Pulsar Timing for GWB: Sta5s5cal Methods

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1 Pulsar Timing for GWB: Sta5s5cal Methods G. Jogesh Babu Penn State Collaborators: Soumen Lahiri (NCSU), Justin Ellis (JPL/Caltech) & Joseph Lazio (JPL)

2 Gravita5onal waves are ripples in space-5me (green grid) produced by accelera5ng bodies such as interac5ng supermassive black holes. These waves affect the 5me it takes for radio signals from pulsars to arrive at Earth. (Credits: David Champion)

3 Pulsars are highly magne5zed neutron stars, the rapidly rota5ng cores of stars les behind when a massive star explodes as a supernova. A pulsar with a strong magne5c field (lines in blue) produces a beam of light along the magne5c axis. As the neutron star spins, the magne5c field spins with it, sweeping that beam through space. If that beam sweeps over Earth, we see it as a regular pulse of light. (Credit: NASA/Goddard Space Flight Center Conceptual Image Lab)

4 Pulsar Timing The best science based on pulsar observa5ons has come from their use as tools via pulsar 5ming. Pulsar 5ming is the regular monitoring of the rota5on of the neutron star by tracking the 5mes of arrival (TOA) of the radio pulses. The pulsar 5ming accounts for every single rota5on of the neutron star over long periods (years to decades) of 5me. This very precise tracking of rota5onal phase allows us to probe the interior physics of neutron stars and test gravita5onal theories. For pulsar 5ming, radio data is folded modulo the instantaneous pulse period P or pulse frequency f=1/p. Averaging over many pulses yields a high signal-to-noise average pulse profile. Although individual pulse shapes vary considerably, the shape of the average profile is quite stable.

5 GW from Pulsar Timing The recent detec5on of gravita5onal waves (GW) by the Laser Interferometer Gravita5onal-Wave Observatory (LIGO) came from two black holes, each about 30 5mes the mass of our sun, merging into one. Gravita5onal waves span a wide range of frequencies that require different technologies to detect. A new study from the North American Nanohertz Observatory for Gravita5onal Waves (NANOGrav) has shown that low-frequency gravita5onal waves could soon be detectable by exis5ng radio telescopes. Nanohertz gravita5onal waves are emi\ed from pairs of supermassive black holes in distant galaxies orbi5ng each other, each of which contain millions or a billion 5mes more mass than those detected by LIGO. Detec5ng this signal is possible if we are able to monitor a sufficiently large number of pulsars spread across the sky and likely seeing the same pa\ern of devia5ons in all of them.

6 Pulsar Timing Array (PTA) A PTA is a set of radio pulsars that can be used in tandem to search for gravita5onal waves. The difference between the measured and predicted TOAs will result in a stream of 5ming residuals, which encode the influence of gravita5onal waves as well as any other random noise in the measurement. By having a PTA, one can correlate the residuals across pairs of pulsars, leveraging the common influence of a gravita5onal-wave background (GWB) against unwanted, uncorrelated noise. The key property of a PTA is that the signal from a GWB will be correlated across pulsars, while that from the other noise processes will not. This makes a PTA func5on as a galac5c-scale, gravita5onal-wave detector.

7 A PTA noise model A GW traversing the Galaxy will affect the period P of an emi\ed pulse train such that the arrival 5me of a par5cular pulse will be perturbed from the arrival 5me expected in wave-free space. The frac5onal change in the pulse frequency ν i =1/P i for i-th pulsar may be modeled as Δν i /ν i =α i h(t)+n i (t), where h (a con5nuous random func5onal) is GW signal common to all pulsars, α i is the angle factor for the i-th pulsar and n i represents all noise sources unique to the i-th pulsar.

8 Time-Domain Implementa5on GWs induce a redshis in the signal from the pulsar that depends on the geometry of the pulsar-earth system. In pulsar 5ming, the observable quan5ty is not the redshis, but the!ming residual, which is just the integral of the redshis. ASer some deriva5ons, the 5ming residuals can be wri\en/approximated as δt = Mβ+ n M is a design matrix (δt need not be linear, but iteration of the base model is a good approximation)

9 By least squares, we get an es5mator of β By assuming Gaussian noise n and evaluating log likelihood ratio (GW present vs. noise), we get optimal cross correlation statistic for a PTA which is an estimate of E(GW amplitude) 2 ): for pulsar pairs ij. S ij are estimates of cross covariances E(r i r jt ) P i is autocovariance matrix of pulsar i, r i are 5ming residuals.

