Astronomy - Cosmology

The Hubble Parameter has a value of approximately 70.1 km/s/Mpc. This is numerically equal to 2.27155 * 10^-18 /s. So its inverse, 1/H₀, has a value of 4.40228 * 10¹⁷ s, or 13.95 billion years. Up to a few technicalities (which reduce the value to 13.73 billion years) this is the age of the universe.

(View Cosmos DVD 1, episode 1, on the Cosmic Calender.)

Dealing with cosmological distances is quite a more difficult matter. As we have seen, the universe is expanding, and the rate of expansion increases with the distance. The expansion is described by the scale factor, which is related to the red shift (denoted "z") by the equations

redshift = apparent wavelength / actual wavelength - 1

and

redshift = scale factor_now / scale factor_past - 1

So we see that wavelengths stretch with the expansion of the universe, and the current redshift has a value of zero. If the Big Bang model of the universe is correct, and the universe began with infinitesimal volume, the red shift at the Big Bang is infinite.

Comparing the equation for redshift to our equation for the Doppler Shift, we see that when the recessional velocity is equal to the speed of light, the redshift is 1. Near this region the Hubble Relation breaks down, and relativistic effects must be taken into account. For this reason, it is most appropriate for us to discuss distances not in terms of meters or light years or parsecs, but instead in terms of redshift, which is based on direct observation, and does not depend on the scale factor of the cosmological model involved.

The Accelerating Universe

This is a plot of the log of the distance (as measured from the apparent and absolute magnitudes) of highly redshifted Type Ia supernovae, as a function of their redshift (source):

The data follows the upper line, which is a fit to a model in which the expansion of the universe accelerates due to something called the Cosmological Constant: a constant energy density which acts like a negative pressure. This was the first evidence that the expansion of the universe is accelerating. As we shall see, further data collected by the Wilkins Microwave Anisotropy Probe (WMAP) indicates that the Cosmological Constant may not have been Einstein's greatest mistake after all!

History of the Universe

The average temperature of the universe is inversely related to the scale factor:

temperature_past / temperature_now = scale factor_now / scale factor_past

So as the universe expanded, the temperature dropped. We know that each particle has an energy associated with it through the equation

energy = mass * c²

Similarly, each force has an energy scale associated with it. By dividing each energy by Boltzmann's constant:

temperature_{freeze out} = energy / k

we find the temperature at which each force becomes effective, and at which each particle condenses from its constituent parts. For instance, above 10¹⁵ K, the particles which mediate the weak force (W and Z bosons) are unstable to pair production, and interactions which change quark flavors (as in radioactive decay) are not possible. When the temperature drops below that associated with the mass of the proton:

1.673 * 10^-27 kg * c² / k
= 10¹³ K

the constituent quarks in the proton are cool enough to be bound together by the strong force:

event temperature (K) scale factor_now / scale factor_then time

strong forces freeze out 10²⁷ 3.7 * 10²⁶ 10^-35 s

weak forces freeze out 10¹⁵ 3.7 * 10¹⁴ 10^-10 s

protons, neutrons freeze out 10¹³ 3.7 * 10¹² 0.0001 s

neutrinos decouple 3 * 10¹⁰ 1.1 * 10¹⁰ 1 s

electrons freeze out 6 * 10⁹ 2.2 * 10⁹ 100 s

primordial ²H, ⁴He form 9 * 10⁸ 3.3 * 10⁸ 2-15 minutes

event	temperature (K)	scale factor_now / scale factor_then	time
strong forces freeze out	10²⁷	3.7 * 10²⁶	10^-35 s
weak forces freeze out	10¹⁵	3.7 * 10¹⁴	10^-10 s
protons, neutrons freeze out	10¹³	3.7 * 10¹²	0.0001 s
neutrinos decouple	3 * 10¹⁰	1.1 * 10¹⁰	1 s
electrons freeze out	6 * 10⁹	2.2 * 10⁹	100 s
primordial ²H, ⁴He form	9 * 10⁸	3.3 * 10⁸	2-15 minutes

When the protons and neutrons froze out, the ratio of protons to neutrons was about 6:1 because of the mass difference between the neutron and proton (the neutron is slightly heavier). The large number of neutrons available, as well as the fact that neutron capture occurs faster than proton fusion, caused the nucleosynthesis reactions here to be somewhat different from those taking place in the Sun. Once the universe cooled enough to allow Deuterium (²H) to exist, the ratio was 7:1 (from neutron decays) and the following reactions occurred:

