Evolution of cosmological perturbations in the universe dominated by multiple scalar fields

Abstract

Recently a Hamiltonian formulation for the evolution of the universe dominated by multiple oscillatory scalar fields was developed by the present author and was applied to the investigation of the evolution of cosmological perturbations on superhorizon scales in the case that scalar fields have incommensurable masses.In the present paper, the analysis is extended to the case in which the masses of scalar fields satisfy resonance conditions approximately. In this case, the action-angle variables for the system can be classified into fast changing variables and slowly changing variables. We show that after an appropriate canonical transformation, the part of the Hamiltonian that depends on the fast changing angle variables can be made negligibly small, so that the dynamics of the system can be effectively determined by a truncated Hamiltonian that describes a closed dynamics of the slowly changing variables. Utilizing this formulation, we show that the system is unstable if this truncated Hamiltonian system has hyperbolic fixed point and as a consequence, the Bardeen parameter for a perturbation of the sytem grows. PACS number(s):98.80.Cq 1 email address: yj4t-hmzk@asahi-net.or.jp §1 Introduction and SummaryThe inflationary universe model is the most successful model in explaining the origin of the present cosmological structures such as galaxies and clusters of galaxies. In this model, quantum fluctuations of an inflaton field, a scalar field driving the inflationary expansion, provide seed perturbations which grow and form the present cosmological structures by gravitational instability. During the slow rolling phase of inflation, these seed perturbations are streched beyond the Hubble horizon and their wavelengths stay larger than the horizon scale until the perturbations come back inside the Hubble horizon during the Friedmann stage after the inflation. The amplitudes of perturbations at this second horizon crossing, which have a direct relevance to the CMB anisotropy observations and provide the initial condition for detailed astrophysical models of galaxy formation, are determined by the so-called Bardeen parameter. Hence, in order to obtain information on the inflationary stage of the universe from observations of the present universe, we have to determine the behavior of the Bardeen parameter of perturbations during the superhorizon stage.If the cosmic matter has a regular equation of state and is dominated by a single component, this Bardeen parameter is conserved with a good accuracy on superhorizon scales [2 ]. However, in a realistic model, during the period between the first horizon crossing in the inflationary regime and the socond horizon crossing in the Friedmann regime, the inflaton field oscillates coherently around a local minimum of a potential and their energy is gradually transformed into matter and radiation which constitute the present universe. During this reheating phase, the equation of state becomes singular periodically [6 ], and entropy modes can be produced [7,11 ]. Further, ...

Theoretical Tools for CMB Physics

“…(5.78) has been pointed out and analyzed. In [153] and [154,155] the analysis of the fluctuations in the case of multiple oscillating (and interacting) fields has been performed. The knowledge of this pathology and of its resolution is now common knowledge in pre-heating studies [156,157,158,159,160,161,162,163,164,165].…”

Long wavelength limit of evolution of nonlinear cosmological perturbations

“…The differential operators in the two groups ðd=d; d=dtÞ and ðð@=@Þ a ; ð@=@aÞ a Þ are commutative in each group, but the differential operators belonging to the different groups, for example, d=d and ð@=@aÞ a , are not commutative. By using the relation In this way, in the LWL formalism where the scale factor a, not t, is used as the evolution parameter [7][8][9], the expressions of the solutions of D n A and D n ð _ A=Þ can be obtained by calculating only a single term written with the higher order ð@=@Þ a derivative of the solution of the corresponding locally homogeneous physical quantity. In the same way, " defined by (2.76) can be expressed in the word of the scale factor a as and all n (n !…”

“…Taruya and Nambu suggested in the multiple scalar field system also the LWL formula exists [3]. Soon in this case the exact LWL formula was constructed [1,2], and it was used to investigate the multiple oscillatory scalar fields [7,8]. In addition, in Ref.…”

Construction and applications of the manifestly gauge invariant expressions of the solutions of the cosmological perturbation theory

“…It was established that the evolutions of the cosmological perturbations in the long wavelength limit are constructed from the derivatives with respect to the solution constants of the solutions of the corresponding exactly homogeneous universe [17 ] [10 ]. This fact has been used and has been developed in the various situations, by the author of the present paper [11 ] [12 ] [13 ] [14 ] [15 ] and with the reinterpretations in the contexts of each authors' researches [18 ] [20 ], [19 ]. In the most general situation in which plural scalar fields and plural perfect fluids coexist, the solution of the evolutions of the cosmological perturbations in the long wavelength limit based on the corresponding exactly homogeneous perturbations are manifestly presented in the paper [13 ] and the paper [14 ] in the linear level and in the nonlinear level, respectively.…”