Tilted flat and untilted non-flat XCDM parameterizations

Observational constraints on the tilted flat-XCDM and the untilted non-flat XCDM dynamical dark energy inflation parameterizations

Chan-Gyung Park11affiliation: Division of Science Education and Institute of Fusion Science, Chonbuk National University, Jeonju 54896, South Korea; e-mail: park.chan.gyung@gmail.com 2 2affiliationmark: and Bharat Ratra22affiliation: Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA
July 16, 2019
Abstract

We constrain tilted spatially-flat and untilted non-flat XCDM dynamical dark energy inflation parameterizations with the Planck 2015 cosmic microwave background (CMB) anisotropy data and recent Type Ia supernovae measurements, baryonic acoustic oscillations data, growth rate observations, and Hubble parameter measurements. Inclusion of the four non-CMB data sets leads to a significant strengthening of the evidence for non-flatness in the non-flat XCDM model from 1.1 for the CMB data alone to 3.4 for the full data combination. In this untilted non-flat XCDM case the data favor a spatially-closed model in which spatial curvature contributes a little less than a percent of the current cosmological energy budget; they also mildy favor dynamical dark energy over a cosmological constant at 1.2. These data are also better fit by the flat-XCDM parameterization than by the standard CDM model, but only at 0.6 significance. Current data is unable to rule out dark energy dynamics. The non-flat XCDM parameterization is more consistent with the Dark Energy Survey constraints on the current value of the rms amplitude of mass fluctuations () as a function of the current value of the nonrelativistic matter density parameter () but does not provide as good a fit to the smaller-angle CMB anisotropy data as does the standard tilted flat-CDM model. Some measured cosmological parameter values differ significantly when determined using the tilted flat-XCDM and the non-flat XCDM parameterizations, including the baryonic matter density parameter and the reionization optical depth.

Subject headings:
cosmological parameters — cosmic background radiation — large-scale structure of universe — inflation — observations — methods:statistical

1. Introduction

In the standard spatially-flat CDM cosmological model (Peebles, 1984) the cosmological constant dominates the current energy budget and powers the currently accelerating cosmological expansion. Cold dark matter (CDM) and baryonic matter are the second and third largest contributors to the cosmological energy budget now, followed by small contributions from neutrinos and photons. For reviews of the standard model see Ratra & Vogeley (2008), Martin (2012), Brax (2018), and Luković et al. (2018). This model is largely able to accommodate most observational constraints, including CMB anisotropy measurements (Planck Collaboration, 2016), baryonic acoustic oscillation (BAO) distance observations (Alam et al., 2017), Hubble parameter data (Farooq et al., 2017),111Hubble parameter values have been measured from low redshift to well past the redshift of the cosmological deceleration-acceleration transition. This transition is between the earlier nonrelativistic-matter-dominated decelerating cosmological expansion and the more recent dark-energy-powered accelerating cosmological expansion. This transition redshift has recently been measured from Hubble parameter observations and is roughly at the value expected in the standard CDM and other dark energy models (Farooq & Ratra, 2013; Moresco et al., 2016; Farooq et al., 2017; Yu et al., 2018). and Type Ia supernova (SNIa) apparent magnitude measurements (Scolnic et al., 2017). Current observational constraints however allow for slightly non-flat spatial geometries and/or mild dark energy dynamics.

The standard spatially-flat CDM inflation model is characterized by six cosmological parameters conventionally chosen to be: and , the current values of the baryonic and cold dark matter density parameters multiplied by the square of the Hubble constant (in units of 100 km s Mpc); , the angular diameter distance as a multiple of the sound horizon at recombination; , the reionization optical depth; and and , the amplitude and spectral index of the power-law primordial scalar energy density inhomogeneity power spectrum.

Observational data are on the verge of being able to place interesting constraints on seven parameter cosmological models. Two more plausible seventh cosmological parameters now under discussion are spatial curvature in non-flat extensions of the standard model and a parameter that governs dark energy dynamics in dynamical dark energy extensions of the standard model.

A simple, and so widely used, dynamical dark energy parameterization is the XCDM one.222Many observations have been used to constrain the XCDM parameterization (see, e.g., Chen & Ratra, 2004; Samushia et al., 2007; Samushia & Ratra, 2010; Chen & Ratra, 2011b; Solà et al., 2017a, 2016, b, c, d; Zhai et al., 2017, and references therein). This parameterizes the equation of state relation between the pressure and energy density of the dark energy fluid through where the equation of state parameter is the additional seventh cosmological parameter. XCDM is not a physically consistent description of dark energy as it is unable to consistently describe the evolution of energy density spatial inhomogeneities. To render XCDM physically consistent requires an eighth cosmological parameter, the square of the speed of sound in the dark energy fluid, . In this paper, as is common practice, we consider a restricted, physically-consistent, modified XCDM parameterization in which is not allowed to vary in space or with time and is arbitrarily set to unity. CDM is the simplest physically consistent dynamical dark energy model (Peebles & Ratra, 1988; Ratra & Peebles, 1988). Here a scalar field with potential energy density is the dynamical dark energy and is the additional seventh cosmological parameter.333While XCDM is often used to model dynamical dark energy, it does not accurately model CDM (Podariu & Ratra, 2001; Ooba et al., 2018).

