Gravitational-wave emission from compact Galactic binaries
Compact Galactic binaries where at least one member is a white dwarf or neutron star constitute the majority of individually detectable sources for future low-frequency space-based gravitational-wave (GW) observatories; they also form an unresolved continuum, the dominant Galactic foreground at frequencies below a few mHz. Due to the paucity of electromagnetic observations, the majority of studies of Galactic-binary populations so far have been based on population-synthesis simulations. However, recent surveys have reported several new detections of white-dwarf binaries, providing new constraints for population estimates. In this article, we evaluate the impact of revised local densities of interacting white-dwarf binaries on future GW observations. Specifically: we consider five scenarios that explain these densities with different assumptions on the formation of interacting systems; we simulate corresponding populations of detached and interacting white-dwarf binaries; we estimate the number of individually detectable GW sources and the magnitude of the confusion-noise foreground, as observed by space-based detectors with 5- and 1-Mkm arms. We confirm earlier estimates of thousands of detached-binary detections, but project only few ten to few hundred detections of interacting systems. This reduction is partly due to our assessment of detection prospects, based on the iterative identification and subtraction of bright sources with respect to both instrument and confusion noise. We also confirm earlier estimates for the confusion-noise foreground, except in one scenario that explains smaller local densities of interacting systems with smaller numbers of progenitor detached systems.
Subject headings:gravitational waves, Galactic binaries, AM CVn, data analysis, LISA
Compact Galactic binaries with periods shorter than approximately five hours are of considerable interest as gravitational wave (GW) sources (Nelemans 2009; Marsh 2011). Such binaries include detached double white dwarf (DDWD) systems, short-period cataclysmic variables (CVs), and AM CVn systems (so named after their prototype, AM Canum Venaticorum; see Solheim 2010). These last are short-period binaries in which a white dwarf (WD) is accreting He-rich material from an H-deficient companion. AM CVn systems are defined spectroscopically by the presence of strong absorption or emission He lines and the absence of H lines, which instead characterize “standard” CVs. In this paper we focus on those binaries that are evolving toward, or have evolved from, extremely short orbital periods, on the order of minutes. These include DDWD and AM CVn systems, sometimes referred to as “ultra-compact” binary systems. Such binaries have received considerable attention, among other reasons, as possible progenitors for Type Ia supernovae, R Coronae Borealis stars (Webbink, 1984; Iben & Tutukov, 1984), and so-called “.Ia” explosions (Bildsten et al., 2007). Our understanding of their formation and evolution remains poor.
Together with ultra-compact X-ray binaries, DDWD and AM CVn systems will be the most numerous and individually brightest GW sources in the Galaxy for low-frequency ( Hz) space-based GW detectors. Within this frequency band, observations suggest DDWD systems and theoretical estimates predict AM CVn systems in our Galaxy (Maxted & Marsh, 1999; Nelemans et al., 2001a, c, 2004).
The aim of this paper is to characterize the detectable GW emission from compact binaries in light of recent population estimates based on new observations. The first estimates were based solely on population-synthesis simulations; see App. A for a detailed discussion and review of the literature. Recent observations of AM CVn systems have provided new information that constrains earlier estimates. For instance, the identification of AM CVn systems among candidates selected from the Sloan Digital Sky Survey (SDSS) spectroscopic catalog allowed Roelofs et al. (2007b) to estimate the density of AM CVn systems by modeling the completeness of the catalog. A follow-up spectroscopic survey of a color-selected subset of the SDSS photometric database seems to find even fewer AM CVn systems than expected (Roelofs et al., 2007b; Rau et al., 2010), leading to local density estimates smaller by than suggested by population-synthesis studies.
To date, 27 AM CVn systems have been presented in the literature; the most recent detections have been achieved in synoptic surveys, including a short-period system from Kepler (Fontaine et al., 2011), and a medium-period, outbursting system from the Palomar Transient Factory (Levitan et al., 2011).
Similarly, surveys of low-mass WD candidates selected from SDSS has led to the discovery of several short-period DDWDs that could be progenitors of AM CVn systems (Badenes et al., 2009; Mullally et al., 2009; Kulkarni & van Kerkwijk, 2010; Marsh et al., 2011). For example, the ELM (Extremely Low Mass) survey (Brown et al., 2010; Kilic et al., 2011a; Brown et al., 2012) searches for companions around extremely low mass WDs (). The ELM survey has approximately quintupled (with 19 new systems) the known sample of DDWDs that will merge within a Hubble time (Kilic et al., 2011b; Brown et al., 2012), providing evidence for at least one plausible AM CVn formation channel (see Sec. 2.1).
In this paper, we consider five scenarios for AM CVn formation that span plausible populations, and we study their consequences for space-based GW detectors by modeling the detection and subtraction of sources in simulated data streams. We consider two representative detectors: the standard LISA developed as a joint NASA–ESA project until 2011 (Prince et al., 2009; Jennrich, 2009), and a descoped configuration with shorter armlengths of 1 Mkm, which is similar, but not equivalent, to the NGO/eLISA design studied by ESA in 2011 (Amaro-Seoane et al., 2012).
We find, first, that the number of detectable AM CVn systems decreases from , as reported in previous assessments of LISA science, to (for one year of observation with a single interferometer—see App. B). Precise numbers depend on the assumptions used to explain how a postulated number of DDWD progenitors can result in the number of AM CVn systems seen in surveys. For the shorter detector, these numbers are reduced by a factor of a few. Second, we confirm that mostly DDWD systems determine the residual confusion foreground left after GW signals from detectable binaries are subtracted. The foreground does not change significantly from previous estimates, except in one scenario in which we explain the fewer observed AM CVn systems by reducing the space density of DDWD progenitors.
The rest of this paper is structured as follows. In Sec. 2 we discuss the current understanding of AM CVn formation. In Sec. 3 we outline the observations of AM CVn and DDWD systems in the Galaxy, and the resulting calibration of local space densities. In Sec. 4 we present our analysis: we introduce our explanatory formation scenarios in Sec. 4.1, discuss the space-based observation of GWs from Galactic binaries in Sec. 4.2, describe our simulations of GW source detection and subtraction in Sec. 4.3, and detail our results in Sec. 5. In Sec. 6 we recapitulate our conclusions and propose possible future work. In the appendices we provide more detailed discussions of specific topics: in App. A we summarize previous analytical and numerical studies of the Galactic foreground for LISA, and give a general semi-analytical derivation of its magnitude and shape; in App. B we discuss GW detection and define our reference detector configurations; in App. C we provide more details about the fits of the GW confusion foreground obtained from our simulations. In the rest of the paper, we use the standard notation where is the gravitational constant and is the speed of light.
