C.4. Homogeneous and isotropic cosmological models¶
Another important application of the general theory of relativity in the context of galaxies is using it to describe the structure and evolution of the Universe on the largest scales. This is the realm of cosmology. In Chapter 17, which describes the growth of initial density fluctuations in the Universe and how they gravitationally collapse to form galaxies, we depend on basic results for the evolution of the Universe. We will describe those here, but refer the reader to more advanced cosmology texts such as Dodelson & Schmidt (2020) for further details.
C.4.1. The Friedmann–Lemaître–Robertson–Walker metric¶
The basic assumption behind the cosmological models that we describe here, and that are generally used to describe our \(\Lambda\)CDM Universe, is that on large scales, the 3D spatial structure of the Universe is both isotropic and homogeneous. Isotropy means that the Universe has the same properties in all directions when viewed from a point, say, the Earth. Homogeneity means that there is no special location in the Universe, that is, each location is statistically equivalent to all other positions. Establishing that isotropy and homogeneity on large scales is a good assumption is not easy and it was taken as an axiom for a long time. By observing the Universe in different directions, it is relatively easy to determine that the Universe appears isotropic as observed from Earth (e.g., the temperature of the cosmic microwave background [CMB] is the same to about one part in \(10^5\) after we account for the motion of the Earth; Smoot et al. 1992), but to determine homogeneity we need to establish the isotropy of the Universe around two separate points. Clever use of distortions in the spectrum of the CMB can be used to do just that (Goodman 1995). But stronger constraints come from large surveys of the redshifts of galaxies, which can also directly determine the homogeneity of the Universe by determining the density distribution around many different galaxies and showing that it is statistically the same. Using surveys of galaxies and quasars, recent surveys have demonstrated that the Universe is statistically-homogeneous on scales \(\gtrsim 100\,\mathrm{Mpc}\) (Scrimgeour et al. 2012; Laurent et al. 2016). Thus, observational evidence is strongly in favor of the simplifying assumptions of isotropy and homogeneity.
To derive a cosmological model, we must then solve Einstein’s field equations (C.50) for metrics that are consistent with the assumptions of isotropy and homogeneity. Starting with the assumption of isotropy, we require that the metric does not change under spatial rotations. We therefore have to build the line element \(\mathrm{d}s^2\) out of components that are invariant under spatial rotations. Because the metric only involves the spatial coordinates as \(x^i\) and \(\mathrm{d} x^i\), we can only use the following rotationally-invariant combinations: \(\tilde{r} = \sum_i (x^i)^2\), \(\sum_i x^i\,\mathrm{d} x^i\), and \(\sum_i \mathrm{d} x^i \mathrm{d} x^i\) (we again explicitly sum over the latin spatial indexes). These we combine with the time coordinate that can appear as \(\tilde{t}\) or \(\mathrm{d}\tilde{t}\) to form the most general metric that is invariant under spatial rotations \begin{equation} \mathrm{d}s^2 = -a(\tilde{r},\tilde{t})\,c^2\mathrm{d}\tilde{t}^2 + b(\tilde{r},\tilde{t})\sum_i{x^i \,c\mathrm{d}\tilde{t}\mathrm{d}x^i} + c(\tilde{r},\tilde{t})\sum_{ij} x^i x^j \mathrm{d}x^i \mathrm{d}x^j + d(\tilde{r},\tilde{t})\sum_i{\mathrm{d}x^i \mathrm{d} x^i}\,. \end{equation} where the \(a(\cdot)\), \(b(\cdot)\), \(c(\cdot)\), and \(d(\cdot)\) are general functions for now. Because rotational invariance is more obvious in spherical coordinates, we can re-write this in spherical coordinates as (see Chapter A.1) \begin{equation} \mathrm{d}s^2 = -a(\tilde{r},\tilde{t})\,c^2\mathrm{d}\tilde{t}^2 + \tilde{r}\,b(\tilde{r},\tilde{t})\,c\mathrm{d}t\mathrm{d}\tilde{r} + \left[\tilde{r}^2\,c(\tilde{r},\tilde{t})+d(\tilde{r},\tilde{t})\right] \mathrm{d}\tilde{r}^2 + \tilde{r}^2\,d(\tilde{r},\tilde{t})\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\,. \end{equation} This