K i Maeda (Waseda univ) Y Misonoh (Waseda univ)

and T Kobayashi (Tokyo univ)

Phys Rev D 82 064024 (2010)

日本物理学会 秋季大会2010 九州工業大学

1

201083 Summer School in Astronomy amp Astrophysics 2010 Toyohashi

Motivation

Horava-Lifshitz gravity (HL gravity) PHorava (2009) [1]

bull 119909119894 rarr 119887119909119894 t rarr 119887119911119905 power-counting renormalizable (119911 = 3)

bull higher curvature term in action singularity avoidance

Discussion in FLRW universe

original complete classification M Minamitsuji (2009)[2]

SVW assumption of the eos is not so clear

Analysis in SVW version without any no clear assumption

detailed balance condition bull restriction on action

impose (original)

relax (SVW)

2

Preceding studies SVW version (with matter)

1 P Wu and H Yu (2009) [3]

unstable static solution expand oscillationhellip

Emergent universe singularity avoidance

2 E J Son and W Kim (2010) [4]

inflation second accelerated expansion

Equation of state in HL gravity eos may be changed

120588119903119886119889 prop 119886minus4 ⟶ 119886minus6 unknown running parameter

Analyze vacuum solution in SVW version

3

Horava-Lifshitz gravity

Anisotropic scaling

119911 = 3 power-counting renormalizable

SVW most general form of potential term for 119911 = 3

T P Sotiriou M Visser and S Weinfurtner (2009)[5]

restriction on 119892119894 stability of flat background

1198921 lt 0 1198929 gt 0 120582 gt 1 1198921 = minus1 (time rescaling) 4

119905

Application to FLRW universe background homogeneous and isotropic

Hamiltonian constraint 1205812 = (1119872119875119871)2 = 1

If 119870 ne 0 119892119904 119892119903 ne 0 discuss only a nonflat universe

original 119892119904 = 0 119892119903 lt 0

SVW 119892119903 119892119904 are arbitrary

119892119903 119892119904 lt 0 corresponding to negative energy density

5

How to Classify Hamiltonian constraint

scale factor particle with zero energy in 119984 119886

possible range for scale factor 119984 119886 le 0

solutions

big bang rarr big crunch oscillation or bounce

6

stable static

unstable static

Classification (Λ gt 0 119870 = 1)

1 Big bang de Sitter

119892 119904 = 5 119892 119903 = 5

singularity is inevitable

Big bang to de Sitter phase

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Preceding studies SVW version (with matter)

1 P Wu and H Yu (2009) [3]

unstable static solution expand oscillationhellip

Emergent universe singularity avoidance

2 E J Son and W Kim (2010) [4]

inflation second accelerated expansion

Equation of state in HL gravity eos may be changed

120588119903119886119889 prop 119886minus4 ⟶ 119886minus6 unknown running parameter

Analyze vacuum solution in SVW version

3

Horava-Lifshitz gravity

Anisotropic scaling

119911 = 3 power-counting renormalizable

SVW most general form of potential term for 119911 = 3

T P Sotiriou M Visser and S Weinfurtner (2009)[5]

restriction on 119892119894 stability of flat background

1198921 lt 0 1198929 gt 0 120582 gt 1 1198921 = minus1 (time rescaling) 4

119905

Application to FLRW universe background homogeneous and isotropic

Hamiltonian constraint 1205812 = (1119872119875119871)2 = 1

If 119870 ne 0 119892119904 119892119903 ne 0 discuss only a nonflat universe

original 119892119904 = 0 119892119903 lt 0

SVW 119892119903 119892119904 are arbitrary

119892119903 119892119904 lt 0 corresponding to negative energy density

5

How to Classify Hamiltonian constraint

scale factor particle with zero energy in 119984 119886

possible range for scale factor 119984 119886 le 0

solutions

big bang rarr big crunch oscillation or bounce

6

stable static

unstable static

Classification (Λ gt 0 119870 = 1)

1 Big bang de Sitter

119892 119904 = 5 119892 119903 = 5

singularity is inevitable

Big bang to de Sitter phase

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Horava-Lifshitz gravity

Anisotropic scaling

119911 = 3 power-counting renormalizable

SVW most general form of potential term for 119911 = 3

T P Sotiriou M Visser and S Weinfurtner (2009)[5]

restriction on 119892119894 stability of flat background

1198921 lt 0 1198929 gt 0 120582 gt 1 1198921 = minus1 (time rescaling) 4

119905

Application to FLRW universe background homogeneous and isotropic

Hamiltonian constraint 1205812 = (1119872119875119871)2 = 1

If 119870 ne 0 119892119904 119892119903 ne 0 discuss only a nonflat universe

original 119892119904 = 0 119892119903 lt 0

SVW 119892119903 119892119904 are arbitrary

119892119903 119892119904 lt 0 corresponding to negative energy density

5

How to Classify Hamiltonian constraint

scale factor particle with zero energy in 119984 119886

possible range for scale factor 119984 119886 le 0

solutions

big bang rarr big crunch oscillation or bounce

6

stable static

unstable static

Classification (Λ gt 0 119870 = 1)

