Are gravitational constant measurement discrepancies linked to galaxy rotation curves?

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1 Ae gvittionl constnt mesuement discepncies linked to glxy ottion cuves? Nobet Klein, Impeil College London, Deptment of Mteils, London SW7 AZ, UK The discepncies between ecently epoted expeimentl vlues of the gvittionl constnt wee nlysed within n ineti intepettion of MOND theoy. Accoding to this scenio the eltive gvittionl cceletion between test mss nd n y of souce msses detemines the mgnitude of post Newtonin coections t smll mgnitudes of cceletion. The nlysis ws pplied to one of the most dvnced ecent Cvendish-type expeiment which eveled n expeimentl vlue fo the gvittionl constnt of 8 ppm bove the cuent CODATA vlue with moe thn five stndd devitions significnce. A emkble geement between this discepncy nd the cceletion nomlies inheent of glxy ottion cuves ws found by consistent extpoltion within the fmewok of MOND. This supising esult suggests tht the two nomlies on totlly diffeent length scles my oiginte fom the sme undelying physics. PACS: 4.5.Kd, 4.8.Cc Moe thn 3 yes fte Newton, gvittion still emins one of the get micles, nd its undestnding in the fmewok of unified theoy with the othe fundmentl foces is still lcking [,]. Newton s gvittionl lw nd its subtle modifictions within the fmewok of Genel Reltivity llow highly ccute pedictions of the motion of plnets nd othe objects within ou sol system. Howeve, s fist epoted by Ve Rubin in 97, the mesued ottion of glxies is incomptible with Newton s lw, if only visible mtte is consideed fo modelling the dynmic behviou [3]. Theefoe, the concept of dk mtte ws suggested s possible explntion of this discepncy, but dk mtte hs not yet been obseved diectly [4]. As n ltentive to dk mtte, the concept of Modified Newtonin Dynmics (MOND) ws intoduced by Milgom in 983 [5], which ccounts fo devitions fom Newton s lw in cse of smll cceletion mgnitudes. The univesl dil cceletion eltion in ottionlly suppoted glxies, s pesented ecently by Mc Gugh et l. [6], cnnot be consistently explined by dk mtte, but stongly fvous the vlidity of MOND theoy [7].Although MOND is still phenomenologicl theoy without ny solid undepinning by physicl pinciples, Milgom pointed out ecently tht the undelying physics of MOND effects my be due to modifiction of ineti, s esult of the intection of cceleted mtte with the quntum vcuum [8]. In contst to MOND intepettions which suggest modifiction of the gvittionl foce lw, s being fomulted by Bekenstein nd Milgom though modified Poisson eqution fo the gvittionl potentil [9], the ineti intepettion would llow the obsevtion of MOND effects in ethbound lbotoies. One equiement fo this is sufficient isoltion of the elevnt test msses fom seismic noise, which ensues tht seismic cceletions (nd othes) e smlle thn the mesued cceletion due to the gvittionl foce to be mesued (see discussion). Accute mesuements of the gvittionl constnt G, most of them by Cvendish-type tosion blnces [,,], Fby - Peot micowve esontos (diect involvement of the utho) [,3] nd opticl intefeometes [4] suspended s line pendulums e not in pefect geement with ech othe, nd the obseved discepncies of up to ten stndd devitions emin unexplined to dte []. Noticebly, the ecently epoted esults by Quinn nd Speke [], s detemined by two independent opetion methods of Cvendish-type G tosion blnce, e 8 ppm bove the esults obtined by Schlmminge et l. [5], which e bsed on bem blnce employing 3 tons of liquid mecuy s souce mss the ltte being consistent with the most ecent CODATA vlue

