Early Universe Models

Transcription

1 Ma148b Spring 2016 Topics in Mathematical Physics

2 This lecture based on, Elena Pierpaoli, Early universe models from Noncommutative Geometry, Advances in Theoretical and Mathematical Physics, Vol.14 (2010) N.5, Daniel Kolodrubetz,, Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology, Phys. Lett. B, Vol.693 (2010) Re-examine RGE flow; gravitational terms in the asymptotic form of the spectral action; cosmological timeline; running in the very early universe; inflation and other gravitational phenomena

3 Asymptotic expansion of spectral action for large energies 1 S = 2κ 2 R g g d 4 x + γ 0 d 4 x 0 + α 0 C µνρσ C µνρσ g d 4 x + τ 0 R R g d 4 x + 1 DH 2 g d 4 x µ 2 0 H 2 g d 4 x 2 ξ 0 R H 2 g d 4 x + λ 0 H 4 g d 4 x + 1 (Gµν i G µνi + Fµν α F µνα + B µν B µν ) g d 4 x 4 A modified gravity model with non minimal coupling to Higgs and minimal coupling to gauge theories

4 Coefficients and parameters: 1 = 96f 2Λ 2 f 0 c 2κ π 2 γ 0 = 1 π 2 (48f 4Λ 4 f 2 Λ 2 c + f 0 4 d) α 0 = 3f 0 10π 2 τ 0 = 11f 0 60π 2 µ 2 0 = 2f 2Λ 2 e a f 0 ξ 0 = 1 12 λ 0 = π2 b 2f 0 a 2 f 0, f 2, f 4 free parameters, f 0 = f (0) and, for k > 0, f k = 0 f (v)v k 1 dv. a, b, c, d, e functions of Yukawa parameters of νmsm a = Tr(Y ν Y ν + Y e Y e + 3(Y u Y u + Y d Y d)) b = Tr((Y ν Y ν ) 2 + (Y e Y e ) 2 + 3(Y u Y u ) 2 + 3(Y d Y d) 2 ) c = Tr(MM ) d = Tr((MM ) 2 ) e = Tr(MM Y ν Y ν ).

5 Two different perspectives on running: parameters a, b, c, d, e run with RGE The relation between coefficients κ 0, γ 0, α 0, τ 0, µ 0, ξ 0, λ 0 and parameters a, b, c, d, e hold only at Λ unif : constraint on initial conditions; independent running There is a range of energies Λ min Λ Λ unif where the running of coefficients κ 0, γ 0, α 0, τ 0, µ 0, ξ 0, λ 0 determined by running of a, b, c, d, e and relation continues to hold (very early universe only) The first perspective requires independent running of gravitational parameters with given condition at unification; the second perspective allows for interesting gravitational phenomena in the very early universe specific to NCG model only

6 The first approach followed in Ali Chamseddine, Alain Connes,, Gravity and the Standard Model with neutrino mixing, ATMP 11 (2007) , arxiv:hep-th/ For modified gravity models ( 1 2η C µνρσc µνρσ ω 3η R2 + θ ) g η R R d 4 x Running of gravitational parameters (Avramidi, Codello Percacci, Donoghue) by β η = 1 (4π) η2 β ω = ω ω 2 (4π) 2 η 60 β θ = 1 7( θ) (4π) 2 η. 90

7 With relations at unification get running like log 10 Μ GeV log 10 Μ GeV Ω log Μ M Z Η log Μ M Z log 10 Μ GeV 0.2 Θ log Μ M Z Plausible values in low energy limit (near fixed points) Look then at second point of view: M.M.-E.Pierpaoli (2009)

8 Renormalization group equations for SM with right handed neutrinos and Majorana mass terms, from unification energy ( GeV) down to the electroweak scale (10 2 GeV) AKLRS S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt Running neutrino mass parameters in see-saw scenarios, JHEP 03 (2005) 024. Remark: RGE analysis in [CCM] only done using minimal SM

