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Decaying-vacuum singularity-free cosmological models based on the Chen-Wu ansatz of a cosmological term varying as R -* where R is the scale factor of the Universe are introduced. They describe a closed ever-expanding universe of density parameter R 1 1 and with no entropy, horizon, or monopole problems. They include and extend the critical density cosmology of 0zer and Taha. The 0zer-aha period of phase transition during part of which the pressure is negative occurs in these models. In its wake and throughout the radiation-dominated era the Universe is Einsteinde Sitter-like with R then and subsequently near unity. Nucleosynthesis proceeds as in the standard model. Consistency with the observed helium abundance and baryon asymmetry allows a maximum vacuum energy close to the radiation energy today. The presence of this vacuum energy could be detrimental to certain theories of galaxy formation. A specific model with initial conditions approaching the hot big bang is studied in detail, particularly as regards the entropy and flatness of the very early Universe.PACS number(s): 98.80.Dr

The consequences of the cosmological constant snssts of Csrvslho, Lima, sad Wsgs (A 3PH + 37R ) sre investigated in an extension of the nonsingulsr Ozer-Tshs cosmology. The considered model describes a closed singularity-kee universe evolving through successive epochs of pure radiation, matter generation, and radiation and matter. The early phase of the last period is shown to be s concrete realization of the postslngulsrity radiation ers scenario of Freeze, Adams, Prieman, and Mottola. PACS number(s): 98.80. Cq, 98.80.Hw X. INTRODUCTION Carvalho, Lima, and Waga [1] have proposed the cosmological constant phenomenological ansatz where p and p are the cosmic energy density and pressure, respectively, and It the curvature index. Combining Eq. (3) and the differentiated form of Eq.(2) one has the energy (= E = pRs) equation II. NONSINGULAR MODELIn a Robertson-Walker»~averse with a perfect-Suid energy-moment»~tensor, Einstein's equations with a variable A give (n = 3/Sz G) a p=~-+ --A(t), (R) Rz 3(2) -" ('+")-R +R R where P and p are dimensionless numbers of the order of unity (natural units being used) R is the RobertsonWs&er scale factor, and H = R/R is Hubble's constant (an overdot denotes time &i&erentiation). Equation (1), a consequence of simple dimensional arguments consistent with quantum gravity [1], generalizes an earlier form, A oc R, suggested by Ozer and Taha [2] and also by Chen and Wu [3].Although the Ozer-Taha (OT) and Chen-Wu (CW) models postulate the same type of variation for A, the resulting cosmological scenarios are not similar. In one case (OT) one has a nonsingular universe with a cold beginning and an early phase transition, in the other (CW) a singular big-bang scenario. These differences are due to the model's difFerent initial conditions and the assignment of opposite signs to a certain integration constant.Carvslbo, Lima, and Waga [1] studied the modifications introduced by the P term in Eq. (1) on the cosmology of Chen and Wu. In this work we investigate the effect of this term in an extended Ozer-Taha cosmology. Also from Eqs. (1) and (2), I~' pp+ 1-In the radiation-(p = p/3) dominated (RD) universe Eqs. (1) -(3) yield (p g 2, p"= so. 'A, and Ao a constant) [1] g2 + + g~-2+4P 1 -2P (6)1-2P R +a(l -P)A R 4+~ ( 7) = ' P"R +PAR-1-2P For Ao & 0, p ( 1, the singular cosmological model based on these equations was studied by Car~&o, Lima, and Waga [1]. Here we investigate a scenario obtained by requiring Ao (0 and for which the model is nonsingular. For simplicity and physical relevance [4] we take p & 0 (tshing P = 0, Ao ( 0, P = h = 1 reProduces the nonsingular OT [2] model). A»mverse with a nonvanishing mi~~mum scale factor Ro at t = 0 arises Rom Eq. (6) if Ao ( 0, P ( z~, and k ( 2p. Then from Eq. (7), po -o.

The gravitational lensing rate in a flat cosmological model of noninteracting matter and vacuum energies is investigated. It turns out to be very close to that in a recently studied modified general relativity model and that it is consistent with observations.Recently one of us (Abdel-Rahman, 2001) investigated the effects of gravitational lensing in a modified general relativity (MGR) model proposed by Al-Rawaf and Taha (1996a, b), building on an old idea of Rastall (1972). It was found that the MGR flat cosmological model is consistent with the current gravitational lensing data.In this paper we consider a flat Robertson-Walker (RW) cosmological model with non-interacting matter and vacuum energies and in which the vacuum energy density varies inversely with the square of the RW scale factor. The vacuum equation of state between the vacuum pressure p v and energy density ρ v is p v = − 1 3 ρ v , placing the cosmology in the class of quintessence (Q) models where p Q = w Q ρ Q with −1 < w Q = const. < 0. The case w Q = −1 corresponds phenomenologically to the cosmological constant.We find that the gravitational lensing probability in the present cosmological model is, for all values of the matter density parameter m , very close to the lensing probability in the MGR model. In particular the two probabilities coincide at m = 0 where both have the value P lens = 2.44. The closeness of the lensing probabilities in the two models is interesting because they are generically different: energy is conserved in the present model but not in the MGR cosmology.For a spatially-flat (k = 0) Robertson-Walker (RW) universe with pressure-less matter of energy density ρ m , and time-dependent vacuum energy of density ρ v , the Astrophysics and Space Science 298: 519-523, 2005. C Springer 2005

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