Ned Wright's Cosmology Tutorial Part 1: Observations of Global PropertiesPart 2: Homogeneity and Isotropy; Many Distances; Scale FactorPart 3: Spatial Curvature; Flatness-Oldness; HorizonPart 4: Inflation; Anisotropy and InhomogeneityBibliography

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 stepin 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.

Critical ObservationsIn 1929 Hubble [ 1, 2, 3] published a claim that the radial velocities of galaxies are proportional to their distance. The redshift of a galaxy is a measure of its radial velocity, and it can be measured using a spectrograph to determine the Doppler shift. The plot below shows Hubble's 1929 data:

The slope of the fitted line is 464 km/sec/Mpc, and is now known as the Hubble constant, Ho. [Sometimes I will use "TeX" mode, so A_x means that x is a subscript, Ax, while A^x means that x

is a superscript, Ax.] Since both kilometers and Megaparsecs (1 Mpc = 3.086E24 cm [the "E24" means multiply the 3.086 by 10 to the 24th power]) are units of distance, the simplified units of Ho

are 1/time, and the conversion is given by 1/Ho = (978 Gyr)/(Ho in km/sec/Mpc)

Thus 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 Ho led to considerable skepticism about cosmological models, and motivated the Steady State model. However, later work found that Hubble had confused two different kinds of Cepheid variable stars used for calibrating distances, and also that what Hubble thought were bright stars in distant galaxies were actually H II regions. Correcting for these errors has led to a lowering of the value of the Hubble constant: there are now primarily two groups using Cepheids: the HST Distance Scale Key Project team (Freedman, Kennicutt, Mould etal) which gets 72+/-8 km/sec/Mpc, while the Sandage team, also using HST observations of Cepheids to calibrate Type Ia supernovae, gets 57+/-4 km/sec/Mpc. Other methods to determine the distance scale include the time delay in gravitational lenses and the Sunyaev-Zeldovich effect in distant clusters: both are independent of the Cepheid calibration and give values consistent with the average of the two HST groups: 65+/-8 km/sec/Mpc. These results are consistent with a combination of results from CMB anisotropy and the accelerating expansion of the Universe which give 71+/-3.5 km/sec/Mpc. With this value for Ho, the "age" 1/Ho is 14 Gyr while the actual age from the consistent model is 13.7+/-0.2 Gyr.

[Huchra's Ho history]

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*D

The 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 redshift. The redshift z is defined such that: 1 + z = lambda(observed)/lambda(emitted)

where lambda is the wavelength of a line or feature in the spectrum of an object. In special relativitywe 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 Ho (pronounced "aitch naught") indicates the current value of a time variable quantity. Since the 1/Ho is approximately the age of the Universe, the value of H depends on time. Another quantity with a naught is to, 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) = D2 quadratic law, on the other hand, transforms into an anisotropic non-quadratic law when changing points of view, as shown below.

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 redshiftsbehind, instead of the same redshifts proportional to distance in all directions. Thus we can measureour 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 perunit area per unit frequency per unit solid angle in MegaJanskies per steradian. 1 Jansky is 10-26 Watts per square meter per Hertz. The error bars have been multiplied by 400 so they can be seen, but the data points are consistent with the radiation from a blackbody with To = 2.725 K.

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:

The Universe is Homogeneous and IsotropicAnother 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 CMBtemperature was higher in the past. When the Universe was only a few minutes old, the temperaturewas 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 veryclose 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.

Homogeneity and IsotropyThe Cosmological Principle:

The Universe is Homogeneous and Isotropic 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.

Many DistancesWith 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 Dnow 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 to since the Big Bang. The distance to Z, D(us to Z), is the sum of all these subintervals: Dnow = D(us to Z) = D(us to A) + D(A to B) + ... D(X to Y) + D(Y to Z)

And the velocity in the Hubble law is just the change of Dnow per unit time. It is close to cz for small redshifts but deviates for large ones. The space-time diagram below repeats the example from Part 1 showing how a change in point-of-view from observer A to observer B leaves the linear velocity vs. distance Hubble law unchanged:

but now showing the lightcones. Note how the lightcones must tip over along with the worldlines ofthe galaxies, showing that in these cosmological variables the speed of light is c with respect to local comoving observers. 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:

The light cones for distant galaxies in the diagram above are tipped over past the vertical, indicatingv > c. The space-time diagram below shows a "zero" (really very low) density cosmological model plotted using the Dnow and t of the Hubble law.

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 flattenthese out to make the previous space-time diagram, the worldlines of the galaxies get flatter and giving velocities v = dDnow/dt that are greater than c. But in special relativistic coordinates the velocities are less than c. We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance x = c*to/2. But the Hubble law distance Dnow, which is measured now, of these most distant galaxies is infinity (in this model). Furthermore, this galaxy with infinite Hubble law distance and hence infinite Hubble law velocity is visible to us, since in this model the observable Universe is the entire Universe. The relationships between the Hubble law distance and velocity (Dnow & v) and the redshift z for the zero density model are givenbelow: v = HoDnowDnow = (c/Ho)ln(1+z)1+z = exp(v/c)

Note that the redshift-velocity law is not the special relativistic Doppler shift law 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 ofgalaxies 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/DA so DA = size/theta

where "size" is the transverse extent of an object and "theta" is the angle (in radians) that it subtendson the sky. For the zero density model, the special relativistic x is equal to the angular size distance, x = DA.

Another important distance indicator is the flux received from an object, and this defines the luminosity distance DL through

Flux = Luminosity/(4*pi*DL2)

A fourth distance is based on the light travel time: Dltt = c*(to-tem). People who say that the greatest distance we can see is c*to are using this distance. But Dltt = c*(to-tem) is not a very useful distance because it is very hard to determine tem, the age of the Universe at the time of emission of the light we see. And finally, the redshift is a very important distance indicator, since astronomers can measure it easily, while the size or luminosity needed to compute DA or DL are always very hard to determine. The redshift is such a useful distance indicator that it is a shame that science journalists conspire to leave it out of stories: they must be taught the "5 w's but no z" rule in journalism school.

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 DL, 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 DL 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 DL relation, and provide more valuable information about the Universe.

The perfect fit of the CMB to a blackbody allows us to determine the DA vs DL 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*R2*sigma*Tem4

where R is the radius, Tem is the temperature of the emitting blackbody, and sigma is the Stephan-Boltzmann constant. If seen at a redshift z, the observed temperature will be Tobs = Tem/(1+z)

and the flux will be

F = theta2*sigma*Tobs4

where the angular radius is related to the physical radius by theta = R/DA

Combining these equations gives

DL2 = L/(4*pi*F)

= (4*pi*R2*sigma*Tem4)/(4*pi*theta2*sigma*Tobs

4) = DA

2*(1+z)4

or DL = DA*(1+z)

2

Models that do not predict this relationship between DA and DL, such as the chronometric model or the tired light model, are ruled out by the properties of the CMB.

Here is a Javascript calculator that takes Ho, OmegaM, 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 D = cz/Ho, but the different types of distances deviate substantially at large redshifts. Also note that these deviations depend on what kind of Universe we live in. Precise measurements of deviations of DL from cz/Ho are what tell us that the expansion of the Universe is accelerating.

Scale Factor a(t)Because the velocity or dDnow/dt is strictly proportional to Dnow, 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 thedistance to any comoving observer as DG(t) = a(t)*DG(to)

where DG(to) is the distance Dnow to galaxy G now, while a(t) is universal scale factor that applies to all comoving objects. From its definition we see that a(to) = 1.

