Confusion noise from LISA capture sources

Captures of compact objects (COs) by massive black holes (MBHs) in galactic nuclei will be an important source for LISA, the proposed space-based gravitational wave (GW) detector. However, a large fraction of captures will not be individually resolvable—either because they are too distant, have unfavorable orientation, or have too many years to go before final plunge—and so will constitute a source of “confusion noise,” obscuring other types of sources. In this paper we estimate the shape and overall magnitude of the GW background energy spectrum generated by CO captures. This energy spectrum immediately translates to a spectral density ${\mathcal{S}}_h^{\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}(f)$ for the amplitude of capture-generated GWs registered by LISA. The overall magnitude of ${\mathcal{S}}_h^{\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}(f)$ is linear in the CO capture rates, which are rather uncertain; therefore we present results for a plausible range of rates. ${\mathcal{S}}_h^{\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}(f)$ includes the contributions from both resolvable and unresolvable captures, and thus represents an upper limit on the confusion noise level. We then estimate what fraction of ${\mathcal{S}}_h^{\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}(f)$ is due to unresolvable sources and hence constitutes confusion noise. We find that almost all of the contribution to ${\mathcal{S}}_h^{\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}(f)$ coming from white dwarf and neutron star captures, and at least $\sim 30\%$ of the contribution from black hole captures, is from sources that cannot be individually resolved. Nevertheless, we show that the impact of capture confusion noise on the total LISA noise curve ranges from insignificant to modest, depending on the rates. Capture rates at the high end of estimated ranges would raise LISA’s overall (effective) noise level $[f{S}_h^{\mathrm{e}\mathrm{f}\mathrm{f}}(f){]}^{1/2}$ by at most a factor $\sim 2$ in the frequency range $1–10\mathrm{m}\mathrm{H}z$, where LISA is most sensitive. While this slightly elevated noise level would somewhat decrease LISA’s sensitivity to other classes of sources, we argue that, overall, this would be a pleasant problem for LISA to have: It would also imply that detection rates for CO captures were at nearly their maximum possible levels (given LISA’s baseline design and the level of confusion noise from galactic white dwarf binaries). This paper also contains, as intermediate steps, several results that should be useful in further studies of LISA capture sources, including (i) a calculation of the total GW energy output from generic inspirals of COs into Kerr MBHs, (ii) an approximate GW energy spectrum for a typical capture, and (iii) an estimate showing that in the population of detected capture sources, roughly half the white dwarfs and a third of the neutron stars will be detected when they still have $\gtrsim 10$ years to go before final plunge.

Figure 1

“Specific binding energy,” $1-E/m$, at plunge, as a function of the plunge eccentricity and inclination angle, for a Kerr BH with spin parameter $a=0.8M$. The quantity $1-E/m$ approximates $\alpha$—the amount of energy (per CO’s mass $m$) emitted to infinity in GWs during the entire inspiral. For the case $a=0.8M$ shown here, and for astrophysically relevant inspirals with $e_{\mathrm{L}\mathrm{S}\mathrm{O}}\lesssim 0.35$, the CO emits between $\sim 4\%$ and $\sim 12\%$ of its mass in GWs (the former value for retrograde, the latter for prograde equatorial orbits).

Figure 2

Specific binding energy at plunge, averaged over (uniformly distributed) inclination angles, as a function of the plunge eccentricity. The plot compares the case $a=0.8$ of Fig. 1, with the case of a nonspinning MBH ($a/M=0$). In the latter case, the plunge energy is given by Eq. (8). An “inclination averaged” inspiral in the astrophysically relevant range of plunge eccentricities ($e_{\mathrm{L}\mathrm{S}\mathrm{O}}\lesssim 0.35$), emits around 5–7% of the CO’s mass in GWs, with only little dependence on the MBH’s spin.

