Unified Superfluid Dark Sector
439
2 Review of Dark Matter Superfluid
Superfluidity is one of the most striking quantum mechanical phenomena on
macroscopic scales. It appears in fluids that when brought to very low temperatures,
form a Bose–Einstein condensate, now described by a single wave-function of
systems coming from the superposition of the de Broglie wavelength of these
bosons. The emergent degree of freedom of this collective system has an emergent
new dynamics: it flows without friction.
We want to use the physics of superfluidity to build a model of DM that on
central region of galaxies DM condenses forming a Bose–Einstein condensation
with a superfluid phase. The necessary conditions for condensation, assuming
weakly coupling, are that de Broglie wavelength λ dB ∼
1
mv must be larger than
the mean inter-particle separation ∼ (m/ρ) 1/3 , and that particle should interact
enough to thermalize. The first condition translates to an upper bound on the mass,
m (ρ/v 3 ) 1/4 , which for a MW-like galaxy (M = 10 12 M ) results in m 4.3 eV.
The second condition requires that the particles interact strongly. An axion-like
particle that obeys these conditions condenses on the central regions of galaxies,
forming a core, which is enveloped by DM particles that are not condensed and
behave like CDM having the usual NFW profile.
After we guaranteed the conditions for DM to condense on galactic scales, we
need to describe the evolution of the superfluid. A superfluid is described by a
weakly self-interacting field theory of a complex field Ψ ∝ ρe iΘ with global U(1)
symmetry. This symmetry is spontaneously broken by the superfluid ground state of
a system at chemical potential μ, so that Θ = mt + θ . At low energy the relevant
degrees of freedom are phonons, which are excitations of the Goldstone boson θ for
the broken symmetry. The effective theory of phonons must be invariant under the
shift symmetry, θ → θ +c, and Galilean symmetry, appropriate for a non-relativistic
superfluid. Therefore, its most general form at leading order in derivatives and zero
temperature is given by:
L phonons = P (X) ;
X = ˙
θ − mΦ − (
∇θ)
2 /2m ,
(1)
where Φ is the gravitational potential. The equation of state of the superfluid is
encoded in the form of P (X), and the phonon sound speed is given by c 2
s =
P ,X
ρ ,X
=
1
m
P ,X
P ,XX
. Superfluids are often described by a polytropic equation of state,
P (X) ∼ X n , corresponding to P (ρ) ∼ ρ
n
n−1 . Written in this form, we can describe
a standard weakly coupled superfluid (BEC DM), for n = 2; for n = 5/2 this
effective theory describes the Unitary Fermi Gas, a gas of ultra-cold fermionic atoms
tuned at unitary.
In the case of the DM superfluid, since we want to reproduce MOND on galactic
scales, this corresponds to n = 3/2, which gives the expected equation of state for
MOND, P ∼ n 3 . One extra ingredient is necessary in order to mediate the MOND
force, is that the phonons couple to the baryon mas density. The action for the DM
superfluid is given by L DM = P DM (X) + L int where
439
2 Review of Dark Matter Superfluid
Superfluidity is one of the most striking quantum mechanical phenomena on
macroscopic scales. It appears in fluids that when brought to very low temperatures,
form a Bose–Einstein condensate, now described by a single wave-function of
systems coming from the superposition of the de Broglie wavelength of these
bosons. The emergent degree of freedom of this collective system has an emergent
new dynamics: it flows without friction.
We want to use the physics of superfluidity to build a model of DM that on
central region of galaxies DM condenses forming a Bose–Einstein condensation
with a superfluid phase. The necessary conditions for condensation, assuming
weakly coupling, are that de Broglie wavelength λ dB ∼
1
mv must be larger than
the mean inter-particle separation ∼ (m/ρ) 1/3 , and that particle should interact
enough to thermalize. The first condition translates to an upper bound on the mass,
m (ρ/v 3 ) 1/4 , which for a MW-like galaxy (M = 10 12 M ) results in m 4.3 eV.
The second condition requires that the particles interact strongly. An axion-like
particle that obeys these conditions condenses on the central regions of galaxies,
forming a core, which is enveloped by DM particles that are not condensed and
behave like CDM having the usual NFW profile.
After we guaranteed the conditions for DM to condense on galactic scales, we
need to describe the evolution of the superfluid. A superfluid is described by a
weakly self-interacting field theory of a complex field Ψ ∝ ρe iΘ with global U(1)
symmetry. This symmetry is spontaneously broken by the superfluid ground state of
a system at chemical potential μ, so that Θ = mt + θ . At low energy the relevant
degrees of freedom are phonons, which are excitations of the Goldstone boson θ for
the broken symmetry. The effective theory of phonons must be invariant under the
shift symmetry, θ → θ +c, and Galilean symmetry, appropriate for a non-relativistic
superfluid. Therefore, its most general form at leading order in derivatives and zero
temperature is given by:
L phonons = P (X) ;
X = ˙
θ − mΦ − (
∇θ)
2 /2m ,
(1)
where Φ is the gravitational potential. The equation of state of the superfluid is
encoded in the form of P (X), and the phonon sound speed is given by c 2
s =
P ,X
ρ ,X
=
1
m
P ,X
P ,XX
. Superfluids are often described by a polytropic equation of state,
P (X) ∼ X n , corresponding to P (ρ) ∼ ρ
n
n−1 . Written in this form, we can describe
a standard weakly coupled superfluid (BEC DM), for n = 2; for n = 5/2 this
effective theory describes the Unitary Fermi Gas, a gas of ultra-cold fermionic atoms
tuned at unitary.
In the case of the DM superfluid, since we want to reproduce MOND on galactic
scales, this corresponds to n = 3/2, which gives the expected equation of state for
MOND, P ∼ n 3 . One extra ingredient is necessary in order to mediate the MOND
force, is that the phonons couple to the baryon mas density. The action for the DM
superfluid is given by L DM = P DM (X) + L int where