Scalar dispersion forces, BE condensation and scalar dark

matter

J A Grifols, UAB

Valparaíso, December 11-15, 2006

Even with no permanent dipole moments,

2-scalar-exchange forces

Spin 0 particles are natural ingredients of the SM and its completions (e.g. Higgs bosons, squarks and sleptons, majorons, …).

A A

B B

)],()(),()([21

),( 000022

TknkTknkiimk

TkDF

In finite temperature QFT, this is described with the propagator (real time formalism)

Matter in a Bose gas (F. Ferrer, JAG)

1

1),(

/)( Te

Tn

sum over real particles in the heat bath

The chemical potential μ(T) is determined from

with volume and temperature fixed, and

∑↔

But in a Bose- Einstein gas this replacement is allowed

only for (condensation temperature)

For , a large macroscopic fraction of the charge resides in the lowest energy state; however, the density of states in the continous representation of the sum over states gives zero weight to the zero mode.

k

nnQ )(

3

3

)2( kd

V

cTT

cTT

22 2/ Vk

mrTT e

r

TgV

c

222

2

64

Let us relate the k=0 piece of the amplitude (hence, of the potential) with the ground state charge density.

Since

we have

(for the chemical potential we used

i.e. in the thermodynamic limit)

0002)()(

kkk

nnnnn

1

121)(

/2

2

0

Tmc

k eT

TQnn

)/( QTOm

m

For a macroscopically large total net charge (much larger than T/m),

hence,

Next, Fourier transform to get the contribution of the condensate to the potential

0

2

01)( Q

T

TQnn

ck

222

2

/1)( ccond TTTrM

grV

c

The thermal part, i.e. the excited states, contribute terms that decay faster than (for T>m) to the potential .

So, finally

The bottom line is:

(short range) (long range)

above B-E condensation below B-E condensation

mre 2

mr

ccTT e

rT

TOTTT

rM

grV

cc

22

222

2

/1)(

mrer

22

1 r

1

2 applications

• In a cosmological setting

• In an astrophysical setting

COSMOLOGY

In our model, light scalars φ constitute the bulk of dark matter and while condensed they are a pressureless fluid and thus constitute CDM.

Suppose they couple to baryons

Exchange of 2 scalars producing a force among baryons

Influence on the CMB

The new long-range interaction affects the tightly coupled photon-baryon fluid. The “mechanical” response to collapse that gives rise to “acoustic” oscillations of sub-horizon sized scales is altered and thus may induce changes in the CMB power spectrum.

Indeed, the effect can be described as a slow down of the “sound” velocity and a corresponding shortening of the sound horizon s at recombination. The acoustic peaks in the CMB correspond to modes that verify

Since multipole moments , changes in s imply changes in l

nsk recn kl

rec

rec

n

n

s

s

l

l

Thus the model produces a shift of the peak locations (towards higher l’s).

If we set (which provides an order of magnitude of present and planned experiments’ ability to reconstruct the first few peaks), we have checked that such variations can be achieved in a model with light scalars of mass ~0.1 eV and effective coupling to baryons g ~ . Furthermore, the resulting model is consistent with the Cosmological Standard Model. In particular, with a DM density of ~30%.

%1/ ll

810

ASTROPHYSICS

We hypothesize that galactic DM is composed (all or in part) by (light) scalar particles.

Focus on a region where ordinary matter has accreted and evolved into a white dwarf. Condensed scalars have been gravitationally trapped by the star.

Because the boson gas sits in an external field, there is condensation in space as well (on top of momentum-space condensation (BEC)). All charge drifts to the center of the star (r=0). Since momentum p and position r cannot be simultaneously sharp (p=r=0), the condensate occupies a finite region of the star. Exactly what fraction of the star depends on the depth and width of the gravitational well and on the scalar mass.

0Q

In the stellar region permeated with the condensate, the pull of our putative long-range 2-scalar force adds to the pull of gravity.

The star attains its equilibrium configuration when the Fermi pressure of the electrons balances the pull of both.

We have solved numerically the hydrostatic equilibrium equations (with a Fermi pressure of a degenerate ideal gas of UR electrons) as a function of strength of the dispersion force and size of the condensed system.

From our analysis, stars fall into two distinct branches:

i) stars with a small core filled with the condensate. Their mass is close to Chandrasekhar’s but their radius is large.

ii) stars with a large fraction of their volume filled with the scalar condensate at the expense of a dramatic reduction in mass and radius.

The relevant parameters that were used in our analysis are the following

Mass of scalars

Effective NNφφ coupling

Critical temperature

These parameter ranges do not conflict with particle physics nor cosmology/astrophysics phenomenology.

eVOm )1010( 17 )1010( 1721 Og

eVOTc )1(

i)

White Dwarf data

Exotic White Dwarfs

Ordinary White Dwarfs

SUMMARY

• Dispersion forces mediated by scalars extend well beyond their Compton wavelength when matter is embedded in a Bose-Einstein condensate composed of those scalars.

• We presented 2 scenarios where this phenomenon could be of physical import.