A 3% Determination of H_{0} at Intermediate Redshifts

A 3% Determination of $H_0$ at Intermediate Redshifts


Recent determinations of the Hubble constant, , at extremely low and very high redshifts based on the cosmic distance ladder (grounded with trigonometric parallaxes) and a cosmological model (applied to Planck 2013 data) respectively, are revealing an intriguing discrepancy (nearly 9% or 2.4) that is challenging astronomers and theoretical cosmologists. In order to shed some light on this problem, here we discuss a new determination of at intermediate redshifts (), using the following four cosmic probes: (i) measurements of the angular diameter distances (ADD) for galaxy clusters based on the combination of Sunyaev-Zeldovich effect and X-ray data (), (ii) the inferred ages of old high redshift galaxies (OHRG) (), (iii) measurements of the Hubble parameter (), and (iv) the baryon acoustic oscillation (BAO) signature (). In our analysis, assuming a flat CDM cosmology and considering statistical plus systematic errors we obtain km s Mpc (1) which is a determination of the Hubble constant at intermediate redshifts. We stress that each individual test adopted here has error bars larger than the ones appearing in the calibration of the extragalactic distance ladder. However, the remarkable complementarity among the four tests works efficiently in reducing greatly the possible degeneracy on the space parameter () ultimately providing a value of that is in excellent agreement with the determination using recessional velocities and distances to nearby objects.

cosmology: cosmological parameters - distance scale - galaxies: distances and redshifts

1 Introduction

One of the most important observational quantities for cosmology is the Hubble constant, , whose value determines the present day expansion rate of the Universe. The determination of remains a very active research topic since the early days of physical cosmology (Jackson 2007; Riess et al. 2011; Suyu et al. 2012; Freedman et al. 2012). Several groups and ongoing missions are now focusing their efforts to a high-accuracy calibration of since it works like a key to quantify many astronomical phenomena in a wide range of cosmic scales. It also plays an important role for several cosmological calculations as the physical distances to objects, the age, size, and matter-energy content of the Universe (Freedman & Madore 2010; Cháves et al. 2012; Farooq and Ratra 2013; Ade et al. 2013).

Currently, the more robust constraints on are being obtained from local tests at low redshifts (i.e., ). The primary method is based on a cosmic distance ladder interlinking different distance indicators. The basic measurements and strategies commonly adopt: Cepheids, tip of the red giant branch, maser galaxies, surface brightness fluctuations, the Tully-Fisher relation, and type Ia supernovae. In the last decade, the Hubble Space Telescope (HST) Key Project did a magnificent job in decreasing significantly the errors on Hubble constant (for a review see Freedman & Madore 2010). Recently, Riess et al. (2011) used the HST observations to determine from Cepheids and Supernovae (SNe Ia). Through a rigorous analysis of the statistical and systematic errors they obtained km s Mpc (1), corresponding to a uncertainty. Later on, using Spitzer Space Telescope, Freedman et al. (2012) re-calibrated the HST Key Project sample and found km s Mpc (1).

The local distance ladder may also be susceptible to unknown sources of systematics like the possible existence of a “Hubble bubble” or other local effects that would affect measurements of the nearby expansion-velocity (Jha et al. 2007; Sinclair et al. 2010; Marra et al. 2013). The possibility of an observational convergence in the near future has increased in the last few years, however, additional progress, say, for a determination with uncertainty will demand a closer scrutiny of the cosmic distance ladder (Suyu et al. 2012; Freedman et al. 2012).

On the other hand, cosmologists desire accurate measurements of the mainly to refine the constraints on the neutrino masses (), density (), and the equation of state parameter () of dark energy based on the cosmic microwave background (CMB) anisotropies data (Macri et al. 2006; Sekiguchi et al. 2010). It is also widely known that CMB data alone can not supply strong constraints on (Spergel et al. 2007, Komatsu et al. 2009), a problem closely related with the degeneracy of the space parameter. Different grouping of the parameters (, , , , etc.) produce the same prediction of the CMB anisotropies. This degeneracy problem can be alliviated only by using a prior on from independent probes (Hu 2005). In this regard, Komatsu et al. (2011), used CMB, Supernovae (SNe Ia) and Baryon acoustic oscillations (BAO) data, to derive a model-dependent value of . It should also be recalled that the phenomenology underlying the CMB test operates on observations made at very high redshifts ( 1070), during the decoupling of radiation and matter. Such determination of involves a combination of physics and phenomena at very different scales and epochs of the cosmic evolution (low, intermediate and high redshifts), and, more important to the present article, due to the inclusion of SNe Ia data, the derived value of is also somewhat dependent on the cosmic distance ladder. Therefore, it is not only important to derive constraints on based on many different kinds of observations, but, also to avoid the combination of phenomena occurring at very different epochs (like SNe Ia and CMB).

