Modified cosmology models from thermodynamical approach
Abstract
We apply the first law of thermodynamics to the apparent horizon of the universe with the power-law corrected and non-extensive Tsallis entropies rather than the Bekenstein–Hawking one. We examine the cosmological properties in the two entropy models by using the CosmoMC package. In particular, the first numerical study for the cosmological observables with the power-law corrected entropy is performed. We also show that the neutrino mass sum has a non-zero central value with a relaxed upper bound in the Tsallis entropy model comparing with that in the \(\Lambda \)CDM one.
1 Introduction
According to the current cosmological observations, our universe is experiencing a late time accelerating expansion. Although the \(\Lambda \)CDM model can describe the accelerating universe by introducing dark energy [1], it fails to solve the cosmological constant problem, related to the “fine-tuning” [2, 3] and “coincidence” [4, 5] puzzles. A lot of efforts have been made to understand these issues. For example, one can modify the gravitational theory to obtain viable cosmological models with dynamical dark energy to explain the accelerating universe [6].
On the other hand, one can reconstruct the Friedmann equations through the implications of thermodynamics. It has been shown that the Einstein’s equations can be derived by considering the Clausius’ relation of a local Rindler observer [7]. In particular, this idea has been applied to cosmology, while the Friedmann equations have been obtained by using the first law of thermodynamics in the horizon of the universe [8]. It has been also demonstrated that the modified Friedmann equations can be acquired from the thermodynamical approach by just replacing the entropy-area relation with a proper one in a wide variety of gravitational theories [8, 9, 10, 11, 12, 13]. Thus, as long as there is a new entropy area relation, thermodynamics gives us a new way to determine the modified Friedmann equations without knowing the underlying gravitational theory. Furthermore, since the entropy area relation obtained from the modified gravity theory can be useful to extract the dark energy dynamics along with the modified Friedmann equations, it is reasonable to believe that even if we do not know the underlying theory of modified gravity, some modifications of the entropy relation will still give us additional information for modified Friedmann equations as well as the dynamics of dark energy, which would be different from \(\Lambda \)CDM. As a result, we expect that the modification of the entropy is also relevant to the cosmological evolutions.
It is known that a power-law corrected term from the quantum entanglement can be included in the black hole entropy near its horizon [14]. Interestingly, one can apply it to cosmology by taking it as the entropy on the horizon of the universe. On the other hand, the universe is regarded as a non-extensive thermodynamical system, so the Boltmann–Gibbs entropy should be generalized to a non-extensive quantity, the Tsallis entropy, while the standard one can be treated as a limit [15, 16, 17]. The Tsallis entropy has been widely discussed in the literature. In the entropic-cosmology scenario [18], the Tsallis entropy model predicts a decelerating and accelerating universe [19]. In addition, a number of works on the Tsallis holographic dark energy have been proposed and investigated [20, 21, 22, 23, 24]. In addition, the Tsallis entropy has also been used in many different dark energy models, such as the Barboza–Alcaniz and Chevalier–Polarski–Linder parametric dark energy and Wang–Meng and Dalal vacuum decay models [25]. Moreover, it is shown that modified cosmology from the first law of thermodynamics with varying-exponent Tsallis entropy can provide a description of both inflation and late-time acceleration with the same parameter choices [27]. In particular, the Tsallis entropy is proportional to a power of the horizon area, i.e. \(S_T \propto A^{\delta }\), when the universe is assumed to be a spherically symmetric system [28].
Although it is possible to modify Friedmann equations by just considering fluid with an inhomogeneous equation of state of the corresponding form [29], we still choose the thermo-dynaimical approach as that in Ref. [30], in which the authors considered the first law of thermodynamics of the universe with fixed-exponent Tsallis entropy and showed that the cosmological evolution mimics that of \(\Lambda \)CDM and are in great agreement with Supernovae type Ia observational data. In this paper, we examine the features of the modified Friedmann equations obtained by replacing the usual Bekenstein–Hawking entropy-area relation, \(S=A/4G\), with the power-law correction and Tsallis entropies [14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30], where G is the gravitational constant.
This paper is organized as follows. In Sect. 2, we consider the power-law corrected and Tsallis entropy models and derive the modified Friedmann equations and dynamical equation of state parameters by applying the first law of thermodynamics to the apparent horizon of the universe. In Sect. 3, we present the cosmological evolutions of the two models and compare them with those in \(\Lambda \)CDM. Finally, the conclusions are given in Sect. 4. The paper is written in units of \(c=\hbar =k_B=1\).
