Title: Part II / Cosmology
Tags: Thermal The::Expanding::Universe Large::Scale::Structure] Large::Scale::Structure Relativistic::Cosmology Introduction The::Accelerating::Universe

Q: What is the Cosmological Principle?
A: On the largest scales, the universe is spatially homogeneous and isotropic.
Homogeneity - the universe looks identical at every point in space
Isotropy - the universe looks the same in every direction

Q: Definition. The scale factor is defined by \[[...]\]
A: \frac{a(t)}{a(t_0)} = \exp(\int_{t_0}^t H(t') dt')
Back Extra: So we have \[H = \frac{\dot{a}}{a}\]

Q: Definition. The peculiar velocity \(\mathbf{v}_{pec}\) is [...]
A: the velocity relative to the comoving frame
Back Extra: \[\mathbf{v} = \dot{a}\mathbf{x} + a \dot{\mathbf{x}} = H \mathbf{r} + \mathbf{v}_{pec}\]

Q: Definition. The redshift of a galaxy's spectral line is given by \[[...]\]
A: z = \frac{\lambda_o}{\lambda_e} - 1
Back Extra: where \(\lambda_o , \lambda_e\) are the obsereved and emitted wavelengths

Q: Given a mass M in a volume V enclosed by a surface S, what is Gauss' Law for gravity
A: \[M = \int_V d^3 \mathbf{r} \frac{\rho(\mathbf{r})}{c^2} = \frac{1}{4 \pi G} \int d^3 \mathbf{r} \nabla^2 \Phi = \frac{1}{4\pi G} \int_S \mathbf{\nabla }\Phi \cdot d\mathbf{S}\]

Q: What is Newton's First Theorem?
A: The gravitational force outside a spherically symmetric mass distribution radius R is the same as if all the mass were concentrated at its centre \(\mathbf{r} = 0\).

Q: What is Newton's second Theorem?
A: Spherical shells at \(r>R\) exert no net gravitational force at \(r = R\)

Q: What is Birkhoff's Theorem?
A: The graviational force at r=R from a spherically symmetric mass distribution is due entirely to the mass inside \(r<R\).

Q: What is the Friedmann equation?
A: \[\left ( \frac{\dot{a}}{a} \right )^2 + \frac{kc^2}{a^2} = \frac{8\pi G}{3c^2} \rho\]

Q: What is the density parameter?
A: \[\Omega(t) = \frac{\rho(t)}{\rho_{\text{crit}}(t)}\]where \[\rho_{\text{crit}}(t) = \frac{3c^2}{8\pi G} H^2\]is the critical density for k = 0 in the Friedmann equation

Q: What is the continuity equation?
A: \[\dot{\rho} = -3 \frac{\dot{a}}{a} (\rho + P)\]where the \(\rho\) is like a dilution term and the \(P\) is a work against expansion

Q: What is the Einstein-de Sitter relation?
A: \[a = \left (\frac{t}{t_0} \right )^{2/3}\]for \(P =0, k = 0\) so a pressure-free matter constant at critical density universe

Q: What is the Raychauduri equation?
A: \[\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^2}(\rho + 3P)\]which comes from differentiating the Friedmann equation and plugging in the continuity equation

Q: What is the general metric for the FLRW spacetime?
A: \[ds^2 = -c^2 dt^2 + a(t)^2 \left [ \frac{dr^2}{1 - kr^2} + r^2 (d\theta^2 + \sin^2 \theta \;d\phi^2)\right ]\]where r is now a comoving radial coordinate and \(k = K/a^2 = 1 / (aR_0)^2\) is the comoving curvature with comoving radius of curvature \(R_0\), so we have \[k = \begin{cases} +1/R_0^2, & S^3(\text{close}) \\ 0, & R^3(\text{flat}) \\ -1/R_0^2, & H^3(\text{open})\end{cases}\]

Q: What is the cosmological horizon?
A: It is the size of the observed universe. Since the Big Bang at t = 0, a photon would have travelled a comoving distance \[\int_0^t \frac{cdt'}{a(t')} = \int_0^r \frac{dr'}{\sqrt{1-kr'^2}}\]or an actual physical distance of \[d_H(t_0) = a(t_0) \int_0^{t_0} \frac{cdt'}{a(t')}\]

Q: What is the Singularity Theorem?
A: In an expanding universe with matter satisying the strong energy condition \(\rho + 3P \ge 0\) there must have been a singularity \((a \to 0 , \rho \to \infty)\) at fninte t in the past \(|t_0 - t| \le H_0^{-1}\)
We can rewrite the acceleration equation as \[\dot{H} + H^2 = \frac{-4\pi G}{3c^2}(\rho + 3P) \leq 0\]which gives \(\frac{\dot{H}}{H^2} \leq -1\) which we can integrate twice to eventually get \(a(t) \leq 1 + H_0 (t - t_0)\) which implies there was a singularity.

