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We study cosmological perturbation spectra using the dynamical equations ofgauge invariant perturbations with a generalized blue/red-shift term. Combinedwith the power-law index of cosmological background, {\nu}, we construct aparameter space, ({\nu}, m), to classify all possible perturbation spectra. Themagnitude-invariant power spectra occupy a two dimensional region in the({\nu}, m) space. We find two groups of scale-invariant solutions, one of whichare also magnitude invariant while the other not, in the expanding andcontracting phases of cosmological evolution. We explore the implications ofmagnitude invariance to the scale-invariance of an underlying spectrum, andunveil a novel duality between the scale-invariant solutions and the boundaryof magnitude-invariant solutions: under two consecutive dualitytransformations, the scale-invariant solutions are mapped onto the boundary ofmagnitude invariant region. The physical origin of such a duality is yet to bestudied. We present cosmological applications of our observations to ade-Sitter universe, an Ekpyrotic background, and a matter-dominatedcontraction. For the first two cases previously known results are re-derived;we also find for a matter-dominated contraction a solution with bothscale-invariance as well as magnitude-invariance--in contrast to Wands'sscale-invariant but magnitude-variant spectrum.

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