In the last few years, several researchers have used timing data from pulsars to search for ultra-low frequency (ULF) gravitational waves (waves at periods from a few days to a few years), especially for the waves making up the stochastic cosmic background such waves. How these limits are obtained are discussed and several precautions are pointed out that must be taken in the analysis of these data

Hellings, Ronald W.

English

NASA Technical Reports Server

N90-19955
PULSAR TIMING AND GRAVITATIONAL WAVES
Ronald W. Hellings
Jet Propulsion Laboratory
Pasadena CA 91109
l
In the last few years, several researcher Detweiler, 1979, Hellings et al. 1983,
Romani, et al, 1983 and Davis, et al., 1985) have used timing data from pulsars to
search for ultra-low frequency (ULF) gravitational waves (waves at periods from a
few days to a few years), especially for the waves making up the stochastic cosmic
background such waves. It is the purpose of this talk to discuss how these limits are
obtained and to point out several precautions that must be taken in the analysis of
these data.
In pulsar timing, the times of arrival of pulses are measured and compared
with a model. The UTC times of arrival t are transformed to TDB times of arrival t via
well-known algorithms (Hellings, 1986 and Backer, et al., 1986). The t's are related to
the TDB times of emission T by
ct = cT + k - (R - r) - (1 + Y) p c 2 k R p + R p
where R =Ro + VT is the location of the pulsar at time T,r is the position of the radio
observatory on the earth, k is a unit vector toward the pulsar, and rp and R p are the
position of intervening body p relative to the earth and the pulsar, respectively, at
the time when the signal passes closest to the body. The (1+_/) term is of course the
Shapiro time delay, with PPN parameter 1, parametrizing the curvature of space. The
position of the observatory may be written as r = q + 5, where q is the position of the
center of the earth, determined from numerically-integrated planetary ephemerides,
and _ is the geocentric position vector of the observatory, determined from
observatory coordinates and from a model of the physical ephemeris of the earth.
The actual times of arrival may be compared with the predicted times of
arrival to give timing residuals, St. Among the noise sources contributing to these
residuals might be the variation of the spacetime metric created by the passage of a
gravitational wave. The rate of change of the timing residuals will be proportional to
the dimensionless amplitude of the wave
d Av
dt (_St) = --=v h(t).
If there is only a single pulsar being observed, then the spectral density of
cosmic gravitational waves is simply less than or equal to the spectral density of the
residuals. However, if there are several pulsars being observed over the same period
of time, it is possible to dig into much larger noise to detect the gravitational wave
noise source since it will be a common signal in the time series for each pulsar. Thus
we may write the frequency residuals from the i th pulsar as (Hellings et al., 1983)
Avi
--= otih(t) + hi(t),
Y
where ai contains geometrical factors, resulting from the relation of the polarization
and propagation vectors of the gravitational wave to the line-of-sight from the earth
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to the pulsar, and hi(t) is the independent noise in the data from each pulsar.
correlating the data from pulsars i and j, one finds
V-2 (AviAvj)= ai_j(h2)+ ai (hnj)+ otj(hni)+ (ninj),
Cross-
where the brackets indicate cross-correlation. Since ni and nj are independent of
each other and of h, all of these terms will tend to zero as the square-root of the
number of data points except for (h2), which is the autocorrelation function of the
gravitational wave amplitude.
Using data from the single millisecond pulsar, PSR1937+21, limits have been
set 4 for gravitational waves of periods less than one year. Using data from several
quiet normal pulsars, limits were set 2 using the cross-correlation technique at
periods from a few months to five years. These limits are compared with other direct
limits and with possible critical energy densities in Figure 1 (Zimmerman et al.,
1980).
There is one caution which must be observed in analysis of pulsar data. This is
that in order to reduce the timing residuals to the levels that appear in the literature,
several deterministic signals have had to be subtracted away. These signals
correspond to unknown (and therefore erroneous) values for the period, period
derivatives, position, proper motion, and possible parallax of the pulsar and, as data
accumulates for the most precise pulsars, the parameters of the earth's orbit and
