The Beginning and Evolution of the Universe

Modern cosmology began with Einstein (1917) when he applied his general relativity theory to cosmology. At this point in time our Galaxy, the Milky Way, was thought by most to be the Universe. To make progress Einstein assumed the Universe was spatially homogeneous and isotropic; this was enshrined as the "Copernican" cosmological principle by Milne (1933). Peebles (1993; § 3) has reviewed the strong observational evidence for large-scale statistical isotropy; observational tests of homogeneity are not as straightforward. Einstein knew that the stars in the Milky Way moved rather slowly and decided, as everyone had done before him, that the Universe should not evolve in time. He could come up with a static solution of his equations if he introduced a new form of energy, now called the cosmological constant. It turns out that Einstein's static model is unstable. In the same year de Sitter (1917) found the second cosmological solution of Einstein's general relativity equations; Lemaître (1925) and Robertson (1928) reexpressed this solution in the currently more familiar form of the exponentially expanding model used in the inflation picture. Weyl (1923) noted the importance of prescribing initial conditions such that the particle geodesics diverge from a point in the past. Friedmann (1922, 1924), not bound by the desire to have a static model, discovered the evolving homogeneous solutions of Einstein's equations; Lemaître (1927) rediscovered these "Friedmann-Lemaître" models. Robertson (1929) initiated the study of metric tensors of spatially homogeneous and isotropic spacetimes, and continuing study by him and A. G. Walker (in the mid 1930s) led to the "Robertson-Walker" form of the metric tensor for homogeneous world models. Of course, in the evolving cosmological model solutions only observers at rest with respect to the expansion/contraction see an isotropic and homogeneous Universe; cosmology thus reintroduces preferred observers!

North (1990) and Longair (2006) provide comprehensive historical reviews. See the standard cosmology textbooks for the modern formalism.

Meanwhile, with some first success in 1912, Slipher (1917)² found that most of the "white spiral nebulae" (so-called because they have a continuum spectrum; what we now term spiral galaxies) emit light that is redshifted (we now know that the few, including M31 [Andromeda] and some in the Virgo cluster, that emit blueshifted light are approaching us), and Eddington (1923) identified this with a redshift effect in the de Sitter (1917) model (not the cosmological redshift effect). Lemaître (1925) and Robertson (1928) derived Hubble's velocity-distance law v = H₀ r (relating the galaxy's speed of recession v to its distance r from us, where H₀ is the Hubble constant, the present value of the Hubble parameter) in the Friedmann-Lemaître models. The velocity-distance Hubble law is a consequence of the cosmological principle, is exact, and implies that galaxies further away than the current Hubble distance r_H = c/H₀ are moving away faster than the speed of light c. Hubble (1925)³ used Leavitt's (Leavitt 1912; Johnson 2005)⁴ quantitative Cepheid variable star period-luminosity relation to establish that M31 and M33 are far away (confirming the earlier somewhat tentative conclusion of Öpik 1922), and did this for more galaxies, conclusively establishing that the white nebulae are other galaxies outside our Milky Way galaxy (there was some other earlier observational evidence for this position but Hubble's work is what convinced people). Hubble got Humason (middle school dropout and one time muleskinner and janitor) to remeasure some Slipher spectra and measure more spectra, and Hubble (1929)⁵ established Hubble's redshift-distance law cz = H₀ r, where the redshift z is the fractional change in the wavelength of the spectral line under study (although in the paper Hubble calls cz velocity and does not mention redshift). The redshift-distance Hubble law is an approximation to the velocity-distance law, valid only on short distances and at low redshifts.

North (1990) provides a comprehensive historical review; Berendzen et al. (1976) and Smith (1982) are more accessible historical summaries. See the standard cosmology textbooks for the modern formalism. Branch (1998) discusses the use of type Ia supernovae as standard candles for measuring the Hubble constant. See Fig. 1 of Leibundgut (2001) for a recent plot of the Hubble law. Harrison (1993), Davis & Lineweaver (2004), and Lineweaver & Davis (2005) provide pedagogical discussions of issues related to galaxies moving away faster than the speed of light.

As one looks out further in space (and so back in time, because light travels at finite speed) wavelengths of electromagnetic radiation we receive now have been redshifted further by the expansion, and so Wien's law tells us (from the blackbody CMB) that the temperature was higher in the past. The younger Universe was a hotter, denser place. Lemaître ("the father of the Big Bang") emphasized the importance of accounting for the rest of known physics in the general-relativistic cosmological models.

Early work on explaining the astrophysically observed abundances of elements assumed that they were a consequence of rapid thermal equilibrium reactions and that a rapidly falling temperature froze the equilibrium abundances. Tolman, Suzuki, von Weizsäcker, and others in the 1920s and 1930s argued that the observed helium-hydrogen ratio in this scenario required that at some point the temperature had been at least 10⁹ K (and possibly as much as 10¹¹ K). Chandrasekhar & Henrich (1942) performed the first detailed, correct equilibrium computation and concluded that no single set of temperature and density values can accommodate all the observed abundances; they suggested that it would be useful to consider a nonequilibrium process. Gamow (1946), building on his earlier work, makes the crucial point that in the Big Bang Model "the conditions necessary for rapid nuclear reactions were existing only for a very short time, so that it may be quite dangerous to speak about an equilibrium state"; i.e., the Big Bang was the place to look for this nonequilibrium process.

Gamow (1948), a student of Friedmann, and Alpher (1948), a student of Gamow, estimated the radiation (photon) temperature at nucleosynthesis, and from the Stefan-Boltzmann law for blackbody radiation noted that the energy budget of the Universe must then have been dominated by radiation. Gamow (1948) evolved the radiation to the much later epoch of matter-radiation equality (the matter and radiation energy densities evolve in different ways and this is the time at which both had the same magnitude), a concept also introduced by Gamow, while Alpher & Herman (1948) predicted a residual CMB radiation at the present time from nucleosynthesis and estimated its present temperature to be 5 K (because the zero-redshift baryon density was not reliably known then, it is somewhat of a coincidence that this temperature estimate is close to the observed modern value). Hayashi (1950) pointed out that, at temperatures about 10 times higher than during nucleosynthesis, rapid weak interactions lead to a thermal equilibrium abundance ratio of neutrons and protons determined by the neutron-proton mass difference, which becomes frozen in as the expansion decreases the temperature, thus establishing the initial conditions for nucleosynthesis. This is fortunate, in that an understanding of higher energy physics is not needed to make firm nucleosynthesis predictions; this is also unfortunate, because element abundance observations cannot be used to probe higher energy physics.

Alpher et al. (1953) concluded the early period of the standard model of nucleosynthesis. By this point it was clear that initial hopes to explain all observed abundances in this manner must fail, because of the lack of stable nuclei at mass numbers 5 and 8 and because as the temperature drops with the expansion it becomes more difficult to penetrate the Coulomb barriers. Cosmological nucleosynthesis can only generate the light elements and the heavier elements are generated from these light elements by further processing in the stars.

Zel'dovich(1963a) and Smirnov (1964) noted that the ⁴He and D abundances are sensitive to the baryon density: the observed abundances can be used to constrain the baryon density. Hoyle & Tayler (1964) carried out a detailed computation of the ⁴He abundance and on comparing to measurements concluded "most, if not all, of the material of our...Universe, has been 'cooked' to a temperature in excess of 10¹⁰ K." They were the first to note that the observed light element abundances were sensitive to the expansion rate during nucleosynthesis and that this could constrain new physics at that epoch (especially the number of light, relativistic, neutrino families).

After Penzias and Wilson measured the CMB (see below), Peebles (1966a,1966b) computed the abundances of D, ³He, and ⁴He and their dependence on, among other things, the baryon density and the expansion rate during nucleosynthesis. The monumental Wagoner et al. (1967) paper established the ground rules for future work.

For a history of these developments see pp. 125–128 and 240–241 of Peebles (1971), the articles by Alpher and Herman and Wagoner on pp. 129–157 and 159–185, respectively, of Bertotti et al. (1990), and chapter 3 and § 7.2 of Kragh (1996). See the standard cosmology textbooks for the modern formalism. Accurate abundance predictions require involved numerical analysis; on the other hand, pedagogy could benefit from approximate semianalytical models (Bernstein et al. 1989; Esmailzadeh et al. 1991).

In the simplest nucleosynthesis scenario, the baryon density estimated from the observed D abundance is consistent with that estimated from WMAP CMB anisotropy data, and higher than that estimated from the ⁴He and ⁷Li abundances. This is further discussed in § 7.2. Fields & Sarkar (2006), Cyburt (2004), and Steigman (2006) are recent reviews of nucleosynthesis.

In addition to residual CMB radiation, there is also a residual neutrino background. Above a temperature of about 10¹⁰ K the CMB photons have enough energy to produce a thermal equilibrium abundance of neutrinos. Below this temperature the neutrinos decouple and freely expand, resulting in about 300 neutrinos per cubic centimeter now (with three families, and this number also includes antineutrinos), at a temperature of about 2 K, lower than that of the CMB because electron-positron annihilation heats the CMB a little. See Dolgov (2002), Hannestad (2006), and the more recent textbooks cited below for more detailed discussions of the (as yet undetected) neutrino background. We touch on neutrinos again in § 5.1.

