Welcome to Ned Wright's Cosmology Tutorial Frequently Asked Questions Cosmology, Religion & Kansas Send me your comments Cosmology is the study of the origin, current state, and future of our Universe. This field has been revolutionized by many discoveries made during the past century. My cosmology tutorial is an attempt to summarize these discoveries. It will be "under construction" for the forseeable future as new discoveries are made. I will attempt to keep these pages up-to-date as a resource for the cosmology courses I teach at UCLA. The tutorial is completely non-commercial, but tax deductible donations to UCLA are always welcome. 

Astronomy and cosmology are very much mathematical sciences, but I have attempted to avoid higher math in these pages. I do use high school algebra and geometry - courses required for admission to UCLA - but I have also included some animations [1, 2, 3, 4], some Java applets [1, 2], and many illustrations in the tutorials, the ABC's of Distances, and the answers to some of the Frequently Asked Questions. 

In addition to the cosmology tutorial, there is also a relativity tutorial and extensive discussions on the age, density and size of the Universe. There is also a bibliography of books at a range of levels, and a Javascript calculator of the many distances involved in cosmology. The course notes (0.7 Mb Postscript, 96 pages, 349 equations, 29 figures) for the upper division undergraduate Stellar Systems and Cosmology course, Astronomy 140, that I taught in spring 2000 are still available on the Web. 

And for a much more technical discussion of cosmology with 335 equations and 29 Figures, see my graduate course Astro 275 lecture notes (0.7 Mb Postscript). Use Ghostview to view this file, or save it to disk and send it to a Postscript printer. 

News of the Universe Faster than light? - No! 20 Jul 2000 - Today's Nature has an article reporting on "superluminal" propagation, but it is just anomalous dispersion. The media are fooled again! Cosmic Microwave Background Anisotropy 9 May 2000 - Hanany et al. announce reults of the August 98 flight of the MAXIMA experiment which are generally in agreement with the BOOMERanG results below, but without any tendency to favor a slightly closed Universe. My CMB angular power spectrum graphs have been updated to include MAXIMA. 27 Apr 2000 - Today's Nature has an article reporting the results from the 10 day long duration balloon flight of the BOOMERanG project. These measurements of very small CMB temperature fluctuations over 1% of the sky with a beam size 40 times smaller than COBE's 7o beam confirm earlier work by a group at Penn & Princeton and data from BOOMERanG's test flight, but provide 3 to 4 times more accuracy. Based on this data, de Bernadis et al. (2000, Nature, 404, 955) conclude that the Universe is flat. The angular size of the characteristics spots on the sky, about 1o, shows that the Universe is flat, or that the total energy density is equal to the critical density. This confirms a prediction of the inflationary scenario. The amplitude of the characteristics spots, about 69 microK, indicates the ratio of ordinary matter (baryonic matter) to dark matter. Detailed studies of the harmonics or overtones of the fundamental oscillations that lead to these spots will provide much more information about the Universe. Stay tuned. The New York Times, the Los Angeles Times, and the Washington Post all covered this story extensively. The first of what will certainly be many interpretations of the BOOMERanG data by outside groups was posted to the LANL preprint server on the same day the Nature paper came out by White, Scott & Pierpaoli. 

Sloan Digital Sky Survey finds z=5.82 Quasar 13 Apr 2000 - SDSS announced a quasar with a redshift of z = 5.82. Of course, the press release called it the "Most Distant Object Ever Observed" which discounts the z = 6.68 galaxy. However, the spectrum of the SDSS object taken by Fan, White, Davis, Becker et al. at the Keck Observatory is much better than the noisy spectrum of the extremely faint z=6.68 object. Notice the very strong and wide Lyman alpha line of hydrogen, redshifted from 122 nm to 829 nm wavelength. The blue side of the line has been absorbed in the Lyman alpha forest. 

New Most Distant Quasar 18 Feb 2000 - Daniel Stern et al. announce the discovery of the most distant known quasar, with a redshift of z = 5.5. The previous record holder has a redshift of z = 5.0. Cosmic Microwave Background Anisotropy 26 Nov 99 - The New York Times had a front page story about new CMB temperature fluctuation data released by the BOOMERanG project. Based on this data, which confirms earlier work by a group at Penn & Princeton, they conclude the Universe is flat. 

Cosmic Near Infrared Backgound 27 Sep 99 - Gorjian, Wright(*) & Chary have found an extragalactic IR background of 22.4 +/- 6.0 nW/m2/sr at 2.2 microns and 11.0 +/- 3.3 nW/m2/sr at 3.5 microns. This result completes the primary goals of the COBE mission. (*) - that's me. And then on 27 Dec 99 Wright & Reese confirmed the cosmic IR background (CIRB) obtaining 23.1 +/- 5.9 nW/m2/sr at 2.2 microns and 12.4 +\- 3.2 nW/m2/sr at 3.5 microns. 

Mystery Object! - NY Times publishes a SPECTRUM!! 21 Aug 99 - The "mystery object", PSS 1537+1227, found on the Digital Palomar Observatory Sky Survey (DPOSS) by Djorgovski et al. has been identified as a Broad Absorption Line quasar (BAL QSO) with a redshift of 1.2. The spectrum of the mystery object shown at right is very similar to the spectrum of the radio-loud BAL QSO 0840+3633 discussed by Becker et al. (1997), redshifted to z = 1.19. 

But the most amazing part of this story is that the NY Times published a spectrum in the 17 Aug 99 print edition - a clear violation of the "5 W's but no Z" rule that I think must be taught in science journalism schools. The graph at right is a plot of the flux received from the mystery object in microJanskies vs. the wavelength of the light in nanometers. Since spectroscopy is the tool most used by astronomers today, this article in the NY times was very significant. Spectra tell astronomers what elements are in distant stars and galaxies, how hot and dense these objects are, and how rapidly they are moving towards us or away from us. 

------------------------------------------ The ABC's of Distances It 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 PARALLAX This methods 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 is used 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 Clusters Not 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 at the 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 Parallax Another 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 Parallax When 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 Distance The 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 Parallax The distance to an expanding object like a supernova remnant like a supernova remnant such as Tycho can be determined by measuring: the angular expansion rate d(theta)/dt using pictures taken many years apart, and 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 remnant a 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 fact the case, with VR = 0.75 V(shock), so the calculated distance can be too small by 25%. 

G. Light Echo Distance The center elliptical ring around SN1987A in the LMC appears to be due to an inclined circular ring around the progenitor. When the pulse of ultraviolet light from the supernova hit the ring, it lit up in ultraviolet 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 the ring are: t1 - t0 = R(1 - sin(i))/c t2 - 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. (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 Binaries If 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 Method The 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 Binaries In 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. 

