1 Introduction

Since RR Lyrae stars are low mass helium core-burning stars, they are believed to be older than 10 Gyr. The study of the period distributions of these variables can therefore provide insights into the original populations of their host system. Field and cluster RR Lyraes show a clear separation between long period/metal poor and short period/metal rich stars, known as the Oosterhoff dichotomy (Oosterhoff 1939). On the other hand, the average period of RR Lyrae populations in Dsph galaxies is intermediate between the two Oosterhoff groups. It has been shown that this intermediate status is not the consequence of a superposition of two populations but rather an intrinsic property of RR Lyrae populations in these systems (e.g. Sextans: Mateo et al. 1995; Leo II: Siegel & Majewski 2000).

Of particular interest among RR Lyrae stars are those pulsating simultaneously in the fundamental and first overtone mode (RRd). These stars offer the opportunity to constrain their mass and luminosity independent of stellar evolution theory (e.g. Bono et al. 1996; Kovács & Walker 1999). The exact status of these variables is still under debate. The intermediate position of RRd stars in the instability strip between RRc (pulsating in the first overtone mode) and RRab stars (pulsating in the fundamental mode) suggests that these stars are in the process of mode switching. This scenario seems however excluded by theoretical calculations (Cox et al. 1980) which yield a much too short duration for this transition state ($\lesssim$10^3-4 years) to account for the high fraction of RRd variables observed in some systems (e.g. IC4499, M68). It seems however that these stars are evolving rapidly (i.e. the changes are observable during a human life-time), and some period and amplitude changes have already been observed (Purdue et al. 1995; Clement et al. 1997; Paparó et al. 1998; Benkö & Jurcsik 2000).

The first RRd star discovered was AQ Leo (Jerzykiewicz & Wenzel 1977), a field RR Lyrae. After this discovery, searches were focused on old stellar systems harboring RR Lyrae stars and, surprisingly, RRd variables were searched successfully in some systems but vainly in others. RRd stars have been found in globular clusters (M 15: Nemec 1985; M 3: Nemec & Clement 1989; NGC 2419 and NGC 2466: Clement & Nemec 1990; M 68: Clement et al. 1993; IC 4499: Walker & Nemec 1996), in Dsph galaxies (Draco: Nemec 1985; Sculptor: Kauzny et al. 1995) and in the Galactic Halo (Clement et al. 1991; Garcia-Melendo & Clement 1997; Clementini et al. 2000). On the other hand, searches for RRd variables were unsuccessful in $\omega$Cen (Nemec et al. 1986), M 80, M 9 and NGC 2298 (Clement & Walker 1991), Ursa Minor (Nemec et al. 1988) and 20 other globular clusters (Clement & Nemec 1990). The parameter(s) driving the occurrence of RRd pulsators in stellar systems is still not clear and more observations are needed before any firm conclusion can be drawn. What is clear, however, is that the two Oosterhoff groups are well separated in a Petersen diagram (a plot of the period ratio versus the fundamental period - Petersen 1973), the OoI RRd pulsators having lower periods and period ratios than their OoII counterparts.

Recently, the collection of known RRd stars has been substantially increased with the discovery of 181 new RRd variables in the Large Magellanic Cloud (Alcock et al. 1997; Alcock et al. 2000b). These stars revealed a new picture because they were spread across the Petersen diagram, filling the gap between the two Oosterhoff groups. Pulsation models show that this distribution may be caused by a mass and/or metallicity spread within the population of RRd stars (Kovács 2000). Spectroscopic measurements on a sample of these stars seems to confirm the metallicity spread (Clementini et al. 2000; Bragaglia et al. 2001).

The search for multi-periodic RR Lyraes in the MACHO data set revealed other surprises. For instance, many RR Lyrae periodograms exhibited two closely spaced frequencies. This frequency pattern, first discovered by Olech et al. (1999a, 1999b) in RR Lyrae stars, cannot be explained by the superposition of radial pulsations and is therefore believed to be related to non-radial modes. Although these kind of stars have been detected in only four different places to date (M 55, M 5, LMC and Galactic Bulge), they seem to be relatively common in their host system. Theoretical modelling of these stars have been proposed by Van Hoolst et al. (1998) and by Dziembowski & Cassisi (1999).

We carried out a period analysis on a sample of $\sim$3700 RR Lyrae stars in Sgr and the Galactic Centre. The paper is structured as follows. Section 2 presents the data and their reduction. In Sect. 3, we describe the selection processes and present the sample of detected multi-periodic RR Lyraes. In Sect. 4, we present the period distributions of the RR Lyrae catalogs and compare them with those observed in other stellar systems. Section 5 is devoted to a presentation of the period-amplitude diagram of the RR Lyrae population in Sgr. In Sect. 6, we use photometric indicators to estimate the metallicity of the RR Lyrae population. Section 7 is devoted to a discussion about the spatial homogeneity of the RR Lyrae population and in Sect. 8 we estimate the RRab content of Sgr. Finally, we summarize our results and conclude in Sect. 9.