Modifications to the signal from a gravitational wave event due to a surrounding shell of matter
NNoname manuscript No. (will be inserted by the editor)
Modifications to the signal from a gravitational wave event due toa surrounding shell of matter
Monos Naidoo · Nigel T. Bishop ∗ · Petrus J. vander Walt
Received: date / Accepted: date
Abstract
In previous work, we established theoretical results concerning the effect of matter shellssurrounding a gravitational wave (GW) source, and we now apply these results to astrophysicalscenarios. Firstly, it is shown that GW echoes that are claimed to be present in LIGO data ofcertain events, could not have been caused by a matter shell. However, it is also shown that thereare scenarios in which matter shells could make modifications of order a few percent to a GW signal;these scenarios include binary black hole mergers, binary neutron star mergers, and core collapsesupernovae.
Keywords
Gravitational waves · Gravitational wave echoes · Bondi-Sachs · Matter shell · Linearized perturbation theory
In previous work [1], we developed a model for the effect of a matter shell around a gravitationalwave (GW) source, obtaining an analytic expression for the modifications to the GWs. In this paper,we apply this model to astrophysical GW sources. The modifications found in [1] included an echo.The idea of a GW echo has received much attention as being a signature of an exotic compactobject(ECO), and it has also been investigated in terms of the astrophysical environment.At the quantum level, Hawking’s information paradox suggests Planck-scale modifications ofblack hole horizons (firewalls [2]) and other modifications to black hole structure (fuzzballs [3]).Dark matter particles have been suggested surrounding star-like objects [4]. Other postulates include
Monos NaidooDepartment of Mathematics, Rhodes University, Grahamstown, 6140, South AfricaE-mail: [email protected] T. Bishop ∗ Department of Mathematics, Rhodes University, Grahamstown, 6140, South AfricaE-mail: [email protected] J. van der WaltDepartment of Mathematics, Rhodes University, Grahamstown, 6140, South AfricaE-mail: [email protected] a r X i v : . [ g r- q c ] J a n Monos Naidoo et al. stars with interiors consisting of self-repulsive, de Sitter spacetime, surrounded by a shell of ordinarymatter (gravastars [5]). Then there are Boson stars, which are macroscopic objects made up of scalarfields [6]. All these objects are compact bodies mimicking black holes, but without a horizon. Oneconsequence of these horizonless structures is that ingoing gravitational waves produced in a mergermay reflect multiple times off effective radial potential barriers. The gravitational waves may be,in effect, trapped between effective radial potential barriers causing the waves to be ‘bounced’ offthese barriers several times with wave packets leaking out to infinity at regular times. These GWsignals, ‘trailing’ the main (outward bound) signal are referred to as GW echoes. [7,8,9]. Muchfurther discussion around GW echoes has been within the context of ‘new physics’ [10,11,12,13,14,15,16].Echoes from a massive, thick shell were considered in [17]. This was a generalisation of theapproach of [18] which was based on an infinitely thin shell. As we have shown in our paper [1]the case for a thin shell can be generalised to that of a thick shell by considering several concentricthin shells and integrating. [17] showed that the deviation from Schwarzschild ringdown in theirastrophysical estimations were relatively small except for a large mass which indicated that forthe majority of astrophysical scenarios the effect would be relatively small. However they did notethat considerations of dark matter around black holes (or compact body mergers) would leavesome parametric freedom for echoes as well. [19], studied both the combination of contributionsof modifications of the Schwarzschild geometry near the surface, and a nonthin shell of mattersurrounding the compact body/merger. They found that a massive shell at a distance could bedistinguished from the purely Schwarzschild evolution of perturbations. However, for the situationof new physics near the surface of a compact object, (a wormhole in their case), the strong echoes ofthe surface dominate the echoes of the distant shell. Furthermore, they found that it would take anextraordinarily large mass, located sufficiently close to the wormhole, to lead to discernable changesin the main echoes of the surface and that these changes would be relatively small. The interactionof GWs with matter has also been studied in