + 0 2 1 τ " " = l + Q 1 + 4 l χ A − 4 1 4 2 χ 1 l, m Q Q KN ω 220 7 1 7 2 τ " 08923 0243 Q Q . . 2 1 2 2 ≡ 0 1 2 1 A A 3 + 16 3 + 16 Q − Q = 3 modes are not available, in the rest we will estimate 1 220 3 3 2 1 . Define also the quality factor and mode 2 ( ω δτ ω ω 2 1 m , KN 1075 + 0 2812 3 + 4 330 A ( ( KN . . 220 = τ τ 0 0 220 l 2 2 ( ω = − 1 1 √ √ = 11 4 2 2 2 π 2 2 − 1 τ τ 2 2 Q M = = = KN 220 Q M 1 2 1 ω τ ω ω 99]. . ∼ ∼ − ≡ 0 ρσ ρσ ρσ , were measured. In principle, using the formulas above, the mass, spin and charge [0 2. For detection of multi-modes having “orthogonal” angular structure (i.e., the two 220 220 220 220 220 is the signal-to-noise ratio (SNR) of the ringdown phase. , ∈ τ ω δτ δω ρ δω KN 220 τ As discussed in appendixA, there is a tight relation between the BH QNMs and the some Let us then assume that the two dominant modes of the gravitational waveform were Consider now two modes, mode 1 ( = 1 We thank Aaron Zimmerman for useful correspondence on this issue and for sharing some data of a/M i 11 ≡ and damping time and damping time j ref. [55]. The small-chargeof results ref. are [56] available and in with ref. the [55] small-spin and expansion agree of very refs. well [52, with].53 the Our full fit numerical is results accurate to within 0 for modes are characterized byof different matrices of different modes. In this case the errors where following expression, and damping time measurements read [63] geodesic properties associatedlimit to [59–61]. the spherical Indeed,for photon it the orbits, turns relations as out (3.16)–(3.17), established that as in the well the geodesics as eikonal correspondence for provides the an correction estimate for the (3 extracted from a GW detection, so that the two frequencies where and damping time ofnumerically a with neutral, high precision spinning [37, BH.57, The58, latter85]. can be found online, or computed numerical data for the from the geodesic correspondence presented in appendixA. time of the BHpresence could of be noise, determined whichfrequencies. precisely. introduces The some proper Unfortunately, uncertainty waya in detection to Bayesian the handle is analysis noise determination always or is ofA done by by the Fisher-matrix either approximating in ringdown study using the the of Monte-Carlo processsingle simulations multi-mode mode, through and ringdown the a is relevant Fisher-matrix entries done analysis are in [62]. shown ref. in [57], the appendixB. which we follow. For a JCAP05(2016)054 3, 9. . / 0 (3.25) (3.24) (3.23) (3.26) (3.22) ∼ = 1 1 χ A / as an upper 2 A Q σ gives a minimum = 0 and Q χ = 1.0 Q σ , as a function of the BH spin and 0.8 Q , , , σ ) ) ) = 7. We assume that the dominant 1 1 1 Q/M ρ A A A ∂Q ∂X / / / . 2 2 2 + 0.6 , the condition A A A χ 7 in the ringdown part46]. [ We stress that , , , ρ 2 1 1 1 100 σ /Q ≈ 1 , τ , τ , τ J/M r 2 2 2 becomes more stringent as the final BH spin ρ ∂χ 1 ∂X ∼ . = 100 and , ω , ω , ω 0 0.4 or higher and, therefore, are only qualitative in the + 1 1 1 ρ Q – 13 – ω ω ω . 4 ( ( ( M ρσ Q/M | 3 2 1 σ f f f /M . This value is shown in figure4, for Q M | 4 ρ = = = ρ=100 ρ=7 Q ∂X ∂M √ 0.2 χ / Q M = ρσ ρσ ρσ . Because X so that it becomes more stringent for higher SNR. Thus, for a ρ σ 2 / 0.0 1 1 − -2 0.1 ρ

detectable M Q ). It is straightforward to solve the system of three equations above for ) |/ (| ∼ 9, our simplified analysis suggests that ringdown tests can set an upper 1 . 0 , τ ) of GW150914 [25] for which 2 ∼ 07 05 . . ; this yields , ω 0 χ 1 Q +0 − ω and use it to estimate the minimum charge that can be measured by a ringdown σ 67 . Q are cumbersome analytical functions. Finally, we can now view = ( . Projected bound on the BH charge-to-mass ratio and i = 0 f X χ 2 We can convert the errors on the frequency and damping time to errors on physical Interestingly, the upper bound on σ , M J/M where bound on detectable charge that scales as 1 detection with a certain SNR quantities by usinglytic a result simple in propagation thecorrelations of between single-mode errors different case (this physical quantities and procedure are we yields small). expect the Specifically, it we correct to impose ana- be also accurate generically, since nearly-extremal limit. our results neglect terms of the order mode has an amplitudethe approximately case three for times many( larger BH than coalescences the]. [63 second, As sub-dominant a mode reference, as the is vertical band denotes the final BH spin σ for two different SNR of the ringdown phase, where Figure 4 which is appropriate for a wide range of BH binaries [51, 63 ]. increases and it can improve by some orders of magnitude between constraint of the order The bound scales as final BH with JCAP05(2016)054 1.0 and Q l=2, s=2 l=3,2 s= l=2,1 s= 7 in the 0.8 ≈ = 2 become ρ l = 1) QNMs of a s 0.6 Q/M of GW150914 [25], our 0.4 05 07 . . 0 +0 − 7, our analysis neglects terms . 0 67 0.2 . ∼ = 2) and Maxwell ( = 0 the Maxwell modes with s 2 = 2 Maxwell modes of a BH are well 0.0 M Q/M l

0.2 0.0 0.8 0.6 0.4

→ J/M ω M R Q – 14 – 1.0 , cf. eq.). (3.16 However, in addition to the shift of falls radially into a static BH with charge = 3, as a consequence of the supersymmetry of extremal Q l µ in the small-spin case [52, 53]. The general case of ) and should be extended to include the nearly extremal χ 2 [55], whereas the modes of the Reissner-Nordstrom BH 0.8 M /M 2 Q and mass ( 0.6 = 3 gravitational modes. In this section we discuss under which O q Q 2 24, which should modify the final result by roughly a factor of 25%.) m . J/M 0 = ∼ when l 0.4 4 2 Q /M 4 l=m=2, s=2 l=m=3, s=2 l=m=2,=1 s 0.2 Q . The frequency of the fundamental gravitational ( . This model does not capture the effects of the angular momentum of the small = 2 and the M 0.0 Figure5 shows the well-known fact that the For this purpose, we consider a simplified model in which a point particle (modelling

