Prospects for Memory Detection with Low-Frequency Gravitational Wave Detectors
Kristina Islo Joseph Simon Sarah Burke-Spolaor Xavier Siemens



Center for Gravitation, Cosmology and Astrophysics, Department of Physics, University of Wisconsin-Milwaukee,
 P.O. Box 413, Milwaukee, WI 53201, USA

Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

Department of Physics and Astronomy, West Virginia University, P.O. Box 6315, Morgantown, WV 26506, USA
Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA

Center for Gravitation, Cosmology and Astrophysics, Department of Physics, University of Wisconsin-Milwaukee,
 P.O. Box 413, Milwaukee, WI 53201, USA
K. Islo
kpislo@uwm.edu



Gravitational wave memory is theorized to arise from the integrated history of gravitational wave emission, and manifests as a spacetime deformation in the wake of a propagating gravitational wave. We explore the detectability of the memory signals from a population of coalescencing supermassive black hole binaries with pulsar timing arrays and the Laser Interferometer Space Antenna (LISA). We find that current pulsar timing arrays have poor prospects, but it is likely that between 1 and 10 memory events with signal-to-noise ratio in excess of 5 will occur within LISA's planned 4-year mission.






§ INTRODUCTION


Non-linearities in Einstein's field equations suggest that during the coalescence of compact objects, the oscillatory gravitational wave (GW) radiation is accompanied by a monotonically increasing or decreasing component of the strain. The result is an enduring change of gravitational potential in the wake of a propagating GW, termed “memory” <cit.>. For example, a GW with memory passing through a system of two isolated, free-falling test masses would permanently stretch or compress the comoving distance between them. 

Like the oscillatory component of a GW, memory is sourced by a changing time derivative of the system's mass multipoles, but only in part. It also grows through the cumulative history of GW emission. As such, the memory signal inherits the radiating system's evolving past: its strength at any time is the result of the integrated history of the system. For a supermassive black hole binary (SMBHB) undergoing coalescence, the memory signal initially displays negligible growth corresponding to the slow time evolution of the binary's inspiral. Once the binary enters its most dynamic phase during coalescence, the system emits a burst of memory signal which propagates outwards. Observations of SMBHB coalescence memory events would shed light on strong-field effects of General Relativity (GR), and provide information about SMBHB properties augmenting that obtained from the oscillatory components. More broadly, a memory observation would provide robust evidence of fundamental symmetries in GR <cit.>.

In this paper, we estimate the current and future potential to detect GW memory from SMBHB coalescence based on a simulation-suite of semi-analytic models for the SMBHB population. Importantly, the models are based in local observables for SMBHBs, and thus encompass only uncertainties from local mass functions, galaxy merger timescales, etc. rather than uncertainties in casting dark matter halo simulations to a binary supermassive black hole population (as has dominated such simulations in the past)<cit.>. This study is also novel from previous studies in that we expand our models to include lower black hole masses, down to M_ BH≳ 10^5, and higher redshifts, such that the memory signals expected in the bands relevant to both Pulsar Timing Arrays (PTAs) and the Laser Interferometer Space Antenna (LISA) can be explored. We also take into account the unknown decoupling radius for binary-host interactions. We look at the likely rate and signal-to-noise ratio (SNR) for memory events detectable within these frequency bands. 

This paper is structured as follows: Section <ref> contains motivation and description of our chosen source population, Section <ref> describes the GW memory signal model, Section <ref> reports the average memory burst rate for memory events of varying strength, and we summarize results in Section <ref>.




§ SUPERMASSIVE BLACK HOLE BINARIES


The largest black holes in the Universe are created by hierarchical evolutionary processes involving the mergers of increasingly massive galaxies. SMBHBs form during major galaxy mergers, and grow more tightly bound through repeated interactions with their galactic environment, which drives orbital evolution to smaller separations <cit.>. The effectiveness of the mechanisms by which these black hole systems are driven to coalescence is an open question in astrophysics, although many studies conclude these binaries will coalesce given sufficiently gas-rich galactic environments or continuous rates of stellar loss-cone refilling <cit.>. SMBHBs that eventually coalesce are candidates for producing strong GW memory bursts. 

