Median statistics cosmological parameter values

We present median statistics central values and ranges for 12 cosmological parameters, using 582 measurements (published during 1990-2010) collected by Croft&Dailey (2011). On comparing to the recent Planck collaboration Ade et al. 2013 estimates of 11 of these parameters, we find good consistency in nine cases.


Introduction
Recent cosmic microwave background anisotropy (see, e.g., Hinshaw et al. 2013;Ade et al. 2013), baryon acoustic oscillation peak length scale (see, e.g., Busca et al. 2013;, supernova Type Ia apparent magnitude versus redshift (see, e.g., Campbell et al. 2013;Liao et al. 2013), and Hubble parameter as a function of redshift (see, e.g., Moresco et al. 2012;) measurements have small enough statistical error bars to encourage the belief that we will soon be in an era of precision cosmology. Of course, there have also been many earlier measurements, most having larger error bars, that have helped the field develop to the current position. In this paper we use statistical techniques to combine the results of the many earlier measurements, and so derive summary estimates of the corresponding cosmological parameters with much tighter error bars than any individual earlier measurement. We then compare these summary results to more precise recent measurements, largely those from the recent analysis of early P lanck space mission cosmic microwave background (CMB) anisotropy data (Ade et al. 2013). Using large-angle CMB anisotropy data to measure cosmological parameters is appealing because, once initial conditions and ionization history are established, it is possible to accurately compute cosmological model CMB anisotropy predictions as a function of cosmological parameter values.
Previous CMB anisotropy experiments, such as W MAP and ground-based ones, along with data from other techniques discussed above, have focussed attention on a "standard" cosmological model (for detailed discussions see Hinshaw et al. 2013;Ade et al. 2013). This model, called the ΛCDM model (Peebles 1984), is a spatially-flat cosmological model with a current energy budget dominated by a time-independent dark energy density in the form of Einstein's cosmological constant Λ that contributes 68.3% of the current energy budget, non-relativistic cold dark matter (CDM) is the next largest contributor at 26.7%, followed by non-relativistic baryonic matter at 4.9% (Ade et al. 2013). For recent reviews see Wang (2012), Tsujikawa (2013), andSolà (2013).
A main goal of the P lanck mission is to measure cosmological parameters accurately enough to check consistency with the ΛCDM model, as well as to possibly detect deviations. However, it is also of interest to find out if previous estimates of cosmological parameters are consistent with the P lanck results. Ade et al. (2013), and references therein, have compared the P lanck results to individual earlier measurements, most notably to the results from the W MAP experiment, from which they find small differences. However, it is also of interest to attempt to derive summary estimates for cosmological parameters from the many earlier measurements that are available, and to compare these summary estimates to the P lanck results. This is what we do in this paper.
To derive our summary estimates of cosmological parameter values we use the very impressive compilation of data of Croft & Dailey (2011). We use 582 (of the 637) measurements for the dozen cosmological parameters collected by Croft & Dailey (2011). These values were published during 1990-2010, and, as estimated by Croft & Dailey (2011), are approximately 60% of the measurements of the 12 cosmological parameters published during these two decades. The main focus of the Croft & Dailey (2011) paper was to compare earlier and more recent measurements and analyze how measuring techniques and results evolve over time. In our paper we use two statistical techniques, namely weighted mean and median statistics, to find the best-fit summary measured value of each of the 12 cosmological parameters. We then compare our summary values to those found from the P lanck data.
In the next section we briefly review the Croft & Dailey (2011) data compilation. Sections 3 and 4 are brief summaries of the weighted mean and median statistics techniques we use to analyze the Croft & Dailey (2011) data. Our analyses and results are described and discussed in Sec. 5, and we conclude in Sec. 6.

Data Compilation
The data we use in our analyses here were compiled by Croft & Dailey (2011). These data were collected from the abstracts of papers listed on the NASA Astrophysics Data System (ADS) 1 . They estimate that by searching abstracts only, about 40% of available measurements were missed. Nevertheless, a great deal of data were collected. Croft & Dailey (2011) searched papers published in a 20 year period  and tabulated 637 measurements. Of the 637 measurements, 582 were listed with a central value and 1σ error bars (these are the data we use in this paper 2 ) while 55 were upper or lower limits with no central value.
1 adsabs.harvard.edu 2 Most of these measurements were listed with two significant figures, so results of our analyses are tabulated to two significant figures (except for ω 0 , which consisted mostly of three significant figure measurements and were so tabulated here). The error bar we use in our analyses is the average of the 1σ upper and lower error bars of Croft & Dailey (2011). 12. ω 0 , the dark energy equation of state parameter in a simplified, incomplete, XCDM-like parameterization.

