Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS)...

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Bouncing Univ er se from Nothing N Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312 [gr-qc]

Transcript of Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS)...

Page 1: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Bouncing Universe from NothingN

Takahiro Terada (CTPU, IBS)

Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312 [gr-qc]

Page 2: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Introduction

N

Page 3: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Cosmological History

Page 4: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Cosmological History

Always Expanding?

Page 5: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Usually, yes.For a flat universe,

H2 =

3<latexit sha1_base64="23XtzSwWEKc8BVGTLrOxgLTtiWo=">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</latexit><latexit sha1_base64="23XtzSwWEKc8BVGTLrOxgLTtiWo=">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</latexit><latexit sha1_base64="23XtzSwWEKc8BVGTLrOxgLTtiWo=">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</latexit><latexit sha1_base64="23XtzSwWEKc8BVGTLrOxgLTtiWo=">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</latexit>

H(t) ⌘ a(t)

a(t)<latexit sha1_base64="R1A5x9w3NSwu7K3BRQDVl4/bfSU=">AAACtXichVG9SgNBEJ6c//EnURvB5jAq2oQ5ESJWQRtLNUYFE8LeZaNLLnfn3SYQj7yAL2BhpWAhNrYKVja+gIWPIJYKNhbOXU5ERZ1ld2e/mW9mP0Z3TOFJxIeY0tHZ1d3T2xfvHxgcSiSHRzY9u+4aPG/Ypu1u68zjprB4Xgpp8m3H5aymm3xLry4H8a0Gdz1hWxuy6fBije1aoiIMJgkqJSdX1Bk5qxb4fl001ELFZYZfKNvSZy3CWz4LzlIyhWkMTf3paJGTgshW7eQNFKAMNhhQhxpwsECSbwIDj9YOaIDgEFYEnzCXPBHGObQgTtw6ZXHKYIRW6dyl106EWvQOanoh26AuJm2XmCpM4T2e4zPe4QU+4tuvtfywRvCXJt16m8udUuJwLPf6L6tGt4S9T9YfjI9cl6IV2n/rk5SzEOoSpNMJkUCx0e7VODh6zi2uT/nTeIpPpPUEH/CW1FqNF+Nsja8fQ5yGpX0fzU9ncy6tYVpbm09ll6Kx9cI4TMAMzSYDWViBVchT30O4hCu4VjJKUSkrlXaqEos4o/DFFPsdKn+gow==</latexit><latexit sha1_base64="R1A5x9w3NSwu7K3BRQDVl4/bfSU=">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</latexit><latexit sha1_base64="R1A5x9w3NSwu7K3BRQDVl4/bfSU=">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</latexit><latexit sha1_base64="R1A5x9w3NSwu7K3BRQDVl4/bfSU=">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</latexit>

ds2 = �dt2 + a2(t)dx2<latexit sha1_base64="9fwvTjFsHLumV9Wys3KlyB7Cp4M=">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</latexit><latexit sha1_base64="9fwvTjFsHLumV9Wys3KlyB7Cp4M=">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</latexit><latexit sha1_base64="9fwvTjFsHLumV9Wys3KlyB7Cp4M=">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</latexit><latexit sha1_base64="9fwvTjFsHLumV9Wys3KlyB7Cp4M=">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</latexit>

Friedmann eq.

Hubble parameterEnergy density

As long as , it keeps expanding.⇢ > 0<latexit sha1_base64="T5+p5A3FDoYhDY387tsz4L146DM=">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</latexit><latexit sha1_base64="T5+p5A3FDoYhDY387tsz4L146DM=">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</latexit><latexit sha1_base64="T5+p5A3FDoYhDY387tsz4L146DM=">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</latexit><latexit sha1_base64="T5+p5A3FDoYhDY387tsz4L146DM=">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</latexit>

(reduced Planck unit: )8⇡G = 1<latexit sha1_base64="AP3BOUvXHfziY4DUyqklyosBZo8=">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</latexit><latexit sha1_base64="AP3BOUvXHfziY4DUyqklyosBZo8=">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</latexit><latexit sha1_base64="AP3BOUvXHfziY4DUyqklyosBZo8=">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</latexit><latexit sha1_base64="AP3BOUvXHfziY4DUyqklyosBZo8=">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</latexit>

Page 6: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Universe can contractIf ⇢ = 0

<latexit sha1_base64="I7d6jdr4jlXq94g3dbpDEd2z+bs=">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</latexit><latexit sha1_base64="I7d6jdr4jlXq94g3dbpDEd2z+bs=">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</latexit><latexit sha1_base64="I7d6jdr4jlXq94g3dbpDEd2z+bs=">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</latexit><latexit sha1_base64="I7d6jdr4jlXq94g3dbpDEd2z+bs=">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</latexit>

is realized, the universe starts to contract.H = 0<latexit sha1_base64="7be2cQtbXt3HQjHQGOe+BFpaXqg=">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</latexit><latexit sha1_base64="7be2cQtbXt3HQjHQGOe+BFpaXqg=">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</latexit><latexit sha1_base64="7be2cQtbXt3HQjHQGOe+BFpaXqg=">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</latexit><latexit sha1_base64="7be2cQtbXt3HQjHQGOe+BFpaXqg=">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</latexit>

(i.e., )

H = �1

2(⇢+ P )

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⇢+ P � 0<latexit sha1_base64="Nn+Mzt1qMILILS7awJY2fh9H/UY=">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</latexit><latexit sha1_base64="Nn+Mzt1qMILILS7awJY2fh9H/UY=">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</latexit><latexit sha1_base64="Nn+Mzt1qMILILS7awJY2fh9H/UY=">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</latexit><latexit sha1_base64="Nn+Mzt1qMILILS7awJY2fh9H/UY=">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</latexit>

Null Energy Condition (NEC):

For example, it is possible for the scalar field with a negative potential.

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">AAAClnichVFLSwJRFD5OL7OHVhuhjSSGbeRMBEWLkCJq6SMfoCIz000vjjPDzCiY9AfatQpyVdAi+g2t2vQHWvgToqVBmxYdx4koyc5w5373O+c7934c2VC5ZSN2PcLY+MTklHfaNzM7N+8PLCxmLb1hKiyj6Kpu5mXJYirXWMbmtsryhsmkuqyynFzb6+dzTWZaXNeO7JbBSnWpovETrkg2UblstGhU+Vo5EMYYOhEaBqILwuBGQg88QBGOQQcFGlAHBhrYhFWQwKKvACIgGMSVoE2cSYg7eQZn4CNtg6oYVUjE1uhfoVPBZTU693tajlqhW1RaJilDEMFnvMMePuE9vuDHn73aTo/+W1q0ywMtM8r+82D6/V9VnXYbqt+qEYqvWpOyJ7RG+7OpZsvxxcmn4TB9x8rgrubpZS+9nYq0V/EGX8nrNXbxkdxqzTflNslSHfDRsMTfoxkG2fWYiDExuRGO77pj88IyrECUZrMJcTiEBGSc113AFXSEoLAj7AsHg1LB42qW4EcIiU+9kpTS</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

However, it keeps contracting, eventually leading to a big crunch.