We can compute the dynamics of the Universe by considering an object with distance D(t) = a(t) Do. This distance and the corresponding velocity dD/dt are measured with respect to us at the centerof 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) H2*D2 = 2*G*(4*pi/3)*rho*D2 or

rho(crit) = 3*H2/(8*pi*G)

For 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 Ho = 71 km/sec/Mpc is 9E-30 = 9*10

-30 gm/cc or 6 protons per cubic meter or 1.4E11

= 1.4*1011 solar masses per cubic Megaparsec. The latter can be compared to the observed 1.85E8 = 1.85*10 8 solar luminosities per Mpc3, requiring a mass-to-light ratio of 760 in solar units to close the Universe. If the density is anywhere close to critical most of the matter must be too dark to be observed. Current density estimates suggest that the matter density is between 0.2 to 1 times the critical density, and this does require that most of the matter in the Universe is dark.

Spatial CurvatureOne consequence of general relativity is that the curvature of space depends on the ratio of rho to rho(crit). We call this ratio = rho/rho(crit). For less than 1, the Universe has negatively curved or hyperbolic geometry. For = 1, the Universe has Euclidean or flat geometry. For greater than 1, the Universe has positively curved or spherical geometry. We have already seen that the zero density case has hyperbolic geometry, since the cosmic time slices in the special relativistic coordinates were hyperboloids in this model.

The figure above shows the three curvature cases plotted along side of the corresponding a(t)'s. These a(t) curves assume that the cosmological constant is zero, which is not the current standard model. > 1 still corresponds to a spherical shape, but could expand forever even though the density is greater than the critical density because of the repulsive gravitational effect of the cosmological constant.

The age of the Universe depends on o as well as Ho. For =1, the critical density case, the scale factor is

a(t) = (t/to)2/3

and the age of the Universe is to = (2/3)/Ho

while in the zero density case, =0, and a(t) = t/to with to = 1/Ho

If o is greater than 1 the age of the Universe is even smaller than (2/3)/Ho.

The figure above shows the scale factor vs time measured from the present for Ho = 71 km/sec/Mpcand for o = 0 (green), o = 1 (black), and o = 2 (red) with no vacuum energy; the WMAP model with M= 0.27 and V = 0.73 (magenta); and the Steady State model with V = 1 (blue). The ages of the Universe in these five models are 13.8, 9.2, 7.9, 13.7 and infinity Gyr. The recollapse of the o = 2 model occurs when the Universe is 11 times older than it is now, and all observations indicate o < 2, so we have at least 80 billion more years before any Big Crunch.

The value of Ho*to is a dimensionless number that should be 1 if the Universe is almost empty and 2/3 if the Universe has the critical density. In 1994 Freedman et al. (Nature, 371, 757) found Ho = 80 +/- 17 and when combined with to = 14.6 +/- 1.7 Gyr, we find that Ho*to = 1.19 +/- 0.29. At face value this favored the empty Universe case, but a 2 standard deviation error in the downward direction would take us to the critical density case. Since both the age of globular clusters used above and the value of Ho depend on the distance scale in the same way, an underlying error in the distance scale could make a large change in Ho*to. In fact, recent data from the HIPPARCOS satellite suggest that the Cepheid distance scale must be increased by 10%, and also that the age of globular clusters must be reduced by 20%. If we take the latest HST value for Ho = 72 +/- 8 (Freedman et al. 2001, ApJ, 553, 47) and the latest globular cluster ages giving to = 13.5 +/- 0.7 Gyr, we find that Ho*to = 0.99 +/- 0.12 which is consistent with an empty Universe, but also consistent with the accelerating Universe that is the current standard model.

Flatness-Oldness ProblemHowever, if o is sufficiently greater than 1, the Universe will eventually stop expanding, and then will become infinite. If o is less than 1, the Universe will expand forever and the density goes down faster than the critical density so gets smaller and smaller. Thus = 1 is an unstable stationary point unless the expansion of the universe is accelerating, and it is quite remarkable that is anywhere close to 1 now.

The figure above shows a(t) for three models with three different densities at a time 1 nanosecond after the Big Bang. The black curve shows a critical density case that matches the WMAP-based concordance model, which has density = 447,225,917,218,507,401,284,016 gm/cc at 1 ns after the Big Bang. Adding only 0.2 gm/cc to this 447 sextillion gm/cc causes the Big Crunch to be right now! Taking away 0.2 gm/cc gives a model with a matter density M that is too low for our observations. Thus the density 1 ns after the Big Bang was set to an accuracy of better than 1 part in2235 sextillion. Even earlier it was set to an accuracy better than 1 part in 1059! Since if the density is slightly high, the Universe will die in an early Big Crunch, this is called the "oldness" problem in cosmology. And since the critical density Universe has flat spatial geometry, it is also called the "flatness" problem -- or the "flatness-oldness" problem. Whatever the mechanism for setting the density to equal the critical density, it works extremely well, and it would be a remarkable coincidence if o were close to 1 but not exactly 1.

Note that the old version of this figure was based on a model with higher current matter density, andalso rounded the true of 0.4 gm/cc to 1 based on rounding the logarithm.

Manipulating Space-Time DiagramsThe critical density model is shown in the space-time diagram below.

Note that the worldlines for galaxies are now curved due to the force of gravity causing the expansion to decelerate. In fact, each worldline is a constant factor times a(t) which is (t/to)

2/3 for this o = 1 model. The red pearshaped object is our past lightcone. While this diagram is drawn from our point-of-view, the Universe is homogeneous so the diagram drawn from the point-of-view of any of the galaxies on the diagram would be identical.

The diagram above shows the space-time diagram drawn on a deck of cards, and the diagram belowshows the deck pushed over to put it into A's point-of-view.

Note that this is not a Lorentz transformation, and that these coordinates are not the special relativistic coordinates for which a Lorentz transformation applies. The Galilean transformation which could be done by skewing cards in this way required that the edge of the deck remain straight, and in any case the Lorentz transformation can not be done on cards in this way because there is no absolute time. But in cosmological models we do have cosmic time, which is the proper time since the Big Bang measured by comoving observers, and it can be used to set up a deck of cards. The presence of gravity in this model leads to a curved spacetime that can not be plotted on a flat space-time diagram without distortion. If every coordinate system is a distorted representation of the Universe, we may as well use a convenient coordinate system and just keep track of the distortion by following the lightcones.

Sometimes it is convenient to "divide out" the expansion of the Universe, and the space-time diagram shows the result of dividing the spatial coordinate by a(t). Now the worldlines of galaxies are all vertical lines.

This division has expanded our past line cone so much that we have to replot to show it all:

If we now "stretch" the time axis near the Big Bang we get the following space-time diagram whichhas straight line past lightcones:

This kind of space-time diagram is called a "conformal" space-time diagram, and while it is highly distorted it makes it easy to see where the light goes. This transformation we have done is analogous to the transformation from the side view of the Earth on the left below and the Mercator chart on the right.

Note that a constant SouthEast course is a straight line on the Mercator chart which is analogous to having straight line past lightcones on the conformal space-time diagram.

Also remember that the o = 1 spacetime is infinite in extent so the conformal space-time diagram can go on far beyond our past lightcone,

as shown above.

Other coordinates can be used as well. Plotting the spatial coordinate as angle on polar graph paper makes the translation to a different point-of-view easy. On the diagram below,

an o = 2 model (which really is "round") is plotted this way with a(t) used as the radial coordinate.The past lightcone of an observer reachs halfway around the Universe in this model.

