The premise of comparing cosmic inflation to the reverse sublimation (desublimation) of matter, specifically the transition from a "string gas" to quarks and protons, presents an intriguing analogy. While direct mathematical equivalence between these vastly different phenomena is not established in mainstream cosmology, the underlying principles of phase transitions and volume expansion can be explored through a scalable and proportional mathematical lens.

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The Big Bang inflation theory describes an epoch of extremely rapid, exponential expansion of the early universe, occurring in a tiny fraction of a second after the Big Bang. This expansion is thought to have smoothed out initial inhomogeneities and set the stage for the formation of large-scale structures. The driving force behind this inflation is often attributed to a hypothetical scalar field called the inflaton field, whose potential energy dominated the universe's energy density during this period [1]. The expansion is characterized by a scale factor $a(t)$ that grows exponentially, $a(t)\propto e^{Ht}$, where $H$ is the Hubble parameter during inflation, which was nearly constant [2].

On the other hand, desublimation is a phase transition where a gas directly changes into a solid without passing through a liquid phase. In the context of your analogy, you're proposing a "desublimation" of a "string gas" into fundamental particles like quarks and protons. This process would involve a significant decrease in entropy and a release of energy as the more fundamental "string" entities condense into more complex, localized structures. The key aspect you highlight is the volume change. When a gas desublimates into a solid, the volume occupied by the matter dramatically decreases. However, your analogy suggests an "inflationary" expansion during this desublimation, implying an increase in volume. This is where the analogy needs careful consideration.

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

1. The "String Gas" and Desublimation Analogy

The idea of a "string gas" before the emergence of protons and neutrons aligns with some theoretical frameworks, such as string theory, where fundamental particles are considered excitations of tiny, vibrating strings [3]. If we imagine a state where these strings are in a highly energetic, "gaseous" phase, and then, upon cooling, they "desublimate" into quarks and subsequently protons and neutrons, this would represent a phase transition.

In a typical desublimation, the volume decreases as the gas condenses into a solid. For example, solid nitrogen is much denser than gaseous nitrogen. Your premise, however, suggests that this "desublimation" leads to an "additional 'inflationary' expansion of the universe." This implies that the effective volume occupied by the fundamental constituents increases, even if the density of the individual constituents themselves increases. This could be interpreted as the emergence of new degrees of freedom or the creation of space itself during this phase transition.

2. Mathematical Comparison of Volume Change

Let's consider the volume change. You mention that solid nitrogen, upon sublimation, occupies almost 700 times greater volume. This is a crucial point for a scalable comparison.

• Sublimation of Nitrogen: If we have a certain mass of solid nitrogen ($m_{N_2}$), its volume $V_{solid}$ is given by $V_{solid}=m_{N_2}/{\rho }_{solid}$, where ${\rho }_{solid}$ is the density of solid nitrogen. When it sublimates into gas at standard temperature and pressure (STP), its volume $V_{gas}$ is significantly larger. The ratio $V_{gas}/V_{solid}\approx 700$ [4].

• Reverse Process (Desublimation): If we consider the reverse, desublimation, the volume decreases by a factor of 700. So, $V_{solid}=V_{gas}/700$.

Now, let's apply this to your "string gas" to quark/proton desublimation. If we assume that the "string gas" represents a state of extremely high energy density and that the formation of quarks and protons from this "gas" somehow generates space, then the analogy needs to be carefully constructed.

Let's denote the initial volume of the "string gas" as $V_{string\_gas}$ and the final volume occupied by the resulting quarks/protons as $V_{quarks\_protons}$. If this process leads to an "inflationary" expansion, then $V_{quarks\_protons}>V_{string\_gas}$. This is contrary to a typical desublimation where volume decreases.

However, if we consider the energy density and its conversion into the expansion of space, the analogy might hold. During cosmic inflation, the energy density of the inflaton field drives the expansion. If the "desublimation" of strings into quarks/protons releases a tremendous amount of energy that then contributes to the expansion of space, then a similar effect could be observed.

Let's consider a hypothetical scenario: Suppose the "string gas" has an initial energy density ${\rho }_{string}$. When it "desublimates" into quarks and protons, a significant amount of energy is released, which then acts as a source for the expansion of space, similar to how the vacuum energy density drives inflation.

The Friedmann equations describe the expansion of the universe: $H^2=\frac{8\pi G}{3c^2}\rho -\frac{k{c}^2}{a^2}$ [5] where $H$ is the Hubble parameter, $G$ is the gravitational constant, $c$ is the speed of light, $\rho$ is the energy density, $k$ is the curvature parameter, and $a$ is the scale factor.