9 1-loop RGE equations: Λ df dλ = β f (Λ) 16π 2 β gi = b i g 3 i with (b SU(3), b SU(2), b U(1) ) = ( 7, 19 6, ) 16π 2 β Yu = Y u ( 3 2 Y u Y u 3 2 Y d Y d + a g g 2 2 8g 2 3 ) 16π 2 β Yd = Y d ( 3 2 Y d Y d 3 2 Y u Y u + a 1 4 g g 2 2 8g 2 3 ) 16π 2 β Yν = Y ν ( 3 2 Y ν Y ν 3 2 Y e Y e + a 9 20 g g 2 2 ) 16π 2 β Ye = Y e ( 3 2 Y e Y e 3 2 Y ν Y ν + a 9 4 g g 2 2 ) 16π 2 β M = Y ν Y ν M + M(Y ν Y ν ) T 16π 2 β λ = 6λ 2 3λ(3g g 2 1 ) + 3g (3 5 g g 2 2 ) 2 + 4λa 8b Note: different normalization from [CCM] and 5/3 factor included in g 2 1

10 Method of AKLRS: non-degenerate spectrum of Majorana masses, different effective field theories in between the three see-saw scales: RGE from unification Λ unif down to first see-saw scale (largest eigenvalue of M) Introduce Y ν (3) removing last row of Y ν in basis where M diagonal and M (3) removing last row and column. Induced RGE down to second see-saw scale Introduce Y ν (2) and M (2), matching boundary conditions Induced RGE down to first see-saw scale Introduce Y ν (1) and M (1), matching boundary conditions Induced RGE down to electoweak energy Λ ew Use effective field theories Y ν (N) and M (N) between see-saw scales

11 Running of coefficients a, b with RGE Coefficients a and b near the top see-saw scale

12 Running of coefficient c with RGE Running of coefficient d with RGE Effect of the three see-saw scales

13 Running of coefficient e with RGE Effect of the three see-saw scales With default boundary conditions at unification of AKLRS...but sensitive dependence on the initial conditions fine tuning

14 Changing initial conditions: maximal mixing conditions at unification ζ = exp(2πi/3) δ ( 1) = U PMNS (Λ unif ) = ζ ζ 2 ζ 1 ζ ζ 2 ζ Y ν = U PMNS δ ( 1)U PMNS Daniel Kolodrubetz,, Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology, arxiv:

15 Effect on coefficients running

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17 Further evidence of sensitive dependence: changing only one parameter in diagonal matrix Y ν get running of top term:

18 Geometric constraints at unification energy λ parameter constraint Higgs vacuum constraint af0 π λ(λ unif ) = π2 2f 0 b(λ unif ) a(λ unif ) 2 = 2M W g See-saw mechanism and c constraint 2f 2 Λ 2 unif f 0 c(λ unif ) 6f 2Λ 2 unif f 0 Mass relation at unification (mν 2 + me 2 + 3mu 2 + 3md 2 ) Λ=Λ unif = 8MW 2 Λ=Λ unif generations

19 Choice of initial conditions at unification: Compatibility with low energy values: experimental constraints Compatibility with geometric constraints at unification It is possible to modify boundary conditions to achieve both compatibilities Example: using maximal mixing conditions but modify parameters in the Majorana mass matrix and initial condition of Higgs parameter to satisfy geometric constraints

20 Cosmology timeline Planck epoch: t s after the Big Bang (unification of forces with gravity, quantum gravity) Grand Unification epoch: s t s (electroweak and strong forces unified; Higgs) Electroweak epoch: s t s (strong and electroweak forces separated) Inflationary epoch: possibly s t s - NCG SM preferred scale at unification; RGE running between unification and electroweak info on inflationary epoch. - Remark: Cannot extrapolate to modern universe, nonperturbative effects in the spectral action and phase transitions in the RGE flow

21 Cosmological implications of the NCG SM Linde s hypothesis (antigravity in the early universe) Primordial black holes and gravitational memory Running effective gravitational constant affecting gravitational waves Gravity balls Varying effective cosmological constant Higgs based slow-roll inflation Spontaneously arising Hoyle-Narlikar in EH backgrounds