2 The models
2.1 Power law corrected entropy (PLCE) model
2.2 Tsallis entropy cosmological evolution model
3 Cosmological evolutions
3.1 Power law corrected entropy model
In Fig. 2, we display the CMB power spectra in the \(\Lambda \)CDM and PLCE models along with the data from Planck 2018. Since the TT spectra of PLCE and \(\Lambda \)CDM are almost identical to the data from Planck 2018 for the high values of the multipole l, we focus on the differences between the two models and the data when \(l<100\) as depicted in Fig. 3. The TT power spectrum in the PLCE model for \(\nu > 0\) is larger than that of \(\Lambda \)CDM when \(l < 100\) with the error in the allowable range of the observational data.
3.2 Tsallis entropy cosmological evolution model
Equation (2.7) of TECE becomes the one in \(\Lambda \)CDM when \(\delta =\tilde{\alpha }=1\). In our study, we only focus on the effects when \(\delta \ne 1\) so we set \(\tilde{\alpha }=1\) and \(\delta =1+\xi \). In Fig. 6, we find that the equation of state, \(w_{DE}\), behaves differently for different values of \(\xi \). In particular, it is larger (smaller) than \(-1\) when \(\xi \) is larger (smaller) than zero without crossing \(-1\) in anytime.
H(z) data points
z | H(z) | Ref. | z | H(z) | Ref. | z | H(z) | Ref. | |||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.07 | 69.0 ± 19.6 | [33] | 13 | 0.4 | 95.0 ± 17.0 | [35] | 25 | 0.9 | 117.0 ± 23.0 | [35] |
2 | 0.09 | 69.0 ± 12.0 | [34] | 14 | 0.4004 | 77.0 ± 10.2 | [38] | 26 | 1.037 | 154.0 ± 20.0 | [36] |
3 | 0.12 | 68.6 ± 26.2 | [33] | 15 | 0.4247 | 87.1 ± 11.2 | [38] | 27 | 1.3 | 168.0 ± 17.0 | [35] |
4 | 0.17 | 83.0 ± 8.0 | [35] | 16 | 0.4497 | 92.8 ± 12.9 | [38] | 28 | 1.363 | 160.0 ± 33.6 | [41] |
5 | 0.179 | 75.0 ± 4.0 | [36] | 17 | 0.4783 | 80.9 ± 9.0 | [38] | 29 | 1.43 | 177.0 ± 18.0 | [35] |
6 | 0.199 | 75.0 ± 5.0 | [36] | 18 | 0.48 | 97.0 ± 62.0 | [39] | 30 | 1.53 | 140.0 ± 14.0 | [35] |
7 | 0.2 | 72.9 ± 29.6 | [33] | 19 | 0.57 | 92.4 ± 4.5 | [40] | 31 | 1.75 | 202.0 ± 40.0 | [35] |
8 | 0.27 | 77.0 ± 14.0 | [35] | 20 | 0.5929 | 104.0 ± 13.0 | [36] | 32 | 1.965 | 186.5 ± 50.4 | [41] |
9 | 0.24 | 79.69 ± 2.65 | [37] | 21 | 0.6797 | 92.0 ± 8.0 | [36] | 33 | 2.3 | 224 ± 8 | [42] |
10 | 0.28 | 88.8 ± 36.6 | [33] | 22 | 0.7812 | 105.0 ± 12.0 | [36] | 34 | 2.34 | 222 ± 7 | [43] |
11 | 0.352 | 83.0 ± 14.0 | [36] | 23 | 0.8754 | 125.0 ± 17.0 | [36] | 35 | 2.36 | 226 ± 8 | [44] |
12 | 0.3802 | 83.0 ± 13.5 | [38] | 24 | 0.88 | 90.0 ± 40.0 | [39] |
3.3 Global fits
Priors for cosmological parameters in the PLCE and TECE models
Parameter | Prior |
---|---|
PLCE model parameter \(\nu \) | \(-0.025 \le \nu \le 1.0\) |
TECE model parameter \(\xi \) | \(-0.01 \le \xi \le 0.02\) |
Baryon density | \(0.5 \le 100\Omega _bh^2 \le 10\) |
