The Local Value of H0_0

We review the local determination of the Hubble constant, H$_0$, focusing on recent measurements of a distance ladder constructed from geometry, Cepheid variables and Type Ia supernovae (SNe Ia). We explain in some detail the components of the ladder: (1) geometry from Milky Way parallaxes, masers in NGC 4258 and detached eclipsing binaries in the Large Magellanic Cloud; (2) measurements of Cepheids with the Hubble Space Telescope (HST) in these anchors and in the hosts of 42 SNe Ia; and (3) SNe Ia in the Hubble flow. Great attention to negating systematic uncertainties through the use of differential measurements is reviewed. A wide array of tests are discussed. The measurements provide a strong indication of a discrepancy between the local measure of H$_0$ and its value predicted by $\Lambda$CDM theory, calibrated by the cosmic microwave background ($Planck$), a decade-long challenge known as the `Hubble Tension'. We present new measurements with the James Webb Space Telescope of $>$320 Cepheids on both rungs of the distance ladder, in a SN Ia host and the geometric calibrator NGC 4258, showing reduced noise and good agreement with the same as measured with HST. This provides strong evidence that systematic errors in HST Cepheid photometry do not play a significant role in the present Hubble Tension. Future measurements are expected to refine the local determination of the Hubble constant.Comment: 14 pages, 14 figures. Invited Review for IAU Symposium 376, Richard de Grijs, Patricia Whitelock and Marcio Catelan, ed

Cluster Cepheids with High Precision Gaia Parallaxes, Low Zeropoint Uncertainties, and Hubble Space Telescope Photometry

We present HST photometry of 17 Cepheids in open clusters and their mean parallaxes from Gaia EDR3. These parallaxes are more precise than those from individual Cepheids (G<8 mag) previously used to measure the Hubble constant because they are derived from an average of >300 stars per cluster. Cluster parallaxes also have smaller systematic uncertainty because their stars lie in the range (G>13 mag) where the Gaia parallax calibration is most comprehensive. Cepheid photometry employed in the period--luminosity relation was measured using the same instrument(WFC3) and filters(F555W,F814W,F160W) as extragalactic Cepheids in SNIa hosts. We find no evidence of residual parallax offset in this magnitude range, zp=-3+/-4 muas, consistent with Lindegren:2021b and most studies. The Cepheid luminosity (P=10d, solar-metallicity) in the HST near-infrared, Wesenheit system derived from the cluster sample is M_{H,1}^W=-5.902+/-0.025 and -5.890+/-0.018 mag with or without simultaneous determination of a parallax offset, respectively. These results are similar to measurements from field Cepheids, confirming the accuracy of the Gaia parallaxes over a broad range of magnitudes. The SH0ES distance ladder calibrated solely from this sample gives H_0=72.8+/-1.3 and H_0=73.2+/-1.1 km/s/Mpc with or without offset marginalization; combined with all anchors we find H_0=73.01+/-0.99 and 73.15+/-0.97, respectively, a 5% or 7% reduction in the uncertainty and a 5.3 sigma Hubble Tension relative to Planck+LambdaCDM. It appears increasingly difficult to reconcile two of the best measured cosmic scales, parallaxes from Gaia and the angular size of the acoustic scale of the CMB, using the simplest form of LambdaCDM to join the two.Comment: 12 pages, submitted to ApJ, comments welcom