can be simplified by re-defining the coordinates, which we have the freedom to do. We can remove the \(d(\tilde{r},t)\) function by re-defining \(\hat{r} = \tilde{r}\sqrt{d(\tilde{r},\tilde{t})}\); then the metric becomes \begin{equation} \mathrm{d}s^2 = -\hat{a}(\hat{r},\tilde{t})\,c^2\mathrm{d}\tilde{t}^2 + \hat{r}\,\hat{b}(\hat{r},\tilde{t})\,c\mathrm{d}t\mathrm{d}\hat{r} + \hat{c}(\hat{r},\tilde{t}) \mathrm{d}\hat{r}^2 + \hat{r}^2\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\,, \end{equation} where the functions \(\hat{a}(\cdot)\), \(\hat{b}(\cdot)\), and \(\hat{c}(\cdot)\) are related to the un-hatted versions by the radial-coordinate transformation, but we do not have to worry about how. We can furthermore get rid of \(\hat{b}(\cdot)\) by redefining the time coordinate to \(\hat{t}\) that is related to \(\tilde{t}\) through \(\tilde{t} = f(\hat{r},\hat{t})\) where \(\hat{b}(\hat{r},\hat{t}) = 2\hat{a}(\hat{r},\hat{t})\partial_{\hat{r}} f(\hat{r},\hat{t})\). The resulting metric can be written in the following form \begin{equation} \mathrm{d}s^2 = -e^{2\alpha(\hat{r},\tilde{t})}\,c^2\mathrm{d}\tilde{t}^2 + + e^{2\beta(\hat{r},\tilde{t})}\,\mathrm{d}\hat{r}^2 + \hat{r}^2\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\,. \end{equation}
So far, we have only used isotropy of the metric around a single point, but next we want to add the additional constraints from spatial homogeneity (as written so far, this metric is, e.g., also the one that leads to the Schwarzschild solution). That is, the metric needs to be the same no matter which spatial point is used as the origin of the spherical coordinate system. This immediately demands that \(\alpha(\hat{r},\tilde{t}) \equiv \alpha(\tilde{t})\) and we can absorb the remaining \(\alpha(\tilde{t})\) through a re-definition of the time coordinate that satisfies \(\mathrm{d}t = e^{\alpha(\tilde{t})} \mathrm{d}\tilde{t}\). Thus, we arrive at \begin{equation}\label{eq-gr-frw-line-element-almostthere} \mathrm{d}s^2 = -c^2\mathrm{d}t^2 + \left\{e^{2\beta(\hat{r},t)}\,\mathrm{d}\hat{r}^2 + \hat{r}^2\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,. \end{equation} The requirement that the spatial part of the metric—the part between the curly braces, which is conventionally denoted as \(\gamma_{ij}\)—is independent of the origin is equivalent to the statement that the Riemann tensor only involves tensors that are invariant under spatial translations, spatial rotations, or Lorentz transformations. The only such tensor that can be part of the Riemann tensor turns out to be the metric itself (the Kronecker tensor and the so-called Levi-Civita tensor are other invariant tensors, but they cannot be included while keeping the symmetries of the Riemann tensor). Thus, the Riemann tensor for the three-dimensional spatial space has to be \begin{equation} \tilde{R}_{ijkl} = \tilde{k}(t)\,\left(\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk}\right)\,, \end{equation} which is the unique combination that has the symmetries of the Riemann tensor and where \(\tilde{k}\) is a constant and we use \(\tilde{R}_{ijkl}\) to distinguish this from the Riemann tensor of four-dimensional spacetime. The Ricci tensor is \begin{equation}\label{eq-gr-frw-Riccitensor-maxsym} \tilde{R}_{ij} = 2\,\tilde{k}(t)\,\gamma_{ij}\,, \end{equation} and the Ricci scalar is \begin{equation} \tilde{R} = 6\,\tilde{k}(t)\,, \end{equation} where we have kept the time-dependence of the curvature scalar \(\tilde{k}\) explicit. Thus, for the metric of Equation (C.108) to be spatially homogeneous, its Ricci tensor has to satisfy Equation (C.110). Computing the Ricci tensor for the metric of Equation (C.108) and equating it to twice \(\tilde{k}(t)\) times the metric gives \begin{align} \tilde{R}_{11}& = {2\over \hat{r}}\partial_{\hat{r}} \beta(\hat{r},t) = 2\tilde{k}(t)\,e^{2\beta(\hat{r},t)} \,,\\ \tilde{R}_{22}= \tilde{R}_{33}/\sin^2 \theta& = e^{-2\beta(\hat{r},t)}\left[\hat{r}\partial_{\hat{r}} \beta(\hat{r},t)-1\right]+1 = 2\tilde{k}(t)\hat{r}^2 \,. \end{align} The first of these equations has the solution \(e^{-2\beta(\hat{r},t)} = C-\tilde{k}(t)\hat{r}^2\) and the other equation fixes \(C = 1\) such that the resulting metric is \begin{equation}\label{eq-gr-frw-line-element-justaboutthere} \mathrm{d}s^2 = -c^2\mathrm{d}t^2 + {1\over 1-\tilde{k}(t)\hat{r}^2}\,\mathrm{d}\hat{r}^2 + \hat{r}^2\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\,. \end{equation}
