Constraints on parameters of dark matter and black hole in the Galactic Center

ISSN 1063-7788, Physics of Atomic Nuclei, 2010, Vol. 73, No. 11, pp. 1870–1877. c© Pleiades Publishing, Ltd., 2010.

ELEMENTARY PARTICLES AND FIELDSTheory

Constraints on Parameters of Dark Matterand Black Hole in the Galactic Center*

A. F. Zakharov1), 2), 3)**, F. De Paolis4), G. Ingrosso4), and A. A. Nucita5)

Received April 12, 2010

Abstract—A concept of dark matter (DM) is introduced. As for other anomalies, we describe two ways tosolve DM problem, namely a conservative way when we have to find substances with DM properties or wehave to change a fundamental gravity law. We discuss constraints on DM concentration near the GalacticCenter from apocenter shift data.

DOI: 10.1134/S1063778810110086

1. INTRODUCTION. URBAIN JEAN JOSEPHLE VERRIER: NEPTUNE DISCOVERY (1846)

& MERCURY’S ANOMALY (1859)(AN INVISIBLE (“DARK”) OBJECT

OR A VIOLATION OF THE NEWTONIANGRAVITY LAW)

A connection between cosmology and Le Ver-rier’s discoveries of anomalies has been pointed outby Juszkiewicz [1]. In 1846, analyzing trajectoriesof known objects (planets) and reconstructing po-tentials and masses and trajectories of all objects inthe game, Le Verrier predicted an existence of extra(initially unknown (dark)) planet [2], Neptune andsoon afterwards the planet was detected by Germanastronomer Galle [3]6).

Le Verrier discovered the Mercury pericenteranomaly and explained 93% of the observed value,but a supplementary advance 38 arcseconds/centurywas without an explanation (later on, the value wascorrected as 43 arcseconds/century) [6].

∗The text was submitted by the authors in English.1)National Astronomical Observatories of Chinese Academy

of Sciences, Beijing, China.2)Institute of Theoretical and Experimental Physics, Moscow,

Russia.3)Bogoliubov Laboratory for Theoretical Physics, JINR,

Dubna, Russia.4)Dipartimento di Fisica, Universita del Salento and INFN

Sezione di Lecce, Italy.5)XMM-Newton Science Operations Centre, ESAC, ESA,

Madrid, Spain.**E-mail: [email protected])Oort and Zwicky used the same scheme leading to the in-

troduction of dark matter (DM) concept; when he discoveredhigh velocity dispersions, they concluded about an existenceof DM rather than a violation of a fundamental gravity law atlarge distances [4, 5].

Le Verrier had analyzed the following options toexplain an anomaly:

A gravitational field of an invisible matter (planet,asteroids near Sun). This means an introduction ofnew objects.

A deviation from the Newtonian law. It means achange of a law of Nature.

A precision of a model is not good enough and amodel has to be clarified.

In 1876 Le Verrier analyzed information about 25transits of Vulcan (according to his opinion 19 tran-sits were reliable) and predicted a transit in March1877 (the planet was not observed) [7]7).

Similarly to cosmological DM and dark energy(DE) problems now, different options were consid-ered such as an existence of an extra planet be-tween the Sun and Mercury (Vulcan’s prediction), amodification of Venus mass by more than 10% andmodification of the Newton gravity law (for exam-ple, such as Newcomb’s modification of the New-ton’s law such as 1/rn [8] (n = 2.0000001574 fordω/century = 42.34′′, earlier Hall (1894) used n =2.00000016 for dω/century = 43′′ [9]). We will showthat, actually, a solution about a modification of theNewton gravity law to explain anomalies such as theMercury one was given by I. Newton in his Principia,really he considered a generalization of a gravitationalforce [10]

F =brp + crm

r3, (1)

then he obtained

dω = 2π

√∣∣∣∣ b + c

mb + pc

∣∣∣∣. (2)

7)Le Verrier died on 23 September 1877.

1870

CONSTRAINTS ON PARAMETERS OF DARK MATTER 1871

Therefore, for p = 1 we have

F =br + crm

r3, (3)

so, if m ≥ 2, we have generalizations of the Newto-nian force with an extra term. On the other hand, fora specific case c = 0, F = brp−3 we obtain

dω = 2π

√∣∣∣∣ b

pb

∣∣∣∣ = 2π√

1p, (4)

and for p = 3 − n we have

dω = 2π

√1

3 − n(5)

and if n = 2 + δ, then

dω = 2π

√1

1 − δ≈ 2π

(1 +

δ

2

). (6)

Therefore, following Le Verrier’s way and analyz-ing carefully trajectories of celestial bodies we canreconstruct gravitational potentials and mass distri-butions governing motions of celestial bodies as it willbe shown for the Galactic Center.