22 Effective gravitational constant G eff = κ2 0 8π = Effective cosmological constant 3π 192f 2 Λ 2 2f 0 c(λ) γ 0 = 1 4π 2 (192f 4Λ 4 4f 2 Λ 2 c(λ) + f 0 d(λ))

23 Conformal non-minimal coupling of Higgs and gravity 1 R gd 4 x 1 R H 2 gd 4 x 16πG eff 12 Conformal gravity 3f 0 10π 2 C µνρσ C µνρσ gd 4 x C µνρσ = Weyl curvature tensor (trace free part of Riemann tensor) C λµνκ = R λµνκ 1 2 (g λνr µκ g µν R λκ +g µκ R λν )+ 1 6 (g λνg µκ g λκ g µν )

24 An example: G eff (Λ ew ) = G (at electroweak phase transition G eff is already modern universe Newton constant) 1/ G = GeV f 2 = G 1 eff (Λ) 96f 2Λ 2 24π 2 Term e/a lower order Dominant terms in the spectral action: ( 1 Λ 2 2 κ 2 R gd 4 x µ H 2 ) gd 4 x κ 0 = Λκ 0 and µ 0 = µ 0 /Λ, where µ 2 0 2f 2Λ 2 f 0

25 Detectable by gravitational waves: Einstein equations R µν 1 2 g µν R = κ 2 0 T µν g µν = a(t) 2 ( δ ij + h ij (x) trace and traceless part of h ij Friedmann equation (ȧ ) ( 4 a 2 (ȧ a ) ) ) ḣ + 2ḧ = κ2 0 Λ 2 T 00

26 Λ(t) = 1/a(t) (f 2 large) Inflationary epoch: a(t) e αt NCG model solutions: Ordinary cosmology: h(t) = 3π2 T f 2 α 2 e2αt + 3α 2 t + A 2α e 2αt + B ( 4πGT 00 α + 3α 2 )t + A 2α e 2αt + B Radiation dominated epoch: a(t) t 1/2 NCG model solutions: Ordinary cosmology: h(t) = 4π2 T 00 t 3 + B + A log(t) f 2 8 log(t)2 h(t) = 2πGT 00 t 2 + B + A log(t) log(t)2

27 Same example, special case: R 2 κ2 0 µ2 0 af 0 π 2 1 and H af 0 /π Leaves conformally coupled matter and gravity S c = α 0 C µνρσ C µνρσ g d 4 x + 1 DH 2 g d 4 x 2 ξ 0 R H 2 g d 4 x + λ 0 H 4 g d 4 x + 1 (Gµν i G µνi + Fµν α F µνα + B µν B µν ) g d 4 x. 4 A Hoyle-Narlikar type cosmology, normally suppressed by dominant Einstein Hilbert term, arises when R 1 and H v, near see-saw scale.

28 Cosmological term controlled by additional parameter f 4, vanishing condition: f 4 = (4f 2Λ 2 c f 0 d) 192Λ 4 Example: vanishing at unification γ 0 (Λ unif ) = Running of γ 0 (Λ): possible inflationary mechanism

29 The λ 0 -ansatz λ 0 Λ=Λunif = λ(λ unif ) π2 b(λ unif ) f 0 a 2 (Λ unif ), Run like λ(λ) but change boundary condition to λ 0 Λ=Λunif Run like λ 0 (Λ) = λ(λ) π2 b(λ) f 0 a 2 (Λ) For most of our cosmological estimates no serious difference

30 The running of λ 0 (Λ) near the top see-saw scale

31 Linde s hypothesis antigravity in the early universe A.D. Linde, Gauge theories, time-dependence of the gravitational constant and antigravity in the early universe, Phys. Letters B, Vol.93 (1980) N.4, Based on a conformal coupling 1 R gd 4 x 1 R φ 2 gd 4 x 16πG 12 giving an effective G 1 eff = G πφ2 In the NCG SM model two sources of negative gravity Running of G eff (Λ) Conformal coupling to the Higgs field

