Galaxy rotation curves and the deceleration parameter in weak gravity
aa r X i v : . [ g r- q c ] J u l September 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 1 Galaxy rotation curves and the deceleration parameter in weak gravity
Maurice H.P.M. van Putten ∗ Sejong University, Seoul 143-747, South Korea ∗ We present a theory of weak gravity parameterized by a fundamental frequency ω = √ − qH of the cosmoloogical horizon, where H and q denote the Hubble and, respec-tively, deceleration parameter. It predicts (i) a C onset to weak gravity across ac-celerations α = a dS in galaxy rotation curves, where a dS = cH denotes the de Sitteracceleration with velocity of light c , and (ii) fast evolution Q ( z ) = dq ( z ) /dz of the decel-eration parameter by Λ = ω satisfying Q > . Q = Q (0), distinct from Q . H ( z ) over 0 < z <
2. A model-independent cubic fit in the secondrules out ΛCDM by 4 . σ and obtains H = 74 . ± . − Mpc − consistent withRiess et al. (2016). Comments on possible experimental tests by the LISA Pathfinderare included. Keywords : galaxy rotation curves; deceleration parameter; dark energy; dark matter
1. Introduction
Modern cosmology shows a Universe that is well described by a three-flatFriedmann-Robertson-Walker (FRW) line-element ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) , (1)that currently experiences accelerated expansion indicated by a deceleration param-eter q = 12 Ω m − Ω Λ < , (2)where q = − H − ¨ a/a with Hubble parameter H = ˙ a/a . It points to a finite darkenergy density Ω Λ = ρ Λ /ρ c , where ρ c = 3 H / πG is the closure density withNewton’s constant G . In this background, galaxies and galaxy clusters formed andevolved by weak gravitational attraction at accelerations α at or below the de Sitterscale a dS = cH, (3)where c denotes the velocity of light. As such, (3) sets a scale to weak gravity common to cosmological evolution and large scale structure.Weak gravity surprises us with more than is expected by Newtonian gravitationalattraction of observed baryonic matter content: Λ = 8 πρ Λ > − and [ √ Λ] =cm − in geometrical units, in which Newton’s constant and the velocity eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 2 of light are set equal to 1, where the latter points to a transition radius r t in rotationcurves of galaxies of mass M b , 4 πr t √ Λ ≃ R g , R g = GM b /c . Just such scale r t of afew kpc is observed in a sharp onset to anomalous behavior in high resolution datafor M b = M M ⊙ at a N = a dS , (4)where a N refers to the acceleration anticipated based on Newton’s law of gravita-tional attraction associated with M b inferred from luminous matter. Further outinto weak gravity, these rotation curves satisfy the emperical baryonic Tully-Fisherrelation or, equivalently, Milgrom’s law. We here describe a theory of weak gravity by spacetime holography, param-eterized by a fundamental frequency of the cosmological horizon ω = p − qH, (5)defined by the harmonic oscillator of geodesic separation of null-generators of thecosmological horizon. This result reflects compactness of the holographic phasespace of spacetime, set by its finite surface area.. By holography, (5) induces a darkenergy in the 3+1 spacetime (1) by the squareΛ = ω . (6)It may be noted that (6) has vanishing contribution in the radiation dominated era( q = 1), whereas it reduces to CDM in a matter dominated era ( q = 1 / p = wρ Λ for total pressure p , w = (2 q − / (1 − q ) vanishesfor q = 1 / m and positions, inertia isdefined by a thermodynamic potential U = mc derived from unitarity of particlepropagators. In weak gravity, this may incur m < m ( α < a dS ) (7)with a C onset to inequality by crossing of apparent Rindler and cosmologicalhorizons. In weak gravity, therefore, a particle’s inertial mass (“weight”) and grav-itational mass may appear distinct. According to the above, weak gravity prediction the following. First, the C onset to (7) is at a transition radius r t = 4 . M / ( H /H ) / . (8)Beyond, asymptotic behavior ( α << a dS ) satisfies Milgrom’s law with a = ω π . (9)These expressions (8-9) may be confronted with data on galaxy rotation curves overan extended redshift range supported by Hubble data H ( z ). Second, the associateddark energy (6) is relevant to late times, when deceleration q ( z ) turns negative. At eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 3 the present epoch, (6) predicts fast cosmological evolution , described by q = q (0)and Q = Q (0), Q ( z ) = dq ( z ) /dz , satisfying q = 2 q , Λ CDM (10)and Q > . , Q , Λ CDM . . (11)In §
2, we discuss the origin of (4) in inertia from entanglement entropy and itsconfrontation with recent high resolution data on galaxy rotation curves. In §
3, wediscuss (6) and its implications (11) in confrontation with heterogeneous data on H ( z ) (0 < z < Sensitivity of galaxy dynamics in weak gravity to the Hubbleparameter H ( z ) is studied in §