2 [6]. As possible explntion fo these discepncies, Andeson et l. suggested sinusoidl time vition of G ccoding to n nlysis of G expeiments being pefomed since ye 98 [7], but ccoding to ecent nlysis of the consequence fo obitl motions within ou sol system by Ioio this scenio is in contdiction to the expeimentl constints fo the obseved obit incese of the LAGEOS stellite nd nomlous peihelion pecession of Stun [8]. Inspied by the nlysis of the cceletion mgnitude dependence of gvittionl foce mesuements by pendulum gvimete within MOND theoy by Meye et l. [3], in this epot post-newtonin nlysis of selected G mesuements is pusued nd comped to glxy ottion cuves using diffeent MOND extpoltion functions. In ode to dpt MOND theoy fo G expeiments, two point msses m nd m being septed by distnce e consideed. In n idelized G expeiment, one of the two msses (clled test mss ) is moved by n incementl distnce fom one equilibium position (souce mss t infinite distnce fom the test mss) to nothe (souce mss t distnce fom the test mss) due to smll cceletion cused by the gvittionl foce F between m nd m. In cente-of-mss fme of efeence, the Newtonin dynmics is descibed in eltive coodintes, the ineti in Newton s second lw is detemined by the educed mss µ. F N N - N mm m m Eq. Although the two msses e nged within non-inetil fme of efeence (the eth), it is pesumed hee tht the post-newtonin effect is solely detemined by the ineti of the educed mss with espect to the gvittionl ttction between m nd m. A tosion pendulum epesent suitble ppoximtion fo locl inetil-fme of efeence with espect to the diection of test mss motion cused by the gvittionl foce between test- nd souce msses. In ode to fit MOND-type post-newtonin coections t low mgnitudes of cceletion, univesl coection function f( ) f ( ); f ( ) Eq. is suggested, with =. - m/s esembling the fundmentl MOND cceletion pmete [5,7,8] nd is fee pmete. The function f( ) epesents coection of the eltive Newtonin cceletion n nd becomes significnt fo smll mgnitudes of the ltte. Fo the cse of negligible test mss (m <<m, µ=m, = N ) Eq. fulfils the equiements fo the symptotic behviou ccoding to MOND theoy, ( N >> ) N (Newtonin limit) nd ( N << ) ( N ) / (deep MOND limit) [5,7,8]. The ltte explins the Tully-Fishe eltion [9], one of the hllmks of MOND theoy. The ecent discovey of Dgonfly 44 [], glxy of the size of the Milky Wy but with only % of its visible mss, celebted in the pess s glxy composed of 99.99% dk mtte, my be consideed s new hllmk of MOND theoy, becuse the mesued velocity dispesion of this glxy gees with the pedicted vlue fo the deep MOND limit, which is independent of the choice of the extpoltion function [].

3 Within modified ineti intepettion of MOND, the coection of the eltive cceletion is equivlent to modifiction of the educed mss ccoding to Eqs. nd. mm MOND f ( ); Eq. 3 m m The coection function ccoding to Eq. will be comped with extpoltion functions epesenting solutions of the modified Poisson eqution [9] fo one point mss fo the most commonly used MOND extpoltion functions MOND (x)=[(+4x) / +]/[(+4x) / -], MOND (x)=x/(+x) nd MOND (x)=x/(+x ) / [8,3], which e N Eq. 4 N MOND MOND N 4 N Eq. 5 MOND N. Eq. 6 N It is woth noting tht MOND,which esults fom the eltivistic geneliztion of MOND theoy (TeVeS) [], is identicl to Eq. fo the pmete choice of = ½. The pcticl dvntge of the suggested extpoltion function (Eq.) is tht the smoothness of the tnsition fom the Newtonin egime to the deep MOND egime cn be vied continuously by the choice of the pmete. The ineti intepettion of MOND does not constin the extpoltion functions to simple solutions of the modified Poison eqution, theefoe Eq. is not in contdiction to ny of the fundmentl MOND tenets. Inseting Newton s lw of gvity into Eq. nd using the definition fo the educed mss (Eq. ), Eq. tuns out to be mthemticlly equivlent to modified gvittionl foce lw F Gm m 3 ( G m m ). Eq. 7 F is symmetic in m nd m, i.e. Newton s st lw is inheently fulfilled. Eq. 7 indictes tht the MOND-coected gvittionl foce exhibits non-line dependence on the msses m nd m by which the foce is geneted. Due to the non-line ntue of the modified gvittionl foce, the supeposition pinciple is violted, i.e. the effective foce on test mss m by n ensemble of souce msses m i, i =,,N cnnot be clculted by simple vecto supeposition of foces ccoding to Eq. 7. As next step to nlyse tosion blnce expeiments within MOND, the post-newtonin coection fo n y of souce msses is evluted. We conside k- point-type souce msses m.m k, 3