K. Expanding Photosphere Method The 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 normally used 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 Fitting When 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 diagram after 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 Parallax When 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 is the 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 luminosity class denoted by a Roman numeral from I to V with I being the lowest gravity and V being the highest (except for class VI which is sometimes seen and for white dwarfs which separately classified.) Stars with high surface gravity (class V) are called 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 = 3-4 g = 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 of stars it is easy to find the normal ones. 

The following methods use the properties of pulsating stars: 

N. RR Lyrae Distance RR Lyrae stars are pulsating stars like Cepheids, but they are low mass stars with short periods (less than a day). They are seen in globular clusters, and appear to all have the same luminosity. Since the masses 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 Distance Cepheid variable stars are pulsating stars, named after the brightest 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 a longer oscillation period than a small pulsating object of the same type. Thus there is a period-luminosity relationship for Cepheids. If 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 Function Planetary 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 Stars When 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 Diameters Hot 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 Fluctuations When 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 illustrate this 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 Supernovae Type 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 come primarily from energy produced by the decay of radioactive nickel and cobalt produced in the explosion. The peak luminosity is correlated with the rate of decay in the light curve. 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 to us 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. 

The following methods use the global properties of galaxies and must be calibrated: U. Tully-Fisher Relation The 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 Relation The 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 Galaxies The 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 Delay When 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 2 quasars: the original double QSO 0957+061, giving a result of Ho = [63 +/- 12] km/sec/Mpc, and 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. 

Y. Sunyaev-Zeldovich Effect Hot 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 replace them with slightly higher energy photons. 

The difference between the CMB seen through the cluster and the unmodified CMB seen elsewhere on 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 is 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. 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 3 of these have adequate X-ray data. 

And finally: 

Z. The Hubble Law The Doppler shift gives the redshift of a distant object which is our best indicator of its distance, but we need to know the Hubble constant. 

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 globular cluster luminosity function can be used to estimate the distance to a galaxy from the observed brightness of its globular clusters. 


Homogeneity and Isotropy

The 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 Distances 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 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: 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 time derivative of D. It is close to cz for small redshifts but deviates for large ones. 

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. The space-time diagram below shows a "zero" (really very low) density cosmological model plotted using the D 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 flatten these out to make the previous space-time diagram, the worldlines of the galaxies get flatter and giving velocities v = dD/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 D, which is measured now, of these most distant galaxies is infinity (in this model). 

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/DA  so  DA = size/theta where "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 = DA. Ask Dr. Science about the angular size distance :)

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: c*(to-tem). People who say that the greatest distance we can see is c*to are using this distance. But 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 rules 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 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*Tobs4)
  = DA2*(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. 

Scale Factor a(t)

Because the velocity or dD/dt is strictly proportional to D, 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 DG(t) = a(t)*DG(to)

where DG(to) is the distance 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 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)
  H2*D2 = 2*(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 = 65 km/sec/Mpc is 8E-30 = 8*10-30 gm/cc or 5 protons per cubic meter or 1.2E11 = 1.2*1011 solar masses per cubic Megaparsec. The latter can be compared to the observed 1.1E8 = 1.1*108 solar luminosities per Mpc3. 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 density is between 0.3 to 1 times the critical density, and this does require that most of the matter in the Universe is dark. 

Spatial Curvature One consequence of general relativity is that the curvature of space depends on the ratio of rho to rho(crit). We call this ratio Omega = rho/rho(crit). For Omega less than 1, the Universe has negatively curved or hyperbolic geometry. For Omega = 1, the Universe has Euclidean or flat geometry. For Omega 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. The age of the Universe depends on Omegao as well as Ho. For Omega=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, Omega=0, and a(t) = t/to     with    to = 1/Ho If Omegao 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 = 65 km/sec/Mpc and for Omegao = 0 (green), Omegao = 1 (black), and Omegao = 2(red). The age of the Universe is 15, 10 and 8.6 Gyr in these three models. The recollapse of the Omegao = 2 model occurs when the Universe is 11 times older than it is now, and all observations indicate Omegao < 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. Taking Ho = 65 +/- 8 and to = 14.6 +/- 1.7 Gyr, we find that Ho*to = 0.97 +/- 0.17. At face value this favors 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 Ho = 60 +/- 7 and to = 11.7 +/- 1.4 Gyr, we find that Ho*to = 0.72 +/- 0.12 which is perfectly consistent with a critical density Universe. It is best to reserve judgement until better data is obtained. 

Flatness-Oldness Problem

However, if Omegao is greater than 1, the Universe will eventually stop expanding, and then Omega will become infinite. If Omegao is less than 1, the Universe will expand forever and the density goes down faster than the critical density so Omega gets smaller and smaller. Thus Omega = 1 is an unstable stationary point, and it is quite remarkable that Omega 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 the critical density case with density = 447,225,917,218,507,401,284,016 gm/cc. Adding only 1 gm/cc to this 447 sextillion gm/cc causes the Big Crunch to be right now! Taking away 1 gm/cc gives a model with Omega 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 in 447 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 Omegao were close to 1 but not exactly 1. Manipulating Space-Time Diagrams 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 Omegao = 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 below shows 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 which has 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 Omegao = 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 Omegao = 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 Problem The 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 = 1000. 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 0.001 at recombination. For the Omegao = 1 model this implies that t/to = 0.00003 so for to about 10 Gyr the time is about 300,000 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. 

Inflation The "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 that the density of the vacuum is less than 1E-30 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 2E91 gm/cc. Thus the vacuum energy density is at least 121 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 = 10 Gyr, and the apparent age of the oldest globular clusters, 16+/-4 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(to-t)). 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 very large 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 Anisotropy

Of 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 and Wolfe (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 less dense 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 by dT/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. The red 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 = 370 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 \etal in 1992. The amplitude was 1 part in 100,000 instead of 1 part in 100:


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 effect very small regions of space, but the huge exponential expansion during the inflationary epoch makes 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. 

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's dT 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 on the 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. This strange conclusion will be greatly strengthened by temperature anisotropy data at smaller angular scales which will be provided by the Microwave Anisotropy Probe (MAP) in the year 2000. 

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 Elements The 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 the solidification 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 calcium in different grains in a differentiated rock. Rubidium and strontium are heavier elements that behave chemically 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 there is 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 elements are 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

Radioactive Dating of an Old Star A 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 it is 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. 

The Age of the Oldest Star Clusters When 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 globular clusters 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 Dwarfs A 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 in the 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. 

-------------------------------- Relativity Tutorial Galilean Relativity Relativity can be described using space-time diagrams. Contrary to popular opinion, Einstein did not invent relativity. Galileo preceded him. Aristotle had proposed that moving objects (on the Earth) had a natural tendency to slow down and stop. This is shown in the space-time diagram below. 

Note the curved worldline above. This shows a variable velocity, or an acceleration. Galileo objected to Aristotle's hypothesis, and asked what happened to an object moving on a moving ship. 

