Gauge Fields and Physics of Inflation -...

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]



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


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





iii) Extra spin-2 degrees of freedom:

iv) Spin-2 field breaks Parity & coupled linearly with gravity waves

Perturb

ation


0 𝜇 = 0




Backgro

un

d

𝛿𝐴𝑖𝑎(𝑡, Ԧ𝑥) = 𝛿𝑆𝑖

𝑎 + 𝐵𝑖𝑗 𝛿𝑖𝑎

Spin-2 field

Scalar and vector d.o.f





iii) Extra spin-2 degrees of freedom:

iv) Spin-2 field breaks Parity & coupled linearly with gravity waves

Perturb

ation


0 𝜇 = 0




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



• Gauge-flation


2−1

4𝐹2 +

𝜅

384(𝐹 ෨𝐹)2



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


2−1

4𝐹2 +

𝜅

384(𝐹 ෨𝐹)2



𝑆𝐶𝑛 = න𝑑4𝑥 −𝑔 −𝑅

2−1

4𝐹2 −

1

2𝜕𝜇𝜒)

2 − 𝜇4 1 + cos(𝜒

𝑓−

𝜆


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.


2−1

4𝐹2 +

𝜅

384(𝐹 ෨𝐹)2



𝑆𝐶𝑛 = න𝑑4𝑥 −𝑔 −𝑅

2−1

4𝐹2 −

1

2𝜕𝜇𝜒)

2 − 𝜇4 1 + cos(𝜒

𝑓−

𝜆


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)


Higgs sector (mass for gauge field)

Scalar inflaton(spectator SU(2))

A. M. and E. Komatsu, [arXiv:1808.09076]



3. A. M. JHEP 07 (2016) 104 [arXiv:1604.03327] 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



𝛼𝐻 = ቊ01

𝛼𝑠 = ቊ01Higgsed (massive SU(2)) Spectator SU(2)




3. A. M. JHEP 07 (2016) 104 [arXiv:1604.03327] 0 0






- 1 1






• 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



𝑆𝐴 is either 𝑆𝐺𝑓 (Gauge-flation) or 𝑆𝐶𝑛 (Chromo-natural

with an arbitrary potential)


0 𝜇 = 0


𝛿𝐴𝑖𝑎 𝑡, Ԧ𝑥 = 𝐵𝑖𝑗 𝛿𝑖

𝑎 + scalar & vector fields


• 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






0 𝜇 = 0


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




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


𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0




order 10, given by BG fields)


• 1) chiral




( 𝜕𝜏 𝜔+ τ,𝑘

𝜔+2 τ,𝑘

)2 ≪ 1 and 𝜕𝜏2𝜔+ τ,𝑘

𝜔+3 τ,𝑘

≪ 1

Deviation from adiabaticity

𝑘

𝑎𝐻


𝛿𝐶 =40

𝛿𝐶 =10/ 2





𝐻2 are two positive, order 10, given by BG fields)


• 1) chiral



( 𝜕𝜏 𝜔+ τ,𝑘

𝜔+2 τ,𝑘

)2 ≪ 1 and 𝜕𝜏2𝜔+ τ,𝑘

𝜔+3 τ,𝑘

≪ 1

Deviation from adiabaticity

𝑘

𝑎𝐻


𝛿𝐶 =40

𝛿𝐶 =10/ 2Chiral particle production

𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0




𝑎 is governed by ( 𝛿𝐶 and 𝑚2



• 1) chiral




( 𝜕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





• 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






• 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


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.





• 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





• 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


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)




𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0

ℎ±′′ + [𝑘2-

𝑎′′

𝑎] ℎ± ≈

2𝜓


Anisotropic stressLinear in 𝐵±

The efficiency of the coupling of 𝐵± to GWs is proportional to the

VEV of the SU(2) gauge field.




• Gravitational waves have two uncorrelated terms

𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0

ℎ±′′ + [𝑘2-

𝑎′′

𝑎] ℎ± ≈

2𝜓


ℎ± = ℎ±𝑣𝑎𝑐 + ℎ±

𝑠

Vacuum GWs

unpolarizedℎ+𝑣𝑎𝑐 = ℎ−

𝑣𝑎𝑐

Sourced by𝐵±

Polarizedℎ+𝑠 ≠ ℎ−

𝑠






𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0

ℎ±′′ + [𝑘2-

𝑎′′

𝑎] ℎ± ≈

2𝜓



𝑠

Vacuum GWs

unpolarizedℎ+𝑣𝑎𝑐 = ℎ−

𝑣𝑎𝑐

Sourced by𝐵±

Polarizedℎ+𝑠 ≠ ℎ−

𝑠


𝛾+𝑠 /𝐻

𝛾−𝑠 /𝐻

𝑘

𝑎𝐻



• That sourced the Gravity waves 𝛿𝑔𝑖𝑗 𝑡, Ԧ𝑥 = 𝑎 ℎ𝑖𝑗 𝑡, Ԧ𝑥 as (ℎ𝑖𝑗 ≡ 𝑎 𝛾𝑖𝑗)


𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0

ℎ±′′ + [𝑘2-

𝑎′′

𝑎] ℎ± ≈

2𝜓



𝑠

The ratio of the power spectra of sourced and vacuum gravitational waves is

𝜓

𝑀𝑃𝑙

𝑛𝑝𝑎𝑖𝑟𝑠𝐻3

𝑃𝑇𝑠

𝑃𝑇𝑣𝑎𝑐






𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0

ℎ±′′ + [𝑘2-

𝑎′′

𝑎] ℎ± ≈

2𝜓



𝑠


𝜓

𝑀𝑃𝑙

𝑛𝑝𝑎𝑖𝑟𝑠𝐻3

𝑃𝑇𝑠

𝑃𝑇𝑣𝑎𝑐


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


𝑎 is governed by


• gravitational waves have two uncorrelated terms

𝐵±′′ + [𝑘2 ∓ 𝛿𝐶𝑘ℋ +

𝑚2

𝐻2 ℋ2 -

𝑎′′

𝑎] 𝐵± ≈ 0

ℎ±′′ + [𝑘2-

𝑎′′

𝑎] ℎ± ≈

2𝜓



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.




Waves in SM Density





𝑛𝐿 ≠ 0

The chiral gravitational waves produced during inflation through the gravitational anomaly in the SM leads to a net lepton number density.




Waves in SM Density





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 =




Waves in SM Density





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!

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