Galactic Bulge Population II Cepheids in the VVV Survey:
  Period-Luminosity Relations and a Distance to the Galactic Center
  We present the near-infrared observations of population II Cepheids in the
Galactic bulge from VVV survey. We identify 340 population II Cepheids in the
Galactic bulge from VVV survey based on their match with OGLE-III Catalogue.
The single-epoch $JH$ and multi-epoch $K_s$ observations complement the
accurate periods and optical $(VI)$ mean-magnitudes from OGLE. The sample
consisting of BL Herculis and W Virginis subtypes is used to derive
period-luminosity relations after correcting mean-magnitudes for the
extinction. Our $K_s$-band period-luminosity relation, $K_s =
-2.189(0.056)~[\log(P) - 1] + 11.187(0.032)$, is consistent with published work
for BL Herculis and W Virginis variables in the Large Magellanic Cloud. We
present a combined OGLE-III and VVV catalogue with periods, classification,
mean magnitudes and extinction for 264 Galactic bulge population II Cepheids
having good-quality $K_s$-band light curves. The absolute magnitudes for
population II Cepheids and RR Lyraes calibrated using Gaia and Hubble Space
Telescope parallaxes, together with calibrated magnitudes for Large Magellanic
Cloud population II Cepheids, are used to obtain a distance to the Galactic
center, $R_0=8.34\pm0.03{\mathrm{(stat.)}}\pm0.41{\mathrm{(syst.)}}$, which
changes by $^{+ 0.05}_{-0.25}$ with different extinction laws. While noting the
limitation of small number statistics, we find that the present sample of
population II Cepheids in the Galactic bulge shows a nearly spheroidal spatial
distribution, similar to metal-poor RR Lyrae variables. We do not find evidence
of the inclined bar as traced by the metal-rich red-clump stars. The number
density for population II Cepheids is more limited as compared to abundant RR
Lyraes but they are bright and exhibit a wide range in period that provides a
robust period-luminosity relation for an accurate estimate of the distance to
the Galactic center.
