Cosmology

All nonzero Christoffel symbols are listed.

\[ \Gamma^r_{tr}=\Gamma^r_{rt}=\frac{1}{c}\frac{\dot a}{a},\qquad \Gamma^\theta_{t\theta}=\Gamma^\theta_{\theta t}=\frac{1}{c}\frac{\dot a}{a},\qquad \Gamma^\phi_{t\phi}=\Gamma^\phi_{\phi t}=\frac{1}{c}\frac{\dot a}{a} \]

\[ \Gamma^t_{rr}=\frac{1}{c}\frac{a\dot a}{1-kr^2},\qquad \Gamma^t_{\theta\theta}=\frac{1}{c}a\dot a\, r^2,\qquad \Gamma^t_{\phi\phi}=\frac{1}{c}a\dot a\, r^2\sin^2\theta \] \[ \Gamma^r_{rr}=\frac{kr}{1-kr^2},\qquad \Gamma^r_{\theta\theta}=-r(1-kr^2),\qquad \Gamma^r_{\phi\phi}=-r(1-kr^2)\sin^2\theta \] \[ \Gamma^\theta_{\phi\phi}=-\sin\theta\cos\theta \]

\[ \Gamma^\theta_{r\theta}=\Gamma^\theta_{\theta r}=\frac{1}{r},\qquad \Gamma^\phi_{r\phi}=\Gamma^\phi_{\phi r}=\frac{1}{r} \] \[ \Gamma^\phi_{\theta\phi}=\Gamma^\phi_{\phi\theta}=\cot\theta \]

Ricci tensor is diagonal tensor.

\[ R_{tt} = -\frac{3}{c^2}\frac{\ddot a}{a} \] \[ R_{rr} = \frac{1}{c^2}\frac{a\ddot a + 2\dot a^2 + 2kc^2}{1-kr^2} \] \[ R_{\theta\theta} = \frac{1}{c^2}r^2\left(a\ddot a + 2\dot a^2 + 2kc^2\right) \] \[ R_{\phi\phi} = \frac{1}{c^2}r^2\sin^2\theta\left(a\ddot a + 2\dot a^2 + 2kc^2\right) \]

The spatial components can also be written as \[ R_{ii} = \frac{1}{c^2}\left(a\ddot a + 2\dot a^2 + 2kc^2\right) \gamma_{ii} \] where \(\gamma_{ii} = - g_{ii}/a(t)^2\).

They add up to Ricci scalar \[ R = - \frac{6}{c^2} \frac{ a\ddot a + \dot a^2 + kc^2}{a^2}. \]

Worldline along constant space coordinates \(r, \theta, \phi\), is geodesic in FLRW metric. For this reason FLRW coordinates \(ct, r, \theta, \phi\) are called comoving coordinates.

The redshift as observed by comoving observer in cosmic rest frame depends on the scale factor: \[ \frac{f_o}{f_e} = \frac{a(t_e)}{a(t_o)} \] where \(f_o\) and \(t_o\) are the frequency and the time when the light is observed, and \(f_e\) and \(t_e\) are the frequency and the time when the light was emitted.

Quantity \(z\) is also defined: \[ 1+z := \frac{a(t_e)}{a(t_o)}. \]

Boundary beyond which galaxies are moving away from Earth faster than light. \[ L_{H}(t) = \frac{c}{H(t)}. \] \(L\) indicates that it is proper length.

\(L_{Hh} \approx 14\, \mathrm{Gly}\)

The maximum comoving distance \(\Delta r\) a light beam emitted from Earth can travel. \[ r_{eh} = \int_{t_{\text{now}}}^{\infty} \frac{1}{a(t)} \dd{ct}. \] The proper length is then \(L_{eh} = a(t) r_{eh}\).

If Hubble parameter \(H(t)\) is constant for all time, Hubble horizon and event horizon are the same.

