Abstract
We introduce the Cosmological Principle, the Friedmann–Robertson–Walker metric, the Friedmann equations, and we construct some simple models for the description of the Universe (Einstein universe, matter dominated universe, radiation dominated universe, vacuum dominated universe). We also discuss the age of the Universe and its possible destiny.
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- 1.
Note that it is common to use the initial capital letter in Universe only when we refer to our Universe. If we mean a generic universe/cosmological model, we write universe.
- 2.
If we impose that the spacetime geometry is also independent of time (“Perfect” Cosmological Principle), we find cosmological models in disagreement with observations.
- 3.
The line element of the spatial 3-metric of the Friedmann–Robertson–Walker spacetime reads
- 4.
In general, ghost images can be potentially seen in any universe (with either trivial or non-trivial topology) with at least one space dimension of finite size. However, it is when the topology is non-trivial that the detection of ghost images seems to be the simplest way to infer if the Universe has at least one space dimension of finite size.
- 5.
The parsec (pc) is a common unit of length in astronomy and cosmology. 1 \(\mathrm{pc} = 3.086 \cdot 10^{16}\) m. 1 Mpc = \(10^6\) pc.
References
C. Bambi, A.D. Dolgov, Introduction to Particle Cosmology: the Standard Model of Cosmology and its Open Problems (Springer-Verlag, Berlin Heidelberg, 2016)
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Problems
Problems
11.1
Write the tt component of the Einstein equations for the Friedmann–Robertson–Walker metric and a perfect fluid and derive the first Friedmann equation.
11.2
With the help of some Mathematica package, verify Eqs. (11.3) and (11.4).
11.3
Check that the Einstein universe is unstable.
11.4
Repeat the discussion in Sect. 11.6 about the age of the Universe in the case of a non-negligible contribution from a radiation component.
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Bambi, C. (2018). Cosmological Models. In: Introduction to General Relativity. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-13-1090-4_11
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DOI: https://doi.org/10.1007/978-981-13-1090-4_11
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