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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 –