Realistic Cosmology is a Consequence of (only) the Co-operation of Physics and Mathematics

In this short note, I wish, because of many years of research and teaching experience, to emphasize the way physics and mathematics, as two entirely different areas of knowledge, should be used in Cosmology, aiming at the study of the Universe as a whole, its beginning, its evolution and its possible end. This studydescription of the cosmological Universe is based on the use of Einstein’s General Relativity Theory (GRT), as the underlying, wellestablished and complete theory for the description of the universal gravitational field.

Up to now, it is true, unfortunately, that large-scale cosmological research, based on GRT, has been only mathematically oriented, but, not, at the same time, also physically oriented. In a little more detail, in describing the Universe as a whole, we have many reasons to assume that, on the very large scales, the content of the Universe is in the form of a continuous gravitational physical system, the so-called cosmic fluid, composed of mass and energy. {In other words, the Universe is not treated as a system of galaxies, each one (practically, a point mass) of linear dimensions small, almost negligible compared to the mutual distances of neighboring galaxies in an otherwise empty spacetime}.

The mass (both baryonic and dark) of this continuous medium is gravitational, namely, it creates a field of the familiar attractive gravitational forces, which tend to shrink the physical system under consideration.

Also, according to the well-known special-relativistic formula E=mc², any amount of energy, E, is dynamically equivalent to a certain amount of mass, m. Therefore, every form of energy, like mass, creates a gravitational field (interestingly, however, not always one of attracting forces), and so both, mass and energy of the cosmic fluid, are responsible for the universal gravitational field, which, in turn, is responsible for and produces all the observed motions and phenomena in the Universe.

Until very-very recently, however, in the so-called Field Equations of the GRT, all forms of energy of the cosmic fluid (except for only the mass density) have not been considered as source of the universal gravitational field; they have been ignored and omitted. Such thermodynamic forms of energy are, e.g., pressure, temperature of matter, heat quantity, heat capacity, thermodynamic transformations, polytropic transformations, internal thermodynamic motions, internal thermodynamic energy, entropy, as well as other statistical- physics characteristics et c.

(In our opinion, emphatically expressed up to now), at exactly this point lies the real, physical problem with the research in Cosmology, remaining, practically, untouched (even during the last fifteen or so years). This most serious problem, since long ago, appeared as follows: About a century ago (in 1932; see [1]), two great scientists (Albert Einstein and Willelm de Sitter) proposed a solution to the field equations of GRT describing the so-called Einstein-de Sitter cosmological model, composed only of mass (then simply baryonic!), nothing else. But (as an expression of the agony for finding, at last, a mathematical cosmological solution to the field equations of GRT!) with the additional (and on purpose!), mathematically simplifying assumption of “no thermodynamic content at all of the model”. And this in the general spirit, of the totally mathematically-oriented cosmological research philosophy of that epoch (rather justified, however, due to the, till then, lack of many observational cosmological data).

From a physical point of view, it is truly disappointing and curious that, until very-very recently, really very-very recently(!), to the above absolutely mathematically oriented assumption of no thermodynamic content at all, not any special attention, either physical or mathematical, was given at all, especially to the fact that the cosmic fluid had no pressure (Since 1932, thermodynamically, the Universe, tacitly, was considered, accepted and treated as dust!).

The important consequences of such assumptions have prevented cosmologists from solving (rather resulted in misinterpreting!) many current cutting-edge cosmological problems. Especially, that, in the currently prevailing ΛCDM (Λ Cold Dark Matter) Cosmological Model, of generally vanishing pressure [with Λ being the (resurrected) unknown cosmological constant], 5% of the mass-energy content of the cosmic fluid is our familiar “baryonic or luminous matter”, 25% is the so-called and unknown “dark matter”, and the rest 70%, exotic and totally unknown, was given the name “dark energy”. And so, 95% of the Universe’s massenergy content remained unspecified!

The way out of this ΛCDM dead end, concerning (at least) the dark energy, was revealed, about 14 years ago (in 2011), only when all the above-mentioned physical thermodynamic characteristics of the cosmic fluid had been consistently taken into account as sources of the universal gravitational field.