4 locted t positions i, i=..k, nd the test mss m locted t. Usully, in G expeiment, only one component of this foce is mesued, detemined by unit vecto n. In Cvendish-type G expeiment, this component genetes the mesued toque. The n-component of the Newtonin cceletion of m cused by the gvittionl foce between the y (m m k ) nd m is given by m cosi Eq. 8 k N G i i i with I epesenting the ngle between i = i - nd n, nd i the distnce between m to m i. Since the souce msses m,,m k e igidly inteconnected, it is ppopite to clculte the n-component of cceletion of the entie y m,,m k due to the gvittionl foce between the y nd m. G m m cosi Eq. 9 k.. kn k i m i i i i Accoding to Eqs., 8 nd 9 the post-newtonin coection is detemined by the eltive cceletion m m cosi. Eq. k.. kn N G k i m i i i i The tem in penthesis devites fom unity fo the cse tht the vlue of the test mss (m ) cnnot be neglected in compison to the souce msses (m,,m k ). Hence, the post-newtonin coection is detemined by Eq., employing Eq. fo the evlution of the Newtonin eltive cceletion. Bsed on the methodology being descibed, Cvendish type expeiments cn be nlysed fo possible MOND effects. As outstnding ecent expeiments, the wok by Schlmminge et l. (efeed s G ) [5] nd Quinn nd Speke [] epesent consideble steps fowd in tems of the chieved ccucy nd epoducibility. Noticebly, thei esults diffe significntly fom ech othe by 8 ppm coesponding to five stndd devitions of the expeimentl uncetinty. G m - m () G (8) Quinn kgs kgs Schlmminge et l. used mecuy souce msses of 3, kg. Theefoe, ccoding to the citeion discussed befoe, this pticul expeiment is well within the deep Newtonin limit - within the climed mesuement eo. We conside G s efeence vlue fo the gvittionl constnt, which is suppoted by the fct tht G lies well within the eo limit of the cuent CODATA vlue of ( ) - m 3 /kgs [6]. Appently, the vlue mesued by Quinn nd Speke is significntly highe. 4

5 In ode to nlyse the dt within the suggested model, the geomety of the expeiment by Quinn nd Speke needs be nlysed cefully. In Quinn s expeiment the tngentil component of the gvittionl foce between fou field msses of m i = kg (i=,3,4,5) ech, which e nged on cicle of dius R = 4 mm, nd fou test msses of m =. kg ech, nged on smlle concentic cicle of dius of R = mm is detemined. The cicle of field msses is otted by n ngle = 8.9 o with espect to the cicle of test msses, in ode to mximize the toque. Although the field msses e shot cylindes in Quinn s expeiment, the point mss ppoximtion is vlid within n ccucy of few pecent, which is sufficient fo the discussion of post-newtonin coections. Fig. : Schemtics of the Cvendish-type G expeiment by Quinn et l. ccoding to []. Using the geomety depicted in Fig., the tngentil component of the dded gvittionl cceletion of ech field mss on one test is clculted fom the geomety of the expeiment. Due to the given symmety, the foce tngentil component is the sme fo ech test mss. The expeiment is pefomed by mesuing the toque geneted by the gvittionl foce due to ottion of the field msses fom + to -. Accoding to Eq., the eltive cceletion between one test mss nd the y of fou field msses is detemined by woking out the physicl distnce i nd the ngles I fo ech test mss fom the geomety of the expeiment depicted in Fig.. The esulting vlue of is ms - = 5. In ode to fit the vlue of the extpoltion function to this esult we employ G G Quinn (.93.45) 4 Eq. to ccount fo possible post-newtonin coections. Bsed on the expeimentl vlue G nd G Quinn nd thei uncetinties, nges between. nd.6, the best fit is chieved fo =.3. Obviously the fitted vlue of depends on the choice of. 5

