Disformal Couplings and Cosmology

Disformal Couplings and 

Cosmology 

Carsten van de Bruck 

University of Sheffield 

Lancaster-Manchester-Sheffield Consortium for Fundamental Physics 

Based on work with Jack Morrice 

and Susan Vu; arXiv: 1303. 1773 (v3) 

Thursday, 5 September 13

Scalar-Tensor Theories: 

Simple extensions of GR 

Describe corners of other extensions of 

GR (e.g. extra dimensions (incl. brane 

worlds), massive gravity, ...,see Danielle 

Wills talk!) 

Useful for phenomenological 

approaches to modified gravity 


Form considered here is: 

S = 

Z p gd 4 x 

apple M 

2 

Pl 

2 R(g) 1 

2 gµ⌫ (@ µ )(@ ⌫ ) V ( ) 

with 

+ X S i [˜g i µ⌫, i], 

˜g (i) 

µ⌫ = C (i) ( )g µ⌫ + D (i) ( ) ,µ ,⌫ 

C is called conformal factor, D is the disformal 

factor. D is not well constrainted by local experiments 

(Brax (2012), Noller (2012), Brax et al (2012)). 


Field equations for only a single coupled field: 

Koivisto et al (2012); CvdB & G. Sculthorpe (2012) 

G µ⌫ = apple 

⇣ 

⌘ 

T µ⌫ ( ) + Tµ⌫ 

matter 

⇤ 

dV 

d 

+ Q =0 

r µ T µ⌫ = Qr ⌫ 

with 

Q = C , 

2C gµ⌫ T µ⌫ + D , 

2C ,µ ,⌫ T µ⌫ r ⌫ 

apple D 

C ,µ T µ⌫ 


FRW , conformal time: 

¨ +2H ˙ + a 

2 dV 

d 

= a 2 ¯Q 

a 2 ¯Q = 

⇢ 

2(C + D(⇢ ˙/a2 )) 

apple 

a 2 dC 

✓ 

d (1 3w) 2D 3H ˙(1 + w)+a 2 dV 

d 

+ C , 

C 

◆ 

˙2 

+ D , 

˙2 

Goal: extend this to multiple 

couplings (radiation + matter) 


Extension: 

¨ +3H ˙ + 

dV 

d 

= X Q i , 

˙⇢ i +3H(⇢ i + p i )= Q i ˙ , 

with 

Q i = C0 i 

T i + D0 i 

, µ , ⌫ T µ⌫ 

2C i 2C 

i 

✓ i 

◆ 

Di 

r µ ⌫T µ⌫ . 

C 

i i 


FRW spacetime, 2 coupled species: 

Q 1 = 

A 1 A 2 

Q 2 = 

A 1 A 2 

A 1 

A 2 

D 1 D 2 ⇢ 1 ⇢ 2 

✓ 

D 1 D 2 ⇢ 1 ⇢ 2 

✓ 

B 1 

B 2 D 1 ⇢ 1 

A 2 

B 2 

B 1 D 2 ⇢ 2 

A 1 

◆ 

◆ 

. 

B i = 

2 

⇢3H 

✓ 

+D i 1+ p ◆ 

i 

⇢ i 

A i = C i + D i (⇢ ˙2 i ) , 

apple ✓ C 

0 

i 

1+3 p ◆ 

i D 

0 

i ˙2 

⇢ i 2 

˙ + V 0 + C0 i 

C i 

˙2 

⇢ i . 


Consider radiation and matter coupled. 

Goal: study consequences of disformal 

couplings on radiation 

Relation between energy momentum tensor in different frames: 

T µ⌫ = 

s 

1 

˙2 

M 4 ˜T µ⌫ , 

p 

⇢ = ˜p˜⇢ 

✓ 

1 

(D i = M 4 

i ) 

leads to 

◆ 

˙2 

M 4 

. 

Equation of state frame dependent! 


Another consequence: distribution function is 

frame-dependent! 

dN = dx 1 dx 2 dx 3 dP 1 dP 2 dP 3 f. 

Lower case momenta frame independent for 

conformal transformations, but not for disformal 

transformations. 


The following holds: 

✓ ◆ 

d d˜ 

dt d f =0. 

d˜/d =(1 ˙2 /M 4 ) 1/2 

For photons: 

˜P µ ˜rµ ˜P ⌫ =0 ⌫ , ˜g µ⌫ ˜P 

µ ˜P ⌫ =0, 

Liouville operator: 

ˆLf = P 0 @f 

@t 

H ij P i P j M 4 @f 

M 4 ˙2 @P 0 

= 

˙ ¨ 

M 4 ˙2 P 0 f. 


We found the same equation for energy density 

as from the action. Consistent kinetic picture! 

Can derive equation for photon number density: 

ṅ +3Hn = 

F˙ 

2F n, F =(d˜/d ) 2 

Specific entropy not constant (Lima et al (2000)): 

nT d 

= d⇢ 

⇢ + p 

C x = Q ˙ = 

n dn ˙ = 

F˙ 

2F n 

nT 

⇣ 

@⇢ 

@T 

⌘ 

n 

apple 

⇢ + p 

nC x 


Going back to two species coupled: 

Measured speed of light (natural units): 

c 2 obs =1 

✓ 1 

M 4 1 

M 4 m 

◆✓ d 

d⌧ m 

◆ 2 

written in Einstein frame quantities: 

c 2 obs = 

1 

1 

˙ 2 

M 4 

˙ 2 

M 4 m 


Distribution function obeys: 

✓ ◆ 

d d˜ 

dt d f =0. 

" # 

d˜ 

Therefore 

d f 

initial 

= 

" 

# 

d˜ 

d f 

final 

In our work we assume scalar field is dark energy 

(exponential potential), so field rolls only very late. 

" 

d˜ 

) f final = 

d 

# 1 

f initial 

final 


Write final distribution function in the form 

1 

f final = 

(exp(⌫/T + µ) 1) 

For small 

µ , we find 

(see also Brax et al 1306.4168) for similar work) 

µ =(c obs 1) (1 e ⌫/T ) 

Also, define an effective temperature 

⇢ J / T 4 

(Jordan frame) 

Temperature does not obey T / (1 + z) 


T(K) / 2.725 K 

Temperature evolution: 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0 0.5 1 1.5 2 2.5 3 

z 

V ( )=V 0 e 


Exclusion region: 

2 

1.5 

1 

0.5 

/ meV) 

0 

log 

10 

(M 

γ 

-0.5 

-1 

-1.5 

-2 

-2.5 

-0.5 0 0.5 1 1.5 2 

log (M / meV) 

10 

m 

V ( )=V 0 e 


Summary: Disformal couplings introduce new 

phenomenological signatures, in particular on 

radiation. If 

M 6= M m 

Measured equation of state is not 1/3, speed 

of light not constant and/or 1. 

1+z 

CMB temperature is not linear in 

in 

general. 

Spectrum does not stay Planckian if scalar 

evolves. 

Effects vanish if M 

= M m

Disformal Couplings and Cosmology

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