[Linde, hep-th/0110195] [Felder et al., hep-th/0202017]

Page 7: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Negative PotentialStandard Model Supergravity String Landscape

[Buttazzo et al., 1307.3536]

V = eK⇣gjiDiWDjW � 3|W |2

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R-symmetry breaking,negative semi-definite

SUSY breaking,positive semi-definite (schematic figure)

De Sitter vacuum construction is nontrivial in superstring theory.

figure from [Kallosh et al., 1808.09428]

(cf. Swampland de Sitter conjecture)[Obied et al., 1806.08362]

Page 8: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Negative PotentialStandard Model Supergravity String Landscape

[Buttazzo et al., 1307.3536]

V = eK⇣gjiDiWDjW � 3|W |2

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R-symmetry breaking,negative semi-definite

SUSY breaking,positive semi-definite (schematic figure)

De Sitter vacuum construction is nontrivial in superstring theory.

figure from [Kallosh et al., 1808.09428]

(cf. Swampland de Sitter conjecture)[Obied et al., 1806.08362]

Once the universe enters the negative region,

is the Big Crunch unavoidable?

Page 9: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Spatial Curvature

ds2 = �dt2 + a2 (t)

dr2

1�Kr2+ r2

�d✓2 + sin2 ✓d'2

��

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Friedmann-Lemaitre-Robertson-Walker metric

H2 =

3� K

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H = �1

2(⇢+ P ) +

Ka2

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positive (closed), zero (flat), or negative (open)spatial curvature

Page 10: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Spatial Curvature

ds2 = �dt2 + a2 (t)

dr2

1�Kr2+ r2

�d✓2 + sin2 ✓d'2

��

<latexit sha1_base64="xB/7hVOqKN9j2pfW2K9Mo3zC440=">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</latexit><latexit sha1_base64="xB/7hVOqKN9j2pfW2K9Mo3zC440=">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</latexit><latexit sha1_base64="xB/7hVOqKN9j2pfW2K9Mo3zC440=">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</latexit><latexit sha1_base64="xB/7hVOqKN9j2pfW2K9Mo3zC440=">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</latexit>

Friedmann-Lemaitre-Robertson-Walker metric

H2 =

3� K

a2<latexit sha1_base64="KfXWjRfSk+8dOmSK9l3O5+PKn6g=">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</latexit><latexit sha1_base64="KfXWjRfSk+8dOmSK9l3O5+PKn6g=">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</latexit><latexit sha1_base64="KfXWjRfSk+8dOmSK9l3O5+PKn6g=">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</latexit><latexit sha1_base64="KfXWjRfSk+8dOmSK9l3O5+PKn6g=">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</latexit>

H = �1

2(⇢+ P ) +

Ka2

<latexit sha1_base64="UMDWRaSU5EUFAKNg0cIXXKGfi3U=">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</latexit><latexit sha1_base64="UMDWRaSU5EUFAKNg0cIXXKGfi3U=">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</latexit><latexit sha1_base64="UMDWRaSU5EUFAKNg0cIXXKGfi3U=">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</latexit><latexit sha1_base64="UMDWRaSU5EUFAKNg0cIXXKGfi3U=">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</latexit>

Contraction → Expansion is possible.simple example of a “bounce”

⇢ = �P = ⇤<latexit sha1_base64="ymPWvW42rLbsZEozNILsCA0yXbA=">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</latexit><latexit sha1_base64="ymPWvW42rLbsZEozNILsCA0yXbA=">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</latexit><latexit sha1_base64="ymPWvW42rLbsZEozNILsCA0yXbA=">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</latexit><latexit sha1_base64="ymPWvW42rLbsZEozNILsCA0yXbA=">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</latexit>

(cosmological constant)

K > 0<latexit sha1_base64="e46tfHgCgPtvJ6wQeiGwhIR83aM=">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</latexit><latexit sha1_base64="e46tfHgCgPtvJ6wQeiGwhIR83aM=">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</latexit><latexit sha1_base64="e46tfHgCgPtvJ6wQeiGwhIR83aM=">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</latexit><latexit sha1_base64="e46tfHgCgPtvJ6wQeiGwhIR83aM=">AAACoHichVHNSgJRFD5Of2Y/Wm2CNkNStJIzERQtQmojtEgtNVKRmelqQ/PHzFWwwRfoBVq4KnIRPUOrNr1Aix4hWha0adFxnIiS7Az33nO/c75z5+NTbF1zOeJTSBgaHhkdC49HJianpqOxmdm8a9UdleVUS7ecQ0V2ma6ZLMc1rrND22GyoeisoJzudOuFBnNczTIPeNNmZUOumVpVU2VOUCUWLRkyP1Fl3dttiVsiVmJxTKAfYn8iBUkcgkhbsTsowTFYoEIdDGBgAqdcBxlc+oogAYJNWBk8whzKNL/OoAUR4tapi1GHTOgp7TW6FQPUpHt3puuzVXpFp+UQU4QlfMQbfMUHvMVn/PhzlufP6P5Lk06lx2V2JXo+v//+L8ugk8PJN2sA46vXoWqV1mB9nHo2fF0a6bR9pKtY7b3VOLt43d/MLnnLeIUvpPUSn/Ce1JqNN7WTYdk2RMgs6bc1/Ul+NSFhQsqsxZPbgW1hWIBFWCFv1iEJKUhDzvevDdfQERaFlLAnZHqtQijgzMGPEI4+Ad43mDY=</latexit>

a(t) =

r3K⇤

cosh

r⇤

3t

!

<latexit sha1_base64="ng6iLG2QbttQjLTfyGF+u883pRw=">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</latexit><latexit sha1_base64="ng6iLG2QbttQjLTfyGF+u883pRw=">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</latexit><latexit sha1_base64="ng6iLG2QbttQjLTfyGF+u883pRw=">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</latexit><latexit sha1_base64="ng6iLG2QbttQjLTfyGF+u883pRw=">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</latexit>

positive (closed), zero (flat), or negative (open)spatial curvature

Page 11: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Expansion, Contraction, and Expansion Again

N

Page 12: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Einstein Gravity & a Scalar Field

S =

Zd4x

p�g

✓1

2R� 1

2gµ⌫@µ�@⌫�� V (�)

<latexit sha1_base64="H5GhMDoFcdS8xHh6Z/nefxJXrQU=">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</latexit><latexit sha1_base64="H5GhMDoFcdS8xHh6Z/nefxJXrQU=">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</latexit><latexit sha1_base64="H5GhMDoFcdS8xHh6Z/nefxJXrQU=">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</latexit><latexit sha1_base64="H5GhMDoFcdS8xHh6Z/nefxJXrQU=">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</latexit>

Action

Eqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

flatterflatter

negative part

Page 13: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

1. Initial Expansion PhaseEqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

flatter

negative partSuppose ⇢ � K

a2> 0

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initially.

