Hubble Constant

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An exercise of the strong-lensing Hubble-constant group at the Lorentz Centre workshop on Gravitational Lensing.

Blandford, Coles, Courbin, Kochanek, Falco, Fassnacht, Hogg, Marshall, Meylan, Moustakas, Momcheva, Suyu

1 Abstract
2 Existing measurements of the scale
3 Data
4 Comments on systems we didn't use
5 Modeling
6 Limits on the world model and the Hubble Constant
7 Presentation for Friday

Abstract

Multiple-image gravitational lens time delays can in principle measure the scale of the Universe (Hubble Constant), but in practice the inferred scale is a strong function of the ill-constrained radial profile of the mass in the lensing galaxy. In this project we use gravitational lenses with measured time delays to put a very conservative lower limit to the scale of the Universe (upper limit on the Hubble Constant) by assuming that the radial profile of the lensing mass is the same as that of the stars in the galaxy, and that no other contributions to the effective projected mass distribution are present. This assumption is conservative because no galaxy with a measured mass profile has shown its mass to be this concentrated, and because we expect lenses to lie on (if only slightly) overdense lines of sight.

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Existing measurements of the scale

Key project: 72 +/- 8 (Freedman et al. 2001). This paper combines several different secondary distance indicators to obtain H_0 = 72 +/- 8 (including both stat and sys errors). The secondary indicators include Type 1a supernovae, Tully-Fisher, surface brightness fluctuations, Type II supernovae, and the fundamental plane. All of the secondary techniques give H_0 in the range 70-72, with the exception of the fundamental plane, which gives 82. The KP scales are all based on measurements of the distances to nearby galaxies, which then act as calibrators for the secondary distance indicators. The distances to the nearby galaxies are set through measurements of Cepheid light curves. This assumes that the period/luminosity relationship is well known and universal, with a zero point set by the Cepheids in the Large Magellanic Cloud (LMC). It also assumes that the variations of the P/L relation does not depend much on environment, i.e., metallicity.

Possible biases due to local calibrators. Most methods based on standard candles rely on local calibrators, like Cepheids. Paturel & Teerikorpi (2005) use an empirical correction to account for the effect of light-curve clipping (among others) due to too bright limiting magnitudes at large distances. Applying this correction to Type 1a supernovae, Tully-Fisher, surface brightness fluctuations, Type II supernovae, and the fundamental plane, the mean value for H_0 is shifted low, from 72 +- 3 to 60 +- 1.5.

Size of HII regions: This method is no longer competitive for determining H_0 because the size of the HII region depends on the size of the parent galaxy. (Tammann 2005)

Globular clusters (CGs): H_0 = 69+/-9 (Kavelaars et al. 2000 [most recent publication]) Globular clusters have bell shaped luminosity functions (LF), and the peak of the luminosity was proposed to be a standard candle (van den Bergh et al. 1985; for review see Tammann & Sandage 1999). However, the formation of GCs may not be a unique process.

Planetary nebulae (PNe): H_0 = 78+/-7 (Ciardullo et al. 2002 [most recent publication]) The method of using brightest planetary nebulae as distance indicators was proposed by Ford & Jenner (1978). However, the method seems to be dependent on the population size (Bottinelli et al. 1991, Tammann 1993), chemical composition, age (Mendez et al. 1993), and dynamics (Sambhus et al. 2005).

The tip of the red-giant branch (TRGB): Da Costa & Armandroff (1990) showed that the TRGBs in globular clusters have constant absolute I-magnitude, independent of metallicity. TRGBs are calibrated using GCs and RR Lyrae stars so they are independent of the Cephieds distance indicators. Some recent (within 5 years) application of the method are done by Karachentsev et al. (2003) and Sakai et al. (2004).

Luminosity class (LC) of spiral galaxies: H_0 = 55+/-3 (Sandage 1999). For spiral galaxies, the luminosities are correlated to the shape of the spiral structures.

Brightest cluster galaxies (BCG): H_0 = 54.2 +/- 5.4 (Sandage & Hardy 1973, Tammann 2005). The method assumes that the BCGs in clusters have the same luminosities.

Surface Brightness Fluctuations (SBF): The method applies to distant galaxies where individual stars are not resolvable. Galaxies that are further away have smaller statistical surface brightness fluctuations because the number of stars in a pixel increases with the distance to the galaxy. Tammann (2005) concluded that the SBF was able to measure relative distances within 13%, but was not to be used for Ho.

