Gravitational Waves from Orphan Memory
GGravitational Waves from Orphan Memory
Lucy O. McNeill, Eric Thrane, ∗ and Paul D. Lasky Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC 3800, Australia
Gravitational-wave memory manifests as a permanent distortion of an idealized gravitational-wave detector and arises generically from energetic astrophysical events. For example, binaryblack hole mergers are expected to emit memory bursts a little more than an order of magni-tude smaller in strain than the oscillatory parent waves. We introduce the concept of “orphanmemory”: gravitational-wave memory for which there is no detectable parent signal. In particular,high-frequency gravitational-wave bursts ( (cid:38) kHz) produce orphan memory in the LIGO/Virgo band.We show that Advanced LIGO measurements can place stringent limits on the existence of high-frequency gravitational waves, effectively increasing the LIGO bandwidth by orders of magnitude.We investigate the prospects for and implications of future searches for orphan memory.
The detection of gravitational waves by LIGO andVirgo [1] has opened up new possibilities for observinghighly-energetic phenomena in the Universe. It was re-cently shown that ensembles of binary black hole detec-tions can be used to measure gravitational-wave memory[2]: a general relativistic effect, manifest as a permanentdistortion of an idealized gravitational-wave detector [3–8]. It is not easy to detect memory. The memory strainis significantly smaller than the oscillatory strain; ∼ τ ∼
10 ms for GW150914 [2]. Forsufficiently short bursts (with timescales that are shortcompared to the inverse frequency of the detector’s sen-sitive band), the memory is well-approximated by a stepfunction, or equivalently an amplitude spectral densityproportional to 1 /f , where f is the frequency. It followsthat the memory of a high-frequency burst introducesa significant low-frequency component which extends tofrequencies arbitrarily below 1 /τ . If the parent burst isabove the detector’s observing band, this can lead to “or-phan memory”: a memory signal for which there is nodetectable parent.There are a number of mechanisms that can lead toorphan memory. In the example above, a high-frequencyburst outside the observing band creates in-band mem-ory. This is the premise of memory burst searches inpulsar timing arrays [9–13], which look for memory frommerging supermassive black holes for which the oscilla-tory signal is out of band. Orphan memory can also besourced by phenomena other than gravitational waves,e.g., neutrinos [14, 15], although the probability of de-tection from known sources is small. In principal, it ispossible for beamed gravitational-wave sources to pro-duce orphan memory signals when the oscillatory signalis beamed away from Earth. In practice, however, thenumber of orphan detections from beaming will be smallcompared to the number of oscillatory detections. Inthis Letter, we focus on memory where high-frequencygravitational-wave bursts produce orphan memory inLIGO/Virgo. Scaling relations.
As a starting point it is useful to in-vestigate scaling relations for gravitational-wave bursts.For a gravitational-wave source with timescale, τ , fre-quency f ≈ /τ , and energy E gw , the strain amplitudescales as h osc0 ∼ E / f / d , (1)where d is the distance to the source, and throughout weuse natural units, c = G = 1. A sine-Gaussian wave-form is well described by these assumptions, and so wework with sine-Gaussian waveforms in the analysis thatfollows. In Figure 1 we show two sine-Gaussian bursts(top panel) with their corresponding memory waveformsapproximated by tanh functions (bottom panel). time (arb. units) -1-0.500.51 time (arb. units) h (t)( a r b . un i t s ) FIG. 1: Strain time series for a gravitational-wave burst.The top panel shows both the burst (solid curves) and mem-ory (dashed curves) strains for two bursts of the same ampli-tude. The high-frequency burst (red) has frequency ten timesthe low-frequency burst (blue). The bottom panel shows anenlarged version of the memory time series’. As the frequencyof the burst increases, the rise time approaches zero and thememory is well-approximated by a step function. a r X i v : . [ a s t r o - ph . I M ] F e b f (Hz) -23 -22 -21 -20 -19 -18 -17 -16 h aLIGOArvanitakiHolometerGoryachev FIG. 2: The dashed black curve shows the sine-Gaussian am-plitude h , with frequency f , required for an average signal-to-noise-ratio of (cid:104) S/N (cid:105) = 5 in Advanced LIGO operating atdesign sensitivity. The solid black curve shows the same sen-sitivity to sine-Gaussian amplitude h except that we includememory in the matched filter calculation, effectively extend-ing the LIGO band to arbitrarily high frequencies. Around afew kHz, the memory becomes comparably important to theoscillatory signal. The memory strength is calculated usingthe fiducial value of κ ; see Eq. 5. The colored curves showthe h sensitivity of dedicated high-frequency gravitational-wave detectors: Arvanitaki [16, green], Goryachev [17, red],Holometer [18, blue]. For the fiducial value of κ , AdvancedLIGO can detect memory bursts with higher significance thandedicated high-frequency detectors. The filled star indicatesthe maximum strain of GW150914 and the frequency of peakemission [1]. The unfilled star indicates the expected memoryfrom GW150914 [2]. On the other hand, the memory scales as [4] h mem0 ∼ E gw d . (2)At first glance it might appear that memory straincan exceed the oscillatory strain for sufficiently large E gw . However, the maximum gravitational-wave fre-quency (for ultra-relativistic systems) is given by f max ∼ /r s ∼ /M s , where r s , M s are the Schwarzschild ra-dius, mass. In order to obtain a conservative estimate ofthe maximum possible memory, we assume that the en-tire remnant black hole mass is radiated away as gravita-tional waves. Thus, the maximum memory occurs when f = 1 /E gw , implying the maximum possible memory is h memmax ∼ E / f / d , (3)which scales like the oscillatory strain, Eq. (1).While h memmax and h osc0 scale the same way, h memmax is al- ways smaller. We define a memory efficiency factor κ ≡ h mem /h osc . (4)where h mem and h osc are respectively the measured mem-ory and oscillatory strain in some detector. The numeri-cal value of κ depends on the inclination and sky positionof the source. For an interferometeric detector such asLIGO, we can estimate the typical value of κ by sim-ulating an ensemble of binary black holes with randominclination angle and sky position, and using the memorywaveforms of Ref. [19]. We findˆ κ ≡ (cid:112) (cid:104) h (cid:105) / (cid:104) h (cid:105) = 0 . . (5)The angled brackets denote ensemble averages over an-gles. This is not far from the estimated value [2] forGW150914 κ = 1 / κ = 1 / κ value, while plausible forefficient memory emission from highly relativistic objectssuch as binary black holes, leads to an overestimation ofthe memory signal from less relativistic systems. For non-relativistic systems, h mem0 can be very small comparedto h osc0 . If a gravitational-wave observatory were everto measure κ >
1, this would be surprising as it wouldseem to indicate a significant quantity of missing energy,perhaps due to beaming.
High-frequency gravitational waves.
At design sensi-tivity, the LIGO/Virgo detectors will operate between ∼ − (cid:38) Hz. A number of astrophysicalsources may emit high-frequency gravitational waves [fora review, see Ref. 21]. These include black hole evap-oration [10 − Hz; 22, 23], dark matter collapsein stars [ ∼ h ≈ × − . Given our fiducial value of κ , the av-erage signal-to-noise ratio in Advanced LIGO is small: (cid:104) S/N (cid:105) ∼ − .We estimate the sensitivity of Advanced LIGO to or-phan memory from generic high-frequency sources by cal-culating the matched filter signal-to-noise ratio for ourfiducial κ = 1 /
20, high-frequency source. The expecta-tion value for the optimal matched filter signal-to-noiseratio is (cid:104)
S/N (cid:105) = 4Re (cid:90) | ˜ h | S h ( f ) df ≈ (cid:34) h S h ( f ) 1 f (cid:35) , (6)where the approximation holds for sine-Gaussians withfrequency f and amplitude h .The dashed black curve in Figure 2 shows the sine-Gaussian amplitude h necessary for an average signal-to-noise ratio (cid:104) S/N (cid:105) = 5 detection in Advanced LIGOoperating at design sensitivity as a function of burst fre-quency f . The solid black curve shows the same h ver-sus f sensitivity curve except that we include memoryin the matched filter calculation. This has the effect ofextending the LIGO observing band to sources for whichthe dominant oscillatory component has arbitrarily highfrequencies. For burst frequencies higher than a few kHz,the memory becomes more easily detectable than the os-cillatory burst. The colored curves show h verus f sen-sitivity for dedicated high-frequency detectors, which wediscuss presently.We compare the Advanced LIGO sensitivity curve toseveral dedicated high-frequency detectors. Fermilab’s“Holometer” (labeled with a blue curve in Figure 2) isa pair of co-located ∼
