[PPT] - The variation of the fine-structure constant from disformal PowerPoint Presentation

SLIDE 1

The variation of the fine-structure constant from disformal couplings

Jurgen Mifsud

Consortium for Fundamental Physics, School of Mathematics and Statistics The University of Sheffield In collaboration with Carsten van de Bruck and Nelson J. Nunes 28th Texas Symposium on Relativistic Astrophysics – Gen` eve

15/12/15

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 1 / 33

SLIDE 2

Outline

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 2 / 33

SLIDE 3

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 3 / 33

SLIDE 4

Introduction–Is α a constant of Nature?

Dirac came up with the idea on the variation of the fundamental constants of Nature in his ’large numbers hypothesis’. Effective (3+1)–dimensional constants can vary in space and time in higher–dimensional theories. Current observations look for variations in the fine–structure constant:

Atomic Clocks [T. Rosenband et al ‘08] ˙ α α

= (−1.6 ± 2.3) × 10−17 yr−1,

Oklo natural reactor [E.D. Davis & L. Hamdan ‘15] |∆α| α < 1.1 × 10−8, z ≃ 0.16,

187Re meteorites [K.A. Olive et al ‘04]

∆α α = (−8 ± 8) × 10−7, z ≃ 0.43,

Dirac came up with the idea on the variation of the fundamental

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 4 / 33

SLIDE 5

Introduction–Is α a constant of Nature?

Dirac came up with the idea on the variation of the fundamental constants of Nature in his ’large numbers hypothesis’. Effective (3+1)–dimensional constants can vary in space and time in higher–dimensional theories. Current observations look for variations in the fine–structure constant:

Atomic Clocks [T. Rosenband et al ‘08] ˙ α α

= (−1.6 ± 2.3) × 10−17 yr−1,

Oklo natural reactor [E.D. Davis & L. Hamdan ‘15] |∆α| α < 1.1 × 10−8, z ≃ 0.16,

187Re meteorites [K.A. Olive et al ‘04]

∆α α = (−8 ± 8) × 10−7, z ≃ 0.43,

Dirac came up with the idea on the variation of the fundamental

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 4 / 33

SLIDE 6

Introduction–Is α a constant of Nature?

Dirac came up with the idea on the variation of the fundamental constants of Nature in his ’large numbers hypothesis’. Effective (3+1)–dimensional constants can vary in space and time in higher–dimensional theories. Current observations look for variations in the fine–structure constant:

Atomic Clocks [T. Rosenband et al ‘08] ˙ α α

= (−1.6 ± 2.3) × 10−17 yr−1,

Oklo natural reactor [E.D. Davis & L. Hamdan ‘15] |∆α| α < 1.1 × 10−8, z ≃ 0.16,

187Re meteorites [K.A. Olive et al ‘04]

∆α α = (−8 ± 8) × 10−7, z ≃ 0.43,

Dirac came up with the idea on the variation of the fundamental

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 4 / 33

SLIDE 7

Introduction–Is α a constant of Nature?

Dirac came up with the idea on the variation of the fundamental constants of Nature in his ’large numbers hypothesis’. Effective (3+1)–dimensional constants can vary in space and time in higher–dimensional theories. Current observations look for variations in the fine–structure constant:

The cosmic microwave background (CMB) radiation [Planck Coll. ‘15] ∆α α = (3.6 ± 3.7) × 10−3, z ≃ 103, Astrophysical data:

Keck/ HIRES–141 absorbers (MM method) [M.T. Murphy et al ‘04] ∆α α

w

= (−0.57 ± 0.11) × 10−5, 0.2 < z < 4.2, VLT/ UVES–154 absorbers (MM method) [J.A. King et al ‘12] ∆α α

w

= (0.208 ± 0.124) × 10−5, 0.2 < z < 3.7,

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 4 / 33

SLIDE 8

Introduction–Is α a constant of Nature?

Dirac came up with the idea on the variation of the fundamental constants of Nature in his ’large numbers hypothesis’. Effective (3+1)–dimensional constants can vary in space and time in higher–dimensional theories. Current observations look for variations in the fine–structure constant:

Astrophysical data:

Keck/ HIRES Si IV absorption systems (AD method) [M.T. Murphy et al ‘01] ∆α α

w

= (−0.5 ± 1.3) × 10−5, 2 < z < 3, Comparison of HI 21–cm line with molecular rotational absorption spectra [M.T. Murphy et al ‘01] ∆α α = (−0.10 ± 0.22) × 10−5, z = 0.25, ∆α α = (−0.08 ± 0.27) × 10−5, z = 0.68,

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 4 / 33

SLIDE 9

Introduction–Is α a constant of Nature?

