Relativistic Cosmology Notes

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The Nature of Cosmology

The Aims of Cosmology

The physical universe is the “maximal set of physical objects which are locally causally connected to each other and to the region of spacetime that is accessible to us by astronomical observation.” Cosmology is concerned with the large scale structure of the observable universe, and its relation to local physics and the rest of the universe. It contemplates the nature and history of the universe. Because we only have one universe to observe, we must be aware of the limitations placed on what we can state with reasonable certainty.

Scientific Cosmology

  • Cosmography, the study of the distribution and motion of matter in the universe, is the starting point for cosmology. This falls under the umbrella of observational cosmology, where the main aim is to describe the universe at large. Observational cosmology often leads to unexpected results, such as the expansion of the universe and the existence of dark matter.
  • Physical cosmology aims to explain the processes that are occurring and how they have led to the structures observed in the universe.
  • Particle and quantum cosmology attempt to explain the observable universe using particle and quantum physics. These fields have special importance to the early universe and inflation.
  • Relativistic cosmology focuses on the implications of curved spacetime on observational and physical cosmology. It usually considers models other than the standard Friedman-Lemaitre-Robertson-Walker (FLRW) model.

Cosmology’s Wider Implications

There are obvious implications for philosophy and the humanities. We must be careful to place the proper constraints on our results and not attempt to explain more than can reliably be achieved by use of the scientific method.

Observational Evidence and its Limitations

There are three broad methods to obtain evidence in cosmology.

Evidence from Astronomical Observations

How do we observe?

  • Electromagnetic radiation, neutrinos, high-energy cosmic rays, gravitational waves
  • The Cosmic Microwave Background (CMB) is the most important observational radiation source for cosmology
    • Relic radiation from the big bang

We are limited to observing our past light cone. It is not feasible for us to observe outside this light cone, so our view of the universe is consequently narrow.

Evidence of a Geological Nature

Studying the history of local objects allows us to access ages much older than what a telescope could probe.

Evidence from Local Physics

We can observe local physics as a consequence of physics that act on cosmological scales. A couple of examples:

  1. Mach proposed that inertia is related to the distribution of mass in the universe. This suggests that the gravitational constant might be time-varying.
  2. “Olbers' paradox” – why is the sky not blindingly bright at all times?
  3. The unique arrow of time, despite the time reversibility of local physical laws

Existence of Horizons

  1. The visual horizon: we cannot see (EM) past the last-scattering surface. It is possible to surpass this horizon with gravitational wave and neutrino observations, but they each have their own horizons
  2. The particle horizon: causality is limited by the speed of light. This presents a horizon that allows us to be causally connected within but not outside of it.
  3. A physics horizon: the energies in the early universe are out of our reach with current and near-future instruments.

A Summary of Current Observations

Through recent advances in availability of data and accuracy of observations, we have entered and era of “precision cosmology.”

Expansion of the Universe - and its Acceleration

  • Hubble discovered in 1929 that the universe was expanding. He related the speed at which Cepheid variable stars were receding to their distance and found a linear relation between the velocity and distance, \(v = H_0 d\). Current measurements of Hubble’s constant give \(H_0 = 73 \textrm{ km/s/Mpc}\) (this was from the Hubble Space Telescope Key Project ~2010).
  • \(\tfrac{1}{H_0}\) gives an estimate of the age of the universe, or at least the time since all galaxies were in the same place.
  • We find that their is a change in the number density of radio sources as we explore further distances, which implies that the universe is evolving as it expands.
    • This was confirmed by the discovery of the CMB
  • Relatively recent observations have shown that the rate of expansion is increasing. This requires dark energy.

Nucleosynthesis and the Hot Big Bang

The observed abundances of elements are well accounted for by the hot big bang theory and further processing in stars.