CDM density | \(0.1 \le 100\Omega _ch^2 \le 99\) |
Optical depth | \(0.01 \le \tau \le 0.8\) |
Neutrino mass sum | \(0 \le \Sigma m_{\nu } \le 2\) eV |
\(\frac{\mathrm {Sound \ horizon}}{\mathrm {Angular \ diameter \ distance}}\) | \(0.5 \le 100 \theta _{MC} \le 10\) |
Scalar power spectrum amplitude | \(2 \le \ln ( 10^{10} A_s ) \le 4\) |
Spectral index | \(0.8 \le n_s \le 1.2\) |
Fitting results for the PLCE and \(\Lambda \)CDM models, where the limits are given at 68\(\%\) and 95\(\%\) C.L., respectively
Parameter | PLCE (68% C.L.) | PLCE (95% C.L.) | \(\Lambda \)CDM (68% C.L.) | \(\Lambda \)CDM (95% C.L.) |
---|---|---|---|---|
\({\varvec{\Omega }}_{{\varvec{ b}}} {{\varvec{ h}}}^{{\varvec{ 2}}} \) | \(0.02237\pm 0.00014 \) | \(0.02237 \pm 0.00027 \) | \(0.02235\pm 0.00014 \) | \(0.02235^{+0.00028}_{-0.00027}\) |
\({\varvec{\Omega }}_{{\varvec{ c}}} {{\varvec{ h}}}^{{\varvec{ 2}}} \) | \(0.1172^{+0.0012}_{-0.0011}\) | \(0.1172^{+0.0022}_{-0.0023}\) | \(0.1173\pm 0.0012 \) | \(0.1173 \pm 0.0023 \) |
\({\varvec{ 100}}{\varvec{\theta }}_{{\varvec{ MC}}} \) | \(1.04101\pm 0.00030 \) | \(1.04101 \pm 0.00059\) | \(1.04100\pm 0.00029 \) | \(1.04100^{+0.00057}_{-0.00058}\) |
\({\varvec{\tau }} \) | \(0.079^{+0.017}_{-0.019} \) | \(0.079^{+0.036}_{-0.034} \) | \(0.078\pm 0.018 \) | \(0.078^{+0.035}_{-0.034} \) |
\({\varvec{\Sigma }} {{\varvec{ m}}}_{\varvec{\nu }} \)/ eV | \(< 0.0982 \) | \(< 0.183 \) | \(< 0.100 \) | \(< 0.195 \) |
\({\varvec{\nu }} \) | \(0.0240^{+0.0110}_{-0.0085} \) | \(0.024^{+0.022}_{-0.033} \) | − | − |
\({{\mathbf{ln}}}(\mathbf{10}^{\mathbf{10}} {\varvec{ A}}_{{\varvec{ s}}})\) | \(3.086^{+0.031}_{-0.035} \) | \(3.086^{+0.068}_{-0.063} \) | \(3.083\pm 0.033 \) | \(3.083^{+0.066}_{-0.064} \) |
\(H_0 \) | \(67.96\pm 0.56 \) | \(68.0 \pm 1.1 \) | \(68.14\pm 0.55 \) | \(68.1 \pm 1.1 \) |
\(\sigma _8 \) | \(0.814^{+0.013}_{-0.011} \) | \(0.814^{+0.023}_{-0.026} \) | \(0.815^{+0.013}_{-0.011} \) | \(0.815^{+0.023}_{-0.025} \) |
\(\chi ^2_{best-fit} \) | 3017.12 | 3018.32 |
Fitting results for the TECE and \(\Lambda \)CDM models, where the limits are given at 68\(\%\) and 95\(\%\) C.L., respectively
Parameter | TECE (68% C.L.) | TECE (95% C.L.) | \(\Lambda \)CDM (68% C.L.) | \(\Lambda \)CDM (95% C.L.) |
---|---|---|---|---|
\({\varvec{\Omega }}_{{\varvec{ b}}} {{\varvec{ h}}}^{{\varvec{ 2}}} \) | \(0.02226\pm 0.00016 \) | \(0.02226^{+0.00033}_{-0.00032}\) | \(0.02236\pm 0.00014 \) | \(0.02236^{+0.00028}_{-0.00027}\) |
\({\varvec{\Omega }}_{{\varvec{ c}}} {{\varvec{ h}}}^{{\varvec{ 2}}} \) | \(0.1174\pm 0.0013 \) | \(0.1174 \pm 0.0025\) | \(0.1173\pm 0.0012 \) | \(0.1173 \pm 0.0023\) |
\({\varvec{ 100}}{\varvec{\theta }}_{{\varvec{ MC}}} \) | \(1.04125\pm 0.00036 \) | \(1.04125^{+0.00073}_{-0.00069}\) | \(1.04099\pm 0.00031 \) | \(1.04099^{+0.00062}_{-0.00060}\) |
