Cosmic voids and the kinetic analysis.
III. Hubble tension and structure formation in the late Universe

The observational indications for the accelerated expansion of the Universe and the positive cosmological constant have essential influence on the theoretical studies of the various phases of the cosmological evolution. The Hubble tension (Verde, Treu and Riess, 2019; Riess, 2020; Riess et al, 2022; Di Valentino et al, 2021; Dainotti et al, 2021, 2023), as the plausible tension between the late and early Universe, has also stipulated the consideration of theories and models (e.g.(Capozziello and Lambiase, 2022; Bouche et al, 2022; Bajardi et al, 2022)) addressing a broad spectrum of relevant issues, including the formation of cosmic structures.

Zeldovich (Zeldovich, 1970; Shandarin and Zeldovich, 1989; Shandarin and Sunyaev, 2009) pancake theory predicts the formation of cosmic structures based on the evolution of initial density perturbations (Peebles, 1993), with the essential use of the Lagrangian singularity theory. The pancake theory originated from cosmology thus linked the ”hydrodynamics” of the Universe to symplectic geometry (Arnold, Shandarin, Zeldovich, 1982; Arnold, 1982). The observational facts on the non-zero cosmological constant or the possible tension of the late (local) and early (global) Universe would obviously also have an influence on structure formation and pancake theory.

The concept of two Hubble flows, the local and global one, with a non-identical Hubble constant, is among the suggested approaches to deal with the Hubble tension and the dark sector (Gurzadyan and Stepanian, 2021a, b). That approach is based on the theorem proved in (Gurzadyan, 1985), positing that the general function for the force satisfying the sphere-point mass gravity’s identity takes the following form:

This function does not satisfy the second statement of Newton’s shell theorem, namely, in this case the gravitational field inside a shell is no more force-free. This formula, that is, taking as the second term the cosmological constant in weak-field general relativity and McCrea-Milne non-relativistic cosmology (McCrea and Milne, 1934) (see also (Zeldovich, 1981)) enables the description of the dynamics of groups and clusters of galaxies, as well as other phenomena (Gurzadyan and Stepanian, 2018; Gurzadyan, 2019; Gurzadyan and Stepanian, 2019, 2020). We also note that there are observational indications (Kravtsov, 2013) for the influence of the halo on the spiral galaxy structure, thus supporting the non-force-free field inside a shell predicted by Eq. 1.

Our present study is a continuation of previous works (Gurzadyan, Fimin, Chechetkin, 2022, 2023), based on the use of the Vlasov kinetic technique applied to gravitating systems and revealing the paths of the formation of the filamentary cosmic structures. The question of the ordered relaxation of perturbations is rather subtle in the system along the selected direction, such as for filaments, along two mutually perpendicular directions. It is clear that the local structure formation essentially differs from the hydrodynamics at large scales, especially regarding the averaging of a self-consistent gravitational field in a system of particles. We show that the deterministic way for emergence of large-scale structures, coherent in the sense of having a translational invariance of substructures, can be crucial in the Local Universe. This approach differs from “order out of chaos” scenario of the pancake theory at global cosmological scales but is complementary to it at local scales. The role of the repulsive second term of Eq. 1 is crucial and also determines the scales of structure formation and sizes of the voids, thus linking the predictions to observations (Böhringer, Chon and Trümper, 1991) (and references therein).

The system of Vlasov-Poisson equations for describing the cosmological dynamics in a system of $N$ particles of identical masses $m_{i}=m\equiv 1$ can be represented in the following form. We note that we implicitly assume that the system is considered in the domain $\Omega\subseteq{\mathbb{R}}^{2}$ space with a smooth boundary $\partial\Omega$, where ${\rm{diam}\>\Omega}\leq\infty$, namely, the size of the region can be continued up to infinity. This system is expressed as follows:

where $F({\bf x},{\bf v},t)$ is the distribution function of gravitationally interacting particles, $A$ is a normalization factor for particle density, and $t_{*}$ is a fixed instant of time. Equation 3 is the Poisson equation for the potential of Eq. 1, taking into account the repulsive cosmological term.