Ooba et al. (2018) have analyzed the Planck 2015 CMB anisotropy data and some BAO distance measurements by using these seven parameter tilted spatially-flat XCDM and CDM dynamical dark energy inflation models and found that both were slightly favored by the data, compared to the standard six parameter flat-CDM model, by 1.1 (1.3) for the XCDM (CDM) case. While these are not significant improvements over the standard model, current data are not able to rule out dark energy dynamics. In addition, both dynamical dark energy models reduce the tension between the Planck 2015 CMB anisotropy and the weak gravitational lensing constraints on , the rms fractional energy density inhomogeneity averaged over 8 Mpc radius spheres,

There have been a number of earlier suggestions that different combinations of observational data favor dynamical dark energy models over the standard CDM model (Sahni et al., 2014; Ding et al., 2015; Solà et al., 2015; Zheng et al., 2016; Solà et al., 2017a, 2016, b; Zhao et al., 2017; Solà et al., 2017c; Zhang et al., 2017a; Solà et al., 2017d; Gómez-Valent & Solà, 2017; Cao et al., 2017; Gómez-Valent & Solà, 2018). As far as we are aware, of these analyses, only those of Zhao et al. (2017) and Zhang et al. (2017a) performed complete CMB anisotropy analyses of the generalized XCDM dynamical dark energy parameterizations they assumed.444Both analyses also included in their data compilation a high value of estimated from the local expansion rate. We do not include this high local value in the data compilation we use here to constrain cosmological parameters, as it is not consistent with the other data we use, in the CDM, XCDM, and CDM models. The other analyses either ignored CMB anisotropy data or only approximately accounted for it.

The standard CDM model assumes flat spatial geometry. In non-flat models non-vanishing spatial curvature introduces a new length scale and so it is incorrect to assume a power-law power spectrum for energy density inhomogeneities in non-flat models (as was assumed for analyses of non-flat models in Planck Collaboration, 2016). Non-flat cosmological inflation provides the only known method for computing a physically consistent power spectrum in non-flat models. For open spatial hypersurfaces the open-bubble inflation model of Gott (1982) is used to compute the non-power-law power spectrum (Ratra & Peebles, 1994, 1995). For closed spatial hypersurfaces Hawking’s prescription for the initial quantum state of the universe (Hawking, 1984; Ratra, 1985) can be used to construct a closed inflation model that gives the non-power-law power spectrum of spatial inhomogeneities (Ratra, 2017). In the non-flat inflation models, compared to the flat inflation model, there is no simple tilt option so is no longer a free parameter and is replaced by the current value of the spatial curvature density parameter .

Using such a physically consistent untilted non-flat inflation model non-power-law power spectrum of energy density inhomogeneities, Ooba et al. (2017a) found that Planck 2015 CMB anisotropy measurements (Planck Collaboration, 2016) do not require flat spatial geometry in the six parameter non-flat CDM model.555Non-CMB observations do not tightly constrain spatial curvature (Farooq et al., 2015; Cai et al., 2016; Chen et al., 2016; Yu & Wang, 2016; L’Huillier & Shafieloo, 2017; Farooq et al., 2017; Li et al., 2016; Wei & Wu, 2017; Rana et al., 2017; Yu et al., 2018; Mitra et al., 2017). In the six parameter non-flat CDM model, compared to the six parameter flat-CDM model, is replaced by . Park & Ratra (2018) used the largest compilation of current reliable observational data to study the non-flat CDM inflation model, confirming the results of Ooba et al. (2017a) and finding stronger evidence for non-flatness, 5.1, favoring a very slightly closed model. The CMB anisotropy data also do not require flat spatial hypersurfaces in the seven parameter non-flat XCDM dynamical dark energy inflation parameterization (Ooba et al., 2017b). Here is the seventh cosmological parameter and again is replaced by . In the simplest seven parameter non-flat CDM dynamical dark energy inflation model (Pavlov et al., 2013) — in which is the seventh cosmological parameter with replaced by Ooba et al. (2017c) again found that CMB anisotropy data do not require flat spatial hypersurfaces. In both the XCDM and CDM inflation cases the data also favor a very slightly closed model. All three slightly closed models are more consistent with constraints from weak lensing observations.

In this paper we examine observational constraints on the seven parameter tilted flat-XCDM and the seven parameter untilted non-flat XCDM dynamical dark energy inflation parameterizations. For this purpose we use an updated version of the Planck 2015 CMB anisotropy, and (almost all currently available reliable) SNIa apparent magnitude, BAO distance, growth factor, and Hubble parameter data compilation of Park & Ratra (2018). Our main update here is the replacement of the Joint Light-curve Analysis (JLA) compilation of apparent magnitude measurements of 740 SNIa (Betoule et al., 2014) by the Pantheon (Scolnic et al., 2017) collection of 1049 SNIa measurements. When used with the Planck 2015 CMB anisotropy data in an analysis of the non-flat CDM case, the Pantheon data place tighter constraints on spatial curvature than do the JLA data. Overall, for the full data compilation, our updated results for the tilted flat-CDM and the non-flat CDM inflation models here are very similar to those of Park & Ratra (2018), with evidence for non-flatness in the non-flat CDM case now becoming 5.2 (from 5.1).

Our first main goal here is to examine the consequences of including a significant amount of recent, reliable, non-CMB data on the finding of Ooba et al. (2018) that the Planck 2015 CMB anisotropy observations and a few BAO distance measurements favor the seven parameter tilted flat-XCDM parameterization over the six parameter standard flat-CDM model. Our second main goal is to examine the effect that the inclusion of this new non-CMB data has on the finding of Ooba et al. (2017b) that the Planck 2015 CMB anisotropy observations and a few BAO distance measurements are not inconsistent with the closed-XCDM inflation parameterization. Our third main goal is to use this large compilation of reliable data to examine the consistency between the cosmological constraints of each type of data and to more tightly measure cosmological parameters than has been done to date, and in particular to also find out which model parameters can or cannot be measured from these data in a cosmological-model-independent manner.

