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Accelerated expansion is a built-in property of the Boltzmann universe

Denis Michel

<figure><img src="image_1.png" /></figure>

To cite this version:

Denis Michel. Accelerated expansion is a built-in property of the Boltzmann universe. Physica A: Statistical Mechanics and its Applications, In press, 10.1016/j.physa.2025.130448. hal-04404580v4

HAL Id: hal-04404580

https://hal.science/hal-04404580v4

Submitted on 22 Feb 2025

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# Accelerated expansion is a built-in property of the Boltzmann universe

Denis Michel

<figure><img src="image_3.png" /></figure>

Université de Rennes1, Irset, Rennes, France. E-mail: denis.michel@live.fr

Large cosmological redshifts show that the expansion of the universe is accelerating, whereas a deceleration was expected under the influence of gravity. Since this discovery, the search for a repulsive force has gone in several directions. The main candidate is the postulated dark energy whose physical support is the subject of intense research, but alternative ideas such as a different physics of gravity on a large scale or the backreaction effect, are not ruled out. However, all of these lines of research fall within the same framework which was established by the pioneers of modern cosmology in the early 20th century. This note proposes to look back at an earlier view of the universe by Boltzmann. Based on his thermodynamic intuition, Boltzmann proposed that our visible universe could be part of a fluctuation within a global universe in equilibrium. This seemingly far-fetched vision is supported here by a minimalist Newtonian modeling which predicts that an acceleration of expansion would be initiated by a fluctuation in density, thus eliminating the objections to the Boltzmann universe regarding its size and maintenance. The strengths and weaknesses of this amusing hypothesis are discussed.

**Keywords :** Density fluctuation, Boltzmann universe, Gravitation, accelerated expansion.

**Abbreviations :** BU : Boltzmann Universe. $\Lambda$DCM : Lambda cold dark matter standard model of cosmology.

## Highlights :

* The Boltzmann hypothesis on the universe is recalled.

* A parsimonious gravity-based mechanism of Boltzmann universe expansion is presented.

* Accelerating expansion is predicted in a Boltzmann universe.

* Dark energy and modifications of GRT are not necessary to explain the accelerated expansion of the BU.

* In this theory, gravity shifts from role of brake to that of driver of expansion.

<figure><img src="image_4.png" /></figure>

> Boltzmann suggested that "In the universe which is in thermal equilibrium throughout, and therefore dead, there will appear here and there regions which will fluctuate noticeably during some time, in which the probability of the state will increase or decrease equally. For the universe, the two directions of time are indistinguishable, just as there is no up or down in space."

The point of this note is not to argue that the universe is infinite or to take the Boltzmann hypothesis for granted, but to show that this hypothesis would naturally imply an acceleration of expansion. This actually measured result in turn gives some substance to Boltzmann's forgotten vision.

# 1 Introduction

Einstein logically introduced a cosmological constant into his equations for a static universe to resist gravitational collapse, but he later agreed that this constant became useless in the dynamic universe [1]. However, the discovery of accelerating expansion [2, 3] revived the idea of a repulsive force, now ascribed to dark energy, necessary to counterbalance gravity which is currently regarded as a brake on expansion. This belief no longer holds when one revisits Boltzmann's forgotten theory of a fluctuating permanent universe in equilibrium [4, 5]. This theory was rejected by the community because it did not fit in with the knowledge of fluctuations in statistical mechanics [5, 6]. In particular, the observed part of the universe is orders of magnitude too large to be considered a fluctuation [5] and the transient nature of such fluctuations would also be incompatible with the Boltzmann hypothesis [6]. But it is shown here that the
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classical reasoning about fluctuations and large deviations does not apply on a scale where gravity comes into play. Moreover, accelerated expansion is naturally expected in a Boltzmann universe through a mechanism based on a local missing mass effect caused by the expansion, which in turn accelerates it. The simple Newtonian modeling of this positive feedback yields a result similar to that of the ΛDCM, except that the hypothetical parameters previously introduced to counteract gravity, the cosmological constant or dark energy, are replaced in the BU by gravity.

# 2 The Boltzmann universe

## 2.1 Thermodynamic features of the universe

The cosmic thermodynamic arrow is singular because it is no longer a question of a system immersed in a large bath, as in ordinary thermodynamics, but the large bath itself, where the global energy is truly global and cannot be buffered by the environment. For Boltzmann, this means that the past universe must have been in a highly improbable state, that the present universe must still be in an improbable state, and that it would end up in the most probable state, i.e. dead and scattered into inert confetti. The primordial universe must have contained an enormous supply of free energy that allowed it to evolve until its thermodynamic death [4]. It is difficult to see where this free energy was hidden in the homogeneous and hot primordial plasma [7], since we know since Carnot that a single source of heat cannot produce work, and that heat is the sign of the most degraded energy. But the combination of expansion, gravitational condensation and atom formation has diversified and organized the constituents of the universe, in a structuring process that seems to defy the second law [5]. Everything happened as if the stock of free energy and its organizing capacity were latent in the physical laws that were not in force at the time of the primordial universe, but were then revealed thanks to the expansion. If we approach the question from the point of view of entropy, the primordial plasma of the universe might at first glance appear more like an isotropic broth in the final stage of evolution in equilibrium, of maximum entropy, with a single temperature and no organized segregation. But conversely we can also admit that an increase in entropy could be due to taken up of gravitational potential energy during the clumping of matter, and therefore, according to general relativity, to the increase in Weyl conformational curvature [8]. It may also be due to some extent to taken up of information into organized structures, including life [9], according to the thermodynamic nature of information [10]. All these phenomena are made possible by the expansion, even if the expansion alone cannot be responsible for the increase in entropy of our universe; otherwise it would lead to the strange consequence of a reversal of the time arrow at maximum expansion in a collapse scenario [11, 8].

Expansion thus appears as the primary and fundamental organizer of the universe through global entropy production, and its determinism is essential to understand. Given the accumulation of *ad hoc* postulates used in current interpretations of expansion, it is instructive to look back at the forgotten Boltzmann hypothesis of a permanent universe in equilibrium. A reexamination of this amazing Boltzmann proposal with a toy Euclidean universe, reveals that the main driver of the expansion and its acceleration may be gravity itself.

## 2.2 Is the universe finite or infinite?

No experiment allows us to distinguish between : (i) the hypothesis of an infinite universe, reminiscent of that of Giordano Bruno, or (ii) the hypothesis of a finite, say for example hyperspherical, universe that is flat on the scale of our observation radius. Einstein's universe is a hypersurface that is globally folded along its entire matter/energy gravitational field [12]. If the universe were hyperspherical, it would necessarily be finite, but determining the global radius of curvature required to know this, is far beyond the possibilities offered by the visible universe. For the same reason, it seems that it will never be possible to prove that the universe is open and infinite. Furthermore, the widespread belief that a self-closed universe is imposed by the theory of general relativity is incorrect. Einstein's equations which establish the relationship between the geometry and content of the universe, are local. According to the analysis of the mathematician Grothendieck : "One of the most striking features that distinguishes the spacetime model of general relativity (curved at every point by the matter within it) from the Euclidean (or Newtonian) model of space and time, and even from Einstein's very first model of special relativity, is that the overall topological shape of spacetime remains undetermined, rather than being imposed by the nature of the model. As a mathematician, the question of what this global form is seems to me to be one of the most fascinating questions in cosmology" [13]. Unless additional actors such as extra dimensions or scalar fields are invoked, the possibility of periodic expansion-contraction bounces, depending on whether the mean density is supracritical or infracritical, has been abandoned for the time being since the discovery of accelerating expansion [2, 3]. In the absence of scientific arguments, the finitude or infinitude of the universe seems undecidable and a matter of opinion. However, the hypothesis of a finite universe containing *N* atoms in all, and not one more, is generally accepted by the community. This preference may be explained by (i) our fascination with Einstein's theory of general relativity, which led us to extend it to the global universe, and (ii) the difficulty of manipulating infinity, especially in physics. Moreover, since we are limited to our visible universe, our reasoning about its structure tends to be internal as well. Boltzmann's proposal requires a capacity of mental deconfinement, an amusing test of which is provided by the game of Fig.1.
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<figure><img src="image_5.png" /><figcaption>**Figure 1.** Draw a continuous broken line (without lifting the pencil) connecting the 9 points of this figure. How many line segments must this zigzag line contain?</figcaption></figure>