2. DARK MATTER IN THE GALACTICCLUSTERS AND IN SPIRAL GALAXIES

In 1930’s Oort [4] (for Galaxy) and Zwicky [5,11] (for the Coma cluster) found that an evaluatedgravitational potential is different from a potential re-constructed with visible matter. Like Le Verrier for theMercury anomaly, Oort used a conservative approachassuming that a gravity law has to be the Newtonianone, therefore, one has to introduce an invisible sub-stance as DM [4]. This approach has been supportedwith studies of rotation curves for spiral galaxies [12–15] (see Fig. 1). From an inspection of the figure,one concludes that initially it is a linear growth ofvelocities with radii, after that velocities are roughlyconstant (or in other words, rotation curves are flat).From the virial theorem for a steady state configura-tion and the Newtonian potential we have [16]

2T + U = 0, (7)

where T is the average kinetic energy, while U is theaverage potential energy or

mV 2 =GmM(R)

R, (8)

where V is an average velocity of stars or cloudsin galaxies at a distance about R from the galacticcenters, M(R) is a total galactic mass concentratedinside a ball with R radius, m is a mass of a star or acloud. If we wish to fit rotation curves with sphericalsymmetrical distributions, assuming ρ = const (thus,

r

v(r)

Fig. 1. The sketch of a typical rotation curve for a spiralgalaxy. Solid line is an observable rotation curve, whiledashed line corresponds to a theoretical rotation curve ifone takes into account only a visible matter.

we have M(R) ∝ R3 and V ∝ R), for the flat part ofrotation curves we suggest ρ = const/R2 (therefore,we have M(R) ∝ R and V = const).

Again, like for the Mercury anomaly, we have thedilemma such as we have to introduce a new sub-stance (DM) (a conservative solution) or to changea fundamental gravity law (a more radical solution).Initially, people prefer the conservative solution andthey try to find DM in different ways but at the mo-ment we cannot exclude a possibility that a mod-ification of gravity law is needed such as MOND,MOG [17–20], conformal cosmological models [21–35] based on the Dirac version of general relativ-ity [36], or f(R) gravity [37–55] (however, there aresevere constraints on these models [56, 57]).

2.1. Different Components of Matter

Counting the luminous sources shows that a frac-tion of luminous matter Ωlum < 1%, while Big Bangnucleosynthesis and CMB data give an estimate forbaryonic matter Ωb ≈ 5%, non-baryonic DM Ωnb ≈20−25% [58, 59].

One could expect an existence of baryonic DMin faint astronomical objects such as brown dwarfs,Jupiter or Earth-like planets, black holes as a resultof stellar evolution, snow balls etc., while for non-baryonic DM we have to introduce new objects suchas primordial black holes or new particles and onesof the most popular of these are neutralinos whichare Weakly Interacting Massive Particles (WIMPs).There are different ways, namely, direct and indi-rect ones [60, 61]. Basically, in direct searches peo-ple measure interactions (recoils) of nuclei with DMparticles, while in indirect searches different sets ofobservational data (γ-radiation, for instance) are usedto find signatures of DM particles. In cases of indirectsearches one can expect alternative explanation with-out an introduction of DM particles.

PHYSICS OF ATOMIC NUCLEI Vol. 73 No. 11 2010

1872 ZAKHAROV et al.

There are claims about discoveries of neutralinoexistence with both direct and indirect searches. TheDAMA Collaboration has claimed that they discov-ered a modulation which is related with neutralinofeatures [62], but other collaborations do not confirmthe DAMA result, moreover, they insist that DAMAresult has to be ruled out with their own data. How-ever, an independent analysis shows that DAMA andother experimental data may be not contradictory, butat the moment it is very hard to evaluate neutralinomass from these data [63]. So, we have to wait forfurther independent confirmations (or disproofs) ofthe DAMA conclusions.

