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 ↓


Large scale structure


Large-scale structure (LSS) refers to the organization and distribution of matter, particularly galaxies, galaxy clusters and voids, in a cosmic web-like pattern across the vast expanse of the universe. This structure reveals patterns that are remnants of the early universe and provide insights into the fundamental workings of gravity, general relativity and the history of the universe.

With the framework of general relativity introduced by Albert Einstein, we understand gravity not as forces, but as distortions of spacetime due to mass. This theory is the cornerstone of our understanding of cosmic structures. The equation that encapsulates the essence of general relativity is:

        Gμν = 8πGTμν

where Gμν is the Einstein tensor characterizing the curvature of spacetime, G is the gravitational constant, Tμν is the energy-momentum tensor, and π is pi, which is approximately 3.14159. This equation tells us that mass and energy determine the curvature of spacetime.

Nature of the universe

The universe, on the broadest scale, looks like a sponge or foam. Galaxies are not evenly distributed. Instead, they form clusters, filaments, and walls around vast, empty voids. This is what we call the cosmic web. One of the greatest maps of this cosmic structure is the Sloan Digital Sky Survey (SDSS), which revealed intricate maps of galaxy distribution.

In the illustration above, imagine that the blue circle represents a supercluster of galaxies, and the smaller red circle represents a cluster, with the lines and other structures representing the filaments of galaxies that form the cosmic web.

Cosmic Microwave Background (CMB)

The cosmic microwave background provides a snapshot of the universe when it was just 380,000 years old. The slight temperature changes observed in the CMB are the seeds that explain the current distribution of large-scale structures. Theoretical advances in cosmology, such as the Lambda Cold Dark Matter (ΛCDM) model, describe how, over billions of years, small disturbances under the influence of gravity evolve into galaxies and clusters.

        ΔT/T ~ 10^-5

This formula shows that the temperature fluctuations ΔT/T in the CMB are remarkably small, about one part in 100,000. Nevertheless, these negligible disturbances eventually grow into the giant structures we see today.

How gravity shapes large-scale structure

Gravity plays a key role in shaping the structure of the universe. Through gravitational attraction, regions with slightly higher densities attract more matter and become denser, while regions with lower densities lose matter. This process, known as gravitational instability, leads to the formation of complex cosmic structures.

        F = G m1 m2 / r^2

This Newtonian formula represents the gravitational force F between two masses, m1 and m2, separated from each other by a distance r. While this is a simpler approach than general relativity, it is useful in understanding elementary gravitational attraction.

Dark matter in large-scale structure

Although invisible and undetectable by conventional means, dark matter is a significant portion of the mass of the universe and significantly affects large-scale structure formation. Unlike baryonic matter, dark matter does not interact with electromagnetic forces; rather, it interacts primarily through gravity.

Dark matter forms a framework around which baryonic matter accumulates, leading to the formation of galaxies. As a cold, non-relativistic form of matter, it helps maintain the structural integrity of the cosmic web on cosmic time scales.

Voids and fibers

The cosmic void is a vast expanse with very few galaxies, surrounded by filaments which are dense regions where galaxies align. The void and filaments are fundamental to our understanding of how matter and light travel in the universe and provide fascinating tests for theories beyond the standard cosmic model.

Imagine the universe as a mixture of foam, in which the spaces are bags of air, and the filaments are the soap that outlines these bags.

In this illustration, consider the gray box to be the cosmic bubble, with the lines being the walls of the bubble, representing filaments.

Measuring the Universe

Astronomers use redshift, which is the change in the wavelength of light due to the motion of galaxies, to measure the structure and dynamics of the universe. The most distant galaxies show the greatest redshift, reflecting their rapid motion away from us due to the expansion of the universe.

        z = (λ_obs - λ_emit) / λ_emit

This formula gives the redshift z, where λ_obs is the observed wavelength, and λ_emit is the emitted wavelength.

The future of large-scale structures

With advances in technology and observational techniques, new projects aim to map the universe with unprecedented accuracy. These approaches offer the potential to discover unknown physics beyond current cosmological paradigms and improve our understanding of fundamental principles, such as the effect of dark energy on structure dynamics.

In conclusion, the vast structure of the universe has given us a profound understanding of cosmic evolution and cosmology. It is an important aspect of modern astrophysics research, and its study continues to unravel the complexities of the universe.


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Large scale structure

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