We investigate primordial black hole formation in the matter-dominated phase of the Universe, where nonspherical effects in gravitational collapse play a crucial role. This is in contrast to the black hole formation in a radiation-dominated era. We apply the Zel'dovich approximation, Thorne's hoop conjecture, and Doroshkevich's probability distribution and subsequently derive the production probability β 0 of primordial black holes. The numerical result obtained is applicable even if the density fluctuation σ at horizon entry is of the order of unity. For σ 1, we find a semi-analytic formula β 0 0.05556σ 5 , which is comparable with the Khlopov-Polnarev formula. We find that the production probability in the matter-dominated era is much larger than that in the radiation-dominated era for σ 0.05, while they are comparable with each other for σ 0.05. We also discuss how σ can be written in terms of primordial curvature perturbations.

It is well known that the vacuum in the Einstein gravity, which is linear in the Riemann curvature, is trivial in the critical (2+1=3) dimension because vacuum solution is flat. It turns out that this is true in general for any odd critical $d=2n+1$ dimension where $n$ is the degree of homogeneous polynomial in Riemann defining its higher order analogue whose trace is the nth order Lovelock polynomial. This is the "curvature" for nth order pure Lovelock gravity as the trace of its Bianchi derivative gives the corresponding analogue of the Einstein tensor \cite{bianchi}. Thus the vacuum in the pure Lovelock gravity is always trivial in the odd critical (2n+1) dimension which means it is pure Lovelock flat but it is not Riemann flat unless $n=1$ and then it describes a field of a global monopole. Further by adding Lambda we obtain the Lovelock analogue of the BTZ black hole.Comment: 3 pages, revised version with two minor changes and an author added. Accepted for publication in Physics Letters

In this paper we have investigated the cosmological dynamics of non-locally corrected gravity involving a function of the inverse d'Alembertian of the Ricci scalar, f ( −1 R)). Casting the dynamical equations into local form, we derive the fixed points of the dynamics and demonstrate the existence and stability of a one parameter family of dark energy solutions for a simple choice,The effective EoS parameter is given by, w eff = (α − 1)/(3α − 1) and the stability of the solutions is guaranteed provided that 1/3 < α < 2/3. For 1/3 < α < 1/2 and 1/2 < α < 2/3, the underlying system exhibits phantom and non-phantom behavior respectively; the de Sitter solution corresponds to α = 1/2. For a wide range of initial conditions, the system mimics dust like behavior before reaching the stable fixed point. The late time phantom phase is achieved without involving negative kinetic energy fields. A brief discussion on the entropy of de Sitter space in non-local model is included.PACS numbers: 98.80.Cq

Further to results in [9], pointing out the role of initial density and velocity distributions towards determining the final outcome of spherical dust collapse, the causal structure of singularity is examined here in terms of evolution of the apparent horizon. We also bring out several related features which throw some useful light towards understanding the nature of this singularity, including the behaviour of geodesic families coming out and some aspects related to the stability of singularity.