¹H + n -> ²H + g

²H + ²H -> ³He + n + g

³He + n -> ⁴He + g

This sequence lasted about 15 minutes. The final ratio of ⁴He to ¹H was about 1:12, so that the universe was about 75% Hydrogen and 25% Helium by mass. A small amount of Deuterium survived; since it does not survive in stars, what Deuterium we observe is primordial. This is a sensitive indicator of the density of normal matter (not dark) in the universe, since a denser universe would have contained more protons and produced more Deuterium during nucleosynthesis.
Heavier elements were not formed because the temperature and density was dropping very quickly; in stars they do form because the temperature and density increase (slowly).
We now take up our history:

event	temperature (K)	scale factor_now / scale factor_then	time
photons decouple, atoms form	3000	1100	400000 years
first stars	60	7-20	10⁹ years
today	2.7	1	1.37 * 10¹⁰ years

"Decoupling" means that those particles are no longer in thermal equilibrium with their environment. When neutrinos decoupled, the universe became transparent to them; similarly for photons.

The scale factors in the table above were obtained by using T_now = 2.73 K, the temperature of the Cosmic Microwave Background Radiation (CMBR). Study of the CMBR has made cosmology an experimental science, as we shall soon see.

The temperatures in our history indicate that the early universe was filled with intense radiation. After the decoupling of photons, the density of matter became greater than the density of radiation (due to red-shifting, as we shall see). Wide field surveys indicate that the universe is now very close to being a homogeneous "dust" of galaxy clusters. The pressure from the radiation has all but vanished. But the Type IA supernovae surveys we discussed above indicate that the expansion of the universe is accelerating. We do not know what causes this acceleration, and for now we simply give it the commonly-used name dark energy. So the universe has passed through three distinct phases:

radiation-dominated,
matter-dominated, and now
dark energy-dominated.

Cosmic Microwave Background Radiation and Cosmology

After collecting five years of data from the Wilkinson Microwave Anisotropy Probe (WMAP), the WMAP Science Team released this image of the Cosmic Microwave Background Radiation (CMBR):

This image is a weighted linear combination of data in five radio frequency bands (23, 33, 41, 61 and 94 GHz) which minimizes the foreground contamination from the Milky Way. It is a snapshot of the photosphere or last scattering surface: the universe when it first became transparent to electromagnetic radiation. The color scale in this image corresponds to temperatures of from 2.7248 to 2.7252 K (blue to red). This represents a deviation from perfect isotropy (equality in all directions) of one part in 13625.

We would like to explain quantitatively how the CMBR came to look like the picture above. To do so we turn to General Relativity (GR), which relates the geometric qualities of spacetime to the matter and energy it is filled with. Spacetime is described by a metric: a rule for how to measure intervals. The usual procedure in GR is to find the most general metric which is consistent with the symmetries of the problem at hand, and to find the most general form of the expressions describing the matter and energy. These are related by Einstein's Equations, which are then solved for relations between the free parameters.

Given our observations above, we need the most general metric which is homogeneous and isotropic. This is the Friedmann-Robertson-Walker (FRW) metric, and it is described by two parameters. The first is "k", the curvature constant; if we choose a time "t" and take all the points in the spacetime which have the same value of t (called a spatial section),

k = 0 means the section is not curved, or "flat" (cross sections in a fixed direction are straight lines),
k = 1 means the section is positively curved (cross sections in a fixed direction are circles), and
k = -1 means the section is negatively curved (cross sections in a fixed direction are hyperbolas).

The other parameter is a function "a(t)"; this is the scale factor we mentioned earlier. It measures the "size" of the universe as a function of time. The Big Bang occurred when a(t) was zero, and the expansion of the universe means that a(t) increases as a function of time. The Hubble Parameter is defined as the rate of change of a(t) divided by a(t). The rate of change of a(t) is denoted a'(t), and it must be positive since the universe is expanding.

Our matter/energy expression must be able to describe our three act history of the universe: radiation-dominated, matter-dominated and dark energy-dominated. This can be done using a perfect fluid, which is described by its energy density "r" and pressure "p" (perfect fluids do not have viscosity or convection). We will usually assume a simple (but reasonable) equation of state (which gives the pressure as a function of the density)

p = w r

with "w" a constant:

if w=0, the equation describes dust;
if w=1/3 it describes radiation, and
if w=-1 it describes the Cosmological Constant.

If the dark energy is not due to a Cosmological Constant, it is usually given the name quintessence, and it has a different equation of state.