1 Big bang de Sitter

119892 119904 = 5 119892 119903 = 5

singularity is inevitable

Big bang to de Sitter phase

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Application to FLRW universe background homogeneous and isotropic

Hamiltonian constraint 1205812 = (1119872119875119871)2 = 1

If 119870 ne 0 119892119904 119892119903 ne 0 discuss only a nonflat universe

original 119892119904 = 0 119892119903 lt 0

SVW 119892119903 119892119904 are arbitrary

119892119903 119892119904 lt 0 corresponding to negative energy density

5

How to Classify Hamiltonian constraint

scale factor particle with zero energy in 119984 119886

possible range for scale factor 119984 119886 le 0

solutions

big bang rarr big crunch oscillation or bounce

6

stable static

unstable static

Classification (Λ gt 0 119870 = 1)

1 Big bang de Sitter

119892 119904 = 5 119892 119903 = 5

singularity is inevitable

Big bang to de Sitter phase

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

How to Classify Hamiltonian constraint

scale factor particle with zero energy in 119984 119886

possible range for scale factor 119984 119886 le 0

solutions

big bang rarr big crunch oscillation or bounce

6

stable static

unstable static

Classification (Λ gt 0 119870 = 1)

1 Big bang de Sitter

119892 119904 = 5 119892 119903 = 5

singularity is inevitable

Big bang to de Sitter phase

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Classification (Λ gt 0 119870 = 1)

1 Big bang de Sitter

119892 119904 = 5 119892 119903 = 5

singularity is inevitable

Big bang to de Sitter phase

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Classification (Λ gt 0 119870 = 1)

2 bouncing universe

119892 119904 = minus1 119892 119903 = minus1

singularity avoidance

Solution with minimum value 119886119879 bouncing universe

big bang big crunch

de Sitter Milne

stable static unstable static

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

3 oscillation and bounce

119892 119904 = minus001 119892 119903 = 05

singularity avoidance

oscillating solution + bouncing solution

119892119904 lt 0 sufficient condition for singularity avoidance

119886119879 119886119898119894119899 119886119898119886119909

119886119879

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

big bang big crunch

de Sitter Milne

stable static unstable static

Classification (Λ gt 0 119870 = 1)

4 bounce or big crunch

119892 119904 = minus5 119892 119903 = 1

possibility of singularity avoidance

initial value 119886 ge 119886119879 bouncing universe

119886119879 119886119898119894119899 119886119898119886119909

119886119879

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Toward a cyclic universe How to escape to macroscopic universe

1 119870 = 1 Λ gt 0 oscillation

2 quantum tunneling to 119886119879

3 de Sitter phase

macroscopic universe

How to obtain cyclic universe

1 Scalar field is responsible for inflation

bull Before tunneling behave as cosmological constant

bull After tunneling slow-roll inflation reheating ( 119984 infin gt0 )

2 upper bound and lower bound cyclic universe

reheating negative ldquostiff matterrdquo

119870 = 1 Λ gt 0 119892 119904lt0

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Summary and future works Classification vacuum FLRW universe in SVW HL gravity

Behavior of the universe is largely dependent on 119892119894

sufficient condition for singularity avoidance 119892119904 lt 0

(oscillation bounce)

With matter condition for singularity avoidance

119892119904 rarr 119892119904 + 119892119904119905119894119891119891 119892119903rarr 119892119903 + 119892119903119886119889 　119892119889 rarr 119892119889119906119904119905

energy density of matter ( ge 0 )

future work

discussion about stability of non-singular solution

bull Inhomogeneity

bull anisotropy Bianchi type IX (in preparation)

12

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

reference [1] P Horava Phys Rev D 79 084008 (2009)

[2] M Minamitsuji Phys Lett B 684 194 (2010)

[3] P Wu and H Yu Phys Rev D 81 103522 (2010)

[4] E J Son and W Kim arXiv10033055

[5] T P Sotiriou M Visser and S Weinfurtner Phys RevLett

102251601 (2009)

[6] S Carloni E Elizalde and P J Silva Classical Quantum

Gravity 27 045 004 (2010)

[7] Y F Cai and E N Saridakis J Cosmol Astropart Phys

10 (2009) 020 G Leon and E N Saridakis J Cosmol

Astropart Phys 11 (2009) 006

13

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

detailed balance condition

Application to FLRW cosmology

14

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

stability of flat background

transverse traceless graviton

bull stability In IR and UV 1198921 lt 0 1198929 gt 0

longitudinal graviton

bull ghost instability 1

3lt 120582 lt 1 120582 gt 1

bull Instability in IR not always pathological

time scale

non-projectable version

15

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

eos of radiation

generalized Maxwell action

dispersion relation for electromagnetic field

depending on scale factor

119899 ∶ number density (~119886minus3)

16

Λ = 0 Λ gt 0 Λ lt 0

17

If Λ ne 0