Dirac came up with the idea on the variation of the fundamental constants of Nature in his ’large numbers hypothesis’. Effective (3+1)–dimensional constants can vary in space and time in higher–dimensional theories. Current observations look for variations in the fine–structure constant:

Astrophysical data:

Recent data [P. Molaro et al ‘13, T.M. Evans et al ‘14] Object z (∆α/α) × 106 Spectrograph Three sources 1.08 4.3 ± 3.4 HIRES HS1549+1919 1.14 −7.5 ± 5.5 UVES/HIRES/HDS HE0515-4414 1.15 −0.1 ± 1.8 UVES HE0515-4414 1.15 0.5 ± 2.4 HARPS/UVES HS1549+1919 1.34 −0.7 ± 6.6 UVES/HIRES/HDS HE0001-2340 1.58 −1.5 ± 2.6 UVES HE1104-1805A 1.66 −4.7 ± 5.3 HIRES HE2217-2818 1.69 1.3 ± 2.6 UVES HS1946+7658 1.74 −7.9 ± 6.2 HIRES HS1549+1919 1.80 −6.4 ± 7.2 UVES/HIRES/HDS

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 4 / 33

SLIDE 10

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 5 / 33

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Disformal Electrodynamics: The Model

We consider the following action: S = Sgrav (gµν, φ) + Smatter

˜

g(m)

µν

+ SEM
Aµ, ˜

g(r)

µν

(1)

such that, ˜ g(m)

µν

= Cmgµν + Dmφ,µφ,ν , (2) ˜ g(r)

µν = Crgµν + Drφ,µφ,ν ,

(3) where Cr,m : conformal factors Dr,m : disformal couplings

both taken to be functions of φ only
J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 6 / 33

SLIDE 12

Disformal Electrodynamics: The Model

We consider the following action: S = Sgrav (gµν, φ) + Smatter

˜

g(m)

µν

+ SEM
Aµ, ˜

g(r)

µν

(1)

such that, ˜ g(m)

µν

= Cmgµν + Dmφ,µφ,ν , (2) ˜ g(r)

µν = Crgµν + Drφ,µφ,ν ,

(3) where Cr,m : conformal factors Dr,m : disformal couplings

both taken to be functions of φ only
J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 6 / 33

SLIDE 13

Disformal Electrodynamics: The Model

We consider the following action: S = Sgrav (gµν, φ) + Smatter

˜

g(m)

µν

+ SEM
Aµ, ˜

g(r)

µν

(1)

such that, ˜ g(m)

µν

= Cmgµν + Dmφ,µφ,ν , (2) ˜ g(r)

µν = Crgµν + Drφ,µφ,ν ,

(3) where Cr,m : conformal factors Dr,m : disformal couplings

both taken to be functions of φ only
J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 6 / 33

SLIDE 14

Disformal Electrodynamics: The Model

The electromagnetic sector is specified by SEM = −1 4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ,

(4) where Fµν is the standard antisymmetric Faraday tensor, jµ is the four–current, The function h(φ) is the direct coupling between the electromagnetic field and the scalar. We aim to work in the Jordan frame The frame in which matter is decoupled from the scalar degree of freedom.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 7 / 33

SLIDE 15

Disformal Electrodynamics: The Model

The electromagnetic sector is specified by SEM = −1 4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ,

(4) where Fµν is the standard antisymmetric Faraday tensor, jµ is the four–current, The function h(φ) is the direct coupling between the electromagnetic field and the scalar. We aim to work in the Jordan frame The frame in which matter is decoupled from the scalar degree of freedom.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 7 / 33

SLIDE 16

Disformal Electrodynamics: The Model

The electromagnetic sector is specified by SEM = −1 4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ,

(4) where Fµν is the standard antisymmetric Faraday tensor, jµ is the four–current, The function h(φ) is the direct coupling between the electromagnetic field and the scalar. We aim to work in the Jordan frame The frame in which matter is decoupled from the scalar degree of freedom.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 7 / 33

SLIDE 17

Disformal Electrodynamics: The Model

The electromagnetic sector is specified by SEM = −1 4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ,

(4) where Fµν is the standard antisymmetric Faraday tensor, jµ is the four–current, The function h(φ) is the direct coupling between the electromagnetic field and the scalar. We aim to work in the Jordan frame The frame in which matter is decoupled from the scalar degree of freedom.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 7 / 33