The CMB radiation contributes of the order of 1% of the static or "snow" seen when switching between channels on a television with a conventional VHF antenna; it is therefore not surprising that it had been detected a number of times before its 1965 discovery and identification. For instance, it is now known that McKellar (1941) deduced a CMB temperature of 2.3 K at a wavelength of 2.6 mm by estimating the ratio of populations in the first excited rotational and ground states of the interstellar cyanogen (CN) molecule (determined from absorption line measurements of Adams). It is now also known that the discrepancy of 3.3 K between the measured and expected temperature of the Bell Labs horn antenna (for communicating with the Echo I satellite) at a wavelength of 12.5 cm found by Ohm (1961) is due to the CMB. Ohm also notes that an earlier measurement with this telescope (DeGrasse et al. 1959) ascribes a temperature of 2 ± 1 K to back and side lobe pick up, that this is " temperature not otherwise accounted for," and that "it is somewhat larger than the calculated temperature expected." Of course, McKellar had the misfortune of performing his analyses well before Gamow and collaborators had laid the nucleosynthesis foundations that would eventually explain the CN measurements (and allow the CMB interpretation), and Ohm properly did not overly stress the discrepancy beyond its weak statistical significance.

While Alpher and Herman (e.g., pp. 114–115 of Alpher & Herman 2001 and p. 130 of Weinberg 1993) privately raised the issue of searching for the CMB, and Hoyle came close to correctly explaining McKellar's CN measurements (see pp. 345–346 of Kragh 1996), Zel'dovich (1963a,1963b, 1965, p. 491, p. 89, and p. 315, respectively), Doroshkevich & Novikov (1964) and Dicke & Peebles (1965, p. 448) are the first published discussions of possible observational consequences of the (then still hypothetical) CMB in the present Universe. The relevant discussions of Zel'dovich and Doroshkevich & Novikov are motivated by the same nucleosynthesis considerations that motivated Gamow and collaborators; Dicke and Peebles favored an oscillating Universe and needed a way to destroy heavy elements from the previous cycle and so postulated an initial hotter stage in each cycle. Both Doroshkevich & Novikov (1964) and Zel'dovich (1965) referred to Ohm (1961) but neither appeared to notice Ohm's 3.3 K discrepancy; in fact Zel'dovich (1965) (incorrectly) argued that Ohm constrains the temperature to be less than 1 K and given the observed helium abundance this rules out the hot Big Bang Model.

Working with the same antenna as Ohm, using the Dicke switching technique to compare the antenna temperature to a liquid helium load at a known temperature, and paying very careful attention to possible systematic effects, Penzias & Wilson (1965) measured the excess temperature to be 3.5 ± 1 K at 7.35 cm wavelength; Dicke et al. (1965) identified this as the CMB radiation left over from the hot Big Bang.

The CMB is the dominant component of the radiation density of the Universe, with a density now of about 400 CMB photons per cubic centimeter at a temperature of about 2.7 K now. As noted in the previous subsection, observed light element abundances in conjunction with nucleosynthesis theory allows for constraints on the density of baryonic matter. Thus, there are a few billion CMB photons for every baryon; the CMB photons carry most of the cosmological entropy.

To date there is no observational indication of any deviation of the CMB spectrum from a Planckian blackbody. Partridge (1995) reviewed early measurements of the CMB spectrum. A definitive observation of the CMB spectrum was made by COBE (see Gush et al. 1990 for a contemporaneous rocket-based measurement), which measured a temperature of 2.725 ± 0.002 K (95% confidence) (Mather et al. 1999) and 95% confidence upper limits on possible spectral distortions: |μ| < 9 × 10^-5 for the chemical potential of early (10⁵ < z < 3 × 10⁶) energy release and |y| < 1.5 × 10^-5 for Comptonization of the spectrum at later times (Fixsen et al. 1996). Wright et al. (1994) have shown that these constraints strongly rule out many alternatives to the Big Bang Model, including the steady state model and explosive galaxy formation.

Anisotropy of the CMB temperature, first detected by COBE (Smoot et al. 1992), reveals important features of the formation and evolution of structure in the Universe. A small dipole anisotropy (discovered in the late 1960s and early 1970s by Conklin and Henry and confirmed by Corey and Wilkinson as well as Smoot, Gorenstein, and Muller) is caused by our peculiar motion; the CMB establishes a preferred reference frame. Higher multipole anisotropies in the CMB reflect the effect of primordial inhomogeneities on structure at the epoch of recombination and more complex astrophysical effects along the past light cone that alter this primordial anisotropy. We discuss these anisotropies, as well as the recently detected polarization anisotropy of the CMB, in § 5.2. The anisotropy signal from the recombination epoch allows precise estimation of cosmological parameters (see § 7.2).

In addition to references cited above, Dicke (1970, pp. 64–70), Wilson (1983), Wilkinson & Peebles (1990), Partridge (1995, chapter 2), and Kragh (1996, § 7.2) review the history. For the modern formalism see the more recent standard cosmology textbooks and Kamionkowski & Kosowsky (1999).

Since the Universe is now expanding, at earlier times it was denser and hotter. A naive extrapolation leads to a (mathematical) singularity at the beginning, with infinite density and temperature, at the initial instant of time, and over all space. This naive extrapolation is unjustified since the model used to derive it breaks down physically before the mathematical singularity is reached. Deriving the correct equations of motion for the very early Universe is an important area of current research. While there has been much work, there is as yet no predictive model that unifies gravity and quantum mechanics—and this appears essential for an understanding of the very early Universe, because as one goes back in time the gravitational expansion of the Universe implies that large cosmological length scales now correspond to tiny quantum-mechanical scales in the very very early Universe. There is a small but active group of workers who believe that only a resolution of this issue (i.e., the derivation of a full quantum theory of gravity) will allow for progress on the modeling of the very early Universe. But most others, perhaps inspired by the wonderful successes of particle physics models that have successfully described shorter and shorter distance physics, now believe that it is important to try to solve some of the "problems" of the Big Bang Model by attempting to model the cosmophysical world at an energy density higher than is probed by nucleosynthesis and other lower redshift physics but still well below the Planck energy density where quantum gravitational effects are important. This is the approach we take in the following discussion, by focusing on problems that could be resolved below the Planck density. Whether Nature has chosen this path is as yet unclear, but at least the simplest versions of the inflation scenario (discussed in the next section) are compatible with current observations and will likely be well tested by data acquired within this decade.

Assuming just nonrelativistic matter and radiation (CMB and neutrinos) in order of magnitude agreement with observations, the distance over which causal contact is possible grows with the age of the Universe. That is, if one assumes that in this model the cosmological principle is now valid because of "initial conditions" at an earlier time, then those initial conditions must be imposed over distances larger than the distance over which causal communication was possible. (And maybe this is what a quantum theory of gravitation will do for cosmology, but in the spirit of the earlier discussion we will view this as a problem of the Big Bang Model that should be resolved by physics at energies below the Planck scale.) Alpher et al. (1953, p. 1349) contains the earliest remarks (in passing) that we are aware of about this particle horizon problem. The terminology is due to Rindler (1956), which is an early discussion of horizons in general. Harrison (1968) also mentions the particle horizon problem in passing, but McCrea (1968) and Misner (1969) contain the first clear statements we are aware of, with Misner stating "These Robertson-Walker models therefore give no insight into why the observed microwave radiation from widely different angles in the sky has...very precisely...the same temperature." Other early discussions are in Dicke (1970, p. 61), Doroshkevich & Novikov (1970), and the text books of Weinberg (1972, pp. 525–526) and Misner et al. (1973, pp. 815–816), This issue was discussed in many papers and books starting in the early 1970s, but the celebrated Dicke & Peebles (1979) review is often credited with drawing prime-time attention to the particle horizon problem.

The large entropy of the Universe (as discussed above, there are now a few billion CMB photons for every baryon) poses another puzzle. When the Universe was younger and hotter there had to have been a thermal distribution of particles and antiparticles and, as the Universe expanded and cooled, particles and antiparticles annihilated into photons, resulting in the current abundance of CMB photons and baryons. Given the lack of a significant amount of antibaryons now, and the large photon to baryon ratio now, at early times there must have been a slight (a part in a few billion) excess of baryons over antibaryons. We return to this issue in the next section.

5.1.1. Gravitational Instability Theory from Newton Onward

5.1.2. Structure Growth in an Expanding Universe

5.1.3. Space Curvature

5.1.4. Dark Energy

5.1.5. Radiation and its Interaction with Baryonic Matter

5.1.6. Possible Matter Constituents

5.1.7. Free Streaming

5.1.8. Initial Density Perturbations and the Transfer Function

5.1.9. Gravitational Waves and Magnetic Fields

As a result of the gravitational growth of inhomogeneities in the matter distribution, when the photons decouple from the baryons at last scattering at a redshift z ∼ 10³ (see § 5.1 above), the photon temperature distribution is spatially anisotropic. In addition, in the presence of a CMB temperature quadrupole anisotropy, Thomson-Compton scattering of CMB photons off electrons prior to decoupling generates a linear polarization anisotropy of the CMB. After decoupling, the CMB photons propagate almost freely, influenced only by gravitational perturbations and late-time reionization. Measurements of the temperature anisotropy and polarization anisotropy provide important constraints on many parameters of models of structure formation. This area of research has seen spectacular growth in the last decade or so, following the COBE discovery of the CMB temperature anisotropy. It has been the subject of recent reviews; see White & Cohn (2002), Hu & Dodelson (2002), Peebles & Ratra (2003, § IV.B.11), Subramanian (2005), Giovannini (2005), and Challinor (2005). Here we focus only on a few recent developments.

The three-year WMAP observations of CMB temperature anisotropies (Hinshaw et al. 2007) are state-of-the-art data. On all but the very largest angular scales, the WMAP data are consistent with the assumption that the CMB temperature anisotropy is well described by a spatial Gaussian random process (Komatsu et al. 2003), consistent with earlier indications (Park et al. 2001; Wu et al. 2001). The few largest-scale angular modes exhibit a lack of power compared to what is expected in a spatially flat CDM model dominated by a cosmological constant (Bennett et al. 2003a), resulting in some debate about the assumptions of large-scale Gaussianity and spatial isotropy. This feature was also seen in the COBE data (Górski et al. 1998). The estimated large-angular-scale CMB temperature anisotropy power depends on the model used to remove foreground Galactic emission contamination. Much work has been devoted to understanding foreground emission on all scales (e.g., Mukherjee et al. 2003a; Bennett et al. 2003b; Tegmark et al. 2003), and the current consensus is that foregrounds are not the cause of the large-angular-scale WMAP effects.