Finally, we re-define the radial coordinate one last time to absorb all but the sign \(k= \{-1,0,+1\}\) of \(\tilde{k}\) and an overall normalization into the radial coordinate with the transformation \(r\,a(t) = r\, R(t)/R_0 = \hat{r}\) with \(R(t) = 1/\sqrt{|\tilde{k}(t)|}\) (the spatial Ricci scalar is then \(\tilde{R} = 6k/R(t)^2\)). We then arrive at the final form of the isotropic and homogeneous metric \begin{equation}\label{eq-gr-frw-line-element} \mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a^2(t)\left\{{1\over 1-kr^2/R_0^2}\,\mathrm{d}r^2 + r^2\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,. \end{equation} This is the famous Friedmann–Lemaître–Robertson–Walker metric (often just Robertson–Walker metric or FLRW metric).
For \(k=0\), it is clear that the FLRW metric is that of flat space, because the spatial part is then simply \(\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2\) written in spherical coordinates \begin{equation}\label{eq-gr-frw-line-element-zerok} \mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a^2(t)\left\{\mathrm{d}\chi^2 + \chi^2\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,, \end{equation} where we have written \(r = \chi\) for consistency with the \(k\neq 0\) spaces below. Note that this does not fix the global structure of space to be \(\mathbb{R}^3\), because it could also, for example, be the product of three one-dimensional tori, which is also a flat space. The Ricci scalar \(\tilde{R} = 0\) at all times and space is therefore flat at all times. For \(k=+1\), the FLRW metric can be written as \begin{equation}\label{eq-gr-frw-line-element-posk} \mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a^2(t)\left\{\mathrm{d}\chi^2 + R_0^2\,\sin^2(\chi/R_0)\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,, \end{equation} where \begin{equation}\label{eq-gr-comoving-sphere} \chi = R_0\,\sin^{-1} \left(r/R_0\right)\,. \end{equation} The spatial metric within the curly braces is that of a sphere in three-dimensions, which has positive curvature. The curvature of the entire space evolves as \(\tilde{R} = 6/R(t)^2\), thus, as space expands, the curvature decreases. The three-dimensional sphere is the only such physically-realistic space; because it is finite, \(k=+1\) spaces are called closed. For \(k=-1\), the FLRW metric can be written as \begin{equation}\label{eq-gr-frw-line-element-negk} \mathrm{d}s^2 = -\mathrm{d}t^2 + a^2(t)\left\{\mathrm{d}\chi^2 + R_0^2\,\sinh^2(\chi/R_0)\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,, \end{equation} where \begin{equation}\label{eq-gr-comoving-hyper} \chi = R_0\,\sinh^{-1} \left(r/R_0\right)\,. \end{equation} Such a space is a three-dimensional generalization of hyperboloids, which can be infinite in size and such spaces are therefore referred to as open. These three versions of the metric are what’s used to compute distances in FLRW Universe models. It is conventional to choose \(R_0\) such that \(a(t)=1\) at the present time. In that case, \(\chi\) is the comoving distance, the distance between objects in the Universe that factors out the overall change \(R(t)\) to the distances of objects because of the expansion or contraction of the Universe. The parameter \(a\) is then the scale factor and it gives the size of the Universe relative to what it is today. The actual distance \(D\) between objects in the Universe at a given time is the proper distance, which is obtained by multiplying the comoving distance with the scale factor: \(D = a\,\chi\). The equations (C.116), (C.117), and (C.119) are also sometimes written in the general form \begin{equation}\label{eq-gr-frw-line-element-zero-pos-negk} \mathrm{d}s^2 = -\mathrm{d}t^2 + a^2(t)\left\{\mathrm{d}\chi^2 + f_K^2(\chi)\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,, \end{equation} where \(f_K^2(\chi)\) follows from Equations (C.116), (C.117), and (C.119).