0.8 0.6 0.4 0.2 0.0 1.4 1.2 1.0

ω M R = 3.3 Excitation ofThe gravitational discussion of and the (hidden) previous section EMNewman relies modes BH on the are in fact affected dynamical that by the processes the gravitational QNMs charge of a Kerr- Figure 5 Kerr BH as athe function BH of charge the (rightisospectral BH panel). to spin the (left In gravitational panel) modes the and with latter of case, a as Reissner-Nordstrom BH as a function of Reissner-Nordstrom BHs [64]. Our analysis is valid up to the gravito-EM modesdiscussed of in a ref. Kerr-Newman [56]. BH for arbitrary spin and charge was recently a small BH) with charge case. Nonetheless,charge it with provides a given anringdown SNR indication ratio. part for]. [46 For example, the From the the SNR SNR of spin necessary GW150914 measurement to is roughly constrain theof the BH order these modes, another feature of theof QNM spectrum a in the new presenceNordstrom of family charge BH is the of when appearance the modes,Maxwell spin mode which of vanishes. reduce afigure5. As (neutral) to a Kerr In the reference, BH the and theproportional standard Kerr-Newman of fundamental to Maxwell case, a gravitational modes static the and acquire Reissner-Nordstrom modes of corrections BH of a are proportional the shown Reissner- to in Kerr BH acquire charge corrections separated from the GW modes.these modes As are we coupled now tocontribute discuss, to the in the gravitational dynamical ringdown sector processes phase andwe like of — a the have if GW. BH excited In neglected merger — thel such ringdown they analysis can possibility of in the and principle previousconditions section assumed the extra that Maxwell modes the can two be dominant neglected. modes were the mass figure4 suggests that the ringdown phasenearly of extremal. GW150914 does (We not note exclude that, that the for final a BH relatively was large JCAP05(2016)054 ) − = or µ Q ≤ e (3.28) (3.29) (3.30) (3.27) = 1 q or , and a , = 0 the ∗ g,e em M Q S Q dr/dr = µ Q of the particle at infinity. , , e g , so only when γ , S S 2 Q | , the source terms ) = = i , I e g ∗ 2 | ψ ) 1 dr dψ → ∞ I 3 I + g → ∞ + ω, r ( ψ g is either a fractional electric charge g ) ψ g ψ q 2 | ω, r V I 2 ( to zero and the problem is effectively mapped e ω ) at hand, the GW and EM energy spectra at − + ψ | 2 µν e – 15 – 2)! ω F ψ ω, r ) ( denote the gravitational and EM master functions, + 2)! − e + 1) or + ( e l l V g,e l ( ( ψ ( g ψ l 2 ∗ µν − π ψ , the coupling functions 1 π 2 2 B 1 dr 32 2 d ω and g,e V g and on the initial Lorentz factor = = + ( ψ µ e g e 2 ∗ ψ dω dω are proportional to the BH charge dE dE 2 dr i d plunging onto a Reissner-Nordstrom BH. The general case in which I q is the tortoise coordinate of a Reissner-Nordstrom BH, and that the BH is charged accordingly (namely, either ∗ r h e , on the mass . The potentials q 2 /r 2 is the frequency, and Q ω + The energy spectra defined above are shown in figure6 for some representative cases. For simplicity, we consider that the charge We employed two different methods to solve the above equations: the first is a standard ). In this case, we can set either h M/r The most salient (andleft more panel) is relevant that for it this has discussion) a feature cutoff of at the roughly the GW lowest spectrum QNM (top frequency of the central BH. A where respectively, where the wavefunctions are evaluated at spatial infinity. respectively, and infinity for each multipole read [65] Green’s function technique whichsystems, makes whereas use the of other thea is solutions shooting a of method. direct the We integration associatedeach explain of other both homogeneous the within procedures full numerical in accuracy, inhomogeneous appendixC.in and system The the as two through we literature. methods explain agree With below with they the reproduce solutions earlier results inspiralling BH. However, in thethe last innermost stages stable of circular coalescence,as orbit, once the it the particle will orbiting crosses plunge particle theof into light reaches the the ring, BH. effect we Since suspect thatOur the that we’re model a ringdown also trying radial is does infall to excited are not will study, subdominant. yield capture which a individual is good spin the estimate effects. relative We can excitation only of hope different that QNMs. these effects 2 detailed derivation of the aboveon equations the are charge given inThe appendixC. The coupling source functions terms depend gravitational and thecontamination EM of perturbations the EM are modes decoupled. into the gravitational In sector the and viceversa. general case we expect a a hidden charge Q into an electric charge so that the effectThe of metric perturbations the can particle bethe can angular analyzed dependence be through of treated a the perturbations harmonicthe perturbatively in decomposition, EM (cf. spherical by harmonics. and appendixC separating gravitational InforBH the radiation details). frequency are due domain, described to by a the charged following particle coupled falling radial radially equations into a charged the particle and thecomputation. BH We consider have the both charge and electric the and mass of hidden the charges particle to is be a small ( simple extension of our JCAP05(2016)054 1.0 1.0 1.0 (3.31) 0.8 = 0.5 = 0.9 = 0.99 . In ap- γ 0.8 0.8 =q =q =q . 0.6 Q Q Q displays two 0.4 q/µ M 0.2 := 9 0.6 0.6 . q 0.0 -2 -4 -6 -8 -10 ωM = 0 10 10 10 10 ωM 10 ) depend on and ¯ q , 0.4 spectrum. Bottom panels: − = 2 fundamental QNMs of 0.4 3 = 0.5 = 0.9 l = 0.9 = 0.5 = a, b, c Q/M = -q =q =q = -q Q Q M Q Q Q Q := = 2 (top right panel) and with 0.2 0.2 l ¯ c Q = 2 gravitational mode, whereas l + 2 gravitational 0.0 0.0 -2 -4 -6 -8 = 1 fundamental EM QNMs of the RN BH Q

M l

10 0.000 10 10 10 0.015 0.010 0.005

e e

μ ω / μ ω / d d dE dE

- - 2 2 b + a – 16 – 1.0 1.0 ∼ e g 1.0 ω ω = = = 0.5 = 0.9 = 0.99 = 2 EM mode. The latter is excited due to the coupling 0.8 ω ω | | l 0.8 0.8 0.6 = -q = -q = -q Q Q Q /dω /dω g g 0.4 9, which also excited in the . 9, cf. table1. The rightmost vertical line in the left panel denotes the . dE dE 0.2 0.6 0.6 0 , = 0 5 0.0 . -2 -5 -8 ωM ωM 10 10 10 R ≡ 99, cf. table1. In the legend of all panels = 0 . Q/M 0.4 0.4 001. The vertical lines denote the 0 . = 0.9 = 0.5 , = 0.9 = 0.5 9 . of the particle, and is well fitted by 0 = 1 Q/M =q =q = -q = -q , γ γ Q Q Q Q 5 . 0.2 0.2 , is even larger than the GW energy flux. are the frequencies at the peaks, and the coefficients ( | = 0 q . Top panels: quadrupolar GW (left panel) and EM (right panel) energy spectra for a particle e,g ω 0.0 0.0 We are now in a position to consider our initial question on the excitation of the EM The ratio between the flux at the two peaks depends on the charge and on the initial | ∼ | Q/M -8 -6 -2 -4

