inflation without a beginning anthony aguirre (ias) collaborator: steven gratton (princeton)

Inflation Without a Beginning

Anthony Aguirre (IAS)

Collaborator: Steven Gratton (Princeton)

Expanding universe Two Classical Cosmologies

The Big BangExpansion dilution.

Extrapolate back in time to initial singularity.

Initial epoch, 13.7 Gyr ago, unknown physics.

‘initial conditions’ postulated shortly after this beginning.

The Steady-StateExpansion new matter creation.

Global state independent of time.

Initial time or singularity absent.

No ‘initial’ conditions.

Our Cosmology

Locally: Big Bang Many observations (most recently WMAP) support hot big-bang for past 13.7 Gyr.

But also flat, homogeneous, isotropic “initial” conditions with scale invariant gaussian density perturbations above horizon scale…

As predicted generically by inflation models.

Simple inflationary picture:

Quasi-flat, Quasi-homogeneous,Quasi-FRW

inflation

Flat, homogeneous, FRW

But inflation does not end all at once (or at all)

Apparently correct inflationary picture:

Rather: approaches a Steady-state distribution of thermalized + inflating regions.

Quasi-flat, Quasi-homogeneous,Quasi-FRW

inflation

Flat, homogeneous, FRW

Semi-eternal Inflation

Only approaches a steady state, leaving some unpalatable qualities:

• Still has a cosmological singularity – born from some ill-defined “quantum chaos”.

• Initial conditions, but unknowable.• Preferred time, but irrelevant.• Other oddities (See Guth)

-e.g. we were born at some finite time, but typical birth-time is infinity!

Semi-eternal Inflation

These might be avoided if inflation, as well as having no end, had no beginning.

Can we have truly (past- and future-) eternal inflation? Apparently not!

Several theorems eternally inflating space- times must contain “singularities”:

Requiring (Borde & Vilenkin 1996).

Requiring (Borde, Guth & Vilenkin 2001).

0min HH

p

Steady-State eternal inflation

Undaunted, let’s analyze the double-well case.

)(V

1V

0V

Steady-State eternal inflation

Undaunted, let’s analyze the double-well case.

Bubbles infinite open FRW universes.

Bubble wallTrue vacuum

Constant slices

Nucleation event

x

t

False vacuum

Steady-State eternal inflation

Undaunted, let’s analyze the double-well case.

Bubbles infinite open FRW universes.

These form at constant rate L/(unit 4-volume).

Constant slices

Nucleation event

True vacuum

x

t

False vacuum

Steady-State eternal inflation

Undaunted, let’s analyze the double-well case.

Bubbles infinite open FRW universes.

These form at constant rate L/(unit 4-volume).

At each time, some bubble distribution.

Inflating region

Steady-State eternal inflation

Strategy: make state approached by semi-eternal inflation exact.

Steady-State eternal inflation

Strategy: make state approached by semi-eternal inflation exact.

Flat spatial sections.

Steady-State eternal inflation

Strategy: make state approached by semi-eternal inflation exact.

Flat spatial sections.

Consider bubbles formed between t0 and t.

t

t0

Steady-State eternal inflation

Strategy: make state approached by semi-eternal inflation exact.

Flat spatial sections.

Consider bubbles formed between t0 and t.

t

t0

Steady-State eternal inflation

Strategy: make state approached by semi-eternal inflation exact.

Flat spatial sections.

Consider bubbles formed between t0 and t.

Send .0 t

t

t0

Steady-State eternal inflation

Strategy: make state approached by semi-eternal inflation exact.

Flat spatial sections.

Consider bubbles formed between t0 and t.

Send

Inflation survives.

.0 t

Steady-State eternal inflation

This eternally inflating universe, based on “open inflation” has no obvious singularities, and was basically described in Vilenkin (1992).

So what about the singularity theorems that ought to forbid it?

Analysis of “singularity”

de Sitter space conformal diagram:

Approach infinite null surface J

– as

t=const.Comoving observers

.0 t

Analysis of “singularity”

Now add bubbles:

Nucleation sites

P

F

Analysis of “singularity”

Singularity theorems

must have incomplete world lines.

