[Paper Review] Quadrupolar radiation in de Sitter: Displacement memory and Bondi metric

Geoffrey Compère, Sk Jahanur Hoque+1 more|arXiv (Cornell University)|Sep 5, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy51 references3 citations

TL;DR

This paper derives the closed-form metric perturbation in de Sitter spacetime due to a localized matter source up to quadrupolar order in both generalized harmonic and Bondi gauges, including both mass and current quadrupoles. It establishes that the linear cosmological displacement memory effect at future null infinity arises from a Λ-BMS transition in the even-parity sector, but not in the odd-parity sector, providing a symmetry-based origin for memory effects in de Sitter spacetime.

ABSTRACT

We obtain the closed form expression for the metric perturbation around de Sitter spacetime generated by a matter source below Hubble scale both in generalized harmonic gauge and in Bondi gauge up to quadrupolar order in the multipolar expansion, including both parities (i.e. both mass and current quadrupoles). We demonstrate that such a source causes a displacement memory effect close to future infinity that originates, in the even-parity sector, from a $Λ$-BMS transition between the two non-radiative regions of future infinity.

Motivation & Objective

• To derive consistent, closed-form metric perturbations in de Sitter spacetime up to quadrupolar order, including both even- and odd-parity moments (mass and current quadrupoles).
• To map the generalized harmonic gauge solution to Bondi gauge, obtaining the full quadrupolar metric in closed form with finite radial expansion.
• To investigate whether the cosmological displacement memory effect at future null infinity arises from Λ-BMS symmetry transitions, as in flat spacetime.
• To correct flaws in prior treatments of quadrupolar radiation and memory in de Sitter, ensuring gauge consistency and proper tail integral evaluation.
• To confirm that boundary conditions with fixed metric are invalid in de Sitter due to metric and stress-energy tensor variations under gravitational wave propagation.

Proposed method

• Solve linearized Einstein equations in generalized harmonic gauge on de Sitter background using multipolar expansion up to quadrupole order.
• Perform explicit evaluation of the retarded tail integral, expressing it as a difference between retarded-time and past-horizon terms.
• Apply a coordinate transformation from generalized harmonic gauge to Bondi gauge, ensuring all logarithmic radial terms cancel, yielding a finite radial expansion.
• Implement boundary gauge conditions compatible with the Λ-BMS groupoid, leading to a well-defined asymptotic symmetry structure.
• Define the displacement memory effect via geodesic deviation at future null infinity and analyze its gauge invariance.
• Use the Starobinski/Fefferman-Graham map to relate Bondi fields to holographic boundary data and confirm dynamic boundary metric evolution.

Experimental results

Research questions

• RQ1 Does the cosmological displacement memory effect in de Sitter spacetime originate from a Λ-BMS transition between non-radiative vacua at future null infinity?
• RQ2 Can a consistent quadrupolar truncation of the metric perturbation be achieved in generalized harmonic gauge, including both even- and odd-parity moments?
• RQ3 How does the tail integral in de Sitter spacetime differ from the flat-space case, and can it be expressed in closed form?
• RQ4 Why do logarithmic terms cancel in the Bondi gauge metric expansion, and does this lead to a finite radial structure as in Minkowski spacetime?
• RQ5 Is the displacement memory effect gauge-invariant, and can it be fully attributed to Λ-BMS symmetry transitions in the even-parity sector?

Key findings

• The metric perturbation in generalized harmonic gauge is derived in closed form up to quadrupolar order, with explicit evaluation of the tail integral as a difference between retarded and past-horizon terms.
• The solution reduces correctly to the standard flat-space quadrupolar metric in the H → 0 limit, confirming consistency with known results.
• In Bondi gauge, the radial expansion contains only a finite number of terms, with all logarithmic divergences canceled, yielding a well-behaved metric.
• The displacement memory effect is gauge-invariant and arises from a change in the boundary metric at future null infinity.
• Only the even-parity (mass quadrupole) component of the memory effect is associated with a Λ-BMS transition; no such transition exists for the odd-parity (current quadrupole) component.
• Boundary conditions with a fixed metric are invalidated by gravitational wave propagation, as both the boundary metric and holographic stress-energy tensor vary, necessitating the use of Λ-BMS gauge.