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Review of Modern Cosmology

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Cosmological Probes of Light Relics

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Cosmology is the quantitative study of the structure and evolution of the universe. In the last few decades, it has emerged as a data-driven field of study which has revolutionized our understanding of the cosmos. In this chapter, we discuss both the theory and the observations underlying modern cosmology. We consider in particular the basics underlying the standard model of cosmology, the thermal history of the universe and the fluctuations around the smooth universe. In addition, we review the main cosmological observables: the cosmic microwave background, the large-scale structure of the universe and the baryon acoustic oscillations.

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Notes

  1. 1.

    For convenience, we invert the otherwise standard notation and use overdots (primes) to denote derivatives with respect to conformal (physical) time \(\tau \) (t), i.e. \(\dot{a} \equiv \mathrm{d}a/ \, \mathrm{d}\tau \) and \(a' \equiv \mathrm{d}a/ \, \mathrm{d}t\). The physical Hubble rate is therefore given by \(H = a'\!/a\).

  2. 2.

    The minimal sum of masses from neutrino oscillation experiments is about 58 meV [17] and in particular cosmological measurements are closing in on this value with a current upper bound of \(\sum _i m_{\nu _i} < 0.23\,\mathrm{{eV}}\) (95 % c.l.) [8]. As the masses are so small that neutrinos have been (ultra-)relativistic for a large part of cosmological history, especially around the time of photon decoupling (\(T_\mathrm {rec}\approx 0.26\,\mathrm{{eV}}\)), their effect on the aspects of interest in this thesis is small. For simplicity, we will therefore treat neutrinos as massless particles throughout, except in the BOSS cosmology of Chap. 7.

  3. 3.

    Although we use the same symbol to denote the optical depth and conformal time, its meaning will always be clear from context.

  4. 4.

    We will come back to this assumption and correct for it in Sect. 3.3.

  5. 5.

    Recombination and (photon) decoupling are often used synonymously, in particular when referring to the time of decoupling, but actually are different processes. We will keep these two notions distinct in those cases where it is important and it will be apparent in all other cases.

  6. 6.

    There may also exist isocurvature perturbations for which the density fluctuations of one species do not necessarily correspond to density fluctuations in other species. These are disfavoured by current observations, in particular those of the CMB anisotropies.

  7. 7.

    The radiation perturbations oscillate on small scales (cf. Sect. 2.3.2). After time-averaging over a Hubble time, these perturbations can however be neglected and the potentials are only sourced by the matter fluctuations [30]. We can therefore neglect the time derivatives of the potentials on subhorizon scales, \(k^2\Phi \gg \ddot{\Psi },\mathcal {H}\dot{\Psi }\).

  8. 8.

    Once dark energy takes over as the main component of the universe, the clustering of matter stops and the growth of structures is halted by the accelerated expansion of the universe.

  9. 9.

    Accurately computing the evolution of all perturbations in the universe requires solving many coupled equations as the interactions between the various species have to be captured by a set of Boltzmann equations. This can only be done numerically which is achieved in the current state-of-the-art Boltzmann solvers CAMB [34] and CLASS [35]. Nevertheless, it is instructive to obtain approximate analytic solutions and get analytic insights in order to deepen our understanding of the underlying physics (cf. e.g. Chap. 5).

  10. 10.

    This is a good approximation in the matter-dominated era. During radiation domination, the time evolution of the gravitational potential around sound-horizon crossing leads to the radiation-driving effect which we will discuss below.

  11. 11.

    Note that we could of course rewrite this solution in terms of an amplitude \(A_{\varvec{k}}\) and a non-zero phase \(\phi _{\varvec{k}}\), i.e. \(\cos (c_s k \tau ) \rightarrow \cos (c_s k \tau + \phi _{\varvec{k}})\), with \(B_{\varvec{k}}= 0\) implying \(\phi _{\varvec{k}}= 0\).

  12. 12.

    As decoupling happens during the epoch of recombination, we use \(\tau _\mathrm {rec}\) instead of \(\tau _F\) or \(\tau _\mathrm {dec}\) to specify the time of photon decoupling. This will allow easier discrimination of other freeze-out events in later parts of this thesis.

  13. 13.

    To be precise, the sound horizon is given by \(r_s(\tau ) = \int _0^\tau \!\mathrm{d}\tau \, c_s(\tau )\), which also captures the small time dependence of \(c_s\) that we however neglect in our analytic treatment.

