# Matter Growth in Imperfect Fluid Cosmology

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## Abstract

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## 1. Introduction

## 2. Energy-Momentum Tensors

## 3. General Conservation Equations

## 4. Perturbation Dynamics

#### 4.1. Metric and Fluid Quantities

#### 4.2. Modified Poisson Equation

#### 4.3. Combination of Conservation and Raychaudhuri Equations

#### 4.4. Relative Perturbations

#### 4.5. Perturbations of (an-)Isotropic Pressures and Energy Flux

#### 4.6. Coupled System of Equations for $\epsilon $ and ${S}_{m}$

#### 4.7. Anisotropic Pressure and Gravitational Potentials

#### 4.8. Matter Perturbations

## 5. ${e}_{\Phi}\Lambda $CDM Cosmology

#### 5.1. Jordan–Brans–Dicke Theory

#### 5.2. JBD Inspired Effective Background Model

#### 5.2.1. General Einstein-Frame Dynamics

#### 5.2.2. Interacting Fluid Approach in Einstein-Frame Dynamics

#### 5.2.3. Effective Hubble Rate

## 6. Growth of Matter Perturbations

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiatti, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, R.P.; et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J.
**1998**, 116, 1009. [Google Scholar] [CrossRef] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Ω and Λ from 42 High-Redshift Supernovae. Astrophys. J.
**1999**, 517, 565. [Google Scholar] [CrossRef] - Bahcall, N.A.; Ostriker, J.P.; Perlmutter, J.P.; Steinhardt, P.J. The cosmic triangle: Revealing the state of the universe. Science
**1999**, 284, 1481–1488. [Google Scholar] [CrossRef] - Amendola, L.; Tsujikawa, S. Dark Energy: Theory and Observations; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Ellis, G.F.R.; Maartens, R.; Maccallum, M.A.H. Relativistic Cosmology; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Martin, J. Everything you always wanted to know about the cosmological constant problem (but were afraid to ask). C. R. Phys.
**2012**, 13, 566–665. [Google Scholar] [CrossRef] [Green Version] - Zlatev, I.; Wang, L.M.; Steinhardt, P.J. Quintessence, cosmic coincidence, and the cosmological constant. Phys. Rev. Lett.
**1999**, 82, 896. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Wang, L.M.; Zlatev, I. Cosmological tracking solutions. Phys. Rev. D
**1999**, 59, 123504. [Google Scholar] [CrossRef] [Green Version] - Malquarti, M.; Copeland, E.J.; Liddle, A.R. k-essence and the coincidence problem. Phys. Rev. D
**2003**, 68, 023512. [Google Scholar] [CrossRef] - Barreira, A.; Avelino, P.P. Anthropic versus cosmological solutions to the coincidence problem. Phys. Rev. D
**2011**, 83, 103001. [Google Scholar] [CrossRef] - Velten, H.E.S.; vom Marttens, R.F.; Zimdahl, W. Aspects of the cosmological “coincidence problem”. Eur. Phys. J. C
**2014**, 74, 3160. [Google Scholar] [CrossRef] - Akrami, Y.; et al. [Planck Collaboration] Planck 2018 results. I. Overview and the cosmological legacy of Planck. arXiv, 2018; arXiv:1807.06205. [Google Scholar]
- Abbott, T.M.C.; et al. [DES Collaboration] Dark Energy Survey Year 1 Results: Constraints on Extended Cosmological Models from Galaxy Clustering and Weak Lensing. arXiv, 2018; arXiv:1810.02499. [Google Scholar]
- Bull, P.; Camera, S.; Kelley, K.; Padmanabhan, H.; Pritchard, J.; Raccanelli, A.; Riemer-Sørensen, S.; Shao, L.; Andrianomena, S.; Athanassoula, E.; et al. Fundamental Physics with the Square Kilometre Array. arXiv, 2018; arXiv:1810.02680. [Google Scholar]
