arXiv:astro-ph/0309240v3 24 Nov 2003Large-scale surveys andcosmic structureBy J.A. PEACOCKInstitute for Astronomy, University of Edinburgh,Royal Observatory, Edinburgh EH9 3HJ, UKThese lectures deal with our current knowledge of the matter distribution in the universe, fo-cusing on how this is studied via the large-scale structure seen in galaxy surveys. We firstassemble the necessary basics needed to understand the development of den sity fluctuations inan expanding universe, and discuss how galaxies are located within the dark-matter density field.Results from the 2dF Galaxy Redshift Survey are presented and contrasted with theoretical mod-els. We show that the combination of large-scale structure an d data on microwave-backgroundanisotropies can eliminate almost all degeneracies, and yield a completely specified cosmologicalmodel. This is the ‘concordance’ universe: a geometrically flat combination of vacuum energyand cold dark matter. The study of cosmic structure is able to establish this in a mannerindependent of external information, such as the Hubble diagram; this extra information canhowever be used to limit non-standard alternatives, such as a variable equation of state for thevacuum.1. Preamble1.1. The perturbed un iverseIt has been clear since the 1930s that galaxies are not distributed at random in theuniverse (Hubble 1934). For decades, our understanding of this fact was limited by thelack of a three-dimensional picture, although some impressive progre ss was made: thededication of pioneers such as Shane & Wirtanen in compiling galaxy catalogues by eyeis humbling to consider. However, studies of the galaxy distribution came of age in the1980s, via redshift surveys, in which Hubble’s v = Hd law is used to turn spectro scopicredshifts into estimates of distance (e.g. Davis & Peebles 1983; de Lapparant, Geller &Huchra 1986; Saunders et al. 1991). We were then able to see clearly (e.g. figure 1) awealth of large-scale structures of size exceeding 1 00 Mpc. The existence of these cosmo-logical structures must be telling us something importa nt about the initial conditions ofthe big bang, and about the physical processes that have operated subsequently. Theselectures c over some of what we have learned in this regard.Throughout, it will be convenient to adopt a notation in which the density (of mass,light, or any property) is expressed in terms of a dimensionless dens ity perturbation δ:1 + δ(x) ≡ ρ(x)/hρi, (1)where hρi is the glo bal mean density. The quantity δ need not be small, but writing thingsin this fo rm na tur ally sugg ests an approach via perturba tion theory in the importantlinear case where δ ≪ 1. As we will see, this was a good approximation at early times.The existence of this field in the universe raises two questions: what generated it, andhow does it evolve? A popula r answer for the first question is inflation, in which quantumfluctuations are able to seed density fluctuations. So fa r, despite some claims, this theoryis no t tested, and we consider later some ways in which this might be accomplished.Mainly, however, we will be concerned here with the question of evolution.12 J.A. Peacock: Surveys and cosmic structureFigure 1. One of the iconic pictures of the large-scale structure in t he galaxy distribution isthis slice made from John Huchra’s ZCAT compilation of galaxy redshifts, reflecting the state ofour knowledge in the mid-1980s. The survey coverage is not quite complete; as well as the h olesdue to the galactic plane around right ascensions 6hand 19h, the rich clusters are somewhatover-represented with respect to a true random sampling of the galaxy population. Nevertheless,this plot emphasizes nicely both the large-scale features such as the ‘great wall’ on the left, thetotally empty void regions, and the radial ‘fingers of God’ caused by virialized motions in theclusters. One of the principal challenges in cosmology is to explain this pattern.1.2. Relativistic viewpoint and gauge issuesMany of the key aspects of the evolution of structure in the universe can be dealt withvia a deceptively simple Newtonian approach, but honesty requires a brief overview ofsome of the difficult issues that will be evaded by taking this route.Because relativistic physics equations are written in a covariant form in which allquantities are indepe ndent of coordinates, relativity does not distinguish between activechanges of co ordinate (e.g. a Lorentz boost) or passive changes (a mathematical change ofvariable, normally termed a gauge transformation). This generality is a problem, sinceit is not trivial to know which coordinates should be used. To see how the problemsarise, ask how tensors of different order change under a gauge transformation xµ→x′µ= xµ+ ǫµ. Consider first a scalar quantity S (which might be density, temperatureetc.). A scala r quantity in re lativity is normally taken to be independent of coordinateframe, but this is only for the case of Lorentz transformations, which do not involve achange of the spacetime origin. A gauge transformation therefore not only induces theusual transfo rmation coefficients dx′µ/dxν, but also involves a translation that relabelsspacetime points. We therefore have to deal with S′(xµ+ ǫµ) = S(xµ), so the rule forthe gauge transformatio n of scalars isS′(xµ) = S(xµ) − ǫα∂S/∂xα. (2)J.A. Peacock: Surveys and cosmic structure 3Similar reasoning yields the g auge transformation laws for higher tensors, although weneed to account not only for the translation of the origin, but also for the usual effect ofthe coordinate transformation on the tensor.Consider applying this to the case of a uniform universe; here ρ only depends on time,so thatρ′= ρ − ǫ0˙ρ. (3)An effective density perturbatio n is thus produced by a local alteration in the timecoordinate: when we look at a universe with a fluctuating density, should we r e allythink of a uniform model in which time is wrinkled? This ambiguity may seem absurd,and in the laborator y it could be resolved empirically by constructing the coordinatesystem directly – in principle by using light signals. This shows that the cosmologica lhorizon plays an importa nt ro le in this topic: perturbations with wavelength λ<∼ctinhabit a regime in which gauge ambiguities can be resolved directly via common sense.The real difficulties lie in the super- horizon modes with λ>∼ct. However, at leastwithin inflationary mo