IS DARK MATTER FACT OR FANTASY – CLUES FROM THE DATA
Philip D. Mannheim
University of Connecticut
Seminar at Miami 2018, Fort Lauderdale
December 2018
1 THE HUNDRED YEAR DARK MATTER PROBLEM
Almost as soon as it was realized the Milky Way was a galaxy, in the 1920s Oort found that the velocities of stars perpendicular to the plane of the galaxy had a missing mass problem.
Almost as soon as it was realized that there were other galaxies, in the 1930s Zwicky and Smith found that the velocity dispersions of galaxies in a cluster of galaxies had a missing mass problem. Three possibilities: more galaxies than one could see, lack of virialization because of corre- lation energy, breakdown of Newton’s Law of Gravity on large distance scales.
In the 1970s HI radio studies of Freeman and of Roberts and Whitehurst and HII opti- cal studies of Rubin, Ford and Thonnard showed the measured rotational velocities in the outskirts of spiral galaxies greatly exceeded the luminous Newtonian expectations.
Then the dam broke, with missing mass problems all the way up to cosmology. And a brand new problem to boot – dark energy.
2 Two possible astrophysical options for dark matter: faint (need better tele- scopes – now ruled out). Dead (black holes, white dwarfs – MACHOS) Ruled out by gravitational lensing off the Magellanic clouds.
One particle physics option for dark matter: particles that are intrinsically unable to emit light (neutrinos, supersymmetric particles, axions, dark pho- tons – WIMPS). Extensive, decades long accelerator and underground searches have yet to find any Wimps. Not ruled out but far from being ruled in, though there is a tension for supersymmetry since no superparticles found at LHC that could solve the elementary Higgs boson hierarchy problem.
3 Additionally no solution to the dark energy/cosmological constant problem found either in the twenty years now since the discovery of the accelerating Universe. Though problem predates the accelerating Universe. 60 And if ΩΛ really is of order 10 then none of the CMB background or fluctuation tests would be successful. In fact they would actually be some of the worst if not the worst fits in the history of physics.
And lurking behind all this is the hope that the standard Newton-Einstein classical gravity expectations are not destroyed by quantum mechanics. (That gravity knows about large scale quantum effects is evidenced by the stabilizing Pauli degeneracy of electrons in a white dwarf and the intrinsically quantum mechanical CMB black body radiation spectrum.)
To see what to do look for clues from the data – focus on galactic rotation curves of spirals since no theory needed – just orbits. Start with solar system wisdom – more sources (Neptune) or new theory (Mercury).
4 Keplerian expectation for planetary orbital velocities – Mercury and Uranus problems
5 Face on view of a spiral galaxy
6 Edge on view of a spiral galaxy
7 70
60
50
40
30
20
10 DDO 154 0 0 2 4 6 8 NGC 3198 and DDO 154 rotation velocities from HI data 8 TO FIX THINGS UP ASSUME DARK MATTER
Need dark matter where there is little luminous matter. Not a small effect at all. Dark to luminous mass ratio in a bright spiral (NGC 3198) is of order two to one, and thus four to one in potential. Dark to luminous mass ratio in a dwarf spiral (DDO 154) is of order three to one, and thus nine to one in potential. Dark to luminous ratio not universal.
Bright spiral rotation curves are flat, dwarf spiral rotation curves are rising. Thus the case where dark matter dominates is the one where rotation curves are not flat but rising, with dark matter being needed in the inner region. Dwarfs thus show the dark matter problem in its starkest form.
To make dark matter halo fits work for bright spirals need two free parameters per halo (e.g. Navarro-Frenk-White NFW dark matter theory profile). One parameter to match asymptotic value of velocity to inner region and second parameter to keep velocity constant in between. Also need two parameters per halo for dwarfs. Thus for 138 well-studied galaxies discussed below (need good HII to get luminous distribution and good HI to get velocity distribution) need 276 free dark matter halo parameters.
9 DIFFICULTIES FOR DARK MATTER THEORY For the moment dark matter halo parameters are not derived from dark matter theory, and basically parameterize the data – you show me the velocity and I will tell you the amount of dark matter. Objective: to make dark matter falsifiable require: YOU SHOW ME THE LUMINOUS DISTRIBUTION AND I WILL TELL YOU THE VELOCITY
Thus dark matter theory does not know a priori how to match any given halo with any given luminous distribution. WHY – BECAUSE GALAXIES OBEY TULLY- FISHER RELATION.
Find that v4/L is universal in spiral galaxies. Thus velocity (due to the full gravitational potential) is correlated with the luminous distribution. Thus to get Tully Fisher would need dark to luminous ratio to be universal, but it is not, so need to tune halo parameters galaxy by galaxy.
So let us look for other regularities, to see what we can pull out of the 138 galaxy sample. Find that v2/c2R at last data point is universal, and not only that, it is of cosmological magnitude.
10 70 100 140 150 60 120 80 50 100 100 60 40 80
30 60 40 50 20 40 20 10 20 DDO 154 IC 2574 NGC 925 NGC 2403 0 0 0 0 0 2 4 6 8 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 5 10 15 20
100 300 250 200
250 80 200 150 200 60 150
150 100 40 100 100 50 20 50 50 NGC 2841 NGC 2903 NGC 2976 NGC 3031 0 0 0 0 0 10 20 30 40 50 0 5 10 15 20 25 30 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 12 14
250
200 150 150 200
150 150 100 100
100 100
50 50 50 50
NGC 3198 NGC 3521 NGC 3621 NGC 3627 0 0 0 0 0 10 20 30 0 5 10 15 20 25 30 35 0 5 10 15 20 25 0 2 4 6 8 200 250 200 200 150 200 150 150 150 100 100 100 100
50 50 50 50
NGC 4736 NGC 4826 NGC 5055 NGC 6946 0 0 0 0 0 2 4 6 8 10 0 5 10 15 0 10 20 30 40 0 5 10 15 20
140
250 120
200 100
80 150 60 100 40
50 20 NGC 7331 NGC 7793 0 0 0 5 10 15 20 0 2 4 6 8 10 FIG. 1: Fitting to the rotational velocities (in km sec−1) of the THINGS 18 galaxy sample
11 140 200 120 150
150 100
80 100 100 60
40 50 50 20 NGC 3726 NGC 3769 NGC 3877 0 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 2 4 6 8 10
200 150
150 150 100 100 100
50 50 50
NGC 3893 NGC 3917 NGC 3949 0 0 0 0 5 10 15 20 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7
250 140 250 200 120 200 100 150 80 150
100 60 100 40 50 50 20 NGC 3953 NGC 3972 NGC 3992 0 0 0 0 5 10 15 0 2 4 6 8 0 10 20 30 40 50
140 200 120 150
100 150
80 100
60 100
40 50 50 20 NGC 4010 NGC 4013 NGC 4051 0 0 0 0 2 4 6 8 10 0 5 10 15 20 25 30 0 2 4 6 8 10
200 150 200
150 150 100
100 100
50 50 50
