Pt Symmetry, Quantum Gravity, and the Metrication of the Fundamental Forces Philip D. Mannheim Department of Physics University

PT SYMMETRY, QUANTUM GRAVITY,

AND THE METRICATION OF THE FUNDAMENTAL FORCES

PHILIP D. MANNHEIM DEPARTMENT OF PHYSICS UNIVERSITY OF CONNECTICUT

PRESENTATION AT DISCRETE 2014

KING’S COLLEGE LONDON, DECEMBER 2014

1 TOPICS TO BE DISCUSSED

1. PT SYMMETRY AS A DYNAMICS

2. HIDDEN SECRET OF PT SYMMETRY–UNITARITY/NO GHOSTS

3. PT SYMMETRY AND THE LORENTZ GROUP

4. PT SYMMETRY AND THE CONFORMAL GROUP

5. PT SYMMETRY AND GRAVITY

6. PT SYMMETRY AND GEOMETRY

7. PT SYMMETRY AND THE DIRAC ACTION

i 8. PT SYMMETRY AND THE METRICATION OF Aµ AND Aµ

9. THE UNEXPECTED PT SURPRISE

10. ASTROPHYSICAL EVIDENCE FOR THE PT REALIZATION OF CONFORMAL GRAVITY

2 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 (arXiv:0001115 [hep-th]). 2. P. D. Mannheim and A. Davidson, Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Phys. Rev. A 71, 042110 (2005). (0408104 [hep-th]). 3. P. D. Mannheim, Solution to the ghost problem in fourth order derivative theories, Found. Phys. 37, 532 (2007). (arXiv:0608154 [hep-th]). 4. C. M. Bender and P. D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100, 110402 (2008). (arXiv:0706.0207 [hep-th]). 5. C. M. Bender and P. D. Mannheim, Giving up the ghost, Jour. Phys. 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, Phys. Rev. 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, Phys. Lett. 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, Phil. Trans. Roy. Soc. A. (arXiv:0912.2635 [hep-th]) 9. C. M. Bender and P. D. Mannheim, PT symmetry in relativistic quantum mechanics, Phys. Rev. D 84, 105038 (2011). (arXiv:1107.0501 [hep-th]) CONFORMAL GRAVITY AND THE COSMOLOGICAL CONSTANT AND DARK MATTER PROBLEMS 10. P. D. Mannheim, Alternatives to dark matter and dark energy, Prog. Part. Nuc. Phys. 56, 340 (2006). (arXiv:0505266 [astro-ph]). 11. P. D. Mannheim, Why do we believe in dark matter and dark energy – and do we have to?, in ”Questions of Modern Cosmology – Galileo’s Legacy”, M. D’Onofrio and C. Burigana (Eds.), Springer Publishing Company, Heidelberg (2009). 12. P. D. Mannheim, Conformal Gravity Challenges String Theory, Proceedings of the Second Crisis in Cosmology Conference, CCC-2, Astronomical Society of the Pacific Conference Series Vol. 413. (F. Potter, Ed.), San Francisco (2009). (arXiv:0707.2283 [hep-th]) 13. P. D. Mannheim, Dynamical symmetry breaking and the cosmological constant problem, Proceedings of ICHEP08, eConf C080730. (0809.1200 [hep-th]). 14. P. D. Mannheim, Comprehensive solution to the cosmological constant, zero-point energy, and quantum gravity problems, Gen. Rel. Gravit. 43, 703 (2011). (arXiv:0909.0212 [hep-th]) 15. P. D. Mannheim, Intrinsically quantum-mechanical gravity and the cosmological constant problem, Mod. Phys. Lett. A 26, 2375 (2011). (arXiv:1005.5108 [hep-th]). 16. P. D. Mannheim and J. G. O’Brien, Impact of a global quadratic potential on galactic rotation curves, Phys. Rev. Lett. 106, 121101 (2011). (arXiv:1007.0970 [astro-ph.CO]). 17. P. D. Mannheim and J. G. O’Brien, Fitting galactic rotation curves with conformal gravity and a global quadratic potential, Phys. Rev. D 85, 124020(2012). (arXiv:1011.3495 [astro-ph.CO]) 18. P. D. Mannheim, Making the case for conformal gravity, Found. Phys. 42, 388 (2012). (arXiv:1101.2186 [hep-th]). 19. J. G. O’Brien and P. D. Mannheim, Fitting dwarf galaxy rotation curves with conformal gravity. MNRAS 421, 1273 (2012). (arXiv:1107.5229 [astro-ph.CO]) 20. P. D. Mannheim, Cosmological perturbations in conformal gravity, Phys. Rev. D. 85, 124008 (2012). (arXiv:1109.4119 [gr-qc]) 21. 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]) 22. P. D. Mannheim and J. G. O’Brien, Galactic rotation curves in conformal gravity, J. Phys. Conf. Ser. 437, 012002 (2013). (arXiv:1211.0188 [astro-ph.CO])

3 TORSION 1. L. Fabbri and P. D. Mannheim, Continuity of the torsionless limit as a selection rule for gravity theories with torsion, May 2014, Phys. Rev. D 90, 024042 (2014). (arXiv:1405.1248 [gr-qc]). 2. P. D. Mannheim and J. J. Poveromo, Gravitational Analog of Faraday’s Law via Torsion and a Metric with an Anti- symmetric Part, General Relativity and Gravitation 46, 1795 (2014). (arXiv:1406.1470 [gr-qc]). 3. P. D. Mannheim, Torsion, Magnetic Monopoles and Faraday’s Law via a Variational Principle, June 2014. (arXiv:1406.2265 [hep-th]). 4. P. D. Mannheim, PT Symmetry, Conformal Symmetry, and the Metrication of Electromagnetism, July 2014. (arXiv:1407.1820 [hep-th]).

