Evidence for Modified Newtonian Dynamics from Cavendish-type gravitational constant experiments

Transcription

1 Evdence for Modfed Newonan Dynamcs from Cavendsh-ype gravaonal consan epermens Norber Klen, Imperal College London, Deparmen of Maerals Souh Kensngon Campus, London SW7 AZ, Uned Kngdom Absrac: Recen epermenal resuls for he gravaonal consan G from Cavendsh-ype epermens were analysed n he framework of MOND Modfed Newonan Dynamcs. The basc assumpon for he analyss s ha MOND correcons apply only o he componen of he gravaonal feld whch leads o an acceleraed moon of he pendulum body accordng o Newon s second law. The analyss s based on numercal soluons of he MOND correced dfferenal equaon for a lnear pendulum a small acceleraon magnudes of he order of Mlgrom s fundamenal acceleraon parameer a 1-1 m/s for he case of a med gravaonal and elecromagnec pendulum resorng force. I was found ha n case of a domnan gravaonal resorng force equvalen o a gravy pendulum he pendulum resonance frequency ncreases wh decreasng acceleraon amplude, whereas n case of a domnan elecromagnec resorng force equvalen o a sprng pendulum only he change of he equlbrum poson of he pendulum body caused by he acceleraon due o an eernal gravaonal feld of a source mass s larger han epeced from Newonan mechancs. The resuls from he pendulum smulaons were employed o f epermenal daa from recen Cavendsh-ype epermens wh repored dscrepances beween G values deermned by dfferen measuremen mehods for a smlar epermenal seup, namely me of swng, angular acceleraon feedback, elecrosac servo and sac deflecon mehods. The analyss revealed ha he repored dscrepances can be eplaned by MOND correcons wh one sngle f parameer. The MOND correced resuls were found o be conssen wh a value of G = m 3 kg -1 s - whn a sandard devaon of 14 ppm. 1. Inroducon Gravy s by far he bgges challenge of fundamenal physcs. Alhough Ensen s heory of general relavy has successfully passed any epermenal es so far, culmnang n he recen drec observaon of gravaonal waves [1], he dynamcs of galaes remans an open ssue. Whereas he mansream eplanaon for galay roaon curves s sll cold dark maer CDM [] - a glue whch holds galaes ogeher whou volang Newon s law of gravy - has emerged recenly ha all galay roaon curves follow a unversal law [3]. Ths unversal relaon beween he observed acceleraon deermned from galay roaon curves - and he calculaed acceleraon due o baryonc.e. vsble maer accordng o Newon s law s conrolled by one fundamenal acceleraon parameer a, s numercal value s abou 1-1 m/s. CDM models so far have faled o provde a conssen eplanaon for hs unversal relaon. Numerous epermens desgned o deec poenal dark maer parcles have produced only null resuls ll dae []. Keepng n mnd he relevance of dark maer for our curren undersandng of he unverse, hese dffcules represen a que subsanal crss of fundamenal physcs, bu a he same me provde a grea challenge for new deas and dscoveres n he fuure. 1

2 As an alernave o he dark maer paradgm, modfcaons of Newon s law have been suggesed o eplan galay roaon curves - as a vable alernave o CDM. The mos noceable approaches are summarzed under Modfed Newonan Dynamcs MOND, nroduced by Mlgrom n 1983 [5,6]. Accordng o he MOND paradgm, whch s purely phenomenologcal, he gravaonal acceleraon feld of an solaed pon mass devaes from he Newonan acceleraon a N accordng o a a F a / a wh F a / a 1 and F a / a a / a Eq. 1 MOND N N N N N wh a =1.1-1 m/s denong he fundamenal acceleraon parameer of MOND [6]. The so-called MOND nerpolaon funcon F depends on he rao of he magnude of he Newonan acceleraon a N o a. The consrans for he choce of F are he Newonan lm a N >> a and he so-called deep MOND lm a N << a. In he deep MOND lm he acceleraon feld of a pon mass M s gven by a MOND=GMa 1/ /r G = gravaonal consan, whch descrbes he observed fla radus ndependen galay roaon curves very well. Fla roaon curves are no conssen wh Newon s law whou he assumpon of vas amouns of dark maer whn a well-mached halo around each galay. The M 1/ dependence of a MOND n he deep MOND lm llusraes ha MOND s a nonlnear heory, whch has mplcaons for he dynamcal behavour of movng masses n a gravaonal feld. Ecep s lms, he choce of he nerpolaon funcon F s no deermned by any physcal law. Accordng o he MOND verson based on a modfed Posson equaon [7] ofen he funcons MOND smple F a a N / MONDsmple or MOND sandard a a N Eq. 1 1 a FMONDsandard an / a 1 an Eq. 3 haven been used o f galay roaon curves. More recenly, Mc Gaugh suggesed F a / a McGaugh N 1 Eq. 4 1 ep a / a N as bes choce o f he plehora of galay roaon curves [3,4]. However, as poned ou n [9] he dfferen funcons look que smlar for he ypcal range of galacc acceleraon magnudes beween 1-1 and 1-1 m/s gven he large errors of epermenal daa. Typcal Cavendsh ype G epermens operae n he range of 1-8 o 1-7 m/s, herefore any possble erapolaon of galay roaon daa va MOND crucally depends on he choce of F. As a possble choce whch makes a good f o galay roaon curves, bu allows o conrol he smoohness of he ranson o he Newonan regme by one parameer