Weighted Mean Statistics
In analyzing data with known errors it is conventional to first consider a weighted mean statistic. This method yields a goodness of fit criterion that can be a valuable diagnostic tool.
The standard formula (see, e.g., Podariu et al. 2001) for the weighted mean of cosmological parameter q is where q i ± σ i are the central values and one standard deviation errors of the i = 1, 2, ..., N measurements. The weighted mean standard deviation of cosmological parameter q is One can also compute the goodness of fit χ 2 , Since this method assumes Gaussian errors, χ has expected value unity and error 1/ 2(N − 1). Hence, the number of standard deviations that χ deviates from unity is a measure of good-fit and is given as A large value of N σ could be an indication of unaccounted-for systematic error, the presence of correlations between the measurements, or the invalidity of the Gaussian assumption.

Median Statistics
The second statistical method we use is median statistics. This method makes fewer assumptions than the weighted mean method, and so can be used in cases when the weighted mean technique cannot. For a detailed description of the median statistics technique see Gott et al. (2001) 3 . In summary, if we assume that the given measurements are: 1) statistically independent; and, 2) have no systematic error for the data set as a whole (as we also assume for weighted mean statistics), then as the number of measurements, N, increases to infinity, the median will reveal itself as a true value. This median is independent of measurement error (Gott et al. 2001), which is an advantage if the errors are suspect. This is also a disadvantage that results in a larger uncertainty for the median than for the weighted mean, because the information in the error bar is not used.
If 1) is true then any value in the data set has a 50% chance of being above or below the true median value. As described in Gott et al. (2001), if N independent measurements M i , where i = 1, ..., N, are taken then the probability of exactly n measurements being higher (or lower) than the true median is It is interesting to note that for large N the expectation value of the distribution width, x, of the true median is x = 0.5, with a standard deviation x 2 − x 2 1/2 = 1/(4N) 1/2 (Gott et al. 2001). Of course, as N increases to infinity, a Gaussian distribution is reached and median statistics recovers the usual standard deviation proportionality to 1/N 1/2 .