Page 14: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

1. Initial Expansion PhaseEqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">AAAClnichVFLSwJRFD5OL7OHVhuhjSSGbeRMBEWLkCJq6SMfoCIz000vjjPDzCiY9AfatQpyVdAi+g2t2vQHWvgToqVBmxYdx4koyc5w5373O+c7934c2VC5ZSN2PcLY+MTklHfaNzM7N+8PLCxmLb1hKiyj6Kpu5mXJYirXWMbmtsryhsmkuqyynFzb6+dzTWZaXNeO7JbBSnWpovETrkg2UblstGhU+Vo5EMYYOhEaBqILwuBGQg88QBGOQQcFGlAHBhrYhFWQwKKvACIgGMSVoE2cSYg7eQZn4CNtg6oYVUjE1uhfoVPBZTU693tajlqhW1RaJilDEMFnvMMePuE9vuDHn73aTo/+W1q0ywMtM8r+82D6/V9VnXYbqt+qEYqvWpOyJ7RG+7OpZsvxxcmn4TB9x8rgrubpZS+9nYq0V/EGX8nrNXbxkdxqzTflNslSHfDRsMTfoxkG2fWYiDExuRGO77pj88IyrECUZrMJcTiEBGSc113AFXSEoLAj7AsHg1LB42qW4EcIiU+9kpTS</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">AAAClnichVFLSwJRFD5OL7OHVhuhjSSGbeRMBEWLkCJq6SMfoCIz000vjjPDzCiY9AfatQpyVdAi+g2t2vQHWvgToqVBmxYdx4koyc5w5373O+c7934c2VC5ZSN2PcLY+MTklHfaNzM7N+8PLCxmLb1hKiyj6Kpu5mXJYirXWMbmtsryhsmkuqyynFzb6+dzTWZaXNeO7JbBSnWpovETrkg2UblstGhU+Vo5EMYYOhEaBqILwuBGQg88QBGOQQcFGlAHBhrYhFWQwKKvACIgGMSVoE2cSYg7eQZn4CNtg6oYVUjE1uhfoVPBZTU693tajlqhW1RaJilDEMFnvMMePuE9vuDHn73aTo/+W1q0ywMtM8r+82D6/V9VnXYbqt+qEYqvWpOyJ7RG+7OpZsvxxcmn4TB9x8rgrubpZS+9nYq0V/EGX8nrNXbxkdxqzTflNslSHfDRsMTfoxkG2fWYiDExuRGO77pj88IyrECUZrMJcTiEBGSc113AFXSEoLAj7AsHg1LB42qW4EcIiU+9kpTS</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

flatter

negative partSuppose ⇢ � K

a2> 0

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initially.

Page 15: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

1. Initial Expansion PhaseEqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

�<latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit><latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit><latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit><latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit>

V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

flatter

negative partSuppose ⇢ � K

a2> 0

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initially.

Page 16: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

2. Contraction PhaseEqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

negative partDuring contraction, H < 0 works as anti-friction.

Page 17: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

2. Contraction PhaseEqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">AAAClnichVFLSwJRFD5OL7OHVhuhjSSGbeRMBEWLkCJq6SMfoCIz000vjjPDzCiY9AfatQpyVdAi+g2t2vQHWvgToqVBmxYdx4koyc5w5373O+c7934c2VC5ZSN2PcLY+MTklHfaNzM7N+8PLCxmLb1hKiyj6Kpu5mXJYirXWMbmtsryhsmkuqyynFzb6+dzTWZaXNeO7JbBSnWpovETrkg2UblstGhU+Vo5EMYYOhEaBqILwuBGQg88QBGOQQcFGlAHBhrYhFWQwKKvACIgGMSVoE2cSYg7eQZn4CNtg6oYVUjE1uhfoVPBZTU693tajlqhW1RaJilDEMFnvMMePuE9vuDHn73aTo/+W1q0ywMtM8r+82D6/V9VnXYbqt+qEYqvWpOyJ7RG+7OpZsvxxcmn4TB9x8rgrubpZS+9nYq0V/EGX8nrNXbxkdxqzTflNslSHfDRsMTfoxkG2fWYiDExuRGO77pj88IyrECUZrMJcTiEBGSc113AFXSEoLAj7AsHg1LB42qW4EcIiU+9kpTS</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

negative partDuring contraction, H < 0 works as anti-friction.

Page 18: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

2. Contraction PhaseEqs. of motion

H2 =

1

3

✓1

2�2 + V (�)

◆� K

a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

negative partDuring contraction, H < 0 works as anti-friction.

If the kinetic energy is sufficiently suppressed on the plateau,the positive curvature can make the universe expand again.

Page 19: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

3. Second Expansion PhaseEqs. of motion

H2 =

1

3

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2�2 + V (�)

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a2,

H = �1

2�2 +

Ka2

,

�+ 3H�+@V (�)

@�= 0 .

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Scalar Potential

�<latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit><latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit><latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit><latexit sha1_base64="6QnDn811/tTWOTSUB7MLtmHvaEg=">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</latexit>

V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">AAAClnichVFLSwJRFD5OL7OHVhuhjSSGbeRMBEWLkCJq6SMfoCIz000vjjPDzCiY9AfatQpyVdAi+g2t2vQHWvgToqVBmxYdx4koyc5w5373O+c7934c2VC5ZSN2PcLY+MTklHfaNzM7N+8PLCxmLb1hKiyj6Kpu5mXJYirXWMbmtsryhsmkuqyynFzb6+dzTWZaXNeO7JbBSnWpovETrkg2UblstGhU+Vo5EMYYOhEaBqILwuBGQg88QBGOQQcFGlAHBhrYhFWQwKKvACIgGMSVoE2cSYg7eQZn4CNtg6oYVUjE1uhfoVPBZTU693tajlqhW1RaJilDEMFnvMMePuE9vuDHn73aTo/+W1q0ywMtM8r+82D6/V9VnXYbqt+qEYqvWpOyJ7RG+7OpZsvxxcmn4TB9x8rgrubpZS+9nYq0V/EGX8nrNXbxkdxqzTflNslSHfDRsMTfoxkG2fWYiDExuRGO77pj88IyrECUZrMJcTiEBGSc113AFXSEoLAj7AsHg1LB42qW4EcIiU+9kpTS</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

negative part

The flat part of the potential, built in for the bounce purpose,lets the universe naturally enter the slow-roll inflationary regime.

