1. Hartle-Hawking wave function and implications to observational cosmology 京都大学基礎物理学研究所 Yukawa Institute for Theoretical Physics 염 동 한 ヨムドンハン Dong-han Yeom

2. Based on - Hwang, Sahlmann and DY, arXiv:1107.4653 - Hwang, Lee, Sahlmann and DY, arXiv:1203.0112 - Hwang, Kim, Lee, Sahlmann and DY, arXiv:1207.0359 - Hwang, Lee, Stewart, DY and Zoe, arXiv:1208.6563 - Sasaki, DY and Zhang, arXiv:1307.5948 - Hwang and DY, arXiv:1311.6872 - Zhang, Saito, Sasaki and DY, in preparation - Hwang, Kim, Lee and DY, in preparation - Sasaki, White and DY, in preparation - …, in preparation

3. Can we see quantum gravitational effects in cosmology?

4. Yes!

5. Can we see quantum gravitational effects in cosmology? Yes! Though, we still need many ‘if’s…

6. P(N *,O|H) = P(N * |H) x P(O|N *,H)

7. Hypothesis Probability to see N * e-foldings in the given hypothesis Our universe Probability to see local observables

8. Quick review of Euclidean quantum cosmology

9. Traditional Problems 1. The singularity theorem: Our universe should begin from the initial singularity. How to resolve? 2. Initial condition of Universe: Is there a principle that uniquely determines our universe? If not, is the hypothesis that explains our universe probabilistically/statistically reasonable?

10. Quantum Cosmology, by introducing One can deal with these problems, by introducing the Schrodinger equation for fields: so-called, Wheeler-DeWitt equation the Wheeler-DeWitt equation. Wave function of Universe (quantized) Hamiltonian constraint

11. No-boundary proposal What is the boundary condition of WDW eqn? Perhaps, the ground state? Hartle-Hawking wave function path integral over regular compact manifold Euclidean action

12. Field direction ~ different initial condition Scale-factor direction ~ time Wave function of Universe

13. Steepest-descent Approximation

14. ~ sum-over instantons ~ sum-over instantons Alternative histories: many-world interpretation Probabilities will be assigned

15. Initial singularity  wave function Present universe: we want to know the probability of here.

16. There can be various analytic continuations. Due to analyticity, the path-integral still makes sense, even though we analytically continue to Euclidean time.

17. Big-bang singularity

18. Find geometry over the complex time, until the geometry to be regular.

19. How to calculate path integral? Mini-superspace Approximation 1: Mini-superspace

20. How to calculate path integral? Steepest-descent Approximation 2: Steepest-descent (sum-over fuzzy instantons)

21. How to calculate path integral? Steepest-descent Approximation 2: Steepest-descent (sum-over fuzzy instantons) 9 parameters determine 1 history. 6 of them are determined by regularity + compactness.

22. How to calculate path integral? Classicality Requirement: Classicality By tuning 2 parameters, we satisfy classicality of matter and metric.

23. How to calculate path integral? Classicality Requirement: Classicality By tuning 2 parameters, we satisfy classicality of matter and metric. (Hartle, Hawking and Hertog, 2007) In the end, we obtain the probability distribution as a one (field) dimensional function.

24. How to calculate path integral? Classicality Requirement: Classicality (Hartle, Hawking and Hertog, 2007) Cutoff due to classicality Large e-folding is exponentially suppressed!

25. Does this prefer inflation? Unfortunately, Euclidean probability does not prefer inflation. Possible answers: 1. Inflation is wrong (Ekpyrotic, big bounce, string gas cosmology, etc.) 2. Ground state is wrong (Vilenkin’s tunneling proposal) 3. Quantum cosmology is wrong (Susskind’s multiverse + anthropic) 4. Small modification (Hartle-Hawking-Hertog’s volume weighting) Is there any better explanation, apart from these unsatisfactory opinions?

26. Criterion of a good model (Hwang and DY, 2013) Typicality Typicality for a given hypothesis: For a given probability cutoff, there is corresponding typicality bound. If the typicality is smaller than the bound, then we reject the hypothesis.

