1. Inflation Without a Beginning
Anthony Aguirre (IAS)
Collaborator: Steven Gratton (Princeton)

2. Expanding universe g Two Classical Cosmologies
The Big Bang
Expansion a 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-State
Expansion a new matter creation.
Global state independent of time.
Initial time or singularity absent.
No ‘initial’ conditions.

3. Our Cosmology Locally: Big Bang
Many observations (most recently WMAP) support hot big-bang for past 13.7 Gyr.
But also a flat, homogeneous, isotropic “initial” conditions with scale invariant gaussian density perturbations above horizon scale…
As predicted generically by inflation models.

4. Simple inflationary picture:
But inflation does not end all at once (or at all)
Flat, homogeneous, FRW
inflation
Quasi-flat, Quasi-homogeneous,Quasi-FRW

5. Apparently correct inflationary picture:
Rather: approaches a Steady-state distribution of thermalized + inflating regions.
Flat, homogeneous, FRW
inflation
Quasi-flat, Quasi-homogeneous,Quasi-FRW

6. 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!
Start 10:38

7. Semi-eternal Inflation
These might be avoided if inflation, as well as having no end, had no beginning.

8. Can we have truly (past- and future-) eternal inflation? Apparently not!
Several theorems a eternally inflating space- times must contain “singularities”:
Requiring (Borde & Vilenkin 1996).
Requiring (Borde, Guth & Vilenkin 2001).

9. Steady-State eternal inflation
Undaunted, let’s analyze the double-well case.

10. Steady-State eternal inflation
Undaunted, let’s analyze the double-well case.
Bubbles a infinite open FRW universes.
True vacuum
Constant slices
Nucleation event x t
False vacuum
Bubble wall

11. Steady-State eternal inflation
Undaunted, let’s analyze the double-well case.
Bubbles a infinite open FRW universes.
These form at constant rate L/(unit 4-volume).
Constant slices
Nucleation event
True vacuum x t
False vacuum

12. Steady-State eternal inflation
Undaunted, let’s analyze the double-well case.
Bubbles a infinite open FRW universes.
These form at constant rate L/(unit 4-volume).
At each time, some bubble distribution.
Inflating region

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

14. Steady-State eternal inflation
Strategy: make state approached by semi-eternal inflation exact.
Flat spatial sections.

15. 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

16. 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

17. Steady-State eternal inflation
Strategy: make state approached by semi-eternal inflation exact.
Flat spatial sections.
Consider bubbles formed between t0 and t.
Send t t0

18. 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.

19. 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?

20. Analysis of “singularity”
de Sitter space conformal diagram:
Approach infinite null surface J – as
t=const.
Comoving observers

21. Analysis of “singularity”
Now add bubbles: P F
Nucleation sites

22. Analysis of “singularity”
Singularity theorems
a must have incomplete world lines. P F

23. Analysis of “singularity”
Singularity theorems
a must have incomplete world lines. P F
And does.
“singularity” found by theorems is J – .
null/timelike geodesics

24. Analysis of “singularity”
What is in the uncharted region? P F
As , all geodesics enter false vacuum.
Continuous fields a J – = pure false vacuum.
13 min
null/timelike geodesics

25. Analysis of “singularity”
What is in the uncharted region? P F
i.e. (J – = pure false vacuum) a (bubble distribution)
null/timelike geodesics

26. Analysis of “singularity”
What is in the uncharted region? i+ i+
J +
Essentially identical.
J +
J –
Inflating bulk
Constant time surfaces
J +

27. We’re done! Singularity free, Eternally inflating Inflating bulk
J +
J +
J –
Inflating bulk
Constant time surfaces
J +

28. 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 –.
Note applies to the thermalizes region also.

29. 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…

30. The Arrow of Time Outside bubbles: no local AOT.
Inside bubbles: AOT away from inflation.
No global AOT.
J +
J +
J – P F
Work on F P
J +
Constant time surfaces

31. The Arrow of Time B.c.s on J – a AOT pointing away from it.
Problem with singularity theorems: “transcendental” AOT. F P
J +
Constant time surfaces

32. The Antipodal Identification
Identification of antipodal points strongly motivated. i+ i+
J +
Maps region I n II.
Maps J – onto itself.
J +
J – P F P
Benefits
-More economical No horizons (in dS)
Don’t forget schroedinger -P F P
J +

33. The Antipodal Identification
Identification of antipodal points strongly motivated. i+ i+
J +
Maps region I n II.
Maps J – onto itself.
J +
J – P F P
Difficulties:
-Does QFT make sense?
Don’t forget schroedinger -P F P
J +

34. Generalizations? Can it be generalized (e.g., to chaotic inflation)?
“bubbles” in background w/
One way: start field at f = 0 on J –. Bubbles of nucleate.
Fluctuating region
Rolling region
Note applies to the thermalizes region also.