- Abbott, B.P.; et al. [LIGO Scientific Collaboration and Virgo Collaboration] GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett.
**2017**, 119, 161101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Madsen, M.S. Scalar fields in curved spacetimes. Class. Quantum Grav.
**2017**, 5, 627. [Google Scholar] [CrossRef] - Pimentel, L.O. Energy-momentum tensor in the general scalar-tensor theory. Class. Quantum Grav.
**1989**, 6, L263. [Google Scholar] [CrossRef] - Battye, R.A.; Pearson, J.A. Effective action approach to cosmological perturbations in dark energy and modified gravity. J. Cosmol. Astropart. Phys.
**2012**, 2012, 019. [Google Scholar] [CrossRef] - Battye, R.A.; Pearson, J.A. Parametrizing dark sector perturbations via equations of state. Phys. Rev. D
**2013**, 88, 061301(R). [Google Scholar] [CrossRef] - Battye, R.A.; Bolliet, B.; Pearson, J.A. f(R) gravity as a dark energy fluid. Phys. Rev. D
**2016**, 93, 044026. [Google Scholar] [CrossRef] - Faraoni, V.; Coté, J. Imperfect fluid description of modified gravities. Phys. Rev. D
**2018**, 98, 084019. [Google Scholar] [CrossRef] - Sawicki, I.; Saltas, I.D.; Amendola, L.; Kunz, M. Consistent perturbations in an imperfect fluid. J. Cosmol. Astropart. Phys.
**2013**, 2013, 004. [Google Scholar] [CrossRef] - Nesseris, S.; Perivolaropoulos, S. Testing LCDM with the Growth Function δ(a): Current Constraints. Phys. Rev. D
**2008**, 77, 023504. [Google Scholar] [CrossRef] - Basilakos, S.; Pouri, A. The growth index of matter perturbations and modified gravity. Mon. Not. R. Astron. Soc.
**2008**, 423, 3761–3767. [Google Scholar] [CrossRef] - Huterer, D.; Kirkby, D.; Bean, R.; Connolly, A.; Dawson, K.; Dodelson, S.; Evrard, A.; Jain, B.; Jarvis, M.; Linder, E.; et al. Growth of Cosmic Structure: Probing Dark Energy Beyond Expansion. Astropart. Phys.
**2015**, 63, 23–41. [Google Scholar] [CrossRef] - Nesseris, S.; Sapone, D. Accuracy of the growth index in the presence of dark energy perturbations. Phys. Rev. D
**2015**, 92, 023013. [Google Scholar] [CrossRef] - Alam, S.; Ata, M.; Bailey, S.; Beutler, F.; Bizyaev, D.; Blazek, J.A.; Bolton, A.S.; Brownstein, J.R.; Burden, A.; Chuang, C.-H.; et al. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Cosmological analysis of the DR12 galaxy sample. Mon. Not. R. Astron. Soc.
**2017**, 470, 2617–2652. [Google Scholar] [CrossRef] - Algoner, W.C.; Velten, H.E.S.; Zimdahl, W. Scalar-tensor extension of the ΛCDM model. J. Cosmol. Astropart. Phys.
**2016**, 2016, 034. [Google Scholar] [CrossRef] - Zimdahl, W.; Velten, H.E.S.; Algoner, W.C. Matter growth in extended ΛCDM cosmology. arXiv, 2017; arXiv:1706.06143. [Google Scholar] [CrossRef]
- Hipólito-Ricaldi, W.S.; Velten, H.E.S.; Zimdahl, W. Non-adiabatic dark fluid cosmology. J. Cosmol. Astropart. Phys.