10 Statistical Tests Our goal is to test the Hypotheses: H 0 : A 2 = 0 vs H 1 : A 2 0. To this end, we need the null distribution of the statistic Â2. One way to accomplish this is to apply a version of the Block Bootstrap method! S.N. Lahiri (NCSU) Lect2 1 / 11

11 Statistical Tests (Re-)Write the timing residual series for the jth pulsar as Y (j) = M (j) β (j) + n (j) Note that under H 0, the noise processes n (j) are uncorrelated! We need to make sure that the Bootstrap construction reproduces this structural restriction! S.N. Lahiri (NCSU) Lect2 2 / 11

12 Block Bootstrap for the null distribution Next, consider the regression residuals ˆn (j) = Y (j) M (j) ˆβ(j) and their centered version ñ (j). Resample BLOCKS of values in ñ (j), independently of the other residual series, to reconstruct the Bootstrap version ˆn (j) of n (j). Define the Bootstrap version of the Y (j) as Y (j) = M (j) ˆβ(j) + n (j) Note that the random vectors Y (j) are independent (and hence uncorrelated) for different j-s. S.N. Lahiri (NCSU) Lect2 3 / 11

13 Block Boostrap caliration of the test Let  2 denote the Bootstrap version of Â2, obtained by replacing the original Y (j) values by Y (j). Then, a test of H 0 : A 2 = 0 vs H 1 : A 2 0 at level α (0, 1) is given by  2 > â α where â α denotes the (1 α)-quantile of the conditional distribution of  2. S.N. Lahiri (NCSU) Lect2 4 / 11

14 A frequency domain test Let d j (ω) denote the discrete Fourier transform (DFT) of ˆn (j), the jth residual series, j = 1,..., m. Let d(ω) denote the vector of the m DFTs. Then, the matrix valued periodogram is I(ω) = d(ω)d(ω). The off-diagonal elements of I(ω) give (raw) estimates of pairwise cross-spectral densities. S.N. Lahiri (NCSU) Lect2 5 / 11

15 A frequency domain test Under H 0 : A 2 = 0, all of these components must be identically equal to zero at ALL frequencies ω. A potential test statistic is T 1 i<j ω J d i (ω)d j (ω) 1 d i (ω) d j (ω) dω where (a + ιb) 1 = a + b, and (a + ιb) = a 2 + b 2, a, b R and ι = 1. An alternative test statistic is: d i (ω)d j (ω) 1 T sup ω J d i<j i (ω) d j (ω). S.N. Lahiri (NCSU) Lect2 6 / 11

16 A frequency domain test Under H 0 : A 2 = 0, all of these components must be identically equal to zero at ALL frequencies ω. A potential test statistic is T 1 i<j ω J d i (ω)d j (ω) 1 d i (ω) d j (ω) dω where (a + ιb) 1 = a + b, and (a + ιb) = a 2 + b 2, a, b R and ι = 1. An alternative test statistic is: d i (ω)d j (ω) 1 T sup ω J d i<j i (ω) d j (ω). S.N. Lahiri (NCSU) Lect2 6 / 11

17 Calibration of the test statistics in FD In practice, we will use a finte set J of frequencies in the definitions of T 1 and T. Under some regularity conditions, if the frequencies in J are well-separated, the real- and the imaginary parts of the DFTs are approximately Gaussian, and independent!! We shall use this fact to find a calibration to T 1 and T. S.N. Lahiri (NCSU) Lect2 7 / 11

18 Calibration of the test statistics in FD Let C j (ω) and S j (ω) are respectively the cosine and the sine transforms of the jth series! Under suitable regularity conditions and under H 0 : ( ) [C j (ω)/σ j (ω)] m j=1; [S j (ω)/σ j (ω)] m j=1 ( ) [Z j (ω)] m j=1; [W j (ω)] m j=1, ω J ω J where Z j (ω)] and W j (ω) are independent and identcally distributed (iid) N(0,1) random variables. Here, the scaling factors σ j (ω) depend on the locations t jk s and the spectral density of the jth series ONLY!!! S.N. Lahiri (NCSU) Lect2 8 / 11

19 Calibration of the test statistics in FD Define the limit analogs of the scaled d j (ω) as D j (ω) = Z j (ω) + ιw j (ω), ω J, j = 1,..., m. Then, it follows that T 1 = d i (ω)d j (ω) /[σ i (ω)σ j (ω)] 1 i<j ω J d i (ω) d j (ω) /[σ i (ω)σ j (ω)] D i (ω)d j (ω) d 1 D i (ω) D j (ω) i<j ω J S.N. Lahiri (NCSU) Lect2 9 / 11

20 Calibration of the test statistics in FD Thus, the test statistic T 1 can be calibrated by simply generating a set of iid N(0,1) random variables!! This is expected to be more accurate than the limit distribution (given by a linear combination of independent Chi-squared rando variables), as m is not very large!! A similar approximation holds for the statistic T. S.N. Lahiri (NCSU) Lect2 10 / 11

21 GWB Testing Thank you!! S.N. Lahiri (NCSU) Lect2 11 / 11