您当前的位置: 首页 > 网页快照
Searching for ultra-light bosons and constraining black hole spin distributions with stellar tidal disruption events - Nature Communications
Methods .
Maximum redshift for TDE observations at LSST .
The maximum redshift up to which a magnitude-limited survey can observe a TDE can be estimated from the luminosity-magnitude relation 43
$$\log \left[\frac{{L}_{{{{{{{{\rm{TDE}}}}}}}}}}{{L}_{\odot }}\right]=2\log \left[\frac{{d}_{L}({z}_{\max })}{{{{{{{{\rm{Mpc}}}}}}}}}\right]-0.4({m}_{\lim }-{{{{{{{{\mathcal{M}}}}}}}}}_{\odot })+10,$$
(10)
where L TDE is the TDE luminosity, \({{{{{{{{\mathcal{L}}}}}}}}}_{\odot }\) and \({{{{{{{{\mathcal{M}}}}}}}}}_{\odot }\) are the Sun’s absolute luminosity and magnitude, \({m}_{\lim }\) is the limiting AB magnitude of the telescope, and d L ( z ) is the luminosity distance. All luminosities and magnitudes must be calculated in a band corresponding to a particular filter. The above parameters are determined as follows. First, we set a magnitude limit \({m}_{\lim }=22.8\) in the LSST g-band as in ref. 6 . The g-band luminosity L TDE is calculated assuming a black-body spectrum and applying the green LSST filter from speclite . The temperature of the TDE black body is fixed to T ?=?2.5?×?10 4 ?K 38 . The amplitude of the black body, on the other hand, is normalized according to two prescriptions. Our baseline prescription, used to produce Figs.? 4 and 5 , corresponds to taking the logarithmic average of the luminosity of observed events in ref. 38 . In this average, we do not include the very bright ASSASN15-lh event, as it is an unusual TDE candidate 39 . We choose to take logarithmic averages to avoid overweighting the large luminosity TDEs. This gives L TDE ?=?10 42 ?erg/s in the LSST g-band. Our second prescription considers very conservative TDE luminosities, and we use it in the next section to evaluate uncertainties for our projected ULB limits. It is obtained by normalizing the black body to a peak spectral density \({L}_{{\nu }_{g}}=1{0}^{42.5}\) ?erg/s at ν g ?=?6.3?×?10 14 ?Hz as suggested by van Velzen 38 (peak spectral densities are defined as L ν ?≡?4 π r 2 ν ?×?( π B ν ( T )), where B ν ( T ) is the black-body spectral radiance and r the black body radius). This gives L TDE ?=?3.5?×?10 41 ?erg/s in the g-band, which is close to the minimum luminosity of all the events observed in ref. 38 . With these choices and including K-corrections, Eq. ( 10 ) results in \({z}_{\max }=0.57\) and \({z}_{\max }=0.31\) for our baseline and conservative luminosity prescriptions.
Discussion of the assumptions for projecting limits .
