Hubble parameter measurement constraints on dark energy
We use 21 Hubble parameter versus redshift data points, from Simon et al. (2005), Gaztañaga et al. (2009), Stern et al. (2010), and Moresco et al. (2012a), to place constraints on model parameters of constant and time-evolving dark energy cosmologies. The inclusion of the 8 new Moresco et al. (2012a) measurements results in constraints more restrictive than those derived by Chen & Ratra (2011b). These constraints are now almost as restrictive as those that follow from current Type Ia supernova (SNIa) apparent magnitude versus redshift data (Suzuki et al., 2012), which now more carefully account for systematic uncertainties. This is a remarkable result. We emphasize however that SNIa data have been studied for a longer time than the data, possibly resulting in a better estimate of potential systematic errors in the SNIa case. A joint analysis of the , baryon acoustic oscillation peak length scale, and SNIa data favors a spatially-flat cosmological model currently dominated by a time-independent cosmological constant but does not exclude slowly-evolving dark energy.
The expansion rate of the Universe changes with time, initially slowing when matter dominated, because of the mutual gravitational attraction of all the matter in it, and more recently accelerating. A number of cosmological observations now strongly support the idea that the Universe is spatially flat (provided the dark energy density is close to or time independent) and is currently undergoing accelerated cosmological expansion. A majority of cosmologists consider dark energy to be the cause of this observed accelerated cosmological expansion.111 Some cosmologists instead view these observations as an indication that general relativity needs to be modified on these large length scales. For recent reviews of modified gravity see Tsujikawa (2010), Bolotin et al. (2011), Capozziello & De Laurentis (2011), Starkman (2011), and references therein. In this paper we assume that general relativity provides an adequate description of gravitation on cosmological scales. This dark energy, most simply thought of as a negative pressure substance, dominates the current cosmological energy budget. For reviews of dark energy see Bass (2011), Jimenez (2011), Li et al. (2011a), Bolotin et al. (2011), and references therein.
Three observational techniques provide the strongest evidence for dark energy: SNIa apparent magnitude measurements as a function of redshift (e.g., Sullivan et al., 2011; Suzuki et al., 2012; Li et al., 2011b; Barreira & Avelino, 2011); cosmic microwave background (CMB) anisotropy data (Podariu et al., 2001b; Komatsu et al., 2011, and references therein) combined with low estimates of the cosmological mass density (e.g., Chen & Ratra, 2003), provided the dark energy density is close to or time independent; and baryon acoustic oscillation (BAO) peak length scale measurements (e.g., Beutler et al., 2011; Blake et al., 2011; Mehta et al., 2012).
The “standard” model of cosmology is the spatially-flat CDM model (Peebles, 1984). In this model about of the current energy budget is dark energy, Einstein’s cosmological constant . Non-relativistic cold dark matter (CDM) is the next largest contributer to the energy budget (around ), followed by non-relativistic baryonic matter (about ). For reviews of the standard model of cosmology see Ratra & Vogeley (2008) and references therein. It has been known for a while that the CDM model is reasonably consistent with most observations (see, e.g., Jassal et al., 2010; Wilson et al., 2006; Davis et al., 2007; Allen et al., 2008, for early indications).222 Note, however, that the “standard” CDM structure formation model, which is assumed in the CDM model, might need modification (see Peebles & Ratra, 2003; Perivolaropoulos, 2010, and references therein). In the CDM model the dark energy density is constant in time and does not vary in space.
Although most predictions of the CDM model are reasonably consistent with the measurements, the CDM model has some curious features. For instance, the measured cosmological constant energy density is 120 orders of magnitude smaller than the energy density naively expected from quantum field theory (this is known as the fine-tuning puzzle). A second curiosity is what is known as the coincidence puzzle: the energy density of a cosmological constant, , is independent of time, but that of matter, , decreases with time during the cosmological expansion, so it is curious why we (observers) happen to live at this (apparently) special time, when the dark energy and the non-relativistic matter energy densities are of comparable magnitude.