Horizon ProblemThe conformal space-time diagram is a good tool use for describing the meaning of CMB anisotropy observations. The Universe was opaque before protons and electrons combined to form hydrogen atoms when the temperature fell to about 3,000 K at a redshift of 1+z = 1090. After this time the photons of the CMB have traveled freely through the transparent Universe we see today. Thus the temperature of the CMB at a given spot on the sky had to be determined by the time the hydrogen atoms formed, usually called "recombination" even though it was the first time so "combination" would be a better name. Since the wavelengths in the CMB scale the same way that intergalaxy distances do during the expansion of the Universe, we know that a(t) had to be 0.0009 atrecombination. For the o = 1 model this implies that t/to = 0.00003 so for to about 14 Gyr the time is about 380,000 years after the Big Bang. This is such a small fraction of the current age that the "stretching" of the time axis when making a conformal space-time diagram is very useful to magnify this part of the history of the Universe.

The conformal space-time diagram above has exaggerated this part even further by taking the redshift of recombination to be 1+z = 144, which occurs at the blue horizontal line. The yellow regions are the past lightcones of the events which are on our past lightcone at recombination. Any event that influences the temperature of the CMB that we see on the left side of the sky must be within the left-hand yellow region. Any event that affects the temperature of the CMB on the right side of the sky must be within the right-hand yellow region. These regions have no events in common, but the two temperatures are equal to better than 1 part in 10,000. How is this possible? This is known as the "horizon" problem in cosmology.

InflationThe "inflationary scenario", developed by Starobinsky and by Guth, offers a solution to the flatness-oldness problem and the horizon problem. The inflationary scenario invokes a vacuum energy density. We normally think of the vacuum as empty and massless, and we can determine thatthe density of the vacuum is less than 10-29 gm/cc now. But in quantum field theory, the vacuum is not empty, but rather filled with virtual particles:

The space-time diagram above shows virtual particle-antiparticle pairs forming out of nothing and then annihilating back into nothing. For particles of mass m, one expects about one virtual particle in each cubical volume with sides given by the Compton wavelength of the particle, h/mc, where h is Planck's constant. Thus the expected density of the vacuum is rho = m4*c3/h3 which is rather large. For the largest elementary particle mass usually considered, the Planck mass M defined by 2*pi*G*M2 = h*c, this density is 2*1091 gm/cc. That's a 2 followed by 91 zeroes! Thus the vacuumenergy density is at least 120 orders of magnitude smaller than the naive quantum estimate, so there must be a very effective suppression mechanism at work. If a small residual vacuum energy density exists now, it leads to a "cosmological constant" which is one proposed mechanism to relieve the tight squeeze between the Omegao=1 model age of the Universe, to = (2/3)/Ho = 9 Gyr, and the apparent age of the oldest globular clusters, 12-14 Gyr. The vacuum energy density can do this because it produces a "repulsive gravity" that causes the expansion of the Universe to accelerate instead of decelerate, and this increases to for a given Ho.

The inflationary scenario proposes that the vacuum energy was very large during a brief period early in the history of the Universe. When the Universe is dominated by a vacuum energy density the scale factor grows exponentially, a(t) = exp(H(t-to)). The Hubble constant really is constant during this epoch so it doesn't need the "naught". If the inflationary epoch lasts long enough the exponential function gets very large. This makes a(t) very large, and thus makes the radius of curvature of the Universe very large. The diagram below shows our horizon superimposed on a verylarge radius sphere on top, or a smaller sphere on the bottom. Since we can only see as far as our horizon, for the inflationary case on top the large radius sphere looks almost flat to us.

This solves the flatness-oldness problem as long as the exponential growth during the inflationary epoch continues for at least 100 doublings. Inflation also solves the horizon problem, because the future lightcone of an event that happens before inflation is expanded to a huge region by the growth during inflation.

This space-time diagram shows the inflationary epoch tinted green, and the future lightcones of two events in red. The early event has a future lightcone that covers a huge area, that can easily encompass all of our horizon. Thus we can explain why the temperature of the microwave background is so uniform across the sky.

Details: Large-Scale Structure and AnisotropyOf course the Universe is not really homogeneous, since it contains dense regions like galaxies and people. These dense regions should affect the temperature of the microwave background. Sachs andWolfe (1967, ApJ, 147, 73) derived the effect of the gravitational potential perturbations on the CMB. The gravitational potential, phi = -GM/r, will be negative in dense lumps, and positive in lessdense regions. Photons lose energy when they climb out of the gravitational potential wells of the lumps:

This conformal space-time diagram above shows lumps as gray vertical bars, the epoch before recombination as the hatched region, and the gravitational potential as the color-coded curve phi(x).Where our past lightcone intersects the surface of recombination, we see a temperature perturbed bydT/T = phi/(3*c2). Sachs and Wolfe predicted temperature fluctuations dT/T as large as 1 percent, but we know now that the Universe is far more homogeneous than Sachs and Wolfe thought. So observers worked for years to get enough sensitivity to see the temperature differences around the sky. The first anisotropy to be detected was the dipole anisotropy by Conklin in 1969:

The map above is from COBE and is much better than Conklin's 2 standard deviation detection. Thered part of the sky is hotter by (v/c)*To, while the blue part of the sky is colder by (v/c)*To, where the inferred velocity is v = 368 km/sec. This is how we measure the velocity of the Solar System relative to the observable Universe. It was another 23 years before the anisotropy predicted by Sachs and Wolfe was detected by Smoot et al. (1992, ApJL, 396, 1). The amplitude was 1 part in 100,000 instead of 1 part in 100, but was perfectly consistent with Lambda-CDM [Wright et al. 1992, ApJL, 396, 13].

The map above shows cosmic anisotropy (and detector noise) after the dipole pattern and the radiation from the Milky Way have been subtracted out. The anisotropy in this map has an RMS value of 30 microK, and if it is converted into a gravitational potential using Sachs and Wolfe's result and that potential is then expressed as a height assuming a constant acceleration of gravity equal to the gravity on the Earth, we get a height of twice the distance from the Earth to the Sun. The "mountains and valleys" of the Universe are really quite large.

Inflation predicts a certain statistical pattern in the anisotropy. The quantum fluctuations normally

affect very small regions of space, but the huge exponential expansion during the inflationary epochmakes these tiny regions observable.

The space-time diagram on the left above shows the future lightcones of quantum fluctuation events. The top of this diagram is really a volume which intersects our past lightcone making the sky. The future lightcones of events become circles on the sky. Events early in the inflationary epoch make large circles on the sky, as shown in the bottom map on the right. Later events make smaller circles as shown in the middle map, but there are more of them so the sky coverage is the same as before. Even later events make many small circles which again give the same sky coverage as seen on the top map.

An animated GIF file showing the spatial part of the above space-time diagram as a function of timeis available here [1.2 MB].

The pattern formed by adding all of the effects from events of all ages is known as "equal power on all scales", and it agrees with the COBE data.

Having found that the observed pattern of anisotropy is consistent with inflation, we can also ask whether the amplitude implies gravitational forces large enough to produce the observed clustering

of galaxies.

The conformal space-time diagram above shows the phi(x) at recombination determined by COBE'sdT data, and the worldlines of galaxies which are perturbed by the gravitational forces produced by the gradient of the potential. Matter flows "downhill" away from peaks of the potential (red spots onthe COBE map), producing voids in the current distribution of galaxies, while valleys in the potential (blue spots) are where the clusters of galaxies form.

