- The document discusses a braneworld model where a 3-brane moves in a 5-dimensional anti-de Sitter bulk. The brane behaves effectively as a tachyon field with an inverse quartic potential. - When the backreaction of the radion field (related to fluctuations of the brane position) is included, the tachyon Lagrangian is modified by its interaction with the radion. This results in an effective equation of state at large scales that describes "warm dark matter". - The model extends the second Randall-Sundrum braneworld model to include nonlinear effects from the radion field, which distorts the anti-de Sitter geometry.

1. Warm Tachyon Matter
Neven Bilić
Ruđer Bošković Institute
Zagreb
Niš, October 2014
AdS Braneworld with Backreaction

2. Basic idea
A 3-brane moving in AdS5 background of the second Randall-Sundrum (RSII) model behaves effectively as a tachyon with the inverse quartic potential.
The RSII model may be extended to include the back reaction due to the radion field. Then the tachyon Lagrangian is modified by the interaction with the radion.
As a consequence, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
Based on the colaboration with Gary Tupper
““Warm” Tachyon Matter from Back-reaction on the Brane” arXiv: 1302.0955 [hep-th].
“Ads Braneworld with Backreaction”, CEJP 12 (2014) 147 arXiv: 1309.6588

3. Outline
1.Randall–Sundrum Model
2.Gravity in the Bulk – the Radion
3.Dynamical Brane – the Tachyon
4.Field Equations
5.Isotropic Homogeneous Evolution
6.Conclusins & Outlook

4. Branewarld cosmology is based on the scenario in which matter is confined on a brane moving in the higher dimensional bulk with only gravity allowed to propagate in the bulk
N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429 (1998)
L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370 (RS I)
L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690 (RS II)
1. Randall – Sundrum model

5.    x
X
x
X
g G
a b
ab 



 ind
p-BRANE moving in p+2 -dimensional bulk
Nambu-Goto action
1 ind
brane ( 1) det( ) p p S d x g      
xμ – coordinates on the brane
σ – brane tension
STRING: in d+1 -dimensional bulk. Nambu-Goto action:
ind
string det( ) ij S  T  d d  g
;  or ind 




 i
j
b
i
a
ij ab x
x
X
x
X
g G
Xa – coordinates in the bulk;
τ, σ– coordinates on the string sheet
induced metric (“pull back”) of the bulk metric Gab

6. In 4+1 dim. bulk (p=3) choose the coordinates
such that
X μ =x μ, μ=0,..3, and let the fifth coordinate
X 4≡ θ be normal to the brane. The simplest
case
Xa
xi
x0 θ
4 44
for 0...3
0 1
G g
G G
 

  
  
yields Dirac-Born-Infeld type of theory

7. which gives
and hence
42DBI1SdxX,,Xg  21pX 2p    21pXpXX     

8. RS I was proposed as a solution to the hierarchy problem, in particular between the Planck scale MPl ~ 1019 GeV and the electroweak scale MEW ~ 103 GeV
RS I is a 5-dim. universe with AdS5 geometry containing two 4-dim. branes with opposite brane tensions separated in the 5th dimension.
The observer is placed on the negative tension brane and the separation is such that the strength of gravity on observer’s brane is equal to the observed 4-dim. Newtonian gravity.
First Randall-Sundrum model (RS I)

9.  x
5x  y
Observers reside on the negative tension brane at y=l
y  0
y  l
  0
  0
The coordinate position y=l of the negative tension brane
serves as a compactification radius so that the effective
compactification scale is
c  1/ l

10. The conventional approach to the hierarchy problem is to
assume n compact extra dimensions with volume Vn
If their size is large enough compared to the Planck scale,
i.e., if
such a scenario may explain the large mass hierarchy
between the electroweak scale MEW and the fundamental
scale M of 4+n gravity. In the simplest case, when
the 4+n dim. spacetime is a product of a 4-dim. spacetime
with an n-dim. compact space, one finds
2 2
Pl
n
n M M V  
1/
c Pl 1/ n
n V M
In this way the fundamental 4+n scale M could be of the
order of MEW if the compactification scale satisfies
2/
c EW EW Pl / ( / ) n  M M M
Unfortunately, this introduces a new hierarchy μc << MEW