P

F

Analysis of “singularity”

Singularity theorems

must have incomplete world lines.

P

FAnd does.

“singularity” found by theorems is J – .

null/timelike geodesics

Analysis of “singularity”

What is in the uncharted region?

P

FAs , all geodesics enter false vacuum.

Continuous fields J – = pure false vacuum.

null/timelike geodesics

0t

Analysis of “singularity”

What is in the uncharted region?

P

Fi.e. (J – = pure false vacuum) (bubble distribution)

null/timelike geodesics

Analysis of “singularity”

What is in the uncharted region?

J +

J –

J –

J –Inflating bulk

Constant time surfaces

J +

J +

i+i+

Essentially identical.

We’re done!

J +

J –

J –

J –Inflating bulk

Constant time surfaces

J +

J +

i+i+•Singularity free,

•Eternally inflating

Steady-State eternal inflation

Like any theory describing a physical system, this model has:Dynamics (stochastic bubble formation).“boundary” conditions. These can be posed as:

Inflaton field in false vacuum on J –.

Other (classical) fields are at minima on J –.

Weyl curvature = 0 on J –.

Steady-State eternal inflation

Nice properties (vs. inflation or semi-eternal inflation):• No cosmological singularity.• Simple B.C.s based on physical principle.• Funny aspects of semi-eternal inflation resolved.• Little horizon problem: all points on boundary surface

are close to all others.

Interesting further features worth studying…

Outside bubbles: no local AOT.

Inside bubbles: AOT away from inflation.

No global AOT.

The Arrow of Time

J +

J –

J –

J –

Constant time surfaces

F

P

P

F

J +

J +

i+i+

Problem with singularity theorems: “transcendental” AOT.

The Arrow of Time

B.c.s on J – AOT pointing away from it.

J +

J –

J –

J –

Constant time surfaces

F

P

P

F

J +

J +

i+i+

Maps region I II.

Maps J – onto itself.

The Antipodal Identification

Identification of antipodal points strongly motivated.

J +

J –

J –

J –

F

P

P

F

J +

J +

i+i+

-P

P

Benefits

-More economical -No horizons (in dS)

Maps region I II.

Maps J – onto itself.

The Antipodal Identification

Identification of antipodal points strongly motivated.

J +

J –

J –

J –

F

P

P

F

J +

J +

i+i+

-P

P

Difficulties:

-Does QFT make sense?

Generalizations?

Can it be generalized (e.g., to chaotic inflation)?

• “bubbles” in background w/

• One way: start field at = 0 on J –. Bubbles of nucleate.

)(V

R .R

R

Rolling region Fluctuating region

R

Summary and ConclusionsSummary and Conclusions

Semi-eternal Semi-eternal cancan be made eternal. be made eternal.

Cosmology defined by simple b.c. on Cosmology defined by simple b.c. on infinite null surface.infinite null surface.

Model resolves Model resolves singularitysingularity, horizon, flatness, , horizon, flatness, initial fluctuation, relic problems of HBB.initial fluctuation, relic problems of HBB.

““Antipodal” identification suggested Antipodal” identification suggested two two universes identified. May be interesting for QFT, universes identified. May be interesting for QFT, string theory studies.string theory studies.

Inflation:Inflation: no end. no end. Also no beginning?Also no beginning?

Inflation Without a Beginning

For more details see:

gr-qc/0301042 and PRD 65, 083507

Generalizations?

Comparison to cyclic universe

Also flat slices, exponential expansion.

de Sitter-like on large scales.

Generalizations?

Comparison to cyclic universe

Two nested quasi-de Sitter branes.Geodesically incomplete.Eaten by bubbles?

Generalizations?

Spacelike boundary surfaces

Any spacelike surface will also do.

But:

Not eternal.

Less unique?

J +

J –J –

F

P

P

F

J +

J +

i+i+

Steady-State eternal inflation

Interesting properties of distribution:

Inflating fractal skeleton global structure.

dN/dr/dV of bubbles satisfies (perfect) CP.