The persisting spacetime offset induced by a passing GW memory signal is proportional to Δ E_ rad/D, where Δ E_ rad is the total energy radiated in the form of GWs and D is the comoving distance to the coalescence. For equal-mass SMBHBs, the energy available for the GW memory burst ranges from 5% to 10% of the total binary energy. The precise value depends on binary inclination and the degree of black hole spin-alignment (see <cit.>). For example, an optimally-oriented binary consisting of two 10^9 black holes coalescing 1   Gpc away from Earth will emit a GW memory burst with amplitude h_ mem∼ 10^-15.

Population synthesis:
To find SMBHB coalescence rates we use the population synthesis model described in <cit.>. The number density of SMBHB coalescences occurring in a time interval dt from galaxy pairs with redshift, mass, and mass ratio range z, M, and q, respectively is


    d^4 Ndt dM_* dq dz_ burst = .d^3 ndz_ gal dM_* dq|_ gal →∙ dV_cdz_ burst d z_ burstdt.


The integrated value N is parameterized by astrophysical, observable quantities that characterize galaxy mergers – namely, galaxy stellar mass function (GSMF), galaxy pair fraction, and galaxy merger timescale. These are cast from galaxy pairs into inferred SMBHBs using the empirical relationship found between host galaxy bulge mass and black hole mass <cit.>. The last factor in Eq. (<ref>) is a cosmological term converting the proper time rate to a redshift rate. Assuming coalescence does not immediately follow binary formation, we must consider consider two redshifts: the redshift at which a galaxy merger forms a hardened binary z_ gal and the redshift of the memory burst upon SMBHB coalescence z_ burst. 

We restrict the parameter space to include only what we know observationally. 
Specifically, we take the redshift to be z < 3, the primary galaxy mass to be 10^8  ≤ M_*≤ 10^12, and the mass ratio to be consistent with `major mergers' with 0.25 < q < 1.

Evolution of SMBHBs:
At the early stages of galaxy merger, we expect dynamical friction to reduce the orbital angular momentum of the individual black holes until they sink to the center of the merger remnant, forming a SMBHB <cit.>. The dominant mode of energy loss below ∼ 10   pc binary separation is not yet understood, although many environmental interactions potentially contribute <cit.>. As such, it is unclear when the environment decouples from the binary, after which GW emission dominates. This has been the center of a growing debate in predicting the strength and spectral shape of the stochastic gravitational wave background in the nanohertz frequency band <cit.>. 

Here, we incorporate varying environmental influence to predict GW memory bursts for populations with different evolutionary timescales. This will dictate the offset between z_ gal and z_ burst.
We introduce a parameter we term the decoupling radius a_d to be the orbital separation at which the binary's evolution typically becomes GW-dominated.
SMBHBs embedded within sparse environments exhaust their environmental interactions earlier and have the potential to stall before reaching a regime where GW-radiation can drive the binary to coalesce (i.e., a_d ≳ 1  pc). The opposite scenario involves a binary strongly coupled to its environment, undergoing extremely efficient orbital shrinking, and reaching coalescence quickly (a_d→ 0).

If we assume circular binaries, the time to coalescence can be expressed in terms of Keplerian parameters and chirp mass ℳ = (M_1M_2)^3/5/(M_1+M_2)^1/5   

    τ_ GW = 5 c^5256 G a_d^4ℳ^5/3 M_ tot^4/3,


Establishing a_d is therefore akin to specifying the total time to binary coalescence, i.e., t_ burst = t_ gal + τ_ GW, where t_ gal is the time between galaxy merger and SMBHB formation.


We adopt a simple power-law model relating total binary mass and decoupling radius to emulate any environmental interaction that is more common among smaller SMBHBs than larger ones:


    a_d = a_8(M_ tot10^8 )^α


where a_8 is the decoupling radius corresponding to an SMBHB with total mass 10^8. 

The least efficient environments consist of a persistently depleted loss-cone and sparse gas in the galaxy merger core. Following , we find the orbital separation at which the binary will stall for a given galaxy-merger-bulge mass. We assume the ratio between bulge radius and bulge mass to be linear, with M87 serving as the fiducial ratio. In this scenario, we find best-fit parameters  a_8= 1.3   pc and α=1.0 to describe this minimally-efficient binary environment.