We find that the seven parameter tilted flat-XCDM inflation parameterization continues to provide a better fit to the data than does the six parameter standard CDM model. However, for the large compilation of data used here we find the XCDM dynamical dark energy case is only 0.57 better than the standard CDM case (compared to the 1.1 Ooba et al. 2018 found with the smaller data compilation). This is not a significant improvement over the standard model but on the other hand the XCDM parameterization cannot be ruled out. Also in agreement with Ooba et al. (2018) we do not find a deviation from (a cosmological constant) for the flat-XCDM case.666These results differ from those of earlier approximate analyses, based on less and less reliable data than we have used here (Solà et al., 2017a, 2016, b, c, d; Gómez-Valent & Solà, 2017, 2018), that favor the flat-XCDM case over the flat-CDM one by 3 or greater and find deviating from by more than 3. The tilted flat-XCDM model continues to better fit the weak lensing constraints in the plane than does the standard flat model.

For the untilted non-flat XCDM inflation parameterization, our results here, determined using much more non-CMB data, are consistent with and strengthen those of Ooba et al. (2017b). For the full data compilation we now find a more than 3.4 deviation from spatial flatness and now, for the first time, we also find a corresponding deviation from a cosmological constant with in the non-flat XCDM case being more than 1.2 away from the cosmological constant value of . The non-flat XCDM parameterization better fits the weak lensing constraints in the plane. For the full data combination we consider here (including CMB lensing data), we find that the observed low- CMB anisotropy multipole number () power spectrum is best fit777Here by best fit we mean that the corresponding model has the lowest of the models under consideration. As discussed elsewhere and below, some of these models are not nested and the number of degrees of freedom of the Planck 2015 data are ambiguous, so in many cases it is not possible to convert the ’s we compute here to a quantitative goodness of fit and so many of our statements here about goodness of fit are qualitative. See below for more detailed discussion of this issue. by the untilted non-flat CDM model, followed by the tilted flat-CDM model, with the tilted flat-XCDM parameterization and the untilted non-flat XCDM parameterization in third and fourth place. The tilted flat-CDM model and the tilted flat-XCDM parameterization best fit the observed higher- ’s, followed by the untilted non-flat XCDM parameterization and the untilted non-flat CDM model in third and fourth place.

We find that is measured in an almost model-independent manner with values that are consistent with most other estimates. However, as also found in Park & Ratra (2018), some measured cosmological parameter values, including those of , , and , differ significantly between the flat and the non-flat models and so care must be exercised when utilizing cosmological measurements of such parameters.

This paper is organized as follows. In Sec. 2 we briefly summarize the cosmological data sets we use in our analyses. In Sec. 3 we briefly summarize the methods we use for our analyses. The observational constraints resulting from these data for the tilted flat-XCDM and the non-flat XCDM inflation parameterizations are presented in Sec. 4. We summarize our results in Sec. 5.

2. Data

As in Park & Ratra (2018) we use the Planck 2015 TT + lowP and TT + lowP + lensing CMB anisotropy data (Planck Collaboration, 2016). Here TT represents the low- () and high- (; PlikTT) Planck temperature-only data and lowP denotes low- polarization , , and power spectra measurements at . The collection of low- temperature and polarization measurements is denoted as lowTEB. For CMB lensing data we use the power spectrum of the lensing potential measured by Planck.

Instead of using the JLA apparent magnitude measurement compilation of 740 SNIa (Betoule et al., 2014), we use the Pantheon collection of 1049 SNIa apparent magnitude measurements over the broader redshift range of (Scolnic et al., 2017), which includes 276 SNIa () discovered by the Pan-STARRS1 Medium Deep Survey and SNIa distance estimates from SDSS, SNLS and low- HST samples. Throughout this paper, we use the abbreviation SN to denote the Pantheon SNIa sample.

We make one change to the BAO compilation of Sec. 2.3 and Table 1 of Park & Ratra (2018), here using Mpc for the BAO data point of Ata et al. (2018), instead of the old value, Mpc, given in the initial version of their preprint (arXiv:1705.06373v1). See Sec. 2.3 of Park & Ratra (2018) for the definitions of the above expressions. Throughout this paper, we use the abbreviation BAO to denote this updated BAO compilation.

We also use the same Hubble parameter, , and growth rate, , measurements listed in Tables 2 and 3 of Park & Ratra (2018).

3. Methods

We use the publicly available CAMB/COSMOMC package (version of Nov. 2016) (Challinor & Lasenby, 1999; Lewis et al., 2000; Lewis & Bridle, 2002) to constrain the tilted flat and the untilted non-flat XCDM dynamical dark energy inflation parameterizations with Planck 2015 CMB and other non-CMB data sets. We use the CAMB Boltzmann code to compute the CMB angular power spectra for temperature fluctuations, polarization, and lensing potential, and COSMOMC, based on the Markov chain Monte Carlo (MCMC) method, to determine model-parameter space that is favored by the data. We use the same COSMOMC settings adopted in Planck Collaboration (2016) and used in Park & Ratra (2018).

The primordial power spectrum in the spatially-flat tilted XCDM inflation case (Lucchin & Matarrese, 1985; Ratra, 1992, 1989) is

(1)

where is wavenumber and is the amplitude at the pivot scale . The primordial power spectrum in the untilted non-flat XCDM inflation case (Ratra & Peebles, 1995; Ratra, 2017) is

(2)

which goes over to the spectrum in the spatially-flat limit (). For scalar-type perturbations, is the wavenumber where is the spatial curvature and is the speed of light. For the spatially-closed model, with negative , the normal modes are characterized by positive integers . We use as the initial power spectrum of perturbations for the non-flat model by normalizing its amplitude at the pivot scale to the value of .