cosmology [20,21], with the objective of using GWobservations to constrain the properties of dark matter.It has been suggested that GW echoes have already been observed in the LIGO data of thebinary black hole merger GW150914 [22,23], and also from the Binary Neutron Star (BNS) mergerGW170917 [24]. These claims have been contested [25,26], sparking a debate and responses indefense of the claims [27,28] with further substantiations [24,29].The plan of this paper is as follows. In previous work [1], we showed that a thin spherical dustshell surrounding a GW source, causes the GW to be modified both in magnitude and phase, butwithout any energy being transferred to or from the dust. That work suggests the possibilty of GWechoes. In Sec. 2 we describe the problem considered in this scenario, the assumptions made andthe key results.The solution of [1] is for a monchromatic GW source. A general waveform may be decomposedinto a sum of Fourier components, and the technical details are given in Sec. 3. The decompositionis implemented within a Matlab script using the Fast Fourier Transform, and validation results arereported in this section.Secs. 4 and 5 are about astrophysics. First in Sec. 4, we investigate whether a matter shell couldexplain the echoes that may exist in the LIGO data of GW150914 and GW170817. It is found thatthe shell would need to have such a large mass that it would constitute a black hole. It follows thatif GW echoes are confirmed in another GW event, and with a relative magnitude and delay timesimilar to that of GW150914 and GW170817, then a matter shell would not be a viable explanation,so strengthening the case that an ECO would have been observed. odifications to the signal from a gravitational wave event due to a surrounding shell of matter 3
It will be shown that a key factor in determining the echo properties of a matter shell is the echodelay time, which in the cases above was of order 1s. If the echo delay time is much smaller, order1ms or smaller, then shell properties that are physically acceptable could lead to measurable effects;however, the short delay time would mean that the effect would not appear as an echo in the usualsense, but rather as a modification to the original signal. Examples of such signal modificationsare given in Sec. 5.1 for a matter shell around an event like GW150914; in Sec. 5.2 for black holequasinormal mode signal following a binary neutron star merger; and in Sec. 5.3 for the case of corecollapse supernovae (CCSNe).Our conclusions are discussed in Sec 6. We also provide, in Appendix A, a summary of theMatlab scripts used; these scripts are available as Supplementary Material.
In [1] we considered the scenario of a thin shell of matter surrounding a gravitational wave sourcesuch as a compact binary merger, as shown schematically in Fig. 1. The spacetime around the GWsource in [1] would be otherwise empty except for the surrounding shell of matter. Confining theinvestigation to a thin shell does not preclude the case of a thick shell. Results can easily be appliedto a series of concentric thin shells and then integrated to give the effect for a thick matter shell. TheEOS is taken, as a start, to be that of dust. The results show that the shell modifies the outgoingGWs in both phase and magnitude without contradicting previous results about energy transfer.The problem is set up within the Bondi-Sachs formalism for the Einstein equations with coordinatesbased on outgoing null hypersurfaces [30,31]. The null hypersurfaces, are labelled by the coordinate x = u , the angular coordinates by x A ( A = 2 ,
3) and the surface area radial coordinate by x = r .The angular coordinates (e.g. spherical polars ( θ, φ )) label the null ray generators of a hypersurface u = constant. The Bondi-Sachs metric then describes a general spacetime, which may be writtenas ds = − (cid:18) e β (cid:18) Wr (cid:19) − r h AB U A U B (cid:19) du − e β dudr − r h AB U B dudx A + r h AB dx A dx B , (1)where, h AB h BC = δ AC . The condition that r is a surface area coordinate implies det( h AB ) =det( q AB ), where, q AB is a unit sphere metric (e.g. dθ + sin θdφ ).The GW strain far from the source is written H M = r ( h + + ih × ), where h + , ih × are the usualpolarization modes in the TT gauge. Now suppose that, in the absence of the matter shell, H M = (cid:60) ( H M exp(2 iπf u )) Z , , (2)where f is the frequency (assumed to be monochromatic) of the GWs; H M is a constant determinedby the physics of the GW source; and Z , is a spin-weighted spherical harmonic related to theusual s Y (cid:96),m as specified in [32,33]. Then, as found in [1], the introduction of a spherical shell aroundthe GW source of mass M S , radius r and thickness ∆ modifies the wave strain to: H = (cid:60) (cid:18) H M (cid:18) M S r + iM S πr f + iM S e − iπr f πr f + O (cid:18) M S ∆r , M S r f (cid:19)(cid:19) exp(2 iπf u ) (cid:19) Z , . (3)Each of the terms containing M S in Eq. (3) represents a correction to the wave strain in the absenceof the shell [1]. The first correction, 2 M S /r , is part of the gravitational red-shift effect, the main Monos Naidoo et al. r Source Δ Fig. 1
