Brief Introduction to the Tenets of Cosmology
The standard cosmological model, in all its simplicity, remains a successful model of the universe because of its remarkable agreement with observations. It is important to know what premises this theory is based on. One of the key assumptions of the cosmological model is that general relativity is an accurate theory in the early universe, and that the basic laws of physics are the same no matter where or when one looks. Another key assumption is known as the Cosmological Principle. It assumes that on large scales the universe is homogeneous and isotropic; there are no preferred reference frames, no matter where one is located, the universe looks the same in all directions. In order to study the evolution of the universe, we need to know its content, we assume the "Standard Model" of particle physics is correct and that the universe contains at least these particles and interactions. There is also observational evidence suggesting the existence of dark matter and dark energy. To summarize, the tenets of cosmology are:
- General Relativity
- Cosmological Principle
- "Standard Model" of Particle Physics
- Dark Matter and Dark Energy.
These ideas make the theory tractable and essentially describe all of the early universe. For more detailed discussions of these points, we recommend the following references [20, 21, 22, 23, 24]
With the cosmological principle in hand we set out to describe our universe. The only measurable large scale motion allowed by the second tenet is universal expansion. It is useful to describe this expansion in terms of a universal scaling factor R(t). We can use this scale factor to relate physical positions with co-moving ones, r = R(t) x. A particle at rest in the co-moving frame (x = 0) will have a physical velocity, v = R(t) x = Hr, which is Hubble's expansion law. Here H = R(t)/R(t) is the expansion rate ( ⋅ = ∂/∂t).
The evolution of the scale factor is determined from the first tenet. Using Einstein's general relativistic field equations, one can attain relations remarkably similar to the conservation of energy and Newton's force law. This similarity arises because of the simplicity of the model. The evolution of the universe is then governed by what are called the Friedmann and Friedmann acceleration equations.
Here, G is Newton's gravitational constant, ρ and P are the mass-energy density and pressure of the universe, c is the speed of light, k is the curvature constant, and is the cosmological constant. The first term on the left-hand side of the Friedmann equation looks like a kinetic energy and the second like a gravitational potential energy, while the right-hand side is essentially the total energy. It is useful to define a "critical" density, ρc; = 3H2=8πG and the relative density Ω = ρ/ρc. Assuming that Λ is zero, we can see that if the kinetic energy of the universe dominates over the potential energy, the universe will expand forever, meaning the universe is "open" with k = -1 and Ω< 1. If the potential energy dominates, the universe will collapse on itself, meaning a "closed" universe with k = +1 and Ω > 1, and if the kinetic and potential energies exactly balance, the universe takes an infinite time to halt its expansion, a "critical" universe with k = 0 and Ω= 1. However, if the cosmological constant term is large, then the universe is electively opened or closed depending on the sign of Λ. A positive cosmological constant is the simplest model for dark energy. In this case, the universe undergoes exponential expansion. For the epochs we are considering, the radiation and matter densities dominate over the curvature and cosmological constant terms, thus we set them to zero.
An important step in knowing how to solve these equations is knowing how the mass- energy of the universe changes with the scale factor. For simplicity, we will consider here two examples of this mass-energy, sticking with a particle description, looking at a universe filled solely with either non-relativistic particles or ultra-relativistic particles. We expect that in an expanding universe the content should get more dilute, changing inversely with the volume (∝1/R3). We also expect additional scale factor dependence due to the redshifting of a particle's momentum, scaling inversely with wavelength (∝1/R). A non-relativistic particle's energy is dominated by its rest mass energy, which has no momentum dependence, thus its mass-energy density scales only with the dilution factor (ρNR∝ 1/R3). Its thermal energy scales like TNR 1/R2, since its kinetic energy scales as KE ∝ p2. A highly relativistic particle's energy is dominated by its kinetic energy, which scales like 1/R since KE ∝ p. Its mass-energy density then combines the dilution and this redshifting, scaling as ρUR ∝ 1/R4. Then relating temperature and density, we find that ρNR ∝ TNR3/2 and ρUR ∝ TUR4. Given a universe with both non- relativistic (matter) and ultra-relativistic (radiation) particle species, it is clear that with the scalings above, that radiation will dominate the mass-energy density of the universe at early times, with matter becoming dominant later in its evolution. Using these scalings and the Friedmann equation, we get the following relations between time, mass-energy density and temperature for radiation and matter dominated universes.
It is also interesting to note here that if these particle species existed in thermal equilibrium with TNR = TUR, the interactions between these species must be much faster
than the Hubble expansion, preventing them from following their prescribed temperature scalings shown above.
The time-temperature relation is then dependent on the
relative number of non-relativistic versus relativistic particles. Observations tell us that
the number of relativistic particles (i.e. photons) greatly out-numbers the number of
non-relativistic particles (i.e. baryons), thus the time-temperature relation is t ∝1/T2
when in thermal equilibrium.
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