Model-Independent Reconstruction of the Cosmological Scale Factor as a Function of Lookback Time

The cosmological scale factor $a(t)$ is one of the most fundamental quantities that describe the smooth background universe, although it is not observable. A whole relation between scale factor $a$ and cosmic time $t$ almost contains all information about cosmological kinematics, such as the expansion history, the Hubble parameter (expansion rate) $H=\dot{a}/a$ ($\dot{a}\equiv da/dt$) and the deceleration parameter $q=-a\ddot{a}/\dot{a}^{2}$. Ringermacher & Mead (2014) introduced a model-independent method of developing a scale factor - lookback time plot from Type Ia supernovae (SNe Ia) and radio-galaxy data, i.e., the Hubble diagram of modulus data against redshift. They used the scale factor plot as a better way to find the transition redshift of the universe at which the universe transitions from decelerating to accelerating.

The assumption that cosmic curvature equals zero was made in the reconstruction process from Type Ia supernovae to scale factor - lookback time data (Daly & Djorgovski, 2004). We reconstruct the scale factor against cosmic time from the Observational Hubble parameter Data (OHD). The expression of the lookback time contains an integral of Hubble parameter, so using OHD to reconstruct cosmic time is model-independent and does not require any assumptions. We use the trapezoidal rule for approximating integrals of Hubble parameters. The reconstruction data errors come from the trapezoidal rule and OHD’s errors.

The scale factor $a$ - cosmic time $t$ has a more basic status than Hubble parameter $H$ - redshift $z$, since the former directly appears in the Friedmann-Robertson-Walker metric. Although the error propagation from OHD to reconstructed data would increase errors, we expect that the $a\text{-}t$ data have a higher sensitivity to model discrimination. We calculate the $H\text{-}z$ and its variance as well as $a\text{-}t$ for several models such as Phantom, $\Lambda$CDM, Chevallier-Polarski-Linder (CPL) parametrization. The result shows $a\text{-}t$ plot has better model discrimination than $H\text{-}z$ plot by comparing the coefficient of variation, which although can not be seen directly from the figure.

The primary purpose of this paper is to present a model-independent approach to reconstruct the scale factor against cosmic lookback time data from OHD. The paper is organized as follows. We introduce the reconstruction method in section 2. The error propagation is also calculated there. The reconstructed $a\text{-}t$ data are shown in section 3. Section 4 discusses the model discrimination ability of original $H\text{-}z$ and reconstructed $a\text{-}t$ data. In the section 5, we use simulated $H(z)$ data to forecast the improvement effects of future $H(z)$ observation on this reconstruction.

The 43 OHD are shown in Figure 2 and Table LABEL:tab:OHDdata, which contains 31 Hubble measurements from the method of cosmic chronometers and 12 from the radial BAO method. The reconstructed $a\text{-}d\tau/da$ and $a\text{-}\tau$ data are shown in Figure 3 and Table 2. The error in the early universe (low $\tau$) is enormous. According to equation (22) and (25), the error is accumulated in calculating high-$z$ or low-$\tau$ data. The main sources of error are the accumulation effect of trapezoidal rule and errors of Hubble measurements. In order to reduce the error from trapezoidal rule, more future OHD data is needed to increase the number of approximate trapezoids to have a more accurate result.

To test the validation of reconstruction data, we use Markov Chain Monte Carlo(MCMC) method to fit the matter density parameter $\Omega_{m}$ in $\Lambda$CDM model:

where we ignore radiation ($\Omega_{R}$) and consider a flat model ($\Omega_{k}=0$).

According to equation (4) and $a=1/(1+z)$, the relation between $\tau$ and $a$ in $\Lambda$CDM is:

where the only free parameter is $\Omega_{m}$.

The result of direct fitting from OHD data is

which is close to the result from Planck Collaboration et al. (2020) $\Omega_{m}=0.315\pm 0.007$ with similar uncertainty, but they are not consistent.

The fitting result from $a\text{-}\tau$ data is:

This result implies an universe with $\Lambda$-dominated accelerating expansion, which is consistent with SNe Ia and BAO results. Compared with OHD fit of $\Lambda$CDM result $\Omega_{m}=0.290\pm 0.008$, this exercise shows the reconstructed $a\text{-}\tau$’s parameter is consistent with the Planck result, which verifies the feasibility of the reconstruction method. However, it also shows its limitation that the trapezoidal rule introduces a much larger uncertainty.

To test the $H\text{-}z$ and $a\text{-}\tau$ plot’s ability of model discrimination, we find the models which are in good agreement with the results obtained from the observations, such as the Planck result (Planck Collaboration et al., 2020), Supernovae Ia (SNe Ia), Baryon Acoustic Oscillations (BAO) and Cosmic Microwave Background (CMB) (Wang et al., 2016). A non-parametric smoothing method — the Gaussian process is also used to get an independent result. The models we choose to distinguish are Phantom, $\Lambda$CDM, Chevallier-Polarski-Linder (CPL) parametrization.