With these parameterizations, Einstein's Equations reduce to two simple equations:

a'²(t) (r(t) / r_c - 1) = k,

and

a''(t) = - (4 p G / 3 c⁴) a(t) (3 p(t) + r(t))
= - (3 w + 1) (4 p G / 3 c⁴) a(t) r(t)

where we have shown explicitly the dependence of the density and pressure on the time (since density and pressure both depend on volume and therefore on the scale factor, which is a function of time). r_c is the "critical density"

3 H² c² / 8 p G

If r = r_c, k must equal zero (the universe must be flat).
If r > r_c, k must be positive (the universe is closed like a sphere), and
if r < r_c, k must be negative (the universe is open like a saddle):

Note that unless w < -1/3, the second of Einstein's Equations implies that the acceleration of the universe must slow down (since a''(t)<0).

Einstein's Equations are supplemented by a conservation equation which guarantees that energy is conserved. With our simple equation of state, the conservation equation has the solution

r(t) is proportional to 1 / a^{3(w + 1)}(t).

This tells us something we already knew: for matter, the density drops as the volume increases, and the radiation pressure drops as a⁴(t) increases, due to the combination of the increase in volume and the redshift.

It is customary to use the ratio r / r_c as a cosmological parameter. This quantity is denoted "W". Using it, our first Einstein Equation (also called the Friedmann Equation) is written

a'²(t) (W - 1) = k

So if W = 1, k must be zero; if W < 1, k must equal -1, and if W > 1, k must equal 1.

With a choice of k and w, we can solve the Friedmann Equation. The plot on the left shows the solutions for radiation (in red) and dust (in green) for each value of k:

The graph on the right shows the solutions for a Cosmological Constant; these solutions are exponential and must be fitted to the appropriate solution from the graph on the right at the appropriate time. From these graphs we see that for smaller values of k, the universe expands more rapidly, and that radiation tends to make the universe expand more rapidly than dust.

A short digression into Particle Physics

According to the Standard Model of Particle Physics (SMPP), the fundamental particles of matter are all either leptons or quarks. There are two main types of leptons: electrons (and their muon cousins), and neutrinos, which are electrically neutral and are very nearly massless. Except for the neutrinos, all quarks and leptons carry electrical charge and exchange photons, which mediate the electromagnetic force. All quarks and leptons can exchange W and Z bosons, which mediate the weak force. The weak force allows one type of quark to change into another, or an electron to change into a neutrino (or vise versa). Only the quarks can exchange gluons, which mediate the strong force and hold quarks together. Individual quarks are never seen; they are always found as a bound pair called a meson, or a bound triplet called a baryon. The most common baryons are protons and neutrons.

When the universe cooled so much that there was insufficient energy to ionize Hydrogen atoms, the universe became transparent to photons. Before then, it consisted of a dense plasma (electrically charged fluid) containing electrons, baryons, and photons. There was also dark matter: massive particles (not yet understood as part of the Standard Model) which do not exchange photons, and so do not interact electromagnetically. Dark matter may come in two forms: cold (CDM) or hot (HDM); HDM travels at speeds close to that of light, while CDM is non-relativistic. There was also dark energy, but its influence appears to have been negligible in the early universe.

Since the baryons are roughly 2000 times more massive than electrons, they were the fundamental source of inertia (resistance to acceleration).
Baryons and dark matter were the fundamental sources of gravitational attraction.
Photons were the fundamental source of pressure, which was therefore experienced by the electrons and baryons but not the dark matter.
Neutrinos contributed to the radiation energy density because they are nearly massless and move at relativistic speeds, but because they are electrically neutral they did not contribute to the pressure: they simply passed through the plasma unimpeded.

Because of their differing physical effects, the contributions of baryons, HDM, CDM and the dark energy to W are distinguished by subscripts:

W = W_b + W_HDM + W_CDM + W_vac

where "vac" denotes the dark energy contribution. Sometimes "L" is used instead of "vac" when we are particularly interested in the cosmological constant. In addition, we sometimes write

W_M = W_b + W_HDM + W_CDM

to denote the contribution of matter to W.

The CMBR Power Spectrum

The minute deviations from perfect isotropy in the CMBR tells us that there were small inhomogeneities in the plasma:

baryon and dark matter density fluctuations created small "pockets" of gravitational attraction into which less dense regions were attracted;
opposing that gravitational infall was the radiation pressure, and these opposing forces caused the plasma to oscillate.

These oscillations occurred at different length scales and left their imprint on the CMBR we observe today. Of course, those imprints have been modified by the expanding universe, and by other factors we have yet to discuss.