In this section, we study the impact of our assumptions on projected limits for ULBs. We concentrate on vector ULBs, given that they can be more effectively probed than scalars using TDE rate measurements. We analyze three variations of the assumptions used in the body of this work for obtaining limits. First, we vary the maximal spins of SMBHs. We take five different spin values, a ?=?0.6, 0.7, 0.8, 0.9, and a ?=?0.998. Lower SMBH spins lead to weaker limits on ULBs, given that limits on ULBs can be set if SMBH spins exceed those allowed by superradiance. Second, we consider two assumptions for the TDE luminosities (discussed in the previous section), denoted as “baseline” and “conservative.” The conservative prescription leads to weaker bounds, as lower TDE luminosities reduce the dataset observed by LSST and thus increases the statistical uncertainties. Finally, we take two possible values for the systematic uncertainties of the analysis, 50% and 75%. While a significant amount of theoretical and observational work is required to evaluate if these are realistic systematics, here we point out that our choices are inspired by the fact that currently around 20% of events classified as TDEs may be impostors 39 . In addition, the SMBH mass function also presents uncertainties that depend on the method used to measure it. For instance, estimates in the local Universe depend on the particular kinematic relation used to infer the SMBH mass from the host galaxy properties 44 . Estimates of the mass function up to high redshift can be extracted from active galactic nuclei emission, but there are uncertainties due to the emission model 33 . For our purposes, a precise estimate of the SMBH mass function in the local Universe suffices, as the maximal redshift at which LSST is expected to see TDEs is ≈0.6. Quoted uncertainties for the corresponding mass function at M BH ?~?10 8 M ⊙ 33 , 44 are about a factor of 2. To estimate the impact of these uncertainties, we have checked that the uncertainties in the SMBH mass functions reported in ref. 33 translate into a ≈50% systematic due to errors in the calculations of our signal estimates. This may be an overestimate, as our knowledge of the SMBH mass function could be improved by using TDE rate measurements at BH masses that lie below our signal range, M BH ? The results of our different spin and systematics prescriptions are shown in Tables? 1 and 2 for our baseline and conservative luminosity assumptions, respectively. In the entries of the table, we indicate the range of ULB masses (in units of 10 ?20 ?eV) that can be ruled out under different assumptions. Note that some entries in the tables indicate disjoint mass exclusion ranges. In these cases, these ranges correspond to the exclusion due to spin-down from the dominant and subdominant superradiant levels.
Table 1 Projected exclusion ranges of spin-1 ULB masses Full size table
Table 2 Projected exclusion ranges of spin-1 ULB masses Full size table
We conclude this section by briefly commenting on the effect of the uncertainties in the M BH measurement in our projected limits. In a binned analysis of the TDE rates as a function of \(\log {M}_{{{{{{{{\rm{BH}}}}}}}}}\) , uncertainties in the BH mass measurements result in migration between TDE events corresponding to different \(\log {M}_{{{{{{{{\rm{BH}}}}}}}}}\) bins. Given that our projected limits are obtained by using TDE event rates in the signal region \(8\le \log {M}_{{{{{{{{\rm{BH}}}}}}}}}/{M}_{\odot }\le 9\) , and given that TDE rates are comparatively larger at lower BH masses, \(\log {M}_{{{{{{{{\rm{BH}}}}}}}}}/{M}_{\odot }\le 8\) , inaccuracies in the BH mass measurement will predominantly result in migration of events from lower mass bins into our signal region. This leads to an increase in our estimate of the number of TDE events in the presence of superradiant effects in the signal region, weakening the projected bounds. An accurate estimate of the overall effects of bin migration on the bounds requires precise knowledge of the M BH measurement uncertainties, but for illustration, we have checked that a ±0.2 dex gaussian uncertainty on M BH allows one to place limits over a wide range of ULB masses, while a?±?0.5 dex uncertainty is likely too large to set meaningful constraints.
Environmental perturbations of the superradiant cloud .
SMBHs are surrounded by a complex environment, which may include a massive accretion disk and a stellar halo. Both of these components gravitationally interact with the superradiant cloud, and may either suppress or favor spin extraction. Here, we estimate these effects to evaluate the robustness of superradiant spin-down in SMBHs. We study four types of perturbations: an accretion disk, a stellar halo, individual stars that approach the cloud on TDE trajectories, and individual stars that approach the cloud on inspiral trajectories (extreme mass ratio inspiral, or EMRIs). For concreteness, we study perturbations on clouds consisting of scalar bosons. We do not expect clouds of vector bosons to be more sensitive to perturbations. For vectors, some superradiant and decaying levels differ by the vector’s spin. Spin-flips, and thus transitions between these two types of levels cannot be induced by gravitational perturbations, such as those from accretion disks. In particular, the dominant superradiant level \(\left1101\right. \rangle\) cannot be mixed by these perturbations with the dominant decaying mode \(\left110-1\right. \rangle\) , so the level \(\left1101\right. \rangle\) is robust against perturbations.
