Table Of Content

The Variable-c Cosmology as a Solution to Pioneer Anomaly ∗ Hossein Shojaie Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 1983963113, Iran ItisshownthatthePioneeranomalyisanaturalconsequenceofvariablespeedoflightcosmolog- ical models wherein thespeed of light is assumed tobea power-law function of thescale factor (or cosmic time). Inotherwords, thePioneer anomaly can beregarded as anon-gravitational effect of the continuously decreasing speed of light which indicates itself as an anomalous light propagation timedelayinlocal frames. Thistimedelayisaccordingly interpretedasanadditionalDopplerblue 2 shift. 1 PACSnumbers: 04.20.Cv,95.55.Pe,98.80.-k 0 Keywords: Pioneeranomaly;varyingspeedoflight. 2 n a I. INTRODUCTION this anomaly and J 13 An unexpected Doppler blue shift in the radio signals H0c0 ≈6.99×10−10sm2, (2) reachingthe earthfromthe Pioneer10& 11 spacecrafts, ] when their distances from the Sun were between 20AU thatis,theproductoftheHubbleconstantandthespeed O and 70AU, shows a constant deceleration of light, where the Hubble constant is assumed to be C H0 = 100hKm/s/Mpc with h ≈ 0.72. For a detailed . A =(8.74±1.33)×10−10m (1) reviewofthisphenomenon,onecanseeTuryshev&Toth h P s2 [68] and references therein. p The aim of this work is to show that the Pioneer - towardstheearth(orthesun),asreportedfirstbyAnder- anomaly can be a natural consequence of those varying o r sonetal.[4]andconfirmedbymoreaccuratedataanaly- speed of light theories in which the speed of light is as- st sis[66,67,69]. Since1998,manyexplanationshavebeen sumed to be a power-law function of cosmic time. The a considered. These explanations can broadly be classified manuscript is arranged as follows. In Section 2, some [ into two classes. The first corresponds to conventional argumentsaboutthe VSL theoriesarediscussed. InSec- 2 physics including both on-board systematic effects [16], tion 3, a VSL model provided by Shojaie & Farhoudi v such as Thermal recoil force [17, 18, 35, 50, 58, 60, 64], [61, 62] is reviewed. According to this model, the Pio- 1 and external forces, such as solar wind [4], effect of the neer anomaly is a local effect of the universal variation 6 expandinguniverse[38],andgravitationaleffectsofmass of speed oflight. This is done in Section 4. In Section 5, 2 distributions, mainly from Kuiper belt [15, 23, 28, 52] without considering a specific model, it is showed that 2 and interplanetary or interstellar dust [51]. The sec- one can use this anomaly to estimate the power of c, . 5 ond class correspondsto new physics, for instance, mod- when it is supposed to be a power-law function of scale 0 ified inertia [45], dark matter distribution [54], cosmo- factor. Section 6 represents summary and remarks. 0 logical constant [44], and modified (Hamiltonian) grav- It is worth noting that beside the Pioneer anomaly, 1 ity [8,14, 20,33, 34,49, 56,59, 65]. But untilnow, none thereareothermoreorlessestablishedanomalieswithin : v ofthemcanexplainthisproblemclearly. Moreover,asno the solarsystems such as the anomaloussecularincrease Xi firmly accepted anomaly has been detected in the orbits of the eccentricity of the lunar orbit [32], the anoma- ofplanetsyet[5,31,63], itestrangesthe phenomenonto lous precession of the perihelion of Saturn [29] and the r a have a gravitational ground. In a review paper by Ni- flyby anomaly [6]. For a review of these anomalies see eto & Anderson [53], it has been claimed that the total references [7, 37] and references therein. Although these estimatedsystematicerrorisabout10percentofthe ob- anomalieshavenotbeenassertedyet,theycanbeverified served un-modeled acceleration. So, it seems that the in the context of the VSL theory in further