2.2. Indirect Searches of DM

In the last years intensive searches for dark mat-ter, especially its non-baryonic component, both ingalactic halos and at galaxy centers have been under-taken (see, for example, [64, 65] for recent results).It is generally accepted that the most promisingcandidate for the DM non-baryonic component isneutralino. In this case, the γ flux from galactichalos (and from our Galactic halo, in particular)could be explained by neutralino annihilation [66–72]. In particular, it was claimed that the excess γ-ray emission �1 GeV found by Energetic Gamma-Ray Experiment Telescope (EGRET) instrumenton the Compton Gamma-Ray Observatory may beexplained by neutralino annihilation in the Galactichalo [73]. However, recent Fermi Large Area Tele-scope (Fermi-LAT) observations do not support theEGRET excess [74].

Since γ-rays are detected not only from highgalactic latitude, but also from the Galactic Center,there is a widespread hypothesis [75] that a DM con-centration might be present at the Galactic Center.In this case the Galactic Center could be a strongsource of γ-rays and neutrinos [64, 67, 76–83] due toDM annihilation. Since it is also expected that DMforms spikes at galaxy centers [84–86], the γ-ray fluxfrom the Galactic Center should increase significantlyin that case.

3. DARK MATTER IN THE GALACTICCENTER

For the black hole in the Galactic Center, Halland Gondolo used estimates of the enclosed massobtained in various ways [87] and tabulated by Ghezet al. [88, 89]. The black hole, stellar cluster, andDM could contribute in the mass inside stellar or-bits. Moreover, if a DM cusp does exist around theGalactic Center, it could modify the trajectories ofstars moving around it in a sensible way dependingon the DM mass distribution. A great progress inmonitoring bright stars near the Galactic Center have

been reached recently [88–90]. The astrometric limitfor bright stellar sources near the Galactic Centerwith 10-m telescopes is today δθ10 ∼ 1 mas and theNext Generation Large Telescope (NGLT) will beable to improve this number at least down to δθ30 ∼0.5 mas [91, 92] or even to δθ30 ∼ 0.1 mas [91–93] in the K-band. Therefore, it will be possible tomeasure the proper motion for about ∼100 stars withastrometric errors several times smaller than errorsin current observations. Recently it was shown [94–97] that it is possible to constrain the parameters ofthe DM distribution around the Galactic Center byconsidering the induced apoastron shift due to thepresence of this DM concentration. Even now onecan get constraints using available data obtained withthe present generation of large telescopes (the so-called conservative limit). Moreover, there are greatexpectations for the constraints with data from futureNGLT observations or with other advanced observa-tional facilities.

3.1. The Mass Concentration at the Galactic Center

Recent advancements in infrared astronomy areallowing to test the scale of the mass profile at thecenter of our galaxy down to tens of AU. With theKeck 10-m telescope, the proper motions of severalstars orbiting the Galactic Center black hole havebeen monitored and almost entire orbits, as, for exam-ple, that of the S2 star, have been measured, allowingan unprecedent description of the gravitational po-tential for the Galactic Center region. Measurementsof the amount of mass M(< r) contained within adistance r from the Galactic Center are continuouslyimproved as more precise data are collected. Recentobservations [88] extend down to the periastron dis-tance (�3 × 10−4 pc) of the S16 star and they corre-spond to a value of the enclosed mass�3.67× 106M�within �3 × 10−4 pc. Here and in the following, weuse the three-component model for the central regionof our galaxy, based on estimates of enclosed massgiven by Ghez et al. [88, 89] recently proposed [87].This model is constituted by the central black hole,the central stellar cluster, and the DM sphere (madeof WIMPs), i.e.,

M(< r) = MBH + M∗(< r) + MDM(< r), (9)

where MBH is the mass of the central black holeSagittarius A∗. For the central stellar cluster, theempirical mass profile is

M∗(< r) =

⎧⎪⎪⎨⎪⎪⎩

M∗

(r

R∗

)1.6

, r ≤ R∗,

M∗

(r

R∗

)1.0

, r > R∗,

(10)



with a total stellar mass M∗ = 0.88 × 106M� and asize R∗ = 0.3878 pc. As far as the mass profile of theDM concentration is concerned, Hall and Gondolohave assumed a mass distribution of the form [87]

MDM(< r) (11)

=

⎧⎪⎨⎪⎩

MDM

(r

RDM

)3−α

, r ≤ RDM,

MDM, r > RDM,

MDM and RDM being the total amount of DM in theform of WIMPs and the radius of the spherical massdistribution, respectively.