Perturbations to the cloud can be computed using elementary techniques from quantum mechanics. As in quantum mechanics, perturbations can be classified as time independent and time dependent, according to the duration of the perturbation relative to some relevant oscillation timescale set by the cloud’s energy eigenvalues. For our purposes, the presence of an accretion disk and stellar halo can be treated using time-independent perturbation theory, given that they are static perturbations to the cloud. Stars on TDE trajectories must be studied using time-dependent perturbation theory, as the timescale for a typical TDE (≈1 year) can be shorter than the timescale set by the inverse energy splittings between different superradiant levels. Finally, stars on EMRI trajectories adiabatically lead to resonant mixing between cloud levels when the inspiral frequency matches the cloud’s energy splittings. As in quantum mechanics, these resonances cannot be captured by standard time-dependent perturbation theory, and must be treated using the Landau-Zener formalism as shown in ref. 45 .
Accretion disk .
Time-independent perturbations lead to level mixing and modify the eigenvalues of the BH-cloud Hamiltonian. Such modifications can lead to a system that does not have any superradiant eigenstates, so that no spin is extracted from the BH. For a perturbing potential V , a level \(\leftnlm\right. \rangle\) remains superradiant after mixing with a decaying level \(\left{n}^{\prime}{l}^{\prime}{m}^{\prime}\right. \rangle\) if the imaginary part of the cloud eigenvalue’s remains positive after mixing. The mixing coefficient between the levels is of order \(\left. \langle {{{\Psi }}}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}\rightV\left{{{\Psi }}}_{nlm}\right. \rangle /({E}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}-{E}_{nlm})\) , where Ψ are the corresponding wave functions, and E n l m the cloud’s energy eigenvalues. Thus, the imaginary component of the perturbed cloud is positive if 12 ,
$$\chi \equiv \frac{{{{\Gamma }}}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}}{{{{\Gamma }}}_{nlm}}{\left\frac{\left. \langle {{{\Psi }}}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}\rightV\left{{{\Psi }}}_{nlm}\right. \rangle }{{E}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}-{E}_{nlm}}\right}^{2} \; < \; 1.$$
(11)
In what follows, we refer to χ as the perturbation estimator. In Eq. ( 11 ), Γ is the superradiant or decaying rate for each state. The energy levels up to \({{{{{{{\mathcal{O}}}}}}}}(\mu {\alpha }^{5})\) are given by 26
$${E}_{nlm}= \mu \left[1-\frac{{\alpha }^{2}}{2{n}^{2}}+{\alpha }^{4}\left(\frac{2l-3n+1}{{n}^{4}(l+1/2)}-\frac{1}{8{n}^{4}}\right)\right.\\ +\left.{\alpha }^{5}\frac{2am}{{n}^{3}l(l+1/2)(l+1)}\right].$$
(12)
The potential V is determined by the density profile of the accretion disk. We estimate the perturbation by assuming that the BH is surrounded by a Shakura-Sunyaev (SS) disk 46 , which corresponds to the disk profile in a phase of significant accretion. We have checked that the less-dense ADAF disks 47 lead to weaker perturbations. The SS disk has azimuthal and reflection symmetry on the plane of the disk, so a spherical harmonic decomposition of such disk profile contains only modes with m disk ?=?0, which cannot induce transitions between superradiant and decaying modes due to selection rules 48 . However, inhomogeneities in the disk can break its azimuthal and reflection symmetries and thus induce transitions. To model these effects, we consider a disk with the radial and vertical profile of the SS disk, and include an order one harmonic perturbation with quantum number m pert on the azimuthal direction. We retain reflection symmetry along the plane of the disk and align the disk axis with the spin of the SMBH for simplicity. The disk mass density profile in spherical coordinates is then given by
