gauge fields and physics of inflation -...
TRANSCRIPT
SU(2) Gauge Fields and Physics of Inflation
Azadeh Maleknejad
Max Planck Institute for Astrophysics
IHP Analytical Methods in Cosmology September 2018
Motivation
• Inflation: Universe at highest observable energy!
• Non-Abelian gauge field theories are the widely
accepted framework for particle physics beyond SM.
• The Primordial SU(2) gauge fields in the physics of inflation?
Energy
𝐸 < 1014𝐺𝑒𝑉~
(𝐸𝐿𝐻𝐶
E> 10−11)~
Motivation
• Inflation: Universe at highest observable energy!
• Non-Abelian gauge field theories are the widely
accepted framework for particle physics beyond SM.
• The Primordial SU(2) gauge fields in the physics of inflation?
Energy
𝐸 < 1014𝐺𝑒𝑉~
(𝐸𝐿𝐻𝐶
E> 10−11)~
This talk is based onarXiv:1808.09076 and arXiv:1805.09318
in collaboration with Kaloian D. Lozanov & Eiichiro Komatsu
Bulk of works have been done on SU(2) fields in Inflation
by A.M., Peter Adshead, Mark Wyman, Marco Peloso, EmaDimastrogiovanni, Eiichiro Komatsu, Aniket Agrawal
and …
outline
I. SU(2) Gauge fields and Inflation
II. Spin-2 Particle Production & Chiral Gravitational Waves
III. Gravitational Leptogenesis
IV. Summary and Outlook
SU(2) Gauge fields and Inflation
• It starts in 2011
• In 4D Einstein gravity, we considered SU(2) gauge fields during inflation for the first time.
• In these papers, we uncovered a number of astonishing features of SU(2) gauge fields in inflation, both in background & perturbation.
𝐴𝜇 = 𝐴𝜇𝑎 𝑇𝑎 [𝑇𝑏 , 𝑇𝑐 ] = i 𝜖𝑎𝑏𝑐𝑇𝑎
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]
A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
SU(2) Gauge fields and Inflation
• It starts in 2011
i) SU(2) gauge fields are FRW friendly: (respect isotropy & homogeneity)
ii) Breaking conformal symmetry & respecting the gauge symmetry
A𝜇𝑎(𝑡) = ቊ
0 𝜇 = 0
𝑎(𝑡)𝜓(𝑡)𝛿𝑖𝑎 𝜇 = 𝑖
𝐴𝜇 = 𝐴𝜇𝑎 𝑇𝑎 [𝑇𝑏 , 𝑇𝑐 ] = i 𝜖𝑎𝑏𝑐𝑇𝑎
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
Backgro
un
d
SU(2) Gauge fields and Inflation
• It starts in 2011
i) SU(2) gauge fields are FRW friendly: (respect isotropy & homogeneity)
ii) Breaking conformal symmetry & respecting the gauge symmetry
iii) Extra spin-2 degrees of freedom:
iv) Spin-2 field breaks Parity & coupled linearly with gravity waves
Perturb
ation
A𝜇𝑎(𝑡) = ቊ
0 𝜇 = 0
𝑎(𝑡)𝜓(𝑡)𝛿𝑖𝑎 𝜇 = 𝑖
𝐴𝜇 = 𝐴𝜇𝑎 𝑇𝑎 [𝑇𝑏 , 𝑇𝑐 ] = i 𝜖𝑎𝑏𝑐𝑇𝑎
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
Backgro
un
d
𝛿𝐴𝑖𝑎(𝑡, Ԧ𝑥) = 𝛿𝑆𝑖
𝑎 + 𝐵𝑖𝑗 𝛿𝑖𝑎
Spin-2 field
Scalar and vector d.o.f
SU(2) Gauge fields and Inflation
• It starts in 2011
i) SU(2) gauge fields are FRW friendly: (respect isotropy & homogeneity)
ii) Breaking conformal symmetry & respecting the gauge symmetry
iii) Extra spin-2 degrees of freedom:
iv) Spin-2 field breaks Parity & coupled linearly with gravity waves
Perturb
ation
A𝜇𝑎(𝑡) = ቊ
0 𝜇 = 0
𝑎(𝑡)𝜓(𝑡)𝛿𝑖𝑎 𝜇 = 𝑖
𝐴𝜇 = 𝐴𝜇𝑎 𝑇𝑎 [𝑇𝑏 , 𝑇𝑐 ] = i 𝜖𝑎𝑏𝑐𝑇𝑎
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
Backgro
un
d
𝛿𝐴𝑖𝑎(𝑡, Ԧ𝑥) = 𝛿𝑆𝑖
𝑎 + 𝐵𝑖𝑗 𝛿𝑖𝑎
Spin-2 field
Scalar and vector d.o.f
Chiral primordial Gravitational waves
How to break the conformal symmetry?