The problem of a GW source which is surrounded by a spherical shell of mass M S located between r = r and r = r + ∆ , where r is the distance from the source. consequence of which is a reduction in the frequency; this effect is well-known, and henceforth wewill assume that GW waveforms to be considered have allowed for this effect. The second term, iM S / ( πr f ), is out of phase with the leading terms 1 + 2 M S /r and hence represents a phase shiftof the GW. This term, to O ( M S ), does not change the magnitude of H and hence has no effecton the energy of the GW. The presence of e − πir f in the third term describes a change in themagnitude of H , as verified in [1]. In this context, the modified signal would then be interpretedas an echo of the main signal. The echo varies from the main signal in both magnitude and phase,with the magnitude of the echo described by R = M S πr f (4)relative to the original signal. The time-delay of the echo is 2 r , but the echo’s magnitude depends on the wave frequency f .The GW sources reported to date are not monochromatic but are burst-like. Such a source maybe decomposed into its Fourier components and Eq. (3) applied to each component, and the echosignal obtained by summing over the transformed components. Because the magnitude of the trans-formation is frequency-dependent, the echo signal will have a form more complicated than simplya time-delay and magnitude change to the original signal. This effect is now analyzed.We replace (cid:60) ( H M exp(2 iπf u )) in Eq. (2) by h ( u ) defined in the interval u ≤ u ≤ u N − ; andthen construct a discrete representation of h ( u ), h k = h ( u k ) ( k = 0 , · · · , N − u k ona regular grid, i.e. u k +1 − u k = δ for k = 0 , · · · , N −
2. Note that if the highest frequency that odifications to the signal from a gravitational wave event due to a surrounding shell of matter 5 needs to be resolved is f m , then N should be chosen so that ( u N − − u ) / ( N − < / (2 f m ), i.e.to satisfy the Nyquist condition. The discrete Fourier transform [34] of { h k } is H n = N − (cid:88) k =0 h k exp (cid:18) πiknN (cid:19) with inverse h k = 1 N N − (cid:88) n =0 H n exp (cid:18) − πiknN (cid:19) . (5)Then defining H ,n and H ,n to be coefficients in the transform domain of the second (phase-shift)and third (echo) terms of Eq. (3), we have H ,n = iM S H n πr f n , H ,n = iM S H n exp( − πir f n )4 πr f n , n = 1 , · · · , N ,H ,n = H ∗ ,N − n , H ,n = H ∗ ,N − n , n = N , · · · , N − ,H , = H , = 0 , (6)where ∗ denotes the complex conjugate, and where we have used the condition that, in the timedomain, all quantities are real. It is being assumed that N is even, and normally N = 2 m (with m an integer) for convenience when using the fast Fourier transform; further f n = n ( N − N ( u N − − u ) . (7)Then h ,k , h ,k are found on applying the inverse discrete Fourier transformation.A Matlab script that implements the calculation of the previous paragraph is available as Sup-plementary Material, and is described in Appendix A. The script was checked by applying it to amonochromatic signal h ( u ) = (cid:60) ( − i exp(2 πif u )). The errors e ,k in h ,k and e ,k in h ,k are e ,k = (cid:12)(cid:12) h ,k − (cid:60) (cid:0) iM S / ( πr f ) exp(2 iπf u ) (cid:1)(cid:12)(cid:12) e ,k = (cid:12)(cid:12) h ,k − (cid:60) (cid:0) iM S / (4 πr f ) e − πir f exp(2 iπf u ) (cid:1)(cid:12)(cid:12) . (8)For the case u = 0ms, u N − = 100ms, f = 1kHz, r = 1 . ≈ km), and M S = 0 . ≈ M (cid:12) ),we found: N || e ,k || || e ,k || . × − . × − . × − . × − . × − . × − (9)where || e k || is defined to be || e k || = (cid:115) (cid:80) N − k =0 e k N . (10)Thus the errors are tending to zero, and N can be chosen so as to attain a desired accuracy.Note that an error of order machine precision is achieved for special values of the frequency f = k ( N − /N/ ( u N − − u ) with k an integer; in this case we would have that the cyclic assumptionof the discrete Fourier transform would be satisfied, i.e. u i = u i + N . Monos Naidoo et al.