4 1 PT SYMMETRY AS A DYNAMICS

Pioneered by Bender (1998). HERMITICITY is only SUFFICIENT to secure reality of eigenvalues. PT SYMMETRY of a Hamiltonian is the NECESSARY condition. Instead of considering the eigenvector equation H|ψi = E|ψi, focus on the eigenvalue equation f(λ) = |H − λI| = 0 instead. Theorem: The condition [H,PT ] = 0 is both a SUFFICIENT (Bender, Berry and Mandilara (2002)) and NECESSARY (Bender and Mannheim (2010)) condition for the reality of f(λ). However three realizations:

1. All λi are real and set of all |ψii is complete. H is then either already Hermitian or by a non-unitary similarity transform on the eigenvector basis can be brought to a basis in which ∗ Hij = Hji. Thus Hermitian in disguise. Example (Bender, Brody and Jones (2003)): r cos θ + ir sin θ s H = r cos θσ + ir sin θσ + sσ = =6 H†. (1) 0 3 1 s r cos θ − ir sin θ

Here P = σ1, T = K (complex conjugation) and [H,PT ] = 0. Energies E± = r cos θ ± (s2 − r2 sin2 θ)1/2. If s2 − r2 sin2 θ = s sin α is positive, both energies are real. 1 + sin α1/2 1 − sin α1/2 G±1 = σ ± σ , GHG−1 = r cos θσ + σ (s2 − r2 sin2 θ)1/2 2 sin α 0 2 2 sin α 0 1 (2)

5 ∗ 2. λi come in complex conjugate pairs (T : E → E ) and set of all |ψii is complete. Thus have decaying states (loss) and growing states (gain), and need both (Mannheim (2013)) for unitarity – as the population of a state is depleted the population of its complex conjugate partner grows. Example: s2 − r2 sin2 θ is negative, energies in a complex conjugate pair.

3. The λi are all equal (and thus all real) but eigenvectors incomplete - Jordan block realization. Example: The above H with s2 − r2 sin2 θ = 0 (G no longer exists), or 1 1 M = =6 M † (3) 0 1

TWO real solutions to |M − λI| = 0, viz. λ1 = 1, λ2 = 1 (Trace = 2, Determinant =1). But only ONE eigenstate – 1 1 p p + q p 1 = = , q = 0, (4) 0 1 q q q 0 Eigenvalues degenerate since they have to share just the one eigenvector. Missing eigenvectors are not stationary. Nonetheless theory is unitary (Bender and Mannheim (2008)) since stationary plus non-stationary states combined form a complete basis.

SUMMARY: PT SYMMETRY OF A HAMILTONIAN IS A DIAGNOS- TIC FOR PROPERTIES OF ITS EIGENVALUES, AND IS A BOTH NEC- ESSARY AND SUFFICIENT CONDITION FOR UNITARITY.

6 2 HIDDEN SECRET OF PT SYMMETRY – UNITARITY/NO GHOSTS

Once the Hamiltonian is not Hermitian one can no longer use the Dirac norm. Given a right eigenvector |Rii of H, which obeys H|Rii = E|Rii, the Dirac norm hRi(t)|Ri(t)i = −iH†t iHt hRi(t = 0)|e e |Ri(t = 0)i is not equal to hRi(t = 0)|Ri(t = 0)i, with the Dirac norm not being preserved in time in the non-Hermitian case. However, if we instead introduce a left eigenvector, which obeys hLi|H = hLi|E, then the norm hLi(t)|Ri(t)i = hLi(t = −iHt iHt 0)|e e |Ri(t = 0)i = hLi(t = 0)|Ri(t = 0)i is preserved in time. With left eigenvectors being related to the PT conjugates of the right eigenvectors, in the non-Hermitian but PT symmetric case one should instead use the PT theory norm, with unitary time evolution then being secured.

In those cases where the Dirac norm is found to be negative, this does not necessarily mean that there are ghost states in the theory, since when the theory can be reinterpreted as a PT theory, the PT theory norm would be positive – case (1) above , where the theory Hamiltonian is Hermitian in disguise, and thus necessarily ghost free, with scalar products not being able to change sign under a similarity transformation.

7 Two cases of this eﬀect have been identiﬁed in the literature. The Lee model (Lee 1954), and the fourth-order Pais-Uhlenbeck oscillator (Pais and Uhlenbeck (1950)) with action, equation of motion and propagator of the form: γ Z I = dt[z¨2 − (ω2 + ω2)z˙2 + ω2ω2z2], PU 2 1 2 1 2 d4z d2z + (ω2 + ω2) + ω2ω2z = 0, dt4 1 2 dt2 1 2 1 1 1 1 G(E) = 2 2 2 2 = 2 2 2 2 − 2 2 . (5) (E − ω1)(E − ω2) ω1 − ω2 E − ω1 E − ω2 In both cases the Dirac norm is found to be negative, but both turn out to be non-Hermitian but PT theories, with their PT theory norms being positive. Thus identify G(E) not with 0 0 hΩR|T (φ(x)φ(x ))|ΩRi, but with hΩL|T (φ(x)φ(x ))|ΩRi instead. See Bender, Brandt, Chen and Wang (2005) for the Lee model (case (1)), Bender and Mannheim (2008 PRL) for the unequal-frequency Pais-Uhlenbeck oscillator (case (1)), and Bender and Mannheim (2008 PRD) for the equal-frequency Pais-Uhlenbeck oscillator (case (3)).

SUMMARY: RATHER THAN BEING A SIGN OF A DISEASE, A NEGATIVE DIRAC NORM COULD BE SIGNIFYING THAT ONE IS IN THE WRONG HILBERT SPACE AND USING THE WRONG NORM, AND THAT ONE IS IN A PT THEORY AND SHOUD BE USING THE UNITARITY-MAINTAINING PT THEORY NORM INSTEAD.

8 3 PT SYMMETRY AND THE LORENTZ GROUP

Under parity x¯ → −x¯. Under time reversal t → −t. Under P T xµ → −xµ. Equivalent to a COMPLEX Lorentz transformation: x0 = x cosh ξ + t sinh ξ, y0 = y cosh ξ + t sinh ξ, z0 = z cosh ξ + t sinh ξ, all with boost angle iπ where cosh(iπ) = −1, sinh(iπ) = 0. Under Lorentz the 6 components of the electromagnetic ﬁeld transform as the reducible D(1, 0) + D(0, 1). Irreducible combinations are E + iB, E − iB. But E and B are both PT odd, so under PT : E(t, x) ± iB(t, x) → −[E(−t, −x) ∓ iB(−t, −x)]. (6) The 6 E and B ﬁelds are thus irreducible under SO(3, 1) × PT . Under Lorentz the 4 components of a Dirac spinor transform as the reducible D(1/2, 0) + 5 5 D(0, 1/2). Irreducible combinations are ψL = (1 − γ )ψ, ψR = (1 + γ )ψ. Under PT : −1 −1 1 3 P ψ(t, x)P = γ0ψ(t, −x). T ψ(t, x)T = iγ γ ψ(−t, x), P T ψ(t, x)T −1P −1 = −γ2γ5ψ(−t, −x),PT ψ¯(t, x)T −1P −1 = −ψ¯(−t, −x)γ2γ5, PT (1 ∓ γ5)ψ(t, x)T −1P −1 = −(1 ± γ5)γ2γ5ψ(−t, −x). (7) So all 4 components of a Dirac spinor are irreducible under SO(3, 1) × PT .

