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In this section, we provide a brief review of the elements of the FRW cosmological model. This model provides the context for interpreting the observational evidence for cosmic acceleration as well as the framework for understanding how cosmological probes in the future will help uncover the cause of acceleration by determining the history of the cosmic expansion with greater precision. For further details on basic cosmology, see, e.g., the textbooks of [Dodelson 2003, Kolb & Turner 1990, Peacock 1999], and [Peebles 1993]. Note that we follow the standard practice of using units in which the speed of light c = 1.

2.1. Friedmann-Robertson-Walker cosmology

From the large-scale distribution of galaxies and the near-uniformity of the CMB temperature, we have good evidence that the Universe is nearly homogeneous and isotropic. Under this assumption, the spacetime metric can be written in the FRW form,

Equation 1 (1)

where r, theta, phi are comoving spatial coordinates, t is time, and the expansion is described by the cosmic scale factor, a(t) (by convention, a = 1 today). The quantity k is the curvature of 3-dimensional space: k = 0 corresponds to a spatially flat, Euclidean Universe, k > 0 to positive curvature (3-sphere), and k < 0 to negative curvature (saddle).

The wavelengths lambda of photons moving through the Universe scale with a(t), and the redshift of light emitted from a distant source at time tem, 1 + z = lambdaobs / lambdaem =1/a(tem), directly reveals the relative size of the Universe at that time. This means that time intervals are related to redshift intervals by dt = -dz / H(z)(1 + z), where H ident dot{a} / a is the Hubble parameter, and an overdot denotes a time derivative. The present value of the Hubble parameter is conventionally expressed as H0 = 100 h km/sec/Mpc, where h approx 0.7 is the dimensionless Hubble parameter. Here and below, a subscript "0" on a parameter denotes its value at the present epoch.

The key equations of cosmology are the Friedmann equations, the field equations of GR applied to the FRW metric,

Equations 2-3 (2)


where rho is the total energy density of the Universe (sum of matter, radiation, dark energy), and p is the total pressure (sum of pressures of each component). For historical reasons we display the cosmological constant Lambda here; hereafter, we shall always represent it as vacuum energy and subsume it into the density and pressure terms; the correspondence is: Lambda = 8pi G rhoVAC = -8pi G pVAC.

For each component, the conservation of energy is expressed by d(a3 rhoi ) = -pi da3, the expanding Universe analogue of the first law of thermodynamics, dE = -pdV. Thus, the evolution of energy density is controlled by the ratio of the pressure to the energy density, the equation-of-state parameter, wi ident pi / rhoi. 1 For the general case, this ratio varies with time, and the evolution of the energy density in a given component is given by

Equation 4 (4)

In the case of constant wi,

Equation 5 (5)

For non-relativistic matter, which includes both dark matter and baryons, wM = 0 to very good approximation, and rhoM propto (1 + z)3; for radiation, i.e., relativistic particles, wR = 1/3, and rhoR propto (1 + z)4. For vacuum energy, as noted above pVAC = -rhoVAC = -Lambda / 8pi G = constant, i.e., wVAC = -1. For other models of dark energy, w can differ from -1 and vary in time. [Hereafter, w without a subscript refers to dark energy.]

The present energy density of a flat Universe (k = 0), rhocrit ident 3H02 / 8pi G = 1.88 × 10-29 h2 gm cm-3 = 8.10 × 10-47 h2 GeV4, is known as the critical density; it provides a convenient means of normalizing cosmic energy densities, where Omegai = rhoi(t0) / rhocrit. For a positively curved Universe, Omega0 ident rho(t0) /rhocrit > 1 and for a negatively curved Universe Omega0 < 1. The present value of the curvature radius, Rcurv ident a / (|k|)1/2, is related to Omega0 and H0 by Rcurv = H0-1 / (|Omega0 - 1|)1/2, and the characteristic scale H0-1 approx 3000 h-1 Mpc is known as the Hubble radius. Because of the evidence from the CMB that the Universe is nearly spatially flat (see Fig. 8), we shall assume k = 0 except where otherwise noted.