Cosmic Microwave Background and the Hot Big Bang

  • The CMB has a temperature \(T \approx 3\textrm{ K}\) and has a precise black body spectrum
  • It was emitted at \(z \approx 1100\) when the universe dipped below 4000 K. Previously, the mean free path for radiation was very small so matter and radiation were tightly coupled. This is what we mean when we say the universe was “opaque.”
  • The precise black body spectrum of the CMB shows that physical laws are unchanged throughout the eons
  • The isotropy of the CMB supports the belief that the universe is homogeneous on the largest scales.
  • There are small fluctuations in temperature maps of the CMB, with \(\abs{\tfrac{\Delta T}{T}} \leq 10^{-5}\). This anisotropy gives rise to the galaxies and clusters we observe. The current best model for the origin of these fluctuations is inflation, which will be discussed later.

Other Background Radiation

In addition to the CMB, we have also observed in detail the radio, microwave, X-ray, and \(\gamma\)-ray bands. In the future, we may observe the background flux of gravitation waves and neutrinos. Today, these sources are at undetectable levels, but they can be indirectly measured (B-mode polarization of the CMB for gravitational waves, and matter power spectrum for neutrinos).

Structure Formation and the Very Early Universe

The idea of inflation allows small quantum fluctuations to be amplified to macroscopic scales where they become the seeds of large-scale structure growth. These fluctuations are visible to us in the CMB, with a particularly large peak at about 1 arcmin.

Baryonic Matter: Galaxy Distribution and Acoustic Peak

The 2dF and SDSS surveys have mapped the distribution of galaxies in detail. In 2005, the confirmation of the theorized Baryon Acoustic Oscillation (BAO) peak was accomplished via the aforementioned surveys.

Dark Matter

We can directly observe luminious baryonic matter and indirectly observe non-luminous matter via absorption and emission. We now have significant evidence that this matter that interacts with light is but a small part of the matter that exists, the rest being dark matter. The prevailing model that includes dark matter is known as Cold Dark Matter, or CDM. A key feature of CDM is its non-baryonic nature. The amount of dark matter detected is much larger than the amount of baryonic matter predicted by nucleosynthesis theory.

Cosmological Concepts

Now that the constituents of the cosmos have been discussed, we must find a theory which represents the matter present, the physics governing it, and the relevant spacetime geometries.

Matter Description

Matter is often modeled as a fluid. This is appropriate on scales where the density is relatively constant. At very large scales, macroscopic gradients become important, and we discard the fluid representation.

Dynamics

What governs dynamics on the largest scales? Electromagnetism isn’t a good candidate, it doesn’t match up with observation. Gravity is believed to be the dominant force on large scales. In much cosmological work, the role of general relativity can be swept under the rug. We really should consider these effects if we want a complete picture.

Cosmological Models

What are the key ingredients or relativistic cosmology?

  1. A spacetime, with Lorentzian metric and associated torsion-free connection
  2. A description of matter and radiation, with thermodynamic, kinetic or field-theoretic models that determine their local physical properties
  3. A unique family of fundamental observers, whose motion represents the average motion of matter. These observers should be expanding during some epoch to correspond with the observable universe.
  4. A set of observational relations that follow from the geometry and interactions between matter and radiation

Averaging Scales

Each attempt at modelling should be based on an implied averaging scale and range of applicability. Cosmological models are only valid descriptions above some scale of averaging. Models are usually only applicable to certain epochs. A full cosmological model is a patchwork of models that apply at the relevant epochs and scales.

Specific Models

  • FLRW Models

    These models are the standard models of cosmology. These universes are spatially homogeneous and isotropic. They are useful in explaining the expansion and history of the universe, but they fail to model a realistic universe because they are exactly homogeneous and isotropic. Perturbations of FLRW models are particularly useful.

  • Spherically Symmetric Inhomogeneous Models

    Lemaitre-Tolman-Bondi (LTB) models are the simplest inhomogeneous, spherically symmetric expanding models. These models have been used to give exact nonlinear models of inohomogeneous cosmologies that do not involve dark energy.

  • Lumpy Inhomogeneous Models

    “Swiss cheese” models have spherical regions interspersed on a smooth background.

  • Spatially Homogeneous Models

    A Bianchi universe is one that is anistropic, expanding, and spatially homogeneous. They are useful for investigations into bounding the anisotropy of the universe.

Last updated on Dec 22, 2020