Our analyses methods are very similar to those described in Sec. 3.2 of Park & Ratra (2018). During the MCMC process we set the same priors for the cosmological parameters as in Park & Ratra (2018). For the seventh parameter , we set . In our analyses with the Pantheon SNIa sample, we do not set priors for the nuisance parameters ( and ) that are related to stretch and color correction of the SNIa light curves, since the stretch and color parameters of the Pantheon SNIa used here are set to zero.888In addition to and , the distance moduli of the Pantheon SNIa are affected by three more nuisance parameters, the absolute -band magnitude (), the distance correction based on the host-galaxy mass (), and the distance correction based on predicted biases from simulation () (Scolnic et al., 2017). Thus, the number of degrees of freedom for the Pantheon sample is less than the number of SNIa. For example, for the flat- model that fits the matter density parameter , the number of degrees of freedom becomes 1043 ().

4. Observational Constraints

We first examine how much more effective the improved Pantheon SNIa data are in constraining cosmological parameters, relative to the JLA data. Figure 1 compares the likelihood distributions of the model parameters for the JLA and the Pantheon data sets, in conjunction with the CMB observations, for the spatially-flat tilted and for the untilted non-flat CDM inflation models. The mean and 68.3% confidence limits of model parameters are presented in Table 1.999The parameters for the tilted flat- model constrained with TT + lowP (+lensing) + JLA are in good agreement with the Planck results. See Planck 2015 cosmological parameter tables base_plikHM_TT_lowTEB_post_JLA for TT + lowP + JLA data and base_plikHM_TT_lowTEB_lensing_post_JLA for TT + lowP + lensing + JLA data (Planck Collaboration, 2015). Without CMB lensing data, the Pantheon data are a little more constraining than the JLA data. When CMB lensing data are included, the largest reduction in error bars occur for the non-flat CDM case, where the error bars for , , and are approximately only 80% as large for the CMB + Pantheon combination when compared to the CMB + JLA case. From this Table we also see that adding CMB lensing data results in a reduction of and in both models.

Note that our six parameter physically-consistent untilted non-power-law power spectrum non-flat model constrained by the CMB and Pantheon data favors non-flat geometry and that the parameter constraints determined using our model are quite different from those presented in Scolnic et al. (2017) that were derived using the seven parameter physically-inconsistent tilted power-law power spectrum non-flat CDM model (with varying spectral index ) which were found to favor spatial flatness (, , for the TT + lowP + Pantheon data combination).

Figure 1.— Likelihood distributions of the tilted flat (left) and untilted non-flat (right) model parameters favored by the Planck 2015 CMB TT + lowP (+ lensing) and SNIa data. Here the parameter constraints are compared for the JLA SNIa data and the Pantheon SNIa data and summarized in Table 1. Two-dimensional marginalized likelihood distributions of all possible combinations of model parameters together with one-dimensional likelihoods are shown as solid and dashed black curves for JLA and filled contours and colored curves for Pantheon data.
{ruledtabular}
Tilted flat- model
Parameter TT+lowP+JLA TT+lowP+lensing+JLA TT+lowP+Pantheon TT+lowP+lensing+Pantheon
[km s Mpc]
Untilted non-flat model
Parameter TT+lowP+JLA TT+lowP+lensing+JLA TT+lowP+Pantheon TT+lowP+lensing+Pantheon
[km s Mpc]
Table 1Mean and 68.3% confidence limits of tilted flat and untilted non-flat model parameters constrained by Planck and SNIa data. JLA versus Pantheon.

Table 2 lists the parameter constraints for the tilted spatially-flat and for the untilted non-flat CDM inflation models, for the updated complete data set we use here. These constraints can be compared to those listed in the bottom right panels of Tables 5–8 of Park & Ratra (2018) that were derived using the JLA SNIa data and the initial preprint value of the Ata et al. (2018) BAO distance measurement. There are very small differences between the constraints derived using our previous and our updated full data sets.

{ruledtabular}
Tilted flat- model
Parameter TT+lowP+SN+BAO++ TT+lowP+lensing+SN+BAO++
[km s Mpc]
Untilted non-flat model
Parameter TT+lowP+SN+BAO++ TT+lowP+lensing+SN+BAO++
[km s Mpc]
Table 2Tilted flat and untilted non-flat model parameters constrained with Planck, SN, BAO, , and data (mean and 68.3% confidence limits).

Our results for the tilted flat and the non-flat XCDM parameterizations are presented in Figs. 25 and Tables 36. In the triangle plots we omit the likelihood contours for TT + lowP (+ lensing) + SN + BAO data (excluding or including the Planck 2015 CMB lensing data) in both the tilted flat and the non-flat XCDM cases because they are very similar to those for TT + lowP (+ lensing) + SN + BAO + data.

The entries for the tilted flat-XCDM parameterization in the TT + lowP panel of Table 3 and in the TT + lowP + lensing panel in Table 4 agree well with the corresponding entries in Table 1 of Ooba et al. (2018), except for those for , , , and . This is because Ooba et al. (2018) use a flat prior non-zero over for while we use a flat prior non-zero over .101010Since the flat prior on adopted here is the same as in the Planck team’s analyses, the parameters for the tilted flat-XCDM case constrained with TT + lowP (+lensing) agree with the Planck results. See base_w_plikHM_TT_lowTEB for TT + lowP data and base_w_plikHM_TT_lowTEB_post_lensing for TT + lowP + lensing data (Planck Collaboration, 2015). The entries for the non-flat XCDM parameterization in the TT + lowP panel of Table 5 and in the TT + lowP + lensing panel in Table 6 agree well with the corresponding entries in Table 1 of Ooba et al. (2017b). Ooba et al. (2017b) and Ooba et al. (2018) used CLASS (Blas et al., 2011) to compute the ’s and Monte Python (Audren et al., 2013) for the MCMC analyses, so it is reassuring that our results agree well with their results. Our estimates of , , and for the tilted flat-XCDM parameterization from the TT + lowP + SN data agree very well with the values presented in Scolnic et al. (2017), , , and , which provides another reassuring check on our analyses.