$$\rho (r,\theta,\phi )={\rho }_{r}(r)\exp (-{(r\cos \theta )}^{2}/{z}_{{{{{{{{\rm{disk}}}}}}}}}^{2})(1+\cos ({m}_{{{{{{{{\rm{pert}}}}}}}}}\phi ))$$
(13)
where ρ r ( r ) is the mass density profile on the radial direction, and z disk is the disk height, which we take from the SS profile 46 . The transition amplitude in Eq. ( 11 ) is thus given by
$$\left\langle {{{\Psi }}}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}\rightV\left{{{\Psi }}}_{nlm}\right\rangle= -\frac{\alpha }{{M}_{{{{{{{{\rm{BH}}}}}}}}}}\mathop{\sum}\limits_{{l}_{{{{{{{{\rm{disk}}}}}}}}}\ge 2}\mathop{\sum}\limits_{-{l}_{{{{{{{{\rm{disk}}}}}}}}}\le {m}_{{{{{{{{\rm{disk}}}}}}}}}\le {l}_{{{{{{{{\rm{disk}}}}}}}}}}\frac{4\pi }{2{l}_{{{{{{{{\rm{disk}}}}}}}}}+1}{I}_{{{\Omega }}}({l}_{{{{{{{{\rm{disk}}}}}}}}},{m}_{{{{{{{{\rm{disk}}}}}}}}},{n}^{\prime},{l}^{\prime},{m}^{\prime},n,l,m)\\ \int drd{r}^{\prime}\,\,{\left(r{r}^{\prime}\right)}^{2}\frac{\min {({r}^{\prime},r)}^{{l}_{{{{{{{{\rm{disk}}}}}}}}}}}{\max {({r}^{\prime},r)}^{{l}_{{{{{{{{\rm{disk}}}}}}}}}+1}}{R}_{{n}^{\prime},{l}^{\prime},{m}^{\prime}}^{}(r){R}_{n,l,m}(r)\,{\rho }_{r}({r}^{\prime})\\ \int d\theta d\phi \sin \theta \exp (-{({r}^{\prime}\cos \theta )}^{2}/{z}_{{{{{{{{\rm{disk}}}}}}}}}^{2})(1+\cos ({m}_{{{{{{{{\rm{pert}}}}}}}}}\phi )){Y}_{{l}_{{{{{{{{\rm{disk}}}}}}}}}}^{{m}_{{{{{{{{\rm{disk}}}}}}}}}\,}(\theta,\; \phi )$$
(14)
where R n , l , m denote the hydrogen atom wave functions and I Ω is an angular integral,
$${I}_{{{\Omega }}}=\int d\phi d\theta \sin \theta {Y}_{{l}^{\prime}}^{{m}^{\prime}}(\theta,\phi ){Y}_{l}^{m}(\theta,\phi ){Y}_{{l}_{{{{{{{{\rm{disk}}}}}}}}}}^{{m}_{{{{{{{{\rm{disk}}}}}}}}}}(\theta,\phi )$$
(15)
Due to the reflection symmetry on the plane of the disk and spherical harmonic orthogonality, only terms with even l disk ?+? m disk , and with m disk ?=?±? m pert contribute to the sum in Eq. ( 14 ) in our simplified estimate. Using Eq. ( 14 ), the superradiant and decaying rates from ref. 25 , and the energy levels in Eq. ( 12 ), we calculate the estimator in Eq. ( 11 ). We show the results in Fig.? 6 for m pert ?=?2. We have checked that our conclusions below do not change if one considers perturbations with other integer numbers of m pert . In Fig.? 6 , we only show the mixings with the decaying levels that lead to the largest perturbation estimator. Our results indicate that the dominant scalar cloud level, \(\left211\right. \rangle\) , is robust against perturbations for α ? ?0.1. The level with the second-largest superradiant growth rate, \(\left322\right. \rangle\) , is robust against perturbations for α ? ?0.2. These conditions on α are satisfied for the ULB masses that can be constrained using TDE measurements (c.f. Fig.? 5 ). Importantly, the perturbation estimator ( χ ) for the most dangerous mixings scales as a high power of α , namely χ ∝ α n with n ? ?5. This indicates that the range of α for which clouds are unstable is rather insensitive to further rescaling of the disk inhomogeneities.