These puzzles could be partially resolved if the dark energy density is a slowly decreasing function of time (Peebles & Ratra, 1988; Ratra & Peebles, 1988). In this case the dark energy density will remain comparable to the non-relativistic matter density for a longer time. For recent discussions of time-varying dark energy models, see Bauer et al. (2011), Chimento et al. (2011), Granda et al. (2011), García-Bellido et al. (2011), Basilakos et al. (2012), Sheykhi et al. (2012), Brax & Davis (2012), Hollenstein et al. (2012), Cai et al. (2012), and references therein. In this paper we will consider two dark energy models (with dark energy being either a cosmological constant or a slowly-evolving scalar field ) as well as a dark energy parameterization.
In the CDM model, time-independent dark energy density (the cosmological constant ) is modeled as a spatially homogeneous fluid with equation of state . Here and are the fluid pressure and energy density. In describing slowly-decreasing dark energy density much use has been made of a parameterization known as XCDM. Here dark energy is modeled as a spatially homogeneous -fluid with equation of state . The equation of state parameter is independent of time and and are the pressure and energy density of the -fluid. When the XCDM parameterization reduces to the complete and consistent CDM model. For any other value of the XCDM parameterization is incomplete as it cannot describe spatial inhomogeneities (see, e.g. Ratra, 1991; Podariu & Ratra, 2000). For computational simplicity, in the XCDM case we assume a spatially-flat cosmological model.
The CDM model is the simplest, consistent and complete model of slowly-decreasing dark energy density (Peebles & Ratra, 1988; Ratra & Peebles, 1988). In this model dark energy is modeled as a scalar field, , with a gradually decreasing (in ) potential energy density . Here we assume an inverse power-law potential energy density , where is a nonnegative constant (Peebles & Ratra, 1988). When the CDM model reduces to the corresponding CDM case. For computational simplicity, we again only consider the spatially-flat cosmological case for CDM.
As mentioned above, for some time now, most observational constraints have been reasonably consistent with the predictions of the “standard” spatially-flat CDM model. CMB anisotropy, SNIa, and BAO measurements provide the strongest support for this conclusion. However, the error bars associated with these three types of data are still too large to allow for a significant observational discrimination between the CDM model and the two simple time-varying dark energy models discussed above. This is one motivation for considering additional kinds of data.
If the constraints from the new data differ considerably from the old ones, this could mean that at least one of the data sets had an undetected systematic error, or it could mean that the model being tested is observationally inconsistent. Either of these is an important result. On the other hand, if the constraints from the new and the old data are consistent, then a joint analysis of all the data could result in tighter constraints, and so might result in significantly discriminating between constant and time-varying dark energy models.
Other measurements that have been used to constrain cosmological parameters333 For reviews see Albrecht et al. (2006), Weinberg et al. (2012), and references therein. include galaxy cluster gas mass fraction as a function of redshift (e.g., Allen et al., 2008; Samushia & Ratra, 2008; Ettori et al., 2009; Tong & Noh, 2011; Lu et al., 2011), galaxy cluster and other large-scale structure properties (Campanelli et al., 2012; De Boni et al., 2011; Mortonson et al., 2011; Devi et al., 2011; Wang, 2012, and references therein), gamma-ray burst luminosity distance as a function of redshift (e.g., Samushia & Ratra, 2010; Wang & Dai, 2011; Busti et al., 2012), lookback time as a function of redshift (Samushia et al., 2010; Dantas et al., 2011; Tonoiu et al., 2011, and references therein), HII starburst galaxy apparent magnitude as a function of redshift (e.g., Plionis et al., 2010, 2011; Mania & Ratra, 2012), angular size as a function of redshift (e.g., Guerra et al., 2000; Bonamente et al., 2006; Chen & Ratra, 2012), and strong gravitational lensing (Chae et al., 2004; Lee & Ng, 2007; Biesiada et al., 2010; Zhang & Wu, 2010, and references therein).444 Future space-based SNIa and BAO-like meassurements (e.g., Podariu et al., 2001a; Samushia et al., 2011; Sartoris et al., 2012; Basse et al., 2012; Pavlov et al., 2012), as well as measurements based on new techniques (Jennings et al., 2011; van de Weygaert et al., 2011; Ziaeepour, 2012, and references therein) should soon provide interesting constraints on cosmological parameters. Of particular interest to us here are measurements of the Hubble parameter as a function of redshift (e.g., Jimenez et al., 2003; Samushia & Ratra, 2006; Samushia et al., 2007; Sen & Scherrer, 2008; Pan et al., 2010; Chen & Ratra, 2011b; Kumar, 2012; Wang & Zhang, 2011; Duan et al., 2011; Bilicki et al., 2012; Seikel et al., 2012). While the constraints from these data are typically less restrictive than those derived from the SNIa, CMB anisotropy, and BAO data, both types of measurements result in largely compatible constraints that generally support a currently accelerating cosmological expansion. This provides confidence that the broad outlines of a “standard” cosmological model are now in place.