SLIDE 18

Disformal Electrodynamics: The Model

The electromagnetic sector is specified by SEM = −1 4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ,

(4) where Fµν is the standard antisymmetric Faraday tensor, jµ is the four–current, The function h(φ) is the direct coupling between the electromagnetic field and the scalar. We aim to work in the Jordan frame The frame in which matter is decoupled from the scalar degree of freedom.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 7 / 33

SLIDE 19

Disformal Electrodynamics: The Model

The electromagnetic sector is specified by SEM = −1 4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ,

(4) where Fµν is the standard antisymmetric Faraday tensor, jµ is the four–current, The function h(φ) is the direct coupling between the electromagnetic field and the scalar. We aim to work in the Jordan frame The frame in which matter is decoupled from the scalar degree of freedom.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 7 / 33

SLIDE 20

Disformal Electrodynamics: The Model

Indeed, we know that ˜ g(r)

µν = Cr

Cm ˜ g(m)

µν +

Dr − CrDm

Cm

φ,µφ,ν ≡ A˜

g(m)

µν + Bφ,µφ,ν .

(5) Then, in terms of this metric, the electromagnetic sector becomes SEM = − 1 4

d4x
−˜

g(m)h(φ)Z

˜

gµν

(m)˜

gαβ

(m) − 2γ2˜

gµν

(m)φ,αφ,β

FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ ,

(6) Gauge invariance: ˜ ∇µ jµ = 0 Variation with respect to respect to Aµ: ˜ ∇ǫ (h(φ)ZF ǫρ) − ˜ ∇ǫ

h(φ)Zγ2φ,β

˜ gǫν

(m)φ,ρ − ˜

gρν

(m)φ,ǫ

Fνβ

= jρ

(7) where we again raise the indices with ˜ g(m)

µν .

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 8 / 33

SLIDE 21

Disformal Electrodynamics: The Model

Indeed, we know that ˜ g(r)

µν = Cr

Cm ˜ g(m)

µν +

Dr − CrDm

Cm

φ,µφ,ν ≡ A˜

g(m)

µν + Bφ,µφ,ν .

(5) Then, in terms of this metric, the electromagnetic sector becomes SEM = − 1 4

d4x
−˜

g(m)h(φ)Z

˜

gµν

(m)˜

gαβ

(m) − 2γ2˜

gµν

(m)φ,αφ,β

FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ ,

(6) Gauge invariance: ˜ ∇µ jµ = 0 Variation with respect to respect to Aµ: ˜ ∇ǫ (h(φ)ZF ǫρ) − ˜ ∇ǫ

h(φ)Zγ2φ,β

˜ gǫν

(m)φ,ρ − ˜

gρν

(m)φ,ǫ

Fνβ

= jρ

(7) where we again raise the indices with ˜ g(m)

µν .

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 8 / 33

SLIDE 22

Disformal Electrodynamics: The Model

Indeed, we know that ˜ g(r)

µν = Cr

Cm ˜ g(m)

µν +

Dr − CrDm

Cm

φ,µφ,ν ≡ A˜

g(m)

µν + Bφ,µφ,ν .

(5) Then, in terms of this metric, the electromagnetic sector becomes SEM = − 1 4

d4x
−˜

g(m)h(φ)Z

˜

gµν

(m)˜

gαβ

(m) − 2γ2˜

gµν

(m)φ,αφ,β

FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ ,

(6) where we raise the indices with the metric ˜ g(m)

µν

and define Z =

1 + B

A ˜ gµν

(m)∂µφ∂νφ

1/2 , γ2 = B A + B˜ gµν

(m)∂µφ∂νφ .

Gauge invariance: ˜ ∇ jµ = 0

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 8 / 33

SLIDE 23

Disformal Electrodynamics: The Model

Indeed, we know that ˜ g(r)

µν = Cr

Cm ˜ g(m)

µν +

Dr − CrDm

Cm

φ,µφ,ν ≡ A˜

g(m)

µν + Bφ,µφ,ν .

(5) Then, in terms of this metric, the electromagnetic sector becomes SEM = − 1 4

d4x
−˜

g(m)h(φ)Z

˜

gµν

(m)˜

gαβ

(m) − 2γ2˜

gµν

(m)φ,αφ,β

FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ ,

(6) Gauge invariance: ˜ ∇µ jµ = 0 Variation with respect to respect to Aµ: ˜ ∇ǫ (h(φ)ZF ǫρ) − ˜ ∇ǫ

h(φ)Zγ2φ,β

˜ gǫν

(m)φ,ρ − ˜

gρν

(m)φ,ǫ

Fνβ

= jρ

(7) where we again raise the indices with ˜ g(m)

µν .

