Limits of Astrophysics with Gravitational Wave Backgrounds
Abstract
The recent Advanced LIGO detection of gravitational waves from the binary black hole GW150914 suggests there exists a large population of merging binary black holes in the Universe. Although most are too distant to be individually resolved by advanced detectors, the superposition of gravitational waves from many unresolvable binaries is expected to create an astrophysical stochastic background. Recent results from the LIGO and Virgo collaborations show that this astrophysical background is within reach of Advanced LIGO. In principle, the binary black hole background encodes interesting astrophysical properties, such as the mass distribution and redshift distribution of distant binaries. However, we show that this information will be difficult to extract with the current configuration of advanced detectors (and using current data analysis tools). Additionally, the binary black hole background also constitutes a foreground that limits the ability of advanced detectors to observe other interesting stochastic background signals, for example from cosmic strings or phase transitions in the early Universe. We quantify this effect.
I Introduction
The first direct detection of gravitational waves was recently announced by the Advanced LIGO (Laser Interferometer Gravitational Wave Observatory) and Virgo Collaborations Abbott et al. (2016a); Aasi et al. (2015); Abbott et al. (2016b); Acernese et al. (2015). The observation of GW150914, a binary black hole (BBH) merger with individual black hole masses of and 29 at a luminosity distance of Abbott et al. (2016c), implies that the masses and coalescence rate of stellar-mass BBHs are at the high end of previous predictions Abbott et al. (2016d, e); Abadie et al. (2010). As a consequence, the astrophysical stochastic gravitational wave background, arising from all coalescing binary black holes too distant to individually resolve Zhu et al. (2011); Rosado (2011); Marassi et al. (2011); Wu et al. (2013); Zhu et al. (2013); Kowalska-Leszczynska et al. (2015), is potentially within reach of advanced detectors. When operating at design sensitivity, Advanced LIGO may detect this binary black hole background with signal-to-noise ratio in as few as 1.5 years Abbott et al. (2016f). However, there is significant uncertainty in the strength of the stochastic signal due to uncertainty in the coalescence rate, currently estimated from only 16 days of double-coincident observation Abbott et al. (2016f, d). In this paper, we build on the LIGO and Virgo results from Ref. Abbott et al. (2016f) and investigate the potential to extract astrophysical information from measurements of the stochastic background.
The detection of an astrophysical stochastic background would be a major accomplishment, providing us with a glimpse of sources at cosmological distances. Given this exciting possibility, we address three key questions concerning the future prospects for gravitational wave science with stochastic backgrounds:
First, how does the information contained in the stochastic signal compare to what we learn from resolvable binaries in the nearby Universe? In Sec. II, we demonstrate that the stochastic signal is dominated by unresolvable sources between redshifts and ; thus, observations of the stochastic background will probe a BBH population that is distinct from directly resolvable sources in the more local Universe.
Second, what astrophysics and cosmology can we explore using results from stochastic searches? In Sec. III, we find that, while second-generation gravitational wave detectors may successfully measure the amplitude of the stochastic background, it is difficult to further distinguish between different models for the binary black hole background.
Third, how does the presence of the expected binary black hole background affect our ability to measure other potentially interesting backgrounds arising, e.g., from cosmic strings Kibble (1976); Caldwell and Allen (1992), the core collapse of population III stars Crocker et al. (2015), or phase transitions in the early Universe Starobinsky (1979); Kosowsky et al. (1992); Giblin and Thrane (2014); Maggiore (2000)? In Sec. IV, we show that the BBH background acts as a limiting foreground, significantly decreasing our sensitivity to other backgrounds of interest.
Ii Information Contained in the Bbh Background
A stochastic background of gravitational waves introduces a correlated signal in networks of terrestrial detectors. Although this signal is much weaker than the detector noise, it is detectable by cross-correlating the strain data from two or more detectors. For a two-detector network and an isotropic, unpolarized, and stationary Gaussian background, the optimal SNR of a cross-correlation search is given by Allen and Romano (1999)
(1) |
where is the noise power spectral density (PSD) of detector , is the normalized isotropic overlap reduction function Christensen (1992), and is the total accumulated coincident observation time. The energy density spectrum of the stochastic background is defined as
(2) |
where is the energy density in gravitational waves per logarithmic frequency interval and is the critical energy density required to close the Universe. Here, is Newton’s constant, is the speed of light, and is the Hubble constant. We assume a standard “737 cosmology,” with , , and .