COBE was not able to see spots as small as clusters or even superclusters of galaxies, but if we use "equal power on all scales" to extrapolate the COBE data to smaller scales, we find that the gravitational forces are large enough to produce the observed clustering, but only if these forces are not opposed by other forces. If the all the matter in the Universe is made out of the ordinary chemical elements, then there was a very effective opposing force before recombination, because the free electrons which are now bound into atoms were very effective at scattering the photons of the cosmic background. We can therefore conclude that most of the matter in the Universe is "dark matter" that does not emit, absorb or scatter light. Furthermore, observations of distant supernovae have shown that most of the energy density of the Universe is a vacuum energy density (a "dark energy") like Einstein's cosmological constant that causes an accelerating expansion of the Universe. These strange conclusions have been greatly strengthened by temperature anisotropy dataat smaller angular scales which was provided by the Wilkinson Microwave Anisotropy Probe (WMAP) in 2003.

Age of the Universe There are at least 3 ways that the age of the Universe can be estimated. I will describe

The age of the chemical elements. The age of the oldest star clusters. The age of the oldest white dwarf stars.

The age of the Universe can also be estimated from a cosmological model based on the Hubble constant and the densities of matter and dark energy. This model-based age is currently 13.75 +/- 0.1 Gyr. But this Web page will only deal with actual age measurements, not estimates from cosmological models. The actual age measurements are consistent with the model-based age which increases our confidence in the Big Bang model.

The Age of the ElementsThe age of the chemical elements can be estimated using radioactive decay to determine how old a given mixture of atoms is. The most definite ages that can be determined this way are ages since thesolidification of rock samples. When a rock solidifies, the chemical elements often get separated

into different crystalline grains in the rock. For example, sodium and calcium are both common elements, but their chemical behaviours are quite different, so one usually finds sodium and calciumin different grains in a differentiated rock. Rubidium and strontium are heavier elements that behavechemically much like sodium and calcium. Thus rubidium and strontium are usually found in different grains in a rock. But Rb-87 decays into Sr-87 with a half-life of 47 billion years. And thereis another isotope of strontium, Sr-86, which is not produced by any rubidium decay. The isotope Sr-87 is called radiogenic, because it can be produced by radioactive decay, while Sr-86 is non-radiogenic. The Sr-86 is used to determine what fraction of the Sr-87 was produced by radioactive decay. This is done by plotting the Sr-87/Sr-86 ratio versus the Rb-87/Sr-86 ratio. When a rock is first formed, the different grains have a wide range of Rb-87/Sr-86 ratios, but the Sr-87/Sr-86 ratio is the same in all grains because the chemical processes leading to differentiated grains do not separate isotopes. After the rock has been solid for several billion years, a fraction of the Rb-87 will have decayed into Sr-87. Then the Sr-87/Sr-86 ratio will be larger in grains with a large Rb-87/Sr-86 ratio. Do a linear fit of Sr-87/Sr-86 = a + b*(Rb-87/Sr-86)

and then the slope term is given by

b = 2x - 1

with x being the number of half-lives that the rock has been solid. See the talk.origins isochrone FAQ for more on radioactive dating.

When applied to rocks on the surface of the Earth, the oldest rocks are about 3.8 billion years old. When applied to meteorites, the oldest are 4.56 billion years old. This very well determined age is the age of the Solar System. See the talk.origins age of the Earth FAQ for more on the age of the solar system.

When applied to a mixed together and evolving system like the gas in the Milky Way, no great precision is possible. One problem is that there is no chemical separation into grains of different crystals, so the absolute values of the isotope ratios have to be used instead of the slopes of a linear fit. This requires that we know precisely how much of each isotope was originally present, so an accurate model for element production is needed. One isotope pair that has been used is rhenium and osmium: in particular Re-187 which decays into Os-187 with a half-life of 40 billion years. It looks like 15% of the original Re-187 has decayed, which leads to an age of 8-11 billion years. But this is just the mean formation age of the stuff in the Solar System, and no rhenium or osmium has been made for the last 4.56 billion years. Thus to use this age to determine the age of the Universe, a model of when the elements were made is needed. If all the elements were made in a burst soon after the Big Bang, then the age of the Universe would be to = 8-11 billion years. But if the elementsare made continuously at a constant rate, then the mean age of stuff in the Solar System is (to + tSS)/2 = 8-11 Gyr

which we can solve for the age of the Universe giving to = 11.5-17.5 Gyr

238U and 232Th are both radioactive with half-lives of 4.468 and 14.05 Gyrs, but the uranium is underabundant in the Solar System compared to the expected production ratio in supernovae. This is not surprising since the 238U has a shorter half-life, and the magnitude of the difference gives an estimate for the age of the Universe. Dauphas (2005, Nature, 435, 1203) combines the Solar System238U:232Th ratio with the ratio observed in very old, metal poor stars to solve simultaneous equations for both the production ratio and the age of the Universe, obtaining 14.5+2.8-2.2 Gyr.

Radioactive Dating of an Old StarA very interesting paper by Cowan et al. (1997, ApJ, 480, 246) discusses the thorium abundance in an old halo star. Normally it is not possible to measure the abundance of radioactive isotopes in other stars because the lines are too weak. But in CS 22892-052 the thorium lines can be seen because the iron lines are very weak. The Th/Eu (Europium) ratio in this star is 0.219 compared to 0.369 in the Solar System now. Thorium decays with a half-life of 14.05 Gyr, so the Solar System formed with Th/Eu = 24.6/14.05*0.369 = 0.463. If CS 22892-052 formed with the same Th/Eu ratio itis then 15.2 +/- 3.5 Gyr old. It is actually probably slightly older because some of the thorium that would have gone into the Solar System decayed before the Sun formed, and this correction depends on the nucleosynthesis history of the Milky Way. Nonetheless, this is still an interesting measure of the age of the oldest stars that is independent of the main-sequence lifetime method.

A later paper by Cowan et al. (1999, ApJ, 521, 194) gives 15.6 +/- 4.6 Gyr for the age based on two stars: CS 22892-052 and HD 115444.

A another star, CS 31082-001, shows an age of 12.5 +/- 3 Gyr based on the decay of U-238 [Cayrel,et al. 2001, Nature, 409, 691-692]. Wanajo et al. refine the predicted U/Th production ratio and get 14.1 +/- 2.5 Gyr for the age of this star.

The Age of the Oldest Star ClustersWhen stars are burning hydrogen to helium in their cores, they fall on a single curve in the luminosity-temperature plot known as the H-R diagram after its inventors, Hertzsprung and Russell.This track is known as the main sequence, since most stars are found there. Since the luminosity of a star varies like M3 or M4, the lifetime of a star on the main sequence varies like t=const*M/L=k/L0.7. Thus if you measure the luminosity of the most luminous star on the main sequence, you get an upper limit for the age of the cluster:

Age < k/L(MS_max)0.7

This is an upper limit because the absence of stars brighter than the observed L(MS_max) could be due to no stars being formed in the appropriate mass range. But for clusters with thousands of members, such a gap in the mass function is very unlikely, the age is equal to k/L(MS_max)0.7. Chaboyer, Demarque, Kernan and Krauss (1996, Science, 271, 957) apply this technique to globularclusters and find that the age of the Universe is greater than 12.07 Gyr with 95% confidence. They say the age is proportional to one over the luminosity of the RR Lyra stars which are used to determine the distances to globular clusters. Chaboyer (1997) gives a best estimate of 14.6 +/- 1.7 Gyr for the age of the globular clusters. But recent Hipparcos results show that the globular clusters are further away than previously thought, so their stars are more luminous. Gratton et al. give ages between 8.5 and 13.3 Gyr with 12.1 being most likely, while Reid gives ages between 11 and 13 Gyr, and Chaboyer et al. give 11.5 +/- 1.3 Gyr for the mean age of the oldest globular clusters.

