bigbang

19. Big-Bang cosmology 1 19. BIG-BANGCOSMOLOGY Written July 2001 by K.A. Olive (University of Minnesota) and J.A. Peacock (University of Edinburgh). Revised September 2005. 19.1. Introduction to Standard Big-Bang Model The observed expansion of the Universe [1,2,3] is a natural (almost inevitable) result of any homogeneous and isotropic cosmological model based on general relativity. However, by itself, the Hubble expansion does not provide sufficient evidence for what we generally refer to as the Big-Bang model of cosmology. While general relativity is in principle capable of describing the cosmology of any given distribution of matter, it is extremely fortunate that our Universe appears to be homogeneous and isotropic on large scales. Together, homogeneity and isotropy allow us to extend the Copernican Principle to the Cosmological Principle, stating that all spatial positions in the Universe are essentially equivalent. The formulation of the Big-Bang model began in the 1940s with the work of George Gamow and his collaborators, Alpher and Herman. In order to account for the possibility that the abundances of the elements had a cosmological origin, they proposed that the early Universe which was once very hot and dense (enough so as to allow for the nucleosynthetic processing of hydrogen), and has expanded and cooled to its present state [4,5]. In 1948, Alpher and Herman predicted that a direct consequence of this model is the presence of a relic background radiation with a temperature of order a few K [6,7]. Of course this radiation was observed 16 years later as the microwave background radiation [8]. Indeed, it was the observation of the 3 K background radiation that singled out the Big-Bang model as the prime candidate to describe our Universe. Subsequent work on Big-Bang nucleosynthesis further confirmed the necessity of our hot and dense past. (See the review on BBN—Sec. 20 of this Review for a detailed discussion of BBN.) These relativistic cosmological models face severe problems with their initial conditions, to which the best modern solution is inflationary cosmology, discussed in Sec. 19.3.5. If correct, these ideas would strictly render the term ‘Big Bang’ redundant, since it was first coined by Hoyle to represent a criticism of the lack of understanding of the initial conditions. 19.1.1. The Robertson-Walker Universe : The observed homogeneity and isotropy enable us to describe the overall geometry and evolution of the Universe in terms of two cosmological parameters accounting for the spatial curvature and the overall expansion (or contraction) of the Universe. These two quantities appear in the most general expression for a space-time metric which has a (3D) maximally symmetric subspace of a 4D space-time, known as the Robertson-Walker metric: ds2 = dt2 − R2(t) [ dr2 1− kr2 + r 2 (dθ2 + sin2 θ dφ2) ] . (19.1) Note that we adopt c = 1 throughout. By rescaling the radial coordinate, we can choose the curvature constant k to take only the discrete values +1, −1, or 0 corresponding to closed, open, or spatially flat geometries. In this case, it is often more convenient to re-express the metric as CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006) available on the PDG WWW pages (URL: http://pdg.lbl.gov/) July 14, 2006 10:37 2 19. Big-Bang cosmology ds2 = dt2 −R2(t) [ dχ2 + S2k(χ) (dθ 2 + sin2 θ dφ2) ] , (19.2) where the function Sk(χ) is (sinχ, χ, sinhχ) for k = (+1, 0,−1). The coordinate r (in Eq. (19.1)) and the ‘angle’ χ (in Eq. (19.2)) are both dimensionless; the dimensions are carried by R(t), which is the cosmological scale factor which determines proper distances in terms of the comoving coordinates. A common alternative is to define a dimensionless scale factor, a(t) = R(t)/R0, where R0 ≡ R(t0) is R at the present epoch. It is also sometimes convenient to define a dimensionless or conformal time coordinate, η, by dη = dt/R(t). Along constant spatial sections, the proper time is defined by the time coordinate, t. Similarly, for dt = dθ = dφ = 0, the proper distance is given by R(t)χ. For standard texts on cosmological models see e.g., Refs. [9–14]. 