The energy density spectrum of a binary black hole background is determined in part by the binary chirp mass , where is the binary’s total mass and its symmetric mass ratio. Figure 1 shows example energy density spectra for stochastic BBH backgrounds of various “average chirp masses,” [more precisely, the background depends on the average ; hereafter, the average chirp mass refers to ], assuming equal mass binaries with . Also shown in Fig. 1 are power-law integrated (PI) curves Thrane and Romano (2013) indicating the sensitivity of the stochastic search after one year of integration with Advanced LIGO at early, middle, and design sensitivity. Power-law integrated curves are defined such that a power-law energy density spectrum drawn tangent to the PI curve will give after one year. More generally, energy density spectra lying above a PI curve have after one year, while those below have .
We adopt the Fiducial model of Ref. Abbott et al. (2016f), with BBH energy density spectra given by Phinney (2001); Zhu et al. (2011); Wu et al. (2012); Abbott et al. (2016f)
(3) |
where is the source-frame energy spectrum of a single BBH source Ajith et al. (2008) (see Appendix A). Since energy and frequency are identically redshifted, is in fact redshift invariant, depending on only through its argument as shown in Eq. (3). describes the evolution of the Hubble parameter in a flat universe. Finally, is the BBH merger rate per comoving volume as measured in the source frame; the factor of in the denominator of Eq. (3) converts this rate into the detector frame.
We assume that traces the star formation rate , subject to a time delay between a binary’s formation and merger Abbott et al. (2016f):
(4) |
Here, is the local coalescence rate at , is the probability of a time delay , and is the formation redshift corresponding to merger at redshift . We take Myr to be the minimum time required for binary evolution through merger, and integrate up to Gyr. We assume for Dominik et al. (2012). For , we adopt the star formation rate presented in Sec. 2.1 of Ref. Vangioni et al. (2015), based on the observed gamma-ray burst rate Kistler et al. (2013). We also assume that binary black holes are born preferentially in low-metallicity environments, multiplying by the fraction of stars formed with metallicities Abbott et al. (2016f), where is the solar metallicity; see Appendix B for details. Below, we also consider the LowMetallicity model of Ref. Abbott et al. (2016f), which instead assumes progenitor metallicities .
Unlike direct searches for binary coalescences, the results of which are dominated by the closest sources, the stochastic background is dominated by distant sources. To explain this simply, we imagine an idealized static Universe with a constant merger rate and no cosmological expansion. The gravitational wave energy density contributed by binaries within a thin spherical shell of radius and thickness scales like , where is gravitational wave strain and is the number of sources within the shell. In our idealized universe, BBH binaries are equally distributed in volume, so , giving . The background contribution from any shell is therefore constant, independent of distance. Since the number of such shells beyond Advanced LIGO’s horizon distance is much greater than the number within, the stochastic background is dominated by distant, unresolvable sources. (This is a reformulation of Olber’s paradox.)
In reality, the BBH population is not uniformly distributed in volume; we assume it traces the star formation rate via Eqs. (3) and (4). In order to more rigorously investigate the SNR contribution from binaries at different redshifts, we define the “SNR density”
(5) |
with
(6) |
SNR density for design-sensitivity Advanced LIGO is plotted as a function of in Fig. 2 for several choices of chirp mass, assuming the Fiducial and LowMetallicity models. Also shown are the cumulative SNRs obtained by integrating the SNR density up to some cutoff . For purposes of comparison, the curves shown are each normalized to total . In each figure, the dashed vertical lines indicate “threshold redshifts” beyond which BBHs of each chirp mass (indicated by the respective colors) are individually resolvable less than 50% of the time (see Appendix C for details). The redshifts therefore indicate the typical range of a direct search for compact binary coalescences – binaries at redshifts are, on average, directly resolvable, while those at are not.
For binaries like GW150914, with and , approximately of the stochastic SNR is due to unresolvable binaries when assuming the Fiducial model. In this case, percent of the stochastic signal is contributed by sources between and , and is due to binaries beyond . These precise values depend on the specific choice of background model. The LowMetallicity model, for instance, predicts that of the SNR is due to unresolvable sources, with percent of the signal contributed by binaries between and . For very high-mass systems (), , and so a much larger fraction of the stochastic SNR is due to resolvable sources. In this case, only approximately of the stochastic signal remains due to unresolvable binaries.