The Age of the Oldest White DwarfsA white dwarf star is an object that is about as heavy as the Sun but only the radius of the Earth. The average density of a white dwarf is a million times denser than water. White dwarf stars form inthe centers of red giant stars, but are not visible until the envelope of the red giant is ejected into space. When this happens the ultraviolet radiation from the very hot stellar core ionizes the gas and produces a planetary nebula. The envelope of the star continues to move away from the central core,and eventually the planetary nebula fades to invisibility, leaving just the very hot core which is now a white dwarf. White dwarf stars glow just from residual heat. The oldest white dwarfs will be the coldest and thus the faintest. By searching for faint white dwarfs, one can estimate the length of

time the oldest white dwarfs have been cooling. Oswalt, Smith, Wood and Hintzen (1996, Nature, 382, 692) have done this and get an age of 9.5+1.1-0.8 Gyr for the disk of the Milky Way. They estimate an age of the Universe which is at least 2 Gyr older than the disk, so to > 11.5 Gyr.

Hansen et al. have used the HST to measure the ages of white dwarfs in the globular cluster M4, obtaining 12.7 +/- 0.7 Gyr. In 2004 Hansen et al. updated their analysis to give an age for M4 of 12.1 +/- 0.9 Gyr, which is very consistent with the age of globular clusters from the main sequence turnoff. Allowing allowing for the time between the Big Bang and the formation of globular clusters(and its uncertainty) implies an age for the Universe of 12.8 +/- 1.1 Gyr.

SummaryMethod Value [Gyr] +Errorbar -ErrorbarElements 14.5 +2.8 -2.5Old Stars 14.4 +2.2 -2.2GC MSTO 12.2 +1.3 -1.3Disk WDs 11.5 +infinity -1GC WDs 12.8 +1.1 -1.1Weighted Mean 12.94 +0.75 -0.75

The ABC's of DistancesIt is almost impossible to tell the distances of objects we see in the sky. Almost, but not quite, and astronomers have developed a large variety of techniques. Here I will describe 26 of them. I will ignore the work that went into determining the astronomical unit: the scale factor for the Solar System, and just consider distances outside of the Solar System.

A. TRIGONOMETRIC PARALLAXThis method rates an A because it is the gold standard for astronomical distances. It is based on measuring two angles and the included side of a triangle formed by 1) the star, 2) the Earth on one side of its orbit, and 3) the Earth six months later on the other side of its orbit.

The top part of the diagram above shows the Earth at two different times, and the triangle formed with a nearby star and these two positions of the Earth. The bottom part shows two pictures of the nearby star projected onto more distant stars taken from the two sides of the Earth's orbit. If you cross your eyes to merge these two pictures, you will either see the nearby star standing in front of the background in 3-D, or else get a headache.

The parallax of a star is one-half the angle at the star in the diagram above. Thus the parallax is the angle at the star in an Earth-Sun-star triangle. Since this angle is always very small, the sine and tangent of the parallax are very well approximated by the parallax angle measured in radians. Therefore the distance to a star is D[in cm] = [Earth-Sun distance in cm]/[parallax in radians]

Astronomers usually say the Earth-Sun distance is 1 astronomical unit, where 1 au = 1.5E13 cm, and measure small angles in arc-seconds. [Note that 1.5E13 is computerese for 15,000,000,000,000]One radian has 648000/pi arc-seconds. If we use these units, the unit of distance is [648000/pi] au =3.085678E18 cm = 1 parsec. A star with a parallax of 1 arc-second has a distance of 1 parsec. No known stars have parallaxes this big. Proxima Centauri has a parallax of 0.76". [The double quote isused to denote arc-seconds (as well as inches).]

The first stellar parallax (of the star 61 Cygni) was measured by Friedrich Wilhelm Bessel (1784-1846) in 1838. Bessel is also known for the Bessel functions in mathematical physics.

B. Moving ClustersNot many stars are close enough to have useful trigonometric parallaxes. But when stars are in a stable star cluster whose physical size is not changing, like the Pleiades, then the apparent motions of the stars within the cluster can be used to determine the distance to the cluster.

The top part of the diagram above shows the space motion of a cluster of stars. Notice that the velocity vectors are parallel so the cluster is neither expanding nor contracting. But when we look atthe motions of the stars projected on the sky we see them converging because of perspective effects.The angle to the convergent point is theta. If the cluster is moving towards us then the convergent point is behind the cluster but there is second convergent point on the opposite side of the sky and we use that. From the motions of the stars on the sky, known as proper motions because they are properties of individual stars, we measure theta and its rate of change, d(theta)/dt. We also need the radial velocity VR of the cluster measured using a spectrograph to see the Doppler shift. The transverse velocity, VT, (sideways motion) of the cluster can be found using VT/VR = tan(theta). The distance of the cluster is then

D[in cm] = VT[in cm/sec]/[d(theta)/dt]

D[in pc] = (VR/4.74 km/sec)*tan(theta)/{d(theta)/dt[in "/yr]}

The odd constant 4.74 km/sec is one au/year. Because a time interval of 100 years can be used to measure d(theta)/dt, precise distances to nearby star clusters are possible. This method has been applied to the Hyades cluster giving a distance of 45.53 +/- 2.64 pc. The average of HIPPARCOS trigonometric parallaxes for Hyades members gives a distance of 46.34 +/- 0.27 pc (Perryman et al.).

C. Secular ParallaxAnother method can be used to measure the average distance to a set of stars, chosen to be all about the same distance from the Earth.

The diagram above shows such a set of stars, but with two possible mean distances. The green stars show a small mean distance, while the red stars show a large mean distance. Because of the mean motion of the Solar system at 20 km/sec relative to the average of nearby stars there will be an average proper motion away from the point of the sky the Solar System is moving towards. This point is known as the apex. Let the angle to the apex be theta. Then the proper motion d(theta)/dt will have a mean component proportional to sin(theta), shown by the lines in the plot of d(theta)/dt vs sin(theta). Let the slope of this line be mu. Then the mean distance of the stars is D[in cm] = V(sun)[in cm/sec]/(mu [in radians/sec])

D[in pc] = 4.16/(mu [in "/yr])

where the odd constant 4.16 is the Solar motion in au/yr.

D. Statistical ParallaxWhen the stars have measured radial velocities, then the scatter in their proper motions can be used to determine the mean distance. It is (scatter in VR)[in cm/sec]

D[in cm] = ---------------------------------------- (scatter in d(theta)/dt)[in radians/sec]

E. Kinematic DistanceThe pattern of differential rotation in our galaxy can be used to determine the distance of a source when its radial velocity is known.

F. Expansion ParallaxThe distance to an expanding object like a supernova remnant such as Tycho can be determined by measuring:

1. the angular expansion rate d(theta)/dt using pictures taken many years apart, and 2. the radial velocity of expansion, VR, using the Doppler shift of lines emitted from the front

and back of the expanding shell. When a spectrograph is pointed at the center of the remnanta double line is seen, with the red shifted emission coming from the back of the shell while the blue shifted emission comes from the front.

The distance is then calculated using D = VR/d(theta)/dt with theta in radians

This method is subject to a systematic error when the velocity of the material behind the shock is less than the velocity of the shock. In supernova remnants in the adiabatic phase this is in fact the case, with VR = 0.75 V(shock), so the calculated distance can be too small by 25%.