• Gauge-flation
𝑆𝐺𝑓 = න𝑑4𝑥 −𝑔 −𝑅
2−1
4𝐹2 +
𝜅
384(𝐹 ෨𝐹)2
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
SU(2) Gauge fields and Inflation
• Gauge-flation
𝑆𝐺𝑓 = න𝑑4𝑥 −𝑔 −𝑅
2−1
4𝐹2 +
𝜅
384(𝐹 ෨𝐹)2
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
SU(2) Gauge fields and Inflation
𝑆𝜒 = න𝑑4𝑥 −𝑔 −1
2𝜕𝜇𝜒)
2 − 𝜇4 1 + cos(𝜒
𝑓−
𝜆
8𝑓𝜒𝐹 ෨𝐹
Effective field theory from integrating out a massive axion field (κ =3𝜆2
𝜇4)
• Gauge-flation
• Chromo-natural
𝑆𝐺𝑓 = න𝑑4𝑥 −𝑔 −𝑅
2−1
4𝐹2 +
𝜅
384(𝐹 ෨𝐹)2
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
SU(2) Gauge fields and Inflation
𝑆𝐶𝑛 = න𝑑4𝑥 −𝑔 −𝑅
2−1
4𝐹2 −
1
2𝜕𝜇𝜒)
2 − 𝜇4 1 + cos(𝜒
𝑓−
𝜆
8𝑓𝜒𝐹 ෨𝐹
P. Adshead, M. Wyman, Phys. Rev. Lett.(2012) [arXiv:1202.2366]
• Gauge-flation
• Chromo-natural
• Inspired by them, several different models with SU(2) fields have been
proposed and studied.
𝑆𝐺𝑓 = න𝑑4𝑥 −𝑔 −𝑅
2−1
4𝐹2 +
𝜅
384(𝐹 ෨𝐹)2
A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513]A. M. and M. M. Sheikh-Jabbari, Phys. Rev. D 84 (2011) [arXiv:1102.1932]
SU(2) Gauge fields and Inflation
𝑆𝐶𝑛 = න𝑑4𝑥 −𝑔 −𝑅
2−1
4𝐹2 −
1
2𝜕𝜇𝜒)
2 − 𝜇4 1 + cos(𝜒
𝑓−
𝜆
8𝑓𝜒𝐹 ෨𝐹
P. Adshead, M. Wyman, Phys. Rev. Lett.(2012) [arXiv:1202.2366]
An incomplete list of models with SU(2) gauge field: 𝛼𝐻 𝛼𝑠
1. A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513] 0 0
2. P. Adshead, M. Wyman, Phys. Rev. Lett.(2012) [arXiv:1202.2366] 0 0
3. A. M. JHEP 07 (2016) 104 [arXiv:1604.03327] 0 0
4. C. M. Nieto and Y. Rodriguez Mod. Phys. Lett. A31 (2016) [arXiv:1602.07197] 1 0
5. E. Dimastrogiovanni, M. Fasiello, and T. Fujita JCAP 1701 (2017) [arXiv:1608.04216] 0 1
6. P. Adshead, E. Martinec, E. I. Sfakianakis, and M. Wyman JHEP 12 (2016) 137 [arXiv:1609.04025] 1 0
7. P. Adshead and E. I. Sfakianakis JHEP 08 (2017) 130 [arXiv:1705.03024] 1 0
8. R. R. Caldwell and C. Devulder Phys. Rev. D97 (2018) [arXiv:1706.03765] 0 0
- 1 1
• These models can be represented in the unified form
𝑆 = 𝛼𝐻 𝑆𝐻(A𝜇 , 𝐻) + 𝛼𝑠 𝑆𝑠(χ) + 𝑆𝐴(A𝜇 , φ)
An incomplete list of models with SU(2) gauge field: 𝛼𝐻 𝛼𝑠
Higgs sector (mass for gauge field)
Scalar inflaton(spectator SU(2))
A. M. and E. Komatsu, [arXiv:1808.09076]
1. A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513] 0 0