The resultant cosmology can be consistent with observation.

Page 20: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Example Model

N

Page 21: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

0 5 10 15-0.4-0.20.00.20.40.60.81.0

-2-1

0

123

V0 η

log 10(a(η)/a(0))

ϕ

Example Model

V (�) =V0

✓tanh2

�p6↵

�+ � tanh

�p6↵

�+ �

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Scalar potential

-10 -5 0 5 10

0.0

0.2

0.4

0.6

0.8

1.0

ϕ

V/V 0

↵ > 0, �1 < � < 1, �1 < � 0<latexit sha1_base64="3G1PD8v0scCrWJMvlTF04cZLIxw=">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</latexit>

Initial conditions for numerical calculation

�(0) =�p6↵, �(0) =0, and

Ka(0)2

=0.05pV0.

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Numerical results

0 5 10 15

0.001

0.010

0.100

1

V0 η

ρ X/V0

↵ = 1, � = 0, � = �0.09143<latexit sha1_base64="xxYHC/PzAVlOvXwadT1yGEHewN0=">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</latexit>

Potential

Curvature

Kinetic

Scale factor (log)

Scalar field

Page 22: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Possible origin of the positive curvature

N

Page 23: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Birth of Closed UniverseMini-superspace approximation

[gµ⌫(t, x),�(t, x)] = [a(t),�(t)]<latexit sha1_base64="SWWc5Y7+XMJ1qfE8RMHhQgBNTJ8=">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</latexit>

Wave function of the universe:

Wheeler-De Whit eq.H(a,�) (a,�) = 0

<latexit sha1_base64="59+JrKD3paaeuPEAoClex+pUSjA=">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</latexit>

where

H(a,�) =1

12a2@

@a

✓a@

@a

◆�

1

2a3@2

@�2� U(a,�)

<latexit sha1_base64="KtcPK1M4IF7KJDtZd6lHQAV6ygo=">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</latexit>

U(a,�) = a3✓3Ka2

� V (�)

<latexit sha1_base64="ej8VNBf9imBxum/OiqsIGmzCrCg=">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</latexit>

Initial conditions

[Kiefer, Sandhoefer, 0804.0672]

a(0) =

s3KV (�)

<latexit sha1_base64="I/INcI1bWULV3vTbEXKfwuq9n/o=">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</latexit>

a(0) =0<latexit sha1_base64="cjdvRNBYO/SFwg0p9gYfHpdP3cs=">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</latexit>

�(0) =const.<latexit sha1_base64="s825m7PtiWDMvx6P4FouzRBcAas=">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</latexit>

�(0) =0<latexit sha1_base64="lS8GefaUn6p7p9f7tMhRfVzG6BM=">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</latexit>

0.0 0.5 1.0 1.5 2.0-4

-3

-2

-1

0

1

2

a/ K /V

U(a)/K[Vilenkin, PRD37, 888 (1988)]

Page 24: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Birth of Closed UniverseMini-superspace approximation

[gµ⌫(t, x),�(t, x)] = [a(t),�(t)]<latexit sha1_base64="SWWc5Y7+XMJ1qfE8RMHhQgBNTJ8=">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</latexit>

Wave function of the universe:

Wheeler-De Whit eq.H(a,�) (a,�) = 0

<latexit sha1_base64="59+JrKD3paaeuPEAoClex+pUSjA=">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</latexit>

where

H(a,�) =1

12a2@

@a

✓a@

@a

◆�

1

2a3@2

@�2� U(a,�)

<latexit sha1_base64="KtcPK1M4IF7KJDtZd6lHQAV6ygo=">AAADdHichVHNahRBEK7Z8SfGn6x6EZJD47KyAbP0bARFEIK5BHLJj5sEMnHp6fTsNumdGXt6F5JhXsAX8OApAQ+SV8jNiy/gIQ/gQTxG9OLBmtnZmDWYdNPd1V/VV/UV5UVKxobSY6tkX7l67frYjfGbt27fmSjfvbcWhz3NRZOHKtQbHouFkoFoGmmU2Ii0YF1PiXVvZz7zr/eFjmUYvDK7kdjqsnYgfcmZQahV/pq4nCmykNbYYzfqyGnygri+ZpwkxCEpXo2EMJK+TkiDpLiHTjdi2kikZkGnH5ZFKOGbGmGXRxJXy3bHTJOZkZKnFWdHKo5kGuo5A2ZmR57xzTSHTbXKFVqn+SLnDacwKlCspbB8BC5sQwgcetAFAQEYtBUwiHFvggMUIsS2IEFMoyVzv4AUxpHbwyiBEQzRHbzb+Nss0AD/Wc44Z3OsovBoZBKo0i/0Iz2hn+kh/UZ//zdXkufItOzi6w24ImpNvH2w+utSVhdfA52/rAs1G/DhWa5VovYoR7Iu+IDf33t3svp8pZo8ogf0O+rfp8f0E3YQ9H/wD8ti5f0FeoZKNHp9PCmOyvl3MOeNtUbdma03lp9U5l4WQxuDSXgINZzMU5iDBViCJnBr0Xpj7VlJ6ac9ZVfs6iC0ZBWc+zCy7PofZRPd2g==</latexit>

U(a,�) = a3✓3Ka2

� V (�)

<latexit sha1_base64="ej8VNBf9imBxum/OiqsIGmzCrCg=">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</latexit>

Initial conditions

[Kiefer, Sandhoefer, 0804.0672]

a(0) =

s3KV (�)

<latexit sha1_base64="I/INcI1bWULV3vTbEXKfwuq9n/o=">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</latexit>

a(0) =0<latexit sha1_base64="cjdvRNBYO/SFwg0p9gYfHpdP3cs=">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</latexit>

�(0) =const.<latexit sha1_base64="s825m7PtiWDMvx6P4FouzRBcAas=">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</latexit>

�(0) =0<latexit sha1_base64="lS8GefaUn6p7p9f7tMhRfVzG6BM=">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</latexit>

0.0 0.5 1.0 1.5 2.0-4

-3

-2

-1

0

1

2

a/ K /V

U(a)/K

tunneling“Creation of the universe from nothing”

[Vilenkin, PLB117, 25 (1982)]

[Vilenkin, PRD37, 888 (1988)]

Page 25: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Cyclic universe

NNNNNNNNNNNN

Page 26: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Cyclic solution

V (�) =V0

✓tanh2

�p6↵

�+ � tanh

�p6↵

�+ �

<latexit sha1_base64="0ftTuH9K6pUkxlY1g/5xXsNg4pM=">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</latexit>