\({\varvec{\tau }} \) | \(0.090\pm 0.021 \) | \(0.090 \pm 0.042 \) | \(0.079\pm 0.018 \) | \(0.079^{+0.037}_{-0.035} \) |
\({\varvec{\Sigma }} {{\varvec{ m}}}_{\varvec{\nu }} \)/eV | \(< 0.186 \) | \(< 0.317 \) | \(< 0.107 \) | \(< 0.195 \) |
\({\varvec{\xi }} \) | \(0.00038\pm 0.00027 \) | \(0.00038^{+0.00055}_{-0.00053}\) | − | − |
\({{\mathbf{ln}}}(\mathbf{10}^{\mathbf{10}} {\varvec{ A}}_{{\varvec{ s}}})\) | \(3.103\pm 0.039 \) | \(3.103 \pm 0.076 \) | \(3.085\pm 0.034 \) | \(3.085^{+0.068}_{-0.065} \) |
\(H_0 \) | \(68.42\pm 0.71 \) | \(68.4 \pm 1.4 \) | \(68.05^{+0.60}_{-0.54} \) | \(68.1^{+1.1}_{-1.2} \) |
\(\sigma _8 \) | \(0.814^{+0.017}_{-0.014} \) | \(0.814^{+0.028}_{-0.032} \) | \(0.814^{+0.014}_{-0.012} \) | \(0.814^{+0.024}_{-0.027} \) |
\(\chi ^2_{best-fit} \) | 3018.96 | 3019.28 |
In Fig. 11, we present our fitting results of PLCE (red) and \(\Lambda \)CDM (blue). Although the PLCE model has been discussed in the literature, it is the first time to illustrate its numerical cosmological effects. In particular, we find that \(\nu \) \(=\) \(( 0.0240^{+0.0110}_{-0.0085})\) in 68\(\%\) C.L., which shows that PLCE and \(\Lambda \)CDM can be clearly distinguished. It is interesting to note that the value of \(\sigma _8=0.814^{+0.023}_{-0.026} \) (95\(\%\) C.L.) in PLCE is smaller than that of \(0.815^{+0.023}_{-0.025}\) (95\(\%\) C.L.) in \(\Lambda \)CDM. As shown in Table 3, the best fitted \(\chi ^2\) value in PLCE is 3017.12, which is also smaller than 3018.32 in \(\Lambda \)CDM. Although the cosmological observables for the best \(\chi ^2\) fit in PLCE do not significantly deviate from those in \(\Lambda \)CDM, it indicates that the PLCE model is closer to the observational data than \(\Lambda \)CDM.
Similarly, we show our results for TECE (red) and \(\Lambda \)CDM (blue) in Fig. 12. Explicitly, we get that \(\xi \) = \(( 3.8\pm {2.7})\times 10^{-4}\) in 68\(\%\) C.L. In addition, the TECE model can relax the limit of the total mass of the active neutrinos. In particular, we have that \(\Sigma m_\nu \) \(< 0.317\) eV, comparing to \(\Sigma m_\nu \) \(< 0.195\) eV in \(\Lambda \)CDM at 95\(\%\) C.L. In addition, the value of \(H_0\) in TECE equals to \(68.42\pm {0.71}\) \((68.4\pm {1.4})\), which is larger than \(68.05^{+0.60}_{-0.54}\) \((68.1^{+1.1}_{-1.2})\) in \(\Lambda \)CDM with 68\(\%\) (95\(\%\)) C.L.
As shown in Table 4, the best fitted \(\chi ^2\) value in the TECE model is 3018.96, which is smaller than 3019.28 in \(\Lambda \)CDM model. Although the difference between the value of \(\chi ^2\) in TECE and \(\Lambda \)CDM is not significant, it still implies that the TECE model can not be ignored. Clearly, more considerations and discussions are needed in the future.
