My notes on the second edition of the excellent Modern Cosmology textbook by Scott Dodelson and Fabian Schmidt. I try to fill in the details of some derivations that the they skipped for brevity, and explain the conceptual intuition behind some formulas that they presented outright. No guarantees when I’ll finish the whole thing.
Additional texts used in these notes are Sean Carroll’s Spacetime and Geometry, Michael Peskin and Daniel Shroeder’s An Introduction to Quantum Field Theory, and Donald Clayton’s Principles of Stellar Evolution and Nucleosynthesis. Astrophysics is truly an applied science; to really understand some of the theoretical bases, you must look elsewhere.
Deepest thanks to Chien-I Chiang and Ori Ganor for teaching me the prerequisite knowledge for tackling cosmology (namely, Lagrangian mechanics and general relativity), and for patiently answering my questions wherever I got stuck on this subject.
You may also be interested in Chung-Pei Ma’s lecture notes for the course at Berkeley, and David Tong’s lecture notes for the course at Cambridge. Besides this, I have notes on some other astrophysics stuff as well.
The expansion of the universe is the expansion of space itself: at the largest scales, objects are physically drifting farther apart because the grid markings are, in a sense, becoming denser. We parameterize this with the scale factor $a=a(t)$ such that the physical distance between objects is $d\equiv ax$, where $x$ is the comoving distance. This allows the physical velocities of (non-relativistic) objects to be separated into \[ v \equiv \dot d = \dot ax + a\dot x, \] where the first term is due to the expansion of space, and the second term is due to a peculiar velocity of the object.
The physical distance $d$ is what gets measured directly by light, through its travel time or (more commonly) its flux’s inverse-square relation. The comoving distance $x$ is determined indirectly, but remains constant ($\dot x = 0$) for objects that do not interact, gravitationally, electrormagnetically, or otherwise.
The effect of expansion is most apparent in the cosmological redshift of photons. Defining the present-day scale factor $a(t_0)=1$, it can be related to the redshift parameter $z$ via \[ 1 + z(t) \equiv \frac{\lambda(t_0)}{\lambda(t)} = \frac1{a(t)}. \tag{1.1} \] Through black-body radiation, it can also be related to temperature via $T(t) = T(t_0) / a(t)$.
The Hubble rate $H(t) \equiv \dot a/a$ measures the rapidity of expansion, with the present-day Hubble rate $H_0 \equiv H(t_0)$ referred to as Hubble’s constant. From the Friedmann equation, \[ \frac{H(t)^2}{8\pi G} = \rho(t) + \frac{\rho_{\text{cr}} - \rho(t_0)}{a(t)^2}, \tag{1.3} \] where $\rho$ is the average density of the universe, and \[ \rho_{\text{cr}} \equiv \frac{3H_0^2}{8\pi G} \tag{1.4} \] is the critical density at which the universe is flat.
Due to tensions in measurements of $H_0$, we parameterize it with a dimensionless number $h$ such that $H_0 \equiv 100\, h$ km/s/Mpc, with $h\sim 0.7$. This way, $H_0$ can be eliminated from calculations by using $h^{-1}$ Mpc as the unit of length, $h^{-1}\, M_\odot$ as the unit of mass, etc.
The ΛCDM model is the concordance model of cosmology, describing a flat universe which grew out of inflationary perturbations, and is presently dominanted by cold dark matter (CDM) and the cosmological constant $\Lambda$.
According to general relativity, particles in the absence of any forces move in geodesics such that in Minkowski coordinates, \[ \f{\d ^2 x^\alpha}{\d \lambda^2} = 0 \tag{2.13} \] for some parameter $\lambda$ that monotonically increases along the particle’s path (such as proper time for massive particles). However, this equation does not necessarily hold true for other coordinate systems, where the basis vectors are not invariant over spacetime. Under a change of coordinates from Minkowski $x^\alpha\mapsto x^{\nu'}$, the geodesic equation becomes \[\begin{aligned} 0 &= \f{\d }{\d \lambda} \f{\d x^{\nu'}}{\d \lambda} \\ &= \f{\d }{\d \lambda}\prn{ \f{\p x^{\nu'}}{\p x^{\alpha}} \f{\d x^{\alpha}}{\d \lambda} } \\ &= \f{\d x^\alpha}{\d \lambda} \f{\d }{\d \lambda} \f{\p x^{\nu'}}{\p x^\alpha} + \underline{ \f{\p x^{\nu'}}{\p x^\alpha} \f{\d ^2 x^\alpha}{\d \lambda^2} }_{\displaystyle\star} \\ &= \f{\d x^\alpha}{\d \lambda} \f{\p}{\p x^\alpha}\f{\d x^{\nu'}}{\d \lambda} + \star\\ &= \f{\d x^\alpha}{\d \lambda} \f{\p}{\p x^\alpha}\prn{ \f{\p x^{\nu'}}{\p x^\beta}\f{\d x^\beta}{\d \lambda} } + \star\\ &= \f{\d x^\alpha}{\d \lambda} \brk{ \f{\p^2 x^{\nu'}}{\p x^\alpha \p x^\beta}\f{\d x^\beta}{\d \lambda} + \f{\p x^{\nu'}}{\p x^\beta} \underline{\f{\d }{\d \lambda}\prn{\f{\p x^\beta}{\p x^\alpha}}}_0 } + \star\\ &= \f{\p^2 x^{\nu'}}{\p x^\alpha \p x^\beta}\f{\d x^\alpha}{\d \lambda}\f{\d x^\beta}{\d \lambda} + \f{\p x^{\nu'}}{\p x^\alpha} \f{\d ^2 x^\alpha}{\d \lambda^2}. \tag{2.18} \end{aligned}\] Applying the inverse transform $x^{\nu'}\mapsto x^\mu$ to both sides yields \[ 0 = \underbrace{\f{\p x^\mu}{\p x^{\nu'}} \f{\p^2 x^{\nu'}}{\p x^\alpha \p x^\beta}} _{\displaystyle \Gamma^\mu_{\alpha\beta}} \f{\d x^\alpha}{\d \lambda}\f{\d x^\beta}{\d \lambda} + \f{\d ^2 x^{\mu}}{\d \lambda^2}, \tag{2.19} \] where $\Gamma^\mu_{\alpha\beta}$ is the Christoffel symbol, related to the metric by \[ \Gamma^\mu_{\alpha\beta} = \f{g^{\mu\nu}}{2} \left[ \p_\beta g_{\alpha\nu} + \p_\alpha g_{\beta\nu} - \p_\nu g_{\alpha\beta} \right]. \tag{2.21} \] The geodesic equation which holds true for all coordinate systems is therefore \[ \f{\d^2 x^\mu}{\d\lambda^2} = -\Gamma^\mu_{\alpha\beta} \f{\d x^\alpha}{\d\lambda} \f{\d x^\beta}{\d\lambda}. \]
The Christoffel symbol is used in defining the covariant derivative $\nabla_\mu$: \[\begin{aligned} \text{scalar} && \nabla_\mu\phi &\equiv \p_\mu\phi,\\ \text{vector} && \nabla_\mu A^\nu &\equiv \p_\mu A^\nu + \Gamma^\nu_{\mu\alpha} A^\alpha,\\ \text{tensor} && \nabla_\mu {T_\nu}^\kappa &\equiv \p_\mu {T_\nu}^\kappa - \Gamma^\lambda_{\mu\nu}{T_\lambda}^\kappa + \Gamma^\kappa_{\mu\lambda}{T_\nu}^\lambda. \tag{2.52} \end{aligned}\] This is necessary because the partial derivative $\p_\mu$ only transforms covariantly when acting on scalars. For instance, the geodesic equation can be rewritten in covariant form as \[ \f{\d x^\alpha}{\d \lambda} \nabla_\alpha \f{\d x^\mu}{\d \lambda} = 0. \tag{2.53} \]
In the flat, expanding ΛCDM universe, distances in spacetime are described with the Euclidean Friedmann–Lamaître–Robertson–Walker (FLRW) metric \[ g_{\mu\nu} \to \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a^2 & 0 & 0\\ 0 & 0 & a^2 & 0\\ 0 & 0 & 0 & a^2 \end{pmatrix}, \tag{2.12} \] with the inverse (contravariant) metric \[ g^{\mu\nu} \to \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a^{-2} & 0 & 0\\ 0 & 0 & a^{-2} & 0\\ 0 & 0 & 0 & a^{-2} \end{pmatrix} \] satisfying the identity $g^{\nu\beta}g_{\beta\alpha} = \delta^\nu_\alpha$.