3 FKlen an / a 1 a an 1 Eq. 5 was suggesed recenly wh o be used as f parameer [8]. As dscussed laer see Fg. 8, values beween ca..8 and cover he smoohness range beween MOND smple and MOND sandard and allow for reasonable fs o galay roaon curves. I s mporan o noe ha no parcular choce of he MOND nerpolaon funcon s ousandng wh regards o any possble physcal eplanaon of MOND effecs. The key dfference beween he dfferen nerpreaons of he MOND paradgm s deermned by he eac meanng of a N n he argumen of F, n case of scenaros where more han one pon mass s presen: Accordng o he MOND verson formulaed by a modfed Posson equaon a N s he magnude of he oal gravaonal feld [7]. Ths ecludes he observably of MOND effecs on earh and even whn our solar sysem. Moreover, leads o he so-called eernal feld effec, whch s mporan for MOND dynamcs of saelle galaes [11]. Accordng o he modfed nera nerpreaon of MOND [1] a N descrbes he componen of he gravaonal feld whch leads o an acceleraed moon of a es mass. In case of an deal orson pendulum see Fg. 1 he plane perpendcular o he pendulum n whch he pendulum body roaes represens an appromave wo-dmensonal neral frame of reference for he lm of nfne pendulum lengh. Gravaonal felds of moon and sun, he cenrfugal acceleraon due o he roaon of he earh and local gravaonal sources lke buldngs conrbue o he local gravaonal feld vecor g local n Fg. 1 whch defnes he pendulum algnmen. Gravy gradens due o massve objecs close o he pendulum cenre may lead o a ne orque, whch s a well-known problem o deal wh for hgh precson G measuremens. However, wh regards o MOND correcons, whch are deermned by he magnude of he acceleraon of he pendulum when he source masses are moved beween far and near poson see Fg. 1, he effec of hese gradens s neglgble. As dscussed n deal n secon 3, n case of he angular acceleraon feedback AAF mehod he pendulum roaes on a urnable, whch leads o a cenrpeal acceleraon of he pendulum body deally composed of wo pon masses conneced by a massless bar, see Fg. 1, whch can be larger han he angular acceleraon of he pendulum due o he movng source masses. Owng o he equvalence of an enforced acceleraed moon and gravy accordng o General Relavy roang spaceshp amng o mmc gravy for asronaus, he cenrpeal acceleraon needs o be consdered for MOND correcons. 3

4 Fg.1: Schemacs of a Cavendsh ype G epermen. A suspended orson wre or orson srp 1 s employed as suspenson of a es mass, deally composed of wo sphercal masses whch are conneced by a massless rgd bar 3. The plane orened perpendcular o he local effecve gravy vecor g local 4 represens an appromave D neral frame of reference 4. Two source masses are arranged wh her cenre-of-mass whn hs plane 4 and can be moved beween near 5 and far poson 6 n order o generae a orque o he orson wre, whch s measured va angular deflecon of he pendulum n order o deermne G. In case of he AFF mehod see e he pendulum s roaed a a consan angular velocy around s as. Torson pendulum epermens are ncredble sensve and have been used o es volaons of Newon s second law a acceleraon magnudes as low as 1 13 m/s : Accordng o he resuls repored by Gundlach e al. [1] devaons from Newon s second law can be ecluded for acceleraon magnudes as low as 1 13 m/s, bu only for elecromagnec forces causng he acceleraon of he pendulum body - n hs case he resorng orque from he orson fbre whch orgnaes from elasc properes of he fbre maeral. Based on hs epermenal consran any reasonable nera nerpreaon of MOND s lmed o MOND modfcaons of he nera for an acceleraon by a gravaonal force only. Accordng o hs lmaon, he MOND nera nerpreaon aken as bass for he analyss dscussed n hs conrbuon s dencal o a MOND correcon of he gravaonal acceleraon whch generaes an acceleraed moon n case of a orson pendulum he acceleraed moon of he pendulum body. In he cone of general relavy hs MOND nerpreaon sugges devaons from Newon s second law jus n case of he magnude of he graden of he space me curvaure n he drecon of he pendulum moon beng small enough. In spe of he hgh sensvy of orson balance epermens and he long hsory gong back o Cavendsh n 1798, he second bg challenge relaed o gravy s he large scaerng of G values deermned from dfferen epermens or even when dfferen modes of operaon are appled n one epermen. Fg., whch s reproduced from a recen revew arcle by Rohlener and Schlammnger [13], llusraes he curren suaon: 4