19.1.2. The redshift : The cosmological redshift is a direct consequence of the Hubble expansion, determined by R(t). A local observer detecting light from a distant emitter sees a redshift in frequency. We can define the redshift as z ≡ ν1 − ν2 ν2 � v12 c , (19.3) where ν1 is the frequency of the emitted light, ν2 is the observed frequency and v12 is the relative velocity between the emitter and the observer. While the definition, z = (ν1− ν2)/ν2 is valid on all distance scales, relating the redshift to the relative velocity in this simple way is only true on small scales (i.e., less than cosmological scales) such that the expansion velocity is non-relativistic. For light signals, we can use the metric given by Eq. (19.1) and ds2 = 0 to write v12 c = R˙ δr = R˙ R δt = δR R = R2 −R1 R1 , (19.4) where δr(δt) is the radial coordinate (temporal) separation between the emitter and observer. Thus, we obtain the simple relation between the redshift and the scale factor 1 + z = ν1 ν2 = R2 R1 . (19.5) This result does not depend on the non-relativistic approximation. July 14, 2006 10:37 19. Big-Bang cosmology 3 19.1.3. The Friedmann-Lemaˆıtre equations of motion : The cosmological equations of motion are derived from Einstein’s equations Rµν − 12gµνR = 8πGNTµν + Λgµν . (19.6) Gliner [15] and Zeldovich [16] seem to have pioneered the modern view, in which the Λ term is taken to the rhs and interpreted as particle-physics processes yielding an effective energy–momentum tensor Tµν for the vacuum of Λgµν/8πGN. It is common to assume that the matter content of the Universe is a perfect fluid, for which Tµν = −pgµν + (p+ ρ)uµuν , (19.7) where gµν is the space-time metric described by Eq. (19.1), p is the isotropic pressure, ρ is the energy density and u = (1, 0, 0, 0) is the velocity vector for the isotropic fluid in co-moving coordinates. With the perfect fluid source, Einstein’s equations lead to the Friedmann-Lemaˆıtre equations H2 ≡ ( R˙ R )2 = 8π GN ρ 3 − k R2 + Λ 3 , (19.8) and R¨ R = Λ 3 − 4πGN 3 (ρ+ 3p) , (19.9) where H(t) is the Hubble parameter and Λ is the cosmological constant. The first of these is sometimes called the Friedmann equation. Energy conservation via Tµν;µ = 0, leads to a third useful equation [which can also be derived from Eq. (19.8) and Eq. (19.9)] ρ˙ = −3H (ρ+ p) . (19.10) Eq. (19.10) can also be simply derived as a consequence of the first law of thermodynamics. Eq. (19.8) has a simple classical mechanical analog if we neglect (for the moment) the cosmological term Λ. By interpreting −k/R2 as a “total energy”, then we see that the evolution of the Universe is governed by a competition between the potential energy, 8πGNρ/3 and the kinetic term (R˙/R)2. For Λ = 0, it is clear that the Universe must be expanding or contracting (except at the turning point prior to collapse in a closed Universe). The ultimate fate of the Universe is determined by the curvature constant k. For k = +1, the Universe will recollapse in a finite time, whereas for k = 0,−1, the Universe will expand indefinitely. These simple conclusions can be altered when Λ �= 0 or more generally with some component with (ρ+ 3p) < 0. July 14, 2006 10:37 4 19. Big-Bang cosmology 19.1.4. Definition of cosmological parameters : In addition to the Hubble parameter, it is useful to define several other measurable cosmological parameters. The Friedmann equation can be used to define a critical density such that k = 0 when Λ = 0, ρc ≡ 3H 2 8πGN = 1.88× 10−26 h2 kg m−3 = 1.05× 10−5 h2 GeV cm−3 , (19.11) where the scaled Hubble parameter, h, is defined by H ≡ 100h km s−1 Mpc−1 ⇒ H−1 = 9.78h−1 Gyr = 2998h−1 Mpc . (19.12) The cosmological density parameter Ωtot is defined as the energy density relative to the critical density, Ωtot = ρ/ρc . (19.13) Note