## 2.3 Boltzmann hypothesis of an universe in equilibrium

Although the idea of infinity is generally disconcerting to the human mind, Boltzmann proposed an eternal universe. This hypothesis is extremely economical, since it eliminates the problems of time zero and end, but in thermodynamics, eternity in a closed system is synonymous with equilibrium, which a priori seems to contradict any possibility of organization. It is ironic that the man who first recognized the need for the universe to evolve, through its entropy, should suggest that it can be in equilibrium; but this apparent contradiction is only a matter of scale. Boltzmann thought of an equilibrium on a very large scale, subject to fluctuations (as any equilibrium is) on a "small" scale, but larger than our limits of observability. In this eminently speculative scenario, matter/energy would always be present and continuously recycled. Each low-density bubble in this background is a new opportunity for matter to mature and organize by local counter-entropic evolution. This idea of Boltzmann was vigorously rejected by Feynman who could not imagine fluctuations as large as the visible universe [6]. The size of the fluctuations relative to that of the whole system decreases progressively with the size of the system, but the exact size of the universe is unknown. Given that fluctuations are inevitable and that no state is forbidden in statistical mechanics, Boltzmann's idea cannot be formally rejected. In particular, if the universe were Euclidean or hyperbolic, it would be infinite; and if it were infinite (in space or time), Poincaré would have easily shown that it necessarily contains fluctuations of any magnitude. But more importantly, in his attacks on the Boltzmann proposal, Feynman certainly had in mind the density fluctuations of ordinary statistical mechanics systems, recalled in the next section, which always converge to the mean with more or less force. But on the scale of the Boltzmann universe, gravitation must be taken into account, which, as we shall see, can cause the system to diverge and increase fluctuations.

# 3 Density fluctuations in equilibrium

## 3.1 Traditional treatment of density fluctuations

In a huge background of free, non-interacting particles of density $\rho_o$, the probability that an enclave of different density $\rho_i$ and volume $V$ will spontaneously appear can be derived from the traditional Poisson function, which itself can be derived either directly from the Gibbs distribution or as an approximation of a Bernoulli distribution [14]:

$$p(V, \rho_i) = \frac{(V\rho_o)^{V\rho_i}}{(V\rho_i)!} e^{-V\rho_o} \quad (1a)$$

which, using the Stirling approximation, takes the forms

$$p(V, \rho_i) \approx e^{V \left[ \rho_i \ln \frac{\rho_o}{\rho_i} - (\rho_o - \rho_i) \right]} \quad (1b)$$

or

$$p(V, \rho_i) \approx \left( \frac{\rho_o}{\rho_i} \right)^{V\rho_i} e^{-V(\rho_o - \rho_i)} \quad (1c)$$

This probability drops as soon as $\rho_i$ moves away from $\rho_o$, and it drops even more abruptly as $V$ increases (Fig.2).

<figure><img src="image_6.png" /><figcaption>**Figure 2.** Probability $P$ of occurrence of a small fluctuation bubble of volume $V$ and density $\rho_i$ in a large bath of density $\rho_o$. Curve drawn to Eq. (1c), valid only for completely independent particles whose positions in the system are all equiprobable.</figcaption></figure>

The mean square fluctuation predicted by Eq. (1a) is

$$\langle (\Delta V \rho_i)^2 \rangle = V \rho_o$$

So the mean fluctuation is just the square root of the same subvolume without fluctuation. The insignificance of the probability of a large density deviation over a large volume seems to vindicate the opponents of Boltzmann's proposal. Moreover, even this result is overestimated under ordinary conditions, and in particular does not hold for ideal gases. It holds only for completely independent particles with neither direct nor indirect mutual interaction, i.e., when the probability of particles entering and leaving $V$ does not depend on the number already present, and when all positions within $V$ are equally accessible. In fact, density fluctuations naturally depend on the forces acting between
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the particles in the system. For example, they are counteracted by short-range mechanical interactions which strongly favor equipartition. In a system of colliding particles, or of voluminous particles whose volume interferes with the available space, the size and lifetime of fluctuations are significantly reduced by exclusion [15] and compression processes [16]. The positive pressure P of an ideal gas is proportional to the particle density $P = \rho k_B T$, so the dimensionless mean square fluctuation is

$$ \langle (\Delta V \rho_i)^2 \rangle = V \rho_o \frac{d\rho_o}{dP} k_B T $$

[16]. But at the cosmic scale the most important interaction to consider is gravity. Unlike the previous interactions, gravity could maintain fluctuations, thus radically overcoming Feynman's objection.

## 3.2 Introducing gravity in the system

Whether thought of indirectly as a deformation of spacetime or directly as a force, gravity is a weak mutual interaction between all particles that can of course be neglected for the usual statistical-mechanical fluctuations, but must be taken into account on large scales. The Boltzmann universe is precisely a fluctuation on a scale where gravity is the most important interaction. The introduction of gravity into an infinite universe raises many physical questions, not all of which will be addressed here, such as the evolution, divergent or not, of fluctuations, the conditions for the application of Jeans instability and the compatibility between gravity and equilibrium. In an infinite homogeneous universe, gravity is expected to be balanced in all directions so as to produce no net curvature, or force in the Newtonian view, but gravity can still influence the local evolution of negative density fluctuations. This point, which will be discussed below, is essential to complete Boltzmann's vision, in which the influence of gravity was missing but would have allowed to refute the main objections raised against it. We will now consider whether a hollow bubble can be inflated by gravity, using a simple Newtonian approach instead of orthodox cosmology, but which, using an approximation, produces explicit solutions including an equation that is eerily similar to Friedmann's except for one sign.

# 4 Gravitational field of an enclave of different density

The gravitational treatment of a cavity depends on the extent of the surrounding mass. In particular, we must be careful not to apply Gauss' shell theorem indiscriminately. The famous Newton/Gauss shell theorem, analogous to Birkhoff's theorem in general relativity, predicts the absence of gravity at any point in a cavity inside a massive shell. It should not be misleadingly generalized to the infinite universe by adding a layer of matter to the shell, then another, and so on and so forth to infinity, conceived as an infinity of nested shells. This reasoning induces an asymmetry by stacking the shells around the first one, whose position is actually arbitrary. In infinite space, any point can be considered the center. This is especially true for the test mass used to demonstrate the shell theorem in Fig.3.

<figure><img src="image_7.png" /><figcaption>**Figure 3.** Opposite surface elements of the shell (corresponding to the black parts of the large ring), intersecting the same solid angle from the test mass (black dot), have gravitational effects on that test mass that cancel out throughout the shell, due to compensation between their surface area (and thus mass) and their gravity (in $r^2$ and $r^{-2}$). But in the case of an infinite surrounding medium, the test mass must move to the right, the closer the truncated cone of matter is.</figcaption></figure>

If the test mass is placed inside a solid shell, it will experience no gravitational attraction. However, if the hollow sphere is a cavity in an infinite medium, then the test mass will move toward the nearest wall. In the case of an infinite surrounding medium, the shell theorem must be replaced by the missing mass principle, whose external gravitational field can be rigorously derived from the superposition of simple gravitational fields shown in Fig.4. According to the superpositions $A = B - C + D$, the gravitational field around the enclave is for A

<figure><img src="image_8.png" /><figcaption>**Figure 4.** Superposition of elementary gravitational fields of an enclave of density $\rho_i$ embedded in an infinite background medium of density $\rho_o$, where the suffixes o and i mean out and in, respectively.</figcaption></figure>
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$$ \vec{\gamma}_r = [0] - \left[ -\frac{4}{3}\pi G \rho_o \frac{R^3}{r^2} \vec{e}_r \right] + \left[ -\frac{4}{3}\pi G \rho_i \frac{R^3}{r^2} \vec{e}_r \right] = \frac{4}{3}\pi G (\rho_o - \rho_i) \frac{R^3}{r^2} \vec{e}_r \quad (2) $$

where G is the gravitational constant. This result satisfies the intuition that the gravitational acceleration in panels B and B-C of Fig.4 cannot be the same. In the infinite medium of panel B, the field is zero for an obvious symmetry reason, since it integrates the gravitational contributions of all matter particles in the universe according to

$$ \vec{g} = -G \int_{r=0}^{\infty} \int_{\theta=0}^{\pi} \int_{\phi=0}^{2\pi} \rho_0 \frac{\hat{r}}{r^2} r^2 \sin\phi \, d\phi \, d\theta \, dr = 0 $$

where $\hat{r}$ is the radial unit vector and $r^2 \sin\phi \, d\phi \, d\theta \, dr$ is the volume element in spherical coordinates. However, this steady state is highly unstable. The particular result for $r = R$ is given by the gravitational field outside and inside the cavity.