5. We summarize our findings in § H .
2. Weak gravity in galaxy rotation curves
Galaxy dynamics is of particular interest as a playground for the equivalence prin-ciple (e.g. ) at small accelerations on the de Sitter scale a dS .Einstein’s Equivalence Principle (EP) asserts that the gravitational field -wherein photon trajectories appear curved - seen by an observer on the surface ofa massive body is indistinguishable from that in an accelerating rocket, at equalweights of its mass . On this premise, general relativity embeds Rindler trajectories- non-geodesics in Minkowski spacetime - by gravitational attraction as geodesicsin curved spacetime around massive bodies, while weight is measured along non-geodesics. With no scale, this embedding is free of surprises, as Rindler accelerationsbecome aribitrarily small. Following such embedding, acceleration vanishes and in-ertia cancels out in the equations of geodesic motion (ignoring self-gravity). Theorigin of inertia is hereby not addressed in Einstein’s EP or general relativity.To address inertia for its potential senstivity to a cosmological background,we take one step back and consider Rindler inertia in non-geodesics of Minkowskispacetime.Inertia is commonly defined by mass-at-infinity in an asymptotically flat space-time (Mach’s principle). The latter is an overly strong assumption in cosmologies(1), whose Cauchy surfaces are bounded by a cosmological horizon at Hubble radius R H = c/H . Any reference to large distance asymptotics is inevitably perturbed ifnot defined by the cosmological horizon. In particular, the apparent horizon h ofRindler spacetime at a distance ξ = c /α colludes with the latter at sufficientlysmall accelerations. Thus, h and H colludes at (4) with corresponding transitionradius r t = p R g R H , (12)giving (8). In what follows, we argue that it sets the onset to reduced inertia (7).In what follows, h and H refer to apparent horizons associated with radialaccelerations. For orbital motions, we appeal to Newton’s separation of particle eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 4 momenta p and associated forces d p /dt in radial and azimuthal components, wherethe former is imparted by the gravitational field of body of mass M b . (Locally, d p /dt is measured as curvature of particle trajectories relative to a tangent plane of null-geodesics.) The radial component of d p /dt hereby carries h as an apparent horizondefined by instantaneous radial acceleration, giving (8). In spiral galaxies, (12) isindeed a typical distance signaling the onset to anamalous galaxy dynamics. Origin of Rindler inertia in entanglement entropy
Unitarity in encoding particle positions by holographic screens satisfies P − + P + ≡ , (13)where, e.g., P ± refer to the probabilites of finding the proverbial cat and, respec-tively, out of a box, as defined by its quantum mechanical propagator. Satisfying(13) requires exact arithmetic , expressing the probability P + = 1 − P − ≃ P − is exponentially small. In a holographic approach, P + = 1 − P − is encoded in information on the box as acompact two-surface with P − ∼ e − ϕ (15)in terms of a Compton phase ∆ ϕ = kξ, (16)expressing the cat’s distance ξ to the walls by Compton wave number k = mc/ ~ fora mass m . Exact arithmetic on (13) requires an information I = 2 π ∆ ϕ. (17)Here, the factor of 2 π in (17) is associated with encoding m on a spherical screen.