that one can now rewrite the Friedmann equation as k/R2 = H2(Ωtot − 1) , (19.14) From Eq. (19.14), one can see that when Ωtot > 1, k = +1 and the Universe is closed, when Ωtot < 1, k = −1 and the Universe is open, and when Ωtot = 1, k = 0, and the Universe is spatially flat. It is often necessary to distinguish different contributions to the density. It is therefore convenient to define present-day density parameters for pressureless matter (Ωm) and relativistic particles (Ωr), plus the quantity ΩΛ = Λ/3H2. In more general models, we may wish to drop the assumption that the vacuum energy density is constant, and we therefore denote the present-day density parameter of the vacuum by Ωv. The Friedmann equation then becomes k/R20 = H 2 0 (Ωm + Ωr + Ωv − 1) , (19.15) where the subscript 0 indicates present-day values. Thus, it is the sum of the densities in matter, relativistic particles and vacuum that determines the overall sign of the curvature. Note that the quantity −k/R20H20 is sometimes referred to as Ωk. This usage is unfortunate: it encourages one to think of curvature as a contribution to the energy density of the Universe, which is not correct. July 14, 2006 10:37 19. Big-Bang cosmology 5 19.1.5. Standard Model solutions : Much of the history of the Universe in the standard Big-Bang model can be easily described by assuming that either matter or radiation dominates the total energy density. During inflation or perhaps even today if we are living in an accelerating Universe, domination by a cosmological constant or some other form of dark energy should be considered. In the following, we shall delineate the solutions to the Friedmann equation when a single component dominates the energy density. Each component is distinguished by an equation of state parameter w = p/ρ. 19.1.5.1. Solutions for a general equation of state: Let us first assume a general equation of state parameter for a single component, w which is constant. In this case, Eq. (19.10) can be written as ρ˙ = −3(1 + w)ρR˙/R and is easily integrated to yield ρ ∝ R−3(1+w) . (19.16) Note that at early times when R is small, the less singular curvature term k/R2 in the Friedmann equation can be neglected so long as w > −1/3. Curvature domination occurs at rather late times (if a cosmological constant term does not dominate sooner). For w �= −1, one can insert this result into the Friedmann equation Eq. (19.8) and if one neglects the curvature and cosmological constant terms, it is easy to integrate the equation to obtain, R(t) ∝ t2/[3(1+w)] . (19.17) 19.1.5.2. A Radiation-dominated Universe: In the early hot and dense Universe, it is appropriate to assume an equation of state corresponding to a gas of radiation (or relativistic particles) for which w = 1/3. In this case, Eq. (19.16) becomes ρ ∝ R−4. The “extra” factor of 1/R is due to the cosmological redshift; not only is the number density of particles in the radiation background decreasing as R−3 since volume scales as R3, but in addition, each particle’s energy is decreasing as E ∝ ν ∝ R−1. Similarly, one can substitute w = 1/3 into Eq. (19.17) to obtain R(t) ∝ t1/2 ; H = 1/2t . (19.18) 19.1.5.3. A Matter-dominated Universe: At relatively late times, non-relativistic matter eventually dominates the energy density over radiation (see Sec. 19.3.8). A pressureless gas (w = 0) leads to the expected dependence ρ ∝ R−3 from Eq. (19.16) and, if k = 0, we get R(t) ∝ t2/3 ; H = 2/3t . (19.19) July 14, 2006 10:37 6 19. Big-Bang cosmology 19.1.5.4. A Universe dominated by vacuum energy: If there is a dominant source of vacuum energy, V0, it would act as a cosmological constant with Λ = 8πGNV0 and equation of state w = −1. In this case, the solution to the Friedmann equation is particularly simple and leads to an exponential expansion of the Universe R(t) ∝ e √ Λ/3t . (19.20) A key parameter is the equation of state of the vacuum, w ≡ p/ρ: this need not be the w = −1 of Λ, and may not even be