6 In Fig., the eltive devition between Quinn s nd Schlmminge s esult fo G is pesented s function of the eltive diffeentil Newtonin cceletion being employed in Quinn s Cvendish expeiment, the ltte nomlized to the MOND cceletion =. - m/s. It is ssumed tht Schlmminge s esult epesents Newton s lw. The e between the ed lines coesponds to the extpolted nge of vlues which e comptible with the nlysis within the expeimentl eo bs. Fo compison, Eq. is lso plotted fo nge of distinct vlues of (full lines), long with MOND extpoltion functions ccoding to Eqs. 3-5 (dshed lines). As diect compison to ecent stophysicl dt, the puple dots epesent individul esolved mesuements long the ottion cuves of nely spil glxies. The oiginl dt in [3] e pesented s tio of the sques of the mesued obitl velocity nd the clculted byonic velocity, s being clculted fom the Newtonin gvittionl cceletion by the byonic (= visible sts nd intestell gs) mss of the glxy. Since the centifugl cceletion is popotionl to gvittionl cceletion, this tio is equl to the tio of mesued cceletion nd Newtonin cceletion. This enbles diect compison with the G dt. Quinn-Schlmminge G expeiments Mc Gugh glxy dt =.5 / MOND ( - )/ - - MOND =. -.6 MOND -3 = -4.. / Fig.: Compison of the post-newtonin cceletion detemined fom the Quinn-Schlmminge G discepncy with glxy ottion dt, in tems of eltive devition fom Newton s lw, ( / )-, s function of the mgnitude of the eltive Newtonin cceletion /. The expeimentl esults e comped with nge of extpoltion functions ccoding to MOND theoies (see explntion in text). The ed dt point coesponds to the Quinn-Schlmminge G discepncy, the puple dt points epesent the glxy dt, nd the e between the ed cuves epesents the extpoltion function bsed on the suggested post-newtonin model fo the nge of the pmete being comptible with the Quinn-Schlmminge discepncy within thei eo mgins. 6

7 The extpoltion function extcted fom the Quinn-Schlmminge discepncy gives n excellent fit to the cloud of glxy ottion cuve dt nd lies between MOND (sometimes clled MOND simple nd MOND (sometimes clled MOND stndd ), both hve been successfully used to fit glxy ottion cuves. Although the lge sctteing nd mesuement eos of the glxy ottion dt is comptible with wide nge of vlues, it is impotnt to emphsize tht the pmete is solely detemined fom the teestil G-nlysis, nd not by fit to the glxy ottion dt. Within ecent compehensive eview bout G expeiments the uthos concluded with the emk The sitution is distubing clely eithe some stnge influence is ffecting most G mesuements o, pobbly moe likely, the mesuements hve unecognized lge systemtic eos [4]. The pesented nlysis descibes possible scenio fo some stnge influence, s seious ltentive to unecognized lge systemtic eos. On the othe hnd, mong the G expeiments being published to dte, the selection of Quinn s nd Schlmminge s dt is subjective, since othe expeiments show tends which e not comptible with the model pesented hee [,,4]. In pticul, one of the moe ecent Cvendish-type expeiment by Gundlch et l. [], which opetes in simil cceletion nge thn the one by Quinn nd Speke, hs eveled vlue close to tht epoted by Schlmminge et l. Howeve, thee is one impotnt peculiity in Gundlch s expeiment: the field msses otte t the extemely low te of md /second on dius of bout 7 cm. In spite of this vey slow te, the coesponding centipetl cceletion is c =6.8-5 m/s, which is 5, times lge thn. Theefoe, the educed mss is cceleted by simil mount of eltive cceletion nd MOND effects my become unmesuble (smlle thn c. ppm) with the given ccucy of the expeiment. In cse of the tosion expeiment by Quinn nd Speke the pendulum motion is eithe supessed (in cses of blncing the toque fom the gvittionl foces by n electosttic toque, dubbed sevo method ) o the mplitude of the oscilltion is not bigge thn the pendulum motion geneted by the gvittionl cceletion of the test mss (Cvendish method) []. Theefoe, this expeiment my be unique in tems of the visibility of MOND effects. The time-of-swing method t HUST (efeed s HUST-9 [5]), which hs esulted in G vlue in geement with the CODATA vlue, my not llow to obseve MOND effects eithe due to lge mplitude of motion (not quoted in [3]) o due to the fct tht the dt cquisition time extends ove lge numbe of pendulum peiods of bout 53 s ech (in contst to s in cse of the Quinn-Speke expeiment): duing the time the pendulum needs to chnge its equilibium position upon moving the souce msses, the gvity cceletion component of moon nd sun which points into the diection of the pendulum motion chnges by n mount which is significntly lge thn. (fo exmple, the cceletion of the moon my chnge by up to 8 times ove time spn of 5 s, depending on the position of the moon, nd the new equilibium position of the pendulum my stbilize fte sevel pendulum oscilltions). Although cceletion by moon nd sun do not esult in ny mesuble toque, they cete n cceleted fme of efeence (with espect to the diection of test mss motion), which my limit MOND effects to n unmesuble mgnitude. Fo line pendulum, the decoupling of seismic noise is much less efficient thn fo tosion pendulum. In cse of pi of line pendulums [-4], the mico-seismic-induced motion of ech individul pendulum is usully much lge thn the distnce of the two pendulum bodies, which is mesued by micowve cvity [,3] o by lse intefeomete [4]. The MOND coection is detemined the cceletion of ech pendulum, theefoe MOND effects e likely to be supessed. 7