G. Light Echo DistanceThe center elliptical ring around SN1987A in the LMC appears to be due to an inclined circular ringaround the progenitor. When the pulse of ultraviolet light from the supernova hit the ring, it lit up inultraviolet emission lines which were observed by the International Ultraviolet Explorer (IUE). The first detection of these lines at time, t1, and also the time when the lines from the last part of the ring to be illuminated, t2, were both clearly evident in the IUE light curve of the UV lines. If t0 is the time that we first saw the supernova, then the extra light travel times to the front and back of thering are: t1 - t0 = R(1 - sin(i))/ct2 - t0 = R(1 + sin(i))/c

where R is the radius of the ring in cm. Thus R = c(t1-t0 + t2-t0)/2

When the HST was launched it took a picture of SN 1987A and saw the ring, and measured the angular radius of the ring, theta. The ratio gives the distance: D = R/theta with theta in radians

Applied to the LMC using SN 1987A one gets D = 47 +/- 1 kpc, based on t1-t0 = 75 +/- 2.6 days, t2- t0 = 390 +/- 1.8 days, and a ring angular semimajor axis of 0.858 arc-seconds. (Gould 1995, ApJ, 452, 189) This method is basically the expansion method applied to the expansion of the shell of supernova radiation that expands at the speed of light. It can be applied to other known geometries, as well.

H. Spectroscopic Visual BinariesIf a binary orbit is observed both visually and spectroscopically, then both the angular size and the physical size of the orbit are known. The ratio gives the distance.

The following methods need the surface brightness of stars. The picture below shows how the surface brightness of stars depends on their colors:

The colors correspond approximately to star temperatures of 5000, 6000 and 7000 K. The color shifts are quite small, but the surface brightness changes are large: in fact, I have cut the surface brightness change in half in order to make the cool star visible. By measuring the ratio of the blue flux of a star to its yellow-green flux, astronomers measure the B-V color of the star. This measure of the blue:visual flux ratio can be used to estimate the surface brightness SB of the star. Since the visual flux is measured as well, the angular radius theta of the star is known from theta = sqrt[Flux/(pi*SB)]. If the physical radius R can be found as well, the distance follows from D = R/theta with theta in radians.

I. Baade-Wesselink MethodThe Baade-Wesselink method is applied to pulsating stars. Using the color and flux light curves, one finds the ratio of the radii at different times: sqrt[Flux(t2)/SB(Color(t2)]R(t2)/R(t1) = --------------------------- sqrt[Flux(t1)/SB(Color(t1)]

Then spectra of the star throughout its pulsation period are used to find its radial velocity Vr(t). Knowing how fast the star's surface is moving, one finds R(t2)-R(t1) by adding up velocity*time during the time interval between t1 and t2. If you know both the ratio of the radii R(t2)/R(t1) from fluxes and colors and the difference in the radii R(t2)-R(t1) from spectroscopy, then you have two equations in two unknowns and it is easy to solve for the radii. With the radius and angle, the distance is found using D = R/theta.

J. Spectroscopic Eclipsing BinariesIn a double-lined spectroscopic binary, the projected size of the orbit a*sin(i) is found from the radial velocity amplitude and the period. In an eclipsing binary, the relative radii of the stars R1/a and R2/a and the inclination of the orbit i are found by analyzing the shapes of the eclipse light curves. Using the observed fluxes and colors to get surface brightnesses, the angular radii of the stars can be estimated. R1 is found from i, a*sin(i) and R1/a; and with theta1 the distance can be found.

For the hot O stars in the binary used to measure the distance to M33 the atmosphere contains a large number of free electrons. These scatter and reflect light without changing the spectrum. Thus the surface brightness can be lower than the surface brightness expected from the colors and line ratios if there is a larger than expected amount of electron scattering. The calculated distance would then be too large.

K. Expanding Photosphere MethodThe Baade-Wesselink method can be applied to an expanding star: the variations in radius do not have to be periodic. It has been applied to Type II supernovae, which are massive stars with a hydrogen rich envelope that explode when their cores collapse to from neutron stars. It can also be applied to Type Ia supernovae, but these objects have no hydrogen lines in their spectra. Since the surface brightness vs color law is calibrated using normal, hydrogen-rich stars, the EPM is normallyused on hydrogen-rich supernovae, which are Type II. The Type II SN1987A in the Large Magellanic Cloud has been used to calibrate this distance indicator.

The following methods use the H-R diagram of stars, which gives the luminosity as a function of temperature. When the luminosity and flux of an object are known, the distance can be found using D = sqrt[L/(4*pi*F)]

L. Main Sequence FittingWhen distances to nearby stars were found using trigonometric parallaxes in the late 19th and early 20th century, it became possible to study the luminosities of stars. Einar Hertzsprung and Henry Norris Russell both plotted stars on a chart of luminosity and temperature. Most stars fall on a single track, known as the Main Sequence, in this diagram, which is now known as the H-R diagramafter Hertzsprung and Russell. Often the absolute magnitude is used instead of the luminosity, and the spectral type or color is used instead of the temperature.

When looking at a cluster of stars, the apparent magnitudes and colors of the stars form a track that is parallel to the Main Sequence, and by correctly choosing the distance, the apparent magnitudes convert to absolute magnitudes that fall on the standard Main Sequence.

M. Spectroscopic ParallaxWhen the spectrum of a star is observed carefully, it is possible to determine two parameters of the star as well as the chemical abundances in the star's atmosphere. The first of these two parameters isthe surface temperature of the star, which determines the spectral type in the range OBAFGKM from hottest to coolest. The hot O stars show ionized helium lines, the B stars show neutral helium lines, the A stars have strong hydrogen lines, the F and G stars have various metal lines, and the coolest K and M stars have molecular bands. The spectral classes are further subdivided with a digit, so the Sun is a G2 star.

The second parameter that can be determined is the surface gravity of the star. The higher the surface gravity, the higher the pressure in the atmosphere, and high pressure leads to line broadening and also reduces the amount of ionization in the atmosphere. The surface gravity is denoted by the dwarfs while stars with medium gravity (class III) are called giants and stars with low gravity (class I) are called supergiants. The use of surface gravity to determine the luminosity of a star depends on three relations:

L = 4*pi*sigma*T4*R2

L = A*Mb Mass-luminosity law with b = 3-4g = G*M/R2

Given the temperature from the spectral type, and the surface gravity from the luminosity class, these equations can be used to find the mass and luminosity. If the luminosity and flux are known the distance follows from the inverse square law.

One warning about this method: it only works for normal stars, and any given single object might not be normal. Main sequence fitting in a cluster is much more reliable since with a large number ofstars it is easy to find the normal ones.

The following methods use the properties of pulsating stars:

N. RR Lyrae DistanceRR Lyrae stars are pulsating stars like Cepheids, but they are low mass stars with short periods (lessthan a day). They are seen in globular clusters, and appear to all have the same luminosity. Since themasses of RR Lyrae stars are determined by the masses of stars which are evolving off the main sequence, this constant luminosity may be caused by the age similarity in globular clusters.

O. Cepheid DistanceCepheid variable stars are pulsating stars, named after the first known member of the class, Delta Cephei. These stars pulsate because the hydrogen and helium ionization zones are close to the surface of the star. This more or less fixes the temperature of the variable star, and produces an instability strip in the H-R diagram.

