Disformal Couplings and Cosmology
Disformal Couplings and Cosmology
Disformal Couplings and Cosmology
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<strong>Disformal</strong> <strong>Couplings</strong> <strong>and</strong><br />
<strong>Cosmology</strong><br />
Carsten van de Bruck<br />
University of Sheffield<br />
Lancaster-Manchester-Sheffield Consortium for Fundamental Physics<br />
Based on work with Jack Morrice<br />
<strong>and</strong> Susan Vu; arXiv: 1303. 1773 (v3)<br />
Thursday, 5 September 13
Scalar-Tensor Theories:<br />
Simple extensions of GR<br />
Describe corners of other extensions of<br />
GR (e.g. extra dimensions (incl. brane<br />
worlds), massive gravity, ...,see Danielle<br />
Wills talk!)<br />
Useful for phenomenological<br />
approaches to modified gravity<br />
Thursday, 5 September 13
Form considered here is:<br />
S =<br />
Z p gd 4 x<br />
apple M<br />
2<br />
Pl<br />
2 R(g) 1<br />
2 gµ⌫ (@ µ )(@ ⌫ ) V ( )<br />
with<br />
+ X S i [˜g i µ⌫, i],<br />
˜g (i)<br />
µ⌫ = C (i) ( )g µ⌫ + D (i) ( ) ,µ ,⌫<br />
C is called conformal factor, D is the disformal<br />
factor. D is not well constrainted by local experiments<br />
(Brax (2012), Noller (2012), Brax et al (2012)).<br />
Thursday, 5 September 13
Field equations for only a single coupled field:<br />
Koivisto et al (2012); CvdB & G. Sculthorpe (2012)<br />
G µ⌫ = apple<br />
⇣<br />
⌘<br />
T µ⌫ ( ) + Tµ⌫<br />
matter<br />
⇤<br />
dV<br />
d<br />
+ Q =0<br />
r µ T µ⌫ = Qr ⌫<br />
with<br />
Q = C ,<br />
2C gµ⌫ T µ⌫ + D ,<br />
2C ,µ ,⌫ T µ⌫ r ⌫<br />
apple D<br />
C ,µ T µ⌫<br />
Thursday, 5 September 13
FRW , conformal time:<br />
¨ +2H ˙ + a<br />
2 dV<br />
d<br />
= a 2 ¯Q<br />
a 2 ¯Q =<br />
⇢<br />
2(C + D(⇢ ˙/a2 ))<br />
apple<br />
a 2 dC<br />
✓<br />
d (1 3w) 2D 3H ˙(1 + w)+a 2 dV<br />
d<br />
+ C ,<br />
C<br />
◆<br />
˙2<br />
+ D ,<br />
˙2<br />
Goal: extend this to multiple<br />
couplings (radiation + matter)<br />
Thursday, 5 September 13
Extension:<br />
¨ +3H ˙ +<br />
dV<br />
d<br />
= X Q i ,<br />
˙⇢ i +3H(⇢ i + p i )= Q i ˙ ,<br />
with<br />
Q i = C0 i<br />
T i + D0 i<br />
, µ , ⌫ T µ⌫<br />
2C i 2C<br />
i<br />
✓ i<br />
◆<br />
Di<br />
r µ ⌫T µ⌫ .<br />
C<br />
i i<br />
Thursday, 5 September 13
FRW spacetime, 2 coupled species:<br />
Q 1 =<br />
A 1 A 2<br />
Q 2 =<br />
A 1 A 2<br />
A 1<br />
A 2<br />
D 1 D 2 ⇢ 1 ⇢ 2<br />
✓<br />
D 1 D 2 ⇢ 1 ⇢ 2<br />
✓<br />
B 1<br />
B 2 D 1 ⇢ 1<br />
A 2<br />
B 2<br />
B 1 D 2 ⇢ 2<br />
A 1<br />
◆<br />
◆<br />
.<br />
B i =<br />
2<br />
⇢3H<br />
✓<br />
+D i 1+ p ◆<br />
i<br />
⇢ i<br />
A i = C i + D i (⇢ ˙2 i ) ,<br />
apple ✓ C<br />
0<br />
i<br />
1+3 p ◆<br />
i D<br />
0<br />
i ˙2<br />
⇢ i 2<br />
˙ + V 0 + C0 i<br />
C i<br />
˙2<br />
⇢ i .<br />
Thursday, 5 September 13
Consider radiation <strong>and</strong> matter coupled.<br />
Goal: study consequences of disformal<br />
couplings on radiation<br />
Relation between energy momentum tensor in different frames:<br />
T µ⌫ =<br />
s<br />
1<br />
˙2<br />
M 4 ˜T µ⌫ ,<br />
p<br />
⇢ = ˜p˜⇢<br />
✓<br />
1<br />
(D i = M 4<br />
i )<br />
leads to<br />
◆<br />
