Ortiz and colleagues in their recent paper have given a physical explanation to the accelerated expansion of the Universe.
The Einstein Field Equations (EFE), describe the fundamental interactions of gravitation as a result of space-time being curved by means of the mass and energy inside this space-time. In this sense the metric and the stress-energy tensor determine the system, and the dynamics of it can be obtained via the variation principle of the Einstein Hilbert (EH) action plus the matter action field.
In 1917, Einstein realized that the dynamics of his new theory predicted a non permanent universe, simply because all the matter attracts gravitationally. Consequently, influenced by the belief, at the time, that the Universe was static and everlasting, he decided to modify his theory, including a constant parameter, called the Cosmological Constant (CC). Short after, in 1922, Alexander Friedmann, sets bases for the theoretical model for an expanding Universe. The accelerated expansion of the Universe was confirmed with observational evidence by George Lamaître in 1927, and by Edwin Hubble, in 1929; establishing the correlation between the redshift and the distance from its sources.
To the moment, the Lambda Cold Dark Matter (ΛCDM) cosmological model is the best candidate to explain the accelerated expansion of the Universe. The power spectrum of the flat ΛCDM model is the one that best fits the cosmological observations, like Cosmic Microwave Background Radiation (CMB), Baryonic Acoustic Oscillations (BAO) and large scale structure formations, among others. Most of the parameters of the model are well understood, except the CC, the one parameter that rules the accelerated evolution of the Universe.
There is a strong disagreement between the predicted dark energy density and the observed one, which is of the order of magnitude of the current matter energy density. This problem is known as the Cosmological Constant Problem.
Under the standard cosmological model ΛCDM, recent measurements of the expansion rate of the Universe, at low redshifts, appear to be in disagreement with the predictions for observations of the CMB, this disagreement is known as the Hubble’s Tension.
Even more so, the fact that the ratio of the baryonic energy density to dark energy density at present time is of the order of one in the ΛCDM model, makes Ortiz and colleagues look up for a physical explanation for such a remarkable coincidence, the “Cosmological Coincidence Problem” in their recent paper.
Scientists have been proposing many alternative models and theories, in order to explain the expansion phenomenon, the Coincidence Problem and Hubble’s Tension. Basically, such explanations can be sorted into two different kinds: one is that the problem depends on the matter content, i.e. the stress-energy tensor, Tµν, and that there must be some extra energy fluid that may fill the space-time, e.g., quintessence; another possibility states that the problem rests on the geometric sector, implying that Einstein’s theory could be wrong or incomplete e.g., Modified Gravity.
In order to address these problems, now Ortiz and colleagues in their paper consider the surface forces of a homogeneous and isotropic cosmological system. Taking into account the surface energy of the analyzed system, a new relativistic effect, never considered before, is derived, modifying the system of equations that governs the evolution of the expansion of the Universe.
“We showed that the inclusion of a surface tension term can explain why the space-time fabric is stretching out, and leads to modify the usual Friedmann equations.”, said Ortiz.
They proposed ST model. The strength of their model resides in its simplicity. Due to the way the model was developed, it preserves all the desirable futures of the ΛCDM model, with one less parameter to be fixed. It also possesses desirable futures of the phantom w = – 4/3 cosmological model, without violating the null energy condition.
Their model ensures homogeneity and isotropy over the whole space-time – it explains: a) the Cosmological Coincidence Problem, b) why there is no need for such a cosmological constant to explain the acceleration of the Universe, and c) why we have not found any particle or fluid responsible for the dark energy component. Wanna have a closer look? So, let’s start with the hypothesis.
Yeah, they introduced one hypothesis based on the surface tension, γS, and its implications in the dynamics of a perfect fluid density. Surface tension, as you know, it is associated with the nature of the chemical bonds of atoms (electromagnetic force) at the surface of an interface between different materials; nevertheless, the surface tension is related to the Euler–Cauchy stress principle which is a basic concept of continuum mechanics. If you don’t know about this principle, let me tell you, this principle states that upon any close surface (real or imaginary) that divides a body, the action of one part of the body on the other is equivalent to its external forces acting on it. For bodies in continuous media, there are two types of external forces:
• Body forces, (f).
