# Alternatives to current dark energy models

Bruno Coppi, Valeria Ricci

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Bruno Coppi, Valeria Ricci. Alternatives to current dark energy models. 2025. [hal-05102920v1](https://hal.science/hal-05102920v1)

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Distributed under a Creative Commons Attribution 4.0 International License
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139

ALTERNATIVES TO CURRENT
DARK ENERGY MODELS

Bruno Coppi and Valeria Ricci

Massachusetts Institute of Technology

MIT (LNS) Report HEP 25/01 (2025)

January 2025

<figure><img src="image_2.png" /></figure>

Physics of High Energy Plasmas

Thermonuclear Plasmas

Plasma Astrophysics
ALTERNATIVES TO CURRENT DARK ENERGY
MODELS

BRUNO COPPI AND VALERIA RICCI

ABSTRACT. The envisioned characteristics of a multi-component, inhomogeneous “cosmological plasma” model are considered whose heavier visible component consists of “loaded galaxy clusters”. These include the optically observed clusters of galaxies, the characteristic electromagnetic radiation (X-rays) emitting plasmas associated with them, whose mass is larger or comparable to that of the galaxies, and the containing dark matter. The other visible component consists of the lighter but far more numerous “field galaxies”. A third component is the “pervading dark matter” in which the visible components are immersed.

On this basis, an energy transport equation, involving a “dark wind”, is proposed for the accelerating expansion energy of the multi-component plasma, rather than a homogeneous, “dark energy” equation of state. This is in analogy to transport equations based on processes relevant to laboratory and space plasmas that are not in thermal equilibrium and for which equations of state cannot be formulated.

# 1. INTRODUCTION

The series of observations by the Dark Energy Spectroscopy Instrument, beginning with those reported in Ref.[1], and the expected results of the Euclid Mission and of other relevant undertakings have been a stimulus for an effort to envision the salient characteristics of an evolving multi-component “cosmological plasma” (CP) model that could be consistent with the available observations.

In particular, the heavier visible component of this plasma is viewed as consisting of “loaded galaxy clusters”, each of them including the optically observed (clustered) galaxies, the associated high temperature plasmas, detected by their X-rays emission and whose total mass can be comparable or higher than that of the galaxies, and the dark matter providing the needed gravitational containment. The other visible component is assumed to consist of the lighter (far more numerous) “field galaxies”. Both components have their own characteristic mass distributions. The third component is the “pervading dark matter” in which the visible components are immersed.
As a start, concepts and descriptions derived from the known physics of laboratory and space plasmas that are not in thermal equilibrium are introduced, with important differences and additional features such as an appropriate distribution of masses in the relevant phase space.

Considering a kinetic energy density for the entire plasma a local transport equation [2] is introduced that involves an inward diffusion and an outward flow ("dark wind") for this energy that could be related to the growing body of relevant observations.

## 2. MULTI-COMPONENT PLASMA MODEL

Collisionless collective processes are envisioned to emerge in the considered cosmological plasmas manifesting themselves as redistribution in space of the component populations, flows, macroscopic spatial structures etc. In view of a relevant analysis, we refer, at first, to the distribution of the loaded clusters in phase space indicated as $f_{hc}(\mathbf{x}, \mathbf{v}, a(t))$, where the considered velocities are limited to be non relativistic and $a(t)$ is the "scale factor" entering the metric

$$ (1) \qquad ds^2 = -(cdt)^2 + a(t)^2 d\mathbf{x}^2. $$

Thus $f_{hc}$ has the slow time evolution of $a(t)$ and the corresponding mass density is

$$ (2) \qquad \rho_{hc} = \int d^3\mathbf{v} m_{hc}(\mathbf{x}, \mathbf{v}, a(t)) f_{hc}(\mathbf{x}, \mathbf{v}, a(t)), $$

where $m_{hc}$ is the mass distribution of the (elementary) loaded clusters and the mass energy density is $\bar{\rho}_{hc} \equiv (\rho_{hc})c^2$.