˙2<br />
M 4<br />
.<br />
Equation of state frame dependent!<br />
Thursday, 5 September 13
Another consequence: distribution function is<br />
frame-dependent!<br />
dN = dx 1 dx 2 dx 3 dP 1 dP 2 dP 3 f.<br />
Lower case momenta frame independent for<br />
conformal transformations, but not for disformal<br />
transformations.<br />
Thursday, 5 September 13
The following holds:<br />
✓ ◆<br />
d d˜<br />
dt d f =0.<br />
d˜/d =(1 ˙2 /M 4 ) 1/2<br />
For photons:<br />
˜P µ ˜rµ ˜P ⌫ =0 ⌫ , ˜g µ⌫ ˜P<br />
µ ˜P ⌫ =0,<br />
Liouville operator:<br />
ˆLf = P 0 @f<br />
@t<br />
H ij P i P j M 4 @f<br />
M 4 ˙2 @P 0<br />
=<br />
˙ ¨<br />
M 4 ˙2 P 0 f.<br />
Thursday, 5 September 13
We found the same equation for energy density<br />
as from the action. Consistent kinetic picture!<br />
Can derive equation for photon number density:<br />
ṅ +3Hn =<br />
F˙<br />
2F n, F =(d˜/d ) 2<br />
Specific entropy not constant (Lima et al (2000)):<br />
nT d<br />
= d⇢<br />
⇢ + p<br />
C x = Q ˙ =<br />
n dn ˙ =<br />
F˙<br />
2F n<br />
nT<br />
⇣<br />
@⇢<br />
@T<br />
⌘<br />
n<br />
apple<br />
⇢ + p<br />
nC x<br />
Thursday, 5 September 13
Going back to two species coupled:<br />
Measured speed of light (natural units):<br />
c 2 obs =1<br />
✓ 1<br />
M 4 1<br />
M 4 m<br />
◆✓ d<br />
d⌧ m<br />
◆ 2<br />
written in Einstein frame quantities:<br />
c 2 obs =<br />
1<br />
1<br />
˙ 2<br />
M 4<br />
˙ 2<br />
M 4 m<br />
Thursday, 5 September 13
Distribution function obeys:<br />
✓ ◆<br />
d d˜<br />
dt d f =0.<br />
" #<br />
d˜<br />
Therefore<br />
d f<br />
initial<br />
=<br />
"<br />
#<br />
d˜<br />
d f<br />
final<br />
In our work we assume scalar field is dark energy<br />
(exponential potential), so field rolls only very late.<br />
"<br />
d˜<br />
) f final =<br />
d<br />
# 1<br />
f initial<br />
final<br />
Thursday, 5 September 13
Write final distribution function in the form<br />
1<br />
f final =<br />
(exp(⌫/T + µ) 1)<br />
For small<br />
µ , we find<br />
(see also Brax et al 1306.4168) for similar work)<br />
µ =(c obs 1) (1 e ⌫/T )<br />
Also, define an effective temperature<br />
⇢ J / T 4<br />
(Jordan frame)<br />
Temperature does not obey T / (1 + z)<br />
Thursday, 5 September 13
T(K) / 2.725 K<br />
Temperature evolution:<br />
4<br />
3.5<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0 0.5 1 1.5 2 2.5 3<br />
z<br />
V ( )=V 0 e<br />
Thursday, 5 September 13
Exclusion region:<br />
2<br />
1.5<br />
1<br />
0.5<br />
/ meV)<br />
0<br />
log<br />
10<br />
(M<br />
γ<br />
-0.5<br />
-1<br />
-1.5<br />
-2<br />
-2.5<br />
-0.5 0 0.5 1 1.5 2<br />
log (M / meV)<br />
10<br />
m<br />
V ( )=V 0 e<br />
Thursday, 5 September 13
Summary: <strong>Disformal</strong> couplings introduce new<br />
phenomenological signatures, in particular on<br />
radiation. If<br />
M 6= M m<br />
Measured equation of state is not 1/3, speed<br />
of light not constant <strong>and</strong>/or 1.<br />
1+z<br />
CMB temperature is not linear in<br />
in<br />
general.<br />
Spectrum does not stay Planckian if scalar<br />
evolves.<br />
Effects vanish if M<br />
= M m<br />
Thursday, 5 September 13