# Visualization of cosmological density fluctuations with phase space analysis: case study: Brans–Dicke theory

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## Abstract

Cosmological perturbation theory is a powerful tool to understanding the large-scale structure of the Universe. However, the set of field equations describes the general linear perturbations for the cosmological models is highly nonlinear and coupled where no analytical solution can be found. It is only after some simplification and numerical computation that we obtain limited solutions. On way around is to employ the phase space analysis and investigate the asymptotic stability of the model. The advantage of using this approach is that the system of field equations become simpler to solve numerically. The algorithm also determines the stability of the system. Here, we apply this approach in Brans–Dicke cosmology and study the attractor solutions after fitting the model with the SNeIa observational data. As a result, the model confirms small fluctuations of energy density. The model also predicts current universe acceleration confirmed by observation and benefits from phase space analysis that the new dynamical variables are independent of the model initial conditions.

## 1 Introduction

Over the past few decades, extensive research has been conducted in order to understand the formation of large scale structures in the universe. More recently, a number of surveys such as the Sloan Digital Sky Survey (SDSS) and the Two-Degree Field (2dF) have have been used to measure the redshifts of millions of galaxies and plot the three-dimensional distribution of matter in the universe. It has been found that smoothing the distribution on the largest scales (larger than several hundred Mpc scales) displays a homogeneous distribution of matter, in accordance with the assumptions that the universe in the large scale is homogeneous and isotropic. At smaller scales, however, there exists density fluctuations; over-dense and under-dense regions like clusters and voids which increase at smaller smoothing scales. In terms of density fluctuation amplitude, the rms is over order unity on average \(\sim \) 10 Mpc scales. Noting that in the past two decades, the cold dark matter (CDM) and dark energy ideas have also contributed into the standard theoretical models of galaxy formations.

A fundamental assumption in structure formation is that it started to grow from weak density fluctuations in the early universe, then, amplified by gravity and eventually turned into the current structure formation. Measurements of the cosmic microwave background (CMB) by the WMAP satellite in combination with the 2dFGRS confirm the assumptions and allow an accurate determination of the geometry and matter content of the Universe about 380000 years after the Big Bang [1, 2, 3, 4].

The perturbation models can in general be divide into scalar, vector and tensor perturbations, based on how they transform under spatial rotations and local time shifts. Based on these, one then study the evolution of small perturbations of homogeneous metric and matter fields. Scalar perturbation models have been widely studied in fourth order gravity mainly in the metric formalism [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. These models originally developed for General relativity by Bardeen [15] and then expanded in the \(1+3\) covariant approach [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Many attempts have been implemented to numerically compute the matter power spectrum and its numerical analysis for classes of modified gravity theories, rather solving the full set of field equations [30, 31, 32, 33]. Modifications to the linear order Einstein equations are thus introduced in terms of general functions of scale and time. In general, all studies have proved that the perturbation’s growth depends on the scale. While in General Relativity the evolution of dust matter is scale-independent [18], the numerical investigation of perturbation theories shows that the matter perturbation growth is scale - dependent [34, 35].

Perturbations in scalar-tensor theories have been considered from different approaches, for example: the reconstruction problem was addressed in [36], the density contrast evolution was studied in [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 37, 38] and the second order perturbations were presented in [39]. In all these approaches the gravitational instability is a fundamental assumption. That is, the early universe has been almost perfectly smooth, with the exception of tiny density deviations with respect to the global cosmic background density and the accompanying tiny velocity perturbations from the general Hubble expansion. The minor density deviations vary from location to location. At one place the density will be slightly higher than the average global density, while a few Mega parsecs further the density may have a slightly smaller value than the average. The observed fluctuations in the temperature of the CMBR are a reflection of these density perturbations, such that the primordial density perturbations have been in the order of \(10^{-5}\). Noting that the origin of this density perturbation field has as yet not been fully understood. The most plausible theory is that the density perturbations are as a result of quantum fluctuations in the very early Universe which during the inflationary phase expanded to macroscopic proportions.

In this work, we consider scalar metric perturbation. We also assume that the matter is made up of a single real scalar field. In the case of General Theory of Relativity that the field equations are extremely coupled and nonlinear, the perturbation become more complicated in which analytical solutions of the governing field equations are impossible to solve or give little understanding about the behaviour of the system. Our novel approach to this problem is to utilise phase space and stability analysis as a fascinating technique to study the dynamics of the system. We apply the technique in the Brans–Dicke (BD) theory as an alternative to General Relativity.