9 4 PT SYMMETRY AND THE CONFORMAL GROUP

Unlike timelike and spacelike intervals, which are only invariant under the 10 Poincare transformations, lightlike intervals are invariant under the full 15-parameter SO(4, 2) conformal group that aﬀects coordinates as xµ + cµx2 xµ → xµ + µ, xµ → Λµ xν, xµ → λxν, xµ → ; ν 1 + 2c · x + c2x2 x2 x2 → λ2x2, x2 → , (8) 1 + 2c · x + c2x2 i.e. 4+6+1+4=15. With the 15 inﬁnitesimal generators acting on the coordinates xµ as µ µ µν µ ν ν µ µ µ 2 µν µ ν P = i∂ ,M = i(x ∂ − x ∂ ),D = ix ∂µ,C = i(x η − 2x x )∂ν, (9) together they close on the 15-parameter SO(4, 2) conformal group, with algebra

[Mµν,Mρσ] = i(−ηµρMνσ + ηνρMµσ − ηµσMρν + ηνσMρµ), [Mµν,D] = 0, [Mµν,Pσ] = i(ηνσPµ − ηµσPν), [Pµ,Pν] = 0, [Mµν,Cσ] = i(ηνσCµ − ηµσCν), [Cµ,Cν] = 0, [Cµ,Pν] = 2i(ηµνD − Mµν), [D,Pµ] = −iPµ, [D,Cµ] = iCµ. (10) The conformal algebra admits of a 4-dimensional spinor representation since the 15 Dirac matrices γ5, γµ, γµγ5,[γµ, γν] also close on the SO(4, 2) algebra. The group SU(2, 2) is the covering group of SO(4, 2) with the 4-dimensional spinor being its fundamental. SUMMARY: UNDER THE CONFORMAL GROUP ALL FOUR COM- PONENTS OF A DIRAC SPINOR ARE IRREDUCIBLE.

10 The fact that 4-component fermions are irreducible under the conformal group means that conformal transformations mix components with opposite PT . We now recognize the PT transformation on the spinors P T ψ(t, x)T −1P −1 = −γ2γ5ψ(−t, −x) as being none other than a conformal transformation since γ2γ5 is one the 15 generators of the conformal group. PT symmetry is thus integrally connected with conformal symmetry. With conformal symmetry neutrinos must be 4-component, and thus right-handed neutrinos must exist. Thus anticipate and will see below, a connection between conformal symmetry and chiral symmetry. 4 × 4∗ = 1 + 15 generates all 15 Dirac matrix generators in the adjoint of SU(2, 2), with there being exactly the needed 16 independent degrees of freedom for an arbitrary 4- ¯ ¯ ¯ ¯ ¯ dimensional space. The 1 + 15 transform as ψψ, ψγ5ψ, ψγµψ, ψγµγ5ψ, ψ(γµγν − γνγµ)ψ, 5 anti (i.e as S, P , Aµ, Aµ, gµν . Similarly, 4 × 4 generates a symmetric 10 and an antisymmetric T T sym 6 that transform as ψ C(γµγν + γνγµ)ψ and ψ C(γµγν − γνγµ)ψ, i.e as gµν , Fµν. Under the conformal group the E and B ﬁelds belong to the same irreducible representation, just as PT requires. Vector ﬁelds transform as the irreducible D(1/2, 1/2) under Lorentz, while continuing to belong to an irreducible representation under the conformal group. SUMMARY: PT IS INTEGRALLY CONNECTED WITH BOTH THE LORENTZ AND CONFORMAL GROUPS, WITH ALL FUNDAMEN- TAL FIELDS NOW BEING IN IRREDUCIBLE REPRESENTATIONS

11 5 PT SYMMETRY AND GRAVITY

Conformal symmetry is a natural symmetry in physics, with it being the full symmetry of the massless fermion and massless gauge boson sector of SU(3) × SU(2) × U(1). Conformal invariance forbids any double-well potential of the form V (φ) = λφ4 − µ2φ2. Hence particle masses must be generated dynamically by fermion condensates hΩ|ψψ¯ |Ωi rather than by fundamental scalar fields – no fundamental scalar fields in QCD but its global chiral symmetry is broken dynamically to produce a Goldstone boson pion. Conformal invariance can be made local with any general field of conformal weight d transforming as Φ(x) → edα(x)Φ(x). Thus natural to take gravity to be conformal too. With the conformal Weyl tensor 1 1 C = R + Rα [g g − g g ] − [g R − g R − g R + g R ] λµνκ λµνκ 6 α λν µκ λκ µν 2 λν µκ λκ µν µν λκ µκ λν (11) 2α(x) 2α(x) behaving as Cµνστ → e Cµνστ under the LOCAL gµν(x) → e gµν(x) (all derivatives of α(x) drop out), the gravitational action Z Z 1 I = −α d4x(−g)1/2C Cλµνκ = −2α d4x(−g)1/2 RµνR − (Rα )2 (12) W g λµνκ g µν 3 α is the unique gravitational action in four spacetime dimensions that is locally conformal invariant. With the conformal invariance requiring the coupling constant αg to be dimen- sionless, as a quantum theory conformal gravity is power counting renormalizable.