2. P. Adshead, M. Wyman, Phys. Rev. Lett.(2012) [arXiv:1202.2366] 0 0
3. A. M. JHEP 07 (2016) 104 [arXiv:1604.03327] 0 0
4. C. M. Nieto and Y. Rodriguez Mod. Phys. Lett. A31 (2016) [arXiv:1602.07197] 1 0
5. E. Dimastrogiovanni, M. Fasiello, and T. Fujita JCAP 1701 (2017) [arXiv:1608.04216] 0 1
6. P. Adshead, E. Martinec, E. I. Sfakianakis, and M. Wyman JHEP 12 (2016) 137 [arXiv:1609.04025] 1 0
7. P. Adshead and E. I. Sfakianakis JHEP 08 (2017) 130 [arXiv:1705.03024] 1 0
8. R. R. Caldwell and C. Devulder Phys. Rev. D97 (2018) [arXiv:1706.03765] 0 0
- 1 1
𝑆𝐴 is either 𝑆𝐺𝑓 (Gauge-flation) or
𝑆𝐶𝑛 (Chromo-natural with an arbitrary potential)
• These models can be represented in the unified form
𝑆 = 𝛼𝐻 𝑆𝐻(A𝜇 , 𝐻) + 𝛼𝑠 𝑆𝑠(χ) + 𝑆𝐴(A𝜇 , φ)
An incomplete list of models with SU(2) gauge field: 𝛼𝐻 𝛼𝑠
𝛼𝐻 = ቊ01
𝛼𝑠 = ቊ01Higgsed (massive SU(2)) Spectator SU(2)
A. M. and E. Komatsu, [arXiv:1808.09076]
1. A. M. and M. M. Sheikh-Jabbari, Phys. Lett. B723 (2013) [arXiv:1102.1513] 0 0
2. P. Adshead, M. Wyman, Phys. Rev. Lett.(2012) [arXiv:1202.2366] 0 0
3. A. M. JHEP 07 (2016) 104 [arXiv:1604.03327] 0 0
4. C. M. Nieto and Y. Rodriguez Mod. Phys. Lett. A31 (2016) [arXiv:1602.07197] 1 0
5. E. Dimastrogiovanni, M. Fasiello, and T. Fujita JCAP 1701 (2017) [arXiv:1608.04216] 0 1
6. P. Adshead, E. Martinec, E. I. Sfakianakis, and M. Wyman JHEP 12 (2016) 137 [arXiv:1609.04025] 1 0
7. P. Adshead and E. I. Sfakianakis JHEP 08 (2017) 130 [arXiv:1705.03024] 1 0
8. R. R. Caldwell and C. Devulder Phys. Rev. D97 (2018) [arXiv:1706.03765] 0 0
- 1 1
Higgs sector (mass for gauge field)
Scalar inflaton(spectator SU(2))
𝑆𝐴 is either 𝑆𝐺𝑓 (Gauge-flation) or
𝑆𝐶𝑛 (Chromo-natural with an arbitrary potential)
𝑆 = 𝛼𝐻 𝑆𝐻(A𝜇 , 𝐻) + 𝛼𝑠 𝑆𝑠(χ) + 𝑆𝐴(A𝜇 , φ)
• The unified action for the inflation models with an SU(2) gauge field
• Due to the SU(2) gauge field with a non-zero VEV, they all share these features
I. The FRW friendly VEV solution
II. New Spin-2 field
Higgs sector (mass for gauge field)
Scalar inflaton(spectator SU(2))
𝑆𝐴 is either 𝑆𝐺𝑓 (Gauge-flation) or 𝑆𝐶𝑛 (Chromo-natural
with an arbitrary potential)
A𝜇𝑎(𝑡) = ቊ
0 𝜇 = 0
𝑎(𝑡)𝜓(𝑡)𝛿𝑖𝑎 𝜇 = 𝑖
𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝛿𝑖
𝑎 + scalar & vector fields
𝑆 = 𝛼𝐻 𝑆𝐻(A𝜇 , 𝐻) + 𝛼𝑠 𝑆𝑠(χ) + 𝑆𝐴(A𝜇 , φ)
• The unified action for the inflation models with an SU(2) gauge field
• Due to the SU(2) gauge field with a non-zero VEV, they all share these features
I. The FRW friendly VEV solution
II. New Spin-2 field
• Massive and always decaying after horizon crossing
• Chiral with a short phase of sizable particle production before horizon crossing
• Sources gravitational waves at linear order and generates chiral GWs
Higgs sector (mass for gauge field)
Scalar inflaton(spectator SU(2))
𝑆𝐴 is either 𝑆𝐺𝑓 (Gauge-flation) or
𝑆𝐶𝑛 (Chromo-natural with an arbitrary potential)
A𝜇𝑎(𝑡) = ቊ
0 𝜇 = 0
𝑎(𝑡)𝜓(𝑡)𝛿𝑖𝑎 𝜇 = 𝑖
𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝛿𝑖
𝑎 + scalar & vector fields
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by (𝛿𝐶 and 𝑚2
𝐻2 are two positive,
order 10, given by BG fields )
Spin-2 Particle Production & Chiral Gravity waves
𝜔𝜎2 τ, 𝑘 effective frequency
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by (𝛿𝐶 and 𝑚2
𝐻2 are two positive,
order 10, given by BG fields)
• due to the derivative interaction, 𝜔𝜎2 τ, 𝑘 is
• 1) chiral
• 2) violates adiabaticity condictiones for
a short period before horizon exit.
Spin-2 Particle Production & Chiral Gravity waves
( 𝜕𝜏 𝜔+ τ,𝑘
𝜔+2 τ,𝑘
)2 ≪ 1 and 𝜕𝜏2𝜔+ τ,𝑘
𝜔+3 τ,𝑘
≪ 1
𝜔𝜎2 τ, 𝑘 effective frequency
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by (𝛿𝐶 and 𝑚2
𝐻2 are two positive,
order 10, given by BG fields)
• due to the derivative interaction, 𝜔𝜎2 τ, 𝑘 is
• 1) chiral
• 2) violates adiabaticity condictiones for
a short period before horizon exit.
Spin-2 Particle Production & Chiral Gravity waves
( 𝜕𝜏 𝜔+ τ,𝑘
𝜔+2 τ,𝑘
)2 ≪ 1 and 𝜕𝜏2𝜔+ τ,𝑘
𝜔+3 τ,𝑘
≪ 1
Deviation from adiabaticity
𝑘
𝑎𝐻
A. M. and E. Komatsu, [arXiv:1808.09076]
𝛿𝐶 =40
𝛿𝐶 =10/ 2
𝜔𝜎2 τ, 𝑘 effective frequency
Spin-2 Particle Production & Chiral Gravity waves
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by (𝛿𝐶 and 𝑚2
𝐻2 are two positive, order 10, given by BG fields)
• due to the derivative interaction, 𝜔𝜎2 τ, 𝑘 is
• 1) chiral
• 2) violates adiabaticity condictiones for
a short period before horizon exit.