Inflating region satisfies perfect “conditional cosmographic principle” of Mandelbrot.

Inflating region

The Antipodal Identification

J –

P

-P J –

J +

J –

J –

J –

P

F

J +

i+i+The identified variant has some nice properties:

1. Economical.

J +

J –

PJ –

-P J –

The Antipodal Identification

The identified variant has some nice properties:

2. The light cones of a P and -P do not intersect no causality violations.

The Antipodal Identification

The identified variant has some nice properties:

3. No event horizons: non-spacelike geodesic connects any point to worldline of immortal observer.

J –

J –

J +

P

PF

O

The Antipodal Identification

QFT in antipodally identified de Sitter space

J –

PJ –

-P J –

Not time-orientable.

The Antipodal Identification

QFT in elliptic de Sitter space

J –

PJ –

-P J –

Not time-orientable.

The Antipodal Identification

QFT in elliptic de Sitter space

QFT in curved spacetime:

Let:

where

Define vacuum by for all k.

Get correlators such as

(note: in dS, get family of vacua .)

þê(x) =P

k aêkþk(x) + aêykþ

ãk(x)

j0i aêkj0i = 0

[aêk;aêk0] = 0; [aêk;aêyk0] = öhî k;k0

G(1)(x;x0) ñ h0jf þê(x);þê(x0)gj0i

jëi

The Antipodal Identification

QFT in elliptic de Sitter space

1. Vacuum level: for choice of can make G(1) antipodally symmetric. But bad at short dist.

2. Field level: symmetrize But : No Fock vacuum.

3. Correlator level: set

but why? And

þS(x) = 2p1 [þ(x) + þ(xA)]

(þk;þk0) = 0

G(1)A (x;x0) = G(1)(x;x0) + G(1)(x;x0

A)

[þê(x);þê(x0)] = 0:

þk:

The Antipodal Identification

QFT in elliptic de Sitter space

4. Way in progress:

1. Define QFT in “causal diamond”.

2. Use antipodal ID to define all correlators using causal diamond correlators.

(Q: will it all work out? Any observable consequences?)

The Antipodal Identification

String theory in elliptic de Sitter space? (see Parikh et al. 2002)

J – =J +

No horizons.

Holography: only one boundary.

J – =J +

-P=P

P=-P

The geodesically complete steady-state models: do they make sense?

De Sitter space could be partitioned by any non-timelike surface B across which the physical time orientation reverses.

Some points suggesting that they do, and are natural:

J +

B

J +

P

P

F

F

The geodesically complete steady-state models: do they make sense?

De Sitter space could be partitioned by any non-timelike surface B across which the physical time orientation reverses. But our J – allows interesting physics everywhere yet no info. coming from B.

More points suggesting that they do, and are natural:

J +

B

J +

P

P

F

F

Eternal inflation

Interesting properties of bubble collisions:

Bubble spatial sections can be nearly homogeneous.

Wait bubble encounter (a new beginning?)

For finite t0, frequency cosmic time t.

t

t0

Analysis of “singularity”

What is in the uncharted region?

P

Fi.e. (J – = pure false vacuum) (bubble distribution)

null/timelike geodesics

Analysis of “singularity”

What is in the uncharted region?

Classically, J – is initial value surface for region I fields.

J +

J –

J –

J –

I

II

i-I

i-II i-

II

p

Analysis of “singularity”

What is in the uncharted region?

Classically, J – is initial value surface for region I fields.

Same for region II!

J +

J –

J –

J –

I

II

i-I

i-II i-

II

p

p

Analysis of “singularity”

What is in the uncharted region?

J +

J –

J –

J –

I

II

i-I

i-II i-

II

Thus, b.c.s on J – both region I and II.

(Tricky bit: i-I vs. i-

II )

For fields constant on J –, regions (classically) are the same!

Analysis of “singularity”

What is in the uncharted region?

Semi-classically:

Form bubbles, none through J – .

J +

J –

J –

J –