Abstract
We determine constraints on spatially-flat tilted dynamical dark energy XCDM and \(\phi \)CDM inflation models by analyzing Planck 2015 cosmic microwave background (CMB) anisotropy data and baryon acoustic oscillation (BAO) distance measurements. XCDM is a simple and widely used but physically inconsistent parameterization of dynamical dark energy, while the \(\phi \)CDM model is a physically consistent one in which a scalar field \(\phi \) with an inverse power-law potential energy density powers the currently accelerating cosmological expansion. Both these models have one additional parameter compared to standard \(\varLambda \)CDM and both better fit the TT + lowP + lensing + BAO data than does the standard tilted flat-\(\varLambda \)CDM model, with \(\Delta \chi ^{2} = -1.26\ (-1.60)\) for the XCDM (\(\phi \)CDM) model relative to the \(\varLambda \)CDM model. While this is a 1.1\(\sigma \) (1.3\(\sigma \)) improvement over standard \(\varLambda \)CDM and so not significant, dynamical dark energy models cannot be ruled out. In addition, both dynamical dark energy models reduce the tension between the Planck 2015 CMB anisotropy and the weak lensing \(\sigma _{8}\) constraints.
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Notes
Using a physically consistent non-flat inflation model (Gott 1982; Hawking 1984; Ratra 1985) power spectrum of energy density inhomogeneities (Ratra and Peebles 1995; Ratra 2017) to analyse the Planck 2015 cosmic microwave background (CMB) anisotropy measurements (Ade et al. 2016), Ooba et al. (2018a) find that these data do not require flat spatial hypersurfaces in the six parameter non-flat \(\varLambda \)CDM model (also see Park and Ratra 2019a,b,d). In the non-flat \(\varLambda \)CDM model, compared to the standard flat-\(\varLambda \)CDM model, there is no simple tilt option so \(n_{\mathrm{s}}\) is no longer a free parameter and it is instead replaced by the current value of the spatial curvature energy density parameter \(\varOmega _{\mathrm{k}}\). CMB anisotropy data also do not require flat spatial hypersurfaces in the seven parameter non-flat XCDM and \(\phi \)CDM inflation models (Ooba et al. 2018b,c; Park and Ratra 2018, 2019b,d). In both these models \(n_{\mathrm{s}}\) is again replaced by \(\varOmega _{\mathrm{k}}\). These models differ from the seven parameter spatially-flat XCDM and \(\phi \)CDM inflation models we study in this paper, in which \(n_{\mathrm{s}}\) is a parameter but \(\varOmega _{ \mathrm{k}}\) is not.
While XCDM is often used to model dynamical dark energy, it is not a physically consistent model as it cannot describe the evolution of energy density inhomogeneities. Also, XCDM does not accurately model \(\phi \)CDM dark energy dynamics (Podariu and Ratra 2001).
Aside from CMB anisotropy measurements, many other observations have been used to constrain the \(\phi \)CDM model (see, e.g., Chen and Ratra 2004, 2011b; Samushia et al. 2007; Yashar et al. 2009; Samushia and Ratra 2010; Farooq and Ratra 2013; Pavlov et al. 2014; Avsajanishvili et al. 2015; Farooq et al. 2017; Solà et al. 2017b,c; Solà and de Cruz Pérez 2017; Zhai et al. 2017; Gómez-Valent and Solà 2017; Avsajanishvili et al. 2017; Ryan et al. 2018, 2019; Park and Ratra 2019c,d; Khadka and Ratra 2019).
We thank C.-G. Park for pointing out a numerical error in our initial CMB only \(\phi \)CDM analyses. Our corrected results here are in very good agreement with those of Park and Ratra (2018).
See Penton et al. (2018) for a discussion of how observed deuterium abundances can be used to constrain spatial curvature.
Closed-\(\phi \)CDM is also the best fitting of the three closed models when BAO data is included (Ooba et al. 2018c).