0.10 0.05 0.00 0.15 10 10 10 10

g e

μ ω / μ ω / dE d d

dE = 1 (bottom panels). The vertical lines in the panels of6 ﬁgure denote the fundamental

= 2 EM mode with - - 2 2 where dipolar EM energy spectrainsets for show the the case samepoint of plot charge opposite in is a (left logarithmic panel)with vertical and scale. equal (right In panel) all charges. cases the The initial Lorentz factor of the QNMs, which are presented inenergy table1 spectrumfor comes completeness. fromof The the largest dipole contribution to (bottom the panels) EM which, formodes sufficiently in large dynamical values processes. Figure6 shows that the flux for Lorentz factor Figure 6 with charge plunging radiallythe on RN a BH RN with BH.l The vertical lines denote the similar cut is alsol present for the EM energy spectrum with peaks: the firstthe one second corresponds peak to corresponds to thebetween the excitation gravitational of and the charged EM BH perturbations spacetimes [cf. [66, information eqs. about67]. (3.27)–(3.28)], the EM which In modes is other of words, a central the feature BH, not GWs of only emitted about in its the gravitational process modes. contain JCAP05(2016)054 = 2 and = 3 and s l 0. 2. Furthermore, . 1 Qq < . γ 1.0 1.0 0.5 M , both the EM and gravitational [cf. figure9 in appendixC]. Thus, I 0.5 0.0 2570 2675 0871 0883 0976 0974 0896 0972 0951 γ ω M 1, at least for ...... Electromagnetic Gravitational − ∼ = 1 of the point particle. An interesting . -0.5 γ Q 1 and the charge-to-mass ratio of the particle = 1 M .

5 4 3 2 1 0 -1.0

R 0.0

4035 0 6787 0 4214 0 4136 0 6194 0 3608 0 3817 0 4937 0 2707 0

g e Q/M ...... / E ω E = 1 γ – 17 – ) 2) 0 2) 0 1) 0 2) 0 2) 0 1) 0 2) 0 2) 0 1) 0 , , , , , , , , , s, l ( -0.5 99 (2 9 (1 5 (1 5 (2 5 (1 (1 9999 (2 99 (1 (1 ...... 0 0 0 0 0 0 0 0 0 Q/M -1.0

0.08 0.06 0.04 0.02 0.00 0.10 = 2. We consider

/ E M l . . Total energy radiated to infinity, summing the gravitational multipoles up to /M | . Fundamental QNMs of a Reissner-Nordstrom BH computed with continued fractions [37] Q | = is nonnegligible only when the BH is near extremality and only when = 1, respectively. radiation are suppressed. Thethe reason is EM that repulsion, instatic leading this solution limit to the of constant gravitational thethe attraction dipole field cancels Majumdar-Papapetrou and equations solution quadrupole describing [68, two moments69]). maximally (there charged is BHs in in equilibrium, fact a aspect of this figure is that in the extremal limit, pendixC we show some supplementalIn results particular, about we the show the energythe ratio fluxes relative (3.31) amplitude emitted of for in the some the EMR values process. peak of is larger when 3.4 Emission of (hidden) EMLet radiation us now in discuss a therepresentative binary total case energy BH of radiated merger an in the initial collision. Lorentz factor This is shown in figure7 for the q/µ Figure 7 the EM multipoles up to Table 1 for different values of the BH charge. Gravitational-led and EM-led modes are denoted by s JCAP05(2016)054 (3.32) (3.34) (3.38) (3.33) (3.35) (3.36) (3.37) in the inset Q . . . 2 the amount of EM radiation ) s [25]. Equation (3.37) shows 2 2 t / q M ( , µ M . 1 . . , s µr erg 2 1 / 033 4 019 . µ 56 . = 0 2 = 1 ≈ 0 M erg 10 Q rr γ γ ∼ 0, we obtain limit we obtain 2 ∼ × Q 6 015 2 for for M . 2 . overestimate M q Q 3 M 0 µ , , a GW150914-like event would produce an , , 30 Qq > 2 which is shown as a function of 2 8 0 ≈ ,D − 4 105 . . M g µ = 2 4 1 2 2 56 | 2 2 M = − 2 q – 18 – /E with the reduced mass of the system, [70 71 ]. This µ /dt rr ¨ 10 r M e g 2 10 3 µ D × E q dt | × for infalls from rest, and we obtain dE 021 01 2 . . 3 dE ∼ 2 3 ˙ 7 2 r 1 0 0 ˙ r 1 ∂ ∂t Q M Q ∼ ∞ ≈ ≈ M ∞ M M/r 2 2 e 1), assuming s, comparable to the luminosity of the weakest gamma-ray 15 2 e g ≈ 2 Z , the EM luminosity can still be very large. In this model / Z dt as expected. M M E E ≈ e g dE = Q p = 2 = E E γ erg g g = e 48 E dt r E /M dE 2 10 Q ≈ /dt e dE = 2 results for the EM channel agree with the numbers reported in ref. [72] l Up to now, our results are formally valid only in a perturbative scheme; however, decades In a binary BH coalescence, the total energy loss in GWs can be enormous. For example, For collisions from rest ( An interesting quantity is the ratio that, even for BHsthe with EM small luminosity of a GW150914-like event is roughly which is of the order of of work shows thatpoint-particle calculations even if fully one replaces nonlinear results for equal-mass BHs can be recovered with during that same process.charge The ratio reason objects, is that, dipoleIn as fact, emission we remarked is our earlier, suppressed for and two equal one mass-to- only gets quadrupole emission. the GW luminosity of GW150914 was once the extrapolation to equal-mass is done. substitution would immediately recoverthe (within gravitational radiation a produced factorcharge during of BHs. the two) head-on However, the collisions our of results point two in equal-mass, particle ref. equal- results [72] for bursts [73]. EM luminosity Using energy conservation ˙ of7. figure From the above relations, in the small- Even for weakly charged BHs with This result can beLarmor formulae compared for with the flat-space GW estimates and using EM simple emission, respectively. quadrupole For and the dipole GW flux we get The same procedure for the flat-space Larmor formula gives Thus, a flat-space Newtonian calculationtwo. agrees with our numerical results to within a factor JCAP05(2016)054 . 3 cm / (3.41) (3.39) (3.42) (3.40) . Low- QNM 4 GeV f . to its typical , 0 . ∼ f Hz do not propagate DM ρ , DM e 2 ρ / em plasma 1 , e/m /m 3 3 h Hz − e f < f 3 cm 10 − / e n DM 2 , / ρ 1 cm 1 4 GeV Hz . , so most of the EM energy released r 3 3 4 0 − 10 cm

