[Paper Review] Wave function of the Universe as a sum over eventually inflating universes

Karthik Rajeev|arXiv (Cornell University)|Dec 8, 2021

Cosmology and Gravitation Theories25 references8 citations

TL;DR

This paper proposes a novel quantum cosmological wave function defined as a sum over universes that eventually inflate, using a Lorentzian path integral in minisuperspace. It shows the resulting wave function is proportional to the Hartle-Hawking wave function, with invariance under initial condition parameter $a_0$; analytic continuation of $a_0$ into the complex plane reveals the no-boundary proposal as a boundary point, and the perturbative sector yields a scale-invariant power spectrum, uniquely fixed by matching to the no-boundary state.

ABSTRACT

We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number $a_0$, can be imposed to realise this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on $a_0$ is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend $a_0$ to an extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find $\bar{\mathcal{D}}$. Interestingly, a special point on the boundary of $\bar{\mathcal{D}}$ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point" in $\bar{\mathcal{D}}$, we converge on a unique quantum state for the perturbations.

Motivation & Objective

• To develop a quantum cosmological framework that treats inflation as a natural outcome of a path integral over spacetimes destined to inflate.
• To address the challenge of initial conditions in quantum cosmology by replacing strict no-boundary or compact geometries with a physically motivated initial condition favoring future inflation.
• To investigate whether the resulting wave function can be analytically continued in the complex $a_0$-plane, and whether this leads to a deeper understanding of the Hartle-Hawking wave function.
• To derive the quantum state of cosmological perturbations from the proposed prescription and determine whether it yields a scale-invariant power spectrum.
• To explore whether an additional matching condition at the no-boundary point uniquely selects the Bunch-Davies vacuum, thus connecting to standard inflationary predictions.

Proposed method

• Define the wave function as a Lorentzian path integral over Lorentzian geometries that eventually enter a de Sitter phase, with an initial condition parametrized by a positive real $a_0$.
• Impose a boundary condition on the scale factor and its momentum that selects only universes with future inflation, ensuring the path integral is well-defined and convergent.
• Use the Kontsevich-Segal theorem and its extension by Witten to determine the domain $\bar{D}$ in the complex $a_0$-plane where analytic continuation of the wave function is valid.
• Compute the background wave function and show it is proportional to the Hartle-Hawking wave function, with dependence on $a_0$ only through an overall phase.
• Quantize linear perturbations around the de Sitter saddle point and derive the corresponding quantum state, showing it leads to a scale-invariant power spectrum in the large-$p_1$ limit.
• Impose a matching condition between the perturbative wave function and the no-boundary state at $a_0 = i/H$, uniquely fixing the quantum state to the Bunch-Davies vacuum.

Experimental results

Research questions

• RQ1 Can a path integral over universes that eventually inflate yield a wave function consistent with the Hartle-Hawking proposal?
• RQ2 Is the wave function invariant under changes in the initial condition parameter $a_0$, and what does this imply for the physical interpretation of the initial slicing?
• RQ3 What is the maximal domain $\bar{D}$ in the complex $a_0$-plane to which the wave function can be analytically continued while preserving its form?
• RQ4 Does the proposed prescription for the background wave function lead to a scale-invariant power spectrum for cosmological perturbations?
• RQ5 Can the Bunch-Davies vacuum be uniquely recovered from the proposed framework by imposing an additional matching condition at the no-boundary point?

Key findings

• The wave function resulting from the sum over eventually inflating universes is proportional to the Hartle-Hawking wave function, with the only dependence on $a_0$ being an overall phase factor.
• The domain $\bar{D}$ in the complex $a_0$-plane, where the path integral converges and analytic continuation is valid, is explicitly determined using convergence criteria and the Kontsevich-Segal theorem.
• The value $a_0 = i/H$ lies on the boundary of $\bar{D}$, and at this point, the wave function exactly reproduces the Hartle-Hawking wave function.
• The perturbative sector of the theory yields a scale-invariant power spectrum for cosmological perturbations in the large-$p_1$ limit, consistent with inflationary predictions.
• By imposing a matching condition between the perturbative wave function and the no-boundary state at $a_0 = i/H$, the Bunch-Davies vacuum is uniquely selected from the one-parameter family of scale-invariant states.
• The entire construction reveals that the Hartle-Hawking wave function arises as a complex analytic continuation of the sum over inflating universes, providing a precise sense in which it is a complex extension of a physically motivated path integral.