5 Fg. reproduced from [13]: Recen measuremens of G wh gven unceranes. The analyss presened n hs conrbuon s resrced o Cavendsh orson pendulum epermens open symbols usng me-of-swng crcles, sac deflecon squares, elecrc servo downwards rangles and angular acceleraon upwards rangles mehods. The 18 resuls by L e al. [14] are no ncluded. In spe of careful error analyss ncludng known sysemac errors, he scaerng of he daa s far bgger han he error bars of ndvdual epermens. Even for he Cavendsh epermens alone he scaerng around he CODATA value s around ppm. Mos of he non-cavendsh ype epermens employ very large source masses, such ha MOND correcons would be n conflc wh he observed consrans for devaons from Newonan gravy whn n our solar sysem see secon 4, and are herefore no consdered for he MOND analyss. In secon 3 I wll presen a daa analyss n he framework of MOND for hose repored Cavendsh epermens where more han one mehod o deermne G was employed. The repored resuls from he BIPM epermen BIPM = Bureau Inernaonal des Pods e Mesures [15] by Qunn e al [16-18] are ousandng wh respec o MOND because hs epermen employs a hn orson srp raher han a orson wre: n case of a orson srp any ws leads o a mnscule lf of he pendulum body n he gravaonal feld of he earh [18]. Therefore, for he srp ca. 97 % of he resorng orque of hs pendulum s of gravaonal raher han of elecromagnec naure, unlke for a orson wre. Ths unque epermen resembles a smple gravy pendulum, bu reans he advanages of a orson pendulum, n parcular weak ecaon by sesmcally drven moon of he pendulum suspenson. Moreover, he lossless resorng orque leads o much hgher qualy facors of he orson resonance of several 1 5, whch reduces he pendulum jer caused by hermal nose [19].. The dynamcal behavour of a MOND pendulum Snce MOND correcons of Newon s law are deermned by a lnear acceleraon scale, s more sraghforward o consder a lnear raher han a orson pendulum. In secon 3 he comparson wh epermenal daa wll be pursued by workng ou he angular acceleraon from he daa gven n he publshed documenaon and by an esmaon of he equvalen mamum lnear acceleraon magnude o be employed for he MOND correcon. 5

6 6 The dfferenal equaon of moon for a lnear pendulum wh a paral gravaonal and elecromagnec resorng force n he Newonan lm can be wren as mg m e em g Eq. 6 wh m denong he mass of he pendulum body, g and em represenng he gravaonal and elecromagnec componen of he resorng force coeffcen, respecvely. s he dampng consan and g e he me dependen acceleraon feld whch s deermned by he mng of moon of he source masses, usually beween wo dfferen angular posons referred as near and far poson, such ha he gravaonal force beween he source masses and he pendulum body generaes a orque. Inroducng as a parameer whch descrbes he relave amoun of elecromagnec resorng force em g em Eq. 7 and he qualy facor Q whch descrbes he number of pendulum perods unl he amplude of free oscllaons decays o 1/e. Eq. 6 can be re-wren as 1 g Q e Eq. 8 wh = g+ em/m denong he angular resonance frequency of he pendulum n case of zero dampng Q. As dscussed n he prevous secon, he MOND correcon s appled o he gravaonal acceleraon erms only: / 1 1 Q a g F g e e Eq. 9 For he case of a pure elecromagnec resorng force = 1 he MOND correcon s appled o he eernal gravaonal feld resulng from he moon of source masses beween far and near poson. In case of a pure gravaonal resorng force = he sum of he eernal and he resorng acceleraon needs o be MOND correced. Snce he gravaonal force s conservave n he Newonan lm, he pendulum dampng erm s consdered as elecromagnec and herefore no MOND correced. For a gven se of sar parameers == and he eraon = o N ma / e e Q a g F g Eq.1 was employed o calculae usng a MATLAB scrp. In order o reduce numercal errors o an accepable level, n parcular for he deermnaon of small changes of he pendulum resonance frequency due o MOND effecs, N ma 1 mllon was found o be suffcen for he calculaon of

7 over 5-1 pendulum perods. For g e a ramp was chosen sarng wh g e= = and a lnear ncrease of g owards a consan value g for T ramp. I urned ou ha he new equlbrum poson aken by he pendulum a > T ramp, whch s equal o g n he Newonan lm, s ndependen of T ramp - n case of MOND correcons beng ncluded. Also, nonlnear g e dd no change he new equlbrum poson. Ths s no a rval saemen because of he nonlnear characer of MOND. Fg. 3 shows seleced eamples of smulaons. In order o vsualze MOND effecs, a low acceleraon value of g = a = m/s was chosen. The chosen MOND nerpolaon funcon accordng o Eq. 5 wh = 1.5 resuls n a MOND enhancemen of he acceleraon a a N = a by a facor The ampludes are presened n uns of g,.e. = 1 represens he Newonan case. I s mporan o noe ha he general pcure s ndependen of he choce of he nerpolaon funcon. a b c d Fg. 3: Resuls of smulaons a g =a. Dsplacemens n uns of a /. Sac enhancemen wh respec o Newon: g MOND/g Newon = a Gravaonal resorng force, shor ramp T ramp=.t, no dampng Q=1 5 b Gravaonal resorng force, long ramp T ramp=t, srong dampng Q=3 c Elecromagnec resorng force, shor ramp T ramp=.t, no dampng Q=1 5 c Elecromagnec resorng force, long ramp T ramp=t, srong dampng Q=3 7