Fig. 6: Evaluation of the stability of the superradiant cloud in the presence of an accretion disk. Colored lines: perturbation estimator χ , defined in Eq. ( 11 ), for the mixings between different superradiant and decaying levels induced by a thin accretion disk with an m pert ?=?2 harmonic perturbation in its density profile, as specified by Eq. ( 13 ). Red and blue lines show the mixing of the \(\left211\right. \rangle\) and \(\left322\right. \rangle\) clouds with selected decaying levels. Colored regions: values of the gravitational parameter α for which the perturbation estimator χ ?>?1 for the aforementioned transitions, in which case the corresponding clouds are unstable against the disk perturbations.
Full size image
Stellar halo .
We now consider the effect of the stellar halo surrounding the black hole. The density profile of stars near a SMBH is expected to have a radial dependence? ∝ r ?7/2 49 , i.e., close to isothermal. To simplify the treatment, we take the radial profile to be isothermal. While the stellar halo is expected to be approximately spherical, and is thus unable to induce cloud transitions, non-spherical perturbations can be expected from the Poissonian nature of stars 50 , and from non-relaxed stellar components. To study cloud transitions, we include an order one harmonic perturbation on top of the spherical profile, so the halo density is taken to be
$$\rho=\frac{{\sigma }^{2}}{2\pi G{r}^{2}}(1+\cos ({m}_{{{{{{{{\rm{pert}}}}}}}}}\phi )).$$
(16)
We follow the same procedure as for the accretion disk to calculate the estimator in Eq. ( 11 ). The results as a function of α are presented in Fig.? 7 for m pert ?=?2. As for the accretion disk perturbations, we find that for α ? ?0.1 the cloud is unlikely to be disrupted.
Fig. 7: Evaluation of the stability of the superradiant cloud in the presence of a stellar halo. Colored lines: perturbation estimator χ , defined in Eq. ( 11 ), for the dominant mixings between different superradiant and decaying levels, induced by a stellar halo with a m pert ?=?2 harmonic perturbation in its density profile as specified by Eq. ( 16 ). Red and blue lines correspond to mixings of the \(\left211\right. \rangle\) and \(\left322\right. \rangle\) levels with decaying levels, respectively. Colored regions: values of the gravitational parameter α for which the perturbation estimator is χ ?>?1 for the shown mixings, in which case the corresponding clouds are unstable against the stellar halo perturbations.
Full size image
Perturbation from stars prior to tidal disruption .
Stars that have orbited SMBHs during the SMBH’s lifetime and that have been tidally disrupted in the past can themselves be transient perturbations on the superradiant cloud before getting tidally disrupted (these TDEs are not necessarily the ones currently observed at the Vera Rubin’s LSST). These perturbations can be studied using time-dependent perturbation theory. Time-dependent perturbations can force transitions of the cloud into decaying modes, which, after being reabsorbed by the cloud, can lead to further spin-down. The time-dependent transition coefficients between states with quantum numbers n l m and \({n}^{\prime},{l}^{\prime},{m}^{\prime}\) is
$${c}_{nlm\to {n}^{\prime}{l}^{\prime}{m}^{\prime}}=-i\int\nolimits_{{t}_{i}}^{{t}_{f}}dt\exp (i{{\Delta }}Et)\left. \langle {{{\Psi }}}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}\rightV(t)\left{{{\Psi }}}_{nlm}\right. \rangle,$$
(17)
where t i , f are the initial and final times of the perturbation, Δ E is the energy splitting between the levels, and V ( t ) the time-dependent perturbation. For a star orbiting the BH in a generic trajectory, the transition matrix is given by
