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Chapter 1 Principles of Cosmology Chapter 1 Principles of cosmology In the following chapter I provide a brief introduction to some of the main tenets of modern cosmology, building from a homogeneous and isotropic model to the formation of structure which is observed with modern surveys. The former sections are discussed in more detail in Peacock (2000). I discuss cosmology as a modern science, with a brief overview of some of the most promising techniques of measurement pursued in the field. I argue that whilst we are at the advent of ‘precision cosmology’, persistent systematics and an incomplete theory necessitate further study. 1.1 The standard model of cosmology All cosmological observations to date corroborate Einstein’s Cosmological Prin- ciple, which describes a universe that is statistically homogeneous (the same at all locations) and statistically isotropic (independent of direction) on large scales. Figure 1.1 shows a snapshot of a simulation of the large-scale structure, depicting this principle. According to the current cosmological model, the Universe underwent a period of rapid expansion 13.8 Gyr ago (Planck Collaboration et al., 2016a). This early period of inflation caused the rapid exponential growth of small causally connected areas, which simultaneously caused the growth of quantum fluctuations in the matter density field which seed early structure formation. The post- inflation, early, hot Universe contains both ionised baryonic matter, dark matter 1 and the dominating radiation, relics of big bang nucleosynthesis during the inflationary period, which are tightly coupled resulting in an opaque Universe. Structure formation on scales smaller than the particle horizon was suppressed until matter-radiation equality, after which the matter content of the Universe dominates. Upon recombination, when ionised nucleii combined with free electrons, the Universe became transparent, allowing the propagation of radiation in all directions and forming the last scattering surface seen today as the cosmic microwave background (CMB). This is a relic blackbody spectrum from the radiation dominated era. During matter domination, the fluctuations, now frozen as classical fluctuations, provide the initial perturbations for gravitational collapse and galaxies and structure form. All the while, the Universe continues to expand such that any two sufficiently close galaxies will perceive the other to move away from them according to Hubble’s law. First evidence from this came from Slipher (1922), who observed the spectral lines of spiral galaxies to be redshifted with respect to their rest frame wavelengths observed in a laboratory. Interpreted as a Doppler shift, this required a recession velocity proportional to distance (Hubble, 1929). Only relatively recently did the vacuum energy density dominate the total energy density of the Universe and begin to drive the late-time, cosmic acceleration that is observed. Einstein’s theory of General Relativity is widely accepted as a fundamental theory to describe the geometric properties of spacetime and how they are determined by matter and energy. In a homogeneous and isotropic framework, Einstein’s field equations give rise to the Friedmann equation which e↵ectively describes the evolution of the Universe through radiation and matter dominated eras. This theory holds a ‘cosmological constant’, denoted as ⇤, which can be associated with the vacuum energy and called ‘dark energy’, accountable for the repulsive- gravity e↵ect on cosmic scales that we observe today. Thus the standard model of cosmology, is known as the ⇤CDM model, owing to the inclusion of a cosmological constant, ⇤, that dominates the total energy density today and cold dark matter (CDM) as the main contributor to the matter density. 1.2 Cosmological spacetime The physics of the expanding Universe is governed by General Relativity (GR). In a gravitational field and in the absence of any other forces, any mass obeys an identical trajectory between two points. GR describes these trajectories through 2 1 Figure 1.1 A 64h− Mpc- thick slice through the Horizon simulation showing the matter density field in the past light cone as a function of redshift and look-back time, all the way to the horizon. The Earth is at the vertex and the Big Bang at a look-back time of 13.6 billion years forms the upper boundary. The large-scale structure reveals a pattern that is homogeneous and isotropic. Image drawn from Kim et al. (2009). 