A maximally-efficient environment allows for even the most massive binaries to reach sub-parsec separations through continual loss-cone refilling and a ready supply of in-flowing gas. In this context, we choose a_8=0.01  pc (per Figure 1 of ) and consider 1.0 ≤α≤ 3.0.

Addition of lower-mass black holes: 
Previous simulations of SMBHB populations (such as that in ) only include progenitor galaxy mass greater than 10^10 which corresponds to individual black hole mass of ∼ 10^7. Lower galaxy masses are likely not relevant PTA-band sources, but 10^5 - 10^7 black hole binaries can produce GW memory signals which enter the LISA band. We use results from the Galaxy And Mass Assembly survey as outlined in <cit.> to estimate the distribution of these lower mass binaries at z < 0.1 along with those from the ULTRAVista survey reported in <cit.> for higher redshifts. 




§ GW MEMORY SIGNAL MODEL


 Over the binary's lifetime memory undergoes a slow growth prior to merger, rapid accumulation of power during coalescence, and eventual saturation to a constant value at ringdown. Thus, in the time domain, the signature of a memory signal from a SMBHB can be approximated by a step-function centered at the moment of coalescence, i.e., 
 

    h_+,×^(mem)(t) = Δ h_+,×^(mem)Θ(t),


where Θ is the Heaviside-step function. In the frequency domain,


    h̃_+,×^(mem)(f) = 
    i Δ h_+,×^(mem)2 π f   for 0 < f < f_c, 
    
    0    for f ≥ f_c,

where f_c, the cut-off frequency, corresponds to twice the orbital frequency at coalescence. Frequencies larger than f_c do not contribute to the GW signal. We can approximate f_c∼τ^-1, where τ is the light crossing time of the merger remnant. τ is also the timescale for the rise of the memory signal during the merger.
(<ref>) is sufficient for PTA signal-to-noise ratio calculations, however, we include a minor correction since LISA may be able to resolve the time varying features of the memory signal between onset of coalescence and ringdown. From <cit.>:

    h̃_+^(mem)(f) ≈ i Δ h^(mem)_+2 π f[1 - π^26(τ f)^2].

The magnitude of the spacetime offset Δ h_ mem is affected by black hole spin-alignment, with the maximally aligned spinning case exhibiting the strongest signal <cit.>. Higher-order spin effects should be incorporated in future simulations to properly reflect the saturated memory amplitude. We use the spin-averaged formula as given by <cit.>, 

    Δ h_+^(mem)≃(1 - √(8)/ 3)24 Gμc^2 Dsin^2 (17+ cos^2 ι),

with dependence on additional binary parameters reduced mass μ, inclination angle ι, and comoving distance D.






 §.§ Signal-to-Noise Ratio

Previous studies expound on ideal strategies for searching and conducting analyses of memory signals (e.g. <cit.>). In this work, rather than discuss best practices, we evaluate the astrophysical motivation for implementing searches of GW memory signals from a cosmic population of SMBHBs, specifically focusing on the signal-to-noise for memory events within LISA and PTAs.



  §.§.§ LISA

Proposed space-based interferometers like LISA would respond to a burst with memory as a permanent change to the proper distance between its free mirrors. In theory, the signal may be stored forever. Ground-based interferometers are not optimized for memory detection, requiring stacking of sub-threshold stellar-mass coalescence signals <cit.>, or waiting on new space-based observatories like the Big Bang Observatory to detect memory from GW150914-like events <cit.>. Here we calculate the SNR for memory from higher mass binaries, relevant for LISA.

The signal-to-noise ratio (SNR) is computed using an optimal match filter to extract a known signal model from noisy data as described in <cit.>. With (<ref>) as our memory signal model, the average SNR is given by


    ⟨ρ^2 ⟩   = 4 Re∫_0^∞⟨h̃_̃+̃^( mem)(f) | h̃_+,( mem)^*(f) ⟩S_f df, 
       = (Δ h_+^(mem))^2π^2∫_0^f_cdff^2(1 - π^26(τ f)^2)^2 1S_n(f).