|(km s Mpc )||(km s Mpc )|
In this paper we use the 21 measurements of Simon et al. (2005), Gaztañaga et al. (2009), Stern et al. (2010), and Moresco et al. (2012a) (listed in Table 1)555 We do not include the 4 recent Zhang et al. (2012) measurements as they have somewhat larger error bars and do not affect our results. to constrain the CDM and CDM models and the XCDM parametrization. The inclusion of the 8 new Moresco et al. (2012a) measurements (with smaller error bars compared to the earlier data) in the analysis results in tighter constraints than those recently derived by Chen & Ratra (2011b) from the previous largest set of measurements considered. The new data constraints derived here are compatible with cosmological parameter constraints determined by other techniques. For the first time, these limits are almost as constraining as those derived from the most recent SNIa data compilation of Suzuki et al. (2012). In addition to the tighter limits resulting from the new data, this is partially also a consequence of the fact that a more careful analysis of the SNIa measurements (Suzuki et al., 2012) has resulted in a larger systematic error estimate and thus weaker SNIa constraints. We emphasize that the study of data is much less mature than that of SNIa apparent magnitude data, so there is the possibility that future error bars might be larger than what we have used in our analysis here. In addition to deriving -data only constraints, we also use these data in combination with recent BAO and SNIa measurements to jointly constrain cosmological parameters in these models.666 See Moresco et al. (2012b) and Wang et al. (2012) for analyses that use most of these data in conjunction with CMB anisotropy and other data to constrain cosmological parameters. Adding the data tightens the constraints, somewhat significantly in some parts of parameter space for some of the models we study.
Our paper is organized as follows. In Sec. 2 we present the basic equations of the three dark energy models we consider. Constraints from the data are derived in Sec. 3. In Sec. 4 we determine constraints from recent SNIa apparent magnitude data. In Sec. 5 we derive constraints from recent BAO data. Joint constraints on cosmological parameters, from a combined analysis of the three data sets, for the three models we consider, are presented in Sec. 6. We conclude in Sec. 7.
2 Dark energy models
In this section we summarize properties of the two models (CDM and CDM) and the one parametrization (XCDM) we use in our analyses of the data.
To determine how the Hubble parameter evolves in these models, we start from the Einstein equation of general relativity
Here is the metric tensor, and are the Ricci tensor and scalar, is the energy-momentum tensor of any matter present, is the cosmological constant, and is the Newtonian gravitational constant.
The energy-momentum tensor for an ideal fluid is , where is the energy density and the pressure. Assuming spatial homogeneity, the Einstein equation reduces to the two independent Friedmann equations
Here is the cosmological scale factor, an overdot denotes a derivative with respect to time, and represents the curvature of the spatial hypersurfaces. These equations, in conjunction with the equation of state,
where is the dimensionless equation-of-state parameter (with corresponding to a cosmological constant and corresponding to the XCDM parametrization), govern the evolution of the scale factor and matter densities.
For a non-relativistic gas (matter) and , and for a cosmological constant and = constant (=0). Solving Eq. (5), the time-dependent energy density is
where and are the current values of the fluid energy density and the scale factor. If there are a number of different species of non-interacting particles, then Eq. (6) holds separately for each of them.