The FLRW metric is independent of the spatial coordinates ($\p_i$) as a consequence of spatial isotropy and homogeneity, but dependent on time ($\p_0$) due to the $a^2$ expansion terms. Its Christoffel symbol is thus \[\begin{aligned} \Gamma^0_{ij} &= \f{g^{0\nu}}{2} \left[ \zero{ \p_j g_{i\nu} } + \zero{ \p_i g_{j\nu} } - \p_0 g_{ij} \right] \\ &= -\f{g^{00}}2\p_0 g_{ij} = \delta_{ij} a\dot a, \\ \Gamma^i_{j0} = \Gamma^i_{0j} &= \f{g^{i\nu}}{2} \left[ \p_0 g_{j\nu} + \zero{ \p_j g_{0\nu} } - \zero{ \p_\nu g_{j0} } \right] \\ &= \delta^{i\nu}a^{-2}\delta_{j\nu}a\dot a = \delta^i_j\f{\dot a}a, \\ \Gamma^\alpha_{00} &= \Gamma^0_{\alpha0} = \Gamma^0_{0\alpha} = \Gamma^i_{jk} = 0. \tag{2.25} \end{aligned}\] Verify for yourself that these equations together resolve every entry of the symbol.
We wish to calculate the effects of cosmic expansion on the energy of a free particle.
If we choose to parameterize a massive particle’s path by $\lambda = \tau/m$, where $\tau$ is the particle’s proper time, then we can write the particle’s comoving momentum as \[ P^\alpha = m\f{\d x^\alpha}{\d\tau} = \f{\d x^\alpha}{\d\lambda}. \tag{2.26} \] We write the same for a massless particle by setting $\lambda = \nu t$, where $\nu$ is the particle’s de Broglie frequency.
For a free particle, $P^0$ is conserved along its geodesic. We identify this as the particle’s energy $E$, so \[ \f{\d }{\d \lambda} = \f{\d x^0}{\d \lambda}\f{\d}{\d x^0} = E\f{\d}{\d t}. \tag{2.27} \]
In the rest frame of the particle, $P^{\hat 0} = E_0 = m$ while $P^{\hat i} = 0$, giving the FLRW dispersion relation \[ g_{\mu\nu}P^\mu P^\nu = -E^2 + \delta_{ij} a^2 P^i P^j = -m^2. \tag{2.29} \] This motivates defining \[ \vec p \leftarrow p^i = a P^i \tag{2.32} \] as the physical momentum $\vec p$ of the particle, since it satisifies the Minkowskian dispersion relation $-E^2 + \delta_{ij} p^i p^j = m^2$.
The geodesic equation for the particle thus simplifies as \[\begin{aligned} \f{\d ^2 x^\mu}{\d \lambda^2} &= -\Gamma^\mu_{\alpha\beta} \f{\d x^\alpha}{\d \lambda} \f{\d x^\beta}{\d \lambda} \\ = \f{\d P^\mu}{\d \lambda}= E\dot P^\mu &= -\Gamma^\mu_{ij} P^i P^j, \end{aligned}\] with the $\mu=0$ component \[\begin{aligned} E\dot E &= -\Gamma^0_{ij} P^i P^j \\ &= -a\dot a \delta_{ij} P^i P^j\\ &= \f{\dot a}{a} \big[m^2 - E^2\big]. \end{aligned}\]
For a massless particle, \[ m = 0 \implies \f{\dot E}{E} = -\f{\dot a}{a} \implies E(t) = \f{E(t_0)}a, \tag{2.31} \] and for a non-relativistic massive particle, \[ E \simeq m \implies \dot E \simeq 0 \implies E(t) \simeq E(t_0). \] This demonstrates that only relativistic particles (i.e. radiation) lose significant energy through cosmological expansion.
A similar method of analysis can be applied to the homogeneous, isotropic background of the universe, characterized by the mean energy density $\rho$ and the mean pressure $\mathcal P$. As with any such ideal fluid, its energy-momentum tensor can be derived to be \[ \renewcommand{\P}{{\mathcal P}} {T^\mu}_\nu \to \begin{pmatrix} -\rho & 0 & 0 & 0\\ 0 & \P & 0 & 0\\ 0 & 0 & \P & 0\\ 0 & 0 & 0 & \P \end{pmatrix} \tag{2.44} \] in the rest frame and obeys the conservation of local energy-momentum $\nabla_\mu {T^\mu}_\nu = 0$, whose $\nu=0$ component tells us that \[\begin{aligned} 0 &= \p_\mu {T^\mu}_0 + \Gamma^\mu_{\alpha\mu}{T^\alpha}_0 - \Gamma^\alpha_{0\mu}{T^\mu}_\alpha\\ &= \p_0 {T^0}_0 + \Gamma^i_{0i}{T^0}_0 - \Gamma^i_{0j}{T^j}_i\\ &= -\p_t \rho - \delta^i_i \f{\dot a}{a} \rho - \delta^i_j \f{\dot a}{a} \delta^j_i \mathcal P\\ \iff 0&= \p_t \rho + 3\f{\dot a}{a}\brk{ \rho + \mathcal P }\\ \iff \f{\p_t(\rho a^3)}{a^3} &= -3\f{\dot a}{a}\mathcal P. \tag{2.57} \end{aligned}\]
In general, the density of a medium with a constant equation of state $w \equiv \mathcal P / \rho$ evolves as $\rho \propto a^{-3-3w}$. For radiation, $\mathcal P_{\text r} \simeq \rho_{\text r}/3 \implies \rho_{\text r} \propto a^{-4}$, whereas for nonrelativistic matter, $\mathcal P_{\text m} \simeq 0 \implies \rho_{\text m} \propto a^{-3}$. The cosmological constant, with $w_\Lambda = -1$, is thus constant through time.
In a small time interval $\d t$, light travels a comoving distance $\d x = \d t/a$, thus the total comoving distance $\chi$ traversed by a light emitted at time $t$ (i.e. the comoving distance from us to an object we observe at time $t$) is \[ \chi(t) = \int_t^{t_0} \f{\d t'}{a(t')} = \int_{a(t)}^1 \f{\d a'}{a'^2 H(a')} = \int_0^z \f{\d z'}{H(z')}. \tag{2.34} \] And the maximum comoving distance $\eta$ that may be causally connected to a point in time $t$ is \[ \eta(t) \equiv \int_0^t \f{\d t'}{a(t')}. \tag{2.35} \] This is also known as the comoving horizon, or as the conformal time, seeing it monotonically increases with $t$.
TODO: Angular sizes and curvature. For the time being, see this xkcd comic for an illustration of this concept.
Many macroscopic statistics of a system are derived from how its particles are distributed in phase-space. We can calculate this phase-space distribution using some quantum mechanics!
First, let us calculate how many particles there are within every infinitesimal phase-space volume $\d^3x\, \d^3p$.
For simplicity, suppose the physical space we are working in is a cube of side length $\Delta x$, with periodic conditions at its boundaries. Through a Fourier decomposition, scalar fields in this physical space can be spanned by basis wavefunctions of the form \[ \psi_{\vec N}(\vec x) = \exp\prn{\f{ 2\pi i \vec N \cdot \vec x }{ \Delta x }}, \] where $\vec N$ is a vector of three nonnegative integers. Each wavefunction $\psi_{\vec N}$ has a wavevector $\vec k_{\vec N} = 2\pi\vec N / \Delta x$, which, by the de Broglie relation, equals the wavefunction’s momentum. (Recall that we work in units where $\hbar = 1$). The integer constraint of $\vec N$ thus induces the quantization of momenta: Within this box, components of momenta are restricted to steps of $2\pi / \Delta x$.