perturbing solar system parameters. The point of this for gravitational wave
analysis is that there might have been enormous gravitational wave signatures in
the data originally, but, if there had been, they would have been subtracted away by
adjusting one of the adjustable parameters of the model.
The method which must be used to take this process into account is to treat the
parameter adjustment process as a data filter and to compute the transfer function of
the filter. A transfer function is the function which multiplies the input spectrum,
frequency be frequency, to produce the output spectrum. The spectrum of the post-
fit residuals must therefore be divided by this transfer function to give the realistic
limits that may be inferred on the original gravitational wave noise in the timing
data records.
Blandford, et al. (1984) computed the transfer function for a filter that adjusted
the pulsar parameters only. We have recently worked out the transfer function for a
combined adjustment of the pulsar parameters and adjustment of solar system
parameters, consistent with the level at which these parameters are known from
other solar system astrometric data. Since the solar system model is based on
numerical integration, it was not possible to produce an analytical expression for
this transfer function. Rather a Monte Carlo analysis was performed in which
twenty years of pulsar timing data were simulated, one point per week, and these data
added to the combined set of solar system data while all parameters, pulsar and solar
system, were adjusted. Twenty such simulated data sets were analysed and the pre-
and post-fit power spectra were compared to get the transfer function for each set. A
mean transfer function was found as a average of the twenty transfer functions. The
results of this analysis are shown in Figure 2. It should be noted that there has been
a noticeable subtraction of power at Mars's orbital period and that other longer
period planetary perturbations combine to subtract almost all power at periods
longer than about five years. The strong absorbtion line at one year combines
94
period planetary perturbations combine to subtract almost all power at periods
longer than about five years. The strong absorbtion line at one year combines
uncertainty in Earth orbital parameters and uncertainty in pulsar position and
proper motion (the latter t sin t parameters acting to keep the line relatively broad).
REFERENCES
Backer, D. C. and HeUings, R. W. 1986, Ann. Rev. Astron. Astrophys. 24, 537.
Blandford, R., Narayan, R., andRomani, R. 1984, J. Astrophys. Astron. 5, 369.
Davis, M. M., Taylor, J. H., Weisberg, J. M., and Backer, D. C. 1985, Nature 315, 547.
Detweiler, S. 1979, Ap. J.,234, 1100.
Hellings, R. W., Astron. J. 1986, 91, 650.
Hellings, R. W. and Downs, G. S. 1983, Ap. J. Lett. 265 L39.
Romani, R. W. and Taylor, J. H. 1983, Ap. J. Left. 265 L35.
Zimmerman, R. L. and Hellings, R. W. 1980, (original of this figure) Ap. J. 241, 475.
95
Figure 1. Limits on the spectrum of cosmic 6
gravitational radiation energy density from
several direct gravitational wave 4
experiments. The line labeled "critical
densities" represents the locus of peaks of a -_
'N 2
set of broad-band spectra, each of which -,-
would provide a critical energy density. The ?
E 0
line labeled "PULSARS" is from the analysis "
of Hellings and Downs (1983). The line
labeled "1937+21" comes from the results of _ -2
Davis et al (1985).
o -4
...i
-6
-8
t ,_i¢ _ I T T I ¥
- PLANETARY ,,,_ORBITS
-\\
I I , l , I , I
-8 -6 -4 -2 0
LOGIof(Hzl
"',, ,...RR_"EARTH
c".p/_...'_'J//.."NORMAL
,_,["/C._',......" MODES W3
_'/ _\ I .EARTH " WI
/_\ L/QUIET WEBER
r "/,x',,c'.,\ ZONE BARS
" v;2
-
\\ .W4
, ,_ I . [
2 4
Figure 2. Mean transfer function of the solar
system data analysis filter. Ordinate is
relative power. Abscissa is frequency in
inverse days.
O.I
O,@
0.4,
O.?
O
-,...-i • •
__rll _.yrll rl' -i,--
_m
O z 4 G
Frequency (XlO -3 yr-1)
96
DISCUSSION
SCHUTZ: Could you clarify one point please? Although an increasing sum of data may
not lower the minimum frequency at which you can set limits, presumably it does
continue to improve limits on the gravitational wave background at higher
frequencies?
HELLINGS: Yes.
TREUHAFT: Would VLBI positions of the millisecond pulsar help eliminate parameters
from your fit of pulsar data?
HELLINGS: Yes. Roger Linfield at JPL has some data to do that in the can, but it hasn't
yet been analyzed. Of course, this assumes a tie between the VLBI reference frame
and the planetary ephemeris reference frame, in which the timing positions will be
given.
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Pulsar timing and gravitational waves

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