8 Fg. 3a shows he resuls for he case of a pure gravaonal resorng force = for nearly zero dampng Q = 1 5 and a shor ramp T ramp =. T wh T denong he pendulum perod T=/ n he Newonan lm. The me s presened n uns of T. The smulaon reveals a snusodal behavour of, bu wh a frequency whch s noceable larger han n he Newonan lm. Lke n he Newonan case, he pendulum sll oscllaes around = 1. Fg. 3b shows he resul of he smulaon for a gravaonal pendulum wh srong dampng Q = 3 and T ramp equal o wo pendulum perods. The resuls reveal a furher ncrease of he pendulum frequency n comparson o Fg. 3a, bu he Newonan value for he equlbrum value of he oscllaon for > T ramp s reaned. The frequency of he oscllaon s drecly relaed o s amplude. Fgs. 3c and d show he smulaed response of he pendulum for he case of an elecromagnec resorng force = 1 for zero c and srong dampng d. In hs case he pendulum resonance frequency reans he Newonan value, bu he equlbrum poson for > T ramp s ncreased by an amoun F MOND g /a, whch s 1.31 for he gven eample. Therefore, a convenonal orson pendulum based on a orson fbre wh domnan elecromagnec resorng force allows o measure MOND effecs from he deermnaon of he sac deflecon or by compensaon of he orque by an elecrosac force elecrc servo accordng o Fg. [18]. In conras, n case of a gravaonal pendulum orson srp wh domnan gravaonal resorng force only he servo should allow o observe he MOND enhancemen of he pendulum ws, because n hs case he pendulum does no oscllae and he naure of he pendulum resorng force s rrelevan. In case of he sac deflecon mehod, he resorng force coeffcen needs o be deermned wh hgh precson n order o work ou he orque from he measured deflecon. Usually s deermned from a measuremen of he pendulum frequency [18]. Snce he pendulum amplude deermnes he MOND-relaed frequency ncrease of he pendulum, he correspondng relave ncrease of s gven by. Eq. 11 The facor n Eq. 11 resuls from 1/. Fg. 4 shows he relave frequency change for a gravaonal pendulum whou dampng as a funcon of he magnude of Newonan acceleraon of he pendulum body n uns of a n comparson o he sac MOND acceleraon enhancemen. Each pon of he curve s deermned by numercal smulaons. The MOND nerpolaon funcon used n Fg. 4 can be used o work ou he epeced dscrepancy beween G measured by sac deflecon and servo mehods for he case of a gravaonal pendulum see secon 3. 8

9 Fg. 4: Smulaed relave change of resonance frequency due o MOND for a pure gravaonal pendulum whou dampng. The resuls are dsplayed as a funcon of he amplude of he harmonc Newonan pendulum acceleraon amplude n uns of a. The relave acceleraon ncrease due o MOND s shown for comparson / Fg. 5: Calculaed relave frequency change of a pendulum due o MOND as a funcon of he relave fracon of he elecromagnec poron of he pendulum resorng force coeffcen for a pendulum acceleraon amplude of a. The case of a med gravaonal/elecromagnec resorng force s of parcular neres for he socalled me-of-swng mehod, whch has ganed populary n recen years. As eplaned n deal n [13], hs mehod reles on measuremens of small changes of he resonance frequency of he pendulum for wo dfferen posons of he source masses: In he near poson he addonal gravaonal force beween he source masses and he pendulum body generaes a small gravaonal componen of he resorng orque coeffcen, whch has o be added o he elecromagnec orque coeffcen of a wre-based orson pendulum. I s of specal neres o evaluae he MOND nduced 9

10 frequency changes as a funcon of see Eq. 7. The resuls of he smulaon, agan for g =a and he same MOND nerpolaon funcon used n Fgs. 3 and 4, are dsplayed n Fg. 5. The predced frequency change shows a srongly non-lnear varaon wh, and nearly dsappears for >.8. As an mporan consequence of hs surprsng resul, one canno epec ha he me-ofswng mehod for he deermnaon of G s sensve o MOND correcons, as long as he gravaonal componen of he resorng orque s a few percen of he oal one only see secon 3. Tll now, no me-of-swng epermens employng a orson srp have been repored. In conras o a orson wre, hese epermens are epeced o be sensve o MOND effecs In summary, he consequences from he smulaons of he MOND pendulum dynamcs for requred MOND correcons of Cavendsh ype G epermens are lsed below: 1. The MOND correcons are ndependen of he mng of he moon of source masses beween near and far poson n spe of he non-lnear naure of MOND.. In case of a pendulum wh domnan gravaonal resorng orque ll dae only appled for he BIPM epermen he gravaonal orque s no enhanced due o MOND effecs f he sac deflecon mehod s used. The reason s ha pendulum oscllaons are enabled: he oscllang resorng angular acceleraon s MOND correced, herefore enhancemens for he plus and mnus drecon cancel ou. 3. In case of a pendulum wh domnan gravaonal resorng orque he pendulum frequency ncreases as a resul of MOND a small ampludes relevan for sac deflecon and me of swng mehods. Ths ncrease n frequency resembles he ncreased roaon frequency of galaes due o MOND effecs. 4. In case of he elecrosac servo mehod he orque s ncreased due o MOND effecs, leadng o enhanced G values. Ths effec s ndependen of he naure of he resorng orque. 5. The same holds rue for he angular acceleraon feedback AAF mehod, because he moon of he pendulum s supressed by an eernal force - lke n case of he elecrosac servo mehod. The only dfference wh respec o he sac deflecon s he acceleraed moon of he pendulum body due o he roaon of he pendulum urnable see secon 3, he correspondng cenrpeal acceleraon needs o be consdered for he MOND correcon. 6. For he case of a pendulum wh domnan elecromagnec resorng orque any fbre he G resuls from me of swng measuremens wll no be affeced by MOND, because a such a small gravaonal poron of he resorng orque a few % he pendulum frequency wll no be ncreased accordng o he smulaons unlke for a pendulum wh domnan gravaonal resorng orque. Therefore, hese epermens are deal for accurae deermnaon of G whou need o pursue any MOND correcon. 3. MOND analyss of Cavendsh epermens for dfferen modes of operaon The BIPM epermen has a long hsory, wh frs G resuls repored n 1 [16-18], see Qunn e al. n Fg.. For he frs me wo dfferen modes of operaon elecrc servo and sac deflecon are compared, wh he am of akng advanage from he fac ha many of he sysemac errors lke 1