2. The formation and evolution of compact binary systems in the Galaxy
In the standard picture of binary-star evolution, compact binaries form following two consecutive mass-transfer episodes; the loss of sufficient angular momentum by friction in the second common-envelope phase ensures sufficiently tight orbits for GW emission to influence dynamics. Here we concentrate on the processes leading to mass transfer and to the formation of AM CVn systems. Due to space limitations, we do not attempt here to provide an extensive literature review, and refer the reader to Solheim (2010) and references therein for a detailed review on the current understanding of the formation and evolution of DDWD and AM CVn systems.
The subsequent orbital evolution of compact binaries is driven by gravitational radiation, by mass transfer, by a dissipative coupling that can feed the angular momentum added through accretion back into the orbit, and by other poorly modeled processes such as wind loss, etc. Therefore, the rate of change of the orbital angular momentum comprises three or more competing components,
Here represents the loss of angular momentum to gravitational radiation,
(for circular orbits) where and are the two binary masses ( is the donor’s mass for mass-transferring systems), their separation, and . Next, , where indicates the donor’s mass-loss rate due to Roche-lobe overflow and defines the orbital radius around the accretor with the same specific angular momentum as the transferred mass (Verbunt & Rappaport 1988).
Mass transfer can proceed either via an accretion disk or via “direct impact” (see, e.g., Marsh & Steeghs 2002; Ramsay et al. 2002; Marsh et al. 2004). “Direct impact” occurs when the donor’s mass stream hits the accreting star directly. The particular route depends on the initial separation in the binary following the ejection of the second envelope, and on the donor’s entropy and equation of state (see, e.g., Gokhale et al. 2007; Dan et al. 2011, 2012). Typically, mass-transferring binaries with initial orbital periods greater than ten minutes form an accretion disk. As discussed below, the presence or absence of an accretion disk is critical to the stability of the system. Whether a disk is present or not, denotes the change in the torque due to dissipative coupling, tidal or magnetic, between the accretor’s spin and the orbital angular momentum during the accretor’s synchronization timescale . We assume that the donor and orbit’s spins are synchronized. In this work, we do not consider the change in angular momentum driven by other physical processes, such as losses due to winds or magnetic braking, because of current limitations in our understanding and modeling of such phenomena.
As the term “detached” in their very name implies, DDWD dynamics are determined solely by gravitational radiation and tides (see e.g. Piro 2011). By contrast, the formation, stability, and outcome of AM CVn systems depend on all three effects in Eq. (1). Three proposed channels exist for the formation of AM CVn systems, and each can vary in its efficiency. We describe them below. Following Nelemans et al. (2001a, b, 2004), we assume that the first two channels listed below can operate with either “optimistic” or “pessimistic” efficiencies, resulting in different AM CVn numbers and period distributions.
2.1. The DDWD channel
A subclass of DDWDs with periods of several minutes becomes semi-detached when angular-momentum loss due to gravitational radiation drives the two bodies sufficiently close that the lower-mass WD fills its Roche lobe and mass transfer begins, either via direct impact or via an accretion disk. The stability of mass transfer depends critically on the mass ratio of the two initial WDs (requiring typically ) and on whether spin–orbit tidal coupling feeds angular momentum accreted by the primary star back to the orbit; the strength of this coupling is characterized by the synchronization timescale (Marsh et al. 2004).
Under optimistic assumptions, strong tidal coupling stabilizes mass transfer, favoring the formation of stable AM CVn systems and resulting in fewer mergers. By contrast, under pessimistic assumptions there is no tidal coupling and many more DDWD systems merge. In the presence of a disk, the accreted material’s angular momentum is mainly transferred back into the orbit by tidal forces at the outer edge of the disk. However, the disc’s material at the inner edge still has sufficient angular momentum to spin up the accretor. Effects not considered here, such as magnetic braking (Farmer & Roelofs 2010), may also impact the formation of AM CVn systems and favor the merger of the WDs.
If stable mass transfer proceeds, the donor WD loses mass and expands in size, while the system evolves to longer orbital periods. The minimum orbital period at which mass transfer can begin is two minutes for a 0.45 and 0.25 binary. Maximum orbital periods for evolved systems in the Galaxy are estimated to be about 90 minutes; these correspond to very old systems whose orbits have slowly evolved over a Hubble time.
Surveys specifically targeted at low-mass WD [such as the ELM survey of WDs: Brown et al. (2010); Kilic et al. (2011a); Brown et al. (2011)] have resulted in the discovery of numerous DDWD systems with , which are expected to get into contact within a Hubble time. Several of these newly discovered low-mass binaries will form stable mass-transferring AM CVn systems, even in the most pessimistic case of no tidal coupling (Brown et al., 2011).
2.2. The Helium star–white dwarf channel
In this channel, a WD accretes from an initially nondegenerate He star with mass 0.4–0.6 before the He has been exhausted in the He-star core. He stars are produced from stars with masses greater than 2 that lose their envelopes on the red-giant branch, and typically have lifetimes comparable to their main-sequence progenitors, allowing for mass transfer before evolution to the WD stage. Mass transfer is stable when the mass ratio of the donor He star to the accretor WD is less than approximately 1.2 (Tutukov & Fedorova 1989; Ergma & Fedorova 1990; see also Nelemans et al. 2001a). Due to mass loss, He burning in the donor’s core quenches and at masses 0.2 the star becomes semi-degenerate. The minimum orbital period reached by these systems is ten minutes.
It remains in question whether 0.1 or 0.3 of He accreted from a He star will destroy a CO WD by edge-lit detonations (henceforth denoted ELDs; see Shen et al. 2010 and references therein). Nelemans et al. (2001a) distinguish between optimistic and pessimistic assumptions for this channel depending on whether double edge-lit detonations remove CO WD systems.
2.3. The evolved Cataclysmic Variable channel
In analogy to traditional CVs, a so-called evolved CV comprises a main-sequence star which transfers mass to a WD or neutron star (Podsiadlowski et al., 2001). The difference is that mass transfer occurs as the donor star evolves off the hydrogen main sequence. However, it is thought that the evolved-CV channel contributes only marginally () to the AM CVn population (Nelemans et al., 2004; Roelofs et al., 2007b) for systems with periods less than 1500s. Therefore, we do not consider this channel in this work, which focuses on the detectability of AM CVn systems by GW interferometers.