Figure <ref> shows the SNR for a memory burst produced by SMBHBs of total binary mass M_ tot coalescing at redshift z. Expected SNR ranges from 100 to 10,000 with the highest SNR events coming from binaries at z < 0.5 and with 10^5 < M_ tot < 10^7. Our approximated memory model only minimally diverges at higher redshift from the LISA SNRs reported in <cit.>. Analysis of LISA mergers within the same range of mass and redshift, indicate that M_ tot < 10^4.2 coalescences will occur beyond the LISA frequency band (≥ 1 Hz), in which case, the memory signal will be the dominant coalescence signature. “Orphan memory” signals, as they are termed in <cit.>, effectively increase the high-frequency limit of the detector, allowing access to previously forbidden binary masses and redshifts.




  §.§.§ Pulsar Timing Arrays

PTAs measure the times of arrival of radio pulses from an array of galactic millisecond pulsars. GW memory manifests in pulsar timing observations as an abrupt increase or decrease in pulsar rotational frequency. A burst passing Earth will affect the timing residuals for all pulsars in the array in a correlated way. Leveraging the data from many pulsars is crucial when claiming a confident memory detection within PTA data as intrinsic noise in individual pulsars can mimic a burst. Below we derive the SNR of a memory event within a single pulsar's residuals – the SNR using an entire array is easy to estimate from this single pulsar derivation. 

If a burst signal like that expressed in (<ref>) spontaneously alters the jth pulsar's measured rotational frequency at time t_ mem, the Fourier transform of the integrated time-series residuals is


    R̃_j(f) = - B_jΔ h_+^( mem)e^ift_ mem2 π f^2,

where B_j is a geometrical factor accounting for pulsar position, GW polarization and propagation direction (see <cit.> or <cit.>).
PTA SNR is similarly determined through a matched filter analysis. From <cit.>:


    ⟨ρ_j^2 ⟩ = 4 Re∫_1/T_ obs^∞|⟨R̃_j(f) | R̃_j^*(f) ⟩|S_r(f)df,

where S_r is the power-spectral density of the timing residuals and T_ obs is the total time span of pulsar observation. Assuming the residuals are white and Gaussian, the one-sided power-spectral density can be written as


    S_r(f) = 2 σ^2 Δ t,

where σ is the white noise root-mean-square errors on the pulsar residuals and 1/Δ t is the observation cadence (typically ∼1/month). Inserting (<ref>) and (<ref>) into (<ref>), and averaging over pulsar and source location gives


    SNR = Δ h^( mem)_+ T_ obs^3/26 πσΔ t.


If a GW memory wavefront were to strike the Earth, we can correlate the entire array's timing residuals over the burst event epoch to uncover frequency changes in every pulsar. Multiple pulsar observations enables more confident GW memory detection by increasing the observed SNR by a factor proportional to √(N_ pulsars) (). 
We incorporate (<ref>) and (<ref>) into the simulation outlined in <ref> to repeatedly generate instances of an ensemble of coalescing SMBHBs, allowing us to arrive at GW memory event statistics for PTAs and LISA. 



§ RESULTS


For clarity, we provide a glossary describing the characteristics of the 4 extremal cases of binary parameters we report on below (see Table <ref>). These include minimal and maximal stalling and the inclusion or exclusion of potential selection bias when constructing an M_ BH-M_ bulge relation. 











Figure <ref> shows Δ h^( mem)_+≥ 5 × 10^-16 can occur between 8.7-4 and 5.2-3 times per century for model A and 2.9-6-1.4-5 times per century for model B. Models C and D approach rates of less than once per Hubble time for these stronger memory bursts. For assembled SMBHB populations initiated from 10^10 ≤ M_*≤ 10^12 and z < 3, <cit.> finds such an event to occur once per century. We recover the results of  if we assume PTA-relevant binaries have decoupling radii less than 0.001   pc. Evidence suggests this assumption may be too optimistic as estimates from Romulus25 point towards a relatively large fraction of major-merger SMBHBs formed by M_ Gal > 10^9 requiring Gyr timescales to evolve to sub-parsec separations <cit.>. 

