Dark Matter Detection Using Pulsar Timing

Dark Matter Detection Using Pulsar Timing ABSTRACT An observation program for detecting and studying dark matter subhalos in our galaxy is propsed. The gravitational field of a massive object distorts space time in such a way that light signals propagating though the field will be slowed down; an effect known as the Shapiro delay. We propose to detect dark matter subhalos by measuring the Shapiro delay caused by a subhalo as it passes in front of a pulsar. We show that such a detection is indeed possible and that measurements of the Shapiro delay can teach us a great deal about subhalo properties. 1. Introduction The identity of dark matter is one of the most profound mysteries of modern science. Although dark matter constitutes roughly 80% of the matter density in the universe, little is known about its properties or spatial distribution. By studying dark matter we can hope to learn about physics beyond the standard model and also increase our understanding of the evolution of structure in the universe. We propose a novel observation program aimed at detecting and understanding clumps of dark matter within our galaxy. The existence of such clumps known as dark matter subhalos is predicted by numerous simulations of galaxy formation (e.g Springel et al. 2008; Diemand et al. 2007; Zentner et al. 2005). The precise structures and distribution of these subhalos are poorly constrained, however. We plan to detect dark matter subhalos and study their structure by monitoring the effects of their gravitational fields on the timing of signals from pulsars. General relativity predicts that time is distorted in the presence of a gravitational field. Consequently, when a subhalo passes between an observer and a pulsar (a rapidly spinning neutron star that emits radio pulses at very even intervals) the radio signal from the pulsar will be modified slightly. We show that this modulation is detectable and that it can be used to probe the density structure of subhalos as well as their distribution in our galaxy. We propose to monitor many pulsars distributed throughout the galaxy in order to detect the signals from dark matter subhalos. In 2 we discuss what we hope to learn from the proposed observation program and what advantages this program has over other approaches to dark matter detection; in 3 we discuss the details of the proposed detection scheme and we give our concluding remarks in 4. 2. Motivation If successful, our proposed observation plan can teach us a great deal about the distribution and properties of dark matter. We hope to learn about how dark matter subhalos are distributed throughout the galaxy and how they are structured internally. Such observations will provide stringent tests of simulations of structure formation. Furthermore, since the clumping properties of dark matter are dependent on its particle properties, studying dark matter subhalos can teach us a lot about the particle nature of dark matter. For instance, the minimum mass of subhalos is predicted to depend strongly on the dark matter s particle interactions. Additionally, the proposed observations will probe the density profiles of dark matter subhalos which will in turn allow us to make constraints on signals from dark matter indirect detection experiments. The proposed observation program has a number of advantages over existing dark matter detection schemes: Because the clumping of cold dark matter (CDM) into subhalos is a generic prediction of many dark matter models, the detection of these subhalos is largely independent of 1