A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team

We report observations from HST of Cepheids in the hosts of 42 SNe Ia used to calibrate the Hubble constant (H0). These include all suitable SNe Ia in the last 40 years at z1000 orbits, more than doubling the sample whose size limits the precision of H0. The Cepheids are calibrated geometrically from Gaia EDR3 parallaxes, masers in N4258 (here tripling that Cepheid sample), and DEBs in the LMC. The Cepheids were measured with the same WFC3 instrument and filters (F555W, F814W, F160W) to negate zeropoint errors. We present multiple verifications of Cepheid photometry and tests of background determinations that show measurements are accurate in the presence of crowding. The SNe calibrate the mag-z relation from the new Pantheon+ compilation, accounting here for covariance between all SN data, with host properties and SN surveys matched to negate differences. We decrease the uncertainty in H0 to 1 km/s/Mpc with systematics. We present a comprehensive set of ~70 analysis variants to explore the sensitivity of H0 to selections of anchors, SN surveys, z range, variations in the analysis of dust, metallicity, form of the P-L relation, SN color, flows, sample bifurcations, and simultaneous measurement of H(z). Our baseline result from the Cepheid-SN sample is H0=73.04+-1.04 km/s/Mpc, which includes systematics and lies near the median of all analysis variants. We demonstrate consistency with measures from HST of the TRGB between SN hosts and NGC 4258 with Cepheids and together these yield 72.53+-0.99. Including high-z SN Ia we find H0=73.30+-1.04 with q0=-0.51+-0.024. We find a 5-sigma difference with H0 predicted by Planck+LCDM, with no indication this arises from measurement errors or analysis variations considered to date. The source of this now long-standing discrepancy between direct and cosmological routes to determining the Hubble constant remains unknown.Comment: 67 pages, 31 figures, replaced to match ApJ accepted version (March 2022), Table 6 distances included here, long form of photometry tables, fitting code, compact form of data, available from Github page, https://pantheonplussh0es.github.i

L'échelle de distance des Céphéides : de l’étalonnage Gaia local aux galaxies lointaines

Cepheids are pulsating variable stars which play a key role as primary distance indicators thanks to the empirical relation between their pulsation period and intrinsic luminosity, the period-luminosity relation. This law is used to calibrate the brightness of type-Ia supernovæ in nearby galaxies, which is in turn used to measure the distance to galaxies in the Hubble flow. This provides an estimate of the current expansion rate of the Universe, known as the Hubble constant (H0). In recent years, a significant tension of at least 4σ has arisen between the early universe measurement of H0 from the Planck satellite, assuming a ΛCDM model, and the late universe direct measurements based on Cepheid distances. The persistence of this tension would imply new physics beyond the standard model of cosmology: it is therefore critical to improve the period-luminosity calibration with precise and reliable Cepheid distance measurements. The Gaia Collaboration recently published trigonometric parallaxes for 1.7 billion stars, allowing for a remarkable improvement in the precision of the distance scale. However, Cepheid parallaxes suffer from calibration issues due to their variability and important brightness. In this thesis, I present an alternative method for calibrating the period-luminosity relation using Cepheid companions and host open clusters, which are not subject to these issues. By adopting these close and unbiased companion stars to determine the distance to Cepheids, I calibrate the period-luminosity relation in the Milky Way and re-evaluate the local value of the Hubble constant. Finally, I study the relation between Cepheid magnitudes and their metal abundance by comparing the Cepheids of the Milky Way and those of the Magellanic Clouds. I conclude that metal-rich Cepheids are brighter than metal-poor ones, with a stronger effect in near-infrared than in optical. This effect may impact the measurement of the Hubble constant and will have to be taken into account more precisely in the future, to better constrain the calibration of the extragalactic distance scale.Les Céphéides sont des étoiles variables pulsantes qui jouent un rôle clé comme indicateurs primaires de distance grâce à la relation empirique entre leur période de pulsation et leur luminosité intrinsèque, la relation période-luminosité. Cette loi est utilisée pour étalonner la luminosité des supernovæ de type Ia dans les galaxies proches, qui est à son tour utilisée pour mesurer la distance aux galaxies dans le flot de Hubble, fournissant une estimation du taux d’expansion actuel de l’Univers: la constante de Hubble (H0). Ces dernières années, une tension significative d’au moins 4σ est apparue entre la mesure de H0 dans l’univers primitif par le satellite Planck, en supposant un modèle ΛCDM, et les mesures directes dans l’univers local basées sur les distances des Céphéides. La confirmation de cette tension pourrait impliquer une nouvelle physique au delà du modèle standard : il est donc essentiel d’améliorer l’étalonnage de la relation période-luminosité grâce à des distances précises et fiables de Céphéides. La collaboration Gaia a récemment publié les parallaxes trigonométriques de plus d’1.7 milliard d’étoiles, permettant une amélioration remarquable de la précision de l’échelle des distances. Cependant, les parallaxes des Céphéides sont affectées par des problèmes de calibration en raison de leur variabilité et de leur importante luminosité. Dans cette thèse, je présente une méthode alternative pour étalonner la relation période-luminosité en utilisant des compagnons de Céphéides et des amas ouverts hôtes, qui ne sont pas soumis à ces problèmes. En utilisant ces compagnons proches non biaisés pour déterminer la distance des Céphéides, j’étalonne la relation période-luminosité dans la Voie Lactée et je réévalue la valeur locale de la constante de Hubble. Enfin, j’étudie le lien entre les magnitudes absolues des Céphéides et leur abondance en métaux en comparant les Céphéides de la Voie Lactée et celles des Nuages de Magellan. J’en déduis que les Céphéides riches en métaux sont plus brillantes que celles qui en sont pauvres, avec un effet plus fort en infrarouge proche qu’en optique. Cet effet peut avoir un impact sur la mesure de la constante de Hubble et devra être pris en compte plus précisément à l’avenir, afin de mieux contraindre l’étalonnage de l’échelle des distances extragalactiques