* For $r \ge R$, according to Eq.(2),

$$ \gamma_r = \frac{4}{3}\pi G (\rho_o - \rho_i) \frac{R^3}{r^2}, \quad (3a) $$

* while for $r \le R$

$$ \gamma_r = \frac{4}{3}\pi G (\rho_o - \rho_i) r. \quad (3b) $$

This last value does not depend on R because of Gauss' theorem. We explained earlier that Gauss' theorem should not hold when a shell is replaced by an infinite environment, but it does hold here for the spherical shell of density $\rho_i$ and thickness ($R - r$) that separates a test mass inside the enclave from the environment outside the enclave. In fact, Gauss' equation for a perfect sphere gives $g(r) = 4\pi r^2 = 4\pi G (\rho_o - \rho_i) \frac{4}{3}\pi r^3$, giving $g(r) = \frac{4}{3}\pi G(\rho_o - \rho_i) r$. The two functions Eq.(3a) and Eq.(3b) meet at the same value for $r = R$, which is the point of maximum acceleration beyond which the law of gravity proportional to radius no longer applies. The orientation of the vectors obviously depends on the sign of $\rho_o - \rho_i$. Since the idea here is to study the expansion of the fluctuation, we will start by considering that $\rho_o > \rho_i$. The previous treatment can be replaced by a minimalist approach, where the gravitational acceleration results from the missing mass related to the difference in density between the general medium and that of the bubble with radius R, according to

$$ \gamma = \frac{mG}{R^2} \quad (4) $$

with

$$ m = \frac{4}{3}\pi R^3 (\rho_o - \rho_i) \quad (5) $$

where $\frac{4}{3}\pi R^3 \rho_i$ is the mass of the inner sphere and $\frac{4}{3}\pi R^3 \rho_o$ is the missing mass of the hollow sphere. A banal prediction is that when the density inside the bubble $\rho_i$ is less than the external density $\rho_o$, the bubble is expected to swell spontaneously because a test mass located at the

edge of the bubble must be pushed outwards by gravity alone. Considering that an internal mass can never catch up R and that the effective radius of the growing bubble is larger, we can approximate that the mass M enclosed in the bubble is constant. If the internal mass affects the expansion, at least initially, the ODE should include the fact that the internal density $\rho_i$ gradually decreases with R, according to

$$ \rho_i = \frac{M}{V(t)} = \frac{M}{\frac{4}{3}\pi R(t)^3} \quad (6) $$

Combining the above equations gives an ODE that simply defines the global gravitational acceleration.

$$ \frac{1}{G} \frac{d^2 R(t)}{dt^2} = \frac{4}{3}\pi \rho_o R(t) - \frac{M}{R(t)^2} \quad (7) $$

The Runge-Kutta 4 numerical simulation of this second-order ODE [17], summarized in Fig.5, shows a dual behavior around $R_o$, the critical value of R which cancels the right-hand side of Eq.(7).

$$ R_o = \left( \frac{3M}{4\pi\rho_o} \right)^{\frac{1}{3}} $$

With gravity alone, the bubble will shrink for an initial radius less than this value (Fig.5C), while it will expand indefinitely for an initial radius greater than $R_o$ (Fig.5B). For large radii, the expansion tends towards a regime where the expansion rate is proportional to the radius (Fig.5A), mimicking a Hubble law. Unfortunately, Eq.(7) is analytically unsolvable, precluding a rigorous comparison with established expansion models, but an alternative solvable ODE can be proposed, that allows further analysis and reveals an unexpected closeness to existing results.

# 5 Analytical solution when approximating quasi-uniform acceleration

In the absence of complex dynamics, the approximation of a quasi-uniform local acceleration allows us to express the rate of increase of the bubble radius R according to the classical law

$$ \left(\frac{dR}{dt}\right)^2 = 2\gamma R \quad (8) $$

which is here

$$ \frac{dR(t)}{dt} = R(t) \sqrt{\frac{8}{3} G \pi (\rho_o - \rho_i)} \quad (9) $$
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Evolution of the bubble size visualized by RK4 numerical resolution of Eq.(7). (A) Instability around $R_o$ visualized by a vector field plot of expansion rate versus radius. With gravity alone, the density fluctuation collapses for $R < R_o$ (orange curve) and explodes for $R > R_o$ (blue curve). For comparison, the behavior of the solvable ODE of Eq.(10) is shown (dashed black curve) using the same arbitrary parameters. (B) Expansion starting at a radius of $1.5 \times R_o$. (C) Collapse obtained starting from a radius of half $R_o$.</figcaption></figure>

which, by introducing the decrease of $\rho_i$, gives,

$$ \frac{dR(t)}{dt} = R(t)\sqrt{2G\left(\frac{4}{3}\pi\rho_o - \frac{M}{R(t)^3}\right)} \quad (10) $$

Not only does this approach yield results close to those of the ODE of Eq.(7), as visualized for the expansion (compare the blue and black expansion curves in Fig.5B), but it is also analytically solvable, as we will now see. Let us begin with simple situations.

## 5.1 The case of an empty bubble $\rho_i = 0$

If starting from an empty bubble,

$$ \frac{dR(t)}{dt} = \sqrt{\frac{8}{3}\pi\rho_o G R(t)} \quad (11) $$

This is simply Hubble's law, which states that the rate of separation is proportional to the distance. If we keep the square root as a constant, we get the radius $R_2$ at time $t_2$, starting from a radius of $R_1$ at $t_1$.

$$ \frac{R(t_2)}{R(t_1)} = e^{\sqrt{\frac{8}{3}\pi\rho_o G}(t_2-t_1)} \quad (12) $$

Identifying the exponent parameter as a sort of Hubble constant $\tilde{H}_o$ would give the density

$$ \rho_o = \frac{3\tilde{H}_o^2}{8\pi G} \quad (13) $$

This expression is similar to that of the critical density derived from the Friedmann-Lemaître equations, with the major difference that in the standard model it is the internal density of the universe including dark matter, whereas here it is the external density of the infinite Boltzmann universe. Any test mass placed inside the empty cavity spontaneously moves towards its nearest edge by a gravitational effect equivalent to that of a fictitious sphere of negative density $-\rho_0$ and with a radius equal to the distance between the center of the cavity and the test mass (schematized in Fig.3). Resuming the previous second-order ODE without approximation, this non-uniform gravity increases linearly with the distance $x$ from the center by $g(x) = \omega^2 x$. We obtain the differential equation $\ddot{x} - \omega^2 x = 0$, whose general solution is

$$ x(t) = x_0 \cosh(\omega t) + \frac{v_0}{\omega} \sinh(\omega t) \quad (14a) $$

$$ v(t) = v_0 \cosh(\omega t) - \omega x_0 \sinh(\omega t) \quad (14b) $$

which tends to an exponential free fall dominated by the term $e^{\omega t}$. The rate of separation between any two test masses inside the cavity is proportional to their relative distance.
## 5.2 In the presence of residual mass in the bubble $\rho_i < \rho_o$

Returning to the complete ODE of Eq.(10) and setting $h = \frac{8}{3}G\pi\rho_o$ and $g = 2GM$ (the letter $a$ is not used as a constant to avoid confusion with the scale factor of standard cosmology), the structure of Eq.(10) can be simplified as follows

$$x' = x \sqrt{h - \frac{g}{x^3}} \qquad (15a)$$

Imposing that $x > (g/h)^{1/3}$ so that the value in the square root is positive, and by changing the variable $y = \sqrt{\frac{h}{g}} x^{\frac{3}{2}} > 1$, Eq.(15a) reads

$$y' = \sqrt{h} \frac{3}{2} \sqrt{y^2 - 1} \qquad (15b)$$

or

$$\frac{dy}{\sqrt{y^2 - 1}} = \sqrt{h} \frac{3}{2} dt \qquad (15c)$$

and since

$$\int \frac{dy}{\sqrt{y^2 - 1}} = \cosh^{-1}(y)$$

we find

$$y(t) = \cosh \left[ \sqrt{h} \frac{3}{2} t + C \right] \qquad (15d)$$

and finally, returning to the starting variable,

$$x(t) = \left(\frac{g}{h}\right)^{\frac{1}{3}} \cosh^{\frac{2}{3}} \left[\sqrt{h} \frac{3}{2} t + C\right] \qquad (15e)$$

