Dark Energy Models and Scalar Fields
a scalar field. From a mathematical point of view, the equations that govern the evolution of these scalar fields, introduced by the theory, are analogous to the state equations of a perfect fluid. In these equations, the pressure acting on space is proportional to the energy density with a parameter $w$ variable from theory to theory. In the case of the cosmological constant, for example, we have seen that the parameter in question is constantly equal to -1 (see box), conversely in other scenarios this can be a time-variable function. If, therefore, the function that governs the state equation of dark energy has a current value close to -1 (derived from observations) this does not absolutely prejudice other radically different values in other cosmic periods, such as during an inflationary period or whatever. In this context there are therefore a plethora of different scenarios each of which with merits and defects, suggestive hypotheses, predictions and risks. The main subdivision, even if it is not the only possible one, is the one based precisely on the state equation presented previously. Considering that to be in agreement with the data of an expanding universe it must be $w < -\frac{1}{3}$, we therefore have three main scenarios, even if completely indicative and which take different names:
- Quintessence Models
- Cosmological Constant Model
- Phantom Energy Models
The Big Rip
In the last of these categories, those of phantom energy, fall all those models that predict an exponential expansion rate of the universe that would lead to the Big Rip. The Big Rip or Great Tear would consist of such a high acceleration of the universe as to progressively and inexorably restrict its observable part. In this catastrophic scenario the pressure exerted by dark energy would be so strong as to literally shatter the universe, reducing its visible horizon more and more. Cosmology, in fact, is not new to this kind of catastrophic predictions, we can in fact remember the scenario that terrorized the physicists of the late nineteenth century and that went under the name of Big Freeze or Great Cold. Subsequently, in the post-Einsteinian era this Great Cold was replaced by the Great Clash or Big Crunch, which postulated the end of the universe in a single black hole due to the contraction of the universe. Once it was seen that this contraction was no longer possible due to the acceleration of galaxies, the current bogeyman became the Big Rip or Great Tear given by excessive acceleration of the universe. Apart from sensationalistic questions, the models that resort to the introduction of a new scalar field have incredibly expanded the range of possible answers and possible questions. How, for example, is cosmic acceleration linked to inflation? Is there a connection between dark energy and dark matter or dark energy and neutrino mass? None of these questions currently has a firm and decisive answer and the introduction of a new scalar field associated with dark energy obviously allows for a greater number of possible interpretations and solutions.
$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$
Cosmological Constant and Vacuum Energy
Let's rewrite Einstein's field equations with the cosmological constant:
$$G_{\mu\nu} + g_{\mu\nu}\Lambda = 8\pi G T_{\mu\nu}$$
After years of reflections, physicists have thought that they could effectively conceive this term simply by transporting it to the other side and thinking of it as a contribution to the Energy tensor:
$$G_{\mu\nu} = 8\pi G \left(T_{\mu\nu} - \frac{\Lambda}{8\pi G}g_{\mu\nu}\right)$$
Now, considering that the metric is:
$$g_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & a^2 & 0 & 0 \\ 0 & 0 & a^2 & 0 \\ 0 & 0 & 0 & a^2 \end{pmatrix}$$
and the energy tensor:
$$T_{\mu\nu} = \begin{pmatrix} \rho & 0 & 0 & 0 \\ 0 & Pa^2 & 0 & 0 \\ 0 & 0 & Pa^2 & 0 \\ 0 & 0 & 0 & Pa^2 \end{pmatrix}$$
The new term can be thought of as:
$$T'_{\mu\nu} = \begin{pmatrix} -\rho - \frac{\Lambda}{8\pi G} & 0 & 0 & 0 \\ 0 & -P + \frac{\Lambda}{8\pi G} a^2 & 0 & 0 \\ 0 & 0 & -P + \frac{\Lambda}{8\pi G} a^2 & 0 \\ 0 & 0 & 0 & -P + \frac{\Lambda}{8\pi G} a^2 \end{pmatrix}$$
In conclusion, vacuum energy can be thought of as a contribution with density:
$$\rho_{vac} = \frac{\Lambda}{8\pi G}$$
and pressure:
$$p_{vac} = -\frac{\Lambda}{8\pi G}$$
therefore with equation of state:
$$p_{vac} = -\rho_{vac}$$
Dirac and the Game of Constants
One of the founding fathers of quantum field theory, Paul Dirac, in a series of subsequent works, one from 1937 and the other from 1974, highlighted a numerical coincidence that, in his opinion, required a causal explanation. Trying to reconcile Gravity and Electromagnetism, Dirac had begun to play with some of the fundamental constants of this field trying to derive some pure numbers. However, continuing in his calculations, Dirac soon realized that the orders of magnitude obtained always turned out to be more or less the same. In particular, Dirac realized that three fundamental ratios shared the same extremely high order of magnitude of $10^{40}$, namely:
- the ratio between gravitational and electric force
- the ratio between electron radius and observable universe
- the ratio between Compton length and Schwarzschild radius
Dirac was so impressed by this numerical coincidence that he believed he could found a new basis for modern Cosmology on these elements. The three numbers in question, in fact, brought together apparently very heterogeneous constants among them, but which, in a certain sense, arose naturally when one tried to reconcile phenomena such as gravitation and electromagnetism.
Ratio between Gravitational and Electric Force
The first of these numbers highlighted by Dirac concerned the ratio between Gravitational Force and Electric Force between atomic particles. An electron and a proton are mutually attracted by two types of forces: the gravitational one expressed by Newton's law, proportional to the two masses of the particles, and the electric one expressed by Coulomb's law proportional to the two charges. Calling therefore $m_p$ the mass of the proton, $m_e$ the mass of the electron and with the letter $e$ the unit electric charge of the two, we have that the gravitational force and the electric force can be simply expressed by the two laws.