L'échelle de distance des Céphéides : de l’étalonnage Gaia local aux galaxies lointaines

Cepheids are pulsating variable stars which play a key role as primary distance indicators thanks to the empirical relation between their pulsation period and intrinsic luminosity, the period-luminosity relation. This law is used to calibrate the brightness of type-Ia supernovæ in nearby galaxies, which is in turn used to measure the distance to galaxies in the Hubble flow. This provides an estimate of the current expansion rate of the Universe, known as the Hubble constant (H0). In recent years, a significant tension of at least 4σ has arisen between the early universe measurement of H0 from the Planck satellite, assuming a ΛCDM model, and the late universe direct measurements based on Cepheid distances. The persistence of this tension would imply new physics beyond the standard model of cosmology: it is therefore critical to improve the period-luminosity calibration with precise and reliable Cepheid distance measurements. The Gaia Collaboration recently published trigonometric parallaxes for 1.7 billion stars, allowing for a remarkable improvement in the precision of the distance scale. However, Cepheid parallaxes suffer from calibration issues due to their variability and important brightness. In this thesis, I present an alternative method for calibrating the period-luminosity relation using Cepheid companions and host open clusters, which are not subject to these issues. By adopting these close and unbiased companion stars to determine the distance to Cepheids, I calibrate the period-luminosity relation in the Milky Way and re-evaluate the local value of the Hubble constant. Finally, I study the relation between Cepheid magnitudes and their metal abundance by comparing the Cepheids of the Milky Way and those of the Magellanic Clouds. I conclude that metal-rich Cepheids are brighter than metal-poor ones, with a stronger effect in near-infrared than in optical. This effect may impact the measurement of the Hubble constant and will have to be taken into account more precisely in the future, to better constrain the calibration of the extragalactic distance scale.Les Céphéides sont des étoiles variables pulsantes qui jouent un rôle clé comme indicateurs primaires de distance grâce à la relation empirique entre leur période de pulsation et leur luminosité intrinsèque, la relation période-luminosité. Cette loi est utilisée pour étalonner la luminosité des supernovæ de type Ia dans les galaxies proches, qui est à son tour utilisée pour mesurer la distance aux galaxies dans le flot de Hubble, fournissant une estimation du taux d’expansion actuel de l’Univers: la constante de Hubble (H0). Ces dernières années, une tension significative d’au moins 4σ est apparue entre la mesure de H0 dans l’univers primitif par le satellite Planck, en supposant un modèle ΛCDM, et les mesures directes dans l’univers local basées sur les distances des Céphéides. La confirmation de cette tension pourrait impliquer une nouvelle physique au delà du modèle standard : il est donc essentiel d’améliorer l’étalonnage de la relation période-luminosité grâce à des distances précises et fiables de Céphéides. La collaboration Gaia a récemment publié les parallaxes trigonométriques de plus d’1.7 milliard d’étoiles, permettant une amélioration remarquable de la précision de l’échelle des distances. Cependant, les parallaxes des Céphéides sont affectées par des problèmes de calibration en raison de leur variabilité et de leur importante luminosité. Dans cette thèse, je présente une méthode alternative pour étalonner la relation période-luminosité en utilisant des compagnons de Céphéides et des amas ouverts hôtes, qui ne sont pas soumis à ces problèmes. En utilisant ces compagnons proches non biaisés pour déterminer la distance des Céphéides, j’étalonne la relation période-luminosité dans la Voie Lactée et je réévalue la valeur locale de la constante de Hubble. Enfin, j’étudie le lien entre les magnitudes absolues des Céphéides et leur abondance en métaux en comparant les Céphéides de la Voie Lactée et celles des Nuages de Magellan. J’en déduis que les Céphéides riches en métaux sont plus brillantes que celles qui en sont pauvres, avec un effet plus fort en infrarouge proche qu’en optique. Cet effet peut avoir un impact sur la mesure de la constante de Hubble et devra être pris en compte plus précisément à l’avenir, afin de mieux contraindre l’étalonnage de l’échelle des distances extragalactiques