Now it is still moving in its final state. Galileo proposed that it is only relative velocities that matter. Thus a space-time diagram can be transformed by painting it on the side of a deck of cards, and then skewing the deck to one side -- but keeping the edges along a straight line: 

Straight worldlines (unaccelerated particles) remain straight in this process. Thus Newton's First Law is preserved, and non-accelerated worldlines are special. This Galilean transformation does not affect the time. Thus two observers moving with respect to each other can still agree on the time, and thus the distance between two objects, which is the difference in their positions measured at equal times, can be defined. This allowed Newton to describe an inverse square law for gravity. 

But Galilean transformations do not preserve velocity. Thus the statement "The speed limit is 70 mph" does not make sense -- but don't try this in court. According to relativity, this must be re-expressed as "The magnitude of the relative velocity between your car and the pavement must be less than 70 mph". Relative velocities are OK. 

Special Relativity But 200 years after Newton the theory of electromagnetism was developed into Maxwell's equations. These equations describe waves with a speed of 1/sqrt(epsilono*muo), where epsilono is the constant describing the strength of the electrostatic force in a vacuum, and muo is the constant describing the strength of the magnetic interaction in a vacuum. This is an absolute velocity -- it is not relative to anything. The value of the velocity was very close to the measured speed of light, and when Hertz generated electromagnetic waves (microwaves) in his laboratory and showed that they could be reflected and refracted just like light, it became clear that light was just an example of electromagnetic radiation. Einstein tried to fit the idea of an absolute speed of light into Newtonian mechanics. He found that the transformation from one reference frame to another had to affect the time -- the idea of sliding a deck of cards had to be abandoned. This led to the theory of special relativity. In special relativity, the velocity of light is special. Anything moving at the speed of light in one reference frame will move at the speed of light in all unaccelerated reference frames. Other velocities are not preserved, so you can still try to get lucky on speeding tickets. 

Because the speed of light is special, space-time diagrams are often drawn in units of seconds and light-seconds, or years and light-years, so a unit slope [45 degree angle] corresponds to the speed of light. The set of all light speed world lines going through an event defines the light cones of that event: the past light cone and the future light cone. An example of light cones is shown above. The fancy light picture on the left shows both the past and future light cones of the event where the two worldlines cross, while the schematic version on the right is easy to use in more complicated diagrams. Thus in the situation shown in 3 space-time diagrams below, the central section shows the worldline of one stationary observer, one observer moving to the right, and two events on the future light cone on the event where the two observers' worldlines cross. 

The left-hand section of the figure shows the Galilean transformation into the frame of reference of the moving observer. The events on the future light cone have shifted to the left, but they are still at the same time. Since the coordinates x and t just provide a way of describing space-time, and are not the space-time themselves, the two events are still on the future light cone. But now slopes of the light rays have changed, so the speed of light has changed. The Lorentz transformation appropriate for special relativity is shown on the right hand of the figure. The events on the future light cone have shifted to the left as before, but now their times have changed, so the slopes of the light rays do not change. The speed of light is invariant in Einstein's special relativity. 

What is the evidence for the invariance of the speed of light? The hypothesis that the speed of light is c relative to its source can easily be disproved by the one-way transmission of light from distant supernovae. When a star explodes as a supernova, we see light coming from material with a large range of velocities dv, at least 10,000 km/sec. Because of this range of velocities, the spectral lines of a supernova are very broad due to the Doppler shift. After traveling a distance D in time D/c, the arrival time of the light would be spread out by dt = (dv/c)(D/c). 


However, this DOES NOT happen. For the Crab supernova, with D/c = 6000 years, dv = 10,000 km/sec would give a range of arrival times of 200 years. But the Crab was only bright for 1 year. For very distant supernovae with D/c = 5 billion years, modern observations with spectrographs show that the redshifted and blueshifted light arrives at the same time: within 10 days. This limit on the spread is 5 billion times smaller than the prediction of the "bullet" model of light. However, light could travel at speed c relative to a medium -- the ether. If it did, then the rate of a "bouncing photon clock" moving with respect to the ether 


would depend on the angle between its photon bouncing axis and its velocity. A stationary bouncing photon clock has a period of P = 2L/c. If it is moving parallel to its axis at velocity v, and light moves at speed c with respect to the ether, then the speed relative to the clock when the photon is moving "upstream" is c-v and the one-way time is L/(c-v). When the photon is moving "downstream" the speed relative to the clock is c+v so the one-way time is L/(c+v). The period is the sum of these times: P(par) = L/(c-v) + L/(c+v) = [2L/c]/(1-v2/c2). If the clock is moving perpendicular to its axis, the light has to move a distance L sideways and a distance vt "upstream" to keep up with the clock, where t is the one-way time. The total distance traveled is ct, which is the hypotenuse of a right triangle with sides L and vt. Thus the period is given by: (ct)2 = L2 + (vt)2  so  t = L/sqrt(c2-v2) P(perp) = 2t = [2L/c]/sqrt(1-v2/c2). Thus the ether model predicts that dP/P = [P(par)-P(perp)]/P = 0.5*v2/c2. Brillet and Hall (1979, PRL, 42, 549) actually built a bouncing photon clock (a laser stabilized to a Fabry-Perot etalon) on a rotating table and compared its rate to an atomic clock (a laser stabilized to a methane line). 


The observed dP/P was (1.5 +/- 2.5)*10-15. For the minimum possible velocity of 30 km/sec, due to the orbit of the Earth around the Sun, this is at least a million times smaller than the ether model prediction. The 370 km/sec velocity of the Solar System with respect to the cosmic background radiation gives an ether model prediction 100 million times larger than the Brillet-Hall limit. For this velocity even fourth order effects (v4/c4) can be strongly ruled out. Michelson and Morley used two bouncing photon clocks at right angles to each other, but without the lasers and counters which didn't exist. This left an L-shaped interferometer. But they were able to show that dP/P was essentially zero instead of the ether model prediction. 

Radar The constancy of the speed of light allows the use of radar (RAdio Detection And Ranging) to measure the position and time of events not on an observer's worldline. All that we need are a clock and the ability to emit and detect radar pulses. 

If we send out a radar pulse at time ts, which is reflected at the event E, and the echo arrives back at time tr, then we know that the light was traveling at c for the entire round-trip journey out to E and back, so the distance of E is D(E) = c*(tr-ts)/2. In our frame of reference we are stationary, and light travels at the same speed coming back from E as it did going out, so the time of the event E is obtained by averaging the send and receive times, t(E) = (tr+ts)/2. 

Time Dilation Armed with radar, we can determine the time of two events on the worldline of an observer moving with respect to us. We can then compare the time interval we measure to the time interval to the time interval measured by the moving observer. Consider the two observers A and B below. 