3. Observations of AM CVn and DDWD systems in the Galaxy
We now turn our attention to recent population estimates of AM CVn systems based on observations. As mentioned, Roelofs et al. (2007b) estimate the total space density of AM CVn systems using observations of a source population identified by emission lines in SDSS. Characterizing selection effects for the six observed systems in SDSS is challenging. The orbital periods of these systems are generally shorter than predicted for the majority of AM CVn systems by population estimates, thereby requiring a significant extrapolation to estimate the space density of the entire population.
Table 1 in Roelofs et al. (2007b) presents the modeled and inferred AM CVn local space densities for different population-synthesis models. The inferred densities range from to , assuming a thin-disk population with a scale height of 300 pc for most systems. A comparison between the inferred and modeled densities immediately disfavors formation models such as the optimistic DDWD channel, and requires an overall reduction in the estimated number of Galactic AM CVn systems by a factor larger than ten. Roelofs et al. (2007b) hypothesize that lower AM CVn densities arise predominantly due to the lower efficiency of the major formation channels. Later in this paper we consider an alternative possibility, namely that systems observed by Roelofs et al. (2007b) may come from a distribution with scale height larger than conventional thin disks, with AM CVn progenitors born early in the history of the Galaxy, before the thin disk was fully developed.
Although the number of known DDWD systems that will merge within a Hubble time has increased to 24, an approximate factor of five, in the last few years (four eclipsing systems are now known; the ELM survey has uncovered the eclipsing DDWD binary with the shortest known period of 12.7 minutes), a detailed population analysis of such systems has yet to be undertaken. More recently, on the basis of the assumed Galactic-disk model, the ELM survey predicts kpc of DDWDs where at least one WD has a mass 0.25 , with a factor of two uncertainty in correction factors (Brown et al., 2011). This is much lower than population-synthesis results (Nelemans et al. 2001c). On the other hand, population synthesis results show good agreement with observations made by the Supernova type Ia Progenitor (SPY) survey. The SPY radial-velocity survey estimated the local space density of WDs to be pc (Holberg et al., 2008), with new DDWDs found (with orbital periods less than ten days) from a sample of . At present, estimating space densities from the ELM and SPY surveys is very challenging, requiring proper understanding of selection biases. Detailed population comparisons between the two surveys are not currently available.
4. Our analysis
We now describe our analysis: in Sec. 4.1 we explain our choices for WD binary populations; in Sec. 4.2 we discuss GW emission from binaries; and in Sec. 4.3 we describe our simulations of binary detection and subtraction in detector data. Sec. 5 below presents the resulting estimates of the number of detectable systems and of the residual confusion-noise foreground.
4.1. WD binary catalogs
In our analysis we considered several different assumptions on AM CVn formation, generated simulated binary catalogs and detector datasets for each, and estimated the resulting binary detection and subtraction prospects. Among the large number of cases that we investigated, here for conciseness we report on five scenarios that we believe are representative of the range of possibilities given the constraints of current observations. Two of the scenarios represent an upper bound (with maximally efficient formation channels) and conversely, a bound derived assuming inefficient formation channels on predictions from population-synthesis models (cases 1 and 2); two others attempt to reproduce SDSS-estimated AM CVn densities (cases 3 and 4 below); a fifth (case 5) explores a Galactic population with a larger scale height at times earlier than 7 Gyr ago. As discussed below, the modified–Galactic-disk scenario (case 5) should be consistent with SDSS observations; however, this agreement is difficult to quantify as the SDSS-estimated densities given in Roelofs et al. (2007b) are derived assuming a standard disk model.
We generate DDWD and AM CVn catalogs with the SeBa population-synthesis code (Portegies Zwart & Verbunt 1996; Nelemans et al. 2001b, 2004; Toonen et al. 2011), with the following assumptions and parameters: i) the time- and position-dependent star formation rate (SFR) based on Boissier & Prantzos (1999) (see Nelemans et al. 2004); ii) initial primary masses distributed with power-law index (see Kroupa et al. 1993); iii) a flat initial mass-ratio distribution; iv) a flat distribution for the logarithm of the semimajor axis, up to a radius of ; v) eccentricities distributed as ; vi) a 50% binary fraction in the initial population of main-sequence stars. We expect a factor of two difference in population-synthesis results obtained using different assumptions and initial parameters than those stated above (Portegies Zwart & Verbunt 1996; Nelemans et al. 2001b, 2004). Each system is described by seven parameters: the binary masses and , the orbital period , the mass-transfer rate , the Galactic latitude and longitude , and the luminosity distance to the Solar System.
Following Nelemans et al. (2004), we distribute
single and binary stars according to a Galactic model
that comprises both disk and bulge components.
where is the cylindrical radius from the Galactic center; results from integrating the time-dependent plane-projected SFR of Boissier & Prantzos (1999) [see Eqs. (1) and (2) of Nelemans et al. (2004)]; is the perpendicular distance from the Galactic plane; and is the disk’s scale height. The density of systems in the spherical bulge is given by
where is the spherical distance from the Galactic center; is the characteristic radius of the bulge; and Eq. (4) is normalized so that there are as many bulge systems as disk systems in the inner 3 kpc of the Galaxy, with a current total mass of .
For the first four scenarios we set and (Jurić et al., 2008; Nelemans et al., 2004; Binney & Tremaine, 1987; Sofue et al., 2009); is fit well by a density with kpc for DDWD systems and kpc for AM CVn systems. In the last scenario we explore a modified model of the Galactic disk with a larger scale height for binaries that are produced at earlier times ( Gyr ago).
For all but one scenarios we use the DDWD catalog analyzed in Nelemans et al. (2004), with local density , and orbital periods as large as 5.5 hrs ( Hz). Binaries at longer periods are not typically detectable by GW instruments because detector noise is higher at low frequencies; they are therefore not considered further in this analysis. All the catalogs of AM CVn systems, by contrast, show the effects of our different assumptions on AM CVn formation:
- Case 1
upper bound (strong tidal coupling and inefficient ELD events). This catalog includes AM CVn systems that have formed via the DDWD channel operating with strong tidal coupling and the He star–WD channel with few ELD events (see Nelemans et al. 2001b, 2004). Although the resulting AM CVn local density is higher () than SDSS-estimated values, past investigations of Galactic WD binaries as LISA sources have all used variants of this scenario (see, e.g., Edlund et al. 2005; Timpano et al. 2006). Thus, this case provides a meaningful comparison between our study and previously published results, which may vary depending on their idealizations of “detection,” “resolution,” etc. (see Sec. 4.3).
