Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.atnf.csiro.au/iau-comm31/pdf/2009_IAUGA_JD6/JD06_o3_Potapovx.pdf Äàòà èçìåíåíèÿ: Fri Aug 6 03:06:37 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Feb 4 08:09:57 2013 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 5

Timing of binary pulsars and the search for low-frequency gravitational waves
Vladimir A. Potapov
Pushchino Radio Astronomical Observatory ASC LPI, Russia

Sergei M. Kopeikin
University of Missouri-Columbia, USA


Contents.

· «Non-ideal pulsar» problem.
· Timing noise and timing model · Filter function and TOA residuals

· Search for GWB using Pulsar Timing Array.
· Influence of "filtering" on the detection significance. · Spectral sensitivity to the detected power noise spectrum.

· Conclusion

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

2


Fitting as an effective "noise filtering".
"Ideal pulsar" case "Non-Ideal pulsar" case TOAs + Noise (including GWB)

TOAs + true values of parameters + Noise (including GWB)

Fitting the pulsar parameters.

p1 ... pn, Err

p1* ... pn*, Err*

Fitting biases parameter's estimates and redefines spectrum of noise.
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 3


Questions:
· How the LSQ-estimations of binary pulsar's parameters and post-fit TOA residuals depend on timing noise? · How the LSQ procedure affects the calculated parameters in the presence of the noise? · Is it mathematically correct to use TOA residuals to estimate the power of the noise? In particular, could we use the binary pulsars data to set an upper limit on energy density of GWB? · To answer these questions an analytical formalism describing behaviour of the variations of the observed pulsar parameters' and TOA residuals in the presence of correlated low-frequency noise was elaborated.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

4


Timing Model. Binary Pulsar on a circular orbit.
Model includes 14 parameters: 6 rotational and 8 orbital. It describes binary pulsars with nearcircular orbit with a good accuracy. (Kopeikin, 1997)

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

5


Influence of the noise on timing results.

Variation of residuals.

K is a filter function f - Fourier frequency, m and N are a number of observations over one orbit and a number of orbital revolutions. Function K was analytically calculated in (Kopeikin, 1997, 1999, Kopeikin & Potapov, 2004) both in time and frequency domains.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

6


Filter function and its influence on the "red" noise power spectrum

Spectrum of GWB from super-massive black hole binaries (hc(f)~f^(-2/3), S(f)~f^(-13/3). The graph shows filter function ( black ), power spectrum ( blue ), and their product ( red ).

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

7


Filter function and its influence on the "red" noise power spectrum

· Low frequencies in power spectrum of the noise are fitted away by the
polynomial fit for the spin-down parameters. · Frequencies being close to the orbital frequency are fitted away by the fit for the orbital parameters. · Hence, parameter's fitting affects GWB power estimation. · Filtering of the noise power near the orbital frequency may be neglected for long-time observations in comparison with the effect of the lowfrequency filtering. ·The longer time span of observations, the more narrow becomes the "spectral lines" near the zero and orbital frequencies.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

8


Influence of a monochromatic gravitational wave on TOA residuals.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

9


Correlation of TOA residuals using PTA data.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

10


Correlation of TOA residuals with the distribution function.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

11


Influence of pulsar's parameter fitting on the detection significance.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

12


Detection significance function for PTA vs. A.

GWB from MBH-MBH binaries, n=-13/3 (=-2/3). Simulation for "target parameters " of PPTA: N = 250, M=20, Res=100 ns, T(obs) = 5 years. For N=8 orbital revolutions. Effect of the filter function is neglected (black curve) / taken into account (red curve). Horizontal lines indicate 32% and 0.3% probability of erroneous detection (1 and 3 confidence).
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 13


Detection significance function for PTA vs g.

At 32% probability of erroneous detection (1 confidence) g 1.8E-10, g(corr) 4E-10. g(corr)/ g 2

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

14


Detection significance function for PTA vs g.

At 32% probability of erroneous detection (1 confidence) g 1.58E-8, g(corr) 3.2E-8

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

15


Detection significance function for PTA.

