No rescue for the no boundary proposal: Pointers to the future of quantum cosmology

In recent work [J. Feldbrugge et al. Phys. Rev. D 95, 103508 (2017). and J. Feldbrugge et al. Phys. Rev. Lett. 119, 171301 (2017).], we introduced Picard-Lefschetz theory as a tool for defining the Lorentzian path integral for quantum gravity in a systematic semiclassical expansion. This formulation avoids several pitfalls occurring in the Euclidean approach. Our method provides, in particular, a more precise formulation of the Hartle-Hawking no boundary proposal, as a sum over real Lorentzian four-geometries interpolating between an initial three-geometry of zero size, i.e., a point, and a final three-geometry. With this definition, we calculated the no boundary amplitude for a closed universe with a cosmological constant, assuming cosmological symmetry for the background and including linear perturbations. We found the opposite semiclassical exponent to that obtained by Hartle and Hawking for the creation of a de Sitter spacetime “from nothing.” Furthermore, we found the linearized perturbations to be governed by an inverse Gaussian distribution, meaning they are unsuppressed and out of control. Recently, Diaz Dorronsoro et al. [Phys. Rev. D 96, 043505 (2017)] followed our methods but attempted to rescue the no boundary proposal by integrating the lapse over a different, intrinsically complex contour. Here, we show that, in addition to the desired Hartle-Hawking saddle point contribution, their contour yields extra, nonperturbative corrections which again render the perturbations unsuppressed. We prove there is no choice of complex contour for the lapse which avoids this problem. We extend our discussion to include backreaction in the leading semiclassical approximation, fully nonlinearly for the lowest tensor harmonic and to second order for all higher modes. Implications for quantum de Sitter spacetime and for cosmic inflation are briefly discussed.

Figure 1

The Morse function for the background is plotted in the complex $N$ plane, for a closed, homogeneous and isotropic $\mathrm{\Lambda }$-dominated cosmology. The solid orange line is the defining integration contour $\mathcal{C}$, and the dashed orange line is the corresponding deformed contour, passing along Lefschetz thimbles. As $N$ tends to infinity in the complex $N$ plane, the real part of the exponent in the integrand (the Morse function) tends to $+\infty$ in the red regions or $-\infty$ in the green regions. It is constant along the blue contours. Upper panel: The real Lorentzian contour $0^{+}<N<\infty$ used for the causal Lorentzian propagator. Lower left panel: The contour for $N$ running from $-\infty$ to $+\infty$ below the origin, as proposed by Diaz Dorronsoro et al. [11]. Lower right panel: The real part of the causal propagator, equivalent to a continuous contour for $N$ running from $-\infty$ to $+\infty$ above the origin.

Figure 2

The classical background geometries appearing in the no boundary path integral. Left panel: The regular, complex saddle point geometry. Middle panel: A real Lorentzian off-shell background geometry appearing at $N_{-}^2\le N^2\le N_{\star }^2$, possessing one strong singularity. Right panel: The real Lorentzian geometry appearing at $N^2>N_{\star }^2$, possessing two strong singularities.

Figure 3

The branch cuts (in red) on the real $N$ axis, for $-N_{+}<N<-N_{-}$ and $N_{-}<N<N_{+}$, form impenetrable barriers for Picard-Lefschetz theory. The classical scale factor squared $q$ crosses zero for a second time (as in the right panel of Fig. 2) on the blue lines. The Hartle-Hawking and Lorentzian-Picard-Lefschetz saddles are indicated HH and L-PL, respectively. The gray lines are the lines of steepest ascent and descent emanating from the four saddle points, with the arrows indicating directions of descent.

Figure 4

Our results imply that a smooth, $\mathrm{\Lambda }$-mediated topology changing transition is ill-defined. Thus, topology change should not be thought of as illustrated in the left panel (where the physical regions of spacetime are blue, space is horizontal and time vertical). Rather, topology change most likely requires passage through a singularity, where massless d.o.f. play a crucial role in enabling the transition and extensions of the semiclassical methods employed in the present paper are needed.

Figure 5

The Morse function $h=\mathrm{Re}(iS/\hslash )$ around a branch cut, in units where $\hslash =1$ and for the parameters $\mathrm{\Lambda }=3$, $q_1=101$, $l=10$, ${\varphi }_1=1$. At the cut, the Morse function reaches its maximum at $N_{\star }=10$ coming from the upper half plane, and its minimum also at $N_{\star }$, though approaching the cut from below.

Figure 6

Picard-Lefschetz theory for a $\mathrm{\Lambda }$-dominated universe with gravitational waves. The solid orange and dashed orange lines are the original and deformed integration contours, respectively, while the zigzag lines denote the branch cuts. The lines denoted by ${\mathcal{J}}_i$ are lines of steepest descent and the lines denoted by ${\mathcal{K}}_i$ are lines of steepest ascent. Left panel: The integration path for the Lorenztian propagator, deformed to run above the cut. Right panel: The integration domain prescribed by Diaz Dorronsoro et al. [11] with the original integration domain above the branch cuts. Note that the contour must be deformed to partially encircle the branch cuts in order to reach the Lefschetz thimbles.