A likelihood analysis has allowed to estimate forthe DM mass the value MDM � 105M�, while theDM sphere size results are in the range 10−4−1 pc.Hall and Gondolo [87] discussed limits on DM massaround the black hole at the Galactic Center. It isclear that present observations of stars around theGalactic Center do not exclude the existence of aDM sphere with mass �4 × 106M�, well containedwithin the orbits of the known stars, if its radius RDMis �2 × 10−4 pc (it is an estimate of the periastrondistance for the S16 [89]). However, if one considersa DM sphere with a larger radius, the correspondingupper value for MDM decreases (although it tendsagain to increase for extremely extended DM con-figurations with RDM � 10 pc). In the following, wewill assume for definiteness a DM mass MDM ∼ 2 ×105M�, which is the upper value for the DM spherein [87] within an acceptable confidence level in therange 10−3−10−2 pc for RDM. As it will be clear inthe following, we emphasize that even such a smallvalue for the DM mass (that is about only 5% ofthe standard estimate (3.67 ± 0.19 × 106)M� for thedark mass at the Galactic Center [89]) may give someobservational signatures.

Evaluating the S2 apoastron shift8) as a functionof RDM, one can further constrain the DM sphereradius since even now we can say that there is noevidence for negative apoastron shift for the S2 starorbit at the level of about 10 mas. In addition, sinceat present the precision of the S2 orbit reconstructionis about 1 mas, we can say that even without futureupgrades of the observational facilities and simplymonitoring the S2 orbit, it will be possible withinabout 15 years to get much more severe constraintson RDM.

8)We want to note that the periastron and apoastron shifts ΔΦas seen from the orbit center have the same value, whereasthey have different values as seen from Earth (see belowEq. (15)). When we are comparing our results with orbitreconstruction from observations, we refer to the apoastronshift as seen from Earth.

Moreover, observational facilities will allow in thenext future to monitor faint infrared objects at theastrometric precision of about 10 μas [98] and, in thiscase, previous estimates will be sensibly improvedsince it is naturally expected to monitor eccentricorbits for faint infrared stars closer to the GalacticCenter with respect to the S2 star.

In the following section, we study the motion ofstars as a consequence of the gravitational potentialΦ(r) evaluated from the mass profile given in Eq. (9).As usual, the gravitational potential can be evaluatedas

Φ(r) = −G

∞∫r

M(r′)r′2

dr′. (12)

3.2. Apoastron Shift Constraints

According to general relativity (GR), the motionof a test particle can be fully described by solvingthe geodesic equations. Under the assumption thatthe matter distribution is static and pressureless, theequations of motion in the Post-Newtonian (PN)approximation become (see, for example, [99, 100])

dvdt

� −∇(ΦN + 2Φ2N ) (13)

+ 4v(v · ∇)ΦN − v2∇ΦN .

We note that the PN approximation is the first rela-tivistic correction from which the apoastron advancephenomenon arises. In the case of the S2 star, theapoastron shift as seen from Earth (from Eq. (15))due to the presence of a central black hole is about1 mas, therefore not directly detectable at present,since the available precision in the apoastron shift isabout 10 mas (but it will become about 1 mas in 10–15 years even without considering possible techno-logical improvements). It is also evident that higher-order relativistic corrections to the S2 apoastron shiftare even smaller and therefore may be neglected atpresent, although they may become important in thefuture.

As it will be discussed below, the Newtonian effectdue to the existence of a sufficiently extended DMsphere around the black hole may cause an apoastronshift in the opposite direction with respect to therelativistic advance due to the black hole. Therefore,we have considered the two effects comparing onlythe leading terms.

For the DM distribution at the Galactic Center wefollow Eq. (11) as done in [87]. Clearly, if in the futurefaint infrared stars (or spots) closer to the black holewith respect to the S2 star will be monitored [98, 101],this simplified model might well not hold and higher-order relativistic corrections may become necessary.



–5

–10 –5

x

, mpc0 5 10

0

5

10

y , mpc

BH S2

ΔΦ

= 580.315 arcsec

–5

–10 –5

x

, mpc0 5 10

0

5

10

y

, mpc

BH + Cluster + DM S2

R

DM

= 10

–3

pc

ΔΦ

= –14249.9 arcsec

Fig. 2. PN orbits for different mass configurations at the Galactic Center. Deviations from elliptical orbit are clearly seen (herewe assume that DM mass MDM � 2 × 105M� and RDM = 10−3 pc).