Consequently, for any phase-space volume $[\Delta x]^3 [\Delta p]^3$, there are $ [\Delta x]^3 [\Delta p]^3 / [2\pi]^3 $ allowed momenta. $[2\pi]^3$ is therefore regarded as the phase-space volume of the fundamental phase-space element \[ \f{ \d^3 x \, \d^3 p }{ [2\pi]^3 } \] in nonrelativistic statistical mechanics, independent of any choice of $\Delta x$ or $\Delta p$.
The distribution function $f_s(\vec x, \vec p, t)$ counts the number of particles of a given species $s$ per such fundamental element around $(\vec x, \vec p, t)$. However, $f_s$ does not distinguish degenerate states in quantum mechanics, so we also have to multiply by the degeneracy factor $g_s$. For example, photons have $g_\gamma = 2$ due to their two spin states.
Thus, every infinitesimal volume $\d^3x\, \d^3p$ in phase-space contains \[ g_s \f{\d^3 x \, \d^3 p}{[2\pi]^3} f_s(\vec x, \vec p, t) \] particles.
Consider some property $A_s$ of a particle which depends on its physical momentum $\vec p$. Integrating it over the phase-space around $\vec x$, weighing with the number of particles per infinitesimal phase-space volume, and dividing by the infinitesimal physical volume $\d x^3$, we obtain the property’s density in physical space \[\begin{aligned} \mathcal A_s(\vec x, t) &= g_s \int \f{\d^3 p}{[2\pi]^3}\, f_s(\vec x, \vec p, t)\, A_s(\vec p). \end{aligned}\] (Dimensionally, this correspondence between phase-space total and physical-space density can be seen through the $[\text{momentum}] = [\text{energy}] = [\text{time}]^{-1} = [\text{length}]^{-1}$ equivalence in natural units.)
Different properties $A_s$ can tell us different microscopic properties of the species at $(\vec x, t)$. For instance, integrating the energy $E_s(p)$ of a particle gives the energy density \[ \rho_s(\vec x, t) = g_s \int \f{\d^3 p}{[2\pi]^3} \, f_s(\vec x, \vec p, t) \, E_s(p), \tag{2.62} \] and integrating two-thirds (by the equipartition principle) of the kinetic energy $K_s(p) = p^2/2E_s(p)$ of a particle gives the pressure \[ \mathcal{P}_s(\vec x, t) = g_s \int \f{\d^3 p}{[2\pi]^3} \, f_s(\vec x, \vec p, t) \, \f{p^2}{3E_s(p)}. \tag{2.64} \]
To model relativistic particles, we have to rewrite the nonrelativistic energy density, momentum density, and pressure fields, derived above, in a relativistically-covariant form. While these properties do not themselves transform covariantly, they can be combined into a energy-momentum tensor field $T{^\mu}_\nu(\vec x, t)$ which does.
As with the non-relativistic case, we will build up the field expressions from the properties of single particles. However, we now have to account for relativistic covariance.
Consider a particle with rest mass $m$ located at position $\vec x_0$. In the rest frame, we can always find a Minkowskian coordinate system (by the local flatness theorem), in which we will mark indices with a hat ($\hat{\ }$). In this coordinate system, the metric is $g_{\hat\mu\hat\nu} \to \mathrm{diag}(-1,1,1,1)$, the particle's relativistic momentum is $P^{\hat \mu} \to (m, 0, 0, 0)$, and its energy-momentum tensor field surrounding a physical-space volume $\d\mathcal V = \d x^{\hat 1} \d x^{\hat 2} \d x^{\hat 3}$ about $\vec x_0$ is \[ {T^{\hat \mu}}_{\hat \nu}(\vec x_0)\,\d\mathcal V \to \begin{pmatrix} -m & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}. \] The tensor field at $\vec x_0$ can thus be written covariantly as \[ T{^\mu}_{\nu}(\vec x_0, t)\, \d\mathcal V = \f{P^\mu(t) P_\nu(t)}{m} = \gamma\cdot\f{P^\mu(t) P_\nu(t)}{P^0}. \] where $\gamma \equiv [1 - v^2]^{-1/2}$ is the Lorentz factor, transforming energy component of relativistic momentum as $P^0 = \gamma m$. Note that this equation holds in the massless limit as well.
The physical-space volume $\d\mathcal V$ transforms due to Lorentz contraction and changes to the metric, \[ \d\mathcal V = \d^3 x \cdot \sqrt{-|g|} \cdot \gamma, \] where $|g| \equiv \mathrm{det}(g_{\mu\nu})$ is the metric determinant. (Note that for diagonal, pseudo-Riemannian metrics such as the FLRW, $|g| = g_{00} \cdot g_{11} \cdot g_{22} \cdot g_{33} < 0$.) The field at any point in spacetime, under any reference frame, is thus \[ T{^\mu}_{\nu}(\vec x, t) = \f{\delta_D^{(3)} (\vec x - \vec x_0)}{\sqrt{-|g|}} \f{P^\mu P_\nu}{P^0}. \]
Generalizing from single particles, the energy-momentum tensor field for some arbitrary species $s$ with distribution function $f_s$ and degeneracy $g_s$ is therefore \[ (T_s){^\mu}_\nu(\vec x, t) = \f{g_s}{\sqrt{-|g|}} \int \f{\d^3 P}{[2\pi]^3} \f{P^\mu P_\nu}{P^0} f(\vec x, \vec p, t). \tag{3.20} \] Identifying with the components of (2.44), the relativistic energy density is then \[ \rho_s(\vec x, t) = -\f{g_s}{\sqrt{-|g|}} \int \f{\d^3 P}{[2\pi]^3} \f{P^0 P_0}{P^0} f_s(\vec x, \vec p, t), \] while the relativistic spatial stress-shear tensor (whose diagonal elements are the pressures) is \[ {(\mathcal P_s)^i}_j(\vec x, t) = \f{g_s}{\sqrt{-|g|}} \int \f{\d^3 P}{[2\pi]^3} \f{P^i P_j}{P^0} f_s(\vec x, \vec p, t). \]
Under thermal equilibrium, the distribution function $f_s$ for every species is only dependent on the momentum magnitude $p$ through the energy $E_s(p) = \sqrt{p^2 + m_s^2}$, with bosons following the Bose–Einstein distribution \[ f_\text{BE}(E_s) = \f1{e^{[E_s - \mu_s]/T} - 1} \tag{2.65} \] and fermions following the Fermi–Dirac distribution \[ f_\text{FD}(E_s) = \f1{e^{[E_s - \mu_s]/T} + 1}, \tag{2.66} \] where $\mu_s$ is the chemical potential and $T$ is the equilibrium temperature. For photons and neutrinos, $\mu_{\text r} \ll T$. Evaluating $\mathcal{P}_{\text r}$ in thermal equilibrium thus gives \[ \f{\p\mathcal{P}_{\text r}}{\p T} = \f{\rho_{\text r} + \mathcal{P}_{\text r}}{T} \propto a^{-3}. \tag{2.67} \] Thus the entropy density of the universe $S \equiv (\rho + \mathcal P)/T$ scales as $a^{-3}$ in thermal equilibrium.
Define the dimensionless density parameter of a species $s$ to be \[ \Omega_s \equiv \f{\rho_s(t_0)}{\rho_{\text{cr}}}, \tag{2.71} \] i.e. its present-day density as a fraction of the universe’s critical density $\rho_{\text{cr}}$, we can write the physical density as \[ \rho_s(a) = \Omega_s \rho_{\text{cr}} a^{-3[1 + w_s]}, \tag{2.72} \] where $a$ is the scale factor and $w_s$ is the equation of state for $s$.
