Antigravity Between Crunch and Bang in a Geodesically Complete Universe

Itzhak BarsUniversity of Southern California

Talk @ PDF-2011 Providence, August 2011

1) I.B. and S.H. Chen, 1004.07522) I.B., and S.H. Chen and Neil Turok, 1105.3606 3) I.B. + Chen + Turok + Steinhardt, to appear (several papers)

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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.

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.

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

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Generic solution is geodesically incomplete in Einstein gravity. Generic solution is geodesically incomplete in Einstein gravity.

6 parameters4=b,c,K,ρ0

2=(4-2) initial values

There is a subset of geodesically complete solutions only with conditions on initial values and parameters of the model.There is a subset of geodesically complete solutions only with conditions on initial values and parameters of the model.

Including relativistic matter and curvature

Including relativistic matter and curvature

V

This is not the whole story: Einstein gauge is valid only in a patch of spacetime,

3/11For 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 ghosts gravitational 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)eλ(x), gge-2λ(x)s)s)eλ(x), gge-2λ(x)

Gauge symmetry leftover from

general coordinate transformations in

extra 1+1 dims.

Gauge symmetry leftover from

general coordinate transformations in

extra 1+1 dims.

Conformal factor of metric =1 for any metric.

Plus the energy constraint: H=0 This is equivalent to the 00 Einstein eq. G00=T00

For all t,x dependence.

FRW

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Positive region

which compensates for the ghost.

BB singularity at aE=0 in E-gauge : gauge invariant factor vanishes in γ-gauge, or any gauge!!

Nothing singular in

γ-gauge

BCRT transformBars Chen Steinhardt Turok

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 Friedmannequationsbecome :

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First integral

Generic solution has

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E(s)=E>0

E(φ)=E+ρ ≥ E(s)

s perform independent oscilationsFor generic initial conditions, the sign of (2-s2) changes over time.

Generic solution is geodesically incomplete in the Einstein gauge.Geodesically complete with the natural extension in sspace.

K=0 casequartic

potentials

s

A x F[sn(z|m),cn(z|m), dn(z|m)]

z=(τ- τ0)/T

A,m,T depend on b,c,K,ρ0,E

There are special solutions that are geodesically complete in the restricted Einstein frame, but must constrain parameter space. There are special solutions that are geodesically complete in the restricted Einstein frame, but must constrain parameter space.

Geodesically complete larger space: φγ,sγ plane

| φ|>|s|

| 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.

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Generic solution is a cyclic universe with

antigravity stuck between crunch and bang! Probably true

for all V(σ).

Generic solution is a cyclic universe with

antigravity stuck between crunch and bang! Probably true

for all V(σ).

s

|z|BCSTTransformE-gauge to γ-gauge

BCSTTransformE-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 ins field space.

big bangs or big crunches in spacetime at the

lightcone ins field space.

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)=nPs (5 parameters)

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b>0

Universe expands to infinite size, turnaround at infinite size

Example n=5 5 nodes in picture

b<0

s

s

aE

σ

Cyclic universeCyclic universe

+

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Anisotropy

K0

If K≠0, ds3=Bianchi IX (Misner)

In Friedmann equations, 2 more fields

Friedman Eqs: kinetic terms for α1, α2 just like σ, plus anisotropy potential if K≠0

Without potentials can find all solutions analytically for any (initial) anisotropy momenta p1,p2; or σ momentum q

including the parameters K,ρ0.

p1=(aE)2∂τα1 etc.Free scalars if K=0, then canonical conjugate momenta p1,p2 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,p1,p2 mixmaster universe is avoided when σ is present (agree with BKL, etc.)

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Antigravity Loop (K=0 case)

This picture for K=0

φ+s=

φ -s=

q= σ momentum, q=(aE)2∂τσ p= anisotropy momentum p1=(aE)2∂τα1 etc.

ALWAYS a period of antigravity sandwiched between

crunch and bang

ATTRACTOR MECHANISM !!if p1 or p2 is not 0 :

both s 0 at the big bang or crunch singularity,

FOR ALL INITIAL CONDITIONS.

Duration of loop : (p2+q2)1/2/ρDuration of loop : (p2+q2)1/2/ρ

K=0: p1,p2 are conserved throughout motion.

q changes during the loop because of V(σ).If small loop, ≈no change.

11/11What 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, s0, 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).

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Antigravity Loop

This picture for K≠0

q,p1,p2 change during each loop (if loop is large) because of the potentials, but trajectory always returns to the origin and connects gravity antigravity regions

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K > 0 case K < 0 case

Higher level E>E*, with Es=E, E=E+ρSimilar behavior to K=0 case.

Lower level E,E<E*, with Es=E, E=E+ρs oscillates in the Vs well, while oscillates outside the V hill.

Then for any initial values there is a

finite bounce at size aE ≠0,

no antigravity.

Higher level E>0, with Es=E, E=E+ρSimilar behavior to K=0 case.

Lower level E*<E<0, with Es=E, E=E+ρ

All solutions are geodesically incomplete in the Einstein gauge. There is no way to

avoid antigravity.

E

E E*=K2/(16b)

.

E

E*=-K2/(16c)

Crunch bang turnaround

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Quantization of mini superspaceThe 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 toWdW eq. is no longer separable, but we can find approximate solution near the singlularity and determine the behavior.

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(ii) Geodesically complete

Quartic potentials

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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-s2/2) changes over time.

semiclassical solution to the WdW wavefunction

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Still open question

If we allow the generic solutions for which (1-s2/φ2) 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)

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Gravity as background in 2T-physics 0804.1585 [hep-th]

Solve kinematics, and impose Q11=Q12=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

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Gravity & SM in 2T-physics Field Theory

Pure gravity has three fields:GMN(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.Chen0811.2510

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

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

a

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Solve Sp(2,R) kinematic constraints (homothety)

Conformal scalar, Weyl symmetry

IB+S.H.Chen0811.2510

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All shadow scalars must be conformal scalars (or similar). NO DIMENSIONFUL CONSTANTS.

Effect on cosmology ?

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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.

.

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