Retrieving the True Masses of Gravitational-wave Sources

Gravitational waves (GWs) encode important information about the mass of the source. For binary black holes (BBHs), the templates that are used to retrieve the masses normally are developed under the assumption of a vacuum environment. However, theories suggest that some BBHs form in gas-rich environments. Here we study the effect of hydrodynamic drag on the chirp signal of a stellar-mass BBH and the impact on the measurement of the mass. Based on theoretical arguments, we show that the waveform of a BBH in gas resembles that of a more massive BBH residing in a vacuum. The effect is important for LISA sources but negligible for LIGO/Virgo binaries. Furthermore, we carry out a matched-filtering search of the best fitting parameters. We find that the best-fit chirp mass could be significantly greater than the real mass if the gas effect is not appropriately accounted for. Our results have important implications for the future joint observation of BBHs using both ground- and space-based detectors.


I. INTRODUCTION
Measuring the mass of a gravitational wave (GW) source is an old but difficult problem because the observable is not mass, but the phase and amplitude of GWs. A model is needed to translate the observables to the mass. However, the standard model that is used in the current GW observations often neglects the astrophysical factors which could affect the dynamical evolution of the source or the propagation of the GWs. Consequently, the standard model could potentially misinterpret the GW signal.
For example, redshift is such a factor [1,2]. It stretches the waveform so that a low-mass source at high redshift looks identical to a massive one at low redshift. Such a "mass-redshift" degeneracy has become a serous issue because the Laser Interferometer Gravitational-wave Observatory (LIGO) and the Virgo detectors have detected seemly over-massive black holes (BHs) [3,4], which are two to three times bigger than those BHs previously detected in X-ray binaries [5,6]. On one hand, the high masses may be real and reflect the peculiarity of the environment in which the BHs form [7][8][9][10][11][12]. Alternative, the BHs may be intrinsically small but appear more massive due to a high redshift.
The high redshift could be explained in two astrophysical scenarios. One possibility is that binary BHs (BBHs) coalesce at high cosmological redshift and get strongly lensed by foreground galaxies or galaxy clusters [13,14]. This scenario, although possible for a small number of of BBHs, could not account for all the massive BHs detected so far by LIGO/Virgo [3,15]. Another possibility is that BBHs merge in places very close to supermassive black holes (SMBHs), so that both the Doppler and * Corresponding author: xian.chen@pku.edu.cn gravitational redshift become significant [2,16,17]. The problem with this scenario is that the event rate is difficult to estimate because the stellar distribution around SMBHs is not well constrained by observations [2].
Is there a way of distinguishing, on a one-to-one basis, the redshifted BBHs from the intrinsically massive ones? It is difficult using ground-based detectors. The corresponding signals usually last no more than one second, too short to reveal any signature of gravitational lensing or a nearby SMBH. However, with a space-borne GW detector, such as the Laser Interferometer Space Antenna (LISA [18]), the answer would be different. Being sensitive to milli-Hertz (mHz) GWs, LISA could detect BBHs at a much earlier evolutionary stage, weeks to millenniums before they enter the LIGO/Virgo band [19][20][21]. The corresponding signals could be as long as the lifetime of LISA, about 4 − 5 years. Earlier studies showed that if a BBH is strongly lensed, LISA could detect multiple images of the source [22,23] or, in some rare cases, detect a shift of the GW phase caused by the wave effect of gravitational lensing [24,25]. If, on the other hand, a BBH is close to a SMBH, LISA could detect a distortion of the waveform caused by either the orbital motion of the binary around the SMBH [26][27][28][29][30] or the tidal force of the SMBH [27,[31][32][33].
Besides redshift, are there other astrophysical factors which could affect the measurement of the masses of GW sources? In this article, we show that the presence of gas around BBHs could also lead to an overestimation of the masses. Investigating this scenario is important because in many theoretical models the merger of stellar-mass BBHs is driven by gas (e.g. [9][10][11][34][35][36][37][38][39] The waveform of a merging BBH can be divided into three parts, corresponding to the inspiral, merger, and ringdown phases [40]. During the inspiral phase, the information of mass is encoded in the GW frequency (f ) and the time derivative of it (ḟ ). For example, consider a BBH whose BH masses are m 1 and m 2 (we assume m 1 ≥ m 2 ). If the binary is in vacuum, the following quantity is equivalent to according to the Newtonian approximation [41], where G is the gravitational constant and c the speed of light. This quantity has the dimension of mass and uniquely determines how the GW frequency increases with time. It is known as the "chirp mass". Gas could make BBHs shrink more rapidly by imposing a hydrodynamical drag on each BH [42]. As a result, the observedḟ would be bigger than that in the vacuum case. Without knowing the gas effect, an observer is likely to apply the vacuum model, i.e., Equation (1), to the observed f andḟ . The derived mass, which we call M o , will be bigger than the real chirp mass.
To see this effect more clearly, we first denote the value ofḟ in the vacuum model asḟ gw and that in the gas model asḟ gas . Furthermore, we define the semi-major axis of a BBH as a and the shrinking rate due to GW radiation asȧ gw . Then we can express the shrinking timescale due to GW radiation as T gw := a/|ȧ gw |. Now suppose the presence of gas causes the binary to shrink at an additional rate ofȧ gas , we can write the gas-drag timescale as T gas := a/|ȧ gas |. Let us further assume for simplicity a circular binary, so that the GW frequency is f = π −1 G(m 1 + m 2 )/a 3 . From the last formula we findḟ gas = (1 + T gw /T gas )ḟ gw . Substitutingḟ gas in Equation (1) forḟ , we find that the "observable" mass is no long the intrinsic chirp mass, but Interestingly, this mass could be much bigger than the real mass when T gas T gw . Now we compare the values of T gas and T gw . For circular binary and in the Newtonian approximation, the GW radiation timescale can be calculated with BBHs with such a semi-major axis could have a gas-drag timescale as short as T gas 10 3 years according to the earlier studies of the BBHs in gaseous environments (e.g. [10]). It is worth noting that T gas is a function of gas density and hence could be even shorter in the most gas-rich environment, such as the innermost part of the accretion disk around a SMBH or the common envelope surrounding a binary [12,35]. From the timescales derived above, we find that for LISA BBHs (f ∼ 1 mHz) it is possible that T gas T gw . For LIGO/Virgo BBHs (f ∼ 10 Hz), gas drag is no longer important because the GW radiation timescale, according to Equation (5), is too short.
To be more quantitative, take T gas = 10 3 years and T gw = 10 4 years for example. We have T gw /T gas = 10. According to Equation (3)