( 𝜕𝜏 𝜔+ τ,𝑘
𝜔+2 τ,𝑘
)2 ≪ 1 and 𝜕𝜏2𝜔+ τ,𝑘
𝜔+3 τ,𝑘
≪ 1
Deviation from adiabaticity
𝑘
𝑎𝐻
A. M. and E. Komatsu, [arXiv:1808.09076]
𝛿𝐶 =40
𝛿𝐶 =10/ 2Chiral particle production
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
𝜔𝜎2 τ, 𝑘 effective frequency
Spin-2 Particle Production & Chiral Gravity waves
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by ( 𝛿𝐶 and 𝑚2
𝐻2 are two positive, order 10, given by BG fields)
• due to the derivative interaction, 𝜔𝜎2 τ, 𝑘 is
• 1) chiral
• 2) violates adiabaticity condictiones for
a short period before horizon exit.
𝜔𝜎2 τ, 𝑘 effective frequency
( 𝜕𝜏 𝜔+ τ,𝑘
𝜔+2 τ,𝑘
)2 ≪ 1 and 𝜕𝜏2𝜔+ τ,𝑘
𝜔+3 τ,𝑘
≪ 1
Virtual 𝐵+ particles
pair production of 𝐵+
BG fields
Chiral particle production
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
Spin-2 Particle Production & Chiral Gravity waves
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by ( 𝛿𝐶 and 𝑚2
𝐻2 are two positive, order 10, given by BG fields)
• The physical number density of 𝐵+ pairs
created up to time τ is (time independent )
where and
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
𝑛𝑝𝑎𝑖𝑟𝑠 ~𝐻3
6𝜋2𝛿𝑐3 𝑒
2− 2 π2 𝛿𝑐
Massless SU(2) models
A. M. and E. Komatsu, [arXiv:1808.09076]
Spin-2 Particle Production & Chiral Gravity waves
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by ( 𝛿𝐶 and 𝑚2
𝐻2 are two positive, order 10, given by BG fields)
• The physical number density of 𝐵+ pairs
created up to time τ is (time independent )
where and
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
𝑛𝑝𝑎𝑖𝑟𝑠 ~𝐻3
6𝜋2𝛿𝑐3 𝑒
2− 2 π2 𝛿𝑐
Massless SU(2) models
A. M. and E. Komatsu, [arXiv:1808.09076]
Since the number density can be large, the backreaction of the 𝐵+ to the background fields can be important.
• In [1], we found analytical formulae for i) the backreaction of 𝐵+ to the background field equations as well as ii) its energy density, and used them to constrain the parameter space of the models with an SU(2) gauge field.
[1] A. M. and E. Komatsu, [arXiv:1808.09076]
The size of backreaction is related to the scale of
Inflation as (𝐻
𝑀𝑃𝑙)2.
Spin-2 Particle Production & Chiral Gravity waves
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by ( 𝛿𝐶 and 𝑚2
𝐻2 are two positive, order 10, given by BG fields)
• Polarization 𝐵+ has a short time of particle production before horizon crossing.
• Polarization 𝐵− is (almost) always very close to its vacuum state, negligible pair production.
• The 𝐵± fields are massive (𝑚2
𝐻2>8) and both polarizations decay after horizon crossing.
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
𝐵+/𝑎
𝐵−/𝑎
𝑘
𝑎𝐻
Spin-2 Particle Production & Chiral Gravity waves
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by ( 𝛿𝐶 and 𝑚2
𝐻2 are two positive, order 10, given by BG fields)
• Polarization 𝐵+ has a short time of particle production before horizon crossing.
• Polarization 𝐵− is (almost) always very close to its vacuum state, negligible pair production.
• The 𝐵± fields are massive (𝑚2
𝐻2>8) and both polarizations decay after horizon crossing.
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
𝐵+/𝑎
𝐵−/𝑎
𝑘
𝑎𝐻
• The energy density of
the 𝐵+ mode is negative in
most of the parameter space
A. M. and E. Komatsu, [arXiv:1808.09076]
Dashed line = negative values
Solid line = positive values
Spin-2 Particle Production & Chiral Gravity waves• Spin-2 field 𝛿𝐴𝑖
𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
Anisotropic stressLinear in 𝐵±
(ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)
Spin-2 Particle Production & Chiral Gravity waves• Spin-2 field 𝛿𝐴𝑖
𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
Anisotropic stressLinear in 𝐵±
The efficiency of the coupling of 𝐵± to GWs is proportional to the
VEV of the SU(2) gauge field.
