Cosmology

Cosmology is the study of the structure and evolution of the universe as a whole.

Olbers’ Paradox #

Olbers’ Paradox: why is the sky dark at night if there are billions of stars? Anywhere you look in the sky there should be a star (just like trying to look through a dense forest).

• By the inverse square law, distance stars look dim: $b \propto \frac{1}{d^2}$.
• Distant stars have a nonzero angular size, since $A = \pi \theta^2 \propto \frac{1}{d^2}$.
• By the two above equations, it appears that brightness of a star is independent of its distance… (if it’s half as bright due to distance, it must also be half the size in the sky. So it should be just as bright as the sun, just a lot smaller.)

Possible solutions to Olbers’ paradox:

• The universe has finite size.
• The universe is infinitely big, but there are few to no stars far away.
• The universe has a finite age: light from most stars has not reached us yet. (This is the primary solution)

Age of the Universe #

How old is the universe? How do we know?

• The universe is about 13.8 billion years old.
• Two ways to estimate the age of the universe are:

1. Measure the distance to the earliest stars: the universe must be as old as the oldest star.
2. Measure the rate of expansion of the universe, and factor in the estimated expansion rate to determine the size of the universe over time. The age of the universe is when the size is estimated to be zero.

Cosmological Principle #

What is the Cosmological Principle?

• The cosmological principle states that on very large scales, the universe appears homogenous (same average density) and isotropic (appears the same in all directions) everywhere.

Shape of the Universe #

What is the shape of the universe?

• Through measurements of the angular sizes of the clusters of variations in the CMBR, we determined that the shape of the universe is roughly flat- meaning that the average mass density of the universe $\Omega_M$ is approximately 1, and Euclidean geometry applies to objects in the universe. In other words, the objects in the universe are distributed across a plane, rather than a sphere or hyperbola.
• If $\Omega_M < 1$, then the universe would be hyperbolic, with many parallel lines, and the angles in a triangle would add up to less than 180 degrees.
• If $\Omega_M > 1$, then the universe would be spherical with no parallel lines, and the angles in a triangle would add up to more than 180 degrees.
• We can measure mass density by summing the angles in a very large triangle, or counting the number of galaxies as a function of distance.

The Cosmological Constant #

What is the Cosmological Constant?

• It was originally thought that the universe was static (stayed the same size over time). In order to offset gravitational attraction, a cosmological constant was needed to balance it out.
  • If $\Lambda = 0$:
    • If $\Omega_M$ (mass density of the universe) $> 1$, then a Big Crunch would occur.
    • If $\Omega_M = 1$ (flat universe), then expansion would continue up to infinity.
    • If $\Omega_M < 1$, expansion would never stop, approaching a constant nonzero value.

Implications of the expansion of the universe:

• The universe will expand forever (unless the force starts becoming attractive).
• The age of the universe is about 14 billion years old given the rate of expansion.
• $\Omega_M - \Omega_\Lambda \approx -0.4$

Big Rip: the repulsive forces of dark energy get arbitrarily large, such that everything (including atoms and their nuclei) get ripped apart.

The Big Bang #

What happened during the Big Bang?

• The universe started out as an extremely hot, compressed, dense ball of gas.
• Over time, it expanded, cooled, and became less dense.
• In the first $10^{-43}$ seconds (Planck time), the universe was so small that classical physics did not apply. We don’t quite understand the mechanics of the universe at this time.
• Between $10^{-35}$ and $10^{-6}$ seconds, particles, antiparticles, and photons were in equilibrium. However, a very small excess of quarks over antiquarks formed, allowing matter to persist.
• Until 100 seconds in, primordial nucleosynthesis occurred, causing the fusion of quarks into protons and neutrons and the formation of hydrogen, helium, and trace amounts of lithium.
• Until 380,000 years in, the universe was opaque and underwent recombination where protons and electrons formed hydrogen atoms.

Inflation #

What are some of the issues with the traditional Big Bang Theory? How were they resolved?

Main problems with the original big-bang theory include:

1. The horizon problem: the temperature of the universe is too consistent: two observed points 13.8 billion light-years from us in opposite directions will have the same temperature, even though it is impossible for any information to have crossed the 26+ billion light-year distance between those two points. In other words, there is seemingly no way for all parts of the Universe to have come into equilibrium with each other.
2. The flatness problem: the original theory provides no explanation for why the universe is spatially flat (average mass density of 1). This is especially problematic because 1 is an unstable solution: if the universe were even a tiny amount denser or less dense, the mass density would either rapidly increase or decrease and we would observe an entirely different universe today.