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 8 / 33

SLIDE 24

Disformal Electrodynamics: The Model

Indeed, we know that ˜ g(r)

µν = Cr

Cm ˜ g(m)

µν +

Dr − CrDm

Cm

φ,µφ,ν ≡ A˜

g(m)

µν + Bφ,µφ,ν .

(5) Then, in terms of this metric, the electromagnetic sector becomes SEM = − 1 4

d4x
−˜

g(m)h(φ)Z

˜

gµν

(m)˜

gαβ

(m) − 2γ2˜

gµν

(m)φ,αφ,β

FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ ,

(6) Gauge invariance: ˜ ∇µ jµ = 0 Variation with respect to respect to Aµ: ˜ ∇ǫ (h(φ)ZF ǫρ) − ˜ ∇ǫ

h(φ)Zγ2φ,β

˜ gǫν

(m)φ,ρ − ˜

gρν

(m)φ,ǫ

Fνβ

= jρ

(7) where we again raise the indices with ˜ g(m)

µν .

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 8 / 33

SLIDE 25

Disformal Electrodynamics: The Model

Indeed, we know that ˜ g(r)

µν = Cr

Cm ˜ g(m)

µν +

Dr − CrDm

Cm

φ,µφ,ν ≡ A˜

g(m)

µν + Bφ,µφ,ν .

(5) Then, in terms of this metric, the electromagnetic sector becomes SEM = − 1 4

d4x
−˜

g(m)h(φ)Z

˜

gµν

(m)˜

gαβ

(m) − 2γ2˜

gµν

(m)φ,αφ,β

FµαFνβ −

d4x
−˜

g(m)˜ gµν

(m) jνAµ ,

(6) Gauge invariance: ˜ ∇µ jµ = 0 Variation with respect to respect to Aµ: ˜ ∇ǫ (h(φ)ZF ǫρ) − ˜ ∇ǫ

h(φ)Zγ2φ,β

˜ gǫν

(m)φ,ρ − ˜

gρν

(m)φ,ǫ

Fνβ

= jρ

(7) where we again raise the indices with ˜ g(m)

µν .

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 8 / 33

SLIDE 26

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 9 / 33

SLIDE 27

Disformal Electrodynamics: Identification of α

Set ˜ g(m)

µν

= ηµν and consider φ to depend on time only. From the field equation (7), and identifying the electric field by E i = F i0, we find the field equation for the electric field to be given by ∇ · E = Zρ h(φ) (8) where ρ = j0 is the charge density. By integrating this equation over a volume V, it is straightforward to derive the electrostatic potential V (r) = ZQ 4πh(φ)r (9) where Q is the total charge contained in V. Comparing this to the standard expression for the tree-level-potential from QED, one finds that α has the following dependence on Z and h: α ∝ Z h(φ) (Note that α ∝ h−1(φ) when ˜ g(m)

µν

≡ ˜ g(r)

µν .)

(10)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 10 / 33

SLIDE 28

Disformal Electrodynamics: Identification of α

Set ˜ g(m)

µν

= ηµν and consider φ to depend on time only. From the field equation (7), and identifying the electric field by E i = F i0, we find the field equation for the electric field to be given by ∇ · E = Zρ h(φ) (8) where ρ = j0 is the charge density. By integrating this equation over a volume V, it is straightforward to derive the electrostatic potential V (r) = ZQ 4πh(φ)r (9) where Q is the total charge contained in V. Comparing this to the standard expression for the tree-level-potential from QED, one finds that α has the following dependence on Z and h: α ∝ Z h(φ) (Note that α ∝ h−1(φ) when ˜ g(m)

µν

≡ ˜ g(r)

µν .)

(10)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 10 / 33

SLIDE 29

Disformal Electrodynamics: Identification of α

Set ˜ g(m)

µν

= ηµν and consider φ to depend on time only. From the field equation (7), and identifying the electric field by E i = F i0, we find the field equation for the electric field to be given by ∇ · E = Zρ h(φ) (8) where ρ = j0 is the charge density. By integrating this equation over a volume V, it is straightforward to derive the electrostatic potential V (r) = ZQ 4πh(φ)r (9) where Q is the total charge contained in V. Comparing this to the standard expression for the tree-level-potential from QED, one finds that α has the following dependence on Z and h: α ∝ Z h(φ) (Note that α ∝ h−1(φ) when ˜ g(m)

µν

≡ ˜ g(r)

µν .)

