Big Bang Cosmology
Big Bang vs. Steady State
Perfect cosmological principle: universe is unchanging in space
and time => Steady-State universe - Bondi, Hoyle, Gold.
True? No!
• Hubble’s Law => expansion means no steady state, unless matter
continually created to preserve density
• preference of AGN/quasars for large distances (early times)
• cosmic microwave background - consistent with Big Bang (BB)
• predominance of light elements (e.g., H, He) consistent with early
hot universe
• Olbers’ paradox (why is night sky dark?) resolved with BB model
Cosmology
Cosmological Principle: At any instant in time, universe is
homogeneous (same at all locations) and isotropic (same in all
directions), i.e., the universe looks the same to all observers.
This despite superclustering on scales up to ~ 100 Mpc;
distribution apparently smoother on larger scales.
It turns out that the cosmological principle is completely
consistent with the Hubble expansion.
Cosmology
If vBA=H0rBA and v CA=H0rCA ,
then vBC= vBA- v CA = H0(rBA- rCA)
= H0rBC.
∴
Hubble’s Law applies to every galaxy if it applies to just one.
Cosmological principle <=> Hubble’s Law
where H0 can be positive, negative, or zero.
Expansion of the Universe
Cosmological principle => no boundary. How to understand this?
Answer: General relativity introduces the idea of space-time
curvature.
Curvature allows us to envision a boundary-free universe that is
not infinite. A 3-D universe curved into a 4th dimension.
Make analogy to a 2-D universe on surface of a 3-D sphere. If the
sphere expands in 3-D, the 2-D surface area expands. A 2-D
observer on the surface infers Hubble’s Law.
Expansion and subsequent
contraction of a hypothetical 2-D
universe on the surface of a sphere.
Expansion of the Universe
In GR cosmology, 3-D space itself expands. All lengths, e.g.,
distance between galaxies, wavelength of light, etc. expand with
the universe. However, this expansion can be opposed locally by
various forces.
Therefore, expansion of space itself is the real explanation for
cosmological redshifts z, not the Doppler effect.
General Relativity
Gravity understood in a new light. Compare with old view.
Newton: Gravitational force causes matter to accelerate. Matter
exerts gravitational force.
Einstein: Gravitational acceleration is due to curved space-time.
Curvature of space-time due to mass-energy (recall E = mc2).
Geometry of Space and Time
Newton: Euclidean geometry where
ds 2 = dx 2 + dy 2 + dz 2 is invariant, i.e., absolute space (and time).
Einstein (SR): No absolute space or time, but
ds2 = c 2 dt 2 − (dx2 + dy 2 + dz 2 )
is invariant.
Einstein (GR): Curvature of space time due to mass-energy yields
4
ds = ∑
2
µ =1
4
µ
ν
g
dx
dx
∑ µν
where dx1,2,3 = dx, dy, dz , and dx 4 = cdt .
ν =1
gµν is a tensor containing information about curvature.
Specifically, where no curvature, get back to SR limit
g µν = 0 for µ ≠ ν and g11 = g 22 = g 33 = −1, g 44 = 1.
Some Effects of Space-Time Curvature
• deflection of light around massive object, e.g., “gravitational
lensing”
• Euclidean geometry not valid on large scale
• Large-scale structure and evolution of universe affected by
curvature
Expansion of Curved Universe
Theory of General Relativity yields an equation for radius of
curvature R,
1 & 2 2GM
1 2
R −
=− c
2
3π R
2
Solutions of this equation, R(t), yield evolution of the universe.
Our measurements of H0 yield current value of R& / R.
Expansion of the Universe
Follow a simpler Newtonian model.
Imagine expansion of a spherical region of radius R(t).
&& = − GM ( R) .
F = ma ⇒ R
R2
Multiply by R& .
GM ( R) &
d  1 & 2  d  GM ( R) 
&
&
&
RR = −
R ⇒
 R  + −
=0
2
R
dt  2  dt 
R 
1 & 2 GM ( R )
⇒
R −
= E = constant,
2
R
i.e., conservation of energy.
Expansion of the Universe
Does the universe expand forever?
Analogy to earlier escape velocity calculation.
Universe unbound (open) if
KE > |PE|, i.e., E > 0.
marginal (flat) if KE = |PE|, i.e., E = 0.
bound (closed) if KE < |PE|, i.e., E < 0.
Rewrite in terms of Hubble constant and density:
v = R& = HR, M ( R) = 4 3π R 3 ρ , and note that ρ = ρ (t ), H = H (t ).
Open universe =>
ρ < ρ crit ≡ 3H 2 8π G.
Flat universe =>
ρ = ρ crit ≡ 3H 2 8π G.
Closed universe =>
ρ > ρ crit ≡ 3H 2 8π G.
At current epoch (t = t0), we measure ρ 0 , H 0 .
Expansion of the Universe
Key question in cosmology:
8π G
8π G
What is the value of Ω = ρ ρ crit = ρ
= ρ0
?
2
2
3H
3H 0
Important parameters
(1) Ω = ρ
ρ crit
Current status of Ω:
R&
(2) expansion parameter H (t ) = , currently H 0
R
R&&R 1 ρ
(3) deceleration parameter q(t ) = − & 2 =
, currently q0
R
2 ρcrit
Observed luminous matter
Ω << 1.
Observed matter and inferred dark matter
Ω ≤ 0.2.
