The premise presented draws an intriguing analogy between cosmic inflation and the phase transition of desublimation, specifically comparing the expansion of the early universe from a "string gas" to the formation of quarks and protons with the volume increase observed when a solid desublimates into a gas. While the analogy is thought-provoking, it's crucial to differentiate between analogies and direct physical equivalences.

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The concept of cosmic inflation describes an extremely rapid, exponential expansion of the early universe, occurring fractions of a second after the Big Bang. This expansion is theorized to have smoothed out initial inhomogeneities, flattened the universe, and explained the origin of large-scale structures observed today [1]. The driving force behind inflation is typically attributed to a hypothetical scalar field, known as the inflaton field, which possessed a large potential energy density [2]. As this field slowly rolled down its potential, it generated a repulsive gravitational force, causing the universe to expand exponentially. When the inflaton field reached the bottom of its potential, it decayed, reheating the universe and producing the particles that make up matter and radiation [3].

The proposed analogy suggests that before inflation, the universe was a "string gas," and the subsequent formation of protons and neutrons (from quarks) is akin to desublimation. Desublimation (or deposition) is a phase transition where a gas directly changes into a solid without passing through a liquid phase. The reverse process, sublimation, is when a solid directly changes into a gas. The example of solid nitrogen sublimating and occupying 700 times its solid volume highlights the significant volume change associated with phase transitions [4].

Let's break down the proposed analogy and its implications:

Inflation and Desublimation Analogy

The core of the analogy lies in the idea of a significant volume increase driven by a change in the fundamental state of matter. In the desublimation of nitrogen, the transition from a highly ordered solid state to a disordered gaseous state involves a substantial increase in the average distance between particles, leading to a large volume expansion.

In the proposed cosmic scenario, the "string gas" represents a highly energetic, pre-inflationary state where fundamental constituents are not yet organized into protons and neutrons. The "desublimation" event would then be the transition from this "string gas" to a state where quarks and eventually protons and neutrons form. The "inflationary" expansion is then linked to the emergence of strong and weak nuclear forces, which are suggested to cause additional expansion.

Mathematical Comparison:

The idea of a "mathematical graph and formula of desublimation being identical to the sublimation curve, only reversed" is fundamentally sound for phase transitions. The Clausius-Clapeyron equation, for instance, describes the relationship between pressure, temperature, and phase transitions, and its principles apply to both sublimation and desublimation, albeit with reversed signs for enthalpy changes [5].

However, directly applying the volume expansion factor of nitrogen desublimation (700 times) to the "desublimation of strings into quarks and protons" requires careful consideration. The physics governing the early universe and string theory is vastly different from the thermodynamics of molecular nitrogen.

Let's consider the energy scales involved. The energy density of the inflaton field during inflation was enormous, far exceeding the binding energies of quarks within protons or the phase transition energies of molecular matter [6]. The expansion during inflation was driven by the vacuum energy density of the inflaton field, not by the kinetic energy of particles or the breaking of intermolecular bonds in the same way as a typical phase transition.

Emergence of Forces and Inflation:

The suggestion that the emergence of weak and strong nuclear forces causes additional inflationary expansion is a novel interpretation. In standard cosmological models, these forces are already present in the very early universe, albeit unified at extremely high energies [7]. The strong and weak forces govern the interactions of quarks and leptons, and their emergence as distinct forces is a consequence of symmetry breaking as the universe cools, not typically a direct cause of further inflationary expansion. Inflation is generally considered to have ended before the universe cooled enough for these forces to become distinct in their modern forms [8].

Scalable Mathematical Comparison and Black Hole Analogy

The mention of "black 'sound' holes" and their scalable mathematical comparison to gravitational black holes is a reference to analogue gravity or dumb holes [9]. These are laboratory systems (e.g., in superfluids, Bose-Einstein condensates, or optical systems) that exhibit phenomena mathematically analogous to those in black holes, such as an event horizon and Hawking radiation [10]. This field of research aims to study aspects of quantum gravity in accessible laboratory settings.

The mathematical similarity between the behavior of convection in the Sun's plasma and nitrogen sublimation on Neptune, as well as the sound black hole analogy, highlights the power of scaling laws and dimensionless numbers in physics. Many physical phenomena, despite vastly different scales and constituent particles, can be described by similar mathematical equations when the relevant dimensionless parameters are matched [11]. For example, fluid dynamics equations can describe both astrophysical phenomena and laboratory experiments.

Connecting Constants in Physics

The question of whether these calculations can connect numerous constants in physics is profound. The standard model of particle physics and cosmology relies on a set of fundamental constants (e.g., gravitational constant, fine-structure constant, masses of fundamental particles) whose values are currently determined experimentally [12]. A grand unified theory or a theory of everything aims to explain these constants from first principles.

If a scalable mathematical framework could indeed unify the description of cosmic inflation with phase transitions of matter, it might offer insights into the origins of these constants. For instance, if the "desublimation of strings" is a fundamental process, the properties of these "strings" and their interactions could potentially dictate the values of fundamental constants that emerge in the lower-energy universe.

However, current understanding of cosmic inflation and string theory does not directly link the inflationary expansion to a phase transition in the same way that water freezes or boils. While string theory proposes that fundamental particles are excitations of tiny vibrating strings, the transition from a "string gas" to quarks and protons is a complex process involving symmetry breaking and particle creation, not a simple thermodynamic phase change in the conventional sense [13].

Conclusion

The proposed analogy is a creative way to think about the immense expansion during cosmic inflation. While the idea of a "desublimation" of fundamental constituents leading to a massive volume increase is intuitively appealing, it's important to recognize that the underlying physics of cosmic inflation, driven by an inflaton field and its potential energy, is distinct from the thermodynamic phase transitions of ordinary matter.

The most relevant answer part is that while the analogy between the volume expansion during desublimation and cosmic inflation is conceptually interesting due to the significant volume change involved in both processes, the underlying physical mechanisms are fundamentally different. Cosmic inflation is driven by the potential energy of a hypothetical inflaton field, leading to an exponential expansion of spacetime itself, whereas desublimation is a thermodynamic phase transition of matter.

The mathematical tools of scaling and dimensionless analysis can indeed reveal similarities across vastly different physical systems, as exemplified by analogue gravity. However, directly equating the volume expansion factor of nitrogen desublimation to the early universe's expansion from a "string gas" into quarks and protons would be an oversimplification, as the energy scales, fundamental forces, and driving mechanisms are distinct. While a unified theory might eventually connect these phenomena, current physics does not support a direct mathematical equivalence between the two processes as described. The question of connecting numerous physical constants through such calculations remains a central goal of theoretical physics, but requires a more fundamental understanding of quantum gravity and the very early universe.

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