In [22], the first of the tentative search for echoes, the authors find evidence for the existence ofechoes in the first detection event [35]
GW150914 . They find further comparable evidence forechoes from the events GW151012 [36] (then referred to as LVT151012) and GW151226 [37]. Thereferences report a number of echo events for GW150914, with the first occurring at about 0.3s aftermerger; therefore, if caused by a matter shell, the radius would be about 45,000km. The magnitudeof the echo was about 0.0992 times that of the original signal. Using 132Hz for the frequency, whichis its value when the amplitude was at its maximum at the end of the merger phase, and applyingEq. (4) gives M S (cid:39) (cid:12) . Such a mass within a radius of 45,000km would constitute a blackhole, so the scenario of an echo caused by a shell can be discounted for GW150914.Extending their investigations to the first BNS detection GW170817 [38], the authors of [22]find evidence again of the existence of echoes in the postmerger event [24]. The echo was reportedto occur at frequency f echo (cid:39)
72 Hz, approximately 1.0 sec after the BNS merger event. The inspiralsignal is at 72Hz about 4.0s before merger, so if the reported echo is caused by a matter shell itmust have a radius of about 2 . (cid:39) . × that of the original signal and applying Eq. (4), it follows that for the echoto be caused by a shell it would have a mass of approximately 10 M (cid:12) . Now, a mass of 10 M (cid:12) inside a radius of 750,000km would constitute a black hole, so the scenario of an echo caused by ashell can be discounted for GW170817.The above two examples illustrate the general difficulty of producing a GW echo by means of amatter shell. An echo, in the usual sense, is a repeat of the original signal after a short time delay,which in practice must be at least hundreds of ms, corresponding to a shell radius of at least ∼ M S = 4 πRr f which, for expected values ofthe frequency f , will be large – either implying a black hole and thus not feasible, or requiring anunexpected astrohysical scenario. A much smaller shell radius would avoid these difficulties, butthe effect of the shell would be seen as a modification of the original signal, rather than as an echo.Some possible scenarios are presented in the next section. M (cid:12) ) and the signal of GW150914 [39,40,41] (Of course, the astrophysical evidence doesnot suggest the existence of such a shell). The results are shown in Fig. 2. The top panel showsthat there is a small but noticeable modification to the template signal, particularly at early times;this is because the frequency is lower at early times and so the modification effects are larger. Thebottom panel shows the contributions of the phase-shift term ( h , blue) and the echo term ( h ,red); it is noticeable that, unlike the template signal, these terms decrease in magnitude as thefrequency increases with time.The accuracy of the results presented in Fig. 2 is limited since the formalism used in [1] assumeda weak field GW source, which is not the case for two black holes at merger. In particular, GWs odifications to the signal from a gravitational wave event due to a surrounding shell of matter 7 Fig. 2
The effect of a matter shell of radius 3ms (about 900km) and mass 0.3ms (about 60 M (cid:12) ) on the signal ofGW150914. The top panel shows the original signal in blue, and the original signal plus modifications due to theshell in red. The lower panel shows the modifications due to the phase-shift term in blue, and due to the echo termin red. reflected by the shell would be partially absorbed by the black holes, so reducing the magnitude ofthe echo contribution to the GW signal.5.2 Binary Neutron Star (BNS) mergersBNS GW events that have been observed include GW170817 [42] and GW190425 [43]. Of these,GW170817 was at a higher signal to noise ratio and the event was observed post-merger in theelectromagnetic spectrum [44], indicating that the post-merger object contained a large amount offree matter; we will therefore focus on this event.The relevant source parameters reported for the event are [45,46]: total mass M + M = 2 . M (cid:12) ,and radii R = 10 . R = 10 . M (cid:12) black hole, which is outside the sensitivity band of the LIGO detectors. Monos Naidoo et al.