From Tables 3 and 4 we see that, when added to the Planck 2015 CMB anisotropy data, for the tilted flat-XCDM case, the BAO measurements mostly prove more restrictive than either the SN, , or data , except for where the SN data are more restrictive than the BAO data. However, when the CMB lensing data are included, Table 4, the CMB + SN limits on , , and are more restrictive than those from the CMB data combined with BAO, or , or measurements, while all four non-CMB data sets used in conjunction with the CMB data provide equally restrictive constraints on and .111111This is not the case in the tilted flat-CDM model, where for the data set including the CMB lensing data, the CMB + BAO constraints on all parameters are more restrictive than those determined by combining the CMB data with either the SN, or , or measurements. For this model we show only the CMB + SN constraints in Table 1. We note that our BAO compilation includes radial BAO measurements as well as the measurements of Alam et al. (2017). It is likely that if these are moved to the and data sets, CMB and BAO, SN, , or constraints will all be about equally restrictive for the flat-XCDM parameterization.

Figure 2.— Likelihood distributions of the tilted flat-XCDM model parameters constrained by Planck CMB TT + lowP, SN, BAO, , and data. Two-dimensional marginalized likelihood distributions of all possible combinations of model parameters together with one-dimensional likelihoods are shown for cases when each non-CMB data set is added to the Planck TT + lowP data (left panel) and when the growth rate, SN, Hubble parameter data, and the combination of them, are added to TT + lowP + BAO data (right panel). For ease of viewing, the cases of TT + lowP (left) and TT + lowP + BAO (right panel) are shown as solid black curves.

Figure 3.— Same as Fig. 2 but now including the Planck CMB lensing data.

The situation in the non-flat XCDM case is more interesting. When the CMB lensing data are included, Table 6, CMB data with either SN, or BAO, or , or data, provide approximately equally restrictive constraints on , , and , while CMB + BAO data provide the tightest constraints on , , , , , and , with CMB + SN setting tightest limits on .121212In the non-flat CDM model (results mostly not shown here, except for CMB + SN shown in Table 1), , , and are about equally well constrained by any of the four non-CMB data sets when used with the CMB (including lensing) data, with CMB + BAO setting tighter limits on , , , , , and .

If we focus on CMB TT + lowP + lensing data, Figs. 3 and 5 and Tables 4 and 6, we see that adding only one of the four non-CMB data sets at a time to the CMB measurements (left triangle plots in the two figures) results in four sets of contours that are quite consistent with each other, as well as with the original CMB alone contours, for both the tilted flat-XCDM case and for the untilted non-flat XCDM parameterization. It is reassuring that the four non-CMB data sets do not pull the CMB constraints in significantly different directions. This is also the case for the tilted flat-XCDM parameterization when the CMB lensing data are excluded (left triangle plot of Fig. 2). However, in the untilted non-flat XCDM case without the lensing data when any of the four non-CMB data sets are added to the CMB data (left triangle plot of Fig. 4), they each pull the results towards a smaller (closer to the flat model) and slightly larger and and smaller than is favored by the CMB data alone, although all five sets of constraint contours are largely mutually consistent. However, there is tension between the TT + lowP + SN and the TT + lowP + BAO contours in the plane (Fig. 4 left and Table 5), where CMB + SN data give constraints on and that deviate from the CMB + BAO data values by over , with also deviating from the cosmological constant () by over for the CMB + SN case and differing from by more than 3 in both cases.

While adding the BAO data to the CMB data typically makes the biggest difference, the other three non-CMB data sets also contribute. Focusing on the TT + lowP + lensing data, we see from Table 4 for the tilted flat-XCDM parameterization that the BAO data tightly constrains model parameters, especially , while the measurements move and to larger values and to a smaller value. In this case is the quantity whose error bar is reduced the most by the full combination of data relative to the CMB + SN data combination, followed by the error bar reduction relative to CMB + BAO data combination. For the untilted non-flat XCDM case, from Table 6, the error bars that shrink the most when CMB (including lensing) data are used in conjunction with the four non-CMB data sets are those on (relative to the CMB + SN case) and and (relative to the CMB + BAO combination).

Again concentrating on the TT + lowP + lensing data, Tables 4 and 6, we see that for the tilted flat-XCDM case, adding the four non-CMB data sets to the mix most affects , , and , with the central value moving down by 1.3 and the and central values moving up by 1.3 and 1.2, all of the CMB data alone error bars; is hardly affected by adding the four non-CMB data sets, changing by only 0.042. The situation for the non-flat XCDM parameterization is a little less dramatic, with , and central values increasing by 0.91 and 0.86 of the CMB data alone error bars, and moving closer to flatness by 0.71; the central value does not change in this case.

Figure 4.— Likelihood distributions of the untilted non-flat XCDM model parameters constrained by Planck CMB TT + lowP, SN, BAO, , and data. Two-dimensional marginalized likelihood distributions of all possible combinations of model parameters together with one-dimensional likelihoods are shown for cases when each non-CMB data set is added to the Planck TT + lowP data (left panel) and when the growth rate, SN, Hubble parameter data, and the combination of them, are added to TT + lowP + BAO data (right panel). For ease of viewing, the result of TT + lowP + BAO is shown as solid black curves in the right panel.

Figure 5.— Same as Fig. 4 but now including the Planck CMB lensing data.

Figure 6 shows marginalized likelihood contours in the plane for the tilted flat-XCDM parameterization and in the plane for the untilted non-flat XCDM case. For CMB TT + lowP + lensing data combined with the non-CMB data sets, the flat-XCDM parameterization prefers , favoring the cosmological constant as dark energy. On the other hand, the non-flat XCDM parameterization, when constrained by the full data, prefers closed spatial hypersurfaces and a dark energy equation of state parameter .

