270 Gravitational instability in Newtonian theory
The Jeans lengthλJ∼cstgris the “sound communication” scale over which
the pressure can still react to changes in the energy density due to gravitational
instability. Gravitational instability is very efficient in a static universe. Even if the
adiabatic perturbation is initially extremely small, say 10−^100 , gravity needs only a
short timet∼ 230 tgrto amplify it to order unity.
Problem 6.1Find and analyze the expression forδvkandδφkfor sound waves and
for the perturbations on scales larger than the Jeans wavelength.
6.2.2 Vector perturbations
The trivial solution of (6.20) withδε=0 andδS=0 can correspond to a nontrivial
solution of the complete system of the hydrodynamical equations. In this case, in
fact, (6.15)–(6.18) reduce to
∇δv= 0 ,∂δv
∂t= 0. (6.28)
From the second equation it follows thatδvcan be an arbitrary time-independent
function of the spatial coordinates,δv=δv(x). The first equation tells us that for
the plane wave perturbation,δv=wkexp(ikx), the velocity is perpendicular to the
wave vectork:
wk·k= 0. (6.29)Thesevector perturbationsdescribe shear motions of the media which do not dis-
turb the energy density. Because there are two independent directions perpendicular
tok,there exist two independent vector modes for a givenk.
6.2.3 Entropy perturbations
In the presence of entropy inhomogeneities (δS=0), the Fourier transform of
(6.20) is
δ ̈εk+(
k^2 c^2 s− 4 πGε 0)
δεk=−σk^2 δSk. (6.30)The general solution of this equation can be written as the sum of its particular
solution and a general solution of the homogeneous equation withδSk=0. The
particular time-independent solution of (6.30),
δεk=−
σk^2 δSk
(
k^2 c^2 s− 4 πGε 0), (6.31)
is called theentropy perturbation.Note, that in the short distance limitk→∞,
when gravity is unimportant,δεk→−σδSk/c^2 s.In this case the contribution to