$$\left. \langle {{{\Psi }}}_{{n}^{\prime}{l}^{\prime}{m}^{\prime}}\rightV(t)\left{{{\Psi }}}_{nlm}\right. \rangle= -\alpha q\mathop{\sum}\limits_{{l}_{\star }\ge 2}\mathop{\sum}\limits_{-{l}_{\star }\le m\le {l}_{\star }}\frac{4\pi }{2{l}_{\star }+1}{Y}_{{l}_{\star }}^{{m}_{\star }\,}(\theta (t),\phi (t)){I}_{{{\Omega }}}({l}_{\star },{m}_{\star },{n}^{\prime},{l}^{\prime},{m}^{\prime},n,l,m)\\ \int dr\,\,{r}^{2}\frac{\min {({r}_{\star }(t),r)}^{{l}_{\star }}}{\max {({r}_{\star }(t),r)}^{{l}_{\star }+1}}{R}_{{n}^{\prime},{l}^{\prime},{m}^{\prime}}^{}(r){R}_{n,l,m}(r),$$
(18)
where r ? ( t ), θ ( t ), ? ( t ) specify the position of the star in spherical coordinates, and q is the star to BH mass ratio
$$q\equiv \frac{{M}_{\star }}{{M}_{{{{{{{{\rm{BH}}}}}}}}}}.$$
(19)
To simplify our estimate of the transition matrix, we limit ourselves to stars on equatorial orbits, and to focus on stars that are on tidal disruption trajectories, we set the orbit’s eccentricity e to be large 34 . For concreteness, we take 1??? e ?=?5?×?10 ?6 . We set the pericenter of the orbit at the BH horizon to constrain ourselves to orbits comfortably within the tidal radius. We take the mass ratio to be q ?=?10 ?8 . We then evaluate the coefficients in Eq. ( 18 ) numerically for two transitions involving the dominant superradiant level \(\left211\right. \rangle\) : hyperfine \(\left211\right. \rangle \to \left21-1\right. \rangle\) and Bohr \(\left211\right. \rangle \to \left31-1\right. \rangle\) transitions. We find transition probabilities ∣ c 211→21?1 ∣ 2 ?~?10 ?15 and ∣ c 211→31?1 ∣ 2 ?~?10 ?19 , approximately independently of the gravitational parameter α . Such small transition probabilities are mostly due to the small mass ratio between the star and the black hole. We conclude that level transitions due to TDE trajectories are highly unlikely.
Stars on EMRI trajectories .
We now consider stars that are on inspiral orbits. EMRIs correspond to orbits that start far from the BH and slowly approach it by emission of GW’s. During the inspiral, these orbits scan a range of orbital frequencies, starting from low frequencies at large semi-major axis, to higher frequencies as they approach the BH. During this scanning phase, the orbital frequency can match the energy splittings between different cloud levels, and induce resonant transitions. These trajectories cannot be treated using standard time-dependent perturbation theory, which does not accurately capture the potentially large effects of resonances. Instead, they must be treated using the Landau-Zener (LZ) formalism 45 .
In what follows, we consider circular EMRIs for simplicity. A circular EMRI leads to resonant transitions between cloud levels with energy differences Δ E when its angular velocity is
$${{{\Omega }}}_{{{{{{{{\rm{res}}}}}}}}}=\frac{{{\Delta }}E}{{{\Delta }}m},$$
(20)
where Δ m is the change in the magnetic quantum number of the cloud levels. From Kepler’s law, the angular velocity of the EMRI is set by the semi-major axis r a ,
$${{\Omega }}=\frac{1}{{r}_{g}}{\left[\frac{{r}_{g}}{{r}_{a}}\right]}^{3/2}.$$
(21)
As mentioned above, for a resonance to happen, Eq. ( 20 ) needs to fall within the scanned range of frequencies. In particular, the minimal angular velocity of the EMRI needs to be smaller than the resonant frequency,
$${{{\Omega }}}_{\min } \; \le \; {{{\Omega }}}_{{{{{{{{\rm{res}}}}}}}}}.$$
(22)
For the supermassive black hole masses of interest to us, M BH ?~?10 8 M ⊙ , EMRI trajectories start at a semi-major axis r a / r g ?~?10 4? 51 , so we set \({{{\Omega }}}_{\min }=3\times 1{0}^{-9}\) ?Hz. Given that the cloud energy splittings that set \({{{\Omega }}}_{{{{{{{{\rm{res}}}}}}}}}\) increase with the gravitational coupling α (c.f., Eq. ( 12 )), the condition in Eq. ( 22 ) can be translated to a condition on the minimal value of α for a given resonance to happen. As examples, for the aforementioned value of \({{{\Omega }}}_{\min }\) , the conditions for three specific transitions are