11. Another problem is that there exist a lower limit on the
fundamental scale M determined by null results in table-top
experiments to test for deviations from Newton’s law
in 4 dimensions, U ~1/ r. These experiments currently
probe sub-millimeter scales,
so that
1/
15
1
0.1
10 TeV
n V mm  
5 5 10 TeV for n=1
3TeV for n=2
M
  
 
 
Stronger bounds for brane-worlds with compact flat extra
dimensions can be derived from null results in particle
accelerators and in high-energy astrophysics
Long et al, Nature 421 (2003).
M. Cavagli`a, “Black Hole and Brane Production in TeV Gravity: A Review”
Int. J. Mod. Phys. A 18 (2003).
S. Hannestad and G.Raffelt, “Stringent Neutron-Star Limits on Large Extra Dimensions”
Phys. Rev. Lett. 88 (2002).

12. In contrast, RS brane-worlds do not rely on
compactification to localize gravity at the brane, but on the
curvature of the bulk (“warped compactification”). What
prevents gravity from ‘leaking’ into the extra dimension at
low energies is a negative bulk cosmological constant
Λ5 acts to “squeeze” the gravitational field closer to the brane.
One can see this in Gaussian normal coordinates XM = (xμ,y)
on the brane at y = 0, for which the AdS5 metric takes the form
2 2 2
(5) (5)
M N ky
MN ds g dX dX e dx dx dy  
     
2
5 2
6
    k ℓ - curvature radius of AdS5
corresponding to the scale k≈1/ℓ
Warp factor

13. The Planck scale is related to the fundamental scale as
 
3
2 3 2 2
Pl
0
1
2 1
8
l
ky kl M
M M e dy e
G k
     
 
So that MPl depends only weakly on l in the limit of large kl.
However, any mass parameter m0 on the observer’s brane
in the fundamental 5-dim. theory will correspond to the
physical mass
0
kl m e m  
If kl is of order 10 – 15, this mechanism produces TeV physical
mass scales from fundamental mass parameters not far from
the Planck scale 1019 GeV. In this way we do not require large
hierarchies among the fundamental parameters
m0, k, M, μc=1/l

14. In RSII observers reside on the positive tension brane at y=0 and the negative tension brane is pushed off to infinity in the fifth dimension. Planck mass scale is determined by the curvature of the five-dimensional space-time
The inverse curvature k serves as the compactification scale (corresponding to V1/n in the standard approach) and hence the model provides an alternative to compactification.
Second Randall-Sundrum model (RS II)
3232Pl02kyMMMedyk  

15.  x
5x  y
As before, the 5-dim bulk is ADS5 with line element
y  0 y  l
5d (x)
y  
  2 2 2
(5) (5) ( ) M N ky
MN ds g X dX dX e g x dx dx dy      

  0

16. Radion
The Randall-Sundrum solution corresponds to an empty observer’s brane at y=0. Placing matter on this brane changes the bulk geometry; this is encoded in the radion field related to the variation of the physical interbrane distance d5(x).
At linear order the radion disapears in RS II

17. At nonlinear level the AdS5 geometry of RSII is distorted by
the radion. To see this, consider the total gravitational action
in the bulk
bulk GH brane S  S  S  S
We choose a coordinate system such that
and we assume a general metric which admits Einstein
spaces of constant 4-curvature with the line element
2 2 2 2
(5) (5) = ( ) = ( , ) ( ) ( , ) , M N
MN ds g X dX dX x y g x dx dx x y dy  
   
   
5
bulk 5 5 5
5
1
2
2
S d x g R
K
    
  
where
At fixed xμ , 5 is the distance along the fifth dimension. d   dy
(5) 5 g 0, 0,1,2,3    
K5=8π/M3
2. Gravity in the Bulk – the Radion