Figure 7

These graphs show the weighting at the saddle points (left panel) and the imaginary part of the saddle point locations (right panel) as a function of the $l=2$ anisotropy mode amplitude ${\varphi }_{1-}$, for $\mathrm{\Lambda }=3$. In the plot of the action (left) the line starting at $+12{\pi }^2/\mathrm{\Lambda }=+4{\pi }^2$ for ${\varphi }_{1-}=0$ corresponds to the saddle points in the lower half $N$ plane, while the line starting at $-4{\pi }^2$ corresponds to the saddle points in the upper half plane. In black (mostly hidden behind the red line) are the linear results without backreaction, in red the results including backreaction but still in linear perturbation theory, and in blue the results stemming from solving the fully backreacted Einstein equations. For values of ${\varphi }_{1-}$ below 1 the linear and nonlinear results agree to high precision, while one can see that at larger values of the anisotropy the nonlinear corrections enhance the instability of the fluctuations, and move the saddle points further towards the real $N$ axis. Note that the weighting of the upper saddle points surpasses that of the lower ones when backreaction is still entirely negligible. Moreover, the nonlinear effects of the full Einstein equations imply that the (unstable) upper saddle points come to dominate already for smaller amplitudes of the fluctuations.

Figure 8

The real and imaginary parts of the mode function $\varphi (t)$ in the saddle point $N_1$ with the boundary conditions $q_0=0$, ${\varphi }_0=0$, $q_1=101$ and ${\varphi }_1=3$ for mode $l=10$. The black dashed lines correspond to the analytic result without backreaction. The red lines correspond to the numerical analysis with backreaction. We observe that the backreaction for ${\varphi }_1=3$, which violates the perturbation theory condition $|\varphi (t)|>1$, leads to a change in the mode functions of approximately 10%, with no qualitative change. For the boundary condition $0\le {\varphi }_1\le 0.8$, which do satisfy the condition $|\varphi (t)|<1$ for all $t\in [0,1]$, the correction due to backreaction is completely negligible.

Figure 9

The Morse function evaluated at the saddle points $h(N_s)$ as a function of the boundary condition ${\varphi }_1$ for which $|\varphi (t)|\le 1$ for all $t$. The lines starting from $h=-4{\pi }^2$ and $h=4{\pi }^2$ correspond to the upper and the lower saddle points. The blue to purple lines correspond to $l=2,3,\dots ,10$. The colored lines are the numerical simulations with backreaction. The dotted lines are the analytic results without backreaction. We observe that the backreaction of the perturbation $\varphi$ on the scale factor $q$ is very small for the boundary condition $0\le {\varphi }_1\le 0.8$. Consequently, the analytic calculations which neglect backreaction are accurate when the upper saddle point starts to dominate over the lower saddle point for $l\ge 5$.

Figure 10

The Morse function $h$ evaluated at $N=N_{\star }+iy$ with the boundary conditions $\mathrm{\Lambda }=3$, $q_1=101$, ${\varphi }_1=0.5$ for the mode $l=10$. The dashed line is the analytic result without backreaction. The red line is the numerical calculation including backreaction. We plot the backreaction calculation for the points for which $|\varphi (t)|<1$ (i.e., for which linear theory is reliable) and the backreaction on the $q$ variable is small, i.e., $\frac{{\dot{\varphi }}^{2}q}{3{\pi }^2}<|\frac{2N^2\mathrm{\Lambda }}{3}|$ for all $t$. Note that the backreaction is extremely small near the saddle point for these boundary conditions. The jump at $y=0$ illustrates the branch cut. The backreaction becomes significant when $N$ approaches the real axis. However, the backreaction does not appear to remove the branch cut.

Figure 11

The Morse function along the thimble for the boundary conditions $\mathrm{\Lambda }=3$, $q_1=101$, ${\varphi }_1=1$ for the mode $l=10$. For these boundary conditions the action of the background and perturbations are comparable in the saddle points. Upper panel: First quadrant of the complex $N$ plane with the lines of steepest ascent and descent of the upper saddle point. The shaded/white regions, defined by $\mathrm{Re}[\gamma ]$ less/greater than 1, respectively, denote the regions in which the finite action condition selects a mode which diverges/vanishes on the initial boundary. The points on the thimble ${\mathcal{J}}_1$ indicate the values of the lapse $N$ at which we have evaluated the importance of backreaction in the lower panels. Lower panels: The Morse functions for the background and perturbation actions ${\overline{S}}^{(0)}$ and ${\overline{S}}^{(2)}$. The blue lines denote the analytic Morse function while the black dots denote the numerical results including backreaction. Note that nonlinear backreaction is completely negligible along the thimble.

Figure 12

The Morse function for the perturbations is plotted along a line $N=x+iy$ in the complex $N$ plane, crossing the real $N$ axis $y=0$ in the vertical direction. The parameters and boundary values are $\mathrm{\Lambda }=3$, $q_0=0$, $q_1=101$, $l=10$, ${\chi }_0=0$, and ${\chi }_1=1$. The black dashed line is the analytic result of perturbation theory without backreaction. The red line is the numerical result including backreaction up to second order in the perturbations.

Figure 13

The Picard-Lefschetz thimbles (solid dark lines, labelled $J_1-J_4$) for our problem, in the complex $N$ plane. Any convergent integral starting and ending at a singularity of the Morse function can be represented as a sum of these thimbles. Also shown are the branch cuts (jagged gray lines) which the no boundary perturbations introduce on the real $N$ axis.