27. Criterion of a good model (Hwang and DY, 2013) Reduced action that does not depend on energy scale of inflation but depend on the shape of potential Ratio of field space that allows sufficient and insufficient e-folding. potential shapeenergy scalefield space Then, the typicality is presented by the competition of three factors: potential shape, energy scale, and the field space of large e-folds

28. 1. The first inflation began at large energy scale. 2. Potential shape is finely tuned. Three ways to prefer inflation 3. There is something new effects.

29. Various results - Hwang, Sahlmann and DY, arXiv:1107.4653 - Hwang, Lee, Sahlmann and DY, arXiv:1203.0112 - Hwang, Kim, Lee, Sahlmann and DY, arXiv:1207.0359 - Sasaki, DY and Zhang, arXiv:1307.5948 - Hwang and DY, arXiv:1311.6872 - Zhang, Saito, Sasaki and DY, in preparation

30. CASE1: First inflation was near-Planck scale? (Hwang and DY, 2013)

31. CASE1: First inflation was near-Planck scale? 1/2 stable vs. 1/2 unstable (Hwang, Sahlmann and DY, 2011; Hwang, Lee, Sahlmann and DY, 2012)

32. CASE1: First inflation was near-Planck scale? (Hwang and DY, 2013)

33. CASE2: Fine-tuning of the potential (Hwang and DY, 2013)

35. CASE2: Starobinski-like model (Kallosh and Linde, 2013)

37. CASE3: New ingredient from multi-field inflation (Hwang, Kim, Lee, Sahlmann and DY, 2012)

38. CASE3: New ingredient from massive gravity (Sasaki, DY and Zhang, 2013)

39. Hypotheses of Quantum Cosmology

40. V

41. V end of inflation

42. V 50 e-f0lding

43. V end of inflation50 e-f0ldingeternal inflation

44. P end of inflation50 e-f0ldingeternal inflation H 0 : Null assumption

45. end of inflation50 e-f0ldingeternal inflation H 1 : Euclidean probability P

46. end of inflation50 e-f0ldingeternal inflation H 2 : HHH Volume weighting P

47. end of inflation50 e-f0ldingeternal inflation H 3 : dRGT model P

48. end of inflation50 e-f0ldingeternal inflation H 4 : Nflation model P

49. end of inflation50 e-f0ldingeternal inflation However, it’s not interesting, since all observables are determined by the unique end of inflation point

50. Implications to observational cosmology

51. end of inflation 50 e-f0lding If the model is more than 2D, then its more interesting and non-trivial!

52. end of inflation 50 e-f0lding If slow-roll approximation holds, then we can find a family of histories that gives the same observables. Histories of each yellow arrows that experience more than 50 e-folding shares the same observables.

53. end of inflation 50 e-f0lding If slow-roll approximation holds, then we can find a family of histories that gives the same observables. Histories of each yellow arrows that experience more than 50 e-folding shares the same observables. 1 2 3

54. 1 2 3 To assign a probability for a set of observables, we have to assign probabilities for the family of histories. x

55. 12 3 P x

56. 12 3 P x Have to integrate for outside the 50 e-folding surface!

57. 12 3 Probabilities of each observables can be approximated by the largest probability, as was in the steepest-descent approximation. P x Have to integrate for outside the 50 e-folding surface!

58. end of inflation 50 e-f0lding Most probable surface is determined. Most probable surface

59. end of inflation 50 e-f0lding Most probable surface is determined. If the most probable surface is inside the 50-efolding surface, then it is better to choose the 50-efolding surface as the probability assignment surface.

60. What determines the probability? For a given inflation model, 1. Hypotheses Hypothesis determines the most probable surface. 2. Attractor Attractor direction of the model prefers a certain end-point. 3. End-of-inflation Surface The shape of the end-of-inflation surface change a preferred point.

61. So what you expect? Still there are too many freedom! If the inflaton potential has reasonably large dof, then there can be gentle direction as well as steep direction. At least, the naive Euclidean weighting (H 1, H 2 ) will prefer the slowest direction, while this can be harmful to observe our universe!

62. So what you expect? This implies that the probability distribution along the most probable surface should be reasonably gentle! Therefore, the Euclidean action should be seriously modified around the inflation era. Perhaps, this can be connected deeper hypothesis of quantum gravity or modified gravity?