Fundamental Plane method (FP): H_0 = 57.0 +/- 4.4 (Federspiel M., 1999, Ph.D. Thesis, Univ. of Basel) calibrated with repect to Virgo and H_0 = 70.4 +/- 9.0 (Jorgensen et al. 1996, Tammann 2005) calibrated with respect to Coma. The method is based on the fact that the velocity dispersions of E/S0 galaxies are correlated to their luminosities.

Tully-Fisher (TF): H_0 = 65.6 +/- 4.1 (Giovanelli et al, 1997; Dale et al. 1999; Tammann 2005). The width of 21-cm line is a measure of the rotation velocity of a spiral galaxy, which is related to the mass and luminosity of the galaxy. The sample used by the authors is calibrated with respect to the Fornax cluster.

SZ: 77 +/- 4 (stat.) +/-10 (syst.) (Bonamente et al 2005). They used 38 clusters, selected to orientation-unbiased, and used Chandra X-ray data, OVRO/BIMA SZ data and various spherical cluster models. Cherry-picking relaxed clusters (Schmidt et al 2004) gave 69 +/- 8 (stat.) with systematic errors claimed to be unimportant. More details can be found here.

CMB: 73 +/- 3 if you believe in flatness and that the dark energy is vanilla cosmological constant (W. Hu's talk -- see Spergel et al 2006 astro-ph/0603449, e.g. Figure 20).
- You get a lower Hubble Constant if w is less negative than -1.
- It goes up and down if the Universe has spatial curvature.
- DGP braneworld requires a large Hubble Constant.
- For these reasons, a measurement of the Hubble constant is a measurement of the properties of the dark energy. This is Hu's heresy.

CMB+LSS+BAO: 69 +/- 2 if you believe in flatness and that the dark energy is vanilla cosmological constant (Eisenstein et al 2005).
- You get lower hubble constants, and much lower precision if you relax flatness or w=-1.
- Really the Eisenstein et al (2005) paper constrains the distance to z=0.35.
- We should also note that Eisenstein et al (2005) is a stunning confirmation of the lambda-CDM paradigm.

Maser distances: Water maser occur in the (outskirts of) accretion discs of central black holes of galaxies. The X-ray emission associated from the accretion stimulates water molecules in cool clouds which emit at radio wavelengths (22 GHz). As the disk rotates so will the masers. One measures the positions, velocities and accelerations and combining them with a model for the disk obtains the distance to the maser-host galaxy. The method can be used both on its own and for callibrating other distance indicators. The first galaxy for which this method was used was NGC4358 (Herrnstein, 1999). Currently there is an ongoing Water Maser Cosmology Project aiming at constraining dark energy and determining the Hubble constant to 1% using water masers. There are currently 63 known galaxies with water masers (a z=0.66 water maser was recently discovered) but only NGC4358 has a measured distance. The main goal of the Water Maser Cosmology Project is to carry out the extensive monitoring required to measure the distances to these galaxies to a 1% accuracy. Some authors suggest that the uncertainly in the models of the accretion disk (which may be very large) may be the main limitation for this method. Conclusion: potentially promising, but no results yet. Recent paper by Macri et al. (2006) quotes H_0= 74 +- 3 (r) +- 6 (s) using 4 well-observed type Ia SNe, Cepheids and water maser distance of NGC4258.

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Data

Notes on references: D = discovery paper, TD = time delay paper, ZL = lens redshift paper, ZS = source redshift paper.