C.4.2. The Friedmann equations¶
The evolution of isotropic and homogeneous Universes is entirely determined by the time evolution of the scale factor \(a\). To solve for this evolution for a given model for the matter and energy content of the Universe, we need to solve Einstein’s field equations (C.50) and for this we need to determine the Einstein tensor. As usual, to do this, we need to compute the Christoffel connection, the Riemann tensor, and then the Ricci tensor and scalar, starting from Equation (C.28). This is straightforward for the FLRW metric in Equation (C.115) and we simply state the result here. The non-zero components of the Ricci tensor are the diagonal elements \begin{align} c^2R_{00} & = -3{\ddot{a} \over a}\,,\\ c^2R_{11} & = {a\,\ddot{a}+2\dot{a}^2+2c^2k/R_0^2 \over 1-kr^2/R_0^2}\,,\\ c^2R_{22} = c^2R_{33}/\sin^2 \theta & = r^2\,\left(a\ddot{a}+2\dot{a}^2+2c^2k/R_0^2\right)\,, \end{align} and the Ricci scalar is \begin{equation} c^2R = 6\left[{\ddot{a} \over a} + \left({\dot{a} \over a}\right) +{c^2k\over a^2 R_0^2}\right]\,. \end{equation}
Next, we need to specify the stress-energy tensor of the matter and energy components that make up the Universe. The assumption of isotropy and homogeneity also applies to \(T^{\mu\nu}\) and similar to how we proceeded for the metric, we can constrain the form of \(T^{\mu\nu}\) by requiring that it is isotropic and spatially homogeneous. Working in the frame in which the matter is at rest, the \(00\) component of \(T^{\mu\nu}\) is the energy density \(\rho c^2\) and because of the requirement of homogeneity, this can only depend on time: \(\rho c^2 \equiv \rho(t) c^2\). The \(0i\) components are the momentum density and for the same reason they can only depend on time, but we saw in Section C.1 that these components are also equal to the energy flux in space and if they are non-zero, they would induce spatial inhomogeneity. We therefore must have that \(T^{0i} = T^{i0} = 0\). Finally, the purely-spatial components \(T^{ij}\) must be built from spatial tensors that are invariant under rotations and translations and again the only option is the spatial metric \(\gamma^{ij}\) itself, multiplied by a function of time. Therefore, the stress energy tensor is given by \begin{align} T^{00} & = \rho(t)c^2\,;\quad T^{ij} = p(t)\,\gamma^{ij}\,, \end{align} where the spatial part of the metric, \(\gamma_{ij}\) is the part of the FLRW metric in Equation (C.115) between the curly braces, \(\rho(t)c^2\) is the energy density, and \(p(t)\) is the pressure. Comparing this to Equation (C.46), we see that the stress-energy tensor has to be that of a perfect fluid. The trace of \(T^{\mu\nu}\) is \begin{equation} T = -\rho(t)c^2 + 3p(t)\,. \end{equation}
We can use this stress-energy tensor to write down the field equations in the form of Equation (C.52); the only non-trivial components are equivalent to the following equations (from now on, we drop the explicit time dependence of \(\rho\) and \(p\) to reduce notational clutter) \begin{align} -3{\ddot{a} \over a} + \Lambda c^2 & = 4\pi G\left(\rho + 3{p\over c^2}\right)\,,\\ {\ddot{a}\over a}+2\left({\dot{a} \over a}\right)^2+2{c^2k \over a^2R_0^2} - \Lambda c^2 & = 4\pi G\left(\rho -\phantom{3} {p\over c^2}\right)\,, \end{align} because the other equations are either trivially satisfied (\(0=0\)) or equivalent to the second one. We can use the first equation to remove the \(\ddot{a}\) term in the second equation to arrive at \begin{equation}\label{eq-gr-friedmann-1} H^2 = \left({\dot{a} \over a}\right)^2= {8\pi G\rho\over 3}+{\Lambda c^2\over 3}-{c^2k \over a^2R_0^2}\,, \end{equation} where \(H = \dot{a}/a\) is the Hubble parameter. Together with the first equation written as \begin{equation}\label{eq-gr-friedmann-2} \dot{H} + H^2 = {\ddot{a} \over a} = -{4\pi G\over 3}\left(\rho + 3{p\over c^2}\right) + {\Lambda c^2\over 3}\,, \end{equation} these two equations make up the Friedmann equations.