We use (<ref>) and (<ref>) to calculate the SNRs associated with each memory burst in Figure <ref>.
Figure <ref> shows the mean number of burst events per year of given SNR specific to both a 12.5 year PTA observation and LISA for 100 realizations of a simulated SMBHB population for models A-D. Given the low rate of strong burst strains originating from PTA target populations, it is unsurprising that memory events with SNR≥5 occur 9.8-8 - 1.2-6 times per century in the optimistic model, and less than once per Hubble time in the most pessimistic model. LISA demonstrates better prospects with SNR≥5 events occurring 0.3 - 2.8 times per year in the most optimistic model, and less than once per million years in the most pessimistic.

We break down the event rates for this simulation by reduced mass and burst redshift in Figure <ref>. We find the most frequent binary coalescences to occur among reduced masses between 10^3 and 10^6, constituting 99% of all memory events. Such a region of binary mass is relatively unprobed. The last few years have seen numerous searches for active galactic nuclei from the black holes in these local, low-mass galaxies and future time-domain surveys, like the Large Synoptic Survey Telescope, may be able to constrain this population <cit.>. Memory detections from these sources would therefore complement our understanding of LISA's target population. As well, studies indicate discernible electromagnetic (EM) counterparts from these coalescences <cit.>. Paired with an EM trigger, memory signal searches could be conducted with restricted parameter priors to boost detection SNRs.

As we compare models A and C (upper and lower panel of Fig. <ref>, respectively), we see that decreasing environmental efficiency results in higher-mass binaries stalled at significant rates. The total number of bursts across our parameter space drops from 3.3 to 0.4 times per year. Lower-mass binaries initiated near the limits of parameter space evolve to closer redshifts, making up for those which may have stalled and keeping 0.1 < z < 1.5 consistently populated. As shown in <cit.>, most MBHBs coalescences whose inspiral signature yields LISA SNR > 5 lie beyond z=3 for a 3-year LISA lifetime, meaning the observation-based results reported here may be interpreted as lower limits on the number LISA memory events. 





§ OUTLOOK AND CONCLUSIONS


We evaluated the prospects for detecting a GW memory burst from source populations within reach of low-frequency GW detectors. We modified the initial inputs from population synthesis to include galaxy masses below 10^10. This probes additional SMBHB masses down to 10^4, and includes a region of observable orphan coalescences from parent mergers outside the LISA band. 
The incorporation of the decoupling radius allowed us to take into account the efficiency of the environmentally-driven orbital decay. Even assuming highly efficient environments, PTAs suffer from a sparser population of coalescing SMBHBs, making memory detection from these sources unlikely. Projections of PTA sensitivity at the advent of the LISA era strongly depend on the likelihood of PTA's access to increasingly sensitive radio telescopes, such as the planned ngVLA, SKA and DSA2000 <cit.>. Detailed explorations of future PTA sensitivities are beyond the scope of this work, although SNRs ≥ 5 remain infrequent at 1.7-3 per century for even a 30 year PTA data set. 

SNRs ≥ 5 can occur on average more than 0.8  yr^-1 for LISA. Considering an M_ BH- selection-bias, rates diminish, although it is likely that expanding our parameter space to include more lower-mass SMBHBs beyond z=3 will boost our estimates. Here, we restricted our restricted ourselves to within currently resolvable astrophysical parameter values. In this context, if < 10^6 binaries are numerous and indeed driven to low separations before their orbital decay is dominated by GW emission, LISA has a compelling science case in GW memory.

Acknowledgments. –
We'd like to extend thanks to those who have helped in the production of this paper. Particularly Marc Favata, Michele Vallisneri, David Kaplan, Luke Zoltan Kelley, Dusty Madison, and Sarah Vigeland for helpful communications and discussions. We are grateful for computational resources provided by the Leonard E Parker Center for Gravitation, Cosmology and Astrophysics at the University of Wisconsin-Milwaukee. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
The NANOGrav project receives support from National Science Foundation (NSF) Physics Frontiers Center award number 1430284.
yahapj