the dark matter particle properties. Many direct detection experiments, however, are aimed at detecting only one type of dark matter candidate. The proposed observations are independent of the local dark matter distribution and velocity spectrum. In contrast, earth based direct detection experiments make a number of assumptions about how the dark matter is distributed near the detector apparatus. If the earth were to reside in a void of dark matter, for instance, then such detection experiments would be doomed. The proposed observations can teach us about the distribution of dark matter in our galaxy, while direct detection experiments cannot. Compared to typical indirect detection experiments, our proposed observations are relatively low background. Our program has the advantage of using a number of existing technologies and observational techniques. 3. The Proposal 3.1. Overview When a light ray passes through the gravitational field of a collection of n massive point objects, it experiences a time delay, t, given by t = n i 2GM i c 3 ln(1 R x i ), (1) where G is the gravitational constant, M i is the mass of the i th object, c is the speed of light, and R and x i are unit vectors pointing in the direction of the source and the i th mass respectively (Larchenkova & Doroshenko 1995). Known as the Shapiro delay, this effect has been confirmed by measurements of the time delay of radar signals reflected off of Venus (Shapiro et al. 1968). In principle, if we could measure the Shapiro delay along various lines of sight, we could constrain the intervening mass distribution, and thereby detect dark matter subhalos. In order to measure the Shapiro delay, however, we need some sort of time keeping device. Such a device is readily available in the form of pulsars. Pulsars are rapidly spinning neutron stars that emit regular radio pulses. These pulses are very uniformly spaced in time. In most circumstances, the timing of a pulsar s pulses can be measured to the 1µs level, and in some cases we can do considerably better. Several authors have considered using pulsar timing to detect dark matter, although this idea has (as far as the author is aware) only been applied to dense dark matter candidates such as Massive Compact Halo Objects rather than to dark matter subhalos (e.g. Larchenkova & Doroshenko 1995). From Eq. 1, it is clear that in the case of a single intervening point mass whose location is known, a measurement of the Shapiro delay can yield the mass of the intervening object. Of course, the application to dark matter detection is complicated by several factors: 1. The location of the dark matter subhalo will likely be unknown. Its position can be constrained somewhat by the fact that it lies somewhere between the observer and the pulsar. 2. The dark matter subhalos are not point masses. Dark matter subhalos will generally have some density profile which is a function of the position within the subhalo. 3. The absolute time delay of the signal from the pulsar cannot be measured, but rather changes in the amount of time delay can be observed. 4. There may be several subhalos along the line of sight to the pulsar. Problems 1 and 2 can be addressed by measuring the Shapiro delay for multiple nearby pulsars. Such measurements would break the degeneracy between the mass and location of the intervening object and probe the density across the subhalo. However, there must be pulsars spaced sufficiently close together on the sky for such measurements to be possible. We show in 3.3 that in many cases there will be several pulsars close to the line of sight of the subhalo. Problems 3 can be addressed by taking into account the velocity of the subhalos with respect to the pulsars. As the subhalo moves past the pulsar, 2

the time delay along the line of sight to the pulsar will change. We show in 3.2 that for reasonable velocities and subhalo properties, the change in the time delay is measurable. Problem 4 is partially addressed by the fact that the odds of two dark matter subhalos being along the line of sight to a pulsar are relatively small if the pulsar is nearby. Again, multiple pulsar measurements can also help to determine whether multiple subhalos are present. 3.2. Time Delay from a Single Subhalo We first characterize the time delay caused by a subhalo passing between an observer and a pulsar. Simulations such as the Aquarius simulation (Springel et al. 2008) have made a number of predictions about the properties of galactic dark matter subhalos. These simulations have shown that the density profile inside of dark matter subhalos is reasonably well described by a Navarro Frenk White (NFW) density profile: ρ(r) = { ρ 0 2 r R 1+ r S R S if r < R C 0 if r > R C (2) where r is the radial coordinate, ρ 0 and R S are parameters of the particular subhalo, and R C is some cut off radius at which the density drops to zero (imposed to make the total mass of the subhalo finite). Following the results of simulations, we assume that R C cr S where we use a value of the concentration parameter c = 10 thoughout this paper ( Lokas & Mamon 2001). The value of c is known to vary between about 4 and 40 depending on the properties of the halo. One disadvantage of the NFW profile is that it predicts an infinite density at zero radius. This divergence is of course not physical and there must be some small radius at which the NFW profile is no longer valid. For the purposes of our analysis, we assume that the density stops increasing below r < 0.01R S. Assuming this profile, the total mass of the subhalo and its R C completely specify its interior mass distribution. Once the mass distribution is known, the total Shapiro delay due to the subhalo can be calculated along any line of sight. In Fig. 1 we plot the total Shapiro delay caused by subhalos of varying mass and radius that are directly along the line of sight. These were calculated by numerically integrating Eq. 1 over the NFW mass distribution described above. For now, we assume that the light ray propogates directly through the center of the subhalo. We have adopted reasonable distances to the subhalo and pulsar of 5 kpc and 10 kpc respectively. Also plotted in Fig. 1 as white lines are the regions in which dark matter subhalos are expected to reside, assuming that the radii of the subhalos are equal to their tidal radii (as predicted by simulations such as Springel et al. (2008)). The tidal radius, r t, is given by ( ) 1/3 M sub r t = r (3) d ln Menc (2 M enc (r) d ln r where r is the distance from the galactic center, M sub is the mass of the subhalo, and M enc (r) is the mass of the galactic halo enclosed within r. Since the tidal radius of a subhalo is a function of its position within the galaxy, there is a range of allowed subhalo masses for each subhalo radius. The upper dashed line corresponds to subhalos that reside at 10 3 pc, roughly the limit below which subhalos are expected to be tidally disrupted, while the lower dashed line corresponds to subhalos at the cut off radius of the galaxy ( 21000 pc). The solid line corresponds to subhalos that reside at 10 4 pc, a region that we might easily observe. As can be seen from the figure, for a wide range of subhalos, the Shapiro delay is actually quite long, greater than about 0.01 sec. The mere fact that the Shapiro delay caused by a subhalo is long does not necessarily mean that it is measurable. Unfortunately, we can only measure changes in the rate of a pulsar s pulses, not their absolute timing. Thus, in order for the subhalo to be detected it must be moving with respect to the pulsar at a rate that is sufficiently fast for the change in delay time to be measurable. We can assume that the velocity of the subhalo with respect to the pulsar is of order 500 km/s, characteristic of galactic orbital velocities in the outer galactic plane. If we further assume a distance to the subhalo of 5 kpc and that the observation time is one year, then the angular shift of the subhalo relative to the pulsar will be roughly 6 10 6 degrees. In Fig. 2 we show how the time delay changes as a function of the location of the pulsar along the line of sight to the subhalo. In generating Fig. 2 we have used a subhalo 3

