[Paper Review] The lensing time delay between gravitational and electromagnetic waves

P. Cremonese, Edvard Mörtsell|arXiv (Cornell University)|Aug 17, 2018

Pulsars and Gravitational Waves Research4 references4 citations

TL;DR

This paper investigates gravitational lensing-induced time delays between gravitational waves (GWs) and electromagnetic (EM) signals from supermassive binary black holes (SMBHBs), showing that for GWs with frequencies between 10⁻⁹ Hz and 10⁻⁶ Hz, lensing by galaxy-scale masses (≈10¹¹ M☉) can produce time delays of days to months. These delays are detectable with future Pulsar Timing Arrays (PTAs), especially with the Square Kilometre Array (SKA), enabling multi-messenger tests of gravity and GW propagation speeds.

ABSTRACT

The recent detection of gravitational waves (GWs) and electromagnetic (EM) waves originating from the same source marks the start of a new multi-messenger era in astronomy. The arrival time difference between the GW and EM signal can be used to constrain differences in their propagation speed, and thus gravitational theories. We study to what extent a non-zero time delay can be explained by gravitational lensing when the line of sight to the source passes near a massive object. For galaxy scale lenses, this delay becomes relevant for GWs with frequencies between $10^{-6}$ and $10^{-9}$ Hz, sourced by super massive binary black-holes. In addition to GWs detectable by Pulsar Timing Arrays (PTAs), we expect to find also a unique and recognizable EM signal. We show that the delay between the GW and EM signal can be of the order of days to months; within reach of future observations. The effect may become important in future multi-messenger astronomy probing of gravitational propagation and interactions.

Motivation & Objective

• To investigate whether gravitational lensing can produce measurable time delays between gravitational waves (GWs) and electromagnetic (EM) signals from the same astrophysical source.
• To assess the feasibility of detecting such time delays using Pulsar Timing Arrays (PTAs) for low-frequency GWs from supermassive binary black holes (SMBHBs).
• To determine the conditions under which lensing-induced time delays become observable, particularly for GWs with wavelengths comparable to or longer than the lens's Schwarzschild radius.
• To evaluate the signal-to-noise ratio (S/N) required for detection, considering realistic PTA configurations and future SKA sensitivity.
• To explore the potential of identifying lensed SMBHB systems via unique EM counterparts and correlating GW and EM emission in the time domain.

Proposed method

• Modeling GW propagation using wave optics for long-wavelength GWs (λ ≥ Rs), where Rs is the lens's Schwarzschild radius, to account for lensing effects.
• Applying the lensing time delay formula derived from general relativity, with time delay Δt ≈ (1/2)D_L θ² / c, where D_L is the lens redshift and θ is the angular Einstein radius.
• Using the lens mass M ≈ 10¹¹ M☉ and source redshift z ≈ 1 to compute time delays in the range of days to months for GW frequencies between 10⁻⁹ Hz and 10⁻⁶ Hz.
• Estimating the signal-to-noise ratio (S/N) for GW detection via PTAs using the formula ρ² ∝ (f² / f_orb²) × (N_p × T_obs / Δt) × (1 / σ²), where f is the GW frequency.
• Evaluating detection feasibility for current (IPTA) and future (SKA) PTA arrays by varying pulsar count (N_p), source mass, and observing cadence.
• Assessing EM counterpart detectability by considering X-ray and optical emission from SMBHBs, with challenges at high redshift (z ≈ 1).

Experimental results

Research questions

• RQ1 Can gravitational lensing by galaxy-scale masses produce measurable time delays between GWs and EM signals from SMBHBs?
• RQ2 What frequency range of GWs is most susceptible to lensing-induced time delays, and what are the expected delay magnitudes?
• RQ3 What signal-to-noise ratio is required for detectable GW signals in PTAs, and can future arrays like SKA achieve this for lensed SMBHBs?
• RQ4 How do lens mass and source redshift affect the magnitude and observability of time delays?
• RQ5 Can the time delay between GW and EM signals be used to constrain gravitational theories or probe GW propagation speeds?

Key findings

• For GWs with frequencies between 10⁻⁹ Hz and 10⁻⁶ Hz, lensing by galaxy-scale masses (≈10¹¹ M☉) can produce time delays of days to months, making them observable with future instruments.
• At redshift z ≈ 1, lensing by a 10¹¹ M☉ galaxy produces time delays of approximately 10–100 days, depending on the lensing configuration and GW frequency.
• With the international PTA (30 pulsars), a signal-to-noise ratio of ρ² ≈ 0.24 is achievable for a 10⁸ M☉ SMBHB at 10⁻⁸ Hz, below the threshold for detection (S/N > 5).
• For a 10⁸ M☉ SMBHB, a signal-to-noise ratio of S/N ≈ 5 is reached with 500 pulsars in the PTA, indicating feasibility with future SKA-based arrays.
• A minimum SMBHB mass of ≈6.3×10⁸ M☉ is required for detectable time delays at frequencies ≤10⁻⁶ Hz, assuming S/N > 5 and 500 pulsars.
• The unique EM signature from SMBHBs, combined with GW time delays, enables identification and targeted follow-up observations, enhancing multi-messenger astronomy potential.