Quantum incompleteness of inflation

Inflation is most often described using quantum field theory (QFT) on a fixed, curved spacetime background. Such a description is valid only if the spatial volume of the region considered is so large that its size and shape moduli behave classically. However, if we trace an inflating universe back to early times, the volume of any comoving region of interest—for example, the present Hubble volume—becomes exponentially small. Hence, quantum fluctuations in the trajectory of the background cannot be neglected at early times. In this paper, we develop a path integral description of a flat, inflating patch (approximated as de Sitter spacetime), treating both the background scale factor and the gravitational wave perturbations quantum mechanically. We find this description fails at small values of the initial scale factor, because two background saddle point solutions contribute to the path integral. This leads to a breakdown of QFT in curved spacetime, causing the fluctuations to be unstable and out of control. We show the problem may be alleviated by a careful choice of quantum initial conditions, for the background and the fluctuations, provided that the volume of the initial, inflating patch is much larger than $H^{-1}$ in Planck units with $H$ the Hubble constant at the start of inflation. The price of the remedy is high: not only the inflating background but also the stable, Bunch-Davies fluctuations must be input by hand. Our discussion emphasizes that, even if the inflationary scale is far below the Planck mass, new physics is required to explain the initial quantum state of the universe.

Figure 1

Left panel: The Penrose diagram of de Sitter spacetime in the flat slicing. The red lines denote time slices and the blue lines denote space slices. The two background solutions relevant to the propagator consist of either pure expansion (a finite portion of the green patch on the left) or first contraction followed by expansion (a portion of the green patch on the right disappearing at the top right, reappearing bottom left and ending up at the same location as the purely expanding solution). Right panel: The two background solutions relevant to the propagator, shown as a function of $t$ for successively decreasing the initial scale factor squared $q_0$ (dotted line, then dashed line, and finally solid line at $q_0=0$). As the initial scale factor is decreased, the two solutions approach each other and merge when $q_0=0$. This last solution would correspond to the limit where one starts at the bottom left corner (or equivalently the top right corner) in the Penrose diagram on the left.

Figure 2

The lines of steepest ascent $\mathcal{K}$ and descent $\mathcal{J}$ for the lapse integral, shown in the complexified plane of the lapse $N$. Left: For generic boundary conditions with $q_0$, $q_1\ne 0$. Right: The limit of vanishing initial boundary $q_0=0$.

Figure 3

The complex $N$ plane with the degenerate saddle point $N_{*}$ and the branch points ${\overline{N}}_{+}$ and ${\overline{N}}_{-}$, relevant to the case where the initial size of the universe is taken to zero. The red line represents the branch cut. The blue line represents the $N$ for which the background $\overline{q}$ passes through 0 for a time $t\in (0,1]$.

Figure 4

Picard-Lefschetz theory for the lapse integral in the limit $q_0\to 0$. The dark lines show the lines of steepest ascent/descent indicated by $\mathcal{K}$ and $\mathcal{J}$. Using Picard-Lefschetz theory the real half-line $(0^{+},\infty )$ is deformed to the red dashed curves. Left: The Lefschetz thimbles for degenerate saddle point $N_{*}$ when setting $q_0=0$. Right: The Lefschetz thimbles in the limit $q_0\to 0$ approached from the upper half plane $\mathrm{Im}{q}_0>0$. We observe that when approaching $q_0=0$ from above, the saddle point $N_{s+}$ is relevant while the saddle point $N_{s-}$ is irrelevant. When approaching $q_0$ from below, the saddle points switch positions.

Figure 5

Picard-Lefschetz theory in $N$ with $q_0=0$, in the limit $K\to 0$, with $\mathrm{\Lambda }=3$ and $q_1=10$. The curves are the lines of steepest ascent and descent emanating from the saddle point at the crossings of the lines. The origin is the point where the two lines meet at the left. Note that the figure is symmetric around the real line. Left panel: Spatial curvature $K=1$: this is the standard no-boundary graph. Middle panel: Spatial curvature $K=1/10$. Right panel: Spatial curvature $K=0$.

Figure 6

The lines of steepest ascent and descent for the integral over $N$ as a function of an increasing initial localization $\sigma$ (the localization $\sigma$ increases from top left to top right, then lower left, and finally lower right). When $\sigma$ becomes larger than the critical value ${\sigma }_c$ (which occurs in the transition from the top right to the lower left graph), a Stokes phenomenon happens: the line of steepest ascent $\mathcal{K}$ from the saddle point $N_{s+}=N_{1,1}^{\sigma }$ corresponding to the bouncing solution no longer intersects the original integration contour on the real $N$ line. Beyond this critical value, only one saddle point (namely the purely expanding saddle point $N_{s-}=N_{1,-1}^{\sigma }$) remains relevant to the semiclassical path integral. Moreover, this saddle point resides on the real $N$ line, so that it describes classical evolution. At this point standard quantum field theory in curved spacetime is recovered.

Figure 7

The probability distribution of the initial wave function in phase space $(q_i,p_i)$. The black line is the Hamiltonian constraint. The horizontal axis is $q_i$ and the vertical axis $p_i$. In order to have only an expanding universe, the wave function needs to have support only in the lower right quadrant.

Figure 8

Left panel: When the initial spread $\sigma$ is small, the relevant thimbles are forced to pass through the “singular curve” where the perturbative action blows up due to regions of vanishing scale factor in the geometries that are summed over. Right panel: For a sufficiently large localization $\sigma \gtrsim {\sigma }_c$, the relevant thimble stays above the singular curve, the path integral for the perturbations is well defined, and, with a judicious choice of initial state, stable fluctuations may be imposed.