40 m, high-powered Michelson in-terferometers, sensitive to gravitational wave frequencies10 − Hz. It has reached an amplitude spectral densityof ∼ × − Hz − / [18]. The Bulk Acoustic Wave (la-beled with a red curve in Figure 2) cavity is a proposedresonant mass detector, sensitive to 10 − Hz pro-jected amplitude spectral density of ∼ − Hz − / [17].The detector is labeled in plots as “Goryachev” using thefirst author from [17]. The final proposed detector thatwe consider here consists of optically levitated sensors,sensitive to frequencies between 50 −
300 kHz, with pro-jected sensitivity to ∼ × − Hz − / [16]. It is labeledin Figure 2 with a green curve and denoted “Arvanitaki”after the first author of [16].Comparing the Figure 2 colored sensitivity curves fordedicated high-frequency detectors with the solid blacksensitivity curve for Advanced LIGO, we see that—givenour fiducial value of κ —Advanced LIGO will detectorphan memory before currently-proposed, dedicated,high-frequency detectors observe an astrophysical burst.Two effects, not included in Figure 2, will tend to makeit harder to detect high-frequency bursts compared tolow-frequency memory detection. First, high-frequencydetectors produce false positives at a higher rate thanAdvanced LIGO. Second, the memory search templatebank is trivially small. All orphan memory looks thesame: like a step function. In order to span the space ofoscillatory bursts, it is likely that many more templatesmust be used.This result has an interesting implication. If high-frequency detectors observe a detection candidate, Ad-vanced LIGO should look for a corresponding memoryburst. A coincident memory burst could provide power-ful confirmation that the high-frequency burst is of as-trophysical origin. Similarly, if Advanced LIGO detectsorphan memory, it may be worthwhile looking for coin-cident bursts in dedicated high-frequency detectors.In Figure 2, we plot sensitivity curves in terms of h :the amplitude of a sine Gaussian burst. It is also useful to frame our results in terms of amplitude spectral den-sity S h ( f ) / . In Figure 3, we show the noise amplitudespectral densities for the three high-frequency detectorsincluded in Figure 2, denoted with dashed red, blue, andgreen curves. The dashed black curve shows the noiseamplitude spectral density of Advanced LIGO.We also plot the amplitude spectral density for threesine-Gaussian bursts with memory. The frequency ofeach burst is matched to the observing bands of differ-ent high-frequency detectors. The colors are chosen sothat, e.g., the red burst spectrum matches with the redGoryachev detector. The burst amplitude is tuned sothat (cid:104) S/N (cid:105) = 5 in the associated high-frequency detec-tor. The solid curves show the memory + oscillatorycomponent of the signal while the dotted curves showonly the oscillatory component.While the oscillatory matched filter signal-to-noise ra-tio is 5 in each high-frequency detector, the associatedLIGO memory signal-to-noise ratio is many times louder:300 for Arvanitaki, 1 . × for Holometer, and 3 . × for Goryachev. This is consistent with the conclusiondrawn from Figure 2: for our fiducial value of κ , Ad-vanced LIGO should be able to easily observe orphanmemory from high-frequency bursts observed in dedi-cated high-frequency detectors. frequency (Hz) -28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 S = h ( f )( H z ! = ) aLIGOArvanitakiHolometerGoryachev FIG. 3: Strain amplitude spectral density. The dashed curvesrepresent the noise in three different detectors: AdvancedLIGO (black) and three dedicated high-frequency detectors(colored). For each dedicated detector, we plot the amplitudespectral density for a sine-Gaussian burst in the middle ofthe observing band (colored dotted peaks). The peak heightis tuned so that the oscillatory burst can be observed with asignal-to-noise ratio (cid:104)