Applied to the parameters of Eq.(10), the solution of Eq.(15e) is therefore, without consideration of the integration constant,

$$R(t) = \left( \frac{3M}{4\pi\rho_o} \right)^{\frac{1}{3}} \cosh^{\frac{2}{3}} \left( \sqrt{6\pi\rho_o G} t \right) \qquad (16)$$

which can be rewritten using the pseudo Hubble constant defined above

$$R(t) = \left( \frac{3M}{4\pi\rho_o} \right)^{\frac{1}{3}} \cosh^{\frac{2}{3}} \left( \tilde{H}_o \frac{3}{2} t \right) \qquad (17)$$

According to this function, expansion is always accelerating ($R'' > 0$), consistent with the title of this article. By this mechanism alone without any additional parameters, the universe is expected to expand indefinitely, with an asymptote at long times corresponding to the exponential expansion of the empty bubble seen in section 5.1, since

$$\cosh^{\frac{2}{3}} \left( \frac{3}{2} H t \right) \underset{t \to \infty}{\longrightarrow} 2^{-\frac{2}{3}} e^{Ht} \qquad (18)$$

While the internal density dilutes with expansion, the external density remains constant due to equilibrium equi-partition. Thus, the exponent parameter is independent of

time, suggesting a kind of era dominated by a constant. Of course, this calculation assumes that the outer density is always greater than the inner density. This is achieved from a fluctuation radius just above $R_o$. Using this radius, the ODE can be rewritten

$$\frac{dR(t)}{dt} = \tilde{H}_o R(t) \sqrt{1 - \left(\frac{R_{o+}}{R(t)}\right)^3} \qquad (19a)$$

of real solution, when imposing $R(t) > R(o)$,

$$\frac{R(t)}{R_{o+}} = \cosh^{\frac{2}{3}} \left( \tilde{H}_o \frac{3}{2} t \right) \qquad (19b)$$

The volume evolves according to a strongly increasing $\cosh^2$ function of time

$$V(t) = \frac{4}{3}\pi R(t)^3 = \frac{M}{\rho_o} \cosh^2 \left( \tilde{H}_o \frac{3}{2} t \right) \qquad (20)$$

$R_o$ is simply the radius of a bubble of the universe occupied by this amount of matter before the fluctuation, i.e. at the global density. One could therefore expect that minimal anisotropies are sufficient to initiate their amplification, and that the frequency of their appearance is high in the infinite Boltzmann background. However, if they are not large enough, they could abort in the same way that instabilities are counteracted by pressure and other parameters. It is technically difficult to combine the purely random fluctuation process with a deterministic gravitational mechanism, so a general equation coupling the initial fluctuation and its subsequent expansion is not attempted.

## 5.3 $\rho_i > \rho_o$

In the case of an overdensity bubble and with the simplified conditions used here, the only deterministic outcome of this minimalist system is collapse. Collapses have been extensively studied in cosmology, but involves many other parameters such as pressure and temperature, and will not be treated here. In the case of our ideal toy universe without pressure, the present approach assumes a decrease in the bubble radius from an initial radius $R_0$ according to the ODE

$$\frac{d[R_0 - R(t)]}{dt} = \sqrt{2G[R_0 - R(t)] \left( \frac{M}{R(t)^2} - \frac{4}{3}\pi\rho_0 R(t) \right)} \qquad (21)$$

whose evolution shape is shown in Fig.6A. In fact, in the case of an overdense bubble and without forces other than the two gravitational accelerations considered here, the expansion could not begin unless an additional parameter is introduced, for example in the form of a Newtonian initial velocity whose origin remains to be determined (non-deterministic impulse, quantum bounce, etc.). The complete ODE in the direction of the expansion would then take the nonlinear form
<figure><img src="image_10.png" /><figcaption>**Figure 6.** (A) Collapse from an overdense bubble according to Eq.(21), keeping the same densities with $h/g = 0.01$ and starting from different initial radii $R_0$. (B) Initiation of expansion. The $\cosh^{2/3}$ expansion model, can start with a random negative fluctuation (left panel. Curve and slope field drawn to Eq.(17)), or a previous cendensation peak and an initial impulse (right panel. Curve and slope field drawn to Eq.(22a) with $g/\epsilon = 2$ and $h/\epsilon = 0.01$).</figcaption></figure>

$$x' = \sqrt{\epsilon + hx^2 - \frac{g}{x}} \qquad (22a)$$

or, adapted to the present parameters,

$$\frac{dR(t)}{dt} = \sqrt{v_0^2 + \frac{8}{3}\pi G\rho_o R(t)^2 - \frac{2GM}{R(t)}} \qquad (22b)$$

with a squared rooted component that is positive, but in which the internal density can transiently exceed the external density (right panel of Fig.6B), when

$$R(t) < \left( \frac{3M}{4\pi\rho_o} \right)^{\frac{1}{3}}$$

Regardless of the initial scenario, the late expansion function predicts an acceleration of the expansion after dilution of the density in matter/energy, and on the other hand, the contribution of the external density is equivalent to a constant, so the gravitational consequences of the Boltzmann proposal are once again similar to those of the dark energy postulate.

## 5.4 The competing $\cosh^{2/3}$ and $\sinh^{2/3}$ functions

The cosmic evolution of the scale factor reported in cosmology textbooks is similar to Eq.(19b), but with a sinh function instead of a cosh function. To see how this function might emerge from the crude Newtonian treatment used here, let us first examine some incongruous scenarios. First, let us imagine that, by some magical inversion of gravity, the densities $\rho_i$ and $\rho_o$ are both repulsive instead of attractive. If $\rho_i > \rho_o$, the ODE is

$$x' = x \sqrt{\frac{g}{x^3} - h} \qquad (23)$$

and, to keep a positive quantity below the square root, we impose now $x < (g/h)^{1/3}$. The method of resolution with a change of variable described above can be repeated, except that the integral to be used is now

$$\int \frac{dy}{\sqrt{1-y^2}} = \sin^{-1}(y)$$

which leads to the solution

$$x(t) = \left(\frac{g}{h}\right)^{\frac{1}{3}} \sin^{\frac{2}{3}}\left[\sqrt{h} \frac{3}{2} t + C\right] \qquad (24)$$

shown in Fig.7A as a red dotted curve. We can see that the supposedly repulsive internal density begins to induce expansion, which slows down when its dilution in $R(t)^{-3}$ increases, allowing the external repulsion to take over. Returning to our previous calculation, the curves corresponding to the equations Eq.(15e) with a $\cosh^{2/3}$ function, and Eq.(24) with a $\sin^{2/3}$ function, obviously cannot merge. The following scenario will allow to bring them together.

## 5.5 When both internal and external densities promote expansion

Of course, solving the previous ODEs would have been easier with a squared rooted component that is never negative or zero regardless of the degree of expansion. So, as a new technical game, imagine now that the internal density is always repulsive and amplifies the expansion instead of slowing it down. This is equivalent to giving the gravitational acceleration vectors in Eq.(2) the same orientation. In this case the ODE is

$$x' = x \sqrt{h + \frac{g}{x^3}} \qquad (25)$$

where $g$ and $h$ are real positive and the value in the square root is always positive. This equation says that the inner and outer densities push in the same direction to promote expansion, but one $(g/x^3)$ is stronger in short times, when $x$ is small, so dividing by $x^3$ gives a large value. In this strange scenario, we expect an expansion curve with a strong start and an initial bulge when the radius is still small and the internal density is high, and then an exponential expansion dominated only by $h$. To solve this ODE, the integral to use is
$$ \int \frac{dy}{\sqrt{1+y^2}} = \sinh^{-1}(y) $$

so that the solution is

$$ x(t) = \left(\frac{g}{h}\right)^{\frac{1}{3}} \sinh^{\frac{2}{3}}\left[\sqrt{h} \frac{3}{2} t + C\right] \quad (26) $$

For large values of $t$, this function rejoins the exponential expansion while for small values of $t$, it approaches a $t^{\frac{2}{3}}$ power law.

$$ \sinh^{\frac{2}{3}}\left(\frac{3}{2}t\right) \underset{t \to 0}{\longrightarrow} \left(\frac{3}{2}t\right)^{\frac{2}{3}} + \mathcal{O}\left(t^{\frac{8}{3}}\right) \quad (27) $$

It is instructive to plot the three functions we have obtained on the same graph (Fig.7A). The curve of Eq.(26) (plain line) interpolates nicely between, at the early stage, the curve obtained with an inversion of the expansive roles of internal and external gravity of Eq.(24) and, at long times, the curve obtained with Eq.(15e) as external gravity becomes increasingly dominant. Care should be taken not to misinterpret the shape of the transition curve of Eq.(26). Its strong initial expansion is due to the concentrated internal mass assumed to be repulsive in this purely theoretical scenario, and the subsequent deceleration before re-acceleration is due to a dilution of this supposedly repulsive internal mass. It happens that, technically speaking, the resolution of Eq.(25) is the one used in the standard model.

