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Basic Physics and Cosmology
from Pulsar Timing Data
J. H. TAYLOR
Princeton University
ABSTRACT
Radio pulsars provide unparalleled opportunities for making measure
ments of astrophysically interesting phenomena. In this paper I concentrate
on two particular applications of high precision tiIning observations of pul
sars: tests of relativistic gravitation theory using the binary pulsar 1913+16,
and tests of cosmological models using timing data from millisecond pul
sars. New upper limits are presented for the energy density of a cosmic
background of low frequency gravitational radiation.
PRODUCTION
Among the nearly 500 radio pulsars that have been discovered since
1967, some of the most rewarding to study have been the binary and
millisecond pulsars. Like all radio pulsars, these spinning neutron stars
are rotationally powered: that is, their energetics are dominated by their
spindown luminosities, E = IBM. Their evolution has been modified,
however, by spinup in a "recycling" process involving mass transfer from a
companion star during its postmainsequence evolution.
Recycled objects make up a small fraction of any sensitIvitylimited
pulsar sample. Fewer than 20 are presently known, and most of them are
still members of gravitationally bound binary systems. Their Apical rotation
periods are substantially shorter Man those of other pulsars, and detailed
measurements of the arrival times of their pubes at Earth have provided
observers with a weals of information on a surprisingly diverse range of
topics. In this paper ~ will summarize the present status of two particular
385
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386
AMERICAN AND SOVIET PERSPECTIVES
applications of timing observations of recycled pulsars: the detection and
quantitative measurement of orbital decay caused by gravitational radiation
from the binary pulsar 1913+16; and experimental limits on the cosmic
gravitational wave background (GWB), based on timing observations of
millisecond pulsars 1855+09 and 1937~21.
TESTING REI^TIV1TY WITH PSR 1913~16
The first binary pulsar was fouled nearly 15 years ago, and its im
portance as a testbed for relativistic gravitation theories was recognized
almost immediately (EIulse and Taylor 1975; see also Brumberg et al. 1975;
Damour and Rufflni 1974; Esposito and Hamson 1975; Wagoner l975~. In
the intervening years, much effort has been put into making increasingly
accurate measurements of its pulse arrival times and comparing the results
with parametrized models. Adequate models must include physics having
lo do with the pulsar's spin and orbit, the interstellar medium, and the
motions of the Earth. Optionally, a model may also include phenomeno
logical parameters designed to distinguish between different theories of
gravitation. Diligent efforts on the part of both theorists and observers,
aimed toward extracting the maximum possible information from the PSR
1913+16 system, have been well rewarded over the 15 years since its dis
covery. The latest results, which I summarize briefly here, have determined
the orbital elements and masses of the pulsar and its companion star with
unprecedented accuracy and established that the orbit is decaying at almost
precisely the rate expected from gravitational radiation damping (Taylor
and Ginsberg 1989~.
The PSR 1913+16 timing experiment is conceptually a simple one.
Observations made over intervals of about five minutes, or 1% of the
~ hour orbital penod, are used to accumulate samples of the periodic
waveforms received Mom the pulsar. The pulse profiles are recorded digi
tally, along with accurate timing information from a reference atomic clock
Subsequent analysis involves determining the equivalent topocentnc time
of arrival, or TOA, for a pulse near the midpoint of each integration The
complete set of TOAs is then analyzed in terms of a set of equations de
scribing the pulsar's spill and orbital motions and the motions of the Earth.
These equations are most naturally expressed in the coordinate system of
an inertial reference frame, for which the solar system barycenter selves as
an adequate approximation.
Necessary steps in the analysis include a relativistic transformation
to convert topocentric TOAs to equivalent coordinate times at the solar
system baIycenter, and then to proper times at the pulsar. Through this
procedure, the rotational phase of the spinning neutron star is computed for
each of the measured arrival times. If reasonably accurate starting values
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HIGHENERGY ASTROPHYSICS
3287
are available for the model parameters, the computed phases (expressed
in cycles) will have integer parts corresponding to the number of pulse
periods elapsed between observations, and fractional parts nearly equal to
zero. Small systematic deviations of the fractional parts from zero are used
to refine the model parameters, using standard linearized least squares
techniques.
Blandford and Teukolsky (1976) derived the first useful formulae for
analyzing TOAs from binary pulsars. They treated the orbit as a slowly
processing Keplerian ellipse; the ejects of relativistic time dilation and
gravitational redshift were grafted onto the nonrelativistic model, and
additional phenomenological parameters were added to allow measurement
of the rate of periastron precession and testing the constancy of other orbital
parameters. More elaborate models have been developed since 1976, and
a thorough discussion of these is given by Taylor and Weisberg (1989~. The
most comprehensive treatment is that of Damour and Deruelle (1986),
which uses a parametrization that cleanly separates several effects expected
to differ in the strongfield limits of distinct gravitation theories.
As is the case for singleline spectroscopic binary stars, a Keplerian
analysis of timing data from a binary pulsar determines the values of five
orbital parameters. However, seven quantities are required to fully specify
the dynamics of an orbiting system (up to uninteresting rotations about the
line of sight). Therefore the measurement of N "postKeplerian" param
eters in a binary pulsar system, in addition to the five readily measured
Keplerian ones, provides the opportunity for N2 distinct tests of any
particular theory of relativistic gravitation.
More than 4000 TOAs for PSR 1913+16 have been recorded at the
Arecibo Observatory since 1974. Taylor and Weisberg (1989) have shown
that the Keplerian parameters of the system are now determined with
fractional accuracies of a few parts per million or better, and that as many
as 5 postKeplerian parameters are measurable with interesting accuracies.
The two largest of these, which measure the rate of periastron advance and
the combined magnitude of time dilation and gravitational redshift effects,
are known with fractional accuracies of about 105 and 2 x 103. Together,
these seven quantities imply that the gravitational masses of the pulsar
and its companion are me = 1.442 ~ 0.003 and m2 = 1.386 ~ 0.003 times
the mass of the Sun, respectively. These masses, together with the orbital
period and eccentricity, can be used to compute an explicit prediction for
the energy losses caused by gravitational radiation within a particular theory
of gravity.
Figure 1 presents a comparison of the observed orbital damping with
that predicted by general relativity. Taylor and Weisberg (1989) show that
the ratio of observed to expected effects is
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388
on
~2