  14. 14.

    The size of the sound horizon imprinted in the baryon perturbations is slightly larger than the size observed in the CMB anisotropies. The latter is set at the time when most photons had decoupled from the baryons. At this point, the baryons however still feel the drag of photons and essentially remain coupled to the photons because there are about \(10^{9}\) times more photons than baryons. The end of the so-called drag epoch, \(\tau _\mathrm {drag} > \tau _\mathrm {rec}\), marks the time when baryons finally loose this contact. The two sizes of the sound horizon inferred from the latest CMB measurements are \(r_s(z_\mathrm {rec}\approx 1090) \approx 145\,\mathrm{{Mpc}}\) and \(r_s(z_\mathrm {drag} \approx 1060) \approx 148\,\mathrm{{Mpc}}\) [8], i.e. they are relatively close, but different at a significance of more than \(8\sigma \) at the current level of precision.

  15. 15.

    In this context, CMB experiments are surveys which map the CMB anisotropies in the sky. These measurements are usually not taken at a single, but at several frequencies. This is to reliably subtract galactic and astrophysical foregrounds, which are other sources of microwave emission and polarization originating, in particular, from galactic dust. We will generally assume that these foregrounds have been accounted for (or their effects can easily be marginalized over) so that we have direct access to the primordial signal.

  16. 16.

    We could have equivalently obtained the angular power spectrum by decomposing the temperature fluctuations \(\delta T\) into spherical harmonics and computing the correlation function of the expansion coefficients. This is how measurements of \(\delta T\) are commonly processed to obtain the power spectrum \(C_\ell ^{TT}\).

  17. 17.

    Cosmic variance refers to the statistical uncertainty inherent in cosmological measurements since we are only able to measure one realization of the true model underlying the universe. In a cosmic variance-limited measurement, the statistical error is dominated by this uncertainty, which is given by \(\Delta C_\ell = \sqrt{2/(2\ell +1)}\,C_\ell \) for a CMB auto-spectrum.

  18. 18.

    The location of the first peak is very sensitive to the curvature of the universe via the distance to last-scattering. The measurement of this peak famously led to the conclusion that our universe has a geometry that is very close to flat [55], which laid the groundwork for the \(\Lambda \mathrm {CDM}\) model. Today, the famous \(\Omega _m\)-\(\Omega _\Lambda \) plot shows that the confidence regions inferred from CMB, BAO and supernovae data, which all have different degeneracy lines, intersect in a single small region that is consistent with a flat universe, \(\Omega _M+\Omega _\Lambda +\Omega _r = 1\).

  19. 19.

    In analogy with the properties of the electric and magnetic fields in electrodynamics, E-mode and B-mode polarization is curl- and divergence-free, respectively.

  20. 20.

    A fraction of the E-modes is converted to B-modes in the late universe through gravitational lensing (see below). In contrast to the primary B-modes, these induced B-modes have been detected and can in principle be used to revert the effects of lensing on the temperature and E-mode spectra in a process referred to as delensing.

  21. 21.

    For simplicity, we set the primordial spectral tilt to unity, \(n_\mathrm {s}= 1\), in this discussion, i.e. assume a perfectly scale-invariant primordial power spectrum.

  22. 22.

    Although we observe the BAO signal as acoustic peaks both in CMB and in LSS measurements, we will usually refer to the latter when we mention BAO observations. Strictly speaking, the spectrum of baryon acoustic oscillations is \(P^\mathrm {w}(k)\). For convenience, we will however also refer to O(k) as the BAO spectrum.

  23. 23.

    Another common dataset that is used to break degeneracies are the local \(H_0\) measurements from supernovae. Having said that, with both global, such as those from the CMB and BAO, and local measurements improving, the inferred values of \(H_0\) are currently statistically discrepant at the \(3\sigma \) level [102]. It is however questionable whether we should be paying too much attention given the vast statistical power of the CMB in particular (see e.g. [103]).

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  1. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA

    Dr. Benjamin Wallisch

  2. Department of Physics, University of California, San Diego La Jolla, CA, USA

    Dr. Benjamin Wallisch

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Wallisch, B. (2019). Review of Modern Cosmology. In: Cosmological Probes of Light Relics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-31098-1_2

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