More precisely, including the four non-CMB data sets in the mix, we find in the tilted flat-XCDM parameterization (bottom right panel of Table 4) that , which is more tightly restricted to and the cosmological constant than the original Ooba et al. (2018) finding of (the last column of their Table 1).131313These results differ from those of earlier approximate analyses, based on less and less reliable data, that indicated evidence for deviating from by more than 3 (Solà et al., 2017a, 2016, b, c, d; Gómez-Valent & Solà, 2017, 2018).

On the other hand, and perhaps the biggest consequence of including the four non-CMB data sets in the analyses here, is the significant strengthening of the evidence for non-flatness in the untilted non-flat XCDM case, with it increasing to , more than 3.4 away from flatness now, for the full data combination in the bottom right panel of Table 6, compared to the 1.1 away from flatness for the CMB alone case. That is now accompanied for the first time by mild evidence favoring dynamical dark energy with that is more than 1.2 away from the cosmological constant. These results are consistent with but strengthen those of Ooba et al. (2017b) who found and from Planck 2015 CMB anisotropy data in combination with a few BAO distance measurements. The stronger results here are driven in part by each of the four non-CMB data sets. CMB + BAO, CMB + SN, and CMB + favor negative values 2.8, 2.0, and 1.9 away from flat, while CMB + and CMB + BAO favor values that are 1.1 more negative and 1 less negative than . On the other hand, CMB + data are consistent with a flat model and CMB + SN and CMB + are consistent with the cosmological constant and . In favoring a closed model with less negative than , the BAO data play the most important role amongst the four non-CMB data sets.

{ruledtabular}
Parameter TT+lowP TT+lowP+SN TT+lowP+BAO
[km s Mpc]
Parameter TT+lowP+ TT+lowP+SN+BAO TT+lowP+SN+BAO+
[km s Mpc]
Parameter TT+lowP+ TT+lowP+BAO+ TT+lowP+SN+BAO++
[km s Mpc]
Table 3Tilted flat-XCDM model parameters constrained with Planck TT + lowP, SN, BAO, , and data (mean and 68.3% confidence limits).

For the full data combination (including CMB lensing data) in Tables 4 and 6, values measured using the tilted flat-XCDM and the non-flat XCDM parameterizations, and km s Mpc, are consistent with each other to within 0.57 (of the quadrature sum of the two error bars).141414Potential systematic errors that might affect the value of , ignored here, have been discussed by Addison et al. (2016) and Planck Collaboration (2017). These values are consistent with the most recent median statistics estimate km s Mpc (Chen & Ratra, 2011a), which is consistent with earlier median statistics estimates (Gott et al., 2001; Chen et al., 2003). Many recent estimates of are also consistent with these measurements (Aubourg et al., 2015; Planck Collaboration, 2016; Semiz & Çamlıbel, 2015; L’Huillier & Shafieloo, 2017; Chen et al., 2017; Luković et al., 2016; Wang et al., 2017; Lin & Ishak, 2017; DES Collaboration, 2017b; Yu et al., 2018; Haridasu et al., 2017; Zhang et al., 2018; Gómez-Valent & Amendola, 2018), but, as is well known, they are lower than the local measurement of km s Mpc (Riess et al., 2018).151515This local measurement is 2.9 (3.3), of the quadrature sum of the two error bars, higher than measured here using the tilted flat-XCDM (untilted non-flat XCDM) parameterization. We note that some other local expansion rate measurements find slightly lower ’s with larger error bars (Rigault et al., 2015; Zhang et al., 2017b; Dhawan et al., 2017; Fernández Arenas et al., 2017).

In our analyses here, and (discussed below) are the only cosmological parameters that are measured in an almost cosmological model (spatial curvature and tilt) independent way. Measurements of other cosmological parameters determined using the two XCDM parameterizations differ more significantly. More precisely, measurements determined using the full data set (including CMB lensing) of , , , , , , and , differ by 0.74, 1.1, 2.0, 2.3, 2.5, 2.7, and 4.8 (of the quadrature sum of the two error bars). For some of these parameters, especially as well as probably and , the cosmological model dependence of the measurement creates a much larger uncertainty than does the statistical error in a given cosmological model. This effect was first noticed in a comparison between measurements made using the tilted flat-CDM and the non-flat CDM model (Park & Ratra, 2018). From Tables 4 and 6, for the full data compilation (including CMB lensing), we find in the tilted flat-XCDM (non-flat XCDM) case () and () at 2, which are almost disjoint. It is not yet possible to measure , , or (and possibly some other cosmological parameters as well) in a model independent way from cosmological data.

{ruledtabular}
Parameter TT+lowP+lensing TT+lowP+lensing+SN TT+lowP+lensing+BAO
[km s Mpc]
Parameter TT+lowP+lensing+ TT+lowP+lensing+SN+BAO TT+lowP+lensing+SN+BAO+
[km s Mpc]
Parameter TT+lowP+lensing+ TT+lowP+lensing+BAO+ TT+lowP+lensing+SN+BAO++
[km s Mpc]
Table 4Tilted flat-XCDM model parameters constrained with Planck TT + lowP + lensing, SN, BAO, , and data (mean and 68.3% confidence limits).

For the full data combination (including CMB lensing data), ’s measured using the two XCDM parameterizations, Tables 4 and 6, agree to 0.32 (of the quadrature sum of the two error bars). Figures 7 and 8 show the marginalized two-dimensional likelihood distribution contours in the plane for the tilted flat and untilted non-flat XCDM parameterizations constrained using the CMB and non-CMB data. For comparison we also show the constraints obtained from a combined analysis of galaxy clustering and weak gravitational lensing based on the first year result of the Dark Energy Survey (DES Y1 All) (DES Collaboration, 2017a), whose 1 confidence limits are and . The marginalized likelihood distribution contours in the plane obtained by adding each non-CMB data set to the Planck 2015 CMB data are consistent with each other, except for the non-flat XCDM case where the TT + lowP + SN contours almost do not overlap with contours derived using any of the other three non-CMB data sets with the TT + lowP data (Fig. 8 top left panel). As expected, the BAO data provide the most restrictive constraints among the four non-CMB data sets.