The diagram above shows the star getting bigger and cooler, then smaller and hotter. Cepheids are brightest when they are hottest, close to the minimum size. Since all Cepheids are about the same temperature, the size of a Cepheid determines its luminosity. A large pulsating object naturally has alonger oscillation period than a small pulsating object of the same type. Thus there is a period-luminosity relationship for Cepheids. If we have two Cepheids with periods that differ by a factor of two, the longer period Cepheid is approximately 2.5 times more luminous than the short period one. Since it is easy to measure the period of a variable star, Cepheids are wonderful for determining distances to galaxies. Furthermore, Cepheids are quite bright, so they can be seen in galaxies as far away as the Virgo cluster, such as M100 The only problem with Cepheids is the calibration of the period-luminosity relation, which must be done indirectly using Cepheids in the Magellanic clouds and Cepheids in star clusters with distances determined by main sequence fitting.And one has to worry that the calibration could depend on the metal abundance in the Cepheids, which is much lower in the LMC than in luminous spirals like M100.

The following methods use the properties of objects in galaxies and must be calibrated:

P. Planetary Nebula Luminosity FunctionPlanetary nebulae are stars which have evolved through the red giant and asymptotic giant phases, and have ejected their remaining hydrogen envelope, which forms an ionized nebula surrounding a very hot and small central star. They emit large amounts of light in the 501 nm line of doubly ionized oxygen [O III] which makes them easy to find. The brightest planetary nebulae seem to have the same brightness in many external galaxies, so their fluxes can be used as a distance indicator. This method is correlated with the Surface Brightness Fluctuation method, which is sensitive to the asymptotic giant branch (AGB) stars before they eject their envelopes.

Q. Brightest StarsWhen a galaxy is very nearby, individual stars can be resolved. The brightness of these stars can be used to estimate the distance to the galaxy. Often people assume that there is a fixed upper limit to the brightness of stars, but this appears to be a poor assumption. Nonetheless, if a large population of bright stars is studied, a reasonable distance estimate can be made.

R. Largest H II Region DiametersHot luminous stars ionize the hydrogen gas around them, producing an H II region like the Orion nebula. The diameter of the largest H II region in external galaxies has been taken as a "standard rod" that can be used to determine distances. But this appears to be a poor assumption.

S. Surface Brightness FluctuationsWhen a galaxy is too distant to allow the detection of individual stars, one can still estimate the distance using the statistical fluctuation in the number of stars in a pixel. A nearby galaxy might have 100 stars projected into each pixel of an image, while a more distant galaxy would have a larger number like 1000. The nearby galaxy would have +/- 10% fluctuations in surface brightness (1/sqrt(N)), while the more distant galaxy would have 3% fluctuations. A figure [75 kB] to illustratethis shows a nearby dwarf galaxy, a nearby giant galaxy, and the giant galaxy at a distance such that its total flux is the same as that of the nearby dwarf. Note that the distant giant galaxy has a much smoother image than the nearby dwarf.

T. Type Ia SupernovaeType Ia supernovae are the explosions of white dwarf stars in binary systems. Accretion from a companion raises the mass above the maximum mass for stable white dwarfs, the Chandrasekhar limit. The white dwarf then starts to collapse, but the compression ignites explosive carbon burning leading to the total disruption of the star. The light output comes primarily from energy produced bythe decay of radioactive nickel and cobalt produced in the explosion. The peak luminosity is correlated with the rate of decay in the light curve: less luminous supernovae decay quickly while more luminous supernovae decay slowly. When this correction is applied, the relative luminosity of a Type Ia SN can be determined to within 20%. A few SNe Ia have been in galaxies close enough tous to allow the Hubble Space Telescope to determine absolute distances and luminosities using Cepheid variables, leading to one of the best determinations of the Hubble constant. Type Ia supernovae can be seen to such great distances that one can measure the acceleration or curvature ofthe Universe using observations of faint supernovae.

The following methods use the global properties of galaxies and must be calibrated:

U. Tully-Fisher RelationThe rotational velocity of a spiral galaxy is an indicator of its luminosity. The relation is approximately

L = Const * V(rot)4

Since the rotational velocity of a spiral galaxy can be measured using an optical spectrograph or radio telescopes, the luminosity can be determined. Combined with the measured flux, this luminosity gives the distance. The diagram below shows two galaxies: a giant spiral and a dwarf spiral, but the small galaxy is closer to the Earth so they both cover the same angle on the sky and have the same apparent brightness.

But the distant galaxy has a greater rotational velocity, so the difference between the redshifted and blueshifted sides of this distant giant galaxy will be larger. Thus the relative distances of the two galaxies can be determined.

V. Faber-Jackson RelationThe stellar velocity dispersion sigma(v) of stars in an elliptical galaxy is an indicator of its luminosity. The relation is approximately

L = Const * sigma(v)4

Since the velocity dispersion of an elliptical galaxy can be measured using an optical spectrograph, the luminosity can be determined. Combined with the measured flux, this luminosity gives the distance.

W. Brightest Cluster GalaxiesThe brightest galaxy in a cluster of galaxies has been used as a standard candle. This assumption suffers from the same difficulties that plague the brightest star and largest H II region methods: rich clusters with many galaxies will probably have examples of the most luminous galaxies even though these galaxies are very rare, while less rich clusters will probably not have such luminous brightest members.

The following methods require no calibration:

X. Gravitational Lens Time DelayWhen a quasar is viewed through a gravitational lens, multiple images are seen, as shown in diagram below.

The light paths from the quasar to us that form these images have different lengths that differ by approximately D*[cos(theta1)-cos(theta2)] where theta is the deflection angle and D is the distance to the quasar. Since quasars are time variable sources, we can measure the path length difference by looking for a time-shifted correlated variability in the multiple images. As of the end of 1996, this time delay has been measured in a few quasars: the original double QSO 0957+061, giving a result of Ho = [63 +/- 12] km/sec/Mpc; PG1115+080, giving a result of Ho = 42 km/sec/Mpc, but another analysis of the same data gives Ho = [60 +/- 17] km/sec/Mpc; B1600+434 giving Ho = [52+14-8] km/sec/Mpc; B1608+656 giving Ho = [63+/-15] km/sec/Mpc; and 0218+357 giving a result of Ho = [71+17-23] km/sec/Mpc.

Y. Sunyaev-Zeldovich EffectHot gas in clusters of galaxies distorts the spectrum of the cosmic microwave background observed through the cluster. The diagram below shows a sketch of this process. The hot electrons in the cluster of galaxies scatter a small fraction of the cosmic microwave background photons and replacethem with slightly higher energy photons.

The difference between the CMB seen through the cluster and the unmodified CMB seen elsewhereon the sky can be measured. Actually only about 1% of the photons passing through the cluster are scattered by the electrons in the hot ionized gas in the cluster, and these photons have their energies increased by an average of about 2%. This leads to a shortage of low energy photons of about 0.01*0.02 = 0.0002 or 0.02% which gives a decrease in the brightness temperature of about 500 microK when looking at the cluster. At high frequencies (higher than about 218 GHz) the cluster appears brighter than the background. Plots of the spectrum can be found here. This effect is proportional to (1) the number density of electrons, (2) the thickness of the cluster along our line of sight, and (3) the electron temperature. The parameter that combines these factors is called the Kompaneets y parameter, with y = tau*(kT/mc2). Tau is the optical depth or the fraction of photons

scattered, while (kT/mc2) is the electron temperature in units of the rest mass of the electron.

The X-ray emission, IX, from the hot gas in the cluster is proportional to (1) the square of the number density of electrons, (2) the thickness of the cluster along our line of sight, and (3) depends on the electron temperature and X-ray frequency. As a result, the ratio

y2/IX = CONST * (Thickness along LOS) * f(T)

If we assume that the thickness along the LOS is the same as the diameter of the cluster, we can use the observed angular diameter to find the distance.

