Inflation and the cosmological constant

A particular compensation-type solution of the main cosmological constant problem has been proposed recently, with two massless vector fields dynamically canceling an arbitrary cosmological constant $\Lambda$. The naive expectation is that such a compensation mechanism does not allow for the existence of an inflationary phase in the very early Universe. However, it is shown that certain initial boundary conditions on the vector fields can in fact give rise to an inflationary phase.

Figure 1

Numerical solution of the reduced field equations for model (1) and Ansatz (2). The dimensionless model parameters are $m=0.01$, $\delta ={10}^{-6}$, $\eta =0$, and $\lambda =0$. The boundary conditions are $a(1)=1$, $\{{\phi }_1(1),{\dot{\phi }}_1(1),{\phi }_2(1),{\dot{\phi }}_2(1)\}=\{10,-0.25,0,-0.433013\}$, and $v(1)=\dot{v}(1)=w(1)=\dot{w}(1)=0$. [The real scalars ${\phi }_n$ are defined by $\sigma \equiv ({\phi }_1+i{\phi }_2)/\sqrt{2}$.] The value of $h(1)$ follows from the Friedmann equation (4a). With these boundary conditions, the reduced field equations have the exact solution $v(\tau )=w(\tau )=0$ for $\tau \ge 1$. The top right panel shows an $h(\tau )$ plateau corresponding to an inflationary phase. The bottom right panel shows that, long after inflation, $h(\tau )$ asymptotically goes as $(2/3){\tau }^{-1}$.

Figure 2

Same as Fig. 1, but now with nonzero cosmological constant, $\lambda =0.01$. Same boundary conditions as Fig. 1, so that the exact solution $v(\tau )=w(\tau )=0$ persists. The Friedmann equation (4a) gives the asymptotic value $h(\infty )=\sqrt{\lambda /3}\approx 0.05774$.

Figure 3

Same parameters and boundary conditions as Fig. 2 (e.g., $\lambda =0.01$), except for small but nonzero starting values of $v$ and $w$. Specifically, the boundary conditions at $\tau =1$ are $\{a,v,\dot{v},w,\dot{w},h,{\phi }_1,{\dot{\phi }}_1,{\phi }_2,{\dot{\phi }}_2\}=\{1,2\times {10}^{-4},2\times {10}^{-4},2\times {10}^{-4},2\times {10}^{-4},0.216025,10,-0.25,0,-0.433013\}$. The total vacuum energy density entering the right-hand side of the Friedmann equation (4a) is given by the sum of the two panels in the middle column and vanishes asymptotically.

Figure 4

Same parameters and boundary conditions as Fig. 3 (e.g., $\lambda =0.01$), except for relatively large starting values of $v$ and $w$. Specifically, the boundary conditions at $\tau =1$ are $\{a,v,\dot{v},w,\dot{w},h,{\phi }_1,{\dot{\phi }}_1,{\phi }_2,{\dot{\phi }}_2\}=\{1,0.0799201,0.01998,0.08,0.02,0.216961,10,-0.25,0,-0.433013\}$.