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We investigate the recently proposed natural double inflation model insupergravity. Chaotic inflation first takes place by virtue of theNambu-Goldstone-like shift symmetry. During chaotic inflation, an initial valueof second inflation (new inflation) is set, which is adequately far from thelocal maximum of the potential due to the small linear term in the K\"ahlerpotential. Then, primordial fluctuations within the present horizon scale maybe produced during both inflations. Primordial fluctuations responsible foranisotropies of the cosmic microwave background radiation and the large scalestructure are produced during chaotic inflation, while fluctuations on smallerscales are produced during new inflation. Because of the peculiar nature of newinflation, they can become as large as $10^{-1}$-$10^{-2}$, which may lead tothe formation of primordial black holes.Comment: 21 pages, 2 figures, to appear in Phys. Rev.

Density fluctuations and primordial black hole formation in natural double inflation in supergravity