It is interesting to see how SNR density changes with average chirp mass. For , the curves are all similar because the knee frequency of is outside the sensitive part of the band; 95% of the SNR is contained between and (see the PI curves in Fig. 1). At , the SNR density distribution shifts to higher because the loud merger signal from high- sources is redshifted into the most sensitive band. Finally, as increases further to , the merger signal from high- signals begins to leave the observing band entirely, leaving mostly signal from low- sources.
Iii Stochastic Model Selection
Valuable astrophysical information is contained in the BBH background, including the masses and merger rates of distant BBH populations inaccessible to direct searches for compact binary coalescences. The degree to which this information can be extracted, however, depends on our ability to perform model selection and parameter estimation. Model selection and parameter estimation has been shown to be difficult for astrophysical backgrounds dominated by low-mass binaries of several Mandic et al. (2012), which only depart from power laws at frequencies above . The low stochastic search sensitivity above Hz suggests that this high-frequency behavior will be extremely difficult to observe.
Backgrounds of more massive BBHs are shifted to lower frequencies (see, e.g., Fig. 1), where non-power-law spectral features are increasingly visible to ground-based detectors. This suggests that black hole backgrounds may be more promising targets for model selection and parameter estimation. In order to evaluate the prospects for model selection on BBH backgrounds, we investigate at what point an astrophysical Fiducial background can be distinguished from a power-law spectrum:
(7) |
where is an arbitrary reference frequency.
The standard stochastic search employs a cross-correlation statistic that is proportional to the strain cross power between the signals and measured by two detectors Allen and Romano (1999). The expectation value and variance of the cross-correlation statistic in a frequency bin of width are, with appropriate normalization,
(8) |
and
(9) |
Here, is the true gravitational wave background. When adopting a particular model for the stochastic background, the likelihood for is the Gaussian Allen and Romano (1999); Mandic et al. (2012)
(10) |
The value measured in any single experiment is a random variable, depending on the particular noise instantiation through . The noise is itself Gaussian distributed about zero with variance . In the absence of real data, we cannot compute Eq. (10), but can instead calculate the ensemble-averaged likelihood
(11) |
obtained by marginalizing over ; this result is similar to Eq. (10), but with an additional factor of in the exponential. Simply assuming produces an overly optimistic estimate of an experimental likelihood. The full (ensemble-averaged) likelihood is the product , given by
(12) |
where is a normalization factor and we have defined the inner product
(13) |
Note that , not , appears on the left-hand side of Eq. (12), since this ensemble-averaged likelihood depends only on the expectation value .
Given an underlying BBH background described by the Fiducial model, we investigate the maximum likelihood ratio between the Fiducial and power-law models. Large values of indicate that the Fiducial model is (correctly) preferenced over the power-law background model; values close to indicate that the two models are indistinguishable. The maximum likelihood when correctly assuming the Fiducial model is , since the background itself is contained within the space of BBH models. The maximum likelihood when incorrectly assuming a power-law model can be derived analytically. The power-law model has one free parameter – the amplitude . The amplitude maximizing the likelihood Eq. (12) satisfies , which is solved to give
(14) |
where . The corresponding maximum likelihood for the power-law model is
(15) |
When formally comparing models with different numbers of parameters (such as the one-parameter power law versus the many-parameter Fiducial model), one could alternatively calculate a Bayes factor rather than a maximum likelihood ratio. However, the Bayes factor is approximated by the maximum likelihood ratio, multiplied by an additional “Occam’s factor” penalizing the more complex of the two models Cornish and Romano (2015). The inclusion of the Occam’s factor here would only serve to penalize the Fiducial model; by neglecting it here, we are showing the most optimistic prospects for discerning the form of an astrophysical BBH background. Additionally, when model parameters are not informative, the associated Occam’s factor is near unity and the maximum likelihood ratio well approximates the Bayes factor.
Figure 3 shows contours of the maximum log-likelihood ratio as a function of the local coalescence rate and chirp mass after three years of observation at design sensitivity. The solid black curve indicates the rates above which a BBH background is detectable with optimal after three years when correctly assuming the Fiducial model. The dashed black curve similarly indicates rates above which BBH backgrounds are detectable with when assuming a power-law model. This is not the optimal SNR, since the space of power-law models does not contain the true BBH spectrum. The best-fit background parameters inferred from GW150914 Abbott et al. (2016d, c, f) are indicated by a star. Over a large region of parameter space, ; in this region, the power-law and Fiducial models cannot be distinguished. Only for chirp masses and local rates much larger than those implied by GW150914 is . While Advanced LIGO is therefore likely to detect the stochastic background associated with GW150914, such a background is indistinguishable from a simple power law. In particular, years of observation at design sensitivity are required to attain .