The CMB temperature anisotropy is conventionally expressed as an expansion in spherical harmonic multipoles on the sky, and for a Gaussian random process the multipole (or angular) power spectrum completely characterizes the CMB temperature anisotropy. The observed CMB anisotropy is reasonably well fit by assuming only adiabatic fluctuations with a scale-invariant power spectrum. These observational results are consistent with the predictions of the simplest inflation models, where quantum-mechanical fluctuations in a weakly coupled scalar field are the adiabatic, Gaussian seeds for the observed CMB anisotropy and large-scale structure.

Smaller-scale inhomogeneities in the coupled baryon-radiation fluid oscillate (see § 5.1 above), and at decoupling some of these modes will be at a maximum or at a minimum, giving rise to acoustic peaks and valleys in the CMB anisotropy angular spectrum. The relevant length scale is the acoustic Hubble length at the epoch of recombination; this may be predicted by linear physics and so provides a standard ruler on the sky. Through the angular diameter distance relation, the multipole numbers ℓ of oscillatory features in the temperature anisotropy spectrum C_ℓ reflect space curvature (Ω_K) and the expansion history (which depends on Ω_M and Ω_Λ) of the Universe. The angular scales of the peaks are sensitive to the value of the matter-density parameter in an open Universe, but not in a spatially flat (Ω_K = 0) Universe dominated by a cosmological constant, where the first peak is at a multipole index ℓ ∼ 220. This provides a useful way to measure the curvature of spatial hypersurfaces. Sugiyama & Gouda (1992) and Kamionkowski et al. (1994a,1994b) presented early discussions of the CMB temperature anisotropy in an open Universe, and Brax et al. (2000), Baccigalupi et al. (2002), Caldwell & Doran (2004), and Mukherjee et al. (2003b) considered the case of scalar-field dark energy in a spatially flat Universe. CMB temperature anisotropy data on the position of the first peak is consistent with flat spatial hypersurfaces (e.g., Podariu et al. 2001; Durrer et al. 2003; Page et al. 2003). Model-based CMB data analysis is used to constrain more cosmological parameters (e.g., Lewis & Bridle 2002; Mukherjee et al. 2003c; Spergel et al. 2007). For example, the relative amplitudes of peaks in this spectrum are sensitive to the mass densities of the different possible constituents of matter (e.g., CDM, baryons, and neutrinos, Ω_CDM, Ω_B, and Ω_ν).

The CMB polarization anisotropy was first detected from the ground by the DASI experiment at the South Pole (Kovac et al. 2002). The three-year WMAP observations are the current state of the art (Page et al. 2007). For a recent review of polarization measurements, see Balbi et al. (2006). The polarization anisotropy peaks at a larger angular scale than the temperature anisotropy, indicating that there are inhomogeneities on scales larger than the acoustic Hubble length at recombination, consistent with what is expected in the inflation scenario. The polarization anisotropy signal is interpreted as the signature of reionization of the Universe. The ability of WMAP to measure polarization anisotropies allows this experiment to probe the early epochs of nonlinear structure formation, through sensitivity to the reionization optical depth τ.

Primordial gravitational waves or a primordial magnetic field can also generate CMB anisotropies. Of particular current interest are their contributions to various CMB polarization anisotropies. (Because polarization is caused by quadrupole fluctuations, these anisotropies constrain properties of the primordial fluctuations, such as the ratio of tensor-to-scalar fluctuations, r.) The effects of gravity waves on the CMB are discussed in the more recent standard cosmology and astroparticle textbooks and by Giovannini (2005). The magnetic field case has been reviewed by Giovannini (2006) and Subramanian (2006); recent topics of interest may be traced from Lewis (2004), Kahniashvili & Ratra (2005, 2007), and Brown & Crittenden (2005).

We continue discussion of the CMB anisotropies and cosmological parameters in § 7.2.

The emission of the first light in the Universe, seen today as the CMB, is followed by a "dark age" before the first stars and quasars form. Bromm & Larson (2004) review formation of the first stars. Eventually, high-energy photons from stars and quasars reionize intergalactic gas throughout the Universe (for reviews, see Fan et al. 2006; Choudhury & Ferrara 2006a; Loeb 2006a,2006b). Observations of polarization of microwave background photons by WMAP (Page et al. 2007) suggest that reionization occurs at redshift z ≈ 11. However, strong absorption of Lyman-α photons by intergalactic neutral hydrogen (Gunn & Peterson 1965), seen in spectra of quasars at redshift z ≈ 6 (Becker et al. 2001; Fan et al. 2002) indicates that reionization was not complete until somewhat later. This is an area of ongoing research (see, e.g., Choudhury & Ferrara 2006b; Gnedin & Fan 2006; Alvarez et al. 2006).

Current models for galaxy formation follow the picture (Hoyle 1953; Silk 1977; Binney 1977; Rees & Ostriker 1977; White & Rees 1978) in which dark matter halos form by collisionless collapse, after which baryons fall into these potential wells, are heated to virial temperature, and then cool and condense at the centers of the halos to form galaxies as we know them. In short, baryons fall into the gravitational potentials of "halos" of dark matter at the same time that those halos grow in size, hierarchically aggregating small clumps into larger ones. The baryons cool by emitting radiation and shed angular momentum, leading to concentrations of star formation and accretion onto supermassive black holes within the dark matter halos.

In addition to the perturbative approach to structure formation discussed in § 5.1, Lemaître also pioneered a nonperturbative approach based on a spherically symmetric solution of the Einstein equations. This spherical accretion model (Gunn & Gott 1972) describes the salient features of the growth of mass concentrations. See Gott (1977), Peebles (1993, Sec. 22), and Sahni & Coles (1995) for reviews of such models.

A phenomenological prescription for the statistics of nonlinear collapse of structure, i.e., the formation of gravitationally bound objects, is given by the Press-Schechter formulae (Press & Schechter 1974; Sheth & Tormen 1999). Attempts to firm up the theoretical basis of such formulae form the "excursion set" formalism, which treats the formation of a gravitationally bound halo as the result of a random walk (Mo & White 1996; White 1996; Sheth et al. 2001). For a review, see Cooray & Sheth (2002). These methods provide probability distributions for the number of bound objects as a function of mass threshold and can be generalized to develop a complementary description of the evolution of voids (Sheth & van de Weygaert 2004). A more rigorous approach assumes structure forms at high peaks in the smoothed density field (Kaiser 1984; Bardeen et al. 1986; Sahni & Coles 1995). Recent reviews of galaxy formation include Avila-Reese (2006) and Baugh (2006). The next subsection, 5.4, describes numerical methods for studying structure formation.

Apparent confirmation of the hierarchical picture of structure formation includes the striking images of galaxies apparently in the process of assembly, obtained by the Hubble Space Telescope (HST) in the celebrated "Hubble Deep Fields" (Ferguson et al. 2000; Beckwith et al. 2006). The detailed properties of galaxies and their evolution are outside the scope of this review. Recent reviews of the observational situation are Gawiser (2006) and Ellis (2007). Texts covering this topic include Spinrad (2005) and Longair (2008).

While the current best model of structure formation, in which CDM dominates the matter-density, works quite well on large scales, current observations indicate some possible problems with the CDM model on smaller scales; see Tasitsiomi (2003), Peebles & Ratra (2003, § IV.A.2), and Primack (2005) for reviews. Simulations of structure formation indicate that CDM model halos may have cores that are cuspier (Navarro et al. 1997; Swaters et al. 2000) and central densities that are higher (Moore et al. 1999a; Firmani et al. 2001) than are observed in galaxies. Another concern is that CDM models predict a larger than observed number of low-mass satellites of massive galaxies (Moore et al. 1999b; Klypin et al. 1999). These issues have led to consideration of models with reduced small-scale power. However, it seems difficult to reconcile suppression of small-scale power with the observed small-scale clustering in the neutral hydrogen at redshifts near 3.

The relationship between the distributions of galaxies (light) and matter is commonly referred to as "biasing." The currently favored dark-energy-dominated CDM model does not require significant bias between galaxies and matter; in the best-fit model the ratio of galaxy to matter clustering is close to unity for ordinary galaxies (Tegmark et al. 2004a).

Cosmological simulations using increasingly sophisticated numerical methods provide a test bed for models of structure formation. Bertschinger (1998) has reviewed methods and results.

Computer simulations of structure formation in the Universe began with purely gravitational codes that directly compute the forces between a finite number of particles (particle-particle or PP codes) that sample the matter distribution. Early results used direct N-body calculations (Aarseth et al. 1979). Binning the particles on a grid and computing the forces using the fast Fourier transform (the particle-mesh or PM method) is computationally more efficient, allowing simulation of larger volumes of space, but has force resolution of the order of the grid spacing. A compromise is the P³M method, which uses PM for large-scale forces supplemented by direct PP calculations on small scales, as used for the important suite of CDM simulations by Davis et al. (1985). For details on these methods, see Hockney & Eastwood (1988).

The force resolution of PM codes and the force resolution and speed of P³M codes may be increased by employing multiple grid levels (Villumsen 1989; Couchman 1991; Bertschinger & Gelb 1991; Gnedin & Bertschinger 1996). Adaptive mesh refinement (AMR; Berger & Collela 1989) does this dynamically to increase force resolution in the PM gravity solver (Kravtsov et al. 1997; Norman & Bryan 1999).

