Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → General relativity and cosmology → Cosmology and the Universe ↓


Dark energy and structure formation


In the grand tapestry of the universe, the intricate dance between matter and energy weaves the vast universe into a coherent entity. At the center of this cosmic ballet lie two profound concepts in cosmology: dark energy and structure formation. While structure formation involves the development of visible cosmic designs such as galaxies, stars, and clusters of galaxies, dark energy introduces a mysterious force that accelerates the expansion of the universe. To truly understand their significance, we must delve deeply into the fundamentals of general relativity and cosmology.

What is dark energy?

Dark energy is a form of energy that pervades all of space and drives the rate at which the universe expands. Unlike dark matter, which forms structures such as galaxies and galaxy clusters, dark energy is thought to be uniformly distributed throughout the universe.

Under general relativity, the effects of dark energy can be modeled by introducing a cosmological constant Λ. The simplest model for dark energy is vacuum energy, which has a constant energy density throughout space. This gives the following density parameter for the universe:

ρ_Lambda = Lambda / (8πG)

> Here, G is the gravitational constant.

Visualizing dark energy

observable universe Dark Energy

In the above image the blue circle represents the observable universe, while the smaller red circle represents the presence of dark energy pervading the universe.

The role of dark energy

The universe has been expanding since the Big Bang. Initially, it was believed that gravity would cause the expansion to slow down. However, observations in the late 20th century showed that the expansion is accelerating. This surprising discovery pointed to an unknown factor: dark energy.

One effect of dark energy is its influence on the fate of the universe. If the density of dark energy remains constant, it will outstrip the gravitational pull of matter, causing the universe to enter a state of infinite accelerating expansion.

Structure formation in the universe

Structure formation deals with how tiny fluctuations in the density of the early universe evolved over time into the galaxies, stars, and larger cosmic structures observed today. These fluctuations are bound by gravity, slowly growing in size, leading to the formation of complex structures in the universe.

In the early times of the universe, matter was more smoothly distributed than it is today. As time went on, gravitational instabilities increased, resulting in the collapse and formation of various cosmic structures.

Cosmic web

Galaxies and cosmic strings

The cosmic web is the vast structure of the universe. The blue circles represent clusters of galaxies while the red lines represent cosmic strings or filaments connecting them.

Quantum fluctuations and density perturbations

Quantum fluctuations during the early universe gave rise to initial density disturbances. These disturbances grew through gravitational collapse and led to the formation of galaxies and large-scale structures.

δ(x,t) = σ * e^[H(t-t_0)]

> Here, δ(x,t) represents the density fluctuation at position x and time t, σ is the amplitude of the fluctuation, and H is the Hubble parameter.

Timeline of the Universe

The formation of structures in the universe can be divided into major epochs, each marked by different dominant forces or processes.

  • Recombination era (about 380,000 years after the Big Bang): Electrons and protons combine to form neutral hydrogen, making the universe transparent to radiation. This period results in the formation of the cosmic microwave background (CMB).
  • Dark Ages: The time when the universe was filled with neutral hydrogen and there were very few other sources of light.
  • Reionization era: The first stars and galaxies formed, and eventually neutral hydrogen was ionized.
  • Galaxy formation and evolution: Galaxies evolved, clumping together to form larger structures.
  • Dominance of dark energy (current era): The influence of dark energy has become more significant, causing the current accelerated expansion of the universe.

General relativity and cosmic evolution

Albert Einstein's general theory of relativity provides a framework for understanding the dynamics of the universe on large scales. The equations governing the expansion of the universe, driven by the distribution of matter and energy, are derived from the Einstein field equations:

G μν + Λg μν = 8πGT μν

> In these equations, G μν represents the Einstein tensor describing spacetime curvature, Λ is the cosmological constant, g μν is the metric tensor, and T μν is the energy–momentum tensor describing the matter and energy densities.

Friedmann equation

The expansion of the universe can be described by the Friedmann equations, which are derived from Einstein's field equations under the assumption of a homogeneous and isotropic universe.

(a dot)^2/a^2 = 8πG/3 * ρ - k/a^2 + Λ/3 2(a double dot)/a = -8πG/3 * (ρ + 3p) + Λ

> Above a represents the scale factor, ρ is the energy density, p represents the pressure, and k represents the curvature of space.

Relation between dark energy and structure formation

While dark energy is expanding the universe on its largest scales, gravitational attraction is bringing matter together to form structures on smaller scales. These processes are interconnected:

  • Dark energy affects the rate of expansion of the universe, which affects the growth rate of cosmic structures.
  • In the early times, when the effect of dark energy was negligible, structures formed primarily through gravity.
  • As the universe expands faster due to dark energy, the growth of new structures is slowing down due to the weakening of gravitational binding in larger cosmic regions.

Conclusion

The interrelationship between dark energy and structure formation underscores the complex nature of the universe. Our understanding of these phenomena is continually enhanced by advanced observations, theoretical frameworks, and technological improvements. As the universe progresses, dark energy and cosmic structures will continue to be important topics of study to unravel the ultimate mysteries of the universe.


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Dark energy and structure formation

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