The observational constraints on the parameters are $\theta_{i}\pm\Delta\theta_{i},\ i=1,\dots,n$, the estimated error in a model quantity $f(\theta_{1},\dots,\theta_{n})$ is (approximately):

In this section, we simulate future OHD to study its influence on $a\textrm{-}\tau$ method. We use the simulation method from Ma & Zhang (2011):

$\Delta H$ is the deviation between simulation value and fiducial value.

Data simulation has to choose a model first. We choose the $\Lambda$CDM model as our fiducial model:

The parameters come from Planck Collaboration et al. (2020), which are

From the spatially flat $\Lambda$CDM model, $\Omega_{\Lambda}$ can be calculated by $\Omega_{m}+\Omega_{\Lambda}=1$. Then, we can obtain a set of fiducial values from equation (43).

Next, we need to estimate the uncertainties of future OHD. 43 Hubble measurements’ uncertainties are shown in Figure 6. After removing one value with an obvious deviation, we draw two outlines to represent the general trend of the uncertainties. Two bounded lines are expressed as one upper line $\sigma_{+}=8.19z+34.31$ and one lower line $\sigma_{-}=2.16z+2.25$. This simulation method believes the future measurements will also conform to this trend, and their value will fall within this error strip between $\sigma_{-}$ and $\sigma_{+}$. In other words, an random uncertainty $\sigma(z)$ of OHD can be estimated by a Gaussian distribution $N(\sigma_{0}(z),(\sigma_{+}-\sigma_{-})/4)$, where $\sigma_{0}(z)$ is the average of $\sigma_{+}$ and $\sigma_{-}$ and the $(\sigma_{+}-\sigma_{-})$ represent $4\sigma$ trip so that the $\sigma(z)$ can fall within the strip. This random uncertainty $\sigma(z)$ can be used to determine the deviation $\Delta H$ by a Gaussian distribution $N(0,\sigma(z))$. 50 Hubble parameters are simulated through the above way, and the result is shown in Figure 7.

Through the previous reconstructed method in section 2 to constrain the matter density parameter $\Omega_{m}$, the corresponding $a\text{-}\tau$ plot is shown in Figure 8. When we only apply the original 43 observed data, the fitting curve describes red points well with the result of $\Omega_{m}=0.372\pm 0.087$, but it can be seen that there is still a certain gap between the fitting curve and the $\Lambda$CDM model curve. When we apply total data, including 43 observed data and 50 simulated data, the error of data points becomes larger due to the integral effect, while the result’s trend is closer to the $\Lambda$CDM model with a result of $\Omega_{m}=0.303\pm 0.047$. These are consistent with our expectations. The $a\text{-}\tau$ plot result should be more biased towards $\Lambda$CDM model after adding 50 data simulated by $\Lambda$CDM model to the sample. In addition, the error of $\Omega_{m}$ decreased from 0.087 to 0.047 after adding more OHD. More data points are conducive to the accuracy of the study. In another word, as more Hubble measurements are added in the future, the $a\text{-}\tau$ diagram can help to constrain cosmological parameters better and help people to study which model can describe our universe more accurately.

We describe a model-independent method to reconstruct the scale factor $a$ against the lookback time $\tau$ from OHD. This is a new way to handle observational Hubble parameter data. 43 $H(z)$ data points are collected for reconstruction, and the redshift range is $(0.0,2.4)$. The original OHD and reconstructed $a\text{-}\tau$ data are both plotted. The $\Lambda$CDM model presents a classic fit through the $a\text{-}\tau$ data, thus validating the reconstructed results. However, one limitation is the large error caused by the nature of the integral.

For the more fundamental position of scale factor than Hubble parameter, their abilities of model discrimination are different. By comparing the coefficient of variations of various cosmology models, we find $a\text{-}\tau$ plot can magnify the differences between models based on the integral effect so that the $a\text{-}\tau$ plot has a better model discriminate ability than the $H\text{-}z$ plot. Note that the differences are not significant enough to be observed directly from the graph. We use the coefficient of variations to compare numerically.

We simulate fifty $H(z)$ data with a fiducial $\Lambda$CDM model based on Ma & Zhang (2011) and reconstruct total $a\text{-}\tau$ data to forecast the improvement effects of future H(z) observation. On the one hand, due to the integration characteristics, errors are also accumulated. Therefore, when the sample takes all data, the error bar increases. On the other hand, $a\text{-}\tau$ plot shows a more accurate result of constraining cosmological parameters. If there are more Hubble measurements in the future, the $a\text{-}\tau$ method can better present the model to help people find a better model to describe our present universe.

We sincerely appreciate the referee’s suggestions, which help us greatly improve our manuscript. This work was supported by the National Science Foundation of China (NSFC) Programs Grants No. 11929301 and National Key R&D Program of China (2017YFA0402600).