Cosmology
Begin with the simplest physical system, adding complexity only when required.
Towards this end, Einstein introduced the Cosmological Principle : the universe
is homogeneous and isotropic on sufficiently large scales…
The universe looks
pretty much like this
everywhere – “walls”
and “voids” are present
but no larger structures
are seen. Redshift
surveys and CMB
confirm this principle.
It follows that the Universe has
no “edge” or center –
extension of Copernican
Principle
Hubble’s Law (Hubble 1929)
the recessional velocity of external galaxies is linearly related to their distance.
recession velocity = Ho x distance
Derive Hubble’s law from the Cosmological Principle
Consider a small triangle. As the universe expands or contracts, the conditions
of homogeneity and isotropy require that the expansion is identical in all
locations. The triangle must grow self-similarly.
Define the present time at to and the scale factor of the expansion as a(t), with
ao = a(to) being the scale factor at to.
Self-similarity requires that any distance x increase by the same. scale factor.
Where Hubble parameter is
Hº
a
a
Ho is value of the Hubble parameter at to
Note that the Cosmological Principle does not require
H > 0 – can have static or contracting universe.
Dynamics of the Universe - Conservation Laws
and Friedmann Equations
To solve for the dynamics of the universe, we will assume the Cosmological
Principle along with General Relativity. Begin with a Newtonian approximation
to derive the evolution of the universe.
Use Lorentzian transformations
within the Newtonian framework
Approximate a region of the universe as a uniform density sphere of
non-relativistic matter. Use the Eulerian equations (like CBE used
for stars in galaxies!) for conservation of mass and momentum to
derive the dynamical evolution of the universe.
+
If mass is conserved, mass density
satisfies EOC
Time dependence of matter density is determined
solely by the evolution of the scale factor - for a
matter dominated Universe ρ ~ a-3
Apply conservation of momentum using Newton’s theory of gravity (though
difficult to define potential in a uniform unbound medium like the Universe!).
Apply Newton’s laws to a universe which is the interior of large sphere
(doesn’t violate CP badly if we consider only regions where x << Runiverse).
Euler’s equation for momentum conservation
or
where v is the local fluid velocity, p is the pressure, and F is the force (in this
case gravitational) per unit mass. If p gradient is zero (for CP) and x  x = x,
then,
Using
For F we use Poisson’s eq.
And mass
conservation
Universe must be changing
velocity if it contains matter!
Einstein used GR to improve upon Newtonian cosmology and provide a
description of the Universe as a whole. He introduced the cosmological
constant Λ into his GR equations since he found (as we did) that if the density
of the Universe is non-zero, the Universe must be expanding. As this was
before Hubble’s work and most believed in a Steady State Universe, this
constant allowed for a non-zero density and a static Universe.
After Hubble’s discovery of an expanding Universe, Einstein famously called
this his “greatest blunder”.
Start with modification to Poisson’s law,
Positive Λ is repulsive force that counteracts gravity.
Now replace density with ρeff = ρ + 3p/c2  total energy density (rest mass + kinetic E)
Get Friedmann solutions for the scale
factor of the Universe by multiplying by
da/dt and integrating wrt time.
For expanding Universe, gravitational term
dominates when scale factor is small.
At later times, first the curvature term then
later the cosmological constant dominate.
Let’s work through one possible solution to the Friedmann equations with a
simplicity assumption of a null cosmological constant, Λ = 0, and recall
Evaluate the constant k in terms of present day observable quantities.
k = 0 only when density is at the critical density, defined as
Cosmological Density
Parameter
Inserting back into Friedmann equation we get
Case 1:
k = 0 and Ωo= 1
Einstein-deSitter universe
Case 2:
k > 0 and Ωo> 1
Case 3:
k < 0 and Ωo< 1
As scale factor
increases, it eventually
reaches a point where
adot = 0. Expansion
stops at amax.
Energy term is now
positive. Solution for
a(t) is analogous to a
rocket shot with
velocity greater than
escape velocity.
a(t) is proportional to t
The Universe expands at
an ever decreasing rate!
After amax is reached,
Universe starts to
collapse!
Borderline Universe or
Marginally Bound
Closed or Bound
Universe
Expansion continues
forever!
Open or Unbound
Universe
The Fate of the Universe
If Λ = 0, the fate of the
universe (whether it
continues to expand
forever or eventually
collapses) depends on
its density.