/ ∼ M M 1 cm

DM 2 ρ 60 e M ≈ e M e 4 GeV e . 60 m n πn 0 – 19 – 4 300 s ≈ 1 250 − m and we normalized the DM density ) GeV ∼ are also phenomenologically available. π QNM 2 h f . Comparison between eq. (3.41) and eq. (3.39) shows that 4 /m . e 3 M 2 h = (2 e DM ∼ m cm ρ / πn Q 4 h ∼ e e em plasma s h f e propagate freely in a hidden plasma of density 1 4 GeV := n − . )

h 0 π M ∼ 60 = (2 DM ∼ ρ M is the electron number density. Photons with frequency h plasma e f n On the other hand, if the BH is charged under a hidden U(1) charge, a sizeable fraction Nonetheless, the spectra presented in figure6 show that there exists a cutoff frequency Charged BHs in these scenarios are interesting GW sources. The GW signal from the local value, the frequency of hidden photons is larger than a typical hidden-plasma frequency whenever where we estimated of the luminosityelectrons, is but would released nevertheless interact in with hidden dark plasma, photons. whose typical frequency The reads latter do not interact with ordinary (the precise value depends on thecharged, spin the and energy on is the released charge infrequency [cf. GWs photons figure5]). and are ordinary If absorbed photons the BHthe with by is frequency the plasma electrically interstellar frequency medium if their frequency is smaller than in the plasma.in In the the process interstellar is medium absorbed by the plasma. The above thresholddark line red is dot-dashed shownBH line, with in the the hidden right photons panel emitted figure1. during the In ringdown the of region a below charged the associated to the fundamentalof EM the QNM of order the of final [cf. BH.1] table This mode has a typical frequency where Interestingly, the phenomenologically-viable regionthreshold in line, the so right in panel this1 figure regionlies hidden below photons the are4 expected to propagate freely. Discussion and finalWe have remarks shown that, in modelshave of large minicharged DM (electric and and/or darkmetric. hidden) radiation, astrophysical charge BHs and In can are thesesome uniquely electric models described charge by within the the Einstein-Maxwellnearly-extremal standard theory Kerr-Newman BHs can arguments with be that circumvented and, prevent in astrophysical particular, BHs to have coalescence of two charged BHssystem. contains In a particular, wealth we ofprovide have information complementary shown about information. that the the The properties inspiral initial of and inspiral the the can post-merger constrain ringdown a stages combination of the JCAP05(2016)054 (where , which h θ e sin ∼ 2 discrepancy [78]. If ultralight, these bosons are – 20 – − g inspiral and ringdown tests might also be performed provided the energy ), so that conversion of dark photons to ordinary photons might occur when h / := Combined Likewise, our analysis of the upper bounds derived from ringdown detections can be BH-based tests provide a unique opportunity to constrain the hidden coupling We have focused on models of massless dark photons. Massive dark photons are an An important open question concerns the formation of charged BHs in these scenarios. θ 6= 0. The frequency of dark photons emitted in BH mergers are typically smaller than the improved in several ways,and for a example more sophisticated bywas statistical considering analysis. not multiple a It detections, is statisticalringdown multiple also fluctuation phase) modes, reasonable and to might that expectwill be even that reach louder detected GW150914 their GW in designentire events the GW (with sensitivity. signals near higher can In provide future, SNRmodels; stringent this in constraints we when on case the hope second-generation the that our charge detectors this of results BHs exciting suggest in prospect minicharged that will DM motivate an further analysis studies of on the this topic. appealing candidate to explain the muon Accretion of charged DM particlesform is in a natural the charging gravitational mechanism. collapseor of Charged hidden) charged BHs charge compact might also stars, byno the DM latter studies capture might on in acquire (electric the their interior. DM accretion To the by best BHs of in our models knowledge of there minicharged are DM and — in light of initial BH charges [cf. eq.are (3.6)] only and a sensitive rescaling to of the Newton’s final constant, BH whereas ringdown total tests lost charge. in GWs andinteresting EM to waves perform duringcoalescence fully the (extending numerical coalescence the simulations is workthe of known. of final charged-BH ref. Kerr-Newman Therefore, binary [72]) BHinspiral it and systems formed and would to close after ringdown be estimate phases to thethe very the merger (together merger. phase) mass, with can spin The estimatesformula. be and combined of used Such charge information to the information disentangle of from massthe would part the masses and of be and the charge crucial spins degeneracy loss forfor of appearing cross during GW150914 future in checks [25, work. the42]. similar dipole to A those more detailed performed analysis for in this direction is left is otherwise challenging to probetion with is EM-based the tests. burst ofwe An low-frequency have interesting dark shown, prospect these photons in dark emittedcan this photons during direc- are the be not merger extremely absorbed [cf.ordinary high. by eq. the)]. (3.38 photons DM In through As plasma a some andtan their kinetic models luminosity mixing of term minicharged [10, DM,18, dark26 ] photons proportionalkilohertz, and are to therefore coupled next-to-impossible to to might detect with be ordinary mechanisms telescopes. in However,tons there which and dark to photons ordinary canBH fermions. convert mergers to and This higher-frequency might conversion leave ordinaryit might a pho- is detectable provide signal in an in principle exoticcharged current possible EM experiments stars, [74, counterpart that via75]. of thehidden the Futhermore, (gigantic) burst radiation same of would mechanisms darkbe we cause photons measurable described. nearby affects and stars nearby extracted Indescribed to (hidden-) from recently such asteroseismology oscillate, for a studies, with