12 Associated gravitational equations of motion are FOURTH-ORDER DERIVATIVE: 1 1 W µν = gµν(Rα );β + Rµν;β − Rµβ;ν − Rνβ;µ − 2RµβRν + gµνR Rαβ 2 α ;β ;β ;β ;β β 2 αβ 2 µν α ;β 2 α ;µ;ν 2 α µν 1 µν α 2 1 µν − g (R α) ;β + (R α) + R αR − g (R α) = T , (13) 3 3 3 6 4αg and being fourth-order, conformal gravity has long been thought to not be unitary, just like its prototype the fourth-order Pais -Uhlenbeck oscillator with action and equation of motion γ Z d4z d2z I = dt[z¨2 − (ω2 + ω2)z˙2 + ω2ω2z2], + (ω2 + ω2) + ω2ω2z = 0. (14) PU 2 1 2 1 2 dt4 1 2 dt2 1 2 The Pais-Uhlenbeck Hamiltonian is constrained, so set x = z˙, and obtain (Mannheim and Davidson (2000)) the two oscillator p2 γ γ H = x + p x + ω2 + ω2 x2 − ω2ω2z2, [z, p ] = 0, [x, p ] = 0. (15) 2γ z 2 1 2 2 1 2 z x 2 2 2 Because of −(γ/2)ω1ω2z term we would appear to have negative energy states with energy spectrum being unbounded from below. However, if z (and thus pz) are anti-Hermitian then all energy states are found to be positive (Bender and Mannheim (2008PRL)). Thus set

πpzz/2 −πpzz/2 πpzz/2 −πpzz/2 y = e ze = −iz, py = e pze = ipz, p2 γ γ H = x − ip x + ω2 + ω2 x2 + ω2ω2y2, (16) 2γ y 2 1 2 2 1 2

13 and with 1 ω1 + ω2 2 2 2 Q = log (pxpy + γ ω1ω2xy), (17) γω1ω2 ω1 − ω2 obtain the completely well-behaved 2 2 ˜ −Q/2 Q/2 px py γ 2 2 γ 2 2 2 ˜ † H = e He = + 2 + ω1x + ω1ω2y = H , (18) 2γ 2γω1 2 2 with eigenstates and norms for H˜ and H of the form

hn˜|m˜ i = δm,n, Σ|n˜ihn˜| = 1, H˜ = Σ|n˜iEnhn˜| (19) −Q −Q −Q −Q hn|e |mi = δm,n, Σ|nihn|e = 1,H = Σ|niEnhn|e , hΩL| = hΩR| . (20) Thus, in parallel, no states of negative energy and no states of negative norm in conformal gravity itself, with conformal gravity being a PT theory with a gµν that is anti-Hermitian. With the Hamiltonian being a PT Hamiltonian one should construct Hilbert space scalar products using the norm of PT theory, and that is how the conformal gravity theory gets to be ghost free and unitary. Thus through the use of both a discrete PT symmetry and a continuous local conformal symmetry, one is now able to construct A CONSISTENT THEORY OF QUANTUM GRAVITY, VIZ. CONFORMAL GRAVITY in four spacetime dimensions, viz. the only spacetime dimensions for which we actually have evidence.

14 6 PT SYMMETRY AND GEOMETRY

6.1 LOCAL TRANSLATION CONNECTIONS

0µ µ ν µ 2 µ ν µ µ µ µ 2 µ 2 In special relativity x = Λν x + a , dτ = ηµνdx dx . Both Λν and a independent of coordinates. x , dx /dτ, d x /dτ all transform as vectors. 0µ µ ν µ 2 µ ν 0λ 0λ µ µ µ In general relativity x = Λν (x)x + a (x), dτ = gµν(x)dx dx . Vectors transform as A = (dx /dx )A . Only dx /dτ transforms as a vector. xµ and d2xµ/dτ 2 do not transform as vectors. xµ is now just a coordinate, so can never be a vector. But with a connection can make d2xµ/dτ 2 a vector. d dx0λ d dx0λ dxµ dx0λ d2xµ d2x0λ dxµ dxν = = + . (21) dτ dτ dτ dxµ dτ dxµ dτ 2 dxµdxν dτ dτ λ Introduce a connection Γ µν which transforms as dx0λ dxβ dxγ d2xρ dx0λ Γ0λ (x0) = Γα (x) + . (22) µν dxα dx0µ dx0ν βγ dx0µdx0ν dxρ It would transform as a tensor if the second term were not there. However this second term is precisely needed to construct a covariant acceleration that does transform as a vector: D2xλ d2xλ dxµ dxν D2x0λ dx0λ D2xµ = + Γλ , = , (23) Dτ 2 dτ 2 µν dτ dτ Dτ 2 dxµ Dτ 2 with D2xλ/Dτ 2 = 0 being geodesic. This gives the equivalence principle. At any point one can transform the connection away and get d2xλ/dτ 2 = 0. However, cannot simultaneously transform the connection away at any adjacent point, if there 2 λ 2 is curvature, i.e. if the Riemann tensor is non-zero. Moreover, mD x /Dτ = 0, to thus enforce minertial = mgravitational if there is curvature, i.e. if there is gravity. λ No constraint on Γ µν except how it transforms as that is all that is needed to make the Riemann tensor λ λ λ η λ η λ R µνκ = ∂κΓ µν − ∂νΓ µκ + Γ µνΓ ηκ − Γ µκΓ ην (24) λ be a tensor. Hence once R µνκ does not vanish in any given coordinate system it does not vanish in any coordinate system. λ ˜λ λ λ This is gravity. If introduce a true rank-3 tensor δΓ µν, then could use Γ µν = Γ µν + δΓ µν as the connection without λ changing any tensorial transformation properties, with R µνκ being replaced by the true rank 4 tensor ˜λ ˜λ ˜λ ˜η ˜λ ˜η ˜λ R µνκ = ∂κΓ µν − ∂νΓ µκ + Γ µνΓ ηκ − Γ µκΓ ην. (25)

15 6.2 LEVI-CIVITA, CONTORSION AND WEYL CONNECTIONS

In standard Riemannian geometry the Levi-Civita connection is used for local translations: 1 Λλ = gλα(∂ g + ∂ g − ∂ g ) = Λλ . (26) µν 2 µ να ν µα α νµ νµ The metric obeys the metricity (metric compatible) condition: λν λν λ αν ν λα ∇µg = ∂µg + Λ αµg + Λ αµg = 0. (27) λ λ Nothing requires that Γ µν be given by Λ µν alone or even be symmetric on its µ, ν indices. λ Introduce the Cartan TORSION tensor Q µν according to λ λ λ λ Q µν = Γ µν − Γ νµ = −Q νµ. (28) Introduce the CONTORSION tensor according to 1 Kλ = gλα(Q + Q − Q ). (29) µν 2 µνα νµα ανµ ˜λ λ λ With Γ µν = Λ µν + K µν the metric obeys λν ∇˜ µg = 0. (30)