Q: What is the scale factor solution for the Radiation Model(\(P = \rho /3, k = 0\))
A: \[a(t) = (t/t_0)^{1/2}\]

Q: What are the Hubble parameter and acceleration in conformal time?
A: \[\tau = \int_0^t \frac{dt'}{a(t')}\]\[\dot{a} = \frac{da}{dt}, a' = \frac{da}{d\tau}\]and \[H \equiv \frac{\dot{a}}{a}, \mathcal{H} \equiv \frac{a'}{a}\]\[\ddot{a} = \frac{d\tau}{dt} \frac{d}{d\tau}(\frac{a'}{a})=\frac{1}{a}\mathcal{H}'\]

Q: What do you add to the Friedmann and acceleration equaltion with the cosmological constant.
A: \[+\frac{1}{3}\Lambda c^2\]

Q: When we include \(\Lambda\) in our equations for the universe and using \(\Omega_{\Lambda_0} = \frac{\Lambda c^2}{3H_0^2}, \Omega_{k_0} = \frac{-kc^2}{H_0^2}\) what do the Friedmann and acceleration equations become.
A: \[\left ( \frac{\dot{a}}{a}\right )^2 = H^2 = H_0^2(\frac{\Omega_{M_0}}{a^3} + \frac{\Omega_{R_0}}{a^4} + \Omega_{\Lambda_0} + \frac{\Omega_{kc}}{a^2})\]\[\left ( \frac{\ddot{a}}{a} \right) = -\frac{1}{2} H_0^2 (\frac{\Omega_{M_0}}{a^3} + \frac{2 \Omega_{R_0}}{a^4} - 2\Omega_{\Lambda_0})\]

Q: What is the scale factor for a flat universe filled with \(\Lambda\) and NR matter?
A: \[a(t) = \left (\frac{\Omega_{M_0}}{\Omega_{\Lambda_0}} \right )^{1/3}\sinh^{2/3}(t/t_{\Lambda})\]where \(t_{\Lambda} = \frac{2}{3}H_0^{-1}/\sqrt{\Omega_{\Lambda_0}}\)

Q: How do we derive the formula for luminosity distance which we can use as evidence for dark energy?
A: We have \[1+z = 1 - H_0(t- t_0) + (1+\frac{q_0}{2})H_0^2(t-t_0)^2+\dots\]from Taylor expanding \(a(t) = \frac{1}{1+z}\)
Then integrating \(r = c\int_t^{t_0}\frac{dt'}{a(t')} = c(t_0 - t) + \frac{cH_0}{2}(t-t_0)^2 + \dots\)
Now revert to find \(t_0 - t = \frac{z}{H_0}(1-(1+\frac{q_0}{2})z + \dots)\)
Substituting gives \(r = \frac{cz}{H_0}(1 - \frac{1}{2}(1+q_0)z + \dots)\)
Hence we obtain the measurable Hubble law \[d_L = (1+z)r = \frac{cz}{H_0}(1+(1-q_0)z + \dots)\]

Q: What is the Virial theorem?
A: A bound gravitational system of massive particles in equilibrium obeys the relation \[2\langle T \rangle = - \langle V \rangle\]where \(\langle T \rangle\) is the time-average of the total KE and \(\langle V\rangle\) of the gravitational potential energy

Q: In thermal equilibrium at temperature T, the probability that the system sits in the state \(\ket{n}\) is given by the Boltzmann distribution, what is it?
A: \[p(n) = \frac{e^{-E_n/k_B T}}{Z}\]Where \(k_B\) is the Boltzmann constant and Z is normalisation constant.

