A&A 443, 883-889 (2005)
G. Paturel1 - P. Teerikorpi2
1 - CRAL-Observatoire de Lyon, 69561 Saint-Genis Laval Cedex, France
2 - Tuorla Observatory, Turku University, Väisäläntie 20, SF21500 Piikkiö, SF, Finland
Received 8 October 2004 / Accepted 18 June 2005
The unique measurements with the Hubble Space Telescope of Cepheid variable stars in nearby galaxies led to extragalactic distances that made the HST Key Project conclude that the Hubble constant is H0 = 72 km s-1 Mpc-1. The idea that H0 is now known is widely spread among the astronomical community. Some time ago, we suggested that a strong selection effect may still exist in the Cepheid method, resulting in too short distances. Using a model similar to traditional bias corrections, we deduce here new estimates of distances from HST and previous ground-based observations which are both affected by this effect, showing the same trend which starts at different distances. The recent measurement of M 83 with the VLT is unbiased. Revisiting the calibration of HSTKP's with our new scale, makes long-range distance criteria more concordant and reduces the value of H0 to 60 km s-1 Mpc-1. Locally, the corrected Cepheid distances give km s-1 Mpc-1and reduce the velocity dispersion in the Hubble flow. These numbers are indicative of the influence of the suggested Cepheid bias in the context of the HSTKP studies and are not final values.
Key words: stars: variables: Cepheids - cosmology: distance scale
The Hubble Space Telescope (HST) has provided astronomers with unprecedented high accuracy measurements free of atmospheric disturbance (Kennicutt et al. 1995). It can detect Cepheid stars in galaxies, up to the Virgo cluster. The determination of distances via the Period-Luminosity (PL) relation (Leavitt & Pickering 1912; Madore & Freedman 1991) made it possible to calibrate several secondary distance indicators (Freedman et al. 2001), which can be used to measure the distances to large numbers of more distant galaxies and to determine the global value of the Hubble constant.
At a given Cepheid period, the cosmic dispersion of the average absolute magnitude is so small (0.2 mag) that the systematic errors in the measured distances to the calibrator galaxies were considered as negligible. Based on this premise, the analysis led several teams (including us) to almost concordant Cepheid distances (Freedman et al. 2001; Saha et al. 2001; Paturel et al. 2002b). The Hubble constant finally derived by the HST Key Project was km s-1 Mpc-1. It seemed to many that the dilemma between high and low H was resolved, especially after the WMAP background radiation analysis led to a similar value. At the same time, the "precision'' global cosmology from the WMAP and (future) PLANCK experiments has emphasized the growing importance of an accurate local H determination to test the physics of the standard cosmological model (Spergel et al. 2003; Blanchard et al. 2003).
The distances from the HST data also confirmed that the Hubble law works at surprisingly small scales as observed in independent studies (Sandage & Tammann 1975; Ekholm et al. 2001; Karachentsev et al. 2002). The Hubble diagram (velocity vs. distance) directly shows the local expansion rate H as given by these remarkable HST observations.
Figure 1 shows an important ingredient of the bias: how a relevant limiting magnitude of Cepheid samples changes with the kinematical distance. Ideally, one would like to see this limit increase in step with distance. But Fig. 1 reveals that the magnitude limit no longer grows properly above a certain distance. We have suggest that the biased region starts there for the HST distance determinations.
|Figure 1: Variation of the deepness of the Cepheid samples with distance. The (normalized) apparent V magnitude limit of the Cepheid samples first grows with distance (radial velocity ) but, after a certain point, tends to bend where we have seen the bias appearing in derived Cepheid distances. We show the positions for an unbiased galaxy (NGC 3351) and a biased one (NGC 1425).|
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Another clue that suggests the presence of the bias, in addition to those already discussed in our previous papers, is given by Fig. 7a. The local Hubble diagram shows a departure from the expected linearity above a distance of 10 Mpc.
Finaly, it should be possible to see the departure from a proper sampling directly on the raw PL relation. As an example we show the V-band PL relation in Figs. 2a and b, for an unbiased galaxy (NGC 3351) and a biased one (NGC 1425). The positions of these galaxies are shown in Figs. 1 and 6, where the bias effect is visible.
|Figure 2: V-band period luminosity relation for NGC 3351, an unbiased galaxy ( top panel) and NGC 1425, a biased galaxy ( bottom panel). These galaxies are shown in Fig. 6, where we compare the corrected and uncorrected distance moduli.|
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Thus, the absolute limiting magnitudes of the Cepheid samples are calculated either as:
As in TP02, we then improve the diagrams with a normalization of the absolute limiting magnitude. It is made using the list of limiting observational periods of Cepheid stars (TP02). Its rationale is to put all individual bias curves on the same mean curve by compressing or expanding the x-axis according to each individual period. This method highlights the phenomenon with less distortion, as shown by observations (TP02) and simulations (PT04).
