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

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Abstract

We constrain tilted spatially-flat and untilted nonflat XCDM dynamical dark energy inflation parameterizations using Planck 2015 cosmic microwave background (CMB) anisotropy data and recent baryonic acoustic oscillations distance measurements, Type Ia supernovae data, Hubble parameter observations, and growth rate measurements. Inclusion of the four non-CMB data sets results in a significant strengthening of the evidence for nonflatness in the nonflat XCDM model from 1.1\(\sigma \) for the CMB data alone to 3.4\(\sigma \) for the full data combination. In this untilted nonflat 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 mildly favor dynamical dark energy over a cosmological constant at 1.2\(\sigma \). These data are also better fit by the flat-XCDM parameterization than by the standard \(\varLambda \)CDM model, but only at 0.3\(\sigma \) significance. Current data is unable to rule out dark energy dynamics. The nonflat XCDM parameterization is compatible with the Dark Energy Survey limits on the present value of the rms mass fluctuations amplitude (\(\sigma _{8}\)) as a function of the present value of the nonrelativistic matter density parameter (\(\varOmega _{m}\)). However, it does not provide as good a fit to the higher multipole CMB temperature anisotropy data as does the standard tilted flat-\(\varLambda \)CDM model. A number of measured cosmological parameter values differ significantly when determined using the tilted flat-XCDM and the nonflat XCDM parameterizations, including the baryonic matter density parameter and the reionization optical depth.

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Notes

  1. Hubble parameter values have been measured from low redshift to well past the redshift of the cosmological deceleration-acceleration transition between the earlier nonrelativistic-matter-dominated decelerating cosmological expansion and the more recent dark-energy-dominated accelerating cosmological expansion. The transition redshift has been measured from Hubble parameter observations and it is roughly at the value expected in dark energy models (Farooq and Ratra 2013; Moresco et al. 2016; Farooq et al. 2017; Yu et al. 2018).

  2. Many observations have been used to constrain the XCDM parameterization (see, e.g., Chen and Ratra 2004, 2011b; Samushia et al. 2007; Samushia and Ratra 2010; Solà et al. 2017a,b,c,d, 2018; Zhai et al. 2017, and references therein).

  3. While XCDM is widely used to model dynamical dark energy, it does not accurately model \(\phi \)CDM (Podariu and Ratra 2001; Ooba et al. 2018d).

  4. Both analyses also included in their data compilation a high value of \(H_{0}\) estimated from the local expansion rate. We do not include this high local \(H_{0}\) 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 \(\varLambda \)CDM, XCDM, and \(\phi \)CDM models.

  5. Non-CMB observations do not tightly constrain spatial curvature (Farooq et al. 2015; Cai et al. 2016; Chen et al. 2016; Yu and Wang 2016; L’Huillier and Shafieloo 2017; Farooq et al. 2017; Li et al. 2016; Wei and Wu 2017; Rana et al. 2017; Yu et al. 2018; Mitra et al. 2018, 2019; Ryan et al. 2018, 2019), except for a compilation of all of the most recent SNIa, BAO and Hubble parameter data, which also (mildly) favors closed spatial hypersurfaces (Park and Ratra 2018c) and for a compilation of primordial deuterium abundance measurements which favors a flat geometry (Penton et al. 2018).

  6. For earlier constraints on the flat-XCDM model, see Zhao et al. (2007), Wang et al. (2007, 2009), and references therein.

  7. These results differ from those of earlier approximate analyses, based on less and less reliable data than we have used here (Solà et al. 2017a,b,c,d, 2018; Gómez-Valent and Solà 2017, 2018), that favor the flat-XCDM case over the flat-\(\varLambda \)CDM one by 3\(\sigma \) or greater and find \(w\) deviating from −1 by more than 3\(\sigma \).

  8. Here by best fit we mean that the corresponding model has the lowest \(\chi ^{2}\) of the models under consideration. As discussed elsewhere and below, a number of these models are not nested and the Planck 2015 data number of degrees of freedom are ambiguous, so in many cases it is not possible to convert the \(\Delta \chi ^{2}\)’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.

