THE WEIERSTRASS CRITERION AND THE LEMAÎTRE–TOLMAN–BONDI MODELS WITH COSMOLOGICAL CONSTANT Λ

Abstract

We analyze Lemaître-Tolman-Bondi models in presence of the cosmological constant Λ through the classical Weierstrass criterion. Precisely, we show that the Weierstrass approach allows us to classify the dynamics of these inhomogeneous spherically symmetric Universes taking into account their relationship with the sign of Λ.

Analogy between equilibrium beach profiles and closed universes

“…Equation (3.1) is a first integral of Newton's second law d 2 h/dx 2 = −dV /dh expressing energy conservation. Following the Weierstrass approach [17,[25][26][27], one obtains a qualitative understanding and a graphical representation of the possible motions (i.e., of the possible solutions of Eq. (2.8)) from the graph of the potential V (h) and its intersections with the horizontal line E = −1/2 (see Fig.…”

Non-Minimally Conformally Coupling Cosmology with Multiple Vacua Potential with Cubic-Quintic-Septic Duffing Oscillator Properties

“…These cycloid oscillations occur due to the presence of the Jacobi elliptic function which is a particular solution of the non-linear Klein-Gordon wave equation. We note that these functions were very useful in solving tricky problems in cosmology and astrophysics [60][61][62][63]. The main difference between both previous cases discussed concerns first the crossing of the phantom divide line and besides the scale field and the scale factor oscillates with comparable amplitude, whereas for the previous case, the scalar field oscillates with amplitude larger than that of the scalar field.…”

Analogues of glacial valley profiles in particle mechanics and in cosmology

“…The solutions y(x) corresponding to the possible motions of the particle are candidates for the description of cross-profiles of glacial valleys. As is well known from mechanics, a qualitative understanding of the motion can be obtained from the study of the potential V (y) and of the intersections between its graph and the horizontal line E = −1/2 (which is called "Weierstrass approach" in various applications [29]- [31]). It is V (y) < 0 ∀y = 0, V (0) = 0, V (y) → −1/(2λ 2 ) as y → ±∞, there is a vertical asymptote y = C/λ > 0, and…”