λ Introduce the WEYL connection W µν according to λ λα λ W µν = g (gναBµ + gµαBν − gνµBα) = W νµ, (31) ˜λ λ λ where Bµ is a vector field. With Γ µν = Λ µν + W µν the metric obeys λν λν ∇˜ µg = 2g Bµ, (32) and is not metric compatible. Geometry is not Riemannian, and is known as a Weyl geometry. 2α(x) α(x) Objective: Invariance under gµν(x) → e gµν(x), Bµ(x) → e Bµ(x), while leaving ∂µBν − ∂µBν invariant. Hence unification of gravitation and electromagnetism – the very first one. Problem: parallel transport is path dependent. Current state of a system depends on its prior history. This problem has never been fully resolved. Also exact conformal invariance implies masslessness. Now solved by Goldstone and Higgs mechanisms.

16 6.3 LOCAL LORENTZ CONNECTIONS

R 4 ¯ a µν The Dirac action d xiψγ ∂aψ is not invariant under local Lorentz transformations of the form exp(wµν(x)M ). So need ab a another connection, known as the SPIN or LORENTZ connection ωµ . Introduce a set of four vierbeins Vµ where the coordinate a refers to a ﬁxed, special relativistic reference coordinate system with metric ηab, with the metric then being a b writable as gµν = ηabVµ Vν . The covariant derivative of the vierbein

aλ aλ λ aν ab λ DµV = ∂µV + Λ νµV + ωµ Vb (33)

a λ a c λ τ will transform as a Lorentz vector under Vµ (x ) → Λ c(x)Vµ (Λ τ x ) provide the spin connection transforms as

0ab a b cd bc a Ωµ = Λ c(x)Λ d(x)Ωµ − Λ (x)∂µΛ c(x). (34)

aλ To get metricity at the vierbein level, i.e. DµV = 0, use 1 1 −ωab = V b∂ V aν + V bΛλ V aν = (V b∂ V aν − V a∂ V bν) + V bαV aν(∂ g − ∂ g ) = ωba. (35) µ ν µ λ νµ 2 ν µ ν µ 2 ν αµ α µν µ ab ab b aν b ˜λ aν Can generalize −ωµ to −ω˜µ = Vν ∂µV + Vλ Γ νµV .

MAJOR CONCERN: IF WE USE ANY GENERALIZED CONNECTION TO METRICATE ANY FUN- DAMENTAL FORCE, THEN HOW DO WE PREVENT THE FIELDS IN THE CONNECTION AND ANY INTERNAL QUANTUM NUMBERS THEY MIGHT CARRY FROM COUPLING TO THE GENERAL- IZED RIEMANN TENSOR R˜µνστ AND THUS TO THE METRIC ITSELF AS WE WILL SEE, PT SYMMETRY AND CONFORMAL SYMMETRY WILL PROTECT US, AND TO SEE IT WILL NEED TO DISCUSS NOT THE METRIC SECTOR BUT THE FERMION SECTOR.

17 7 PT SYMMETRY AND THE DIRAC ACTION

7.1 STANDARD CURVED SPACE DIRAC ACTION

With γaγb + γbγa = 2ηab need spin connection and vierbeins to generalize ﬂat space Dirac action 1 Z I = d4xψγ¯ ai∂ ψ + H.c. (36) D 2 a to standard Levi-Civita-based Riemannian curved space: 1 Z I = d4x(−g)1/2iψγ¯ aV µ(∂ + Σ ωbc)ψ + H.c. (37) D 2 a µ bc µ where Σab = (1/8)[γa, γb]. Could equally use CPT conjugate rather than Hermitian conjugate. In either case on integrating by parts obtain Z 4 1/2 ¯ a µ bc ID = d x(−g) iψγ Va (∂µ + Σbcωµ )ψ. (38)

18 7.2 GENERALIZED CURVED SPACE DIRAC ACTION

When one has a generalized connection the Dirac action is given by 1 Z I˜ = d4x(−g)1/2iψγ¯ aV µ(∂ + Σ ω˜bc)ψ + H.c. (39) D 2 a µ bc µ

Following an integration by parts I˜D is found to take the form 1 Z I˜ = I + d4x(−g)1/2iψV¯ aµV bλV cν(δΓ γ [γ , γ ] + (δΓ )†[γ , γ ]γ )ψ, (40) D D 16 λνµ a b c λνµ b c a R 4 1/2 ¯ a µ bc with integration by parts not simply giving back d x(−g) iψγ Va (∂µ + Σbcω˜µ )ψ. Recalling that γa[γb, γc] − [γb, γc]γa = 4ηabγc − 4ηacγb a b c b c a abcd 5 γ [γ , γ ] + [γ , γ ]γ = 4i γdγ , γ5 = iγ0γ1γ2γ3, abcd µ ν σ τ −1/2 µνστ Va Vb Vc Vd = (−g) , (41) we can rewrite I˜D as 1 Z I˜ = I + d4x(−g)1/2iψγ¯ δΓdψ (42) D D 4 d where 1 1 δΓd = [δΓ + (δΓ )†](−g)−1/2µλντ iγ5V d + [δΓ − (δΓ )†][gµλV dν − gµνV dλ]. (43) 2 µλν µλν τ 2 µλν µλν With CP T ψ(t, x)T −1P −1C−1 = iψ†(−t, −x)γ5, CPT ψ¯(t, x)T −1P −1C−1 = −iγ0γ5ψ(−t, −x), if we use the CPT conjugate instead of Hermitian conjugate then get 1 Z I˜ = I + d4x(−g)1/2iψV¯ aµV bλV cν(δΓ γ [γ , γ ] − (δΓ )CPT [γ , γ ]γ )ψ, (44) D D 16 λνµ a b c λνµ b c a

CPT † with equivalence requiring (δΓµλν) = −(δΓµλν) . If does not hold, then CPT invariance goes further than Hermiticity.

