Simple Harmonic Motion:Geometrical Solutions of Equations of Motions

Conventionally while we talk about geometry associated with a simple harmonic oscillator, we draw a circle with a radius equal to the amplitude of Oscillator and imagine a particle moving along the perimeter with a frequency same as that of Oscillator and then the x coordinate of the position of the particle gives the value of the displacement of the Oscillator. Here we discuss another kind of diagram (polar plot) depicting dynamics of Oscillator and come up with a solution to boundary and initial value problem geometrically.Comment: 9 pages, 4 figure

Seeing Higher-Dimensional Grand Unification In Primordial Non-Gaussianities

The observed low-energy values of the $SU(3)\times SU(2)\times U(1)$ gauge couplings, extrapolated via the minimal Standard Model Renormalization Group evolution, hint at the exciting possibility of a Grand Unified Theory (GUT) at $M_U \sim 10^{14}$ GeV --- a scale, however, too high to probe directly via collider searches. Fortunately, since the Hubble scale H can be as high as $5 \times 10^{13}$ GeV $\sim M_U$ during the inflationary era, such GUT scale states can be cosmologically produced at that time and leave direct on-shell signatures such as their masses and spins, via primordial non-Gaussianity (NG). We explore this possibility in one of its simplest realizations given by the extra-dimensional framework of orbifold GUTs, in which proton decay can be straightforwardly suppressed to be within the stringent bounds. Here, along with the massive GUT states there must also be H-mass spin-2 Kaluza-Klein (KK) gravitons, collectively giving rise to striking NG signatures. In our set-up we localize the inflaton on one of the boundaries of an extra dimension. The inflationary vacuum energy can readily lead to formation of a horizon in the bulk, where the KK modes then form a continuum above a mass gap of $\sim \mathcal{O}(H)$. We find that the optimal case for observable NG signals is when the extra dimension is stabilized close to the onset of this horizon, ensuring a discrete KK spectrum such that the lightest KK modes can be cosmologically produced without significant Boltzmann suppressions. Although we mostly focus on the case where there is no higher-dimensional cosmological constant, we also obtain considerable holographic insights from the $\text{AdS}_5/\text{CFT}_4$ correspondence when such a cosmological constant is included.Comment: v1: 30+10 pages, 6 figures; v2: discussion added at the end of subsection 4.3, reference added, conclusions unchanged, journal versio