Another approach to achieving both speed and good force resolution in gravitational N-body simulation is use of the hierarchical tree algorithm (Barnes & Hut 1986). Large cosmological simulations have used a parallelized version of this method (Zurek et al. 1994). Significant increase in speed was found with the tree particle-mesh algorithm (Bode et al. 2000). GOTPM (Dubinski et al. 2004), a parallelized hybrid PM + tree scheme, has been used for simulations involving up to 8.6 × 10⁹ particles. PMFAST (Merz et al. 2005) is a recent parallelized multilevel PM code.

Incorporation of hydrodynamics and radiative transfer in cosmological simulations has made it possible to study not only the gravitational formation of dark matter halos but also the properties of baryonic matter, and, thus, the formation of galaxies associated with those halos. Methods for solving the fluid equations include smooth-particle hydrodynamics (SPH; see Monaghan 1992 for a review), which is an inherently Lagrangian approach, and Eulerian grid methods. Cosmological SPH simulations were pioneered by Evrard (1988) and Hernquist & Katz (1989). To date, the cosmological simulation with the largest number of particles (10¹⁰) employs SPH and a tree algorithm (GADGET; Springel et al. 2001). Grid-based codes used for cosmological simulation include that described by Cen (1992) and Ryu et al. (1993).

To date, no code has sufficient dynamic range to compute both the large-scale cosmological evolution on scales of many hundreds of megaparsecs and the formation of stars from baryons, but physical heuristics have been successfully incorporated into some codes to model the conversion of baryons to stars (see, e.g., Cen 1992).

The Millenium Run simulation (Springel et al. 2005) represents the current state of the art in following the evolution of both the dark matter and baryonic components on scales from the box size, 500h^-1 Mpc, down to the resolution limit of roughly 5h^-1 kpc. See this article and references therein for discussion of the many pieces of uncertain physics necessary for producing the observed baryonic structures.

Another approach to modeling the properties of the galaxies associated with dark matter halos is to use the history of halo mergers together with semianalytic modeling of galaxy properties (Lacey et al. 1993; Kauffmann et al. 1993; Cole et al. 1994; Somerville & Primack 1999). When normalized to the observed luminosity function of galaxies and Tully-Fisher relation for spiral galaxies, these semianalytic models (SAMs) reproduce many of the observed features of the galaxy distribution. A common approach is to use SAMs to "paint on" the properties of galaxies that would reside in the dark matter halos found in purely gravitational simulations. See Avila-Reese (2006) and Baugh (2006) for recent reviews. Related to the SAMs approach are halo occupation models (Berlind & Weinberg 2002; Kravtsov et al. 2004) that parameterize the statistical relationship between the masses of dark matter halos and the number of galaxies resident in each halo.

Studies of the global spacetime of the Universe assume the "cosmological principle" (Milne 1933), which is the supposition that the Universe is statistically homogeneous when viewed on sufficiently large scales. The angular distribution of radio galaxies provides a good test of this approach to homogeneity, because radio-bright galaxies and quasars may be seen in flux-limited samples to nearly a Hubble distance, c/H₀. Indeed, the ∼31,000 brightest radio galaxies observed at a wavelength of 6 cm (Gregory & Condon 1991) are distributed nearly isotropically, and similar results are found in the FIRST radio survey (Becker et al. 1995). (For a review of other evidence for large-scale spatial isotropy see § 3 of Peebles 1993.) In contrast, the Universe is clearly inhomogeneous on the more modest scales probed by optically selected samples of bright galaxies, For example, significant clustering is observed among the roughly 30,000 galaxies in the Zwicky et al. (1961–1968) catalog.

Maps of the distribution of nebulae revealed anisotropy in the sky before astronomers came to agree that many of these nebulae were distant galaxies (Charlier 1925). The Shapley & Ames (1932) catalog of galaxies clearly showed the nearby Virgo cluster of galaxies. Surveys of selected areas on the sky using photographic plates to detect distant galaxies clearly revealed anisotropy of the galaxy distribution and were used to quantify this anisotropy (Mowbray 1938). de Vaucouleurs (1953) recognized in this anisotropy the projected distribution of the local supercluster of galaxies.

Rubin (1954) used two-point correlations of galaxy counts from Harvard College Observatory plates to detect fluctuations on the scale of clusters of galaxies. The Shane & Wirtanen (1954) Lick Survey of galaxies used counts of galaxies found on large-format photographic plates taken at Lick Observatory to make the first large-scale map of the angular distribution of galaxies suitable for statistical analysis. Early analysis of these data included methods such as counts-in-cells analyses and the two-point correlation function (Limber 1954; Totsuji & Kihara 1969). The sky map of the Lick counts produced by Seldner et al. (1977) visually demonstrated the rich structure in the galaxy distribution. Peebles and collaborators used these data for much of their extensive work on galaxy clustering (Groth & Peebles 1977); for a review see Peebles (1980, § III).

The first Palomar Observatory Sky Survey (POSS) yielded two important catalogs: the Abell (1958) catalog of clusters and the Zwicky et al. (1961–1968) catalog of clusters and galaxies identified by eye from the photographic plates. Abell (1961) found evidence for angular "superclustering" (clustering of galaxy clusters) that was confirmed statistically by Hauser & Peebles (1973). Photographic plates taken at the United Kingdom Schmidt Telescope Unit (UKSTU) were digitized using the automatic plate measuring (APM) machine to produce the APM catalog of roughly two million galaxies. Calibration with CCD photometry made the APM catalog invaluable for accurate study of the angular correlation function of galaxies on large scales (Maddox et al. 1990). Perhaps the last large-area galaxy photometric survey to employ photographic plates was the Digitized Palomar Observatory Sky Survey (DPOSS) (Gal et al. 2004).

The largest imaging survey that employs a camera with arrays of charge-coupled devices (CCDs) is the Sloan Digital Sky Survey (SDSS; Stoughton et al. 2002). The imaging portion of this survey includes five-color digital photometry of 8000 deg² of sky, with 215 million detected objects. Imaging for the SDSS is obtained using a special-purpose 2.5 m telescope with a 3° field of view (Gunn et al. 2006).

Important complements to optical surveys include large-area catalogs of galaxies selected in the infrared and ultraviolet. Nearly all-sky source catalogs were produced from infrared photometry obtained with the Infrared Astronomical Satellite (IRAS; Beichman et al. 1988) and the ground-based Two Micron All Sky Survey (2MASS; Jarrett et al. 2000). The ongoing Galaxy Evolution Explorer satellite (GALEX; Martin et al. 2005) is obtaining ultraviolet imaging over the whole sky.

Systematic surveys of galaxies using spectroscopic redshifts to infer their distances began with observations of galaxies selected from the Shapley-Ames catalog (Humason et al. 1956; Sandage 1978). Important early mapping efforts include identification of superclusters and voids in the distribution of galaxies and Abell clusters by Jôeveer et al. (1978), the Gregory & Thompson (1978) study of the Coma/Abell1367 supercluster and its environs that identified voids, and the Kirshner et al. (1981) study of the correlation function of galaxies and discovery of the giant void in Boötes. Early targeted surveys include the Giovanelli & Haynes (1985) survey of the Perseus-Pisces supercluster.

Redshift surveys of large areas of the sky began with the first Center for Astrophysics redshift survey (CfA1; Huchra et al. 1983), which includes redshifts for 2401 galaxies brighter than apparent magnitude m_B = 14.5 over a wide area toward the North Galactic Pole. CfA2 (Falco et al. 1999) followed over roughly the same area, extending to apparent magnitude m_B = 15.5. At this depth, the rich pattern of voids, clusters, and superclusters were strikingly obvious (de Lapparent et al. 1986). Giovanelli & Haynes (1991) review the status of galaxy redshift surveys ca. 1991.

Both CfA redshift surveys used the Zwicky catalog of galaxies to select targets for spectroscopy. The Southern Sky Redshift Survey (SSRS; da Costa et al. 1998) covers a large area of the southern hemisphere (contiguous with CfA2 in the northern galactic cap) to similar depth, using the ESO/Uppsala Survey to select galaxy targets and a spectrograph similar to that employed for the CfA surveys. The Optical Redshift Survey (ORS) supplemented existing redshift catalogs with 1300 new spectroscopic redshifts to create a nearly all-sky survey (Santiago et al. 1995).

Deep "pencil-beam" surveys of narrow patches on the sky revealed apparently periodic structure in the galaxy distribution (Broadhurst et al. 1990).

The Las Campanas Redshift Survey (LCRS; Shectman et al. 1996), the first large-area survey to use fiber optics, covered over 700 deg² near the South Galactic Pole. This survey was important because it showed that structures such as voids and superclusters found in shallower surveys are ubiquitous, but the structures seen by LCRS were no larger than those found earlier. The Century Survey (Geller et al. 1997) and the ESO Deep Slice survey (Vettolani et al. 1998) were likewise useful for statistically confirming this emerging picture of large-scale structure.

Sparse surveys of galaxies to efficiently study statistical properties of the galaxy distribution include the Stromlo-APM redshift survey (Loveday et al. 1996) based on 1/20 sampling of the APM galaxy catalog and the Durham/UKSTU redshift survey (Ratcliffe et al. 1998).

While optically selected surveys are relatively blind to structure behind the Milky Way, redshift surveys based on objects detected in the infrared provide nearly all-sky coverage. A sequence of surveys of objects detected by IRAS were carried out, flux-limited to 2 Jy (Strauss et al. 1992), 1.2 Jy (Fisher et al. 1995), and 0.6 Jy (Saunders et al. 2000). The 6dF Galaxy Survey (Jones et al. 2004) will measure redshifts of 150,000 galaxies photometrically identified by 2MASS (Jarrett et al. 2000).

The Two Degree Field Galaxy Redshift Survey (2dFGRS) of 250,000 galaxies (Colless et al. 2001) was selected from the APM galaxy catalog and observed using the Two Degree Field multifiber spectrograph at the Anglo-Australian 4 m telescope. The survey is complete to apparent magnitude m_J = 19.45 and covers about 1500 deg².