Fig. 1 shows the evolution of the radiation, matter, and dark energy densities with redshift. The Universe has gone through three distinct eras: radiation-dominated, z gtapprox 3000; matter-dominated, 3000 gtapprox z gtapprox 0.5; and dark-energy dominated, z ltapprox 0.5. The evolution of the scale factor is controlled by the dominant energy form: a(t) propto t2/3(1 + w) (for constant w). During the radiation-dominated era, a(t) propto t1/2; during the matter-dominated era, a(t) propto t2/3; and for the dark energy-dominated era, assuming w = -1, asymptotically a(t) propto exp(Ht). For a flat Universe with matter and vacuum energy, the general solution, which approaches the latter two above at early and late times, is a(t) = (OmegaM / OmegaVAC)1/3 (sinh[3(OmegaVAC)1/2 H0 t / 2])2/3.

Figure 1

Figure 1. Evolution of radiation, matter, and dark energy densities with redshift. For dark energy, the band represents w = -1 ± 0.2.

The deceleration parameter, q(z), is defined as

Equation 6 (6)

where Omegai(z) ident rhoi(z) / rhocrit(z) is the fraction of critical density in component i at redshift z. During the matter- and radiation-dominated eras, wi > 0 and gravity slows the expansion, so that q > 0 and ddot{a} < 0. Because of the (rho + 3p) term in the second Friedmann equation (Newtonian cosmology would only have rho), the gravity of a component that satisfies p < -rho / 3, i.e., w < -1/3, is repulsive and can cause the expansion to accelerate (ddot{a} > 0): we take this to be the defining property of dark energy. The successful predictions of the radiation-dominated era of cosmology, e.g., big bang nucleosynthesis and the formation of CMB anisotropies, provide evidence for the (rho + 3p) term, since during this epoch ddot{a} is about twice as large as it would be in Newtonian cosmology.

2.2. Distances and the Hubble diagram

For an object of intrinsic luminosity L, the measured energy flux F defines the luminosity distance dL to the object, i.e., the distance inferred from the inverse square law. The luminosity distance is related to the cosmological model through

Equation 7 (7)

where r(z) is the comoving distance to an object at redshift z,

Equations 8-9 (8)


and where chi(x) = sin(x) for k > 0 and sinh(x) for k < 0. Specializing to the flat model and constant w,

Equation 10 (10)

where OmegaM is the present fraction of critical density in non-relativistic matter, and OmegaR appeq 0.8 × 10-4 represents the small contribution to the present energy density from photons and relativistic neutrinos. In this model, the dependence of cosmic distances upon dark energy is controlled by the parameters OmegaM and w and is shown in the left panel of Fig. 2.

Figure 2a Figure 2b

Figure 2. For a flat Universe, the effect of dark energy upon cosmic distance (left) and volume element (right) is controlled by OmegaM and w.

The luminosity distance is related to the distance modulus µ by

Equation 11 (11)

where m is the apparent magnitude of the object (proportional to the log of the flux) and M is the the absolute magnitude (proportional to the log of the intrinsic luminosity). "Standard candles," objects of fixed absolute magnitude M, and measurements of the logarithmic energy flux m constrain the cosmological model and thereby the expansion history through this magnitude-redshift relation, known as the Hubble diagram.

Expanding the scale factor around its value today, a(t) = 1 + H0 (t - t0) - q0 H02 (t - t0)2 / 2 + ··· , the distance-redshift relation can be written in its historical form

Equation 12 (12)

The expansion rate and deceleration rate today appear in the first two terms in the Taylor expansion of the relation. This expansion, only valid for z << 1, is of historical significance and utility; it is not useful today since objects as distant as redshift z ~ 2 are being used to probe the expansion history. However, it does illustrate the general principle: the first term on the r.h.s. represents the linear Hubble expansion, and the deviation from a linear relation reveals the deceleration (or acceleration).