For photons ($g_\gamma = 2, \mu_\gamma \ll T, E_\gamma(p) = p$), we have \[\begin{aligned} \rho_\gamma &= \f{2}{[2\pi]^3} \int \d^3 p \, \f{p}{e^{p/T} - 1}\\ &= \f{2}{[2\pi]^3} \f{4\pi}3 \int_0^\infty \d(p^3) \, \f{p}{e^{p/T} - 1}\\ &= \f{8\pi}{[2\pi]^3} \int_0^\infty \d p \, \f{p^3}{e^{p/T} - 1}\\ &= \f{8\pi T^4}{[2\pi]^3} \int_0^\infty \d u \, \f{u^3}{e^{u} - 1}\\ &= \f{8\pi T^4}{[2\pi]^3} \, 3!\zeta(4)\\ &= \f{\pi^2}{15} T^4 \tag{2.75} \end{aligned}\] where the Riemann Zeta function \[\begin{aligned} \zeta(n) &\equiv \sum_{j=1}^\infty \f1{j^n} \\ &= \f1{[n-1]!} \int_0^\infty \d x \f{x^{n-1}}{e^x - 1}\\ &= \f1{[n-1]![1 - 2^{1-n}]} \int_0^\infty \d x \f{x^{n-1}}{e^x + 1}, \tag{C.29} \end{aligned}\] with $\zeta(4) = \pi^4/90$. This lets us calculate the present-day photon density parameter \[ \Omega_\gamma = 2.47 \times 10^{-5} h^{-2} \] from the present-day temperature of the CMB. This is very insignificant.
Because $\rho_{\text r} \propto a^{-4}$ (2.57), the temperature of photon radiation must scale as $a^{-1}$. Since $E_{\text r}$ also scales as $a^{-1}$ (2.31), the equilibrium photon distribution function is unaffected by the cosmic expansion. Note that in reality, there are small perturbations around this equilibrium, corresponding to anisotropies in the CMB.
Baryons (i.e. nucleons and electrons), on the other hand, come in many different phases (e.g. diffuse gas, ionized plasma, compact objects, etc.) throughout cosmic history, thus cannot be modeled by an equilibrium distribution. Instead, their density parameter is estimated with methods such as starcounting, mapping of the intergalactic medium, comparisons of light elements abundances, and measurements of baryon acoustic oscillations (BAOs). Elemental abundances and BAO measurements constrains the present-day baryon density parameter to \[ \Omega_{\text b} = 0.0222(5) h^{-2},\, 0.0225(3) h^{-2} \] respectively. Thus $\rho_{\text b} \simeq 0.05\rho_{\text{cr}}$, hinting a significant amount of non-baryonic matter.
TODO Dark matter: Anisopies in the CMB. Late universe distance-redshift. LSS.
TODO: neutrinos, dark energy
The Einstein equations (plural because each tensor entry was considered a separate “equation”) \[ G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T^{(\text m)}_{\mu\nu} \tag{3.1} \] relates Newton’s gravitational constant $G$, the matter energy-momentum tensor $T^{(\text m)}_{\mu\nu}$, the cosmological constant $\Lambda$, the metric tensor $g_{\mu\nu}$, and the Einstein tensor \[ G_{\mu\nu} \equiv R_{\mu\nu} - \f12 g_{\mu\nu}R, \tag{3.2} \] where \[ R_{\mu\nu} = \p_{\alpha} \Gamma^\alpha_{\mu\nu} - \p_{\nu} \Gamma^\alpha_{\mu\alpha} + \Gamma^\alpha_{\beta\alpha} \Gamma^\beta_{\mu\nu} - \Gamma^\alpha_{\beta\nu} \Gamma^\beta_{\mu\alpha}. \tag{3.3} \] is the Ricci tensor, and $R\equiv g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar.
We can reinterpret the cosmological constant as a “dark energy” by folding it into an overall energy-momentum tensor $T_{\mu\nu} \equiv T^{(\text m)}_{\mu\nu} + T^{(\Lambda)}_{\mu\nu}$, where the dark energy’s energy-momentum tensor \[ T^{(\Lambda)}_{\mu\nu} \equiv -\f{\Lambda}{8\pi G} g_{\mu\nu}. \] Under this interpretation, the Einstein equation simplifies to $G_{\mu\nu} = 8\pi G T_{\mu\nu}$. Furthermore, matching \[ {(T^{(\Lambda)})^\mu}_\nu = -g^{\mu\alpha} T^{(\Lambda)}_{\alpha\nu} = -\f{\Lambda}{8\pi G} \delta^\mu_\nu \] with (2.44) gives \[ \rho_\Lambda = -\mathcal{P}_\Lambda = \f{\Lambda}{8\pi G}, \tag{3.10} \] constraining the dark-energy equation of state $w_\Lambda$ to exactly $-1$. This notation will also facilitate the generalization of $\Lambda$ to a time-varying “quintessence”.
In the Euclidean FLRW metric, the Christoffel symbol $\Gamma^\mu_{\alpha\beta}$ is given by (2.25). Note that it is independent of the spatial cordinates ($\p_i$). The Euclidean FLRW Ricci tensor is thus \[\begin{aligned} R_{00} &= \p_{\alpha} \zero{\Gamma^\alpha_{00}} - \p_{0} \Gamma^\alpha_{0\alpha} + \Gamma^\alpha_{\beta\alpha} \zero{\Gamma^\beta_{00}} - \Gamma^\alpha_{\beta0} \Gamma^\beta_{0\alpha} \\ &= -\p_0\prn{ \zero{\Gamma^0_{00}} + \Gamma^a_{0a} } \\ &\quad - \brk{ \zero{\Gamma^0_{00}} \zero{\Gamma^0_{00}} + \zero{\Gamma^a_{00}} \zero{\Gamma^0_{0a}} + \zero{\Gamma^0_{b0}} \zero{\Gamma^b_{00}} + \Gamma^a_{b0} \Gamma^b_{0a} } \\ &= -\p_0\prn{\delta^a_a \f{\dot a}{a}} - \delta^a_b \f{\dot a}a \delta^b_a \f{\dot a}a \\ &= -3\brk{\frac{a\ddot a + \dot a^2 - \dot a^2}{a^2}} = -3\frac{\ddot a}{a}, \\ R_{0j} = R_{j0} &= \p_{\alpha} \Gamma^\alpha_{0j} - \zero{\p_{j} \Gamma^\alpha_{0\alpha}} + \Gamma^\alpha_{\beta\alpha} \Gamma^\beta_{0j} - \Gamma^\alpha_{\beta j} \Gamma^\beta_{0\alpha} \\ &= \brk{ \p_{0} \zero{\Gamma^0_{0j}} + \zero{\p_{a} \Gamma^a_{0j}} } \\ &\quad -\brk{ \zero{\Gamma^0_{00}} \zero{\Gamma^0_{0j}} + \Gamma^a_{0a} \zero{\Gamma^0_{0j}} + \zero{\Gamma^0_{b0}} \Gamma^b_{0j} + \zero{\Gamma^a_{ba}} \Gamma^b_{0j} } \\ &\quad -\brk{ \zero{\Gamma^0_{0j}} \zero{\Gamma^0_{00}} + \Gamma^a_{0j} \zero{\Gamma^0_{0a}} + \Gamma^0_{bj} \zero{\Gamma^b_{00}} + \zero{\Gamma^a_{bj}} \Gamma^b_{0a} } \\ &= 0, \\ R_{ij} &= \p_{\alpha} \Gamma^\alpha_{ij} - \zero{\p_{j} \Gamma^\alpha_{i\alpha}} + \Gamma^\alpha_{\beta\alpha} \Gamma^\beta_{ij} - \Gamma^\alpha_{\beta j} \Gamma^\beta_{i\alpha} \\ &= \brk{ \p_{0} \Gamma^0_{ij} + \zero{\p_{a} \Gamma^a_{ij}} } \\ &\quad +\brk{ \zero{\Gamma^0_{00}} \Gamma^0_{ij} + \Gamma^a_{0a} \Gamma^0_{ij} + \zero{\Gamma^0_{b0}} \zero{\Gamma^b_{ij}} + \zero{\Gamma^a_{ba}} \zero{\Gamma^b_{ij}} } \\ &\quad -\brk{ \zero{\Gamma^0_{0j}} \zero{\Gamma^0_{i0}} + \Gamma^a_{0j} \Gamma^0_{ia} + \Gamma^0_{bj} \Gamma^b_{i0} + \zero{\Gamma^a_{bj}} \zero{\Gamma^b_{ia}} } \\ &= \p_0\prn{ \delta_{ij} a\dot a } + \delta^a_a \f{\dot a}a \delta_{ij} a\dot a - \delta^a_j \f{\dot a}a \delta_{ia} a\dot a - \delta_{bj} a\dot a \delta^b_i \f{\dot a}a \\ &= \delta_{ij}\brk{a\ddot a + \dot a^2} + 3 \delta_{ij} \dot a^2 - \delta_{ij} \dot a^2 - \delta_{ij} \dot a^2 \\ &= \delta_{ij}\brk{ a\ddot a + 2\dot a^2 }. \\ \implies R_{\mu\nu} &\to \begin{pmatrix} -3\ddot a/a & 0 & 0 & 0\\ 0 & a\ddot a + 2\dot a^2 & 0 & 0\\ 0 & 0 & a\ddot a + 2\dot a^2 & 0\\ 0 & 0 & 0 & a\ddot a + 2\dot a^2 \end{pmatrix}. \tag{3.7} \end{aligned}\] With the corresponding Ricci scalar \[\begin{aligned} R &= g^{00} R_{00} + g^{ij}R_{ij} \\ &= 3\f{\ddot a}a + \f{\delta^{ij}}{a^2} \delta_{ij} \brk{a\ddot a + 2\dot a^2} \\ &= 6 \brk{\f{\ddot a}a + \brk{\f{\dot a}a}^2}. \end{aligned}\]
