Testingthegravitationtheorywithshort-periodstars

aroundourGalacticCenterA.Hees,GalacticCenterGroup,UCLA

PI:A.GhezGCGMembers:T.Do,E.Becklin,S.Chappell,D.Chu,Z.Chen,A.Ciurlo,A.

Dehghanfar,A.Gautam,K.Kosmo,G.Martinez,M.Morris,S.Sakai,G.Witzel (UCLA)K.Matthews(Caltech),J.Lu,S.Jia (Berkeley),L.Gallego-Cano,R.Schödel (CSIC,

Spain)

LesRencontresdeMoriond,GravitationsessionMarch30th,2017

Globalpictureandmotivations• Searchingformodifiedgravitymotivatedby:• Quantumgravityand/orunificationtheories• DarkMatterandDarkEnergy

PictureinspiredbyAltschul etal,Adv,inSpaceRes.55,501,2015

Localphysics(SolarSystem,labtests,GNSS,…)

Astrophysics/cosmo(SNIa,CMB,structureformation,galacticdynamics,…)

UnificationQuantumgravityDMandDE

Highenergy(particlephysics:CERN-LHC,Fermilab,DESY,…)

GCobservationsallowtotestthegravitationtheoryinanewregion

• GRverywelltestedatsmallscalesandinweakgravitationalfield• GCoffersanewdifferentregiontotestGR.- isgravitationworkingasexpectedaroundaBH?- ifscreeningmechanismintheSolarSystem- strongfieldeffects- …ËË

ËË

ËËËË

ËË

¯̄

-5 0 5 10-10

-8

-6

-4

-2

0

2

log HMassêSolar MassLlogHPo

tentialêc2L

S0-2 closest approach

LIGO GW150914

Light deflection by the Sun

Precession of Mercury

Hulse-Taylor pulsar

Microlensing

Strong LensingGalactic rotation

Terrestrial Labs

Figure inspiredbyD.Psaltis,2004

DifferentobservationscanbeusedintheGCtotestthegravitationaltheory• ObservingtheBHshadow:EventHorizonTelescope(VLBI)

• ObservinganhypotheticalpulsarorbitingtheBH:nonedetectedsofar

• MonitoringthemotionofstarsorbitingtheBH:formorethan20years• UCLAobservations@Keckobservatory• MPIobservations@VLT

Forareview,seeJohannsen,CQG,2016

UCLAGCGhasbeenmonitoringthemotionofstarssince1995• Keckobservatory:• SpeckleandAdaptiveOpticsimagingAccuracy@0.15mas

• SpectroscopicmeasurementsAccuracy@30km/s

• Trackingthemotionof∼1000stars:• Usedtoconstructthereferenceframe• Keplerian motionforcentralstarsobserved

1- Galactic Center Group

3

Members:Andrea Ghez, Tuan Du, Eric Becklin, Samantha Chappell, Devin Chu, Zhuo Chen, Anna Ciurlo, Arezu Dehghanfar, Laly Gallego-Cano, Abhimat Gautam, Siyao Jia , Aurélien Hees, Kelly Kosmo, Jessica Lu, Greg Martinez, Keith Matthews, Mark Morris, Rainer Schoedel, Shoko Sakai, Gunther Witzel

Instrumentation: W.M.Keck observatory

• Speckle and AO imaging• Spectroscopy

A2starsfitprovidesthebestestimatesfortheSMBHmassanddistance

• Only2starswithhighenoughphasecoverage• Bestestimates:

seeBoehle etal,ApJ,2016

• Non-closingorbitsNOTduetoGRprecessionornewphysics:referenceframesystematics

S0-2’sclosestapproachin2018willenablerelativisticmeasurements

• Relativisticredshift:testoftheequivalenceprinciple

• pN orbitaldynamics:measurespace-timecurvature

Anticipatedbyalotoftheorists, seee.g.Rubilar andEckart,ApJ 2001,Zucker etal,ApJ,2006,Will,ApJ 2008,etc…

[RV ]rel =v2

2c+

GM

rc

Amplitude∼ 200km/s

h!̇iGR =6⇡GM

Pc2a(1� e2)

∼ 1.6x10-4 rad/yr

Wearenowoptimizingour2017/2018observationscampaigntodetecttherel.redshift• Withdataupto2016:sensitiveto∼ 1.5𝜎 totherel.redshift• Adaptiveschedulingtooldevelopedtooptimizeourtelescopetime.Whenandwhattoobserve?

