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CALTECH AY 21 - Cosmology: Standard Model

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eaa.iop.orgDOI: 10.1888/0333750888/1632 Cosmology: Standard ModelJohn Peacock FromEncyclopedia of Astronomy & AstrophysicsP. Murdin © IOP Publishing Ltd 2006 ISBN: 0333750888Downloaded on Tue Jan 31 17:13:13 GMT 2006 [127.0.0.1]Institute of Physics PublishingBristol and PhiladelphiaTerms and ConditionsCosmology Standard ModelENCYCLOPEDIA OF ASTRONOMY AND ASTROPHYSICSCOSMOLOGY in the modern sense of quantitative study ofthe large-scale properties of the universe is a surprising-ly recent phenomenon. The first galaxy RADIAL VELOCITY(a blueshift, as it turned out) was only measured in 1912,by Slipher. It was not until 1924 that Hubble was able toprove that the ‘nebulae’ were indeed large systems ofstars at vast distances, by which time it was clear thatalmost all galaxies had spectral lines displaced to longerwavelengths. Subsequent observations increasingly veri-fied Hubble’s (1929) linear relation between distance dand the recessional velocity inferred if redshift was inter-preted as a Doppler shift:The theoretical groundwork for describing the uni-verse via GENERAL RELATIVITY was already in place bythe mid-1920s, so that it was not long before the basicobservational fact of an expanding universe could begiven a relatively standard interpretation. The mainobservational and theoretical uncertainties in thisinterpretation concern the matter and energy content ofthe universe. Different possibilities for this contentgenerate very different COSMOLOGICAL MODELS. The pur-pose of this article is to outline the key concepts andpractical formulae of importance in understandingthese models, and to show how to apply them to astro-nomical observations.Isotropic spacetimeModern observational cosmology has demonstrated thatthe real universe is highly symmetric in its large-scaleproperties, but it would in any case make sense to startby considering the simplest possible mass distribution:one whose properties are homogeneous (uniform densi-ty) and isotropic (the same in all directions). The nextstep is to solve the gravitational field equations to findthe corresponding metric. Many of the features of themetric can be deduced from symmetry alone—andindeed will apply even if Einstein’s equations arereplaced by something more complicated. These generalarguments were put forward independently by H PRobertson and A G Walker in 1936.Consider a set of ‘fundamental observers’, in differ-ent locations, all of whom are at rest with respect to thematter in their vicinity. We can envisage them as each sit-ting on a different galaxy, and so receding from eachother with the general expansion (although real galaxieshave in addition random velocities of order 100 km s–1and so are not strictly fundamental observers). A globaltime coordinate t is supplied by the time measured withthe clocks of these observers—i.e. t is the proper timemeasured by an observer at rest with respect to the localmatter distribution. The coordinate is useful globallyrather than locally because the clocks can be synchro-nized by the exchange of light signals between observers,who agree to set their clocks to a standard time when forexample the universal homogeneous density reachessome given value. Using this time coordinate plusisotropy, we already have enough information to con-clude that the metric must take the following form:Here, we have used the equivalence principle to say thatthe proper time interval dτbetween two distant eventswould look locally like special relativity to a fundamen-tal observer on the spot: for them, c2dτ2 = c2dt2 – dx2 –dy2 – dz2. Since we use the same time coordinate as theydo, our only difficulty is in the spatial part of the metric:relating their dx etc to spatial coordinates centered on us.Distances have been decomposed into the productof a time-dependent scale factor R(t) and a time-independent comoving coordinate r. It is clear that thismetric nicely incorporates the idea of a uniformlyexpanding model with no center. For small separations,where space is Euclidean, we have a simple scaling ofvector separations: x(t): R(t) x(t0). The same law appliesirrespective of the origin we choose: x1(t) – x2(t) : R(t)[x1(t0) – x2(t0)], and so every observer deduces v = H rBecause of spherical symmetry, the spatial part ofthe metric can be decomposed into a radial and a trans-verse part (in spherical polars, the angle on the skybetween two events is dψ2 = dθ2 + sin 2θdφ2). The func-tions f and g are arbitrary; however, we can choose ourradial coordinate such that either f = 1 or g = r2, to makethings look as much like Euclidean space as possible.Furthermore, the remaining function is determined bysymmetry arguments.Consider first the simple case of the metric on thesurface of a sphere. A balloon being inflated is a commonpopular analogy for the expanding universe, and it willserve as a two-dimensional example of a space of con-stant curvature. If we call the polar angle in sphericalpolars r instead of the more usual θ, then the element oflength, dσ, on the surface of a sphere of radius R isIt is possible to convert this to the metric for a 2-space ofconstant negative curvature by the device of consideringan imaginary radius of curvature, R→iR. If we simulta-neously let r→ir, we obtainThese two forms can be combined by defining a newradial coordinate that makes the transverse part of themetric look Euclidean:Cosmology Standard ModelCopyright © Nature Publishing Group 2002Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998and Institute of Physics Publishing 2002Dirac House, Temple Back, Bristol, BS21 6BE, UK1Cosmology Standard ModelENCYCLOPEDIA OF ASTRONOMY AND ASTROPHYSICSwhere k = +1 for positive curvature and k = –1 for nega-tive curvature.This is in fact the general form of the spatial part ofthe Robertson–Walker metric. To prove this in 3D, con-sider a 3-sphere embedded in four-dimensionalEuclidean space, which is defined via the coordinaterelation x2 + y2 + z2 + w2 = R2. Now define the equiva-lent of spherical polars and write w = R cos α, z = R sinα cos β, y = R sin α sin β cos γ, x = R sin α sin β sin γ,where α, βand γare three arbitrary angles.Differentiating with respect to the angles gives a four-dimensional vector (dx,dy,dz,dw), and it is a straight-forward exercise to show that the squared length of thisvector which is the Robertson–Walker metric for the case of pos-itive spatial curvature.This k = +1 metric


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