Analysis
Since both weighted mean and median statistics techniques have individual benefits, we analyze the compilation of data for 12 parameters from Croft & Dailey (2011) using both methods. Our results are shown in Table 1. Among other things, the table lists our computed weighted mean and corresponding standard deviation σ wm value for the cosmological parameters, as well as the computed median value and the 1σ and 2σ intervals around the median.
Column 5 of Table 1 lists N σ , the number of standard deviations the weighted mean goodness-of-fit parameter χ deviates from unity, see Eq. (4). In all cases N σ is much greater than unity, indicating that the weighted mean results cannot be trusted. In the case of the Hubble constant this is likely due to the fact that the observed error distribution is non-Gaussian, see . 4 Perhaps a similar effect explains the large N σ values for some of the other parameters here. In any case, for our purpose here, the important point is that the weighted mean technique cannot be used to derive a summary estimate by combining together the different measurements tabulated by Croft & Dailey (2011) for each cosmological parameter.
In a situation like this the median statistic technique can be used to combine together the measurements to derive an effective summary value of the cosmological quantity of interest (e.g., Podariu et al. 2001;. Column 6 of Table 1 lists the computed medians of the 12 cosmological parameters; the corresponding 1σ and 2σ ranges of these parameters are listed in columns 7 & 8. The median statistics estimate for the Hubble parameter here, h = 0.68 +0.08 −0.14 , is consistent with that estimated earlier by Chen & Ratra (2011) from 553 measurements of h tabulated by Huchra, h = 0.68 ± 0.028 (with understandably much tighter error bars as a consequence of the many more measurements than the 124 we have used here). 5 Interestingly, from many fewer Ω m measurements than considered here,  determine consistent, but somewhat tighter median statistics constraints on Ω m by discarding the most discrepant, ∼ 5%, of the measurements (those which contribute the most to χ 2 ).
Also of interest, the median statistics estimates in Table 1 of Ω m = 0.29 and σ 8 = 0.84 result in Ω 0.6 m σ 8 = 0.40, which is significantly smaller than the median statistics estimate Ω 0.6 m σ 8 = 0.52 listed in Table 1 that was determined directly from the 11 measurements of Croft & Dailey (2011). 6 On the other hand, Γ = Ω m h computed using the median statistics estimates of Ω m = 0.29 and h = 0.68 is Γ = 0.20, and is in very good agreement with the Table 1 median statistics value of Γ = 0.19 from the 17 measurements of Croft & Dailey (2011).
In most cases the median statistics results of Table 1 provide reasonable (2010) summary estimates for the cosmological parameters. The one exception, perhaps, is that for h, which is estimated to be h = 0.68 ± 0.028 by Chen & Ratra (2011) from very many more measurements than the 124 used to derive the h value in Table 1. Perhaps the best current estimate of cosmological parameter values are those determined from the initial cosmic microwave background anisotropy measurements made by the P lanck satellite (Ade et al. 2013). The last two columns of Table 1 lists the P lanck estimates for most of these parameters. Here, the estimated cosmological constrained value and 1σ standard deviation range (with the exception of Ω k and ω 0 that have 2σ ranges, and m ν that has a 2σ upper limit) are listed. 7 Comparing our computed median results to the recent P lanck values, one finds that 5 For earlier, very consistent, estimates of h using median statistics see Gott et al. (2001) and . 6 It is likely that the larger Ω 0.6 m σ 8 = 0.52 found here is mostly a consequence of the higher Ω 0.6 m σ 8 values of a number of earlier analyses based on large-scale peculiar velocity measurements. While there are not enough measurements tabulated for us to more carefully examine this, it might be relevant that Croft & Dailey (2011) in the fifth paragraph of their Sec. 3.4, when discussing their Fig. 13, note that peculiar velocity measurements have not had a great track record when used to measure cosmological parameters. 7 The variance for parameters Γ and Ω 0.6 m σ 8 were not given in Ade et al. (2013), but were calculated by adding their component's errors in quadrature (see the last footnote in Table 1). All parameter estimates use both P lanck temperature power spectrum data as well as W M AP polarization measurements at low multipoles. Ade et al. (2013) do not provide a P lanck estimate for β. almost all of the P lanck central value results fall within the 1σ range of our median results. One exception is Ω 0.6 m σ 8 , possibly because of reasons discussed above; our estimates of Ω m = 0.29 and σ 8 = 0.84 results in a Ω 0.6 m σ 8 value which is very consistent with the P lanck estimate of Ω 0.6 m σ 8 = 0.415. The other exception is ω 0 which P lanck estimates to be -1.49. Our median statistics 2σ range is −1.25 ≤ ω 0 ≤ −0.808 computed from the 36 measurements of Croft & Dailey (2011). Croft & Dailey (2011) note that the number of measurements for ω 0 are still increasing with time 8 , unlike the case for the other parameters. As such, the estimation of ω 0 is an area still under development and so we should not give much weight to the difference in our estimate from that of P lanck.
More provocatively, it is instructive to compare our median statistics central estimates to the 1σ (or 2σ) P lanck ranges. As expected, we see that our estimate of Ω m (Ω Λ ) lies somewhat above (below) the corresponding P lanck 1σ range. Our estimates of Ω b and Γ are below the corresponding P lanck 1σ ranges. Our estimate of n is well above the P lanck 1σ range, being quite consistent with the simplest scale-invariant spectrum (Harrison 1970;Peebles & Yu 1970;Zeldovich 1972) while P lanck data strongly favors a non-scale-invariant spectrum, also readily generated by quantum fluctuations during inflation (see, e.g., Ratra 1992). And as might have been anticipated, our median statistics central Ω 0.6 m σ 8 value is well above the P lanck 1σ range. f Median statistics range. In several cases for the 2σ range there were not enough measurements to determine a 2σ lower limit. In these cases, the lowest data point was used to represent the 2σ lower limit. This is the case for Ω Λ , Ω b , n, β, m ν , Γ, Ω 0.6 m σ 8 , and Ω k . g Estimated Constrained Value using P lanck+WP (W MAP polarization) data. These are from the last column of Table 2 of Ade et al. (2013), except for m ν , Ω k , and ω 0 which are from the third column of Table 10 in Ade et al. (2013). For m ν there was no central value listed and so a 2σ upper limit is given. h Values are taken from Tables listed in the previous footnote. A 1σ range was given for all parameters except for m ν , Ω k , and ω 0 where a 2σ upper limit or range is given. i Here we have added in quadrature the errors on Ω m and h to get the range of Γ. To get the range for Ω 0.6 m σ 8 we have taken the error on Ω 0.6 m which is given as 0.6Ω 0.4 m σ Ωm and added it in quadrature with the error on σ 8 .

Conclusion
From the measurements compiled by Croft & Dailey (2011), the median statistics technique can be used to compute summary estimates of 12 cosmological parameters. On comparing 11 of these values to those recently estimated by the P lanck collaboration, we find good consistency in 9 cases. The two exceptions are the parameters Ω 0.6 m σ 8 and ω 0 . It is likely that the P lanck estimate of Ω 0.6 m σ 8 is more accurate, while ω 0 estimation is still in its infancy and so one should not give much significance to this current discrepancy.
It is very reassuring that summary estimates for a majority of cosmological parameters considered by Croft & Dailey (2011) are very consistent with corresponding values estimated from the almost completely independent P lanck + W MAP polarization data. This provides strong support for the idea that we are now converging on a "standard" cosmological model.
We are grateful to Rupert Croft for giving us the Croft & Dailey (2011) data and for useful advice. We also thank Omer Farooq for helpful discussions and useful advice. This work was supported in part by DOE grant DEFG03-99EP41093 and NSF grant AST-1109275.