This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Rikpratik Sengupta, Department of Physics, Aliah University, Kolkata 700 160, West Bengal, India (E-mail addresses: rikpratik.sengupta@gmail.com(RS))

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## 3. Discussion

When the universe is contracting, the energy density grows and ultimately diverges making the scalar curvature and the Hubble parameter to diverge as well. This can be understood by the fact that the scale factor vanishes in the Friedmann equation. Bounce is the mechanism that simply prevents the initial singularity from forming by making the scale factor begin to increase before it can reach zero or making the energy density drop off before it can diverge. The condition for bounce is achieved through ¨a > 0, such that the contracting universe starts expanding. Alternatively, in turnaround, the expanding universe must begin to start contracting so that both the scale factor and energy density do not diverge in the finite future and this can be achieved through the condition ¨a < 0. Both at the bounce and turnaround, the Hubble parameter vanishes rather than diverging. The the scale factor neither reaches zero or infinite value as the effective energy density on the brane remains finite. Thus, the universe transits smoothly through both the bounce and turnaround. For cosmology on the brane, both the mechanisms can be achieved by a minimum of requisite components- a scalar field with an inflationary potential and a DE component that violates the NEC.

The problem with the phantom is that such an exotic fluid has a number of theoretical inconsitencies and pathologies at the quantum level that makes their existence questionable. The problem with the future singularities can be cured from the correction term on the brane as we have found, but such a fluid may also lead to the vacuum being unstable. Attempts to construct dynamical scalar field models of the phantom have led to a negative kinetic term[25] that in turn results in quantum instabilities[30]. However, there exists a cosmological model of DE with vanishing Λ where the vacuum energy obtained from the quantization of a free scalar field having low mass is described by a supernegative EoS and the model is free of pathologies at the quantum level[31]. Two problems in general feature in most oscillating cosmology models. We shall discuss them very briefly without going into much details before concluding the letter. The first problem is posed by the continued existence of singular objects like black holes from the area theorems of Hawking. However, before the turnaround leading up to the next bounce (via a contraction phase) in a phantom dominated universe, such structures may well be dissolved due to the extremely large gravitationally repulsive effects[32], thus being prevented from disrupting the evolution of the universe during the contracting phase following the turnaround. Infact, it has been shown that[33] the Hawking area theorems may not hold true if the NEC (ρ + p ≥ 0) is violated, as is the case for a phantom dominated universe. Any surviving remnant microscopic black holes may act as possible dark matter candidates. Moreover, the black hole singularity may also be resolved in the UV corrected picture just like the initial big bang and the big rip singularities and also, there may exist non-singular black hole mimickers like gravastar on the brane[34], leading to a complete resolution of the problem at once. It is worth mentioning in this context that the RSII braneworld has also been used in explaining a recent GW event GW170817[35] and the recent observation of the dark shadow of M87∗ [36]. The second problem is associated with the entropy of the universe, which we think remains the same in a periodic manner after the bounce over each cycle, such that the possible increase in entropy during the expanding phase being compensated by a possible decreasing during the radiation/matter dominated expanding phase. This prevents the enrtopy to increase to infinitely large values limiting the number of cycles. We are however concerned mainly with the bounce and turnaround in a single cycle in this letter.

This is the first model that can avert the initial singularity using a single brane approach with a positive brane tension. Braneworlds that have a space-like extra dimension like the one we have considered here are characterized by a positive brane tension (as the effective gravitational constant on the brane needs to be positive to explain the attractive nature of gravity) but such a setup could not resolve the big bang singularity. The non-singular models of brane cosmology till date have either resorted to a single brane with time-like extra dimension where no scalar field has to be invoked to generate the bounce which happens naturally from the cosmological dynamics[10], but the brane tension must be negative for the same reason of obtaining a positive effective gravitational constant, or alternatively, introduce a second braneworld with a negative tension parallel to the positive tension brane with finite separation between the branes. The advantage of introducing the parallel negative tension braneworld is two-fold: firstly, negative tension branes have the unique feature of reduced inertia on matter with positive energy density being dumped onto it helping the dynamical realization, and secondly the two brane setup comes with the benefit of a scalar field known as the radion which modulates the inter-brane separation and can both source the bounce at early times as well as behave like phantom dark energy at late times dure to the non-canonical kinetic term evolving to have a negative value[37]. However, there are some tachyonic instabilities associated with negative tension braneworlds which can be possibly resolved in M-theory but has not been explored well enough and requires further formal developments in M-theory (although the properties are really appealing). On the contrary, the ingredients of our model are well explored and the physics is more well understood in terms of a single positive tension brane. The phantom dark energy also does not lead up to the big rip as the quadratic correction to stress energy becomes significant before the singularity can is reached.

Also, the scalar field we have used is a physically well motivated one, since it can accommodate the inflationary scenario naturally and its potential need not be reconstructed to explain the generation of seed cosmological perturbations. Most models of cosmology with a nonsingular bounce have either to resort to alternative mechanisms to generate seed perturbations that are not very well understood physically, or have to reconstruct the potential on an ad hoc basis to generate the perturbations, but in our model an inflationary epoch following the bounce driven by the scalar field with an inflationary emergent potential already has all the ingredients responsible for generating these perturbations, and is well understood. We may think of it as a toy model, not because the scenario is physically ill-motivated, but since we have not tested the model against observations. We plan to analyze the primordial observables like the amplitude of scalar perturbations, tensor to scalar ratio and the spectral index and test them against latest observations in a follow up work in the recent future.