A5𝜎 detectionattheendof2018isexpected… staytuned!

Onthemid-term,weexpecttomeasurepN orbitaldynamics

Multiplestarsfitneeded:technicaldifficultieswithsolutionsunderstudies(Bayesianalgorithms,priors,analyticalmodeling,…)

Improvementsintheobservationsexpected

Onthelong-term,the“nohair”theoremcanbetested

• “ABHisfullydescribedbyitsmass(M),charge(Q)andangularmomentum(J)”

• Testedbymeasuringspinandquadrupole moment(Q2):quadraticrelation?

distribution but also to understand the biases it introduces to themeasurement of the black hole properties.

Second, each of the measurements uses a very differentobservational technique (e.g., astrometric positions of stars,timing of radio pulsars, sub-millimeter images of the accretionflow) and is therefore subject to very different systematicuncertainties. Comparing the results from the three types ofobservations will allow us to identify the systematics inherentto each.

Finally, each type of observation is expected to lead tocorrelated uncertainties (or even degeneracies) between theblack hole spin and quadrupole moment. However, thecorrelations and degeneracies in each method are alongdifferent directions in the parameter space (see Figure 14).The orbital precession of stars and pulsars will primarilymeasure the spin of the black hole. The timing of pulsars willindependently measure the quadrupole moment of the space-time. A detection of an asymmetry of the black hole shadowwill measure deviations of the quadrupole moment from theKerr value. Combining all these measurements will lead touncorrelated measurements of the black hole spin andquadrupole moment and hence provide a test of the gravita-tional no-hair theorem.

Even though our focus in this article has been on testing theno-hair theorem, it is also important to emphasize thatcombining these three types of measurements will also haveother important implications for the astrophysics of accretionflows and of supermassive black holes in the centers ofgalaxies. For example, as discussed above, differences amongthe enclosed mass inferred at different radii with the EHT, withstars, and with pulsars, will allow us to constrain thedistribution of stellar objects and dark matter at the very center

of our Galaxy. Furthermore, measurement of the relativeorientation of the black hole spin and the angular momentum ofthe inner accretion flow will inform our understanding of blackhole feeding and alignment of black hole spins (see, e.g., thediscussion in Psaltis et al. 2015a).It is true that the EHT and GRAVITY experiments still need

to demonstrate that they can operate at their designedspecifications and a pulsar, as well as at least two stars, needto be discovered in sufficiently close orbits around Sgr A* forthe three types of observations discussed here to be realized atthe required accuracy. However, all these are possible in thevery near future and promise to revolutionize our probes andunderstanding of strong-field gravity and accretion flows in thevicinity of black holes.

D.P. thanks the Max-Planck-Institut für Radioastronomie fortheir hospitality during the visit in which this project wasconceived. D.P. acknowledges support from NASA/NSFTCAN award NNX14AB48G and NSF grant AST 131203.N.W. acknowledges valuable discussions with Tal Alexanderand Kuo Liu. This work has been supported by the ERCSynergy Grant BlackHoleCam under grant agreement No.610058.

REFERENCES

Abdujabbarov, A., Atamurotov, F., Kucukakca, Y., Ahmedov, B., & Camci, U.2013, Ap&SS, 344, 429

Amarilla, L., & Eiroa, E. F. 2013, PhRvD, 87, 044057Angélil, R., & Saha, P. 2014, MNRAS, 444, 3780Angélil, R., Saha, P., & Merritt, D. 2010, ApJ, 720, 1303Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214Bambi, C., & Freese, K. 2009, PhRvD, 79, 043002Bambi, C., & Yoshida, N. 2010, CQGra, 27, 205006Barausse, E., & Sotiriou, T. P. 2008, PhRvL, 101, 099001Bardeen, J. M. 1973, in Black Holes, ed. C. DeWitt & B. S. DeWitt (New