works. origin of this phenomenon may be a not-yet-discovered physical evidence. In addition, it should be mentioned that although some authors believe that the Pioneer II. VSL THEORIES anomaly cannot have a cosmological origin [36, 37, 47], some othersargue thatit can be derivedfromconformal Varying speed of light theories, have been provided as gravity considerations [71, 72]. It is worth noting that alternativestoinflationaryscenariosinordertosolvethe thereisanapproximatecoincidencebetweenthevalueof standard big bang (SBB) problems [1, 2, 9–12, 21, 22, 25,26,30,39–41,48,73]. Asextensivelydiscussedinthe literature,the lackof a consistentfield theoryfor VSL is a major difficulty of it. Besides, at least, two objections ∗ [email protected] are against VSL theories. 2 The first is about the dimensionality of the speed of equippedwithavariablec,simplyreplacingc0,thecoef- light,c. Itisclaimedthatvariationsofdimensionalquan- ficientofdtinametricbyc(t),isnotenoughtointroduce tities do not affect the physics and such variations can a VSL model. Diffeomorphism tells us that this change be removed by redefinitions of the units. However, any doesnotcontributetoanewphysicsandcanberemoved variationofadimensionlessquantityisaunit-freescalar, by redefinition of time coordinate. From Lagrangian andsinceitismerelybuiltupofdimensionalparameters, viewpoint, terms with c˙ which arise from metric, are ab- its variation can be regarded as an indicator of physical sorbed to an exact differential term and will not cause (observer-independent)variationofoneormoreofitsdi- any variation. This conclusion is valid until there is no mensional constituents. For instance, any probable vari- otheroriginforc˙exceptmetric. Forinstance,inascalar- ation of fine structure constant, α, with respect to the tensor theory of gravitation, provided the scalar field φ cosmic time, is the indication of the actual time evolu- carries the dynamics of c instead of G (e.g. φ ≡ c4), c˙ tionofoneormorefundamentalconstantsthatconstruct appears explicitly in the equations. (However, it should it, namely c, ~ or e. Moreover, there are many theories be mentioned here that a varying constant model is not which regard variations of dimensional constants other automatically equivalent to a scalar-tensor theory.) It than the speed of light, for example, scalar-tensor the- is worth noting that in a scalar-tensortheory, φ−1 is the ories of gravitation [27] which concern probable varia- coefficientwhichrelatesenergy-momentumtensorT to µν tion of gravitational constant [19], or Beckenstein’s pro- the Einstein tensor G , and determines the strengthby µν posalin which the fundamental electric chargehas time- which the ordinary matter affects geometry. dependency [13]. Switchingbacktothedynamicsequations,namelythe The second objection is that the constancy of speed Einstein equation of light is one of the principles of the special relativity 4 G =(8πG/c )T (5) (SR).Itcanbeshownthatonthebackgroundofflat(Eu- µν µν clidean) geometry, relativity principle guaranties the ex- when restricting ourselves to GR, the speed of light in istence of a frame-independent upper limit speed. If this thedenominatorofleft-handside,hasnotcomefromthe speed is assumed to be infinite, then Galilean transfor- metric and its time evolution has a physicalsignificance. mations and Newtonian absolute time arise. Otherwise, In addition, it is evident that in this way, c˙ does not thisinvariantspeedleadstoLorentztransformationsand contribute to any vacuum solution of GR. This, for ex- SR. In other words, the existence of an invariant upper ample, guarantees the static nature of the Schwarzschild limit speed in an Euclidean space-time, is a result of the solution. relativity principle. Firm evidences, specially from elec- Summarizing the above arguments, to construct a trodynamics, claim that this speed must be the speed of VSL model via generalizing GR, one should carry out massless particles, or particularly the speed of light. 