1. Antigravity Between Crunch and Bang in a Geodesically Complete Universe Itzhak Bars University of Southern California Talk @ PDF-2011 Providence, August 2011 1)I.B. and S.H. Chen, 1004.0752 2)I.B., and S.H. Chen and Neil Turok, 1105.3606 3)I.B. + Chen + Turok + Steinhardt, to appear (several papers) 1/11 This talk is about solving cosmological equations analytically, no approximations. Found all the solutions for a specific model, and then discovered model independent phenomena that could not be noticed with approximate solutions. Among them is the notion of geodesic completeness, from which it follows there is a period of antigravity in the history of the universe. Also new general lessons for cosmology.

2. Analytically solved with this V: found ALL solutions Cosmology with a scalar coupled to gravity Also anisotropic metrics : Kasner, Bianchi IX. Two more fields in metric important only near Big Bang 2/11 Generic solution is geodesically incomplete in Einstein gravity. 6 parameters 4=b,c,K,ρ 0 2=(4-2) initial values There is a subset of geodesically complete solutions only with conditions on initial values and parameters of the model. Including relativistic matter and curvature V

3. This is not the whole story: Einstein gauge is valid only in a patch of spacetime, 3/11 For geodesic compleness: a slight extension of Einstein gravity (with gauge degrees of freedom) Local scaling symmetry (Weyl): allows only conformally coupled scalars (generalization possible) A prediction of 2T-gravity in 4+2 dims. no ghostsgravitational parameter Fundamental approach: Gauge symmetry in phase space I.B. 0804.1585, I.B.+Chen 0811.2510 Khury + Seiberg +Steinhardt + Turok McFadden + Turok 0409122 Can dynamics push this factor to negative values? ANTIGRAVITY in some regions of spacetime? (Plus gauge bosons, fermions, more conformal scalars, in complete Weyl invariant theory.) Also motivated by colliding branes scenario.  s)   s)e λ(x), g   g  e -2λ(x) Gauge symmetry leftover from general coordinate transformations in extra 1+1 dims.

4. Conformal factor of metric =1 for any metric.  Plus the energy constraint: H=0 This is equivalent to the 00 Einstein eq. G 00 =T 00 For all t,x dependence. FRW  4/11 Positive region which compensates for the ghost. BB singularity at a E =0 in E-gauge : gauge invariant factor vanishes in γ-gauge, or any gauge!! Nothing singular in γ-gauge BCRT transform Bars Chen Steinhardt Turok

5. Analytic solutions – all of them!! Special case: Particle in a potential problem, intuitively solved by looking at the plot of the potential. Completely decoupled equations, except for the zero energy condition. Solutions are Jacobi elliptic functions, with various boundary conditions. BCST transform Friedmann equations become : 5/11 First integral Generic solution has

6. 6/11 E(s)=E>0 E(φ)=E+ρ ≥ E(s)  s  perform independent oscilations For generic initial conditions, the sign of (  2 -s 2 )  changes over time. Generic solution is geodesically incomplete in the Einstein gauge. Geodesically complete with the natural extension in  s  space. K=0 case quartic potentials  s   A x F[sn(z|m), cn(z|m), dn(z|m )] z=(τ- τ 0 )/T A,m,T depend on b,c,K,ρ 0,E There are special solutions that are geodesically complete in the restricted Einstein frame, but must constrain parameter space.

7. Geodesically complete larger space: φ γ,s γ plane | φ|>|s| | φ|<|s| | φ|>|s| Generic solution:  s  periodic parametric plot using Mathematica A smooth curve that spans the various quadrants (not shown) Closed curve if periods relatively quantized. 7/11 Generic solution is a cyclic universe with antigravity stuck between crunch and bang! Probably true for all V(σ).  s |z| BCST Transform E-gauge to γ-gauge BCST Transform E-gauge to γ-gauge No signature change Recall Kruskal-Szekeres versus Schwarzchild; now in field space. big bangs or big crunches in spacetime  at the lightcone in  s field space.

8. Geodesically complete solutions in the Einstein gauge, without antigravity Conditions on 6 parameter space: (1) Synchronized initial values  s  (2) Relative quantization of periods P  (5 parameters) =nP s (5 parameters) 8/11 b>0 Universe expands to infinite size, turnaround at infinite size Example n=5  5 nodes in picture b<0 s   ss   aEaE σσ Cyclic universe

9. + 9/11 Anisotropy K0K0 If K≠0, ds 3 =Bianchi IX (Misner) In Friedmann equations, 2 more fields Friedman Eqs: kinetic terms for α 1, α 2 just like σ, plus anisotropy potential if K≠0 Without potentials can find all solutions analytically for any (initial) anisotropy momenta p 1,p 2 ; or σ momentum q including the parameters K,ρ 0. p 1 =(a E ) 2 ∂ τ α 1 etc. Free scalars if K=0, then canonical conjugate momenta p 1,p 2 are constants of motion. Near singularity, kinetic terms dominate, so all potentials, including V(σ) negligible. Then σ momentum q is also conserved near the singularity. For a range of q,p 1,p 2 mixmaster universe is avoided when σ is present (agree with BKL, etc.)

