Abstract
The most prominent effect of gravity in our current universe appears to be a uniform repulsion. In this chapter we shall introduce some of the basic concepts for a theoretical description of this astonishing discovery made at the end of the 20th century.
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Notes
- 1.
Note: \(\sinh x = -i \sin (i x)\) and thus \(\text {arcsinh} (\sqrt{|K|} r_e)/\sqrt{|K|} = \arcsin (\sqrt{K} r_e)/\sqrt{K}\) for \(K<0\).
- 2.
Note:
$$\begin{aligned} \int _{a+\delta a}^{b + \delta b} f(x) \mathrm{d}x = F(b+\delta b) -F(a+\delta a) \approx F(b) -F(a) + f(b)\,\delta b -f(a)\, \delta a. \end{aligned}$$ - 3.
- 4.
Actually, this is not quite correct. But the underlying problem can be fixed (see for instance [27]).
- 5.
This is quite crude. For instance, we do not expect \(C(\gamma )\) to diverge in the limit \(\gamma \rightarrow 0\) as in the case of the power law. The opposite limit is also problematic, because we are dealing with a finite geometry. We also like to add the following caveat. In statistical mechanics fluctuation correlation functions at critical points, where the fluctuation correlation length is infinite, are usually described by power laws. In the continuum limit a system at criticality is described as scale invariant. So in this context power laws are quite generally associated with scale invariance. In cosmology this term is special and we return to scale invariance in Chap. 11. The general point, i.e. scale invariance in statistical mechanics versus scale invariance in cosmology, is discussed in [31].
- 6.
Note that the \(l=0\) term is a constant, which can and should be eliminated by a proper definition of the average temperature. The \(l=1\) or dipole term is mainly determined by the Doppler shift due to our motion through space and therefore subtracted in the present context.
- 7.
Note that \(\delta \phi \) is the deviation from the average potential. Overdensity implies \(\delta \phi <0\), as we can see from the Poisson equation in Fourier space, i.e. \(-k^2 \delta \phi _k \sim \delta \rho _k\).
- 8.
Which we evaluate via
$$\begin{aligned} D_{h, p}(t) = \frac{1}{H_o (1+z(t))} \int _{0}^{1/(1+z(t))} \frac{\mathrm{d}x}{x^2 \sqrt{\Omega _v +\frac{\Omega _K}{x^2} + \frac{\Omega _m}{ x^3} +\frac{\Omega _r}{ x^4}}} \end{aligned}$$(10.42)(Problem 3).
- 9.
By choosing the \(\cos \) solution we have tacitly implied an initial condition for the phases of all oscillators, namely that they start from their respective maxima. This is a physically sensible starting condition for an extremely hot plasma where radiation is dominant after inflation. We will see in the next chapter that inflation basically blows up quantum fluctuations of density by a large factor without providing significant relative momentum that would be characteristic of a \(\sin \) mode. We also make the reasonable assumption (known as adiabatic) that density fluctuations for all species align. In principle the overdensity of one species could also initially be compensated by an underdensity of another. This would result in no curvature perturbation initially and thus the modes are called isocurvature. The latest Planck results constrain their contribution to below \(\sim \)2\(\%\).
- 10.
The time t should really be conformal time.
- 11.
The most recent numbers from the PLANCK satellite imply \(\Omega _b\simeq 0.045\).
- 12.
At very small \(l\lesssim 5\) you will notice that the error bar drastically increases. The reason for this is small statistics: there are only \(2l+1\) spherical harmonics for a given l. Physically this means that we cannot measure a lot of unrelated large scale fluctuations from our one observation point in the universe and thus, statistically speaking, our sample size is limited. The phenomenon is referred to as cosmic variance (cf. the remark in Sect. 10.3.1).
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Hentschke, R., Hölbling, C. (2020). Accelerated Expansion of the Universe. In: A Short Course in General Relativity and Cosmology. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-46384-7_10
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DOI: https://doi.org/10.1007/978-3-030-46384-7_10
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