Inflation in an external four-form gauge field

We consider the possibility that the Universe, viewed as a three-brane, originated in a region of strong external field strength due to a four-form gauge field in the bulk. It is shown that in a scenario of this kind inflation is generic for a wide range of initial conditions. This is true even for small field inflation with a simple quadratic inflaton potential, as well as for Higgs potentials with the initial field well away from the local maximum, not necessarily starting from rest. The power spectrum, spectral index, and $r$ parameter help to constrain parameters in this scenario, and $r<0.1$ favors Higgs potentials.

Figure 1

“Force” diagrams for the slow spiral approximation, where the spiral is counterclockwise inward towards the origin for the quadratic potential (a), and clockwise outward towards $\rho =m$ for the Higgs potential (b). The approximation is that ${\partial }_t^2\mathit{\phi }$ towards the origin (solid dot) is simply the centripetal acceleration required for circular motion in the $\mathit{\phi }$ plane, neglecting any acceleration beyond that.

Figure 2

Numerical solution of the equations of motion (16) and (17) in a quadratic potential with initial ${\rho }_0=3m,{\upsilon }_0=0$, over a period of $2\times {10}^6$ Planck times.

Figure 3

The initial trajectory of the previous figure, projected into the ${\phi }_5-{\phi }_6$ plane, which illustrates the initial transient behavior that dies out after roughly 2000 Planck times.

Figure 4

Same as Fig. 2, but with a random initial velocity ${\upsilon }_0$. The transient behavior is clearly seen in this example.

Figure 5

For the example of the quadratic potential with a random initial ${\upsilon }_0$: (a) The expansion rate $H=\dot{a}/a$. (b) Comparison of the computed $\theta (t)$ in the ${\phi }_5-{\phi }_6$ plane with the slow spiral prediction (24). (c) Comparison of the speed $\upsilon (t)$ in the ${\phi }_5-{\phi }_6$ plane with the slow spiral prediction (26).

Figure 6

Numerical solution of the equations of motion (16) and (17) in a Higgs potential with initial ${\rho }_0=0.2m$ and random ${\upsilon }_0$, over a period of $1\times {10}^6$ Planck times. The initial transient behavior is shown in green.

Figure 7

For the example of the Higgs potential with a random initial ${\upsilon }_0$: (a) The expansion rate $H=\dot{a}/a$. (b) Comparison of the computed $\theta (t)$ in the ${\phi }_5-{\phi }_6$ plane with the slow spiral prediction (24). (c) Comparison of the speed $\upsilon (t)$ in the ${\phi }_5-{\phi }_6$ plane with the slow spiral prediction (26).

Figure 8

The number of $e$-foldings $N$, in the Higgs potential case, as a function of the inflaton field $\rho =|\mathit{\phi }|$ at the start and end of inflation. The latter are displayed as fractions $f_0={\rho }_{\text{initial}}/m$ and $f_f={\rho }_{\text{final}}/m$ of the minimum of the Higgs potential at $\rho =m$.