They both set their clocks to zero at the event Z where their worldlines cross. A sends out a radar pulse at the event S at time tA(S) = 1. This is received by observer B at event R with time tB(R) = k. The factor k depends on the relative velocity of A and B, but since the light takes some time to travel between A and B, we know that k will be larger than 1. The notation tA means times determined by A, while tB are times determined by B. If A sends out a pulse at some other time, tA = x, it will be received by B at time tB = x*k by the principle of similar triangles. In particular, if A sends out a pulse at time tA = k it will be received by B at time tB = k*k. Now consider the radar pulse reflected by B at event R. It starts at time tB = k. When will A receive it? Since the speed of A with respect to B is the same as the speed of B with respect to A, this time has to be tA(T) = k*k. We can now compute the distance of event R from A's worldline and time of event R according to A, tA(R). These are DA(R) = c(k*k-1)/2 and tA(R) = (k*k+1)/2. Thus the speed of B according to A is v = D/t = DA(R)/tA(R) = c(k*k-1)/(k*k+1). We can solve for k giving k = sqrt((1+v/c)/(1-v/c)) which is the relativistic Doppler shift formula. But we also find that tA(R) > tB(R), so A says that B's clock is running slow. The amount of this time dilation is (1+k*k)/(2*k) = 1/sqrt(1-v2/c2). Thus moving clocks run slow. Note that B will also find that A's clock is running more slowly than his. There is a symmetric disagreement about clock rates. This slow down factor is exactly the slow down calculated above in the ether model for a bouncing photon clock moving perpendicular to its bounce axis. The clock moving parallel to the axis slows down by the same amount under special relativity because of the Lorentz-Fitzgerald contraction of moving objects in the direction of motion.



The space-time diagrams above both show a rod moving past an observer. On the left the rod is moving, while on the right the same situation is shown in the rod's frame of reference. The observer moving with respect to the rod makes a radar determination of its length, as does an observer moving along with the rod. The observer on the rod sees a length of 5 light-ticks because it takes 10 ticks for light to make the round trip to the end of the rod and back. The observer moving with respect to the rod at v = 0.6*c measures only 8 ticks for the round trip and thus gives the length of the rod as 4 light-ticks. Thus the length of a moving rod appears to be reduced by a factor of sqrt(1-v2/c2). Thus length contraction changes P(par) for the bouncing photon clock to P(par) = [2L*sqrt(1-v2/c2)/c]/(1-v2/c2)
    = [2L/c]/sqrt(1-v2/c2) = P(perp) so the rate of a bouncing photon clock does not depend on the angle between its velocity and its bouncing axis. Because the clocks of different observers run at different rates, depending on their velocities, the time for a given observer is a property of that observer and his worldline. This time is called the proper time because it is "owned" by a given particle, not because it is the "correct" time. Proper time is invariant when changing reference frames because it is the property of a particle, not of the reference frame or coordinate system. In general, given any two events A and B with B inside the future light cone of A, there is one unaccelerated worldline connecting A and B, just as there is one straight line connecting two points in space. In the frame of reference of the observer following this unaccelerated worldline, his clock is always stationary, while clocks following any other worldline from A to B will be moving at least some of the time. Because moving clocks run slow, these observers will measure a smaller proper time between events A and B than the unaccelerated observer. Thus the straight worldline between two events has the largest proper time, and all other curved worldlines connecting the two events have smaller proper times. This is exactly analogous to the fact that the straight line between two points has the smallest length of all possible curves between the points. Thus the "twin paradox" is no more paradoxical than the statement that a man who drives straight from LA to Las Vegas will cover fewer miles than a man who drives from LA to Las Vegas via Reno. 


The pair of space-time diagrams above show quintuplets separated at birth. The middle worldline shows the quint who stays home. The space-time diagram on the left is done from the point of view of the middle quint. Each dot on a worldline is a birthday party, so the middle quint is 10 years old when they all rejoin each other, while the other quints are 6 and 8 years old. The space-time diagram on the right shows the same events from the point of view of an observer initially moving with one of the moving quints. When the quints come together their ages are still 6, 8, 10, 8, and 6 years. Thus the straight worldline between two events can be found by maximizing the proper time, just as the straight line between two points can be found by minimizing the length. 

General Relativity Now we come to a matter of gravity: how can gravity be an inverse square law force, when the distance between two objects can not even be defined in Einstein's special relativity? Special relativity was constructed to satisfy Maxwell's equations, which replaced the inverse square law electrostatic force by a set of equations describing the electromagnetic field. So gravity was the only remaining action-at-a-distance inverse square law force. And gravity has a unique property; the acceleration due to gravity at a given place and time is independent of the nature of the body. 


The space-time diagram above shows a proton and an antiproton moving under the influence of an electric field on the left and a gravitational field on the right. Gravity accelerates all objects equally. This fact was known to Newton, and tested to an accuracy of 1 part in 100 million by Eotvos. Later work by Dicke and by Braginsky has improved the accuracy of the test to 1 part in a trillion. Thus through any event in space-time, in any given direction, there is only one worldline corresponding to motion solely influenced by gravity. Compare this to the geometric fact that through any point, in any given direction, there is only one straight line. We are led to propose that worldlines influenced only by gravity are really straight worldlines. But how can an accelerating body have a straight worldline? It all depends on how you measure it. Suppose we plot a straight line on polar graph paper, and then make a plot of radius vs angle as shown below? 


In the radius vs theta plot the straight line is curved. I have shown adjacent two lines dashed and labeled just like the proton and antiproton shown earlier to emphasize that while there are an infinite number of curved lines in the radius vs theta plot, there is only one straight line through the initial point with the initial direction. Principle of Equivalence Einstein proposed that the effects of gravity (in a small region of spacetime) are equivalent to the effect of using an accelerated frame of reference without gravity. As as example, consider the famous "Einstein elevator" thought experiment. If an elevator far out in space accelerates upward at 10 meters/second2, it will feel like a downward acceleration of gravity at 1 g = 10 m/s2. If a clock on the ceiling of the elevator emits flashes of light f times per second, an observer on the floor will see them arriving faster than f times per second because of the Doppler shift due to the acceleration of the elevator during the light transit time. 


The space-time diagram on the left above shows the clock on the elevator ceiling emitting flashes of light. The light transit time is h/c where h is the height of the ceiling, and the velocity change is a*h/c so the Doppler shift increases the rate of flash arrival by a factor of (1+a*h/c2), so the flash arrival rate is f' = (1+a*h/c2)*f. On the right is the same situation with stationary clocks in a gravitational field. In order to have the flash arrival rate faster by a factor of (1+g*h/c2), the clock on the ceiling must run faster by this factor. In other words, clocks run faster when they are high up in a gravitational field. This effect has been seen in the laboratory by Pound and Rebka (1960, PRL, 4, 337) who used the Mossbauer effect to measure a frequency shift (f'/f -1) = (2.57+/-0.20)*10-15 after dropping photons a distance of 22.6 meters. The expected shift was 2.46*10-15. The effect of gravity on clocks was tested to greater precision by Vessot etal (1980, PRL, 45 2081) who launched a hydrogen maser straight up at 8.5 km/sec, and watched its frequency change as it coasted up to 10,000 km altitude and then fell back to Earth. The frequency shift due to gravity was (f'/f -1) = 4*10-10 at 10,000 km altitude, and the experimental result agreed to within 70 parts per million of this shift. 