Although the constraints from the tilted flat and untilted non-flat XCDM analyses (excluding and including CMB lensing data) are similar to the DES Y1 All result, our constraints here favor a larger value by about (of the quadrature sum of the two error bars) for the flat-XCDM case for the full data combination. We emphasize that the best-fit point for the non-flat XCDM parameterization constrained by the Planck CMB data (including lensing) combined with all non-CMB data enters well into the 1 region of the DES Y1 All constraint contour (Fig. 8 lower right panel), unlike the case for the tilted flat XCDM case (Fig. 7 lower right panel).

{ruledtabular}
Parameter TT+lowP TT+lowP+SN TT+lowP+BAO
[km s Mpc]
Parameter TT+lowP+ TT+lowP+SN+BAO TT+lowP+SN+BAO+
[km s Mpc]
Parameter TT+lowP+ TT+lowP+BAO+ TT+lowP+SN+BAO++
[km s Mpc]
Table 5Untilted non-flat XCDM model parameters constrained with Planck TT + lowP, SN, BAO, , and data (mean and 68.3% confidence limits).

Table 7 lists the individual and total values for the best-fit tilted flat and untilted non-flat models. This is an updated version of Table 9 of Park & Ratra (2018), for the updated data sets we use here. Table 8 lists the corresponding quantities for the tilted flat and the untilted non-flat XCDM parameterizations. The best-fit position in the parameter space is found directly by searching for the minimum total from the converged MCMC chains. We emphasize that the best-fit position found in this approximate way can differ from the true best-fit position.161616For example, values presented here for the tilted flat- model constrained using TT + lowP data differ from the Planck results that were obtained by Powell’s minimization method (, , with total ; Planck Collaboration 2015), which is an efficent algorithm to find the location of the minimum but has a limitation that it sometimes stays in a local minimum. The discrepancy comes from the fact that the MCMC chains, though converged, are not sufficiently long enough to densely search around the best-fit position in the multi-dimensional parameter space. One way to alleviate this discrepancy is to estimate the best-fit position by the weighted average of model parameter values in the converged MCMC chains, where the weight is given in proportion to , and then apply Powell’s minimization method by varying the foreground nuisance parameters while fixing the best-fit model parameters. For the same model and data above, this method gives , , with total . In these Tables we list the individual contribution of each data set used to constrain model parameters. The total is the sum of those from the high- CMB TT likelihood (), the low- CMB power spectra of temperature and polarization (), lensing (), SN (), BAO (), (), data (), and the contribution from the foreground nuisance parameters (). Due to the nonstandard normalization of the Planck 2015 CMB data likelihoods, the number of Planck 2015 CMB degrees of freedom is ambiguous. Since the number of degrees of freedom of the Planck CMB data is not available and the absolute value of is arbitrary, only the difference of of one model relative to the other is meaningful for the Planck CMB data. In Table 7, for the non-flat model, we list , the excess over the value of the tilted flat- model constrained with the same combination of data sets. For the non-CMB data sets, the numbers of degrees of freedom are 1043, 15, 31, 10 for the SN, BAO, , data sets, respectively, for a total of 1099 degrees of freedom. The reduced ’s for the individual non-CMB data sets are . There are 189 points in the TT + lowP Planck 2015 data (binned angular power spectrum) and 197 when the CMB lensing observations are included.

Conclusions about the qualitative relative goodness of fit of the tilted flat and non-flat models drawn from the updated data here are not very different from those found earlier (Park & Ratra, 2018) from the original data. For the non-flat CDM case relative to the flat-CDM model, we have for TT + lowP + lensing and the full non-CMB compilation (last column in the last row of Table 7). As discussed above and in Ooba et al. (2017a, b, c) and Park & Ratra (2018), it is unclear how to turn this into a quantitative relative probability as the two six parameter models are not nested (and the number of degrees of freedom of the Planck CMB anisotropy data is not available). It is clear however that the non-flat CDM model does a worse job in fitting the higher- ’s than it does in fitting the lower- ones. We note that there has been discussion about systematic differences between constraints derived using the higher- and the lower- Planck 2015 CMB data (Addison et al., 2016; Planck Collaboration, 2017). In addition, in the context of the flat-CDM model, there appear to be inconsistencies between the higher- Planck 2015 CMB anisotropy data and the South Pole Telescope CMB anisotropy data (Aylor et al., 2017). It is possible that, if real, when these differences are resolved this could result in a reduction of the ’s found here.

{ruledtabular}
Parameter TT+lowP+lensing TT+lowP+lensing+SN TT+lowP+lensing+BAO
[km s Mpc]
Parameter TT+lowP+lensing+ TT+lowP+lensing+SN+BAO TT+lowP+lensing+SN+BAO+
[km s Mpc]
Parameter TT+lowP+lensing+ TT+lowP+lensing+BAO+ TT+lowP+lensing+SN+BAO++
[km s Mpc]
Table 6Untilted non-flat XCDM model parameters constrained with Planck TT + lowP + lensing, SN, BAO, , and data (mean and 68.3% confidence limits).