Figure 3

Energy spectrum of a single inspiral, for $M={10}^6{M}_{\odot }$, $a=0$, and for a range of plunge eccentricities $e_{\mathrm{L}\mathrm{S}\mathrm{O}}$. ${\nu }_{\mathrm{L}\mathrm{S}\mathrm{O}}$ is the orbital frequency at the LSO. These spectra are based on approximate, post-Newtonian waveforms as described in the text. Each of the curves represents the sum of contributions from the first 20 harmonics of the orbital frequency. Discontinuities appear whenever an $n$ harmonic reaches frequency $n\times {\nu }_{\mathrm{L}\mathrm{S}\mathrm{O}}$ and cuts off.

Figure 4

Our model spectrum [Eq. (15)] for a single inspiral with $M={10}^6{M}_{\odot }$, averaged over inclination angle and final eccentricity, and normalized to $\alpha =0.06$. The nth GW harmonic cuts off when $f/n$ reaches the plunge frequency, $f_p$, causing the spectrum to flatten out for $f_p\lesssim f\lesssim 4f_p$. We cut off our model spectrum at $f>4f_p$, where we no longer find emission from any of the dominant harmonics ($n\sim 2–4$).

Figure 5

Model energy spectrum for a single capture, averaged over MBH mass and normalized by $\alpha =0.06$—see Eq. (22). Note the apparent sharp drops in the spectrum at frequencies below $\sim {10}^{-3}\mathrm{H}\mathrm{z}$ and above $\sim 2\times {10}^{-2}\mathrm{H}\mathrm{z}$ are simply an artifact of our restricting attention to MBH masses in the range ${10}^5–{10}^7{M}_{\odot }$.

Figure 6

Comparison of the background ${\mathcal{S}}_h^{\mathrm{W}\mathrm{D}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$ from all WD captures (dash-dotted lines) with LISA’s instrumental noise (solid line) and with LISA’s instrumental plus confusion noise from WD binaries (dashed line). We show three cases, corresponding to WD capture rates of ${\kappa }^{\mathrm{W}\mathrm{D}}=4\times {10}^{-6},4\times {10}^{-7}$, and $4\times {10}^{-8}$—cf. Eqs. (2, 3).

Figure 7

Same as in Fig. 6, but this time showing the background ${\mathcal{S}}_h^{\mathrm{N}\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$ from all NS captures. We show two cases, corresponding to NS capture rates of ${\kappa }^{\mathrm{N}\mathrm{S}}=6\times {10}^{-7}$ and $6\times {10}^{-8}$.

Figure 8

Same as in Fig. 6, but this time showing the background ${\mathcal{S}}_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$ from all BH captures. We show two cases, corresponding to BH capture rates of ${\kappa }^{\mathrm{B}\mathrm{H}}=6\times {10}^{-7}$ and $6\times {10}^{-8}$.

Figure 9

Illustration of how the detection rate $\mathcal{D}$ for some source class $\mathcal{C}$ scales with the event rate $\mathcal{R}$ (number of sources per unit time per unit space volume). For sufficiently small $\mathcal{R}$, we have $\mathcal{D}\propto \mathcal{R}$, whereas for very large $\mathcal{R}$ we have $\mathcal{D}\propto {\mathcal{R}}^{-1/2}$. Hence, the detection rate must peak at a certain event rate, ${\mathcal{R}}_c$. ${\mathcal{R}}_c$ is roughly the rate where confusion noise from $\mathcal{C}$-type sources begins to dominate the total noise level.

Figure 10

Illustration of how the confusion noise $S_h^{\mathcal{C}}$ scales with the event rate density $\mathcal{R}$. $S_h^{\mathcal{C}}$ is linear in $\mathcal{R}$ in both the very small $\mathcal{R}$ and very large $\mathcal{R}$ regimes. It depends on $\mathcal{R}$ nonlinearly around the rate ${\mathcal{R}}_c$, where confusion noise begins to dominate the total noise level.

Figure 11

Percentage of GW energy radiated (out of the total energy radiated during the entire inspiral) as a function of the time $\tau$ left to plunge, for typical captures of a WD (upper left panel), a NS (upper right panel), and a BH (lower panel), for a range of MBH masses. The masses of the WD, NS, and BH are taken to be $m=0.6,1.4$, and $10M_{\odot }$, respectively, and in all cases the CO plunges at eccentricity $e_{\mathrm{L}\mathrm{S}\mathrm{O}}=0.15$. A substantial fraction of the energy is emitted long before plunge, when the source is still not detectable by LISA. This energy is mostly emitted in relatively high-frequency bursts near periastron passage, and so contributes to the confusion noise.