By comparing the CMBR temperatures at different angles, we can construct the power spectrum: a measure of the correlations between temperature as a function of angle. The power spectrum which best fits the 5-year WMAP data is shown here:

The horizontal scale is the multipole moment, denoted l; the angular scale is actually p/l, so larger values of l denote smaller angles. The vertical scale is the temperature correlation function C_l, multiplied by l(l+1)/2p, and is measured in mK², or (10^-6K)². The blue region represents the range of data values within one standard deviation of the fit line.

The peaks in the power spectrum are called acoustic peaks, and indicate the length scales of the oscillations in the plasma before the temperature fluctuations froze out in the last scattering surface. The height and positioning of the peaks are influenced by a number of cosmological parameters. We will focus on the following:

W_b - additional baryon mass provides additional gravitational instability and increases the amplitudes (sizes) of the acoustic peaks - if W_b is a larger fraction of W_M, there is more inertia and the speed of sound in the plasma is smaller; this can make the sound horizon smaller (the region reachable by a given oscillation) and therefore the acoustic peaks occur at smaller scales

W_CDM - increases gravitational density fluctuations

W_vac - changes transition to dark energy-domination; at that time, density perturbations cease growing

t_optical - after the last scattering surface freezes out (t_optical = 1), matter structures begin to form in the early universe; when the first stars begin burning, much of the universe is re-ionized (from somewhere between z=30 and z=7, until about z=6; t_optical < 1); this is the optical depth at (time of) re-ionization; the re-ionization smooths out temperature fluctuations (earlier re-ionization produces more smoothing)

In addition, any parameter which affects k (primarily the Ws) changes the angular scales we observe. Photons of the CMBR follow geodesics; the Geodesic Equations for the FRW metric tell us that the rate an angle changes is inversely proportional to the square of the scale factor. From the graphs above, we can see that smaller values of k cause the power spectrum to shift to higher values of l.

Note that the length scales corresponding to the acoustic peaks should correlate with the distances between large structures in the universe. Survey results have recently begun to confirm that expectation.

Comparing CMBFAST Results

We can use CMBFAST to explore regions of the space of possible cosmological parameters. The following applet will allow an easy comparison of the effects of various parameter values with the current best fit to the data. When you run CMBFAST, use the default parameter values for everything except the parameter(s) you have been assigned. Using CMBFAST, we can investigate the following:

the effects of altering the ratio of W_b to W_cdm without changing W_M;
the effects of changing the ratio of W_vac to W_M;
the effects of making W > 1 or < 1 without changing the ratio of W_b to W_cdm to W_vac, and
the effects of varying the optical depth.

After you run CMBFAST (which may take a few minutes), your results will be returned to you. Look for the number on the "Output Files" (ie., "02036736" in "cmb_02036736.fcl"). Enter that number in the applet below and it will load the power spectrum data and plot it. Each time you enter a different number, it will be plotted in a different color (from red to violet). The best fit plot is plotted in black.

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

We have ignored changes in the temperature anisotropy due to gravitational lensing. We have also ignored ionization from early hot clusters, and gravitational redshifting from the galaxy foreground (between us and the last scattering surface). Finally, all of our deliberations are strictly with the temperature anisotropies; there are also polarization anisotropies: deviations in the electric field and magnetic field directions in the CMBR. These promise to be important sources of information in future studies.

Portfolio Exercise 6: Using CMBFAST and the comparison applet, perform the 4 investigations listed above the applet. Do at least 4 trials for each investigation. Be sure that Omega_b is greater than 0 for each trial.
Capture images of each of the comparison plots for the 4 investigations, and annotate them, describing which plots belong to which parameter values. Include all 4 in your portfolio.

Theory vs Experiment - a reality check

Experiment typically drives theory...

Experiment Theory

1814 - Fraunhofer discovers absorption spectra 1913-5 - Bohr explains them quantum mechanically

1845 - Leverrier measures perihelion shift of Mercury 1916 - Einstein explains using General Relativity (GR)

1887 - Michelson and Morley determine c is a constant 1905 - Einstein explains with Special Relativity

1893 - Eotvos determines M_inertial = M_grav 1916 - Einstein incorporates this into GR as the Equivalence Principle

(1800s) - measurements of black body radiation 1900 - Planck derives black body distribution

1929 - Hubble discovers expansion of universe 1933 Robertson finds GR solution for expanding universe

1965 - Penzias and Wilson discover Cosmic Microwave Background Radiation (CMBR)

1965 - Rubin begins measuring galaxy rotation curves no consensus yet on what dark matter is

1998 - Filippenko and Riess discover expansion of universe is accelerating no consensus yet on what dark energy is