The radius, $r_{crit}$, determines the mutual contribution of the gravitational attraction and repulsive $\Lambda$-term. That is to say, at relevant scales, it defines the local Hubble flow given by the equation:

where $H_{local}$ and $\rho_{local}$ are the local values of the Hubble constant and the matter density, respectively (Gurzadyan and Stepanian, 2021b). The critical radius, $r_{crit}$, also defines the scaling between the semi-periodic coherent structures (i.e., the voids) formed as a result of action of attraction and repulsion (Gurzadyan, Fimin, Chechetkin, 2022, 2023).

We go on to consider the stationary dynamics, thus we set $F=F({\bf x},{\bf v})$. The subsequent analysis mainly concerns the second equation of the system Eqs. 2-3, which is the Poisson equation relative to the potential with no explicit time dependence. That is why varying the Hilbert-Einstein-Poisson action (Vedenyapin, Fimin, Chechetkin, 2021; Vedenyapin et al, 2011) includes a separate variation of fields (for a fixed particle distribution) and the distribution function. Thus, the considered below approach is applicable for adiabatic processes at quasi-equilibrium (weakly varying) particle distribution functions. In this case, for the distribution function, we can use the energy substitution (Vedenyapin et al, 2011), namely: $F({\bf x},{\bf v})=f(\varepsilon)\in C^{1}_{+}({\mathbb{R}}^{1})$, where $\varepsilon={\bf v}^{2}/2+\Phi({\bf x})$.

Introducing the (kinetic) temperature according to the general definition (Vlasov, 1961, 1978), $\theta\equiv-({\partial\Phi/\partial x_{j}})/({\partial\rho/\partial x_{j}})\big{|}_{j=\overline{1,d}}$, we get its dependence on the local properties of the self–consistent potential. We note that the temperature, generally speaking, can be considered as a tensor quantity, but for simplicity, we confine ourselves to the scalar temperature in the above formula, implying summation over the repeated indices.

Therefore, if there are appropriate relations between the parameters of the problem, the system of Vlasov–Poisson equations has solutions of the type of distribution functions that admit the energy substitution. The potentials of the gravitational field, which have the property of convexity (cf. the difference with tidal fields Gurzadyan and Ozernoi (1981)) in the general case, for an arbitrary $R_{\Omega}\leq\infty$ (in contrast to the case of the attraction potential), there is a limitation $R_{\Omega}<\big{(}q_{0}\theta^{2}/(4\pi\lambda^{2})\big{)}^{1/4}$.

Now we go on to examine in detail the properties of the gravitational potential and the influence of the cosmological term, as well as its influence at a given point of boundary conditions for region of space $\Omega$. As already noted in (Gurzadyan, Fimin, Chechetkin, 2022, 2023), in the formulation of the Dirichlet problem for the Poisson equation with a constant right-hand side on the boundary of the $\Omega$ region, according to averaging gravitational field outside the compact subdomain $\Omega_{0}$ containing system of particles and located inside domain $\Omega$ (${\rm meas}\,{\Omega_{0}}\ll{\rm meas}\,{\Omega}$), we can assume that the data on the $\partial\Omega$ boundary is given by in accordance with the theorem in Gurzadyan (1985). Then we consider what happens in the context of other similar equations.

We should first consider the properties of the boundary value problem for Eq. 6 in the following:

If we assume a priori the quantity $\phi$ to be small in norm, we obtain from this a linear equation; in principle, a linearization near any norm-finite solution $\phi_{0}$ homogeneous equation gives a similar result, although with the change $\lambda\to\lambda\exp(\phi_{0});$ for simplicity, we further assume $\phi_{0}=0$): $\Delta\phi+\lambda\phi={c^{2}\Lambda}/({2\theta}).$ We note that the boundary condition at $r\to R_{\Omega}$ must take the form: $\sim k_{1}(\theta)/R_{\Omega}+k_{2}(\theta)$).