References
Abbott, T.M.C., Abdalla, F.B., Alarcon, A., et al. (DES Collaboration): (2017a). arXiv:1708.01530
Abbott, T.M.C., Abdalla, F.B., Annis, J., et al. (DES Collaboration): (2017b). arXiv:1711.00403
Ade, P.A.R., Aghanim, N., Arnaud, M., et al. (Planck Collaboration): Astron. Astrophys. 594, A13 (2016). arXiv:1502.01589
Anderson, L., Aubourg, É., Bailey, S., et al.: Mon. Not. R. Astron. Soc. 441, 24 (2014). arXiv:1312.4877
Aubourg, É., Bailey, S., Bautista, J.E., et al.: Phys. Rev. D 92, 123516 (2015). arXiv:1411.1074
Audren, B., Lesgourgues, J., Benabed, K., Prunet, S.: J. Cosmol. Astropart. Phys. 1302, 001 (2013). arXiv:1210.7183
Avsajanishvili, O., Samushia, L., Arkhipova, N.A., Kahniashvili, T.: (2015). arXiv:1511.09317
Avsajanishvili, O., Huang, Y., Samushia, L., Kahniashvili, T.: (2017). arXiv:1711.11465
Beutler, F., Blake, C., Colless, M., et al.: Mon. Not. R. Astron. Soc. 416, 3017 (2011). arXiv:1106.3366
Blas, D., Lesgourgues, J., Tram, T.: J. Cosmol. Astropart. Phys. 1107, 034 (2011). arXiv:1104.2933
Brax, P.: Rep. Prog. Phys. 81, 016902 (2018)
Brax, P., Martin, J., Riazuelo, A.: Phys. Rev. D 62, 103505 (2000). arXiv:astro-ph/0005428
Calabrese, E., Archidiacono, M., Melchiorri, A., Ratra, B.: Phys. Rev. D 86, 043520 (2012). arXiv:1205.6753
Cao, S.-L., Duan, X.-W., Meng, X.-L., Zhang, T.-J.: Eur. Phys. J. C 78, 313 (2018). arXiv:1712.01703
Chen, G., Ratra, B.: Astrophys. J. 612, L1 (2004). arXiv:astro-ph/0405636
Chen, G., Ratra, B.: Publ. Astron. Soc. Pac. 123, 1127 (2011a). arXiv:1105.5206
Chen, Y., Ratra, B.: Phys. Lett. B 703, 406 (2011b). arXiv:1106.4294
Chen, Y., Kumar, S., Ratra, B.: Astrophys. J. 835, 86 (2017). arXiv:1606.07316
Ding, X., Biesiada, M., Cao, S., Li, Z., Zhu, Z.-H.: Astrophys. J. 803, L22 (2015). arXiv:1503.04923
Farooq, O., Ratra, B.: Astrophys. J. 766, L7 (2013). arXiv:1301.5243
Farooq, O., Madiyar, F.R., Crandall, S., Ratra, B.: Astrophys. J. 835, 26 (2017). arXiv:1607.03537
Fixsen, D.J.: Astrophys. J. 707, 916 (2009). arXiv:0911.1955
Gómez-Valent, A., Solà, J.: Europhys. Lett. 120, 39001 (2017). arXiv:1711.00692
Gott, J.R.: Nature 295, 304 (1982)
Haridasu, B.S., Luković, V.V., Vittorio, N.: J. Cosmol. Astropart. Phys. 1805, 033 (2018). arXiv:1711.03929
Hawking, S.W.: Nucl. Phys. B 239, 257 (1984)
Khadka, N., Ratra, B.: (2019). arXiv:1909.01400
Lin, W., Ishak, M.: Phys. Rev. D 96, 083532 (2017). arXiv:1708.09813
Lucchin, F., Matarrese, S.: Phys. Rev. D 32, 1316 (1985)
Martin, J.: C. R. Phys. 13, 566 (2012). arXiv:1205.3365
Mitra, S., Choudhury, T.R., Ratra, B.: Mon. Not. R. Astron. Soc. 479, 4566 (2018). arXiv:1712.00018
Mitra, S., Park, C.-G., Choudhury, T.R., Ratra, B.: Mon. Not. R. Astron. Soc. 487, 5118 (2019). arXiv:1901.09927
Mukherjee, P., Banday, A.J., Riazuelo, A., Górski, K.M., Ratra, B.: Astrophys. J. 598, 767 (2003). arXiv:astro-ph/0306147
Ooba, J., Ratra, B., Sugiyama, N.: Astrophys. J. 864, 80 (2018a). arXiv:1707.03452
Ooba, J., Ratra, B., Sugiyama, N.: Astrophys. J. 869, 34 (2018b). arXiv:1710.03271
Ooba, J., Ratra, B., Sugiyama, N.: Astrophys. J. 866, 68 (2018c). arXiv:1712.08617
Park, C.-G., Ratra, B.: Astrophys. J. 868, 83 (2018). arXiv:1807.07421
Park, C.-G., Ratra, B.: Astrophys. J. 882, 158 (2019a). arXiv:1801.00213
Park, C.-G., Ratra, B.: Astrophys. Space Sci. 364, 82 (2019b). arXiv:1803.05522
Park, C.-G., Ratra, B.: Astrophys. Space Sci. 364, 134 (2019c). arXiv:1809.03598
