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Most of our present understanding of the Universe is concisely and beautifully summarized in the hot big-bang cosmological model (see e.g., Weinberg 1972; Peebles, 1993). This mathematical description is based upon the isotropic and homogeneous Friedmann-Lemaitre-Robertson-Walker solution of Einstein's general relativity. The evolution of the Universe is embodied in the cosmic scale factor R(t), which describes the scaling up of all physical distances in the Universe (separation of galaxies and wavelengths of photons). The conformal stretching of the wavelengths of photons accounts for the redshift of light from distant galaxies: the wavelength of the radiation we see today is larger by the factor R (now) / R (then). Astronomers denote this factor by 1 + z, which means that an object at ``redshift z'' emitted the light seen today when the Universe was a factor 1 + z smaller. Normalizing the scale factor to unity today, Remission = 1 / (1 + z).

It is interesting to note that the assumption of isotropy and homogeneity was introduced by Einstein and others to simplify the mathematics; as it turns out, it is a remarkably accurate description at early times and today averaged over sufficiently large distances (greater than 100 Mpc or so).

The evolution of the scale factor is governed by the Friedmann equation for the expansion rate:

Equation 3 (3)

where rho = Sigmai rhoi is the total energy density from all components of mass-energy, and Rcurv is the spatial curvature radius, which grows as the scale factor, Rcurv propto R(t). [Hereafter we shall set c = 1.] As indicated by the ± sign in Eq. (3) there are actually three FLRW models; they differ in their spatial curvature: The plus sign applies to the negatively curved model, and the minus sign to the positively curved model. For the spatially flat model the curvature term is absent.

The energy density of a given component evolves according to

Equation 4 (4)

where pi is the pressure (e.g., pi << rhoi for nonrelativistic matter or pi = rhoi / 3 for ultra-relativistic particles and radiation). The energy density of matter decreases as R-3, due to volume dilution. The energy density of radiation decreases more rapidly, as R-4, the additional factor arising because the energy of a relativistic particle ``redshifts'' with the expansion, E propto 1 / R(t). (This of course is equivalent to the wavelength of a photon growing as the scale factor.) This redshifting of the energy density of radiation by R-4 also implies that for black body radiation, the temperature decreases as T propto R-1.

It is convenient to scale energy densities to the critical density, rhocrit ident 3H02 / 8pi G = 1.88h2 x 10-29 g cm-3 appeq 8.4 x 10-30 g cm-3 or approximately 5 protons per cubic meter,

Equation 5 (5)

Note that the critical-density Universe (Omega0 = 1) is flat; the subcritical-density Universe (Omega0 < 1) is negatively curved; and the supercritical-density Universe (Omega0 > 1) is positively curved.

There are at least two components to the energy density: the photons in the 2.728 K cosmic microwave background radiation (number density ngamma = 412 cm-3); and ordinary matter in the formation of neutrons, protons and associated electrons (referred to collectively as baryons). The theory of big-bang nucleosynthesis and the measured primordial abundance of deuterium imply that the mass density contributed by baryons is OmegaB = (0.02 ± 0.002)h-2 appeq 0.05. In addition, the weak interactions of neutrinos with electrons, positrons and nucleons should have brought all three species of neutrinos into thermal equilibrium when the Universe was less than a second old, so that today there should be three cosmic seas of relic neutrinos of comparable abundance to the microwave photons, nnu = (3 / 11) ngamma appeq 113 cm-3 (per species). (BBN provides a nice check of this, because the yields depend sensitively upon the abundance of neutrinos.) Together, photons and neutrinos (assuming all three species are massless, or very light, << 10-3 eV) contribute a very small energy density Omeganu gamma = 4.17h-2 x 10-5 appeq 10-4.

There is strong evidence for the existence of matter beyond the baryons, as dynamical measurements of the matter density indicate that it is at least 20% of the critical density (OmegaM > 0.2), which is far more than ordinary matter can account for. The leading explanation for the additional matter is long-lived or stable elementary particles left over from the earliest moments (see Section 5).

Finally, although it is now known that the mass density of the Universe in the form of dark matter exceeds 0.2 of the closure density, there are even more exotic possibilities for additional components to the mass-energy density, the simplest of which is Einstein's cosmological constant. Seeking static solutions, Einstein introduced his infamous cosmological constant; after the discovery of the expansion by Hubble he discarded it. In the quantum world it is no longer optional: the cosmological constant represents the energy density of the quantum vacuum (Weinberg 1989; Carroll et al. 1992). Lorentz invariance implies that the pressure associated with vacuum energy is pVAC = -rhoVAC, and this ensures that rhoVAC remains constant as the Universe expands. Einstein's cosmological constant appears as an additional term Lambda / 3 on the right hand side of the Friedmann equation [Equ. 3]; it is equivalent to a vacuum energy rhoVAC = Lambda / 8pi G.

All attempts to calculate the cosmological constant have been unsuccessful to say the very least: due to the zero-point energies the vacuum energy formally diverges (``the ultraviolet catastrophe''). Imposing a short wavelength cutoff corresponding to the weak scale (~ 10-17 cm) is of little help: OmegaVAC ~ 1055! The mystery of the cosmological constant is a fundamental one which is being attacked from both ends: Cosmologists are trying to measure it, and particle physicists are trying to understand why it is so small.

Because the different contributions to the energy density scale differently with the cosmic scale factor, the expansion of the Universe goes through qualitatively different phases. While today radiation and relativistic particles are not significant, at early times they dominated the energy, since their energy density depends most strongly on the scale factor (R-4 vs. R-3 for matter). Only at late times does the curvature term (propto R-2) become important; for a negatively curved Universe it becomes dominant. For a positively curved Universe, the expansion halts when it cancels the matter density term and a contraction phase begins.

The presence of a cosmological constant, which is independent of scale factor, changes this a little. A flat or negatively curved Universe ultimately enters an exponential expansion phase driven by the cosmological constant. This also occurs for a positively curved Universe, provided the cosmological constant is large enough,

Equation 8 (8)

If it is smaller than this, recollapse occurs. Einstein's static Universe obtains for rhoM = 2rhoVAC and Rcurv = 1/sqrt(8pi G rhoVAC).

The evolution of the Universe according to the standard hot big-bang model is summarized as follows:

Finally, a comment on the expansion rate and the size of the ``observable Universe.'' The inverse of the expansion rate has units of time. The Hubble time, H-1, corresponds to the time it takes for the scale factor to roughly double. For a matter-, radiation-, or curvature-dominated Universe, the age of the Universe (time back to zero scale factor) is: 2/3 H-1, 1/2 H-1, and H-1 respectively. The Hubble time also sets the size of the observable (or causally connected) Universe: the distance to the ``horizon,'' which is equal to the distance that light could have traveled since time zero, is 2t = H-1 for a radiation-dominated Universe and 3t = 2H-1 for a matter-dominated Universe. Paradoxically, although the size of the Universe goes to zero as one goes back to time zero, the expansion rate is larger, and so points separated by distance t are moving apart faster than light can catch up with them.

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