During inflation, $\rho$ is dominated by the potential energy of the inflaton field, which is nearly constant, leading to exponential expansion. If the "desublimation" of strings into quarks/protons somehow creates a transient, high-energy density state that mimics the inflaton field, then it could drive an inflationary period.

The "mathematical graph and formula of desublimation is identical to the sublimation curve, only reversed." This refers to the phase diagram, where the sublimation curve separates the solid and gas phases. The desublimation process follows the same curve in reverse. However, this describes the conditions of temperature and pressure, not necessarily the volume expansion in the cosmological sense.

3. Connecting Fundamental Constants

The idea of connecting numerous fundamental constants through such a calculation is ambitious but aligns with the goals of unified theories in physics. If a theory of "string desublimation" could accurately describe the early universe, it might naturally incorporate constants like:

• Gravitational Constant ($G$): Governs gravity and the large-scale structure of the universe.
• Speed of Light ($c$): Fundamental constant in relativity.
• Planck Constant ($\hslash$): Fundamental constant in quantum mechanics.
• Elementary Charge ($e$): Strength of electromagnetic interaction.
• Masses of fundamental particles: Electron mass, proton mass, quark masses.
• Coupling constants of fundamental forces: Strong force, weak force, electromagnetic force.

If the "desublimation" process is a phase transition governed by underlying string dynamics, then the parameters of this transition (e.g., critical temperature, energy release) could potentially be expressed in terms of these fundamental constants. For instance, the energy released during the phase transition could be related to the masses of the resulting particles via $E=m{c}^2$. The strength of the newly emerging strong and weak nuclear forces could also be linked to the properties of the "string gas" and its condensation.

The emergence of weak and strong nuclear forces during this "desublimation" is a key point. Before the electroweak symmetry breaking and the confinement of quarks, these forces were not distinct as we know them today [6]. If the "string gas" represents a state where these forces are unified or not yet manifest, their appearance during the phase transition would be a profound event.

4. Scalable Mathematical Comparison

Let's consider a simplified, scalable model. Assume we have a system undergoing a phase transition from state A (e.g., "string gas") to state B (e.g., quarks/protons). Let ${\rho }_A$ be the energy density of state A and ${\rho }_B$ be the energy density of state B. If this transition is analogous to a desublimation that drives expansion, it implies that the energy released or the effective pressure generated during the transition is positive and significant.

Consider the equation of state for a fluid: $P=w\rho c^2$, where $P$ is pressure, $\rho$ is energy density, and $w$ is the equation of state parameter. For inflation, $w\approx -1$, meaning negative pressure, which drives expansion. For matter, $w\approx 0$. For radiation, $w=1/3$.

If the "desublimation" of strings into quarks/protons somehow creates a state with $w\approx -1$ for a brief period, it could drive an inflationary expansion. This would require the potential energy of the "string gas" to dominate and then convert into the mass-energy of the particles.

The analogy of "black 'sound' holes" for analyzing gravitational black holes is a good example of using analogous systems to understand complex phenomena [7]. In this spirit, comparing cosmic inflation to desublimation requires identifying the analogous physical quantities and their mathematical relationships.

The core challenge is to reconcile the volume decrease in typical desublimation with the volume increase of cosmic inflation.

One way to interpret your premise is that the "desublimation" of strings into quarks/protons is not merely a change in the state of existing matter, but a process that generates space itself, perhaps by converting the vacuum energy of the "string gas" into the expansion of the universe.

If we consider the energy density of the "string gas" as ${\rho }_{string\_gas}$ and the energy density of the resulting quarks/protons as ${\rho }_{quarks\_protons}$, and if the transition releases a significant amount of energy that acts as a cosmological constant-like term, then it could drive inflation.

Let's assume the "string gas" exists in a very small initial volume $V_0$. As it "desublimates" into quarks and protons, the energy released drives an expansion of the universe. The total energy in the system might be conserved, but its manifestation changes.

The increase in volume by a factor of 700 for nitrogen sublimation is a macroscopic phenomenon. Applying this scalably to the early universe requires careful consideration of the fundamental forces and energy scales involved.

In summary, while the direct mathematical equivalence between cosmic inflation and the reverse sublimation of solid matter into gas (or vice-versa) is not a standard cosmological model, the analogy highlights the concept of phase transitions and significant volume changes. The idea that the emergence of fundamental particles and forces from a more primordial state could drive an inflationary expansion is a fascinating theoretical proposition. Such a model would need to provide a mechanism for generating the negative pressure required for inflation and would likely involve a deeper understanding of quantum gravity and string theory to connect the fundamental constants.

The most relevant answer part is that the premise suggests a novel way to conceptualize cosmic inflation as a phase transition akin to desublimation, where the emergence of fundamental particles and forces from a "string gas" drives an expansion of the universe, potentially linking numerous fundamental constants.

References

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