The ratio is called the Hubble parameter. The present value of the Hubble parameter is known as the Hubble constant and is denoted by . Defining the redshift , and the present value of the density parameters,
in the CDM model we can rewrite Eq. (2) as
where we have made use of . This is the Friedmann equation of the CDM model with spatial curvature. In this model the cosmological parameters . Here is the non-relativistic (baryonic and cold dark) matter energy density parameter at the present time. Below we shall have need for the dimensionless Hubble parameter .
It has become popular to parametrize time-varying dark energy as a spatially homogeneous -fluid, with a constant equation of state parameter . With this XCDM parametrization the Friedmann equation takes the form
where for computational simplicity we consider only flat spatial hypersurfaces, and the model parameters . The XCDM parametrization is incomplete, as it cannot describe the evolution of energy density inhomogeneities.
The simplest complete and consistent dynamical dark energy model is CDM. In this model dark energy is a slowly-rolling scalar field with an, e.g., inverse-power-law potential energy density where is the Planck mass and is a non-negative free parameter that determines . The scalar field part of the CDM model action is
with corresponding scalar field equation of motion
In the spatially-flat case the Friedmann equation is
with scalar field energy density given by
Solving the coupled differential Eqs. (11)—(13), with the initial conditions described in Peebles & Ratra (1988), allows for a numerical computation of the Hubble parameter . In this case the model parameter set is .
3 Constraints from the data
We use 21 independent data points (Simon et al., 2005; Gaztañaga et al., 2009; Stern et al., 2010; Moresco et al., 2012a), listed in Table 1, to constrain cosmological model parameters. The observational data consist of measurements of the Hubble parameter at redshifts , with the corresponding one standard deviation uncertainties .
To constrain cosmological parameters p of the models of interest we compute the function
depends on the model parameters p as well as on the nuisance parameter whose value is not known exactly. We assume that the distribution of is a Gaussian with one standard deviation width and mean . We can then build the posterior likelihood function that depends only on the p by integrating the product of exp and the prior likelihood function exp (see, e.g., Ganga et al., 1997),
the integral can be expressed in terms of the error function,777.
We maximize the likelihood , or equivalently minimize , with respect to the parameters p to find the best-fit parameter values . In the models we consider depends on two parameters. We define 1, 2, and 3 confidence intervals as two-dimensional parameter sets bounded by , and , respectively.
Even though the precision of measurements of the Hubble constant have greatly improved over the last decade, the concomitant improvement in the precision of other cosmological measurements means that in some cases the Hubble constant uncertainty still significantly affects cosmological parameter estimation. For a recent example see Calabrese et al. (2012). The values of that we use in this paper are 68 2.8 km s Mpc and 73.8 2.4 km s Mpc. The first is from a median statistics analysis (Gott et al., 2001) of 553 measurements of (Chen & Ratra, 2011a); this estimate has been remarkably stable for over a decade now (Gott et al., 2001; Chen et al., 2003). The second value is the most precise recent one, based on HST measurements (Riess et al., 2011). Other recent measurements are not inconsistent with at least one of the two values we use as a prior (see, e.g., Freedman et al., 2012; Sorce et al., 2012; Tammann & Reindl, 2012).
|Model and prior||B.F.P||B.F.P||B.F.P|
Figures 1—3 show the constraints from the data for the three dark energy models we consider, and for the two different priors. Table 2 lists the best fit parameter values. Comparing these plots with Figs. 1—3 of Chen & Ratra (2011b), whose 1, 2 and 3 constraint contours are reproduced here as dot-dashed lines in the left panels of Figs 1—3, we see that the contours derived from the new data are more constraining, by about a standard deviation, because of the 8 new, more precise, Moresco et al. (2012a) data points used here. On comparing the left and right panels in these three figures, we see that the constraint contours are quite sensitive to the value of used, as well as to the uncertainty associated with the Hubble constant measurement.