For a more detailed analysis of the situation, it is expedient to pass to the integral representation of the equation for gravitational potential. The equation for the potential with Maxwell–Boltzmann particle distribution, corresponding to internal Dirichlet problem in a bounded domain $\Omega$ (under boundary conditions corresponding to the McCrea-Milne model), takes the following form:

In essence, the above is the explicit form of the equation for the potential introduced in the expression (Eq. 3), where ${\mathcal{G}}({\bf x},{\bf x}^{\prime})$ is the Green function for the inner boundary value problem in the domain $\Omega$; in this case, due to symmetry of the latter, we have:

We go on to introduce a new variable $U({\bf x})\equiv(\Phi({\bf x})-C_{0})/\theta,$ for a symmetric problem, where $C_{1}$ will enter the constant $C_{0}$, the above equation can be written as the Hammerstein integral equation of the form:

The linear equation (Eq. 14) above does not contain any inhomogeneity. However, it does carry information about the Dirichlet conditions and contains Green’s function associated with the original Poisson equation. As the next step, we can consider inhomogeneous linear equation taking into account quadratic repulsion $\alpha(\theta,\Lambda){\bf x}^{2}$.

The solution in the form of a series (23) was obtained under the assumption that it exists. Let’s check that expression (Eq. 23) is indeed a solution, that is, it satisfies Eq. 20, with the right side $-\widehat{\beta}({\bf x})$; to do this, we substitute into this inhomogeneous equation (Eq. 23):

We represent $\phi({\bf x})$ (solution of the inhomogeneous equation) using (Eq. 24) as:

Thus, the solution of the inhomogeneous Fredholm equation for $\lambda^{{\dagger}}\neq\lambda_{\bf n}$ expressed using the resolvent $\Gamma({\bf x},{\bf x}^{\prime},\lambda^{{\dagger}})$ for the integral kernel $-|{\bf x}-{\bf x}^{\prime}|^{-1}$ (this representation of the solution, generally speaking, is valid also in the case of deviation from the spherical symmetry of the problem). For the case $\lambda^{{\dagger}}=\lambda_{\bf n}$, there is no solution to the inhomogeneous equation, since $\langle\widehat{\beta}({\bf x}),\sin({\bf n}{{\bf x}})\rangle\neq 0$ (according to Fredholm’s $3$rd theorem).

For an integral kernel that differs from the classical Newtonian potential, a representation of the solution of the Fredholm equation in terms of a resolvent with sinusoidal eigenfunctions is impossible. We go on to demonstrate the technique for constructing the resolving resolvent for semi-symmetric integral Schmidt–type kernels.

The above representation of $\delta U({\bf x})$ as a regular series in the neighborhood of $U_{0}({\bf x})$ due to the presence of majorants of the coefficients only. However, if the value of the Fredholm determinant for the linearized equation $D(\lambda)\to 0$ (at the point $(\lambda_{0},U_{0}$)), then the second branch of the Hammerstein equation solution becomes relevant.

Thus, we get an analytical representation of the solution the Hammerstein equation for the potential, which is unique by construction and whose existence can be guaranteed in some neighborhood of equilibrium solution, $U_{0}$. This means the resistance of the system to external influences (the presence of external gravitational fields, varying within a certain spatial region) and final changes in the most self-consistent system field. The solutions of the linearized and nonlinear equations for the potential, define of a quasi-periodic nature of the structures, when the difference in the amplitudes of the local maxima of the potential depends on the relation $\alpha/\lambda_{II}$. If we accept that extrema density distribution of matter correspond to nodes and anti-nodes on solutions of equation (Eq. 13), it becomes obvious that one-dimensional (1D) and two-dimensional (2D) structures, observable in cosmological scales can be explained as completely deterministic objects, without the involvement of stochastic dynamics.

Above, we consider the case of Fredholm equations linearized near the point of constant potential. In doing so, we assumed that the kinetic temperature parameter has a fixed value. Clearly, there is a question of clarification of the equilibrium function and of the subsequent behavior of the system of gravitating particles.

The discussion that follows mainly refers to the first variant of accounting for inhomogeneity, but it can be extended in a completely trivial way to the second option (if we are further taking $U_{0}\to U_{0}-\alpha{\bf x}^{2}$).