1.3 Advanced LIGO interferometers
1.3.4 Data analysis techniques
Matched filtering is a data analysis technique that correlates a data stream with a known signal to try to detect the presence of that signal in the data stream. It is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) of a known waveform. In GW data analysis, matched filtering is used to search for deterministic signals that can be calculated analytically or numerically, like those from BBHs and BNSs [55, 56, 57], and it played a vital role in the first detection of those signals [5, 14, 15, 16, 17, 18]. Matched filtering is also used in theF-statistic searches and Bayesian searches for signals from rapidly rotating neutron stars [58, 59].
19 7
(a) LIGO Livingston Observatory
(b) LIGO Hanford Observatory
FIG. 5. Noise budget plots for the gravitational wave channels of the two LIGO detectors. The strain sensitivities are sim- ilar between the two sites. Plot (a) shows the low-frequency curves for L1, whereas Plot (b) shows the high-frequency curves for H1 detector. Quantum noise is the sum of the quantum radiation pressure noise and shot noise. Dark noise refers to electronic noise in the signal chain with no light in- cident on the readout photodetectors. Thermal noise is the sum of suspension and coating thermal noises. Gas noise is the sum of squeezed film damping and beam tube gas phase noises. The coupling of the residual motion of the Michelson (MICH) and signal recycling cavity (SRCL) degrees of free- dom to gravitational wave channel is reduced by a feedforward cancellation technique. At low frequencies, there is currently a significant gap between the measured strain noise and the root-square sum of investigated noises. At high frequencies, the sensitivity is limited by shot noise and input beam jitter.
and alignment purposes. These very narrow lines are eas- ily excluded from the data analysis, while the broadband noise inevitably limits the instrument sensitivity. The latter is therefore a more important topic of investiga- tion.
A. Seismic and thermal noises
Below 10 Hz, there is significant displacement noise from residual seismic motion. On average, at both the Livingston and Hanford sites, the ground moves by ⇠ 10 9 m/pHz at 10 Hz—ten orders of magnitude larger
than the Advanced LIGO target sensitivity at this fre- quency. To address this di↵erence, seismic noise is fil- tered using a combination of passive and active stages. The test masses are suspended from quadruple pendu- lums [25]. These passive filters have resonances as low as 0.4 Hz and provide isolation as 1/f8 in the detection
bandwidth. The pendulums are mounted on multistage active platforms [41, 42]. These systems use very-low- noise inertial sensors to provide the required isolation in the detection band and at lower frequencies (below 10 Hz). This isolation is crucial for bringing the interfer- ometer into the linear regime and allowing the longitu- dinal control system to maintain it on resonance. The active platforms combine feedback and feedforward con- trol to provide one order of magnitude of isolation at the microseism frequencies (around 0.1 Hz) and three or- ders of magnitude between 1 Hz and 10 Hz. Most of the suspension resonances are located in this band, where ground excitation from anthropogenic noise and wind is significant.
Fluctuations of local gravity fields around the test masses—caused by ground motion and vibrations of the buildings, chambers, and concrete floor—also couple to the gravitational wave channel as force noise [43] (grav- ity gradient noise). The coupling to the di↵erential arm length displacement is given by
L(f ) = 2Ngrav(f ) (2⇡f )2
Ngrav(f ) = G⇢Nsei(f ),
(8)
where Ngrav is the fluctuation of the local gravity field
projected on the arm cavity axis, the factor of 2 ac- counts for the incoherent sum of noises from the four test masses, G is the gravitational constant, ⇢' 1800 kg m 3
is the ground density near the mirror, ' 10 is a geo- metric factor, and Nsei is the seismic motion near the
test mass. Since the ground near the test masses moves by ' 10 9m/pHz at 10 Hz, local gravity fluctuations at
this frequency are Ngrav ⇡ 10 15m s 2/
p
Hz and the to- tal noise coupled into the gravitational wave channel at 10 Hz is L⇡ 5 ⇥ 10 19m/pHz. Gravity gradient noise is
one of the limiting noise sources of the Advanced LIGO design in the frequency range 10–20 Hz. However, the typical sensitivity measured during O1 is still far from this limitation.
Thermal noises arise from finite losses present in me- chanical systems and couple to the gravitational wave channel as displacement noises. Several sources of ther- mal noise can be identified. Suspension thermal noise [45] causes motion of the test masses due to thermal vibra- tions of the suspension fibers. Coating Brownian noise Figure 1.5: Top: noise sources at low frequencies at the LIGO Livingston interferometer.
Bottom: noise at high frequency in the LIGO Hanford interferometer. Both plots show noise at the start of the first Advanced LIGO observation run in September, 2015. At low frequencies noise is dominated by seismic noise, noise from the angular optical control systems and an unknown source of noise believed to be light scattering noise. At high frequencies the noise is dominated by quantum noise in the form of shot noise. This plot is reproduced from [33].
Coincident bursts
When searching the data for transient GW signals that do not have a deterministic waveform, it is common to search for coincident “bursts” of strain power in multiple detectors. These types of searches are used for sources like supernovae and any other signal that does not have a deterministic waveform [27, 60, 26, 61]. Often these searches either cross-correlate data between the detectors and look for larger-than-expected cross- power, or they look for large, simultaneous excursions in the power in the individual detectors. A combination of both can also be used [62].
Long-duration cross-correlation
Searches for long-lived signals that are not well-modeled, like the SGWB, typically rely upon cross-correlating the data in pairs of detectors [63] to search for a signal common to both instruments. This method can also be used to search for rapidly rotating neutron stars [64, 24, 25]. We will discuss cross-correlation based searches for long-duration signals extensively in chapters 2 and 3.
1.4
Concluding remarks
In this chapter we discussed GWs and interferometric detectors. We introduced the mathematical formalism used when discussing GWs, and how GWs interact with inter- ferometric detectors like Advanced LIGO and Advanced Virgo. We also briefly discussed several different sources of GWs, as well as a cursory overview of different data analysis techniques used in searching for them. In the next chapter, we expand on searches for persistent GWs like the SGWB or rapidly rotating neutron stars, and in the subsequent chapter we present results for those searches using data from O1.
Chapter 2
Cross-correlation searches for
persistent gravitational waves
The stochastic gravitational-wave background (SGWB) is a promising source of gravita- tional waves (GWs) that is expected to arise due to the superposition of many individ- ually unresolvable GW sources. It carries information about unresolved stellar sources of GWs and potentially sources of GWs from the earliest epochs in the evolution of the universe, such as inflation. To be consistent with the way different types of energy in the universe are treated in cosmology, the SGWB is usually discussed in terms of a dimensionless energy density parameter
ΩGW(f ) = 1 ρc dρGW d ln f (2.1) where ρc = 3H 2 0c2
8πG is the critical energy density to close the universe, and dρGW/d ln f
is the energy density in GWs per logarithmic frequency bin.
The rest of this chapter will discuss sources of an SGWB and methods of searching for it using the Advanced LIGO and Advanced Virgo detectors. In section 2.1, we discuss sources of an SGWB before moving on to current direct and indirect limits on the SGWB in several different frequency bands. In section 2.3, we discuss the cross-correlation strategy used to search for an isotropic SGWB in LIGO detectors. In section 2.4, we discuss how the isotropic assumption about the SGWB can be relaxed so that we can search instead for an anisotropic background of GWs. Finally, in section 2.5 we discuss
an unmodeled, directed search for GWs in each frequency bin and present a new method for setting limits on the strain amplitude of a rapidly rotating neutron star using that search.