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

Bibliography

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

Until a few hundred years ago, the Solar System and the Universe were equivalent in the minds of scientists, so the discovery that the Earth is not the center of the Solar System was an important step in the development of cosmology. Early in the 20th century Shapley established that the Solar System is far from the center of the Milky Way. So by the 1920's, the stage was set for the critical observational discoveries that led to the Big Bang model of the Universe.

The slope of the fitted line is 464 km/sec/Mpc, and is now known as the Hubble constant, H

1/HThus Hubble's value is equivalent to approximately 2 Gyr. Since this should be close to the age of the Universe, and we know (and it was known in 1929) that the age of the Earth is larger than 2 billion years, Hubble's value for H_{o}= (978 Gyr)/(H_{o}in km/sec/Mpc)

Hubble's data in 1929 is actually quite poor, since individual
galaxies have peculiar velocities of several hundred km/sec,
and Hubble's data only went out to 1200 km/sec. This has led
some people to propose
quadratic redshift-distance laws,
but the data shown below on Type Ia SNe from
Riess, Press and Kirshner (1996)

extend beyond 30,000 km/sec and provide a dramatic confirmation of the Hubble law,

v = dD/dt = H*DThe fitted line in this graph has a slope of 64 km/sec/Mpc. Since we measure the radial velocity using the Doppler shift, it is often called the

1 + z = lambda(observed)/lambda(emitted)where lambda is the wavelength of a line or feature in the spectrum of an object. In special relativity we know that the redshift is given by

1 + z = sqrt((1+v/c)/(1-v/c)) so v = cz + ...but the higher order corrections (the "...") in cosmology depend on general relativity and the specific model of the Universe.

The subscript "o" in H_{o} (pronounced "aitch naught") indicates the
current value of a time variable quantity. Since the 1/H_{o} is
approximately the age of the Universe, the value of H depends on
time. Another quantity with a naught is t_{o}, the age of the
Universe.

The linear distance-redshift law found by Hubble is compatible with
a Copernican view of the Universe: our position is not a special one.
First note that the recession velocity is symmetric: if A sees B receding,
then B sees that A is receding, as shown in this diagram:

which is based on a sketch by Bob Kirshner. Then consider the following space-time diagram showing several nearby galaxies moving away from us from our point of view (galaxy A, the blue worldline) on the top and from galaxy B's (the green worldline) point of view on the bottom.

The diagrams from the two different points of view are identical except for the names of the galaxies. The v(sq) = D

Thus if we saw a quadratic velocity vs. distance law, then an observer in a different galaxy would see a different law -- and one that would be different in different directions. Thus if we saw v(sq), then B would see much higher radial velocities in the "plus" direction than in the "minus" direction. This effect would allow one to locate the "center of Universe" by finding the one place where the redshift-distance law was the same in all directions. Since we actually see the same redshift-distance law in all directions, either the redshift-distance law is linear or else we are at the center which is anti-Copernican.

The Hubble law generates a homologous expansion which does not change the shapes of objects, while other possible velocity-distance relations lead to distortions during expansion.

The Hubble law defines a special frame of reference at any
point in the Universe. An observer with a large motion with
respect to the Hubble flow would measure blueshifts in front
and large redshifts behind, instead of the same redshifts
proportional to distance in all directions.
Thus we can measure our motion relative to the Hubble flow,
which is also our motion relative to the observable
Universe. A *comoving*
observer is at rest in this special frame of reference. Our
Solar System is not quite comoving: we have a velocity of 370
km/sec relative to the observable Universe. The Local Group
of galaxies, which includes the Milky Way, appears to be moving
at 600 km/sec relative to the observable Universe.

Hubble also measured the number of galaxies in different directions
and at different brightness in the sky. He found approximately
the same number of faint galaxies in all directions, even though
there is a large excess of bright galaxies in the Northern part of
the sky. When a distribution is the same in all directions,
it is *isotropic*.
And when he looked for galaxies with fluxes brighter
than F/4 he saw approximately 8 times more galaxies than he
counted which were brighter than F. Since a flux 4 times smaller
implies a doubled distance, and hence a detection volume that
is 8 times larger, this indicated that the Universe is close
to *homogeneous* (having uniform density) on large scales.

The figure above shows a homogeneous but not isotropic pattern on the left and an isotropic but not homogeneous pattern on the right. If a figure is isotropic from more than 1 (2 if spherical) points, then it must also be homogeneous.

Of course the Universe is not really homogeneous and isotropic,
because it contains dense regions like the Earth. But it can still
be statistically homogeneous and isotropic, like this
24 kB simulated galaxy field, which is
homogeneous and isotropic after smoothing out small scale details.
Peacock and Dodds (1994, MNRAS, 267, 1020) have looked at the
fractional density fluctuations in the nearby Universe as a function of
the radius of a top-hat smoothing filter, and find:

Thus for 100 Mpc regions the Universe is smooth to within several percent. Redshift surveys of very large regions confirm this tendency toward smoothness on the largest scales, even though nearby galaxies show large inhomogeneities like the Virgo Cluster and the supergalactic plane.

The case for an isotropic and homogeneous Universe became much stronger
after Penzias and Wilson announced the discovery of the Cosmic Microwave
Background in 1965. They observed an excess flux at 7.35 cm wavelength
equivalent to the radiation from a blackbody with a temperature of
3.5+/-1 degrees Kelvin. [The Kelvin temperature scale has degrees of the
same size as the Celsius scale, but it is referenced at absolute zero,
so the freezing point of water is 273.15 K.]
A blackbody radiator is an object that absorbs any radiation that hits it,
and has a constant temperature.
Many groups have measured
the intensity of the CMB at different wavelengths.
Currently the best information on the spectrum of the CMB comes from the
FIRAS instrument on the
COBE
satellite, and it is shown below:

The x axis variable is the wavenumber or 1/[wavelength in cm]. The y axis variable is the power per unit area per unit frequency per unit solid angle in MegaJanskies per steradian. 1 Jansky is 10

The temperature of the CMB is almost the same all over the sky.
The figure below shows a map of the temperature on a scale where
0 K is black and 3 K is white.

Thus the microwave sky is extremely isotropic. These observations are combined into the Cosmological Principle:

Another piece of evidence in favor of the Big Bang is the abundance of the light elements, like hydrogen, deuterium (heavy hydrogen), helium and lithium. As the Universe expands, the photons of the CMB lose energy due to the redshift and the CMB becomes cooler. That means that the CMB temperature was higher in the past. When the Universe was only a few minutes old, the temperature was high enough to make the light elements by nuclear fusion. The theory of Big Bang Nucleosynthesis predicts that about 1/4 of the mass of the Universe should be helium, which is very close to what is observed. The abundance of deuterium is inversely related to the density of nucleons in the Universe, and the observed value of the deuterium abundance suggests that there is one nucleon for every 4 cubic meters of space in the the Universe.

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© 1996-2017 Edward L. Wright. Last modified 21 Jul 2017