(For a single flat screen, I = 2∆ ϕ and for a cubic box I = 12∆ ϕ .)We next consider a Rindler observer O of mass m , i.e., a non-geodesic trajec-tory of constant acceleration in 1+1 Minkowski spacetime ( t, x ). The light cone atthe origin delineates an apparent horizon h , | ct | = x , to a Rindler observer withworldline ( t, x ) = (sinh( λτ ) , cosh λτ ), where λ = α/c . O ’s distance ξ to h is Lorentzinvariant, which can be attributed an Unruh temperature k B T = α ~ πc . (18)With h null, the entropy S , putting Boltzmann’s constant k B equal to 1, dS = − dI (19)gives rise to a thermodynamic potential dU = − T U dS = (cid:18) ~ α πc (cid:19) (cid:16) π mc ~ dξ (cid:17) . (20) eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 5 Conform EP, O identifies a uniform gravitational field that it may attribute tosome massive object well beyond h , featuring a divergent redshift towards h . Inthis gravitational field and relative to h , O assumes a potential energy U = Z ξ dU = m c . (21)By a thermodynamical origin to inertia, Rindler observers experience fluctuationstherein equivalently to momentum fluctations by detection of photons from a warmUnruh vacuum.Emperically, inertia is instantaneus with no associated time scale. Correspond-ingly, (21) is not an ordinary thermodynamic potential, but one arising from non-local entanglement entropy S associated with the apparent horizon h in (19).In the absence of any length scale in Minkowksi spacetime, (21) establishes aNewtonian identity between mass-energy and inertia. In three-flat cosmology, theresulting m = m will hold whenever α > a dS , ensuring that h falls within H (Fig.1). Fig. 1. In three-flat FRW universes with Hubble radius R H , the Rindler horizon h at distance ξ = c /α may fall inside (left) or outside the cosmological the cosmological horizon. Collusion at α = a dS of Rindler and cosmological horizon defines a sharp onset to weak gravity. When H fallswithin h , it interferes with the phase space of Rindler observers. Rindler inertia m then dropsbelow its Newtonian value set by rest mass energy m c . (Reprinted from .) C onset to weak gravity at a dS In (7), H drops inside h (Fig. 1), and the integral leading to (21) is cut-off at H ,leaving U smaller than m c . By a dS , EP is no longer scale free, i.e., inertia m measured by U (gravitational pull in an equivalent gravitational field) may deviatefrom the Newtonian value m . Reduced inertia m < m when h falls beyond H in(7) gives rise to enhanced acceleration at a given Newtonian gravitational forcing F N = m M b /r . (This is prior to a covariant embedding in geodesic motion in eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 6 curved spacetime, alluded to above.) Arising from crossing of the apparent Rinderand cosmological horizon surfaces, the onset to m < m is C sharp. Just suchbehavior is apparent in high resolution galaxy rotation data (Fig. 2). Fig. 2. High resolution data of galaxy rotation curves, here plotted as m/m = a N /α as afunction of a N /a dS , where α denotes the observed centripital acceleration and a N is the expectedNewtonian acceleration based on the observed baryonic matter. The results show a sharp onsetto weak gravity across ( a N , m ) = ( a dS , m ). (Reprinted from .) Fig. 2 shows binned rotation curve data from many spiral galaxies, after nor-malizing the independent variable to a N (rather than the coordinate radius r ). Plotting α/a N as a function of a N (or a N /a dS ), the results α > a N are often re-ferred to as the “missing mass problem.” In light of our focus on Rindler horizons eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 7 falling beyond the cosmological horizon, Fig. 2 shows mm = a N α . (22)Observationally, the right hand side of (22) is inferred from the ratio( V /r ) / ( V b /r ) = ( V /V b ) of observed ( V ) and anticipated ( V c ) circular velocities,the latter based on luminous matter. Hence, E k E k, = mV m V b = mαm a N = 1 , (23)showing that orbital kinetic energies are unchanged.In weak gravity (7), holographic representations of a particle of mass m involvestwo low energy dispersion relations of image modes in 3+1 spacetime and of Plancksized surface elements on the cosmological horizon, satisfying ~ ω = p ~ Λ + c p ( r < R H ) , T = q T U + T H ( r = R H ) , (24)where k B T H = ~ a H / (2 πc ) denotes the cosmological horizon temperature at internalsurface gravity a H = (1 / − q ) H . At equal mode counts, we have B ( p ) mm = R H ξ (25)with the momentum dependent ratio B ( p ) = p ~ Λ + c p k B p T U + T H . (26)Rindler’s relation ξ = c /α hereby obtains α = p B ( p ) a dS a N , (27)where a N refers to the Newtonian acceleration a N = M b /r . An effective descriptionof weak gravity (7) now follows, upon taking a momentum average h B ( p ) i of (26)in (27). Averaging over a thermal distribution obtains the green curve in Fig. 2.In the asymptotic regime, a N << a dS , (27) reduces to Milgrom’s law α = √ a a N (28)with Milgrom’s parameter (9) directly asociated with the background cosmology asanticipated based on dimensional analysis in geometrical units. In light of the q -gradients Q in (11), A = A (0), A ( z ) = a − da ( z ) /dz , satisfies A ≃ − . , A , Λ CDM > . (29)This discrepant outcome of (6) and ΛCDM may be tested observationally in surveysof rotation curves covering a finite redshift range. eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 8
3. Accelerated expansion from cosmologial holography
The holographic principle proposes a reduced phase space of spacetime, matterand fields to that of a bounding surface in Planck sized degrees of freedom.
As the latter is generally astronomically large in number, and hence excited atcommensurably low energies. In weak gravity, these low energies readily reach alow energy scale of the cosmological vacuum, set by the cosmological horizon atHubble radius R H = c/H based on the unit of luminosity L = c /G . As a null-sphere, the cosmological horizon features closed null-geodesics. Thegeodesic deviation of a pair of null-geodesics satisfies a harmonic oscillator equa-tion . By explicit calculation of the Riemann tensor in a three-flat FRW universewith deceleration parameter q , the angular frequency satisfies (5). A holographicextension to spacetime within obtains a wave equation with dispersion relation ω = √ k + Λ, where k refers to the wave number of modes orthogonal to the cos-mological horizon and Λ in (6). Dark energy is hereby described in terms of thecanonical cosmological parameters ( H, q ), allowing a detailed comparison of cosmicevolution with (6) versus ΛCDM.
Evolution by
Λ = ω and Λ CDM
FRW universes with Λ in (6) leave baryon nucleosynthesis invariant, since q = 1 ina radiated dominated epoch. For most of the subsequent evolution of the universe,this dark energy is relatively small compared to matter content. In recent epochs,however, the derivative Q ( z ) = dq ( z ) dz (30)varies rather strongly compared to what is expected in ΛCDM, expressed in (11)and shown in Fig. 3.Cosmological evolution in the approximation of (1) is described by an ordinarydifferential equation for the scale factor a ( t ). For (6), this is described by an ordinarydifferential equation (ODE) as a function for y = log h , h = H/H , as a function of z , y ′ ( z ) = 3(1 + z ) ω m e − y − (1 + z ) − (31)derived