Table 8 lists the individual and total values for the best-fit tilted flat and untilted non-flat XCDM parameterizations. In the last column we list , the excess of the seven parameter XCDM case over the value of the corresponding six parameter model constrained using the same combination of data sets.171717We note that the ’s of Table 8 are significantly smaller than those of Table 7. As a result of the additional uncertainity in computing ’s discussed in the previous footnote, it is likely that the ’s of Table 8 could change if longer MCMC chains are used to compute them, which could change the relative goodness of fit probablities discussed later in this paragraph, which are perhaps best viewed as being only semi-quantitative. These models are nested; the seven parameter tilted flat-XCDM (untilted non-flat XCDM) parameterization reduces to the six parameter tilted flat-CDM (untilted non-flat CDM) model when . In this case the ambiguity in the number of Planck 2015 degrees of freedom is not an obstacle to converting the values to a relative goodness of fit. From , for the full data set (including CMB lensing), for one additional free parameter, we find that the tilted flat-XCDM (untilted non-flat XCDM) parameterization is a 0.57 (0.87) better fit to the data than is the tilted flat-CDM (untilted non-flat CDM) model. (We emphasize that non-flat CDM does not fit the data as well as flat-CDM, although the difference in the goodness of fit cannot yet be precisely quantified.) These results are consistent with those of Ooba et al. (2018) and Ooba et al. (2017b).

Of all these four models, the tilted flat-XCDM parameterization best fits the combined data, but at a lower level of significance than the 1.1 of Ooba et al. (2018), and not close to the 3 or 4 significance found in earlier approximate analyses (Solà et al., 2017a, 2016, b, c, d; Gómez-Valent & Solà, 2017, 2018). While the tilted flat-XCDM parameterization does not provide a significantly better fit to the data, current data cannot rule out dynamical dark energy.

Figure 6.— 1 and 2 likelihood contours in the plane for the tilted flat-XCDM parameterization (left panel) and in the plane for the untilted non-flat XCDM parameterization (right panel), constrained by Planck CMB TT + lowP + lensing and non-CMB data sets. The horizontal and vertical dashed lines indicate (the cosmological constant) or . Contours in both panels follow the color scheme shown in the left panel.

Figures 9 and 10 show the CMB high- TT, and the low- TT, TE, EE power spectra of the best-fit tilted flat and untilted non-flat XCDM dynamical dark energy inflation parameterizations, excluding and including the lensing data, respectively. The best-fit tilted flat-XCDM models favored by the Planck CMB and non-CMB data agree well with the observed CMB power spectra at all . However, similar to the non-flat case studied in Park & Ratra (2018), the non-flat XCDM model constrained with the Planck 2015 CMB anisotropy data and each non-CMB data set generally gives a poorer fit to the low- EE power spectrum while it better fits the low- TT power spectrum (see the bottom left panel of Figs. 9 and 10). The shape of the best-fit power spectra of various models relative to the Planck CMB data points are consistent with the values listed in Table 8. For example, the best-fit untilted non-flat XCDM parameterization constrained by using the TT + lowP + BAO measurements has a low- EE power spectrum that deviates the most from the Planck data and the corresponding value of that is larger by 7.83 relative to the best-fit tilted flat- model for the TT + lowP data (see Tables 7 and 8). However, the poor fit to the Planck EE power spectrum data is alleviated when the full combination of non-CMB data is used.

Figure 11 shows the best-fit initial power spectra of scalar-type fractional energy density perturbations for the untilted non-flat XCDM parameterization constrained by the Planck TT + lowP (left) and TT + lowP + lensing (right panel) data together with other non-CMB data sets. The reduction in power at low in the best-fit closed-XCDM inflation parameterization power spectra shown in Fig. 11 is partially responsible for the low- TT power reduction of the best-fit closed model ’s (shown in the lower panels of Figs. 9 and 10) relative to the best-fit tilted flat model ’s.181818Other effects, including the usual and integrated Sachs-Wolfe effects, also play a role in affecting the shape of the low- ’s. The case of the best-fit non-flat XCDM parameterization for the TT + lowP data is the most dramatic one, consistent with the reduced low- TT power (Figs. 9).


Figure 7.— 1 and 2 likelihood contours in the plane for the tilted flat-XCDM parameterization constrained by Planck CMB TT + lowP (+lensing), SNIa, BAO, , and data. In each panel the model 1 and 2 constraint contours obtained from the first-year Dark Energy Survey (DES Y1 All) (DES Collaboration, 2017a) are shown as thick solid curves for comparison.


Figure 8.— Same as Fig. 7 but for the untilted non-flat XCDM parameterization.

5. Conclusion

We use the tilted flat-XCDM and the untilted non-flat XCDM dynamical dark energy inflation parameterizations to measure cosmological parameters from an updated, reliable, large compilation of observational data.

Our main results, in summary, are:

  • We confirm, but at lower significance, the Ooba et al. (2018) result that the tilted flat-XCDM parameterization provides a better fit to the data than does the standard tilted flat-CDM model. The improvement is not significant, but on the other hand current data are unable to rule out dynamical dark energy.

  • In the untilted non-flat XCDM case, we confirm, with greater significance, the Ooba et al. (2017b) result that cosmological data does not demand spatially-flat hypersurfaces for this parameterization, and that the non-flat XCDM parameterization provides a better fit to the data than does the non-flat CDM model (qualitatively it is clear that the standard tilted flat-CDM model is a better fit to the data than is the untilted non-flat CDM model). In the non-flat XCDM case, these data (including CMB lensing measurements) favor a closed model at more than 3.4 significance, with spatial curvature contributing a little less than a percent to the current cosmological energy budget, and favor dark energy dynamics (over a cosmological constant) at a little more than 1.2.

  • measured in both models are very similar, and consistent with most other measurements of . However, as is well known, an estimate of the local expansion rate (Riess et al., 2018) is about 3 larger.

  • measured in both models are almost identical and consistent with the recent DES measurement (DES Collaboration, 2017a).

  • The measured is more model dependent than the measured and the value measured using the non-flat XCDM parameterization is more consistent with the recent DES measurement (DES Collaboration, 2017a).

  • ,