**2009**, 2009, 016. [Google Scholar] [CrossRef] - Hipólito-Ricaldi, W.S.; Velten, H.E.S.; Zimdahl, W. Viscous dark fluid universe. Phys. Rev. D
**2010**, 82, 063507. [Google Scholar] [CrossRef] - del Campo, S.; Fabris, J.C.; Herrera, J.C.; Zimdahl, W. Cosmology with Ricci dark energy. Phys. Rev. D
**2013**, 87, 123002. [Google Scholar] [CrossRef] - Romero Fuño, A.; Hipólito-Ricaldi, W.S.; Zimdahl, W. Matter perturbations in scaling cosmology. Mon. Not. R. Astron. Soc.
**2016**, 457, 2958–2967. [Google Scholar] [CrossRef] [Green Version] - Kunz, M.; Sapone, M. Dark Energy versus Modified Gravity. Phys. Rev. Lett.
**2007**, 98, 121301. [Google Scholar] [CrossRef] [PubMed] - Cardona, W.; Hollenstein, L.; Kunz, M. The traces of anisotropic dark energy in light of Planck. J. Cosmol. Astropart. Phys.
**2014**, 2014, 032. [Google Scholar] [CrossRef] - Blas, D.; Floerchinger, S.; Garny, M.; Tetradis, N.; Wiedemann, U.A. Large scale structure from viscous dark matter. J. Cosmol. Astropart. Phys.
**2015**, 2015, 049. [Google Scholar] [CrossRef] - Jordan, P. Zum gegenwärtigen Stand der Diracschen kosmologischen Hypothesen. Z. Physik
**1959**, 157, 112–121. [Google Scholar] [CrossRef] - Brans, C.; Dicke, R.H. Mach’s principle and a relativistic theory of gravitation. Phys. Rev.
**1961**, 124, 925. [Google Scholar] [CrossRef] - Dicke, R.H. Mach’s principle and invariance under transformation of units. Phys. Rev.
**1962**, 125, 2163. [Google Scholar] [CrossRef] - Agarwal, N.; Bean, R. The Dynamical viability of scalar-tensor gravity theories. Class. Quantum Grav.
**2008**, 25, 165001. [Google Scholar] [CrossRef] - Batista, C.E.M.; Zimdahl, W. Power-law solutions and accelerated expansion in scalar-tensor theories. Phys. Rev. D
**2010**, 82, 023527. [Google Scholar] [CrossRef] - Clifton, T.; Ferreira, P.; Padilla, A.; Skordis, C. Modified Gravity and Cosmology. Phys. Rep.
**2012**, 513, 1–189. [Google Scholar] [CrossRef] - Chiba, T.; Yamaguchi, M. Conformal-Frame (In)dependence of Cosmological Observations in Scalar-Tensor Theory. J. Cosmol. Astropart. Phys.
**2013**, 2013, 040. [Google Scholar] [CrossRef] - Song, Y.-S.; Percival, W.J. Reconstructing the history of structure formation using redshift distortions. J. Cosmol. Astropart. Phys.
**2009**, 2009, 009. [Google Scholar] [CrossRef] - Nesseris, S.; Pantazis, G.; Perivolaropoulos, L. Tension and constraints on modified gravity parametrizations of Geff(z) from growth rate and Planck data. Phys. Rev. D
**2017**, 96, 023542. [Google Scholar] [CrossRef] [Green Version] - Gil-Marín, H.; Guy, J.; Zarrouk, P.; Burtin, E.; Chuang, C.-H.; Percival, W.J.; Ross, A.J.; Ruggeri, R.; Tojerio, R.; Zhao, G.-B.; et al. The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey DR14 quasar sample: Structure growth rate measurement from the anisotropic quasar power spectrum in the redshift range 0.8<z<2.2. Mon. Not. R. Astron. Soc.
**2018**, 477, 1604–1638. [Google Scholar] - Hou, J.; Sánchez, A.G.; Scoccimarro, R.; Salazar-Albornoz, S.; Burtin, E.; Gil-Marín, H.; Percival, W.J.; Ruggeri, R.; Zarrouk, P.; Zhao, G.-B.; et al. The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey DR14 quasar sample: Anisotropic clustering analysis in configuration-space. Mon. Not. R. Astron. Soc.
**2018**, arXiv:1801.02656480, 2521–2534. [Google Scholar] [CrossRef] - Zhao, G.-B.; Wang, Y.; Saito, S.; Gil-Marín, H.; Percival, W.J.; Wang, D.; Chuang, C.-H.; Ruggeri, R.; Mueller, E.-M.; Zhu, F.; et al. The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey DR14 quasar sample: A tomographic measurement of cosmic structure growth and expansion rate based on optimal redshift weights. Mon. Not. R. Astron. Soc.
**2018**, 482, 3497–3513. [Google Scholar] [CrossRef]