High density = lots of
mass = enough matter
to gravitationally halt
expansion and cause
gravitational collapse
Low density = little mass
= not enough
gravitational attraction to
stop expansion…it goes
on forever
If Ho = 70 km/s/Mpc, then ρcrit ≈ 1 x 10-29 g/cm3
(about one H atom in 200 L volume of space)
Cosmology and General Relativity
Einstein’s Principle of Equivalence  space-time is curved in the
presence of a gravitational field (the mass density of the
Universe tells us about the geometry of space-time
Interpret the solutions for the scale factor obtained from
Newtonian theory within the framework of GR (Friedmann
equations can be derived from Einstein’s field equations).
k is positive or o > 1, Bound/Closed Universe
k is negative or o < 1, Open Universe
k = 0 or o = 1, Flat/Marginally Bound/Critical
Universe
Use 2D analogy to visual curvature of space-time
• Detected by measuring the sum of the angles of any triangle
• Locally, space may appear flat (Euclidean)
• Define a metric along the surface
A metric, or distance measure, describes the distance, ds, between two
points in space or space-time. For a unit sphere
A geodesic is the shortest distance between two points and can be found by
minimizing the space or space-time interval,
Space-time metric:
For special relativity, in a Lorentz
frame we define a distance in
spacetime as
For light, the metric yields ds2 = 0. Light
follows a null geodesic - meaning that the
physical distance travelled is equal to ct.
Everything that we see in the universe by
definition lies along null geodesics, as the
light has just had enough time to reach us.
3 Types of spacetime intervals:
ds2 = zero - lightlike (photons travel along these lines - this is a lightcone in
2-d space)
ds2 > zero - timelike (positions are close enough in space that a photon
would have had more than enough time to travel from one event to the
other)
ds2 < zero - spacelike (photon cannot traverse the distance in the time given
- one event could not have caused the other)
To construct a metric that is valid in a cosmological context, assume
• the cosmological principle is true
• each point in spacetime has one and only one co-moving, timelike geodesic
passing through it
If the expansion of the universe is homogeneous and
isotropic, co-moving coordinates are constant with time.
For a co-moving observer, there is a metric for the universe called the
Robertson-Walker metric, or the Friedmann-LeMaitre-Robertson-Walker metric
Where a is the scale factor of the Universe, r is the co-moving distance, k is the
sign of curvature (0, -1, +1) and dη is the solid angle.
This metric specifies the geometry of the universe with just one undetermined
factor, a(t), which is determined from the Friedmann equations. Together, the
Friedmann equations and RW metric completely describe the geometry.
Co-moving distance DC = ro
Proper distance DP = (a/ao)ro (Proper distance does change with time)
Cosmological Redshift
We now have a way to describe the evolution of the size of the universe
(Friedmann equation) and measuring distances within the universe (RobertsonWalker metric). Recast these in terms of observable quantities  we don’t
directly observe a(t), but we can observe the cosmological redshift of objects due
to the expansion of the universe.
Recall the Doppler shift of light (redshift or blueshift) is defined as
Since ao is a(to) then,
ao=1
As Universe expands, photon’s λ expands proportionally to the scale factor a(t)
Since redshift arises due to expanding wavelength of all photons traveling
through an expanding Universe, it is called a cosmological redshift.
As a consequence, we don’t normally convert redshift to distance, since we need
to assume a particular model for how a(t) has evolved.
Friedmann equation in observable quantities
Friedmann equation
Critical value of Λ in flat, empty universe
Rewrite Friedmann with these substitutions
At time to
Fundamental component
of the Friedmann
equation upon which our
measures for the
distance and evolution of
other quantities will be
based.
Expressing Distances in an Expanding Universe
The geometry and expansion rate of the Universe effects angular sizes and
distances measured. Integrate over components of RW metric.
DH = c/Ho  Hubble Distance (distance light travels in Hubble time, tH = 1/Ho)
DC=
 Radial Co-moving Distance
DM = DC (flat)  Transverse Co-moving Distance, differs for curved space
(see Hogg 2000)
DA = L(proper length)/θ(angular size) = DM/(1+z)  Angular Distance
DL = sqrt (L/4π*flux) = DM(1+z) = DA(1+z)2  Luminosity Distance
If Λ = 0 and flat geometry, then
DL = 2c/Ho [z/(G+1)] {1+[z/(G+1)]} where G = (1 + z)1/2
See Ned Wright’s Javascript Cosmology Calculator for DL for different
cosmologies:
http://www.astro.ucla.edu/~wright/CosmoCalc.html
Angular Size vs Redshift
As an object is moved to higher
redshifts its angular size first
decreases (as expected) but soon
begins to increase after passing
through a minimum value. The
appearance of a minimum angular
size at a given redshift zmin is a
generic feature of cosmological
models with Ωm > 0.