NGC 4085 NGC 4088 NGC 4100 0 0 0 0 1 2 3 4 5 6 0 5 10 15 0 5 10 15 20 25 FIG. 2: Fitting to the rotational velocities of the Ursa Major 30 galaxy sample – Part 1
12 140 200 200 120
150 150 100
80
100 100 60
40 50 50 20 NGC 4138 NGC 4157 NGC 4183 0 0 0 0 5 10 15 0 5 10 15 20 25 30 0 5 10 15 200 120 100
100 150 80 80 60 100 60 40 40 50 20 20 NGC 4217 NGC 4389 UGC 6399 0 0 0 0 5 10 15 0 1 2 3 4 0 2 4 6 8
100 100 80
80 80 60 60 60 40 40 40
20 20 20
UGC 6446 UGC 6667 UGC 6818 0 0 0 0 2 4 6 8 10 12 0 2 4 6 8 0 2 4 6 8
100 120 120
100 80 100
80 80 60
60 60 40 40 40
20 20 20 UGC 6917 UGC 6923 UGC 6930 0 0 0 0 2 4 6 8 10 0 1 2 3 4 5 0 5 10 15
100 120 200 100 80
150 80 60
60 100 40 40
50 20 20 UGC 6973 UGC 6983 UGC 7089 0 0 0 0 2 4 6 8 10 0 5 10 15 0 1 2 3 4 5 6 7 FIG. 3: Fitting to the rotational velocities of the Ursa Major 30 galaxy sample – Part 2
13 120 80 120 120
100 100 100 60 80 80 80
60 60 40 60
40 40 40 20 20 20 20 DDO 64 F 563-1 F 563-V2 F 568-3 0 0 0 0 0.0 0.5 1.0 1.5 2.0 0 5 10 15 0 1 2 3 4 5 6 0 2 4 6 8 10 50 100 100 80 40 80 80 60 60 30 60 40 40 20 40
20 20 20 10
F 583-1 F 583-4 NGC 959 NGC 4395 0 0 0 0 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8
120 140 120 80 120 100 100
60 100 80 80 80 60 40 60 60 40 40 40 20 20 20 20 NGC 7137 UGC 128 UGC 191 UGC 477 0 0 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 120 50 100 120 100 40 80 100 80 30 60 80 60 60 20 40 40 40
10 20 20 20 UGC 1230 UGC 1281 UGC 1551 UGC 4325 0 0 0 0 0 5 10 15 20 25 30 35 0.0 0.5 1.0 1.5 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0
120 120 80 100 150 100
80 60 80 100 60 60 40
40 40 50 20 20 20 UGC 5005 UGC 5750 UGC 5999 UGC 11820 0 0 0 0 0 5 10 15 20 25 0 2 4 6 8 0 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 FIG. 4: Fitting to the rotational velocities of the LSB 20 galaxy sample
14 50 60
80 300 50 40 250 60 40 200 30 30 150 40 20 20 100 20 10 50 10 ESO 140040 ESO 840411 ESO 1200211 ESO 1870510 0 0 0 0 0 5 10 15 20 25 30 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 70 140 100 60 120 150
80 50 100
40 80 60 100
60 30 40 40 20 50 20 20 10 ESO 2060140 ESO 3020120 ESO 3050090 ESO 4250180 0 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 1 2 3 4 5 0 2 4 6 8 10 12 14
120 150 140 150 100 120
100 80 100 100 80 60 60 40 50 50 40
20 20 ESO 4880490 F 571 - 08 F 579 -V1 F 730- V1 0 0 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 60 250 100 150 50 200 80 40 150 100 60 30
100 40 20 50
10 50 20 UGC 4115 UGC 6614 UGC 11454 UGC 11557 0 0 0 0 0.0 0.5 1.0 1.5 0 10 20 30 40 50 60 0 2 4 6 8 10 12 0 1 2 3 4 5 6 60 150 150 250 50
200 40 100 100 150 30
100 20 50 50
10 50 UGC 11583 UGC 11616 UGC 11648 UGC 11748 0 0 0 0 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 0 2 4 6 8 10 12 0 5 10 15 20
150
100
50
UGC 11819 0 0 2 4 6 8 10 12 FIG. 5: Fitting to the rotational velocities15 of the LSB 21 galaxy sample 80 80 120 100
100 60 80 60 80 60 40 40 60 40 40 20 20 20 20 DDO 168 DDO 170 M 33 NGC 55 0 0 0 0 0 1 2 3 4 0 2 4 6 8 10 12 0 2 4 6 8 0 2 4 6 8 10 12 120 140 120 250 100 120 100 200 100 80 80 80 150 60 60 60 100 40 40 40
50 20 20 20 NGC 247 NGC 300 NGC 801 NGC 1003 0 0 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 0 10 20 30 40 0 5 10 15 20 25 30 100 250 250 80 80 200 200 60 60 150 150 40 40 100 100
20 20 50 50
NGC 1560 NGC 2683 NGC 2998 NGC 3109 0 0 0 0 0 2 4 6 8 10 0 5 10 15 20 25 30 35 0 10 20 30 40 0 1 2 3 4 5 6 7
300 350 100 250 250 300 80 200 250 200 60 150 200 150 150 40 100 100 100
50 50 20 50 NGC 5033 NGC 5371 NGC 5533 NGC 5585 0 0 0 0 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 50 0 2 4 6 8 10 12 14
300 350 140 100
250 120 300 80 250 200 100
80 200 60 150 60 150 40 100 40 100
50 20 20 50 NGC 5907 NGC 6503 NGC 6674 UGC 2259 0 0 0 0 0 10 20 30 40 0 5 10 15 20 0 10 20 30 40 50 60 0 2 4 6
350 300 300 250 250 200 200 150 150
100 100
50 50 UGC 2885 Malin 1 0 0 0 10 20 30 40 50 60 70 0 20 40 60 80 100 FIG. 6: Fitting to the rotational velocities16 of the Miscellaneous 22 galaxy sample 140 120 80 120 100
100 80 60 80 60 40 60 40 40 20 20 20 F 568-V1 F 574-1 UGC 731 0 0 0 0 5 10 15 0 2 4 6 8 10 12 14 0 2 4 6 8 10 120 100 80 100 80 60 80
60 60 40 40 40
20 20 20 UGC 3371 UGC 4173 UGC 4325 0 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7
80 100 80
80 60 60 60
40 40 40
20 20 20
UGC 4499 UGC 5414 UGC 5721 0 0 0 0 2 4 6 8 0 1 2 3 4 0 2 4 6 8
100 60 100 50 80 80 40 60 60 30 40 40 20
20 10 20 UGC 5750 UGC 7232 UGC 7323 0 0 0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5
FIG. 7: Fitting to the rotational velocities (in km17 sec−1) of the 24 dwarf galaxy sample – Part 1 100 50 120
100 80 40
80 60 30
60 40 20 40
20 10 20 UGC 7399 UGC 7524 UGC 7559 0 0 0 0 5 10 15 20 25 30 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 40 80 100
80 30 60
60
20 40 40
10 20 20
UGC 7577 UGC 7603 UGC 8490 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 0 2 4 6 8 10
120 80 100 150
60 80 100 60 40
40 50 20 20 UGC 9211 UGC 11707 UGC 11861 0 0 0 0 2 4 6 8 0 5 10 15 20 0 2 4 6 8 10 12
100 120 80 100 80
60 80 60 60 40 40 40
20 20 20 UGC 12060 UGC 12632 UGC 12732 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 12 14
FIG. 8: Fitting to the rotational velocities18 of the 24 dwarf galaxy sample – Part 2 60 50 80 50 40
60 40 30 30 40 20 20
20 10 10 UGC 3851 UGC 4305 UGC 4459 0 0 0 0 2 4 6 8 10 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 60 60 100
50 50 80
40 40 60
30 30 40 20 20
20 10 10 UGC 5139 UGC 5423 UGC 5666 0 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10
FIG. 9 Fitting to the rotational velocities of the 6 dwarf galaxy sample
19 Properties of the THINGS 18 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0154 LSB 4.2 0.007 0.8 8.1 0.03 0.003 0.45 1.12 [?][?][?][?] IC 2574 LSB 4.5 0.345 4.2 13.1 0.19 0.098 0.28 1.69 [?][?][?][?] NGC 0925 LSB 8.7 1.444 3.9 12.4 0.41 1.372 0.95 4.17 [?][?][?][?] NGC 2403 HSB 4.3 1.647 2.7 23.9 0.46 2.370 1.44 2.89 [?][?][?][?] NGC 2841 HSB 14.1 4.742 3.5 51.6 0.86 19.552 4.12 5.83 [?][?][?][?] NGC 2903 HSB 9.4 4.088 3.0 30.9 0.49 7.155 1.75 3.75 [?][?][?][?] NGC 2976 LSB 3.6 0.201 1.2 2.6 0.01 0.322 1.60 10.43 [?][?][?][?] NGC 3031 HSB 3.7 3.187 2.6 15.0 0.38 8.662 2.72 9.31 [?][?][?][?] NGC 3198 HSB 14.1 3.241 4.0 38.6 1.06 3.644 1.12 2.09 [?][?][?][?] NGC 3521 HSB 12.2 4.769 3.3 35.3 1.03 9.245 1.94 4.21 [?][?][?][?] NGC 3621 HSB 7.4 2.048 2.9 28.7 0.89 2.891 1.41 3.18 [?][?][?][?] NGC 3627 HSB 10.2 3.700 3.1 8.2 0.10 6.622 1.79 15.64 [?][?][?][?] NGC 4736 HSB 5.0 1.460 2.1 10.3 0.05 1.630 1.60 4.66 [?][?][?][?] NGC 4826 HSB 5.4 1.441 2.6 15.8 0.03 3.640 2.53 5.46 [?][?][?][?] NGC 5055 HSB 9.2 3.622 2.9 44.4 0.76 6.035 1.87 2.36 [?][?][?][?] NGC 6946 HSB 6.9 3.732 2.9 22.4 0.57 6.272 1.68 6.39 [?][?][?][?] NGC 7331 HSB 14.2 6.773 3.2 24.4 0.85 12.086 1.78 9.61 [?][?][?][?] NGC 7793 HSB 5.2 0.910 1.7 10.3 0.16 0.793 0.87 3.61 [?][?][?][?]