11 source mass nhomogeney and measured dsances are dencal for boh mehods. Ths approach was nended o rule ou sysemc errors, provded ha boh mehods show he same resuls for G. Anoher more recen epermen employng wo modes of operaon wll be dscussed n he second par of hs secon. Fg. 6: Geomercal arrangemen of source masses m s and es masses m for he BIPM epermen accordng o he dealed descrpon of he epermen gven n [18]. The geomery of he BIPM epermen s skeched n Fg. 6. As menoned earler, n hs epermen a copper beryllum orson srp raher han a orson wre has been used. Ths leads o he already menoned nearly 1% gravaonal characer of he resorng orque. In fac, he values repored n [18] for he gravaonal and elecromagnec componens are g= Nm/rad and em= Nm/rad, respecvely, leadng o =.33 accordng o Eq. 7. The use of he orson srp has anoher mporan consequence: whereas orson wre epermens employ lghwegh pendulum bodes of several 1 grams only a rade-off amng o acheve a hgh sensvy he orson srp enables he use of a heaver pendulum body of several klograms. As ndcaed n Fg. 6, he pendulum body used n he BIPM epermen s composed of four cylndrcally shaped Cu-Te cylnders of m =1. kg arranged n a quadrupole confguraon on op of an alumnum plae. The radus of he crcle whch nersecs wh he as of each cylnder s R 1 = 1 mm. The source mass s formed by an array of four cylnders of ca. m s=11 kg each, whch are arranged on a urnable. A an angle = 18.9 o beween feld and source masses he orque creaed by he gravaonal force beween source and feld masses s a s mamum. In order o conduc G measuremens, he source mass urnable s moved perodcally beween o and o, leadng o orque of Nm appled o he pendulum [18]. Wh he gven momen of nera of he pendulum body of I= kgm he angular acceleraon of he pendulum body s d /d = orque I = s -. Gven he radus R 1 = 1 mm of he es mass carousel accordng o Fg. 7 es masses are he major conrbuon o he mass of he pendulum body, he relevan lnear acceleraon magnude for hs epermen s a N R 1 d /d = s - = 396a. In case of he servo mode of he epermen, he pendulum does no move, herefore he - erm n he argumen of he MOND nerpolaon funcon Eq. 9 s zero a any me, n spe of. Consequenly, he MOND correcon has o be appled o he measured elecrosac orque accordng o 11

12 orque orque ES, MOND ES, Newon a 1 an 1 Eq. 1 for he case of he fleble MOND nerpolaon funcon. In case of he sac deflecon or Cavendsh mode he sac deflecon s deermned by he equlbrum value of he oscllaon afer movng he source mass urnable beween - and + or va versa. Snce s very small n fac here was no noceable dfference from he smulaon beween = and.3, he measured pendulum deflecon s no affeced by MOND. However, accordng o Eq. 11 he calculaed orque s ncreased due o he ncrease of pendulum frequency. The pendulum oscllaon for hs measuremen has an amplude deermned by he gravaonal orque resulng from he urnable movemen beween he wo mamum orque posons [18]. Accordng o Eq. 11 he epeced MOND correcon for he orque n he sac deflecon mode s orque orque an Eq. 13 SD, MOND SD, Newon wh a n aken from he smulaon of he pendulum dynamcs, whch s depced n Fg. 4 for one choce of - n case of he fleble MOND nerpolaon funcon. For boh cases Eq.1 and Eq. 13 he MOND orque correcon s equal o he relave sysemac measuremen error for G. Afer frs resuls from he BIPM epermen repored n 1 [16], he epermen was re-bul, and some sysemac errors were correced [17]. The laes resul shown n Fg. ndcaes ha G dffers for he wo mehods, alhough he auhors quoe an average value, because he error bars overlap [18]. The repored G values for he wo mehods are G ES= m 3 kg/s and G SD = m 3 kg/s, respecvely [13,18]. Boh values are sgnfcanly hgher han he 14 CODATA value G CODATA14 = m 3 kg/s [] see also graphc dsplay n Fg. 1. The resul of he smulaons were used o fnd he requred amoun of MOND correcon paramerzed by, whch gves he bes f o boh he elecrc servo and he sac deflecon daa. Fg. 7 shows he relave sandard devaon beween smulaons and epermens for boh mehods square sum of sandard devaons for boh mehods as a funcon of, under he assumpon of he valdy of he CODATA value. Snce he 13/14 BIPM resuls conrbued o he 14 revson of he CODATA value, he f was also performed for he 1 CODATA [1] value G CODATA1= m 3 kg/s, whch s 36 ppm lower han he 14 revson. Ths dfference s margnal gven he amoun of daa scaerng shown n Fg.. Under he assumpon of he valdy of he 1 CODATA value he bes f was acheved for = 1.65, he correspondng epermenal and predced devaons from Newon are dsplayed n able 1. 1

