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Early Universe Cosmology

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Book cover The B−L Phase Transition

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Abstract

The main intention of this thesis is to motivate and investigate the \(B\)–\(L\) phase transition as the possible origin for the thermal phase of the hot early universe. Before we are ready to do so, we have to acquaint ourselves with the observational evidence for this phase and understand which physical processes have or may have taken place in it. For this reason we shall provide a brief review of early universe cosmology in this chapter, thereby compiling the background material for the further discussion. We will first discuss the present composition of the universe (cf. Sect. 2.1) and then some of the main events in the thermal history of the universe in reverse chronological order (cf. Sect. 2.2). We would like to emphasize that in this introductory chapter we will crudely restrict ourselves to aspects which are relevant for our purposes. More balanced and comprehensive presentations of the topic are for instance provided in standard textbooks [1–3] or dedicated review articles.

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Notes

  1. 1.

    Note that in the recent cosmic past, shortly after the onset of star formation, the entropy contained in black holes has come to dominate over the entropy in radiation [18].

  2. 2.

    In the following discussion we shall restrict ourselves to the relic abundance of primordial neutrinos. If neutrinos are Dirac fermions, the abundance of antineutrinos should at each time be approximately the same as the abundance of neutrinos.

  3. 3.

    In order to ensure that the universe as a whole is electrically charge neutral, there has to be present one electron for each proton in the universe. As a single proton is, however, roughly \(1,\!800\) times heavier than an electron, the contribution from electrons to the total energy presently stored in matter is negligibly small, which is why we will not consider it any further.

  4. 4.

    It is also large compared to the observed abundance of luminous matter. The density parameter of stars is smaller than \(\Omega _b^0\) by one order of magnitude, \(\Omega _{\text {stars}} \simeq 2.7 \times 10^{-3}\) [27]. Most baryons are thus optically dark, probably contained in some diffuse intergalactic medium [28].

  5. 5.

    Antiparticles of cosmic origin such as antiprotons and positrons are seen in cosmic rays. Their fluxes are, however, consistent with the assumption that they are merely secondaries produced in energetic collisions of cosmic rays with the interstellar medium rather than primordial relics.

  6. 6.

    For recent reviews on dark matter, cf. for instance Refs. [32–35]. Another ansatz to account for the various observed, but unexplained gravitational effects is to modify the theory of general relativity. While modifications of gravity (cf. in particular Refs. [36, 37]) are often able to explain isolated phenomena, they usually struggle to give a consistent description of all observed phenomena, which is why we will not consider them any further in this thesis.

  7. 7.

    Seminal works in this field have been the observations by Vera Rubin and Kent Ford, who measured the rotation curve of the Andromeda Nebula in 1970 [38], as well as by Sandra Faber and Robert Jackson, who studied stellar velocities in elliptical galaxies in 1976 [39].

  8. 8.

    The first astronomer to stumble upon the problem of the missing mass in galaxy clusters was Fritz Zwicky. In 1933, observations of the Coma Cluster led him to conclude that the galaxies in the cluster should actually fly apart, if there were not large amounts of invisible matter present in it, holding them together [41]. Zwicky is hence usually credited as the discoverer of dark matter.

  9. 9.

    As light neutrinos turn nonrelativistic only at very late times in the cosmological evolution, they represent, in fact, a form of hot dark matter in the current universe.

  10. 10.

    Later on we shall use a rounded version of the value in Eq. (2.15), namely \(\Omega _{\text {DM}}^{\text {obs}}h^2 = 0.11\).

  11. 11.

    Certain scenarios of warm dark matter or mixed dark matter which is composed of a mixture of cold, warm and or hot components, are also admissible [43, 44]. Likewise, also small amounts of baryonic matter in the form of massive compact halo objects (MACHOs) [45, 46] and or cold molecular gas clouds [47] may well contribute to the dark matter in galaxy halos.

  12. 12.