8 Finlly, the extpoltion function fvoued by the pesented nlysis is consistent with ecent plnety obsevtions. As shown by Hees et l., MOND theoies bsed on the modified Poisson eqution cn be uled out fom ecent Cssini dt on the obitl dynmics in ou sol system fo the nge of extpoltion function being discussed hee [6]. Howeve, this nlysis only efes to the specific effects of the modified Poisson eqution nd not to the modifiction of ineti t low mgnitudes. A citicl test which my llow to distinguish between the ineti nd field intepettion of MOND theoy is the velocity dispesion of the ecently discoveed Milky Wy stellite glxy CRATER II [7]. As pointed out by Mc Gugh [] the MOND pediction fo the velocity dispesion with nd without consideing the extenl field effect, which does not llow the obsevtion of MOND effects in teestil expeiment in cse of the field intepettion of MOND, diffes by fcto of two. Hence, CRATER II is not only citicl test lbotoy fo MOND vs dk mtte, but it should lso llow to distinguish between these two intepettions of MOND. Fo the ineti cse the extenl field effect is ielevnt nd the velocity dispesion should be ound 4 km/s []. The expeimentl confimtion of this pediction would be stongly suppotive gument fo the discussed intepettion of G discepncies by MOND theoy. Conclusion The pesented nlysis hs eveled fist indiction tht the obseved discepncies between G- vlues detemined by diffeent teestil expeiments my hve the sme physicl oigin s the nomlies of glxy ottion cuves. Since unknown o undeestimted systemtic eos in cuent G expeiment cnnot be uled out, this finding my just be coincidence. In ode to test this hypothesis, ccute mesuements of smll foces (gvittionl nd non-gvittionl) in diffeent egimes of cceletion by diffeent methods e pivotl. It is ecommended to un Cvendish expeiments using viety of toque mgnitudes eithe by compison of diffeent souce msses o by using diffeent ngul positions fo given souce mss y. Mesuements of toque tios my be less sensitive to some systemtic eos thn bsolute G mesuements. The dynmics of the expeimentl pocedue (mgnitude of pendulum oscilltion, pendulum peiod, souce mss movement, position of moon nd sun duing dt cquisition) my hve n influence on the esults due to the subtle non-line ntue of MOND effects. Line pendulum expeiments seem to be less suited thn Cvendish expeiments, becuse they e moe pone to stong nd uncontolled pendulum motions diven by mico-seismic ctivities. The pesented model povides tool nd guideline fo dt nlysis within the ineti intepettion of MOND theoy. Acknowledgements: The utho likes to expess his thnks to Clive Speke, Univesity of Biminghm, UK, fo checking the consistency of the nlysis of his G expeiment, Stcy Mc Gugh fom Cse Westen Reseve Univesity fo poviding the glxy dt nd Hinich Meye, Csten Niebuh nd Ebehd Wuensch, DESY, Gemny, fo motivting discussions. 8

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