3 the geometry of spacetime itself, rather than as a gravitational force experienced by a given particle. An invarient interval of proper time, d⌧ between two distant events in a homogeneous and isotropic Universe can be described by a space- time metric, as the square of the physical distance between two infinitesimally separated points, mapped by coordinates xµ, to give ds2 = c2d⌧ 2 g dxµdx⌫ . (1.1) − ⌘ µ⌫ The value of ds2 should be independent of the coordinate choice and agreed upon by any observer. Here, gµ⌫ is the metric tensor which is determined by solving Einstein’s gravitational field equations and subscripts run over one temporal and three spatial coordinates. This metric tensor describes the geometry of the Universe and is specifically determined by a given matter distribution, according to Einstein’s field equation: 8⇡G Gµ⌫ +⇤gµ⌫ = − T µ⌫ , (1.2) c4 where Gµ⌫ is the stress-energy tensor, which determines gravity for all matter and energy, and ⇤is the cosmological constant, discussed in Section 3.5.5. The distortions from Euclidian geometry are sourced by the matter distribution via the energy-momentum tensor, Tµ⌫, which is diagonal for the Universe and composed of energy density and pressure components. The distinction has been made between the left-hand side of this equation, which describes the geometrical properties of space-time and the right-hand side, which describes the properties of matter. As such, this equation can be summarised by “spacetime tells matter how to move; matter tells spacetime how to curve” Wheeler & Ford (1998). On the left hand side, the Gµ⌫ term, geometric information enters through contractions over the Ricci Tensor, Rµ⌫, which describes the curvature of space-time using the metric. 1 Gµ⌫ Rµ⌫ Rgµ⌫ , (1.3) ⌘ − 2 where R is the curvature scalar, which is obtained by the remaining contraction of the Ricci tensor. The Newtonian limit to the field equation (v c) is ⌧ 4⇡G 2Φ= (⇢c2 +3p) , (1.4) r c2 where Φ= c2g /2, 2 = δ δ is the Laplacian in space, p is the pressure, ⇢c2 00 r i j is the rest frame energy density and ⇢, the mass density. This shows that relativistic fluids have an enhanced gravitational influence in GR due to the 4 pressure contribution to the e↵ective density. For a non-relativistic case, we arrive at Poisson’s equation, defined as 2Φ=4⇡G⇢. (1.5) r Einstein’s field equation ( 1.3) shows geodesics to be determined by ⇢c2. The vacuum contribution to the energy density must then have an associated energy- µ⌫ momentum tensor, Tv . Assuming this representation is invariant under Lorentz transforms in locally inertial frames. T µ⌫ = ⇢ c2gµ⌫, (1.6) vac − v in an arbitrary frame. Therefore the vacuum contribution enters the field equation in an identical manner to the cosmological constant, possessing a w = 1 − 2 equation-of-state and constant energy density, ⇢vc . Substitution of the vacuum energy-momentum tensor into Einstein’s field equation, 1 8⇡G Rµ⌫ Rgµ⌫ +⇤gµ⌫ = − (T µ⌫ + T µ⌫ ) , (1.7) − 2 c4 matter vac Hence, the vacuum energy density is associated with the cosmological constant, 1 8⇡G Rµ⌫ Rg +⇤gµ⌫ = − T µ⌫, (1.8) − 2 µ⌫ c4 as long as it is identified with, ⇤ ⇢ = . (1.9) v 8⇡G The geometry of an expanding universe satisfying the Cosmological Principle, that the Universe is homogeneous and isotropic, is described by a Friedmann- Robertson-Walker (FRW) metric. Under these assumptions, equation 1.33 can be simplified in terms of spherical polar coordinates χ, ✓ and φ,to, ds2 = c2dt2 + a(t)2[dχ2 + S2(χ)2(d✓2 +sin2(✓)dφ2)] , (1.10) − k where χ is the comoving radial distance, a(t) is the dimensionless scale factor 2 and Sk(χ) is the angular diameter distance, which depends on the geometry or global curvature, k. Three cases of a spatially flat, Euclidian universe, an open spherical and a closed, saddle-like universe are specified by a curvature constant 5 of k =0, 1, 1, respectively, defining the angular diameter distance as, − sinh(r)(k = 1) − Sk(r)=8r (k =0) (1.11) > > <sin(r)(k =1), > > The radial distance to a galaxy at a: coordinate χ is s(t)=a(t)χ and di↵erentiating with respect to time,s ˙ =˙aχ, which recovers Hubble’s law, (s/s ˙ )=(˙a/a)=H(t). For χ 1, the metric describes the Hubble expansion. The isotropic velocity ⌧ field, as observed by galaxies at a distance r and with a velocity, v, is observed to obey Hubble’s Law, v = H0r , (1.12) where H0 is the Hubble parameter today, which can be recast into a dimensionless 1 1 form, h = H/100 km s− Mpc− . 1.2.1 Expansion dynamics The Friedmann equation, which governs the evolution of the scale factor with the matter content of the Universe, in its first form is given by a˙ 2 8⇡G kc2 = H2(a)= ⇢(a) , (1.13) a 3 − a2 ⇣ ⌘ Di↵erentiating with respect to time and appealing to an adiabatic expansion of the Universe dE + PdV = 0, we obtain an equation of motion for the scale factor, a¨ 4⇡G 3p = ⇢(a)+ , (1.14) a − 3 c2 3⇣p ⌘ where the active mass density, ⇢(a)+ c2 , appears. These equations both contain contributions from either the matter density or pressure, highlighting that the expansion history of the Universe is dependent on the dominant contribution to the energy density at a given time.
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