- Case 2
inefficient formation channels (weak tidal coupling and efficient ELD events). This catalog includes AM CVn systems that have formed via the DDWD channel operating with minimal tidal coupling and the He star–WD channel with many ELD events (see Nelemans et al. 2001b, 2004). The AM CVn local density is , a factor of two greater than SDSS-estimated values (Roelofs et al., 2007a), and fall within the factor of two uncertainty of population-synthesis results.
- Case 3
mild tidal coupling and no ELD events. This catalog includes AM CVn systems that have formed via only the DDWD channel, where mild tidal coupling ( = 2 years) between the WDs allows for stable mass transfer in up to 10% of systems compared to the optimistic channel of Case 1. We assume that the He star–WD channel is suppressed. [However, observations discussed in (Roelofs et al., 2007a) suggest the presence of hot donors, which imply either that the He star–WD channel is active with core He burning having been quenched only recently, or that following the second common-envelope ejection, the initial DDWD orbital separation is so small that the donor WD cannot cool within the gravitational-radiation timescale.] The AM CVn local density is .
- Case 4
fewer progenitor DDWDs. In this scenario we explain the SDSS-estimated AM CVn densities by postulating a 1/5 reduction in the number of their DDWD progenitors. This reduction in DDWD systems is consistent with existing observations and with the factor of two uncertainty introduced in population-synthesis models by our limited understanding of binary evolution and Galactic structure. This catalog also includes AM CVn systems that have formed both via the DDWD channel with mild tidal coupling ( = 1 year) and via the pessimistic He star–WD channel (see Nelemans et al. 2001b, 2004); their densities are also reduced by 1/5 (across all periods). The DDWD and AM CVn local densities are and .
- Case 5
modified Galactic-disk model. In this scenario we place older systems in a thicker disk. With the caveat of small-number statistics, AM CVn systems identified by emission lines in SDSS have inferred projected velocities km s, hinting that AM CVn systems with longer periods may originate from a disk with a thick component, or at least from a thin disk with a larger scale height. Indeed, such a distribution of AM CVns at birth is consistent with our current (if everchanging) understanding of the spatial and temporal evolution of thin and thick galaxy disks (Scannapieco et al., 2011; Jurić et al., 2008; Schönrich & Binney, 2009a, b; Siebert et al., 2011; Bovy et al., 2012). Observational constraints are unfortunately limited (see Napiwotzki 2009 for a discussion on the thin-disk, thick-disk, and halo populations of single WDs). As an illustrative example, we altered the disk component of Eq. 3 to have for systems older than 7 Gyr, for systems younger than 7 Gyr, and as for cases 1–4. This effectively places all systems older than 7 Gyr in a thick disk, which includes the great majority of longer-period systems. In addition, we assumed that AM CVn systems only formed via the DDWD channel. The DDWD and AM CVn local densities are and .
For each scenario, Table 1 shows the numbers of AM CVn and DDWD systems in the catalogs, as well as their local density (i.e., the Galactic mid-plane density at 8.5 kpc from the Galactic center).
|number in catalog / local space density |
|DDWD||AM CVn (DDWD ch.)||AM CVn (He-star–WD ch.)|
|case 3||/||/||none /||0|
|case 5||/||/||none /||0|
4.2. Detecting gravitational waves from WD binaries
Compact Galactic binaries in the low-frequency band emit quasi-monochromatic GWs, with frequency drifts of (at most) several frequency bins for year-long observation. In this paper, we model compact-binary waveforms by including only the dominant quadrupolar emission (Peters & Mathews, 1963), with instantaneous strain amplitude
where is the redshifted chirp mass (with the total mass, the reduced mass, the chirp mass, and the redshift), is the luminosity distance to the source, and is the orbital period of the binary.
Figure 1 shows the number density as a function of [the GW strain of Eq. (5)], and [the GW frequency], for the different DDWD and AM CVn catalogs of each of our scenarios. The colored dots indicate detected binaries (see Sec. 4.3): green shows DDWDs, red shows AM CVn systems (from the DDWD channel) and blue shows AM CVn systems (from the He-star–WD channel).
Binaries with orbital periods shorter than approximately 20 minutes will appear as isolated signals in year-long datasets of space-based GW observatories, and will be detectable when the signals are sufficiently strong compared to instrument noise at the same frequency. By contrast, most binaries with longer orbital periods will not be detected individually, because they will be too dense in frequency space to tell apart. These will produce a noise-like confusion foreground that will affect the detection and parameter estimation of extra-Galactic GW sources such as massive–black-hole binaries. Nevertheless, the brightest binaries will be detected above the confusion foreground.
Looking at Fig. 1, we see that DDWD systems have larger densities than AM CVn systems, and larger GW strains (because they have greater chirp masses). Thus it is DDWD systems that are principally responsible for the shape and strength of the GW foreground from WD binaries (see Sec. 5.2). We note also that all AM CVn populations show an overall low-frequency cutoff at , due to the age of the Universe.
A simplified (but roughly correct) description of GW data analysis is that GW signals will be detectable when their signal-to-noise ratio, SNR, is greater than a detection threshold chosen to yield a minimal number of false alarms. SNR is the ratio of signal strength to rms instrument noise (plus confusion noise, if present) at the same frequency. For LISA-like interferometric detectors, the yearly orbit and rotation of the spacecraft constellation imprint the signals with frequency and amplitude modulations that encode the sky position of the source. Detector response is further affected in frequency-dependent fashion by Time-Delay Interferometry (TDI), the technique used to subtract the otherwise overwhelming laser frequency noise (thus, SNR would be evaluated for the TDI signal with respect to TDI noise). See App. B for an overview of LISA-like detectors, their response to GWs, and the theory of probabilistic signal detection in noise.
4.3. Simulations of detection and subtraction
To estimate the number of Galactic-binary systems that we will be able to detect individually with our reference detectors, as well as the confusion noise from the residual unsubtracted systems, we performed simulations of detection and subtraction using a variant of the method introduced by Timpano et al. (2006) (see Sec. 5.1). To wit:
For each of our scenarios we compute the frequency-domain TDI response for each system in the corresponding catalog, using an optimized variant (Vallisneri, 2011) of the Mock LISA Data Challenge fast-binary code (Cornish & Littenberg, 2007); we then superpose the responses to produce a full year-long dataset.
We run through the catalog and evaluate the SNR of each system (see Sec. B) with respect to a background of instrumental noise plus confusion noise from all nearby systems, estimated as a running median of their power spectral density. We flag as detected the systems with .
We subtract the signals detected in step 2 from the dataset. We then go back to step 2 and iterate, using the resulting lower estimate of confusion noise.