For a spherically symmetric mass distribution(such as that described above) and for a gravitationalpotential given by Eq. (12), Eq. (13) may be rewrittenin the form (see for details [102])

dvdt

� −GM(r)r3

(14)

×[(

1 +4ΦN

c2+

v2

c2

)r− 4v(v · r)

c2

],

r and v being the vector radius of the test particlewith respect to the center of the stellar cluster andthe velocity vector, respectively. Once the initial con-ditions for the star distance and velocity are given, therosetta shaped orbit followed by a test particle canbe found by numerically solving the set of ordinarydifferential equations in Eq. (14). In Fig. 2, as anexample, assuming that the test particle orbiting theGalactic Center region is the S2 star, we show thePN orbits obtained by the black hole only and theblack hole plus the stellar cluster plus the contri-bution of DM mass density with RDM = 10−3 pc.In each case the S2 orbit apoastron shift is given(for black hole (without stellar cluster or DM con-centration) ΔΦ = 580.315 arcsec, for black hole plusstellar cluster ΔΦ = 460.047 arcsec, for black holeplus stellar cluster plus DM (with RDM = 10−2 pc)we have ΔΦ = −298.592 arcsec, while for black holeplus stellar cluster plus DM (with RDM = 10−3 pc)we have ΔΦ = −14 292.9 arcsec). As one can see,for selected parameters for DM and stellar clustermasses and radii the effect of the stellar cluster isalmost negligible, while the effect of the DM dis-tribution is crucial, since it enormously overcomesthe shift due to the black hole (for RDM = 10−3 pc).Moreover, as expected, DM contribution is opposite

in sign with respect to that of the black hole [100].We note that the expected apoastron (or, equivalently,periastron) shifts (mas/revolution), ΔΦ (as seen fromthe center) and the corresponding values Δφ±

E as seenfrom Earth (at the distance R0 � 8 kpc from theGalactic Center) are related by

Δφ±E =

d(1 ± e)R0

ΔΦ, (15)

where the signs “±” indicate the angle shits of theapoastron (+) and periastron (−), respectively. TheS2 star’s semi-major axis and eccentricity are d =919 AU and e = 0.87 [89]. In Fig. 3, the S2 apoastronshift as a function of the DM distribution size RDMis given for α = 0 and MDM � 2 × 105M�. Takinginto account that the present day precision for theapoastron shift measurements is of about 10 mas, onecan say that the S2 apoastron shift cannot be largerthan 10 mas. Therefore, any DM configuration thatgives a total S2 apoastron shift larger than 10 mas(in the opposite direction due to the DM sphere) isexcluded. The same analysis is done for two differ-ent values of the DM mass distribution slope, i.e.,α = 1 and α = 2. In any case, we have calculatedthe apoastron shift for the S2 star orbit assuming atotal DM mass MDM � 2 × 105M�. As one can see,the upper limit of about 10 mas on the S2 apoastronshift may allow to conclude that DM radii in therange about 10−3−10−2 pc are excluded by presentobservations for DM mass distribution slopes. Wenotice that the results of the present analysis allowto further constrain the results of Hall and Gondolo(2006), who have concluded that if the DM sphereradius is in the range 10−3−1 pc, configurations withDM mass up to MDM = 2 × 105M� are acceptable.



–20

–2510

–1

R

DM

, pc

Δφ

, mas

10

–2

10

0

10

1

10

–3

–15

–10

–5

0

Fig. 3. Apoastron shift as a function of the DM radiusRDM for α = 0 and MDM � 2.0 × 105M�. Taking intoaccount the present-day precision for the apoastron shiftmeasurements (about 10 mas), one can say that DM radiiRDM in the range 8 × 10−4−10−2 pc are not acceptable.

The present analysis shows that DM configurationsof the same mass are acceptable only for RDM out ofthe range 10−3−10−2 pc, almost irrespectively of theα value.

We have considered the constraints that the upperlimit (presently of about 10 mas) of the S2 apoastronshift may put on the DM configurations at the galac-tic center considered earlier [87]. When (in about 10–15 years, even without considering improvements inobservational facilities) the precision of S2 apoastronshift will be about 1 mas (that is equal to the presentaccuracy in the S2 orbit reconstruction), our analysiswill allow to further constrain the DM distributionparameters. In particular, the asymmetric shapes ofthe curves in Fig. 3 imply that any improvement inthe apoastron shift measurements will allow to extendthe forbidden region especially for the upper limitfor RDM. Quantitatively, we have a similar behaviorcurves for other choices of slope parameters α forDM concentrations. In this context, future facilitiesfor astrometric measurements at a level 10 μas offaint infrared stars will be extremely useful [98] andthey give an opportunity to put even more severeconstraints on DM distribution. In addition, it is alsoexpected to detect faint infrared stars or even hotspots [101] orbiting the Galactic Center. In this case,a consideration of higher-order relativistic correctionsfor an adequate analysis of the stellar orbital motionhas to be needed for correct modeling. Due to a greatprogress in precision of measurements, one could notexclude a possibility that matter density will be so lowthat alternative scenarios (to DM annihilation model)will be needed to explain γ flux from the Galactic