constant [17,18,19]. It is now common to use w to stand for this vacuum equation of state, rather than of any other constituent of the Universe, and we use the symbol in this sense hereafter. We generally assume w to be independent of time, and where results relating to the vacuum are quoted without an explicit w dependence, we have adopted w = −1. The presence of vacuum energy can dramatically alter the fate of the Universe. For example, if Λ < 0, the Universe will eventually recollapse independent of the sign of k. For large values of Λ (larger than the Einstein static value needed to halt any cosmological expansion or contraction), even a closed Universe will expand forever. One way to quantify this is the deceleration parameter, q0, defined as q0 = − RR¨ R˙2 ∣∣∣∣∣ 0 = 1 2 Ωm +Ωr + (1 + 3w) 2 Ωv . (19.21) This equation shows us that w < −1/3 for the vacuum may lead to an accelerating expansion. Astonishingly, it appears that such an effect has been observed in the Supernova Hubble diagram [20–23] (see Fig. 19.1 below); current data indicate that vacuum energy is indeed the largest contributor to the cosmological density budget, with Ωv = 0.72± 0.05 and Ωm = 0.28± 0.05 if k = 0 is assumed [23]. The nature of this dominant term is presently uncertain, but much effort is being invested in dynamical models (e.g., rolling scalar fields), under the catch-all heading of “quintessence.” 19.2. Introduction to Observational Cosmology 19.2.1. Fluxes, luminosities, and distances : The key quantities for observational cosmology can be deduced quite directly from the metric. (1) The proper transverse size of an object seen by us to subtend an angle dψ is its comoving size dψ Sk(χ) times the scale factor at the time of emission: d = dψ R0Sk(χ)/(1 + z) . (19.22) (2) The apparent flux density of an object is deduced by allowing its photons to flow through a sphere of current radius R0Sk(χ); but photon energies and arrival rates are July 14, 2006 10:37 19. Big-Bang cosmology 7 redshifted, and the bandwidth dν is reduced. The observed photons at frequency ν0 were emitted at frequency ν0(1 + z), so the flux density is the luminosity at this frequency, divided by the total area, divided by 1 + z: Sν(ν0) = Lν([1 + z]ν0) 4πR20S 2 k(χ)(1 + z) . (19.23) These relations lead to the following common definitions: angular-diameter distance: DA = (1 + z)−1R0Sk(χ) luminosity distance: DL = (1 + z) R0Sk(χ) (19.24) These distance-redshift relations are expressed in terms of observables by using the equation of a null radial geodesic (R(t)dχ = dt) plus the Friedmann equation: R0dχ = 1 H(z) dz = 1 H0 [ (1− Ωm − Ωv − Ωr)(1 + z)2 + Ωv(1 + z)3+3w + Ωm(1 + z)3 +Ωr(1 + z)4 ]−1/2 dz . (19.25) The main scale for the distance here is the Hubble length, 1/H0. The flux density is the product of the specific intensity Iν and the solid angle dΩ subtended by the source: Sν = Iν dΩ. Combining the angular size and flux-density relations thus gives the relativistic version of surface-brightness conservation: Iν(ν0) = Bν([1 + z]ν0) (1 + z)3 , (19.26) where Bν is surface brightness (luminosity emitted into unit solid angle per unit area of source). We can integrate over ν0 to obtain the corresponding total or bolometric formula: Itot = Btot (1 + z)4 . (19.27) This cosmology-independent form expresses Liouville’s Theorem: photon phase-space density is conserved along rays. 19.2.2. Distance data and geometrical tests of cosmology : In order to confront these theoretical predictions with data, we have to bridge the divide between two extremes. Nearby objects may have their distances measured quite easily, but their radial velocities are dominated by deviations from the ideal Hubble flow, which typically have a magnitude of several hundred km s−1. On the other hand, objects at redshifts z >∼ 0.01 will have observed recessional velocities that differ from their ideal values by <∼ 10%, but absolute distances are much harder to supply in this case. The