Fig. 3
The effect of a matter shell of radius 25km and mass 0.7 M (cid:12) on a quasinormal mode (QNM) signal of a 2 M (cid:12) remnant of a binary neutron star merger. The original signal is in blue, and the original signal plus modificationsdue to the shell is in red. In order to estimate the possible effect of matter on GWws emitted from a central remnant, weconsider the model of a spherical shell of mass M S = 0 . M (cid:12) and radius r = 25km around a GWsource at either 6kHz or 2kHz. We find that the phase-shift term 2 iM S / ( r ν ) evaluates to iM S r πf = 0 . i or 0 . i , (11)for 6kHz or 2kHz, respectively. The echo effect would be 1 / . – The model in [1] assumed that the shell is static, but the aftermath of a BNS meger will behighly dynamical. – The hypothesis of a shell forming is not supported by a detailed numerical simulation; indeed,since the system started as an inspiral, the matter outside the remnant should have a ring-likestructure. Thus, Eq. (11) may overestimate the matter effect for an observer on the axis ofrotation of the system, but may be appropriate for an observer in the equatorial plane. – The comment at the end of Sec. 5.1 about absorption of GWs by a black hole applies here.So the quantitative values in Eq. (11) should be interpreted as indicative of the order of magnitudeof the interaction of GWs with matter, rather than as precise estimates. It should also be notedthat if the numerical modeling includes all the matter, and if the simulation run period includesthe quasinormal mode ringdown, then shell effects would already be included in the simulation. odifications to the signal from a gravitational wave event due to a surrounding shell of matter 9 M (cid:12) , evolution normally proceeds through several stages of coreburning and then to core collapse once nuclear fusion halts when there are no further burningprocesses to balance the gravitational attraction. Typically, these cores are iron cores, with thecritical mass signalling the onset of core collapse ranging from 1 . M (cid:12) to 1 . M (cid:12) . The core breaksinto two during the collapse, with the inner core of 0 . M (cid:12) to 0 . M (cid:12) in sonic contact and collapsinghomologously and the outer core collapsing supersonically. The inner core reaches supranucleardensities of ∼ × g/cm where the nuclear matter stiffens, resulting in a bounce of the innercore. The resulting shock wave is launched into the collapsing outer core. However, the shock losesenergy to dissociation of iron nuclei, stalling at ∼
150 km within ∼
10 ms after formation. Manycomputationally demanding simulations exist [51,52,53] for generation of GWs from CCSNe.
