A Quick Review of Cosmology: The Geometry of Space, Dark Matter, and the Formation of Structure. Goals D escribe the structure, mass-energy content and the evolution of the universe as a whole. .... a nd how we know this. I nterpret observations of objects „at cosmological distances“.
Literature: J. Peacock, Cosmological Physics; P. Schneider, Extragalactic Astronomy & Cosmology; Mo, van den Bosch & White, Galaxy Formation
a) Homogeneity: Averaged over sufficiently large scales ( 1000 Mpc), the universe is approximately homogeneous and isotropic ("cosmological principle").
b) Hubble Law: The universe is expanding so that the distance (to defined precisely later) between any two widely separated points increases as: dl/dt = H(t) xl
c) GR: Expansion dynamics of the universe are determined by the mass and energy content (General Relativity).
d) Big Bang: universe had early hot and dense state
e) Structure Formation: On small scales ( 100 Mpc), a great deal of structure has formed, mostly through "gravitational self-organization": stars, galaxy clusters.
Starting point: What is the universe expanding into?
2.1. The Robertson Walker Metric
= Robertson-Walker Metric (RW-M)
R = present-day curvature
r = comoving radial coordinates
a(t) = expansion or scale factor
NB: a(t) subsumes all time dependence that is compatible with the cosmological principle.
Demanding isotropy and homogeneity, the time dependent solution family to Einstein‘s field equation is quite simple:
with , R = (H0 R)-2, H0 = const.
a=1/(1+z): z is the „redshift“, with lobs = (1+z) lemm
and mass_and_radiation + R + = 1.
Note: this Equation links the expansion history a(t) to the mass energy content of the Universe
mass ~ a-3radiation~ a-4L ~ a0
“equiv. Omega curvature”
=invariant under expansion
Angular size of objects ~ constant 0.5<z<5 !!
qgalaxy ~ 0.5”
= phys. size of object /
observed angular size
*with discrete sources removed
When did recombination occur, or what is the redshift of the CMB radiation?
Tnow 3 K
Fluctuations strongest at harmonic “peak” scales
Can we explain quantitatively the observed "structure" (galaxy clusters, superclusters, their abundance and spatial distribution, and the Lyman- forest) as arising from small fluctuations in the nearly homogeneous early universe?
Growth from fluctuations/ ~ 10-5 to / 1unity, worked out by Jeans (1910) and Lifshitz (1946).
But: We (humans) are overdense by a factor of 1028!
Galaxies are overdense by a factor of 100 – 1000.
We need to work out the rate of growth of / as a function of a(t) [only depends on a(t)!]
To study the non-linear phase, we will look at
Simple analytic approximations (Press-Schechter)
4.1. Linear Theory of Fluctuation Growth
We start with the continuity equation and neglect radiation and any pressure forces for now:
and the equation of motion:
pis the derivative with respect to the proper (not co-moving) coordinate.
In addition, we have Poisson's Equation:
Derivation of (simplified) linear structure growth
= comoving position; rp = proper position
vp = proper velocity = comoving (peculiar) velocity =
with , accounting for the Hubble expansion Note that :
a) temporal derivatives a coordinate frame that ex-pands with the Hubble flow, we consider these equations in
taken in the co moving coordinates
b) spatial derivative
and assuming and are small
this is a continuity equation for perturbations!
where and is the peculiar velocity
Define the a coordinate frame that ex-pands with the Hubble flow, we consider these equations in potential perturbation, f(x,t), through
differs by a²
perturbative Poisson‘s Equation in co-moving coordinates
Similar operations for the equation of motion
in co-moving coordinates!
Note: because velocities are assumed to be small, the term has been dropped on the left.
This equation describes the evolution of the fractional density contrast in an expanding universe!
Mapping from early to late fluctuations = f(a(t))!
Note: at z>>1 almost all cosmologies have m ~ 1
a,b > 0 yields:
A = growing mode; B = decaying mode (uninteresting)
Fractional fluctuations grow linearly with the overall expansion!
constant with time, i.e. all perturbations are „frozen in“
Fractional density contrast decreases (in linear approximation)
for Mass 1.
In the pressure-less limit the growth rate is independentof the spatial structure.
1) d(z)=d(z=0)/Dlin(z) linear growth factor Dlin
2) d~a(t)~1/(1+z), or slower
0.6% in stars
For these cosmological parameters + GR
one can explain …
.. the degree of structure seen at very different scales at very different epochs
See also Spergel et al 2007 (WMAP 3yr data)
x4.2. Structure growth beyond linear perturbations:The ‘top-hat model’ (spherical collapse)
di = r(ti)/rb(ti)-1
M = rb(4pri3/3)(1+ di)
from the Friedman Equation
= 0 at
dc(in linear approx.) =1.69
e.g. Padmanabhan p. 282
p(d>dc | R) = ½ [1-erf(dc/(21/2s(R))]
dn/dM (M, z)
Analytic approx. (R. Somerville)
Numerical simulations (V. Springel)
Mmtiny galaxies today