The Advanced LIGO network consists of two interferometers at Hanford, Washington and Livingston, Louisiana. The sensitivity of the Hanford-Livingston (H1-L1) network to a BBH background is ultimately limited at high frequencies by the overlap reduction function , which rapidly approaches zero for Hz Christensen (1992). During Initial LIGO, a third interferometer (H2) was present at Hanford, colocated and co-oriented with H1 Aasi et al. (2015). With a constant overlap reduction function of , the H1-H2 pair is significantly more sensitive at high frequencies than H1-L1. While there are currently no plans to reinstall a second interferometer at Hanford during Advanced LIGO, it is interesting to consider the performance of a hypothetical H1-H2 network of colocated 4-km aLIGO interferometers. Fig. 3 also shows maximum likelihood ratios between the Fiducial and power law models for this hypothetical H1-H2 network. Although the BBH background is detectable by the H1-H2 network after three years, it remains indistinguishable from a power law. Approximately 50 years of observation with design-sensitivity colocated detectors are required to reach . Although this represents a factor improvement over the H1-L1 performance above, it is nevertheless an impractically long time.
In Fig. 4, contours of are instead shown in terms of the background amplitude at 10 Hz (which scales as ) and the frequency at which the background’s energy density is at a maximum (; see Fig. 1). From this figure, it is apparent that the only backgrounds distinguishable from power laws using H1-L1 are those for which Hz, which corresponds to the most sensitive frequency band for the isotropic stochastic search. The H1-H2 network shows sensitivity across a broader frequency band, as this configuration avoids the penalty associated with the overlap reduction function at high frequencies.
Any configuration of advanced detectors appears unlikely to differentiate an astrophysical BBH model from a simple power law. Hence, parameter estimation and model selection on the BBH background is limited to studying only its amplitude rather than its shape, and efforts to simultaneously constrain multiple parameters (e.g., and ) from the stochastic background alone will be thwarted by large degeneracies. Some sensitivity can be gained, however, by applying direct CBC measurements as priors on stochastic background parameters. With tight priors on the chirp mass and local rate, for instance, the stochastic search becomes increasingly sensitive to amplitude differences between different models of the BBH merger rate and redshift distribution.
For a GW150914-like background, for instance, Fig. 5 shows likelihood ratios between the Fiducial and LowMetallicity models as a function of observation time, for both the H1-L1 and H1-H2 detector networks (solid and dashed curves, respectively). We take the Fiducial model as the “true” background, and assume delta-function priors on the average chirp mass and local coalescence rate. Even in this most optimistic case, at least 25 years of observation with H1-L1 are required to distinguish (with log-likelihood ratio ) between these models. Colocated detectors, however, begin to distinguish between the Fiducial and LowMetallicity models in only three years. In a more careful treatment using realistic priors on the average chirp mass and local coalescence rate, we find that approximately 30 years of observation are required to distinguish between background models using H1-L1, while 10 years are required with H1-H2.
As a rule of thumb, it is only possible to distinguish between two astrophysical scenarios if the difference between their predicted spectra exceeds the sensitivity of the detector network. For the Advanced LIGO Hanford-Livingston network operating at design sensitivity, a deviation of
(16) |
is required in order to select between two models with significance.
The above analysis assumes a standard cross-correlation pipeline, optimal for a stochastic background that is stationary, isotropic, and Gaussian. However, the BBH stochastic background is non-Gaussian Abbott et al. (2016f), and it may be possible to improve upon the above results with future pipelines optimized for non-Gaussian backgrounds Thrane (2013); Martellini and Regimbau (2014); Cornish and Romano (2015).
Iv Resolving Additional Backgrounds
Once a stochastic signal is observed by advanced detectors, a natural question will be: is it consistent with the expected background from binary black hole mergers (the “BBH-only” hypothesis), or is there a contribution from something else, e.g. cosmic strings or cosmological sources (the “BBH+” hypothesis)? In this sense, the BBH background now becomes a limiting foreground, obscuring the presence of additional, weaker background components. As a simple scenario, consider the combined signal from a Fiducial background of GW150914-like black holes (chirp mass and local rate ) and a flat background of cosmological origin. How loud must the cosmological background be in order to be detectable against the BBH background? This question is equivalent to asking: how loud must the stochastic signal be in order to detect a spectral index that is inconsistent with the BBH scenario? Since we know that a potentially detectable background from BBHs is expected, thanks to the observation of GW150194, only observation of a spectral index inconsistent with can provide evidence of a distinct cosmological background.