The theory of inflation (the universe started extremely small, then expanded exponentially) resolves the two problems above:

1. Horizon problem: since all matter in the universe originally existed in a small space, it had enough time to achieve equilibrium before expansion began.
2. Flatness problem: all curvature flattens out after extreme expansion. A large enough universe would appear flat over a sufficiently small region.

• Inflation occurred between $t=10^{-37}s$ and $10^{-35}s$, during which the size of the universe increased by a factor of $10^{100}$ (faster than the speed of light!).

Cosmic Microwave Background Radiation #

What is cosmic background radiation?

• The cosmic microwave background radiation (CMBR) is electromagnetic radiation (light) from the big bang, filling the Universe. It looks the same in all directions. It was discovered with a radio telescope (in 1965, by Arno Penzias and Robert Wilson).
• CMBR is like a wall at redshift = 1000 after which we are unable to observe photons.
• CMBR is likely a result of the homogenous collection of particles in the early universe just after the Big Bang. Over time, the source of radiation expanded and cooled as the universe also expanded, eventually becoming the roughly 3K temperature we observe today.
• The small ripples in the cosmic microwave background radiation allowed for variations in temperature, which then resulted in the formation of early stars and galaxies. Over time, the clusters grouped together, creating the current structure of the universe with clusters of galaxies in filaments.

Multiverse and Theory of Everything #

Is it possible for there to be multiple universes?

What is the Theory of Everything?

• The four fundamental forces are:
  • Gravitational force, which weakly attracts matter together,
  • Electromagnetic force, which causes charged particles to interact,
  • Strong nuclear force, which binds subatomic particles together,
  • Weak nuclear force, which describes interactions between bosons and fermions.
• So far, we have discovered that electromagnetic and weak nuclear force were once unified into electroweak force. Grand Unified Theories (GUTs) attempt to combine electroweak force and strong nuclear force, which may have happened very early in the universe’s age. The Theory of Everything attempts to unify all four of the forces into one single entity. Once such theory is superstring theory.
• Shortly after the big bang, three of the four forces (electromagnetism, strong nuclear, and weak nuclear) should have been unified into the “grand unified theory (GUT)” force — they would have been symmetric. As the temperature fell below about $10^{29}$ K (around $t = 10^{-37}$ s), the strong and electroweak forces should have broken off and become different manifestations of the GUT force; the symmetry of the Universe should have been broken.
• If the universe somehow cooled below the critical temperature of $10^{29}$ K while the Grand Unified Force was still unbroken, then we could say that the universe is supercooled, similarly to how an undisturbed bottle of water below the freezing point only freezes if moved. If the universe acted in a similar way, a disturbance in the supercooled state could have prompted it to rapidly expand and cool down to the “normal” state.

Anthropic Principle and the Drake Equation #

How likely is it that the universe could have sustained life?

• Not likely at all:
  • The cosmological constant needed to be exactly right so that the universe didn’t immediately implode or not become dense enough to form stars and galaxies.
  • The strength of the fundamental forces needed to be just right.
  • The masses of particles needed to be exactly right to form the elements.
• It’s possible that the universe was designed specifically for us (anthropic principle), but it’s also possible that our universe is one of many (possibly even infinite) multiverses, and we just happen to exist on one that can support life.

How likely is it for intelligent life to exist outside of Earth?

• The Drake equation provides a possible estimate:

  $$ N = R f_s f_p n_e f_l f_i f_c L $$
  • $R$ is the average rate of star formation: about 10 stars per year.
  • $f_s$ is the proportion of stars that can support life: between 0.1 and 0.5.
  • $f_p$ is the proportion of good stars with planetary systems: between 0.1 and 1.
  • $n_e$ is the number of planets or moons per star in the habitable zone: between 0.1 and 2.
  • $f_l$ is the fraction of $n_e$ that life develops on: between $10^{-3}$ and 1.
  • $f_i$ is the fraction of living species that develop intelligence: $10^{-6}$ to 1.
  • $f_c$ is the fraction of intelligent life that can communicate: $10^{-3}$ to 1.
  • $L$ is the lifetime of the communicative phase: $100$ to $10^9$ years.

  Pessimistically, $N = 10^{-12}$ (so we might be the only ones!!!). Optimistically, $N = 10^{10}$.

• Regardless of $N$, interstellar travel is very impractical because of the massive challenges of accelerating to the speed of light (energy requirements, making sure we don’t die by the acceleration…). In addition, even if we do go fast enough, time dilation would mean thousands of years would pass before reaching the nearest star system.