Experiment	Theory
1814 - Fraunhofer discovers absorption spectra	1913-5 - Bohr explains them quantum mechanically
1845 - Leverrier measures perihelion shift of Mercury	1916 - Einstein explains using General Relativity (GR)
1887 - Michelson and Morley determine c is a constant	1905 - Einstein explains with Special Relativity
1893 - Eotvos determines M_inertial = M_grav	1916 - Einstein incorporates this into GR as the Equivalence Principle
(1800s) - measurements of black body radiation	1900 - Planck derives black body distribution
1929 - Hubble discovers expansion of universe	1933 Robertson finds GR solution for expanding universe
1965 - Penzias and Wilson discover Cosmic Microwave Background Radiation (CMBR)
1965 - Rubin begins measuring galaxy rotation curves	no consensus yet on what dark matter is
1998 - Filippenko and Riess discover expansion of universe is accelerating	no consensus yet on what dark energy is

but sometimes theory comes first:

Theory Experiment

1916 - Einstein formulates General Relativity 1920 - Dyson, Eddington and Davidson report gravitational deflection of light

1960 - Pound and Rebka measure gravitational redshift

1969 - Jenkins measures gravitational time effects

1978 - Taylor and Hulse measure energy loss in binary pulsars due to gravitational waves

1980 - Hubble Space Telescope discovers gravitational lensing

1916 - Schwarzschild discovers spherical vacuum solution to GR (describes black holes) no direct evidence yet

1917 - Einstein proposes Cosmological Constant 2007 - 5 years of WMAP data favors nonzero Cosmological Constant

1921/6 - Kuluza and Klein propose extra dimensions no evidence yet

1930 - Dirac predicts antimatter 1932 - Anderson discovers the positron

1931 - Fermi predicts the neutrino 1956 - Reines and Cowan discover it

1961 - Gell-Mann and Ne'eman explain particle zoo using quark model 1974 - Richter and Ting discover the J/y

1963 - Kerr discovers vacuum solution to GR with angular momentum 2007 ? - Gravity Probe B searches for frame dragging

1967 - Glashow, Weinberg and Salam formulate Electroweak theory still need a Higgs!

1972 - Gell-mann develops Quantum Chromodynamics (QCD) no contradictions yet...

1974 - Pati and Salam formulate first Grand Unified Theory no evidence yet

1975 - Hawking proposes that black holes emit radiation no evidence yet

1971 - Ramond, Neveu and Schwartz formulate Supersymmetry no evidence yet

1975/81 - Scherk, Green and Schwarz develop String Theory no evidence yet

1981-2 - Guth, Linde, Albrecht and Steinhardt propose inflation models no contradictions yet...

1986 - Ashtekar develops loop quantum gravity no evidence yet

Theory	Experiment
1916 - Einstein formulates General Relativity	1920 - Dyson, Eddington and Davidson report gravitational deflection of light
	1960 - Pound and Rebka measure gravitational redshift
	1969 - Jenkins measures gravitational time effects
	1978 - Taylor and Hulse measure energy loss in binary pulsars due to gravitational waves
	1980 - Hubble Space Telescope discovers gravitational lensing
1916 - Schwarzschild discovers spherical vacuum solution to GR (describes black holes)	no direct evidence yet
1917 - Einstein proposes Cosmological Constant	2007 - 5 years of WMAP data favors nonzero Cosmological Constant
1921/6 - Kuluza and Klein propose extra dimensions	no evidence yet
1930 - Dirac predicts antimatter	1932 - Anderson discovers the positron
1931 - Fermi predicts the neutrino	1956 - Reines and Cowan discover it
1961 - Gell-Mann and Ne'eman explain particle zoo using quark model	1974 - Richter and Ting discover the J/y
1963 - Kerr discovers vacuum solution to GR with angular momentum	2007 ? - Gravity Probe B searches for frame dragging
1967 - Glashow, Weinberg and Salam formulate Electroweak theory	still need a Higgs!
1972 - Gell-mann develops Quantum Chromodynamics (QCD)	no contradictions yet...
1974 - Pati and Salam formulate first Grand Unified Theory	no evidence yet
1975 - Hawking proposes that black holes emit radiation	no evidence yet
1971 - Ramond, Neveu and Schwartz formulate Supersymmetry	no evidence yet
1975/81 - Scherk, Green and Schwarz develop String Theory	no evidence yet
1981-2 - Guth, Linde, Albrecht and Steinhardt propose inflation models	no contradictions yet...
1986 - Ashtekar develops loop quantum gravity	no evidence yet

Please send comments or suggestions to the author.