← Back to CoursesCosmology: PhD Level

Neuroanatomy Explorer

Drag to rotate · scroll to zoom · click regions to explore

View
Loading 3D model…

Click a region
to explore it

Memory Deck

Flip each card and rate whether you knew it. Your score is saved.

Term
Definition

Deck complete — score saved.

Match the Pairs

Match each term to its definition. Finish the board to earn your score.

All matched — score saved.

Concept Constellation

Every key idea in this course, mapped as an explorable 3D constellation. Drag to rotate, scroll to zoom, click a node.

Click a node to read its definition.

Relativistic Cosmology and the Friedmann Equations

Manual: General · Subject: Cosmology

Derive the background dynamics of an expanding universe from general relativity.

Einstein Equations in Cosmology

From Stress-Energy to Expansion

Assuming the FLRW metric and a perfect fluid with energy density ρ\rho and pressure pp, Einstein's equations reduce to the Friedmann equations: H2=8πG3ρ−ka2+Λ3H^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}+\frac{\Lambda}{3} and a¨a=−4πG3(ρ+3p)+Λ3.\frac{\ddot a}{a}=-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}. Together with the continuity equation ρ˙+3H(ρ+p)=0,\dot\rho+3H(\rho+p)=0, these determine the background evolution once an equation of state is specified.

Common Cosmic Fluids

ComponentEquation of stateScaling with aa
Radiationw=p/ρ=1/3w=p/\rho=1/3ρ∝a−4\rho\propto a^{-4}
Matterw=0w=0ρ∝a−3\rho\propto a^{-3}
Curvatureeffective w=−1/3w=-1/3ρk∝a−2\rho_k\propto a^{-2}
Vacuum energyw=−1w=-1ρΛ=const\rho_\Lambda=\text{const}
💡

Acceleration Condition

Cosmic acceleration occurs when ρ+3p<0\rho+3p<0, so a sufficiently negative pressure component can drive a¨>0\ddot a>0.

For pressureless matter, how does the density scale with the scale factor?

Critical Density and Density Parameters

The critical density is ρc=3H2/(8πG)\rho_c=3H^2/(8\pi G). Dimensionless density parameters are defined by Ωi=ρi/ρc\Omega_i=\rho_i/\rho_c, with curvature encoded by Ωk=−k/(a2H2)\Omega_k=-k/(a^2H^2). The Friedmann equation can then be written as 1=Ωm+Ωr+ΩΛ+Ωk1=\Omega_m+\Omega_r+\Omega_\Lambda+\Omega_k at a given epoch.

What condition defines a spatially flat universe in the Friedmann equation?

Interpreting the Equation of State

Dust-like matter

  • Pressure is negligible compared with energy density.
  • Dominates structure formation at late times.
  • Gives decelerating expansion in standard GR.

Dark energy

  • Has sufficiently negative pressure.
  • Can cause accelerated expansion.
  • May be a cosmological constant or dynamical field.

Which term in the Friedmann equation can mimic a negative curvature-like contribution but is not geometric curvature?

Deriving the First Friedmann Equation

  1. 1

    Start from the FLRW metric and compute the Einstein tensor.

  2. 2

    Insert a perfect-fluid stress-energy tensor into Einstein's equations.

  3. 3

    Use the 0000-component to isolate the expansion rate.

  4. 4

    Identify the curvature, matter, radiation, and vacuum-energy contributions.

⚠️

Research-Level Caveat

The background equations are exact only for an idealized homogeneous and isotropic spacetime; real cosmology requires perturbations, backreaction considerations, and observational averaging.