This question can be cast as a model selection problem. The simplest BBH-only model is an power law parametrized only by an amplitude :
(17) |
This model is valid if we restrict our attention to the regime, where a power law is indistinguishable from a realistic background as demonstrated in Sec. III. For the BBH+ model, assume a power law plus a constant :
(18) |
As in Sec. III, we consider the maximum likelihood ratio between these models, with likelihoods defined as in Eq. (12). The “BBH-only” likelihood is maximized by the amplitude given in Eq. (14) [replacing with the combined background considered here]. The “BBH+” likelihood is maximized by
(19) | ||||
Contours of the maximum log-likelihood ratio are shown in Fig. 6 as a function of the cosmological background amplitude and the total integration time, for the H1-L1 detector network (left-hand panel) and for two colocated detectors (right-hand panel). In each panel, the black solid (dashed) curves indicate the observation times necessary to detect the combined astrophysical and cosmological background with optimal ; note that these curves become vertical as approaches zero, corresponding to the fixed detection time of the BBH background alone. The gray solid (dashed) curves indicate the cosmological background amplitudes that would otherwise be detectable with optimal , if there existed no BBH background.
The fact that the gray curves lie deep within the region implies that the presence of a BBH background serves to obscure any cosmological background that would otherwise be detectable. If no BBH background were present, for instance, Advanced LIGO could detect a cosmological background of amplitude with after three years of observation. When a BBH background is present, however, a much larger amplitude of (corresponding to ) is required to resolve an additional flat background component. After three years of observation at design sensitivity, Advanced LIGO will therefore be able to constrain the amplitudes of additional background components to . A network of colocated detectors performs somewhat better, constraining additional background components to after one year of observation and to after three years of observation.
In the above, we treat the power-law amplitudes and of Eqs. (17) and (18) as entirely free parameters. In reality, we will likely be able to place a prior on these parameters, using CBC estimates of the average chirp mass and local coalescence rate as well as estimates of the theoretical uncertainty in background modeling Abbott et al. (2016f). However, even if we assume the amplitudes and are known to within a factor of 2 (an optimistic assumption given the uncertainty in the merger rate evolution with redshift Abbott et al. (2016f)), we find little change in the results presented in Fig. 6. With optimistic priors, the ability of H1-L1 and H1-H2 to resolve a cosmological background is improved only for observation times yr. After yr of integration, the experimental uncertainty on the stochastic background amplitude falls below the width of the prior distribution, and so the priors are no longer informative.
V Conclusion and Future Outlook
In this paper, we seek to address three questions concerning an astrophysical binary black hole background. First, how does the information contained in a stochastic background compare with what can be learned from nearby, individually resolvable binary mergers? Direct searches for binary black hole coalescences are, on average, sensitive to redshifts less than for events like GW150914. The stochastic background, on the other hand, is dominated by binary mergers in the far more distant Universe, with 90% of the stochastic SNR due to sources between redshifts and . The stochastic background therefore encodes astrophysical information (e.g., the mass distribution and rate of BBH mergers as a function of redshift) about a population of black hole binaries that is distinct from the local population visible to CBC searches.
Second, what astrophysics can we hope to extract from future observations of the binary black hole background? In principle, the functional form of the background’s energy density spectrum depends upon the precise characteristics of the underlying binary black hole population (mean chirp mass, local coalescence rate, star formation history, etc). We demonstrate, however, that for realistic chirp masses and coalescence rates, the form of the stochastic background is indistinguishable from a simple power law with Advanced LIGO. In the near future, parameter estimation and model selection on the stochastic background are therefore limited to measuring only the overall amplitude of the background.
Finally, how is our ability to measure other stochastic backgrounds affected by the presence of an astrophysical BBH background? We find that an astrophysical BBH background obscures the presence of any underlying cosmological background that might otherwise be detected with Advanced LIGO. For such a cosmological background to be resolvable, it must be strong enough to overcome our uncertainty in the amplitude of the BBH background. In this sense, the BBH background now acts as a foreground, limiting Advanced LIGO’s sensitivity to additional, weaker background components.