York: Gordon and Breach), 229Bates, S. D., Johnston, S., Lorimer, D. R., et al. 2011, MNRAS, 411, 1575Bower, G. C., Deller, A., Demorest, P., et al. 2014, ApJ, 780, 2Bower, G. C., Deller, A., Demorest, P., et al. 2015, ApJ, 798, 120Bramante, J., & Linden, T. 2014, PhRvL, 113, 191301Broderick, A. E., Fish, V. L., Doeleman, S. S., & Loeb, A. 2009, ApJ, 697, 45Broderick, A. E., Fish, V. L., Doeleman, S. S., & Loeb, A. 2011, ApJ, 735, 110Broderick, A. E., Johannsen, T., Loeb, A., & Psaltis, D. 2014, ApJ, 784, 7Broderick, A. E., & Loeb, A. 2006, MNRAS, 367, 905Carter, B. 1971, PhRvL, 26, 331Chan, C.-K., Psaltis, D., Özel, F., Narayan, R., & Sad¸owski, A. 2015, ApJ,

799, 1Chennamangalam, J., & Lorimer, D. R. 2014, MNRAS, 440, L86Cordes, J. M., & Lazio, T. J. W. 2001, ApJL, 562, L157Cordes, J. M., & Lazio, T. J. W. 2002, arXiv:astro-ph/0207156Damour, T., & Deruelle, N. 1985, AnIHP, 43, 107Damour, T., & Deruelle, N. 1986, AnIHP, 44, 263Deneva, J. S., Cordes, J. M., & Lazio, T. J. W. 2009, ApJL, 702, L177Dexter, J., Agol, E., Fragile, P. C., & McKinney, J. C. 2010, ApJ, 717, 1092Dexter, J., & O’Leary, R. M. 2014, ApJL, 783, L7Doeleman, S., Agol, E., Backer, D., et al. 2009a, astro2010: The Astronomy

and Astrophysics Decadal Survey, 2010, 68Doeleman, S. S., Fish, V. L., Broderick, A. E., Loeb, A., & Rogers, A. E. E.

2009b, ApJ, 695, 59Eatough, R. P., Falcke, H., Karuppusamy, R., et al. 2013, Natur, 501, 391Eatough, R. P., Kramer, M., Klein, B., et al. 2013, in IAU Symp. 291, ed.

J. van Leeuwen (Cambridge: Cambridge Univ. Press), 382Eatough, R., Lazio, T. J. W., Casanellas, J., et al. 2015, Advancing

Astrophysics with the Square Kilometre Array, Fundamental Physics withPulsars (Trieste: SISSA), 45

Einstein, A. 1915, Sitzungsberichte der Königlich Preußischen Akademie derWissenschaften (Berlin)

Eisenhauer, F., Perrin, G., Brandner, W., et al. 2011, Msngr, 143, 16Falcke, H., Melia, F., & Agol, E. 2000, ApJL, 528, L13Faucher-Giguère, C.-A., & Loeb, A. 2011, MNRAS, 415, 3951

Figure 14. Comparison of the posterior likelihood of measuring the spin andquadrupole moment of Sgr A* using the orbits of two stars (blue), timing ofthree periapsis passages of a low-precision pulsar (red), and the shape of itsshadow (gold). The solid curve shows the expected relation between these twoquantities for the Kerr metric. The filled circle marks the assumed spin andquadrupole moment ( 0.6D � , q 0.36∣ ∣ � ). Combining these threeindependent types of measurements, each of which suffers from differentbiases and potential systematic uncertainties, will significantly increase ourconfidence in the inference of these two black hole properties and in the test ofthe no-hair theorem.