## 5.6 Comparison with the Friedmann equation

The degree of expansion in the FLRW model is quantified by the scale factor $a$ which depends on cosmic time. In general, the high nonlinearity of Einstein's equations forbids complete analytical solutions. Indeed, several densities coexist, whose effects vary strongly with expansion: the mass density, which goes as the inverse of the cube of the scale factor ($a^{-3}$), relativistic matter and radiation as $a^{-4}$, the curvature density which goes as $a^{-2}$, and dark energy, or the cosmological constant, which is independent of $a$. However, by limiting the number of players, analytical calculations are possible for certain combinations. For example, considering only matter and the cosmological constant is relevant because the other parameters have become negligible in today's universe. For a flat ($k=0$) FLRW universe with only matter and a positive cosmological constant and according to the Friedmann equation, the evolution of $a$ obeys

$$ \frac{da(t)}{dt} = a(t) \sqrt{\frac{8}{3} \pi G (\rho_{\Lambda} + \rho_{m}(t))} \quad (28) $$

where $\rho_{\Lambda,0}$ is related to Einstein's cosmological constant $\rho_{\Lambda} = \Lambda/(8\pi G)$ (which is repulsive) and $\rho_{m}$ is the total

matter density including dark and baryonic matter (which is attractive) [18, 19]. Interestingly $\rho_{\Lambda}$ in Eq.(28) and $\rho_{o}$ in Eq.(9), have the same positive sign, consistent with the expected fact that "external attraction" or "internal repulsion" both have a pro-expansion effect. In standard cosmology, all densities are necessarily internal, since the finite universe is expanding as a whole, so $\rho_{m}$ is equivalent to $\rho_{i}$ of the BU. Written in the usual terms, normalizing the densities by the critical density $\rho_{c,0} = 3H^2/(8\pi G)$ (note the apparent analogy with Eq.(13)) and assuming that the non-relativistic matter density evolves according to $\rho_{m}(t) = \frac{\rho_{m,0}}{a(t)^3}$, Eq.(28) becomes

$$ a' = H_0 a \sqrt{\Omega_{\Lambda,0} + \frac{\Omega_{m,0}}{a^3}} \quad (29) $$

It includes the normalized inner matter density ($\Omega_{m,0} = \rho_{m,0}/\rho_{c,0} \sim 0.3$) and the normalized dark energy density ($\Omega_{\Lambda,0} = \rho_{\Lambda,0}/\rho_{c,0} \sim 0.7$). According to the previous resolution (Eq.(26)), this ODE gives the following evolution of $a$ in the $\Lambda$DCM with very low curvature and radiation density [18, 19, 20]:

$$ a(t) = \left( \frac{\Omega_{m,0}}{\Omega_{\Lambda,0}} \right)^{\frac{1}{3}} \sinh^{\frac{2}{3}} \left( \sqrt{\Omega_{\Lambda,0}} H_0 \frac{3}{2} t \right) \quad (30) $$

The similarity between Eq.(30) and Eq.(26) is technically due to the fact that both $\Omega_{\Lambda,0}$ and $\Omega_{m,0}$ are given a positive sign in Eq.(29). Eq.(30) evokes the popular flared, vase-shaped silhouette of the evolution of expansion, with an initial strong expansion, a deceleration and then an acceleration [21, 22](Fig.7B), although in fact this silhouette should rather be seen as a sequence of different functions: $a(t) \propto \sqrt{t}$, then $a(t) \propto t^{\frac{2}{3}}$ and then $a(t) \propto e^{\sqrt{\Lambda}/3}$, during the phases of radiation (not introduced in Eq.(28)), matter and dark energy dominance respectively. Anyway, for large values of $t$, in itself the single Eq.(30) approaches the exponential asymptote, while for small $t$ it gives the $t^{\frac{2}{3}}$ power law of the matter-dominated era [19]. The shape shown in Fig.7B has been used to illustrate the acceleration of the expansion because it smoothly switches between the matter-dominated era for small times and the asymptotic de Sitter universe for large times [22]. According to this shape, the transition between the deceleration and acceleration phases occurs at the inflection point (Fig.7B), at

$$ t_i = \frac{\ln(2 + \sqrt{3})}{3\sqrt{\Omega_{\Lambda,0}} H_0} \quad (31) $$

The acceleration can decrease or increase depending on the sign of $\ddot{a}$ in the second Friedmann equation [20, 21], but the two densities $\rho_{m}$ and $\rho_{\Lambda}$ combine their effects to increase the norm of the expansion velocity in the first Friedmann equation. In the standard treatment, matter controls the expansion through

$$ \frac{dR}{dt} = R \sqrt{\frac{8}{3} \pi G \rho} $$
<figure><img src="image_11.png" /><figcaption>**Figure 7.** Formal solutions of the ODEs. (A) Solutions obtained for inverse gravity, with repulsive internal and external densities (Eq.(24), dotted red curve), obtained for normal gravity, with attractive internal and external densities (Eq.(15e), dashed green curve), and for repulsive internal density and attractive external density, both promoting expansion (Eq.(26), black smooth curve). (B) Cosmic time dependence of the scale factor according to the FLRW model drawn to Eq.(30) with $\Omega_{\Lambda,0} = 0.7$, $\Omega_{m,0} = 0.3$ and $H_0 = 70$ km/s Mpc in seconds $(2.27 \times 10^{-18} \text{ s}^{-1})$. Using these values, and without introducing any parameters other than matter and dark energy, the transition between the convex and concave parts is predicted to be about $2.3 \times 10^{17} \text{ s}$.</figcaption></figure>

without the minus sign given to $\rho_i$ in Eq.(9). The mathematical consequence of the first Friedmann equation is that the expansion starts spontaneously and strongly for any positive value of $\rho_m$ and for $R = 0+$, even if $\rho_m > \rho_\Lambda$ and in the absence of an initial impulse. In contrast, with the gravitational vector approach used above, expansion is impossible as long as $\rho_i > \rho_o$ unless a dose of initial momentum is added. The first Friedmann equation therefore cannot be used for the present kinetic model of the BU where the inner mass density should oppose expansion, and where the cause of the initial expansion should be explicitly included in the equations, either as an inner subdensity or as an initial velocity $v_0$ as in Eq.(22b). However, the expansion functions of time in sinh$^{2/3}$ or in cosh$^{2/3}$ both predict an acceleration of the expansion after dilution of the density in matter/energy, and on the other hand, the contribution of the external density is equivalent to a constant, so the gravitational consequences for the BU and the $\Lambda$CDM are similar in the long term.

# 6 Gravity-driven expansion

To introduce dark energy, we read in cosmology textbooks: "It is obvious that gravity can only slow down the expansion of the universe". This intuition stems from the same difficulty we have in considering the solution to the puzzle in Fig.1. It is surprising that the present rudimentary calculation, which could date back to the 18th century, purely Newtonian and unrelated to the subsequent cosmology, yields the currently accepted accelerating expansion. It is predicted here in a direct and compact way, without recourse to any cosmological constant and, since matter is sufficient, without recourse to a "dark energy era", suggesting that a rigorous analysis of the BU in cosmology would be worth pursuing. A precise quantification of the Boltzmann hypothesis is of course delicate at this stage given the number of unknown parameters,

in particular the nature of the initial fluctuation. In fact, we can imagine different scenarios depending on whether it is positive (particle concentration) or negative (particle dilution).

* In the second hypothesis of a local density decrease, the mechanism proposed above is complete and would drive the entire expansion once the critical size is randomly exceeded. However, this hypothesis would be more consistent with a relatively high general density $\rho_o$ and Hubble constant, and could limit the compressibility of the surrounding medium.

* In the first hypothesis where the expansion is preceded by a random condensation peak, the universe could have a moderate $\rho_o$ density. In this case, the mechanism proposed here would become dominant only for the secondary phases of the expansion, essentially that attributed to dark energy in the standard model. The primary condensation could be initiated by a rebound or by local instabilities amplified by gravity, or result from an oscillatory mechanism like that described below in section 8.6, which would be responsible for the reprogramming of matter in a primordial magma and for the first phases of the expansion, possibly after a quantum bounce.

