A FAST DATA PROCESSING TECHNIQUE FOR CONTINUOUS GRAVITATIONAL WAVE SEARCHES

: This article discusses the potential advantages of a data processing technique for con- 1 tinuous gravitational wave signals searches in the data measured by ground-based gravitational 2 wave interferometers. Its main advantage over other techniques is that it does not need to search 3 over the signal’s direction of propagation. Although it is a “ coherent method” (i.e. it coherently 4 processes year-long data), it is applied to a data set obtained by multiplying the original time- 5 series with a (half-year) time-shifted copy of it. As a result, the phase modulation due to the 6 interferometer motion around the Sun is automatically canceled in the signal of the synthesized 7 time-series. Although the resulting signal-to-noise ratio is not as high as that of a coherent search, 8 it equals that of current hierarchical methods. In addition, since the signal search is performed 9 over a parameters space of smaller dimensionality, the associated false-alarm probability should 10 be smaller than those characterizing hierarchical methods and result in an improved likelihood of 11 detection.


Introduction
The first direct observation of a gravitational wave (GW) signal announced by 16 the LIGO project [1] on February 11, 2016 [2], represents one of the most important 17 achievements in experimental physics today, and marks the beginning of GW astronomy 18 [3]. By simultaneously measuring and recording strain data with two interferometers at 19 Hanford (Washington) and Livingston (Louisiana), LIGO scientists were able to reach 20 an extremely high level of detection confidence and infer unequivocally the GW source 21 of the observed signal to be a coalescing binary system containing two black-holes of 22 masses M 1 = 36 +5 −4 M ⊙ and M 2 = 29 +4 −4 M ⊙ at a luminosity distance of 410 +160 −180 Mpc 23 corresponding to a red-shift z = 0.09 +0.03 −0.04 (the above uncertainties being at the 90 percent 24 confidence level). 25 The network of the two United States LIGO interferometers could constrain the di- 26 rection to this binary system only to a broad region of the sky because the Italian-French 27 VIRGO detector [4] was not operational at the time of the detection and no electromag- 28 netic counterparts could be uniquely associated with the observed signal. Ground-based 29 observations inherently require use of multiple detectors widely separated on Earth and 30 operating in coincidence. This is because a network of GW interferometers operating 31 at the same time can (i) very effectively discriminate a GW signal from random noise 32 and (ii) provide enough information for reconstructing the parameters characterizing 33 the wave's astrophysical source (such as its sky-location, luminosity distance, mass(es), 34 dynamic time scale, etc.) [3,5,6,7,8].

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The GW signals emitted by millisecond pulsars are continuous, narrow band, and 77 quasi-sinusoidal, and can be modeled by a family of parametrized template waveforms.

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The long observation times, needed to guarantee their detection by optimal filtering, 79 also require to account for temporal changes of the wave's frequency due to the pulsar's  Their expressions can be written in the following forms [21] 87 (1) where f 0 is the signal frequency at the pulsar's rest frame, f 1 is the first pulsar's frequency 88 spin-down term, 1n is the unit vector to the pulsar's sky position relative to the SSB, (⃗ r(t),

