aa r X i v : . [ h e p - ph ] J u l Testing fifth forces from the Galactic dark matter ∗ Lijing Shao † Kavli Institute for Astronomy and Astrophysics,Peking University, Beijing 100871, China (Dated: July 5, 2019)
Abstract
Is there an unknown long-range force between dark matter (DM) and ordinary matters? Whensuch a fifth force exists and in the case that it is ignored, the equivalence principle (EP) is violated apparently . The violation of EP was severely constrained by, for examples, the E¨ot-Wash laboratoryexperiments, the lunar laser ranging, the
MICROSCOPE satellite, and the long-term observationof binary pulsars. We discuss a recent bound that comes from PSR J1713+0747. When it iscombined with the other bounds, a compelling limit on the hypothetical fifth force is derived. Forthe neutral hydrogen, the strength of such a fifth force should not exceed 1% of the gravity.
Keywords: dark matter; equivalence principle; binary pulsars
I. INTRODUCTION
Over the past hundreds of years, while physicists have established a sophisticated pictureto delineate the ordinary world around us, we are still lacking a coherent description ofthe dark world. Two notable substances, the dark matter (DM) and the dark energy, wereconjectured, though we do not know much detail of them [1]. In this proceeding we focuson the DM. Up to now, the DM was solely discovered via its gravitational interaction withthe ordinary matters. By using the word “ discovery ”, we mean to look for interactions withour experimental instruments, either directly or indirectly. The primary example for directsearches is to look for the interaction of DM with nucleons in underground laboratories.As an example of indirect searches, by looking for γ -ray excess in the direction of theGalactic Center, we aim to detect DM particles that, via some portal, decay or annihilateinto some standard-model particles which eventually couple to photons. Although various ∗ Presented at the meeting
Recent Progress in Relativistic Astrophysics , 6-8 May 2019 (Shanghai, China). † [email protected] λ larger than the typical length scale of the systems under discussion [4, 5]. Here we make useof the Galactic distribution of DM, hence λ ≫ O (10 kpc) and the mass of the force mediator m ≪ − eV /c [6]. We assume m → A and the DM, from scalar (“ − ” sign) orvector (“+” sign) exchange, is [4, 5], V ( r ) = ∓ g q ( A )5 q (DM)5 πr , (1)where g is the coupling constant and q is the dimensionless fifth-force charge [4, 5]. If sucha fifth force was ignored by the experimenter, she/he will “discover” an apparent violation ofthe equivalence principle (EP) between body A and body B when performing her/his gravityexperiments in the gravitational field of the DM, with an E¨otv¨os parameter η ( A,B )DM [3], η ( A,B )DM = ± g πGu q (DM)5 µ DM " q ( A )5 µ A − q ( B )5 µ B , (2)where G is the Newtonian gravitational constant, and ( q /µ ) is an object’s charge per atomicmass unit u . This is true even that the gravity is still described by the general relativity(GR), and EP is valid if the experimenter is aware of the fifth force. From observations, ifEP is observed to hold, one can put a limit on the fifth force. Such tests were performedwith the E¨ot-Wash laboratory experiment [3, 5] and the lunar laser ranging [7, 8]. TheE¨otv¨os parameter was constrained to be | η DM | . − . Here we discuss an independenttest from the binary pulsar PSR J1713+0747 [6], which has some specific distinctions fromSolar-system experiments.The proceeding is organized as follows. In the next section the relevant observationalcharacteristics of PSR J1713+0747 are introduced [9]. In section III, we review the EP-violating signal in the orbital dynamics of a binary pulsar [10, 11]. The method is applied to2ut a limit on the fifth force in section IV and the advantages of using neutron stars (NSs)are outlined. The last section discusses the possibility of finding a suitable binary pulsarclose to the Galactic Center, that will boost the test significantly. II. PSR J1713+0747
PSR J1713+0747 is a 4.5 ms pulsar in a binary system with an orbital period P b = 68 d.Its companion is a white dwarf (WD) with mass m c = 0 . M ⊙ . Due to its narrow pulseprofile and stable rotation, PSR J1713+0747 is monitored by the North American NanohertzGravitational Observatory (NANOGrav), the European Pulsar Timing Array (EPTA), andthe Parkes Pulsar Timing Array (PPTA). Splaver et al. [12], Zhu et al. [13], and Desvignes et al. [14] have published timing solutions for this pulsar, and the latest timing parametersfrom combined datasets are given in Zhu et al. [9]. Some relevant parameters for thisproceeding are collected in Table I.Because of mass transfer activities in the past, this binary has a nearly circular orbit.Nevertheless, its eccentricity, e . − , can still be measured. For the purposes in this TABLE I. Parameters for PSR J1713+0747 [9]. The numbers in parentheses indicate the uncer-tainties on the last digit(s).