Noting that the gravitational constant in Brans Dicke theory is non-constant. It dependents on the scalar field where its contribution to the Lagrangian density is through its own kinetic term. The evolution of the scalar field as a source term affects the matter distribution in the universe. This in turn makes the gravitational constant dependent of the matter field as well. This provides a manifestation of the Mach’s principle as interpreted by Dicke. (see [40] for a recent review). The authors in [41, 42, 43] have extensively studies different formulation of Brans Dicke theories and their applications in modern cosmology

The authors in [44] have already studied the evolution of the density contrast for Jordan-Fierz–Brans–Dicke theories in a Friedmann–Lemaitre–Robertson–Walker Universe. Following their work , we are going to investigate the density perturbation of the model based on the phase plane analysis. In particular, we focus on the late time stable solutions and give a qualitative description of the model. This has been done by using proper transformations in which the complicated equations become simplified. In general, there are two approach to examine the stability analysis of the model by simultaneously solving the dynamical system and best fitting the stability parameters with the observational data.

1-we find the critical points and their eigenvalues in terms of parameters of the model, find the stability conditions for each critical point and then constrain the parameters with observation.

2-We first constrain the model parameters and initial conditions with observation. Then find the critical points and eigenvalues for the best fitted parameters and investigate the stability of the model [41, 45].

This approach is more appropriate for complex systems that finding the critical points and eigenvalues in terms of model parameters is difficult. This is the approach we followed in this paper

Our work is organized as follows: In Sect. 2, Brans–Dicke theory has been extracted from Scalar-tensor theories of gravity, we summarize the background field equations of BD theories in both Jordan and Einstein frame. In Sect. 3 following [44], we implement the metric perturbation and drive the perturbed equations. In Sect. 4, we best fit our model with the observational data. In Sect. 6, phase space analysis of the model is discussed. The perturbed second order, coupled and nonlinear equations are transformed into simple ones by introducing new dynamical variables. The stability of the model and attractor solutions are discussed. Section is devoted to discuss the solution in more details. We further derive equation of state parameter in terms of the best fitted stability and model parameters and test the model against observational data. Section 7 is the summary.

## 2 Extracting Brans–Dicke theory from Scalar-tensor theories of gravity

*R*the scalar curvature of \(g_{\mu \nu }\), and

*g*its determinant. The variation of action (1) gives,

## 3 Perturbations in BD theories

*v*is the potential for velocity perturbations. In the following, we also assume that both perturbed and unperturbed matter obey the same equation of state.

*ij*) components which yields \(\partial (\Phi -\Psi )=0\) and indicates that both scalar potentials are equal as in GR. We finally yield the following set of equations in Fourier space:

## 4 Constraining the cosmological parameters

The Eqs. (31)–(36) describe the general linear perturbations for the model. The differential equations are second order, nonlinear and coupled in a large number of variables and are impossible to solve them analytically. Our purpose in this section is first to simplify these perturbed equations without losing the generality by converting the second order equations to the first order ones. This makes the equations simpler to solve numerically and and prepare them for phase space analysis in the next section.

The phase space is in particular useful in visualizing the behavior of the dynamical systems and studying the stability of the system.

*k*is replaced by \(\chi _{2}\mathcal {H}\).

*m*of a supernova that is related to the distance modulus \(\mu \) and luminosity distance \(d_L\) of the supernova by

*M*is the absolute magnitude of the supernova. In flat FRW cosmology, the luminosity distance is given by

In the following section we use the best fitted value of \(H_0\) and \(\alpha \) to conduct the stability analysis in the phase space for our model.

## 5 Phase space analysis

Critical points for the perturbed system

| \(\chi _{2}\) | \(\chi _{3}\) | \(\chi _{4}\) | \(\chi _{5}\) | \(\chi _{6}\) | \(\chi _{7}\) | \(\chi _{8}\) |
---|---|---|---|---|---|---|---|

\(P_1\) | 0 | 0.4 | 0 | \(-\) 0.3 | 3 | 0 | 0 |

\(P_2\) | 0 | 0.6 | 0 | 0.3 | 3 | 0 | 0 |

\(P_3\) | 0 | 1.2 | 0 | 0.6 | 3 | 0 | 0 |

\(P_4\) | 0 | \(-\) 0.6 | 0 | \(-\) 0.3 | 3 | 0 | 0 |

\(P_5\) | 0 | \(-\) 0.9 | 0 | 0.7 | 3 | 0 | 0 |

\(P_6\) | 0 | \(-\) 2 | 0 | 1.5 | 3 | 0 | 0 |

\(P_7\) | 0 | 0.5 | 0 | \(-\) .1 | 30.2 | 0 | 0 |

\(P_8\) | 0 | 2 | 0 | 8.8 | 1.1 | 0 | 0 |

\(P_9\) | 0 | \(-\) 0.5 | 0 | \(-\) 0.1 | 23 | 0 | 0 |

\(P_{10}\) | 0 | \(-\) 1.5 | 0 | \(-\) 5.1 | 0.2 | 0 | 0 |

\(P_{11}\) | 0 | \(-\) 21.3 | 0 | 1 | 452.1 | 0 | 0 |

\(P_{12}\) | 19.8 | 20 | 0 | 78.1 | .1 | 0 | 0 |

\(P_{13}\) | \(-\) 19.8 | 20 | 0 | 78.1 | .1 | 0 | 0 |

*z*for both best fitted and arbitrary initial conditions.The graphs show that the solutions are independent of the initial conditions. This is because, in the infinite time limit, the system approaches the attractor, independent of the initial conditions.

