[Paper Review] Probing massive gravitons in $f(R)$ with lensed gravitational waves

Vipin Kumar Sharma, Sreekanth Harikumar+3 more|arXiv (Cornell University)|Oct 16, 2023

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper investigates modified gravitational wave (GW) dispersion and lensing in $f(R) = R^{1+\epsilon}$ gravity, showing that massive scalar modes—particularly longitudinal and breathing polarizations—remain independent of mass for $\epsilon \ll 1$. It demonstrates that lensed GWs in the milli-Hertz band, especially from $10^3$–$10^6\,M_\odot$ lenses, exhibit detectable deviations from general relativity, enabling future constraints on scalaron mass $m_\phi \approx \mathcal{O}(10^{-15})$ eV via LISA.

ABSTRACT

We investigate the novel features of gravitational wave solutions in $f(R)$ gravity under proper gauge considerations in the shifted Ricci scalar background curvature ($R^{1+ε}$). The solution is further explored to study the modified dispersion relations for massive modes at local scales and to derive constraints on $ε$. Our analysis yields new insights as we scrutinize these dispersion effects on the polarization (modified Newman-Penrose content) and lensing properties of gravitational waves. It is discovered that the existing longitudinal scalar mode, and transverse breathing scalar mode are both independent of the mass parameter for $ε<<1$. Further, by analysing the lensing amplification factor for the point mass lens model, we show that lensing of gravitational wave is highly sensitive to these dispersion effects in the milli-Hertz frequency (wave optics regime). It is expected that ultra-light modes, having mass about $\mathcal{O} (10^{-15})$ eV for $ε<<1 (\approx 10^{-7})$ lensed by ($10^3\leq M_{Lens}\leq 10^6$)$M_\odot$ compact objects are likely to be detected by the advanced gravitational wave space-borne detectors, particularly within LISA's (The Laser Interferometer Space Antenna) sensitivity band.

Motivation & Objective

• To investigate the behavior of massive graviton modes in $f(R) = R^{1+\epsilon}$ gravity under shifted Ricci scalar background curvature.
• To derive modified dispersion relations for gravitational waves in the $f(R)$ framework, particularly for $\epsilon \ll 1$.
• To analyze the polarization structure of GWs using the Newman-Penrose formalism, focusing on scalar modes.
• To assess the sensitivity of gravitational wave lensing to massive modes, especially in the wave optics regime.
• To constrain the scalaron mass using lensing amplification in the LISA frequency band.

Proposed method

• Adopt a $f(R) = R^{1+\epsilon}$ modification to the Einstein-Hilbert action, with $\epsilon \ll 1$, to model deviations from general relativity.
• Apply gauge-invariant perturbation theory in a shifted Ricci scalar background ($R^{1+\epsilon}$) to derive modified GW solutions.
• Use the modified Newman-Penrose formalism to analyze polarization states, including longitudinal and breathing modes.
• Compute the lensing amplification factor in the wave optics regime for point mass lenses, incorporating frequency-dependent dispersion from massive modes.
• Derive the dimensionless frequency dependence of the amplification factor and phase, comparing $f(R)$ with GR predictions.
• Constrain the scalaron mass by comparing predicted lensing deviations with LISA’s sensitivity band ($\sim$ mHz) and lens masses $10^3$–$10^6\,M_\odot$.

Experimental results

Research questions

• RQ1 How do massive scalar modes in $f(R) = R^{1+\epsilon}$ gravity affect the dispersion relations of gravitational waves at local scales?
• RQ2 What is the polarization structure of GWs in $f(R)$ gravity, particularly the behavior of longitudinal and breathing modes, and how does it depend on $\epsilon$ and scalaron mass?
• RQ3 How does the lensing amplification factor of GWs differ between $f(R)$ gravity and general relativity in the wave optics regime?
• RQ4 Can lensed GWs from compact lenses ($10^3$–$10^6\,M_\odot$) in the mHz band reveal deviations from GR due to massive graviton modes?
• RQ5 What is the minimum detectable scalaron mass in $f(R)$ gravity that could be constrained by future space-based GW detectors like LISA?

Key findings

• For $\epsilon \ll 1$, the longitudinal and transverse breathing scalar modes in $f(R)$ gravity are independent of the scalaron mass, indicating a robust polarization structure in the weak-field limit.
• The scalaron mass is constrained to $m_\phi \approx \mathcal{O}(10^{-15})$ eV when $\epsilon \approx 10^{-7}$, consistent with Solar System curvature background ($R^{(B)} \approx 10^{-35}$ eV$^2$).
• In the wave optics regime, lensing amplification factors for $f(R)$ gravity deviate significantly from unity at low frequencies, unlike in GR where they tend to 1.
• The amplification factor in $f(R)$ depends on both the lens mass and the dimensionless frequency, unlike in GR where it depends only on the impact parameter.
• Deviations in the amplification factor and phase are most pronounced for $f(R)$ with $m_\phi \approx \mathcal{O}(10^{-15})$ eV and lens masses $10^3$–$10^6\,M_\odot$, making them detectable by LISA.
• The fractional variation in GW speed due to massive modes falls within the operational sensitivity range of LISA, enabling direct testing of $f(R)$ gravity through lensed GW observations.