Parameter Value
Spin frequency, ν (s − ) 218.8118438547250(3)Orbital period, P b (d) 67.8251299228(5)Time derivative of P b , ˙ P b (10 − s s − ) 0.34(15)Corrected ˙ P b (10 − s s − ) 0.03(15)Orbital inclination, i (deg) 71.69(19)ˆ x component of the eccentricity vector, e x − . y component of the eccentricity vector, e y . e x , ˙ e x (s − ) 0 . × − Time derivative of e y , ˙ e y (s − ) − . × − Companion mass, m c ( M ⊙ ) 0.290(11)Pulsar mass, m p ( M ⊙ ) 1.33(10) e x ≡ e cos ω and e y ≡ e sin ω where ω is the longitude of periastron. Usingdata from 1993 to 2014, the timing precision of PSR J1713+0747 has achieved to be sub- µ s. It renders a previous timing model for small-eccentricity binary pulsars, ELL1 [15], notaccurate enough. Zhu et al. [9] developed an extended model,
ELL1+ , by including higher-order contributions from the eccentricity. The
ELL1+ model includes terms up to O ( e ) inthe R¨omer delay [9]. The measured values for e x and e y are listed in Table I. In addition,the first time derivatives of e x and e y are also given in the table, assuming that the changesare linear in time [9].The measurements of the orbital decay and the eccentricity evolution were used to putconstraints on different aspects of gravitational symmetries [9, 16], including1. the gravitational constant G ’s constancy, (cid:12)(cid:12)(cid:12) ˙ G/G (cid:12)(cid:12)(cid:12) . − yr − ;2. the universality of free fall for strongly self-gravitating bodies in the gravitationalpotential of the Milky Way, | η Gal | < . α , | ˆ α | . − .The first one is based on the orbital decay measurement, and the rest are based on theeccentricity evolution measurements [9]. The second test is our focus here and it is to bediscussed below. III. EP-VIOLATING SIGNALS IN A BINARY PULSAR
Damour and Schaefer [10] proposed to use small-eccentricity binary pulsars in testingthe strong EP. When the EP is violated, a “gravitational Stark effect” takes place andpolarizes the binary orbit in a characteristic way. A related phenomenon, the so-called“Nordtvedt effect”, takes place in the Earth-Moon-Sun system when the EP is violated [17].It also happens for binary pulsars when the PPN preferred-frame parameters α [18, 19] and α [9, 20] are nonzero. In the current case, the relative acceleration between the pulsar andits companion star reads [10, 11],¨ R = − GMR ˆ R + A PN + A η , (3)where R is the binary separation, G is the gravitational constant, M is the total mass, andˆ R ≡ R /R . The post-Newtonian (PN) corrections are collected in A PN , and the EP-violating4bnormal acceleration is denoted as A η . At leading order, A η ≃ η DM a DM where a DM is thegravitational acceleration generated by the DM. To be explicit, here we take GR as thegravity, and the A η term comes from an unknown “fifth force” instead of some modifiedgravity. If A η comes from some modified gravity, then there will be extra considerations;for example, in that case the gravitational constant G should be replaced with an effectivegravitational constant, G [11].We define the eccentricity vector e ( t ) ≡ e ˆ a to have a length of e , and a direction fromthe center of mass of the binary towards the periastron, ˆ a . In the Newtonian gravity, e ( t )is a constant vector due to the fact that the Newtonian interaction has a larger symmetrygroup than SO(3). In GR, there is the famous periastron advance where, at leading order, e ( t ) rotates uniformly at a rate,˙ ω PN = 31 − e (cid:18) πP b (cid:19) / (cid:18) GMc (cid:19) / . (4)Under the relative acceleration (3), equations of motion get modified. After averaging overan orbit, the most important ones read [11], (cid:28) d P b d t (cid:29) =0 , (5) (cid:28) d e d t (cid:29) = f × l + ˙ ω PN ˆ k × e , (6) (cid:28) d l d t (cid:29) = f × e , (7)where ˆ k is the direction of orbital norm, l ≡ √ − e ˆ k , and f ≡ A η / (16 πGM/P b ) / .These differential equations can be integrated to give [10, 11], e ( t ) = e η + e GR ( t ) , (8)where e GR ( t ) is a uniformly rotating vector with a rate according to GR’s prediction (4), and e η ≡ f ⊥ / ˙ ω PN is a constant vector with f ⊥ representing the projection of f on the orbitalplane. IV. CONSTRAINTS ON THE FIFTH FORCE
From the theoretical side, we have a characteristic evolution of the eccentricity vector,dictated in Eq. (8), while from the observational side, we have measured the linear changes in5he eccentricity vector, decomposed to ˙ e x and ˙ e y in Table I. Therefore, by comparing them,we can perform a test of the existence of the A η term. Notice that the DM acceleration a DM comes from the Galactic DM distribution. It is different from that of Zhu et al. [9] where theauthors, considering a different scenario, used the total acceleration from the whole MilkyWay to obtain η Gal . We used an updated Galactic model [21] to calculate the accelerationfrom the DM. This choice does not change the relative strength in constraining the fifthforce from different experiments.The values of ˙ e x and ˙ e y from PSR J1713+0747 are consistent with e η = 0 in Eq. (8),which means that A η = 0 and η DM = 0. A careful analysis gives | η DM | < × − at the95% confidence level [6]. This limit is weaker than those from the E¨ot-Wash laboratoryexperiments and the lunar laser ranging. However, due to the very nature of the celestialbinary system, PSR J1713+0747 has multiple advantageous aspects [6]. • Driving force.
