Warm Tachyon Matter

Neven Bilić

Ruđer Bošković Institute

Zagreb

Niš, October 2014

AdS Braneworld with Backreaction

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

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

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

x

X

x

XGg

ba

ab

ind

p-BRANE moving in p+2 -dimensional bulk

Nambu-Goto action

1 ind

brane ( 1) det( )p pS d x g

xμ – coordinates on the brane

σ – brane tension

STRING: in d+1 -dimensional bulk. Nambu-Goto action:

ind

string det( )ijS T d d g

or;ind

i

j

b

i

a

abij xx

X

x

XGg

Xa – coordinates in the bulk;

τ, σ– coordinates on the string sheet

induced metric (“pull back”) of the bulk metric Gab

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

which gives

and hence

4 2

DBI 1S dx X , ,X g

21p X

2

p

21

pX p

X X

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)

x5x y

Observers reside on the negative tension brane at y=l

0y 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

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

nM M V

1/

c Pl1/ n

nV 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/ ( / ) nM M M

Unfortunately, this introduces a new hierarchy μc << MEW

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

10.1

10 TeV

nV mm

55 10 TeV for n=1

3TeV for n=2M

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).

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

MNds g dX dX e dx dx dy

2

5 2

6k ℓ - curvature radius of AdS5

corresponding to the scale k≈1/ℓ

Warp factor

The Planck scale is related to the fundamental scale as

3

2 3 2 2

Pl

0

12 1

8

l

ky klMM 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

klm 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

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)

32 3 2

Pl

0

2 ky MM M e dy

k

x

5x y

As before, the 5-dim bulk is ADS5 with line element

0y y l5( )d x

y

2 2 2

(5) (5) ( ) M N ky

MNds g X dX dX e g x dx dx dy

0

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

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 braneS 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

MNds g X dX dX x y g x dx dx x y dy

5

bulk 55 5

5

12

2S d x g R

K

where

At fixed xμ , is the distance along the fifth dimension. 5d dy

(5) 5 0, 0,1,2,3g

K5=8π/M3

2. Gravity in the Bulk – the Radion

2 2

4 2 4

b , , 5

5

( )1= 3 ( ) 6

2

y

ulk

RS 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 50R

2 2( )W y

g

bulk bulk GHS S S

(5) (5)5

10

2MNMN

R R g

We arrive at the 5-dimensional line element

22

2 2 2

(5) 2

( )( )

( )

W yds 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)

( ) kyW y e

This metric is a solution to Einsten’s equations

provided

precisely as it is in the RSII model

( ) kyW y e

where the field ϕ represents the radion.

2 5=6

k

Integration over y fom 0 to ∞ yields

, ,4

bulk

5 5 5

3 6= (1 2 )

2 4 (1 )

gR kS d x g

kK kK K

The brane action is defined as a Nambu-Goto action

4 i

brane = det ndS 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 )yg g ind 2| = ( )kl

y lg e g

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 bulkS

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

Then, the total action action takes a simple form

4

b brane , ,

1=

16 2ulk

RS S d x g g

G

where we have identified the 4-dimensional gravitational

constant

2

5 50

1 2 1=

8W dy

G K kK

13= sinh

4 G

and the canonically normalized radion

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

1g g

1

Plk l

5

1 1ln

2d

k

0y y

Consider a 3-brane moving in the 5-d bulk spacetime

with metric 2

22 2 2

(5) 2= [ ( )]

( )

kyky

ky

eds e x g dx dx dy

e x

3. Dynamical Brane – the Tachyon

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 ,MX x Y x

22 2 3

ind

(5) , , , ,2 2 2

( )= =

( )

kY kYM N

MN kY kY

e eg g X X g Y Y

e e

yields 1/2

2 24 2 2

b , ,2 3

( )= ( ) 1

( )

kYkY

rane kY

eS d x g e g Y Y

e

with the induced metric

4 i

b = det nd

raneS d x g

Changing Y to a new field we obtain

the effective brane action = /kYe k

2 2 2, ,4

b 4 4 2 2 3

(1 )= 1

(1 )rane

gkS 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= 1raneS d x g g

k

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.

, ,

1V g

L

nV

A. Sen, JHEP 0204 (2002); 0207 (2002).

L.R. Abramo and F. Finelli, PLB 575(2002).

Tachyon as CDM

0p L

For n=0, i.e., V=V0 , one gets the Chaplygin gas

the first definite model for a DE/DM unification

0Vp

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)

E.g., for positive power law potential

2

0( ) nV 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

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

Instead of the tachyon field defined previously as

it is convenient to introduce a new field

= /kYe k

22

)(2

)(

3=3=)(

xkex xkY

Then the combined radion and brane Lagrangian becomes

, ,2 2

, , 2 3

1= 1

2

gg

L

2

2

4 6= 2 6sinh , = , =

3 6

G

k k

where

4. Field Equations

We introduce canonical conjugate momentum fields

,

, 2 3, , , ,

= = = =1 /

gL Lg

g

For timelike and we may also define the norms

= =g g

Then the Lagrangian may be expressed as

22

2 2 2 2

1 1=

2 1 / ( )

L

,,

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

LL L

1 2= =u u

22

1 2 2 2 2 2

1 1= =

2 1 / ( )p p

22 2 2 2

1 2 2

1= = 1 / ( )

2

The Hamiltonian may be identified with the total energy

density

1 2= 3 =T

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

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

, ,

, ,

= =

= =

H H

L L

2u1u

We obtain a set of four 1st order diff. equations

= 2 2 2

=1 / ( )

2 2 2

1 2 2 2 2

3 4 33 =

1 / ( )H

2 2 2

2 2 2 2 2

1 4 33 =

1 / ( )H

4 4= sinh 2

3 3

G G

where

1 1 ; 2 2 ;3 = 3 =H u H u

1 , 2 , 1 , 1 ,, , ,u u u u

2 4= 2 6sinh

3

G

To exhibit the main features we solve our equations

assuming spatially flat FRW spacetime with line

ellement

1 2=a

H H Ha

and we have in addition the equation for the scale a(t)

8=

3

a G

a

H

in which case

2 2 2 2 2 2( )ds dt a dr r d

We evolve the radion-tachyon system from t=0 with

some suitably chosen initial conditions

5. Isotropic Homogeneous Evolution

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

After the transient period the equation of state w=p/ρ

becomes positive and oscillatory

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

230.017

4

Aw

where we have estimated A by comparing the exact and

approximate solutions for tΦ

In the asymptotic regime ( ) t

22 2

2 3/2

3sin 2

2 | | / 2

p tw A

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

D

D

=M

M

Tw

m

We take and identify

This gives mDM ≈ 0.43 keV

D e| =< >= 0.017M qw w

e eq= = 7.4 eV at qT T t t

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.

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

Ta a w a

T

.

The horizon mass before equality evolves as

,

3

e

eH

q

qa

aMM

where for a spatially flat universe.

Thus, at a=aNR we obtain

15

e 2 10qM M

3 10

H eq(0.017) 10M M 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)

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)

Hvala

Approximate asymptotic solution to second order in the

amplitude A of tΦ

23 2 24 2

2 cos3 2

A tA

2 22 2

2

31 2cos

2

A tA

t

In the asymptotic regime ( )

2

2 3/22 | | /

pw

t

so 2 23sin 2

2

tw A

cos 2A t

t