The spectroscopic component of the SDSS (Stoughton et al. 2002) includes medium-resolution spectroscopy of 675,000 galaxies and 96,000 quasars identified from SDSS photometry. These spectra are obtained with dual fiber-optic CCD spectrographs on the same 2.5 m telescope. The main galaxy redshift survey is complete to m_r = 17.77 and is complemented by a deeper survey of luminous red galaxies. The ongoing extension of this survey (SDSS-II) will expand the spectroscopic samples to more than 900,000 galaxies and 128,000 quasars.

Spectroscopic surveys that trace structure in the galaxy distribution at much larger redshift include the DEEP2 survey (Coil et al. 2004) and others (Steidel et al. 2004) employing the Keck Observatory, and the VIMOS VLT Deep survey (Le Fèvre et al. 2005).

Mapping of the Universe using galaxy clusters as tracers began with study of the Abell catalog (Abell 1958; Abell et al. 1989). Studies of the angular clustering of Abell clusters includes Hauser & Peebles (1973). Several redshift surveys of Abell clusters have been conducted, including those described by Postman et al. (1992) and Katgert (1996). Important cluster samples have also been identified from digitized photographic plates from the UKSTU, followed up by redshift surveys of cluster galaxies (Lumsden et al. 1992; Dalton et al. 1992). More distant samples of clusters have been identified using deep CCD photometry (see, e.g., Postman et al. 1996; Gladders & Yee 2005). In X-ray bandpasses, cluster samples useful for studying large-scale structure have been identified using data from ROSAT (Romer et al. 1994; Böhringer et al. 2004). The SDSS is producing large catalogs of galaxy clusters using a variety of selection methods (Bahcall et al. 2003). Use of the Sunyaev-Zel'dovich effect (the microwave decrement caused by Thomson-Compton scattering of the CMB photons by the intracluster gas) holds great promise to identify new deep samples of galaxy clusters (Carlstrom et al. 2002). General reviews of clusters of galaxies include Rosati et al. (2002), Voit (2005), and Borgani (2006).

The advent of multiobject wide-field spectrographs has made possible the collection of very large samples of spectroscopically confirmed quasars, as observed by the 2dF QSO Redshift Survey (Croom et al. 2004) and the SDSS (Schneider et al. 2005). For a ca. 1990 review of the field, see Hartwick & Shade (1990). While quasars themselves are too sparsely distributed to provide good maps of the large-scale distribution of matter, their clustering in redshift space has been measured (Osmer 1981) and generally found to be similar to that of galaxies (Outram et al. 2003). Similar results have been obtained from clustering analyses of active galactic nuclei in the nearby universe (Wake et al. 2004), although this clustering depends in detail on the type of active galactic nucleus (AGN; Constantin & Vogeley 2006).

The distribution of absorption lines from gas, particularly from the Lyman-α "forest" of neutral hydrogen clouds along the line of sight toward bright quasars (Lynds 1971; Rauch 1998) provides an important statistical probe of the distribution of matter (see, e.g., McDonald et al. 2005) on small scales and at large redshift.

When measured over sufficiently large scales, the peculiar motions of galaxies or clusters simply depend on the potential field generated by the mass distribution (see Peebles 1980; 1993; Davis & Peebles 1983). Techniques for measuring distances to other galaxies are critically reviewed in Rowan-Robinson (1985), Jacoby et al. (1992), Strauss & Willick (1995), and Webb (1999). Together with the galaxy or cluster redshifts, these measurements yield maps of the line-of-sight component of the peculiar velocity. From such data it is possible to reconstruct a map of the matter-density field (e.g., Bertschinger & Dekel 1989; Dekel 1994) or to trace the galaxy orbits back in time (e.g., Peebles 1990; Goldberg & Spergel 2000). Analyses of correlations of the density and velocity fields also yield constraints on the cosmic matter-density (e.g., Willick et al. 1997).

Rubin et al. (1976) were the first to find evidence for bulk flows from galaxy peculiar velocities. Dressler et al. (1987) found evidence for a bulk flow toward a large mass concentration, dubbed the "Great Attractor." Lauer & Postman (1994) found surprising evidence for motion of the Local Group on a larger scale. However, analysis of subsequent peculiar velocity surveys indicates that the inferred bulk flow results, including those of Lauer and Postman, are consistent within the uncertainties (Hudson et al. 2000). The status of this field ca. 1999 is surveyed by Courteau & Willick (2000); recent results include Hudson et al. (2004), and Dekel (1994) and Strauss & Willick (1995) review this topic. Comparison of peculiar velocity surveys with the peculiar velocity of our Galaxy implied by the CMB dipole indicates that a significant component of our motion must arise from density inhomogeneities that lie at rather large distance, beyond 60h^-1 Mpc (Hudson et al. 2004).

The clustering pattern of extragalactic objects reflects both the initial conditions for structure formation and the complex astrophysics of formation and evolution of these objects. In the standard picture described above, linear perturbation theory accurately describes the early evolution of structure; thus measurement of clustering on very large scales, where the clustering remains weak, closely reflects the initial conditions. On these scales the density field is very nearly Gaussian random phase, therefore the two-point correlation function of the galaxy number density field (also called the autocorrelation or covariance function) or its Fourier transform, the power spectrum, provides a complete statistical description. (Temperature anisotropies of the CMB discussed in § 5.2 arise from density fluctuations at redshift z ∼ 10³ that evolve in the fully linear regime.) On the scales of galaxies and clusters of galaxies, gravitational evolution is highly nonlinear, and the apparent clustering depends strongly on the detailed relationship between mass and light in galaxies. In between the linear and nonlinear regimes lies the "quasi-linear" regime in which clustering growth proceeds most rapidly. A wide range of statistical methods have been developed to quantify this complex behavior. Statistical properties of the galaxy distribution and details of estimating most of the relevant statistics are described in depth by Peebles (1980), Martínez & Saar (2002), and Bernardeau et al. (2002). Methods of using galaxy redshift surveys to constrain cosmology are reviewed by Lahav & Suto (2004) and Percival (2006). Constraints on cosmological parameters from such measurements are discussed below in § 7.2.

The simplest set of statistical measures are the n-point correlation functions, which describe the joint probability in excess of random of finding n galaxies at specified relative separation. Early applications of correlation functions to galaxy data include Limber (1954), Totsuji & Kihara (1969), and Groth & Peebles (1977). The n-point functions may be estimated by directly examining the positions of n-tuples of galaxies or by using moments of galaxy counts in cells of varying size. Tests of scaling relations among the n-point functions are discussed in detail by Bernardeau et al. (2002).

Power spectrum analyses of large galaxy redshift surveys (Vogeley et al. 1992; Fisher et al. 1993; Tegmark et al. 2004b) yield useful constraints on cosmological models. Closely related to power spectrum analyses are estimates of cosmological parameters using orthogonal functions (Vogeley & Szalay 1996; Pope et al. 2004). Tegmark et al. (1998) discuss the merits of different methods of power spectrum estimation. Verde et al. (2002) describe a measurement of the galaxy bispectrum.

A number of statistics have been developed to quantify the geometry and topology of large-scale structure. The topological genus of isodensity contours characterizes the connectivity of large-scale structure (Gott et al. 1987). Measurements of the genus are consistent with random phase initial conditions (as predicted by inflation) on large scales (Gott et al. 1989), with departures from Gaussianity on smaller scales where nonlinear gravitational evolution and biasing of galaxies are evident (Vogeley et al. 1994; Gott et al. 2006). Similar techniques are used to check on the Gaussianity of the CMB anisotropy (Park et al. 2001; Wu et al. 2001; Komatsu et al. 2003), as well as identify foreground emission signals in CMB anisotropy data (Park et al. 2002).

The void probability function, which characterizes the frequency of completely empty regions of space (White 1979), has been estimated from galaxy redshift surveys (Maurogordato & Lachièze-Rey 1987; Hoyle & Vogeley 2004). Catalogs of voids have been constructed with objective void-finding algorithms (El-Ad et al. 1996; Hoyle & Vogeley 2002).

Early investigations of the pattern of galaxy clustering dating back to Charlier (1925) suggested a clustering hierarchy. The fractal model of clustering introduced by Mandelbrot (1982, and references therein) further motivated investigation of the possibility of scale-invariant clustering of galaxies. Results of such analyses of galaxy survey data were controversial (compare, e.g., Sylos Labini et al. 1998 with Hatton 1999 and Martínez et al. 2001 and references therein). While fractal behavior is seen on small scales, there is fairly strong evidence for an approach to homogeneity in galaxy redshift and photometric surveys on very large scales. Thus, a simple scale-invariant fractal description seems to be ruled out. A multifractal description of clustering continues to provide a useful complement to other statistical descriptors (Jones et al. 2005). Consideration of modified forms of the fractal picture are of interest for providing slight non-Gaussianity on very large scales that might be needed to explain the very largest structures in the Universe.

Anisotropy of galaxy clustering in redshift space results from bulk flows on large scales that amplify clustering along the line of sight to the observer and from motions of galaxies in virialized systems such as clusters that elongate those structures along the line of sight (Kaiser 1987). Hamilton (1998) provides an extensive review and Tinker et al. (2006) describe recent methods for estimating cosmological parameters from redshift-space distortions of the correlation function or power spectrum.

The dependence of clustering statistics on properties of galaxies provides important clues to their history of formation and reflects the complex relationship between the distributions of mass and luminous matter. The amplitude of galaxy clustering is seen to vary with galaxy morphology (e.g., Davis & Geller 1976; Guzzo et al. 1997) and with luminosity (e.g., Hamilton 1988; Park et al. 1994). In recent analyses of the SDSS and 2dFGRS, these and similar trends with color, surface brightness, and spectral type are seen (Norberg et al. 2002; Zehavi et al. 2005).