Scalar potential

-10 -5 0 5 10

0.0

0.2

0.4

0.6

0.8

1.0

ϕ

V/V 0

↵ > 0, �1 < � < 1, �1 < � 0<latexit sha1_base64="3G1PD8v0scCrWJMvlTF04cZLIxw=">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</latexit>

Initial conditions for numerical calculation

Numerical results

a(0) =

s3KV (�)

<latexit sha1_base64="I/INcI1bWULV3vTbEXKfwuq9n/o=">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</latexit>

a(0) =0<latexit sha1_base64="cjdvRNBYO/SFwg0p9gYfHpdP3cs=">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</latexit>

�(0) =0<latexit sha1_base64="lS8GefaUn6p7p9f7tMhRfVzG6BM=">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</latexit>

�(0) = �3p6↵

<latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">AAACqXichVHNLgNRFD7Gf/10sJHYiKZSEc0ZBJFIhI2lqv5E2zQz49Ib05kxc9uEhgfwAhZWhIUIb2Bl4wUs+ghiWYmNhdPpiNDg3Nx7z/3O+c65X45mG9wViNUWqbWtvaOzqzvQ09vXH5QHBpOuVXJ0ltAtw3LSmuoyg5ssIbgwWNp2mFrUDJbS9lbr8VSZOS63zE1xYLNcUd01+Q7XVUFQXh7O2gUewYmlqZmsu++IylxWNeyCepSXQxhFz0abHcV3QuDbuiXfQxa2wQIdSlAEBiYI8g1QwaWVAQUQbMJyUCHMIY97cQZHECBuibIYZaiE7tG5S6+Mj5r0rtd0PbZOXQzaDjFHIYxPeI01fMQbfMb3X2tVvBr1vxzQrTW4zM4HT4bjb/+yinQLKHyx/mB85joU3aH9tz5BOQueLk46bQ+pK9YbvcqHp7X44ka4Mo4X+EJaz7GKD6TWLL/qVzG2cQYBGpbyczTNTnI6qmBUic2Gllf8sXXBCIxBhGYzD8uwBuuQoL7HcAm3cCdNSjEpLW01UqUWnzME30zSPwCIxZva</latexit><latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit><latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit><latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit>

↵ = 5⇥ 10�5, � = � = 0,<latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit><latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit><latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit><latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit>

K = 0.05V0<latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit>

0 100 200 300 400 5001.0

1.2

1.4

1.6

V0 ηa(η)/a(0)

(Details in the next slide)

Page 27: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Summary and conclusions

N

Page 28: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Summary & Conclusion

Page 29: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Summary & Conclusion• We find new nontrivial cosmological solutions.

- (Creation from Nothing →) Expansion → Contraction → Inflationary Expansion- (Creation from Nothing →) Cyclic

Page 30: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Summary & Conclusion• We find new nontrivial cosmological solutions.

- (Creation from Nothing →) Expansion → Contraction → Inflationary Expansion- (Creation from Nothing →) Cyclic

• 4dim, Einsteing gravity, a real canonical scalar.No violation of Null Energy Condition. No singularity.

Page 31: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Summary & Conclusion• We find new nontrivial cosmological solutions.

- (Creation from Nothing →) Expansion → Contraction → Inflationary Expansion- (Creation from Nothing →) Cyclic

• 4dim, Einsteing gravity, a real canonical scalar.No violation of Null Energy Condition. No singularity.

• Our solutions open interesting possibilities for General Relativity and Our Universe (past and future).

Page 32: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Summary & Conclusion• We find new nontrivial cosmological solutions.

- (Creation from Nothing →) Expansion → Contraction → Inflationary Expansion- (Creation from Nothing →) Cyclic

• 4dim, Einsteing gravity, a real canonical scalar.No violation of Null Energy Condition. No singularity.

• Our solutions open interesting possibilities for General Relativity and Our Universe (past and future).

• Many things to be explored.

Page 33: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Discussion• Not restricted to our past. It may occur in future.

• Tuning is necessary. Anthropic justification?

• Eternal inflation or “just enough” inflation. Observable effects?

• CMB data slightly favor positive curvature. It can relax the H_0 tension.

• In contrast with bubble universes in the string landscape.

• Vacuum energy can change during the contraction phase.

• Bounce at an arbitrarily higher energy scale may be possible.

Page 34: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Appendix

N

Page 35: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Dynamics of Scalar Field

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@�= 0

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approximation regime relevance

“No friction” e.g.) oscillation relevant around H=0

“Slow-Roll regime” Potential energy dominate.

attractor solution during expansion

“Ultra-Slow-Roll regime” Kinetic energy is important.

attractor solution during contraction

“Scaling solution” (special situation)

Page 36: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Dynamics of Scalar Field

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approximation regime relevance

“No friction” e.g.) oscillation relevant around H=0

“Slow-Roll regime” Potential energy dominate.

attractor solution during expansion

“Ultra-Slow-Roll regime” Kinetic energy is important.

attractor solution during contraction

“Scaling solution” (special situation)

How to suppress the kinetic energy

is the key for a successful bounce.

Page 37: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

0 5 10 15-0.4-0.20.00.20.40.60.81.0

-2-1

0

123

V0 η

log 10(a(η)/a(0))

ϕ

0 5 10 150.001

0.0050.010

0.0500.100

V0 η

TermsinEoM

Tuning of the offset

�+ 3H�+@V

@�= 0

<latexit sha1_base64="QHwtnVXeHZbu5mr/cQq5DrXekSQ=">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</latexit>

3H2 =1

2�2 + V � 3K

a2<latexit sha1_base64="n/nKuh5nwUmhHkwaWN4N7ezrX8k=">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</latexit>

Kinetic energy

Potential energy

“Curvature energy”

AccelerationSlope

FrictionAcceleration

Friction

Slope

Scale factor (log)

Scalar field

0 5 10 15

0.001

0.010

0.100

1

V0 η

ρ X/V0

Potential

Curvature

Kinetic

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

ϕ

V(ϕ)/V0

↵ = 1, � = 0, � = �0.09143<latexit sha1_base64="xxYHC/PzAVlOvXwadT1yGEHewN0=">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</latexit>

Page 38: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

0 5 10 15

0.0

0.5

1.0

1.5

-2-1012345

V0 η

log 10(a(η)/a(0))

ϕ

0 5 10 15

0.0050.010

0.0500.100

0.500

V0 η

TermsinEoM

Tuning of the left-right asymmetry

AccelerationFriction

Slope

↵ = 1, � = �0.3805885, � = �0.05<latexit sha1_base64="UgpxbAm8vBM91TVO9IQTTTveMFk=">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</latexit>

Scale factor (log)