The relevant mass flow is

$$ (3) \qquad \Gamma_{hc} = \int d^3\mathbf{v} \, v \, m_{hc}(\mathbf{x}, \mathbf{v}, a(t)) f_{hc}(\mathbf{x}, \mathbf{v}, a(t)). $$

A significant special case could be that of spherical symmetry around an observer at $r=0$ and where the velocities of all components have a radial outward velocity $V_r$ only. In this case it is convenient to express the metric in spherical coordinates as

$$ (4) \qquad ds^2 = -(cdt)^2 + a(t)^2(dr^2 + r^2 d\Omega^2), $$

where $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the standard metric on the unit 2-sphere $S^2$.

Referring to the loaded clusters populations, the distribution function in the $r, V_r$ space can be represented by $f_{hc*}(r, V_r, a(t))$ and

$$ (5) \qquad \rho_{hc} = \int_{0}^{V_M} dV_r m_{hc}(r, V_r, a(t)) f_{hc*}(r, V_r, a(t)). $$
Thus the loaded cluster density becomes

$$ (6) \qquad n_{hc}(r, a(t)) = \int_{0}^{V_M} dV_r f_{hc*}(r, V_r, a(t)) $$

and the kinetic energy density component is proportional to

$$ (7) \qquad \int_{0}^{V_M} dV_r V_r^2 f_{hc*}(r, V_r, a(t), m_{hc}(r, V_r, a(t))). $$

Referring to the lighter visible component, that has different densities and masses, we may define the equivalent distributions $f_{lc}(\mathbf{x}, \mathbf{v}, a(t))$ and $f_{lc*}(r, V_r, a(t))$ leading to $\rho_{lc}$ and $n_{lc}$ following Eqs. (2) and (6).

The dark matter component is limited to be characterised by a mass density $\rho_{dm}$, a mass energy density $\bar{\rho}_{dm} = (\rho_{dm})c^2$ and a comprehensive energy density $\mathcal{E}_{dm}$.

Then a global energy density, that includes all form of energy associated with the three populations, may be represented by an effective pressure $\mathcal{E}_p$. In order to relate to Ref.[1] we may introduce the dimensionless generic function $\bar{W}$ as

$$ (8) \qquad \mathcal{E}_p = \bar{W} \bar{\rho}, $$

where $\bar{\rho} = (\rho_{hc} + \rho_{lc} + \rho_{dm})c^2$.
### 3. “CONVENTIONAL” TRANSPORT EQUATIONS

We start with a model thermal energy balance equation that includes components reminiscent of features suggested by laboratory experiments on plasmas that are relatively close to thermal equilibrium and for which a temperature $T$ can be defined. In particular, for an inhomogeneous spherically symmetric plasma with a density $n = n(r)$ and $T = T(r)$, we consider

$$ (9) \qquad \frac{3}{2} n \frac{dT}{dt} + T n \nabla \cdot \mathbf{u} \approx - \nabla \cdot (\Gamma_T) + S_T. $$

Here $\mathbf{u}$ is the mass fluid velocity and $\Gamma_T$ is the heat flow vector that is assumed to be of the order of the thermal diffusion component.

Clearly, an adiabatic equation of state is found, once the particle conservation equation is added, when $S_T \equiv 0$ and $|n(dT/dt)| \gg |\nabla \cdot (\Gamma_T)|$. If we consider, instead, the limit of large thermal conductivity, the only component of $\Gamma_T$ is

$$ (10) \qquad (\Gamma_T)_r \approx -n D_T \frac{\partial T}{\partial r}, $$

and $T \approx \text{const.}$ would be the result.

On the other hand, in analogy to what had been found to work for the cases of particle transport [3] and angular momentum transport [4] in magnetically confined plasmas, a composite form of $\Gamma_T$ may be suggested. In particular a counterdiffusion flux [3] may be introduced in $\Gamma_T$ as follows

$$ (11) \qquad (\Gamma_T)_r \approx -n D_T \frac{\partial T}{\partial r} + \text{sign} \left( \frac{\partial T}{\partial r} \right) r \Omega_{CD} n T_S. $$