# 7 Simplifying consequences of this hypothesis

The hypothesis of Hubble bubbles generated by fluctuations in a permanent universe could make the postulated dark energy unnecessary to explain a continuously accelerating expansion of the BU. Even better, gravity, which is usually seen as a restoring force that is expected to slow down the expansion, here becomes THE force of expansion. Fitting this scenario to astrophysical obser-
vations to derive the conditions of the initial tear and the structure of the universe, is beyond the scope of this paper, but the idea of expansion by gravity is already appealing and deserves some attention. Just think of the many advantages it offers :

1) No need to speculate about the origin of the expansion energy

• The vacuum energy estimated as Planck energy per Planck volume (c⁴/G), seems to be quantitatively divorced from its measurable macroscopic effects, and at the moment is difficult to correlate with dark energy. In the mechanism described here, there is no need for a *ad hoc* hypothesis of repulsive energy, whether in the form of vacuum energy, dark energy or of a cosmological constant Λ. The simple amplification of the fluctuations by gravity is sufficient.

• Einstein explained that he introduced his cosmological constant to theoretically account for the existence of a finite mean density in a static universe, but that it is no longer necessary in the dynamical case [1]. For the sake of completeness, it is worth adding that this constant is also not necessary in a static universe, provided it is infinite.

• The alternative hypothesis that antimatter could produce antigravity has been experimentally disproved [23].

2) Remove the uncomfortable assumption that the whole can be finite.

• In a fully equilibrium Boltzmann scenario, the total energy of the universe is expected to be constant, with gravitational energy exactly compensating for all the kinetic and organizational phenomena it induces. By definition, the universe is everything, and so it is a definitively isolated system necessarily in equilibrium, unless it is not everything.

• With eternal spacetime, we can stop thinking in vain about the absence of spacetime that preceded spacetime. Boltzmann's hypothesis does not require a beginning or an end or any special creation. In the BU, there are only transformations but no creation, neither of matter/energy nor of space.

3) The importance of expansion.

The fact that the acceleration of expansion is included in the Boltzmann model of the universe would be consistent with the importance of this phenomenon which ensures, among other effects, the following roles :

• The absorption of all entropy in the form of photons necessarily produced by all processes of organization in the universe, from the formation of atoms (recombination, baryogenesis, nucleosynthesis) to life. In this respect, the cosmic arrow and the thermodynamic arrow of the universe are intimately connected and clearly point in the same direction.

• Combined with the speed limit of information, the expansion of the universe leads to the formation of virtual information spheres between which communication is impossible when the speed of recession approaches the speed

of light, which could, for example, prevent Darwinian competition between different life forms [24].

4) The relative constancy of the acceleration factor.

The fact that the cosmological constant, identified with dark energy, is approximately constant, implies that dark energy is not a content of spacetime - otherwise it would be subject to dilution like internal matter - but a property of spacetime itself, which would explain why expansion creates more dark energy, so that its density remains constant. Such a postulate is unnecessary in the BU hypothesis, where the external density is naturally constant.

5) Agreement with certain astrophysical observations :

• The large-scale flatness (measured by WMAP and its successors) is generally explained by the hypothesis of a powerful initial inflation of a hyperspherical universe, which would have made it completely flat on the visible scale. In the present model, this would no longer be an approximation, but an expected property.

• The cosmic microwave background radiation (CMBR) left over from the early universe also provides some support for the possibility that our universe may have originated from a random fluctuation in a larger, high-entropy universe [5].

• The observation of structures whose formation time seems incompatible with the apparent age of the universe, or the presence of complex atoms in the early galaxies, could be consistent with the pre-existence of some elaborate structures that were not erased during the fluctuation.

• If the present model is not to remain just another theory, specific testable predictions would have to be precisely identified. As a first step in this direction, let us try to distinguish between the two competing expansion functions of time examined above, in cosh²/³(Eq.(19b)) and sinh²/³ [18, 19, 20, 21, 22]. In Fig.8A, where both curves are fitted to their central exponential asymptote, we see that the cosh²/³ curve has the slowest initial growth and then accelerates sharply. The expansion model obtained above predicts that the expansion rate follows

$$ \frac{R'(t)}{R(t)} = \tilde{H}_o \tanh \left( \tilde{H}_o \frac{3}{2} t \right) \quad (32) $$

Note that the numerical resolution of the ODE of Eq.(7) gives the same result (not shown). By comparison, that of the sinh²/³ function gives the expansion rate

$$ \frac{\dot{a}}{a} = H_0 \coth \left( H_0 \frac{3}{2} t \right) \quad (33) $$

Eq.(33) predicts that the expansion rate was higher in the past whereas it was lower in the past according to Eq.(32) (Fig.8B). Thus, Eq.(32) would be more consistent with the comparison of the apparent expansion rates determined from near and far measurements of [25]. However, this result has more to do with the assignment of op-
posite signs to repulsive and attractive effects than with
the model itself.

<figure><img src="image_12.png" /><figcaption>**Figure 8.** (A) Comparative expansion curves corresponding to the functions of time in cosh<sup>2/3</sup> of Eq.(19b) and in sinh<sup>2/3</sup> of Eq.(30). The two curves are fitted by their common exponential asymptote. (B) Comparative expansion rates defined as R'(t)/R(t). The cosh<sup>2/3</sup>-shaped expansion function has an increasing expansion rate while that of the sinh<sup>2/3</sup> function is decreasing.</figcaption></figure>

8 A very preliminary model

All of the above advantages should not obscure the
limitations and problems of the model presented here.

**8.1 Ways to introduce new space**

The BU expansion proceeds by a generalized and iso-
tropic free fall in vacuum, which is imperceptible but mi-
mics the properties of the Hubble cosmological expansion.
This removes the doubts about the persistence of initial
fluctuations raised by Boltzmann’s hypothesis. But a ma-
jor difference between the standard model and the BU
concerns precisely the way in which space is introduced
into the universe. In the standard universe, all conventio-
nal matter-energy is present from the big bang, while space
is introduced from the big bang by geometric dilation. This
universal creation of new space is sometimes thought of as
a stretching of pre-existing space, if that term has any
meaning, between all objects in distant gravitational re-
lationship, but not between objects in close gravitational
relationship. In contrast, the much more mundane BU co-
lonizes existing space rather than creating new space. The
difference may seem subtle at first, but it is actually funda-
mental. Indeed, space creation frees the standard universe
from the laws of relativity by allowing arbitrary recession
velocities, since objects do not really move in space, whe-
reas in the BU the laws of relativity are expected to still
apply and asymptotically limit recession velocities to the
speed of light. The redshift in the BU would therefore
be a combination of the gravitational redshift and of the
relativistic Doppler effect whose laws differ from the cos-
mological redshift for all expansion modes [24]).

8.2 A Newtonian toy universe

This article is merely a "proof of principle" of its title
using a toy universe model, with no ambition to challenge
the superior relativistic approaches. It would be very inter-
esting to model the Boltzmann fluctuation with the space-
time curvature tools of general relativity, to see if the re-
sult is consistent with the basic treatment used here. But
the debates about how to model the backreaction suggest
that the issue is not a simple one. In fact, Newton's laws
were recovered by Einstein as a first-order approximation
to his field equations, and general relativity usually tends
to refine the precision of Newtonian models rather than
radically destroy them. In addition to its simplicity, the
Newtonian approach presented here avoids certain theo-
retical difficulties, such as the combined introduction of
gravitational and antigravitational parameters into gene-
ral relativity. The determination of the expansion velocity
by the addition of the repulsive and attractive gravita-
tional influences, is suitable for the determination of the
cumulative energy, while the calculation of the velocity by
the subtraction of the opposing thrust and braking forces
(but not as a primitive of the vectorial sum of accelera-
tions), is better suited to the present purely kinetic and
directional approach. In the energy approach, if the accele-
ration depends on the position, such as in the present case
$\gamma(t) = h x(t) - \frac{g}{x(t)^2}$, we can write $\gamma = \frac{dv}{dt} = v \frac{dv}{dx}$ where
$v dv = \gamma(x)dx$, and we have $\frac{1}{2}v^2 = \int \gamma(x)dx + C$, directly
relating velocity to position without using time. But wi-
thout coupling to potential energy, this method leads to
cumulative effects of opposing forces $\int \gamma(x)dx = \frac{hx^2}{2} + \frac{g}{x}$
(because the restoring force becomes positive by integra-
tion). In contrast, in the present kinetic approach which
should respect the local direction of each contribution to
the acceleration, the velocity should be obtained by inte-
gration with respect to time by solving the second-order
ODE as a function of $t$, $\frac{d^2x}{dt^2} = \gamma(x(t))$. However, since
this type of ODE generally does not allow explicit reso-
lution, an alternative "energetic-kinetic" approach, which
approximates a quasi-uniform local acceleration, is allowed
by the fact that $\gamma(x(t))$ varies smoothly, with no inversions
or oscillations and no critical dependence on local accelera-
tion fluctuations. Of course, this preliminary model needs
to be refined and completed and does not immediately
address the many unanswered questions in cosmology.