$$\left211\right. \rangle \to \left21-1\right. \rangle \qquad \alpha \; \gtrsim \; 0.2\\ \left211\right. \rangle \to \left32-2\right. \rangle \kern1.4pc \alpha \; \gtrsim \; 0.03\\ \left211\right. \rangle \to \left43-3\right. \rangle \kern1.4pc \alpha \; \gtrsim \; 0.03.$$
(23)
After the trajectory has passed through a resonance, a fraction of the ULBs will transit from the initial state to the resonantly excited state. The fraction of the ULBs that remains in the initial state is given by
$${\left{c}_{sr}\right}^{2}=\exp (-2\pi z),$$
(24)
where z is the LZ parameter. When the EMRI perturbs the cloud strongly, z ? ?1 and the cloud transits entirely into the resonantly excited level. More precisely, the LZ parameter is proportional to the strength of the perturbation η 2 , and inversely proportional to the rate of detuning γ ,
$$z\equiv \frac{{\eta }^{2}}{\gamma }.$$
(25)
The rate of detuning γ is defined by the time evolution of the inspiral frequency, Ω( t )?=?Ω γ t . For a circular orbit, the rate of detuning from GW emission is 45
$$\gamma=\frac{96}{5}\frac{q}{{(1+q)}^{1/3}}{({M}_{{{{{{{{\rm{BH}}}}}}}}}{{\Omega }})}^{5/3}{{{\Omega }}}^{2},$$
(26)
where q is the ratio of the stellar to BH mass defined in Eq. ( 19 ). The strength of the perturbation η , on the other hand, depends on the stellar trajectory. In what follows, we limit ourselves to orbits on the equatorial plane. In this case, the strength of the inspiral perturbation is 45
$$\eta= -\alpha q\mathop{\sum}\limits_{{l}_{\star }\ge 2}\mathop{\sum}\limits_{-{l}_{\star }\le m\le {l}_{\star }}\frac{4\pi }{2{l}_{\star }+1}{Y}_{{l}_{\star }}^{{m}_{\star }}(\pi /2,0){I}_{{{\Omega }}}({l}_{\star },{m}_{\star },{n}^{\prime},{l}^{\prime},{m}^{\prime},n,l,m)\\ \int dr\,\,{r}^{2}\frac{\min {({r}_{\star }(t),r)}^{{l}_{\star }}}{\max {({r}_{\star }(t),r)}^{{l}_{\star }+1}}{R}_{{n}^{\prime},{l}^{\prime},{m}^{\prime}}^{}(r){R}_{n,l,m}(r),$$
(27)
where R n l m are the hydrogenic radial wave functions, l ? , m ? are the quantum numbers associated with the spherical harmonic decomposition of the star’s perturbing potential, and r ? is the time-dependent radius of the circular stellar orbit. To estimate the size of the inspiral perturbation, we take the orbit radius r ? to be equal to the on-resonance radius, which by Kepler’s law is given by
$${r}_{\star }={r}_{g}{({{{\Omega }}}_{{{{{{{{\rm{res}}}}}}}}}{r}_{g})}^{-2/3}.$$
(28)
We show the probability of transition between the superradiant n l m ?=?211 level to three selected decaying levels in Fig.? 8 . In the figure, we also show in dotted lines the minimum value of α required for the EMRI to pass through the resonance, according to Eq. ( 23 ). From the plot, we note that all the transitions are suppressed at large α , mostly due to the sharp and monotonically increasing dependence of the rate of detuning \(\gamma \sim {{{\Omega }}}_{{{{{{{{\rm{res}}}}}}}}}^{11/3}\) with α . This leads to strongly suppressed transitions to the \(\left21-1\right. \rangle\) level, 1??? ∣ c s r ∣ 2 ? ?10 ?4 , since these transitions require large values of α ? ?0.2 for the EMRI to pass through the resonance. For transitions into the decaying \(\left32-2\right. \rangle\) and \(\left43-3\right. \rangle\) levels, only α ? ?0.03 is required. For α ?≈?0.03 the transition probability is large and of \({{{{{{{\mathcal{O}}}}}}}}(10\%)\) . However, the TDE signals discussed in this work reside in the region α ? ?0.1, for which the probability of transition is small, \({{{{{{{\mathcal{O}}}}}}}}(1{0}^{-3})\) . Thus, these transitions are also not of concern for our purposes.