18

The Astrophysical Journal, 818:121 (19pp), 2016 February 20 Psaltis, Wex, & KramerFig.fromPsaltis,Wex,Kramer,ApJ,2016

Q2 = �J2

M

seealsoWill,ApJL,2008

Inthemeantime:currentobservationscanbeusedtoconstraindeviationsfromNewtoniangravity

Futureobservationswillenabledetection/testsofGReffects

Afifthforceisawell-motivatedphenomenology• ModificationoftheNewtonianpotential

�(r) = �GM

r

h1 + ↵e�r/�

i

• Phenomenologymotivatedby• newinteractionwithamassivegaugeboson• highdimensionstheories• braneworld scenarios• tensorscalartheories• Massivegravity• …

Fischbach andTalmadge,Nature,1989

KrauseandFischbach,2001Arkani-Hamed,etal,PRD,1999Alsingetal,PD,2012Will,PRD,1998

• ProducesomenoticeableeffectsonS0-2’sorbit seeBorka etal,JCAP,2012

S0-2andS0-38constrainafifthforce• Analysisextremelycarefulwithcorrelations(GM,R0)andwithsystematics

• Willbeimprovedsignificantlyinthefuture(after2018…)

��-� ��� ��� ��� ��� ��� ���� ���� ������-��

��-�

��-�

��-�

��-�

���

��� ��-�� ��-� ��-� ��-� ��� ���

λ [�]

��%�������������

����

|α|

λ [��]

��

��� �������

������

�����������

���

��� ��� ���� ����

λ [��/��]

��

������� ���

������

����

�������� �� ���������� ����

FigureupdatedfromKonopliv etal,Icarus,2011

ConstraintontheadvanceofS0-2’speriastron• Modelindependentconstraint• Analysisextremelycarefulwithcorrelations(ref.frame)andsystematics

-� -� -� � � � �ω��-� [��-� ���/��]

3

We set the outer radius cuto↵ r

0

to 0.011 parsec suchthat it encloses S0-2 and S0-38 at apoapse. In total, ourmodel includes 21 parameters consisting of the six orbitalparameters for each of the two stars: P , e, the time ofclosest approach T

0

, the argument of periastron !, the in-clination i and the longitude of the ascending node ⌦ andnine global parameters: the SMBH gravitational param-eter GM , the strength of the fifth force ↵, the amount ofextended mass, M

ext

(< r

0

), the distance to our GalacticCenter, R

0

, and the positions (x0

and y

0

) and velocities(v

x

0

, vy

0

, vz

0

) of the SMBH. The SMBH positions andvelocities are important in order to take into accountimperfections in the construction of the reference frame.The observations are assumed to be independent and nor-mally distributed and we use a Gaussian likelihood an-alytically marginalized with respect to the SMBH posi-tions and velocities, which reduces the number of param-eters e↵ectively sampled to 16. We use flat priors for allthe parameters except for the extended mass M

ext

(< r

0

)for which we use an exponential prior characterized bya standard deviation of �

M

ext

(<r

0

)

= 100M�. This prioris motivated by observations of the stellar cusp [69–72].The extended mass power-law slope � is fixed to 0.5. Wehave checked that our results are not sensitive to the ac-tual value of �

M

ext

(<r

0

)

and �. In particular, �M

ext

(<r

0

)

can be increased by two orders of magnitude without im-pacting our results. Di↵erent fits are performed for fixedvalues of � ranging from 5 AU (astronomical units) to104 AU.

From the sampling of the posterior probability distri-bution function of ↵ produced by each fit, we determinea statistical 95% confidence upper limit on the absolutevalue of ↵. It has been shown in [43] that our orbital fitssu↵er from systematic e↵ects related to the constructionof the absolute reference frame. In order to assess thesesystematics, we used a Jackknife resampling method [73].We used the seven di↵erent reference frames created in[43] in which each one has one SiO maser excluded. Theresults of the orbital fits of S0-2 performed using theseseven subset reference frames are then used in order toinfer a systematic uncertainty given by

�

2

sys

n� 1

n

nX

i=1

(xi

� x̄

i

)2 , (3)

where n is the sample size (here 7), xi

is the estimatorof a parameter derived by excluding the ith maser andx̄

i

is the mean of the values x

i

(see also Appendix C of[43] for more details about this procedure). The inferredsystematic uncertainty is then added in quadrature tothe statistical uncertainty derived from the orbital fit.The values of our analysis before and after the Jackknifeprocedure can be found in Tab. 1 from the Supplementalmaterial.

