In this work we describe a model-independent method of developing a plot of scale factor a(t) versus lookback time tL from the usual Hubble diagram of modulus data against redshift. This is the first plot of this type. We follow the model-independent methodology of Daly & Djorgovski used for their radio-galaxy data. Once the a(t)data plot is completed, any model can be applied and will display as described in the standard literature. We then compile an extensive data set to z = 1.8 by combining Type Ia supernovae (SNe Ia) data from SNLS3 of Conley et?al., high-z SNe data of Riess et?al., and radio-galaxy data of Daly & Djorgovski to validate the new plot. We first display these data on a standard Hubble diagram to confirm the best fit for ΛCDM cosmology, and thus validate the joined data set. The scale factor plot is then developed from the data and the ΛCDM model is again displayed from a least-squares fit. The fit parameters are in agreement with the Hubble diagram fit confirming the validity of the new plot. Of special interest is the transition time of the universe, which in the scale factor plot will appear as an inflection point in the data set. Noise is more visible in this presentation, which is particularly sensitive to inflection points of any model displayed in the plot, unlike on a modulus-z diagram, where there are no inflection points and the transition-z is not at all obvious by inspection. We obtain a lower limit of z ≥ 0.6. It is evident from this presentation that there is a dearth of SNe data in the range z = 1-2, exactly the range necessary to confirm a ΛCDM transition-z around z = 0.76. We then compare a "toy model" wherein dark matter is represented as a perfect fluid with an equation of state p = –(1/3) ρ to demonstrate the plot sensitivity to model choice. Its density varies as 1/t 2 and it enters the Friedmann equations as Ωdark/t 2, replacing only the Ωdark/a 3 term. The toy model is a close match to ΛCDM, but separates from it on the scale factor plot for similar ΛCDM density parameters. It is described in the Appendix. A more complete transition time analysis will be presented in a future paper.
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机译：在这项工作中，我们描述了一种与模型无关的方法，可从模数数据相对于红移的通常哈勃图绘制比例因子a（t）与回溯时间tL的关系图。这是该类型的第一幅情节。我们遵循Daly和Djorgovski用于其射电星系数据的与模型无关的方法。数据图绘制完成后，可以应用任何模型，并将按照标准文献中的说明进行显示。然后，我们将来自Conley等人的SNLS3的Ia类超新星（SNe Ia）数据，Riess等人的高z SNe数据以及Daly和Ile的射电星系数据组合在一起，将z = 1.8的大量数据汇总Djorgovski验证了新的情节。我们首先将这些数据显示在标准的哈勃图上，以确认最适合ΛCDM宇宙学，从而验证合并的数据集。然后根据数据绘制比例因子图，并再次从最小二乘拟合中显示ΛCDM模型。拟合参数与哈勃图拟合一致，确认了新图的有效性。特别令人感兴趣的是宇宙的跃迁时间，它在比例因子图中将作为数据集中的拐点出现。在此演示文稿中，噪声更明显，这对图中显示的任何模型的拐点都特别敏感，这与模量-z图不同，后者没有拐点，并且通过检查根本看不到过渡-z。我们获得z≥0.6的下限。从该演示中可以明显看出，在z = 1-2的范围内缺少SNe数据，恰好是确认ΛCDM过渡z大约在z = 0.76所需的范围。然后，我们将“玩具模型”与状态方程p = –（1/3）ρ的暗物质表示为理想流体进行比较，以证明情节对模型选择的敏感性。它的密度变化为1 / t 2，并以Ωdark/ t 2的形式进入Friedmann方程，仅替换Ωdark/ a 3项。玩具模型与ΛCDM紧密匹配，但在比例因子图上针对类似的ΛCDM密度参数与之分离。在附录中有描述。将来的论文中将提供更完整的过渡时间分析。
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