GWs in case, [76, r.m.s a a77]. phenomena fluctuations passing similar that to burst could that of known to turn spinning BHsan unstable overreview). [79–81] due A to more a rigorousof superradiant analysis instability BH of (cf. physics this ref. instability in [82] is the for a context further of interesting application dark-radiation models. JCAP05(2016)054 (A.3) (A.2) (A.1) , 2 dφ φφ g , 2 + L tt g 0 φφ φφ dtdφ g g + tφ 0 tt rr g g tφ g only. The radial motion of null particles tt − g θ + 2 2 2 ELg − 0 tφ 0 φφ g g dθ 2 tφ and + 2 g q θθ r g rr – 21 – φφ + g g + 2 0 tφ 2 g E dr − , where a prime denotes a radial derivative. The orbital 0 rr := g V V + Ω = 2 = 2 dt case in which the analysis is remarkably straightforward, since ˙ r = 0 = tt g m V ) and some geodesic properties is more involved [61]. For simplicity, = = 2 l l, m ds are the (conserved) specific energy and angular momentum of the geodesic, L are defined by and E L/E Let us start with the stationary and axisymmetric line element correspondence frequency of the light ring is simply the azimuthal frequency, the metric coefficients arewith respect evaluated to on the affine theratio parameter of equatorial the plane, null geodesic. and The a light ring dot and denotes the corresponding derivative where In this appendix weof discuss spherical the photon correspondence orbitsrelated [59–61]. between to the the In azimuthal BH the orbitalsponds static QNMs frequency, whereas to case, and the the the imaginary the Lyapunov part real properties exponentmodes of part of the with of frequency the generic corre- the orbit ( QNM [60].here frequency In we is the focus rotating on case the the relation between our results — itwork). would GW-based be bounds very on interestingto the to constrain BH fill the charge might this parameter be gap space combined (see of to minicharged refs. realistic DM [83, accretion models. models84] forAcknowledgments some related We are indebted toof Leonardo this Gualtieri project for many andthank useful Aaron to Zimmerman discussions for Kent during useful Yagi correspondenceus. the and and development V.C. for Nico sharing acknowledges Yunes some financialConsolidator data for support of valuable Grant ref. provided [55] comments “Matter under with ongrant the and the agreement European strong-field draft. Union’s no. gravity: H2020selho We ERC Nacional MaGRaTh-646597. new de e Desenvolvimento frontiers Cient´ıfico Grant Tecnológicothrough C.M.1. No. in acknowledges 232804/2014- Einstein’s financial Research theory” supportIndustry at from Canada Perimeter and Con- Institute byopment the is & Province Innovation. supported of ThisStronGrHEP-690904 Ontario by work was and through the supported by the Government by FCT-Portugal Ministry through the of of H2020-MSCA-RISE-2015 the Economic Canada Grant project Devel- No. IF/00293/2013. through A Quasinormal modes of Kerr-Newman BHs from the geodesic these modes are associated only with equatorial motion [61]. where all metric coefficients areon functions the of equatorial plane is governed by JCAP05(2016)054 pre ) is = 3 1 the (A.6) (A.4) (A.5) (A.7) (A.8) l, m m Ω. = 2 (bottom l . m l = 2 [55], but ∼ = = 0 l l R limit the above ω m Q/M = l , ) tφ pre 4% for any spin, both in the φφ g , Lg ≈ | tt + Ω g + λ , | θ − 2 2) φφ U (Ω 2 g / l 2 tφ 2 ∂θ ∂ g E ∼ + 1 r − n tφ pre ( , g i = Ω θθ θ represents the frequencies of small oscillations ˙ t g and the expressions above are evaluated at the − Ω m θ 2 l 1. Interestingly, in the + Ω – 22 – φφ − √ + Ω tt g θ g 2 , ∼ ) 00 I = = Ω 2 m 2)Ω V iω / θ L/E = pre Ω r + ˙ t 1 Ω l + 1 + ( R l − ( ω tφ g = ∼ ) λ R ω L/E 2( 1. However, in8 figure we show that the analytical result (A.8) agrees − tt g l = 2. Relative errors are always smaller then ) = m r, θ is the overtone number. Note that the above expression formally coincides with that = n l Although not relevant for our analysis, for completeness we discuss the geodesic corre- Reference [61] shows that the QNM frequency in the eikonal limit reads Strictly speaking, the geodesic prediction (A.8) should only be valid in the eikonal In the main text, we have used this striking agreement to estimate the spondence of the dampingLyapunov time coefficient of of the the modes. orbit In [60] the eikonal limit the latter is related to the where in theexpression last coincides step with we that derived used for static spacetimes in ref. [60], i.e. around quasicircular equatorial orbits alongis the the angular precession direction, frequency respectively, of whereas Ω the orbital plane. light-ring location on the equatorial plane. Ω evaluated at the light-ring location. Other relevant orbital frequencies are where U( again evaluated at thecomplex light-ring QNM location can on be the written equatorial as plane.–61] [59 Thus, for modes of a weakly-chargedpredictions Kerr-Newman are BH. likely Noteapproximation smaller should that than not the the affect deviations our observational from analysis errors the significantly. on geodesic theseB modes, therefore this Fisher matrix analysis We follow ref. [57] for thewaveforms analysis in of uncertainties noise. associated with measurements We of ringdown assume that the GW signal during the ringdown phase can be limit, i.e. when where obtained in refs. [60]valid only for for static slowly-rotating spacetimesderived BHs. in and ref. A it [61]. more extends involved the result results for of QNMs ref. with [59] generic ( whichremarkably are well with the exactwhen numerical results forneutral the case QNMs (top of panels apanels of Kerr-Newman figure8) of BH and figure8). even in Intheir the Kerr-Newman deviation the case from latter with the case geodesic the predition exact is results always are smaller only than available 3%. for JCAP05(2016)054 1.0 (B.1) (B.2) (B.3a) (B.3b) = 3 and 0.8 m 0.8 . , (in the latter = S S l ψ ψ 1. We show the M 2 Q/M=0,Δ=2.% . Q/M=0.2,Δ=3.% 0.6 0.6 sin 2 cos 2 m = 0 S S χ χ . φ = φ Q l 0.4 lmn 0.4 , sin 2 sin 2 S ) S S S θ θ lmn , ψ l=m=2 exact, geodesics geodesics l=m=2 exact, l=m=3 exact, l=m=4 exact, S t/τ ) are evaluated at the (complex) cos 0.2 0.2 − , φ e − + cos ) lmn S S θ S ( ψ aω lmn ψ ( × φ 0.0 0.0 F + lm t

