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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 post-main-sequence evolution. Recycled objects make up a small fraction of any sensitIvity-limited 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
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
HIGH-ENERGY 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 non-relativistic 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 strong-field limits of distinct gravitation theories. As is the case for single-line 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 "post-Keplerian" param- eters in a binary pulsar system, in addition to the five readily measured Keplerian ones, provides the opportunity for N-2 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 post-Keplerian 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 10-5 and 2 x 10-3. 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
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 post-Keplenan measurables have fractional accuracies of only ~ 10-50%, 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 ~ 10-2°. 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 (~ 10-4 to 10-3) of a period, observations of millisecond pulsars
HIGH-ENERGY ASTROP~SICS 389
390 AMERICAN AND SOVIET PERSPECTIVES 5 o 1 ' 1 ~' 1 1937+21 . _ ~ ~ ~ ~ t. ~ . ·~ ~ ~ . . . · - 1 - ~:, ._ c, o 5 1 _ 1 1 ' 1 ' ' 1 - 1 - 185StO9 . . · I , I ,_l I , I I I , I , . I 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 least-squares 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 10-36 g am~3,
HIGH-ENERGY 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 10-7 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 instrumentally-modified 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, solar-system 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 10-8 and 1.4 x 10-8, respectively, and their positions
392 AMERICAN AND SOVIET PERSPECTIVES 2 >A ~2 Q - C) a. V] - oo o - , I,,,, I,,,, I I r 1 l T ' ' ' ' ~ _ - - - c, O - , - c -2 c: - - cat En o ~ _ n en I I I i I I I I I I I I 1 i I I I i I I I _ -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 power-law 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).
HIGH-ENER~ ~TROP~ICS Table I: Upper limits on the energy density of the cosmic GWB. 393 Pulsar Data 1855+09 1984.8-1989.3 0.30 < 1.2 x 10-36 < 6 x 10-8 1937+21 1984.8-1989.3 0.22 ~ 1.6 x 10-36 < ~ x 10-8 1937+21 1982.9~1989.3 0.16 < 0.8 x 10-36 <4 x 10-8 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. REFERENCES Albrecht, A, and N. Ibrolc 1989. Phys Rev. Lett., submitted. Bennett, D.P., and F.R. Bouchet. 1989. Phys Rev. Lett., submitted. Blandford, R. R. Narayan, and R. Romani. 1984. J. Astrophys. Astr. 5: 369. Blandford, Al)., and S.A. Teukolsky. 1976. Astrophys. J. 205: 580 (FIT). Brumberg, V4, Ya.B. Zel'donch, I.D. Nonkov, and N.I. Shaken. 1975. Astr. Lettem 1: 5. Damour, T., and N. Deruelle. 1986. Ann. Inst. H. Poincare (Physique ThEonque) 44: 263 (DD). Damour, I, and R Ruffini. 1974. C. R Acad. Sal. (Pans) 279: A971. Davis, M.M., J.H. Taylor, J.M. Weisberg, and D.C Backer. 1985. Nature 315: 547. Detweiler, S. 1979. Astrophys. J. 234: 1100. Esposito, LOO., and E.R Hamson. 1975. Astrophys J. Otters) 196: L1. Hellings, R.W., and G.S. Downs 1983. Astrophys. J. (Letters) 265: L39. Hogan, CJ., and MJ. Rees. 1984. Nature 311: 109. Hulse, Rid, and J.H. Taylor. 1975. Astrophys. J. (Letters), 195: LS1. Rawley, It, J.H. Taylor, M.M. Davis, and D.W. Allan. 1987. Science 238: 761. Romani, R.W., and J.H. Taylor. 1983. Astrophys. J. 265: I As. urchin, M.V. 1978. Soviet Astronom~AJ 22 36. Starobinsly, AA 1979. Sonet Physic~JETP Letters 30: 682. Taylor, J.H., and J.M. Weisberg. 1989. Astrophys. J. In press. ~goner, R.V. 1975. Astrophys. J. fitters) 196: L63. Written, E. 1984. Phys. Rev. D30 272.
