Cosmology and Dark Matter I: Einstein & the Big Bang

by Jerry Sellwood

The Expanding Universe

• Hubble discovered that galaxies move away at a rate proportional to their distance

v = H0 d

where H0 is known as “Hubble’s constant”

• HST key project measured

H0 = 72 8 km/s/kpc

Our position is not special

• Every other observer would see the same

• The entire universe is expanding and there is no center

• All galaxies were closer together in the past– at time 1 / H0 distances were zero

Origin of the redshift

• Galaxies are not really moving– the space between them is expanding

• Photons from distant galaxies were emitted when the universe was smaller and have been stretched by the subsequent expansion of the universe

• Distances in an expanding universe scale everywhere with time by a scale factor a(t)– We choose a(t0) = 1

– and the expansion rate today is da/dt = H0 a

• If the redshift is z, then a = 1 / ( 1 + z )

General Relativity

• Principle of equivalence– A gravitational field and an accelerating reference

frame are indistinguishable

• Implies light must curve in a gravitational field

• But light always travels by the shortest path

• So mass causes space to become curved

• A uniformly dense universe must be uniformly curved

Is the universe uniform?• We will assume the universe is

– (a) homogeneous– (b) isotropic

• At least on very large scales• Supporting evidence:

– Radio galaxies Baleisis et al (1998)

• The distribution of radio sources brighter than 70mJy from combined surveys of the N & S Aitoff projection (equal area) in RA & dec

3-D distribution• SDSS redshift survey

– Every dot is a galaxy in a slice (incomplete)

– fewer distant galaxies because they are fainter

• Clustering hierarchy reveals big density variations

• Averages over very large volumes seem constant

Curved Spacetime

• A uniform & isotropic universe must be one of 3 possible curved spacetimes– positively curved

(4-sphere)– negatively

curved (4-saddle)– flat

Friedmann’s equation• In a uniform universe, Einstein’s equations

can be reduced to a single differential equation

• where is the energy density (matter + radiation), is a possible “cosmological constant”, – R is the radius of curvature, and

= 1 for an open universe = –1 for a closed universe = 0 for a flat universe

• Note that the geometry cannot change

Newton & Einstein

• Both men worried:– if gravity is always attractive– why hasn’t the universe collected into one glob?

• Einstein added the repulsive term to his field equations in order to allow a static universe

• Expanding universe discovered later– declared his cosmological constant “my greatest

blunder”

Critical density

• Neglect the term for the time being• If H2 = 8G/3, the equation requires = 0

– a flat universe

• Thus if crit = 3H2 / 8G– space is negatively curved ( = 1)– the universe is open and expands forever

• whereas if crit – space is positively curved ( = 1)– the universe is closed and will eventually recollapse

Equations of state

• The energy density decreases as the universe expands.

• Two cases:– non-relativistic matter: a-3 = (1+z)3

– relativistic matter + radiation: a-4 = (1+z)4

• Also curvature term varies as (1+z)2

• So at high redshift, the energy term is always the largest

The Hot Big Bang• The universe

expands adiabatically

• Photons lose energy because of the redshift

• t0 approx 13.70.2 Gyr 4.31018s

Nucleosynthesis I

• There was a very slight excess of baryons over anti-baryons

• Annihilations left a vast excess of photons over particles

• After 0.5s, lose thermal equilibrium between neutrons and protons

• Neutrons start to decay

Nucleosynthesis II• A few seconds later,

photons are no longer able to dissociate deuterium

• In next three minutes almost all available neutrons are absorbed into 4He nuclei

• Reactions stop as the temperature and density continue to drop

Nucleosynthesis III

• Final abundances depend on:– the photon to baryon ratio– the expansion rate, which places a limit on the

number of relativistic species (e.g. neutrinos)

Tests of BBN• Hard to find

unpolluted primordial material that is accessible spectroscopically

• Best work is by primordial 2H abundance by Tytler – in absorption lines in QSO spectra

Baryon fraction

• 2H, 4He, 3He & 7Li abundances must all be consistent with one value of the ratio of photons to baryons

• Know the number of CMB photons

• Therefore baryon fraction is 4% of the closure density today

Matter-radiation equality

• Next milestone

• Very little happens

• Expansion rate changes

Formation of the CMB• For 300,000 years, the intense radiation

keeps hydrogen ionizedH+ + e H +

• Constant scattering of photons – maintains thermal equilibrium with matter – makes the universe opaque

• But as photons are redshifted, atoms are quite suddenly able to survive

• The universe quickly becomes neutral and transparent

Cosmic Microwave Background

• Universe is filled with the relic radiation from the big bang

• It is redshifted now to T=2.725 K

• Spectrum is a near perfect black body with a peak in the microwave

Re-ionization

• Diffuse gas in the universe did not stay neutral for long

• First stars and/or quasars emitted enough UV radiation to ionize all the diffuse gas

• Density is far too low by that time for the ionized gas to be opaque