13 Elecrc servo: a/a N = 197 ppm epermenal a/a N = 3 ppm smulaon Sac deflecon: a/a N = 33 ppm epermenal a/a N = 96 ppm smulaon Table 1: Measured and smulaed devaons from Newon relave o he 1 CODATA value for he wo operaonal modes of he BIPM epermen. The ny dfferences beween epermens and smulaons abou 6-7 ppm are well below he epermenal errors of abou 55 ppm. Under he assumpon of he valdy of he 14 CODATA value he mnmum of he sandard devaon s hgher, bu sll whn he epermenal errors F o CODATA 1 F o CODATA 14.8 sandard devaon Fg. 7: Sandard devaon beween he predced MOND correcon and epermenal devaon from Newon for he wo mehods used n he BIPM epermen as a funcon of he f parameer. The mnma of he curves ndcae he value whch gves he bes f o he CODATA value for boh mehods. Recenly L e al. repored a dscrepancy for G measured by wo operaon mehod of a convenonal orson epermen whch employs a orson wre raher han he orson srp [14]. Ths resul breaks prevous records n erms of he quoed measuremen error. In spe of he fac ha he epermen has been operaed and mproved over many year, he averaged G values deermned by he angular acceleraon feedback AAF mehod whch were adaped from Gundlach e al. [] and refned s sgnfcanly hgher han he one deermned by he me-of-swng ToS mehod: G L, ToS = m 3 kg -1 s -, G L,AAF = m 3 kg -1 s - In fac, G deermned by AAF s 45 ppm hgher han G deermned by ToS, alhough he quoed measuremen error s 11.6 ppm for boh. In order o pursue he MOND correcon for he AAF G value, he amplude of he Newonan acceleraon needs o be worked ou carefully. As dscussed n [14], he AAF mehod allows a drec 13

14 measuremen of he angular acceleraon. Accordng o [14], he peak-o-peak amplude s 94 nrad/s. The pendulum body s of recangular shape, he wdh whch deermnes he srengh of he measured angular deflecon s b = 91 mm. In conras o he epermen by Qunn e al. wh crcular es masses he magnude of he lnear acceleraon can only be esmaed. A characersc value for he radus s b/4 =.75 mm, whch represens he cenre of mass of one plae secon measured from he edge owards he suspenson pon n he mddle. The resulng Newonan lnear acceleraon magnude s a N,source masses = 175a, whch s lower han n case of he BIPM epermen. However, n case of he AAF epermen he pendulum suspenson s mouned on a urnable whch roaes wh a quoed angular frequency =.44 mrad/s, he correspondng cenrpeal acceleraon a N,cenr = b/4 = 119a. Snce a N,source masses and a N,cenr are perpendcular o each oher, he esmaed oal magnude of acceleraon a N = a N,source masses + a N,cenr 1/ =114a. Therefore, owng o he roang urnable, MOND effecs are largely supressed, bu sll no neglgble because of he hgh accuracy of he epermen. In case of he ToS mehod he perod of he pendulum s decreased by 1.7 s n he near poson wh respec o he far poson of he source masses [14]. Gven he average pendulum perod of T 43 s and he fac ha he resorng force of he pendulum s 1% elecromagnec n he far poson, comes ou o be.9961 accordng o Eq. 7. Calculaons of he pendulum frequency wh hs value of do no show any measurable devaons from Newon. Therefore, he G value measured by ToS does no requre any MOND correcon. The f of he fleble MOND nerpolaon funcon Eq.5 o he measured dscrepancy of 45 ppm beween he AAF and ToS G values leads o β = 1.87, whch s surprsngly close o he value of 1.65 obaned from he f of he G values deermned by wo mehods employed by Qunn e al.. Ths resul represens a raher asonshng concdence. Fg. 8 shows a/a N as a funcon of he magnude of a N n uns of a for he sac deflecon and elecrc servo mode of he BIPM epermen, and for he average AAF resuls repored by L e al. The error bars represen he errors repored for he epermens, he error of 6 ppm for he 1 CODATA value s no aken no accoun. The a/a N daa pon represenng he sac deflecon mode of he BIPM epermen was correced by he rao of he smulaed a/a N values from able 1, owng o he MOND correcon accordng o Eq.13. For a drec comparson wh recen asrophyscal daa, he purple dos represen ndvdually resolved measuremens along he roaon curves of nearly 1 spral galaes. The orgnal daa n [5] are presened as rao of he squares of he measured and calculaed orbal veloces - he laer from he Newonan gravaonal acceleraon by he baryonc = vsble sars and nersellar gas mass of he galay. The full lnes represen he MOND nerpolaon funcon accordng o Eq. 5 for dfferen values of he f parameer, ncludng fs o he measured daa from he wo G epermens. Fg. 8 wll be furher dscussed n secon 4. 14

15 Fg. 8: Relave devaon from Newonan gravaonal acceleraon as a funcon of he magnude of he Newonan acceleraon n uns of a for he wo operaon modes of he BIPM epermen [18] and he AFA mode of he epermen by L e al. [14]. In order he vsualze he BIPM resuls deermned by he wo mehods, he elecrc servo and sac deflecon resuls are dsplayed wh her dencal a n/a value moved o he lef and rgh, respecvely. The elecrc servo daa pon represens he relave dfference beween he epermenal G value for hs mehod and he CODATA 1 value, he deflecon daa pon represens he correspondng dfference - mulpled by he rao of he smulaed values for boh mehods aken from able 1. The nser shows a magnfcaon of he dagram around he daa pons from he G epermens. The purple daa pons represen daa eraced from galay roaon curves accordng o [5]. The MOND nerpolaon funcons F Klen Eq. 5 for several choces of he parameer full lnes, ncludng = 1.65 and = 1.87 obaned from fs o he G epermens are compared wh wo common MOND nerpolaon funcons Eqs. and 3 and Mc Gaugh s unversal radal acceleraon relaon RAR, Eq Dscusson The analyss of a pendulum a small acceleraon ampludes whn a MOND nera scenaro presened n hs conrbuon shows ha dfferen operaonal modes of a gven G epermen lead o dfferen resuls for he measured G values. Ths unque qualy of he MOND paradgm dsngushes MOND from oher suggesed modfcaons of Newon s law, such as shor range correcons by a Yukawa erm o be added o he Newonan gravaonal poenal. The laer has been dsproved wh hgh precson by a varey of epermens - as par of he Eö-Wash campagn [6]. 15