## 6 Discussion

## 7 Conclusion

This paper is devoted to study the conventional metric perturbation in the Brans–Dicke thoery using the phase space algorithm. in practice, we integrated two different perturbation model; the standard metric perturbation and the perturbation based on phase space analysis of the newly constructed variables from the original geometric, matter and other fields. The conventional scalar perturbation in the metric introduces the scalar field, \(\Phi \), that together with the brans-Dicke scalar field \(\varphi \), the matter density \(\rho \) and Hubble parameter constitute the model that can potentially explain the large scale structure of the universe.

In phase space formalism, the newly constructed variables are used to manifest the attractor characteristic of the model. It is shown that the scalar perturbation of \(\Phi \), \(\delta \), \(\upsilon \) and \(\varphi \), strongly depend on the initial conditions while the newly constructed variables are independent of them. By infinitesimal perturbation of these new variables, the solutions approach the same state corresponding to the attractor state. Noting that for those initial conditions largely different from the one that create the attractor point, the solutions will be represented as an oscillation with large amplitude while for the initial conditions close to the one that create attractor point, the oscillation is with small amplitude.

In standard metric perturbation, the complexity and nonlinearity of equations in addition to exceeding the number of unknown parameter’s, and increasing the sensitivity of the solutions to the initial conditions make it difficult to to interpret the large scale structure of the universe. However, its application in phase space creates a novel mechanism that simplifies both the complexity of the problem and system analysis. here, we also study the stability of the system and visualise the behaviour of the dynamical variable. More importantly, by fitting the parameters of the model, we supported our result with observational data. Though, non of the original variables in metric perturbation or newly constructed variables in phase space are directly or indirectly observable, the EoS parameter resulted from model fitting is in good match with observational data. Also It is important to note that while phase space analysis provides a qualitative description of the model, the fitting algorithm create some quantitative variables compatible with observation.