0 0 all derivatives with respect to dx (or ∂x ) assuming However,inthecontextofthewholeuniverse,SRonly g00 =1, and then replace dx0 (or ∂x0) with cdt (or c∂t) defines local inertial frames. In other words, it is a local in the final relations wherever they appear provided the approximation of the global manifold obeying a gravita- left-hand side of the Einstein equation is non-zero. This tionaltheorysuchasgeneralrelativity(GR).Thismeans procedurecanpreservethegeneralcovarianceofatypical that, although there is an invariant upper limit speed in VSL theory and lets it reduce to SR in inertial frames. each local inertial patch of the universe, which is ruled However, this procedure gives the same results as if one by SR, its value maydiffer frompatchto patch, andany assumes g00 =c2(t) and Γ000 =c˙/c at first step. globalgravitationaltheory which governsthe whole uni- versemaycontainthederivativesofthislocallyinvariant speed. So, a variable speed of light should enter in a III. A VARIABLE-c COSMOLOGICAL MODEL gravitational theory minimally, in such a way that one canretrieveSRandLorentztransformationsinanylocal Cosmologicalprinciple andWeyl postulate areofvital inertialframe. As an example,this canbe easily done in importance for the standard cosmology. These princi- a metric like ples,whicharein agreementwith observations,arecom- ds2 =c2dt2−g dxidxj , (3) pletelyindependentofGR.Moreprecisely,thecosmologi- ij calprinciple,namelythehomogeneityandisotropyofthe 2 where g0i = 0, by assuming g00 = c (t) which conse- large scale universe, makes the Friedmann-Robertson- quently, adds a new non-zero connection component Walker(FRW)metricapreferableframewithleastnum- ber of parameters which leads one to derive the Fried- c˙ 0 Γ = . (4) mannequations. On the other hand, by the Weylpostu- 00 c late, the matter content ofthe universe canbe chosento This component, like other connection components, is be a perfect fluid. set to zero in any inertial frame, and one works with a InaVSLcosmologicalmodel,introducedbyShojaie& constant c in local frames. Farhoudi[61,62],theuniverseisassumedtoobeytwoad- Althoughtheaboveexamplemayberegardedasamin- ditional principles, besides those accepted in GR. These imal extensionof SR to a generalizedgeometricaltheory two principles are: 3 Principle(1):: The norm of a four-vector (E,pc), is an consequentlyshowsthatthegeodesicsarederivedfroma invariantinanypropervolumeundergeneraltrans- Schwarzschildmetric (with the sun at the center). How- formations. Specially, in FRW frame ofcosmology, ever,the effect of varying-c,at leastfar enoughfrom the andaccordingto the Weyl postulate andthe prop- sun,maybesignificantmerelybyimposingananomalous erty of the comoving frame, this norm reduces to timedelay. Thiscanbeseen,forexample,bynotingthat theconnectioncomponentΓ0 canbenomorezerowhen 2 00 Mc =const., (6) onedealswithaVSLtheory. AMoredetaileddiscussion is postponed to the Appendix. in the proper volume within any arbitrary proper Toanalyzethe Pioneeranomalyinthe VSL modelde- radius, where M is the enclosed rest mass in this scribed in the last section, let us consider the behav- volume. ior of a signal from a spacecraft at comoving distance Principle(2):: The relation r reaching the earth in FRW metric. Since the spacial distance between the spacecrafts and the sun are much M more smaller than the size of the visible universe, one =const., (7) Rc2 canneglecttheeffectofthecurvatureandsimplyassume dr ≈dr/(1−kr2). Doing that, one has holds in any proper volume with proper radius R. In other words, the ratio M/Rc2 remains constant t0 c(t) in any arbitrary proper volume. r = dt (9) Z a(t) te Accordingto this model [62], the universebegins from a hot big bang and expands and decelerates forever, like where te and t0 denote the time of emission from the a flat model, despite the kind of its topology. That is, spacecraft and reaching the earth, respectively. By ex- theflatuniverse,incontrasttoEinstein-deSittermodels, pandingcandauptofirsttermaroundt0 andusing(8), is a stable case and, the Hubble parameter, H(t) ≡ a˙/a one can write tends to zero az t → ∞. The extra term related to the variationofthe speedoflight, c˙, cancelsoutthe effectof r ≈ t0 c0+c˙0(t0−t)dt thecurvature,k,inthedynamicsequations. Inaddition, Zte a0+a˙0(t0−t) the principles lead matter content of the universe, if as- t0 1 5 sumedtobe aperfectfluid, tohavetheequationofstate ≈ c0− H0c0(t0−t) dt p = −1ρc2, similar to the value predicted from string Zte a0 (cid:20) 4 (cid:21) 3 5 models. It is worthmentioning thatthe speed of lightin =c0∆t− (H0c0)(∆t)2 , (10) FRW frame varies as 8 a −1 t −1 where in the last step, it is assumed that a0 = 1 and c=c0(a0) 4 =c0(t0) 5, (8) ∆t = t0 − te. Obviously, the last term shows an extra unexpected term that can be explained as a negative where t is the cosmic time. The detailed derivations and constant acceleration, A, which value is explanations can be found in Ref. [61, 62]. theOhthoerirzoandvaanndtatgheesreolficthaibsumndoadnecleasrperothbaletmtshefafldaetanwesasy, A= 45H0c0 ≈8.73×10−10sm2 (11) from the SBB, without need to an add-on inflationary in a very good agreement with the Pioneer anomaly, scenario. Also,itprovidesabetterdescriptionaboutthe where (2) is used. In deriving this result, the way the fainter than expected flux of the supernovae type Ia via object moves does not affect the result. It is completely decreasing-c, instead of an accelerating epoch. In this independentofthe velocity,accelerationanddirectionof case, one does not confront with more complicated sub- the motion, and instead, depends on the Hubble expan- jects such as the cosmological constant problem. More- sion rate and the speed of light. The direction of the over, the speed of light, c, can be regarded as a scalar deceleration is always towards the observer, that is, in fieldwhichdecayscontinuouslytoproduce matterinthe directionofthesignalray. Moreover,asthisdeceleration universe. isrelatedtothe receivingsignalandnottothe objectit- self,itdoesnotcontributetoanyprecessionintheorbits IV. PIONEER ANOMALY IN VSL THEORY of the planets. A key question should be answered here, before pro- V. CONVERSED VIEW ceeding. Which metric determines the geodetic motion of a freely-falling object like the Pioneer spacecrafts, Schwarzschild metric or FRW metric? Nonetheless, the There can be a conversedapproachto this problem in effect of the universal expansion is much smaller than VSL context, but this time, without assuming any spe- the gravitational effects within galactic scales and this cific model. Supposing the validity of the cosmological 4 principle, one simply ends with FRW metric (3). More- ACKNOWLEDGMENT over,ifoneassumesthatthespeedoflightinthiscosmo- logical frame is a power-law function of the scale factor, The author would like to appreciate Dr. Mehrdad that is c ∝ an, then one can estimate the range of n Farhoudi for his hospitality and useful comments. This by derivation method similar to (10), using (1) and (2). workhasbeenfinanciallysupportedbytheresearchcoun- This means that the relation (10) is modified to cil of Shahid Beheshti University. t0 1 r ≈ [c0+(n−1)H0c0(t0−t)]dt APPENDIX Zte a0 n−1 2 =c0∆t+ H0c0(∆t) (12) There are three main differences between the 2 Schwarzschild metric and FRW metric, which make the up to first order. This evidently implies a constant de- combination of these two metrics difficult: celeration • The Schwarzschild metric is static but FRW is a dynamical metric. A=−(n−1)H0c0 . (13) • The Schwarzschild metric is a vacuum solution Equating (13) and (1), and using (2) then leads to while FRW, at least when applied to the real universe, concerns the perfect fluid as its energy- n=−0.25±0.19 . (14) momentum tensor. • TheSchwarzschildmetricisinhomogeneoussinceit Not surprisingly, equation (14) is a manifestation of describes the spacetime outside a spherically sym- decreasing-c in an expanding universe. metric object. On the other hand, homogeneity is one of two assumptions in deriving FRW metric. However,to describe the evolution of black holes and VI. SUMMARY AND REMARKS other local systems such as the solar system, collapsing stars, galaxies, clusters and super clusters of galaxies in The Pioneer anomaly is an un-modeled Doppler blue anexpandinguniverse,oneneeds ametric whichasymp- shift, equivalent to a constant deceleration, which has totically tends to FRW metric at large radius while at not had a conventional physical explanation yet. In this thesametimereducestoSchwarzschildspacetimeatsig- work, it has been shown that a VSL cosmologicalmodel nificantly small distances. In addition, the conjunction can generally predict this anomaly, provided that firstly conditions between two metrics, according to each spe- thecosmologicalprincipleappliesandsecondlythespeed cificproblem,shouldbeappropriatelyapplied. Attempts of light is a power-law function of the scale factor, with to obtain such a solution has been dated back to 1933 its power lying in the range (−0.44,−0.06). by McVittie [46] and followed by the others [24, 55, 57]. Accordingtothisscenario,ithasbeenarguedthatthis Butuntilnow,thereisnodefinitiveandwell-definedmet- negative acceleration is not due to a true acceleration, ric for this purpose and all attempts have failed. butmerelyisananomalouslightpropagationtimedelay, Despite the abovefact, the McVittie’s metric seemsto interpreted as an unexpected blue shift. Hence, it does be the best prototype for embedding a spherically sym- not depend on the velocity, acceleration or direction of metric object in an expanding universe. In its general the motionofthe object. Thenegativityofthis anomaly form, it is written as implies that the Hubble constant and c˙ have opposite signs. That is, in an expanding universe, the speed of ds2 = 1−µ(t,r) 2c2dt2−(1+µ(t,r))4a2(t)dr2+r2dΩ2 , light is decreasing. (cid:18)1+µ(t,r)(cid:19) (1+ 1kr2)2 4 A complaint against the procedure followed in (10) (15) and (12), may be the issue of embedding a central body where solution,thatisaSchwarzschildmetric,withinanoverall Gm 1 cosmological background, namely the FRW metric. The µ(t,r)= 1+ kr2 . (16) situation somehow is similar to considering local inertial 2a(t)rc20r 4 frames of freely-falling objects within the background of The constant parameters m and k can be related to the a Schwarzschild metric; the freely-falling observer can- massofthecentralobjectandthecurvatureofthespace, notdetectgravity,butthe tidalforcecanbeobservedby respectively,anda(t)istheasymptoticcosmologicalscale him/her. ThePioneeranomalycanbesimilarlyregarded factor. The solution is exactly FRW for m = 0 and asa localdetectable effectofa universalevolutionofthe wholly Schwarzschild metric when k = 0 and a = 1. speed oflight. Nonetheless, the authorbelieves that this Moreover,for µ≪1 the metric reduces to phenomenon is the only evidence that exhibits the expansion of the universe locally. ds2 =(1+2µ)c2dt2−(1−2µ)a2(t) dr2+r2dΩ2 , (17) (cid:0) (cid:1) 5 which is obviously the perturbed FRW metric in New- in local systems in a different way. In this method, the tonian gauge, that is, µ can be regarded as the Newto- local system is considered in a local inertial frame and niangravitationalpotential. Forasun-sizestar,thisper- the effect of the background FRW metric is shown in turbed metric is well worth applicable outside the star; deviation from the geodetic motion. In this sense, and at much larger distances, it is fully FRW metric. It can using the relation be easily shown that according to metric (15) (and consequently the perturbed metric (16)), six components of d2xi dxj dxk connections (excluding those whicharethe samein both dt2 +Γijk dt dt =−Ri0j0xj , (20) metrics), namely, one has a˙ 1 1 2 1 3 1 Γ01 =Γ10 =Γ02 =Γ20 =Γ03 =Γ30 = a , (18) d2r − a¨ − c˙a˙ r =0 . (21) dt2 (cid:18)a ca(cid:19) are not negligible even for small µ. If one also assumes thatthe speedoflightis afunctionoft,then inaddition Obviously, the third term in the above relation is due to the above connection components, to VSL theory, that is, an additional deviation from the standard geodesic deviation (second term). Not surpris- c˙ 0 Γ00 = (19) ingly, a¨ is also affected by varying c. Adding the gravi- c tational effect of the sun, relation (21) becomes cannot be disregarded for small values of µ, too. That is, concerning the McVittie’s metric, these components d2r a¨ c˙a˙ Gm L2 − − r+ − o =0 , (22) of connection are capable of bringing the effect of global dt2 (cid:18)a ca(cid:19) r2 m r3 o expansiontolocalsystems. Itshouldbementionedthat, there have always been doubts about the lowest scale at where m and L are the mass and the angular momen- o o which expansion can be observed. As it has been stated tum of the object, respectively. Actually, this deviation in the literature [3], in principle, there is no restriction ina localinertialframe is a lightpropagationtime delay aboutthis scaleandthe expansionproceedsinallscales. in FRW, which is shown in the text. However, in practice, local systems do describe the ge- It should be mentioned that a natural connection can ometry and consequently there is a scale that beyond it be seen between Schwarzschild and FRW metrics in the objects do not feel expansion. contextof conformalgravity[42,43, 70] where appropri- The lack of an appropriate metric in this situation, ate conformal transformations can smoothly interpolate leadsauthorstoconsidertheeffectofuniversalexpansion between the two metrics. [1] Albrecht,A.1999, arxiv:astro-ph/9904185 [15] Bertolami,O.&Vieira,P.2006,Class.Quant.Grav.,23, [2] Albrecht, A. & Magueijo, J. 1999, Phys. Rev. D, 59, 4625 043516 [16] Bertolami, O.&Pa´ramos,J.2007,Int.J.Mod.Phys.D, [3] Anderson,J.L. 1995, Phys. Rev.Lett., 75, 3602 16, 1611 [4] Anderson,J.D., et al. 1998, Phys.Rev.Lett., 81, 2858 [17] Bertolami, O., Francisco, F., Gil, P.J.S., & Pa´ramos, J. [5] Anderson,J.D., et al. 2002, Phys.Rev.D, 65, 082004 2008, Phys. Rev.D,78, 103001 [6] Anderson, J.D., Campbell, J.K., & Nieto, M.M. 2007, [18] Bertolami, O., Francisco, F., Gil, P.J.S., & Pa´ramos, J. NewAstron., 12, 383 2010, Space Sci. Rev.,151, 75 [7] Anderson, J.D. & Nieto, M.M. 2010, Relativity in Fun- [19] Brans, C. & Dicke,R.H. 1961, Phys. Rev.,124,925 damental Astronomy: Dynamics, Reference Frames, and [20] Brownstein, J.R. & Moffat, J.W. 2006, Class. Quant. Data Analysis,inKlionerS.A.,SeidelmannP.K.,Soffel Grav., 23, 3427 M. H., eds, Proc. IAU Symp. 261, pp. 189, Cambridge [21] Clayton, M.A. & Moffat, J.W. 1998, Phys. Lett. B, 480, Univ.Press, Cambridge 263 [8] Avramidi,I.G.