18. 2 2
4 2 4
b , , 5
5
1 ( )
= 3 ( ) 6
2
y
ulk
R
S d x g dy g
K

 
    
           
  
 
where R is the 4-d Ricci scalar associated to the 4-d metric .
For consistency with 5-dim Einsten’s equations with
cosmological constant
The 5-dim bulk action may be put in the form
we impose
In addition, we impose the gauge condition
(the ‘Einstein’s frame’ gauge) so the coefficient of R is
a function of y only
5 5 R  0 
2 2   W( y)
g
bulk bulk GH S  S  S
5 (5) (5)
1
0
2 MN MN R R g
 
    
 

19. We arrive at the 5-dimensional line element
   
 
2
2
2 2 2
(5) 2
( )
( )
( )
W y
ds x W y g x dx dx dy
x W y
 
         
 
 

By choosing and neglecting the radion we
recover the AdS5 geometry. Hence, the choice
corresponds to the original RSII model.
J.E. Kim, G. Tupper, and R. Viollier, PLB 593 (2004)
( ) ky W y e 
This metric is a solution to Einsten’s equations
provided
precisely as it is in the RSII model
( ) ky W y e 
where the field ϕ represents the radion.
2 5 =
6
k



20. Integration over y fom 0 to ∞ yields
4 , ,
bulk
5 5 5
3 6
= (1 2 )
2 4 (1 )
R g k
S d x g
kK kK K

     
      
    

The brane action is defined as a Nambu-Goto action
4 i
brane = det nd S d x g  
Where gind is a 4-dim metric induced on the brane.
For the two branes at y=0 and y=l we find
ind
0 | = (1 ) y g g      ind 2 | = ( ) kl
y l g e g 
    

21. the last term in is canceled by the two brane actions
in the limit
0
5
6
= =l
k
K
 
With the RSII fine tuning for the brane tensions
l  
bulk S
brane =0 brane =
4 2 4 2 2
0
| |
= (1 ) ( )
y y l
kl
l
S S
d x g d x g e

          
The total brane contribution is

22. Then, the total action action takes a simple form
4
b brane , ,
1
=
16 2 ulk
R
S S d x g g
G

 
 
       
   
where we have identified the 4-dimensional gravitational
constant
2
5 0 5
1 2 1
=
8
W dy
G K kK


 
  1 3
= sinh
4 G
  

and the canonically normalized radion

23. is no longer infinite so the physical size of the 5-th dimension is of the order although its coordinate size is infinite
The appearance of a massless mode ϕ – the radion – causes 2 backreaction effects
1.Matter on observer’s brane sees the (induced) metric
2.The physical distance to the AdS5 horizon at coordinate infinity
1gg 1Plkl 511ln2dk     

24. y  0 y
Consider a 3-brane moving in the 5-d bulk spacetime
with metric
2
2
2 2 2
(5) 2 = [ ( )]
( )
ky
ky
ky
e
ds e x g dx dx dy
e x

   

 
   
   
3. Dynamical Brane – the Tachyon

25. The points on the brane are parameterized by
. The 5-th coordinate Y is treated as a
dynamical field that depends on x. The brane action
 ,   M X x Y x  
2
2 2 3
ind
(5) , , 2 2 2 , ,
( )
= =
( )
kY kY
M N
MN kY kY
e e
g g X X g Y Y
e e
 
       
     
    
     
yields
1/2
2 2
4 2 2
b 2 3 , ,
( )
= ( ) 1
( )
kY
kY
rane kY
e
S d x g e g Y Y
e

 
  
 
      
   

with the induced metric
4 i
b = det nd
rane S d x g  

26. Changing Y to a new field we obtain
the effective brane action
= / kY  e k
2 2 2
4 , ,
b 4 4 2 2 3
(1 )
= 1
(1 ) rane
k g
S d x g
k k

      
  