32 1 Example of effective Geff (Λ, f2 ) near the top see-saw scale Example: fixing Geff (Λunif ) = G gives a phase of negative gravity with conformal gravity becoming dominant near sign change of Geff (Λ) 1 at 1012 GeV

33 Gravity balls G eff,h = G eff 1 4π 3 G eff H 2 combines running of G eff with Linde mechanism Suppose f 2 such that G eff (Λ) > 0 G eff,h < 0 for H 2 > G eff,h > 0 for H 2 < Unstable and stable equilibrium for H: l H (Λ, f 2 ) := µ2 0 (Λ) = 2 f 2 Λ 2 f 0 2λ 0 (with λ 0 -ansatz) Negative gravity regime where e(λ) a(λ) λ(λ) π2 b(λ) f 0 a 2 (Λ) l H (Λ, f 2 ) > 3 4πG eff (Λ), 3 4πG eff (Λ). = (2f 2Λ 2 a(λ) f 0 e(λ))a(λ) π 2 λ(λ)b(λ) 3 4πG eff (Λ, f 2 )

34 An example of transition to negative gravity Gravity balls: regions where H 2 0 unstable equilibrium (positive gravity) surrounded by region with H 2 l H (Λ, f 2 ) stable (negative gravity): possible model of dark energy

35 Primordial black holes (Zeldovich Novikov, 1967) I.D. Novikov, A.G. Polnarev, A.A. Starobinsky, Ya.B. Zeldovich, Primordial black holes, Astron. Astrophys. 80 (1979) J.D. Barrow, Gravitational memory? Phys. Rev. D Vol.46 (1992) N.8 R3227, 4pp. Caused by: collapse of overdense regions, phase transitions in the early universe, cosmic loops and strings, inflationary reheating, etc Gravitational memory: if gravity balls with different G eff,h primordial black holes can evolve with different G eff,h from surrounding space

36 Evaporation of PBHs by Hawking radiation dm(t) dt (G eff (t)m(t)) 2 with Hawking temperature T = (8πG eff (t)m(t)) 1. In terms of energy: M 2 dm = 1 Λ 2 G 2 eff (Λ, f 2) dλ With or without gravitational memory depending on G eff behavior Evaporation of PBHs linked to γ-ray bursts

37 Higgs based slow-roll inflation dshw A. De Simone, M.P. Hertzberg, F. Wilczek, Running inflation in the Standard Model, hep-ph/ v2 Minimal SM and non-minimal coupling of Higgs and gravity. ξ 0 R H 2 gd 4 x Non-conformal coupling ξ 0 1/12, running of ξ 0 Effective Higgs potential: inflation parameter ψ = ξ 0 κ 0 H inflationary period ψ >> 1, end of inflation ψ 1, low energy regime ψ << 1

38 In the NCG SM have ξ 0 = 1/12 but same Higgs based slow-roll inflation due to κ 0 running (say κ 0 > 0) ψ(λ) = π ξ 0 (Λ)κ 0 (Λ) H = 2 96f 2 Λ 2 f 0 c(λ) H Einstein metric gµν E = f (H)g µν, for f (H) = 1 + ξ 0 κ 0 H 2 Higgs potential λ 0 H 4 V E (H) = (1 + ξ 0 κ 2 0 H 2 ) 2 For ψ >> 1 approaches constant; usual quartic potential for ψ << 1

39 Conclusion: Various possible inflation scenarios in the very early universe from running of coefficients of the spectral action according to the relation to Yukawa parameters: phases and regions of negative gravity, variable gravitational and cosmological constants, inflation potential from nonmininal coupling of Higgs to gravity Main problem: these effects depend on choice of initial conditions at unification (sensitive dependence) and several of these scenarios are ruled out when moving boundary conditions

40 Other cosmological aspects of the Spectral Action The problem of cosmic topology The Poisson summation formula Nonperturbative computation of the spectral action on 3-dimensional space forms Slow-roll inflation: potential, slow-roll coefficients, power spectra Slow-roll inflation from the nonperturbative spectral action and cosmic topology