Scalar field

0 5 10 15

0.001

0.010

0.100

1

V0 η

ρ X/V0

PotentialCurvature

Kinetic

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ϕ

V(ϕ)/V0

�+ 3H�+@V

@�= 0

<latexit sha1_base64="QHwtnVXeHZbu5mr/cQq5DrXekSQ=">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</latexit>

3H2 =1

2�2 + V � 3K

a2<latexit sha1_base64="n/nKuh5nwUmhHkwaWN4N7ezrX8k=">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</latexit>

Kinetic energy

Potential energy

“Curvature energy”

AccelerationSlope

Friction

Page 39: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

0 5 10 15-0.4-0.20.00.20.40.60.8

-3-2-101234

V0 η

log 10(a(η)/a(0))

ϕ

0 5 10 15

0.001

0.010

0.100

1

V0 η

ρ X/V0

0 5 10 150.005

0.010

0.050

0.100

V0 η

TermsinEoM

PotentialCurvature

Kinetic

Tuning of the width

Acceleration

Friction

Slope

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

ϕ

V(ϕ)/V0

↵ = 0.8924, � = 0, � = �0.1<latexit sha1_base64="LxL98qW1M6GXKf0cPqSnyWWmJbI=">AAACw3ichVFNSxtRFD0ZW6u2amw3QjdDQ0oXdrgTA1qhIJZCl1GbKBgJb8aXOGS+OvMSsCF/oH+gUFctdCH+hS4KbtR9F/4EcangxoU3k2lFRXuHN++8c++57zyuFbpOrIiOMtrAg4eDj4aGRx4/GR0bz048rcRBK7Jl2Q7cIFq1RCxdx5dl5ShXroaRFJ7lyhWr+a6XX2nLKHYC/6PaCuW6Jxq+U3dsoZiqZYtV4YabQn+rkzH7plCc0qufWmJDr1pSJew/oiE8j5nXZJi1bI4MSkK/DcwU5JBGKcj+QhUbCGCjBQ8SPhRjFwIxf2swQQiZW0eHuYiRk+QluhhhbYurJFcIZpv8b/BpLWV9Pvd6xona5ltcXhErdeTpD+3QKe3TLh3TxZ29OkmPnpct3q2+Voa18S+Ty+f/VXm8K2xeqe71rFDHbOLVYe9hwvReYff17c9fT5fnlvKdl/SDTtj/dzqiPX6B3z6zfy7Kpe17/Px1EnG2zqvLozJvDuY2qBQMc9ooLBZz8wvp0IbwHC/wiiczg3l8QAllvukbfuMAh9p7ralFmuqXaplU8wzXQuteAn9Ao4c=</latexit>

Scale factor (log)

Scalar field

�+ 3H�+@V

@�= 0

<latexit sha1_base64="QHwtnVXeHZbu5mr/cQq5DrXekSQ=">AAAC0XichVFBSxtBFP7cqrW2alovQi+LwVIohBcttBQKohePakwUXAmzm4kZ3Owus5NAGgKlR/sDevBkQbD0L/TmxZM3af0J0qOCFw++bFZsK9W3zM4333vfm294buSr2BCd9FkP+gcGHw49Gn78ZGR0LPP0WSkOG9qTRS/0Q73milj6KpBFo4wv1yItRd315aq7Nd/NrzaljlUYrJhWJDfqYjNQVeUJw1Q5M+dUKqFpO1FNdexX9oy9YDt/EE5VC48PQhslfLvUucG9ivc2lTNZylES9m2QT0EWaSyGmR9wUEEIDw3UIRHAMPYhEPO3jjwIEXMbaDOnGakkL9HBMGsbXCW5QjC7xf9NPq2nbMDnbs84UXt8i89Ls9LGFB3TNzqjQ/pOp3T5317tpEfXS4t3t6eVUXlse6Jwca+qzrtB7UZ1p2eDKt4mXhV7jxKm+wqvp29++HJWeLc81X5BX+k3+9+lEzrgFwTNc29vSS7v3OHn2onmbJVXh0eV/3cwt0FpOpefyU0vvc7OzqVDG8JzTOIlT+YNZrGARRT5pn0c4Sd+WQWrZX20PvVKrb5UM46/wvp8BYu4rJs=</latexit>

3H2 =1

2�2 + V � 3K

a2<latexit sha1_base64="n/nKuh5nwUmhHkwaWN4N7ezrX8k=">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</latexit>

Kinetic energy

Potential energy

“Curvature energy”

AccelerationSlope

Friction

Page 40: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Birth of Closed UniverseInitial conditions

a(0) =

s3KV (�)

<latexit sha1_base64="I/INcI1bWULV3vTbEXKfwuq9n/o=">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</latexit>

a(0) =0<latexit sha1_base64="cjdvRNBYO/SFwg0p9gYfHpdP3cs=">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</latexit>

�(0) =const.<latexit sha1_base64="s825m7PtiWDMvx6P4FouzRBcAas=">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</latexit>

�(0) =0<latexit sha1_base64="lS8GefaUn6p7p9f7tMhRfVzG6BM=">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</latexit>

[Vilenkin, PRD30, 509 (1984)]

[Vilenkin, PRD37, 888 (1988)]

How this is determined is controversial.

Two proposals for nucleation probability

[Vilenkin, PRD37, 888 (1988)]

P (a,�) / exp

✓⌥24⇡2M4

P

V (�)

<latexit sha1_base64="pJ8VjSSQRoznCygwGKLtsFhYYjg=">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</latexit>

[Hartle, Hawking, PRD28, 2960 (1983)]“no-boundary proposal”

Even if the universe is born with tiny energy density by the Hartle-Hawking process,the energy density can increase in the contraction phase, which makes the scenario viable.

Page 41: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Cyclic solution

V (�) =V0

✓tanh2

�p6↵

�+ � tanh

�p6↵

�+ �

<latexit sha1_base64="0ftTuH9K6pUkxlY1g/5xXsNg4pM=">AAADRnicpVHLahRBFL3d8RHjI5NkI4hQOCRMEIaaUZIQEILZZKGQhzMJTI9DdVk9XaT6keqagdjMD/gDLlwZyCL4AW7cuckPZJFPCIKbKNm48HZPB9FgXFjd1XXuuffcPsV1YyUTQ+mxZY9cuXrt+uiNsZu3bt8ZL01MNpOop7lo8EhFestliVAyFA0jjRJbsRYscJXYdLeXs/xmX+hERuELsxuLdsC6ofQkZwapTuljs+LEvpwlT2ZIs0OJo4RnKsQxLPTJy5TUySAdki3ieJpxgiEKSM4nO9ogMecwFfsMufx1tOz6pp3hh44rDCva/V+jLgsCds7NdkplWqX5IhdBrQBlKNZqVPoEDryCCDj0IAABIRjEChgk+LSgBhRi5NqQIqcRyTwvYABjqO1hlcAKhuw2frsYtQo2xDjrmeRqjn9RuDUqCUzTI3pAT+kh/UBP6I+/9krzHpmXXTzdoVbEnfE3dzfO/qkK8DTg/1Jd6tmABwu5V4ne45zJbsGH+v7rt6cbi+vT6Qzdo1/Q/3t6TD/jDcL+N76/JtbfXeLn3InGrId7gKOq/TmYi6BZr9YeVetrj8tLT4uhjcI9eAAVnMw8LMEKrEIDuHXfWraeWc/tQ/ur/d0+G5baVqGZgt/WCPwEty7P7A==</latexit>