The $00$-component of the Einstein equation gives the first Friedmann equation \[\begin{aligned} G_{00} = R_{00} - \f12 g_{00} R &= 8\pi G g_{00} {T^0}_0 \\ \iff -3\f{\ddot a}{a} + 3\brk{\f{\ddot a}a + \brk{\f{\dot a}a}^2} &= 8\pi G\rho \\ \iff \brk{\f{\dot a}a}^2 &= \f{8\pi G}3 \rho, \tag{3.12} \end{aligned}\] and any $ii$-component gives the second Friedmann equation \[\begin{aligned} {G^1}_1 = g^{11} R_{11} - \f12 R &= 8\pi G {T^1}_1 \\ \iff \f{\ddot a}a + 2\brk{\f{\dot a}{a}}^2 - 3\brk{\f{\ddot a}a + \brk{\f{\dot a}a}^2} &= 8\pi G\mathcal{P} \\ \iff -2\f{\ddot a}a &= 8\pi G\mathcal{P} + \brk{\f{\dot a}a}^2 \\ \iff \f{\ddot a}a &= -\f{4\pi G}3 \brk{\rho + 3\mathcal{P}}. \tag{3.90} \end{aligned}\]
Written in terms of the Hubble rate and the density parameters, the first Friedmann equation becomes \[ \brk{\f{H(t)}{H_0}}^2 = \f{\rho(t)}{\rho_{cr}} = \sum_s a^{-3[1+w_s]} \Omega_s. \tag{3.13} \]
In a curved FLRW universe, there is an additional curvature parameter $\Omega_K \equiv 1 - \Omega(t_0)$ in the Friedmann equations. For instance, (3.13) becomes \[ \brk{\f{H(t)}{H_0}}^2 = a^{-2}\Omega_K + \sum_s a^{-3[1+w_s]} \Omega_s. \tag{3.14} \]
The physical trajectory of a particle can be computed from the time evolution of its physical momentum $\vec p$. We start with the dispersion relation, \[ P^2 \equiv g_{\mu\nu} P^\mu P^\nu = -E^2 + p^2 = -m^2, \tag{3.27} \] where we have identified $p \equiv \sqrt{g_{ij} P^i P^j}$ as the physical momentum magnitude, such that the physical momentum vector $\vec p = p\uvec p$ has components \[ p^i \equiv p\hat p^i = \sqrt{g_{ii}} P^i = aP^i. \tag{3.32} \] While the direction of motion $\uvec p$ is constant in a homogenous and isotropic universe, the momentum magnitude can vary over time owing to the background expansion. From the geodesic equation and the dispersion relation, we obtain \[\begin{aligned} \f{\d P^0}{\d\lambda} &= \f1{2 P^0} \f{\d(P^0 P^0)}{\d t} \f{\d t}{\d\lambda} = \f12\f{\d(E^2)}{\d t} \\ &= -\f12\f{\d(p^2)}{\d t} = -p\f{\d p}{\d t}, \\ \f{\d P^0}{\d\lambda} &= -\Gamma^0_{\alpha\beta} P^\alpha P^\beta = a\dot a\delta_{ij} P^i P^j \\ &= \f{\dot a}a g_{ij} P^i P^j = H p^2 \\ \implies \f{\d p}{\d t} &= -Hp. \tag{3.37} \end{aligned} \] Since \[ \f{\d x^i}{\d t} = \f{\d x^i}{\d\lambda} \f{\d\lambda}{\d t} = \f{P^i}{P^0} = \f{p^i}{aE}, \] the equations of motion are thus \[\begin{aligned} \f{\d\vec x}{\d t} &= \f{\d\vec x}{\d\lambda} \f{\d\lambda}{\d t} = \f{\vec p}{aE}, \\ \f{\d\vec p}{\d t} &= \f{\d p}{\d t}\uvec p = -H\vec p. \end{aligned}\]
When particle number is locally conserved in some phase-space distribution $f(\vec x, \vec p, t)$, the continuity equation $\d f / \d t = 0$ holds, where \[\begin{aligned} \f{\d}{\d t} &= \f{\p}{\p t} + \f{\d\vec x}{\d t}\cdot\f{\p}{\p\vec x} + \f{\d\vec p}{\d t}\cdot\f{\p}{\p\vec p} \\ &= \f{\p}{\p t} + \f{\d\vec x}{\d t}\cdot\f{\p}{\p\vec x} + \f{\d p}{\d t}\f{\p}{\p p} + \f{\d\uvec p}{\d t}\cdot\f{\p}{\p\uvec p} \\ \tag{3.17} \end{aligned}\] is the total time derivative. When there are particle-particle interactions that do not conserve particle number, we introduce a collision term \[ \f{\d f}{\d t} = C[f]. \tag{3.19} \] This is the most general form of the Boltzmann equation. Note that $C$, the collision functional, is a differential operator.
Although homogeneity and isotropy together means $\p f / \p\vec x = \p f/\p\uvec p = 0$, in order to facilitate later calculations of perturbation, we will still keep a small $\p f / \p \vec x$. In the FRLW metric for a slightly inhomogenous particle distribution, the Boltzmann equation thus simplifies to \[\begin{aligned} \f{\d f}{\d t} &= \f{\p f}{\p t} + \f{\d\vec x}{\d t}\cdot\f{\p f}{\p\vec x} + \f{\d p}{\d t}\f{\p f}{\p p} + \f{\d\uvec p}{\d t}\cdot\zero{\f{\p f}{\p\uvec p}} \\ &= \f{\p f}{\p t} + \f{p}{aE} [\uvec p\cdot\nabla]f - Hp \f{\p f}{\p p} = C^{(0)}[f]. \tag{3.38} \end{aligned}\] (Note that we mark this and the following collision terms with a superscript $^{(0)}$ to distinguish them from the collision terms in the next section, which incorporate with perturbations to the FLRW metric.)
In the relativistic limit ($p \gg m$), we have $E \simeq p$ and thus \[ \f{\p f}{\p t} + \f1a [\uvec p\cdot\nabla]f - Hp\f{\p f}{\p p} = C^{(0)}[f], \tag{3.39} \] while in the nonrelativistic limit ($p \ll m$), we have $E\simeq m$ and thus \[ \f{\p f}{\p t} + \f p{am} [\uvec p\cdot\nabla]f - Hp\f{\p f}{\p p} = C^{(0)}[f], \tag{3.40} \] with a very suppressed $\p f / \p\vec x$ term.
We will first approach the fully homogenous/nonrelativistic case ($\p f / \p\vec x = 0$). Integrating the Boltzmann equation over all physical momenta, we obtain \[\begin{aligned} &\quad \int \f{\d^3 p}{[2\pi]^3} \f{\p f}{\p t} - H \int \f{\d^3 p}{[2\pi]^3} p \f{\p f}{\p p} \\ &= \f{\p}{\p t}\int \f{\d^3 p}{[2\pi]^3} f - H \brk{ \int \f{\d^2 \hat p}{[2\pi]^3} p^3 f\Bigg|_{p=0}^{p=\infty} - 3\int \f{\d^3 p}{[2\pi]^3} f } \\ &= \f{\p n}{\p t} + 3 H n = \int \f{\d^3 p}{[2\pi]^3} C[f]. \tag{3.43} \end{aligned}\] In the absence of collisions $(C[f] = 0)$, this confirms the $n\propto a^{-3}$ relation from earlier. However, collisions can alter the dynamics of $n$.
