Unified Superfluid Dark Sector
441
ρ =
1
2
(m 1 + m 2 )n
ρ +
+
1
2
ΔE
P 1 ,X 1 − P 2 ,X 2
ρ −
+V (ΔE t) .
(4)
The adiabatic sound speed of each species, governing the growth of perturbations,
is c 2
s i = P i ,X i /(m i P i ,X i X i ), where P i ,X i X i ≥ 0 to ensure the c 2
s i > 0.
With that, the Friedmann equations for a spatially flat universe can be written, in
the non-relativistic approximation as:
3H
2 M
2
Pl = ρ + + ρ − + V (ΔE t) ,
˙
H M
2
Pl −(1/2) (ρ + + ρ − ) .
(5)
The energy density ρ + redshifts like matter and represents the conservation of
number density of DM particles, ρ − evolves under the influence of the potential
and the potential term evolves as dark energy. In the case n = 2, the BEC DM,
P (X i ) = Λ 4
i
X 2
i
m 2
i
, the “+” can be thought as the energy density for the sum of the
phases and “−” for the difference. This can be recast in the canonical variables 3
representing the two states of the superfluid: ξ = (1/N)(N 2
1 θ 1 + N 2
2 θ 2 ) and
χ = (N 1 N 2 /N)(θ 1 − θ 2 ), with N i = Λ 2
i /m i and N =
N 2
1 + N 2
2 .
The Friedmann equations can be combined leading to a universal equation for
the Hubble parameter:
2 ˙
H + 3H
2
= V (ΔE t)/M
2
Pl .
(6)
From this equation, we can see that during matter domination the potential
is not important and the density redshifts as matter; for late times, when the
potential becomes dominant, so to ensure that the slow-roll approximation holds
for the acceleration we need ΔE/2H 0 λ1, which implies, for n = 2, that Λ
M pl (m 1 + m 2 )/2 and that the decay constant, the scale of the spontaneous
symmetry breaking f χ , is super-Planckian as in the case of pNGB models.
We can see in Fig. 1 the evolution of the unified model in comparison to
the concordance model, choosing M 4 = 2M 2
pl H 2
0 ∼ meV
4 , in order to have
acceleration today; and ΔE/2H 0 = 0.2. Our model evolves like ΛCDM until times
close to today, describing the matter era and the late-time acceleration, evolving in
a distinct way in the future.
4 Growth of Density Inhomogeneities
A viable alternative to the ΛCDM model must not only reproduce the evolution of
the background, but it should be able to describe the growth of density perturbations
3 Coming from the diagonalization of the Lagrangian at leading order in ΔE/m i λ1.
441
ρ =
1
2
(m 1 + m 2 )n
ρ +
+
1
2
ΔE
P 1 ,X 1 − P 2 ,X 2
ρ −
+V (ΔE t) .
(4)
The adiabatic sound speed of each species, governing the growth of perturbations,
is c 2
s i = P i ,X i /(m i P i ,X i X i ), where P i ,X i X i ≥ 0 to ensure the c 2
s i > 0.
With that, the Friedmann equations for a spatially flat universe can be written, in
the non-relativistic approximation as:
3H
2 M
2
Pl = ρ + + ρ − + V (ΔE t) ,
˙
H M
2
Pl −(1/2) (ρ + + ρ − ) .
(5)
The energy density ρ + redshifts like matter and represents the conservation of
number density of DM particles, ρ − evolves under the influence of the potential
and the potential term evolves as dark energy. In the case n = 2, the BEC DM,
P (X i ) = Λ 4
i
X 2
i
m 2
i
, the “+” can be thought as the energy density for the sum of the
phases and “−” for the difference. This can be recast in the canonical variables 3
representing the two states of the superfluid: ξ = (1/N)(N 2
1 θ 1 + N 2
2 θ 2 ) and
χ = (N 1 N 2 /N)(θ 1 − θ 2 ), with N i = Λ 2
i /m i and N =
N 2
1 + N 2
2 .
The Friedmann equations can be combined leading to a universal equation for
the Hubble parameter:
2 ˙
H + 3H
2
= V (ΔE t)/M
2
Pl .
(6)
From this equation, we can see that during matter domination the potential
is not important and the density redshifts as matter; for late times, when the
potential becomes dominant, so to ensure that the slow-roll approximation holds
for the acceleration we need ΔE/2H 0 λ1, which implies, for n = 2, that Λ
M pl (m 1 + m 2 )/2 and that the decay constant, the scale of the spontaneous
symmetry breaking f χ , is super-Planckian as in the case of pNGB models.
We can see in Fig. 1 the evolution of the unified model in comparison to
the concordance model, choosing M 4 = 2M 2
pl H 2
0 ∼ meV
4 , in order to have
acceleration today; and ΔE/2H 0 = 0.2. Our model evolves like ΛCDM until times
close to today, describing the matter era and the late-time acceleration, evolving in
a distinct way in the future.
4 Growth of Density Inhomogeneities
A viable alternative to the ΛCDM model must not only reproduce the evolution of
the background, but it should be able to describe the growth of density perturbations
3 Coming from the diagonalization of the Lagrangian at leading order in ΔE/m i λ1.