19 7.3 DIRAC EQUATION WITH TORSION

˜λ λ λ λ With Γ µν = Λ µν + K µν, on taking K µν to be Hermitian, 1 Z I˜ = d4x(−g)1/2iψγ¯ aV µ(∂ + Σ ω˜bc)ψ + H.c. (45) D 2 a µ bc µ takes the form Z ˜ 4 1/2 ¯ a µ bc 5 ID = d x(−g) iψγ Va (∂µ + Σbcωµ − iγ Sµ)ψ, (46) where 1 1 Sµ = (−g)−1/2µαβγQ , (−g)−1/2 Sµ = [Q + Q + Q ]. (47) 8 αβγ µαβγ 4 αβγ γαβ βγα iγ5β(x) This action has local axial gauge invariance under: ψ → e ψ, Sµ → Sµ + ∂µβ(x). Thus could have generated this same term via minimal coupling in ID, viz. Z 4 1/2 ¯ a µ bc 5 ID = d x(−g) iψγ Va (∂µ + Σbcωµ − iγ Sµ)ψ (48)

λ in a strictly Riemannian geometry with connection Λ µν. THUS AXIAL SYMMETRY IS EQUIVALENT TO TORSION - A DUAL DESCRIPTION - AND RATHER THAN BEING SOMEWHAT UNCONVENTIONAL OR EXOTIC, TORSION IS SOMETHING WE SEE EVERYDAY AS THE AXIAL SECTOR OF GAUGE THEORIES SUCH AS SU(3) × SU(2) × U(1).

MOREOVER, EVEN THOUGH WE DID NOT DEMAND IT A PRIORI, THE CURVED SPACE DIRAC ACTION ID IS BOTH LOCALLY CONFORMAL INVARIANT AND PT INVARIANT, WITH THE AXIAL Sµ HAVING CONFORMAL WEIGHT ZERO AND BEING PT ODD.

20 iα(x) Moreover, could introduce electromagnetism via minimal coupling on ID by requiring invariance under ψ → e ψ, Aµ → Aµ + ∂µα(x), to obtain:

Z ˜ 4 1/2 ¯ a µ bc 5 JD = d x(−g) iψγ Va (∂µ + Σbcωµ − iAµ − iγ Sµ)ψ, (49) with the local conformal invariance and PT invariance being maintained since Aµ has conformal weight zero and is PT even. R ¯ Do path integration over the fermions DψDψ exp(iJ˜D) to obtain an eﬀective action of the form (’t Hooft (2010), Shapiro (2002)): Z 1 1 1 1 I = d4x(−g)1/2C R Rµν − (Rα )2 + F F µν + S Sµν , (50) EFF 20 µν 3 α 3 µν 3 µν

µν α 2 where C is a log divergent constant, Fµν = ∂µAν − ∂νAµ, Sµν = ∂µSν − ∂νSµ. We recognize the RµνR − (1/3)(R α) term µν µν to be that of conformal gravity, and the FµνF and SµνS terms to be those of a chiral electromagnetism.

EVEN THOUGH THE CONNECTION CONTAINS TORSION, THE CURVATURE TERM THAT IS GENERATED IS BASED ON THE STANDARD WEYL TENSOR AND NOT ON SOME GENERALIZED TORSION-DEPENDENT GENERALIZATION OF IT. THIS IS BECAUSE EVEN THOUGH THE WEYL TENSOR ITSELF IS LOCALLY CONFORMAL INVARIANT THE GENERALIZED WEYL TENSOR IS NOT, WITH THE LOCAL CONFORMAL INVARIANCE AND PT INVARIANCE OF THE DIRAC ACTION J˜D REQUIRING THAT IEFF POSSESS THESE INVARIANCES TOO. THUS THE GRAVITY SECTOR DOES NOT SEE THE TORSION AND IS BASED ON THE STAN- DARD LEVI-CIVITA CONNECTION ALONE.

21 Moreover, can even extend the minimal coupling to SU(N) × SU(N) with SU(N) generators T i that obey [T i,T j] = ijk k i i i i i i if T . Replace Aµ by gT Aµ and Qαβγ by gT Qαβγ, and thus replace Sµ by gT Sµ, to obtain a locally U(N) × U(N) invariant Dirac action of the form Z ˜ 4 1/2 ¯ a µ bc 5 i i 5 i i JD = d x(−g) iψγ Va (∂µ + Σbcωµ − iAµ − iγ Sµ − igT Aµ − igγ T Sµ)ψ. (51)

R ¯ For the fermion path integration DψDψ exp(iJ˜D) yields an eﬀective action of the form (’t Hooft (2010)): Z 1 1 1 1 1 1 I = d4x(−g)1/2C R Rµν − (Rα )2 + F F µν + S Sµν + F i F iµν + Si Siµν , (52) EFF 20 µν 3 α 3 µν 3 µν 3 µν 3 µν

i i i ijk j k i i i ijk j k where C is the same log divergent constant, where Fµν = ∂µAν −∂νAµ +gf AµAν, and where Sµν = ∂µSν −∂νSµ +gf SµSν . Thus we obtain conformal gravity coupled to a chiral U(N) × U(N) Yang-Mills theory.

2α(x) a α(x) a i Full local conformal invariance if gµν(x) → e gµν(x), Vµ (x) → e Vµ (x), Aµ(x) → Aµ(x), Sµ(x) → Sµ(x), Aµ(x) → i i i Aµ(x), Sµ(x) → Sµ(x).

i Need to understand why massless Sµ and Sµ have never been seen. Could be Grassmann (torsion is antisymmetric), could acquire large Higgs mechanism masses by chiral symmetry breaking, or could be conﬁned.

22 7.4 INVARIANCES

Under PT a generic ψ¯(xµ)Γψ(xµ) will transform into ψ¯(−xµ)γ2γ5Γ∗γ2γ5ψ(−xµ). Thus ψψ¯ , ψγ¯ µψ, ψ¯(γµγν − γνγµ)ψ are PT even, while ψγ¯ 5ψ and ψγ¯ µγ5ψ are PT odd. µ aν µ µ aν µ λ bc Under PT we have [∂/∂x ]V (−x ) = −[∂/∂(−x )]V (−x ) (and analogously for Λ νµ) to make wµ act as an odd PT R 4 R 4 bc operator in the d x = d (−x) integration. Thus i∂µ and iwµ both act as PT even operators. Thus with Aµ being PT even and Sµ being PT odd Z ˜ 4 1/2 ¯ a µ bc 5 i i 5 i i JD = d x(−g) iψγ Va (∂µ + Σbcωµ − iAµ − iγ Sµ − igT Aµ − igγ T Sµ)ψ. (53) is PT symmetric. The full set of invariances of J˜D is (1) local translations (2) local Lorentz transformations (3) local conformal invariance (4) local U(1) gauge invariance (5) local axial U(1) gauge tinvariance (6) local chiral Yang-Mills (6) PT invariance (7) CPT invariance

i To complete the circle need to ﬁnd a geometric origin for Aµ and Aµ.