cos 2 sin 2

0.8 0.6 0.4 0.2 0.0 0.8 0.6 0.4 0.2 0.0 1.0 1.0 S

I I I I h 0 0 ) = χ ( ω / ) = χ ( ω / ω ω S S lmn = ω φ φ i( ) + e S lmn – 23 – 1.0 S , ψ lmn ) cos 2 ) cos 2 = 2 [55]). The blue band brackets a percentage variation S S S A l θ θ , φ 2 2 S 0.8 lmn X θ 0.8 ( r + M F + = (1 + cos (1 + cos Q/M=0,Δ=4.% h Q/M=0.2,Δ=3.% 0.6 × 2 2 1 1 0.6 h = χ χ h + i ) = ) = S S + 0.4 h 0.4 , ψ , ψ S S , φ , φ S S geodesics l=m=2 exact, geodesics 2 l=m= exact, 3 l=m= exact, l=m=4 exact, θ θ 0.2 are pattern functions that depend on the orientation of the detector and the 0.2 ( ( ). × + × , F F . Comparison between the exact gravitational QNMs of a Kerr-Newman BH and those lmn + S F = 4 modes of a Kerr BH are almost identical. 0.0 0.0 The waveform measured at a detector is given by

1.0 0.5 0.0 1.0 0.5 0.0 2.5 2.0 1.5 2.5 2.0 1.5

R R R R 0 0 ω / ω ω / ω ) = χ ) = χ ( ( = QNM frequencies, so they areon complex the numbers (henceforth we drop the angular dependence expressed as aspacetime. linear The superposition gauge of invariant waveforms exponentially are decaying then given sinusoids, by the QNMs of the Figure 8 obtained from a geodesic analysis [cf. eq. (A.8)] which is formally valid for case the exact QNMs are available only for l ∆ relative to the geodesic prediction. With the adopted normalization, the curves for ratio between the realfunction (left of panels) the and BH thecase spin imaginary of and (right a panels) normalized Kerr part by BH, their of bottom the value panels in QNM show the frequency the nonspinning as case case. a of a Top panels Kerr-Newman show BH the with In this expansion the spheroidal functions direction of the source, namely where JCAP05(2016)054 , lmn lmn Q (B.4) (B.5) (B.7) (B.6) , f = 0, and × lmn i × . We assume , φ F ρ + + lmn F h , φ , 5, × )]. In the large / π can then be calculated, = 0 ,A (2 a / + = 1 θ A i × lmn ψ , φ 2 . lmn × × ω ψ , F A ), one defines the inner product h cos b f . := ( sin 2 df , δθ = 1 ) h a 2 1 = Γ i ) − S ˜ h , 2 δθ + lmn ∗ 2 2 lmn ) f ˜ F h + lmn ab 2 lmn f h , Q φ ψ , b Γ ( ψ , + Q + h 2 1 Σ = Γ A 2 ∂h ∂θ S sin cos ˜ ) is characterized by a given set of values of h − | t are then given by ∗ 1 2 lmn ( (1 + 4 ˜ h = Γ a s Q (1 + 4 ab ψ , , ψ , ∂h 2 lmn 2 lmn ∞ ∂θ 2 2 – 24 – 0 aa + lmn ) exp Q Q 2 lmn 2 lmn Z φ θ Σ 1 + 4 sin cos × ( Q 2 Q 1 1 ≡ √ A 2 (0) ≡ 1 + 4 1 + 4 γ ab = p , in measuring the parameter A ) 2 lmn 2 lmn Γ 2 2 1 + 4 = Γ 1 + 4 a ) by / Q Q t h θ 1 = | ( ) ) = lmn lmn 1 2 i ∆ × s × lmn γ γ lmn lmn 2 | h h ) represents the distribution of prior information. An estimate A φ ) ( γ γ θ Af Af Q θ a + × 1 + 4 1 + 4 ( ( 2 2 A A AQ A p δθ and average the source over sky position and over detector and 2 2 γ γ − − (0) ( is h p ) and a t = = = = = Γ = = = Γ ( θ lmn ) are the Fourier transforms of the respective gravitational waveforms 1 = ( + × f h Q A A a × lmn + lmn lmn lmn lmn lmn ( , and f f θ + + φ φ 2 Q Q a + × A A ˜ h ˆ θ + × . A A Γ Γ × lmn + lmn A A π − Γ Γ φ φ Γ Γ 4 a Γ Γ / θ ) and = f are the source parameters. In the limit of large SNR, if the noise is stationary and = 1 ( a 1 i a 2 ˜ h θ δθ | ), were computed and presented in ref. [57] [here We will follow the prescription outlined in ref. [57] to compute the SNR The Fisher matrix components in the parameter basis of ( With a given noise spectral density for the detector, ). The components of the Fisher matrix Γ lmn t lmn S ( limit, they read large quality factor where where Gaussian, the probability that the GW signal in the limit ofof large the SNR, Fisher by matrix, taking the square root of the diagonal elements of theQ inverse BH orientations, making use of the angle averages: the source parameters h of the rms error, ∆ where h| between two signals JCAP05(2016)054 . ab h (B.9) (C.1) (C.2) being (B.10) ) basis 12 f lmn Q , , and 02 lmn f f = 0 . lmn , /dχ Q ) , , with 2 × lmn , lmn lm φ θ, φ ( Y Q ) θdφ 2 d ) , lm 2 = Γ dr Y . := 12 sin 2 lmn . lmn 3 , and     f 2 lmn ) Q Q ) r Q k 0 lmn + /A θK Γ + 2 lmn × Q 2 lmn dt + lmn 2 2 A 0 Q φ Q 02 0 lmn (1 + 4 dθ sin f ≡ and 2 ( Q 2 r = Γ lmn r ψ − + (1 + 4 2 lmn Q + (1 + 4 , 2 = 2 K /dχ lmn 2 00 0 0 Q ) 2 2 1 f A lmn , sin ) ) only. Additionally, the vector potential can be r – 25 – lmn dr δA lmn f k f v 2 2 lmn × lmn is the background vector potential of the RN BH. 2 /A t, r 2 lmn Γ − φ df 1 1 + 4 H π Q 1 + e v Q H A 40 0 − := , is freely available online [85]. lmn + = Γ e lmn f 2 ≡ ). Finally, the transformation from the ( = 0 Q/rdt Q (1 + 4 = dt 0 β 1 γ lmn ψ lmn γ (1 + 4 H − v 0 0 0 0 0 f f v H e − . Here we consider the Regge-Wheeler decomposition of lmn k j e 2 = 2 lmn f − Γ 2 γ is a small perturbation to the background Reissner-Nordstrom lmn + lmn γ 2 , cos     φ Q f /r 2 lmn 2 A = 2 2 2 − ) (0) ab Q = Q × g Mathematica (0) ab = = Γ = = A ab g are decomposed as + h ) are functions of ( (1 + 4 + ( lmn lmn × lmn lmn , and where ab δA f γ φ , where 2 j Q Q ,K h ) 2 M/r = 2 lmn + 6= δA + lmn lmn lmn f φ 2 f ,H A k Q − Γ ρ + 1 Γ Γ Γ ) basis reads A = ( ,H , where = 1 0 2 can be written as v ab Q, χ A H e h ab h + Here, (0) ab where where ( C Technical details on theIn radial infall this of appendix a wesystem point (3.27)–(3.28). give charge into some We a follow detailspossible charged the about BH procedure errors. outlined the Our in derivation integrationindependent ref. technique and codes, [65], is integration as but fully