Because gravity is a manifest of the curved spacetime, free-fall statesare ideal in performing gravity tests. Though the measurement precision is not as goodas that of the E¨ot-Wash group, the
MICROSCOPE satellite gains a factor of 500 inthe driving force by putting the experiment in space in a free-fall state, thus achievinga better bound on η [22]. On the contrast, binary pulsars are usually worse in testing η due to the smaller driving force from the Milky Way. Nevertheless, if considering theDM as the attracting source, binary pulsars do not have such a disadvantage, for all theexperiments performed in the Solar system have the same attraction from the GalacticDM distribution. Even better, if a suitable binary pulsar is found in a region that hasa larger DM attraction, it may outperform the other tests (see the next section). It isinteresting to note that, the triple pulsar, PSR J0337+1715, though being excellent intesting the strong EP [23, 24], does not probe the fifth force from the Galactic DM. • Measurement precision.
The precision in measuring ˙ e is proportional to σ/ √ ¯ N T where σ is the rms of time-of-arrival (TOA) residuals, ¯ N is the average number ofTOAs per unit time, and T is the observational baseline [11]. Therefore, the test frombinary pulsars will improve as a function of time, especially with the new instruments,like the Five-hundred-meter Aperture Spherical radio Telescope (FAST) in China [25],and the Square Kilometre Array (SKA) in Australia and South Africa [26–28]. • Material sensitivity.
Unlike the majority of solid materials on the Earth that have6imilar portions of protons and neutrons, NSs are almost 100% made of neutrons whichare different from its WD companion. This gains a factor of O (10 ) when interpretingthe result | η DM | < × − to more fundamental theory quantities. Thus, though themeasurement of η DM from PSR J1713+0747 is worse than the other measurements, ithas a comparable power when being translated into fundamental theory parameters(see Fig. 1 in Ref. [6] for details). • Binding energy.
Ordinary materials that were used in the EP test have a massdeficit about O (0 . O (10%) due to the gravitational binding. Thisbenefits a lot in probing some specific parameter space that is very hard to investigatewith solely terrestrial experiments (see Table 1 and Fig. 1 in Ref. [6]).By combining all existing EP experiments, we reach the following conclusion: if there isa long-range fifth force between the DM and the ordinary matters, its strength should notexceed of the gravitational force for neutral hydrogens [6]. V. GALACTIC CENTER BINARY PULSARS
As is discussed in the previous section, the driving force from the DM is important inthis test. The experiments in the Solar system, by definition, cannot be done elsewherebut in the Solar system. Due to the static large-scale DM distribution in the Galaxy, theacceleration a DM in the Solar system is a fixed quantity, almost zero variation from placeto place inside the Solar system. Therefore, for these experiments, one cannot enlarge itsdriving force.However, binary pulsars in principle can be distributed anywhere in the Galaxy, and inthe future that the SKA is to discover all pulsars in the Milky Way that point towards theEarth [26]. Among them, it is likely that there are suitable binary pulsars for this test inthe region where the driving force is much larger. In particular, we consider the GalacticCenter region where the DM density is much denser. Gondolo and Silk [29] argued that inthe inner region of our Galaxy, there might be a DM spike. Such a spike will indeed enhancethe driving force significantly when a binary pulsar has a distance smaller than ∼
10 parsecto the Galactic Center. Studies on the pulsar population suggested that the inner parsec7ould harbor as many as thousands of active radio pulsars that beam at the Earth [30].Current and future searching plans are on their way (see e.g. Ref. [31]).
