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
present epoch Hubble constant
“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
All-sky map at radio wavelengths*
*with discrete sources removed
B. (Some) Physics of the Microwave Background
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 / ~ 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
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 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.
As for the acoustic waves, these equations can be combined to:
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!
(2) low-density universe
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)
di = r(ti)/rb(ti)-1
M = rb(4pri3/3)(1+ di)
What is the Equation of Motion for a ‘spherical top hat’?
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