Hinshaw et al. (2012) using CMB experiments from WMAP-9 found km s Mpc (1) whereas Ade et al. (2013) using Planck mission analysis, reported km s Mpc (1), corresponding to a 2.1% uncertainty. It is worth noting that the current results from different experiments working with very high redshifts physics are consistent with each other; however, the CMB experiments are discrepant with the local measurements. In particular, the PLANCK mission tension is at confidence level (Ade et al. 2013). In the present “era of precision cosmology”, investigating this tension more fully may bring new insights into improving the determination of the Hubble constant. Naturally, if such tension is not a mere consequence of systematics, and it is further strengthened by the incoming data, a new cosmology beyond CDM may also prove necessary.

Recently, some cosmological tests at intermediate redshifts () have emerged as a promising technique to estimate (Simon et al. 2005; Deepak & Dev 2006; Cunha et al. 2007; Lima et al. 2009; Stern et al. 2010). The main advantage of these methods is their independence of local calibrators (Carlstrom et al. 2002; Jones et al. 2005). In principle, such methods provide a crosschecking for local direct estimates which are free from Hubble bubbles, since the majority of galaxy clusters are well inside the Hubble flow. All these methods are dependent on the assumptions about the astrophysical medium properties, as well as, of the cosmological model adopted in their analysis, and, individually, are not yet competitive with the traditional methods based on the cosmic distance ladder (Freedman et al. 2001).

In this Letter, we identify a remarkable complementarity involving four different cosmological probes at intermediate redshifts () whose combination provides a valuable cross-check for . The first test is based on the angular diameter distance from galaxy clusters via Sunyaev-Zeldovich effect (SZE) combined with measurements of the X-ray flux (SZE/X-ray technique). Although suggested long ago, only recently it has been applied for a fairly large number of clusters (Bonamente et al. 2006; Cunha et al. 2007; Holanda et al. 2012). The second one is the age estimates of galaxies and quasars at intermediate redshifts. It became possible through the new optical and infrared techniques together the advent of large telescopes (Lima et al. 2009). The third possibility arises from measurements of the Hubble parameter, , from differential ages of galaxies (Simon et al. 2005; Gaztañaga et al. 2009; Stern et al. 2010), while the fourth is the BAO signature (Eisenstein et al. 2005). The cooperative interaction among these independent tests reduce greatly the errors on , and, more interesting, located just on a redshift zone distinct both from CMB anisotropies and the methods defined by the cosmic distance ladder (Cepheid, SNe Ia, etc.). As we shall see, these four probes act in concert to predict constraints on that are in excellent agreement with local measurements.

Figure 1: Old galaxies data at intermediate redshifts as a cosmic probe. a) Age-redshift plane and the total sample of galaxies. Black points for and correspond, respectively, to the Ferreras et al. (2009) and Longhetti et al. (2007) samples. Black, red and blue bars represent the statistical, statistical+incubation and statistical+incubation+systematic errors, respectively. Data points correspond to 11 galaxies of our selected subsample (see the main text). b) Effect of . Age of the Universe for some selected values of the density parameter. For smaller values of the ages of the Universe for a given redshift increase, thereby accommodating the oldest selected objects. c) The effect on the Age-redshift relation. Dotted curves are the predictions of the cosmic concordance model () and different values of . For values smaller than or bigger than the curves predicted by the models move away from the data.

2 Basic Equations, Probes and Samples

In the CDM cosmology, the angular diameter distance (AD), ), can be written as (Lima & Alcaniz 2002; Lima et al. 2003; Holanda et al. 2010)


where km s Mpc (subscript denotes present day quantities), and the dimensionless function is given by .

Bonamente and collaborators (2006) determined the ADD distance to 38 galaxy clusters in the redshift range using X-ray data from Chandra and Sunyaev-Zeldovich effect data from the Owens Valley Radio Observatory and the Berkeley-Illinois-Maryland Association interferometric arrays. Assuming spherical symmetry, the cluster plasma and dark matter distributions were analyzed by using a hydrostatic equilibrium model accounting for radial variations in density, temperature and abundances. The common statistical contributions for such galaxy clusters sample are: % from galactic (and extragalactic) X-ray background, galactic N , CMB anisotropy , % associated to Sunyaev-Zeldovich effect from point sources, while statistics due to cluster asphericity amounts to %. The estimates of systematic effects include: X-ray temperature calibration %, radio halos %, while calibrations of SZE and X-ray background flux contribute, respectively, with % and %. Different authors (Mason et al. 2001; Reese et al. 2002, 2004) also believe that typical statistical errors amounts for nearly % with + 12.4% and - 12% for systematics (see table 3 in Bonamente et al. 2006).