Scalar potential

-10 -5 0 5 10

0.0

0.2

0.4

0.6

0.8

1.0

ϕ

V/V 0

↵ > 0, �1 < � < 1, �1 < � 0<latexit sha1_base64="3G1PD8v0scCrWJMvlTF04cZLIxw=">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</latexit>

Initial conditions for numerical calculation

Numerical results

a(0) =

s3KV (�)

<latexit sha1_base64="I/INcI1bWULV3vTbEXKfwuq9n/o=">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</latexit>

a(0) =0<latexit sha1_base64="cjdvRNBYO/SFwg0p9gYfHpdP3cs=">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</latexit>

�(0) =0<latexit sha1_base64="lS8GefaUn6p7p9f7tMhRfVzG6BM=">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</latexit>

�(0) = �3p6↵

<latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit><latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit><latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit><latexit sha1_base64="P8dHHAuG9zO6GHjQNa4gk+Usb6k=">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</latexit>

↵ = 5⇥ 10�5, � = � = 0,<latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit><latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit><latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit><latexit sha1_base64="le1K67YAepRxV3PIsSUT0SCu7c0=">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</latexit>

K = 0.05V0<latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit>

0 100 200 300 400 5001.0

1.2

1.4

1.6

V0 ηa(η)/a(0)

(Details in the next slide)

Page 42: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Cyclic solution↵ = 5⇥ 10�5, � = � = 0,

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K = 0.05V0<latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit><latexit sha1_base64="of6jswWC/kSVzQWApJMym5AxL4k=">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</latexit>

0 100 200 300 400 5001.0

1.2

1.4

1.6

V0 η

a(η)/a(0)

22.2 22.4 22.6 22.8

1.62150

1.62160

1.62170

1.62180

1.62190

V0 η

a(η)/a(0)

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

V0 η

ρ X/V0

22.2 22.4 22.6 22.80.0

0.1

0.2

0.3

V0 η

ρ X/V0

Scale factor Energy densities

Potential

Curvature

Kinetic

Page 43: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Mechanism of CyclesCoarse graining oscillations of the scalar field

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-0.04 -0.02 0.00 0.02 0.040.0

0.2

0.4

0.6

0.8

1.0

ϕ

V/V 0

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wave ⇠ �1<latexit sha1_base64="cnXbXMXe3IYv87GagtT7rb9Ei6w=">AAACpXichVE7SwNBEB7Pd3zkfBSCTTBEbQxzIihWoo2N4CMPwYTj7lzj4r2420ST4B/wD1jYqGCh/gYrG/+ARX6CWEawsXByORENxll2d/ab+Wb2Y3TX5L5ArHVInV3dPb19/ZGBwaHhqDwymvGdomewtOGYjreraz4zuc3SgguT7boe0yzdZFn9aK0Rz5aY53PHTomyy/KWVrD5ATc0QZAqjx+rOcFORFUrsdNYzudWbE5R5TgmMbBYq6OEThxC23TkB8jBPjhgQBEsYGCDIN8EDXxae6AAgktYHqqEeeTxIM7gFCLELVIWowyN0CM6C/TaC1Gb3o2afsA2qItJ2yNmDBL4jLdYxye8xxf8+LNWNajR+EuZbr3JZa4aPZvYef+XZdEt4PCb1YbxletR9IB2e32CcpYCXZx0ugHSUGw0e5Uq5/Wd5e1EdRqv8ZW0XmENH0mtXXozbrbY9gVEaFjK79G0Opn5pIJJZWshvrIajq0PJmEKZmk2i7AC67AJaepbgUu4hTtpRtqQUlKmmSp1hJwx+GGS+gmysJqv</latexit><latexit sha1_base64="cnXbXMXe3IYv87GagtT7rb9Ei6w=">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</latexit><latexit sha1_base64="cnXbXMXe3IYv87GagtT7rb9Ei6w=">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</latexit><latexit sha1_base64="cnXbXMXe3IYv87GagtT7rb9Ei6w=">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</latexit>

oscillates around -1/3.wave<latexit sha1_base64="ktjma3WlcSymErt3yOjT9JYyY7U=">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</latexit><latexit sha1_base64="ktjma3WlcSymErt3yOjT9JYyY7U=">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</latexit><latexit sha1_base64="ktjma3WlcSymErt3yOjT9JYyY7U=">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</latexit><latexit sha1_base64="ktjma3WlcSymErt3yOjT9JYyY7U=">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</latexit>

Cont

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ion

Expa

nsio

n

Page 44: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Discussion• Not restricted to our past. It may occur in future.

• Tuning is necessary. Anthropic justification?

• Eternal inflation or “just enough” inflation. Observable effects?

• CMB data slightly favor positive curvature. It can relax the H_0 tension.

• In contrast with bubble universes in the string landscape.

• Vacuum energy can change during the contraction phase.

• Bounce at an arbitrarily higher energy scale may be possible.

Page 45: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Constraints on Curvature⌦K = �0.056+0.028

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(68%, Planck TT+lowE)

⌦K = �0.044+0.018�0.015

<latexit sha1_base64="ugYu6NtyCHPgtNOAlFXacdBJY+g=">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</latexit><latexit sha1_base64="ugYu6NtyCHPgtNOAlFXacdBJY+g=">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</latexit><latexit sha1_base64="ugYu6NtyCHPgtNOAlFXacdBJY+g=">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</latexit><latexit sha1_base64="ugYu6NtyCHPgtNOAlFXacdBJY+g=">AAACwHichVHLLgRBFL3ae7wGG4lNx4RIxLgtI0QiETYSC89BYuhUtzLT0S/dNZPQmR+wFwsrEgvxCWKFxA9Y+ASxJLGxcLunRRBUp6pOnXvPvX1yNdc0fIH4UCVV19TW1Tc0JpqaW1rbku0dK75T9HSe1R3T8dY05nPTsHlWGMLka67HmaWZfFXbmQ7jqyXu+YZjL4s9l29YLG8b24bOBFFqcig3Z/E8U3MWEwWdmcFsWZ6QBzGNmYy8GQwQUMbKshqElDJSVpMpAtGSfwIlBimI17yTvIIcbIEDOhTBAg42CMImMPDpWwcFEFziNiAgziNkRHEOZUiQtkhZnDIYsTt05um1HrM2vcOafqTWqYtJ2yOlDL14j+f4jHd4gY/49mutIKoR/sse3VpFy1217aBr6fVflUW3gMKn6g/FR65H0W3af/sTlDMW+TLIpxsxoWO90qu0f/S8NL7YG/ThKT6R1xN8wGtya5de9LMFvngMCRqW8n00P8HKcFrBtLKQSU1OxWNrgG7ogX6azShMwgzMQ5b6HsIl3MCtNCUVJEfaraRKVbGmE74saf8d4d2iPA==</latexit>