8.3 Endless expansion ?

The Boltzmann hypothesis of a global universe in equi-
librium poses a major problem in the face of the irre-
versibility of gravitational instabilities. The problem with
Boltzmann fluctuations based on statistical thermodyna-
mics alone, was that they didn't seem big enough. This
problem is solved here by introducing gravity but an in-
verse problem arises : how to stop their growth! The ex-
ponential growth of R(t) could generate drag forces, de-
pending on the properties of the medium. We can imagine
an inertial effect of the dynamic fluid and the formation
of a peripheral shell of overdensity that could generate an
overpressure, but it is not clear how these mechanisms can
ensure the average global invariance of the universe. The
massive redistribution of the surrounding matter caused
by the exponential rate of advance of the R(t) front should
have no gravitational effect. In fact, (i) for r < R, the
gravitational field is determined solely by the central sub-
density, by virtue of Gauss's theorem; and (ii) for r > R,
since the total mass of the system (cavity + shell) remains
constant, the overall gravitational field does not change
significantly at long distances. What's more, all particles,
regardless of their local density, follow the same trajecto-
ries under the effect of the gravitational field, by virtue of
the equivalence principle. It is hard to imagine how non-
gravitational effects - due to pressure, viscosity, or even
nuclear reactions - could take over from gravity to slow
down the expansion of the cavity, since any stationary phe-
nomenon would tend to disappear if R(t) stops growing.
In other words, these effects could temporarily modify the
local dynamics, but would not constitute a fundamental
restoring mechanism as found in thermodynamics or elec-
tromagnetism.

8.4 Gravitational instability in equili-
brium

A major weakness of the Boltzmann model supplemen-
ted by gravity is the formal incompatibility between the
notions of instability and equilibrium. While in an equi-
librium universe we might expect any density fluctuation
to resolve itself, gravity, which is always attractive and
irreversible, amplifies these fluctuations instead of dissi-
pating them. This leads to a fundamental instability : an
initial sub-density fluctuation can grow exponentially and
never resolve. Any anisotropy or fluctuation is condem-
ned to irreversibility, either towards collapse for a positive
fluctuation or towards expansion for a negative fluctua-
tion, contradicting the idea of a statistically homogeneous
global universe. This would imply that a dense universe
in equilibrium cannot exist in the long term because of
the lack of time symmetry of gravity. Newton had al-
ready noted the instability inherent in a force without a
large-scale counterpart. Gravity is the weakest but most
stubborn of all forces because unlike the other funda-
mental interactions, it has no intrinsic restoring force. In
terms of charge, we consider the universe to be globally
neutral, containing an equal number of positive and ne-
gative charges. This is not the case with gravity, which is
always attractive and never repulsive. The gravitational
collapse of black holes and the growth of voids seem to
be two opposite manifestations of the same irreversible
process. How, then, can we reconcile the idea of a ho-
mogeneous, equilibrium universe imagined by Boltzmann
with the growth of sub-density fluctuations ? Could a sca-
lar field coupled to matter density modify the evolution
of voids and prevent their infinite expansion ? Conside-
ring the universe as a whole, we can also imagine shock
waves caused by compression walls separating adjacent,

unsynchronized bubble universes. In this spirit, a solution
to this problem could be, for example, a finite and curved
but globally static universe in equilibrium (section 8.6).

8.5 Is infinity physical ?

Physicists are less comfortable with infinity than ma-
thematicians. Einstein did not even mention the case of
an infinite universe when discussing the conditions of ap-
plication of his cosmological constant [1]. Did not he think
of it, or did he think of it but did not even think it worth
writing about because it seemed so unphysical? Einstein
explained in 1921 that he did not believe in an infinite uni-
verse because of Mach's principle. Since the inertia of each
body increases as more masses are in proximity, it seemed
natural to him that the universe should be spatially fi-
nite in order to reduce the effects of inertia on the actions
and reactions between the different bodies in the universe
[26]. Lemaître also wrote his aversion to infinity when he
stated that it is impossible to fill an infinite space with
matter, which can only be finite [27]. The last sentence of
his justification remains to be justified itself, and besides,
he was talking about de Sitter's universe resulting from
the growth of a finite universe, which therefore can never
become infinite. Leading scientists still claim that infinity
is not a physical concept, but in fact would not it be more
accurate to say that it is not a concept that our human
brains can comprehend? The all-conquering champion of
infinity manipulation, the mathematician Georg Cantor,
endured periods of deep despondency, but he taught us
that some infinities can be greater than others, and that
infinity is expandable. Moreover, if the idea of a much lar-
ger amount of mass is undeniably uncomfortable, is it, on
reflection, any more so than the idea of the spontaneous ge-
neration of energy, matter, space and time out of nothing?
In addition, there may be situations where a universe can
be conceived as a Boltzmann fluctuation in a finite frame,
preventing irreversible gravitational instability.

8.6 Alternative to infinity : Oscillating universes

Fluctuations can be imagined in a vast, unbounded but
finite universe, where 3D space would be folded in on itself
like the surface of a hypersphere, but with no global ex-
pansion (unlike the standard model where $k = +1$ implies
expansion). The fixity of this hypersphere would allow the
universe to be in equilibrium, according to the proposal
of Boltzmann. An equilibrium, which is inherently dy-
namic, would not prevent internal transient fluctuations.
Of course, Boltzmann was thinking in terms of thermal
fluctuations, but now gravitational fluctuations can also
be considered. In the hyperspherical space containing a
constant total amount of matter/energy, negative density
fluctuations appear and grow, forming expanding bubbles ;
but these bubbles cannot grow indefinitely. For example
for a single void, matter ejected at the periphery will
inevitably and violently re-concentrate at its antipode, creating a gravitational shock and rebound, sending matter back into the original void and creating a new central sub-density, starting the process all over again in reverse. Gravitational irreversibility would no longer be a problem in a fixed, closed 3D surface. It is likely that many cavities coexist, but bouncing remains perfectly plausible in a multi-cavity global universe. In addition to causing kinetic reversals, these bounces would play an important role in the cyclical reprogramming of matter through all the stages of standard cosmology, including phases of nucleosynthesis and star formation. They could also prevent black holes from becoming irreversible matter traps, if, at the time of rebound, they have not completely evaporated. In this chaotic and turbulent global system, the numerous out-of-sync expanding fluctuations would be in constant competition for space, with some bubbles aborting and others merging, but the overall steady state would retain the absolute robustness of a classical equilibrium. Observers inside an expanding cavity do not perceive the overall structure of the hypersphere. They see their region as a typical Hubble universe and an apparent event horizon. From their point of view, the expansion appears to be real and "driven from within" by dark energy, whereas it is merely a transient free-fall in a static global universe.

# 9 Conclusions

The hypothesis of expansion acceleration under the sole effect of attractive gravity simplifies its theoretical framework and compared to the standard model, requires neither the dark energy hypothesis nor the repulsive cosmological constant. The difference between these actors is that the nature of dark energy is still unknown, while gravity is a known phenomenon. However, like the standard universe, the BU scenario does not escape the question of the thermal fate of the universe. The voids could evolve towards an asymptotic state of global vacuum, but the highly speculative finite version with incessant gravitational oscillations (section 8.6), would be a robust alternative. The details of the BU equilibrium scenario remain to be worked out, but Boltzmann, who proved his visionary genius, did not seem frightened by the idea of a much larger universe in equilibrium. This brings us to the answer to Fig.1 which is given in Fig.9. This solution requires us to break out of our self-imposed mental framework, but it is not forbidden. When this solution is revealed to them, some people retort that it is cheating. Cheating is breaking a rule, but if there are no rules other than the ones we unconsciously make for ourselves, there is no cheating. In the same way, no scientific argument has ever formally forbidden a universe in equilibrium, of course much larger than our visible universe. Coincidentally, Fig.9 is the logo chosen by the Heidelberg Institute for Theoretical Studies (HITS, https://www.h-its.org/) to illustrate its slogan : "Think beyond the limits!" (C. Dullemond, personal communication).