Our results show that the value ↵ = 0 is always withinthe 68% confidence interval meaning that no significant

deviation from Newtonian gravity is found. The redcurve from Fig. 2 shows our 95% confidence upper limiton |↵|. Our best constraint is at the level of � ⇠ 150AU which corresponds roughly to the S0-2 distance atperiapse. For this value of �, our dataset gives a 95%confidence upper limit of |↵| < 0.016. For higher valuesof �, the upper limit on |↵| evolves proportionally to �

2

(similarly to the curves obtained by LLR and planetaryephemerides, see Fig. 2) up to when it reaches |↵| ⇠ 1where it diverges at � ⇠ 6000 AU. We note that the limiton ↵ is primarily driven by S0-2 in the median range of� and S0-38 helps for small and large �.As shown in Fig. 1, the constraints obtained in this

work probe a new part of the parameter space and arecomplementary to Solar System measurements. In par-ticular, short-period stars are probing space-time in ahigher potential and around a central body much moremassive than in the other experiments. This is high-lighted on the right panel of Fig. 2 where � is expressed interm of the gravitational radius of the central body. Fur-thermore, short-period stars probe the space-time arounda SMBH, which is conceptually di↵erent from Solar Sys-tem tests where the space-time curvature is generatedby weakly gravitating bodies. In particular, some non-perturbative e↵ects may arise around strongly gravitat-ing bodies (see e.g., [74]). In addition, in models of grav-ity exhibiting screening mechanisms, deviations from GRmay be screened in the Solar System (see e.g., [75]). Inthis context, searches for alternative theories of gravita-tion in other environments are important.A specific theoretical model covered by the fifth force

framework is a massive graviton. In that context, wefound a 90% confidence limit � > 5000 AU for ↵ = 1,which can be interpreted as a lower limit on the gravi-ton’s Compton wavelength �

g

> 7.5⇥ 1011 km or equiv-alently as an upper bound on the graviton’s mass m

g

<

1.6 ⇥ 10�21 eV/c2 (see also [35]). This constraint is oneorder of magnitude less stringent than the recent boundobtained by LIGO [76], which nevertheless does not ap-ply for all models predicting a fifth force.From an empirical prospective, one of the e↵ects pro-

duced by a fifth force is a secular drift of the argument ofperiastron ! [31, 77]. Several theoretical scenarios pre-dict such an e↵ect, which can be constrained by observa-tions. We produced a new orbital fit using a model thatincludes seven global parameters (the SMBH GM , R

0

and the positions and velocities of the SMBH) and sevenorbital parameters for each star, the additional parame-ter being a linear drift of the argument of the periastron!̇. As a result of our fit including the Jackknife analysis,we obtained an upper confidence limit on a linear driftof the argument of periastron for S0-2 given by

|!̇S0-2

| < 1.7⇥ 10�3 rad/yr at 95 % C.L. . (4)

This limit is currently one order of magnitude largerthan the relativistic advance of the periastron !̇

GR

=canbeusedtoconstrainvarious

theoreticalscenarios

Tosummarize• GRisprobablynottheultimatetheoryofgravitation.

Needforobservationstoconstrain/detectnewphysics• Newwindowtostudygravitationisopening:

usingorbitaldynamicsaroundaSMBH• DeviationfromNewton:fifthforceandS0-2advanceofperiastron areconstrained• OurdatasetisjuststartingtobesensitivetoGReffectsandallowstosearchfor/constrainnewphysics

S0-2’sclosestapproachin2018willopenanewwindowtotestGR:staytuned!

BACKUPSLIDES

Fifthforceanalysis

• Evaluationofsystematicsduetotheref.frame:• Jacknife resamplingalgorithm:“drop1maser”analysis

• Parametersfitted:• Fifthforceparameter• BHGM:highlycorrelatedwith𝛼• DistancetoGC:𝑅%• SMBHpositionandvelocity:importanttotakeintoaccountimperfectionsinthereferenceframe• Extendedmass• 6orbitalparametersforeachstar(P,e,T0,𝜔,Ω, 𝑖)