from ¨ a ( τ ) = a (cid:0) h − ω m a − (cid:1) ( a (0) = 1, h (0) = 1), d/dt = − (1 + z ) Hd/dz , as a function of time τ = tH . For ΛCDM, we have h ( z ) = p (1 − ω m ) + ω m (1 + z ) for ΛCDM. Fig. 2 shows illustrative numerical solutionsfor various values ω m of dark matter content at z = 0. At late times, the dynamicaldark energy (6) features a fast expansion compared to ΛCDM. In particular, q and Q defined in (10-11) are markedly distinct in these two models. Heterogeneous data on H ( z ) Measurements of the Hubble parameter H ( z ) by various methods of observationsnow extends over an increasingly large redshift range. For recent compilations, see, eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 9 -0.5 0 0.5 1 1.5 2 z -101234 Q ( z ) gap ω m = 0.22 ω m = 0.3 ω m = 0.38 ω m ( Λ CDM) = 0.22 ω m ( Λ CDM) = 0.3 ω m ( Λ CDM) = 0.38 -0.5 0 0.5 1 1.5 2 z -2-1.5-1-0.500.51 q ( z ) -0.5 0 0.5 1 1.5 2 z h ( z ) Fig. 3. Evolution of the deceleration parameter q ( z ) with redshift in a three-flat FRW universewith dark energy Λ = ω and ΛCDM for various values q = q (0). They show pronounceddifferences in slope Q at z = 0 associated with different curvatures of h ( z ) = H ( z ) /H .Table 1. Binned data { z k , H ( z k ) } ( k = 1 , , · · · on the Hub-ble parameter H ( z ) (cid:2) km s − Mpc − (cid:3) over an extended range of red-shiftz z , and inferred estimates of H ′ ( z k ′ ) and q ( z k ′ ) at midpoints z k ′ = ( z k + z k − ) / k = 2 , , · · · , Q ( z k ) ( k = 2 , , · · · , k redshift z k , z k ′ H ( z k ) σ k H ′ ( z k ′ ) q ( z k ′ ) Q ( z k )1 0.166 75.7 3.350.2605 78.20 - 26.46 -0.57362 0.355 80.7 1.70 - - 1.97580.3910 82.65 - 54.17 -0.08843 0.427 84.6 4.80 - - 0.26590.4725 87.35 - 60.44 -0.01894 0.518 90.1 1.75 - - 0.14920.5755 93.90 - 66.09 0.10085 0.633 97.7 1.90 - - 0.09131.0015 127.85 - 81.82 0.28096 1.37 158 8.50 - - -0.14701.60 177 - 82.61 0.21357 1.83 196 31.0 - - -0.23282.09 210 - 53.85 -0.20778 2.35 224 5.0 e.g., Sola et al. covering 0 < z < .
936 and Farooq et al. covering 0 < z ≤ . eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 10 Table 1 shows N = 8 binned data on H ( z ) with a mean of normalized standarddeviation ˆ σ k = σ k /H ( z k ) satisfying ( N N X k =1 ˆ σ − k ) − ≃ . (32)Included are the estimates of q ( z ) obtained from H ′ ( z ) by central differencing.As heterogeneous data sets, these compilations require tests against physicalconstraints before they can be used in regression analysis. Different (often unknown)systematics can potentially create trend anomalies that violate essential priors. Ifso, the data set contains incompatibilities .We recall that the Universe entered an essentially matter dominated epoch when z appreciably exceeds the transition redshift z t defined by the vanishing of thedeceleration paramer q ( z ), when the Hubble flow of galaxies passing through thecosmological horizon changed sign. The constraint z t < .For z >
1, therefore, the positivity conditions H ( z ) > , q ( z ) = − z ) H − ( z ) H ′ ( z ) > essential physical priors to any data set.Table 1 points to a violation of (33) with q ( z ) = − . z = 2 .