L'échelle de distance des Céphéides : de l’étalonnage Gaia local aux galaxies lointaines

Cepheids are pulsating variable stars which play a key role as primary distance indicators thanks to the empirical relation between their pulsation period and intrinsic luminosity, the period-luminosity relation. This law is used to calibrate the brightness of type-Ia supernovæ in nearby galaxies, which is in turn used to measure the distance to galaxies in the Hubble flow. This provides an estimate of the current expansion rate of the Universe, known as the Hubble constant (H0). In recent years, a significant tension of at least 4σ has arisen between the early universe measurement of H0 from the Planck satellite, assuming a ΛCDM model, and the late universe direct measurements based on Cepheid distances. The persistence of this tension would imply new physics beyond the standard model of cosmology: it is therefore critical to improve the period-luminosity calibration with precise and reliable Cepheid distance measurements. The Gaia Collaboration recently published trigonometric parallaxes for 1.7 billion stars, allowing for a remarkable improvement in the precision of the distance scale. However, Cepheid parallaxes suffer from calibration issues due to their variability and important brightness. In this thesis, I present an alternative method for calibrating the period-luminosity relation using Cepheid companions and host open clusters, which are not subject to these issues. By adopting these close and unbiased companion stars to determine the distance to Cepheids, I calibrate the period-luminosity relation in the Milky Way and re-evaluate the local value of the Hubble constant. Finally, I study the relation between Cepheid magnitudes and their metal abundance by comparing the Cepheids of the Milky Way and those of the Magellanic Clouds. I conclude that metal-rich Cepheids are brighter than metal-poor ones, with a stronger effect in near-infrared than in optical. This effect may impact the measurement of the Hubble constant and will have to be taken into account more precisely in the future, to better constrain the calibration of the extragalactic distance scale.Les Céphéides sont des étoiles variables pulsantes qui jouent un rôle clé comme indicateurs primaires de distance grâce à la relation empirique entre leur période de pulsation et leur luminosité intrinsèque, la relation période-luminosité. Cette loi est utilisée pour étalonner la luminosité des supernovæ de type Ia dans les galaxies proches, qui est à son tour utilisée pour mesurer la distance aux galaxies dans le flot de Hubble, fournissant une estimation du taux d’expansion actuel de l’Univers: la constante de Hubble (H0). Ces dernières années, une tension significative d’au moins 4σ est apparue entre la mesure de H0 dans l’univers primitif par le satellite Planck, en supposant un modèle ΛCDM, et les mesures directes dans l’univers local basées sur les distances des Céphéides. La confirmation de cette tension pourrait impliquer une nouvelle physique au delà du modèle standard : il est donc essentiel d’améliorer l’étalonnage de la relation période-luminosité grâce à des distances précises et fiables de Céphéides. La collaboration Gaia a récemment publié les parallaxes trigonométriques de plus d’1.7 milliard d’étoiles, permettant une amélioration remarquable de la précision de l’échelle des distances. Cependant, les parallaxes des Céphéides sont affectées par des problèmes de calibration en raison de leur variabilité et de leur importante luminosité. Dans cette thèse, je présente une méthode alternative pour étalonner la relation période-luminosité en utilisant des compagnons de Céphéides et des amas ouverts hôtes, qui ne sont pas soumis à ces problèmes. En utilisant ces compagnons proches non biaisés pour déterminer la distance des Céphéides, j’étalonne la relation période-luminosité dans la Voie Lactée et je réévalue la valeur locale de la constante de Hubble. Enfin, j’étudie le lien entre les magnitudes absolues des Céphéides et leur abondance en métaux en comparant les Céphéides de la Voie Lactée et celles des Nuages de Magellan. J’en déduis que les Céphéides riches en métaux sont plus brillantes que celles qui en sont pauvres, avec un effet plus fort en infrarouge proche qu’en optique. Cet effet peut avoir un impact sur la mesure de la constante de Hubble et devra être pris en compte plus précisément à l’avenir, afin de mieux contraindre l’étalonnage de l’échelle des distances extragalactiques