The process converges after a few iterations, with the number of detected systems decreasing better than exponentially. We then evaluate and fit the power spectral density from the residual confusion foreground.
This process represents a rather optimistic model of source detection and subtraction that neglects two important effects: source detection will be confused by the presence of nearby signals beyond what is predicted by the increased noise level, and subtraction will be degraded by the imperfect estimation of parameters in noise (see, e.g., Crowder & Cornish, 2004). Nevertheless, we can adjust the level of optimism by varying our assumptions regarding the duration of observation , the detection threshold , and the number of TDI observables that are available (two, in effect, for the standard LISA configuration with five or six operating links; one for a two-arm configuration).
It turns out that our results for the number of detected sources and for the residual confusion foregrounds are fit remarkably well by power laws of a single effective-SNR parameter . As a baseline we take a standard assumption, consistent with prior literature, of one year, one observable (the standard unequal-arm Michelson ), and . A conservative assumption would correspond to taking one year, one observable, and , yielding ; an optimistic assumption to five years, two independent observables, and , yielding . Results are not sensitive to details such as the choice of running-median window, and whether signals are subtracted immediately or after each full catalog run-through.
In this section we present our results on the impact that different Galactic populations of DDWD and AM CVn systems have on the number of individually detectable sources (Sec. 5.1) and on the total and confusion GW foregrounds (Sec. 5.2), for our two representative space-based GW detectors and for each of our five population scenarios.
Figure 2 recapitulates the findings of our simulations: it shows the frequency-space density of catalog sources; the power spectral density of the unsubtracted (blue) and subtracted (red) confusion foreground, compared to the instrument noise (black, all in the bottom panel); and the location and numbers of detected sources (dots in the bottom panel, histograms in the top panels). The detected sources lie approximately a factor of above the composite of instrument noise and subtracted confusion noise.
5.1. Individually detected sources
Tables 2 (for the 5 Mkm detector) and 3 (for the 1 Mkm detector) show the number of recovered systems of each type in our five scenarios. The numbers quoted correspond to simulation results for one year, one interferometric observable, and ; the power laws give approximate fits as functions of .
Consider first the number of detectable sources as seen by the longer, standard-LISA–like detector. In all cases except the “fewer progenitors” scenario 4, the number of detected DDWDs remains comparable to prior optimistic projections (). Even in case 4, where the DDWD systems in the catalog are reduced by a factor of five, detections drop only by half, thanks to a weaker confusion foreground. By contrast, the number of detected AM CVn varies considerably depending on their density and formation scenarios (). In cases 2 and 3, more DDWDs are detectable at relatively high frequencies because of the reduced confusion noise from fewer AM CVn.
Case 1 was chosen to be directly comparable to the standard population catalogs of earlier works (Nelemans et al., 2001a, b, c, 2004), but the claimed number of detections for AM CVn systems is significantly lower than those earlier estimates, derived with a simple resolvability criterion of a single source per frequency bin with instrument-noise (Nelemans et al., 2001b, 2004). This is unsurprising, given that in our analysis we model detection by identifying and subtracting bright sources iteratively, and that we require a higher SNR, defined with respect to both instrument and confusion noise. Furthermore, in this paper we use proper interferometric observables instead of “raw” GW strain.
Downscaling the detector’s armlength to 1 Mkm approximately halves the number of DDWD systems detected for , except for case 4, where the decrease is . The number of AM CVn system detections is likewise reduced by factors . Note however that the scalings of Tables 2 and 3 are slightly different. The ESA eLISA/NGO study (Amaro-Seoane et al., 2012) finds numbers compatible with ours, once we take into account our noise assumptions for the 1 Mkm detector, which are somewhat more favorable than NGO’s.
|DDWD||AM CVn (DDWD)||AM CVn (He–WD)|
|DDWD||AM CVn (DDWD)||AM CVn (He–WD)|
5.2. Unsubtracted and residual compact-binary foregrounds
In Appendix A, we apply a familiar analytical treatment of the compact-binary GW foreground to a broad range of Galactic models examined in the literature, and we conclude that the foreground depends strongly on the Galactic binary merger rate and on their characteristic chirp masses, but only weakly on other parameters. Namely,
where is the characteristic distance from Earth to Galactic binaries (a weak function of the model); is the binary merger rate in the Galaxy; and is the characteristic chirp mass . The slope describes the power emitted at GW frequency by a single binary evolving due to GWs, and it applies to a stationary ensemble where new binaries are born with constant rate (equal to ) at frequencies below the detector bandwidth. Equation (6) is normalized to values representative of our assumptions; as shown in the left panel of Fig. 3, it is in good agreement with the results of our simulations for cases 1–3 and 5, while the coefficient is reduced by a factor of five for case 4.
The residual confusion foreground consists of the unresolvable DDWD and AM CVn systems that are left once relatively brighter or isolated sources have been subtracted iteratively from the original population catalog. The residual foregrounds found in our simulations are very similar among cases 1–3 and 5, while they are a factor of five weaker for case 4. We note that the overall number of DDWD systems in cases 1–3 and 5 differ by a factor of five, thereby confirming earlier work and our predictions that the DDWD population provides the dominant component to the strength of the confusion foregrounds (both unsubtracted and subtracted). As shown in the middle and right panels of Fig. 3, for both the 5 Mkm and 1 Mkm detectors, the total equivalent strain noise (instrument noise plus confusion foreground) is fit well by the expression
where , ,
is the frequency-domain density of binary systems, which modulates the transition from the confused to the resolved regime. Equation (7) is formulated in analogy to the standard expression Eq. (C1) [see (Barack & Cutler, 2004a, b)], with a different functional form that fits our simulations better. In App. C we give more precise and fits as functions of , and we provide expressions for the total TDI noise and for the seasonal modulation of the GW foreground due to detector motion (Giampieri & Polnarev, 1997; Seto, 2004; Edlund et al., 2005).
In this work, we have examined the impact that different observationally inferred populations of Galactic white dwarf binaries have on the number of individually detectable sources and on the confusion foreground for future low-frequency GW observations. Earlier studies of Galactic white dwarf binaries as GW sources relied primarily on population synthesis, because of the small number of observed systems, in particular mass transferring ones (i.e., AM CVn systems). However, the past several years have seen a flurry of new observations, quintupling the known detached systems that will merge within a Hubble time, doubling the known AM CVn systems, and leading to new, lower estimates for the local densities of the latter.
