AVOIDING THE PHANTOM MENACE
Júlio C. Fabris
Departamento de Física – UFES
Barcelona - 2006
Inflation requires:
a0

  3p  0
Strong energy condition is violated!
However, the null energy condition
does not need to be violated:
 
T k k  0

k k  0
 p0
A phantom field (or fluid) is characterized by
p  
  1
All energy conditions are violated!
The phantom case became quite fashion due to the
recent observational results
Of course, the predictions depend on which data are
taken into account and how the statistics is made
Combining CMB (WMAP), matter clustering (SDSS
e 2dFGRS) and supernovae:
10
  0.93 00..13
Matts Roos, astrop-ph/0509089
Other estimations:
CMB, SNIa, large scale
structures

 1.39    0.79
S. Nannestad e E. Mortsell, JCAP 0409, 001(2004)
Including X-Rays

  1,20
0.24
 0.28
S.W. Allen et al., Mon. Not. R. Astron. Soc. 353, 457(2004)
8G 2
 a' 
a 
  
3
a
2
a'
 '3 (1   )   0
a
  0a
a  a0
3(1 )
2
(1 3 )
dt  ad
1
 
3
1
    1
3
  1



a  

  0
 

0
a  
 
  0
0
a  

  

Inevitable consequence for a Universe dominated by
an exotic fluid where all the energy conditions are
violated:
A “big rip”: a curvature singularity in a future finite
proper time
This requires, however, homogeneity and isotropy
Some remarks on the perturbative behaviour of
the phantom fields
J.C.F and S.V.B. Gonçalves, PRD, 2006
Considering a barotropic equation of state
p  
The following equation governing the gravitational
potential is found:


' '3H (1   )' q  2H '(1  3 ) H   0
2
2
The scale factor behaves as
a 
2 /(13 )
The equation for the potential becomes
(1  3 ) 
 ' '6
 q 2   0
(1   )  '
The solution depends on the sign of the pressure:
 0
 
 


c J
1 
c I (
1 

(  q)  c2 J  (  q)
 0

  q)  c2 K (   q)
Asymptotic behaviour
 0
q  0
  c1  c2
q     

3(1 )
(1 3 )
cos
5  3

2(1  3 )

2
 q  

 0
q  0
q  
  c1  c2
 
3(1 )

(1 3 )
e
2
  q
The instability at small scale may be solved using a field
representation:

1
1

, 
R  g  R    ,  ,  g   ,     g  V ( )
2
2


a 
2
(1 3 )
 3(1   )
  2
ln 
1  3
2 (1   ) 
V ( ) 
e
2
3 (1   )
3(1 )
The perturbed equation is

''  2
''
' '2 H  ' q  2 H ' H   0
'  
' 

Using the background solution,
3(1   ) '
2
' '2
q  0
1  3 
The solution is:
 

c1 J (q )  c 2 J  (q )
5  3

2(1  3 )

Asymptotic behaviour:
q  0
q  
  c1  c2
 
6
(1 )
(1 3 )
2
cos( q   )
 0
q  0
t 
 0
q  
t 
1
   0
3
q  0
t 
1
   0
3
q  
t 
1
1    
3
q  0
 t
1
1    
3
q  
 t
5
    1
3
q  0
 t
5
    1
3
q  
 t
5
 
3
q  0
 t
5
 
3
q  
 t
The Hubble horizon is given by
lH

3(1 ) /(1 3 )
It grows for normal fields, but it decreases when the
phanton field dominates the matter content of the universe
Considering now local configurations:
K.A. Bronnikov e J.C.F, PRL(2006)
2
d

ds 2  A(  )dt 2 
 r 2 (  )d 2
A(  )
( A ' r 2 ) '  2r 2V ( )
r ''
2
  '2
r
A(r 2 ) '' r 2 A ''  2
 0

A “normal” scalar field
 0

An “anomalous” scalar field
 0
A static region
 0
A horizon
An expanding non-singular universe
The horizon can be the border between regular regions
Phantom inflation may be very attractive today
And it may not be so dangerous!
Moreover, local configurations are very attractive