Effect of the filter function is neglected (black curve) / taken into account for N= 8 (red line), N=10 ( yellow ), N=16 ( green ), and N=34 ( blue ) orbital revolutions.
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 16


Influence of filter function on the detection significance.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

17


"Filter function" for PTA.

Depth of the "orbital spectral lines" will fall as a N/2 , where N is a number of pulsars in PTA ,when "zero spectral line" still filters out ultra-low frequencies.
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 18


Filtering the power of a "red" noise spectrum with PTA.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

19


Conclusion

.

· Influence of the fitting procedure on the detection significance for timing noise with power-law spectrum (e.g. GWB from MBH-MBH binaries) becomes inessential for PTA when the time of observation > 25 orbital periods. · Most of millisecond binary pulsars chosen for PTA have orbital periods < 2 weeks (more than 25 orbital revolutions per year). Hence, the influence of the fitting procedure on the detection significance becomes negligible when the total time of observations exceeds 1.5 years. ·The effect of filtering of the near-orbital frequencies on GWB power spectra is reduced as N, where N is the number of pulsars in PTA. · It seems reasonable to exclude from the set of PTA objects used for TOA correlation analysis binary pulsars with relatively long orbital periods (e.g. J1643-1224) to prevent possible distortion of the index of the power spectrum in the low frequency domain.
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 20


Thank you.

This work was supported by RFBR grant 09-02-00922
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 21


Influence of the parameters fitting procedure.

CALCULATION OF AN UPPER LIMIT OF ULTRA-LOW FREQUENCY GWB
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 22


Limiting of GWB at ultra-low frequency.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

23


Limiting of GWB at ultra-low frequency.

(Kopeikin, 1997)
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 24


Limiting of GWB at ultra-low frequency.

Limits for gh^2 Forecast for B1913+16: < 10^(-4) (effect of the fitting of orbital parameters ignored) Bertotty, Carr, Rees, 1983; B1913+16: < 1600, B1855+09: < 2.7x10^(-4) Kopeikin, 1997; J1640+2224: < 8.5x10^(-4), < 4.2x10(-4) Potapov et. al., 2003 , Potapov, 2004

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

25


Conclusions:

· Variations of the orbital parameters are dominated by low-frequency
noise with spectral indexes n>=5 (S(f) ~ f^(-n)) ( GWB in particular) in case of the long-time observations (N >> 1). · The stability of pulsar time on a long intervals (decades and more) has a fundamental cosmological limitation caused by low-frequency "red" noises of GWB with spectral indexes n>=5.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

26


Timing noise model
Timing noise is modelled by a sequence of random jumps in pulsar phase. The noise is the Poisson process with a known response function and the Gaussian distribution. The noise has a power spectrum S(f) that consists of the sum of functions having infrared 1/f^n divergence (n=1,2,...). Physical interpretation of the "red" noise: ·Random walks: S(f)~ 1/f^2, 1/f^4, 1/f^6 ­ generated by irregularities in pulsar rotation. ·Flickers: S(f) ~ 1/f ­ irregularities of the atomic clock and inaccuracy of the planetary ephemeris, ~1/f^3 - perturbations of the interstellar medium, ~1/f^5 ­ stochastic GWB from early Universe. ~1/f^(-13/3) ­ stochastic GWB from Massive Black Hole binaries.
07.09.2009 IAU GA XXVII JD6 Rio de Janeiro 2009 27


07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

28


Ways to evaluate the stochastic GWB.
At frequency F > 1/T, where T is the total lenght of observations. (F ~ 10^(-7) ­ 10^(-9) Hz).

· Evaluation of the TOA residuals variation. · Investigation of the power noise spectrum of TOA residuals. · Investigation of the power noise spectrum of an "ensemble pulsar"
(investigation of the stability of an "ensemble pulsar time").

· Search for the correlation of TOA residuals of pulsars in the "Pulsar Timing
Array" (PTA).

· Frequency 1/L < F < 1/T, where L ­ light-time from pulsar to observer
(10^(-7) ­ 10^(-12) Hz).

· Investigation of the long-time variations of the binary pulsars parameters.

07.09.2009

IAU GA XXVII JD6 Rio de Janeiro 2009

29