A collision is an umbrella term for any sort of direct particle-particle interaction, including scattering, pair production, annihilation, and decay.
Consider, for instance, a reversible reaction \[ (1) + (2) \leftrightarrow (3) + (4) \tag{3.44} \] between species $1$, $2$, $3$, and $4$. Ignoring relativistic and quantum-mechanical corrections for the moment, the reaction’s net effect on the dynamics of species $1$ can be broken down into its effect on the set of possible species $1$ physical momenta \[ \f{\d f_1}{\d t} = C[f_1] = \int \f{\d^3 p_1}{[2\pi]^3} C[f_1(\vec p_1)] \, f_1(\vec p_1). \]
Now, for a given species $1$ physical momentum $\vec p_1$, the other species’ physical momenta $\vec p_2$, $\vec p_3'$, and $\vec p_4'$ are constrained by the the conservation of net comoving momentum \[ (P_1)^\alpha + (P_2)^\alpha = (P_3')^\alpha + (P_4')^\alpha, \] which is component-wise equivalent to \[\begin{aligned} \vec p_1 + \vec p_2 &= \vec p_3' + \vec p_4', \\ E_1(p_1) + E_2(p_2) &= E_3(p_3') + E_4(p_4'), \tag{3.45} \end{aligned}\] where $E_s(p) = \sqrt{p^2 + m_s^2}$ is the dispersion relation for species $s$. When integrating over all possible $\vec p_2$, $\vec p'_3$, and $\vec p'_4$, we will use Dirac delta functions to enforce these constraints. Note to satistify the normalization condition, every $n$-dimensional Dirac delta function $\delta_D^{(n)}$ carries with it a prefactor of $[2\pi]^n$. One can think of it as accounting for the uncertainty in the constraint.
Then, given these physically possible momenta, their reaction rate is controlled by the square of the scattering amplitude $\mathcal M(\vec p_1, \vec p_2, \vec p'_3, \vec p'_4)$, which is computed via Feynman diagrams. In the integrand, this is weighted by the probabilities: $-f_1(\vec p_1)\, f_2(\vec \p_2)$ for the forward reaction and $+f_3(\vec p'_4)\, f_4(\vec p'_4)$ for the reverse reaction, with the signs respectively corresponding to the annihilation and creation of species $1$.
Putting it all together, the collision term for a species $1$ particle traveling with momentum $\vec p_1$ is semi-classically \[\begin{aligned} C[f_1(\vec p_1)] &= \int\f{\d^3 p_2}{[2\pi]^3} \int\f{\d^3 p_3'}{[2\pi]^3} \int\f{\d^3 p_4'}{[2\pi]^3} \\ &\quad\cdot [2\pi]^3\, \delta_D^{(3)}(\vec p_3' + \vec p_4' - \vec p_1 - \vec p_2) \\ &\quad\cdot 2\pi\, \delta_D^{(1)}(E_3(p_3') + E_4(p_4') - E_1(p_1) - E_2(p_2)) \\ &\quad \cdot |\mathcal M(\vec p_1, \vec p_2, \vec p_3', \vec p_4')|^2 \\ &\quad\cdot \brk{ f_3(\vec p_3')\, f_4(\vec p_4') - f_1(\vec p_1)\, f_2(\vec p_2) }. \tag{3.46} \end{aligned}\] This derivation can be easily generalized for other types of collisions. However, it breaks down at high speeds and high densities, at which point a fully relativistic and quantum treatment is necessary.
In relativity, the space of momenta is four-dimensional. But because all real particles obey the dispersion relation $-m^2 = g_{\mu\nu} P^\mu P^\nu = -E^2 + p^2$, for any given rest mass $m$, the set of physically possible relativistic momenta is confined to a relativistically invariant, hyperbolic three-manifold in this phase space, known as the mass shell. (Real particles are thus said to be “on the mass-shell”.)
Because the fundamental elements of physical momentum space $\d^3 p / [2\pi]^3$ change volume under Lorentz transformations, the integrals in (3.46) are not relativistically invariant. To obtain an invariant integral measure, we have to work on the mass shell embedded in relativistic momentum space instead.
Since energy and time, like physical position and momentum, are canonical conjugates in quantum mechanics, one can repeat the demonstration in the Nonrelativistic distribution statistics section to show that in any time interval $\Delta t$, energy eigenvalues are separated by steps of $2\pi / \Delta t$. The fundamental elements of relativistic momentum space are thus \[ \f{\d^3 p\, \d E}{[2\pi]^4}. \] To constrain ourselves to the mass-shell of this space, we integrate over the dispersion relation, finding that the fundamental element of physical momentum space is really \[\begin{aligned} \f{\d^3 p}{[2\pi]^3}\Bigg|_\text{shell} &= \int_\text{shell} \f{\d^3 p\, \d E}{[2\pi]^4} \\ &= \f{\d^3 p}{[2\pi]^3} \int_0^\infty \f{\d E}{2\pi} \cdot 2\pi\, \delta^{(1)}_D(E^2 - p^2 - m^2) \\ &= \f{\d^3 p}{[2\pi]^3} \int_0^\infty \d E\, \f{\delta^{(1)}_D(E - \sqrt{p^2 + m^2})}{2E} = \frac{\d^3 p}{[2\pi]^3 \cdot 2E(p)}, \tag{3.47} \end{aligned}\] where the additional $2\pi$ factor in front of the Dirac delta again follows from the normalization condition, accounting the uncertainty in energy.
When product particles are concentrated at high densities ($f \gtrsim 1/e$, i.e., high likelihood of occupancy in some fundamental phase-space elements), their interparticle quantum effects strengthen, and the differing spin statistics of bosons and fermions start to significantly affect the reaction rates as well. For fermions, the Pauli exclusion principle suppresses reactions when there is a high density, adding a Pauli blocking factor of $[1-f_s]$ to the reaction that produces a fermonic species $s$ in (3.46). Bosonic reactions, on the other hand, are favored at high density (this is the cause of stimulated emissions), requiring a Bose enhancement factor of $[1+f_s]$ instead.
When corrected for relativistic and quantum effects, the collision factor is thus \[\begin{aligned} C[f_1(\vec p_1)] &= \f1{2E_1(p_1)} \int\f{\d^3 p_2}{[2\pi]^3 \cdot 2E_2(p_2)} \int\f{\d^3 p_3'}{[2\pi]^3 \cdot 2E_3(p_3')} \int\f{\d^3 p_4'}{[2\pi]^3 \cdot 2E_4(p_4')} \\ &\quad\cdot [2\pi]^3\, \delta_D^{(3)}(\vec p_3' + \vec p_4' - \vec p_1 - \vec p_2) \\ &\quad\cdot 2\pi\, \delta_D^{(1)}(E_3(p_3') + E_4(p_4') - E_1(p_1) - E_2(p_2)) \\ &\quad \cdot |\mathcal M(\vec p_1, \vec p_2, \vec p_3', \vec p_4')|^2 \\ &\quad\cdot \Big[ f_3(\vec p_3')\, f_4(\vec p_4')\, [1 \pm f_1(\vec p_1)] \, [1 \pm f_2(\vec p_2)] - f_1(\vec p_1)\, f_2(\vec p_2) \, [1 \pm f_3(\vec p_3')] \, [1 \pm f_4(\vec p_4')] \Big]. \tag{3.48} \end{aligned}\] This derivation can again be easily generalized to other types of collisions.
Small inhomogeneities and anisotropies in spacetime, either due to slightly uneven distributions of mass-energy, or due to gravitational waves in spacetime itself, are reflected in small perturbations $h_{\mu\nu}$ to the FLRW metric $g^{(0)}_{\mu\nu}$.