• Surface forces or stress. (F).
Thus, the total force ‘F’ applied to a body or to a portion of the body is the contribution of all the forces. The interactions between pairs of atoms or molecules, are usually modeled with the Lennard-Jones potential. This is a good approximation for short interaction distances, since it is related to the Pauli exclusion principle and the Van der Waals force due to intermolecular forces. The gravitational potential, being several orders of magnitude less than the electromagnetic potential, is neglected.
“We know that the gravitational potential is the one that acts at large distances,
so we worked under the hypothesis that surface tension due to this potential is relevant at large scales.”, said Ortiz.
They applied the Euler-Cauchy stress principle to the thermodynamic equation, and arrived to a density evolution equation:
The resulting density equation takes into account the matter content inside the system and the surface tension of the given system. It is worth noting that the initial density ρ0 in above equation is the same for both RHS terms, since it is a homogeneous system.
So as to consider some extra different density fluids that do not interact among them, the superposition principle is valid, so the resulting density equation would be a linear combination of the different fluids densities. Based on the Euler-Cauchy stress principle, the above density equation is true for any spherical surface chosen in the cosmological domain.
After that, they stated, the equation of the dynamics of the Universe, by taking into account both effects: the gravitational one, due to the inner region, plus a general relativistic effect due to the surface tension. They actually obtained resulting acceleration equation from modified Friedmann equation. I mentioned both equations below just have a look:
So, what they actually note here is that the positive sign on the last term of the acceleration equation, is responsible for the accelerated expansion of the Universe; this acceleration is due to the surface energy of the considered system. Yeah, what you heard is right.. They also note that this term is time dependent.
Under these circumstances, the surface energy acts much like the cosmological constant. This term gives the information on how the surface tension stretches out the space-time fabric, which explains why there is no such a dark energy fluid or particle.
These equations resemble the Friedmann Equations of a ΛCDM model, with the advantage of one less parameter to be fixed in the model, since ρ0 is obtained by the system conditions.
Their model satisfies the Dominant Energy Condition (DEC) ρ ± P ≥ 0, since ρ > 0 which was seen by us on density evolution equation, and yes, they only considered dust and radiation matter content. Under this hypothesis, the accelerated expansion of the Universe is the result of the surface tension, rather than the consequence of a negative pressure.
Have I missed something?? Yeah, right, explanation they had given on ‘cosmological Coincidence problem’.
Well friends, the information that our Universe is at an accelerated expansion epoch, comes from the observation of the redshifts of the frequency of light emitted by distant sources. In order to contrast proposed model with the cosmological observations, Ortiz and colleagues rewrite the Friedmann Equations (i.e. Modified Friedmann and resulting acceleration) in terms of the redshift parameter and found one critical density relation:
They note that the first term on the RHS of the critical density relation, is the matter energy density term, Ω0m, and that this term is of the same order of magnitude as the third term, the energy density of the surface tension, Ω0mπ, which is responsible of the accelerated expansion of the Universe. The nature of this last term gives an explanation to the Cosmological Coincidence Problem.
“Under our stated hypothesis, that the surface tension is responsible for the accelerated expansion of the Universe, we can conclude that there is no such Hubble’s tension; the problem is not in the measurements data technique, but in the employed cosmological model.”, said Ortiz.
They concluded that the proposed model is in excellent agreement with the observations. The model could be further developed to include additional degrees of freedom, which could be done by adding extra fluids or considering a non flat-space.
Reference: C. Ortiz et al., “Surface Tension: Accelerated Expansion, Coincidence Problem & Hubble Tension”, International Journal of Modern Physics D, 2020. https://www.worldscientific.com/doi/10.1142/S0218271820501151 https://doi.org/10.1142/S0218271820501151
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