well of correcting consistent, asof typos the and with the and coupled has previous codes, been results written in validated in the by literature two for unchargedC.1 BHs. One EM andThe metric perturbations spacetime metric dueg to a point charge falling into a charged BH can be written as In the case ofand a radially falling particle, the metric perturbations have even (polar) parity written as The perturbations geometry, for any index We also have to the ( JCAP05(2016)054 (C.6) (C.3) (C.4) (C.5) (C.7) (C.8) (C.9) (C.13) (C.10) (C.12) (C.11) , , 0 v (1) v − v e A , e 0 2 3 j r 12 2 √ , + 2 iπ πr v + 4 0 4 − 2 iQωf − e rv = 4 2 Q 12 0 0 = 2 2 − v r v 0 0 v Q + e 02 3 2) + 2 iQωf 00 , QK r + 2 2 ) Qf , − rv 2 + v 2 ) as r +2 v + e . , θ, φ e − 2 v 1 12 0 ) ( v 2 j 2) t, r Q e e 2 02 = ( lm f t, r − 12 ψ − ( v Y πr 2 irωf 2 − r 2 ψ 2 r      2)+2 = 4 + 1) + 4 − + 1) rω iωt Qe − , iQωf 0 l l 2 + 2 + v 2 0 0 0 0 ( ( v v 0 + 2 l e l 12 e ( 0 ( − + ) only. In a stationary background, we can − dte rv e 2 , (1) 2 12 ωf 0 0 rv r v r 2 iQωK 0 v A t, r A Z πA 02 − 4 . v 0 2 − Qf ir i v f e rH 2 + r π e 2 – 26 – 2 √ 4 e v 0 +1)+2 K 2 1 2 + 2 2 r 1 r − 0 l = 8 v 02 πj r v ( H 2 00 = v √ l e Qe H + (1) ( re − 2 Qf rv + 2 (0) e 00 0 0 0 0 0 0 A v 1 2 = 4 + 2 r 2 0 − A − i ) = 02 v + 1) iωK e v + √ 4 f 1 + 1 r 1 = l v 1 2 r re 0 ( (0) e 0 2) H l ω, t 2      2 1 + 6) + 2 + rv 12 0 2 K 0 − + 1) + − 0 + 2 πA f = Ψ( r 2 + v 0 l r v 0 2 2 rv − ( 0 v v 2) 2 P ( − l ab ω e 0 v rv − 0 0 = 8 H 2 T − 1 + 1) e − ’s are functions of ( 0 v + r 12 K 2 v l K e 1 0 H r v A f ( 1 ω re rv l 2 2 1 2 v H 02 2 ( + ( 12 e e H Ke r − f v + rf 2 2 1 2 ) − − − 0 − 2) 2 iω irω + 00 1 e v iωK v 00 2 0 0 + 2 1 − − ) only. The stress-energy tensor of the particle can also be decomposed in + r − − H r − − K 02 H 2 0 e 2 0 v 00 2 e v 0 1 v e r 2 t, r Q 2 02 − 02 0 ωf iωH 2 1 e r f ω + 1) 2 00 2 − H 2 ωH l − + iωH iωf r ir 2 1 2 ( K − l 0 0 ( − − − ( 0 − K ir K 0 v + 0 0 e H r 0 r + + − + 2 v 0 H r iωH 2 2 1 e 2 2 H 1 iωK − where, once again, the explicitly eliminate thefunctions: time we dependence define by the Fourier a transform Fourier of transformation a function of the perturbation Below we shall consider alreadyfor the each perturbation transformed function. Fourier quantities,potential By using into substituting the the the Einstein same expressions equations notation forset and the of expanding metric equations up and to the first vector order, we obtain the following whereas the perturbed Maxwell equations yield functions of ( terms of spherical harmonics, JCAP05(2016)054 ) , v θ, φ 2 , e.g. ), the 2 (C.18) (C.15) (C.14) (C.16) (C.17) (C.19) 1 re ω H 4 , and an 1 , − − K ] 6 iω Q M − 4 5 and − in the differential − K, 1 ) r ω r H = ( 2 ) and their derivatives. r z ) are linear in + Λ ψ , required by the ( 0 ,S M g H + 6Λ [2(Λ + 3) (3 v r obeys the following differential = e 4 g Mr α, β, γ, δ , . Q 2 ψ z ] + H S r 8Λ) 3) + 6 , = − − 3 g S v , r f, g, h, k, n, V e ψ ¯ ) = S g (Λ + 2) V + 2(3 + ψ − 2 ¯ and manipulating eqs. (C.12) and (C.13), we − ψ 3)) (Λ + 1) 2Λ + 3 2 e M M 2 A F − − ω . ψ – 27 – . By defining the vector ( 27 v − v v = 2 e e K − + Λ − + ( r d e ψ 0 dr 2 = ) ) 2 0 g r , and (Λ + 1)[ r 2 12 M ), and the coefficients ( v Mr f , and 2 ), such that the function 2 nψ − Q 2Λ + 3 1 0 g − 1 ( − ,S re − n r H 2 ( 1 , ψ ( ) , − 2 g 2 S 0 + 3Λ r ψ 2 + 2))] Q = r H 2 Q + 2 − = ( k 2) + 1) + (Λ + 1) = ( 2 v − , M v e v − ¯ Q ) S e ) + ψ ( e r 4 v , 2 r e M = r − 2 + 9Λ and + Λ n Λ + 2 3 − (Λ − + 2Λ γ δ r α β ( Λ + 2 M 2 M . To simplify the system further, we wish to find a transformation ( 9 2 r will also involve EM perturbations, since these appear as a source terms in the r − Q (3 = 1, M ( α h k f g r z 3 3 + 2 g ( r v 2 S r = ω e 6 + 2 r M + = ( + A r 0 = = M Simplification of the gravitational sector is more involved [65]. By manipulating eqs. Due to the Bianchi identities, not all of the above equations are independent. Let us first F α e g V V = [ = = h f differential equations can be written as algebraic relation between Note that We obtain first-order equations. By comparing the coefficients of different order in (C.6)–(C.9) we can obtain a system of two differential equations for obtain eq.). (3.28 α with equation equations it is possible to obain a relation involving ( The remaining functions are given below. C.2 Final equations Through the procedure describedsecond-order in differential the equations, previous section, namelygiven we eqs. by obtain) (3.27 a and system (3.28), of where two the coupled coefficients are look into the Maxwelland equations is (C.11)–(C.13). a consequence Equationit of) (C.11 is (C.11)–(C.12) is easy and automatically the to satisfied, continuity see equation that, for by the defining currents. Indeed, In the abovecomponent equations of the we perturbed already Einstein used equations. the fact that where JCAP05(2016)054 is γ (C.24) (C.30) (C.20) (C.26) (C.27) (C.28) (C.21) (C.25) (C.29) (C.22) (C.23) 3)) , − v v 0 1 2 e , ) is the outer , A 2 − v r r 2 v ( e 2 iQωe 3 e 2Λ+3 + + 3) r − = ( . Note that a point πr v 2 e 2 + 3 2 3 r v Q √ e 8 ,I + − 2 is the particle charge, 2 Q 3 ( q , M 0 ω − 1) (2Λ + 3 p T v , − 3(Λ + 2) 3 , 2 iωT , whose columns are independent v , ) − e 2 e ± e ) − 3) 3 + v 2 X + r ) iωT e r ). The system (3.27)–(3.28) can be − 3)) 3)) r M ( ∗ . + v − l µr 4 r − e − − S v − 2 