16 The mos mporan resul of he presened analyss of recen G epermens by a MOND pendulum model s a conssen eplanaon of he observed dscrepances of G resuls deermned from dfferen epermens and by 4 dfferen mehods. Ths represens a remarkable concdence and herefore provdes evdence for Modfed Newonan Dynamcs from erresral epermens for he frs me. The comparson of he observed dscrepances wh galay roaon curves and MOND nerpolaon funcons dsplayed n Fg. 8, alhough appealng n erms of he perfec machng of he fed nerpolaon funcons wh he average of he roaon curve resuls, should be aken wh a pnch of sal: As menoned before, he choce of a parcular MOND nerpolaon funcon s no movaed by any known physcal mechansm. Therefore, he machng of he fed nerpolaon funcon wh he galay daa does sugges - bu no prove - ha he observed devaons from Newon have he same physcal orgn han he roaon curves of galaes. Fg. 8 should be aken as workng hypohess, amng o refne and desgn epermens whch operae closer o he acceleraon range of galaes. In fac, he resuls publshed by Gundlach e al. n 7 sugges ha hs s possble [1]. A sraghforward way o provde furher evdence based on esng epermens s o run he AAF epermen by L e al. a lower speed of he pendulum urnable, whch should lead o hgher values of G, f MOND effecs are aken no accoun. In case of he gravaonal orson pendulum used by Qunn e al. precse measuremens of he pendulum perod as a funcon of amplude owards as close as possble owards a / - may prove or dsprove he MOND hypohess. However, hs s no as rval as sounds, because he nfluence of parasc effecs lke emperaure drf, nose and envronmenal gravaonal gradens may cause ncreasng measuremen errors. Referrng o Fg., he bg queson s wheher MOND correcons lead o sgnfcan mprovemens of he puzzlng dscrepances beween G epermens. The answer s subjecve, and dfferen people may have dfferen opnons abou he maury of gven epermens. I would argue ha over he las 1 years here has been a convergence of G resuls by well-esablshed and mos horoughly nvesgaed mehods owards a value slghly above he 14 CODATA value of abou G = m 3 kg -1 s -. G s very close o he value repored by Schlammnger e al. whch was eraced from a unque epermen employng several ons of lqud mercury as es mass, oally unaffeced by any MOND correcon due o he srengh of gravaonal neracon [7]. Whou preendng ha hs resul represens a gold sandard, s noceable ha for a number of hghprecson ToS Cavendsh epermens wh record sensvy ncludng L e al. [14] and Newman e al [4] he resuls have merged around G. I would argue ha earler resuls repored by he HUST eam L e al. [8] and Tu e al. [3] represen predecessors of he recen resuls repored by L e al. [14] and herefore may be affeced by sysemac errors, whch have been elmnaed by he eam snce hen. The recen epermenal G value repored by Newman e al. [4] does no requre any MOND correcon because of he employed ToS mehod and he large magnude of he Newonan acceleraon and herefore represens anoher recen evdence for G beng close o G. Fg. 9 shows repored G values from recen Cavendsh ype G epermens ncludng MOND correcons based on he fleble MOND nerpolaon funcon Eq. 4 and one common f parameer β = 1.3. Wh hs choce of β, all correced G resuls are conssen wh G whn he error margn of each epermen. The analyss ncludes he earler AAF epermen by Gundlach e al. [], whch was MOND-correced n he same way as he AAF G resuls by L e al. Due o he hgher angular velocy of he pendulum urnable he magnude of he Newonan acceleraon s hgher han n case 16

17 of L e al., herefore he correcon reduces he G value jus by abou 7 ppm. The sandard devaon of he correced epermenal resuls shown n Fg. 9 from G s only 13.9 ppm. Fg. 9 provdes evdence ha he epermenal groups have done an ousandng job o elmnae sysemac errors and o refne her G epermens o a very hgh level of performance and confdence. More sgnfcan for he wder scenfc communy, hs resul provdes srong evdence for Modfed Newonan Dynamcs. Fg. 9. Repored G-values from recen Cavendsh-ype epermens from [14, 4, 18, ] before black and afer MOND correcon red for he fleble MOND nerpolaon funcon Eq. 4 and one common f parameer = 1.3. In case of ToS me-of-swng based daa pons no MOND correcon s requred,.e. correced and uncorreced daa pons are dencal. The dashed lne represens G = m 3 kg -1 s -. The sandard devaon of he correced daa pons from G s 13.9 ppm. The observed devaons from Newon are no n drec conflc wh he epermenal lms for possble devaons from Newon whn our solar sysem: The acceleraon magnude of a N/a 4 for he BIPM epermen s equal o he gravaonal acceleraon of he sun a a dsance of 38 AU asronomcal uns, whch s abou en mes he dsance beween sun and Pluo. However, he erapolaon of hese resuls va he fed MOND nerpolaon funcon o he dsance of Pluo and Saurn leads o a MOND correcon of and , respecvely. Ths s jus abou equal o he 17