20 Properties of the Ursa Major 30 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI NGC 3726 HSB 17.4 3.340 3.2 31.5 0.60 3.82 1.15 3.19 [?][?][?][?] NGC 3769 HSB 15.5 0.684 1.5 32.2 0.41 1.36 1.99 1.43 [?][?][?][?] NGC 3877 HSB 15.5 1.948 2.4 9.8 0.11 3.44 1.76 10.51 [?][?][?][?] NGC 3893 HSB 18.1 2.928 2.4 20.5 0.59 5.00 1.71 3.85 [?][?][?][?] NGC 3917 LSB 16.9 1.334 2.8 13.9 0.17 2.23 1.67 4.85 [?][?][?][?] NGC 3949 HSB 18.4 2.327 1.7 7.2 0.35 2.37 1.02 14.23 [?][?][?][?] NGC 3953 HSB 18.7 4.236 3.9 16.3 0.31 9.79 2.31 10.20 [?][?][?][?] NGC 3972 HSB 18.6 0.978 2.0 9.0 0.13 1.49 1.53 7.18 [?][?][?][?] NGC 3992 HSB 25.6 8.456 5.7 49.6 1.94 13.94 1.65 4.08 [?][?][?][?] NGC 4010 LSB 18.4 0.883 3.4 10.6 0.29 2.03 2.30 5.03 [?][?][?][?] NGC 4013 HSB 18.6 2.088 2.1 33.1 0.32 5.58 2.67 3.14 [?][?][?][?] NGC 4051 HSB 14.6 2.281 2.3 9.9 0.18 3.17 1.39 8.52 [?][?][?][?] NGC 4085 HSB 19.0 1.212 1.6 6.5 0.15 1.34 1.11 10.21 [?][?][?][?] NGC 4088 HSB 15.8 2.957 2.8 18.8 0.64 4.67 1.58 5.79 [?][?][?][?] NGC 4100 HSB 21.4 3.388 2.9 27.1 0.44 5.74 1.69 3.35 [?][?][?][?] NGC 4138 LSB 15.6 0.827 1.2 16.1 0.11 2.97 3.59 5.04 [?][?][?][?] NGC 4157 HSB 18.7 2.901 2.6 30.9 0.88 5.83 2.01 3.99 [?][?][?][?] NGC 4183 HSB 16.7 1.042 2.9 19.5 0.30 1.43 1.38 2.36 [?][?][?][?] NGC 4217 HSB 19.6 3.031 3.1 18.2 0.30 5.53 1.83 6.28 [?][?][?][?] NGC 4389 HSB 15.5 0.610 1.2 4.6 0.04 0.42 0.68 9.49 [?][?][?][?] UGC 6399 LSB 18.7 0.291 2.4 8.1 0.07 0.59 2.04 3.42 [?][?][?][?] UGC 6446 LSB 15.9 0.263 1.9 13.6 0.24 0.36 1.36 1.70 [?][?][?][?] UGC 6667 LSB 19.8 0.422 3.1 8.6 0.10 0.71 1.67 3.09 [?][?][?][?] UGC 6818 LSB 21.7 0.352 2.1 8.4 0.16 0.11 0.33 2.35 [?][?][?][?] UGC 6917 LSB 18.9 0.563 2.9 10.9 0.22 1.24 2.20 4.05 [?][?][?][?] UGC 6923 LSB 18.0 0.297 1.5 5.3 0.08 0.35 1.18 4.43 [?][?][?][?] UGC 6930 LSB 17.0 0.601 2.2 15.7 0.29 1.02 1.69 2.68 [?][?][?][?] UGC 6973 HSB 25.3 1.647 2.2 11.0 0.35 3.99 2.42 10.58 [?][?][?][?] UGC 6983 LSB 20.2 0.577 2.9 17.6 0.37 1.28 2.22 2.43 [?][?][?][?] UGC 7089 LSB 13.9 0.352 2.3 7.1 0.07 0.35 0.98 3.18 [?][?][?][?]
21 Properties of the LSB 20 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0064 LSB 6.8 0.015 1.3 2.1 0.02 0.04 2.87 6.05 [?][?][?][?] F563-1 LSB 46.8 0.140 2.9 18.2 0.29 1.35 9.65 2.44 [?][?][?][?] F563-V2 LSB 57.8 0.266 2.0 6.3 0.20 0.60 2.26 6.15 [?][?][?][?] F568-3 LSB 80.0 0.351 4.2 11.6 0.30 1.20 3.43 3.16 [?][?][?][?] F583-1 LSB 32.4 0.064 1.6 14.1 0.18 0.15 2.32 1.92 [?][?][?][?] F583-4 LSB 50.8 0.096 2.8 7.0 0.06 0.31 3.25 2.52 [?][?][?][?] NGC 0959 LSB 13.5 0.333 1.3 2.9 0.05 0.37 1.11 7.43 [?][?][?][?] NGC 4395 LSB 4.1 0.374 2.7 0.9 0.13 0.83 2.21 2.29 [?][?][?][?] NGC 7137 LSB 25.0 0.959 1.7 3.6 0.10 0.27 0.28 3.91 [?][?] ES [?] UGC 0128 LSB 64.6 0.597 6.9 54.8 0.73 2.75 4.60 1.03 [?][?][?][?] UGC 0191 LSB 15.9 0.129 1.7 2.2 0.26 0.49 3.81 15.48 [?][?][?][?] UGC 0477 LSB 35.8 0.871 3.5 10.2 1.02 1.00 1.14 4.42 [?][?] ES [?] UGC 1230 LSB 54.1 0.366 4.7 37.1 0.65 0.67 1.82 0.97 [?][?][?][?] UGC 1281 LSB 5.1 0.017 1.6 1.7 0.03 0.01 0.53 3.02 [?][?][?][?] UGC 1551 LSB 35.6 0.780 4.2 6.6 0.44 0.16 0.20 3.69 [?][?][?][?] UGC 4325 LSB 11.9 0.373 1.9 3.4 0.10 0.40 1.08 7.39 [?][?][?][?] UGC 5005 LSB 51.4 0.200 4.6 27.7 0.28 1.02 5.11 1.30 [?][?][?][?] UGC 5750 LSB 56.1 0.472 3.3 8.6 0.10 0.10 0.21 1.58 [?][?][?][?] UGC 5999 LSB 44.9 0.170 4.4 15.0 0.18 3.36 19.81 5.79 [?][?][?][?] UGC 11820 LSB 17.1 0.169 3.6 3.7 0.40 1.68 9.95 8.44 [?][?][?][?]
