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Dualities between Scale Invariant and Magnitude Invariant Perturbation Spectra in Inflationary/Bouncing Cosmos

We study cosmological perturbation spectra using the dynamical equations of gauge invariant perturbations with a generalized blue/red-shift term. Combined with the power-law index of cosmological background, ν, we construct a parameter space, (ν, m), to classify all possible perturbation spectra. The magnitude-invariant power spectra occupy a two dimensional region in the (ν, m) space. We find two groups of scale-invariant solutions, one of which are also magnitude invariant while the other not, in the expanding and contracting phases of cosmological evolution. We explore the implications of magnitude invariance to the scale-invariance of an underlying spectrum, and unveil a novel duality between the scale-invariant solutions and the boundary of magnitude-invariant solutions: under two consecutive duality transformations, the scale-invariant solutions are mapped onto the boundary of magnitude invariant region. The physical origin of such a duality is yet to be studied. We present cosmological applications of our observations to a de-Sitter universe, an Ekpyrotic background, and a matter-dominated contraction. For the first two cases previously known results are re-derived; we also find for a matter-dominated contraction a solution with both scale-invariance as well as magnitude-invariance--in contrast to Wands's scale-invariant but magnitude-variant spectrum.