Because of the gravitational speedup for uphill clocks, an observer moving between two events can achieve a larger proper time by shifting his worldline upward in the middle. Going too far upward requires moving so fast that time dilation due to motion reduces the proper time more than the gravitational speedup, so there is an optimum curvature to the worldline that maximizes the proper time. 


The space-time diagram on the left above shows 9 observers moving between two events with different accelerations. The third from the right has the correct balance between going uphill to get a faster clock rate and avoiding motion to avoid time dilation. As a result, this observer has the largest proper time between the two events. Note that the accelerations are negative for paths that have a maximum height, so the third worldline from the right is plotted as the third dot from the left on the chart. The optimum worldline curvature is the acceleration of gravity, and it is negative because things fall down, not up. 

Curved Spacetime Curved coordinates alone, such as the polar graph, do not provide a satisfactory model for gravity. Two straight lines through the same point but with different directions will never cross again, while two worldlines influenced only by gravity which pass through the same event with different velocities can cross again. Consider the Galileo spacecraft, which made two Earth flybys. In between the flybys, Galileo was on an elliptical orbit with a 2 year period. In order to allow "straight" lines to cross multiple times, a curved space-time is needed. As a familiar example of a curved space, consider the surface of the Earth and the great circle arc connecting two cities. The great circle is the shortest distance between two points on the surface of the Earth, and it is the path followed by airliners. 
The great circle path from Los Angeles (34 N, 118 W) to Tel Aviv (32 N, 35 E) goes all the way to 70 N latitude. Plotting latitude vs longitude, as if longitude were time and latitude position, gives the pseudo-spacetime diagram below.

The two great circles through Los Angeles, one to Tel Aviv and one to Singapore, are both "straight" lines, but they intersect in two places. This is impossible in plane geometry but it does occur in non-Euclidean geometry. The pseudo-spacetime diagram above is almost identical to a real spacetime diagram for objects moving in a tunnel drilled through the center of a massive sphere. Gravity produces oscillatory motions so worldlines for different objects, each influenced only by gravity, can cross at many events. Thus the fundamentals of relativity that are important for cosmology are: The speed of light is a constant independent of the velocity of the source or the observer. Events that are simultaneous as seen by one observer are generally not simultaneous as seen by other observers, so there can be no absolute time. Each observer can define his own proper time -- the time measured by a good clock moving along his worldline. Observers can assign times and positions to events not on their worldlines using radar observations. Every observer will see his clock running faster than other clocks which are moving with respect to him, and this is a mathematically consistent pattern required by the properties of radar observations. As a result, the unaccelerated worldline between two worldlines will have the longest proper time of all worldlines connecting these events. In the presence of gravity, the worldlines of objects accelerated only by gravity have the longest proper times. Gravity requires that spacetime have a non-Euclidean geometry, and this curvature of spacetime must be created by matter. There are many books on relativity available, but two that stick to a simple level of mathematics are: "Relativity and Common Sense" by Hermann Bondi "General Relativity from A to B" by Robert Geroch 