config	name	zl	zs	Images	comments	references	time delays (days)
A C B D	B1608+656	1.394	0.630	4R	Binary lens.	[1], Fassnacht et al. 2002, ApJ 581, 823, TD [2], Myers et al. 1995, ApJ 447, L2, D, ZL [3] Fassnacht et al. 1996, ApJ 460, L103, ZS	BA=32pm2 BC=36pm2 BD=77pm2
A C B D	PG1115+080	1.722	0.31	4R	Group.	[4]Weymann et al. 1980, Science, 118, 4, D, ZS [5], Schechter et al. 1997, ApJ 475, L85, TD [6] Tonry 1998, AJ 115, 1, ZL	AB=13pm2 BC CD=11pm2
B A	HE1104-1805	2.32	0.73	2R-		[7], Wisotzki et al. 1993, AA 278, L15, D, ZS Kochanek et al. 2006, in prep, TD [8], Lidman et al. 2000, AA 364, L62, ZL	AB=161pm7
B A	HE2149-2745	2.03	0.49 0.50?	2	redshifts unreliable?	[9], Wisotzki et al. 1996, AA 315, 405, D, ZS [10], Burud et al. 2002, AA 383, 71, TD, ZL	AB=103pm12
A B	B1600+434	1.59	0.42	2R-		[11] Jackson et al. 1995, MNRAS, 274, 25L, (D and ZS) [12] Burud et al. 2000, ApJ, 544, 117(TD) [13] Fassnacht, Cohen, 1998, AJ, 115, 377 (ZL)	AB=51pm4
A B	FBQ J0951+2635	1.24	(0.24)	2	redshifts unreliable?	[14] Schechter et al. 1998, ApJ, 115, 1371 (D) [15] Jakobsson et al. 2005, A&A, 431, 103, (TD and ZL)	AB=16pm2
B A C D	HE0435-1223	1.689	0.46	4R		[16] Wisotzki et al. 2002, A&A 395, 17 (D) [17] Kochanek et al. 2006, (TD) [18] Morgan et al. 2005, AJ, 129, 2531 (ZL)	AB=-8.0p0.7m0.8 AC=-2.1p0.8m0.7 AD=-14.4p0.8m0.9
B D A C	RX J1131-1231	0.658	0.295	4R+		[19] Sluse et al. 2003, (D and ZL) [20], Morgan et al 2006 astro-ph/0605321(TD)	AB=11.98p1.52m1.27 AC=9.61p1.97m1.57
B A	SBS1520+530	1.86	0.72	2	lens redshift may be wrong	[21], Chavushyan et al 1997 AA 318, L67, D, ZS [22], Burud et al. 2002, AA 391, 481, TD, ZL	AB=130pm3

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Comments on systems we didn't use

config	name	zl	zs	Images	comments	references	time delays (days)
	SDSS J1650+4251	1.54		2	no imaging	Morgan 2003
	RX J0911+0551	2.80	0.77	4	cluster outskirts, no way	Bade et al 1997 discovery Burud et al 1998
no way	Q0957+561	1.41	0.36	2R	cluster	Walsh et al 1979
	B0218+357	0.96	0.68	2V	lens is a tiny spiral galaxy	O'Dea et al 1992
	RXJ1131-1231	0.658	0.295	4R+	may be a satellite galaxy under the cusp
	SDSS J1004+4112	1.734	0.68	4R	cluster
	PKS1830-211	2.51	0.89	2ER	Low galactic latitude with lots of confusing stars in the field. Controversy over whether there are one or two lensing galaxies.

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Modeling

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Limits on the world model and the Hubble Constant

The analysis of the lens data was all within the context of one particular model - that which has the most concentrated mass profile one could stomach, and which assumes that the additional convergence due to galaxies, groups and clusters either along the line of sight or in the lens plane is zero. It is this choice of model that will give us an upper limit on H_0: no special statistical manipulations are needed. The figure below shows how the relative time delay changes as the surface mass density changes (Courbin, Saha & Schechter, 2002).

Given the assumptions listed above, the lens data, and the assertion of a single Hubble constant, we should combine the data from our lens sample in the usual way, by taking the product of the likelihood functions. However, it is more convenient to analyse each lens individually, and then work with the individual marginalised posterior densities Pr(H0,Om,Ol|data). Provided we assigned uniform priors on these three parameters in each analysis, we can compute the joint posterior by just multiplying the individual distributions. We do this on a fine three-dimensional grid, and then project onto various planes for presentation purposes. Note that we do not allow for a variable w on the grounds that at present it seems improbable.

One important issue in this analysis is that of having suspect data. In some cases it has turned out (after the fact) that the measured time delay was somewhat innaccurate. Not having access to the original time series the only option open to us is to appeal for consistency between H0 estimates. This can be done in a number of ways, all of which boil down to weighting the datasets by their evidence values (given the model). Equivalently, the relative weights can be estimated directly from the data (as "hyperparameters").

The combination of the lensing H0 upper limit with WMAP is interesting: the degeneracy in the CMB data is such that H0 values of less than 75 or so (check!) indicate a slightly closed Universe. We can quantify this constraint... Likewise, we can compare with the HST key project and tighten down its unsightly long upper tail. The figure below shows the results from Freedman et al (blue), and Spergel et al (solid and dashed red). The strong lens 99% limit is marked on the plot in green.