Dynamical systems of cosmological models for different possibilities of G and $$\rho _{\Lambda }$$
Authors: Sonia Chingtham , Singh S. Surendra 
Published: October 2022

The present paper deals with the dynamics of spatially flat Friedmann–Lemaître–Robertson–Walker $$(FLRW)$$

(
F
L
R
W
)

cosmological model with a time varying cosmological constant $$\Lambda $$
Λ
where $$\Lambda $$
Λ
evolves with the cosmic time t through the Hubble parameter H, that is, $$\Lambda (H)$$

Λ
(
H
)

. We use the expression of $$\Lambda (H)$$

Λ
(
H
)

in the form of Taylor series with respect to H keeping only the even powers of H because of the general covariance of the effective action of quantum field theory in a curved background. Dynamical systems for three different cases based on the possibilities of gravitational constant G and the vacuum energy density $$\rho _{\Lambda }$$

ρ
Λ

have been analysed. In Case I, both G and $$\rho _{\Lambda }$$

ρ
Λ

are taken to be constant. We analyse stability of the system by using the notion of spectral radius, behavior of perturbation along each of the axes with respect to cosmic time and Poincaré sphere. In Case II, we have dynamical system analysis for $$G=\hbox {constant}$$

G
=
constant

and $$\rho _{\Lambda } \ne $$

ρ
Λ

≠

constant where we study stability by using the concept of spectral radius and perturbation function. In Case III, we take $$G \ne $$

G
≠

constant and $$\rho _{\Lambda } \ne $$

ρ
Λ

≠

constant where we introduce a new set of variables to set up the corresponding dynamical system. We find out the fixed points of the system and analyse the stability from different directions: by analysing behaviour of the perturbation along each of the axes, Center Manifold Theory and stability at infinity using Poincaré sphere respectively. Phase plots and perturbation plots have been presented. We deeply study the cosmological scenario with respect to the fixed points obtained and analyse the late time behavior of the Universe. The effective equation of state parameter $$\omega _{eff}$$

ω

eff

, total energy density $$\Omega _{tt}$$

Ω

tt

are also evaluated at the fixed points for each of the three cases and these values are in agreement with the observational values in Aghanim et al. (Astron Astrophys 641(A6): 2020, 2018). We have also presented the EoS parameter for dark energy sector $$\omega _{de}(z_{r})$$

ω

de

(

z
r

)

, the Hubble parameter $$H(z_{r})$$

H
(

z
r

)

and the deceleration parameter $$q(z_{r})$$

q
(

z
r

)

as functions of redshift $$z_{r}$$

z
r

for all the three cases and their plots over redshift are also provided. We analyse the quintessence-like, phantom-like or the purely cosmological-constant type dark energy, etc behavior when the EoS approaches the fixed point value near $$-1$$

-
1

. The present values of $$\omega _{de}(z_{r})$$

ω

de

(

z
r

)

, $$H(z_{r})$$

H
(

z
r

)

, $$q(z_{r})$$

q
(

z
r

)

and $$z_{rt}$$

z

rt

have been tabulated in Table 4 and they fall within the range of cosmological observations. The transition redshift value $$(z_{rt})$$

(

z

rt

)

for each of the three cases have also been evaluated. In each of the cases the developed model agrees with the fact that the Universe is in the epoch of accelerated expansion for suitable values of free parameters chosen. The developed cosmological models associated with each of the three cases have a deep connection with the accelerated expansion phenomena of the evolving Universe.