The perturbed metric $g_{\mu\nu} = g^{(0)}_{\mu\nu} + h_{\mu\nu}$ is still symmetric, so $h_{\mu\nu}$ carries $10$ independent components in total. They can be rewritten as a combination of two $3$-scalars (rank $0$), two $3$-vectors (rank $1$), and one $3$-tensor (rank $2$) via a scalar-vector-tensor decomposition. Custom constraints (gauge choices) can be imposed on the decomposition to reduce them back to $10$ degrees of freedom.
Under any three-dimensional spatial rotation with matrix $\hat R$, the scalar perturbations are invariant, the vector perturbations each transform as $\vec v \to \hat R\vec v$, and the tensor perturbation transforms as $\hat h \to \hat R \hat h \hat R^T$. There are few physical mechanisms that can generate vector perturbations, while scalar perturbations correspond to concentrated mass-energy, and tensor perturbations correspond to gravitational waves.
We will focus on scalar perturbations for now, and delay the discussion of tensor perturbations to Chapter 6. In the conformal Newtonian gauge, the scalar-perturbed FLRW metric is \[ g_{\mu\nu} \to \begin{pmatrix} -[1 + 2\Psi] & 0 & 0 & 0\\ 0 & a^2 [1 + 2\Phi] & 0 & 0\\ 0 & 0 & a^2 [1 + 2\Phi] & 0\\ 0 & 0 & 0 & a^2 [1 + 2\Phi] \end{pmatrix}, \tag{3.49} \] where $\Psi$ and $\Phi$ are functions of spacetime. With this choice of gauge, the temporal perturbation $\Psi$ corresponds to the Newtonian potential $-GM/r$, and governs the motion of all particles, while the spatial perturbation $\Phi$ corresponds to a local perturbation in the spatial curvature (or equivalently, in the scale factor) and predominantly affects the motion of relativistic particles. In general, $\Psi$ and $\Phi$ are tightly coupled to each other, and for static solutions, $\Phi = -\Psi$. Because they are typically small and slow-varying, with magnitudes within $10^{-4}$ in most of our universe, we will calculate anything involving them or their derivatives to linear order only.
To linear order, the inverse metric is \[ g^{\mu\nu} \to \begin{pmatrix} -[1 - 2\Psi] & 0 & 0 & 0\\ 0 & a^{-2} [1 - 2\Phi] & 0 & 0\\ 0 & 0 & a^{-2} [1 - 2\Phi] & 0\\ 0 & 0 & 0 & a^{-2} [1 - 2\Phi] \end{pmatrix}. \] The Christoffel symbol is thus \[\begin{aligned} \Gamma^0_{00} &= g^{00} \p_0 g_{00} \\ &= [1 - 2\Psi] \dot\Psi \simeq \dot\Psi, \\ \Gamma^0_{i0} = \Gamma^0_{0i} &= \f{g^{00}}2 \brk{ \zero{\p_0 g_{0i}} + \p_i g_{00} - \zero{\p_0 g_{0i}} } \\ &= [1 - 2\Psi] \p_i \Psi \simeq \p_i\Psi, \\ \Gamma^0_{ij} &= \f{g^{00}}2 \brk{ \zero{\p_i g_{j0}} + \zero{\p_j g_{0i}} - \p_0 g_{ij} } \\ &= \f{1 - 2\Psi}2 \p_0(a^2[1 + 2\Phi] \delta_{ij}) \\ &\simeq \brk{ a\dot a [1 + 2\Phi] + a^2 \dot\Phi -2a\dot a\Psi } \delta_{ij} \\ &= a^2 \brk{ H + 2H[\Phi - \Psi] + \dot\Phi } \delta_{ij}, \\ \Gamma^i_{00} &= \f{g^{ii}}2 \brk{ \zero{\p_0 g_{0i}} + \zero{\p_0 g_{i0}} - \p_i g_{00} } \\ &= \f{1 - 2\Phi}{a^2} \p_i \Psi \simeq \f{\p_i \Psi}{a^2}, \\ \Gamma^i_{j0} = \Gamma^i_{0j} &= \f{g^{ii}}2 \brk{ \p_0 g_{ji} + \zero{\p_j g_{0i}} - \zero{\p_i g_{0j}} } \\ &= \f{1 - 2\Phi}{a^2} \p_0(a^2[1 + 2\Phi] \delta_{ij}) \\ &\simeq a^{-2} \brk{ a\dot a [1 + 2\Phi] + a^2\dot\Phi - 2a\dot a\Phi } \delta_{ij} \\ &= [ H + \dot\Phi ]\delta_{ij}, \\ \Gamma^i_{jk} &= \f{g^{ii}}2 \brk{ \p_j g_{ik} + \p_k g_{ij} - \p_i g_{jk} } \\ &= [1 - 2\Phi] \brk{ \p_j\Phi\delta_{ik} + \p_k\Phi \delta_{ij} - \p_i\Phi \delta_{jk} } \\ &\simeq \brk{ \delta_{ik}\p_j + \delta_{ij}\p_k - \delta_{jk}\p_i }\Phi, \tag{3.56} \end{aligned}\] where $H = \dot a/a$ is the Hubble rate.
With the Christoffel symbol computed, we can determine the paths of geodesics in this peturbed spacetime.
For a particle with comoving momentum $P^\mu$, we again identify its physical momentum magnitude as $p = \sqrt{g_{ij} P^i P^j}$, and its energy as $E = \sqrt{ m^2 + p^2 }$. The spatial components of its comoving momentum can thus be written as \[ P^i = \f{p^i}{\sqrt{ g_{ii}}} \simeq \f{1 - \Phi}a p^i, \tag{3.60} \] where $p^i$ are the components of $\vec p = p\uvec p$. (And conversely, $p^i = a [1 + \Phi] P^i$.) Contracting this comoving momentum, \[ -m^2 = g_{\mu\nu} P^\mu P^\nu = -[1 + 2\Psi] [P^0]^2 + p^2, \tag{3.57} \] we also find its temporal component \[ P^0 = \f{E}{\sqrt{1 + 2\Psi}} \simeq [1 - \Psi] E. \tag{3.59} \] (And conversely, $1/P^0 \simeq [1 + \Psi]/E$.)
Recall from Energy and momentum of a particle that we can always choose a path parametrization $\lambda$ such that $P^\alpha = \d x^\alpha / \d\lambda$. Applying the geodesic equation then gives \[\begin{aligned} \f{\d P^i}{\d t} &= \f{\d\lambda}{\d t} \f{\d P^i}{\d\lambda} = -\f1{P^0} \Gamma^i_{\alpha\beta} P^\alpha P^\beta \\ &= -\brk{ \Gamma^i_{00} P^0 + 2\Gamma^i_{0j} P^j + \f1{P^0} \Gamma^i_{jk} P^j P^k } \\ &\simeq -\brk{ \f E{a^2} [1 - \Psi]\p_i\Psi + \f{2}a [H + \dot\Phi] [1 - \Phi] p^i + \f1E [1+\Psi]\brk{\f{1-\Phi}a}^2 \brk{ 2p^i p^k \p_k- \delta_{jk} p^j p^k \p_i}\Phi } \\ &\rightsquigarrow -\brk{ \f E{a^2} \nabla\Psi + \f{2}a [H + \dot\Phi - H\Phi] \vec p + \f1{a^2 E} \brk{ 2\vec p [\vec p\cdot\nabla] - p^2 \nabla }\Phi } \\ &= -\brk{ \f E{a^2} \nabla\Psi + \f{2}a [H + \dot\Phi - H\Phi] \vec p + \f{p^2}{a^2 E} \brk{ 2 \nabla_\parallel - \nabla }\Phi }, \tag{3.66} \end{aligned}\] where $\nabla_\parallel \equiv \uvec p[\uvec p\cdot\nabla] = p^i p^k \p_k / p^2$ gives the gradient vector in the direction of motion.