3. The Standard Model of the Universe 3.1 Line spectrum
  When gases are heated they absorb energy. Electrons move to higher energy levels (greater potential energy) Electromagnetic radiation is emitted when an electron returns to a lower energy level. The energy emitted is equal to the difference in energy levels between the two states. This is the photon energy given by E = hf , and the corresponding wavelength is given by w = c/f .
  When light from a heated gas is analysed using a spectrometer the different wavelengths are clearly visible as a set of distinct lines. This is the emission spectrum of that gas. Each gas is characterised by a unique set distinct lines (wavelengths) . This is a result of the differences in the atomic structure of each element. No single wavelength can identify an element , but the overall pattern clearly identifies the element (or compound) This is of immense importance in studying spectra from distant star.
  All stars emit a continuous spectrum of all wavelengths through the electromagnetic spectrum. The relative intensity of these wavelengths is given by the black body distribution. Gases in the surrounding 'atmosphere' absorb amounts of electromagnetic energy which correspond to the differences between energy levels for the electrons . As electrons return to the lower state the wavelengths are emitted in all directions. Therefore very little of the original electromagnetic energy at these wavelengths actually reaches the observer. The continuous spectrum of a star appears to be missing these wavelengths . This pattern of missing wavelengths in a continuous spectrum is called an absorption spectrum. For a given gas the position of these absorption lines coincides exactly with the wavelengths of the emission spectrum for that gas. Therefore , the study of the spectra from distant stars can be used to identify gases surrounding the star, by looking for characteristic 'fingerprint' pattern of gases and elements
  The absorption spectrum of a distant galaxy can be used to determine the velocity v at which the galaxy is moving relative to our galaxy. This is because each distinct pattern of wavelengths will have been 'red shifted'...a consequence of the Doppler effect.The degree of red shift is directly proportional to the recessional velocity.
  3.2 The Doppler Effect The change of wavelength due to the relative motion between the source and observer is known as the Doppler Effect. We will make the assumption that v is much smaller than c . Let w be the wavelength when there is no relative velocity between source and observer, The change of wavelength is directly proportional to the relative velocity v between the source and the observer. The change of wavelength, = (v x w)/c and the fractional change of wavelength = v/c As v becomes larger this approximation is no longer valid, due to the greater influence of relativity.
  3.3 Hubble's Law One of the most important discoveries occured in 1929, when Edwin Hubble discovered that the Universe is expanding. He found that the red-shifts of a set of local galaxies were directly proportional to the distance of these galaxies from our galaxy. This has became known as Hubble's Law, and is given by
  v = Hd
  where H is known as Hubble's constant. The value of H is measured in km/s per Megaparsec. A value of 50 km/s per Mpc would mean that a galaxy 1Mpc away would be receding at 50km/s. A galaxy 10 Mpc away would be receding at 500 km/s.
  This equation does not take account of the effect of gravity on the outward expansion. The presence of gravity will oppose the outward expansion,so the velocity of expansion reduces with time. The value of H is the value of H as measured today. It is sometimes called the Hubble Parameter...since its value is not constant with time. ( A graph of distance between two galaxies against time is not a straight line)
  The value of H will always give a calculated value of the age of the Universe (1/H) which is higher than the actual value. An accurate determination of H depends on the accuracy to which we can determine distances to distant galaxies.Unfortunately, the greater the distance of a galaxy the greater the uncertainty in measurements of that distance.Most estimates give values between 40 and 80 km/s per Megapasec.
  The Fate of the Universe The gravitational attraction between masses slows down the outward expansion of the Universe. If we knew the mass of the Universe with a high degree of accuracy , we would be able to predict confidently just what the future of the Universe will be. Unfortunately much of the mass of the Universe appears missing...or at least it hasn't been found yet. This missing mass, which would enable the density of the Universe to be calculated , is difficult to observe, unlike stars which emit their own light. Yet an accurate determination is absolutely essential if we are to predict with some accuracy the fate of the Universe.
  3.5 The Cosmological Principle According to the Cosmological Principle there are no preferred places in the Universe. Measurements of the Universe made from Earth, disregarding local irregularities, can be considered to be identical to those made in any other part of the Universe. A classic illustration of this is in the way in which the expansion of the Universe, according to Hubble's Law, occurs.
  Take a length of elastic.Mark dots at even intervals to indicate galaxies. Gradually pull the elastic apart.At any time the distances from a galaxy will vary according with Hubble's Law, irrespective of which galaxy (dot) is chosen as the reference point. A similar pattern can be observed by using a series of dots on a balloon which is gradually inflated.
  Therefore, irrespective of where we are in the Universe, we should observe the same expansion properties , and the same laws of physics in operation. Earth has therefore no special significance, other than to humans, on a cosmic scale !
  The Universe before t=0.01s : Some Problems to Solve ! The early Universe,according to the Big Bang, involved extremely high temperatures. So high are these temperatures, that it is extremely difficult to reach them in scientific experiments. Fundamental investigations into particles involve accelerating particles at very high speeds, which would normally be associated with particles at very high temperatures. It is possible to reach speeds in particle accelerators which correspond to 10^15 K , which would occur about 10^-15 s after the Big Bang. For times earlier than this, we need to rely on theoretical models,used to explain our present observations ,and 'extrapolate' these backwards to predict the mechanisms which occur in the very early stages of the Big-Bang. This is an incredible difficult task, because a number of complex interactions which take place . Protons and neutrons are thought to consist of smaller particles, called quarks At temperatures above 10^7 these quarks cannot be held together as protons and neutrons. At higher temperature the four fundamental forces become almost indistinguishable from one another . Different 'messenger' particles are believed to be responsible for these four different forces. Above 10^5 K photons, massless particles, are being converted into particles,and their anti-particles.The reverse process is also happening . The theories are therefore very different to theories involving everday experiences of forces,and the synthesis of the four fundamental forces makes them extraordinarily complex theories. These are known as Grand Unified Theories (GUTs). These attempt to explain the physical behaviour of particles and forces during the very early stages of the Universe, in a single set of equations.The more recent theories which try to reach the goal of TOE (Theory of Everything) are called String Theories. These have gained some success in providing a desription of gravity, but the theory itself appears to some scientists as a mathematical instrument or 'toy', with very little foundation based on physics (as most of us know it !)
  At time t=0 a singularity is predicted, where all laws as we know them break down.Space and time are infinitely distorted. The uncertainty principle of quantum mechanics prevents any accurate predictions to be made for times less than 10^-45 s.
  Microwave Background Radiation In 1965 , Arno Penzias and Robert Wilson discovered , quite accidently, the existence of a low level microwave radiation. This was found to be virtually of the same intensity no matter which way they looked with their radio telescope. This was one of the greatest discoveries for the support of the Big Bang Theory. After billions of years the great fireball would have cooled to give a black-body peak temperature of approximately 2.7 K. Furthermore, this temperature would be associated with a peak wavelength of 1.07mm . Analysis of the radiation supported this prediction. Until recently, the uniformity of the background radiation was of some concern to many cosmologists.The apparent uniformity of the background radiation was not consistent with the requirements for the formation of galaxies. Small variations in the density are required in order that galaxies can be formed. This is due to the effect of gravitational attraction between particles of gases in these higher density regions. Therefore, this should be reflected in slight variations in the background radiation. The measurement of such variations was performed by Cosmic Background Explorer (COBE) in 1992.These measurements amounted to variations of just 30 millionths of a degree Kelvin.
  Frequently Asked Questions in Cosmology
  What is the evidence for the Big Bang? The evidence for the Big Bang comes from many pieces of observational data that are consistent with the Big Bang. None of these prove the Big Bang, since scientific theories are not proven. Many of these facts are consistent with the Big Bang and some other cosmological models, but taken together these observations show that the Big Bang is the best current model for the Universe. These observations include: The darkness of the night sky - Olbers' paradox.