Q: What is the Maxwell-Boltzmann distribution and what does it tell us?
A: \[f(v)dv = 4 \pi v^2 \left ( \frac{m}{2\pi k_B R} \right )^{3/2}e^{-mv^2 / 2k_B T}dv\]it tells us the distribution of the speeds of gas molecules.

Q: What is the equation of state for an ideal gas?
A: \[P = nk_B T\]

Q: Definition. An isolated system is in equilibrium if [...]
A: its state variables do not change in time.

Q: What are the laws of thermodynamics?
A: 0th: Two systems in equilibrium with each other have the same temperature, pressure and chemical potential
1st: Energy is conserved: \(dE = TdS - PdV + \mu dN(\text{ heat - mechanical + chemical})\)
2nd: For an isolated system, entropy change is always non-negative.

Q: What is entropy defined as?
A: \[S(E,V,N) = k \log \Omega(E,V,N)\]where \(\Omega\) is the number of microstates that yield the given mactostate with \(E,V,N\).

Q: What is the density of states and how does it relate to the average number of particles in the range \(p \to p + dp\)? What are the Bose-Einstein and Fermi-Dirax distributions?
A: \[f(p) = \frac{4 \pi V}{h^3} p^2\]\[\overline{n}(p)dp = \frac{4 \pi V g_s}{h^3} \frac{p^2 dp}{e^{\beta(E(p) - \mu)} \mp 1}\]with - for bosons and + for fermions

Q: What is Ehrenfest's principle?
A: The probability for a transition between states approaches zero for an infinitely slowly varying perturbation

Q: What is recombination?
A: \[p + e^- \leftrightarrow H + \gamma\]

Q: What is Saha's equation?
A: \[n_e^2 = n_H \left ( \frac{2 \pi m_e k T}{h^2}\right )^{3/2} e^{-I/kT}\]where \(I = (m_p + m_e - m_H)c^2\) is the ionisation energy of hydrogen

Q: Definition. We describe inhomogeneity with the density perturbation \(\delta(\mathbf{r},t)\) \[[...]\]
A: \delta(\mathbf{r},t) = \frac{\rho(\mathbf{r},t) - \overline{\rho}(t)}{\overline{\rho}(t)}
Back Extra: where \(\overline{\rho}(\mathbf{r},t) = \langle \rho(\mathbf{r},t)\rangle = V^{-1}\int \rho(\mathbf{r}, t) d^3 r\)

Q: Definition. The clustering of galaxies is described by the two-point correlation function \[[...]\]where \(r = |\mathbf{r}|\)
A: \xi(r) = \langle \delta(\mathbf{r}', t), \delta(\mathbf{r + r'}, t) \rangle = \frac{1}{\overline{\rho}(t)^2} \langle \rho(\mathbf{r}', t)\rho(\mathbf{r + r'}, t) \rangle - 1
Back Extra: Can be interpreted as the probability in excess of random of finding one galaxy in volume \(\delta V_1\) and another in volume \(\delta V_2\) with the volumes seperated by distance r.
We have \(\xi (\mathbf{r}) = \xi (r)\) as we assume that \(\xi\) should be independent of \(\mathbf{r'}\) by statistical homogeneity and independent of the direction by statistical isotropy.

Q: Definition. The variance of the density perturbation is given by \[[...]\]In practice we define the variance with respect to a length scale R as \[[...]\]
A: \sigma^2 = \langle \delta(\mathbf{r},t)^2 \rangle = \xi(r = 0); \sigma_R^2 = \langle \delta_R (\mathbf{r},t)^2 \rangle
Back Extra: where we have smoothed the density perturbation using a window function with a cut off.

Q: Definition. The power spectrum \(P(k)\) is the Fourier transform of the two-point correlation function is defined by \[[...]\]
A: P(k) = \int d^3 \mathbf{x} \xi(r) e^{- \mathbf{k \cdot x} } = 4 \pi \int r^2 dr \frac{\sin kr}{kr} \xi(k)
Back Extra: It can be interepreted as a decomposition of the density perturbation into different Fourier modes which tells us the power on a given physical lengthscale.