Ref. [53] presents GW waveforms from simulations of CCSNe for various zero age main sequence(ZAMS) masses in the range 9 M (cid:12) , · · · , M (cid:12) . The GW signal starts with an initial burst of durationabout 50ms and frequency about 100Hz, followed by a quiescent period. We model this part of thewaveform as h + + ih × ∼ sin(0 . πu ) sin(0 . πu ) Z , , ≤ u ≤ h + + ih × = 0 , ≤ u ≤ . (12)GWs are generated by aspherical motions in the inner core, commencing just after the bounce. Theinner core is surrounded by the outer core, treated as a thick matter shell, and we now model itsmodifications to the GW signal. The shell has an inner radius r in , an outer radius r out , and densityat r = r in of ρ with density fall-off ρ ∝ r − − a with 1 / ≤ a ≤
2. The effect of the whole shell isobtained by decomposing it into thin shells and then integrating. The result for the echo term isnot a simple analytic expression and will be evaluated numerically. However, the phase shift termdoes give a simple analytic result (cid:90) r out r in iM s r πf dr = i (cid:90) r out r in πr ρ ( r in ) a r a r πf dr = 4 iρ r in af (cid:18) − (cid:18) r in r out (cid:19) a (cid:19) , (13)where f is the GW frequency. Fig. 4 shows the original signal given by Eq. (12) in blue, and theoriginal signal plus shell modifications in red, for the case r in = 0 . ≈ r out = 0 . Time in ms -1.5-1-0.500.511.5 G W a m p li t ude CCSNe: Initial burst (blue), Initial burst + shell effects (red)
Fig. 4
The effect of a matter shell as specified in the text, on the initial burst (Eq. (12)) in a CCSNe GW signal.The original signal is in blue, and the modified signal is shown in red. ( ≈ a = 1, and ρ = 0 . / ms ( ≈ . × g/cm ). These values model: the inner coreas a proto-neutron star (PNS) of radius 30km, and whose oscillations generate the GWs; the shockboundary as having a radius of 150km; and the density at the inner radius ( ≈ . × g/cm )at a couple of orders below the supranuclear density. For this model, Eq. (13) evaluates to 0 . i ,and the total mass of the shell to 0 . M (cid:12) .There is some uncertainty in the parameter values that should be used in modeling the mattershell around the inner core, and Eq. (13) shows how varying the parameters would change themagnitude of the shell effect. A numerical simulation of a CCSNe entails modeling gravity as wellas a number of other physical process, and requires extensive computational resources. It is notfeasible to model the whole star using general relativity (GR): approximations to GR are used, andGWs may be estimated using the quadrupole formula [53]. Thus corrections to the quadrupolarsignal due to shell effects are necessary. odifications to the signal from a gravitational wave event due to a surrounding shell of matter 11 There are astrophysical scenarios which can be regarded as comprising a shell of matter around aGW source, and this paper has investigated in what way the GW signal would be affected. Theinvestigation started with GW events for which echoes have been claimed to exist in the LIGO data,and it was found that such echoes could not be caused by a matter shell. Thus, an unambiguousobservation of GW echoes in the future would favour the existence of ECOs.We investigated the effect of matter shells in three specific example cases. The first was a binaryblack hole merger analogous to GW150914, surrounded by a hypothetical matter shell at radius900km and mass 60 M (cid:12) . Astrophysically, such a shell is highly unlikely to exist, but this case isuseful as it well illustrates some of the features of the shell effects on the waveform. The next caseconsidered was the quasinormal mode signal from the remnant of a binary neutron star mergerlike GW170917. In this case, it is known that there is a substantial amount of matter around theremnant, although the extent to which the shell model is appropriate is unclear. The final caseconsidered was that of a core collapse supernova. Although GWs from such events have not beenobserved, they are regarded as potential sources; and here it is clear that the proto neutron star,in which the GWs are generated, is surrounded by shells of matter. Of the three cases, the corecollapse supernova is that which yielded the largest shell modifications to the GWs, and for whichthe predictions are most reliable.The effects of matter shells are small but measurable if the signal to noise ratio is sufficientlyhigh. As GW observations become more accurate, through both hardware developments and, astime passes, the increasing chance of observing nearby events, these effects will need to be takeninto account. A Matlab scripts
The Matlab scripts NormTest.m (see Eq. (9)), BH.m used in Sec. 5.1, BNS.m used in Sec. 5.2, and CCSN.m used inSec. 5.3.1 are plain text files; the file clean.xlsx is a spreadsheet file containing input data for BH.m. All the files areavailable as online supplementary material.
Acknowledgements
This work was supported by the National Research Foundation, South Africa, under grantnumber 118519.
Conflict of interest
The authors declare that they have no conflict of interest.
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