  The Hubble Law - the linear distance vs redshift law. The data are now very good.
  Homogeneity - fair data showing that our location in the Universe is not special.
  Isotropy - very strong data showing that the sky looks the same in all directions to 1 part in 100,000.
  Time dilation in supernova light curves.
  The observations listed above are consistent with the Big Bang or with the Steady State model, but many observations support the Big Bang over the Steady State: Radio source and quasar counts vs. flux. These show that the Universe has evolved. Existence of the blackbody CMB. This shows that the Universe has evolved from a dense, isothermal state.
  Variation of TCMB with redshift. This is a direct observation of the evolution of the Universe.
  Deuterium, 3He, 4He, and 7Li abundances. These light isotopes are all well fit by predicted reactions occuring in the First Three Minutes.
  Finally, the angular power spectrum of the CMB anisotropy that does exist at the several parts per million level is consistent with a dark matter dominated Big Bang model that went through the inflationary scenario.
  Why do we think that the expansion of the Universe is accelerating? The evidence for an accelerating expansion comes from observations of the brightness of distant supernovae. We observe the redshift of a supernova which tells us by what the factor the Universe has expanded since the supernova exploded. This factor is (1+z), where z is the redshift. But in order to determine the expected brightness of the supernova, we need to know its distance now. If the expansion of the Universe is accelerating due to a cosmological constant, then the expansion was slower in the past, and thus the time required to expand by a given factor is longer, and the distance NOW is larger. But if the expansion is decelerating, it was faster in the past and the distance NOW is smaller. Thus for an accelerating expansion the supernovae at high redshifts will appear to be fainter than they would for a decelerating expansion because their current distances are larger. Note that these distances are all proportional to the age of the Universe [or 1/Ho], but this dependence cancels out when the brightness of a nearby supernova at z close to 0.1 is compared to a distant supernova with z close to 1.
  If the Universe is only 10 billion years old, why isn't the most distant object we can see 5 billion light years away? This question makes some hidden assumptions about space and time which are not consistent with all definitions of distance and time. One assumes that all the galaxies left from a single point at the Big Bang, and the most distant one traveled away from us for half the age of the Universe at almost the speed of light, and then emitted light which came back to us at the speed of light. By assuming constant velocities, we must ignore gravity, so this would only happen in a nearly empty Universe. In the empty Universe, one of the many possible definitions of distance does agree with the assumptions in this question: the angular size distance, and it does reach a maximum value of the speed of light times one half the age of the Universe. See Part 2 of the cosmology tutorial for a discussion of the other kinds of distances which go to infinity in the empty Universe model since this gives an unbounded Universe.
  If the Universe is only 10 billion years old, how can we see objects that are now 30 billion light years away? When talking about the distance of a moving object, we mean the spatial separation NOW, with the positions of both objects specified at the current time. In an expanding Universe this distance NOW is larger than the speed of light times the light travel time due to the increase of separations between objects as the Universe expands. This is not do to any change in the units of space and time, but just caused by things being farther apart now than they used to be. What is the distance NOW to the most distant thing we can see? Let's take the age of the Universe to be 10 billion years. In that time light travels 10 billion light years, and some people stop here. But the distance has grown since the light traveled. The average time when the light was traveling was 5 billion years ago. For the critical density case, the scale factor for the Universe goes like the 2/3 power of the time since the Big Bang, so the Universe has grown by a factor of 22/3 = 1.59 since the midpoint of the light's trip. But the size of the Universe changes continuously, so we should divide the light's trip into short intervals. First take two intervals: 5 billion years at an average time 7.5 billion years after the Big Bang, which gives 5 billion light years that have grown by a factor of 1/(0.75)2/3 = 1.21, plus another 5 billion light years at an average time 2.5 billion years after the Big Bang, which has grown by a factor of 42/3 = 2.52. Thus with 1 interval we got 1.59*10 = 15.9 billion light years, while with two intervals we get 5*(1.21+2.52) = 18.7 billion light years. With 8192 intervals we get 29.3 billion light years. In the limit of very many time intervals we get 30 billion light years.
  Another way of seeing this is to consider a photon and a galaxy 30 billion light years away from us now, 10 billion years after the Big Bang. The distance of this photon satisfies D = 3ct. If we wait for 0.1 billion years, the Universe will grow by a factor of (10.1/10)2/3 = 1.0066, so the galaxy will be 1.0066*30 = 30.2 billion light years away. But the light will have traveled 0.1 billion light years further than the galaxy because it moves at the speed of light relative to the matter in its vicinity and will thus be at D = 30.3 billion light years, so D = 3ct is still satisifed.
  If the Universe does not have the critical density then the distance is different, and for the low densities that are more likely the distance NOW to the most distant object we can see is bigger than 3 times the speed of light times the age of the Universe. Back to top.
  How can the oldest stars in the Universe be older than the Universe? Of course the Universe has to be older than the oldest stars in it. So this question basically asks: which estimate is wrong - The age of the Universe The age of the oldest stars
  Both The age of the Universe is determined from its expansion rate: the Hubble constant, which is the ratio of the radial velocity of a distant galaxy to its distance. The radial velocity is easy to measure, but the distances are not. Thus there is currently a 15% uncertainty in the Hubble constant.
  Determining the age of the oldest stars requires a knowledge of their luminosity, which depends on their distance. This leads to a 25% uncertainty in the ages of the oldest stars due to the difficulty in determining distances.
  Thus the discrepancy between the age of the oldest things in the Universe and the age inferred from the expansion rate is within the current margin of error. In fact, in 1997 improved distances from the HIPPARCOS satellite suggested that this discrepancy has vanished.
  Can objects move away from us faster than the speed of light? Again, this is a question that depends on which of the many distance definitions one uses. However, if we assume that the distance of an object at time t is the distance from our position at time t to the object's position at time t measured by a set of observers moving with the expansion of the Universe, and all making their observations when they see the Universe as having age t, then the velocity (change in D per change in t) can definitely be larger than the speed of light. This is not a contradiction of special relativity because this distance is not the same as the spatial distance used in SR, and the age of the Universe is not the same as the time used in SR. In the special case of the empty Universe, where one can show the model in both special relativistic and cosmological coordinates, the velocity defined by change in cosmological distance per unit cosmic time is given by v = c ln(1+z) which clearly goes to infinity as the redshift goes to infinity, and is larger than c for z > 1.718. For the critical density Universe, this velocity is given by v = 2c[1-(1+z)-0.5] which is larger than c for z > 3 .
  What is the redshift? The redshift of an object is the amount by which the spectral lines in the source are shifted to the red. That is, the wavelengths get longer. To be precise, the redshift is given by z = [WL(obs)-WL(em)]/WL(em) where WL(em) is the emitted wavelength of a line, which is known from laboratory measurements, and WL(obs) is the observed wavelength of the line. In an expanding Universe, distant objects are redshifted, with z = Ho D/c for small distances. This law was discovered by Hubble and Ho is known as the Hubble constant.
  Are quasars really at the large distances indicated by their redshifts? The short answer is Yes! Stockton (1978, ApJ, 223, 747) observed faint galaxies near in the sky to bright quasars at moderate redshifts. He chose quasars with moderate redshifts so he would still be able to see galaxies at the redshift of the quasar. He found that a good fraction of the redshifts of the faint galaxies agreed with the redshifts of the quasars. In other words, quasars are associated with galaxies that have the same redshift as the quasar and have just the brightness expected if the quasars are at their cosmological distances. Thus at least some quasars are at the distance indicated by their redshifts, and this includes some of the most luminous quasars: for example 3C273. Thus the simple answer selected by Occam's razor is that all quasars are at the distances indicated by their redshifts.
  The statistical arguments advanced by Arp and others in favor of anomalous quasar redshifts are often incorrect.
  What about objects with discordant redshifts, like Stephan's Quintet? One famous example of objects with different redshifts appearing in the same part of the sky is Stephan's Quintet. But the low redshift galaxy (in the lower left) is obviously more resolved into stars and looks "bumpier". By the surface brightness fluctuation method of distance determination, this bumpiness means that the low redshift galaxy is indeed much closer to us than the other four members of the quintet.