    
In the absence of the radion field ϕ we have a pure
undistorted AdS5 background and
This action describes a tachyon with inverse quartic
potential.
(0) 4
b 4 4 , , = 1 rane S d x g g
k

 

    
 

27. More generally an effective Born-Infeld type lagrangian
for the tachyon field θ describes unstable modes in string
theory
The typical potential has minima at . Of particular
interest is the inverse power law potential .
For n > 2 , as the tachyon rolls near minimum, the pressure
very quickly and one thus apparently gets
pressure-less matter (dust) or cold dark matter.
  , ,  V 1  g
  L   
  
n V  
A. Sen, JHEP 0204 (2002); 0207 (2002).
L.R. Abramo and F. Finelli, PLB 575(2002).
Tachyon as CDM
p L0

28. For n=0, i.e., V=V0 , one gets the Chaplygin gas
the first definite model for a DE/DM unification
0 V
p

 
A. Kamenshchik, U. Moschella, V. Pasquier, PLB 511 (2001)
In general any tachyon model can be derived as a map from
the motion of a 3-brane moving in a warped extra dimension.
N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)

29. E.g., for positive power law potential
2
0 ( ) n V  V 
One can get dark matter and dark energy as a single entity
i.e., another model for DE/DM unification
N. B., G. Tupper, R.Viollier, Cosmological tachyon condensation.
Phys. Rev. D 80 (2009)
Tachyon has been heavily exploited in almost any
cosmological context: as inflaton, DM, DE

30. 1.The geometric tachyon is seen on our brane as a form of matter and hence it affects the bulk geometry in which it moves.
2.The back-reaction qualitatively changes the geometric tachyon: the tachyon and radion form a composite substance with a modified equation of state.
The dynamical brane causes two back-reaction effects

31. Instead of the tachyon field defined previously as
it is convenient to introduce a new field
= / kY  e k
2 2
2 ( )
( )
3
( ) = 3 =
k x
x e kY x

 
Then the combined radion and brane Lagrangian becomes
2 2 , ,
, , 2 3
1
= 1
2
g
g

  
 
  
    

L
2
2
4 6
= 2 6sinh , = , =
3 6
G
k k
 
    
where
4. Field Equations

32. We introduce canonical conjugate momentum fields
,
, 2 3
, , , ,
= = = =
1 /
L L g
g
g

   
   
   
   
  
      
For timelike and we may also define the norms
= g = g    
             
Then the Lagrangian may be expressed as
2
2
2 2 2 2
1 1
=
2 1 / ( )



 
   
L
, ,  

33. The corresponding energy momentum tensor
may be expressed as a sum of two ideal fluids
1 1 1 1 2 2 2 2 1 2 T = ( p )u u ( p )u u ( p p )g             
with
2
2 2
= 2
1 / ( )
T g g
g
 
     

   
     
    
L
L L
1 2 u = u =          
2
2
1 2 2 2 2 2
1 1
= =
2 1 / ( )
p p 


 
   
2
2 2 2 2
1 2 2
1
= = 1 / ( )
2  

      

34. The Hamiltonian may be identified with the total energy
density
1 2 = T 3 = 
 H  L  
which yields
2
2 2 2 2
2
1
= 1 / ( )
2  

H      
It may be easily verified that H is related to L through the
Legendre transformation
, , , , ( , ) = ( , )              
    H     L
Here the dependence on Φ and Θ is suppressed. The rest
of the field variables are constrained by Hamilton’s
equations

35. Hamilton’s equations
Now we multiply the first and second equations by and , respectively, and we take a covariant divergence of the next two equations
,, ,, == ==             HHLL2u 1u

36. We obtain a set of four 1st order diff. equations
=   
2 2 2
=
1 / ( )


 

    
2 2 2
1 2 2 2 2
3 4 3
3 =
1 / ( )
H 
 

   
    
    
2 2 2
2 2 2 2 2
1 4 3
3 =
1 / ( )
H 
 

   
   