This technique is very difficult, and years of hard work by pioneers like Mark Birkinshaw yielded only a few distances, and values of Ho that tended to be on the low side. Recent work with close packed radio interferometers operating at 30 GHz has given precise measurements of the radio brightness decrement for 18 clusters, but only a few of these have adequate X-ray data. A recent Sunyaev-Zeldovich determination of the Hubble constant gave 77 +/- 10 km/sec/Mpc from 38 clusters.

Big new instruments for measuring the Sunyaev-Zeldovich effect include the South Pole Telescope the SZA, and the Atacama Cosmology Telescope.

And finally:

Z. The Hubble LawThe Doppler shift gives the redshift of a distant object which is our best indicator of its distance, butwe need to know the Hubble constant, Ho. Then

D = VR/Ho

But the measured value of the Hubble constant has changed by a factor of 8 since Hubble's work, asdiscussed in Huchra's Ho history.

But wait, there's MORE! Pulsar dispersion measures and interstellar extinction increase with distance along a given line of sight and can be used to determine distances. The peak luminosity of a classical nova can be estimated from its rate of decay, but the variation has the opposite sense to that of Type Ia SNe: more luminous novae decay more rapidly. The globular cluster luminosity function can be used to estimate the distance to a galaxy from the observed brightness of its globular clusters.

Popular Cosmology Books "The Big Bang" by Simon Singh. History of cosmology from the ancient Greeks through the

cosmic microwave background. "The Fabric of the Cosmos" by Brian Greene. A non-technical look at all the hard problems

in physics and cosmology, by a very good writer. Greene also wrote "The Elegant Universe" which is all about string theory,

"Echo of the Big Bang", by Michael Lemonick. A non-technical history of the WMAP mission.

"The Extravagant Universe" by Bob Kirshner. A "useful and polite" and always entertaining look at the supernova evidence for the cosmological constant and the accelerating expansionof the Universe.

"The Origin and Evolution of the Universe" edited by Zuckerman and Malkan. A UCLA Center for the Study of the Evolution and Origin of Life (CSEOL) Symposium. 1996, Jones and Bartlett Publishers. I wrote the leadoff article in this collection.

"The Very First Light", by John Mather and John Boslough, 1996. BasicBooks. The true history of the COBE project written by the Project Scientist.

"Afterglow of Creation" by Marcus Chown. Good science journalism. "The Five Ages of the Universe : Inside the Physics of Eternity" by Fred Adams & Greg

Laughlin. -- The very distant future of our Universe. "Our Evolving Universe" by Malcolm Longair. -- A book with beautiful pictures and a good

non-technical introduction to cosmology. "Einstein's Greatest Blunder? : The Cosmological Constant and Other Fudge Factors in the

Physics of the Universe" by Donald Goldsmith. I haven't read "The Runaway Universe : TheRace to Discover the Future of the Cosmos" by the same author, but it covers the accelerating expansion seen in the recent supernova data.

"Quintessence : The Mystery of the Missing Mass in the Universe" by Lawrence Krauss. An update of "The Fifth Essence". Dark Matter.

"The Origin of the Universe" by John D. Barrow. Covers a wide range of topics: inflation, including the chaotic and perpetual versions, wormholes and space-time foam, the anthropic principle, with nice illustrations.

"The Origins of our Universe" by Malcolm Longair. A small book based on the Christmas lectures at the Royal Society. Out of print.

"Before the Beginning : Our Universe and Others" by Martin Rees. There are no equations, but also no graphs, figures or illustrations of any kind.

"Ripples in the Cosmos" by Michael Rowan-Robinson. Has the same erroneous T vs t graph as "Shadows of Creation". Out of print.

"The Shadows of Creation", by Michael Riordan and David Schramm. Dark matter. Has the same erroneous T vs t graph as "Ripples in the Cosmos". Out of print.

"Coming of Age in the Milky Way" by Timothy Ferris. A more extensive coverage of the material in the "Creation of the Universe" show on PBS.

"The Alchemy of the Heavens: Searching for Meaning in the Milky Way", by Ken Croswell. A good book about our Galaxy.

"The First Three Minutes" by Steven Weinberg. Old but good treatment of light element synthesis. Out of print.

"The Big Bang" by Joseph Silk. Many typos, unfortunately. "A Short History of the Universe" by Joseph Silk. Many typos, unfortunately.

College Introductory Astronomy Textbooks "Astronomy: The Evolving Universe", Michael Zeilik. "Astronomy: the Cosmic Perspective", Zeilik & Gaustad "Journey Through the Universe", Jay Pasachoff "Astronomy: From the Earth to the Universe", Jay Pasachoff "Abell's Exploration of the Universe", David Morrison, Sidney Woolf & Andrew Fraknoi "Realm of the Universe", George O. Abell, Sidney Woolf & David Morrison

College Astrophysics Textbooks "Introductory Astronomy and Astrophysics", Zeilik, Gregory & Smith "An Introduction to Modern Astrophysics", Bradley Carroll & Dale Ostlie

College Cosmology Textbooks "Cosmology", 4th edition, Michael Rowan-Robinson. A recently updated classic. "Introduction to Cosmology", Barbara Ryden. A new upper division textbook. "Foundations of Modern Cosmology", Hawley and Holcomb. For upper division

non-science majors.

Graduate Cosmology Textbooks "Spacetime and Geometry: An Introduction to General Relativity" by Sean Carroll. GR with

chapters on applications including cosmology. "Gravity: An Introduction to Einstein's General Relativity" by James Hartle. Good

introduction to GR. "Gravitation and Cosmology" by Steven Weinberg. "Cosmological Physics", John Peacock. All of physics relevant to cosmology, which is all of

physics. "Principles of Physical Cosmology" by P. J. E. Peebles. "The Early Universe" by Kolb and Turner. "Gravitation" by Misner, Thorne and Wheeler. Also known is the "Bible" or the "telephone

book", this is primarily a very good textbook on general relativity. 1215 pages, 2.5 kg in paperback!

Ned Wright's Cosmology TutorialCritical ObservationsThe Universe is Homogeneous and Isotropic

Homogeneity and IsotropyThe Universe is Homogeneous and IsotropicMany DistancesScale Factor a(t)Spatial CurvatureFlatness-Oldness ProblemManipulating Space-Time DiagramsHorizon ProblemInflationDetails: Large-Scale Structure and Anisotropy

Age of the UniverseThe Age of the ElementsRadioactive Dating of an Old Star

The Age of the Oldest Star ClustersThe Age of the Oldest White DwarfsSummary

The ABC's of DistancesA. TRIGONOMETRIC PARALLAXB. Moving ClustersC. Secular ParallaxD. Statistical ParallaxE. Kinematic DistanceF. Expansion ParallaxG. Light Echo DistanceH. Spectroscopic Visual BinariesI. Baade-Wesselink MethodJ. Spectroscopic Eclipsing BinariesK. Expanding Photosphere MethodL. Main Sequence FittingM. Spectroscopic ParallaxN. RR Lyrae DistanceO. Cepheid DistanceP. Planetary Nebula Luminosity FunctionQ. Brightest StarsR. Largest H II Region DiametersS. Surface Brightness FluctuationsT. Type Ia SupernovaeU. Tully-Fisher RelationV. Faber-Jackson RelationW. Brightest Cluster GalaxiesX. Gravitational Lens Time DelayY. Sunyaev-Zeldovich EffectZ. The Hubble Law

Popular Cosmology BooksCollege Introductory Astronomy TextbooksCollege Astrophysics TextbooksCollege Cosmology TextbooksGraduate Cosmology Textbooks