23 i 8 PT SYMMETRY AND THE METRICATION OF Aµ AND Aµ

It had been suggested by Weyl that one could metricate electromagnetism by introducing the Bµ-dependent Weyl connection λ W µν with a real Bµ. However, if we now use 1 Γ˜λ = Λλ + W λ + Kλ = gλα(∂ g + ∂ g − ∂ g ) + gλα(g B + g B − g B ) + Kλ (54) µν µν µν µν 2 µ να ν µα α νµ να µ µα ν νµ α µν in the spin connection, then on recalling that 1 Z I˜ = I + d4x(−g)1/2iψγ¯ δΓdψ (55) D D 4 d 1 1 δΓd = [δΓ + (δΓ )†](−g)−1/2µλντ iγ5V d + [δΓ − (δΓ )†][gµλV dν − gµνV dλ]. (56) 2 µλν µλν τ 2 µλν µλν the Bµ dependence is found to drop out identically. THE WEYL FIELD Bµ DOES NOT COUPLE TO FERMIONS. Now if Bµ is the regular electromagnetic field it would be PT even, while Sµ is PT odd. However ∂µ is PT odd. Thus λ λ λ Λ µν + W µν + K µν would mix fields with differing PT assignments. With the development of quantum mechanics it was realized that the phase of a wave function of an electrically charged particle could be complex, and Weyl noted that if one dropped the connection to gravitation one could use Aµ → Aµ +∂µα(x) iα(x) to compensate for a phase change ψ(x) → e ψ(x) on the wave function by minimally coupling as ∂µ − iAµ. Now ∂µ − iAµ λ λ is okay since both terms are PT odd. So let us do the same thing for W µν, and make it PT odd too. Thus replace W µν by 2 2 V λ = − igλα (g A + g A − g A ) − igλα g gT iAi + g gT iAi − g gT iAi (57) µν 3 να µ µα ν νµ α 3 να µ µα ν νµ α Find that its insertion into the spin connection with its convenient −2/3 charge normalization leads to none other than the i ˜ Aµ-dependent and Aµ-dependent contributions to JD precisely as given and normalized above, viz.: Z ˜ 4 1/2 ¯ a µ bc 5 i i 5 i i JD = d x(−g) iψγ Va (∂µ + Σbcωµ − iAµ − iγ Sµ − igT Aµ − igγ T Sµ)ψ. (58) Now no longer any path dependent parallel transport problem since the geometry obtained from the fermion path integration µν α 2 is the strictly Riemannian RµνR −(1/3)(R α) . Thus by taking Aµ to have conformal weight zero and the Weyl connection to µν µν be pure imaginary, then even though ∇˜ σg = −(4/3)g iAσ 6= 0, we convert Weyl geometry into Riemannian geometry, and obtain a DUAL DESCRIPTION (MINIMAL COUPLING OR METRICATION) OF THE FUNDAMENTAL FORCES.

24 9 THE UNEXPECTED PT SURPRISE

The Pais-Uhlenbeck fourth-order oscillator describes the quantum-mechanical limit of the conformal gravity, with action γ Z I = dt z¨2 − ω2 + ω2 z˙2 + ω2ω2z2 , (59) PU 2 1 2 1 2 where γ, ω1, and ω2 are all positive constants. The associated equation of motion and propagator are 4 2 d z 2 2 d z 2 2 1 1 1 1 4 + (ω1 + ω2) 2 + ω1ω2z = 0,G(E) = 2 2 2 2 = 2 2 2 2 − 2 2 , (60) dt dt (E − ω1)(E − ω2) ω1 − ω2 E − ω1 E − ω2 and theory would appear to have a negative Dirac norm ghost state. However, when reinterpreted as a PT theory and the PT theory norm is used (hψCPT |ψi), there are then no states of negative norm (Bender and Mannheim 2008). The Hamiltonian is constrained, so set x =z ˙, and obtain (Mannheim and Davidson 2000) the two oscillator p2 γ γ H = x + p x + ω2 + ω2 x2 − ω2ω2z2. (61) 2γ z 2 1 2 2 1 2 2 2 2 Because of −(γ/2)ω1ω2z term we would appear to have negative energy states with energy spectrum being unbounded from below. However, if z is anti-Hermitian then all energy states are found to be positive (Bender and Mannheim 2008). p2 γ γ y = eπpzz/2ze−πpzz/2 = −iz, p = eπpzz/2p e−πpzz/2 = ip ,H = x − ip x + ω2 + ω2 x2 + ω2ω2y2, (62) y z z 2γ y 2 1 2 2 1 2

2 2 1 ω1 + ω2 2 2 2 ˜ −Q/2 Q/2 px py γ 2 2 γ 2 2 2 Q = log (pxpy + γ ω1ω2xy), H = e He = + 2 + ω1x + ω1ω2y , (63) γω1ω2 ω1 − ω2 2γ 2γω1 2 2 Thus, by analog, no states of negative energy and no states of negative norm in conformal gravity if the conformal gravity theory is a PT theory with a gµν that is anti-Hermitian. With a metrication of Aµ being associated with PT symmetry and an Aµ that is anti-Hermitian, we see that gµν and Aµ are treated identically, just as one might even anticipate given µν µν ∇˜ σg = −(4/3)g iAσ. SUMMARY: THROUGH THE WEYL AND TORSION CONNECTIONS, PT SYMMETRY AND LOCAL CONFORMAL SYMMETRY WE ARE ABLE TO METRICATE ALL THE FUNDAMENTAL FORCES, IN A MANNER THAT TREATS gµν AND Aµ IDENTICALLY.