4 Constraints from the SNIa data
While the data provide tight constraints on a linear combination of cosmological parameters, the very elongated constraint contours of Figs. 1—3 imply that these data alone cannot significantly discriminate between cosmological models. To tighten the constraints we must add other data to the mix.
The second set of data that we use are the Type Ia supernova data from the Suzuki et al. (2012) Union2.1 compilation of 580 SNIa distance modulus measurements at measured redshifts (covering the redshift range of 0.015 to 1.414) with associated one standard deviation uncertainties . The predicted distance modulus is
where km s Mpc and is the dimensionless coordinate distance,
As the SNIa distance modulus measurements are correlated, is defined as
Here is a vector of differences , and is the inverse of the 580 by 580 Union 2.1 compilation covariance matrix. In index notation,
The covariance matrix is symmetric so this can be written as
The corresponding likelihood function, when considering a flat prior, is
the above integral takes the form
where is a constant that does not depend on the model parameters p,
and so can be ignored. We minimize with respect to the model parameters p to find the best-fit parameter values and constraint contours.
Figure 4 shows constraints from the SNIa data on the three dark energy models we consider here. For the CDM model and the XCDM parametrization the constraints shown in Fig. 4 are in very good agreement with those in Figs. 5 and 6 of Suzuki et al. (2012). The CDM model SNIa data constraints shown in Fig. 4 have not previously been computed. Comparing the SNIa constraints of Fig. 4 to those which follow from the data, Figs. 1—3, it is clear that SNIa data provide tighter constraints on the CDM model. For the XCDM case both SNIa data and data provide approximately similar constraints, while the SNIa constraints are somewhat more restrictive than the ones for the CDM model. However, in general, the SNIa constraints are not very significantly more restrictive than the constraints, which is a remarkable result. It is also reassuring that both data favor approximately similar regions of parameters space, for all three models we consider. However, given that the degeneracy in parameter space is similar for the and SNIa data, a joint analysis of just these two data sets is unlikely to greatly improve the constraints.
5 Constraints from the BAO data
In an attempt to further tighten the cosmological parameter constraints, we now include BAO data in the analysis. To constrain cosmological parameters using BAO data we follow the procedure of Blake et al. (2011). To derive the BAO constraints we make use of the distance parameter , a combination of the angular diameter distance and the Hubble parameter, given by
Here is the angular diameter distance
where is the dimensionless coordinate distance given in Eq. (20).
which is independent and where is defined in Sec. 2.
Using the WiggleZ data from Table 3 of Blake et al. (2011), we compute
Here is a vector consisting of differences and is the inverse of the 3 by 3 covariance matrix given in Table 3 of Blake et al. (2011).
where is the sound horizon at the drag epoch, is given in Eq. (6) of Eisenstein & Hu (1998). The correlation coefficients for this case are also given in Table 3 of Blake et al. (2011). Using the covariance matrix we define
where is a vector consisting of differences and is the the covariance matrix (Blake et al., 2011). We then marginalize over a flat prior for to get
Since and correspond to independent data, the combined BAO data
We can maximize the likelihood by minimizing with respect to the model parameters p to get best-fit parameter values and constraint contours. Figure 5 show the constraints from the BAO data on the three dark energy models we consider here. The XCDM parametrization constraints shown in this figure are in good agreement with those shown in Fig. 13 of Blake et al. (2011). The constraints shown in the other two panels of Fig. 5 have not previously been computed. Comparing to the and SNIa constraint contours of Figs. 1—4, we see that the BAO contours are also very elongated, although largely orthogonal to the and SNIa ones. Consequently, a joint analysis of these data will result in significantly tighter constraints than those derived using any one of these data sets.