Q: What are the continuity(energy conservation) and Euler(momentum conservation) equations?
A: \[\delta' + \nabla_{\mathbf{x}} \cdot [\mathbf{v}(1 + \delta)] = 0\]\[\mathbf{v}' + \mathcal{H}\mathbf{v} + (\mathbf{v} \cdot \nabla_{\mathbf{x}})\mathbf{v} = -\nabla_{\mathbf{x}}\phi - \frac{1}{\rho} \nabla_{\mathbf{x}}P\]

Q: What are the linear density perturbation equations in conformal and cosmic time?
A: \[\delta''(\mathbf{x}, \tau) + \mathcal{H}(\tau) \delta'(\mathbf{x},\tau) - \frac{3}{2}\Omega_M (\tau) \mathcal{H}(\tau)^2 \delta(\mathbf{x},\tau) = 0\]\[\ddot{\delta}(\mathbf{x},t) + 2H(t) \dot{\delta}(\mathbf{x},t) - \frac{4 \pi G}{c^2} \overline{\rho}_M(t)\delta(\mathbf{x},t) = 0\]

How does the evolution of energy densities in our universe look?

Q: What evidence is there for dark energy and dark matter?
A: - Acceleration of expansion of the universe, type 1A supernovae evidence
- Using Virial theorem galaxies rotate much faster than expected assuming there is no dark matter

Q: What are the fluids equations?
A: - Continuity equation: \(\frac{\partial n}{\partial t} + \nabla \cdot (n \mathbf{u}) = 0\)
- Euler equation: \(mn(\frac{\partial }{\partial t} + \mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla P\)
- Equation of state: \(P = P(n,T)\)

Q: What are the Jeans length and Jeans wavenumber, and how are they defined?
A: The Jeans wavenumber \( k_J \) and Jeans length \( \lambda_J \) determine the threshold at which gravitational effects dominate over pressure forces in a fluid. \[ k_J = \sqrt{\frac{4\pi G m \bar{n}}{c_s^2}} \] \[ \lambda_J = \frac{2\pi}{k_J} = c_s \sqrt{\frac{\pi}{G m \bar{n} }} \] If a perturbation's size is larger than \( \lambda_J \), gravitational collapse occurs instead of oscillations.

Q: What is the transfer function in cosmology?
A: The transfer function \( T(k) \) describes the evolution of density perturbations from an initial time \( t_i \) to the present, defined as: \[ \delta(k, t_0) = T(k) \delta(k, t_i) \] It accounts for the effects of horizon entry during radiation or matter-dominated eras.

Q: What are adiabatic and Gaussian perturbations?
A: - Adiabatic Perturbations: All components of the universe fluctuate in phase, maintaining constant entropy per particle.
- Gaussian Perturbations:The initial perturbations follow a Gaussian probability distribution, meaning they are statistically characterized by their power spectrum.

Q: What is the Harrison-Zeldovich power spectrum?
A: It is the scale-invariant spectrum with spectral index \( n = 1 \), meaning the dimensionless power spectrum \( \Delta_{\Phi}(k) \) remains constant. It suggests initial density perturbations with equal power across scales.

Q: What is the observed power spectrum of the universe?
A: The observed power spectrum \( P(k) \) evolves from the initial conditions due to the transfer function and is influenced by processes such as baryonic acoustic oscillations (BAOs) and dark matter evolution.

Q: What are baryonic acoustic oscillations?
A: BAOs are periodic fluctuations in the density of baryonic matter in the universe, resulting from sound waves in the photon-baryon fluid before recombination. They leave an imprint on the large-scale structure at a characteristic scale of \(\sim 150 \text{ Mpc}\).

Q: What is blackbody radiation and how does it relate to the cosmic microwave background (CMB)?
A: Blackbody radiation is the thermal electromagnetic radiation emitted by an idealized object in equilibrium at temperature \( T \). The CMB is the remnant radiation from the early universe, with a nearly perfect blackbody spectrum at temperature \( T_{\text{CMB}} \approx 2.73 \) K, representing the universe's thermal history.

Q: What are ultra-relativistic particles and how is their energy related to their momentum?
A: Ultra-relativistic particles are those with energy \( E \approx pc \), where the rest mass is negligible compared to the kinetic energy. This condition occurs when \( k_B T \gg mc^2 \), meaning the thermal energy dominates the rest mass energy.

Q: How does the temperature of the universe evolve with time?
A: The temperature of the universe evolves as \( T \propto \frac{1}{a(t)} \), where \( a(t) \) is the scale factor. As the universe expands, the temperature decreases, maintaining a blackbody spectrum.