4 Conclusions
We have calculated the cosmological evolutions of \(\rho _{DE}\) and \(w_{DE}\) in the PLCE and TECE models. We have found that the EoS of dark energy in PLCE (TECE) does not cross \(-1\). We have shown that the CMB TE power spectrum of the PLCE model with a positive \(\nu \) is closer to the Planck 2018 data than that in \(\Lambda \)CDM, while the CMB TT spectrum in the TECE model has smaller values around \(l\sim 20-27\), which are lower than that in \(\Lambda \)CDM, but close to the data of Planck 2018. By using the Newton method in the global fitting, we have obtained the first numerical result in the PLCE model with \(\nu =0.0240^{+0.0110}_{-0.0085} \) in 68\(\%\) C.L., which can be distinguished well with \(\Lambda \)CDM. Our Fitting results indicate that the PLCE model gives a smaller value of \(\sigma _8\) with a better \(\chi ^2\) value than \(\Lambda \)CDM. In the TECE model, we have gotten that \(\xi =(3.8\pm 2.7)^{-4}\) and \(\Sigma m_\nu \) \(< 0.186\) eV in 68\(\%\) C.L., while \(H_0\) is closer to 70. The best fitted value of \(\chi ^2\) is 3018.96 in the TECE model, which is smaller than 3019.28 in \(\Lambda \)CDM. These results have demonstrated that the TECE model deserves more attention and research in the future.
Notes
Acknowledgements
This work was supported in part by National Center for Theoretical Sciences and MoST (MoST-107-2119-M-007-013-MY3).
References
- 1.L. Amendola, S. Tsujikawa, Dark Energy : Theory and Observations (Cambridge University Press, Cambridge, 2015)zbMATHGoogle Scholar
- 2.S. Weinberg, Rev. Mod. Phys. 61, 1 (1989)ADSCrossRefGoogle Scholar
- 3.S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)Google Scholar
- 4.N. Arkani-Hamed, L.J. Hall, C.F. Kolda, H. Murayama, Phys. Rev. Lett. 85, 4434 (2000)ADSCrossRefGoogle Scholar
- 5.P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003)ADSCrossRefGoogle Scholar
- 6.E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)ADSCrossRefGoogle Scholar
- 7.T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995)ADSMathSciNetCrossRefGoogle Scholar
- 8.R.G. Cai, S.P. Kim, JHEP 0502, 050 (2005)ADSCrossRefGoogle Scholar
- 9.M. Akbar, R.G. Cai, Phys. Lett. B 635, 7 (2006)ADSMathSciNetCrossRefGoogle Scholar
- 10.M. Akbar, R.G. Cai, Phys. Rev. D 75, 084003 (2007)ADSCrossRefGoogle Scholar
- 11.M. Jamil, E.N. Saridakis, M.R. Setare, Phys. Rev. D 81, 023007 (2010)ADSCrossRefGoogle Scholar
- 12.Z.Y. Fan, H. Lu, Phys. Rev. D 91(6), 064009 (2015)ADSMathSciNetCrossRefGoogle Scholar
- 13.Y. Gim, W. Kim, S.H. Yi, JHEP 1407, 002 (2014)ADSCrossRefGoogle Scholar
- 14.S. Das, S. Shankaranarayanan, S. Sur, Phys. Rev. D 77, 064013 (2008)ADSCrossRefGoogle Scholar
- 15.C. Tsallis, J. Stat. Phys. 52, 479 (1988)ADSCrossRefGoogle Scholar
- 16.M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80, 53 (1998)ADSCrossRefGoogle Scholar
- 17.G. Wilk, Z. Wlodarczyk, Phys. Rev. Lett. 84, 2770 (2000)ADSCrossRefGoogle Scholar
- 18.D.A. Easson, P.H. Frampton, G.F. Smoot, Phys. Lett. B 696, 273 (2011)ADSCrossRefGoogle Scholar
- 19.N. Komatsu, S. Kimura, Phys. Rev. D 88, 083534 (2013)ADSCrossRefGoogle Scholar