To understand the consequences of these new estimates on GW detection, we have generated a range of plausible populations of white dwarf binaries, assuming different AM CVn formation scenarios. For example, in one model (case 3) we postulate that fewer AM CVn systems are observed by SDSS than predicted because the tidal coupling between the spin of the accretor and the orbit is not as efficient as expected, so more systems are lost to mergers. We have then investigated the resulting numbers of individually resolvable GW sources and the magnitude of the total and residual GW foregrounds.
In four out of five models, we find that the number of detectable detached systems is 12,000–14,000, for one year of observation by a LISA-like GW detector with 5 Mkm arms (using an optimistic detection threshold, but considering a single interferometric observable). These four models leave unchanged the number of detached binaries used in earlier estimates, but make various assumptions about the number of systems lost to mergers or to edge-lit detonations, or (in case 5) about the spatial distribution of binaries. In one model (case 4), which instead explains the observed local AM CVn density with five times fewer detached progenitor systems, we find that 6,000 of the latter should be detectable. For the 1-Mkm-arm detector, DDWD detections would be 6,000 and 1,500 for cases 1-3 and 5, and case 4, respectively. The number of detectable AM CVn systems ranges from several tens to a few hundred (for cases 2–4), or even as high 2,000 (for cases 1 and 5), again for the 5-Mkm detector. For the 1-Mkm detector these numbers are smaller by approximately a factor of three to four. See Tables 2 and 3 for details.
Now, why is it that even in our upper-bound model (case 1), where the total number of AM CVn systems is comparable to earlier estimates, we project far fewer AM CVn detections than the 10,000 quoted by Nelemans et al. (2001b, 2004)? Conversely, in four out of five models we project more detached-binary detections (12,000–14,000 vs. 10,000) than Nelemans et al. (2001b). Such differences arise because we define detection by way of an iterative identify-and-subtract process, where a source’s signal-to-noise ratio is evaluated with respect to noise from both the instrument and the partially subtracted foreground. Our approach is similar to that used by Timpano et al. (2006). By contrast, according to Nelemans et al. (2001b, 2004), a source is detectable if it has a signal-to-noise ratio (relative to instrument noise alone) greater than five, or a signal-to-noise ratio greater than one with no other source in the same frequency bin. Our more stringent definition impacts the number of detected AM CVn systems more than their detached counterparts at the same frequencies, because AM CVn systems have smaller chirp masses, and therefore relatively weaker GW-strain amplitudes.
Indeed, we confirm that the dominant contribution to the diffuse GW foreground comes from detached binaries, which have larger chirp masses. In App. A, we show that the spectrum of the unsubtracted foreground can be derived robustly by simple analytical arguments, and depends mainly on the Galactic merger rate and the characteristic binary chirp mass [see Eq. (6)]. These analytical estimates agree well with our numerical results. In addition, for both the 5 Mkm and 1 Mkm detectors we provide fitting expressions [Eqs. (7)–(9)] for the residual confusion foreground due to unresolvable sources. Our estimates are comparable to (but measurably different from) the standard formulas discussed by Barack & Cutler (2004b) and used by many authors in the past.
The projected detection numbers and residual confusion foregrounds change significantly if we use different assumptions on the detection threshold, the duration of observation, and the number of available interferometric observables. We have simulated the identify-and-subtract process for several such assumptions, and we find empirically that the detection numbers and confusion-foreground fitting parameters scale as power laws of a single effective threshold .
In one of our models (case 5), we have examined (to our knowledge, for the first time) how GW observations would change if we modify the evolution and structure of the Galactic disk to match the observed local AM CVn density, while keeping overall binary numbers as in case 1. We find no significant impact on the number of detectable systems, nor on the strength or shape of the confusion foreground. However, on average the detected systems will have measurably smaller GW amplitudes, because of their larger distances from the Solar system. We conjecture that such distance measurements could be used to infer the structure of the Galactic disk.
Finally, we expect that the advent of wide-field synoptic and targeted surveys (such as the Palomar Transient Factory and the ELM survey respectively), as well as ESA’s GAIA mission, will continue to provide new data to this field of research, and will enable more stringent constraints on binary populations. So far, the observed density of detached systems appears to be consistent with theoretical estimates, but a detailed population analysis based on the ELM survey has yet to be undertaken. Furthermore, all estimates of AM CVn densities based on surveys are made uncertain by the intricate selection effects. Because of these uncertainties, in this paper we chose to examine a variety of population models that span the range of likely outcomes. Our overall conclusion is that the Galactic-binary science enabled by low-frequency GW detectors is very robust, for a broad range of possible binary populations and detector sensitivities.
Appendix A Previous estimates of GW emission from Galactic binaries
|reference||model||source classes||input population|
|this work, Nelemans et al. (2004)||RS + CB||DDWD + AMCVn||SeBa|
|Nelemans et al. (2001b)||RS + CB||DDWD + AMCVn||SeBa|
|Edlund et al. (2005)||RS + CB||DDWD + AMCVn||SeBa|
|Timpano et al. (2006)||RS + CB||DDWD||SeBa, Webbink (1984)|
|Seto (2002)||RS + CB||DDWD||Webbink (1984)|
|Bender & Hils (1997)||RS + CB||DDWD||Webbink (1984)|
|Hils et al. (1990)||CB||DDWD||Webbink (1984)|
|Hils & Bender (2000)||CB||AMCVn||Iben & Tutukov (1984); Tutukov & Yungelson (1996)|
|Hils (1998)||RS + CB||AMCVn||Iben & Tutukov (1984); Tutukov & Yungelson (1996)|
|Yu & Jeffery (2010)||RS + CB||DDWD (+ halo)||BSE + Han (1998)|
|Liu (2009); Liu et al. (2010)||RS + CB||DDWD||BSE + Han (1998)|
|Farmer & Phinney (2003)||CB||extragalactic DDWD||BSE|
|+ AM CVn|
|Webbink & Han (1998)||CB||DDWD||Han (1998)|
|Ruiter et al. (2010)||RS + CB||DDWD + AMCVn||StarTrack, Belczynski et al. (2002, 2008)|
|Ruiter et al. (2009)||(halo)|
|Benacquista et al. (2004)||RB + CB||DDWD (+ globular)||Specified in paper|
|Postnov & Prokhorov (1998)||CB||DDWD||See Lipunov et al. (1996)|
|Lipunov et al. (1995)||CB||DDWD (+ halo)||specified in paper|