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⃗ v(t)) are the position and velocity at time t of the detector relative to the SSB respectively, Although the GW strain measured at the interferometer, s(t), is a linear combination of the wave's two polarization components of the form it can been shown that the time-dependence of the amplitude A and phase Φ 0 is due to 97 the detector's beam-pattern functions through its orientation to the source. As these two interest; this yields the detection statistics commonly known in the field as the F-statistics 106 [22,24]. This method is, however, computationally intensive when processing year-long 107 data sets over a large parameter space. Since any signal present in the data will possess 108 parameters that are different from any one we might include in the template search 109 bank, it becomes essential to identify a criterion by which the bank is selected so that 110 the maximum degradation in SNR would still result in a detection. The current adopted 111 approach to select the templates, needed to search for a GW signal characterized by 112 a set of parameters, is to introduce a metric in the parameter space to quantify how 113 closely the templates must be spaced [23]. This approach, when applied to searches becomes computationally impossible with year-long data sets [25].
where we have denoted with Ψ 0 the constant associated with the integrations of the 149 anti-symmetric condition fulfilled by the velocity vector at times t and t + T. If we 150 now multiply the two data streams, after some simple algebra we find the following 151 expression for the GW signal in the resulting new time-series 152 Q gw (t) ≡ s(t)s(t + T) The GW signal in the new time-series is equal to the sum of two terms, one of which no A non-zero eccentricity e, a non-negligible pulsar's spin-down f 1 , together with the presence of quadratic noise terms in the newly formed time-series, temper the above analysis and require additional considerations. Let's first focus on the noise terms. Let R k ≡ R(t k ) = s k + η k be the detector measurement at time t k , with η k representing the random process associated with the detector's noise and k = 1, ...N. By multiplying the detector's data at time t k with itself at time t k + N/2 ∆t (with ∆t being the sampling time) we now find 2 Q k ≡ R k R k+N/2 = s k s k+N/2 + s k η k+N/2 + s k+N/2 η k + η k η k+N/2 .
In what follows we will assume the noise auto-correlation time to be significantly smaller 158 than T and η to be a stationary, Gaussian distributed random process of zero-mean and 159 variance σ 2 η . Note that the expression for Q k in Eq. (7) tells us that the noise affecting this 160 newly synthesized data set is no longer Gaussian because of the presence of quadratic 161 noise terms. 162 We now define our detection statistics to be the Fourier transform of Q k where i is the imaginary unit. Since the Fourier transform is typically taken over a

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This means that it is entirely characterized by its mean and variance, which may be 166 obtained by estimating the expectation value of S k , µ S , and its quadratic deviation from 167 its mean, σ 2 S . By denoting with l the integer corresponding to the frequency 2 f 0 , it is easy 168 to find the following expressions for the mean and variance associated with the above where we have denoted with ⟨. . . ⟩ the operation of expectation value. The expression above for σ 2 S reflects the observation that GW amplitudes emitted by pulsars are expected to be much smaller than the noise level; also we assumed an integration time equal to the data off-set time T ≃ 6 months = N/2 × ∆t. From Eqs.(9,10) we can then derive the expression of the amplitude SNR achievable by our data processing technique Note the SNR dependence on the fourth-root power of the number of data samples.

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Although such dependence is worse than that achievable by a fully-coherent search (for 172 which the SNR grows as the square-root of the number of samples), it is well known 173 that a fully-coherent search of GW signals from an isolated pulsar whose parameters 174 are unknown is computationally impossible [22]. Although the dependence of the SNR  To complete our analysis, in what follows we first include the pulsar's spin-down The GW signal in the new time-series is again equal to the sum of two terms. Although now the first term shows a dependence on the parameters (θ, ϕ), such a dependence is nevertheless suppressed by the spin-down parameter f 1 . This suggests that, if we would break the new time-series in a number n of contiguous data chunks of duration τ properly selected, we could neglect the signal's dependence on the sky position. The time duration, τ, during which we can neglect the dependence over the sky-position parameters, is given by the following inequality relating the frequency shift experienced by the gravitational wave signal in Q gw (t) to the frequency resolution τ −1 The above expression includes only the larger of the two quantities depending on (θ, ϕ) 183 entering the first term of the signal expression given in Eq. (12). we would conclude that there would not be any need to break up the data since τ > T. 193 We will now quantify the effect of the Earth's eccentricity by focusing on the interferometer's geometrical configuration associated with the apogee/perigee points.

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To avoid or further minimize the effects associated with the eccentricity, the reference 205 frame of our coordinate system could be moved away from the SSB to the center of the 206 elliptical trajectory of the Earth around the Sun. Also, the time shift T could be treated as 207 a slowly varying function of time so as to make the velocities at times t and t + T (away 208 from the apogee/perigee) as "close" as possible.

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Finally, if the break-up of the data in n chunks results necessary, one can regain SNR by "stacking-up" the n spectra. Since the noise in each set is uncorrelated with all the others, it is easy to show that the resulting amplitude SNR is equal to Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 October 2021 after using the identity n × m = N.

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The proposed data processing technique has the appealing feature of not requiring a 212 search over the parameters (θ, ϕ) associated with the source location. This gain, however, 213 results in a SNR that grows as N 1/4 rather than N 1/2 characteristic of coherent searches.

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Nevertheless, the SNR dependence of our technique on the data length is equal to that 215 shown by semi-coherent searches. Since this is achieved by requiring a significantly 216 smaller number of templates due to a smaller-size parameter space, it offers some