Abstract
Have you ever wondered how the Universe was born, or, at least, how it looked billions and billions of years ago, in times very far from our present epoch? Was the Universe already existing in the very far past, and was it the same as today, or was it much different, or, maybe, had it not yet come into existence?
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Notes
- 1.
- 2.
It is a background of electromagnetic radiation, filling the whole space with an almost uniform distribution of very weak intensity. Detected for the first time by Penzias e Wilson [58], is characterized by a spectral distribution of Planckian type, typical of radiation in thermal equilibrium. Its temperature gradually decreases as the Universe expands, and its present value is about 2. 7 K. According to the standard model, the cosmic background of relativistic radiation should also contain a component of relic neutrinos in thermal equilibrium, at a temperature slightly larger than that of photons (see e.g., [2, 3, 57]). However, because of its very faint intensity, such a neutrino component has never been (so far) detected.
- 3.
Direct observations tell us that the temperature of the cosmic radiation may vary from point to point, deviating from its average value (2. 7 K) with small percent variations not larger than about one part in a hundred thousand. Such tiny inhomogeneities have been directly measured for the first time by the satellite COBE [59] in the Nineties, and later (with always increasing accuracy) by the WMAP experiment in the first decade of this century. Currently, we are obtaining even more precise information thanks to the data collected by the PLANCK satellite, whose first results have been released on March 2013.
- 4.
- 5.
The inflationary phase does not produce new intrinsic inhomogeneities, but simply amplifies the tiny quantum fluctuations unavoidably present in the matter fields and in the cosmic geometry.
- 6.
The inflationary phase, in fact, “pumps up” the sizes of those spatial domains which were small enough to have allowed reciprocal interactions, and makes them larger than the horizon size typical of that epoch (i.e., larger than the maximum distance a light ray would have had time to travel up to that moment). See Fig. 6.1.
- 7.
Such an accelerated phase is qualitatively similar, from a dynamical point of view, to the inflationary phase that the Universe has (or should have) experienced in its very early epochs. The typical acceleration of the inflationary regime, however, is by far more intense than the current cosmic acceleration.
- 8.
The acronym CDM means Cold Dark Matter, and the name ΛCDM emphasizes the fact that, in this model, the cosmic gravitational field is currently generated by two main sources: cold (i.e., non-relativistic) dark matter and a cosmological constant Λ.
- 9.
As discussed, for instance, in the paper by Borde, Guth, and Vilenkin [62].
- 10.
We should keep in mind that such a number (14 billion years), like any other number expressing the duration of a time interval, is always to be referred to some particular observer and some particular choice of the parameter used to measure the time coordinate. In this case the 14 billion years are referred to the so-called cosmic time parameter, which is the time coordinate used by a “comoving” observer, i.e., by an observer placed at a fixed spatial position which gets “carried away” unresisting through space–time by the expansion of the cosmic geometry. What is important, anyway, in our context, is that the temporal distance between the Big Bang and the present epoch is finite: hence, according to the standard model, our Universe is not an “infinitely old” creature (namely, it has not existed forever).
- 11.
The action for the gravitational field, in the theory of general relativity, is determined by a geometric quantity called “scalar curvature,” integrated over the whole space–time region we are considering, and divided by the Planck length squared.
- 12.
The cosmological “pre-big bang” scenario, suggested by string theory, has been formulated and discussed in the works by Gasperini and Veneziano (see for instance [63]). See also the book [64] for a detailed but qualitative introduction to the pre-big bang scenario, and the textbook [42] for a more technical discussion.
- 13.
With the name “string cosmology” we will denote, generically, a cosmological scenario based on (or at least inspired by) string theory.
- 14.
Like the inflationary phase described at the beginning of this chapter, with the only difference that the space–time curvature is growing instead of being decreasing, since inflation occurs before the Big Bang.
- 15.
To be consistent with the conservation of the total charge, of the total angular momentum, as well as with all the existing physical conservation laws, particles have always to be produced in pairs: each produced particle must be associated to a produced antiparticle with opposite charge, opposite angular momentum, and so on.
- 16.
The rank of a tensor is given by the number of indices which characterize its explicit representation, and “counts” the number of its components. For instance, a tensor of rank 1 is represented by an object with a single index: A μ . A tensor of rank 2 is represented by an object with two indices: F μ ν . And so on.
- 17.
From the initial of the names of Bogolmon’y, Prasad, and Sommerfeld.
- 18.