Figure 10 from Sahni and Starobinsky (2000) The angular size as a
function of cosmological redshift z for flat cosmological models ΩM+ΩΛ= 1.
Heavier lines correspond to larger values of ΩM.
Angular diameter distance vs z
(plotting DA/DH where DA=L/θ)
Luminosity distance vs z
(plotting DL/DH)
DH=c/Ho= 3000h-1Mpc
At high z, angular diameter
distance is such that 1 arcsec is
about 5 kpc.
flat, Λ=0 – solid
open, Λ=0 – dotted
flat, non-zero Λ - dashed
(from Hogg 2000 astro-ph 9905116)
Surface Brightness Dimming
SB is flux per unit solid angle
Recall Flux = L/(4πDL2) and the solid angle subtended by a source of
projected area A is Ω = A/DA2
Since DL = DA(1+z)2 we can show the surface brightness is
Observed surface brightness of objects decreases very rapidly as one
moves to high redshifts purely due to cosmology.
Cosmological dimming is independent of cosmological
parameters.
How does the age of the Universe differ in different
cosmologies?
Friedman models (Λ = 0) give
to = (2/3)Ho-1
0 < to < (2/3)Ho-1
(2/3)Ho-1 < to < Ho-1
k=0
k = +1
k = -1
What is the relationship between redshift and age of the
Universe?
Lookback time
tL = 2/3Ho-1[1-(1+z)-3/2] for flat Universe
Age of the Universe when light left source at redshift z
te(Gyr) = 10.5 (Ho/65) (1+z)-3/2 for flat Universe
Lookback times vs
z plotting tL/tH and
age vs z plotting
t/tH
tL is difference
between age of
Universe now and
age tU when
photons left
emitting source
flat, Λ=0 – solid
open, Λ=0 – dotted
flat, non-zero Λ - dashed
Age of the Universe as calculated from various models. A flat universe
with a significant cosmological constant (ΩM=0.3, ΩΛ=0.7) yields an age
close to what you get with a constant value of Ho (tH or 1/Ho).
14
Age of ΩM=1, ΩΛ=0 universe is about 9 billion years
Age of ΩM=0.3, ΩΛ=0.7 universe is about 14 billion years
Comparison of
Different
Cosmological
Models
qo is “deceleration parameter”
Classical Cosmological Tests (or How to determine if the
Universe is Open, Closed, Flat or Accelerating?)
1) Add up all matter in the Universe to determine mass density
Luminous matter:
only ~1% of ρcrit
Dark matter?
Still a factor of ~5 to low to
close the Universe with DM
This only constrains ΩM and not
the value of the cosmological
constant ΩΛ which also impacts
fate/age of the Universe
 From Big Bang Nucleosynthesis, we will see that baryonic matter is about 45% of the critical density with all matter (baryonic+DM) totaling ~30%.
2) Measure the curvature of space-time by surveying the Universe on large
scales. Volume of space is a function of cosmological parameters. Thus,
for a given class of objects the redshift distribution, N(z), will depend upon
Ω0 (or ΩM) and ΩΛ.
Assume a uniformly distributed population of objects with mean density n0.
Then,
Measurements attempted with QSOs, though must account for
evolutionary effects. Results indicate that if the Universe is curved, radius
of curvature would have to be comparable to or greater than the Hubble
distance – essentially flat…but why…. This is a challenge for Big Bang
Cosmology
3) Look for dynamical effects on the expansion rate of the
Universe (impacted by both ΩM and ΩΛ ).
Models showing unbound, decelerating
universe – more distant objects should be
moving faster than nearby objects if the
universe is decelerating.
Objects were
receding
less rapidly
in the past!
70 km/s/Mpc
The rate of cosmic expansion can
be determined by probing objects
at great distances where the
geometry and Hubble parameter
changes can be detected. Use
Type 1 supernovae – standard
candles that can be seen in distant
galaxies (several Gpc)  galaxy
distance can be determined
independent from Universal
expansion.
Results from Type 1a Supernovae surveys: First evidence
for accelerating Universal expansion
Flatness criterion – dashed line
Acceleration =0 - solid diagonal
Big Crunch below solid horizontal