S/N (cid:105) = 5. The solid colored lines showsthe amplitude spectral density when we include the memorycalculated with our fiducial value of κ . The memory burstsproduce large signals in Advanced LIGO, with (cid:104) S/N (cid:105) rangingfrom 300 to 10 . We also investigate the stochastic background from or-phan memory. Stochastic backgrounds arise from the in-coherent superposition of many unresolved signals. Theyare parameterized by the ratio of energy density in gravi-tational waves to the total energy density needed to closethe Universe [28],Ω gw ( f ) = 1 ρ c dρ gw d ln f = 2 π H f S h ( f ) , (7)where ρ c is the critical energy density of the Universe, dρ gw is the gravitational-wave energy density between f and f + df , S h ( f ) is the strain power spectral density ofan ensemble of sources, and H is the Hubble parameter.For an ensemble of sine-Gaussian bursts, S h ( f ) ispeaked at f , leading to a peaked distribution of Ω gw ( f );see Fig. 4. Gravitational-wave memory also creates astochastic background. For frequencies f (cid:28) f , S h ( f ) ∝ f − and so Ω gw ( f ) ∝ f . The dashed green curve inFig. 4 shows the stochastic background from an ensem-ble of 100 kHz sine-Gaussian bursts while the solid greencurve shows the stochastic background including memorycontributions.In this case, we have tuned the peak height so that thestochastic memory is just barely detectable by AdvancedLIGO operating for one year at design sensitivity withcross-correlation signal-to-noise ratio of one [28]. Thisis illustrated graphically by the fact that the solid greenline intersects the Advanced LIGO “power-law integratedcurve”; see [29] for additional details. We also includepower-law integrated curves for the high-frequency de-tectors included in previous figures. In each case, we as-sume a cross-correlation search using a pair of colocateddetectors operating for one year. We find that—givenour fiducial choice of κ —Advanced LIGO is likely to ob-serve an orphan memory background before the peak isobserved in high-frequency detectors. Conclusion.
In this Letter, we show that orphan mem-ory from high-frequency gravitational-wave sources canbe detected when the oscillatory component of the signalis outside of LIGO’s frequency band. Moreover, assumingefficient memory emission, Advanced LIGO is orders ofmagnitude more sensitive to these bursts than dedicateddetectors. Although no memory-specific LIGO/Virgosearches have been implemented, non-detections by pre-vious “burst” searches can be converted to limits on thegravitational-wave memory strain, e.g., [30].A dedicated gravitational-wave memory search is de-sirable. It will have enhanced sensitivity compared tocurrent burst searches. Further, a dedicated search canbe used to determine whether a detection candidate isconsistent with a memory burst by checking to see ifthe residuals (following signal subtraction) are consistentwith Gaussian noise.High-frequency gravitational-wave sources are conjec-tural. It is possible that there are no high-frequency f (Hz) -10 -8 -6 -4 -2 + (f) aLIGOArvanitakiHolometerGoryachev FIG. 4: Gravitational-wave energy density spectra. Solidgreen shows the spectra for an ensemble of high-frequency100 kHz bursts with memory. The dashed green shows justthe oscillatory signal. The peak height is chosen so thatthe signal is just visible by Advanced LIGO after one yearof integration. The black curve shows the “power-law inte-grated curve” [29] for Advanced LIGO at design sensitivity.A stochastic background spectra that intersects this curveis probably detectable with a cross-correlation signal-to-noiseratio exceeding one. Background spectra that do not intersectthe power-law integrated curve are not detectable. We alsoinclude power-law integrated curves for several other high-frequency detectors mentioned in this paper, all calculatedassuming a co-located detector pair operated for one year ofintegration. For sources with our fiducial value of κ , it is pos-sible to detect the stochastic background from orphan mem-ory before the signal is observable in dedicated high-frequencydetectors. sources in the Universe strong enough to produce or-phan memory detectable by Advanced LIGO and succes-sor experiments such as the Einstein Telescope [31] andCosmic Explorer [32]. However, the generic productionof memory by high-frequency sources makes it a usefulprobe of new physics, which might be revealed by cur-rent or planned detectors. Memory from lower-frequency(LIGO-band) sources provides a compelling astrophysicaltarget [2]. Acknowledgements
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