Boltzmann was a specialist in this kind of thinking, and was therefore often misunderstood by his colleagues.

<figure><img src="image_13.png" /><figcaption>**Figure 9.** A line divided into 4 segments is enough. This solution requires us to break out of the framework we tend to impose on ourselves.</figcaption></figure>

## 9.1 The usual rejection of Boltzmann's ideas

Boltzmann's hypotheses were regularly challenged. Statistical thermodynamics was vehemently rejected at the time, but is now recognized as one of the most brilliant advances in science. The debate on irreversibility, illustrated by the famous dialogues with Zermelo and Loschmidt, is now in Boltzmann's favor, especially since the discoveries of chaos and strange attractors. But Boltzmann's theory of the permanent universe has yet to convince anyone [5]. The additional mechanism proposed here overturns Feynman's argument for the impossibility of Boltzmann density fluctuations on the scale of the observable universe [6], since the fluctuation is merely the priming of a gravitational pump that can increase the lifetime of the fluctuation bubbles. The simple introduction of Newtonian gravity into the Boltzmann equilibrium fluctuation is sufficient to predict the observed behavior of the expansion of the universe, discovered long after his suicide.

## 9.2 A change in the paradigm

Evangelista Torricelli caused a paradigm shift in 17th-century Florence with his water and mercury barometers. Previously, it had been thought that a liquid in a tube would rise by suction because nature abhors a vacuum. According to this law, the liquid, which belongs to nature, obediently performs its task by trying to reduce the vacuum created by the suction. Torricelli clearly showed that the precise limit of this rise, which depends on the density of the liquid, proves that its rise is actually a push from outside, due to atmospheric pressure. In other words, the liquid is no longer pulled from the inside, but pushed from the outside. Such a paradigm shift applies to BU, whose expansion is not the result of internal expansion, but of a free fall outward. The postulated force of horror of the void has never been confirmed but it is intriguing to note
that this law seems to turn into a "love of the void" in the large-scale universe.

## 9.3 A new role for gravity

In astrophysics, when an observation does not fit into an unquestioned model, convenient parameters can be added to that model to adjust the results to the observations, such as a mathematical constant or a dose of dark stuff that pulls or pushes, as the case may be. Compared to the dark physics of the universe, the number of parameters in the BU is minimal. Many theories have been proposed to explain the observed acceleration of the expansion, such as modifications of general relativity [28, 29], new scalar fields, negative pressure of various origins [30, 31], or a backreaction effect resulting from density inhomogeneities [32, 33, 34]. In this respect, the mechanism described here is somewhat similar to the kinetic aspect of backreaction beside its standard geometric expansion mode [32] for GRT modeling of the expansion of voids in our universe. However, the overall effect of the backreaction appears to be attenuated on large scales and does not quantitatively reproduce the observed acceleration of the universe. The hypothesis described here does not require the postulation of unknown forces or new physics, but relies solely on good old-fashioned gravity. It thus obeys the principle of parsimony that has often proven fruitful in physics. The Boltzmann idea explored here may seem a bit crazy at first glance, but its implications shown here suggest that it may be relevant, leads to a substantial saving of necessary assumptions and deserves some attention.

## 10 References

[1] A. Einstein, W. de Sitter, On the relation between the expansion and the mean density of the Universe, Proc. Natl. Acad. Sci. U.S.A. 18 (1932) 213-214. link

[2] A.G. Riess, A.V.Filippenko, P. Challis, et al., Observational evidence from supernovae for an accelerating universe and cosmological constant. Astron. J. 116 (1998) 1009-1038. link

[3] S. Perlmutter, Supernovae, dark energy, and the accelerating universe: The status of the cosmological parameters. Proceedings of the XIX International Symposium on Lepton and Photon Interactions at High Energies. Stanford, California (1999) link

[4] L. Boltzmann, Vorlesungen Über Gastheorie. Vol I, Leipzig, J.A. Barth, 1896. Translated by S.G. Brush, Lectures on Gas Theory, Berkeley. University of California Press, 1964.

[5] J.L. Lebowitz, Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropy. Phys. A Stat. Mech. 194 (1993) 1-27. link

[6] R. Feynman, The character of physical law. MIT Press, Cambridge, MA., 1967.

[7] G. Marx, The emergence of free energy. Phys. A Stat. Mech. 221 (1995) 361-371. link

[8] R. Penrose, Singularities and time-asymmetry. In Hawking S.W., Israel W. (eds.). General Relativity: An Einstein Cen-tenary Survey. Cambridge University Press. 581-638, 1979.

[9] D. Michel, Life is a self-organizing machine driven by the informational cycle of Brillouin. Orig. Life Evol. Biosph. 43 (2013) 137-150. link

[10] L. Brillouin, Science and information theory. Academic Press Inc. New York 1956.

[11] S. Hawking, A brief history of time. Bantam Books, 1988.

[12] L. Barnett, The Universe and Dr Einstein. Foreword by Albert Einstein. William Sloane Associates, New York, 1950.

[13] A. Grothendieck, Recoltes et Semaines - Réflexions et témoignage sur un passé de mathématicien, 1928-2014. link

[14] L.D. Landau, E.M. Lifshitz, Statistical Physics - Vol.5 Part 1 3ed. Pergamon Int. Lib. 1976.

[15] D. Michel, New treatments of density fluctuations and recurrence times for re-estimating Zermelo's paradox. Phys.A. Stat. Mech. 407 (2014) 128-134. link

[16] G. Müller, Grandcanonical ensemble, In Equilibrium statistical physics, 2015. link

[17] J.R. Dormand, P.J. Prince, New Runge-Kutta algorithms for numerical simulation in dynamical astronomy. Celestial Mechanics. 18 (1978) 223-232. link

[18] M. Bartelmann, C. Dullemond, Cosmology, 2011. link

[19] M.V. Sazhin, O.S. Sazhin, The scale factor in a Universe with dark energy. Astron. Rep. 60 (2016) 425-437. link

[20] J.B. Hartle, Gravity, An Introduction To Einstein's General Relativity. Addison Wesley, 2003.

[21] W. Rindler. Relativity, special, general and cosmological. Oxford University Press 2d Edition, 2006.

[22] F. Steiner. Solution of the Friedmann equation determining the time evolution, acceleration and the age of the universe. Conderence in Evolution Equations - the State of the Art, Reisensburg, Germany, December 2007. link

[23] E.K. Anderson, C.J. Baker, W. Bertsche, N.M. Bhatt, G. Bonomi et al. Observation of the effect of gravity on the motion of antimatter. Nature 621 (2023) 716-722. link

[24] D. Michel, Analytical relationships between source-receiver distances, redshifts and luminosity distances under pure modes of expansion. Adv. Astrophys. 2 (2017) 217-230. link

[25] A.G. Riess, W. Yuan, L.M. Macri, et al. A comprehensive measurement of the local value of the Hubble constant with 1 km s⁻¹ Mpc⁻¹ uncertainty from the Hubble space telescope and the SH0ES Team. ApJL 934 (2022) L7. link

[26] A. Einstein. Geometry and Experience. Lecture at the Prussian Academy of Sciences in Berlin. 27 January 1921. Published by Methuen & Co. Ltd, London, 1922.

[27] G. Lemaître, Note on De Sitter's Universe. Journal of Mathematics and Physics 4 (1925) 188-192. link

[28] S.M. Carroll, V. Duvvuri, M. Trodden, M.S. Turner, Is cosmic speed-up due to new gravitational physics? Phys. Rev. D. 70 (2004) 043528. link

[29] A. Joyce, L. Lombriser, F. Schmidt, Dark energy versus modified gravity. Ann. Rev. Nucl. Part. Sci. 66 (2016) 95-122. link
[30] E. Oks, Brief review of recent advances in understanding dark matter and dark energy. New Astron. Rev. 93 (2021) 101632. link

[31] W.J. Wolf, P.G. Ferreira. Underdetermination of dark energy. Phys. Rev. D 108 (2023) 103519. link

[32] T. Buchert. On average properties of inhomogeneous fluids in general relativity : Perfect fluid cosmologies. Gen. Rel. Grav. 33 (2001) 1381-1405. link

[33] E.W. Kolb, Backreaction of inhomogeneities can mimic dark energy. Class. Quantum Grav. 28 (2011) 164009 link

[34] S. Räsänen, Backreaction : directions of progress. Class. Quantum Grav. 28 (2011) 164008. link