|Lipunov & Postnov (1987)||CB||compact binaries||specified in paper|
|Lipunov et al. (1987)|
|Giampieri & Polnarev (1997)||CB||generic||generic|
|Evans et al. (1987)||CB||DDWD||specified in paper|
|BSE: Hurley et al. (2000, 2002).|
|SeBa: Portegies Zwart & Verbunt (1996); Portegies Zwart & Yungelson (1998); Nelemans et al. (2001c); Toonen et al. (2011).|
|Notes: RS = resolved sources; CB = confusion background.|
|this work, Nelemans et al. (2004)||see Sec. 4.1||0.3||2.5||6.4||0.5|
|this work (thick disk)||1.25||2.5||6.8||0.5|
|Nelemans et al. (2001b)||0.2||2.5||5.3||–|
|Edlund et al. (2005)|
|Timpano et al. (2006)||see Hils & Bender (2000) and Nelemans et al. (2004)|
|Bender & Hils (1997)||0.09||3.5||4.8||–|
|Hils et al. (1990)||0.09||3.5||4.8||–|
|Hils & Bender (2000)||ns||ns||ns||–||ns|
|Yu & Jeffery (2010)||0.352||2.5||5.6||0.5|
|Liu et al. (2010); Liu (2009)||0.09||2.5||5.0||–|
|Farmer & Phinney (2003)||–||–||–||–||–|
|Webbink & Han (1998)||0.09||4.0||4.7||–|
|Ruiter et al. (2010)||0.2||2.5||5.4||0.5|
|Ruiter et al. (2009)||(halo)|
|Benacquista et al. (2004)||0.2||2.5||5.3||–|
|Postnov & Prokhorov (1998)||4.2||5||7.2||–|
|Lipunov et al. (1995)||0.46||4.6||5.7||see paper|
|Lipunov & Postnov (1987)||4.2||10||9.6||–|
|Lipunov et al. (1987)|
|Giampieri & Polnarev (1997)||0.1||3.5||4.5||ns|
|Evans et al. (1987)||constant density||0.2||10||–|
|computed for disk + bulge for this case.|
|Numerical fit, DDWD systems only.|
|Exponential squared bulge unless otherwise specified: with .|
|Notes: ns = not specified. Distance to the Galactic Center is assumed to be 8.5 kpc for all models.|
More than two dozen papers have made estimates of the GW signal from individually detectable Galactic compact binaries and from the continuum of unresolvable sources at low frequencies (variously called “confusion background” or “Galactic foreground”) for a LISA-like GW detector with 5-Mkm arms. An earlier informative review of the literature is given by Ruiter et al. (2010). We note that for a 1-Mkm detector the foreground decreases substantially, to a level comparable to instrument noise (see Fig. 3 in Section 5.2).
Table 4 compares a subset (large, but possibly not complete) of relevant published estimates, grouped by population model. For each entry, we show whether the estimates include resolved sources (RS), the continuum background (CB), or both; we show also which classes of binaries are included. In Table 5 we provide relevant scale lengths for the Galactic disk and bulge distributions, where available; these are indicative of the range of assumed Galactic models, but they do not provide a complete picture since different density laws also appear. We do not give quantitative information on possible halo components, which are also frequently used. The major differences between the various estimates arise from two sets of assumptions:
Assumptions about the relative numbers of systems in terms of total mass, mass ratio, and separation. These arise from different treatments of compact-binary progenitor evolution and of the stability of mass-transferring systems. Approaches include both analytic estimates and Monte Carlo population-synthesis calculations.
Assumptions about Galactic structure and population space densities. These can affect estimates in two ways: first, the magnitude of GW signals will depend on the scale size assumed for the Galactic population; second, the “calibration” of the local number density of sources using observations of a particular source type will depend sensitively on the assumptions about Galactic structure.
We now present a semi-analytical derivation of the amplitude and shape of the GW continuum. Similar models have been used extensively in GW data analysis, see for example works by Hughes (2002); Barack & Cutler (2004a, b). Our results show that the magnitude and spectral shape of the continuum are a robust function of the merger rate and characteristic chirp mass of Galactic binary systems, and depend only weakly on Galactic structure.
We follow the arguments presented in Evans et al. (1987) and Giampieri & Polnarev (1997). The GW luminosity of a circular binary and the change in its orbital energy as a function of GW frequency are given by (Peters & Mathews, 1963)
where is the energy of the system and is the GW frequency. Combining these expressions yields the frequency evolution of the system,
If is the birth rate per unit frequency and volume of systems with , in a stationary regime the density of sources must be
By continuity, must also be equal to the merger rate, , of systems which merge at . DDWD systems, which generate most of the GW foreground, merge at Hz, and most of them are born below a few Hz (although estimates vary). Thus, if we restrict Eq. (A3) between those two frequencies, we may just set , defined as the total merger rate above Hz.
Now, the direction-averaged GW strain from a circular binary is given by
see for instance (Giampieri & Polnarev, 1997), where is the luminosity distance to the binary; the corresponding detector response is
where and are the antenna-pattern functions (see, e.g., Maggiore 2007) and is the polarization- and inclination-averaged antenna pattern in the direction .
We combine Eqs. A3, A4, and A5 to obtain the one-sided detector response density . The yearly orbital motion of LISA-like detectors modulates the amplitude of signals to individual sources; this effect averages out for isotropic source distributions, but not for the anisotropic Galactic-binary foreground (Giampieri & Polnarev, 1997; Edlund et al., 2005). We therefore average over a year (while assuming for now that all binaries have the same chirp mass ), to obtain
Of the two quantities defined in the last line of Eq. (A6), accounts for the time-, merger-rate- and direction-averaged instrument response, and it depends on the orbit and geometry of the detector. For an isotropic distribution of sources and a triangular interferometer with angles, , which is often taken as the average isotropic response for LISA-like detectors (Giampieri & Polnarev 1997). The factor 1/5 is the average of over source locations and polarizations (again, e.g., in Maggiore 2007). Again for a LISA-like detector, but for typical disk populations, is found empirically to vary by from 3/20.
The quantity is the merger-rate–averaged inverse squared distance. For a Galactic disk, it depends on the scale height and radial scale length, and of course on the position of the detector within the disk. A typical disk density distribution considered in many papers is the squared–hyperbolic-secant disk model,
where is the total merger rate in the Galaxy and is the volume of the Galaxy. We find empirically that varies slowly for and . Table 5 summarizes the wide range of Galactic models used in the literature. For most disk models the characteristic distance to Galactic sources is approximately 5 to 6 kpc, and is close to 8.5 kpc, the assumed distance to the Galactic center.