Fig. 1. The Shapiro time delay experienced by a light ray passing through the center of a subhalo as a function of the mass and radius of the subhalo. The subhalo is assumed to have a density distribution specified by Eq. 2 and the following discussion. The distance to the subhalo is 5 kpc and the distance to the light source (i.e. the pulsar) is 10 kpc. The region between the dotted white lines corresponds to the region of parameter space where we expect to observe subhalos. 4

with M sub = 10 5 M and radius 300 pc. Comparing to Fig. 1, we see that a subhalo with these parameters lies comfortably within the bounds of the allowed subhalos. From Fig. 2 it is clear that an angular change of 6 10 6 degrees corresponds to a delay time change of about 4.2µs, roughly five times greater than the level of precision with which we can time pulsars. This result shows that detecting the time delay of pulsar signals produced by subhalos is a real possibility. Note that increasing the mass of the subhalo will make the detection even larger. 3.3. Detection Probability In order for a dark matter subhalo to be observed by the techniques described above, it must pass sufficiently close to the line of sight between the observer and the pulsar. The probability of observing such an event depends on three factors: the size of the subhalo, the distribution of pulsars in the galaxy and the distribution of subhalos in the galaxy. We consider each of these factors in the following discussion. In 3.2 we showed that for a subhalo with M sub = 10 5 M, the time delay effect is detectable. Of course, far enough away from the subhalo, the magnitude of the time delay will drop to zero and the effect will vanish. For the M sub = 10 5 M subhalo located 5 kpc away, the angular radius at which the effect becomes undetectable is roughly 5 degrees. Thus, in order to break the degeneracy between the mass and position of the subhalo and to begin exploring its density properties, we will need to have at least two pulsars that we are timing within 5 degrees of one another. Current estimates of the number of detectable pulsars in our galaxy are around 50000 while the number of currently detected pulsars is roughly 5000 (van Albada 1999). If the detected pulsars were spaced uniformly on the sky, then the angular separation between them would be roughly 2.9 degrees. Of course, pulsars are not distributed uniformly on the sky but rather cluster towards the galactic plane. Thus, in many cases the pulsars are space more closely together than 2.9 degrees (and in many cases further apart). Given the clustering of pulsars, it should be possible with existing pulsar data to find many regions on the sky where the spacing between several pulsars is on the order of 1 degree. Such regions would be ideal for detecting dark matter subhalos. Although the number of currently known pulsars is likely enough to detect dark matter subhalos, there is also the possibility of discovering new pulsars to improve our ability to constrain subhalo properties and increase our detection probability. A number of searches are currently underway to discover and time new pulsars. The proposed Square Kilometer Array, for instance, if constructed would likely detect many thousands of new pulsars (van Albada 1999). It may be useful to look for pulsars in globular clusters. Such clusters are expected to contain a number of pulsars, and since the angular size of these clusters is small, they are ideally suited for dark matter detection. The frequency of subhalo detections also depends on how many massive subhalos are nearby. According to simulations like the Aquarius simulation (Springel et al. 2008), subhalos have a mass function that goes roughly as dn dm sub dv = 3 104 M 1 kpc 3 (M sub / M ) 1.9 ( )( ) (4) r R S 1 + r R S (Koushiappas 2010). Using Eq. 4, we find that there should be roughly 240 subhalos with mass between 10 5 M < M sub < 10 6 M within 5 kpc of earth. Since each of these subhalos is detectable within an angular radius of 5 degrees, we expect about 15% of the sky to be covered by these subhalos (making the simple approximation that they are all located 5 kpc away and that the subhalos are isotropically distributed). Consequently, the probability of several pulsars being in front of subhalos is quite high. 3.4. Backgrounds A number of factors can effect the observed pulse rate of pulsars in addition to the presence of dark matter subhalos along the line of sight to the pulsar. For example, gravitational radiation emitted by the pulsar will cause its spin rate to decrease, while dispersion of the pulse signal in the interstellar medium will cause a delay in the arrival times of these signals. There are a number 5