6 Joint constraints
To constrain cosmological parameters from a joint analysis of the , SNIa, and BAO data we compute
for each of the three cosmological models considered here. We minimize with respect to model parameters p to get best-fit parameter values and constraint contours.
|Model and prior||B.F.P||B.F.P||B.F.P|
Figures 6—8 show constraints on the cosmological parameters for the CDM and CDM models and the XCDM parametrization, from a joint analysis of the BAO and SNIa data, as well as from a joint analysis of the BAO, SNIa and data. Table 3 lists information about best-fit parameter values. Including the data in the analysis tightens the constraints by more than one standard deviation, in parts of the parameter spaces.
|Model and prior||SNIa+BAO||+BAO||+SNIa+BAO|
|CDM||0.25 0.36||0.25 0.36||0.26 0.36|
|0.53 0.89||0.45 0.85||0.55 0.83|
|0.25 0.36||0.23 0.38||0.25 0.35|
|0.53 0.89||0.60 0.92||0.62 0.88|
|0.30 0.38||0.26 0.37||0.29 0.37|
|0.30 0.38||0.24 0.35||0.27 0.35|
|0.25 0.35||0.25 0.36||0.26 0.35|
|0 0.54||0 1.01||0 0.54|
|0.25 0.35||0.23 0.35||0.25 0.33|
|0 0.54||0 0.57||0 0.35|
Adding the data for the km s Mpc prior case improved the constraints most significantly in the CDM case (by more than 1 on in parts of parameter space), Fig. 6, and least significantly for the CDM model, Fig. 8. For the case of the km s Mpc prior, adding again tightens up the constraints the most for the CDM model (by more than 1 on ), Fig. 6, and least so for the XCDM parametrization, Fig. 7.
Figures 9—11 show the constraints on the cosmological parameters of the three models, from a joint analysis of the BAO and data, as well as from a joint analysis of the three data sets. Table 3 lists the best-fit parameter values. Comparing these figures to Figs. 6—8 allows for a comparison between the discriminating power of the SNIa and data.
Figure 9 shows that adding SNIa data to the and BAO data combination for the km s Mpc prior case tightens up the constraints by more than 1 on from below, while addition of SNIa data for the km s Mpc prior case tightens up the constraints by more than 1 on from above. Addition of SNIa data to the and BAO combination doesn’t much improve the constraints on for either prior.
Figures 9—11 show that adding SNIa data to the and BAO combination results in the most prominent effect for the XCDM case, Fig. 10. Here for the km s Mpc prior it tightens up the constraints by more than 1 on from above and below while for the km s Mpc prior it tightens up the constraints by more than 2 on from below. Addition of SNIa data to the and BAO combination doesn’t much improve the constraints on for either prior in this case.
In the CDM case, Fig. 11, adding SNIa data to and BAO combination affects the constraint on the most for the km s Mpc prior case. The effect on is little stronger than what happens in the CDM and XCDM cases but still less than 1 .
Table 4 lists the two standard deviation bounds on the individual cosmological parameters, determined from their one-dimensional posterior probability distributions functions (which are derived by marginalizing the two-dimensional likelihood over the other cosmological parameter) for different combinations of data set.
The constraints on the cosmological parameters that we derive from only the BAO and SNIa data are restrictive, but less so than those shown in Fig. 4 of Chen & Ratra (2011b). This is because the new Suzuki et al. (2012) SNIa compilation data we use here is based on a more careful accounting of the systematic errors, which have increased. Consequently, including the data, in addition to the BAO and SNIa data, in the analysis, more significantly tightens the constraints: compare Figs. 6—8 here to Figs. 4—6 of Chen & Ratra (2011b). We emphasize, however, that this effect is prominent only in some parts of the parameter spaces.
In summary, the results of a joint analysis of the , BAO, and SNIa data are very consistent with the predictions of a spatially-flat cosmological model with energy budget dominated by a time-independent cosmological constant, the standard CDM model. However, the data are not yet good enough to strongly rule out slowly-evolving dark energy density. More, and better quality, data are needed to discriminate between constant and slowly-evolving dark energy density.
It is probably quite significant that current data constraints are almost as restrictive as those from SNIa data. The acquisition of data has been an interesting backwater of cosmology for the last few years. We hope that our results will help promote more interest in this exciting area. Since the technique has not been as much studied as, say, the SNIa apparent magnitude technique, a little more effort in the area is likely to lead to very useful results.
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