The normalized absolute limiting magnitude is calculated as:
|Figure 3: Observed Hubble values in log scale against normal ( top panel) and normalized ( bottom panel) limiting absolute magnitude. These absolute magnitudes are calculated from Eq. (1), using corrected radial velocity . Top panel: the observed depends on the limiting absolute magnitude. One may recognize the classical shape of a bias curve with a plateau of constant near the faint limiting absolute magnitudes ( . Bottom panel: after the normalization, all the data are put on a same bias curve. The plateau is more clearly visible.|
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In our first paper TP02 we realized that the trend in the parameter could be due to a correlation of errors. Indeed, in Fig. 3a, we calculate the limiting absolute magnitude from the radial velocity, according to Eq. (1). Thus, the radial velocity appears on both axes. If the observed velocity of a given object is too large, due, for instance, to random velocity, the corresponding point will be higher on the y-axis and lower on the x-axis. The point will be displaced in a direction with a negative slope. If the velocity is too small, the point will be displaced in the opposite direction, i.e., along the same line. This effect could thus produce something like the effect we have shown in Fig. 3a.
In TP02, we argued that the dispersion in the local Hubble law is small enough so that the systematic distance error dominates in the diagram. Indeed, at small distances, less than 5 to 10 Mpc, there is evidence for a low dispersion 50 km s-1 (e.g. Ekholm et al. 1999; Karachentsev et al. 2002, 2003; Thim et al. 2003), as we discussed in TP02. Also, we checked that the trend in the vs. diagram is also visible for cluster members, when the cluster velocity is used for the members, which should be at the same (cluster) distance.
If we repeat the test with (Eq. (2)), instead of , there is also a correlation of errors but in the opposite direction. We thus expect a positive slope. The result is shown in Fig. 4a. There is no evidence of such a positive slope (on the contrary, the slope still seems slightly negative). The real trend is probably intermediate between these two extreme cases. If a true bias exists, its signature is expected to diminish in the vs. diagram, when the kinematic distance is replaced by the biased photometric distance. Figure 4b shows the same diagram, but with the normalization. Thus, we conclude that the observed trend may well be real. Because such a trend can result from a statistical bias (see the Appendix, and also PT04), we will apply the old precepts to analyze it, by fitting a classical bias curve.
|Figure 4: Observed Hubble values in log scale against normal ( top panel) and normalized ( bottom panel) limiting absolute magnitude. These absolute magnitudes are calculated from Eq. (2), using observed distance moduli . Top panel: the correlation of the observed with the limiting absolute magnitude is still visible, as in Fig. 3a, although it should be canceled by the inverse correlation of errors (see text). However it is difficult to recognize the classical shape of a bias curve because of the uncertainties on . Bottom panel: after the normalization, all the data are put on the same bias curve.|
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We emphasize that this adopted value of the parameter for the bias curve does not affect the global value of H0, as derived in Sect. 5. Only the relative form of the bias curve is important there, giving an estimate of the distance bias for each calibrator galaxy.
We have added in Figs. 5a and b (filled star) the measurement of the galaxy M 83 (NGC5236) by Thim et al. (2003). They used the 8.2 m Unit Telescope 1 of the ESO VLT under subarcsecond seeing and could reach and d. This very good ground observation puts M 83 at 4.5 Mpc, directly in the unbiased plateau. This demonstrates the capability of the Very Large Telescope to measure Cepheid distances up to at least 5 Mpc.
In spite of the complexity of the bias, it is remarkable that the simple bias curve represents quite well the behaviour of real data. From the adopted bias curve we deduce the change and derive a new corrected distance for each calibrator galaxy.
|Figure 5: Application of the cluster incompleteness bias curve fitting to the Cepheid bias. Top panel: the fit of Fig. 3b with the bias curve following the method described in the text. Bottom panel: the same figure as the previous one, when the bias fit is applied to Fig. 4b. We have added on both figures the recent measurement of the galaxy M 83 (filled star) by Thim et al. (2003). It is in the unbiased plateau.|
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Only about twenty percent of galaxies can be considered as unbiased. This seems surprising, but in fact the fraction is similar to typical cases of galaxy cluster incompleteness or field galaxy Malmquist biases (Bottinelli et al. 1987; Theureau et al. 1997). Fortunately, most of the biased galaxies are still very valuable when corrected.
|Figure 6: Comparison of corrected and uncorrected distance moduli for HST (light gray or red) and ground-based (dark gray or blue) observations. The ground-based (open circles) and HST observations (filled circles) show the same trend (dashed lines) at different distances as explained by the systematic error in Cepheid distances. From the ground, the bias starts soon beyond the closest galaxy (M 31), except for the new VLT result on M 83 (filled star). From space, the bias appears beyond the distance of closest galaxy groups (M 81). At the distance of the closest large galaxy cluster (Virgo) it has a large incidence. The identified galaxies (NGC 3351 and NGC 1425) were used above (Fig. 2) to show the PL relation for an unbiased and biased galaxy.|
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The Hubble diagrams with uncorrected and corrected distances (Figs. 7a,b) show that the correction improves the linearity of the Hubble relation (the linear correlation coefficient increases from to ) and reduces the dispersion (from 120 to 84 km s-1). The local value of drops from 69 to 56 km s-1 Mpc-1. It was surprising in Freedman et al. (2001) that the local Hubble diagram for Cepheids was quite scattered. Now we propose that this was due to the variable distance bias.