22 Properties of the LSB 21 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI ESO 0140040 LSB 217.8 7.169 10.1 30.0 20.70 3.38 8.29 [?][?][?] NA ESO 0840411 LSB 82.4 0.287 3.5 9.1 0.06 0.21 1.49 [?][?] ES NA ESO 1200211 LSB 15.2 0.028 2.0 3.5 0.01 0.20 0.66 [?][?] ES NA ESO 1870510 LSB 16.8 0.054 2.1 2.8 0.09 1.62 2.02 [?][?][?] NA ESO 2060140 LSB 59.6 0.735 5.1 11.6 3.51 4.78 4.34 [?][?][?] NA ESO 3020120 LSB 70.9 0.717 3.4 11.2 0.77 1.07 2.37 [?][?] ES NA ESO 3050090 LSB 13.2 0.186 1.3 5.6 0.06 0.32 1.87 [?][?] ES NA ESO 4250180 LSB 88.3 2.600 7.3 14.6 4.79 1.84 5.17 [?][?][?] NA ESO 4880490 LSB 28.7 0.139 1.6 7.8 0.43 3.07 4.34 [?][?] ES NA F571-8 LSB 50.3 0.191 5.4 14.6 0.16 4.48 23.49 5.10 [?][?][?][?] F579-V1 LSB 86.9 0.557 5.2 14.7 0.21 3.33 5.98 3.18 [?][?][?][?] F730-V1 LSB 148.3 0.756 5.8 12.2 5.95 7.87 6.22 [?][?][?] NA UGC 04115 LSB 5.5 0.004 0.3 1.7 0.01 0.97 3.42 [?][?][?] NA UGC 06614 LSB 86.2 2.109 8.2 62.7 2.07 9.70 4.60 2.39 [?][?][?][?] UGC 11454 LSB 93.9 0.456 3.4 12.3 3.15 6.90 6.79 [?][?][?] NA UGC 11557 LSB 23.7 1.806 3.0 6.7 0.25 0.37 0.20 3.49 [?][?][?][?] UGC 11583 LSB 7.1 0.012 0.7 2.1 0.01 0.96 2.15 [?][?][?] NA UGC 11616 LSB 74.9 2.159 3.1 9.8 2.43 1.13 7.49 [?][?][?] NA UGC 11648 LSB 49.0 4.073 4.0 13.0 2.57 0.63 5.79 [?][?][?] NA UGC 11748 LSB 75.3 23.930 2.6 21.6 9.67 0.40 1.01 [?][?][?] NA UGC 11819 LSB 61.5 2.155 4.7 11.9 4.83 2.24 7.03 [?][?][?] NA
23 Properties of the Miscellaneous 22 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0168 LSB 4.5 0.032 1.2 4.4 0.03 0.06 2.03 2.22 [?][?][?][?] DDO 0170 LSB 16.6 0.023 1.9 13.3 0.09 0.05 1.97 1.18 [?][?][?][?] M 0033 HSB 0.9 0.850 2.5 8.9 0.11 1.13 1.33 4.62 [?][?][?][?] NGC 0055 LSB 1.9 0.588 1.9 12.2 0.13 0.30 0.50 2.22 [?][?][?][?] NGC 0247 LSB 3.6 0.512 4.2 14.3 0.16 1.25 2.43 2.94 [?][?][?][?] NGC 0300 LSB 2.0 0.271 2.1 11.7 0.08 0.65 2.41 2.69 [?][?][?][?] NGC 0801 HSB 63.0 4.746 9.5 46.7 1.39 6.93 2.37 3.59 [?][?][?][?] NGC 1003 LSB 11.8 1.480 1.9 31.2 0.63 0.66 0.45 1.53 [?][?][?][?] NGC 1560 LSB 3.7 0.053 1.6 10.3 0.12 0.17 3.16 2.16 [?][?][?][?] NGC 2683 HSB 10.2 1.882 2.4 36.0 0.15 6.03 3.20 2.28 [?][?][?][?] NGC 2998 HSB 59.3 5.186 4.8 41.1 1.78 7.16 1.75 3.43 [?][?][?][?] NGC 3109 LSB 1.5 0.064 1.3 7.1 0.06 0.02 0.35 2.29 [?][?][?][?] NGC 5033 HSB 15.3 3.058 7.5 45.6 1.07 0.27 3.28 3.16 [?][?][?][?] NGC 5371 HSB 35.3 7.593 4.4 41.0 0.89 8.52 1.44 3.98 [?][?][?][?] NGC 5533 HSB 42.0 3.173 7.4 56.0 1.39 2.00 4.14 3.31 [?][?][?][?] NGC 5585 HSB 9.0 0.333 2.0 14.0 0.28 0.36 1.09 2.06 [?][?][?][?] NGC 5907 HSB 16.5 5.400 5.5 48.0 1.90 2.49 1.89 3.44 [?][?][?][?] NGC 6503 HSB 5.5 0.417 1.6 20.7 0.14 1.53 3.66 2.30 [?][?][?][?] NGC 6674 HSB 42.0 4.935 7.1 59.1 2.18 2.00 2.52 3.57 [?][?][?][?] UGC 2259 LSB 10.0 0.110 1.4 7.8 0.04 0.47 4.23 3.76 [?][?][?][?] UGC 2885 HSB 80.4 23.955 13.3 74.1 3.98 8.47 0.72 4.31 [?][?][?][?] Malin 1 LSB 338.5 7.912 84.2 98.0 5.40 1.00 1.32 1.77 [?][?][?][?]
24 Properties of the 30 Dwarf Galaxy Sample 2 2 Galaxy Distance LB i (R0)disk Rlast MHI Mdisk (M/LB)disk (v /c R)last 9 B ◦ 9 9 B −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) F568-V1 78.20 2.15 40 3.11 17.07 2.32 16.00 7.45 2.95 F574-1 94.10 3.42 65 4.20 13.69 3.31 14.90 4.35 2.77 UGC 731 11.80 0.69 57 2.43 10.30 1.61 3.21 4.63 1.91 UGC 3371 18.75 1.54 49 4.53 15.00 2.62 4.49 2.91 1.78 UGC 4173 16.70 0.33 –5 4.43 12.14 2.24 0.07 0.20 1.21 UGC 4325 11.87 1.71 41 1.92 6.91 1.04 6.51 3.82 4.37 UGC 4499 12.80 1.01 50 1.46 8.38 1.15 1.80 1.79 2.37 UGC 5414 9.40 0.49 55 1.40 4.10 0.57 1.13 2.29 3.31 UGC 5721 7.60 0.48 +10 0.76 8.41 0.57 1.90 3.96 2.27 UGC 5750 56.10 4.72 64 5.60 21.77 1.00 3.68 0.78 1.03 UGC 7232 3.14 0.08 59 0.30 0.91 0.06 0.14 1.76 7.64 UGC 7323 7.90 2.39 47 2.13 5.75 0.70 4.19 1.75 4.59 UGC 7399 24.66 4.61 2.32 32.30 6.38 5.42 1.18 1.32 UGC 7524 4.12 1.37 46 3.02 9.29 1.34 5.29 3.86 2.67 UGC 7559 4.20 0.04 61 0.87 2.75 0.12 0.05 1.32 1.43 UGC 7577 2.13 0.05 –15 0.51 1.39 0.04 0.01 0.20 1.18 UGC 7603 9.45 0.80 78 1.24 8.24 1.04 0.41 0.52 1.81 UGC 8490 5.28 0.95 +10 0.71 11.51 0.72 1.53 1.61 1.47 UGC 9211 14.70 0.33 44 1.54 9.62 1.43 1.23 3.69 1.55 UGC 11707 21.46 1.13 68 5.82 20.30 6.78 9.89 8.76 1.77 UGC 11861 19.55 9.44 50 4.69 12.80 4.33 45.84 4.86 6.55 UGC 12060 15.10 0.39 40 1.70 9.89 1.67 3.45 8.94 2.00 UGC 12632 9.20 0.86 46 3.43 11.38 1.55 4.26 4.97 1.82 UGC 12732 12.40 0.71 39 2.10 14.43 3.23 4.11 5.76 2.40 UGC 3851 4.85 2.33 2.07 11.70 1.32 0.47 0.20 1.37 UGC 4305 2.34 0.41 –5 0.68 4.75 0.31 0.08 0.20 1.18 UGC 4459 3.06 0.03 27 0.60 2.47 0.04 0.01 0.20 0.97 UGC 5139 4.69 0.20 14 0.96 3.58 0.21 0.05 0.24 1.19 UGC 5423 7.14 0.14 45 0.61 1.97 0.05 0.28 2.01 1.82 UGC 5666 3.85 2.53 56 3.56 11.25 1.37 1.96 0.77 1.88
25 UNIVERSALITY AT THE LAST DATA POINTS Mannheim and O’Brien [PRL 106, 121101(2011), PRD 85, 124020 (2012), MNRAS 421, 1273 (2012)] found that one can fit the last data point in each of the 138 galaxies with:
2 ∗ ∗ ∗ ∗ v N β N γ γ0 = + + − κR, (1) 2 2 c R LAST R 2 2 N ∗β∗c2 N ∗γ∗c2R γ c2R v2 = + + 0 − κc2R2, (2) R 2 2 where N ∗ = visible mass in solar mass units, β∗ = 1.48 × 105 cm, ∗ −41 −1 −30 −1 −54 −2 γ = 5.42 × 10 cm , γ0 = 3.06 × 10 cm , κ = 9.54 × 10 cm (3) Get mass to light ratios that are typical of the local solar neighborhood. Previously (Mannheim, ApJ 479, 659 (1997)) the κ term was not needed since data did not go out as far. Characteristic of 1/r, r and r2 potentials. 2 Numerically, γ0 is cosmological and κ ∼ 1/(100 Mpc) is a cluster scale. For bright spirals ∗ ∗ ∗ N γ is of order γ0. On solar system distance scales only the β term is important. This formula has not been shown to follow from dark matter theory and challenges it, especially since it does embody distance scales that are rele- vant to CDM theories.