To solve the Friedmann equations, we need to specify the functions \(\rho(t)\) and \(p(t)\). The standard approach is to use an equation of state to relate the energy density to the pressure and the typically-used equations of state can all be written as (see the discussion in Chapter 17.1.3) \begin{equation}\label{eq-gr-eq-state-constant-w} p = w \rho c^2\,, \end{equation} where \(w\) is a constant. For example, for pressureless dust, which describes the behavior of non-relativistic matter (ordinary and dark) on large scales well, we have that \(w = 0\) (see Equation C.47). Light has \(w = 1/3\). It is also common to convert the cosmological constant term \(\propto \Lambda\) in the Friedmann equations into an energy component by using \(\rho = \Lambda c^2 / [8\pi G]\) and \(w=-1\); in this case the cosmological constant is referred to as dark energy. To determine how the density and pressure of a perfect fluid with equation of state (C.132) depends on the scale factor \(a\), we can use the conservation of the stress-energy tensor, \(\nabla_\mu T^{\mu\nu} = 0\). This gives \begin{align} 0 = c \nabla_\mu T^{\mu\nu} & = -\dot{\rho}c^2 -3{\dot{a} \over a}\left(\rho c^2 + p\right) = -\dot{\rho}c^2 -3{\dot{a} \over a}\,\rho c^2\left(1+w\right)\,, \end{align} or \begin{equation} {\dot{\rho}\over \rho} =-3{\dot{a} \over a}\left(1+w\right)\,, \end{equation} with solution \begin{equation}\label{eq-gr-rho-perfectfluid-constantw-intime} \rho \propto a^{-3\left(1+w\right)}\,. \end{equation} This has the expected behavior that the energy density of non-relativistic matter, which is largely rest mass, dilutes as \(1/a^3\) and that the energy density of light dilutes as \(1/a^4\), because of the additional redshifting effect of expanding space. Dark energy with \(w=-1\) has a constant density \(\rho(a) = \mathrm{constant}\). Because light’s wavelength is \(\propto 1/a\), light emitted at earlier times and observed today shows a redshift \(z\) that is given by \(1+z = 1/a\). Because of this equation and because observations of the earlier Universe generally involve redshifted light, the redshift \(z\) is often used instead of \(a\) when discussing cosmology. Any function below that is a function of the scale factor can be equivalently considered a function of the redshift \(z = 1/a-1\).
Writing the cosmological constant as a density component and assuming that there are \(N\) perfect-fluid energy components following Equation (C.132) indexed by \(i\), we can write the first Friedmann equation (C.130) as \begin{align}\label{eq-gr-friedmann-density-parameter} {H^2 \over H_0^2} & = \Omega_{0,k}\,a^{-2} + \sum_i \Omega_{0,i} a^{-(1+w_i)}\,, \end{align} where \(H_0\) is the value of the Hubble parameter today (the Hubble constant), the density parameters \(\Omega_{0,i}\) are the energy density of component \(i\) expressed in terms of today’s critical density \begin{equation} \rho_{0,c} = {3H_0^2\over 8\pi G} \end{equation} as \begin{equation} \Omega_{0,i} = {\rho_{0,i} \over \rho_{0,c}}\,, \end{equation} and \begin{equation} \Omega_{0,k} = 1-\sum_i \Omega_{0,i} = -{c^2k\over H^2_0 R_0^2} \end{equation} represents the curvature component as a density parameter as well. We have defined the critical density and curvature parameter here at the present time, but these definitions can be generalized to any time; all of these quantities in general change with time.