    
4 4
= sinh 2
3 3
G  G 
  
 
where
1 1 ; 2 2 ; 3H = u 3H = u  
 
1 , 2 , 1 , 1 , u , u , u , u    
                   
2 4
= 2 6sinh
3
G
   

37. To exhibit the main features we solve our equations
assuming spatially flat FRW spacetime with line
ellement
1 2 =
a
H H H
a
 
and we have in addition the equation for the scale a(t)
8
=
3
a G
a
 H
in which case
ds2  dt2  a2(dr2  r2d2 )
We evolve the radion-tachyon system from t=0 with
some suitably chosen initial conditions
5. Isotropic Homogeneous Evolution

38. Evolution of the radion-tachyon system for λl2=1/3 with the initial
conditions at t=0: 1.01, 0.1, 0        
Time is taken in units of l

39. After the transient period the equation of state w=p/ρ becomes positive and oscillatory

40. Since the oscillations in w are rapid on cosmological
timescales, it is most useful to time average co-moving
quantities. By averaging the asymptotic w over long
timescales we find
2 3
0.017
4
A
w
where we have estimated A by comparing the exact and
approximate solutions for tΦ
In the asymptotic regime ( t   )
2
2 2
2 3/2
3
sin 2
2 | | / 2
p t
w A

  
   
      

41. CDM is rather successful in explaining the large-scale power spectrum and CMB spectrum. However there is some inconsistency of many body simulations with observations:
1.Overproduction of satellite galaxies
2.Modelling halos with a central cusp.
Warm dark matter
It is believed that these problems may be alleviated
to some extent with the so called Warm DM
Besides, it has been recently argued that cosmological data favor a dark matter equation of state wDM ≈ 0.01
A. Avelino, N. Cruz and U. Nucamendi, arXiv:1211.4633

42. D
D
= M
M
T
w
m
We take and identify
This gives mDM ≈ 0.43 keV
D e| =< >= 0.017 M q w w
e T = T q = 7.4 eV at t  teq
To demonstrate that our asymptotic equation of state is
associated with WDM, we assume that our equation of state
w≈0.017 corresponds to that of (nonrelativistic) DM thermal
relics of mass mDM at the time of radiation-matter equality teq
a mass typical of WDM. Besides, it may be shown that
the horizon mass at the time when the equivalent DM
particles just become nonrelativistic is typically of the order
of a small galaxy mass.

43. These DM particles have become non relativistic at the
temperature T≡TNR =mDM , corresponding to the scale
eq
N e e
NR
< > R q q
T
a a w a
T

.
The horizon mass before equality evolves as
,
3
e
e H  


 


q
q a
a
M M
where for a spatially flat universe.
Thus, at a=aNR we obtain
15
e 2 10 q M  M
3 10
H eq M (0.017) M 10 M
the mass scale typical of a small galaxy and therefore
the DM may be qualified as warm.
P. Coles and F. Luchin, Cosmology
(Wiley 2002)

44. Conclusions & Outlook
•We have demonstrated that back-reaction causes the brane – radion – tachyon system to behave as “warm” dark matter with a linear barotropic equation of state
•The ultimate question regards the clustering properties of the model. At the linear level one expects a suppression of small-scale structure formation: initially growing modes undergo damped oscillations once they enter the co-moving acoustic horizon
•Perturbation theory is not the whole story – it would be worth studying the nonlinear effects, e.g., using the Press-Schechter formalism as in the pure tachyon model of
•Finally, it would be of interest to investigate the brane- radion-tachyon system as a model for inflation
N. B., G. Tupper, R.Viollier, Phys. Rev. D 80 (2009)

45. Hvala

46. Approximate asymptotic solution to second order in the
amplitude A of tΦ
2
3 2 2 4 2
2 cos
3 2
A t
A 
 
    
 
 
2 2
2 2
2
3
1 cos 2
2
A t
A
t
   
    
   

In the asymptotic regime ( )
2
2 3/2 2 | | /
p
w



    
t  
so 2 2 3
sin 2
2
t
w A
 
 
 
cos 2
A t
t
 
  
 