👽Galaxies and the Universe Unit 8 – Cosmological Models: The Expanding Universe

Key Concepts and Definitions

• Cosmology studies the origin, evolution, and ultimate fate of the universe on the largest scales
• The Big Bang theory proposes that the universe began in an extremely hot, dense state approximately 13.8 billion years ago and has been expanding ever since
• Redshift occurs when light from distant galaxies is stretched to longer wavelengths due to the expansion of the universe
• Hubble's law describes the relationship between a galaxy's distance and its recessional velocity, with more distant galaxies moving away faster
• Dark matter is a hypothetical form of matter that does not interact with electromagnetic radiation but has gravitational effects on visible matter
• Dark energy is a mysterious form of energy that permeates all of space and is thought to be responsible for the accelerating expansion of the universe
• The cosmological principle states that on large scales, the universe is homogeneous (uniform density) and isotropic (looks the same in all directions)

Historical Background

• In 1915, Albert Einstein published his theory of general relativity, which laid the groundwork for modern cosmology
• In the 1920s, Edwin Hubble discovered that distant galaxies are moving away from us, providing evidence for an expanding universe
• Georges Lemaître, a Belgian priest and physicist, proposed the idea of a "primeval atom" in 1931, which later became known as the Big Bang theory
• In the 1940s, George Gamow and his colleagues developed the theory of Big Bang nucleosynthesis, explaining the origin of light elements in the early universe
• The discovery of the cosmic microwave background (CMB) radiation in 1965 by Arno Penzias and Robert Wilson provided strong evidence for the Big Bang theory
• In the late 1990s, observations of distant supernovae revealed that the expansion of the universe is accelerating, leading to the concept of dark energy

The Big Bang Theory

• The Big Bang theory is the prevailing cosmological model for the observable universe
• It states that the universe began as an extremely hot, dense, and rapidly expanding singularity approximately 13.8 billion years ago
• The early universe underwent a period of cosmic inflation, exponentially expanding in a fraction of a second
• As the universe expanded and cooled, it allowed for the formation of subatomic particles, atoms, stars, and galaxies
• The theory explains the origin of the cosmic microwave background radiation, the abundance of light elements, and the large-scale structure of the universe
• The Big Bang theory does not address the initial cause of the expansion or what preceded the singularity

Observational Evidence

• Hubble's law and the redshift of distant galaxies provide evidence for an expanding universe
• The cosmic microwave background radiation is a remnant of the early universe, with a nearly uniform temperature of 2.7 Kelvin
• The abundance of light elements (hydrogen, helium, and lithium) in the universe matches predictions from Big Bang nucleosynthesis
• The large-scale structure of the universe, with galaxies arranged in clusters and filaments, is consistent with the growth of density fluctuations in the early universe
• Observations of distant supernovae indicate that the expansion of the universe is accelerating, suggesting the presence of dark energy
• Gravitational lensing and the rotation curves of galaxies provide evidence for the existence of dark matter

Mathematical Models

• The Friedmann equations, derived from Einstein's field equations, describe the expansion of the universe in the context of general relativity
  • The first Friedmann equation relates the rate of expansion to the density and curvature of the universe: $\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2}$
  • The second Friedmann equation describes the acceleration of the expansion: $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right)$
• The scale factor, $a(t)$, represents the relative size of the universe as a function of time
• The density parameter, $\Omega$, relates the actual density of the universe to the critical density required for a flat universe
  • $\Omega < 1$ corresponds to an open universe with negative curvature
  • $\Omega = 1$ corresponds to a flat universe with zero curvature
  • $\Omega > 1$ corresponds to a closed universe with positive curvature
• The Lambda-CDM model, which includes dark energy (represented by the cosmological constant, $\Lambda$) and cold dark matter (CDM), is the most widely accepted cosmological model

Expansion Rate and Hubble's Law

• Hubble's law states that the recessional velocity of a galaxy is proportional to its distance from the observer
• The Hubble constant, $H_0$, represents the current expansion rate of the universe and is approximately 70 km/s/Mpc
• The Hubble time, defined as the inverse of the Hubble constant, provides an estimate for the age of the universe (assuming a constant expansion rate)
• The redshift, $z$, of a galaxy is related to its distance and the scale factor of the universe at the time the light was emitted: $1 + z = \frac{a_0}{a_e}$
• The Hubble-Lemaître law is a more accurate version of Hubble's law that accounts for the expansion of space itself
• Measuring the expansion rate and its change over time helps constrain the properties of dark energy and the ultimate fate of the universe

Dark Matter and Dark Energy

• Dark matter is a hypothetical form of matter that does not interact with electromagnetic radiation but has gravitational effects on visible matter
  • It is thought to make up approximately 27% of the universe's total energy density
  • Evidence for dark matter includes the rotation curves of galaxies, gravitational lensing, and the large-scale structure of the universe
  • Candidates for dark matter include weakly interacting massive particles (WIMPs), axions, and primordial black holes
• Dark energy is a mysterious form of energy that permeates all of space and is thought to be responsible for the accelerating expansion of the universe
  • It is estimated to make up approximately 68% of the universe's total energy density
  • The cosmological constant, $\Lambda$, is the simplest explanation for dark energy, representing a constant energy density throughout space and time
  • Alternative theories include quintessence, phantom energy, and modified gravity
• The nature and properties of dark matter and dark energy are among the most significant open questions in modern cosmology

Future of the Universe

• The ultimate fate of the universe depends on its density, expansion rate, and the nature of dark energy
• In a flat universe with a cosmological constant, the expansion will continue accelerating, leading to a "Big Freeze" scenario
  • Galaxies beyond the local group will eventually become unreachable as they recede faster than the speed of light
  • Stars will exhaust their fuel, leaving behind white dwarfs, neutron stars, and black holes
  • The universe will become increasingly cold, dark, and empty as entropy increases
• In an open universe, the expansion will continue indefinitely, but at a decreasing rate, also leading to a "Big Freeze"
• In a closed universe with insufficient dark energy, the expansion will eventually reverse, leading to a "Big Crunch" scenario
  • The universe will collapse back into a singularity, potentially leading to a new Big Bang (oscillating universe model)
• Other speculative scenarios include the "Big Rip," where the expansion becomes so rapid that it tears apart structures, and the "Big Bounce," where the universe undergoes cycles of expansion and contraction
• Observational evidence suggests that the universe is flat and will likely continue expanding indefinitely due to dark energy