**Figure 1.**Matter fraction ${\Omega}_{m}$ and geometric energy fraction ${\Omega}_{x}$ for negative (left) and positive (right) values of m.

**Figure 2.**Total EoS parameter $w=\frac{p}{\rho}$ for various negative (left) and positive (right) values of m.

**Figure 3.**Dependence of $f{\sigma}_{8}\left(z\right)$ on z for a non-vanishing sound-speed parameter ($m=\alpha =\beta =\mu =\nu =0$).

**Figure 4.**Matter growth for various values of the parameter m with $\alpha =\beta =\mu =\nu ={c}_{\epsilon}^{2}=0$.

**Figure 5.**Dependence of $f{\sigma}_{8}\left(z\right)$ on z for different values of the parameter m with $\alpha =\beta =\mu =\nu ={c}_{\epsilon}^{2}=0$.

**Figure 6.**Matter growth in the presence of anisotropic stresses ($m=\alpha =\beta =\nu ={c}_{\epsilon}^{2}=0$).

**Figure 7.**Dependence of $f{\sigma}_{8}\left(z\right)$ on z in the presence of anisotropic stresses ($m=\alpha =\beta =\nu ={c}_{\epsilon}^{2}=0$).

**Figure 8.**Dependence of $f{\sigma}_{8}\left(z\right)$ on z in the presence of heat fluxes ($m=\beta =\mu =\nu ={c}_{\epsilon}^{2}=0$).

Index | Data set | z | $\mathit{f}{\mathit{\sigma}}_{8}\left(\mathit{z}\right)$ | Year | Notes |
---|---|---|---|---|---|

1 | 6dFGS + SnIa | 0.02 | 0.428 ± 0.0465 | 2016 | (${\Omega}_{0m}$, h, ${\sigma}_{8}$) = (0.3, 0.683, 0.8) |

2 | SnIa + IRAS | 0.02 | 0.398 ± 0.065 | 2011 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.3, 0) |

3 | 2MASS | 0.02 | 0.314 ± 0.048 | 2010 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.266, 0) |

4 | SDSS-veloc | 0.10 | 0.370 ± 0.130 | 2015 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.3, 0) |

5 | SDSS-MGS | 0.15 | 0.490 ± 0.145 | 2014 | (${\Omega}_{0m}$, h, ${\sigma}_{8}$) = (0.31, 0.67, 0.83) |

6 | 2dFGRS | 0.17 | 0.510 ± 0.060 | 2009 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.3, 0) |

7 | GAMA | 0.18 | 0.360 ± 0.090 | 2013 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.27, 0) |

8 | GAMA | 0.38 | 0.440 ± 0.060 | 2013 | |

9 | SDSS-LRG-200 | 0.25 | 0.3512 ± 0.0583 | 2011 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.25, 0) |

10 | SDSS-LRG-200 | 0.37 | 0.4602 ± 0.0378 | 2011 | |

11 | BOSS-LOWZ | 0.32 | 0.384 ± 0.095 | 2013 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.274, 0) |

12 | SDSS-CMASS | 0.59 | 0.488 ± 0.060 | 2013 | (${\Omega}_{0m}$, h, ${\sigma}_{8}$) = (0.307115, 0.6777, 0.8288) |

13 | WiggleZ | 0.44 | 0.413 ± 0.080 | 2012 | (${\Omega}_{0m}$, h) = (0.27, 0.71) |

14 | WiggleZ | 0.60 | 0.390 ± 0.063 | 2012 | |

15 | WiggleZ | 0.73 | 0.437 ± 0.072 | 2012 | |

16 | Vipers PDR-2 | 0.60 | 0.550 ± 0.120 | 2016 | (${\Omega}_{0m}$, ${\Omega}_{b}$) = (0.3, 0.045) |

17 | Vipers PDR-2 | 0.86 | 0.400 ± 0.110 | 2016 | |

18 | FastSound | 1.40 | 0.482 ± 0.116 | 2015 | (${\Omega}_{0m}$, ${\Omega}_{K}$) = (0.270, 0) |

19 | SDSS-IV | 1.52 | 0.420 ± 0.076 | 2018 | (${\Omega}_{0m}$, ${\Omega}_{b}{h}^{2}$, ${\sigma}_{8}$) = (0.26479, 0.02258, 0.8) |

20 | SDSS-IV | 1.52 | 0.396 ± 0.079 | 2018 | (${\Omega}_{0m}$,${\Omega}_{b}{h}^{2}$, ${\sigma}_{8}$) = (0.31, 0.022, 0.8225) |

21 | SDSS-IV | 1.23 | 0.385 ± 0.099 | 2018 | (${\Omega}_{0m}$, ${\sigma}_{8}$) = (0.31, 0.8) |

22 | SDSS-IV | 1.526 | 0.342 ± 0.070 | 2018 | |

23 | SDSS-IV | 1.944 | 0.364 ± 0.106 | 2018 |

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**MDPI and ACS Style**

Zimdahl, W.; Velten, H.E.S.; Algoner, W.C.
Matter Growth in Imperfect Fluid Cosmology. *Universe* **2019**, *5*, 68.
https://doi.org/10.3390/universe5030068

**AMA Style**

Zimdahl W, Velten HES, Algoner WC.
Matter Growth in Imperfect Fluid Cosmology. *Universe*. 2019; 5(3):68.
https://doi.org/10.3390/universe5030068

**Chicago/Turabian Style**

Zimdahl, Winfried, Hermano E.S. Velten, and William C. Algoner.
2019. "Matter Growth in Imperfect Fluid Cosmology" *Universe* 5, no. 3: 68.
https://doi.org/10.3390/universe5030068