See for instance the paper by Khoury, Ovrut, Steinhardt, and Turok [65]. The name of this scenario comes from the ancient Greek language, and denotes something which is “emerging from fire” (with reference to the old myth of the Arabian Phoenix).
- 19.
As suggested in the paper by Steinhardt and Turok [66].
- 20.
See for instance the paper by Goheer, Kleban, and Susskind [67].
- 21.
Such a maximum entropy is determined by the area of a spherical surface of Hubble radius L H , measured in units of Planck length L P . Hence it is a number of order \(L_{H}^{2}/L_{P}^{2}\).
- 22.
The entropy density of the thermal radiation is proportional to the cubic power of its temperature T H . The entropy of the radiation stored inside a cosmological volume of radius L H is thus a number of order \(T_{H}^{3}L_{H}^{3}\).
- 23.
Such a difference, as we shall see below, has non-trivial impact on various properties of the produced gravitational radiation.
- 24.
See for instance the work by Burgess et al. [68].
- 25.
This optical lattice is due to the topology, because the gravitational force generated by the antibrane can act on the brane propagating along the circles of the torus in one direction an in the opposite one, thus reaching the brane from many points, as if many antibranes were present as sources of the total effective gravitational field.
- 26.
See for instance the paper by Kachru et al. [69].
- 27.
It is a geometry characterized by a negative cosmological constant, typically obtained as a possible solution of the gravitational equations describing the vacuum state of supersymmetric models (including, for instance, superstrings).
- 28.
Described in a recent book by Penrose [70].
- 29.
A black hole is a concentrate of matter so dense as to be contained within a portion of space of radius smaller than its Schwarzschild radius r s , a typical distance which for a static black hole is given by \(r_{s} = 2\mathit{ML}_{P}^{2}\), where M is its total mass. If we consider the Earth, for instance, we find for the Schwarzschild radius a distance slightly less than 1 cm. For an observer outside the black hole, the surface of the Schwarzschild sphere of radius r s represents the so-called event horizon: the gravitational attraction inside this surface is so strong that no classical object or signal is able to propagate to the outside, crossing the horizon.
- 30.
It is the so-called Hawking radiation. Taking into account quantum effects one finds, in fact, that the horizon of a static black hole behaves as a hot body radiating energy at a temperature which is inversely proportional to the horizon radius \(2ML_{P}^{2}\). Because of this energy loss the mass of the black hole decreases in time and, consequently, the horizon radius becomes smaller and smaller, until it totally disappears at the end of the evaporation process.
- 31.
This effect is explained, for instance, in a recent paper by Gurzadyan e Penrose [71].
- 32.
With the name “spectrum”, or spectral distribution, we shall precisely denote the mean energy of the produced gravitons per unit volume and per unit of logarithmic interval of frequency. This quantity represents, from a physical point of view, the energy density of the graviton background at any given fixed value of frequency.
- 33.
There are also inflationary models where the horizon radius L H does not grow in time. For instance, it stays constant in models based on the de Sitter geometry, while it decreases in time in pre-big bang models based on string theory (see Sect. 6.1).
- 34.
Growing spectra are also called “blue”, while decreasing spectra are called “red”.
- 35.
- 36.
See, however, the NOTE ADDED IN PROOF at the end of this chapter.
- 37.
Were it larger, in fact, it would have affected the cosmological dynamics since the nucleosynthesis epoch, in contrast with the results of current observations.
- 38.
They are instruments able to respond to the passage of a gravitational wave, with the function of amplifying the electromechanical effects produced by the wave, and providing a signal strong enough to be detectable. The available gravitational antennas are currently of two types, based, respectively, on the mechanism of the resonant mechanical bar and of the interferometer. There are projects, already in advanced phase, of interferometric antennas to be launched in space and put in orbit around the Sun, in order to achieve a better sensitivity in the low-frequency bands of the gravitational spectrum (see e.g., the book by Maggiore [74]; see also [42] for a discussion focalized on the detection of relic gravitons, or [3] for a textbook in Italian).
- 39.
The anisotropies and inhomogeneities of the cosmic electromagnetic radiation are currently measured with the highest precision at angular scales of the order of one degree (or slightly lower), corresponding to the fluctuations of wavelengths ranging from about λ = L H to λ = 0. 01 L H , where L H is the Hubble radius. The corresponding frequencies range from 10−18 to 10−16 Hz.
- 40.
See, however, the NOTE ADDED IN PROOF at the end of this chapter.
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Gasperini, M. (2014). The Very Early Past of Our Universe. In: Gravity, Strings and Particles. Springer, Cham. https://doi.org/10.1007/978-3-319-00599-7_6
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