[Paper Review] M-theory observables for cosmological space-times

T. Banks, Willy Fischler|ArXiv.org|Feb 14, 2001

Cosmology and Gravitation Theories15 references113 citations

TL;DR

This paper proposes a gauge-invariant framework for defining M-theory observables in cosmological spacetimes with a positive cosmological constant, generalizing the S-matrix to asymptotically de Sitter universes. It shows that multiple S-matrices—related by holographic gauge transformations—emerge due to cosmological horizons, with their equivalence restored in the limit Λ → 0, offering a novel resolution to the horizon problem without inflation.

ABSTRACT

We discuss the construction of the analog of an S-matrix for space-times that begin with a Big-Bang and asymptote to an FRW universe with nonnegative cosmological constant. When the cosmological constant is positive there are many such S-matrices, related mathematically by gauge transformations and physically by an analog of the principle of black hole complementarity. In the limit of vanishing $Λ$ these become (approximate) Poincare transforms of each other. Considerations of the initial state require a quantum treatment of space-time, and some preliminary steps towards constructing such a theory are proposed. In this context we propose a model for the earliest semiclassical state of the universe, which suggests a solution for the horizon problem different from that provided by inflation.

Motivation & Objective

• To define gauge-invariant, mathematically well-posed observables in cosmological spacetimes that begin with a Big Bang and asymptote to an FRW universe with non-negative Λ.
• To address the breakdown of standard S-matrix formalism in cosmology due to singular initial conditions and multiple cosmological horizons in asymptotically de Sitter (AsDS) spacetimes.
• To propose a quantum treatment of spacetime geometry that allows for a consistent description of the earliest semiclassical state of the universe.
• To explore the implications of holographic gauge invariance—generalizing black hole complementarity—for the structure of quantum states in AsDS universes.
• To provide an alternative solution to the horizon problem distinct from inflation, based on the structure of M-theory observables in early-time cosmology.

Proposed method

• Construct a generalized S-matrix formalism for asymptotically expanding FRW universes with non-negative Λ, using boundary conditions at the Big Bang singularity and future null infinity or cosmological horizons.
• Introduce a holographic gauge invariance principle, where different choices of holographic screens (null hyperplanes or cosmological horizons) correspond to gauge-equivalent descriptions of the same physics.
• Use Bousso’s entropy bound and the finite-dimensional Hilbert space structure of AsDS spacetimes (with dimension ~exp(S_HD)) to argue that the number of physical states is finite and consistent with causal disconnection.
• Model the early universe as a state in a conformal field theory (CFT), suggesting a non-inflationary resolution to the horizon problem via the structure of M-theory observables.
• Apply the principle of black hole complementarity to cosmological horizons, showing that different observers see different effective physics due to non-commuting observables.
• Demonstrate that in the limit Λ → 0, the multiple S-matrices become Poincaré-transformed versions of each other, recovering Lorentz invariance in the flat-space limit.

Experimental results

Research questions

• RQ1 How can one define a gauge-invariant, mathematically well-defined observable in cosmological spacetimes with a Big Bang and positive cosmological constant?
• RQ2 What is the role of holographic gauge invariance in relating different S-matrices in asymptotically de Sitter spacetimes?
• RQ3 How does the principle of black hole complementarity extend to cosmological horizons, and what does it imply for the structure of the Hilbert space in AsDS universes?
• RQ4 Can the horizon problem be resolved without inflation, using the structure of M-theory observables in the early universe?
• RQ5 What is the relationship between the finite-dimensional Hilbert space of AsDS spacetimes and the indefinite growth of horizon volumes in such spaces?

Key findings

• In asymptotically de Sitter spacetimes with positive Λ, multiple S-matrices exist, related by gauge transformations that generalize the holographic screen choice, with no unique physical S-matrix.
• These S-matrices become equivalent under Poincaré transformations in the limit Λ → 0, recovering the standard S-matrix formalism of flat spacetime.
• The finite-dimensional Hilbert space of AsDS universes—of size ~exp(S_HD), where S_HD is the Hawking-De Sitter entropy—is compatible with causal disconnection of horizon volumes via a cosmological complementarity principle.
• The early universe's semiclassical state is proposed to be described by an integrable CFT, offering a non-inflationary resolution to the horizon problem.
• Observables measured by low-energy observers in different horizon volumes are non-commuting, analogous to infalling vs. asymptotic observers in black hole physics, implying a fundamental complementarity in cosmology.
• The probabilistic nature of quantum mechanics in cosmology arises not from intrinsic quantum indeterminacy but from the incompatibility of measurable observables with the global unitary evolution of the universe’s wave function.