Figure 12

GW output from the last 1000 years of inspiral of a captured WD at 1 Gpc. Shown (solid convex curves) are the characteristic amplitudes $h_{c,n}$ [see Eq. (46)] of each of the first 20 harmonics (labeled $n=1,\dots ,20$), for a capture of a $0.6M_{\odot }$ WD by a ${10}^6{M}_{\odot }$ nonspinning MBH, with plunge eccentricity 0.15. The signal curves are superposed on LISA’s instrumental noise (solid concave curve) and LISA’s noise curve including confusion from WD binaries (dashed curve). In each $h_{c,n}$ curve, the right end corresponds to the instant of plunge, the left end corresponds to 1000 years before plunge, and dots along the curve correspond (from right to left) to 10, 100, and 500 years before plunge. Note that even 500 years before plunge, most of the GW energy is radiated “in band”—i.e., in the frequency range where LISA is most sensitive. Although at such early times the orbital frequency is well “below band,” most of the energy radiated then is carried by high harmonics of the orbital frequency.

Figure 13

Combined SNR (from two synthetic Michelsons) as a function of time $\tau$ left to plunge for the three cases shown in Fig. 11, at a fiducial distance of 1 Gpc. Each SNR data point represents the matched-filter SNR for a 3-yr-long waveform segment centered at $\tau$.

Figure 14

Total LISA noise (dash-dotted line), including instrumental noise, confusion noise from WD binaries, and confusion noise from captures of only white dwarfs. The two total noise curves correspond to the higher and lower end of the estimated event rate for WD captures. (For the lower rate, the level of WD-capture confusion noise is low enough that the corresponding total noise curve appears to coincide in this plot with the dashed curve, representing instrumental and WD-binaries confusion noise only.) By virtue of Eq. (48), we have assumed here that the entire capture background ${\mathcal{S}}_h^{\mathrm{W}\mathrm{D}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$ contributes to the confusion noise $S_h^{\mathrm{W}\mathrm{D}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$. The total noise curve has been calculated using Eq. (32).

Figure 15

Same as in Fig. 14, but this time showing the total LISA noise when capture confusion from only neutron star sources is taken into account. By virtue of Eq. (49), we have assumed here that the entire background ${\mathcal{S}}_h^{\mathrm{N}\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$ contributes to the confusion noise $S_h^{\mathrm{N}\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$.

Figure 16

Same as in Fig. 14, but with capture confusion from only black hole sources taken into account. The three dash-dotted lines show the total noise curve under different assumptions as to the event rate ${\kappa }^{\mathrm{B}\mathrm{H}}$ and the fraction of the background that is subtractable. The lower total noise curve refers to the lower end of event rate estimates (${\kappa }^{\mathrm{B}\mathrm{H}}=6\times {10}^{-8}$), with $S_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$ assumed to be $0.3\times {\mathcal{S}}_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$. The middle total noise curve corresponds to the upper end of rate estimates (${\kappa }^{\mathrm{B}\mathrm{H}}=6\times {10}^{-7}$), again with $S_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}=0.3\times {\mathcal{S}}_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$. The upper curve assumes the upper end of rate estimates, but with $S_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}={\mathcal{S}}_h^{\mathrm{B}\mathrm{H}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{t}}$. Thus the upper curve represents an upper limit on the effect of confusion noise from BH captures.

Figure 17

The detection volume $V_{\det }$ times $\tau$ (the time from the middle of the 3-yr observation until plunge), as a function of $\ln \tau$, for our three fiducial sources. The total number of detected captures is proportional to the integral $\int V_{\det }\tau d\ln \tau$. Roughly half (one-third) of the WDs (NSs) that LISA detects will have $\tau >10\mathrm{y}\mathrm{r}$. BH detections will be dominated by sources with $\tau <3\mathrm{y}\mathrm{r}$, but roughly $20\%$ will have $\tau >5\mathrm{y}\mathrm{r}$.