The Cepheid distance scale : from the local Gaia calibration to distant galaxies

Les Céphéides sont des étoiles variables pulsantes qui jouent un rôle clé comme indicateurs primaires de distance grâce à la relation empirique entre leur période de pulsation et leur luminosité intrinsèque, la relation période-luminosité. Cette loi est utilisée pour étalonner la luminosité des supernovæ de type Ia dans les galaxies proches, qui est à son tour utilisée pour mesurer la distance aux galaxies dans le flot de Hubble, fournissant une estimation du taux d’expansion actuel de l’Univers: la constante de Hubble (H0). Ces dernières années, une tension significative d’au moins 4σ est apparue entre la mesure de H0 dans l’univers primitif par le satellite Planck, en supposant un modèle ΛCDM, et les mesures directes dans l’univers local basées sur les distances des Céphéides. La confirmation de cette tension pourrait impliquer une nouvelle physique au delà du modèle standard : il est donc essentiel d’améliorer l’étalonnage de la relation période-luminosité grâce à des distances précises et fiables de Céphéides. La collaboration Gaia a récemment publié les parallaxes trigonométriques de plus d’1.7 milliard d’étoiles, permettant une amélioration remarquable de la précision de l’échelle des distances. Cependant, les parallaxes des Céphéides sont affectées par des problèmes de calibration en raison de leur variabilité et de leur importante luminosité. Dans cette thèse, je présente une méthode alternative pour étalonner la relation période-luminosité en utilisant des compagnons de Céphéides et des amas ouverts hôtes, qui ne sont pas soumis à ces problèmes. En utilisant ces compagnons proches non biaisés pour déterminer la distance des Céphéides, j’étalonne la relation période-luminosité dans la Voie Lactée et je réévalue la valeur locale de la constante de Hubble. Enfin, j’étudie le lien entre les magnitudes absolues des Céphéides et leur abondance en métaux en comparant les Céphéides de la Voie Lactée et celles des Nuages de Magellan. J’en déduis que les Céphéides riches en métaux sont plus brillantes que celles qui en sont pauvres, avec un effet plus fort en infrarouge proche qu’en optique. Cet effet peut avoir un impact sur la mesure de la constante de Hubble et devra être pris en compte plus précisément à l’avenir, afin de mieux contraindre l’étalonnage de l’échelle des distances extragalactiques.Cepheids are pulsating variable stars which play a key role as primary distance indicators thanks to the empirical relation between their pulsation period and intrinsic luminosity, the period-luminosity relation. This law is used to calibrate the brightness of type-Ia supernovæ in nearby galaxies, which is in turn used to measure the distance to galaxies in the Hubble flow. This provides an estimate of the current expansion rate of the Universe, known as the Hubble constant (H0). In recent years, a significant tension of at least 4σ has arisen between the early universe measurement of H0 from the Planck satellite, assuming a ΛCDM model, and the late universe direct measurements based on Cepheid distances. The persistence of this tension would imply new physics beyond the standard model of cosmology: it is therefore critical to improve the period-luminosity calibration with precise and reliable Cepheid distance measurements. The Gaia Collaboration recently published trigonometric parallaxes for 1.7 billion stars, allowing for a remarkable improvement in the precision of the distance scale. However, Cepheid parallaxes suffer from calibration issues due to their variability and important brightness. In this thesis, I present an alternative method for calibrating the period-luminosity relation using Cepheid companions and host open clusters, which are not subject to these issues. By adopting these close and unbiased companion stars to determine the distance to Cepheids, I calibrate the period-luminosity relation in the Milky Way and re-evaluate the local value of the Hubble constant. Finally, I study the relation between Cepheid magnitudes and their metal abundance by comparing the Cepheids of the Milky Way and those of the Magellanic Clouds. I conclude that metal-rich Cepheids are brighter than metal-poor ones, with a stronger effect in near-infrared than in optical. This effect may impact the measurement of the Hubble constant and will have to be taken into account more precisely in the future, to better constrain the calibration of the extragalactic distance scale