On the other hand, the age-redshift relation, , for a flat (CDM) model has also only two free parameters ()


Note that for the above expression reduces to the well known result for Einstein-de Sitter model (CDM, ) for which . The above expression means that by fixing from observations one may derive limits on the cosmological parameters and (or equivalently ). It is also worth noticing that the quantities appearing in the product defining the age parameter, , are estimated based on different (independent) observations (Lima & Alcaniz 2000; Friaça et al. 2005).

In this context, Ferreras et al. (2009) catalogued 228 red galaxies on the interval using HST/ACS slitless grism spectra from the PEARS program thereby studying the stellar populations of morphologically selected early-type galaxies from GOODS North and South fields. The first subsample adopted here consists of six passively evolving old red galaxies selected from the original Ferreras et al. sample. The second data set of old objects is also a subsample of the Longhetti et al. (2007) sample. Only nine field galaxies spectroscopically classified as early-types at were selected from a complete sample of the Munich Near-IR Cluster Survey (MUNICS) with known optical and near-IR photometry (the age of each galaxy estimated from its stellar population). It should be stressed that for both samples, the selected data set (eleven galaxies) provide the most accurate ages and the most restrictive galaxy ages.

In Figure 1a, we display the sample constituted by the eleven data points chosen from two distinct subsamples of old objects. As discussed by Lima et al. (2009), we have added an incubation time with a conservative error bar for all galaxies. It is defined by the amount of time interval from the beginning of structure formation process in the Universe until the formation time () of the object itself. It will be assumed here that varies slowly with the galaxy and redshift in our sample thereby associating a reasonable uncertainty, , in order to account for the present ignorance on this kind of “nuisance” parameter (Fowler 1987; Sandage 1993). Here we consider Gyr, and following Lima et al. (2009) we also combine statistical and systematic errors.

In Figures 1b and 1c, we compare the age of these old objects at intermediate redshifts with the predictions of the CDM models for different values of the free parameters ( and ). It is also worth noticing that the present status of systematic uncertainties in this context is still under debate. Jimenez et al. (2004) studied sources of systematic errors in deriving the age of a single stellar population and concluded that they are not larger than per cent. Others authors consider systematic errors around % (Percival & Salaris 2009). In the present study we have adopted for all data.

The third observational probe comes from data, obtained from differential ages of galaxies and radial BAO (Simon et al. 2005; Gaztañaga et al. 2009). Some years ago, Jimenez and collaborators suggested an independent estimator for the Hubble parameter (differential ages of galaxies) and used it to constrain the equation of state of dark energy (Jimenez et al. 2003). Later on, the differential ages of passively-evolving galaxies were used to obtain in the range of (Simon et al. 2005), and this sample was further enlarged by Stern et al. (2010). In addition, Gaztañaga et al. (2009) took the BAO scale as a standard ruler in the radial direction (”Peak Method”) thereby obtaining three more additional data: , , and , which are model and scale independent. Now, by comparing the theoretical expression, , with the observational data the corresponding bounds can be readily derived. We remark that the relative age difference (the key to the method) is only of the order of (Stern et al. 2010). However, to be more conservative we are assuming here systematic errors of for the relative age difference and radial BAO data.

A joint analysis based on the above 3 different probes already provides tight constraints on the value of . However, as shown by Cunha et al. (2007), the analysis of data (from SZE/X-ray technique) leads to more stringent constraints on the space parameter () when combined with the BAO signature (Eisenstein et al. 2005; Percival et al. 2010), and the same happens when the ages of high redshift objects are considered (Lima et al. 2009). Therefore, instead to fix a definite flat CDM cosmology, it is natural to leave free which will be more accurately fixed by adding the BAO signature as a fourth probe to the complete joint analysis performed here. The remnant BAO peak can be interpreted as a consequence of the baryon acoustic oscillations in the primordial baryon-photon plasma prior to recombination. It was detected from a large sample of luminous red galaxies and can be characterized by a dimensionless parameter:


where is the redshift at which the acoustic scale has been measured, and is the dimensionless comoving distance to . The above quantity is independent of the Hubble constant, and, as such, this BAO signature alone constrains only the parameter.