Since \[\begin{aligned} \f{\d \Phi}{\d t} &= \p_0\Phi + \f{\d\lambda}{\d t} \f{\d x^i}{\d\lambda} \p_i\Phi \\ &= \dot\Phi + \f{P^i}{P^0} \p_i\Phi \simeq \dot\Phi + \f1{aE} [\vec p\cdot\nabla] \Phi, \end{aligned}\] the equations of motion are thus \[\begin{aligned} \f{\d\vec x}{\d t} &\leftarrow \f{\d x^i}{\d\lambda} \f{\d \lambda}{\d t} = \f{P^i}{P^0} \rightsquigarrow \f1{aE} [1 - \Phi + \Psi] \vec p, \\ \f{\d \vec p}{\d t} &\leftarrow \f{\d }{\d t}\prn{ a[1+\Phi]P^i } \\ &= \brk{ \dot a[1 + \Phi] + a\dot\Phi + \f1E [\vec p\cdot\nabla]\Phi } P^i +a[1+\Phi] \f{\d P^i}{\d t} \\ &\rightsquigarrow [ H + \dot\Phi ]\vec p + \f{p^2}{aE} \nabla_\parallel\Phi -\brk{ \f Ea \nabla\Psi + 2[ H + \dot\Phi ]\vec p + \f{p^2}{aE} \brk{ 2\nabla_\parallel - \nabla }\Phi } \\ &= - \brk{ \f Ea \nabla\Psi + [H + \dot\Phi]\vec p - \f{p^2}{aE} \nabla_\perp\Phi }, \tag{3.69} \end{aligned}\] where $\nabla_\perp \equiv \nabla - \nabla_\parallel$ gives the gradient vector transverse to the direction of motion.
As corollaries, the time evolution of momentum magnitude is \[\begin{aligned} \f{\d p}{\d t} &= \uvec p\cdot\f{\d \vec p}{\d t} \\ &\simeq - \brk{ \f Ea [\uvec p\cdot\nabla]\Psi + [H + \dot\Phi]p - \f{p^2}{aE} [\zero{ \uvec p\cdot\nabla_\perp}] \Phi } \\ &= - \brk{ \f E{ap} [\uvec p\cdot\nabla]\Psi + H + \dot\Phi }p, \tag{3.71} \end{aligned}\] while the time evolution of momentum direction is \[\begin{aligned} \f{\d\uvec p}{\d t} &= \f{\d}{\d t} \f{\vec p}p = \f1p \brk{\f{\d\vec p}{\d t} - \uvec p\f{\d p}{\d t}} \\ &\simeq - \brk{ \f E{ap}\nabla\Psi + [H + \dot\Phi]\uvec p - \f{p}{aE} \nabla_\perp \Phi } + \brk{ \f E{ap} \nabla_\parallel\Psi + [H + \dot\Phi]\uvec p } \\ &= -\f1a \brk{ \f Ep \nabla_\perp\Psi - \f pE \nabla_\perp \Phi }. \tag{3.72} \end{aligned}\] Notice how the curvature in $\Psi$ deflects nonrelativistic particles ($p \ll E$) significantly more than relativistic particles ($p \simeq E$), while the curvature in $\Phi$ only significantly deflects relativistic particles. Because $\Phi \simeq -\Psi$, the contribution from $\Phi$ is responsible for radiation being deflected approximately twice as much by gravitational lensing as the Newtonian potential $\Psi$ alone would predict.
To linear order in the perturbations, the Boltzmann equation is \[\begin{aligned} \f{\d f}{\d t} &= \f{\p f}{\p t} + \f{\d\vec x}{\d t} \cdot \f{\p f}{\p\vec x} + \f{\d p}{\d t} \f{\p f}{\p p} + \zero{\f{\d\uvec p}{\d t} \cdot \f{\p f}{\p\uvec p}} \\ &= \f{\p f}{\p t} + \f{p}{aE} [\uvec p\cdot\nabla]f - \brk{ \f E{ap} [\uvec p\cdot\nabla]\Psi + H + \dot\Phi }p \f{\p f}{\p p} = C[f]. \end{aligned}\] (Note that although the distribution function may contain small anisotropies $\p f / \p\uvec p$, they are of the same order as the metric perturbations. The $[\d\uvec p/\d t]\cdot[\p f/\p\uvec p]$ term is thus quadratic order in overall perturbations, and consequently vanishes in our approximation.)
In the relativistic limit ($E\simeq p$), this reduces to \[ \f{\p f}{\p t} + \f1a [\uvec p\cdot\nabla]f - \brk{ \f 1a [\uvec p\cdot\nabla]\Psi + H + \dot\Phi } p \f{\p f}{\p p} = C[f], \tag{3.74} \] while in the non-relativistic limit ($E \simeq m$), this reduces to \[ \f{\p f}{\p t} + \f p{am} [\uvec p\cdot\nabla]f - \brk{ \f m{ap} [\uvec p\cdot\nabla]\Psi + H + \dot\Phi } p \f{\p f}{\p p} = C[f]. \tag{3.76} \]
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In order to develop a basic understanding of the cosmic microwave background, we will first consider the behavior of photons and electrons near the era of recombination. At that time, they are approximately in thermal equilibrium, with small inhomogeneities that we will treat with linear-order perturbation theory.
In thermal equilibrium without gravitational perturbations, photons follow the Bose-Einstein distribution (2.65) \[ f^{(0)}(p, t) = \f 1{ e^{p / T(t) } - 1 }, \tag{5.4} \] which satisfies the Boltzmann equation for relativistic particles in the unperturbed FLRW metric (3.39), \[ \begin{aligned} C^{(0)}[f^{(0)}] &= \f{ \p f^{(0)} }{ \p t } - H p \f{\p f^{(0)}}{\p p} \\ &= -\brk{ \f{\dot T}{T} + H } p \f{\p f^{(0)}}{\p p} = 0, \tag{5.5} \end{aligned} \] with a vanishing collision term due to the equilibrium condition.
With a perturbation $ \Theta(\vec x, \uvec p, t) \equiv \delta T(\vec x, \uvec p, t) / T(t) $ to the mean temperature, $|\Theta| \ll 1$, we introduce small inhomogeneities and anisotropies into the distribution. To linear order in $\Theta$, they are \[ \delta f(\vec x, p, \uvec p, t) \simeq \f{ \p f^{(0)} }{ \p T } \delta T = T \f{ \p f^{(0)} }{ \p T } \Theta = - p \f{ \p f^{(0)} }{ \p p } \Theta, \tag{5.3} \] where the last equality holds for any function of the ratio $p / T$.
The thermal perturbation $\Theta$ thus induces gravitational perturbations $\Phi$ and $\Psi$ of the same order. The perturbed distribution $f = f^{(0)} + \delta f$ must then satisfy the general Boltzmann equation for relativistic particles in the perturbed FLRW metric, \[ \begin{aligned} C[f] &\simeq \zero{ C^{(0)}[f^{(0)}] } + C^{(0)}[ \delta f ] + \delta C[ f^{(0)} ] \\ &= -\zero{ p\f{\p}{\p p} \prn{ C^{(0)}[f^{(0)}]} \Theta } - p \f{\p f^{(0)}}{\p p} C^{(0)}[ \Theta ] + \delta C[ f^{(0)} ] \\ &= -\brk{ \brk{ \f{\p}{\p t} + \f1a [\uvec p\cdot\nabla] - Hp \f{\p}{\p p} }\Theta + \f1a [\uvec p\cdot\nabla]\Psi + \dot\Phi } p \f{\p f^{(0)}}{\p p} \\ &\simeq \brk{ \dot\Theta + \f1a [\uvec p\cdot\nabla] (\Theta + \Psi) + \dot\Phi } p \f{\p f^{(0)}}{\p p}. \end{aligned} \tag{5.9} \] (Note that here, $C^{(0)}$ denotes the collision operator in the absence of $\Phi$ and $\Psi$ perturbations, and $f^{(0)}$ denotes the distribution function in the absence of $\Theta$ perturbations.)
Now perturbed from thermal equilibrium, the collision term no longer vanishes. In the low-energy regime near recombination, photons (with momentum $\vec p$ and energy $E = p$) predominantly interact through Thomson scattering with non-relativistic electrons (with momentum $\vec p_e$ and energy $E_e \simeq m_e$), \[\begin{aligned} e^- + \gamma &\leftrightarrow e^- + \gamma, \\ \vec p_e + \vec p &= \vec p_e' + \vec p'. \end{aligned}\] Thomson scattering is approximately elastic, preserving kinetic energies $p = p'$. (Incidentally, this is why we neglected a $\p\Theta/\p p$ dependency in the thermal perturbation.)
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