\(L_{eh} \approx 16\, \mathrm{Gly}\)

The maximum distance \(\Delta r\) a light beam has travelled since the beginning of the univrse. The region within particle horizon is called the observable universe. \[ r_{ph} = \int_0^{t_{\text{now}}} \frac{1}{a(t)} \dd{ct}. \]

\(L_{ph} \approx 45\, \mathrm{Gly}\)

The simplest equation of state is chosen for the matter and radiation in the universe: \[ p = w\rho c^2 \] assuming that it is barotropic fluid.

Matter and energy are assumed to be perfect fluid. If \(w = 0\), we get a energy-momentum tensor for dust-like matter \[ T^{\mu}{}_{\nu} = \begin{bmatrix} \rho c^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}. \]

The energy-momentum tensor for electromagnetic radiation is given by \[ T^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\alpha}F^{\nu}{}_{\alpha} - \frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta} \right). \] This tensor has zero trace, therefore we require \(w = 1/3\), in order to describe electromagnetic radiation.

If \(w=-1\) we find that \[ T_{\mu\nu} = \rho c^2 g_{\mu\nu} \] which resembles the cosmological constant term \(\Lambda g_{\mu\nu}\) in Einstein field equation, that describing the thing that accelerates the expansion of the universe.

The energy density \(\rho_c\) at which the curvature \(k = 0\).

We can express current energy density \(\rho\) in terms of critical density using density parameter \(\Omega_{\rm M/R}\): \[ \Omega_{\rm M/R} := \frac{\rho}{\rho_c}. \] \(\rm M/R\) subscript means that the energy density is considering the matter and radiation.

Critical density in flat universe \(k=0\) with no cosmological constant \(\Lambda=0\): \[ \rho_c = \frac{3H^2}{8\pi G}, \] where \(H\) is the Hubble parameter.

The first Friedmann equation can be rewritten as \[ 1 = \frac{8\pi G \rho}{3 H^2} + \frac{\Lambda c^2}{3 H^2} - \frac{k c^2}{a^2 H^2} \] in which we can define each term to be density parameter \(\Omega_{\rm M/R}\), dark energy parameter \(\Omega_{\Lambda}\), and curvature parameter \(\Omega_{\rm k}\): \[ 1 = \Omega_{\rm M/R} + \Omega_{\Lambda} + \Omega_{\rm k}. \]

From the 00-component of the Einstein field equation:

From the trace of the Einstein field equation:

and replacing with the first Friedmann equation, \[ \implies \frac{\ddot{a}}{a} = \frac{1}{3} \Lambda c^2 - \frac{4\pi G}{3} \left( \rho + \frac{p}{c^2} \right). \]

• The name is not widely accepted

Obtained by taking derivative of first Friedmann equation, and substituting the second Friedmann equation in. \[ \dot{\rho} = -3 \frac{\dot{a}}{a} \left( \rho + \frac{p}{c^2}\right). \]

The same equation can be obtained by assuming that the universe expands adiabatically, or using the local conservation of energy-momentum.

The energy density can be expressed in terms of the scaling factor, by combining equation of state with the third Friedmann equation: \[ \rho(a) = K a^{-3(1+w)}. \]

• SIDM

Hypothesis that dark matter interact with itself, in addition to gravity.

• MOND

Hypothesis that the gravity changes its strength based on length scale.

It is two colliding clusters of galaxies, where the distribution mass obtained based on the gravitational lensing is outpacing the movement of the intergalactic gas which are colliding in the interface of the collision and stagnating behind.

It is known that the majority of the mass comes from the gas. But, the mass distribution aligns with the distribution of galaxies which did not interact with each other much. This suggest the existence of non-interacting matter.

The cosmological constant \(\Lambda\) in the Einstein field equation is may be the result of dark energy, as the equation can be rewritten in trace-reversed form that indicates \(\Lambda\) can be some kind of source of curvature: \[ R_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu} - \frac{1}{2}\left( T + \frac{c^4}{4\pi G}\Lambda\right) g_{\mu\nu} \right). \]