Part 1: Observations of Global Properties

Part 2: Homogeneity and Isotropy; Many Distances; Scale Factor

Part 3: Spatial Curvature; Flatness-Oldness; Horizon

Part 4: Inflation; Anisotropy and Inhomogeneity

FAQ | Tutorial : Part 1 | Part 2 | Part 3 | Part 4 | Age | Distances | Bibliography | Relativity

The Cosmological Principle:

To say the Universe is homogeneous means that any measurable
property of the Universe is the same everywhere. This is only
approximately true, but it appears to be an excellent approximation
when one averages over large regions. Since the age of the
Universe is one of the measurable quantities, the homogeneity
of the Universe must be defined on a surface of constant proper
time since the Big Bang. Time dilation causes the proper time measured
by an observer to depend on the velocity of the observer, so we
specify that the time variable t in the Hubble law is the
proper time since the Big Bang *for comoving observers*.

With the correct interpretation of the variables, the Hubble law
(*v = HD*) is
true for all values of D, even very large ones which give v > c.
But one must be careful in interpreting the distance and velocity.
The distance in the Hubble law must be defined so that if A and B
are two distant galaxies seen by us in the same direction, and A and B
are not too far from each other, then the difference in distances from
us, D(A)-D(B), is the distance A would measure to B. But this
measurement must be made "now" -- so A must measure the distance to B
at the same proper time since the Big Bang as we see now.
Thus to determine D_{now} for a distant galaxy Z we would
find a chain of galaxies ABC...XYZ
along the path to Z, with each element of the chain close to its neighbors,
and then have each galaxy in the chain measure the distance to the next
galaxy at time t_{o} since the Big Bang.
The distance to Z, D(us to Z), is the sum of all these subintervals:

DAnd the velocity in the Hubble law is just the change of D_{now}= D(us to Z) = D(us to A) + D(A to B) + ... D(X to Y) + D(Y to Z)

but now showing the lightcones. Note how the lightcones must tip over along with the worldlines of the galaxies, showing that in these cosmological variables the speed of light is c

The time and distance used in the Hubble law are not the same as the
x and t used in special relativity, and this often leads to confusion.
In particular, galaxies that are far enough away from us necessarily
have velocities greater than the speed of light:

Worldlines of comoving observers are plotted and decorated with small, schematic lightcones. The red pear-shaped object is our past light cone. Notice that the red curve always has the same slope as the little light cones. In these variables, velocities greater than c are certainly possible, and since the open Universes are spatially infinite, they are actually required. But there is no contradiction with the special relativistic principle that objects do not travel faster than the speed of light, because if we plot exactly the same space-time in the special relativistic x and t coordinates we get:

The grey hyperbolae show the surfaces of constant proper time since the Big Bang. When we flatten these out to make the previous space-time diagram, the worldlines of the galaxies get flatter and giving velocities v = dD

v = HNote that the redshift-velocity law is_{o}D_{now}D_{now}= (c/H_{o})ln(1+z) 1+z = exp(v/c)

1+z = sqrt[(1+v/c)/(1-v/c)]which only applies to special relativistic coordinates, not to cosmological coordinates.

While the Hubble law distance is in principle measurable, the need for
helpers all along the chain of galaxies out to a distant galaxy makes
its use quite impractical. Other distances can be defined and measured
more easily. One is the *angular size distance*, defined by

theta = size/Dwhere "size" is the transverse extent of an object and "theta" is the angle (in radians) that it subtends on the sky. For the zero density model, the special relativistic x is equal to the angular size distance, x = D_{A}so D_{A}= size/theta

Another important distance indicator is the flux received from an
object, and this defines the *luminosity distance*
D_{L} through

Flux = Luminosity/(4*pi*DA fourth distance is based on the light travel time: D_{L}^{2})

The predicted curve relating one distance indicator to another depends
on the cosmological model. The plot of redshift vs distance for Type Ia
supernovae shown earlier is really a plot of cz vs D_{L},
since fluxes
were used to determine the distances of the supernovae.
This data clearly rule out models that do not give a linear
cz vs D_{L}
relation for small cz.
Extension of these observations to more
distant supernovae
have started to allow us to measure the
curvature of the cz
vs D_{L} relation, and provide more valuable
information about the Universe.

The perfect fit of the CMB to a
blackbody allows us to determine the
D_{A} vs D_{L} relation.
Since the CMB is produced at great distance but still looks like a
blackbody, a distant blackbody must look like a blackbody (even though
the temperature will change due to the redshift).
The luminosity of blackbody is

L = 4*pi*Rwhere R is the radius, T^{2}*sigma*T_{em}^{4}

Tand the flux will be_{obs}= T_{em}/(1+z)

F = thetawhere the angular radius is related to the physical radius by^{2}*sigma*T_{obs}^{4}

theta = R/DCombining these equations gives_{A}

DModels that do not predict this relationship between D_{L}^{2}= L/(4*pi*F) = (4*pi*R^{2}*sigma*T_{em}^{4})/(4*pi*theta^{2}*sigma*T_{obs}^{4}) = D_{A}^{2}*(1+z)^{4}or D_{L}= D_{A}*(1+z)^{2}

Here is a
Javascript calculator that takes
H_{o},
Omega_{M},
the normalized
cosmological constant *lambda*
and the redshift z
and then computes all of the these distances.
Here are the technical formulae
for these distances.
The graphs below show these
distances vs. redshift for three models: the critical density matter
dominated Einstein - de Sitter model (EdS), the empty model, and the
accelerating
Lambda-CDM
model (LCDM) that is the current consensus model.

Note that all the distances are very similar for small distances, with

Because the velocity or dD_{now}/dt is strictly proportional to
D_{now}, the
distance between any pair of comoving objects
grows by a factor (1+H*dt) during a time interval dt.
This means we can write the distance to any comoving observer as

Dwhere D_{G}(t) = a(t)*D_{G}(t_{o})

We can compute the dynamics of the Universe by considering an object
with distance D(t) = a(t) D_{o}. This distance and the corresponding
velocity dD/dt are measured with respect to us at the center of the
coordinate system. The gravitational acceleration due to the spherical
ball of matter with radius D(t) is g = -G*M/D(t)^{2} where the mass
is M = 4*pi*D(t)^{3}*rho(t)/3. Rho(t) is the density of matter which
depends only on the time since the Universe is homogeneous. The mass
contained within D(t) is independent of the time since the interior
matter has slower expansion velocity while the exterior matter has
higher expansion velocity and thus stays outside. The gravitational
effect of the external matter vanishes: the gravitational acceleration
inside a spherical shell is zero, and all the matter in the Universe
with distance from us greater than D(t) can be represented as union of
spherical shells. With a constant mass interior to D(t) producing the
acceleration of the edge, the problem reduces to the problem of a
body moving radially in the gravitational field of a point mass.
If the velocity is less than the escape velocity, the expansion
will stop and recollapse. If the velocity equals the
escape velocity
we have the critical case. This gives

v = H*D = v(esc) = sqrt(2*G*M/D) HFor rho less than or equal to the critical density rho(crit), the Universe expands forever, while for rho greater than rho(crit), the Universe will eventually stop expanding and recollapse. The value of rho(crit) for H^{2}*D^{2}= 2*G*(4*pi/3)*rho*D^{2}or rho(crit) = 3*H^{2}/(8*pi*G)

FAQ | Tutorial : Part 1 | Part 2 | Part 3 | Part 4 | Age | Distances | Bibliography | Relativity

© 1996-2009 Edward L. Wright. Last modified 12 Jun 2009