26 -8 -7
-8 ] ] -9 - 2 - 2
-9 -10
-10
-11 -11
-12 log [ g ( OBS )][ ms log [ g ( OBS )][ ms -12
-13 -13 0 2 4 6 8 10 12 6 7 8 9 10 11 12 13
log[M/M ⊙ ] log[M/M ⊙ ]
Figure 1: g(OBS) versus M/M for the last data point and for all data points in each galaxy.
-9 ] - 2
-10
-11 log [ g ( OBS )][ ms -12
-12 -11 -10 -9 log[g( NEW)][ms -2 ]
Figure 2: g(OBS) versus g(NEW ) for the last data point in each galaxy. Red line is g(NEW ) versus g(NEW ).
27 PLOTING ALL THE DATA ON A SINGLE GRAPH McGaugh, Lelli and Schombert, Phys. Rev. Lett. 117, 201101 (2016). O’Brien, Chiarelli and Mannheim, Phys. Lett. B 782, 433 (2018). Plot centripetal accelerations - g(OBS) versus g(NEW ), i.e. observed versus Newtonian expectation. Sample consists of 207 galaxies with 5791 data points. We will find it instructive to break the sample up into high surface brightness (HSB) and low surface brightness and dwarf galaxies (collectively LSB), with 56 HSB galaxies with 2870 points and 151 LSB galaxies with 2921 points. −R/R ∗ For an exponential Freeman disk Σ(R) = Σ0e 0 with scale length R0 and N stars: N ∗β∗c2R N ∗β∗ R gNEW (R) = 3 [I0 (x) K0 (x) − I1 (x) K1 (x))] → 2 , x = . (4) 2R0 R 2R0
A simple formula to consider is Milgrom’s MOND. It involves a universal acceleration parameter a0 = 0.6 × −10 −2 10 m s . It recovers Newton at accelerations a that obey a a0 and yields asymptotically flat rotation curves that obey Tully-Fisher at a a0
2 2 2 1/2 1/2 4 ∗ ∗ v 1 [g(NEW ) + 4a ] 1/2 v a0N β 0 1/2 g(MON) = = g(NEW ) + → a0 g(NEW ) , 2 → 2 .(5) R MON 2 2g(NEW ) R R
Another candidate is the curve considered by McGaugh, Lelli and Schombert with a fundamental g0 = 0.6 × 10−10 m s−2
g(NEW ) 1/2 1/2 g(MLS) = 1/2 → g0 g(NEW ) . (6) 1 − exp(−(g(NEW )/g0) )
28 -8 ]
- 2 -9
-10
-11
-12 log [ g ( OBS )][ ms
-13
-13 -12 -11 -10 -9 -8 log[g( NEW)][ms -2 ]
Figure 3: MLS fit (dotted curve) and MOND fit (solid curved line) to measured g(OBS) versus g(NEW ). Red line is g(NEW ) versus g(NEW ).
29 IMPLICATIONS
Recover Milgrom’s observation that departures from the luminous Newto- nian expectation begin when acceleration a falls below a0 – Milgrom’s Law, a true law of nature, regardless in fact of the validity of the MOND formula itself in a a0.
Implies that g(OBS) is correlated with luminous g(NEW ) without reference to dark matter at all.
What does it mean for dark matter theory and Newton’s Law of Gravity.
Is MLS curve or MOND single curve the correct interpretation of the data, i.e. is it tantamount to a law of nature, or is width significant too.
To see where it all might come from we look at an open issue in gravity theory.
30 LACK OF NECESSITY OF THE SECOND ORDER POISSON EQUATION
β ∇2φ = ρ, φ = − , r β ∇4φ = ρ, φ = − + γr, r β ∇6φ = ρ, φ = − + γr + δr3, (7) r Second Order Poisson Equation SUFFICIENT to give Newton’s law of Gravity– but not NECESSARY. Assuming it to be necessary causes the dark matter problem.
LACK OF NECESSITY OF THE EINSTEIN EQUATIONS
1 Z 1 1 ! I = − d4x(−g)1/2Rα , − R − g Rα = T (8) EH 16πG α 8πG µν 2 µν α µν has exterior Ricci flat vacuum Schwarzschild solution with Tµν = 0, Rµν = 0:
1 2MG g00 = = 1 − (9) grr r Non-relativistically Einstein equations reduce to the second order Poisson equation and Newton’s Law of Gravity. v2/c2 corrections give perihelion of Mercury and gravitational bending of light.
Eddington’s objection (1922): If action based on higher power of Ricci scalar or tensor, Rµν = 0 is still a vacuum solution. Einstein Equations SUFFICIENT to give Schwarzschild solution – but not NECESSARY. Assuming them to be necessary causes the dark matter problem. So how do we make necessary – need a principle – conformal invariance – leads to fourth order Poisson equation and eliminates need for dark matter.
31 CONFORMAL GRAVITY – FORMAL STRUCTURE Weyl introduced the conformal Weyl tensor 1 1 C = R + Rα [g g − g g ] − [g R − g R − g R + g R ] , (10) λµνκ λµνκ 6 α λν µκ λκ µν 2 λν µκ λκ µν µν λκ µκ λν λν which obeys g Cλµνκ = 0. It has the property that under a local rescaling of the metric (local conformal 2α(x) transformation) gµν(x) → e gµν(x) 2α(x) Cλµνκ → e Cλµνκ (11) with all derivatives of α(x) dropping out (just like a gauge transformation). Under conformal transformation ds2 → e2α(x)ds2, ds2 = 0 → ds2 = 0, (12) so light cone is left invariant. Gravitational action is UNIQUE: Z 4 1/2 λµνκ Z 4 1/2 µν 1 α 2 I = −α d x(−g) C C = −2α d x(−g) R R − (R ) , (13) W g λµνκ g µν 3 α where αg is dimensionless gravitational coupling constant. Cosmological constant is forbidden at the level of the action, as is Einstein-Hilbert action. With 1 1 2 W µν = gµν(Rα );β + Rµν;β − Rµβ;ν − Rνβ;µ − 2RµβRν + gµνR Rαβ − gµν(Rα );β 2 α ;β ;β ;β ;β β 2 αβ 3 α ;β 2 2 1 + (Rα );µ;ν + Rα Rµν − gµν(Rα )2 = 2Cµλνκ − CµλνκR , (14) 3 α 3 α 6 α ;λ;κ λκ equations of motion are of the form µν µν µν µν µν µν µν TUNIV = TGRAV + TMAT = −4αgW + TMAT = 0, gµνW = 0, gµνTMAT = 0 (15)
Conformal theory has Rµν = 0 as a vacuum solution, so immediately recover Schwarzschild metric for solar system, and induce GN dynamically. But there are other solutions.