traditional solution to this problem is the construction of the distance ladder: an interlocking set of methods for obtaining relative distances between various classes of July 14, 2006 10:37 8 19. Big-Bang cosmology object, which begins with absolute distances at the 10 to 100 pc level and terminates with galaxies at significant redshifts. This is reviewed in the review on Global cosmological parameters—Sec. 21 of this Review. By far the most exciting development in this area has been the use of type Ia Supernovae (SNe), which now allow measurement of relative distances with 5% precision. In combination with Cepheid data from the HST key project on the distance scale, SNe results are the dominant contributor to the best modern value for H0: 72 km s−1Mpc−1 ± 10% [24]. Better still, the analysis of high-z SNe has allowed the first meaningful test of cosmological geometry to be carried out: as shown in Fig. 19.1 and Fig. 19.2, a combination of supernova data and measurements of microwave-background anisotropies strongly favors a k = 0 model dominated by vacuum energy. (See the review on Global cosmological parameters—Sec. 21 of this Review for a more comprehensive review of Hubble parameter determinations.) 19.2.3. Age of the Universe : The most striking conclusion of relativistic cosmology is that the Universe has not existed forever. The dynamical result for the age of the Universe may be written as H0t0 = ∫ ∞ 0 dz (1 + z)H(z) = ∫ ∞ 0 dz (1 + z) [(1 + z)2(1 + Ωmz)− z(2 + z)Ωv ]1/2 , (19.28) where we have neglected Ωr and chosen w = −1. Over the range of interest (0.1 <∼ Ωm <∼ 1, |Ωv| <∼ 1), this exact answer may be approximated to a few % accuracy by H0t0 � 23 (0.7Ωm + 0.3− 0.3Ωv)−0.3 . (19.29) For the special case that Ωm + Ωv = 1, the integral in Eq. (19.28) can be expressed analytically as H0t0 = 2 3 √ Ωv ln 1 + √ Ωv√ 1− Ωv (Ωm < 1) . (19.30) The most accurate means of obtaining ages for astronomical objects is based on the natural clocks provided by radioactive decay. The use of these clocks is complicated by a lack of knowledge of the initial conditions of the decay. In the Solar System, chemical fractionation of different elements helps pin down a precise age for the pre-Solar nebula of 4.6 Gyr, but for stars it is necessary to attempt an a priori calculation of the relative abundances of nuclei that result from supernova explosions. In this way, a lower limit for the age of stars in the local part of the Milky Way of about 11 Gyr is obtained [25]. The other major means of obtaining cosmological age estimates is based on the theory of stellar evolution. In principle, the main-sequence turnoff point in the color-magnitude diagram of a globular cluster should yield a reliable age. However, these have been July 14, 2006 10:37 19. Big-Bang cosmology 9 Figure 19.1: The type Ia supernova Hubble diagram [20–22]. The first panel shows that for z � 1 the large-scale Hubble flow is indeed linear and uniform; the second panel shows an expanded scale, with the linear trend divided out, and with the redshift range extended to show how the Hubble law becomes nonlinear. (Ωr = 0 is assumed.) Comparison with the prediction of Friedmann-Lemaˆitre models appears to favor a vacuum-dominated Universe. controversial owing to theoretical uncertainties in the evolution model, as well as July 14, 2006 10:37 10 19. Big-Bang cosmology observational uncertainties in the distance, dust extinction and metallicity of clusters. The present consensus favors ages for the oldest clusters of about 12 Gyr [26,27]. These methods are all consistent with the age deduced from studies of structure formation, using the microwave background and large-scale structure: t0 = 13.7± 0.2 Gyr [28], where the extra accuracy comes at the price of assuming the Cold Dark Matter model to be true. Figure 19.2: Likelihood-based confidence contours [28] over the plane ΩΛ (i.e. Ωv assuming w = −1) vs Ωm. The SNe Ia results very nearly constrain Ωv − Ωm, where