III. MATCHED FILTERING AND PARAMETER ESTIMATION
In practice, LISA employs a technique called the "matched filtering" to estimate the parameters of a GW source [44]. In this technique, the similarity of two waveforms, say h 1 (t) and h 2 (t), is quantified by the "fitting factor" (FF), defined as where h 1 |h 2 denotes an inner product, which can be calculated with In the last equation, the tilde symbols stand for the Fourier transformation and the stars stand for the complex conjugation. The quantity S n (f ) is the spectral noise density of LISA (see details in [17]). An exact match, in principle, would mean FF = 1.
In reality, noise exists and consequently FF is not unity even when h 1 and h 2 are identical. A more practical definition of "match" is that δh|δh < 1, where δh :=h 1 (f ) −h 2 (f ) [45]. There is a close relationship between δh|δh , FF, and the signal-to-noise ratio (SNR) defined as ρ 2 := h|h , which could help simplifying our calculation. We note that in GW observations often we are in the situation where h 1 h 2 . Therefore, we have ρ 2 h 1 |h 1 h 2 |h 2 . Using the last equation, we find that the condition for match is equivalent to Since LISA will claim a detection when ρ 10 [20,46], only those temples with F F > 0.995 are acceptable. In our particular problem, h 1 (t) is the GW signal from a BBH embedded in a gaseous environment, and h 2 (t) is the waveform template that we use to match with h 1 (t) and extract physical parameters. To prepare a template bank for h 2 (t), a model is needed. So far, only the vacuum model has been considered in the literature. In the following, we show that even though the vacuum model is a wrong one, the resulting FF could still be very high. Consequently, using this model will confuse the estimation of the mass of a BBH.
In this work, we compute the waveforms using where A is a normalization factor depending on the source distance but not important for matched filtering, and is the phase of GW. For h 2 (t), i.e., the vacuum model, a(t) and f (t) are computed following a post-Newtonian approximation ( [41]). For h 1 (t), i.e., when there is gas, we add a term 1.5f /T gas to the equation ofḟ to mimic the effect of gas drag. To find the maximum FF, we explore the parameter space of m 1 and φ c while keeping q fixed to 0.7, for simplicity. In a future work, we will complete the analysis by searching in the full parameter space of (m 1 , q, φ c ). Our fiducial parameters are M = 8.7 M , q = 0.7, a(0) = 0.002 AU, and T gas 10 3 years. The values of the first three parameters are chosen such that in vacuum the BBH would merge on a timescale of T gw 10 4 years. Figure 1 shows the resulting FF as a function of the LISA observing time, T s . We find that FF is above 0.995 during the first 1.1 years of observation. The high FF means that the vacuum template gives a reasonable fit to the signal, even though it is the wrong template to use here. The best-fit M o is about 36.94 M , much larger than M. This result agrees well with what we have envisioned in Section II.
When the observing time is longer than 1.1 years, the FF decays to a value below 0.995. This result indicates that LISA would be able to distinguish the BBHs in gaseous environments from those in vacuum, given that the observing time is long enough. The exact time that is needed to reveal the difference depends on the parameters of the BBH, as well as the properties of the surrounding gas. This issue deserves further investigation.

IV. CONCLUSIONS
We have shown that the presence of gas around BBHs could affect the chirp signal and lead to a significant overestimation of the mass of the binaries. This effect is important for LISA observation but negligible for LIGO/Virgo sources. Our results have important implications for the future joint observation of BBHs using both ground and space-borne detectors [47,48].