The angular-diameter distance dA, the distance inferred from the angular size deltatheta of a distant object of fixed diameter D, is defined by dA ident D / deltatheta = r(z) / (1 + z) = dL / (1 + z)2. The use of "standard rulers" (objects of fixed intrinsic size) provides another means of probing the expansion history, again through r(z).

The cosmological time, or time back to the Big Bang, is given by

Equation 13 (13)

While the present age in principle depends upon the expansion rate at very early times, the rapid rise of H(z) with z — a factor of 30,000 between today and the epoch of last scattering, when photons and baryons decoupled, at zLS appeq 1100, t(zLS) appeq 380,000 years — makes this point moot.

Finally, the comoving volume element per unit solid angle dOmega is given by

Equation 14 (14)

For a set of objects of known comoving density n(z), the comoving volume element can be used to infer r2(z) / H(z) from the number counts per unit redshift and solid angle, d2 N / dzdOmega = n(z) d2 V / dzdOmega. The dependence of the comoving volume element upon OmegaM and w is shown in the right panel of Fig. 2.

2.3. Growth of structure and LambdaCDM

A striking success of the consensus cosmology is its ability to account for the observed structure in the Universe, provided that the dark matter is composed of slowly moving particles, known as cold dark matter (CDM), and that the initial power spectrum of density perturbations is nearly scale-invariant, P(k) ~ knS with spectral index nS appeq 1, as predicted by inflation [Springel, Frenk & White 2006]. Dark energy affects the development of structure by its influence on the expansion rate of the Universe when density perturbations are growing. This fact and the quantity and quality of large-scale structure data make structure formation a sensitive probe of dark energy.

In GR the growth of small-amplitude, matter-density perturbations on length scales much smaller than the Hubble radius is governed by

Equation 15 (15)

where the perturbations delta(x, t) ident deltarhoM(x, t) / bar{rho}M(t) have been decomposed into their Fourier modes of wavenumber k, and matter is assumed to be pressureless (always true for the CDM portion and valid for the baryons on mass scales larger than 105 Modot after photon-baryon decoupling). Dark energy affects the growth through the "Hubble damping" term, 2Hdot{delta}k.

The solution to Eq. (15) is simple to describe during the three epochs of expansion discussed earlier: deltak (t) grows as a(t) during the matter-dominated epoch and is approximately constant during the radiation-dominated and dark energy-dominated epochs. The key feature here is the fact that once accelerated expansion begins, the growth of linear perturbations effectively ends, since the Hubble damping time becomes shorter than the timescale for perturbation growth.

The impact of the dark energy equation-of-state parameter w on the growth of structure is more subtle and is illustrated in Fig. 3. For larger w and fixed dark energy density OmegaDE, dark energy comes to dominate earlier, causing the growth of linear perturbations to end earlier; this means the growth factor since decoupling is smaller and that to achieve the same amplitude by today, the perturbation must begin with larger amplitude and is larger at all redshifts until today. The same is true for larger OmegaDE and fixed w. Finally, if dark energy is dynamical (not vacuum energy), then in principle it can be inhomogeneous, an effect ignored above. In practice, it is expected to be nearly uniform over scales smaller than the present Hubble radius, in sharp contrast to dark matter, which can clump on small scales.

Figure 3

Figure 3. Growth of linear density perturbations in a flat universe with dark energy. Note that the growth of perturbations ceases when dark energy begins to dominate, 1 + z = (OmegaM / OmegaDE)1/3w.

1 A perfect fluid is fully characterized by its isotropic pressure p and energy density rho, where p is a function of density and other state variables (e.g., temperature). The equation-of-state parameter w = p / rho determines the evolution of the energy density rho; e.g., rho propto V1 + w for constant w, where V is the volume occupied by the fluid. Vacuum energy or a homogeneous scalar field are spatially uniform and they too can be fully characterized by w. The evolution of an inhomogeneous, imperfect fluid is in general complicated and not fully described by w. Nonetheless, in the FRW cosmology, spatial homogeneity and isotropy require the stress-energy to take the perfect fluid form; thus, w determines the evolution of the energy density. Back.

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