32 THE GENERAL AND EXACT CONFORMAL GRAVITY POTENTIAL
Mannheim and Kazanas, Ap J 342, 635 (1989), Gen. Rel. Gravit. 26, 337 (1994). With B(r) = g00, for a static, spherically symmetric source, the conformal gravity equations of motion reduce without approximation to the FOURTH order Poisson equation: 000 000 3 0 r 0000 4B (rB) 4 3 0 r (W 0 − W r) = B + = = ∇ B = (T 0 − T r) = f(r). (16) B(r) r r 4αgB(r)
For a source localized to a region r < r0 the solution is 2β B(r) = 1 − + γr (17) r where 1 Z r 1 Z r γ = − 0 dr0r02f(r0), 2β = 0 dr0r04f(r0). (18) 2 0 6 0 Thus for a star the gravitational potential is β∗c2 γ∗c2r V ∗(r) = − + (19) r 2 ∗ −R/R and for a spiral galaxy with N stars and surface brightness Σ(R) = Σ0e 0 we obtain the local contribution
2 ∗ ∗ 2 vLOC N β c R R R R R = 3 I0 K0 − I1 K1 R 2R0 2R0 2R0 2R0 2R0 N ∗γ∗c2R R R N ∗β∗c2 N ∗γ∗c2 + I1 K1 → 2 + , (20) 2R0 2R0 2R0 R 2 But this is not the whole story.
33 SECOND ORDER IS LOCAL – FOURTH ORDER IS GLOBAL Newton proved a famous theorem: for an 1/r potential and a spherical distribution of matter, the force due to material outside a shell containing the point of observation is zero. Thus NEWTONIAN GRAVITY IS LOCAL. Thus if there is a galactic missing mass problem you have to put dark matter where the problem is, i.e. within galaxies. The reason for this is that the solid angle grows as r2 and the force falls as r2, so they cancel. But suppose we change the potential. We do not change the solid angle, and now there is no cancellation. Thus for linear potentials we cannot ignore the outside, i.e. cannot ignore the rest of the Universe, and so CONFORMAL GRAVITY IS GLOBAL. Now conformal potential grows with distance. Thus the most significant contributions come from the material that is the furthest away, viz cosmology. For second order Poisson equation
1 Z r Z ∞ dφ(r) 1 Z r ∇2φ = g(r), φ(r) = − dr0r02g(r0) − dr0r0g(r0), = , dr0r02g(r0). (21) r 0 r dr r2 0 Force only depends on interior matter For fourth order Poisson equation with h(r) = f(r)c2/2
2 r Z r 1 Z r 1 Z ∞ r Z ∞ ∇4φ = h(r), φ(r) = − dr0r02h(r0) − dr0r04h(r0) − dr0r03h(r0) − dr0r0h(r0), (22) 2 0 6r 0 2 r 6 r
dφ(r) 1 Z r 1 Z r r Z ∞ = − dr0r02h(r0) + dr0r04h(r0) − dr0r0h(r0). (23) dr 2 0 6r2 0 3 r Force depends on both interior and exterior matter.
34 THE CONTRIBUTION OF THE REST OF THE UNIVERSE Rest of universe has homogeneous and inhomogeneous matter For homogeneous matter have Robertson-Walker Hubble flow. How does Hubble flow look to an observer at rest in a galaxy. Need to transform from comoving coordinates to Schwarzschild coordinates and make a conformal transformation.
4r Z ρ = 1/2 , τ = dtR(t) (24) 2(1 + γ0r) + 2 + γ0r
2 2 1 1 + γ0ρ/4 R (τ) 2 2 2 2 2 c dτ − 2 2 2 dρ + ρ dΩ2 R (τ) 1 − γ0ρ/4 [1 − γ0ρ /16] 2 2 2 2 dr 2 γ0 = (1 + γ0r)c dt − − r dΩ2, k = − < 0. (25) (1 + γ0r) 4 2 In rest frame an open (k < 0) RW geometry acts like a universal linear potential γ0c R/2 and universal 2 acceleration γ0c /2 (analogous to MOND a0). But also inhomogeneous matter – clusters and superclusters of galaxies
v2 v2 γ c2 N ∗β∗c2 N ∗γ∗c2 γ c2 TOT = LOC + 0 − κc2R → + + 0 − κc2R. (26) R R 2 R2 2 2 Mannheim and O’Brien fit 138 (later 207) galaxies (fits given above) with VISIBLE N ∗ ∗ ∗ ∗ of each galaxy as only variable, β , γ , γ0 and κ are all universal, and with NO DARK MATTER, and with 276 (later 414) less free parameters than in dark matter calcula- tions. DARK MATTER IS JUST AN ATTEMPT TO DESCRIBE GLOBAL PHYSICS IN LOCAL TERMS AND THUS DOES NOT EXIST.
35 PREDICTION
2 ∗ ∗ 2 vLOC N β c R R R R R gLOC = = 3 I0 K0 − I1 K1 R 2R0 2R0 2R0 2R0 2R0 N ∗γ∗c2R R R + I1 K1 . (27) 2R0 2R0 2R0 γ c2 g(CG) = g(LOC) + 0 − κc2R. (28) 2 N ∗β∗c2 N ∗γ∗c2R γ c2R v2 → + + 0 − κc2R2. (29) TOT R 2 2 At largest distances −κc2R2 term dominates. Velocities must thus start to fall. But v2 cannot go negative, and thus galaxies must have a limiting size, with none bigger than 150 kpc or so. Caused by an interplay between local and global physics.
100 140 300 80 120 250 100 60 200 80 150 40 60 100 40 20 20 50 DDO 154 UGC 128 Malin 1 0 0 0 0 10 20 30 40 50 0 20 40 60 80 0 50 100 150 FIG. 10: Extended distance predictions for DDO 154, UGC 128, and Malin 1
36 CONFORMAL GRAVITY FITING TO g(OBS) VERSUS g(NEW ) PLOT Instead of g(OBS) versus g(NEW ), plot g(OBS) versus the conformal gravity g(LOC). Thus if anything correlate g(OBS) with the luminous content of a galaxy, not its Newtonian contribution.
-8 ]
- 2 -9
-10
-11
-12 log [ g ( OBS )][ ms
-13
-13 -12 -11 -10 -9 -8 log[g( LOC)][ms -2 ]
Figure 4: g(OBS) versus the conformal gravity g(LOC).
To show that conformal gravity does fit the data, plot g(OBS) versus g(CG) for 207 galaxies with 5791 points.
-8 ]
- 2 -9
-10
-11
-12 log [ g ( OBS )][ ms
-13
-13 -12 -11 -10 -9 -8 log[g(CG)][ms -2 ]
Figure 5: g(OBS) versus the conformal gravity g(CG). The solid diagonal is the line g(OBS) = g(CG).
37 To show that conformal gravity does fit the data, overlay g(CG) on a plot of g(OBS) versus g(NEW ) for 207 galaxies with 5791 points. Fit the width and not just a single curve. Width is physical.
-8 ]
- 2 -9
-10
-11
-12 log [ g ( OBS )][ ms
-13
-13 -12 -11 -10 -9 -8 log[g( NEW)][ms -2 ]
Figure 6: g(CG) overlay of g(OBS) versus g(NEW ). The solid lines other than the diagonal are the g(CG) expectations.
38 To explore the width, overlay g(CG) on a plot of g(OBS) versus g(NEW ) for 56 HSB galaxies with 2870 points. Fit the width and not just a single curve.
-8 ]
- 2 -9
-10
-11
-12 log [ g ( OBS )][ ms
-13
-13 -12 -11 -10 -9 -8 log[g( NEW)][ms -2 ]
Figure 7: g(CG) overlay of the HSB g(OBS) versus g(NEW ). The lines other than the diagonal are the g(CG) expectations.
39 To explore the width, overlay g(CG) on a plot of g(OBS) versus g(NEW ) for 151 LSB galaxies with 2971 points. Now just a single curve, one with a very interesting continuation to very small g(NEW ).
-8 ]
- 2 -9
-10
-11
-12 log [ g ( OBS )][ ms
-13
-13 -12 -11 -10 -9 -8 log[g( NEW)][ms -2 ]
Figure 8: g(CG) overlay of the LSB g(OBS) versus g(NEW ). The lines other than the diagonal are the g(CG) expectations.
