We have characterized the pulsation properties of 164 candidate RR Lyrae variables (RRLs) and 55 candidate Anomalous and/or short-period Cepheids in Leo I dwarf spheroidal galaxy. On the basis of its RRLs Leo I is confirmed to be an Oosterhoff-intermediate type galaxy, like several other dwarfs. We show that in their pulsation properties, the RRLs representing the oldest stellar population in the galaxy are not significantly different from those of five other nearby, isolated dwarf spheroidal galaxies. A similar result is obtained when comparing them to RR Lyrae stars in recently discovered ultra-faint dwarf galaxies. We are able to compare the period distributions and period-amplitude relations for a statistically significant sample of ab type RR Lyrae stars in dwarf galaxies (~1300stars) with those in the Galactic halo field (~14,000stars) and globular clusters (~1000stars). Field RRLs show a significant change in their period distribution when moving from the inner (dG<14 kpc) to the outer (dG>14kpc) halo regions. This suggests that the halo formed from (at least) two dissimilar progenitors or types of progenitor. Considered together, the RRLs in classical dwarf spheroidal and ultra-faint dwarf galaxies-as observed today-do not appear to follow the well defined pulsation properties shown by those in either the inner or the outer Galactic halo, nor do they have the same properties as RRLs in globular clusters. In particular, the samples of fundamental-mode RRLs in dwarfs seem to lack High Amplitudes and Short Periods (HASP:AV>1.0mag and P <0.48d) when compared with those observed in the Galactic halo field and globular clusters. The observed properties of RRLs do not support the idea that currently existing classical dwarf spheroidal and ultra-faint dwarf galaxies are surviving representative examples of the original building blocks of the Galactic halo.
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