Even in the nearer distance range, for galaxies observed from the ground with km s-1 (open circles in Fig. 7), the dispersion decreases from 35 km s-1 to 31 km s-1. These numbers again highlight the remarkable smoothness of the Hubble flow in the local inhomogeneous environment, for long recognized as a puzzle (e.g. Sandage 1999), but which has recently obtained theoretical clarification from CDM N-body simulations by Klypin et al. (2003) and Macciò et al. (2005), and other considerations by Teerikorpi et al. (2005). These studies support the suggestion by Chernin (2001) and Baryshev et al. (2001) that the low velocity dispersion may be a local signature of the dominating cosmological vacuum or dark energy.
|Figure 7: Local Hubble diagrams for uncorrected ( top) and corrected ( bottom) distances. Top panel: the clear curvature in the uncorrected relation is now understood to be caused by the bias that moves the galaxies up from the Hubble line. Open circles show the galaxies observed from the ground with a velocity km s-1. Bottom panel: the linearity, which is the property of the large-scale Hubble law as shown by secondary distance indicators (e.g., supernovae Ia), is revealed here, locally, by Cepheid stars. The line corresponds to a local value H= 56 km s-1 Mpc-1. The filled star represents the recent VLT measurement of the galaxy M 83 (see text).|
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Table 1 below summarizes the result. The value of H0 from a given secondary distance indicator is calculated as a weighted mean ( ) of the contribution of each calibrator. The mean values of H0 are calculated by weighting individual values according to the mean errors ( ). Two important facts are seen: i) the range of individual values is reduced. In particular the Fundamental Plane method is no longer an enigmatic outlier, but is in good agreement with the two main distance indicators (SNIa and Tully-Fisher). ii) The Hubble constant is reduced by about 15 percent. The mean value from different distance indicators is H0=60 km s-1 Mpc-1.
Table 1: Influence of the Cepheid bias on the Hubble constant from long-range distance criteria.
How some astronomers (most notably A. Sandage and G. Tammann) in the pre-HST era, consistently derived values of H0 of less than 60 km s-1 Mpc-1, even though the number of calibrator galaxies was much smaller. With hindsight, we see that the distances to those nearby calibrators were less biased, while efforts were made to avoid the selection biases in field and cluster galaxy samples (Sandage & Tamman 1975; Bottinelli et al. 1986).
We summarize the new results from this study:
This work has been supported by the Academy of Finland (the project ``Fundamental questions of observational cosmology''). We thank G. Tammann for proposing a test to see the influence of the correlation of errors. We thank the referee for reading the manuscript with extreme attention.
Table A.1: The sample used in this paper.
The true distances are derived from the radial velocity and an arbitrary Hubble constant. Then, from a Gaussian distribution ( ) of ,we obtain the absolute magnitudes through realistic V- and I-band PL relations. Then, from distances and absolute magnitudes we obtain apparent magnitudes V and I. Each quantity is combined with realistic errors and an extinction is added to the apparent magnitudes from random color excess and proper extinction coefficients. Thus we constructed for true galaxies an artificial catalog of Cepheids with known periods and apparent magnitudes, like a real one. However, for these data we know the true distances and realistic observational limits for periods and apparent magntiudes.
Thus, making classical derivations of distances from the V and I Period-luminosity relations, we construct the bias diagram (as in Fig. 3), for two cases:
|Figure A.1: Simulation of the bias. Top panel: calculation of from a simulation, in the case where there is no selection effect on magnitudes. Bottom panel: the same calculation as above, but a with a realistic selection on magnitudes (the limiting magnitude includes the amplitude of the Cepheid light curve).|
To see the effect of the velocity dispersion, we added to the simulated galaxies a random velocity dispersion increasing with distance, in order to reproduce the observations that show a very small velocity dispersion in the nearby universe. The adopted velocity dispersion is: (in km s-1). We obtained Fig. A.2a, b that reproduce quite well Figs. 5a, b based on real data. This also confirms that the bias is visible, even when it should be cancelled by an inverse correlation of errors (Bottom panel). The velocity dispersion contributes to hide the bias. The bias being visible with real data, this suggests again that the local velocity dispersion is very small.
|Figure A.2: Simulation of the bias when we add a velocity dispersion increasing with distance. Top panel: case where the correlation of errors due to the velocity reinforces the bias trend. Bottom panel: case where the correlation of errors due to should produce a trend opposite to the trend produced by the bias.|