PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → General relativity and gravity → Cosmology ↓


Friedmann equation


The Friedmann equations are fundamental in cosmology for describing the expansion of the universe. Derived by Russian physicist Alexander Friedmann in 1922 and 1924, these equations come from Einstein's field equations of gravity in the context of a homogeneous and isotropic universe. They serve as important tools in understanding the dynamics of the universe, predicting the behavior of the universe under different conditions such as matter dominance, radiation dominance or the influence of dark energy.

Understanding the cosmological principle

The cosmological principle assumes that the universe is homogeneous and isotropic on large scales. "Homogeneous" means that the universe looks the same at every point (i.e., every location in the universe is statistically the same). "Isotropic" means that the universe looks the same in every direction. Based on these assumptions, we can model the universe using the Friedmann-Lemaître-Robertson-Walker (FLRW) metric.

Mathematically, these concepts are represented in the FLRW metric, which describes a 4-dimensional spacetime universe that expands or contracts uniformly:

ds² = -c²dt² + a(t)² [ dr² / (1 - kr²) + r²(dθ² + sin²θ dφ²) ]

Components of the metric

  • ds² is the spacetime interval.
  • a(t) is the scale factor, a function of time that describes the expansion of the universe.
  • k is the curvature parameter, which can be -1 (open universe), 0 (flat universe) or +1 (closed universe).
  • c is the speed of light.
  • r, θ, and φ are spherical coordinates.

The FLRW metric leads us to obtain the Friedmann equations by estimating the Einstein field equations under these assumptions of a uniform universe.

Einstein field equations

At the heart of general relativity and cosmology are the Einstein field equations, which relate the geometry of spacetime to the distribution of matter and energy. In simple terms, they describe how matter and energy affect the curvature of spacetime. The equations are given as:

Gμν + Λgμν = (8πG / c⁴) Tμν

Components of the Einstein field equation

  • Gμν is the Einstein tensor, representing the spacetime curvature due to matter.
  • Λ is the cosmological constant proposed by Einstein, which characterizes the energy density of empty space.
  • gμν is the metric tensor from the FLRW metric.
  • G is the gravitational constant.
  • c is the speed of light.
  • Tμν is the stress–energy tensor, representing the matter and energy content.

Using the FLRW metric, we can simplify the Einstein field equations into a more manageable form, the Friedmann equations, which primarily describe the dynamics of the scale factor a(t).

Friedmann equation

The Friedmann equations relate the expansion of the universe, determined by the scale factor a(t), to the matter-energy content of the universe. These equations allow us to model different stages of the evolution of the universe.

First Friedmann equation

The first Friedmann equation relates the rate of expansion to the energy density of the universe:

(H(t))² = (8πG / 3) ρ - (kc² / a(t)²) + (Λ / 3)
  • H(t) is the Hubble parameter, defined as H(t) = ȧ/a, where ȧ is the derivative of a(t) with respect to time.
  • ρ is the average energy density of the universe.
  • k is the curvature parameter.
  • Λ is the cosmological constant.

The first Friedmann equation shows how the expansion rate is affected by the density of matter, the curvature of space, and the effect of dark energy represented by the cosmological constant.

Second Friedmann equation

The second Friedmann equation describes how the acceleration of the expansion of the universe is determined by pressure and energy density:

ȧ̈ + (4πG / 3)(ρ + 3p / c²) a + (Λc² / 3) a = 0
  • ȧ̈ is the second derivative of a(t) with respect to time, which represents the acceleration of the expansion of the universe.
  • p is the pressure of the material of the universe.
  • The other terms are the same as defined in the first equation.

This equation shows that increasing the pressure can slow the expansion, while a sufficiently large cosmological constant can lead to accelerated expansion.

Example applications of the Friedmann equations

Closed, open and flat universes

The Friedmann equations classify universes into three types depending on the curvature parameter k:

  • A closed universe (k=1) is one with positive curvature, like a 3D sphere, where the universe will eventually stop expanding and collapse in a "Big Crunch".
  • An open universe (k=-1) resembles a hyperbolic geometry, which is forever expanding.
  • A flat universe (k=0) expands, slowing down over time, but never completely stopping, which aligns with observations of our current universe.

Example of a flat universe (k=0):

In a flat universe with no cosmological constant, the Friedmann equations predict that only the matter density governs the general expansion dynamics:

(H(t))² = (8πG / 3) ρ

This scenario defines a perfectly balanced state within an infinite, ever-expanding universe.

Understanding the cosmological constant

The cosmological constant, Λ, introduced by Einstein, represents the energy density of empty space or dark energy. In recent cosmological models, dark energy is used to explain the accelerated expansion of the universe observed in supernova data.

Example of an accelerating universe with dark energy:

Including the cosmological constant can significantly change the Friedmann equations. Consider the importance of λ:

ȧ̈ = (Λc² / 3) a - (4πG / 3)(ρ + 3p / c²) a

It predicts that if dark energy dominates matter density, the expansion of the universe accelerates, potentially leading to an infinite cosmic inflation scenario.

Visual explanation using graphs and equations

The following are graphical representations of the dynamics of the expansion of the universe using mathematical interpretations of the Friedmann equations:

Time But) Open universe Closed universe Flat Universe

These simple curves depict the evolution of the universe over time with respect to different curvature scenarios. Each curve represents one of several cosmological possibilities derived from the Friedmann equations.

Conclusion

The Friedmann equations are central in cosmology's efforts to understand the evolution and structure of the universe. These equations allow scientists to fit observational data into theoretical models and better understand phenomena such as the Big Bang, cosmic inflation, and dark energy.

By applying these equations, researchers have gained important insights into past universe ages and potentially predicted its future behavior based on current observations and theoretical advances. These models, although built on simplifications such as symmetry and isotropy, continue to provide a strong working framework in the search for cosmic truths.


PHD → 6.2.1


U
username
0%
completed in PHD

← Prev (6.2)
Cosmology
Next (6.2.2) →
Cosmic inflation

Comments 



Friedmann equation

https://www.physicsflow.com/phd/6.2.1?pfal=en