ACKNOWLEDGMENTS
We thank Zhoujian Cao, Michael Kramer, and Norbert Wex for helpful discussions. Thiswork was supported by the Young Elite Scientists Sponsorship Program by the China Associ-ation for Science and Technology (2018QNRC001), and partially supported by the NationalNatural Science Foundation of China (11721303), and the Strategic Priority Research Pro-gram of the Chinese Academy of Sciences through the Grant No. XDB23010200. [1] V. Sahni, Dark matter and dark energy, Lect. Notes Phys. , 141 (2004),arXiv:astro-ph/0403324.[2] G. Bertone, D. Hooper, and J. Silk, Particle dark matter: evidence, candidates and constraints,Phys. Rept. , 279 (2005), arXiv:hep-ph/0404175.[3] C. W. Stubbs, Experimental limits on any long range nongravitational interaction betweendark matter and ordinary matter, Phys. Rev. Lett. , 119 (1993).[4] E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl, and S. Schlam-minger, Torsion balance experiments: a low-energy frontier of particle physics,Prog. Part. Nucl. Phys. , 102 (2009).[5] T. A. Wagner, S. Schlamminger, J. H. Gundlach, and E. G. Adelberger, Torsion-balance testsof the weak equivalence principle, Class. Quant. Grav. , 184002 (2012), arXiv:1207.2442.[6] L. Shao, N. Wex, and M. Kramer, Testing the universality of free fall towards dark matterwith radio pulsars, Phys. Rev. Lett. , 241104 (2018), arXiv:1805.08408.[7] K. L. Nordtvedt, Cosmic acceleration of Earth and the Moon by dark matter,Astrophys. J. , 529 (1994).[8] J. G. Williams, S. G. Turyshev, and D. H. Boggs, Lunar Laser Ranging tests of theequivalence principle with the Earth and Moon, Int. J. Mod. Phys. D , 1129 (2009),arXiv:gr-qc/0507083.[9] W. W. Zhu et al. , Tests of gravitational symmetries with pulsar binary J1713+0747, on. Not. Roy. Astron. Soc. , 3249 (2019), arXiv:1802.09206 [astro-ph.HE].[10] T. Damour and G. Schaefer, New tests of the strong equivalence principle using binary pulsardata, Phys. Rev. Lett. , 2549 (1991).[11] P. C. C. Freire, M. Kramer, and N. Wex, Tests of the universality of free fall forstrongly self-gravitating bodies with radio pulsars, Class. Quant. Grav. , 184007 (2012),arXiv:1205.3751.[12] E. M. Splaver, D. J. Nice, I. H. Stairs, A. N. Lommen, and D. C. Backer, Masses, parallax,and relativistic timing of the PSR J1713+0747 binary system, Astrophys. J. , 405 (2005),arXiv:astro-ph/0410488.[13] W. W. Zhu et al. , Testing theories of gravitation using 21-year timing of pulsar binaryJ1713+0747, Astrophys. J. , 41 (2015), arXiv:1504.00662 [astro-ph.SR].[14] G. Desvignes et al. , High-precision timing of 42 millisecond pulsars with theEuropean Pulsar Timing Array, Mon. Not. Roy. Astron. Soc. , 3341 (2016),arXiv:1602.08511 [astro-ph.HE].[15] C. Lange, F. Camilo, N. Wex, M. Kramer, D. C. Backer, A. G. Lyne, and O. Doroshenko, Preci-sion timing measurements of PSR J1012+5307, Mon. Not. Roy. Astron. Soc. , 274 (2001),arXiv:astro-ph/0102309.[16] L. Shao and N. Wex, Tests of gravitational symmetries with radio pulsars,Sci. China Phys. Mech. Astron. , 699501 (2016), arXiv:1604.03662.[17] K. Nordtvedt, Equivalence principle for massive bodies. II. Theory,Phys. Rev. , 1017 (1968).[18] T. Damour and G. Esposito-Far`ese, Testing local Lorentz invariance of gravity with binarypulsar data, Phys. Rev. D46 , 4128 (1992).[19] L. Shao and N. Wex, New tests of local Lorentz invariance of gravity with small-eccentricitybinary pulsars, Class. Quant. Grav. , 215018 (2012), arXiv:1209.4503.[20] J. F. Bell and T. Damour, A New test of conservation laws and Lorentz invariance in relativisticgravity, Class. Quant. Grav. , 3121 (1996), arXiv:gr-qc/9606062.[21] P. J. McMillan, The mass distribution and gravitational potential of the Milky Way,Mon. Not. R. Astron. Soc. , 76 (2017), arXiv:1608.00971 [astro-ph.GA].[22] P. Touboul et al. , MICROSCOPE mission: first results of a space test of the equivalenceprinciple, Phys. Rev. Lett. , 231101 (2017), arXiv:1712.01176 [astro-ph.IM].
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