Center. Electromagnetic processes in plasma witha presence of a strong gravitational field near theGalactic Center may be important components ofsuch alternative scenarios for the detected γ flux.In our considerations we adopted simple analyticalexpression and reliable values for RDM and MDM pa-rameters following [87] just to illustrate the relevanceof the apoastron shift phenomenon in constrainingthe DM mass distribution at the Galactic Center. Ifother models for the DM distributions are considered(see, for instance, [103] and references therein) thequalitative aspects of the problem are preserved, al-though, of course, quantitative results on apoastronshifts may be different.

One can find a discussion of a DM problem withgravitational microlensing in reviews [104–107].

ACKNOWLEDGMENTS

We thank academicians A.A. Logunov andS.S. Gershtein for useful discussions of the results atthe Conference at the State Scientific Center—IHEP(Protvino, Russia) and organizers of the meeting fortheir attention to our contribution. We also thankreferees and editors for their valuable remarks9).

AFZ is grateful to the National Natural ScienceFoundation of China (NNSFC) (grantsnos. 10703007, 10873020, G10573025, 40674081,10603008, 40890161, and 10733020) and NationalBasic Research Program of China (G2006CB806303)for a partial financial support of the work.

REFERENCES1. R. Juszkiewicz, private commun. (2008).2. U. J. J. Le Verrier, Comptes Rendus (31 Aug. 1846).3. J. G. Galle, Mon. Not. R. Astron. Soc. 7, 153 (1846).4. J. H. Oort, Bull. Astron. Inst. Neth. 6, 249 (1932).5. F. Zwicky, Helv. Phys. Acta 6, 110 (1933).6. U. J. J. Le Verrier, R. Acad. Sci. Paris 59, 379

(1859).7. U. J. J. Le Verrier, R. Acad. Sci. Paris 83, 583

(1876).8. S. Newcomb, The Elements of the Four Inner

Planets and the Fundamental Constants of As-tronomy, Suppl. Am. Ephem. Naut. Alm. (Govt.Printing Office, Washington, 1895).

9. A. Hall, Astron. J. 14, 49 (1894).10. I. Newton, Philosophia Naturalis Principia

Mathematica (Imprimatur, S. Pepys, London,1686), p. 141.

11. F. Zwicky, Astrophys. J. 86, 217 (1937).12. V. Rubin and W. K. Ford, Jr., Astrophys. J. 159, 379

(1970).

9)The paper was presented at the Science Session–Conference of the Nuclear Physics Section of the Branch ofPhysical Sciences of Russian Academy of Sciences (IHEP,Protvino, December 2008).



13. V. Rubin, N. Thonnard, and W. K. Ford, Jr., Astro-phys. J. Lett. 217, L1 (1977).

14. V. Rubin, N. Thonnard, and W. K. Ford, Jr., Astro-phys. J. Lett. 225, L107 (1978).

15. V. Rubin, N. Thonnard, and W. K. Ford, Jr., Astro-phys. J. 238, 471 (1980).

16. H. Goldstein, Classical Mechanics 2nd ed.(Addison-Wesley, Reading, 1980).

17. M. Milgrom, Astrophys. J. 270, 365 (1983).18. J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004);

Phys. Rev. D 71, 069901(E) (2005).19. J. W. Moffat, Class. Quantum Grav. 23, 6767 (2006).20. L. Blanchet and A. Le Tiec,

arXiv:0804.3518v2[astro-ph].21. D. Behnke et al., Phys. Lett. B 530, 20 (2002).22. D. Behnke, PhD Thesis, Rostock Report MPG-VT-

UR 248/04 (2004).23. D. B. Blaschke et al., Phys. At. Nucl. 67, 1050

(2004).24. V. N. Pervushin, A. F. Zakharov, and V. A. Zinchuk,

in Proc. of the INTAS Summer School and In-tern. Conference on New Trends in High-EnergyPhysics (Experiment, Phenomenology, Theory),(Yalta, Crimea, Ukraine, 2005), p. 271.

25. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, Int. J. Mod. Phys. A 21, 5957(2006).

26. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, Phys. Lett. B 633, 458 (2006).

27. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, in Proc. of the Workshop onNuclear Science and Safety in Europe, Ed. byT. Cechak et al. (Springer, 2006), p. 125.

28. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, in Proc. of the 8th Intern. Work-shop on Relativistic Nuclear Physics: from Hun-dreds MeV to TeV, JINR, Dubna, Russia, 2006,p. 11.

29. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, AIP Conf. Proc. 841, 362 (2006).

30. V. N. Pervushin, A. F. Zakharov, and V. A. Zinchuk,in Proc. of the Workshop on Nuclear Science andSafety in Europe, Ed. by T. Cechak et al. (Springer,2006), p. 201.

31. A. F. Zakharov, V. N. Pervushin, and V. A. Zinchuk,Phys. Part. Nucl. 37, 104 (2006).

32. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, Int. J. Geom. Methods Mod.Phys. 4, 171 (2007).

33. B. M. Barbashov, V. N. Pervushin, A. F. Zakharov,and V. A. Zinchuk, Phys. At. Nucl. 70, 191 (2007).

34. A. B. Arbuzov et al., Phys. At. Nucl. 72, 744 (2009).35. A. B. Arbuzov et al., Gravit. Cosmol. 15, 199

(2009); Arbuzov et al., Phys. Lett. B 691, 230(2010); A. F. Zakharov and V. N. Pervushin, ArXiv:1006.4745[qr-qc].

36. P. A. M. Dirac, Proc. R. Soc. London, Ser. A 333,403 (1973).

37. S. Capozziello, Int. J. Geom. Methods Mod. Phys.4, 53 (2007).

38. S. Capozziello and M. Francaviglia, Gen. Relativ.Gravity 40, 357 (2008).

39. S. Capozziello, V. F. Cardone, and M. Francaviglia,Gen. Relativ. Gravity 38, 711 (2006).

40. S. Capozziello, V. F. Cardone, and A. Troisi, Phys.Rev. D 71, 043503 (2005).

41. S. Capozziello, V. F. Cardone, and A. Troisi, JCAP0608, 001 (2006).

42. S. Capozziello, V. F. Cardone, and A. Troisi, Phys.Rev. D 73, 104019 (2006).

43. S. Capozziello, V. F. Cardone, and A. Troisi, Mon.Not. R. Astron. Soc. 375, 1423 (2007).

44. S. Capozziello et al., Phys. Lett. B 639, 135 (2006).45. S. Capozziello et al., Phys. Rev. D 73, 043512

(2006).46. S. Capozziello et al., Class. Quantum Grav. 23, 1205

(2006).47. S. Capozziello and R. Garattini, Class. Quantum

Grav. 24, 1627 (2007).48. S. Capozziello, S. Nojiri, and S. D. Odintsov, Phys.

Lett. B 634, 93 (2006).49. S. Capozziello, A. Stabile, and A. Troisi, Phys. Rev.

D 76, 104019 (2007).50. S. Capozziello, A. Stabile, and A. Troisi, Class.

Quantum Grav. 24, 2153 (2007).51. S. Capozziello and A. Troisi, Phys. Rev. D 72,

044022 (2005).52. S. Capozziello, A. Troisi, and V. F. Cardone, New

Astron. Rev. 51, 341 (2007).53. S. Carloni et al., Class. Quantum Grav. 22, 4839

(2005).54. A. Borowiec, W. Godlowski, and M. Szydlowski,

Phys. Rev. D 74, 043502 (2006).55. A. Borowiec, W. Godlowski, and M. Szydlowski, Int.

J. Geom. Methods Mod. Phys. 4, 183 (2007).56. A. F. Zakharov, A. A. Nucita, F. De Paolis, and

G. Ingrosso, Phys. Rev. D 74, 107101 (2006).57. A. F. Zakharov et al., AIP Conf. Proc. 966, 173

(2008).58. N. Bahcall and X. Fan, Astrophys. J. 504, 1 (1998).59. M. Roos, Introduction to Cosmology (Wiley, New

York, 2003).60. P. F. Smith and J. D. Lewin, Phys. Rep. 187, 203

(1990).61. H.-V. Klapdor-Kleingrothaus and K. Zuber, Particle

Astrophysics (IOP, Bristol, Philadelphia, 2000).62. R. Bernabei et al., arXiv:0804.2741 [astro-ph].63. C. Savage et al., arXiv:0808.3607v2 [astro-ph].64. G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405,

279 (2005).65. G. Bertone and D. Merritt, Mod. Phys. Lett. A 20,

1021 (2005).66. A. V. Gurevich and K. P. Zybin, Phys. Lett. A 225,

217 (1997).67. L. Bergstrom, P. Ullio, and J. H. Buckley, Astropart.

