Inspecting the Cepheid parallax of pulsation using Gaia EDR3 parallaxes. Projection factor, period-luminosity and period-radius relations.

As primary anchors of the distance scale, Cepheid stars play a crucial role in our understanding of the distance scale of the Universe because of their period-luminosity relation. Determining precise and consistent parameters (radius, temperature, color excess, and projection factor) of Cepheid pulsating stars is therefore very important. With the high-precision parallaxes delivered by the early third Gaia data release, we aim to derive various parameters of Cepheid stars in order to calibrate the period-luminosity and period-radius relations and to investigate the relation of period to p-factor. We applied an implementation of the parallax-of-pulsation method through the algorithm called Spectro-Photo-Interferometry of Pulsating Stars, which combines all types of available data for a variable star in a global modeling of its pulsation. We present the SPIPS modeling of a sample of 63 Galactic Cepheids. Adopting Gaia EDR3 parallaxes as an input associated with the best available dataset, we derive consistent values of parameters for these stars such as the radius, multiband apparent magnitudes, effective temperatures, color excesses, period changes, Fourier parameters, and the projection factor. We then derive new calibrations of the period-luminosity and period-radius relations. After investigating the dependences of the p-factor on the parameters of the stars, we find a high dispersion of its values and no evidence of its correlation with the period or with any other parameters. Statistically, the p-factor has an average value of p=1.26$\pm$0.07, but with an unsatisfactory agreement. In absence of any clear correlation between the p-factor and other quantities, the best agreement is obtained under the assumption that the p-factor can take any value in a band with a width of 0.15. This result highlights the need for a further examination of the physics behind the p-factor.