10. 10/11 Antigravity Loop (K=0 case) This picture for K=0 φ+s= φ -s= q= σ momentum, q=(a E ) 2 ∂ τ σ p= anisotropy momentum p 1 =(a E ) 2 ∂ τ α 1 etc. ALWAYS a period of antigravity sandwiched between crunch and bang ATTRACTOR MECHANISM !! if p 1 or p 2 is not 0 : both  s  0 at the big bang or crunch singularity, FOR ALL INITIAL CONDITIONS. Duration of loop : (p 2 +q 2 ) 1/2 /ρ Duration of loop : (p 2 +q 2 ) 1/2 /ρ K=0: p 1,p 2 are conserved throughout motion. q changes during the loop because of V(σ). If small loop, ≈no change.

11. 11/11 What have we learned? Found new techniques to solve cosmological equations analytically. Found all solutions for several special potentials V(σ). Several model independent general results: geodesic completeness, and an attractor mechanism to the origin,  s  0, for any initial values. Antigravity is very hard to avoid. Anisotropy + radiation + KE requires it. Studied Wheeler-deWitt equation (quantum) for the same system, can solve some cases exactly, others semi-classically. Same conclusions. Will this new insight survive the effects of a full quantum theory (very likely yes). Should be studied in string theory. These phenomena are direct predictions of 2T-physics in 4+2 dimensions. Open: What are the observational effects today of a past antigravity period? This is an important project. Study of small fluctuations and fitting to current observations of the CMB (under investigation).

12. 12/14 Antigravity Loop This picture for K≠0 q,p 1,p 2 change during each loop (if loop is large) because of the potentials, but trajectory always returns to the origin and connects gravity  antigravity regions

13. 13/14 K > 0 caseK < 0 case Higher level E>E*, with E s =E, E   =E+ρ Similar behavior to K=0 case. Lower level E,E   <E*, with E s =E, E   =E+ρ s oscillates in the V s well, while  oscillates outside the V  hill. Then for any initial values there is a finite bounce at size a E ≠0, no antigravity. Higher level E>0, with E s =E, E   =E+ρ Similar behavior to K=0 case. Lower level E*<E  <0, with E s =E, E   =E+ρ All solutions are geodesically incomplete in the Einstein gauge. There is no way to avoid antigravity. E E E*=K 2 /(16b). E E*=-K 2 /(16c)

14. Crunch  bangturnaround 14/14

15. Quantization of mini superspace The Wheeler- deWitt Equation Completely separable when Radiation adds just a constant, so it amounts to a new energy level, rather than zero on the right hand side, therefore easily handled. When f(s/φ)=0, this is the quantum relativistic harmonic oscillator in 1+1 dimensions Anisotropy adds a term proportional to WdW eq. is no longer separable, but we can find approximate solution near the singlularity and determine the behavior. 15/26

16. (ii) Geodesically complete Quartic potentials 16/26 E(s)=E>0 E(φ)=E+ρ≥E(s)  s perform independent oscilations. For arbitrary b,c and/or arbitrary initial conditions, the sign of (1-s 2 /  2 ) changes over time.

17. semiclassical solution to the WdW wavefunction 17/26

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29. Still open question If we allow the generic solutions for which (1-s 2 /φ 2 ) changes sign, what is the observable effect in our patch of the universe? These solutions are not geodesically complete in the Einstein frame, but are non-singular, and geodesically complete in the gamma frame (and also in general frame) 29/26

30. 30/26 Gravity as background in 2T-physics 0804.1585 [hep-th] Solve kinematics, and impose Q 11 =Q 12 =0: Get all shadows, e.g. conformal shadow. Compare to flat case Sp(2,R) algebra puts kinematic constraints on background geometry : homothety Lie derivative

31. 31/26 Gravity & SM in 2T-physics Field Theory Pure gravity has three fields: G MN (X), metric  (X), dilaton W(X), appears in δ(W), and more … Gauge symmetry and consistency with Sp(2,R) lead to a unique gravity action in d+2 dims, with no parameters at all. IB: 0804.1585 IB+S.H.Chen 0811.2510 No scale, The equations of motion reproduce the Sp(2,R) constraints, called kinematic equations, (proportional to δ’(W), δ’’(W)), and also the dynamical equations (proportional to δ(W)). a a

32. 32/26 2T Gravity triplet has unique couplings to matter: scalars, spinors & vectors. Imposes severe constraints on scalar fields coupled to gravity. a a

33. 33/20 Solve Sp(2,R) kinematic constraints (homothety) Conformal scalar, Weyl symmetry IB+S.H.Chen 0811.2510 a

34. 34 All shadow scalars must be conformal scalars (or similar). NO DIMENSIONFUL CONSTANTS. Effect on cosmology ? a a Can it change sign as a result of dynamics? In the history of the universe? Or in some patches of space-time? If so, are there observational effects in our patch of spacetime? Coupling many scalars. Generating scales. Effects on the gravitational constant and on cosmology.. 34/26