&Fucci,G.2009,Can.J.Phys.,87,1089 [22] Clayton, M.A. & Moffat, J.W. 2000, Phys. Lett. B, 477, [9] Barrow, J.D. 1999, Phys. Rev.D,59, 043515 269 [10] Barrow, J.D. & Magueijo, J. 1998, Phys. Lett. B, 447, [23] deDiego,J.A.,Nunez,D.,&Zavala,J.2006,Int.J.Mod. 246 Phys. D, 15, 533-544 [11] Barrow, J.D. & Magueijo, J. 1999, Class. Quant. Grav., [24] Einstein, A. & Straus, E.G. 1945, Rev. Mod. Phys., 17, 16, 1435 120 [12] Bassett, B.A., et al. 2000, Phys. Rev.D,62, 103518 [25] Ellis, G.R. 2007, Gen. Rel.Grav., 39, 511 [13] Beckenstein, J.D. 1982, Phys.Rev.D, 25, 1527 [26] Ellis, G.R. & Uzan, J.-P 2005, Am.J. Phys., 73, 240 [14] Bertolami, O. & Pa´ramos, J. 2004, Class. Quant. Grav., [27] Fujii,Y.&Maeda,K.-I.2004,TheScalar-Tensor Theory 21, 3309 of Gravitation, Cambridge University Press, UK [28] Iorio, L. 2007, Mon. Not. Roy.Astron.Soc., 375, 1311 6 [29] Iorio, L. 2009, Astron. J., 137, 3615 [48] Moffat, J.W. 1993, Int.J. Mod. Phys. D,2, 351 [30] Iorio, L. 2010, Gen. Rel. Grav., 42,199 [49] Moffat, J.W. 2004, arXiv:gr-qc/0405076 [31] Iorio, L. 2010, Mon. Not. Roy.Astron. Soc., 405, 2615 [50] Murphy,E.M. 1999, Phys. Rev.Lett., 83, 1890 [32] Iorio, L. 2011, Mon. Not. Roy.Astron. Soc., 415, 1266 [51] Nieto, M.M., Turyshev, S.G., & Anderson, J.D. 2005, [33] Jaekel, M.-T. & Reynaud, S. 2005, Mod. Phys. Lett. A, Phys. Lett. B, 613, 11 20, 1047 [52] Nieto, M.M. 2006, Phys. Rev.D, 72, 083004 [34] Jaekel, M.-T. & Reynaud,S. 2006, Class. Quant. Grav., [53] Nieto,M.M.&Anderson,J.D.2007,Contemp.Phys.,48, 23, 7561 No. 1, 41 [35] Katz, J.I. 1999, Phys.Rev.Lett., 83, 1892 [54] Nieto, M.M. 2008, Phys. Lett. B, 659, 483 [36] Lachièze-Rey,M. 2007, Class. Quant.Grav., 24, 2735 [55] Nolan, B.C. 1998, Phys. Rev.D, 58, 064006 [37] Lämmerzahl, C., Preuss, O., & Dittus, H. 2008, Is the [56] Ostvang, D. 2002, Class. Quant.Grav., 19, 4131 Physics Within the Solar System Really Understood?, in [57] Pachner, J. 1963, Phys. Rev.,132, 1837 Dittus, H., Lämmerzahl, C., & Turyshev, S.G., eds., [58] Rievers,B.&Lämmerzahl,C.2011,Ann.Phys.,523,439 “Lasers, Clocks and Drag-Free Control: Exploration of [59] Saffari, R.& Rahvar,S.2008, Phys. Rev.D,77, 104028 Relativistic Gravity in Space”, Astrophysics and Space [60] Scheffer, L.K. 2003, Phys. Rev.D, 67, 084021 Science Library,Vol. 359, pp.75-104, Springer, Berlin [61] Shojaie, H.& Farhoudi,M. 2006, Can. J. Phys., 84, 933 [38] Lämmerzahl,C.2009,ThePioneerAnomalyandthe So- [62] Shojaie,H.&Farhoudi,M.2007,Can.J.Phys.,85,1395 lar System in the Expanding Universe, presentation at [63] Tangen, K.2007, Phys.Rev. D,76, 042005 the workshop “Intertwining Theory and Observational [64] Toth, V.T. & Turyshev, S.G. 2009, Phys. Rev. D, 79, EvidenceinContemporaryCosmology”(WuppertalUni- 043011 versity,Germany,February 2009) [65] Trencevski, K. 2005, Gen. Rel. Grav., 37, 507 [39] Magueijo, J. 2000, Phys.Rev.D, 62, 103521 [66] Turyshev,S.G., et al. 2006, Int.J. Mod. Phys.D, 15, 1 [40] Magueijo., J. 2003, Rept.Prog. Phys., 66, 2025a [67] Turyshev,S.G.&Toth,V.T.2009, SpaceSci.Rev.,148, [41] Magueijo, J. & Moffat, J.W. 2008, Gen. Rel. Grav., 40, 149 1797 [68] Turyshev, S.G. & Toth, V.T. 2010, Living Rev.Relativ- [42] Mannheim,P.D.&Kazanas,D.1989,Astrophys.J.,342, ity, 13, 4 635 [69] Turyshev,S.G.,Toth,V.T.,Ellis,J.,&Markwardt,C.B. [43] Mannheim, P.D. 2010, Prog. Part. Nucl.Phys., 56, 340 2011, Phys. Rev.Lett.107, 081103 [44] Massa, C. 2008, Astrophys.& SpaceSci., 317, 139 [70] Varieschi, G.U. 2010, Gen. Rel. Grav., 42, 929 [45] McCulloch,M.E.2007,Mon.Not.Roy.Astron.Soc.,376, [71] Varieschi, G.U. 2010, arXiv:1010.3262 [astro-ph] 338 [72] Varieschi, G.U. 2011, ISRNAstron. Astrophys., 806549 [46] McVittie, G.C. 1933, Mon. Not. Roy. Astron. Soc., 93, [73] Youm, D.2001, Phys.Rev.D, 63, 125011 325 [47] Mizony, M. & Lachièze-Rey, M. 2005, Astron. Astro- phys.,434, 45