Theory of early universe
Ω = 1.
Age of the Universe
Earlier, we argued t0 < H0-1 if universe decelerating.
Solve expansion equations for Ω = 1 => find t0 = 2/3 H0-1.
∴
open universe
Ω < 1 ⇒ 2 3 H 0−1 < t0 < H 0−1
flat universe
Ω = 1 ⇒ t0 = 2 3 H 0−1
closed universe Ω > 1 ⇒ 0 < t < 2 3 H −1
0
0
Ultimate fate?
If Ω ≤ 1, expansion continues; all stars eventually die, > 1012 yr;
universe becomes dark.
If Ω > 1, recontraction of universe; followed by rebound?
Olbers’ Paradox
In an infinite static universe, every line of
sight eventually intercepts a star => night
sky is everywhere bright!
Resolution in Big Bang model:
Finite age => can’t see beyond a distance
r = c∆t.
Also, light from within this distance is
increasingly redshifted as we approach the
edge, the “cosmic event horizon”.
Light Elements
Can trace expansion back to an early hot dense state.
At high energies, particles exist in an unbound state.
As universe expands and cools, synthesis of elements, then atoms.
Given presence of protons (1H) and neutrons, light elements 2H, 3He,
4He, 6Li, 7Li produced in early universe - these elements are also not
produced efficiently in stars.
Cosmic abundances:
75% H, 25% He, trace Li, Be are all explained by Big Bang model.
High H, He content implies a high temperature past, since such
matter prefers less binding energy, more light elements.
Cosmic Microwave Background
Back in time, at z ~ 103 (when T ~ 3000 K), electrons and protons
combine to form H atoms => matter is no longer opaque to radiation,
since free electrons were good at absorbing photons.
Blackbody radiation from this epoch flies out unhindered by matter.
Should see this relic radiation, but redshifted so that
λmax, 0
λmax
=
T
= 1 + z ≈ 10 3
T0
T
⇒ T0 =
≈ 3 K.
1+ z
First observed by Penzias & Wilson
(1965). Newer data from COBE
satellite (1992).
Note: we can see galaxies/quasars back to z < 5, but CMB comes
from z ~ 103! Cannot “see” any further back.
Cosmic Microwave Background
COBE’s measurement of the
CMB spectrum.
T = 2.726 ± 0.005 K.
COBE all-sky map of
CMB. See fluctuations
∆T
~ 10 −5
T
which could have lead to
supercluster structure.
Seeing Through the Distance
Extrapolating to Earliest Phases
Before z ~ 103, guided only by theory. However, cannot go back
arbitrarily far.
Limit of current knowledge:
Gm
∆L ~ 2 .
c
h
h
=
.
Quantum Mechanics => can’t observe within ∆L <
∆p mc
Equate two ∆L’s.
Gravity => can’t detect events within
1/ 2
photons
GM
h
 hc  Planck mass, a combination of 3
~
⇒ m = mp =   ,
2
c
mc
 G  fundamental constants.
Gm p  Gh 1 / 2
h
Also, L p =
= 2 =  3  , Planck length.
m pc
c
c 
tp =
1/ 2
Planck time. Plug in #’s =>
 Gh 
= 5  ,
c c 
t p = 1.35 × 10−43 s. Can’t describe t ≤ t p .
Lp
Big Bang Model
Expansion from highly condensed initial state. Theory combines
general relativity and particle physics.
Four fundamental forces:
strong nuclear - weak nuclear - electromagnetic - gravity
electroweak at high energies
combines at even higher energies
?
Nucleons composed of quarks. Quarks composed of …? All
particles have corresponding antiparticles.
Big Bang Model
Brief history:
time
10 −43 s
Planck time tp. Don’t know what precedes this. Need a
quantum theory of gravity.
10 −35 s
Strong nuclear force decouples from electroweak. Inflation
begins - rapid exponential growth. Most quarks and
antiquarks annihilate. Small asymmetry => some quarks
remain. Baryon (made of quarks) to photon ratio 10-9-10-10.
10 −32 s
Inflation ends. Observable universe went from 10-23 cm to
10 cm.
10 −12 s
Weak and electromagnetic force separate.
10 −6 s
Nucleons form.
Big Bang Model
Brief history:
time
10 2 -103 s Cosmic nucleosynthesis - light nuclei form, e.g., He, Li.
1013 s
(z ~ 103) electrons and protons combine => atoms form.
Photons now able to stream freely.
1016 s
Galaxies, stars, planets begin to form.
1016 s
The present.
10 40 s
Protons decay (perhaps). Atomic matter ceases to exist.
Universe heads toward darkness/heat death.
Last Word
Observed luminous matter
Dark matter
Inflation theory
Ω < 0.01.
Ω ≈ 0 .2 .
Ω = 1.
Where is the rest of the mass-energy?
(1) In matter? Nucleons (e.g., brown dwarfs, white dwarfs, planets,
rocks) can only account for up to Ω = 0.2, according to cosmic
nucleosynthesis arguments. So look for exotic particles: massive
ν’s, axions, supersymmetric particles.
(2) In energy? Dark energy may make up the required deficit. An
unknown energy so that Ω eff= Ω + Λ = 1, where Λ is the
cosmological constant (dark energy term) that makes the Hubble
expansion accelerate at the current epoch! Recent evidence supports
this hypothesis.