Cosmology

Prerequisites: General Relativity. Full path (16 pages)

• Intuitive
• Concrete
• Abstract
• FAQ

Intuitive

Cosmology studies the universe as a whole: its large-scale structure, its history and what it contains. The basic observation is that distant galaxies move away from us, and the farther away they are, the faster they recede. The natural reading is that space itself is expanding, so that the distances between all galaxies grow with time. A common picture is a balloon with dots painted on it. When the balloon is inflated every dot sees every other dot move away, and no dot is the center.

Running the expansion backwards means that the universe was once much denser and hotter. In that early phase it was a hot soup of particles and light. As it expanded it cooled, nuclei of light elements formed in the first minutes, and after about 380,000 years it became cool enough for electrons and nuclei to form neutral atoms. From then on light could travel freely. We still see that light today, stretched by the expansion into microwaves: the cosmic microwave background.

On the largest scales the universe looks the same in every direction and at every place. This makes the description simple: the whole geometry is captured by a single function of time, the scale factor, which says how much distances have grown. General relativity relates its evolution to the matter and energy content. Fitting this description to observations gives the composition of the universe today. Ordinary matter is only about five percent of the total energy density. The rest is dark matter, which clumps but does not shine, and dark energy, which makes the expansion accelerate. The small irregularities in the early universe grew under gravity into the galaxies and clusters we see today.

Concrete

• The expanding universe by Markus Pössel; "An introduction to the physics and mathematics of the expanding universe, using no more than high-school level / undergraduate mathematics. "

Recommended Textbooks:

• The best introductions are Relativity, Gravitation and Cosmology by Ta-Pei Cheng and
• The Early Universe by Edward Kolb and Michael S. Turner
• Edward R. Harrison, Cosmology, the Science of the Universe

• M. Berry, Cosmology and Gravitation

• Another nice introduction to the CMB and related effects is A Poor Man’s CMB Primer and Student Friendly Guide to the Cosmic Microwave Background by Robert D. Klauber

Abstract

We use signature $(-,+,+,+)$ and $c = 1$. The cosmological principle is the assumption that, on large scales, spacetime admits a foliation by spacelike hypersurfaces $\Sigma_t$ that are homogeneous and isotropic. Then spacetime is (locally) $M = \mathbb R \times \Sigma$, each $\Sigma_t$ is a maximally symmetric Riemannian 3-manifold with a 6-dimensional isometry group, and the metric takes the Friedmann-Lemaître-Robertson-Walker (FLRW) form $$ds^2 = -dt^2 + a(t)^2\left[\frac{dr^2}{1 - k r^2} + r^2 d\Omega^2\right] , \qquad k \in \{-1, 0, +1\} .$$ The universal covers of the spatial slices are $H^3$, $\mathbb R^3$ and $S^3$, with isometry groups $SO(3,1)$, $E(3)$ and $SO(4)$ (up to discrete factors). The global topology of $\Sigma$ is not fixed by these local symmetries, since quotients by discrete subgroups of the isometry group are allowed.

Symmetry forces the energy-momentum tensor into perfect-fluid form, $T_{\mu\nu} = (\rho + p)u_\mu u_\nu + p\, g_{\mu\nu}$ with $u = \partial_t$, and the Einstein equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu}$ reduce to two ordinary differential equations, the Friedmann equations $$H^2 \equiv \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}, \qquad \frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho + 3p) + \frac{\Lambda}{3} .$$ The contracted Bianchi identity gives the continuity equation $\dot\rho + 3H(\rho + p) = 0$, so a component with $p = w\rho$ and constant $w$ scales as $\rho \propto a^{-3(1+w)}$.

The real universe is FLRW only on average. Cosmological perturbation theory writes $g = \bar g + \delta g$ and expands the Einstein equations to linear order. Because the background is invariant under spatial rotations and translations, the perturbations decompose into scalar, vector and tensor parts with respect to the spatial isometry group, and the three sectors decouple at linear order. Infinitesimal diffeomorphisms act on $\delta g$ as gauge transformations, $\delta g \to \delta g + \mathcal L_\xi \bar g$, so physical statements are made with gauge-invariant combinations. In the conformal Newtonian gauge, with conformal time $\eta$ and $k = 0$, the scalar sector reads $$ds^2 = a(\eta)^2\left[-(1 + 2\Psi)\,d\eta^2 + (1 - 2\Phi)\,\delta_{ij}\,dx^i dx^j\right] ,$$ and $\Phi = \Psi$ in the absence of anisotropic stress. A further gauge-invariant variable, the comoving curvature perturbation $\mathcal R$ (closely related to $\zeta$), is conserved on scales larger than the Hubble radius for adiabatic perturbations. This conservation is what lets predictions from inflation, usually stated as a nearly scale-invariant power spectrum of $\mathcal R$, be connected to the anisotropies of the cosmic microwave background and to large-scale structure.

Recommended Books:

• John A. Peacock, Cosmological Physics

FAQ

Many questions that frequently pop up in cosmology are answered here and here

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Manifold
5. Newton's Second Law
6. Vector Calculus
7. Curvature
8. Fiber Bundles
9. Newton's law of Gravity
10. Newtonian Mechanics
11. Parallel Transport
12. Connections
13. Special Relativity
14. Levi-Civita Connection
15. Einstein Equation
16. General Relativity

Leads to: Dark Matter, Standard Model of Cosmology.