35. Summary and Conclusions
Semi-eternal can be made eternal.
Cosmology defined by simple b.c. on infinite null surface.
Model resolves singularity, horizon, flatness, initial fluctuation, relic problems of HBB.
“Antipodal” identification suggested a two universes identified. May be interesting for QFT, string theory studies.
Inflation: no end. Also no beginning?

36. Inflation Without a Beginning
For more details see:
gr-qc/ and PRD 65,
10:59

37. Generalizations? Comparison to cyclic universe
Also flat slices, exponential expansion.
ade Sitter-like on large scales.
Note applies to the thermalizes region also.

38. Generalizations? Comparison to cyclic universe
Two nested quasi-de Sitter branes.
Geodesically incomplete.
Eaten by bubbles?
Note applies to the thermalizes region also.

39. Generalizations? Spacelike boundary surfaces
Any spacelike surface will also do.
But:
Not eternal.
Less unique?
J +
J + P F
Note applies to the thermalizes region also.
J –
J – F P
J +

40. Steady-State eternal inflation
Interesting properties of distribution:
Inflating fractal skeleton a global structure.
dN/dr/dV of bubbles satisfies (perfect) CP.
Inflating region satisfies perfect “conditional cosmographic principle” of Mandelbrot.
Inflating region

41. The Antipodal Identification
The identified variant has some nice properties:
1. Economical.
J – P -P
J + F i+ 46

42. The Antipodal Identification
The identified variant has some nice properties:
2. The light cones of a P and -P do not intersect a no causality violations.
J +
J – P -P

43. 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 + P F O

44. The Antipodal Identification
QFT in antipodally identified de Sitter space
Not time-orientable.
J – P -P

45. The Antipodal Identification
QFT in elliptic de Sitter space
Not time-orientable.
J – P -P 47

46. 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 )
Note measure factor \sqrt(-g)g^{00} in norm.

47. The Antipodal Identification
QFT in elliptic de Sitter space
Vacuum level: for choice of a can make G(1) antipodally symmetric. But bad at short dist.
Field level: symmetrize But : No Fock vacuum.
Correlator level: set but why? And

48. The Antipodal Identification
QFT in elliptic de Sitter space
4. Way in progress:
Define QFT in “causal diamond”.
Use antipodal ID to define all correlators using causal diamond correlators.
(Q: will it all work out? Any observable consequences?) 57

49. The Antipodal Identification
String theory in elliptic de Sitter space? (see Parikh et al. 2002)
No horizons.
Holography: only one boundary.
J – =J +
P=-P
-P=P
J – =J +

50. The geodesically complete steady-state models: do they make sense?
J +
Some points suggesting that they do, and are natural: F P B P F
J +
De Sitter space could be partitioned by any non-timelike surface B across which the physical time orientation reverses.

51. The geodesically complete steady-state models: do they make sense?
J +
More points suggesting that they do, and are natural: F P B P F
J +
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.

52. Eternal inflation Interesting properties of bubble collisions:
Bubble spatial sections can be nearly homogeneous.
Wait a bubble encounter (a new beginning?)
For finite t0, frequency a cosmic time t. t t0

53. Analysis of “singularity”
What is in the uncharted region? P F
i.e. (J – = pure false vacuum) a (bubble distribution)
null/timelike geodesics

54. Analysis of “singularity”
What is in the uncharted region?
Classically, J – is initial value surface for region I fields.
i-II
J +
J –
i-II I p II
i-I

55. Analysis of “singularity”
What is in the uncharted region?
Classically, J – is initial value surface for region I fields.
Same for region II!
i-II
J +
J –
i-II I p p II
i-I

56. Analysis of “singularity”
What is in the uncharted region?
Thus, b.c.s on J – a both region I and II.
(Tricky bit: i-I vs. i-II )
For fields constant on J –, regions (classically) are the same!
J +
J –
i-II
i-II I II
i-I

57. Analysis of “singularity”
What is in the uncharted region?
Semi-classically:
Form bubbles, none through J – .
J +
J –