The energy density of our Universe is dominated by matter (both ordinary and dark with combined density parameter \(\Omega_{0,m}\)), dark energy (or, equivalently, a non-zero cosmological constant; density parameter \(\Omega_{0,\Lambda}\)), and radiation (with density parameter \(\Omega_{0,r}\)). Thus, for our Universe, we can write Equation (C.136) as \begin{align}\label{eq-gr-friedmann-matter-de-radiation-curvature} E^2(a) = {H(a)^2 \over H_0^2} & = \Omega_{0,m}\,a^{-3} + \Omega_{0,\Lambda}+\Omega_{0,r}\,a^{-4}+\Omega_{0,k}\,a^{-2}\,, \end{align} where we have introduced the function \(E(a)\) that we will use further below. The density parameters of the various energy components can be constrained using observations of the CMB and of the large-scale structure of galaxies. These constrain the curvature to be very small (Planck Collaboration et al. 2020b) \begin{equation} \Omega_{0,k} = 0.001\pm0.002\,. \end{equation} Assuming then that the Universe is flat, the values of the other density parameters are \begin{align} \Omega_{0,m} & = 0.3111 \pm 0.0056\,,\\ \Omega_{0,\Lambda} & = 0.6889 \pm 0.0056\,,\\ \Omega_{0,r} & = (9\pm0.4)\times 10^{-5}\,. \end{align} Thus, at the present time, the energy density of the Universe is dominated by the dark energy component, but with a substantial contribution from matter.
Because of the different behavior of the energy density with scale factor for different \(w\) (Equation C.135), different components dominate the energy density at different times. While the energy density in radiation is negligible today, because it behaves as \(a^{-4}\), at early times, radiation dominates the energy budget. This only lasts until a redshift of \(z_{\mathrm{eq}} \approx 3,400\), the redshift at which the energy density of radiation and matter are equal (found from \(\Omega_{0,m} [1+z]^{3} = \Omega_{0,r}[1+z]^{4}\)). This is before \(z\approx 1,000\) when the CMB forms and baryons decouple from light. At \(1 \lesssim z \lesssim 3,400\), matter (ordinary and dark) dominates the energy budget and the cosmological constant (or dark energy) term in the Friedmann equations is negligible. At \(z \lesssim 1\), the cosmological constant becomes important.
C.4.3. The angular diameter and luminosity distances¶
In an expanding (or contracting) Universe, the distance in static, flat space is replaced by a whole bevy of different distances depending on the context. For example, the distance \(D_A\) that relates an object’s small angular size \(\delta\theta\) to its physical size \(\delta s\) through \begin{equation} \delta\theta = {\delta s \over D_A}\,, \end{equation} is known as the angular diameter distance and we need this in our discussion of gravitational lensing in Chapter 15. Placing the origin at the observer’s position at \(a=1\) and assuming that we are observing an object at scale factor \(a\), we can then determine the angular diameter distance by first converting the object’s size \(\delta s\) to the observer’s time by dividing by \(a\) to account for the change in the Universe’s size. Then we have that \(\delta \theta = (\delta s / a) /r\), where \(r\) is the coordinate distance. Therefore \begin{equation} D_A = a\,r\,. \end{equation}