(10)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 10 / 33

SLIDE 30

Disformal Electrodynamics: Identification of α

Set ˜ g(m)

µν

= ηµν and consider φ to depend on time only. From the field equation (7), and identifying the electric field by E i = F i0, we find the field equation for the electric field to be given by ∇ · E = Zρ h(φ) (8) where ρ = j0 is the charge density. By integrating this equation over a volume V, it is straightforward to derive the electrostatic potential V (r) = ZQ 4πh(φ)r (9) where Q is the total charge contained in V. Comparing this to the standard expression for the tree-level-potential from QED, one finds that α has the following dependence on Z and h: α ∝ Z h(φ) (Note that α ∝ h−1(φ) when ˜ g(m)

µν

≡ ˜ g(r)

µν .)

(10)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 10 / 33

SLIDE 31

Disformal Electrodynamics: Identification of α

Set ˜ g(m)

µν

= ηµν and consider φ to depend on time only. From the field equation (7), and identifying the electric field by E i = F i0, we find the field equation for the electric field to be given by ∇ · E = Zρ h(φ) (8) where ρ = j0 is the charge density. By integrating this equation over a volume V, it is straightforward to derive the electrostatic potential V (r) = ZQ 4πh(φ)r (9) where Q is the total charge contained in V. Comparing this to the standard expression for the tree-level-potential from QED, one finds that α has the following dependence on Z and h: α ∝ Z h(φ) (Note that α ∝ h−1(φ) when ˜ g(m)

µν

≡ ˜ g(r)

µν .)

(10)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 10 / 33

SLIDE 32

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 11 / 33

SLIDE 33

Disformal Electrodynamics: Evolution of α

Using α ∝ Z h(φ), Z =

1 + B

A ˜ gµν

(m)∂µφ∂νφ

1/2 (11) We define the redshift evolution of α by the quantity ∆α α (z) ≡ α(z) − α(z = 0) α(z = 0) = h(φ0)Z(z) h(φ(z))Z0 − 1, (12) where φ0 is the field value today and Z0 is the value of Z evaluated today. In a spatially–flat FRW gravitational metric, the temporal variation of α reduces to the following ˙ α α = 1 Z ∂Z ∂φ ˙ φ + ∂Z ∂ ˙ φ ¨ φ

− 1

h dh dφ ˙ φ. (13)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 12 / 33

SLIDE 34

Disformal Electrodynamics: Evolution of α

Using α ∝ Z h(φ), Z =

1 + B

A ˜ gµν

(m)∂µφ∂νφ

1/2 (11) We define the redshift evolution of α by the quantity ∆α α (z) ≡ α(z) − α(z = 0) α(z = 0) = h(φ0)Z(z) h(φ(z))Z0 − 1, (12) where φ0 is the field value today and Z0 is the value of Z evaluated today. In a spatially–flat FRW gravitational metric, the temporal variation of α reduces to the following ˙ α α = 1 Z ∂Z ∂φ ˙ φ + ∂Z ∂ ˙ φ ¨ φ

− 1

h dh dφ ˙ φ. (13)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 12 / 33

SLIDE 35

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 13 / 33

SLIDE 36

Disformal Electrodynamics: Cosmology

We now specify our gravitational-scalar action, which leads us to the EF theory described by the following action S =

d4x√−g

1 2R − 1 2gµν∂µφ∂νφ − V (φ)

+ Smatter
˜

g(m)

µν

− 1

4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ ,

(14) where the last term in the action above describes the dynamics of the CMB photons. Field equations G µν = T µν

φ

+ T µν

(m) + T µν (r),

(15)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 14 / 33

SLIDE 37

Disformal Electrodynamics: Cosmology

We now specify our gravitational-scalar action, which leads us to the EF theory described by the following action S =

d4x√−g

1 2R − 1 2gµν∂µφ∂νφ − V (φ)

+ Smatter
˜

g(m)

µν

− 1

4

d4x
−˜

g(r)h(φ)˜ gµν

(r) ˜

gαβ

(r) FµαFνβ ,

(14) where the last term in the action above describes the dynamics of the CMB photons. Field equations G µν = T µν

