A Newtonian Approach to the Cosmological Dark Fluids - Selected Topics of Computational and Experimental Fluid Mechanics - page 504

= ( ) ˙ =˙ ( ) +

and the peculiar velocity v v ; that is, v is the velocity

of the fluid with respect to the background comoving ( Hubble -)flow. By the chain
rule, the derivatives transform as

a t x , such that u a t x

a − 1

∇ r ₒ ∇ x and (∂/∂ t ) r ₒ (∂/∂ t ) x − H x ·∇ x .

(In what follows we will omit the subindex x from the spatial gradients and ∂/∂ t

should be understood as being taken at fixed x .)
We now consider perturbations to the quantities ˁ , P and ʦ ,

ˁ( x , t ) = ˁ( t )( 1 + ʴ( x , t )) (13)

c s ʴˁ + ˃ʴ

ʴ P = S (14)

) = ʦ(

ʦ( x , t t ) + ˆ( x , t ) (15)

where a bar denotes background quantities that only depend on the time coordinate.
We introduced also c s = (∂ P /∂ˁ) S , the squared adiabatic sound speed and ˃ ≡

(∂ .Interms

of the perturbed variables, the continuity, Euler and Poisson equation become

P /∂ S ) ˁ . Note also that the perturbation to the energy density is ʴˁ = ˁʴ

a ∇· (

v =

∂ʴ
∂

1

t +

1 + ʴ) 0 , (16)

∂ ∇ ʴ

v 1

a (

1
a ∇ ˆ −

P

t +

+ ·∇ ) =−

+ ʴ) ,

H v v v (17)

∂ ˁ(

a 1

and

2

Ga 2 ∇ ˆ = 4 ˀ ˁʴ. (18)

The first two equations are quadratic in the perturbed variables, therefore, in the
following we treat them as small and linearize Eqs. ( 16 ) and ( 17 ) to obtain

∂ʴ
∂

1
a ∇· t +

v = 0 , (19)

∂ v 1

a ∇ ˆ +

1
a ∇

c s ʴ =

t +

H v + 0 . (20)

∂

Note that in the last equation we used Eq. ( 14 ) and considered adiabatic perturbations
only. By appealing the conservation of angular moment in an expanding universe, it is
expected that the divergence-free piece of the peculiar velocity should decays quickly
with time. This can be easily seen by taking the rotational of Eq. ( 20 ), arriving at
∇×

a − 1 , whichmeans that in the absence of sources of vector perturbations these
modes are not relevant in first order perturbed cosmology, allowing us to consider
only the curl-free piece of the velocity in the following discussion; moreover, any
initial large vector perturbation would break the isotropy of the background, and thus
it is not compatible with the Cosmological Principle.
Now, we are in position to give a closed linear second order equation for the
density contrast

v ∝

ʴ , taking the partial time derivative of Eq. ( 19 ) and using Eqs. ( 18 )