Figure 2: determination at intermediate redshifts. a) from three different probes. The () plane provided by the SZE/X-ray + galaxy age + data including statistical plus systematic errors. The best-fitting values are and . b) The BAO razor. Contours in the () plane from a joint analysis involving SZE/X-ray + galaxy age + + BAO data. The best-fitting model converges to and . c) Final values of . Likelihood functions for the parameter in a flat CDM Universe. The solid black curve corresponds to a joint analyses involving SZE/X-ray + galaxy age + Hubble parameter whereas the red dashed curve includes the BAO signature. The horizontal lines are cuts in the regions of and probability. The joint analysis performed with the four different probes (including statistical plus systematic errors) provides (1). This constraint on is in excellent agreement with the latest determinations based on the extragalactic distance ladder (Table I).

3 Complementarity for

Let us now consider a joint analysis based only on the combination of the first three probes, namely: (i) SZE/X-ray distances, (ii) the age of the oldest intermediate redshift objects, and (iii) the measurements of the Hubble Parameter . Further, a complete joint analysis including the BAO signature from the SDSS catalog it will be performed. For both analysis, we stress that a specific flat CDM cosmology has not a priori been fixed.

All the observational expression adopted here have only two free parameters (). In this way, we perform the statistics over the plane. Combining the three tests discussed above the value reads:


where the quantities with subindex “” are the observational quantities, is the uncertainty in the individual distance, is the contribution of the statistical errors, is the incubation time error, are the contribution of the systematic errors for each sample added in quadrature and the complete set of parameters is given by .

In Figure 2a, we show dimensionless parameter versus the matter density parameter (). Our analysis combining statistical and systematic errors from galaxy ages are in black lines with , and confidence levels (c.l.). In the same way, the green lines represent the confidence levels for the SZE/X-ray sample whereas the corresponding constraints from Hubble parameter sample are show in orange lines. As should be expected, the constraints of each sample on plane are too weak when individually considered. However, due to the complementarity among them, our joint analysis for these three samples predicts km s Mpc and ( c.l.) for two free parameters with . The reduced values are (including systematics) and (no systematics).

In Figure 2b, we display the contours on the space parameter obtained through a joint analysis involving the combination of all four probes. The dashed lines are cuts on plane from BAO signature. The red, green and blue contours are constraints with , and c.l., respectively. This complete analysis provides km.s.Mpc ( uncertainty) whereas the density parameter is for two free parameters with a ().

Reference Method () Epoch Hinshaw et al. 2012 WMAP-9 Ade et al. 2013 PLANCK Riess et al. 2011 Cepheid+SNe+Maser Freedman et al. 2012 HST Key Project This letter SZE/X-ray+Age+H(z) This letter SZE/X-ray+Age+H(z)+BAO

Table 1: Constraints on for different methods and epochs.

In Figure 2c, we show the likelihood function for the parameter in a flat CDM universe. Both curves were obtained by marginalizing on the matter density parameter. The shadow lines are cuts in the regions of and probability. For the solid black line the BAO signature was not considered. The constraints for the black line are () with (), respectively. For the red line the BAO signature has been included. The constraints are (corresponding to error in ) and () with and , respectively. For all these analyses, the statistical and systematic errors were added.

In Table 1, we compare the results derived here with the latest determinations of . Note that competitive constraints were obtained using only probes at intermediate redshifts, but the present results are nicely consistent with the latest determinations based on the HST Key Project.

4 Conclusions

Several ongoing and future experiments (HST Key Project, SES, PLANCK, GAIA, Spitzer/CHP, JWST, and others) are dedicated to more accurately measuring the value of , partially, because an accuracy better than will provide critical information on several cosmological parameters.

We have demonstrated here that a joint analysis involving four independent cosmological tests at intermediate redshifts () provides an independent determination of (statistical and systematic errors combined). In the framework of a flat CDM model we have obtained (including statistical plus systematics), as shown in table I, this value is not only consistent, but has the same precision of a recent determination using nearby Cepheids and Supernovae (Riess et al. 2011; Freedman et al. 2012). It is also suggested here that the present determination of the Hubble constant based only on probes at intermediate redshifts provides a competitive cross-check for any determination of .

The main advantage of the present treatment is that it does not rely on extragalactic distance ladder being fully independent of local calibrators. Naturally, its basic disadvantage rests on the large and different systematic uncertainties appearing in each probe when separately applied. However, the remarkable complementarity among the four tests works in concert thereby reducing greatly the possible degeneracy and uncertainties appearing in the present determination of the Hubble constant.

Finally, concerning the intriguing tension between the determinations from nearby objects and current CMB data, our analysis at intermediate is clearly favoring the local methods.

The authors acknowledge the referee for his/her suggestions that improved the manuscript. JASL and JVC are partially supported by CNPq (Brazilian Research Agency).


  1. affiliation: limajas@astro.iag.usp.br
  2. affiliationmark:
  3. affiliation: jvcunha@ufpa.br
  4. affiliationmark:


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