φ

+ T µν

(m) + T µν (r),

(15)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 14 / 33

SLIDE 38

Disformal Electrodynamics: Cosmology

Klein-Gordon equation φ − V ′ = −Qm − Qr, (16) Conservation equations ∇µT µ

(m)ν = Qmφ,ν ,

∇µT µ

(r)ν = Qrφ,ν ,

(17) where, Qm = C ′

m

2Cm T(m) + D′

m

2Cm φ,µφ,νT µν

(m) − ∇µ

Dm Cm φ,νT µν

(m)

,

(18) Qr = C ′

r

2Cr T(r) + D′

r

2Cr φ,µφ,νT µν

(r) + h′

h C 2

r

1 + Dr

Cr gµνφ,µφ,ν ˜ LEM − ∇µ Dr Cr φ,νT µν

(r)

.

(19)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 15 / 33

SLIDE 39

Disformal Electrodynamics: Cosmology

Klein-Gordon equation φ − V ′ = −Qm − Qr, (16) Conservation equations ∇µT µ

(m)ν = Qmφ,ν ,

∇µT µ

(r)ν = Qrφ,ν ,

(17) where, Qm = C ′

m

2Cm T(m) + D′

m

2Cm φ,µφ,νT µν

(m) − ∇µ

Dm Cm φ,νT µν

(m)

,

(18) Qr = C ′

r

2Cr T(r) + D′

r

2Cr φ,µφ,νT µν

(r) + h′

h C 2

r

1 + Dr

Cr gµνφ,µφ,ν ˜ LEM − ∇µ Dr Cr φ,νT µν

(r)

.

(19)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 15 / 33

SLIDE 40

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 16 / 33

SLIDE 41

Disformal Electrodynamics: Cosmology–FRW

We shall now consider perfect fluid energy-momentum tensors for radiation and matter in the EF, radiation in the RF and matter in the JF as follows T µν

(r) = (ρr + pr)uµuν + prgµν,

T µν

(m) = (ρm + pm)uµuν + pmgµν, (20)

˜ T µν

(r) = (˜

ρr + ˜ pr)˜ uµ˜ uν + ˜ pr ˜ gµν

(r),

˜ T µν

(m) = (˜

ρm + ˜ pm)˜ uµ˜ uν + ˜ pm˜ gµν

(m),

(21) Furthermore, we will now consider a zero curvature FRW EF metric, ds2 = gµνdxµdxν = −dt2 + a2(t)δijdxidxj, leading to ¨ φ + 3H ˙ φ + V ′ = Qm + Qr, (22) ˙ ρm + 3H(ρm + pm) = −Qm ˙ φ, (23) ˙ ρr + 3H(ρr + pr) = −Qr ˙ φ, (24) where H = ˙ a/a is the Hubble parameter and dot represents an EF time

derivative. We introduce η ≡ ˜

LEM/˜ ρr in what follows.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 17 / 33

SLIDE 42

Disformal Electrodynamics: Cosmology–FRW

We shall now consider perfect fluid energy-momentum tensors for radiation and matter in the EF, radiation in the RF and matter in the JF as follows T µν

(r) = (ρr + pr)uµuν + prgµν,

T µν

(m) = (ρm + pm)uµuν + pmgµν, (20)

˜ T µν

(r) = (˜

ρr + ˜ pr)˜ uµ˜ uν + ˜ pr ˜ gµν

(r),

˜ T µν

(m) = (˜

ρm + ˜ pm)˜ uµ˜ uν + ˜ pm˜ gµν

(m),

(21) Furthermore, we will now consider a zero curvature FRW EF metric, ds2 = gµνdxµdxν = −dt2 + a2(t)δijdxidxj, leading to ¨ φ + 3H ˙ φ + V ′ = Qm + Qr, (22) ˙ ρm + 3H(ρm + pm) = −Qm ˙ φ, (23) ˙ ρr + 3H(ρr + pr) = −Qr ˙ φ, (24) where H = ˙ a/a is the Hubble parameter and dot represents an EF time

derivative. We introduce η ≡ ˜

LEM/˜ ρr in what follows.

