Introduction to Cosmology

224 Cosmic Structures

in the last scattering. The angular scale subtended by progenitors corresponding to

the largest cosmic structures known, of size perhaps 200ℎ−^1 Mpc, is of the order of

3 ∘, corresponding to CMB multipoles around퓁=20.

Viscous Fluid Approximation. The common approach to the physics of matter in

the Universe is by the hydrodynamics of a viscous, nonstatic fluid. With this nonrel-

ativistic (Newtonian) treatment and linear perturbation theory we can extract much

of the essential physics while avoiding the necessity of solving the full equations of

general relativity. In such a fluid there naturally appear random fluctuations around

the mean density휌(푡), manifested by compressions in some regions and rarefactions

in other regions. An ordinary fluid is dominated by the material pressure but, in the

fluid of our Universe, three effects are competing: radiation pressure, gravitational

attraction and density dilution due to the Hubble flow. This makes the physics differ-

ent from ordinary hydrodynamics: regions of overdensity are gravitationally amplified

and may, if time permits, grow into large inhomogeneities, depleting adjacent regions

of underdensity.

The nonrelativistic dynamics of a compressible fluid under gravity is described

by three differential equations, theEulerian equations. Let us denote the density of

the fluid by휌, the pressure푝, and the velocity field풗, and use comoving coordi-

nates, thus following the time evolution of a given volume of space. The first equation

describes the conservation of mass: what flows out in unit time corresponds to the

same decrease of matter in unit space. This is written

d휌

d푡

=−휌∇⋅풗. (10.1)

Next we have the equation of motion of the volume element under consideration,

d풗

d푡

=−

1

휌

∇푝−∇휙, (10.2)

where휙is the gravitational potential obeying Poisson’s equation, which we met in
Equation (3.35),
∇^2 휙= 4 휋퐺휌. (10.3)
Equation (10.2) shows that the velocity field changes when it encounters pressure
gradients or gravity gradients.
The description in terms of the Eulerian equations is entirely classical and the grav-
itational potential is Newtonian. The Hubble flow is entered as a perturbation to the
zeroth-order solutions with infinitesimal increments훿풗,훿휌,훿푝and훿휙. Let us denote
the local density휌(r,푡)at comoving spatial coordinaterand world time푡. Then the
fractional departure atrfrom the spatial mean density휌(푡)is the dimensionlessmass
density contrast
훿m(r,푡)=

휌m(r,푡)−휌푚(푡)

휌푚(푡)

. (10.4)

The solution to Equations (10.1)–(10.3) can then be sought in the form of waves,

훿m(r,푡)∝ei(k⋅r−휔푡), (10.5)