u:
U]A
s5
_
~
o
8
10
AMERICAN AND SOVIET PERSPECTIVES
21 1 1 ' ' ' 1 ' ' '
' 1 ' l , 1

_
1 1 , 1 1 1 1
O ~




, , , , 1 , 1 1~
75
80
Date
B5
90
FIGURE 1 Filled circles represent the measured shifts of the times of PSR 191316's
penastron passage relative to a non~issipative model in which the orbital period remains
fixed at its 1974.78 value. The smooth curve illustrates the prediction of general relate fir.
(After Taylor and Weisberg 1989.)
Pb (Observed)
POOR Theory) = 1.010 ~ 0.011
where Pi is the rate of change of orbital period. This I% agreement is an
impressive confirmation of Einstein's theory, in a regime where gravitation
theories have not previously been testable. The remaining postKeplenan
measurables have fractional accuracies of only ~ 1050%, but it is notable
that they, too, have been found to have values in accord with general
relativity.
T~ COSMIC GRAVITATIONAL WAVE BACKGROUND
Millisecond pulsars, the most extreme examples of the recycled class,
have periods as short as P ~ 1.5 ms and spindown rates as small as
P ~ 102°. According to conventional models, these parameters sug
gest unusually large ages and weak magnetic fields. Otherwise, however,
the millisecond pulsars appear to be quite similar to their more slows
rotating cousins. Because pulsar timing accuracies tend to be a fixed frac
tion (~ 104 to 103) of a period, observations of millisecond pulsars
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HIGHENERGY ASTROP~SICS
389
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390
AMERICAN AND SOVIET PERSPECTIVES
5
o
1 ' 1
~' 1
1937+21
.
_ ~ ~ ~ ~
t. ~
.
·~ ~
~ . . .
· 
1