r l µe = r 2 2 8Λ + 3 v v /dr ( − r ) iωT γr e = e e 3 ± 3)) r ) )( − Ae , r γr 1 + 2 − 3 − − )( 2 Ψ l qe − 1 + 2 2Λ+3 4 matrix r r , dψ √ . − − − v ∗ ) πr V √ + 2) + r ( 4Λ × qQ e + 2) − − ) 2 v 2Λ + 3 2Λ + 3 , l 16 rγ + − – 28 – − qQ r r qQ r e 1 + 2 /dr v rγ ( ( ( − − r g 3) ( − ( (Λ + 1)(2Λ + 3) + 4 Ψ √ π π − ( 1)( re − 2 p r 2 2 2 2 ∗ − v r r 2 d − 2Λ + 3 πr Q e rγ , dψ √ √ qQ v r v dr l 2 e ( 2 2 2 qQ e (Λ + 3 + + − ( − e ( ( = = 2 2 2 2 i 1 1 − − , ψ 2 r 0 g Q Q r 2 + 1) + 1) + 3)) T ( ( = = = dt ψ dτ v Q 3 + − 1 1 − ω e Λ = 2Λ + 3 A r j v 2 v A πQrA e 1 = ( (2 − e + 2 2 Q ( 2 ( 4 rj 2 √ r Ψ v i r ω 2 Q 2 8 3 Q − 2Λ+3 r + Λ + 5 ) can be integrated analytically, but the result is cumbersome and we 2 − 2 ( r − v ( ( − 2 iπQe e Q 2 r 1 T 4 16 − Q + Q 2 πA v 2 v 2 v Q e ( 1 √ 8 ) is the function that describes the particle motion, ω iQe πj r iQωe 8 ( − − T = = = = 4 The first method relies on the Green’s function approach, also called method of variation e 2 1 g I I S S where charge will not follow a geodesic of the background metric and its motion is defined by [86] We start by constructingsolutions the fundamental of 4 the associated homogeneous differential equations. The independent solutions the energy per unit of(inner) mass horizon as of measured by the an Reissner-Nordstrom asymptotic observer, BH, and The equation for don’t show it. C.3 Numerical procedure To solve the perturbationThe equations latter (3.27) agree with and each (3.28) other we within employed numericalof two accuracy. different parameters methods. [87]. Let written as with JCAP05(2016)054 ∼ g ψ (C.33) (C.34) (C.35) (C.31) (C.32) (C.36) 0) and , . In this = 0, and, . With the can be ob- + in g,e r g,e → ∞ A ) = (1 A X γ + r e ,A + r g A can be obtained by inte- X . . ∗ iωr XI e , . ∗ out g,e := ∗ , , A 3 4 iωr S iωr I I 1 − + e ∗ ∗ e − ∗ + ∞ g,e iωr iωr r g,e X e e iωr A ∗ A − – 29 – ∼ ∼ dr e ∼ ∼ e g 1). The general solution can be written in terms of in g,e Z g,e ψ , ψ g,e A ψ ψ X ∼ = ) = (0 g,e are constants. The first two columns of Ψ ψ ∞ e , respectively [cf. eqs. (3.29)–(3.30)]. At the horizon, ∞ 4 , I ,A + r g,e . ∞ g 2 A I A ∗ and 3 . iωr , we can compute the amplitude of the GW and EM waves at infinity, I γ − + e r g,e 1). The other two columns of the matrix A , 0) and ( ∼ , e ψ 0 and when the BH is highly charged, cf. figure6. Note that the ratio depends ) = (0 ) = (1 and + r e ∞ e 1 I Qq < Let us now expose the second method that we used to solve the system (3.27)–(3.28). Finally, for completeness in figure 10 we present some representative cases for the GW ∗ ,A ,A + r g ∞ g iωr − A A The limits ofthe the proper above integral boundaryhorizon are conditions and suitably at outgoing chosen waves thegravitational at such and horizon infinity. that EM and From master the the at functions general final is infinity, solution, solution given i.e., the by obeys asymptotic ingoing form waves of at the the and therefore theintegral GW component and EM fluxes can be obtained through the absolute value of the can be obtained inhave the the following following way. form: We notice that at the horizon,and the at infinity required we solutions have In the abovetained equations by integrating theary conditions homogeneous (C.31), equations by( from choosing the two horizon independentgrating outwards solutions, with from say bound- infinity ( ( inwards, with boundarythe conditions fundamental matrix (C.32), as by choosing, once again, e The shooting method reliesFirst, on we an impose integration that of the thethe solutions full near full the system horizon of equations are inhomogeneousingoing up of equations. the and to form outgoing (C.31). infinity modes, We where namely then integrate the general solution will be aThe superposition physical solutions of corresponding the therefore, to this a becomes particle a falling two-parameterproper into shooting values the problem of for BH the require amplitudes regime our results for the EM flux are in perfect agreement with those presented in ref. [88]. which enable us to compute the GWC.4 and EM energy Supplemental spectra results throughIn eqs. this (3.29)–(3.30). sectionradial we infall present of a some point supplemental chargethe into results two a Reissner-Nordstrom peaks on BH. Figure9 of theAsshows the GW the discussed ratio quadrupolar in and between the GW EMwhen main flux emission text, [cf. in the eq. the relative)] (3.31 amplitude as of a the function EM of peak the is BH nonnegligible charge. only on the boost factor and EM energy spectra for the radial infall of a high-energy point charge, i.e. JCAP05(2016)054 , ]. , 1.0 , SPIRE IN 0.8 Metric of a 0.6 (1969) 869[ ωM , cf. eq. (3.31). 157 0.4 □ △ ○ = 0.99 = 0.5 = 0.8 = 0.9 = 0.99 Q/M □ △ ○ =q =q =q =q = -q ]. -0.80 , Cambridge University Press, Q Q Q Q Q 0.2 □ △ ○ γ= 1.01 γ= 1.1 γ= 1.2 ]. □ △ ○ SPIRE □ △ ○ Astrophys. J. IN . ]. , SPIRE -0.85 □ △ ○ 0.0 -5 -7 -9 -11 IN q/µ 10 10 10 10

□ △

0.100 0.001 ○

SPIRE

e := ) γ μ ( ω / d

dE ]. - Q/M 2 IN q □ △ ○ -0.90 (1965) 918[ 1.0 – 30 – and ¯ □ △ ○ 6 SPIRE ]. (1975) 51[ Theory of pulsars: Polar caps, sparks and coherent IN □ △ ○ Q/M 196 0.8 (1977) 433 [ -0.95 Electromagnetic extractions of energy from Kerr black holes SPIRE □ △ ○ := IN Pulsar electrodynamics ¯ □ △ Q ○ 179 □ △ ○ ○ (1975) 245[ 0.6 J. Math. Phys. -1.00

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