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 17 / 33

SLIDE 43

Disformal Electrodynamics: Cosmology–FRW

Qm = Ar ArAm − DrDmρrρm

Bm − DmBr

Ar ρm

,

(25) Qr = Am ArAm − DrDmρrρm

Br − DrBm

Am ρr

,

(26) where Ar = Cr + Dr

ρr − ˙

φ2 , Am = Cm + Dm

ρm − ˙

φ2 , (27) Br = 1 2C ′

r (3wr − 1) ρr − 1

2D′

r ˙

φ2ρr + h′ h

Cr − Dr ˙

φ2 ηρr + Drρr C ′

r

Cr ˙ φ2 + V ′ + 3H ˙ φ (1 + wr)

,

Bm = 1 2C ′

m (3wm − 1) ρm − 1

2D′

m ˙

φ2ρm + Dmρm C ′

m

Cm ˙ φ2 + V ′ + 3H ˙ φ (1 + wm)

(28)
J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 18 / 33

SLIDE 44

Disformal Electrodynamics: Cosmology–FRW

Qm = Ar ArAm − DrDmρrρm

Bm − DmBr

Ar ρm

,

(25) Qr = Am ArAm − DrDmρrρm

Br − DrBm

Am ρr

,

(26) where Ar = Cr + Dr

ρr − ˙

φ2 , Am = Cm + Dm

ρm − ˙

φ2 , (27) Br = 1 2C ′

r (3wr − 1) ρr − 1

2D′

r ˙

φ2ρr + h′ h

Cr − Dr ˙

φ2 ηρr + Drρr C ′

r

Cr ˙ φ2 + V ′ + 3H ˙ φ (1 + wr)

,

Bm = 1 2C ′

m (3wm − 1) ρm − 1

2D′

m ˙

φ2ρm + Dmρm C ′

m

Cm ˙ φ2 + V ′ + 3H ˙ φ (1 + wm)

(28)
J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 18 / 33

SLIDE 45

Disformal Electrodynamics: Cosmology–FRW

EF Friedmann equations H2 = 1 3 (ρm + ρr + ρφ) , ˙ H = −1 6

3
ρm + ˙

φ2 + ρr

4 − Dr

Cr ˙ φ2

(29)

The scalar field characterizing the disformal couplings is also responsible for the current acceleration of the Universe, i.e., it is the dark energy. Non–interacting dark sector (Type Ia supernova) ˙ ρeff

DE = −3H(1 + weff)ρeff DE, H2 = 1

3

a−4ρ0,r + a−3ρ0,m + ρeff

DE

weff =

pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρm + ρr + ρφ − a−4ρ0,r − a−3ρ0,m

= pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρeff

DE

(30)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 19 / 33

SLIDE 46

Disformal Electrodynamics: Cosmology–FRW

EF Friedmann equations H2 = 1 3 (ρm + ρr + ρφ) , ˙ H = −1 6

3
ρm + ˙

φ2 + ρr

4 − Dr

Cr ˙ φ2

(29)

The scalar field characterizing the disformal couplings is also responsible for the current acceleration of the Universe, i.e., it is the dark energy. Non–interacting dark sector (Type Ia supernova) ˙ ρeff

DE = −3H(1 + weff)ρeff DE, H2 = 1

3

a−4ρ0,r + a−3ρ0,m + ρeff

DE

weff =

pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρm + ρr + ρφ − a−4ρ0,r − a−3ρ0,m

= pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρeff

DE

(30)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 19 / 33

SLIDE 47

Disformal Electrodynamics: Cosmology–FRW

EF Friedmann equations H2 = 1 3 (ρm + ρr + ρφ) , ˙ H = −1 6

3
ρm + ˙

φ2 + ρr

4 − Dr

Cr ˙ φ2

(29)

The scalar field characterizing the disformal couplings is also responsible for the current acceleration of the Universe, i.e., it is the dark energy. Non–interacting dark sector (Type Ia supernova) ˙ ρeff

DE = −3H(1 + weff)ρeff DE, H2 = 1

3

a−4ρ0,r + a−3ρ0,m + ρeff

DE

weff =

pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρm + ρr + ρφ − a−4ρ0,r − a−3ρ0,m

= pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρeff

DE

(30)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 19 / 33

SLIDE 48

Disformal Electrodynamics: Cosmology–FRW

EF Friedmann equations H2 = 1 3 (ρm + ρr + ρφ) , ˙ H = −1 6

3
ρm + ˙

φ2 + ρr

4 − Dr

Cr ˙ φ2

(29)

The scalar field characterizing the disformal couplings is also responsible for the current acceleration of the Universe, i.e., it is the dark energy. Non–interacting dark sector (Type Ia supernova) ˙ ρeff

DE = −3H(1 + weff)ρeff DE, H2 = 1

3

a−4ρ0,r + a−3ρ0,m + ρeff

DE

weff =

pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρm + ρr + ρφ − a−4ρ0,r − a−3ρ0,m

= pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρeff

DE

(30)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 19 / 33

SLIDE 49

Disformal Electrodynamics: Cosmology–FRW

EF Friedmann equations H2 = 1 3 (ρm + ρr + ρφ) , ˙ H = −1 6

3
ρm + ˙

φ2 + ρr

4 − Dr

Cr ˙ φ2

(29)

The scalar field characterizing the disformal couplings is also responsible for the current acceleration of the Universe, i.e., it is the dark energy. Non–interacting dark sector (Type Ia supernova) ˙ ρeff

DE = −3H(1 + weff)ρeff DE, H2 = 1

3

a−4ρ0,r + a−3ρ0,m + ρeff

DE

weff =

pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρm + ρr + ρφ − a−4ρ0,r − a−3ρ0,m

= pφ + ρr

wr − 1

3a−4 ρ0,r ρr

ρeff

DE

(30)

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 19 / 33

SLIDE 50

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 20 / 33

SLIDE 51

Disformal Electrodynamics: Examples

We specify the following form of couplings and potential: Ci(φ) = βiexiφ , Di(φ) = M−4

i

eyiφ, V (φ) = M4

V e−λφ,

h(φ) = 1 − ζ(φ − φ0), such that the introduced parameters are tuned in order to be in agreement with the observational data on the variation of α together with the cosmological parameters. Parameter Estimated value w0,φ −1.006 ± 0.045 H0 (67.8 ± 0.9) km s−1Mpc−1 Ω0,m 0.308 ± 0.012

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 21 / 33

SLIDE 52

Disformal Electrodynamics: Examples

We specify the following form of couplings and potential: Ci(φ) = βiexiφ , Di(φ) = M−4

i

eyiφ, V (φ) = M4

V e−λφ,

h(φ) = 1 − ζ(φ − φ0), such that the introduced parameters are tuned in order to be in agreement with the observational data on the variation of α together with the cosmological parameters. Parameter Estimated value w0,φ −1.006 ± 0.045 H0 (67.8 ± 0.9) km s−1Mpc−1 Ω0,m 0.308 ± 0.012

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 21 / 33

SLIDE 53

Disformal Electrodynamics: Examples

We specify the following form of couplings and potential: Ci(φ) = βiexiφ , Di(φ) = M−4

i

eyiφ, V (φ) = M4

V e−λφ,

h(φ) = 1 − ζ(φ − φ0), such that the introduced parameters are tuned in order to be in agreement with the observational data on the variation of α together with the cosmological parameters. Ex Mr = Mm Mm βm xm |ζ| MV λ * ∼ meV ∼ meV 1 < 5 × 10−6 2.69 meV 0.45 ∼ meV 15 meV 8 0.14 2.55 meV 0.45 * ∼ meV 15 meV 8 0.14 < 5 × 10−6 2.55 meV 0.45

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 22 / 33

SLIDE 54

Disformal Electrodynamics: Examples

We specify the following form of couplings and potential: Ci(φ) = βiexiφ , Di(φ) = M−4

i

eyiφ, V (φ) = M4

V e−λφ,

h(φ) = 1 − ζ(φ − φ0), such that the introduced parameters are tuned in order to be in agreement with the observational data on the variation of α together with the cosmological parameters. Ex ( ˙ α/α)|0 × 1017 |∆α/α|zCMB * −2.14 ∼ −1.62 10−8 ∼ 10−6 −2.41 ∼ 0.70 10−8 ∼ 10−7 * −2.10 ∼ −1.24 10−7 ∼ 10−6

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 23 / 33

SLIDE 55

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 24 / 33

SLIDE 56

*Disformal/ Disformal & electromagnetic couplings

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 25 / 33

SLIDE 57

*Disformal/ Disformal & electromagnetic couplings

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 26 / 33

SLIDE 58

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 27 / 33

SLIDE 59

**Disformal & conformal couplings

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 28 / 33

SLIDE 60

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 29 / 33

SLIDE 61

***Disformal, conformal & electromagnetic couplings

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 30 / 33

SLIDE 62

1

Introduction–Is α a constant of Nature?

2

Disformal Electrodynamics The Model Identification of α Evolution of α Cosmology

FRW

Examples

Disformal/ Disformal & electromagnetic couplings Disformal & conformal couplings Disformal, conformal & electromagnetic couplings

3

Conclusion

J. Mifsud

The variation of the fine-structure constant from disformal couplings 15/12/15 31 / 33

SLIDE 63