Spectroscopy obtained with 8–10 m class telescopes has recently made it possible to accurately study structure in the galaxy distribution at higher redshift (Coil et al. 2004; Adelberger et al. 2005; Le Fèvre et al. 2005).

As mentioned in § 4, observations of Type Ia supernovae (SNeIa) provide strong evidence that the expansion of the Universe is accelerating. Type Ia supernovae have the useful property that their peak intrinsic luminosities are correlated with how fast they dim, which allows them to be turned into standard candles. At redshifts approaching unity, observations indicate that they are dimmer (and so farther away) than would be predicted in an unaccelerating Universe (Riess et al. 1998; Perlmutter et al. 1999). In the context of general relativity this acceleration is attributed to dark energy that varies slowly with time and space, if at all. A mass-energy component that maintains constant (or nearly constant) density has negative pressure. Because pressure contributes to the active gravitational mass density, negative pressure, if large enough, can overwhelm the attraction caused by the usual (including dark) matter mass density and result in accelerated expansion. For a careful review of the early supernova tests see Leibundgut (2001). For discussions of the cosmological implications of this test see Peebles & Ratra (2003) and Perivolaropoulos (2006). Current supernova data show that models with vanishing cosmological constant are more than four standard deviations away from the best fit.

The supernova test assumes general relativity and probes the geometry of spacetime. The result is confirmed by a test using the CMB anisotropy that must, in addition, assume the CDM model for structure formation discussed in § 4 (see § 5.3 for apparent problems with this model). As discussed in § 5.2, CMB anisotropy data on the position of the first peak in the angular power spectrum are consistent with the curvature of spatial hypersurfaces being small. Many independent lines of evidence indicate that the mass density of nonrelativistic matter (CDM and baryons)—a number also based on the CDM structure formation model—is about 25% or 30% of the critical Einstein–de Sitter density (see §§ 4 and 7.2). Because the contemporary mass density of radiation and other relativistic matter is small, a cosmological constant or dark energy must contribute 70% or 75% of the current mass budget of the Universe. For reviews of the CMB data constraints, see Peebles & Ratra (2003), Copeland et al. (2006), and Spergel et al. (2007).

The model suggested by the SNeIa and CMB data, spatially flat and with contemporary mass-energy budget split between a cosmological constant or dark energy (∼70%), dark matter (∼25%), and baryonic matter (∼5%), is broadly consistent with the results of a large number of other cosmological tests. In this subsection we present a very brief discussion of some of these tests and the constraints they impose on the parameters of this "standard" cosmological model. Two nice reviews of the cosmological tests are § 13 of Peebles (1993) and Sandage (1995). Hogg (1999) provides a concise summary of various geometrical measures used in these tests. Section IV of Peebles & Ratra (2003) reviews more recent developments and observational constraints. Here we summarize some of these as well as the significant progress of the last four years. Numerical values for cosmological parameters are listed in Lahav & Liddle (2006), although in some cases there is still significant ongoing debate.

There have been many—around 500—measurements of the Hubble constant H₀, (Huchra 2007), the current expansion rate. Since there is debate about the error estimates of some of these measurements, a median statistics meta-analysis estimate of H₀ is probably the most robust estimate (Gott et al. 2001). At two standard deviations this gives H₀ = 100h km s^-1 Mpc^-1 = 68 ± 7 km s^-1 Mpc^-1 = (14 ± 1 Gyr)^-1 (Chen et al. 2003), where the first equation defines h. It is significant that this result agrees with the estimate from the HST Key Project (Freedman et al. 2001), the HST estimate of Sandage and collaborators (Sandage et al. 2006), and the WMAP three-year data estimate (which assumes the CDM structure formation model; Spergel et al. 2007).

A measurement of the redshift dependence of the Hubble parameter can be used to constrain cosmological parameters (Jimenez & Loeb 2002; Simon et al. 2005). For applications of this test using preliminary data, see Samushia & Ratra (2006) and Sen & Scherrer (2007).

Expansion time tests are reviewed in Peebles & Ratra (2003, § IV.B.3). A recent development is the WMAP CMB anisotropy data estimate of the age of the Universe, t₀ = 13.7 ± 0.3 Gyr at two standard deviations (Spergel et al. 2007), which assumes the CDM structure formation model. This WMAPt₀ estimate is consistent with t₀ estimated from globular cluster observations (Krauss & Chaboyer 2003; Gratton et al. 2003; Imbriani et al. 2004) and from white dwarf star measurements (Hansen et al. 2004). The above values of H₀ and t₀ are consistent with a spatially flat, dark-energy dominated Universe.

As discussed in § 2.3, Peebles & Ratra (2003), Fields & Sarkar (2006), and Steigman (2006), ⁴He and ⁷Li abundance measurements favor a higher baryon density than the D abundance measurements and the WMAP CMB anisotropy data. (This difference is under active debate.) However, it is remarkable that high-redshift (z ∼ 10³) CMB data and low-redshift (z ≲ few) abundance measurements indicate a very similar baryon density. A summary range of the baryonic density parameter from nucleosynthesis is Ω_B = (0.0205 ± 0.0035)h^-2 at two standard deviations (Fields & Sarkar 2006).

As mentioned above, Type Ia supernovae apparent magnitudes as a function of redshift may be used to constrain the cosmological model. See Peebles & Ratra (2003, § IV.B.4) for a summary of this test. Riess et al. (1998) and Perlmutter et al. (1999) provided initial constraints on a cosmological constant from this test, and Podariu & Ratra (2000) and Waga & Frieman (2000) generalized the method to constrain scalar-field dark energy. Developments may be traced back from Wang & Tegmark (2005), Clocchiatti et al. (2006), Astier et al. (2006), Riess et al. (2007), Nesseris & Perivolaropoulos (2005), Jassal et al. (2006), and Barger et al. (2007). Proposed satellite experiments are under active discussion and should result in tight constraints on dark energy and its evolution. See Podariu et al. (2001), Perlmutter et al. (2006), and Réfrégier et al. (2006) for developments in this area.

The angular size of objects (e.g., quasars, compact radio sources, radio galaxies) as a function of redshift provides another cosmological test. These observations are not as numerous as the supernovae, so this test is much less constraining, but the results are consistent with those from the SNeIa apparent magnitude test. Developments may be traced back through Chen & Ratra(2003a) and Podariu et al. (2003). Daly & Djorgovski (2006) describe a way of combining the apparent magnitude and angular size data to more tightly constrain cosmological parameters.

"Strong" gravitational lensing, by a foreground galaxy or cluster of galaxies, produces multiple images of a background radio source. The statistics of strong lensing may be used to constrain the cosmological model. Fukugita et al. (1990) and Turner (1990) have noted that for low nonrelativistic matter-density the predicted lensing rate is significantly larger in a cosmological-constant dominated spatially flat model than in an open model. The scalar-field dark-energy case is discussed in Ratra & Quillen (1992) and lies between these two limits. For reviews of the test see Peebles & Ratra (2003, § IV.B.6) and Kochanek (2006). Recent developments may be traced back from Fedeli & Bartelmann (2007). Cosmological constraints from the CLASS gravitational lens statistics data are discussed in Chae et al. (2002, 2004), and Alcaniz et al. (2005). These constraints are consistent with those derived from the supernova apparent magnitude data, but are not as restrictive.

Galaxy motions respond to fluctuations in the gravitational potential, and, thus, peculiar velocities of galaxies may be used to estimate the nonrelativistic matter-density parameter Ω_M (as discussed in §§ 4 and 6.5 above and in Peebles 1999 and Peebles & Ratra 2003, § IV.B.7) by comparing the pattern of flows with maps of the galaxy distribution. Note that peculiar velocities are not sensitive to a homogeneously distributed mass-energy component. For a summary of recent results from the literature, see Pike & Hudson (2005). Measurements of the anisotropy of the redshift-space galaxy distribution that is produced by peculiar velocities also yield estimates of the matter-density (Kaiser 1987; Hamilton 1998). Most methods measure this anisotropy in the galaxy autocorrelation or power spectrum (see, e.g., Tinker et al. 2006). Recent analyses include Hawkins et al. (2003) and da Ângela et al. (2005) from the 2dFGRS and 2QZ surveys. Also of interest are clustering analyses of the SDSS that explicitly take into account this redshift-space anisotropy either by using the predicted distortions when constructing eigenmodes (Pope et al. 2004) or by constructing modes that are sensitive to radial vs. angular fluctuations (Tegmark et al. 2004b).

A median statistics analysis of density estimates from peculiar velocity measurements and a variety of other data indicates that the nonrelativistic matter-density parameter lies in the range 0.2 ≲ Ω₀ ≲ 0.35 at two standard deviations (Chen & Ratra 2003b). This is consistent with estimates based on other data, e.g., the WMAP CMB data result in a very similar range (Spergel et al. 2007).

"Weak" gravitational lensing (which mildly distorts the images of background objects), in combination with other data, should soon provide tight constraints on the nonrelativistic matter-density parameter. For reviews of weak lensing see Réfrégier (2003), Schneider (2006), and Munshi et al. (2006). See Schimd et al. (2007), Hetterscheidt et al. (2007), and Kitching et al. (2007) for recent developments. Weak gravitational lensing also provides evidence for dark matter (see, e.g., Clowe et al. 2006; Massey et al. 2007).

Rich clusters of galaxies are thought to have originated from volumes large enough to have fairly sampled both the baryons and the dark matter. In conjunction with the nucleosynthesis estimate of the baryonic mass density parameter, the rich cluster estimate of the ratio of baryonic and nonrelativistic (including baryonic) matter—the cluster baryon fraction—provides an estimate of the nonrelativistic matter-density parameter (White et al. 1993; Fukugita et al. 1998). Estimates of Ω_M from this test are in the range listed above. A promising method for measuring the cluster baryonic gas mass fraction uses the Sunyaev-Zel'dovich effect (Carlstrom et al. 2002).

