Abstract
The attentive reader must have noticed that, in fact, we have given no actual initial conditions on cosmological perturbations in Chap. 6. What we have done is to give the form of the primordial modes and show how, in the 5 different cases that we investigated, all scalar perturbations are sourced by a single scalar potential. For example, in the adiabatic case we showed in Eq. 6.77 how \(\Phi (\mathbf k)\) is related to \(C_\gamma (\mathbf k)\), which we proved to be equal to \(\zeta (\mathbf k)\) and \(\mathcal R(\mathbf k)\).
The more the universe seems comprehensible, the more it also seems pointless
Steven Weinberg, The first three minutes
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Notes
- 1.
This calculation is essentially identical to the one which leads to the definition of the Fermi momentum of a gas of fermions, which might be more familiar to the reader. See e.g. Huang (1987)
- 2.
We drop here the subscript of \(\delta \) in order to keep a light notation. The treatment can be in principle applied to any \(\delta \), but the most interested case is that for matter, i.e. CDM plus baryons.
- 3.
The expansion is usually considered for \(\Delta T\), the temperature fluctuation. In this case, the \(a_{T,\ell m}\)’s carry dimensions of temperature. The only difference between the coefficients of the two expansions is a factor \(T_0\), i.e. the temperature of the CMB.
- 4.
The notation \(Y_{\ell m}\) usually denotes spherical harmonics in the real form, obtained by combining \(Y^m_\ell \) in a suitable way in order to trade the imaginary exponential \(\exp (im\phi )\) for a sine or cosine function of \(m\phi \).
- 5.
According to the Copernican principle we do not occupy a special position in the universe. Therefore, its average properties that we are able to determine should be the same as those determined by any other observer.
References
Ade, P.A.R., et al.: Planck 2015 results. XVII. Constraints on primordial non-Gaussianity. Astron. Astrophys. 594, A17 (2016)
Huang, K.: Statistical Mechanics, 2nd edn. Wiley-VCH, New York (1987)
Lyth, D.H., Liddle, A.R.: The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure (2009)
Schwarz, D.J., Copi, C.J., Huterer, D., Starkman, G.D.: CMB anomalies after Planck. Class. Quant. Grav. 33(18), 184001 (2016)
Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008)
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Piattella, O. (2018). Stochastic Properties of Cosmological Perturbations. In: Lecture Notes in Cosmology. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-95570-4_7
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