An alternative to the Big Bang cosmologies is obtained by the Big Bounce cosmologies. In this paper, we study a bounce cosmology with a Type IV singularity occurring at the bouncing point, in the context of $F(R)$ modified gravity. We investigate the evolution of the Hubble radius and we examine the issue of primordial cosmological perturbations in detail. As we demonstrate, for the singular bounce, the primordial perturbations originating from the cosmological era near the bounce, do not produce a scale invariant spectrum and also the short wavelength modes, after these exit the horizon, do not freeze, but grow linearly with time. After presenting the cosmological perturbations study, we discuss the viability of the singular bounce model, and our results indicate that the singular bounce must be combined with another cosmological scenario, or should be modified appropriately, in order that it leads to a viable cosmology. The study of the slow-roll parameters leads to the same result, indicating the singular bounce theory is unstable at the singularity point, for certain values of the parameters. We also conformally transform the Jordan frame singular bounce, and as we demonstrate, the Einstein frame metric leads to a Big Rip singularity. Therefore, the Type IV singularity in the Jordan frame, becomes a Big Rip singularity in the Einstein frame. Finally, we briefly study a generalized singular cosmological model, which contains two Type IV singularities, with quite appealing features.Comment: IJMPD accepte

The development of dark energy models has stimulated interest to cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We review a broad class of phenomena connected with soft cosmological singularities in classical and quantum cosmology. We discuss the classification of singularities from the geometrical point of view and from the point of view of the behaviour of finite size objects, crossing such singularities. We discuss in some detail quantum and classical cosmology of models based on perfect fluids (anti-Chaplygin gas and anti-Chaplygin gas plus dust), of models based on the Born-Infeld-type fields and of the model of a scalar field with a potential inversely proportional to the field itself. We dwell also on the phenomenon of the phantom divide line crossing in the scalar field models with cusped potentials. Then we discuss the Friedmann equations modified by quantum corrections to the effective action of the models under considerations and the influence of such modification on the nature and the existence of soft singularities. We review also quantum cosmology of models, where the initial quantum state of the universe is presented by the density matrix (mixed state). Finally, we discuss the exotic singularities arising in the brane-world cosmological models.Comment: final version, published in Classical and Quantum Gravity as a topical revie

Energy Momentum Squared Gravity (EMSG) is a cosmological model where the scale factor is non vanishing at all times and hence does not favor big bang cosmology. However, the profile of density in the radiation dominated universe shows that EMSG supports inflationary cosmology. Inflationary cosmological models are successful in providing convincing answers to major cosmological issues like horizon problem, flatness problem and small value of cosmological constant but hitherto no model of inflation has been observationally confirmed. Owing to this, Varying Speed of Light (VSL) were introduced which are a class of cosmological models which disfavor inflation and propose an alternative route to solve these cosmological issues by just allowing the speed of light (and Newtonian Gravitational constant) to vary. VSL theories were motivated to address the shortcomings of inflation but do not address the shortcomings related to the initial big bang singularity. In this spirit, we present here a novel cosmological model which is free from both the "initial big bang singularity" and "inflation" by incorporating a mutually varying speed of light $c(t)$ and Newtonian gravitational constant $G(t)$ in the framework of EMSG. We report that in EMSG, for a dust universe ($\omega=0$), cosmological models for a time varying $c(t)$ and $G(t)$ and constant $c$ and $G$ are indistinguishable, whereas for a radiation dominated universe ($\omega = 1/3$), a mutually varying $c(t)$ and $G(t)$ provides an exiting alternative to inflationary cosmology which is also free from initial big bang singularity. We further report that for an ansatz of scale factor representing a bouncing cosmological model, the VSL theory can be applied to a quadratic $T$ gravity model to get rid of "inflation" and "big bang singularity" and concurrently solve the above mentioned cosmological enigmas.Comment: 8 pages, 6 figures, Accepted version in The European Physical Journal Plu