40 Blow up the small g(NEW ) region to show flattening
] 3 - 2
2
1
0
-1
[ g ( OBS )- g ( NEW )][-ms2 10.5 -3 10 -13 -12 -11 -10 -9 -8 log[g( NEW)][ms -2 ]
Figure 9: Scaled g(OBS) − g(NEW ) versus g(NEW ).
In a follow up paper Lelli, McGaugh, Schombert, and Pawloski (AJ 836, 152 (2017)) augmented the spi- ral galaxy data with some dwarf spheroidal data and some late type galaxy data, which showed a flatten- ing off at very low g(NEW ). They even considered changing the g(MLS) formula by adding on a term 2 1/2 gˆ exp(−g(NEW )g0/gˆ ) whereg ˆ is a new free parameter, and they characterized the data as exhibiting a possible acceleration floor. In conformal gravity such an acceleration floor is natural.
41 PUTTING ALL THE DATA ON A SINGLE GRAPH A DISTANCE-DEPENDENT REGULARITY By 10 kpc get departures from luminous Newtonian expectation in every one of the 207 galaxies. ] - 2
[ ms 5 11
0
-5 ( g OBS )- g ( NEW ))* 2.76 × 10 0 20 40 60 80 100 R[ kpc]
2 Figure 10: g(OBS) − g(NEW ) scaled by γ0c versus R.
2 ∗ ∗ 2 ∗ ∗ 2 vTOT N β c N γ c Moreover, the shortfall is almost constant in distance, just as expected from R → R2 + 2 + 2 γ0c 2 2 − κc R.
42 TULLY-FISHER RELATION 4 Plot v = AM/M for the last data point in each of the 207 galaxies is provided as the continuous curve in the figure, a fit that gives an extracted value of A = 0.0098 km4s−4.
12 ]
4 10 - s 4
km 8 ][ 4 v [ 6 log
4
4 6 8 10 12
log [ M / M⊙ ]
4 4 4 Figure 11: v versus M for the last data point point in each of the 207 galaxies. Overlaid are v = AM/M (continuous curve) and v = ∗ B(M/M )(1 + N /D) (dashed curve).
2 ∗ 2 ∗ ∗ ∗ 2 At the crossing point between Newton and linear, one can set v = β c N /R + (γ N + γ0)c R/2 and ∗ 2 ∗ ∗ ∗ 2 4 β c N /R = (γ N +γ0)c R/2 for an R that depends on each galaxy, and thus at that point one can set v = ∗ 2 4 −4 ∗ 10 B(M/M )(1 + N /D) where B = 2c M Gγ0 = 0.0074 km s , where D = γ0/γ = 5.65 × 10 , and where here N ∗ includes all galactic baryonic sources. Since the velocities at the last data points do not differ much 4 ∗ from those at the crossover points in each galaxy, at the last data points we plot v = B(M/M )(1 + N /D) as the dashed curve in the figure.
43
(ΩΜ,ΩΛ) = 26 ( 0, 1 ) (0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) 24 (1.5,–0.5) (2, 0) Supernova Cosmology Flat 22 Project Λ = 0 B m 20 Calan/Tololo (Hamuy et al, A.J. 1996) effective 18
16
14 0.02 0.05 0.1 0.2 0.5 1.0 redshift z Supernovae data as log plot
44 3 No Big Bang 99% 95% 90% 2 68%
1 Λ Ω
expands forever 0 lly recollapses eventua
closed Flat Λ = 0 flat -1 Universe open
0 1 2 3
ΩΜ Hubble plot confidence contours in ΩM (t0), ΩΛ(t0) plane.
45 CONFORMAL COSMOLOGY
Z 1 1 I = − d4x(−g)1/2 S;µS − S2Rµ + λS4 + iψγ¯ µ[∂ + Γ ]ψ − hSψψ¯ (30) M 2 ;µ 12 µ µ µ 1 2 1 α 4 T = (ρ + p )U U + p g − S R − g R − g λS (31) µν M M µ ν M µν 6 0 µν 2 µν α µν 0 4 Cλµνκ = 0,Wµν = 0,Tµν = 0, Λ = λS0 (32)
1 2 1 α S R − g R = (ρ + p )U U + p g − g Λ (33) 6 0 µν 2 µν α M M µ ν M µν µν Mannheim ApJ 391, 429 (1992): cosmological Geff < 0, i.e. repulsive, and that there is no flatness problem.
3 ˙ 2 2 ˙ 2 h ¯ ¯ i Geff = − 2 , R + kc = R ΩM + ΩΛ , (34) 4πS0 ¯ 4 ¯ 8πGeffρM ¯ 8πGeffΛ ΩΛ cΛ TV ΩM = 2 2 , ΩΛ = 2 , = = − 4 (35) 3c H 3cH Ω¯ M ρM T Λ is negative (free energy released in early universe phase transition) and overwhelmingly larger than ρM . With Geff < 0, Ω¯ Λ is positive. Then with k < 0 (from rotation curves) and q = −Ω¯ Λ, we obtain
0 ≤ Ω¯ Λ ≤ 1, − 1 ≤ q ≤ 0 (36) and luminosity function [Mannheim, Prog. Part. Nucl. Phys. 56, 340 (2006)]:
2 1/2 c (1 + z) q0 d = − 1 − 1 + q − . (37) L 0 2 H0 q0 (1 + z )
46 24
22
18
16
14 0.0 0.2 0.4 0.6 0.8 1.0 redshift
Figure 12: The q0 = −0.37 conformal gravity fit (upper curve) and the ΩM (t0) = 0.3, ΩΛ(t0) = 0.7 standard model fit (lower curve) to the z < 1 supernovae Hubble plot data. 47 30
28
26
24
22 apparent magnitude
20
18
16
14 0 1 2 3 4 5 redshift
Figure 13: Hubble plot expectations for q0 = −0.37 (highest curve) and q0 = 0 (middle curve) conformal gravity and for ΩM (t0) = 0.3, ΩΛ(t0) = 0.7 standard gravity (lowest curve). 48 SUMMARY All the big problems have a common origin: The extrapolation of standard Newton-Einstein Gravity beyond its solar system origins. 1. Continue to galaxies get dark matter problem 2. Continue to cosmology get the cosmological constant/dark energy problem 3. Continue to strong gravity get the singularity/black hole problem 4. Continue to quantum field theory far off the mass shell get the renormalization and vacuum zero- point energy problems The Standard Solution: Supersymmetry, Extra Dimensions, String Theory, The Multiverse, The Anthropic Principle. No evidence for any of them. But until recently no evidence against any of them either. Recent evidence against supersymmetry. Not found at the Large Hadron Collider. Should have been found in same energy region as the recently found Higgs boson. Solution: Change the extrapolation: get conformal gravity. All these problems are solved, with no need for any of the dark fixes. MORAL OF THE STORY At the beginning of the 20th century studies of black-body radiation on microscopic scales led to a paradigm shift in physics. Thus it could that at the beginning of the 21st century studies of phenomena such as black-body radiation, this time on macroscopic cosmological scales, might be presaging a paradigm shift all over again.
49 CONFORMAL GRAVITY AND THE COSMOLOGICAL CONSTANT AND DARK MATTER PROBLEMS
1. P. D. Mannheim, Conformal cosmology with no cosmological constant, General Relativity and Gravitation 22, 289 (1990).
2. P. D. Mannheim and D. Kazanas, Exact vacuum solution to conformal Weyl gravity and galactic rotation curves, Astrophysical Journal 342, 635 (1989).
3. D. Kazanas and P. D. Mannheim, General structure of the gravitational equations of motion in conformal Weyl gravity, Astrophysical Journal Supplement Series 76, 431 (1991).
4. P. D. Mannheim and D. Kazanas, Solutions to the Reissner-Nordstrom, Kerr and Kerr-Newman problems in fourth order conformal Weyl gravity, Physical Review D 44, 417 (1991).
5. P. D. Mannheim, Conformal gravity and the flatness problem, Astrophysical Journal 391, 429 (1992).
6. P. D. Mannheim, Dynamical mass and geodesic motion, General Relativity and Gravitation 25, 697 (1993).
7. P. D. Mannheim, Linear potentials and galactic rotation curves, Astrophysical Journal 419, 150 (1993). (hep-ph/9212304)
8. P. D. Mannheim and D. Kazanas, Newtonian limit of conformal gravity and the lack of necessity of the second order Poisson equation, General Relativity and Gravitation 26, 337 (1994).