09. Associated with a drop in H ′ ( z ), the last data point at k = 8 appearsto be incompatible with k = 1 , , · · · , Detecting Q in H ( z ) data Table 2 lists results of nonlinear model regression by our two cosmological models tothe Hubble data of Table 1, parameterized by ( H , ω m ) with the MatLab function fitnlm using weights w k ∝ σ − k . Fig. 4 shows the fits and 95% prediction limits(pl, very close but slightly wider than 95% confidence limits).Fig. 4 shows the match of evolution with (6) including the normalized errors,further for ΛCDM and a model-independent cubic fit.For (6) these errors are essentially the same as obtained by a model-independentcubic fit H ( z ) = H (cid:18) q ) z + 12 ( Q + (1 + q ) q ) z + b z (cid:19) + O (cid:0) z (cid:1) (34)with free coefficients ( H , q , Q , b ), and they are about one-half the normalizederrors in the fit by ΛCDM.For the two models, q and Q obtain according to their repective evolutionequations, i.e., Ω M = (1 / q ) for (6), (10) for ΛCDM, and Q = (2 + q )(1 − q ) > . ω ) , (1 + q )(1 − q ) . , (35) eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 11 z H ( z ) (C) Data(H , ω m ) = (74.02,0.2981)95% pl z -1.5-1-0.500.511.5 N o r m a li z ed r e s i dua l s STD = 0.18189 z H ( z ) (A) Data(H , ω m ) = (74.11,0.2821)95% cl z -1.5-1-0.500.511.5 N o r m a li z ed r e s i dua l s STD = 0.24287 z H ( z ) (B) Data(H , ω m ) = (65.71,0.3594)95% pl z -1.5-1-0.500.511.5 N o r m a li z ed r e s i dua l s STD = 0.50327 Fig. 4. Nonlinear regression model fits (continuous line) to the binned data (circles) of Table 1( k = 1 , , · · · ,
7) by the MatLab function fitnlm to (A) Λ = ω , (B) ΛCDM, (C) a cubic polynomial.Included are the 95% prediction limits (pl, dashed lines; very similar to the 95% confidence limits,not shown). Included is the incompatible data point k = 8 (red) in Table 1. whereas Q in our cubic fit (34) is determined directly by nonlinear regression.Included in Fig. 4) is the k = 8 data point in Table 1. As expected, it isinconsistent with (6) and only marginally consistent with ΛCDM. Table 2. Results of nonlinear model regression on the binned compatible data { z k , H ( z k ) } ( k = 1 , , · · ·
7) of Table 1.model H (cid:2) km s − Mpc − (cid:3) ω m q Q Λ = ω . ± . . ± . − . ± . . ± . . ± . . ± . − . ± . . ± . . ± . . ± . − . ± . . ± . In Table 2, Q = 2 . ± . . σ . eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 12
4. Sensitivity to H ( z ) in high redshift galaxy rotation curves By continuity in the onset to weak gravity at α = a dS (8), we have y ,h = (cid:18) a N a dS (cid:19) h = (cid:18) r t R h (cid:19) (36)whereby (27) takes the form αa dS = √ µ r t R h (37)with µ = 2 h B ( p ) i . In weak gravity α . a dS , anomalous behavior in galacticdynamics may be expressed by an apparent equivalent dark matter fraction f ′ DM = α − a N α , (38)conform the definition of f DM in .Table 3 lists data on high redshift sample of rotation curves and associateddata on (28-38). Fig. 5 shows the apparent ( f DM ) and predicted ( f ′ DM ) darkmatter fractions. Based on r t /R H , this sample of galaxies probes weak gravity (27)in α < a dS but not the asymptotic regime α << a dS , a point recently empasizedby . Table 3.
Analysis on apparent ( f ′ DM ) versus observed ( fDM ) dark matter fractions in high z rotation curves 24with baryonic mass Mb = M M ⊙ and rotation velocities Vc in units of km s − Rh .Galaxy z H ( z ) /H q ( z ) M rt/Rh µ f ′ DM Vc fDM (95% c.l.)COS4 01351 0.854 1.5986 0.0853 1.7 0.6740 0.7050 0.2031 276 0.21 ± . < . ± . < . < . < . Fig. 5 shows quantitative agreement of f DM and f ′ DM , except for cZ 4006690( V c = 301 km s − ). We note that its observed rotation curve is asymmetric, whichmay indicate systematic errors unique to this galaxy. (The other galaxies all haveessentially symmetric rotation curves.) Apart from this particular galaxy, f ′ DM ≃ f DM in over a broad range of redshifts confirms that (8), and hence weak gravityin galaxy dynamics is co-evolving with H ( z ) in the background cosmology.