  9. In addition to \(\alpha _{\textrm{SN}}\) and \(\beta _{\textrm{SN}}\), the distance moduli of the Pantheon SNIa are affected by three more nuisance parameters, the absolute \(B\)-band magnitude (\(M_{B}\)), the distance correction based on the host-galaxy mass (\(\Delta _{M}\)), and the distance correction based on predicted biases from simulation (\(\Delta _{B}\)) (Scolnic et al. 2017). Consequently, the number of degrees of freedom of the Pantheon sample is less than the number of SNIa. For example, for a flat-\(\varLambda \textrm{CDM}\) model analysis that fits \(\varOmega _{m}\), the number of degrees of freedom becomes 1042 (\(=1048-6\)).

  10. The parameter values of the tilted flat-\(\varLambda \textrm{CDM}\) model constrained by using TT + lowP (+ lensing) + JLA data 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).

  11. Since the flat prior on \(h\) 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).

  12. This is not the case in the tilted flat-\(\varLambda \)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 \(H(z)\), or \(f\sigma _{8}\) measurements. For this model we show only the CMB + SN constraints in Table 1.

  13. In the nonflat \(\varLambda \)CDM model (results mostly not shown here, except for CMB + SN shown in Table 1), \(\varOmega _{b} h^{2}\), \(\varOmega _{c} h^{2}\), and \(\theta _{\textrm{MC}}\) 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 \(\tau \), \(A_{s}\), \(\varOmega _{k}\), \(H_{0}\), \(\varOmega _{m}\), and \(\sigma _{8}\).

  14. These results differ from those of earlier approximate analyses, based on less and less reliable data, that indicated evidence for \(w\) deviating from −1 by more than 3\(\sigma \) (Solà et al. 2017a,b,c,d, 2018; Gómez-Valent and Solà 2017, 2018).

  15. Potential systematic errors that might affect the value of \(H_{0}\), ignored here, have been discussed by Addison et al. (2016) and Aghanim et al. (2017).

  16. This local measurement is 2.9\(\sigma \) (3.3\(\sigma \)), of the quadrature sum of both error bars, higher than \(H_{0}\) measured here using the tilted flat-XCDM (untilted nonflat XCDM) parameterization. Other local expansion rate estimates find slightly lower \(H_{0}\)’s with larger error bars (Rigault et al. 2015; Zhang et al. 2017b; Dhawan et al. 2018; Fernández Arenas et al. 2018).

  17. Our \(\chi ^{2}\) values presented here for the tilted flat-\(\textrm{XCDM}\) model constrained with TT + lowP data are similar to the Planck results (\(\chi _{\textrm{PlikTT}}^{2}=761.9\), \(\chi _{\textrm{lowTEB}}^{2}=10495.14\), \(\chi _{\textrm{prior}}^{2}=1.86\) with total \(\chi ^{2}=11258.91\); Planck Collaboration 2015).

  18. Other effects, including both the usual and integrated Sachs-Wolfe effects, also affect the shape of the low-\(\ell \) \(C_{\ell }\)’s.

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Acknowledgements

We thank D. Scolnic for providing us the Pantheon data. We acknowledge valuable discussions with C. Bennett, J. Ooba, and D. Scolnic. C.-G.P. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1D1A1B03028384). B.R. was supported in part by DOE grant DE-SC0019038.

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  1. Division of Science Education and Institute of Fusion Science, Chonbuk National University, Jeonju, 54896, South Korea

    Chan-Gyung Park

  2. Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS, 66506, USA

    Chan-Gyung Park & Bharat Ratra

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  1. Chan-Gyung Park
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Park, CG., Ratra, B. Observational constraints on the tilted flat-XCDM and the untilted nonflat XCDM dynamical dark energy inflation parameterizations. Astrophys Space Sci 364, 82 (2019). https://doi.org/10.1007/s10509-019-3567-3

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