Fig. 8: Landau-Zener transitions. Colored lines: probability of resonant Landau-Zener transitions induced by EMRIs from the superradiant level \(\left211\right. \rangle\) to selected decaying levels. Transition probabilities to \(\left21-1\right. \rangle\) , \(\left32-2\right. \rangle\) , and \(\left43-3\right. \rangle\) are shown in blue, red, and gray, respectively. Dashed-colored vertical lines: minimum value of the gravitational coupling required for the EMRI to scan the corresponding resonant transition frequency, according to Eq. ( 23 ) (red and gray-dashed lines overlap).
Full size image
For completeness, we briefly discuss the case where α is small enough for the transitions into \(\left32-2\right. \rangle\) and \(\left43-3\right. \rangle\) to happen with \({{{{{{{\mathcal{O}}}}}}}}(10\%)\) probabilities. To study this case, we must include the cloud’s backreaction on the orbit that up to now has been neglected, given that it can significantly suppress the resonant transitions 45 . A simple estimate of the backreaction effects can be obtained by comparing the angular momentum of the star with the angular momentum required for the cloud to transition into a decaying level. The angular momentum of a star in a circular resonant trajectory is of order
$${L}_{c}= {M}_{\star }\sqrt{G{M}_{{{{{{{{\rm{BH}}}}}}}}}{r}_{\star }}\\= 1{0}^{85}{\left[\frac{{M}_{{{{{{{{\rm{BH}}}}}}}}}}{1{0}^{8}{M}_{\odot }}\right]}^{1/2}\left[\frac{{M}_{\star }}{{M}_{\odot }}\right]{\left[\frac{{r}_{\star }}{{{{{{{{\rm{mpc}}}}}}}}}\right]}^{1/2},$$
(29)
while the angular momentum of a fully grown cloud that has extracted spin Δ a from the SMBH is
$${L}_{{{{{{{{\rm{cloud}}}}}}}}}= {G{M}_{{{{{{{{\rm{BH}}}}}}}}}}^{2}{{\Delta }}a\\= 1{0}^{91}{\left[\frac{{M}_{{{{{{{{\rm{BH}}}}}}}}}}{1{0}^{8}{M}_{\odot }}\right]}^{2}\left[\frac{{{\Delta }}a}{0.1}\right].$$
(30)
The angular momentum required to induce a transition into a decaying mode is of order L cloud , which is roughly six orders of magnitude larger than the angular momentum of a typical EMRI. As a result, the inspiral orbit can be significantly affected by the cloud and could lead to trajectories of the “sinking” or “floating” types discussed in refs. 45 , 52 , in which case resonant transitions in the cloud are likely suppressed. Studying these orbits in detail is beyond the scope of this work, but we point out that for 10 5 ? M BH ? ?10 7 M ⊙ these effects could have a severe impact on the EMRI rates at LISA, given that the cloud’s backreaction on the stellar orbit has the potential to dramatically affect the orbits of stars on EMRI trajectories. .
From:
系统抽取对象
机构     
(5)
(1)
系统抽取主题     
(1)  
(1)  
(1)  
(1)  
(1)  
(1)  
(1)  
(1)  
(1)  
(1)  
(1)  
(1)