18 upper ecluson boundary for anomalous radal acceleraon n case of Saurn and Uranus, and no n conflc wh oher planes of our solar sysem see able n [9]. Wh regards o he assumed nerpreaon of MOND, he epermenal evdence from hs conrbuon shows clear evdence ha MOND effecs can ndeed be observed n he presence of a gravaonal feld whch s much larger han Mlgom s acceleraon parameer a, as long as he dynamcal degrees of freedom are srcly confned o a D plane wh normal vecor srcly parallel orened o he local eernal gravaonal feld see Fg. 1. Ths new evdence rules ou fully fledged MOND heores based on a modfed Poson equaon [7], whch ofen have been used for smulaons of he MOND dynamcs of mul-body sysems such as galay clusers. As a second consran, he evdence from hs conrbuon ha elecromagnec forces resorng orque of a orson wre are no subjec o MOND correcons confrms he fndngs from prevous work [11,] ha a general MOND modfcaon of he neral mass can be ruled ou. Fnally, he evdence ha he cenrpeal acceleraon needs o be consdered for MOND correcons n order o successfully f G daa from AAF epermens leads o he concluson ha General Relavy s an mporan prerequse for any aemp o undersand he Physcs behnd he MOND phenomenology. 5. Conclusons The observed dscrepances beween values of he gravaonal consan deermned by dfferen operaonal modes of recen Cavendsh-ype epermens were found o be conssen wh Modfed Newonan Dynamcs. MOND correcons appled o measured G values have reduced daa scaerng of repored values of he gravaonal consan for recen Cavendsh ype epermens sgnfcanly and sugges ha he real G s m 3 kg -1 s - - wh a sandard devaon for he correced epermenal daa of only 14 ppm. Fuure epermens wh mproved sensvy for small acceleraon ampludes should be pursued o suppor hese nal fndngs and o fll he gap beween he acceleraon magnude of galay roaon curves and erresral G epermens. The paper descrbes he mehodology of daa analyss accordng o he MOND paradgm and pus resrcons on possble physcal nerpreaons of he MOND phenomenology. The ndcaed evdence for he drec observaon of gravaonal effecs a he acceleraon scale of galaes n earh-bound laboraores provdes an amazng perspecve for solvng one of he mos burnng quesons relaed o our undersandng of he unverse. 6. Acknowledgemens I would lke o epress my sncere hanks o Hnrch Meyer, now emerus Professor a Bergsche Unversae Wupperal Germany, where I graduaed from. Hs group s sll runnng a lnear double pendulum epermen a DESY n Germany, whch I orgnally desgned n 1987 as par of my PhD hess. Hnrch Meyer has been lookng for MOND effecs n hs double pendulum epermen and publshed daa whch eclude MOND, bu whou akng no accoun he dynamc effecs of MOND [3]. Ths resul has nspred me o ake a closer look a Cavendsh ype G epermens. I would lke o hank Clve Speake, Professor a Unversy of Brmngham, who s runnng he BIPM epermen, whch s currenly locaed a NIST. Clve gave me he opporuny o vs hs ousandng epermen n hs 18

19 laboraores n Brmngham durng 16, and whou hs grea eperence and hs eplanaons I would no have been able o pursue hs analyss n a meanngful way. 7. References [1] B. P. Abbo e al. LIGO Scenfc Collaboraon and Vrgo Collaboraon, Phys. Rev. Le. 119, [] A. Bauds, European Revew 6, 7 17 [3] S. McGaugh, F. Lell, J.M. Schomber, Phys. Rev. Le. 117, [4] P. L e al., Asronomy and Asrophyscs 615, A3 18 [5] M. Mlgrom, Asrophys. J. 7, [6] B. Famaey, S. McGaugh, Lvng Rev. Relavy 15, 1 1 [7] J.D. Bekensen, M. Mlgrom, Asrophys. J. 86, [8] N. Klen, arxv:154.76v4 [gr-qc] 16 [9] D.C. Rodrgues e al, Naure Asronomy, [1] M. Mlgrom, Annals of Physcs 19, [11] E. Garald e al., Phys. Rev. Le. 1, [1] J.H. Gundlach e al., Phys. Rev. Le. 98, [13] C. Rohlener, S. Schlammnger, Revew of Scenfc Insrumens 88, [14] Q. L e al., Naure 56, [15] see hps:// [16] T.J. Qunn e al., Phys. Rev. Le. 87, [17] T. Qunn e al., Phys. Rev. Le. 111, and Phys. Rev. Le. 113, 3991 Erraum 14 [18] T. Qunn e al., Trans. R. Soc. A 37, [19] J. Luo e al., Class. Quanum Grav. 6, [] P.J Mohr e al., Journal of Physcal and Chemcal Reference Daa 45, [1] P.J Mohr e al, Revews of Modern Physcs 84,

20 [] J.H. Gundlach, S.M. Merkowz, Phys. Rev. Le. 85, 869 [3] L.C. Tu e al., Phys. Rev. D 8, 1 1 [4] R.Newman e al., Phl. Trans. R. Soc. A 37, [5] B. Famaey, S. Mc Gaugh, Lvng Rev. Relavy 15, 1 1, Fg 1, boom panel, daa on hp://asroweb.case.edu/ssm/daa [6] T.A. Wagner, S.Schlammnger, J.H. Gundlach, E.G. Adelberger, Classcal and Quanum Gravy 9, 18 1 [7] S. Schlammnger e al. Phl. Trans. Roy. Soc. 37, [8] Q. L e al., Phl. Trans. R. Soc. A 37, [9] M. Sereno, Ph. Jezer, Mon. No. R. Asron. Soc. 371, 66 6 [3] H. Meyer e al., Gen. Relav. Grav. 44, 537 1