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approximation to the gravitational wave signal is sufficient for detection1. Higher mass systems merge in LIGO’s most sensitive band 100–300 Hz. For these systems, we use templates that include each of the inspiral, merger, and ringdown phases of the coalescence.
All candidate gravitational wave events identified in both searches were consistent with back- ground. In the absence of a confident detection, we place upper limits on the rates of CBCs in the nearby Universe. The inferred upper limits on event rates depend on the assumed distributions of compact binary parameters, which presently are not well-constrained. We therefore considered sev- eral compact binary mass and spin distributions in our calculations, including non-spinning binaries, aligned-spin binaries, precessing-spin binaries (where possible), and a focused study of BNS, NSBH, BBH systems with masses Gaussian-distributed about nominal values for the neutron star and black hole masses (mNS = 1.34±0.05M and mBH = 5±1M). For each population distribution, we measured the mean detectable volume of the search to a simulated population derived from that distribution . The absence of any event above the loudest event in the observed volume then implies an upper limit on the coalescence rate density [77, 78]. The upper limits for CBC rates computed from our gravitational wave observations do not yet constrain even the most optimistic estimates of event rates obtained through other approaches, which were discussed in Sec. 1.4.
In the final section of this chapter, we take a look at the results of the searches in terms of sensitivity to spinning binaries. We note that the searches performed quite well in this regard, matching closely with back-of-the-envelope calculations for the expected sensitive volume. Still there is room for improvement, which we will quantify in Chap. 6. The remainder of this thesis will then focus on demonstrating that some of this sensitive volume can in fact be recovered with the use of spinning templates.
(a) Power spectra (b) Horizon distance
Figure 4.1: Detector sensitivities in S6 and VSR2/3. noise for the LIGO and Virgo detectors in the S6 and Virgo VSR2/3 runs [2]. To the right, we show the horizon distance computed from these spectra as a function ofMtotal for equal-mass non-spinning binaries in S6/VSR2-3 (solid lines). Using the spectra shown in Fig. 2.6, we also compute the horizon distance for the LIGO and Virgo detectors in S5 and VSR1 (dashed lines). The horizon distance is computed with theIMRPhenomBwaveform approximation. We note that while the strain sensitivity improved between S5 and S6 by up to a factor of two, the horizon distance to CBCs increased only∼15% because most of the sensitivity improvement occurred at high frequencies, where the CBC signal spends very little time. On the other hand, the Virgo detector achieved more than a factor of two improvement in horizon distance between VSR1 and VSR2/3, since its greatest improvements were at low frequencies.
a new output mode cleaner on an aLIGO seismic isolation table [79]. In addition, the hydraulic seismic isolation system was improved by fine tuning its feed-forward path. In the period between Virgo’s first science run (VSR1) and Virgo’s second science run (VSR2), several enhancements were made to the Virgo detector. Specifically, a more powerful laser was installed, along with a thermal compensation system and improved scattered light mitigation. During early 2010, monolithic sus- pensions were installed, which involved replacing Virgo’s test masses with new mirrors hung from fused-silica fibers [80]. Virgo’s third science run (VSR3) followed this upgrade. In Fig. 4.1a, we show “representative” (see Ref. [2]) strain spectral densities for each of the three detectors operating during the S6 and VSR2/3.
A convenient measure of the sensitivity of a detector to CBC signals is thehorizon distance, which is the distance at which an optimally oriented and optimally sky-positioned compact binary would produce a signal with expected SNR of hρi= 8 in that detector. The horizon distance combines the detector strain sensitivity, encoded in its power spectrum, with the expected form of the CBC signals to produce a single quantity that summarizes the sensitivity of the detector to those signals at a given time. The horizon distance also helps establish a benchmark for the performance of a CBC search pipeline. If we assume an isotropic distribution of sky position and binary orientation
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angles, then the mean sensitive distance athρi= 8 is given in terms of the optimal horizon distance by
DSM= Dhoriz
2.26 . (4.1)
The angle-averaged sensitive distance DSM is called the SenseMon distance. From the SenseMon distance, we can compute a naive expectation for the sensitive volume
VSM= 4π
3 DSM3 . (4.2)
In actual searches, wemeasurethe sensitive volume by performing the analysis with simulated signals added to the data. Comparing the measured sensitive volume to the SenseMon sensitive volume gives us a sense of whether the pipeline is performing as expected.
The expected SNR for a signalhin a detector with spectral densitySn(f) is given by
hρi=p
hh|hi, (4.3)
where the inner producth|iis defined by Eqn. 3.3. Ifh1Mpcdenotes the signal from the same binary system located at 1 Mpc, and D is the distance to the source in Mpc, thenh=h1Mpc/D. Setting hρi= 8 in Eqn. 4.3, we obtain the expression
Dhoriz=
phh1Mpc|h1Mpci
8 Mpc. (4.4)
In Fig. 4.1b, we show the horizon distance for equal mass non-spinning binaries computed from the spectra in Fig. 4.1a. Our calculation uses theIMRPhenomBapproximation, so as to include the effects of merger and ringdown. We show the horizon distance only up to Mtotal = 100M, since this value is the upper mass cutoff in the searches described below, but the detectors have sensitivity to systems up to roughly
Mmax≈ c3 6√
6Gflow ≈350M, (4.5)
where we have taken flow= 40 Hz. At masses beyond 350M, the entire CBC signal is out of the LIGO band2.
Since the noise in the LIGO and Virgo detectors is not stationary, the horizon distances shown
2Actually, the GW signal consists of a superposition of waves with frequencies that are integer multiples of the orbital frequency. The dominant frequency isfGW= 2forb, but other multiples also contribute to the signal, bringing part of the signal for masses beyondMtotal= 350Mback into band.
(a) (b)
Figure 4.2: Variation of sensitivity over the course of the S6/VSR2-3 science runs. We show (left) the distribution of horizon distances for am1 =m2 = 1.4M binary neutron star signal, computed from each of the 2048 s blocks of data analyzed in the S6/VSR2-3 low mass CBC search. Each point in the time series (right) indicates the average sensitive distance over an interval of seven days and the error bar indicates the variance in the sensitivity during that week. The detector sensitivity is a strong function of time, changing in response to local conditions and periodic maintenance and commissioning of the detectors.
in Fig. 4.1b conceal important information about the variability of the detector sensitivities over the course of S6/VSR2-3. In the S6/VSR2-3 CBC searches, the spectral density was computed on 2048-second blocks of contiguous data [81], as explained in Sec. 3.4. We account for non-stationary detector behavior by recomputing the spectral density for every 2048 second block of data. In Fig. 4.2, we show the distribution of the horizon distance towards a canonical m1 =m2 = 1.4M binary neutron star source for each of the 2048 second blocks of data analyzed in the low mass CBC search We note the wide variation in sensitivity over the course of such a long run. The Hanford site finished S6 operating at a sensitivity with horizon distance of 45 Mpc. At the beginning of the S6 observational run, Hanford had an observable distance of only 30 Mpc. This improvement corresponds to an increase is detection rate by a factor of (45/30)3, which is slightly more than a factor of three. The decrease in horizon distance for the Virgo detector in VSR3 is due to a mirror with an incorrect radius of curvature being installed during the conversion to monolithic suspensions.