Gravity in the dynamical approach to the cosmological constant

One of the most disturbing difficulties in thinking about the cosmological constant is that it is not stable under radiative corrections. The feedback mechanism proposed by Mukohyama and Randall is a dynamical way to protect a zero or small cosmological constant against radiative corrections. Hence, while this by itself does not solve the cosmological constant problem, it can help in solving the problem. In the present paper we investigate stability and gravity in this approach and show that the feedback mechanism is both classically and quantum mechanically stable and has self-consistent, stable dynamics.

Figure 1

The behavior of $\mathcal{B}\equiv {\kappa }^{3}H\pi /c$ for four different sets ofinitial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. This figure indeed shows that $-(3-\dot{H}/H^2)\mathcal{B}$ quicklyconverges to 1.

Figure 2

The evolution of $\pi$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 3

The asymptotic evolution of $\pi$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. Four lines are degenerate because of the attractor behavior.

Figure 4

The evolution of ${\kappa }^2{\Lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}={\kappa }^{4}V(\varphi )=c\kappa (\varphi -{\varphi }_0)$ for four different setsof initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 5

The asymptotic evolution of ${\kappa }^2{\Lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}={\kappa }^{4}V(\varphi )=c\kappa (\varphi -{\varphi }_0)$ for four different setsof initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. Four lines are degenerate because of the attractor behavior.

Figure 6

The evolution of the Hubble parameter $H$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 7

The asymptotic evolution of the Hubble parameter $H$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. Four lines are degenerate because of the attractor behavior.

Figure 8

The evolution of the dimensionless evolution rate $\dot{H}/H^2$ of $H$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 9

The asymptotic evolution of the dimensionless evolution rate $\dot{H}/H^2$ of $H$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. Four lines are degenerate because of the attractor behavior.

Figure 10

The early stage of the recovery of the standard Friedmann equation(${\mathcal{F}}_0\to 0$)for four different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 11

Recovery of the standard dynamical equation (${\mathcal{G}}_0\to 0$)for four different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 12

The early stage of the recovery process of the standard Friedmannand dynamical equations (${\mathcal{F}}_0\to 0$, ${\mathcal{G}}_0\to 0$)for four different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 13

The late stage of the recovery of the standard Friedmann equation(${\mathcal{F}}_0\to 0$)for four different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. Four lines are degenerate because of the attractor behavior.

Figure 14

The evolution of the extra degree $\phi$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$.

Figure 15

The asymptotic evolution of the extra degree $\phi$ forfour different sets of initial values. In the figure, $a$ is thescale factor and $a_0=a(t=0)$.The initial values of $\varphi$ and $\dot{H}/H^2$ are $c\kappa (\varphi -{\varphi }_0){|}_{t=0}=0.01$ and $\dot{H}/H^2{|}_{t=0}=-0.02$. Two more independent conditionsat $t=0$ are given by setting ${\mathcal{F}}_0$ and ${\mathcal{G}}_0$ in four different ways as $({\mathcal{F}}_0,{\mathcal{G}}_0{)}_{t=0}=(0.1,0.1)$, $(-0.1,0.1)$, $(0.1,-0.01)$, $(-0.1,-0.01)$. Four lines are degenerate because of the attractor behavior.