    The accelerated expansion of our universe became evident for the first time in measurements of the distance-redshift relation of high-redshift type Ia SNe in 1998 [48, 49].

  13. 13.

    Naively one might expect the energy density of the vacuum to be related to the Planck scale, \(\rho _\Lambda \sim M_P^4\). Interpreting dark energy as the energy of the vacuum, one then has to explain why \(\rho _\Lambda \simeq 0.73 \rho _c^0 \sim 10^{-123} M_P^4\). For a classic discussion of this so far unsolved problem cf. Ref. [55].

  14. 14.

    Given the allowed range of the total neutrino mass (cf. Eq. (2.7)), matching the two expressions for \(\rho _{\nu _i}\) in Eqs. (2.18) and (2.19) and solving for \(z\) shows that the heaviest neutrino, which eventually contributes most to \(\Omega _\nu ^0\), turns nonrelativistic at a redshift of \(\mathcal {O}(10..100)\).

  15. 15.

    Curiously enough, the matter-dominated era lasts sufficiently long to allow for the formation of such complex structures as galaxies, solar systems and human beings, which, from the perspective of mankind, appears to be a fortunate cosmic coincidence. The question of why dark energy becomes relevant exactly at the present time, i.e. why presently \(\Omega _\Lambda \sim \Omega _m\) rather than \(\Omega _\Lambda \ll \Omega _m\) or \(\Omega _\Lambda \gg \Omega _m\), is one of the greatest puzzles of modern cosmology. Cf. e.g. Ref [56].

  16. 16.

    Prior to hydrogen recombination, at \(T \sim 0.5\,\text {eV}\), helium decouples in a similar way. As hydrogen is still fully ionized at this time, the universe remains opaque after helium recombination.

  17. 17.

    For reviews on the physics of the CMB and its potential to constrain cosmological models, cf. for instance Refs. [58, 59].

  18. 18.

    Perturbations in the photon-baryon fluid can only evolve causally as long as they extend over scales smaller than the sound horizon. This explains the position of the first acoustic peak in the CMB power spectrum. It is located at an angular scale of roughly \(1^\circ \) or equivalently at \(\ell \sim 200\), which corresponds to the angular diameter of the sound horizon at last scattering.

  19. 19.

    Given a scale factor \(a \propto t^p\), \({\Omega _{\text {tot}} - 1}\) scales like \(\dot{a}^{-2} \propto t^{2(1-p)}\). During the phases of radiation and matter domination we respectively have \(p = 1/2\) and \(p = 2/3\).

  20. 20.

    For reviews on BBN, cf. for instance Refs. [62, 63].

  21. 21.

    Data on helium-3 solely derives from the solar system and high-metallicity regions of ionized hydrogen in our galaxy, which makes it difficult to infer its primordial abundance. On top of that, the theory of stellar helium-3 synthesis is in conflict with observations. For these two reasons, helium-3 is usually not used as a cosmological probe.

  22. 22.

    Recall that BBN enables us to trace the evolution of the hot thermal phase up to temperatures as high as \(T_{\text {RH}}^{\text {min}} \simeq 4\,\text {MeV}\) or equivalently cosmic times as early as \(t \simeq 0.05\,\text {s}\) (cf. Sect. 2.2.2).

  23. 23.

    The QCD scale \(\Lambda _{\text {QCD}}\) corresponds to the energy scale at which, according to its renormalization group running in perturbative QCD, the strong coupling constant \(g_s\) formally diverges.

  24. 24.

    Likewise, when referring to some Higgs product operator \(s^\dagger s\) acquiring a VEV \(v\), we will also sometimes write \(v = \left\langle s\right\rangle \), although we actually mean \(v = \left\langle s^\dagger s\right\rangle ^{1/2}\).

  25. 25.

    Gauge configurations belonging to different homotopy classes are transformed into each other via large gauge transformations.

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Schmitz, K. (2014). Early Universe Cosmology. In: The B−L Phase Transition. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-00963-6_2

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