Thus, in the large thermal thermal conductivity limit, considering plasmas with negative radial gradients of their temperature profiles, i.e. $\text{sign}(\partial T / \partial r) = -1$, we are led to consider the transport equation $\Gamma_T \approx 0$ that implies

$$ (12) \qquad \frac{\partial T}{\partial r^2} \approx -\frac{\Omega_{CD}}{2D_T} T_S. $$

In the special case where $T_S \approx T$ this would give the peaked $T$ profile represented by

$$ (13) \qquad T \approx T_0 \exp \left\{ - \int_0^{r^2} dr^2 \frac{\Omega_{CD}}{2D_T} \right\}. $$

In fact, these considerations can be applied to a toroidal axisymmetric, magnetically confined plasma with a large aspect ratio $R_0/r$ where
$R_0$ is the torus major radius and $r$ is the minor radial coordinate. In the simple case where the heating source is only Ohmic, that is

$$ (14) \qquad S_T = E_\phi J_\phi, $$

where $\phi$ is the toroidal coordinate and $E_\phi$ is about constant [5],

$$ (15) \qquad J_\phi \approx \sigma_\parallel E_\phi $$

where $\sigma_\parallel$ is the electrical conductivity. Then

$$ (16) \qquad S_T \approx E_\phi^2 \sigma_\parallel \propto T_e^{3/2}, $$

where $T_e$ is the electron temperature and we may assume that $T_s \propto T_e$ leading to a radial profile for $T$ of the kind represented by Eq. (13). This is consistent with the profiles that have been systematically observed [5] by relevant experiments.
## 4. QUASI-CLASSICAL (BY ANALOGY) TRANSPORT EQUATION

Following the considerations made for laboratory plasmas where an equation of state cannot be formulated, a model transport equation can be proposed that involves the component of $\bar{W}$, that is $\bar{W}_U$, representing the kinetic expansion energy of the cosmological plasma rather than $T(r)$. An important difference is that, referring to distant regions and to the evidence [7, 8] of an accelerating expansion, we are led to consider

$$ (17) \qquad \frac{d\bar{W}_U}{dr} = \mathcal{F}(r^2, t) > 0, $$

where $\mathcal{F}$, in general, should be an increasing function of $r^2$. In the envisioned transport equation an effective diffusion coefficient $D_*$ and a “counter diffusion” flow velocity $U_{CD}$, for the CP accelerating expansion energy, will have a primary role.

In particular we consider

$$ (18) \qquad (\bar{\Gamma}_*)_r \approx -D_* n_{cr} \frac{d\bar{W}_U}{dr} - n_{cr} U_{CD} \bar{W}_S, $$

where, consistently with Eq. (11), we take

$$ (19) \qquad U_{CD} \approx r \Omega_{CD} > 0. $$

The inward diffusion term included in Eq. (18) can be attributed to the onset of “population mixing modes” that are the analogues of the “impurity mixing modes” identified for magnetically confined plasmas and described in Ref. [6].

Then we are led to analyse the transport equation

$$ (20) \qquad D_* \frac{d\bar{W}_U}{dr} - r\Omega_{CD}\bar{W}_S \approx 0, $$

where $\bar{W}_S$ is representative of the source for the $U_{CD}$ flow velocity, we may refer to $U_{CD}\bar{W}_S$ as “Dark Wind” and $\frac{\Omega_{CD}\bar{W}_S}{D_*}$ is considered to be an increasing function of $r$.

Introducing the scale factor $a(t)$, Eq. (20) may be extended as

$$ (21) \qquad \frac{\partial \bar{W}_U}{\partial \bar{r}^2} - \frac{\Omega_{CD}}{2D_*} \bar{W}_S \approx 0, $$

where $\bar{r}^2 = r^2 a^2(t)$, and the solution

$$ (22) \qquad \bar{W}_U - \bar{W}_0 \approx \frac{1}{a^2(t)} \left[ \int_{\bar{r}_0^2}^{\bar{r}^2} d\bar{r}^2 \left( \frac{1}{2D_*} \Omega_{CD} \bar{W}_S \right) \right], $$

may be suggested.
Clearly, the present analysis has been limited to considering positive pressures. On the other hand if an inverted $U_{CD}$ flow velocity could be forced to appear in an equation for $\bar{W}$, that is $U_{CD} \simeq (1 - \eta)r\Omega_{CD}^0$, we would have