- 20.E.M.C. Abreu et al., Physica A 441, 141 (2016)ADSMathSciNetCrossRefGoogle Scholar
- 21.S. Ghaffari et al., Eur. Phys. J. C 78, 706 (2018)ADSCrossRefGoogle Scholar
- 22.M.A. Zadeh, A. Sheykhi, H. Moradpour, K. Bamba, Eur. Phys. J. C 78, 940 (2018)ADSCrossRefGoogle Scholar
- 23.E.N. Saridakis, K. Bamba, R. Myrzakulov, F.K. Anagnostopoulos, JCAP 1812, 012 (2018)ADSCrossRefGoogle Scholar
- 24.M. Tavayef, A. Sheykhi, K. Bamba, H. Moradpour, Phys. Lett. B 781, 195 (2018)ADSCrossRefGoogle Scholar
- 25.E.M. Barboza et al., Physica A 436, 301 (2015)ADSMathSciNetCrossRefGoogle Scholar
- 26.M. Abdollahi Zadeh, A. Sheykhi, H. Moradpour, Mod. Phys. Lett. A 34, 1950086 (2019)CrossRefGoogle Scholar
- 27.S. Nojiri, S.D. Odintsov, E.N. Saridakis, Eur. Phys. J. C 79(3), 242 (2019)ADSCrossRefGoogle Scholar
- 28.C. Tsallis, L.J.L. Cirto, Eur. Phys. J. C 73, 2487 (2013)ADSCrossRefGoogle Scholar
- 29.K. Bamba, S. Capozziello, S. Nojiri, S.D. Odintsov, Astrophys. Space Sci. 342, 155 (2012)ADSCrossRefGoogle Scholar
- 30.A. Lymperis, E.N. Saridakis, Eur. Phys. J. C 78, 993 (2018)ADSCrossRefGoogle Scholar
- 31.W.H. Press et al., Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University Press, Cambridge, 2007)zbMATHGoogle Scholar
- 32.A. Lewis, S. Bridle, Phys. Rev. D 66, 103511 (2002)ADSCrossRefGoogle Scholar
- 33.C. Zhang, H. Zhang, S. Yuan, T.J. Zhang, Y.C. Sun, Res. Astron. Astrophys. 14, 1221 (2014)ADSCrossRefGoogle Scholar
- 34.R. Jimenez, L. Verde, T. Treu, D. Stern, Astrophys. J. 593, 622 (2003)ADSCrossRefGoogle Scholar
- 35.J. Simon, L. Verde, R. Jimenez, Phys. Rev. D 71, 123001 (2005)ADSCrossRefGoogle Scholar
- 36.M. Moresco et al., JCAP 1208, 006 (2012)ADSCrossRefGoogle Scholar
- 37.E. Gaztanaga, A. Cabre, L. Hui, Mon. Not. R. Astron. Soc. 399, 1663 (2009)ADSCrossRefGoogle Scholar
- 38.M. Moresco et al., JCAP 1605, 014 (2016)ADSCrossRefGoogle Scholar
- 39.D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, S.A. Stanford, JCAP 1002, 008 (2010)ADSCrossRefGoogle Scholar
- 40.B.A. Reid et al., Mon. Not. R. Astron. Soc. 426, 2719 (2012)ADSCrossRefGoogle Scholar
- 41.M. Moresco, Mon. Not. R. Astron. Soc. 450, L16 (2015)ADSCrossRefGoogle Scholar
- 42.N.G. Busca et al., Astron. Astrophys. 552, A96 (2013)CrossRefGoogle Scholar
- 43.Y. Hu, M. Li, Z. Zhang, arXiv:1406.7695 [astro-ph.CO]
- 44.A. Font-Ribera et al., [BOSS Collaboration], JCAP 1405, 027 (2014)Google Scholar
- 45.R. Adam et al., [Planck Collaboration], Astron. Astrophys. 594, A10 (2016)Google Scholar
- 46.N. Aghanim et al., [Planck Collaboration], Astron. Astrophys. 594, A11 (2016)Google Scholar
- 47.P.A.R. Ade et al., [Planck Collaboration], Astron. Astrophys. 594, A15 (2016)Google Scholar
- 48.C. Heymans et al., Mon. Not. R. Astron. Soc. 432, 2433 (2013)ADSCrossRefGoogle Scholar
- 49.F. Beutler et al., Mon. Not. R. Astron. Soc. 416, 3017 (2011)ADSCrossRefGoogle Scholar
- 50.L. Anderson et al., [BOSS Collaboration], Mon. Not. R. Astron. Soc. 441, 24 (2014)Google Scholar
Copyright information
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP^{3}