Spin-2 Particle Production & Chiral Gravity waves• Spin-2 field 𝛿𝐴𝑖
𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as
• Gravitational waves have two uncorrelated terms
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
ℎ± = ℎ±𝑣𝑎𝑐 + ℎ±
𝑠
Vacuum GWs
unpolarizedℎ+𝑣𝑎𝑐 = ℎ−
𝑣𝑎𝑐
Sourced by𝐵±
Polarizedℎ+𝑠 ≠ ℎ−
𝑠
(ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)
Spin-2 Particle Production & Chiral Gravity waves• Spin-2 field 𝛿𝐴𝑖
𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as
• Gravitational waves have two uncorrelated terms
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
ℎ± = ℎ±𝑣𝑎𝑐 + ℎ±
𝑠
Vacuum GWs
unpolarizedℎ+𝑣𝑎𝑐 = ℎ−
𝑣𝑎𝑐
Sourced by𝐵±
Polarizedℎ+𝑠 ≠ ℎ−
𝑠
(ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)
𝛾+𝑠 /𝐻
𝛾−𝑠 /𝐻
𝑘
𝑎𝐻
Spin-2 Particle Production & Chiral Gravity waves• Spin-2 field 𝛿𝐴𝑖
𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as (ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)
• Gravitational waves have two uncorrelated terms
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
ℎ± = ℎ±𝑣𝑎𝑐 + ℎ±
𝑠
The ratio of the power spectra of sourced and vacuum gravitational waves is
𝜓
𝑀𝑃𝑙
𝑛𝑝𝑎𝑖𝑟𝑠𝐻3
𝑃𝑇𝑠
𝑃𝑇𝑣𝑎𝑐
A. M. and E. Komatsu, [arXiv:1808.09076]
Spin-2 Particle Production & Chiral Gravity waves• Spin-2 field 𝛿𝐴𝑖
𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as (ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)
• Gravitational waves have two uncorrelated terms
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
ℎ± = ℎ±𝑣𝑎𝑐 + ℎ±
𝑠
The ratio of the power spectra of sourced and vacuum gravitational waves is
𝜓
𝑀𝑃𝑙
𝑛𝑝𝑎𝑖𝑟𝑠𝐻3
𝑃𝑇𝑠
𝑃𝑇𝑣𝑎𝑐
A. M. and E. Komatsu, [arXiv:1808.09076]
In the presence of the primordial SU(2) gauge fields
i) The tensor power spectrum is not entirely specified by the scale of inflation!
ii) Sizable tensor to scalar ratio without large field.
iii) The tensor power spectrum is partially chiral and parity odd correlations
𝑇𝐵 and 𝐸𝐵 are non-zero!G
• Spin-2 field 𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝑡, Ԧ𝑥 𝛿𝑖
𝑎 is governed by
• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as (ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)
• gravitational waves have two uncorrelated terms
𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +
𝑚2
𝐻2 ℋ2 -
𝑎′′
𝑎] 𝐵± ≈ 0
ℎ±′′ + [𝑘2-
𝑎′′
𝑎] ℎ± ≈
2𝜓
𝑀𝑃𝑙ℋ2 𝛱±[𝐵±]
The ratio of the power spectra of sourced and vacuum gravitational waves is
In the presence of the primordial SU(2) gauge fields
i) The tensor power spectrum is not entirely specified by the scale of inflation!
ii) Sizable tensor to scalar ratio without large field.
iii) The tensor power spectrum is partially chiral and parity odd correlations
𝑇𝐵 and 𝐸𝐵 are non-zero!
iv) Possible large tensor non-Gaussianity.
v) Unlike U(1) gauge fields, the SU(2) field in the FRW friendly ansatz does not receive any sizable backreaction from charged scalar and fermions coupled to it.
vi) Naturally explains the matter asymmetry in our Universe!