(68%, Planck TT,TE,EE+lowE)

⌦K = �0.0106± 0.0065<latexit sha1_base64="kF3+ygiVw0I/lZZQlAlC/a9A3Ao=">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</latexit><latexit sha1_base64="kF3+ygiVw0I/lZZQlAlC/a9A3Ao=">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</latexit><latexit sha1_base64="kF3+ygiVw0I/lZZQlAlC/a9A3Ao=">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</latexit><latexit sha1_base64="kF3+ygiVw0I/lZZQlAlC/a9A3Ao=">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</latexit>

(68%, Planck TT,TE,EE+lowE+lensing)

⌦K = 0.0007± 0.0019<latexit sha1_base64="IeM7hFutvZVHMymDGiAhlZ9Wba4=">AAACt3ichVE7SwNBEB7Pd3xFbQSbw2CwCnMi+ABBtBEsfEaFXDj2zk1y5F7ebQIa/ANib2GlYCF2tmpl4x+wyE8Qywg2Fk4uJ6JB3WN3v/1mvpn7GN2zzEAgVluk1rb2js6u7lhPb1//QHxwaDtwS77B04Zruf6uzgJumQ5PC1NYfNfzObN1i+/oxaV6fKfM/cB0nS1x4PGszfKOmTMNJojS4kl11eZ5pqk2EwWDWZWVI3lexhQiTsuqZ4dQmdXiiZCjJTcDJQIJiNaaG78HFfbABQNKYAMHBwRhCxgE9GVAAQSPuCxUiPMJmWGcwxHESFuiLE4ZjNginXl6ZSLWoXe9ZhCqDepi0fZJKcM4PuEV1vARr/EZ33+tVQlr1P/lgG69oeWeNnA8svn2r8qmW0DhS/WH4jPXp2iO9t/+BOXMhL5M8umFTN2x0ehVPjytbc5tjFeSeIEv5PUcq/hAbp3yq3G5zjfOIEbDUn6OphlsT6YUTCnrU4mFxWhsXTAKYzBBs5mGBViGNUhT3xO4gVu4k2YlTcpJhUaq1BJphuHbkvY/AJDKn1U=</latexit><latexit sha1_base64="IeM7hFutvZVHMymDGiAhlZ9Wba4=">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</latexit><latexit sha1_base64="IeM7hFutvZVHMymDGiAhlZ9Wba4=">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</latexit><latexit sha1_base64="IeM7hFutvZVHMymDGiAhlZ9Wba4=">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</latexit>

(68%, Planck TT,TE,EE+lowE+lensing+BAO)

[Planck collaboration 2018: Cosmological parameters] [Riess, Casertano, Yuan, Macri, Scolnic, 1903.07603]

Page 46: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Uplifting Vacuum Energy

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Another minimum for φ may not be necessary.

Potential of light (axionic) field

[Graham, Kaplan, Rajendran, 1902.06793]

Page 47: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Uplifting Vacuum Energy

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Another minimum for φ may not be necessary.

Potential of light (axionic) field

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Page 48: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Uplifting Vacuum Energy

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Another minimum for φ may not be necessary.

Potential of light (axionic) field

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit> [Graham, Kaplan, Rajendran, 1902.06793]

Uplift of the cosmological constant is possible.

Page 49: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Increase Energy before BounceKinetic-potential scaling solution

1

2�2(t) / V (�(t))

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V (�) ⇠ exp⇣�p

3(1 + w)�⌘

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Page 50: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Increase Energy before BounceKinetic-potential scaling solution

1

2�2(t) / V (�(t))

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V (�) ⇠ exp⇣�p

3(1 + w)�⌘

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

bounce around here inflation

(w < -1/3)

Page 51: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Increase Energy before BounceKinetic-potential scaling solution

1

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V (�)<latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit><latexit sha1_base64="IaAfaNI+HMQeXonNPy/J76Wkukg=">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</latexit>

bounce around here inflation

(w < -1/3)

This may be useful for the Hartle-Hawking

“no-boundary” scenario.

Page 52: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

Related literatureOur scenario as a whole gives us a new interesting possibility,

but each part has been studied well in the literature.

Contraction by a negative potential Bounce with positive curvature

Cyclic universe

[Linde, hep-th/0110195]

[Felder et al., hep-th/0202017]

[Martin, Peter, hep-th/0307077]

[Gordon, Turok, hep-th/0206138]

[Falciano, Lilley, Peter, 0802.1196]

[Haro, 1511.05048]

[Parker, Fulling, PRD7, 2357 (1973)]

[Starobinsky, SAL4, 82 (1978)]

[Barrow, Matzner, PRD21, 336 (1980)]

[Hawking, Les Houches 1983]

[Page, CQG 1, 417 (1984)]

[Schmidt, gr-qc/0108087]

[Cornish, Shellard, gr-qc/9708046]

[Kardashev, MNRAS 243, 252 (1990)]

[Dabrowski, gr-qc/9503017]

[Graham, Horn, Kachru, Rajendran, Torroba, 1109.0282]

[Graham, Horn, Rajendran, Torroba, 1405.0282]

[Biswas, 0801.1315]See also

[Biswas, Alexander, 0812.3182]

[Barrow, Ganguly, 1703.05969]

[Ganguly, Barrow, 1710.00747]

Note: We do NOT violate the null energy condition. We consider only NON-singular bounce.

Page 53: Bouncing Universe from N Nothing · Bouncing Universe from NNothing Takahiro Terada (CTPU, IBS) Hiroki Matsui, Fuminobu Takahashi, TT, Phys. Lett. B795 (2019) 152, arXiv:1904.12312

What is new?Conditions (common)(A) 4d Einstein gravity, FLRW universe with positive spatial curvature, A single real canonical scalar(B) No violation of null energy condition, No singularity

conditions for “N-shaped” bouncing scenario(1) Expansion → Contraction → Expansion(2) The last expansion can be an arbitrarily long inflation phase

conditions for cyclic scenario(i) No need of negative potential(ii) No need of fine tuning

To best of our knowledge, our scenarios are the first bouncing/cyclic scenarios satisfying the following conditions.