~:,
._
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_ 1 1 ' 1 ' ' 1  1 
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83 84 85
86 87 88 89
Date
FIGURE 2 Timing residuals for PSRs 1937~21 and 1855109 Mom obsenrabons made
at Arecibo, Pueblo Rico. Data quality has been unifollll since the introduction of new
equipment and procedures in October, 1984.
furler important advantage: it facilitates making quantitative corrections
for the complex instrumental response caused by irregular sampling and
finite length of the data sets. Following Blandford et aL (19843, I define
an instrumental response function T(f) such that the observed spectrum
of timing residuals is S(f) = T(f)P(f): the product of the instrumental
response and the intrinsic noise spectrum, P(f). As illustrated in Figure 3,
the most important features of T(f) are a low frequency cutoff below
f ~ r~t and a deep notch centered at f = 1 Arm, caused by the necessity
to measure the celestial coordinates of the pulsar as part of the leastsquares
fitting process.
~ make it easy to compare the observed fluctuation spectra with
hypothetical intrinsic spectra, I have multiplied the computed TO f) curves
by power law spectra with s = 0, 2, 3, and 5. For convenience in plotting,
the results were then normalized to the mean power level in the two
lowest frequenter channels of the observed spectra, S(f). At the top of
Figure 4, we see mat the spectrum for PSR 1855109 is reasonably well
approximated by the spectrum labeled s = 0. In other words, there is no
evidence for a significant contribution to these residuals beyond that of the
random measurement errors. The equivalent mass density in a cosmic GWB
corresponding to the dashed curve labeled s = 5 is p = 2.2 x 1036 g am~3,
OCR for page 385
HIGHENERGY ASTROPHYSICS
..f
/ ,
/
391
I1~1~111111l111l111l1~ll111l111l111l111l111
41937+21 (1984.8  1989.3)
' 1855+09
1937+21 (1~9  1989.3)
6 t~ 1 1 ~ I 1 ~ I I 1 1,, 1,,, 1,, ,`1i, ', 1 1 1 1 1 1 1 1 1 1, I 1,,,
L2 1.0 .8 .6
.4 ~ .0
log [Frequency (c/y)]
.4 6
.8 1.0
FIGURE 3 Instrumental transmission functions, T(f), corresponding to the data sets
illustrated in Figure ~
or Q. = 1.1 x 107 for a Hubble constant Ho = 100 km s~i Mpc~. The
observed power in the lowest spectral channel is well below this cuIve, and
yields the conservative upper limit for the GWB quoted in the first line of
Bible 1.
In the center and bottom portions of Figure 4 are similar plots show
ing the observed spectra for the data from PSR 1937+21, together with
instrumentallymodified power laws. The spectrum at the center of the fig
ure corresponds to the uniform, high~uality data obtained since October
1984, while the bottom plot corresponds to the entire 6.4 yr data span.
In both of these spectra the frequencies f > 2 yr~i are clearly dominated
by white noise (s = 0), while lower frequencies show clear signs of con
tributions with s > 0. Although further analysis remains lo be done, my
colleagues and I believe that at frequencies f < 1 yr~i these spectra are
dominated by a combination of clock errors, solarsystem ephemeris errors,
and umnodeled interstellar propagation effects. In any event, it is clear that
if a cosmic GWB makes any significant contn~ution, it must be greatest
in the lowest frequency channel of each spectrum. The dashed cuIves la
beled s = 5 ~ the middle and bottom of Figure 4 correspond to tractional
densities Q. = 3.8 x 108 and 1.4 x 108, respectively, and their positions
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392
AMERICAN AND SOVIET PERSPECTIVES
2
>A
~2
Q

C)
a.
V]

oo
o

, I,,,, I,,,, I I r 1 l T ' ' ' '
~ _



c, O
 ,

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En
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n
en
I I I i I I I I I I I I 1 i I I I i I I I
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to  5 .0 S 1.0
1°B [Frequency (c/y)]
FIGURE 4 Solid lines and filled circles observed residual spectra, S(f), corresponding
to the data sets illustrated in Figure ~ Dashed cuNes: hypothetical powerlaw spectra
modified by the instrumental response functions T(f), and arbitrarily normalized to the
mean power level in the lowest two bins of S(f).
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HIGHENER~ ~TROP~ICS
Table I: Upper limits on the energy density of the cosmic GWB.
393
Pulsar Data
1855+09 1984.81989.3 0.30 < 1.2 x 1036 < 6 x 108
1937+21 1984.81989.3 0.22 ~ 1.6 x 1036 < ~ x 108
1937+21 1982.9~1989.3 0.16 < 0.8 x 1036 <4 x 108
relative to the measured spectra lead to the remaining conservative upper
limits quoted in Bible 1.
The detailed implications of the numbers quoted in liable 1, particularly
for model universes in which cosmic strings help to seed galaxy formation,
are still somewhat controversial (Albrecht and lbrok 1989; Bennett and
Bouchet 1989~. It is already clear, however, that the experimental limits are
difficult to reconcile with the GWB energy density expected from cosmic
string simulations, particularly when the strings retain sizes large enough
to be useful in aiding galaxy formation.
ACKNOWLEDGMENTS
Parts of this work were carried out in collaboration with ~ ~ Raw
ley, M. F. Ryba, D. R Stinebring, and J. M. Weisberg. Our research is
supported, in part, by Me U.S. National Science Foundation.
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