$$ (23) \qquad \bar{W}_U - \bar{W}_0 \simeq \frac{1 - \alpha_N}{a^2(t)} \left[ \int_{\bar{r}_0^2}^{\bar{r}^2} d\bar{r}^2 \left( \frac{1}{2D_*} \Omega_{CD}^0 \bar{W}_S \right) \right] $$

and $\bar{W}_U$ would be allowed to become negative at the cost of giving a different interpretation for $\mathcal{E}_p$. In this case Eq.(23) could be related to the Dark Energy equation of state adopted in Refs.[9] and [1], that is

$$ (24) \qquad \bar{w} = (\bar{w}_0 + \bar{w}_a) - a\bar{w}_a, $$

where $\bar{w}(a) < 0$, $\bar{w}_0$ and $\bar{w}_0/\bar{w}_a > 0$.
## 5. FINAL CONSIDERATIONS

A notable feature of Eq.(22) is that it represents a theoretical model involving an increase of the inhomogeneous expansion of the plasma visible components, while the plasma density can be allowed to decrease. This would not be equally well represented by an *ad hoc* equation of state.

The components of the considered cosmological plasma are expected to interact gravitationally, the Einstein's equation replacing the Poisson equation when comparing to non-relativistic theories of gravitational plasma.

Following the example of the adoption of Eq.(24) [1, 9], considering that this involves a homogeneous model, an appropriate procedure needs to be followed in order to relate equation (22) to the available observations.

As indicated earlier plasma modes, involving a redistribution in space of the field galaxies population and of the loaded clusters population, can be found as in the case of electromagnetic plasmas with a heavy (impurity ion”) minority population [6] . Clearly, whether the timescales for the growth of this kind of modes that can be estimated are realistic, is an important question to consider.

## ACKNOWLEDGMENTS

It is a pleasure to thank G.Bertin and M.M. Ivanov for their valued and timely comments.

## REFERENCES

[1] A.G. Adame et al. DESI 2024 VII: Cosmological Constraints from the Full-Shape Modeling of Clustering Measurements. arXiv:2411.12022, 55 p. (2024).

[2] S.I. Braginskii. Transport processes in a plasma (1965), Reviews of Plasma Physics, 1, 205-311

[3] B. Coppi and C.Spight Ion-Mixing Mode and Model for Density Rise in Confined Plasmas. Phys. Rev. Lett., 41 n.8, 551-554 (1978).

[4] B.Coppi Accretion theory of 'spontaneous' rotation in toroidal plasmas. Nucl.Fus., 42 n.1, 1 (2002)

[5] B.Coppi Nonclassical transport and the 'principle of profile consistency'. Comments Plasma Phys. Controll. Fus., 5 n.6, 261-269 (1980)

[6] B.Coppi, H.P.Furth, M.N.Rosenbluth, R.Z.Sagdeev Drift Instabilities due to impurity ions Phys. Rev. Lett. 17, 377-379 (1966)

[7] M.M. Ivanov Effective Field Theory for Large Scale Structure. arXiv:2212.08488, 46 p. (2024)

[8] E.V.Linder Exploring the Expansion History of the Universe. Phys. Rev. Lett., 90, 091301 (2003)
[9] A. Vikhlinin, *Chandra Cluster Cosmology Project III: Cosmological Parameter Constraints*. The Astrophysical Journal, **692** n.2, 1060-1074 (2009)

(B.C.) MASSACHUSETTS INSTITUTE OF TECHNOLOGY, DEPARTMENT OF PHYSICS
& AMERICAN ACADEMY OF ARTS AND SCIENCES & INAF, 77, MASS. AVE.
CAMBRIDGE, MA 02139, U.S.A.

Email address: coppi@psfc.mit.edu

(V.R.) UNIVERSITÀ DEGLI STUDI DI PALERMO & INAF, DIPARTIMENTO DI
MATEMATICA E INFORMATICA, VIA ARCHIRAFI 34, 90123 PALERMO, ITALIA

Email address: valeria.ricci@unipa.it