Particularly, the matter, baryonic and dark, of the cosmic fluid is not standing there immovable and frozen, as in the case of the zeropressure ΛCDM model. Instead, and quite physically, the motions of the fluid are treated as polytropic flows (surely, not geodesic motions!) and the fluid’s particles are characterized by their mutual collisions. So, the thermodynamic kinetic energy of these particles is distributed, with, as a result, the polytropic fluid-Universe to acquire the properties of a continuous thermodynamic medium.

This model, (in contrast to the adiabatic and isothermal cosmological models) is the polytropic cosmological model with collisions and thermodynamic content, and, in the international bibliography, it is known as the Polytropic Collisional Cold Dark Matter (PC/CDM) Model. In its context, so far, almost all the current cutting-edge cosmological problems have been confronted successfully, in a self-consistent and classical way, based on only Classical General Relativity Theory and Classical Thermodynamics. In other words, as acknowledged officially by the International Astronomical Union themselves since 2022, neither Quantum Theory nor Modified Gravity are necessary for the dynamical description of the observationally approachable and realistic Universe. And, as is generally accepted, in this thermodynamic way of thinking, the polytropic gas behaves as the (exotic and totally unknown) dark energy, and the polytropic model leads to a suitable fitting with the observational data about the current expanding era (after the recombination cosmological epoch).

a. Quite interestingly, the current cutting-edge cosmological problems, definitely solved in the PC/CDM model, are, e.g.,
-the spatial flatness of the Universe,
-the dark energy (as the thermodynamic internal energy of the polytropic cosmic fluid),
-the cosmic acceleration {because of the relative importance of the (changing in time due to cosmic expansion) mass density and thermodynamic characteristics},
-the transition from the initially cosmic deceleration to cosmic acceleration,
-the coincidence problem (why the cosmic acceleration happened “now”),
-the deceleration parameter,
-the age of the Universe,
-the (dead and unnecessary anymore) cosmological constant,
-the Hubble Diagram of the distant SNe Ia supernovae,
-the non-existing distant SNe Ia supernova dimming (as an illusion in the ΛCDM model),
-the resulting range of values of the polytropic index,
-the behavior of the total Equation-of-State (EoS) parameter,
-the behavior of the associated jerk parameter,
-the rest-mass energy and dark energy equality etc.

b. Additionally, the PC/CDM model
-makes some new predictions for the properties of matter (baryonic and dark), to be verified or not by observations (e.g., the so-called Hubble Tension),
-it is compatible with the measurements concerning the Cosmic Microwave Background (CMB)-shift parameter,
-it explains the spectral index of cosmological perturbations and, finally,
-it possesses a well-defined ΛCDM-limit, in which case all the results of the ΛCDM Cosmology are recovered.

In view of all the above and of the many problems solved, the PC/CDM model emerges as a new cosmological reality, thus arising as (possibly, one of the very few such models up to now) viable alternative to the ΛCDM model! (For a recent review, details and further extended international bibliography, see [2]; see also [3] for a simple exposition).

In conclusion therefore, as in the Summary above: Ambitious cosmologists, especially the younger ones, must always remember that, their research effort towards realistic cosmology, is not enough to be based on the use of simply Mathematics, or simply impressive photographs, or simply powerful electronic computers, or, even, simply particle accelerators, but, beyond and above all these, firstly and mainly, it is imperative that it is based on the philosophy and on the use of Physics!

The author declares no conflict of interest.

1. Einstein A, de Sitter W (1932) On the relation between the expansion and the mean density of the Universe. Proc Natl Acad Sci 18(3): 213-214.
2. Kleidis K, Spyrou NK (2023) The dark energy as a natural property of cosmic polytropes - A Tutorial. 2023, Review Article, in Dynamics 3(1): 71-95.
3. Spyrou NK Physics (2025) Mathematics and Cosmology: A Brief Informative Note. Academic Journal of Physics Research 1(2): 555556.