5. Conclusions and outlook
By volume, structure formation and cosmology represent the most common gravi-tational interactions at the scale of a dS or less. Relatively strong interactions in oursolar system and its general relativistic extension to compact objects are, in fact,the exception. A principle distinction between the two is sensitivity to a dS in weakgravitation, otherwise ignorable in the second. This potential is completely absentin Einstein’s EP and classical formulations of general relativity, but becomes a realpossibility by causality and compactness of the cosmological horizon. eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 13 -3 -2 -1 a N /a dS α a d S -
260 280 300 320 340 360 V c [km s -1 ] -0.4-0.200.20.40.60.81 A ppa r en t f D M ′ COS4 01351 D3a 6397GS4 43501 zC 406690 zC 400569D3a 15504
Fig. 5. (Upper panel.) Apparent dark matter fractions f ′ DM (blue squares) and observationaldata on f DM . Results agree except for cZ 406690 ( V c = 301 km/s). (Lower panel.) Locationof the data in Table 3 in weak gravity a N < a dS and associated predicted accelerations α/a dS .(Reprinted from .) Inspired by spacetime holography, weak gravity in galactic dynamics and cosmo-logical evolution is parameterized by a dS and the fundamental requency ω of thecosmological horizon, set by the canonical parameters ( H, q ) in (1). Observationalconsequences are (i) a perturbed inertia at small accelerations ( α < a dS ) acrossa sharp onset at a N = a dS with asymptic behavior ( a N << a dS ) described by aMilgrom’s parameter a = ω / π ; and (ii) Q > . ω .Our Λ = ω is completely independent of the bare cosmological constant Λ aris-ing from vacuum fluctuations in quantum field theory. Complementary to unitaryholography based on particle propagators, vacuum flucations with no entanglementhave a trivial Poincare invariant propagator. By translation invariance, vacuum eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 14 fluctuatons do not drag spacetime like matter does, whereby Λ has no inertia andcarries no gravitational field conform Mach’s principle.Predictions (i-ii) are supported by the results of Fig. 1 and Table 2: • Rotation curve data point to a C onset to weak gravity in Fig. 1 at (cid:18) a N a dS , αa N (cid:19) = (1 , . (39)We attribute this behavior to inertia originating in entanglement entropy,of the apparent Rindler or cosmological horizon, whichever is more nearby.At small accelerations, inertia is reduced when h falls beyond H , enhancingacceleration for a given gravitational forcing. By (8), (9) and (29), weakgravity is sensitive to background cosmology, manifest in galaxy rotationcurves over an extended redshift range listed in Table 3 and shown in Fig.5. • A model-independent analysis of the latter by a cubic polynonial identifies Q ≃ . ± .
29. ΛCDM with Q . . σ .Normalized errors in the cubic fit and those to Λ = ω are about one-halfof the normalized errors in the fit to ΛCDM. • Estimates of H in the first two are consistent with recent measurement H = 73 . ± .
74 km s − Mpc − in local surveys . Combining our resultfor Λ = ω with the latter obtains relatively fast expansion compared toΛCDM with H = 73 . ± . − Mpc . (40)The above suggests a strong form of the EP, by inisting on equivalent fluctuationsin inertia of Rindler observers (non-geodesics in Minkowski spacetime) with fluc-tuations in measuring weight on a scale (non-geodesics in curved spacetime). Mo-mentum fluctuations between two bodies in mutual gravitational attraction hereinare fully entangled, ensuring preservation of total momentum. If so, curvature frommass must follow very similar rules giving rise to (21). Consequently, any scenariofor entropic gravity should include the origin of inertia in entanglement entropy,the details of which remain to be spelled out.Recently, the LISA Pathfinder has conducted an extensive set of a high precisionfree fall gravity and inertia experiments at accelerations well below a dS . It useslaser-interferometry and electrostatic forcing to track and perturb test masses abouttheir geodesic motion, in the spacecraft’s self-gravitational field. In probing the deSitter scale of 1 ˚A s − or less, we note that (8) conventiently rescales to the size ofa small spacecraft, i.e., r t ≃
30 cm m / (41)about a gravitating mass m = m kg, representative for the 2 kg test masses in theLISA Pathfinder mission. It seems worthwhile to look at data covering α < a N . eptember 26, 2018 2:37 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ms page 15 Acknowledgements.
The author thanks stimulating discussions with S.S. Mc-Gaugh, J. Binney, K. Danzmann, T. Piran, G. Smoot, C. Rubbia, J. Sol`a, L. Smolinand M. Milgrom. This report was supported in part by the National Research Foun-dation of Korea under grant No. 2015R1D1A1A01059793 and 2016R1A5A1013277.