Fig. 2. The Shapiro time delay (in seconds) as a function of the angular position of the background pulsar. The line of sight to the subhalo corresponds to an angular position of (0, 0). The distance to the pulsar is 10 kpc and the distance to the subhalo is 5 kpc. The mass of the subhalo is 10 5 M and its radius is 300 pc. 6

of techniques that can be used to separate these time delays from those caused by intervening subhalos. For one, the rate of pulsar spin down due to a number of sources is well understood and can be fitted out of the observed variation in the pulsar spin rate. See Taylor & Weisberg (1989) for an example of how factors such as the amount of pulsar spin down due to gravitational radiation can be precisely determined. Secondly, if multiple pulsars reside behind the subhalo, then the timing changes of each pulsar will be correlated to one another, allowing us to distinguish these changes from those due to other sources. Similarly, the presence of multiple pulsars located at different distances along the line of sight can be used to eliminate backgrounds. Light travelling from a pulsar that is located in front of a subhalo will of course not experience any Shapiro delay, while every pulsar that is behind the the subhalo will experience the delay. Since the distances to pulsars can be measured very accurately, we should be able to use this positional information to further constrain the properties of the subhalos. Additionally, almost all other sources of time delay will increase with time whereas the time delay due to the presence of subhalos may decrease. If the subhalo and pulsar velocities are such that the density along the line of sight to the pulsar is decreasing, then the pulsar signals will be seen to arrive faster and faster. Such a detection would be an almost unambiguous detection of a dark matter subhalo. Finally, other large masses passing between the earth and the pulsar may lead to a time delay signal that is confused with that from a dark matter subhalo. However, in all cases other than a dark matter subhalo, if the object is massive enough to cause a time delay it will likely be a compact object. A compact object would yield a much more quickly time varying signal and would also not be seen to affect the timing of other nearby pulsars. subhalos. Our proposed detection program has a number of significant advantages of over current dark matter searches. We have shown that the observations necessary for this program are feasible and could yield a great deal of valuable information about dark matter. Since these measurements can be carried out using existing telescopes and because the payoff is so great, we believe that this program is well worth the necessary effort. REFERENCES Diemand, J., Kuhlen, M., & Madau, P. 2007, ApJ, 667, 859 Koushiappas, S. M. 2010, Private Communication Larchenkova, T. I., & Doroshenko, O. V. 1995, A&A, 297, 607 Lokas, E. L., & Mamon, G. A. 2001, MNRAS, 321, 155 Shapiro, I. I., Pettengill, G. H., Ash, M. E., Stone, M. L., Smith, W. B., Ingalls, R. P., & Brockelman, R. A. 1968, Phys. Rev. Lett., 20, 1265 Springel, V., et al. 2008, MNRAS, 391, 1685 Taylor, J. H., & Weisberg, J. M. 1989, ApJ, 345, 434 van Albada, T. 1999, http://www.skatelescope.org /science/science.html Zentner, A. R., Berlind, A. A., Bullock, J. S., Kravtsov, A. V., & Wechsler, R. H. 2005, ApJ, 624, 505 4. Conclusions We have presented an observational program that if carried out could potentially detect the presence of dark matter subhalos in our galaxy and allow us to probe the internal structures of these This 2-column preprint was prepared with the AAS LATEX macros v5.2. 7