9. P. D. Mannheim, Open questions in classical gravity, Foundations of Physics 24, 487 (1994). (gr-qc/9306025)
10. P. D. Mannheim, Linear potentials in galaxies and clusters of galaxies, April 1995. (astro-ph/9504022)
11. P. D. Mannheim, Cosmology and galactic rotation curves, November 1995. (astro-ph/9511045)
12. P. D. Mannheim, Conformal cosmology and the age of the universe, January 1996. (astro-ph/9601071)
13. P. D. Mannheim and J. Kmetko, Linear potentials and galactic rotation curves - detailed fitting, February 1996. (astro-ph/9602094)
14. P. D. Mannheim, Are galactic rotation curves really flat?, Astrophysical Journal 479, 659 (1997). (astro-ph/9605085)
15. P. D. Mannheim, Local and global gravity, Foundations of Physics 26, 1683 (1996). (gr-qc/9611038)
16. P. D. Mannheim, Curvature and cosmic repulsion, March 1998. (astro-ph/9803135)
17. P. D. Mannheim, Implications of cosmic repulsion for gravitational theory, Physical Review D 58, 103511 (1998). (astro-ph/9804335)
18. P. D. Mannheim, Cosmic acceleration as the solution to the cosmological constant problem, Astrophysical Journal 561, 1 (2001). (astro-ph/9910093)
19. P. D. Mannheim, Attractive and repulsive gravity, Foundations of Physics 30, 709 (2000). (gr-qc/0001011)
20. P. D. Mannheim, How recent is cosmic acceleration?, International Journal of Modern Physics D 12, 893 (2003). (astro-ph/0104022)
21. P. D. Mannheim, Alternatives to dark matter and dark energy, Progress in Particle and Nuclear Physics 56, 340 (2006). (astro-ph/0505266)
50 22. P. D. Mannheim, Schwarzschild limit of conformal gravity in the presence of macroscopic scalar fields, Physical Review D 75, 124006 (2007). (gr- qc/0703037)
23. P. D. Mannheim, Comprehensive Solution to the cosmological constant, zero-point energy, and quantum gravity problems, General Relativity and Gravi- tation 43, 703 (2011). (arXiv:0909.0212 [hep-th])
24. P. D. Mannheim, Intrinsically quantum-mechanical gravity and the cosmological constant problem, Modern Physics Letters A 26, 2375 (2011). (arXiv:1005.5108 [hep-th])
25. P. D. Mannheim and J. G. O’Brien, Impact of a global quadratic potential on galactic rotation curves, Physical Review Letters 106, 121101(2011). (arXiv: 1007.0970 [astro-ph.CO])
26. P. D. Mannheim and J. G. O’Brien, Fitting galactic rotation curves with conformal gravity and a global quadratic potential, Physical Review D 85, 124020 (2012). (arXiv: 1011.3495 [astro-ph.CO])
27. P. D. Mannheim, Making the case for conformal gravity, Foundations of Physics 42, 388 (2012). (arXiv: 1101.2186 [hep-th])
28. J. G. O’Brien and P. D. Mannheim, Fitting dwarf galaxy rotation curves with conformal gravity. Monthly Notices of the Royal Astronomical Society 421, 1273 (2012). (arXiv:1107.5229 [astro-ph.CO])
29. P. D. Mannheim, Cosmological perturbations in conformal gravity, Physical Review D 85, 124008 (2012). (arXiv:1109.4119 [gr-qc])
30. P. D. Mannheim and J. G. O’Brien, Galactic rotation curves in conformal gravity, Journal of Physics: Conference Series 437, 012002 (2013). (arXiv:1211.0188 [astro-ph.CO])
31. P. D. Mannheim, Living without supersymmetry – the conformal alternative and a dynamical Higgs boson, Journal of Physics G 44, 115003 (2017). (arXiv:1506.01399 [hep-ph]).
32. P. D. Mannheim, Comment on ”Problems with Mannheim’s conformal gravity program”, Physical Review D 93, 068501 (2016)). (arXiv:1506.02479 [gr-qc])
33. P. D. Mannheim, Mass generation, the cosmological constant problem, conformal symmetry, and the Higgs boson, Progress in Particle and Nuclear Physics 94, 125 (2017). (arXiv:1610.08907 [hep-ph])
34. P. D. Mannheim, Is the cosmological constant problem properly posed?, International Journal of Modern Physics D 26, 1743009 (2017). (arXiv:1703.09286 [hep-th]).
35. J. G. O’Brien, T. L. Chiarelli and P. D. Mannheim, Universal properties of galactic rotation curves and a first principles derivation of the Tully-Fisher relation, Physics Letters B 782, 433 (2018). (arXiv:1704.03921 [astro-ph.GA])
36. A. Amarasinghe, M. G. Phelps and P. D. Mannheim, Cosmological perturbations in conformal gravity II, arXiv:1805.06807 [gr-qc], May 2018.
51 GHOST PROBLEMS, UNITARITY OF FOURTH-ORDER THEORIES AND PT QUANTUM MECHANICS 1. P. D. Mannheim and A. Davidson, Fourth order theories without ghosts, January 2000. (hep-th/0001115)
2. P. D. Mannheim and A. Davidson, Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Physical Review A 71, 042110 (2005). (hep- th/0408104)
3. P. D. Mannheim, Solution to the ghost problem in fourth order derivative theories, Foundations of Physics 37, 532 (2007). (hep-th/0608154)
4. C. M. Bender and P. D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Physical Review Letters 100, 110402 (2008). (arXiv:0706.0207 [hep-th])
5. C. M. Bender and P. D. Mannheim, Giving up the ghost, Journal of Physics A 41, 304018 (2008). (arXiv:0807.2607 [hep-th])
6. C. M. Bender and P. D. Mannheim, Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart, Physical Review D 78, 025022 (2008). (arXiv:0804.4190 [hep-th])
7. C. M. Bender and P. D. Mannheim, PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues, Physics Letters A 374, 1616 (2010). (arXiv:0902.1365 [hep-th])
8. P. D. Mannheim, PT symmetry as a necessary and sufficient condition for unitary time evolution, Philosophical Transactions of the Royal Society A 371, 20120060 (2013). (arXiv:0912.2635[hep-th])
9. C. M. Bender and P. D. Mannheim, PT symmetry in relativistic quantum mechanics, Physical Review D 84, 105038 (2011). (arXiv:1107.0501 [hep-th])
10. P. D. Mannheim, Astrophysical evidence for the non-Hermitian but PT -symmetric Hamiltonian of conformal gravity, Fortschritte der Physik 61, 140 (2013). (arXiv: 1205.5717 [hep-th])
11. P. D. Mannheim, PT symmetry, conformal symmetry, and the metrication of electromagnetism, Foundations of Physics 47, 1229 (2017). (arXiv:1407.1820 [hep-th])
12. P. D. Mannheim, Advancing the case for PT symmetry – the Hamiltonian is always PT symmetric, arXiv:1506.08432 [quant-ph], June 2015.
13. P. D. Mannheim, Extension of the CPT Theorem to non-Hermitian Hamiltonians and unstable states, Physics Letters B 753, 288 (2016). (arXiv:1512.03736 [quant-ph])
14. P. D. Mannheim, Antilinearity rather than Hermiticity as a guiding principle for quantum theory, Journal of Physics A 51, 315302 (2018). (arXiv:1512.04915 [hep-th])
15. P. D. Mannheim, Appropriate inner product for PT-Symmetric Hamiltonians, Physical Review D 97, 045001 (2018). (arXiv:1708.01247 [quant-ph])
2 2 2 2 16. P. D. Mannheim, Unitarity of loop diagrams for the ghostlike 1/(k −M1 )−1/(k −M2 ) propagator, Physical Review D 98, 045014 (2018). (arXiv:1801.03220 [hep–th])
17. P. D. Mannheim, Goldstone bosons and the Higgs mechanism in non-Hermitian theories, arXiv:1808:00437 [hep-th], August 2018.
52