An Improved Calibration of the Wavelength Dependence of Metallicity on the Cepheid Leavitt law

The Cepheid period-luminosity (PL) relation (or Leavitt law) has served as the first rung of the most widely used extragalactic distance ladders and is central to the determination of the local value of the Hubble constant ($H_0$). We investigate the influence of metallicity on Cepheid brightness, a term that significantly improves the overall fit of the distance ladder, to better define its wavelength dependence. To this aim, we compare the PL relations obtained for three Cepheid samples having distinct chemical composition (in the Milky Way and Magellanic Clouds) and focusing on the use of improved and recent data while covering a metallicity range of about 1 dex. We estimate the metallicity effect (hereafter $\gamma$) in 15 filters from mid-infrared to optical wavelengths, including 5 Wesenheit indices, and we derive a significant metallicity term in all filters, in accord with recent empirical studies and models, in the sense of metal-rich Cepheids being brighter than metal-poor ones. We describe the contribution of various systematic effects in the determination of the $\gamma$ term. We find no evidence of $\gamma$ changing over the wavelength range 0.5 - 4.5 $\mu$m indicating the main influence of metallicity on Cepheids is in their luminosity rather than color. We identify factors which sharpen the empirical constraints on the metallicity term over past studies including corrections for the depth of the Magellanic Clouds, better calibrated Cepheid photometry, improved Milky Way extinction estimates and revised and expanded metallicity measurements in the LMC.Comment: 19 pages, 6 figures, submitted to Ap

An Improved Calibration of the Wavelength Dependence of Metallicity on the Cepheid Leavitt Law

International audienceThe Cepheid period-luminosity (PL) relation (or Leavitt law) has served as the first rung of the most widely used extragalactic distance ladder and is central to the determination of the local value of the Hubble constant (H 0). We investigate the influence of metallicity on Cepheid brightness, a term that significantly improves the overall fit of the distance ladder, to better define its wavelength dependence. To this aim, we compare the PL relations obtained for three Cepheid samples having distinct chemical composition (in the Milky Way and Magellanic Clouds) and focusing on the use of improved and recent data while covering a metallicity range of about 1 dex. We estimate the metallicity effect (hereafter γ) in 15 filters from mid-IR to optical wavelengths, including five Wesenheit indices, and we derive a significant metallicity term in all filters, in agreement with recent empirical studies and models, in the sense of metal-rich Cepheids being brighter than metal-poor ones. We describe the contribution of various systematic effects in the determination of the γ term. We find no evidence of γ changing over the wavelength range 0.5-4.5 μm, indicating that the main influence of metallicity on Cepheids is in their luminosity rather than color. Finally, we identify factors that sharpen the empirical constraints on the metallicity term over past studies, including corrections for the depth of the Magellanic Clouds, better-calibrated Cepheid photometry, improved Milky Way extinction estimates, and revised and expanded metallicity measurements in the LMC

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

29 pages, 23 figures, 5 tables. Accepted for publication in Astronomy & Astrophysiscs journal (30/08/2021)International audienceContext. 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. Aims. With the high-precision parallaxes delivered by the early third Gaia data release (EDR3), 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. Methods. We applied an implementation of the parallax-of-pulsation method through the algorithm called spectro-photointerferometry of pulsating stars (SPIPS), which combines all types of available data for a variable star (multiband and multicolor photometry, radial velocity, effective temperature, and interferometry measurements) in a global modeling of its pulsation. Results. 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. Conclusions. Using the best set of data and the most precise distances for Milky Way Cepheids, we derive new calibrations of the period-luminosity and period-radius relations: M K S = −5.529 ±0.015 −3.141 ±0.050 (log P−0.9) and log R = 1.763 ±0.003 +0.653 ±0.012 (log P− 0.9). After investigating the dependences of the projection 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 such as radial velocity, temperature, or metallicity. Statistically, the p−factor has an average value of p = 1.26 ± 0.07, but with an unsatisfactory agreement (σ = 0.15). 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