## References

- 1.P.J.E. Peebles, Astron. J.
**75**, 13 (1970)ADSCrossRefGoogle Scholar - 2.
- 3.S.D.M. White, Mon. Not. R. Astron. Soc
**177**, 717 (1976)ADSCrossRefGoogle Scholar - 4.S.J. Aarseth, E.L. Turner, J.R. Gott, Astrophys. J.
**228**, 664 (1979)ADSCrossRefGoogle Scholar - 5.J. Matsumoto, Phys. Rev. D
**87**, 104002 (2013)ADSCrossRefGoogle Scholar - 6.R. Gannouji, B. Moraes, D. Polarski, JCAP
**0902**, 034 (2009)ADSCrossRefGoogle Scholar - 7.D. Polarski, R. Gannouji, Phys. Lett. B
**660**, 439 (2008)ADSCrossRefGoogle Scholar - 8.
- 9.A. Hojjati, L. Pogosian, A. Silvestri, S. Talbot, Phys Rev. D
**86**, 0123503 (2012)ADSCrossRefGoogle Scholar - 10.P. Zhang, Phys Rev. D
**73**, 123504 (2006)ADSCrossRefGoogle Scholar - 11.G.B. Zhao, L. Pogosian, A. Silvestri, J. Zylberberg, Phys Rev. D
**79**, 083513 (2009)ADSCrossRefGoogle Scholar - 12.T. Giannantonio, M. Martinelli, A. Silvestri, A. Melchiorri, JCAP
**1004**, 030 (2010)ADSCrossRefGoogle Scholar - 13.R. Bean, D. Bernat, L. Pogosian, A. Silvestri, M. Trodden, Phys. Rev. D
**75**, 064020 (2007)ADSMathSciNetCrossRefGoogle Scholar - 14.A. de la Cruz-Dombriz, A. Dobado, A.L. Maroto, Phys. Rev. D
**77**, 123515 (2008)ADSCrossRefGoogle Scholar - 15.J.M. Bardeen, Phys. Rev. D
**22**, 1882 (1980)ADSMathSciNetCrossRefGoogle Scholar - 16.S. Carloni, P.K.S. Dunsby, A. Troisi, Phys. Rev. D
**77**, 024024 (2008)ADSMathSciNetCrossRefGoogle Scholar - 17.K.N. Ananda, S. Carloni, P.K.S. Dunsby, Phys. Rev. D
**77**, 024033 (2008)ADSCrossRefGoogle Scholar - 18.K.N. Ananda, S. Carloni, P.K.S. Dunsby, Class. Quant. Grav.
**26**, 235018 (2009)ADSCrossRefGoogle Scholar - 19.A. Abebe, M. Abdelwahab, A. de la Cruz-Dombriz, P.K.S. Dunsby, Class. Quant. Grav.
**29**, 135011 (2012)ADSCrossRefGoogle Scholar - 20.F. Perrotta, C. Baccigalupi, Nucl. Phys. Proc. Suppl.
**124**, 72 (2003)ADSCrossRefGoogle Scholar - 21.S. Tsujikawa, Phys. Rev. D
**76**, 023514 (2007)ADSCrossRefGoogle Scholar - 22.F. Perrotta, S. Matarrese, M. Pietroni, C. Schimd, Phys. Rev. D
**69**, 084004 (2004)ADSCrossRefGoogle Scholar - 23.F. Perrotta, C. Baccigalupi, Phys. Rev. D
**59**, 123508 (1999)ADSCrossRefGoogle Scholar - 24.R. Gannouji, D. Polarski, JCAP
**0805**, 018 (2008)ADSCrossRefGoogle Scholar - 25.J.C.B. Sanchez, L. Perivolaropoulos, Phys. Rev. D
**81**, 103505 (2010)ADSCrossRefGoogle Scholar - 26.A. De Felice, T. Kobayashi, S. Tsujikawa, Phys. Lett. B
**706**, 123 (2011)ADSCrossRefGoogle Scholar - 27.A. De Felice, S. Mukohyama, S. Tsujikawa, Phys. Rev. D
**82**, 023524 (2010)ADSCrossRefGoogle Scholar - 28.A. De Felice, R. Kase, S. Tsujikawa, Phys. Rev. D
**83**, 043515 (2011)ADSCrossRefGoogle Scholar - 29.P. Brax, P. Valageas, Phys. Rev. D
**86**, 063512 (2012)ADSCrossRefGoogle Scholar - 30.A. Hojjati, L. Pogosian, G.-B. Zhao, JCAP
**1108**, 005 (2011)ADSCrossRefGoogle Scholar - 31.E. Bertschinger, P. Zukin, Phys. Rev. D
**78**, 024015 (2008)ADSCrossRefGoogle Scholar - 32.R. Bean, M. Tangmatitham, Phys. Rev. D
**81**, 083534 (2010)ADSCrossRefGoogle Scholar - 33.E.V. Linder, R.N. Cahn, Astropart. Phys.
**28**, 481 (2007)ADSCrossRefGoogle Scholar - 34.A. de la Cruz-Dombriz, A. Dobado, A.L. Maroto, Phys. Rev. Lett
**103**, 179001 (2009)ADSCrossRefGoogle Scholar - 35.A. Abebe, A. de la Cruz-Dombriz, P.K.S. Dunsby, Phys. Rev. D
**88**, 044050 (2013)ADSCrossRefGoogle Scholar - 36.G. Esposito-Farèse, D. Polarski, Phys. Rev. D
**63**, 063504 (2001)ADSCrossRefGoogle Scholar - 37.B. Boisseau, G. Esposito-Farese, D. Polarski, A.A. Starobinsky, Phys. Rev. Lett.
**85**, 2236 (2000)ADSCrossRefGoogle Scholar - 38.L. Amendola, Phys. Rev. D
**69**, 103524 (2004)ADSCrossRefGoogle Scholar - 39.T. Tatekawa, S. Tsujikawa, JCAP
**0809**, 009 (2008)ADSCrossRefGoogle Scholar - 40.C. H. Brans, The Roots of scalar-tensor theory: An Approximate history, [gr-qc/0506063]Google Scholar
- 41.H. Farajollahi, A. Salehi,
*JCAP*(07), 036 (2011)Google Scholar - 42.H. Farajollahi, A. Salehi,
*JCAP*(11), 006 (2010)Google Scholar - 43.H. Farajollahi, A. Salehi,
*JCAP*(11), 002 (2012)Google Scholar - 44.J. A. R. Cembranos, A. de la Cruz Dombriz, L. Olano Garcia
*Phys. Rev. D***88**123507 (2013)Google Scholar - 45.H. Farajollahi, A. Salehi, Phys. Rev. D
**83**, 124042 (2011)ADSCrossRefGoogle Scholar - 46.M. Giovannini, Int. J. Mod. Phys. D
**22**, 363 (2005)ADSCrossRefGoogle Scholar - 47.A.G. Riess et al., Astron. J.
**116**, 1009 (1998)ADSCrossRefGoogle Scholar - 48.R. Amanullah et al., ApJ.
**716**, 712 (2010)ADSCrossRefGoogle Scholar

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