An extension of this cluster test makes use of measurements of the rich cluster baryon mass fraction as a function of redshift. For relaxed rich clusters (not those in the process of collapsing) the baryon fraction should be independent of redshift. The cluster baryon fraction depends on the angular diameter distance (Sasaki 1996; Pen 1997), so the correct cosmological model places clusters at the right angular diameter distances to ensure that the cluster baryon mass fraction is independent of redshift. This test provides a fairly restrictive constraint on Ω_M, consistent with the range above; developments may be traced back through Allen et al. (2004), Chen & Ratra (2004), Kravtsov et al. (2005), and Chang et al. (2006). When combined with complementary cosmological data, especially the restrictive SNeIa data, the cluster baryon mass fraction versus redshift data provide tight constraints on the cosmological model, favoring a cosmological constant but not yet ruling out slowly varying dark energy (Rapetti et al. 2005; Alcaniz & Zhu 2005; Wilson et al. 2006).

The number density of rich clusters of galaxies as a function of cluster mass, both at the present epoch and as a function of redshift, constrains the amplitude of mass fluctuations and the nonrelativistic matter-density parameter (see § IV.B.9 of Peebles & Ratra 2003 and references therein). Current cluster data favor a matter-density parameter in the range discussed above (Rosati et al. 2002; Voit 2005; Younger et al. 2005; Borgani 2006).

The rate at which large-scale structure forms could eventually provide another direct test of the cosmological model. The cosmological constant model is discussed in Peebles (1984) and some of the more recent textbooks listed below. The scalar-field dark-energy model is not as tractable; developments may be traced from Mainini et al. (2003), Mota & van de Bruck (2004), and Maio et al. (2006).

Measurements of CMB temperature and polarization anisotropies (see § 5.2 above and § IV.B.11 of Peebles & Ratra 2003) provide some of the strongest constraints on several cosmological model parameters. These constraints depend on the assumed structure formation model. Current constraints are usually based on the CDM model (or some variant of it). As discussed in § 5.2, the three-year WMAP data (Hinshaw et al. 2007) provide state-of-the-art constraints (Spergel et al. 2007).

Data on the large-scale power spectrum (or correlation function) of galaxies complement the CMB measurements by connecting the inhomogeneities observed at redshift z ∼ 10³ in the CMB to fluctuations in galaxy density close to z = 0, and by relating fluctuations in gravitating matter to fluctuations in luminous matter (which is an additional complication). For a recent discussion of the galaxy power spectrum, see Percival et al. (2007), from which earlier developments may be traced. It is a remarkable success of the current cosmological model that it succeeds in providing a reasonable fit to both sets of data. The combination of WMAP data with clustering measurements from SDSS or the 2dFGRS reduces several of the parameter uncertainties. For recent examples of such analyses, see Tegmark et al. (2004a) and Doran et al. (2007a).

The peak of the galaxy power spectrum reflects the Hubble length at matter-radiation equality and so constrains Ω_M h. The overall shape of the spectrum is sensitive to the densities of the different matter components (e.g., neutrinos would cause damping on small scales) and the density of dark energy. The same physics that leads to acoustic peaks in the CMB anisotropy causes oscillations in the galaxy power spectrum—or a single peak in the correlation function. Eisenstein et al. (2005) report a three standard deviation detection of this "baryon acoustic oscillation" peak at ∼100h^-1 Mpc in the correlation function of luminous red galaxies (LRGs) measured in the SDSS. The resulting measurement of Ω_M is independent of and consistent with other low-redshift measurements and with the high-redshift WMAP result. This is remarkable given the widely different redshifts probed (LRGs probe z = 0.35) and notable because possible systematics are different. For discussions of the efficacy of future measurements of the baryon acoustic oscillation peak to constrain dark energy, see Wang (2006), McDonald & Eisenstein (2006), and Doran et al. (2007b). For constraints from a joint analysis of these data with supernovae and CMB anisotropy data, see Wang & Mukherjee (2006).

Tegmark et al. (2006) include a nice description of how the large-scale galaxy power spectrum provides independent measurement of Ω_M and Ω_B, which breaks several parameter degeneracies and thereby decreases uncertainties on Ω_M,h and t₀. A combined WMAP + SDSS analysis reduces uncertainties on the matter-density, neutrino density, and tensor-to-scalar ratio by roughly a factor of 2. See Sánchez et al. (2006) for an analysis of the 2dFGRS large-scale structure data in conjunction with CMB measurements.

Measurements of the clustering of Lyman-α forest clouds complement larger-scale constraints, such as those from the CMB and large-scale structure, by probing the power spectrum of fluctuations on smaller scales (McDonald et al. 2005). Combining observations of 3000 SDSS Lyman-α forest cloud spectra with other data, Seljak et al. (2006) constrain possible variation with the length scale of the spectral index of the primordial power spectrum and find that Lyman-α cloud clustering may indicate a slightly higher power spectrum normalization, σ₈ (the fractional mass density inhomogeneity smoothed over 8h^-1 Mpc), than do the WMAP data alone, or the WMAP data combined with large-scale structure measurements.

The presence of dark energy or nonzero spatial curvature causes time evolution of gravitational potentials as CMB photons traverse the Universe from their "emission" at z ∼ 10³ to today. The resulting net redshifts or blueshifts of photons cause extra CMB anisotropy, known as the Integrated Sachs-Wolfe (ISW) contribution. This contribution has been detected by cross-correlation of CMB anisotropy and large-scale structure data. The resulting constraints on dark energy are consistent with the model discussed above (and references cited in Boughn & Crittenden 2005; Gaztañaga et al. 2006). In principle, measurements of the ISW effect at different redshifts can constrain the dark-energy model. Pogosian (2006) discusses recent developments and the potential of future ISW measurements.

The plethora of observational constraints on cosmological parameters has spawned interest in statistical methods for combining these constraints. Lewis & Bridle (2002), Verde et al. (2003), and Tegmark et al. (2004a) have discussed statistical methods employed in some of the recent analyses described above. Use of such advanced statistical techniques is important because of the growing number of parameters in current models and possible degeneracies between them in fitting the observational data. Developments may be traced back through Alam et al. (2007), Zhang et al. (2007), Zhao et al. (2007), Davis et al. (2007), Wright (2007), and Kurek & Szydłowski (2007).

To describe large-scale features of the Universe (including CMB anisotropy measured by WMAP and some smaller-angular-scale experiments, large-scale structure in the galaxy distribution, and the SNeIa luminosity-distance relation) the simplest version of the "power-law-spectrum spatially flat ΛCDM model" requires fitting no fewer than six parameters (Spergel et al. 2007): nonrelativistic matter-density parameter Ω_M, baryon density parameter Ω_B, Hubble constant H₀, amplitude of fluctuations σ₈, optical depth to reionization τ, and scalar perturbation index n. This model assumes that the primordial fluctuations are Gaussian random phase and adiabatic. As suggested by its name, this model further assumes that the primordial fluctuation spectrum is a power law (running power-spectral index independent of scale dn/d ln k = 0), the Universe is flat (Ω_K = 0), the bulk of the matter-density is CDM (Ω_CDM = Ω_M - Ω_B) with no contribution from hot dark matter (neutrino density Ω_ν = 0), and that dark energy in the form of a cosmological constant comprises the balance of the mass-energy density (Ω_Λ = 1 - Ω_M). Of course, constraints on this model assume the validity of the CDM structure formation model.

Combinations of observations provide improved parameter constraints, typically by breaking parameter degeneracies. For example, the constraints from WMAP data alone are relatively weak for H₀, Ω_Λ, and Ω_K. Other measurements such as from SNeIa or galaxy clusters are needed to break the degeneracy between Ω_K and Ω_Λ, which lies approximately along Ω_K ≈ -0.3 + 0.4Ω_Λ. The degeneracy between Ω_M and σ₈ is broken by including weak lensing and cluster measurements. The degeneracy between Ω_M and H₀ can be removed, of course, by including a constraint on H₀. As a result, including H₀ data restricts the geometry to be very close to flat. A caveat regarding this last conclusion is that it assumes that the dark-energy density does not evolve.

CDM-model-dependent clustering limits on baryon density (Ω_B = (0.0222 ± 0.0014)h^-2 from WMAP and SDSS data combined at 95% confidence; Tegmark et al. 2006) are now better than those from light element abundance data (because of the tension between the ⁴He and ⁷Li data and the D data). It is important that the galaxy observations complement the CMB data in such a way as to lessen reliance on the assumptions stated above for the power-law flat ΛCDM model. If the SDSS LRG P(k) measurement is combined with WMAP data, then several of the prior assumptions used in the WMAP-alone analysis (Ω_K = 0, Ω_ν = 0, no running of the spectral index n of scalar fluctuations, no inflationary gravity waves, no dark-energy temporal evolution) are not important. A major reason for this is the sensitivity of the SDSS LRG P(k) to the baryon acoustic scale, which sets a "standard ruler" at low redshift.

The SNeIa observations are a powerful complement to CMB anisotropy measurements because the degeneracy in Ω_M versus Ω_Λ for SNeIa measurements is almost orthogonal to that of the CMB. Without any assumption about the value of the Hubble constant but assuming that the dark energy does not evolve, combining SNeIa and CMB anisotropy data clearly favors nearly flat cosmologies. On the other hand, assuming the Universe is spatially flat, combined SNeIa and cluster baryon fraction data favors dark energy that does not evolve—a cosmological constant—see Rapetti et al. (2005), Alcaniz & Zhu (2005), and Wilson et al. (2006).

