Wolfram Alpha:

```Hubble's Law
------------

Hubble compared recession velocities of galaxies
by measuring their redshift and comparing this
with their distance.  The distances were obtained
using Cepheid variable stars found in the galaxies.
Cepheid variable stars were discovered by Henrietta
Leavitt and are objects whose period of variability
is related to their average luminosity (brightness).
By comparing stars at assumed comparable distances
in the Small Magellanic Cloud, she found that the
ones with longer periods had higher luminosities.
Subsequent to Leavitt's discovery, Harlow Shapley
used parallax methods to measure the distance to
one of these cepheid variables.  This allowed him
to determine its actual or absolute brightness.
Knowing the relationship between distance, apparent
brightness and absolute brightness, allowed for the
distance to any cepheid variable to be calculated.

Hubble discovered the following relationship
between the recession velocity, v, and the proper
distance, D, and associated this with the expansion
of the universe:

v = H0D

Where H0 is the HUBBLE CONSTANT.  This is HUBBLE'S
LAW.

The proper distance, D, is the separation measured
at a specific time.  It can be considered as the
physical or actual measurement made by a giant
cosmological-sized ruler between the star and the
Earth.  Since the Universe is expanding, the proper
distance between two distant objects (which aren’t
held together by gravity) will increase with time.

Hubble's law tells us that velocity is dependent
on distance.  Gravitationally, interacting galaxies
move relative to each other independent of the
expansion of the Universe.  These relative velocities
need to be accounted for separately.

Hubble's Law can be derived in terms of the expansion
of space in the following manner:

O  a    a    a    a   a
o----o----o----o----o----> x
x         y

Point O is fixed.  a is the COSMIC SCALE FACTOR and
is a function of t.  It represents the relative expansion
of the universe.  It is easy to see that as the system
is stretched in the x direction, the further away the
points are from O, and the faster they must be moving.

D = an     where n is an integer that represents the
number of a's.  So x->y = 2.  D is the
distance between the objects.
.       .
D = v = an
.
= a(a/a)n
.
= (a/a)D

= H0D where H0 is the HUBBLE CONSTANT

In 3D:

D = a√{(Δx)2 + (Δy)2 + (Δz)2}
.    .
D = (a/a)D as before.

Since a is function of time, the coordinate mesh
expands with time.

---------------
/    /    /    /
/----/----/----/
/    /    /    /
---------------
^
------------             |
/   /   /   /             | time
/---/---/---/              |
/   /   /   /
------------

a(t)
---------
/  /  /  / a(t)
/--/--/--/
/  /  /  /
---------

The current estimate of H0 is about 72 (km/s)/Mpc
(1 Mpc ~ 3.26 x 106 light years).  So at 3.26
Mly the velocity ~ 72 km/s, at 7.52 mly it is
144 km/s and so on, i.e.

<- 3.26 Mly ->
o              o -> v ~ 72 km/s
O

Note:  H0 is a function of time and therefore
is not actually a constant in the traditional
sense.

Comoving Distance
-----------------

There is another distance commonly referred used by
cosmologists to define distances between objects.
This is called to comoving distance.  The comoving
distance, DCM, is defined as the proper distance
divided by the scale factor.  Thus,

DCM = DP(t)/a(t)

For objects which only get further apart as a result
of the expansion of the Universe, the comoving distance
between them does not change over time.  The comoving
distance changes if the proper distances change due
to a factor other than the expansion of the Universe
(i.e. gravitational attraction).  Comoving distance
and proper distance are defined to be equal at the
present time.

Relativistic Redshift
---------------------

Consider a stationary source and an observer
moving away from it with velocity, v.  Therefore,

λs + vts = cts

Where ts is the period of the light in the
source's frame.  Rearranging gives:

ts = λs/(c - v)

= λs/c(1 - β) where β = v/c

= 1/fs(1 - β)

The observer sees clocks in the source's frame
running slower relative to clocks in his own
frame.  The observer will measure this time to
be:

to = ts/γ where γ = 1/√(1 - β2)

Now fo = 1/t0 = γ/ts = 1/ts√(1 - v2/c2)

= fs(1 - β)/√(1 - v2/c2)

fo/fs = √[(1 - β)/((1 + β)]

The redshift, z, is defined as:

z = (fo - fs)/fs

= (fo/fs) - 1

= (λs/λo) - 1  (f = c/λ)

= (1 - β)/√(1 - v2/c2) - 1

=  √[(1 - β)/(1 + β)] - 1

Using the binomial expansion, this can be written
as:

z = (1 - β)(1 + v2/2c2) - 1

For v << c

z = -β

= -v/c

Cosmological Redshift
---------------------

The above analysis describes Doppler shift at
relativistic velocities.  In Doppler shift, the
wavelength of the emitted radiation depends on
the motion of the object at the instant the photons
are emitted.  In cosmological redshift, the
wavelength at which the radiation is originally
emitted is stretched as it travels through expanding
space.  Cosmological redshift results from the
expansion of space itself and NOT from the motion
of an individual body.

Using the v << c aproximation, z can be written
in terms of the Hubble constant as:

z = v/c

= H0D/c  (v = H0D)

= H0cΔt/c   (D = c.dt)

= H0Δt
.
= (a/a)Δt

= (Δa/Δt)Δt/a

= Δa/a

= (a(t0) - a(ts))/a(ts)

= a(t0)/a(ts) - 1

∴ z + 1 = a(t0)/a(ts)

Distance Measurement
--------------------

We can use the above relationships to estimate the
distance to objects.

D = v/Ho

= βc/Ho

= [((z + 1)2 - 1)/((z + 1)2 + 1)](c/Ho)

If v << c we can use:

v = HoD ∴ D = v/Ho

z = v/c ∴ v = cz

D = cz/H0

The modern discovery of TYPE 1A supernovae has
enabled further refinement of these techniques by
allowing measurements at much greater distances.
Type Ia supernovae occur in a binary system where
2 stars orbit one another.  One of the stars is a
white dwarf.  The other can be a giant star or even
a smaller white dwarf.  The white dwarf pulls
material off its companion star, adding that matter
to itself.  Eventually, when the white dwarf reaches
a certain mass, a nuclear reaction occurs causing
the white dwarf to explode and produce a burst of
light 5 billion times brighter than the Sun.
Because the chain reaction always happens in the
same way, and at the same mass, the brightness of
these Type Ia supernovae are also always the same.
Using this knowledge, the distance to the galaxy
containing the supernova can be computed by comparing
the observed and absolute intensities and again
applying the inverse square law (Distance Modulus
Equation).

Age of Universe
---------------

The reciprocal of H0 provides an estimate of the
age of the universe.

Note [H0] = [(L/t)/L] = [1/t] ∴ [1/H0] = [t]

Using the current estimate of H0 ~ 72 (km/s)/Mpc

1 Mpc = 3.086 x 1019 km

So,

1 km/1 MPc = 1/3.086 x 1019 = 3.24 x 10-20

1/H0 = 1/(72 x 3.24 x 10-20) s

= 1/233 x 10-20 s

= 0.0043 x 1020 s

= 4.3 x 1017 s

~ 13.6 x 109 years.```