Phys. 9, 137 (1998).68. A. Tasitsiomi and A. V. Olinto, Phys. Rev. D 66,

083006 (2002).69. F. Stoehr et al., Mon. Not. R. Astron. Soc. 345, 1313

(2003).70. F. Prada et al., Phys. Rev. Lett. 93, 241301 (2004).



71. S. Profumo, Phys. Rev. D 72, 103521 (2005).72. Y. Mambrini et al., hep-ph/0509300.73. W. de Boer et al., Phys. Rev. Lett. 95, 209001 (2005).74. A. A. Abdo et al., Phys. Rev. Lett. 103, 251101

(2009).75. N. W. Evans, F. Ferrer, and S. Sarkar, Phys. Rev. D

69, 123501 (2004).76. A. Bouquet, P. Salati, and J. Silk, Phys. Rev. D 40,

3168 (1989).77. F. W. Stecker, Phys. Lett. B 201, 529 (1988).78. V. Berezinsky, A. Bottino, and G. Mignola, Phys.

Lett. B 325, 136 (1994).79. G. Bertone et al., Phys. Rev. D 70, 063503 (2004).80. O. Y. Gnedin and J. R. Primack, Phys. Rev. Lett. 93,

061302 (2004).81. L. Bergstrom et al., Phys. Rev. Lett. 95, 241301

(2005).82. D. Horns, Phys. Lett. B 607, 225 (2005).83. G. Bertone and D. Merritt, Phys. Rev. D 72, 103502

(2005).84. P. Gondolo and J. Silk, Phys. Rev. Lett. 83, 1719

(1999).85. P. Ullio, H.-S. Zhao, and M. Kamionkowski, Phys.

Rev. D 64, 043504 (2001).86. D. Merritt, arXiv: astro-ph/0301365.87. J. Hall and P. Gondolo, Phys. Rev. D 74, 063511

(2006).88. A. M. Ghez et al., Astron. Nachr. 324 (S1), 527

(2003).89. A. M. Ghez et al., Astrophys. J. 620, 744 (2005).90. R. Genzel et al., Astrophys J. 594, 812 (2003).91. N. N. Weinberg, M. Miloslavljevic, and A. M. Ghez,

astro-ph/0512621.

92. N. N. Weinberg, M. Miloslavljevic, and A. M. Ghez,Astrophys J. 622, 878 (2005).

93. G. Ames et al., http://tmt.ucolick.org/reports_and_notes/reports/Web_final_Greenbook.pdf (2002)

94. A. F. Zakharov, A. A. Nucita, F. De Paolis, andG. Ingrosso, Phys. Rev. D 76, 062001 (2007).

95. A. F. Zakharov, A. A. Nucita, F. De Paolis, andG. Ingrosso, J. Phys. Conf. Ser. 133, 012032 (2008).

96. A. F. Zakharov et al., Space Sci. Rev. 148, 301(2009); A. F. Zakharov et al., in Probing the Natureof Gravity: Confronting Theory and Experimentin Space, Ed. by C. W. F. Everitt et al. (Springer,2010), p. 353.

97. A. M. Ghez et al., Astrophys. J. 689, 1044 (2008).98. F. Eisenhauer, G. Perrin, S. Rabien, et al., astro-

ph/0508607.99. S. Weinberg, Gravitation and Cosmology: Prin-

ciples and Applications of the General Theory ofRelativity (Wiley, New York, 1972).

100. A. A. Nucita et al., Publ. Astron. Soc. Pacif. 119, 349(2007).

101. R. Genzel and V. Karas, arXiv:0704.1281v1[astro-ph].

102. G. G. Rubilar and A. Eckart, Astron. Astrophys.374, 95 (2001).

103. D. Merritt, S. Harfst, and G. Bertone, Phys. Rev. D75, 043517 (2007).

104. A. F. Zakharov and M. V. Sazhin, Physics–Uspekhi,41, 945 (1998).

105. A. F. Zakharov, Phys. Part. Nucl. 39, 1176 (2008).106. A.F. Zakharov, New Astron. Rev. 53, 202 (2009).107. A. F. Zakharov, Gen. Rel. Grav. 42, 2301 (2010)


Constraints on parameters of dark matter and black hole in the Galactic Center

Documents

Transcript of Constraints on parameters of dark matter and black hole in the Galactic Center