If we are observing the object using light (which we generally are!), then we can compute \(r\) on a null geodesic where \(\mathrm{d} s = 0\) and thus \(c\mathrm{d}t = a\,\sqrt{1/(1-kr^2/R_0^2)}\mathrm{d} r = a\,\mathrm{d} \chi\) from Equation (C.115). Then we can obtain \(r = r(\chi)\) by inverting the relevant equation (\(r=\chi\), Equation C.118, or Equation C.120 depending on whether \(k=0,+1,-1\); in general we can write this as \(r = f_K(\chi)\) from the general form of the FLRW metric from Equation C.121) where the comoving distance is \begin{equation}\label{eq-gr-comoving-distance} \chi = c\int{\mathrm{d}t \over a} = c\int{\mathrm{d}a \over \dot{a} a}\,. \end{equation} To solve this integral, we use the expression for \(\dot{a}/a\) for our Universe from Equation (C.140). The comoving distance from Equation (C.147) is given by \begin{align} \chi & = {c\over H_0} \int_{1}^a{\mathrm{d}a' \over a'^2\,E(a')}\\ & = {c\over H_0} \int_{1}^a{\mathrm{d}a \over a^2}{1 \over \sqrt{\Omega_{0,m}\,a^{-3} + \Omega_{0,\Lambda}+\Omega_{0,r}\,a^{-4}+\Omega_{0,k}\,a^{-2}}}\\ & = {c\over H_0} \int_{0}^z\mathrm{d}z\,{1 \over \sqrt{\Omega_{0,m}\,(1+z)^{3} + \Omega_{0,\Lambda}+\Omega_{0,r}\,(1+z)^{4}+\Omega_{0,k}\,(1+z)^{2}}}\,,\label{eq-gr-comovingdist-explicit} \end{align} where we have written the integral either as an integral over scale factor or redshift. The angular diameter distance is then \begin{equation}\label{eq-gr-angulardist-explicit} D_A = a\,f_K(\chi) = a \times \begin{cases} {c \over H_0\sqrt{\Omega_{0,k}}}\,\sin\left(\chi\,{H_0\sqrt{\Omega_{0,k}}\over c}\right)\,, & k =+1\\ \chi\,, & k = 0\\ {c \over H_0\sqrt{-\Omega_{0,k}}}\,\sinh\left(\chi\,{H_0\sqrt{-\Omega_{0,k}}\over c}\right)\,, & k = -1\end{cases}\,. \end{equation}
To compute the angular diameter distance between two scale factors \(a_1\) and \(a_2\) (or redshifts \(z_1\) and \(z_2\)) observing an object at \(a_2\) from \(a_1\), compute the integral in Equation (C.150) between these two scale factors and replace the prefactor \(a\) in Equation (C.151) by \(a_2/a_1\).
The distance \(D_L\) that relates a source’s flux \(f\) to its intrinsic luminosity \(L\) through \begin{equation} f={L \over 4\pi\,D_L^2} \end{equation} is known as the luminosity distance. In terms of a source’s apparent and absolute magnitude \(m\) and \(M\), respectively, this is \begin{equation} m-M = 5\log_{10}\left({D_L \over 10\,\mathrm{pc}}\right)\,. \end{equation}
To work out the luminosity distance to a certain scale factor, we consider a sphere centered on the source with radius equal to the comoving distance \(\chi\). From Equation (C.121), it is clear that this sphere has a radius of \(f_K(\chi)\), so the luminosity is spread out as \(L / [4\pi\,f^2_K(\chi)]\). However, the expansion of the Universe between the time of emission (assumed to be at scale factor \(a\)) and the time of observation (assumed to be at the present time, \(a=1\)) introduces two factors of \(a\), because the energy of each photon is redshifted as \(E \rightarrow aE\) and the rate at which photons are detected is decreased \(\propto a\) by time dilation, such that overall \begin{equation} f={a^2\,L \over 4\pi\,f_K^2(\chi)}\,. \end{equation} The luminosity distance is therefore given by \begin{equation} D_L = {f_K(\chi) \over a} = {1\over a} \times \begin{cases} {c \over H_0\sqrt{\Omega_{0,k}}}\,\sin\left(\chi\,{H_0\sqrt{\Omega_{0,k}}\over c}\right)\,, & k =+1\\ \chi\,, & k = 0\\ {c \over H_0\sqrt{-\Omega_{0,k}}}\,\sinh\left(\chi\,{H_0\sqrt{-\Omega_{0,k}}\over c}\right)\,, & k = -1\end{cases}\,. \end{equation} Comparing this to Equation (C.151), we find the following general relation between the angular-diameter and luminosity distances \begin{equation} D_L = {D_A \over a^2} = (1+z)^2\,D_A\,. \end{equation}
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