The bottom line is that statistical analyses of these complementary observations strongly support the flat ΛCDM cosmological model. It is remarkable that many of the key parameters are now known to better than 10%. However, several weaknesses remain, as discussed in the following, and final, section of this review. Time and lots of hard work will tell if these weaknesses are simply details to be cleaned up, or if they reveal genuine failings of the model, the pursuit of which will lead to a deeper understanding of physics and/or astronomy. It is worth recalling that, at the beginning of the previous century just before Einstein's burst of 1905 papers, it was thought by most physicists that classical physics fit the data pretty well.

As discussed in §§ 4 and 7, there is strong evidence that the dominant component of mass-energy is in the form of something like Einstein's cosmological constant. In detail, does the dark energy vary with space or time? Data so far are consistent with a cosmological constant with no spatial or temporal evolution, but the constraints do not strongly exclude other possibilities. This uncertainty is complemented by the relatively weak direct evidence for a spatially flat universe; as Wright (2006), Tegmark et al. (2006), and Wang & Mukherjee (2007) point out, it is incorrect to assume Ω_K = 0 when constraining the dark-energy time dependence, because observational evidence for spatial flatness assumes that the dark energy does not evolve.

More precisely, dark energy is often described by the XCDM parameterization, where it is assumed to be a fluid with pressure p_X = ω_Xρ_X, where ρ_X is the energy density and ω_X is time independent and negative but not necessarily -1 as in the ΛCDM model. This is an inaccurate parameterization of dark energy; see Ratra (1991) for a discussion of the scalar-field case. In addition, dark energy and dark matter are coupled in some models now under discussion, so this also needs to be accounted for when comparing data and models; see Amendola et al. (2007), Bonometto et al. (2007), Guo et al. (2007), and Balbi et al. (2007) for recent discussions.

On the astronomy side, the evidence is not ironclad; for example, inference of the presence of dark energy from CMB anisotropy data relies on the CDM structure formation model and the SNeIa redshift-magnitude results require extraordinary nearly standard candle-like behavior of the objects. Thus, work remains to be done to measure (or reject) dark-energy spatial or temporal variation and to shore up the observational methods already in use.

With tighter observational constraints on "dark energy," one might hope to be guided to a more fundamental model for this construct. At present, dark energy (as well as dark matter) appears to be somewhat disconnected from the rest of physics.

Astronomical observations currently constrain most of the gravitating matter to be cold (small primeval free-streaming velocity) and weakly interacting. Direct detection would be more satisfying and this probably falls to laboratory physicists to pursue. The Large Hadron Collider (LHC) may produce evidence for the supersymmetric sector that provides some of the most-discussed current options for the culprit. As mentioned in the previous question, some models allow for coupling between the dark matter and dark energy. On the astronomy side, observations may provide further clues and, perhaps already do; there are suggestions of problems with "pure" CDM from the properties of dwarf galaxies and galactic nuclear density profiles. Better understanding of the complex astrophysics that connect luminous (or, at least, directly detectable) matter to dark matter will improve such constraints.

In contrast to various proposed candidates for the more dominant "cold" component of dark matter, we know that neutrinos exist. While there are indications from underground experiments of nonzero neutrino mass (Eguchi et al. 2003) and the cosmological tests discussed above yield upper bounds on the sum of masses of all light neutrino species, there has yet to be a detection of the effect of neutrinos on structure formation. A highly model-dependent analysis of Lyman-α forest clustering (Seljak et al. 2006) results in an upper bound of (95% confidence; the sum is over light neutrino species).

Using the standard theory for nucleosynthesis to constrain the baryon density from observations of light element abundances, measurements of ⁴He and ⁷Li imply a higher baryon density than do D measurements, see §§ 2.3 and 7.2 and Fields & Sarkar (2006) and Steigman (2006). Constraints on the baryon density from WMAP CMB anisotropy data are consistent with that from the D abundance measurements. It is possible that more and better data will resolve this discrepancy. On the other hand, this might be an indication of new physics beyond the standard model.

Matter is far more common than antimatter. It is not yet clear how this came to be. One much-discussed option is that grand unification at a relatively high temperature is responsible for the excess. An alternate possibility is that the matter excess was generated at much lower temperature during the electroweak phase transition.

The observational constraints we have reviewed are local; they do not constrain the global topology of space. On the largest observable scales, CMB anisotropy data may be used to constrain models for the topology of space (see, e.g., Key et al. 2007 and references cited therein). Current data do not indicate a real need for going beyond the simplest spatially flat Euclidean space with trivial topology.

The exact nature of the primordial fluctuations is still uncertain. The currently favored explanation posits an inflationary epoch that precedes the conventional Big Bang era (see § 3). The simplest inflation models produce nearly scale-invariant adiabatic perturbations. A key constraint on inflation models is the slope of the primordial spectrum; WMAP data (Spergel et al. 2007) suggest a deviation from the scale-invariant n = 1 value, but this is not yet well measured. At present, the most promising method for observationally probing this early epoch is through detection of (the scale-invariant spectrum of) inflationary gravity waves predicted in a number of inflation models. Detection of these waves or their effects (e.g., measuring the ratio of tensor-to-scalar fluctuations via CMB anisotropy data), would constrain models for inflation; however, nondetection would not rule out inflation because there are simple inflation models without significant gravity waves.

Another critical area for studying the initial fluctuations regards the possibility of non-Gaussian perturbations or isocurvature (rather than adiabatic) perturbations. The evidence indicates that these are subdominant, but that does not exclude a nonvanishing and interesting contribution.

Some models of inflation also predict primordial magnetic field fluctuations. These can have effects in the low-redshift Universe, including on the CMB anisotropy. Observational detection of some of these effects will place interesting constraints on inflation.

Probably (although we would not be astonished if the answer turned out to be no). With tighter observational constraints on the fossil fluctuations generated by quantum mechanics during inflation one might hope to be guided to a more fundamental model of inflation. At present, inflation is more of a phenomenological construct; an observationally consistent, more fundamental model of inflation could guide the development of very high-energy physics. This would be a major development. Another pressing need is to have a more precise model for the end of inflation, when the Universe reheats and matter and radiation are generated. It is possible that the matter excess is generated during this reheating transition.

Discovery of evidence for the epoch of reionization, from observations of absorption line systems toward high-redshift quasars and the polarization anisotropy of the CMB, has prompted intense interest, both theoretical and observational, in studying formation of the first objects. See § 5.3 above.

Carrying on from the previous question, the details of the process of turning this most familiar component of mass-energy into stars and related parts of galaxies remains poorly understood. Or so it seems when compared with the much easier task of predicting how collisionless dark matter clusters in a Universe dominated by dark matter and dark energy. Important problems include the effects of "feedback" from star formation and active galactic nuclei, cosmic reionization, radiative transfer, and the effect of baryons on halo profiles. High-resolution hydrodynamic simulations are getting better, but even Moore's law will not help much in the very near future (see comment in Gott et al. 2006). Solving these problems is critical, not only for understanding galaxy formation, but also for using galaxies—the "atoms of cosmology"—as a probe of the properties of dark matter and dark energy.

Clues to the relationship between mass and light and, therefore, strong constraints on models of galaxy formation include the detailed dependence of galaxy properties on environment. Outstanding puzzles include the observation that, while galaxy morphology and luminosity strongly vary with environment, the properties of early-type (elliptical and S0) galaxies (particularly their colors) are remarkably insensitive to environment (Park et al. 2007).

It is now clear that nearly every sufficiently massive galaxy harbors a supermassive black hole in its core. The masses of the central supermassive black holes are found to correlate strongly with properties of the host galaxy, including bulge velocity dispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000). Thus, galaxy formation and the formation and feeding of black holes are intimately related (see, e.g., Silk & Rees 1998; Kauffmann & Haehnelt 2000; Begelman & Nath 2005).

This model works quite well on large scales. However, on small scales it appears to have too much power at low redshift (excessively cuspy halo cores, excessively large galactic central densities, and too many low-mass satellites of massive galaxies). Modifications of the power spectrum to alleviate this excess small-scale power cause too little power at high redshift and, thus, delay formation of clusters, galaxies, and Lyman-α clouds. Definitive resolution of this issue will require more and better observational data as well as improved theoretical modeling. If the CDM structure formation model is found to be inadequate, this might have significant implications for a number of cosmological tests that assume the validity of this model.

The small amplitude of the quadrupole moment observed by COBE persists in the WMAP observations even after many rounds of reanalysis of possible foreground contributions (see Park et al. 2007 and references cited therein). Although one cannot, by definition, rule out the possibility that it is simply a statistical fluke (with significance of about 95% in flat ΛCDM), this anomaly inspires searches for alternative models, including multiply connected Universes (see above).

The largest superclusters, e.g., the "Sloan Great Wall" (Gott et al. 2005), seen in galaxy redshift surveys are not reproduced by simulations of the concordance flat ΛCDM cosmology (Einasto et al. 2006). Perhaps we need larger simulations (see discussion in Gott et al. 2006) or better understanding of how galaxies trace mass.

Because we see the Universe from only one place, at only one time, we must wrestle with questions related to whether or not we (or at least our location) is special.

Peebles (2005) notes the remarkable coincidences that we observe the Universe when (1) it has just begun making a transition from being dominated by matter to being dominated by dark energy, (2) the Milky Way is just running out of gas for forming stars and planetary systems, and (3) galaxies have just become useful tracers of mass. While anthropic arguments have been put forward to answer the question of why we appear to live at a special time in the history of the Universe, a physically motivated answer might be more productive and satisfying. Understanding of the details of structure formation, including conversion of baryons to stars (mentioned above), and constraints on possible evolution of the components of mass-energy in the Universe may provide clues.

Progress in cosmology is likely to come from more and higher-quality observational and simulation data as well as from new ideas. A number of ground-based, space-based, and numerical experiments continue to collect data and new near-future particle physics, cosmology, astronomy, and numerical experiments are eagerly anticipated. It is less straightforward to predict when a significant new idea might emerge.