Aniket Agrawal, Tomohiro Fujita, Eiichiro Komatsu, [arXiv:1802.09284] & [arXiv:1707.03023]
Kaloian D. Lozanov, A. M., Eiichiro Komatsu [arXiv:1805.09318]
A. M. Phys. Rev. D (2014) & JCAP 1612 (2016)
• Inflato- Leptogenesis: a leptogenesis model during inflation, based on chiral
gravitational waves and SM.
Chiral Gravitational Gravitational Anomaly Net Lepton Number
Waves in SM Density
∇𝜇𝐽𝑙𝜇 =
(𝑁𝐿 − 𝑁𝑅)
384𝜋2෨𝑅𝑅RL hh
S. H. -S. Alexander, M. E. Peskin and M. M. Sheikh-Jabbari (2006).
𝑛𝐿 ≠ 0
The chiral gravitational waves produced during inflation through the gravitational anomaly in the SM leads to a net lepton number density.
• Inflato- Leptogenesis: a leptogenesis model during inflation, based on chiral
gravitational waves and SM.
Chiral Gravitational Gravitational Anomaly Net Lepton Number
Waves in SM Density
∇𝜇𝐽𝑙𝜇 =
(𝑁𝐿 − 𝑁𝑅)
384𝜋2෨𝑅𝑅RL hh
S. H. -S. Alexander, M. E. Peskin and M. M. Sheikh-Jabbari (2006).
𝑛𝐿 ≠ 0
The chiral gravitational waves produced during inflation through the gravitational anomaly in the SM leads to a net lepton number density.
• Inflato- Leptogenesis: a leptogenesis model during inflation, based on chiral
gravitational waves and SM.
Chiral Gravitational Gravitational Anomaly Net Lepton Number
Waves in SM Density
∇𝜇𝐽𝑙𝜇 =
(𝑁𝐿 − 𝑁𝑅)
384𝜋2෨𝑅𝑅RL hh
S. H. -S. Alexander, M. E. Peskin and M. M. Sheikh-Jabbari (2006).
𝑛𝐿 ≠ 0
Inflationary models involving SU(2) gauge fields produce intrinsic chiral gravitational waves. Natural setting for Inflationary leptogenesis!
A. M. Phys. Rev. D (2014) & JCAP 1612 (2016)
𝑛𝐿 =
• Inflato- Leptogenesis: a leptogenesis model during inflation, based on chiral
gravitational waves and SM.
Chiral Gravitational Gravitational Anomaly Net Lepton Number
Waves in SM Density
∇𝜇𝐽𝑙𝜇 =
(𝑁𝐿 − 𝑁𝑅)
384𝜋2෨𝑅𝑅RL hh
S. H. -S. Alexander, M. E. Peskin and M. M. Sheikh-Jabbari (2006).
𝑛𝐿 ≠ 0
Inflationary models involving SU(2) gauge fields produce intrinsic chiral gravitational waves. Natural setting for Inflationary leptogenesis!
A. M. Phys. Rev. D (2014) & JCAP 1612 (2016)
96𝜋Net Lepton number density
Mass of heavy
right-handed
neutrinosEnergy density of SU(2)
Gauge field
Presence of gauge fields in the matter content of inflation leads to
o New spin-2 field with sizable particle production before horizon exit.
o Relaxing the direct relation between tensor power spectrum
and the scale of inflation 𝑃𝑇 = (1+𝐶
𝜋2)𝐻2
𝑀𝑃𝑙2
o Violation of the Lyth bound,
o Violation of the consistency relation (changing 𝑛𝑇).
Summary and Outlook
Presence of gauge fields in the matter content of inflation leads to
o Chiral gravitational waves and parity odd correlations 𝑇𝐵 and 𝐸𝐵
o Possible large tensor non-Gaussianity.
o Unlike U(1) gauge fields, the SU(2) field in the FRW friendly ansatz does not receive any sizable backreaction from charged scalar and fermions coupled to it.
o Naturally explains the matter asymmetry in our Universe!
Summary and Outlook
Thank You!