Inflation Without a Beginning

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

2. 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. Expanding universe g Two Classical Cosmologies

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

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!

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

11. Constant slices True vacuum t x Nucleation event 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). False vacuum

12. Inflating region 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.

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

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

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

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

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

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

27. J + J – J – J – We’re done! i+ i+ • Singularity free, • Eternally inflating 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 –.

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

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

32. J + J – J – J – P F F P The Antipodal Identification Identification of antipodal points strongly motivated. i+ i+ J + Maps region I n II. Maps J – onto itself. P Benefits -More economical -No horizons (in dS) -P J +

33. J + J – J – J – P F F P The Antipodal Identification Identification of antipodal points strongly motivated. i+ i+ J + Maps region I n II. Maps J – onto itself. P Difficulties: -Does QFT make sense? -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

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/0301042 and PRD 65, 083507

37. Generalizations? Comparison to cyclic universe • Also flat slices, exponential expansion. • ade Sitter-like on large scales.

38. Generalizations? Comparison to cyclic universe • Two nested quasi-de Sitter branes. • Geodesically incomplete. • Eaten by bubbles?

39. P F F P Generalizations? Spacelike boundary surfaces i+ i+ • Any spacelike surface will also do. • But: • Not eternal. • Less unique? J + J + J – J – 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. i+ i+ J + P J + J – J – -P J – J – J – F P The Antipodal Identification The identified variant has some nice properties: 1. Economical.

42. J + J – P J – J – -P 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.

43. J + O J – P F J – P 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.

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

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

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 .)

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?)

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 + • 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: F P B P F J +