Chameleon mechanism
Lecture 2
Cosmic acceleration


Many independent data sets indicate
that the expansion of the Universe is
accelerating
Similar to precise tests of GR?
Dark energy v Dark gravity

Standard model based on GR fits the date very well
assuming the existence of dark matter/ dark energy

We have never tested GR on
cosmological scales
cf. precession of perihelion
dark planet v GR
Is cosmology probing breakdown of GR on large scales ?
Problems

Modifications to general relativity inevitably introduce
additional degrees of freedom (often scalar)
cf BD gravity
(3  2BD )2  8 G

Imposing the solar system constraints often kills the
modifications completely   40,000
BD

We need to suppress the fifth force mediated by the new
degree of freedom by some mechanisms without killing the
modification on all scales
Screening mechanism

“Screening” mechanisms
The fifth force should act only on large scales and it should be
hidden on small scales
This implies that these mechanisms should work in environmental
dependent ways

Two well know examples

Chameleon mechanism
The mass of the scalar mode depends on density thus changes
according to environments

Vainshtein mechanim
Compton wavelength

Suppose the scalar has a mass
   m   4 G
2
2
r ( r )
Spherically symmetric solutions
 d2 2 d
2


m
 2
   4 G
r dr
 dr

GM
 (r)  
exp  mr 
r
m 1
The scalar potential decays exponentially above the Compton
wavelength m 1
r
Chameleon mechanism

(Khoury&weltman)
Scalar is usually very light on cosmological scales
to derive the acceleration today m2  H 02
no suppression of the fifth force

Scalar is coupled to matter
Depending on density, the mass can change
It is easier to understand the dynamics in Einstein frame
  



BD
2
4 
g   exp  
g ,
S   d x   g  R 
 M  
    V ( )   Lm [ g  ]
pl 






log  2
2
1


 / M pl


SE   d x   g  R      V ( )   Lm [e
g  ]
2




4


M pl
1
3  2BD
Effective potential

Consider non-relativistic matter
Analytic solution

Analytic solution - spherically symmetric case
Analogy - a ball rolling on upside down potential
Low density
High density
Analytic solution
(i) Inside high density region 0  r  Rroll
the field is trapped at the minimum
 c 
V '
 c exp 
  0

M pl
 M pl 

  c
(ii) “rolling region” R  r  R
roll
c
d 2 2 d



c
2
dr
r dr M pl
 (r) 
 c r 2
3M pl
(iii) outer region
d 2 2 d

V '  0
2
dr
r dr
 (r)  
A
 B
2 r
C exp  m ( r  Rc ) 
 c
r
Analytic solution

Matching  ,  ' at the transition
thin shell condition Rc  Rc  Rroll  Rc
    3Rc  M exp  m ( r  Rc 
 (r)  


 4 M pl
 

R
 c 
Rc
 
  c  1
Rc
6 M pl  c
r

Thin shell

Physical interpretation
If the thin shell condition is satisfied, only the shell of the size
Rc contributes to the fifth force
 
 (r)   
 4 M
pl

  3Rc  M exp  m ( r  Rc 
 
 

R
r
 c 
Rc
  c

 1
Rc
6 M pl  c
If this shell condition is not satisfied (thick shell), the fifth force is
unsuppressed
   M exp  m ( r  Rc 
 (r)   
 4 M 
pl 

r
 
Solar system constraints

Solar system constraints
 10 g cm 3
 gal
1024 g cm 3
gal  
Rc

Rc
6 M pl 
gal
6 M pl 
The sun has a potential 
106
The thin shell suppression eases the constraints

Rc
gal

 4.7  106 (BD  40000)
 5  1011
Rc
M pl
This is a model (potential) independent constraint
1
3  2BD
From galaxy to cosmology

Example
V  V0  M  / M pl 
1/2
4
gal
M pl
Solar system constraints
gal
M pl
 10
11
cos mo
M pl
 M4 



 crit 
 M

 
 gal
4
2
10
10



gal
M pl
 gal
2
1024 g cm3
crit 1029 g cm3
 101
M
103 eV
This is a severe fine-tuning but interestingly related to the energy
scale of dark energy
Galaxy
Rc cos mo  gal
cos mo

Rc
6 M pl  gal 6 M pl  gal
6
The Milky way galaxy  Milk 10
in order to screen the Milky way, we need
cos mo
M pl
 106
From solar system to cosmology
f(R) gravity example

f(R) gravity
 R  f ( R)

S   d x g 
 Lm 
 16 G

4
1
2   4 G   2 f R
2
1
8 G
2 f R    R( f R ) 

3
3
     f R
The potential is determined by
the function f(R)
ex)
2 n
c1  R / M 
2
f ( R)   M
2 n
c2  R / M   1
fR 
df
 f R0   f R
dR
Hu-Sawicki model

No cosmological constant f ( R)  0, R  0
“high curvature” regime ( R / M )  1
M 
f ( R)  2  M 

 R 
2
n

2
set the mass scale as
the scalar
c1
M2
2c2
M 2  H 02m
df
nc1  M 
fR 
 2 

dR
c2  R 
2
c1

6 
c2
m
n 1
the background R0  3M 2  1  4  
m 

(no 3D curvature)
c1
1 
 
  3  1  4

c2
n 
m  
n 1
f R0
Background cosmology

Background cosmology
 , f R 0
modified Freedman equation
1
8 G
2
H  f R ( HH ' H )  f  H f RR R ' 

6
3
2
1  weff  
2
1 yH '
3 yH
yH  H 2  m a 3
background is indistinguishable
from LCDM for small | f R 0 | 1
weff  1, | f R 0 | 1
d
'
d ln a
Linear perturbations

Linear perturbations

n 1
Linearise the potential term  R
quasi-static approximations  f R   f R
01
| f R0 |
Mpc
6
10
3.2
1/2
n 1

8 Ga
1
R R 
2
2 2

  fR  a   fR 
 m m ,   


 3(n  1) | f R 0 |  R0  
3


2
In Fourie space, the solution for the Newton potential is
large scales
2 2
2


4

a

/
k


2
k   4 G 
  m m
2 2
2
 3   a  / k  
k   4 Gm m
2
16
 G m m
3
Small scales
BD  0,  PPN  1/ 2
GR
 1
k 2 
ST
Growth function

Growth of density perturbations
The energy-momentum conservation
T00    m (1   m ),
1
a
 m   m  3  0
Ti 0   mvmi ,  i vmi  m
k2
 m  H m  2   0
a
quasi-static approximation
 m  2 H m  k 2
2 2
2


4

a

/
k


2
k   4 G 
  m m
2 2
2
 3   a  / k  
m
 mLCDM

The pattern of the growth
f(R)
LCDM
Chameleon mechanism

Correspondence  / M pl  f R ,   2 / 3
Cosmological field V  V0  M  / M pl  for n=1
4
1/2
cos mo / M pl  102 | f R 0 | 102 solar system constraints
cos mo / M pl  105 | f R 0 | 105 screening of the milky way

Cosmological N-body simulations
procedure
1) solve the non-linear scalar field equation by relaxation method
2) move particles including the fifth force
3) recalculate the density field and go back to 1)
Snapshots at z=0

If the fifth force is not suppressed, we have
Fifth force is not
suppressed
Chameleon
is working
Compton
wavelength
is short
From solar system to cosmology
= From early universe to today
z  1010
z  20
z0
t
Time evolution
| f R 0 | 104