Park, C.-G., Ratra, B.: (2019d). arXiv:1908.08477
Pavlov, A., Westmoreland, S., Saaidi, K., Ratra, B.: Phys. Rev. D 88, 123513 (2013). arXiv:1307.7399
Pavlov, A., Farooq, M., Ratra, B.: Phys. Rev. D 90, 023006 (2014). arXiv:1312.5285
Peebles, P.J.E.: Astrophys. J. 284, 439 (1984)
Peebles, P.J.E., Ratra, B.: Astrophys. J. 325, L17 (1988)
Penton, J., Peyton, J., Zahoor, A., Ratra, B.: Publ. Astron. Soc. Pac. 130, 114001 (2018). arXiv:1808.01490
Podariu, S., Ratra, B.: Astrophys. J. 532, 109 (2001). arXiv:astro-ph/9910527
Ratra, B.: Phys. Rev. D 31, 1931 (1985)
Ratra, B.: Phys. Rev. D 40, 3939 (1989)
Ratra, B.: Phys. Rev. D 45, 1913 (1992)
Ratra, B.: Phys. Rev. D 96, 103534 (2017). arXiv:1707.03439
Ratra, B., Peebles, P.J.E.: Phys. Rev. D 37, 3406 (1988)
Ratra, B., Peebles, P.J.E.: Phys. Rev. D 52, 1837 (1995)
Ratra, B., Vogeley, M.: Publ. Astron. Soc. Pac. 120, 235 (2008). arXiv:0706.1565
Riess, A.G., Casertano, S., Yuan, W., et al.: (2018). arXiv:1801.01120
Ross, A.J., Samushia, L., Howlett, C., et al.: Mon. Not. R. Astron. Soc. 449, 835 (2015). arXiv:1409.3242
Ryan, J., Doshi, S., Ratra, B.: Mon. Not. R. Astron. Soc. 480, 759 (2018). arXiv:1805.06408
Ryan, J., Chen, Y., Ratra, B.: Mon. Not. R. Astron. Soc. 488, 3844 (2019). arXiv:1902.03196
Sahni, V., Shafieloo, A., Starobinsky, A.A.: Astrophys. J. 793, L4 (2014). arXiv:1406.2209
Samushia, L., Ratra, B.: Astrophys. J. 714, 1347 (2010). arXiv:0905.3836
Samushia, L., Chen, G., Ratra, B.: (2007). arXiv:0706.1963
Sievers, J.L., Hlozek, R.A., Nolta, M.R., et al.: J. Cosmol. Astropart. Phys. 1310, 060 (2013). arXiv:1301.0824
Solà, J., de Cruz Pérez: (2017). arXiv:1703.08218
Solà, J., Gómez-Valent, A., de Cruz Pérez, J.: Astrophys. J. 811, L14 (2015). arXiv:1506.05793
Solà, J., Gómez-Valent, A., de Cruz Pérez, J.: Astrophys. J. 836, 43 (2017a). arXiv:1602.02103
Solà, J., Gómez-Valent, A., de Cruz Pérez, J.: Mod. Phys. Lett. A 32, 1750054 (2017b). arXiv:1610.08965
Solà, J., Gómez-Valent, A., de Cruz Pérez, J.: Phys. Lett. B 774, 317 (2017c). arXiv:1705.06723
Solà, J., de Cruz Pérez, J., Gómez-Valent, A.: Europhys. Lett. 121, 39001 (2018). arXiv:1606.00450
Yashar, M., Bozek, B., Abrahamse, A., Albrecht, A., Barnard, M.: Phys. Rev. D 79, 103004 (2009). arXiv:0811.2253
Yu, H., Ratra, B., Wang, F.-Y.: Astrophys. J. 856, 3 (2018). arXiv:1711.03437
Zhai, Z., Blanton, M., Slosar, A., Tinker, J.: Astrophys. J. 850, 183 (2017). arXiv:1705.10031
Zhang, Y.-C., Zhang, H.-Y., Wang, D.-D., et al.: Res. Astron. Astrophys. 17, 6 (2017). arXiv:1703.08293
Zhao, G.-B., Raveri, M., Pogosian, L., et al.: Nat. Astron. 1, 627 (2017). arXiv:1701.08165
Zheng, X., Ding, X., Biesiada, M., Cao, S., Zhu, Z.-H.: Astrophys. J. 825, 17 (2016). arXiv:1604.07910
Acknowledgements
We thank G. Horton-Smith and C.-G. Park for helpful discussions. We thank the referee for comments that helped us improve the paper. This work is supported by Grants-in-Aid for Scientific Research from JSPS (Nos. 16J05446 (J.O.) and 15H05890 (N.S.)). B.R. is supported in part by DOE grant DE-SC0019038.
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Ooba, J., Ratra, B. & Sugiyama, N. Planck 2015 constraints on spatially-flat dynamical dark energy models. Astrophys Space Sci 364, 176 (2019). https://doi.org/10.1007/s10509-019-3663-4
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DOI: https://doi.org/10.1007/s10509-019-3663-4