25 10 ASTROPHYSICAL EVIDENCE FOR THE PT REALIZATION OF CONFORMAL GRAVITY

Fourth order equation for potential (Mannheim and Kazanas 1994):

000 4 0000 4B 3 0 r ∇ B(r) = B + = (T 0 − T r) ≡ f(r), (64) r 4αgB(r) with solution r Z r 1 Z r 1 Z ∞ r2 Z ∞ B(r) = − dr0 r02f(r0) − dr0 r04f(r0) − dr0 r03f(r0) − dr0 r0f(r0) + Bˆ(r), (65) 2 0 6r 0 2 r 6 r where Bˆ(r) obeys ∇4Bˆ(r) = 0. ∗ −R/R Take each spiral galaxy to be a thin disk with N stars distributed with surface brightness Σ(R) = Σ0e 0 and luminosity 2 L = 2πR0Σ0. The net contribution to circular velocities due the local material within galaxies is found to be given by ∗ ∗ 2 2 ∗ ∗ 2 2 2 N β c R R R R R N γ c R R R vLOC(disk) = 3 I0 K0 − I1 K1 + I1 K1 . (66) 2R0 2R0 2R0 2R0 2R0 2R0 2R0 2R0 ∗ The ratio N M /L is recognized as the galactic mass to light ratio M/L. From the rest of the universe γ c2R v2 (R) = v2 (disk) + 0 − κc2R2, (67) TOT LOC 2 M G β∗ = = 1.48 × 105 cm, γ∗ = 5.42 × 10−41 cm−1, γ = 3.06 × 10−30 cm−1. κ = 9.54 × 10−54 cm−2. (68) c2 0 Mannheim and O’Brien (2011, 2012, 2013) ﬁt 141 galaxies with VISIBLE N ∗ of each galaxy as only variable, ∗ ∗ ∗ β , γ , γ0 and κ are all universal, and with NO DARK MATTER, and with 282 less free parameters than in dark matter calculations.

26 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 velocities of the Miscellaneous 22 galaxy sample 32 140 120 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

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UGC 4499 UGC 5414 UGC 5721 0 0 0 0 2 4 6 8 0 1 2 3 4 0 2 4 6 8

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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 km sec−1) of the 24 dwarf galaxy sample – Part 1 33 100 50 120

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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

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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

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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

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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 velocities of the 24 dwarf galaxy sample – Part 2 34 60 50 80 50 40

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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

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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

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

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40 30 30

20 20 20 10 10 NGC5291N NGC5291S NGC5291SW 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 FIG. 11: Rotation curves for the tidal dwarf galaxies NGC 5291N, NGC 5291S, and NGC 5291SW

35 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 [?][?][?][?]

36 ALL YOU NEED TO KNOW ABOUT STANDARD COSMOLOGY

dr2 ds2 = −c2dt2 + R2(t) + r2dθ2 + r2sin2θdφ2 (69) 1 − kr2

R˙ RR¨ H = , q = − (70) R R˙ 2

A T = (ρ + p )U U + p g − Λg , ρ = (71) µν M M µ ν M µν µν M Rn

2 2 2 R˙ + kc = R˙ [ΩM + ΩΛ] (72)

FLATNESS PROBLEM IF BIG BANG – INFLATION

RR¨ n q = − = − 1 ΩM − ΩΛ (73) R˙ 2 2

8πGρ 8πGΛ Ω = M , Ω = , (74) M 3c2H2 Λ 3cH2

4 ΩΛ cΛ TV = = 4 (75) ΩM ρM T

37 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 [?][?][?][?]

38 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 [?][?][?][?]

39 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

40 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 [?][?][?][?]

41 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

42 Table 1: Properties of the 3 Tidal Dwarf Galaxies

tot 2 2 Galaxy D LB i (R0)gas Rlast Mgas Mdisk (M/L)disk (v /c R)last 8 B ◦ 8 8 B −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) NGC 5291N 62.0 9.5 45 0.8 4.7 7.7 3.4 0.35 5.7 NGC 5291S 62.0 10.7 45 0.8 5.2 8.6 2.1 0.20 1.9 NGC 5291SW 62.0 5.7 45 0.8 2.7 4.6 1.1 0.20 3.4 NGC 5291N 62.0 9.5 55 0.8 4.7 7.7 1.9 0.20 4.3

(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

14 0.02 0.05 0.1 0.243 0.5 1.0 redshift z Supernovae data as log plot – Nobel Prize 2011 to Perlmutter, Riess and Schmidt 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.

44 ALL YOU NEED TO KNOW ABOUT CONFORMAL COSMOLOGY

Z 1 1 I = − d4x(−g)1/2 S;µS − S2Rµ + λS4 + iψγ¯ µ[∂ + Γ ]ψ − hSψψ¯ (76) M 2 ;µ 12 µ µ µ 1 1 T = (ρ + p )U U + p g − S2 R − g Rα − g λS4 (77) µν M M µ ν M µν 6 0 µν 2 µν α µν 0 Cλµνκ = 0,Tµν = 0 (78)

1 1 S2 R − g Rα = (ρ + p )U U + p g − g λS4 (79) 6 0 µν 2 µν α M M µ ν M µν µν 0 3 Geff = − 2 (80) 4πS0 ¨ 2 2 2 RR n R˙ + kc = R˙ Ω¯ M + Ω¯ Λ , q = − = − 1 Ω¯ M − Ω¯ Λ (81) R˙ 2 2 ¯ 4 ¯ 8πGeffρM ¯ 8πGeffΛ ΩΛ cΛ TV ΩM = 2 2 , ΩΛ = 2 , = = − 4 (82) 3c H 3cH Ω¯ M ρM T 2 −1 2 2 −1 4 ¯ T T Tmax ¯ T ¯ ΩΛ = 1 − 2 1 + 4 , ΩM = − 4 ΩΛ (83) Tmax TV TV

0 ≤ Ω¯ Λ ≤ 1, − 1 ≤ q ≤ 0 (84)

2 1/2! c (1 + z) q0 dL = − 1 − 1 + q0 − 2 (85) H0 q0 (1 + z )

45 24

20 apparent magnitude

14 0.0 0.2 0.4 0.6 0.8 1.0 redshift

Figure 1: 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.

46 30

22 apparent magnitude

14 0 1 2 3 4 5 redshift

Figure 2: 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).

47 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.