  Has the time dilation of distant source light curves predicted by the Big Bang been observed? This time dilation is a consequence of the standard interpretation of the redshift: a supernova that takes 20 days to decay will appear to take 40 days to decay when observed at redshift z=1. The time dilation has been observed, with 4 different published measurements of this effect in supernova light curves. These papers are: Leibundgut etal, 1996, ApJL, 466, L21-L24 Goldhaber etal, in Thermonuclear Supernovae (NATO ASI), eds. R. Canal, P. Ruiz-LaPuente, and J. Isern. Riess etal, 1997, AJ, 114, 722. Perlmutter etal, 1998, Nature, 391, 51. These observations contradict tired light models of the redshift.
  Are galaxies really moving away from us or is space just expanding? This depends on how you measure things, or your choice of coordinates. In one view, the spatial positions of galaxies are changing, and this causes the redshift. In another view, the galaxies are at fixed coordinates, but the distance between fixed points increases with time, and this causes the redshift. General relativity explains how to transform from one view to the other, and the observable effects like the redshift are the same in both views. Part 3 of the tutorial shows space-time diagrams for the Universe drawn in both ways. Also see the Relativity FAQ answer to this question.
  Why doesn't the Solar System expand if the whole Universe is expanding? This question is best answered in the coordinate system where the galaxies change their positions. The galaxies are receding from us because they started out receding from us, and the force of gravity just causes an acceleration that causes them to slow down. Planets are going around the Sun is fixed size orbits because they are bound to the Sun. Everything is just moving under the influence of Newton's laws (with very slight modifications due to relativity). [Illustration] For the technically minded, Cooperstock et al. computes that the influence of the cosmological expansion on the Earth's orbit around the Sun amounts to a growth by only one part in a septillion over the age of the Solar System. This effect is caused by the cosmological background density within the Solar System going down as the Universe expands, which may or may not happen depending on the nature of the dark matter. The mass loss of the Sun due to its luminosity and the Solar wind leads to a much larger [but still tiny] growth of the Earth's orbit which has nothing to do with the expansion of the Universe. Even on the much larger (million light year) scale of clusters of galaxies, the effect of the expansion of the Universe is 10 million times smaller than the gravitational binding of the cluster. Also see the Relativity FAQ answer to this question.
  Is the Universe expanding or is it just that our definitions of length and time are changing? The definitions of length and time are not changing in the standard model. The second is still 9192631770 cycles of a Cesium atomic clock and the meter is still the distance light travels in 9192631770/299792458 cycles of a Cesium atomic clock.
  What is meant by a flat Universe? The Universe appears to be homogeneous and isotropic, and there are only three possible geometries that are homogeneous and isotropic as shown in Part 3. A flat space has Euclidean geometry, where the sum of the angles in a triangle is 180o. A curved space has non-Euclidean geometry. In a positively curved, or hyperspherical space, the sum of the angles in a triangle is bigger than 180o, and this angle excess gives the area of the triangle divided by the square of the radius of the surface. In a negatively curved or hyperbolic space, the sum of the angles in a triangle is less than 180o. When Gauss invented this non-Euclidean geometry he actually tried measuring a large triangle, but he got an angle sum of 180o because the radius of the Universe is very large (if not infinite) so the angle excess or deficit has to be tiny for any triangle we can measure. If the radius is infinite, then the Universe is flat.
  Bolyai developed this geometry and published it, whereupon Gauss wrote to Bolyai's father: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years." And Lobachevsky had published very similar work in the obscure Kazan Messenger.
  What is the Universe expanding into? This question is based on the ever popular misconception that the Universe is some curved object embedded in a higher dimensional space, and that the Universe is expanding into this space. This misconception is probably fostered by the balloon analogy which shows a 2-D spherical model of the Universe expanding in a 3-D space. While it is possible to think of the Universe this way, it is not necessary, and there is nothing whatsoever that we have measured or can measure that will show us anything about the larger space. Everything that we measure is within the Universe, and we see no edge or boundary or center of expansion. Thus the Universe is not expanding into anything that we can see, and this is not a profitable thing to think about. Just as Dali's Corpus Hypercubicus is just a 2-D picture of a 3-D object that represents the surface of a 4-D cube, remember that the balloon analogy is just a 2-D picture of a 3-D situation that is supposed to help you think about a curved 3-D space, but it does not mean that there is really a 4-D space that the Universe is expanding into. Or you can ask Dr. Science :)
  What came before the Big Bang? The standard Big Bang model is singular at the time of the Big Bang, t = 0. This means that one cannot even define time, since spacetime is singular. In some models like the chaotic or perpetual inflation favored by Linde, the Big Bang is just one of many inflating bubbles in a spacetime foam. But there is no possibility of getting information from outside our own one bubble. Thus I conclude that: "Whereof one cannot speak, thereof one must be silent." From Bruce Margon and Craig Hogan at the Univ. of Washington
  Why is the sky dark at night? If the Universe were infinitely old, and infinite in extent, and stars could shine forever, then every direction you looked would eventually end on the surface of a star, and the whole sky would be as bright as the surface of the Sun. This is known as Olbers' Paradox after Heinrich Wilhelm Olbers [1757-1840] who wrote about it in 1823-1826 but it was also discussed earlier. Absorption by interstellar dust does not circumvent this paradox, since dust reradiates whatever radiation it absorbs within a few minutes, which is much less than the age of the Universe. However, the Universe is not infinitely old, and the expansion of the Universe reduces the accumulated energy radiated by distant stars. Either one of these effects acting alone would solve Olbers' Paradox, but they both act at once.
  Will the Universe expand forever or recollapse? This depends on the ratio of the density of the Universe to the critical density. If the density is higher than the critical density the Universe will recollapse in a Big Crunch. But current data suggests that the density is less than or equal to the critical density so the Universe will expand forever. See Part 3 of the tutorial for more information.
  What is the dark matter? When astronomers add up the masses and luminosities of the stars near the Sun, they find that there are about 3 solar masses for every 1 solar luminosity. When they measure the total mass of clusters of galaxies and compare that to the total luminosity of the clusters, they find about 300 solar masses for every solar luminosity. Evidently most of the mass in the Universe is dark. If the Universe has the critical density then there are about 1000 solar masses for every solar luminosity, so an even greater fraction of the Universe is dark matter. But the theory of Big Bang nucleosynthesis says that the density of ordinary matter (anything made from atoms) can be at most 10% of the critical density, so the majority of the Universe does not emit light, does not scatter light, does not absorb light, and is not even made out of atoms. It can only be "seen" by its gravitational effects. This "non-baryonic" dark matter can be neutrinos, if they have small masses instead of being massless, or it can be WIMPs (Weakly Interacting Massive Particles), or it could be primordial black holes. My nominee for the "least likely to be caught" award goes to hypothetical stable Planck mass remnants of primordial black holes that have evaporated due to Hawking radiation. The Hawking radiation from the not-yet evaporated primordial black holes may be detectable by future gamma ray telescopes, but the 20 microgram remnants would be very hard to detect. Also see the Relativity FAQ answer to this question, Jonathan Dursi's tutorial on dark matter, and the Center for Particle Astrophysics on dark matter. Dr. Science on dark matter :). See CDM
  What is the value of the Hubble constant? This is the question that professional astronomers ask the most frequently, and the answer is: Ho = 65 +/- 8 km/sec/Mpc but I would rather see
  What can a layperson do in cosmology? Stay in school! There is a lot to learn about the Universe. Keep taking math and science courses! The book of nature lies continuously open before our eyes (I speak of the Universe) but it can't be understood without first learning to understand the language and characters in which it is written. It is written in mathematical language, and its characters are geometrical figures. - Galileo Galilei
  That was true 400 years ago and it is much more true today! If you are out of school, check out the bibliography. Tell your Congressman and Senators to support astrophysics research at NASA, NSF, and DOE.
  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.
 
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