It should be noted that the Fiducial and LowMetallicity background models we consider here make specific assumptions about the metallicities of black hole progenitors and the masses and formation times of black hole binaries, properties that are currently only poorly understood. Different models of the BBH background will yield different numerical results for the above three questions. Qualitatively, however, the above conclusions are robust.
Future developments, however, may brighten these prospects. It may be possible to achieve better model selection than we show here through the development of a non-Gaussian stochastic pipeline, optimized for signals like the expected BBH background. Future stochastic measurements will also be strongly aided by any developments in instrumentation or data analysis that improve detector sensitivities at high frequencies, as it is only at high frequencies that the BBH background deviates substantially from a power law. To this end, one strategy is the use of colocated detectors, such as the H1-H2 configuration of Initial LIGO, to avoid the penalty associated with the overlap reduction function at high frequencies (at the potential cost of introducing correlated environmental noise) Aasi et al. (2015). As see in Sec. II, it may in fact be possible for an Advanced LIGO H1-H2 configuration to differentiate between astrophysical background models on a more practical time scale of yr (as opposed to hundreds or thousands of years with the H1-L1 configuration).
In the more distant future, third-generation detectors like the Einstein Telescope (ET) Punturo et al. (2010) will be able to probe black hole binaries at cosmological distances. ET is projected to resolve individual events like GW150914 to redshifts of Sathyaprakash et al. (2012), allowing for precision observation of the binary black hole population over the entire history of star formation. The ability of ET to resolve such events raises the exciting possibility of the individual identification and subtraction of each BBH coalescence from the data, opening the way for the detection of weaker, underlying stochastic backgrounds of astrophysical or even cosmological origin.
Acknowledgements
We thank Christopher Berry, Nelson Christensen, Eric Howell, Vuk Mandic, Duncan Meacher, Tania Regimbau, and Alan Weinstein. T. C. is a member of the LIGO Laboratory, supported by funding from the U.S. National Science Foundation. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement Grant No. PHY-0757058. ET is supported by ARC FT150100281. IM was partially supported by STFC and by the Leverhulme Trust.
This paper has been assigned the LIGO document number LIGO-P1600059.
Appendix A Binary Black Hole Energy Spectrum
We adopt the BBH model described in Ref. Ajith et al. (2008), which presents a phenomenological description of the inspiral, merger, and ringdown of a black hole coalescence. The corresponding energy spectrum for a single binary is Zhu et al. (2011)
(20) |
where
(21) |
Definitions for , , , and are given in Sec. IV of Ref. Ajith et al. (2008).
Appendix B Star Formation and Mean Metallicity Evolution
Following the Fiducial model of Ref. Abbott et al. (2016f), we adopt the star formation rate Vangioni et al. (2015); Kistler et al. (2013)
(22) |
with , , , and .
In the Fiducial model, the rate of binary black hole formation is proportional to the rate of star formation with metallicity below . The mean stellar metallicity is given as a function of redshift by Abbott et al. (2016f); Madau and Dickinson (2014)
(23) |
with stellar metal yield , return fraction , baryon density , and . The star formation rate used in calibrating and is Madau and Dickinson (2014)
(24) |
Assuming that stellar metallicity is log-normally distributed with a standard deviation of , the fraction of stars with is
(25) |
The rate of binary black hole formation is assumed to be proportional to .
Appendix C Threshold Redshifts
The optimal signal-to-noise ratio of a single-detector matched filter search is
(26) |
where is the measured strain signal. Using the phenomenological BBH model of Ref. Ajith et al. (2008), the signal-to-noise ratio of an optimally positioned and oriented binary is
(27) |
where is the luminosity distance, the proper distance to the source, and
(28) |
The values of , , , and are given in Ref. Ajith et al. (2008). The source distance is given in terms of redshift by
(29) |
In general, the squared signal-to-noise ratio of an arbitrarily positioned and oriented source is reduced from the optimal value Eq. (27) by an antenna factor , which depends on the source’s sky position, polarization angle, and inclination. Given an ensemble of randomly positioned and oriented sources, the median value of is . The threshold redshifts quoted in Sec. II are obtained by numerically solving Eqs. (27) and (29) for the redshift at which the squared signal-to-noise ratio of an optimally positioned and oriented binary is . Beyond redshift , less than of binaries are directly resolvable.
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