Inserting numerical values appropriate for our simulations (cases 1–3) in Eq. (A6) yields
where is the characteristic chirp mass . The power spectral density of the GW strain is obtained by dropping the factor (which is close to 3/20 for LISA-like detector). Equation (A8) is in good agreement with the results of our simulations. Note however that the spectra of Fig. 2 are plotted for the TDI observable (rather than for strain or strain response ), so they must be compared with
where , and is the detector armlength.
Our analysis indicates that and are relatively insensitive to the structure of the Galactic disk, consistent with the results of our simulations, and instead depend strongly on the merger rate and the characteristic chirp mass . By constrast, we expect that the local population density of sources, as well as the number of resolvable source as a function of their GW strain, will depend roughly linearly on the disk’s scale height.
The transition between the confused and resolved detection regimes is characterized by , which is determined by the evolution of the chirp frequency and by the absolute merger-rate density of sources. Integrating Eq. (A3) over we find
consistent with our simulations.
This simple analytical derivation of the GW continuum shows that the major differences between the various estimates listed in Table 4 arise primarily from two sources: the total merger rate of systems and the composition of the source population, leading to a different characteristic chirp mass . is also relevant, but less important. (Remember however that our derivation is only predictive for frequencies high enough that most binaries form at longer periods—for LISA-like detectors, this is true above Hz. For lower frequencies, the shape of the continuum depends on the detailed accounting of birth rate vs. period.)
Appendix B Low-frequency GW interferometers and their response to binary waves
LISA (Prince et al., 2009; Jennrich, 2009), the space-based GW observatory studied jointly by NASA and ESA until early 2011, comprises three spacecraft orbiting the Sun in a quasi-equilateral configuration with armlengths km. LISA measures GW strain by monitoring the oscillating distance between freely-falling test masses, using heterodyne laser interferometry. The spacecraft hover around the test masses to protect them from external perturbations, performing slight orbital corrections with extremely precise micro-Newton thrusters.
Although LISA is sometimes described as a large Michelson interferometer in space, the lasers exchanged by spacecraft are neither split nor reflected. Rather, two lasers on each spacecraft are used to establish six one-way links; each link provides a test-mass–referenced interferometric measurement of the phase (or equivalently, the frequency) of the incoming laser, compared to the local, outgoing beam. In terms of fractional laser frequency, the effect of a plane GW on the links is given by the two-pulse response (Estabrook & Wahlquist, 1975)
where is the unit vector from spacecraft to ; the GW propagation vector; the TT-gauge GW strain (see, e.g., Maggiore 2007) at the spacecraft position and at time ; and the photon time of flight along the arm. Thus, a GW pulse is registered twice in : once when it impinges on spacecraft , and a time after it has impinged on spacecraft . In the limit of long GW wavelengths , the response to plane GWs is therefore proportional to L.
Each measurement is also affected by displacement noise from residual test-mass accelerations, position noise from optical-path and shot noise in the heterodyne measurements, and, most importantly, by oscillations in the central laser frequency, which are many orders of magnitude stronger than expected GW imprints, and which appear as , with the frequency noises for the six lasers. Laser frequency noise is removed by combining multiple one-way measurements with appropriate time delays so that each laser appears in canceling pairs, as in the unequal-arm Michelson observable
This noise-suppression technique, specific to LISA-like interferometers, is known as time-delay interferometry (TDI; see, e.g., Vallisneri 2005 for a review). TDI observables can be seen as synthesized interferometers, since their components trace the paths (and reproduce the phase delays) of light across recognizable interferometric configurations (hence ’s “Michelson” designation). Many different observables are possible with LISA’s six laser links, but all can be reconstructed from a set of three observables, such as the unequal-arm Michelson , , and centered on the three spacecraft (Vallisneri et al., 2008).
In this paper, we assume an rms displacement noise
for each of the proof masses, and a total rms position noise
for each measurement.
The GW mission currently under study as an ESA-led project (Amaro-Seoane et al., 2012) would decrease cost primarily by reducing spacecraft and propellant mass, allowing for launch on smaller rockets than envisaged for LISA. Propellant is saved by placing the spacecraft closer together and closer to Earth. At the low-frequency end, the shorter armlengths reduce the response to GWs proportionally; for the same displacement noise, sensitivity also decreases by the same ratio. At the high-frequency end, the laser power available for position measurement increases as , since beams are broadly defocused at millions of kms, improving shot noise (but not other optical noises) by an rms factor . As a consequence, the sweet spot of the LISA sensitivity shifts to higher frequencies.
Spacecraft mass may also be saved by adopting a two-arm “mother–daughters” configuration with only four laser links, two each between the middle “mother” spacecraft and one of the “daughters”. With four links, only one independent TDI observable can be formed, with a loss in sensitivity averaging at compact-binary frequencies. Because no trade studies have yet been performed to optimize the payload for the new orbits, in this paper we consider a “short LISA” configuration with aggressively reduced -km armlengths, and with the LISA displacement and position noises (except for the power rescaling of the position-noise component, which we assume as five times smaller in rms). The actual sensitivity of a cost-constrained space-based GW observatory is likely to fall somewhere between this short LISA and the classic configuration.
Searches for GWs of known shape proceed by correlating detector data with theoretical waveform templates computed for the expected range of source parameters. The statistical confidence of detection for a GW signal immersed in Gaussian additive noise is described by the signal-to-noise ratio (SNR) of the data after filtering by the best-fitting normalized template . For a single TDI observable , this is given by (Cutler & Flanagan, 1994)
where and are the Fourier transforms of the TDI data and of the TDI response to waveform respectively; “” denotes the complex conjugate; and is the noise power spectral density of TDI observable . Normalization amounts to requiring .
With this noise weighting, the false-alarm probability of SNR exceeding a chosen detection threshold is proportional to . Because the many templates used in typical searches amount to repeated independent trials, is routinely set to relatively large values (e.g., 5–10) to yield one false alarm per many years of data. The detectability of a signal is characterized by its optimal SNR , which is the SNR that would be obtained on average over noise realizations for a perfectly matching template.
The noise includes the confusion-noise foreground from unresolved sources (see Sec. 5), as well as instrument noise . For the unequal-arm Michelson observable in the (rather reasonable) limit of equal interferometer armlengths, is given by (Vallisneri et al., 2008)
In these units , .
To obtain the TDI response to a GW signal Eq. (5), we
insert Eq. (5), multiplied by the appropriate phasing
function and sky-position–dependent polarization tensors, into Eq. (B2). The response is modulated further by the
heliocentric motion of the detector (which yields a time-dependent
Doppler shift) and its yearly rotation in a plane inclined 60
from the ecliptic (which generates sidebands at frequency multiples of