Finite distance effects on the Hellings-Downs curve in modified gravity

arXiv:2407.21567 · gr-qc, astro-ph.CO · Submitted 2024-07-31 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "Finite distance effects on the Hellings-Downs curve in modified gravity".

Jocelyn: The gist The overlap reduction function for subluminal gravitational wave propagation in modified gravity is sensitive to finite distance effects,

Vera: First, who's behind it and why it matters.

Paper summary: Vera: We started by looking at how to reformulate harmonic coefficients using spherical Bessel functions, which is a bit technical but necessary to get past some issues with current approximations for the Hellings-Downs curve in subluminal gravitational wave propagation.

Jocelyn: And the core thesis of this paper, "Finite distance effects on the Hellings-Downs curve in modified gravity," is that these finite distance effects introduce an effective cut-off in the spherical harmonics decomposition at multipole numbers around about sqrt one - v squared kL <ref:2407.21567#pg1,finite distance effects introduce an effective cut-off in the spherical harmonics>.

Subrahmanyan: So they are arguing that this cut-off is necessary because current approximations to the Hellings-Downs curve for subluminal propagation, specifically when v < one diverge at small angular pulsar separation <ref:2407.21567#pg1,current approximations to the Hellings-Downs curve for subluminal>.

Vera: They then show that this introduced cut-off actually reconciles the overlap reduction function with an autocorrelation value from a previous reference, ref twenty-six, by showing an exponential suppression for greater than that cut-off <ref:2407.21567#pg2>.

Jocelyn: It matters because they then compute the ORF (two point one) by focusing on the coincident pulsar limit, where previous approximations gave a divergent result, and they show that for v < one the ORF (four point three) is well approximated by summing only up to cut = sqrt one - v squared kL.

Subrahmanyan: This means that when v is less than one, we don't need to sum all the way to infinity in the spherical harmonic expansion; we can stop at that specific multipole number determined by k and L.

Vera: They also confirm that the asymptotic value for small angles, where xi goes to zero, matches an analytical approximation given by Eq. (four point seven), which is (kL, xi to zero) about three pi kL v squared (one - v two) squared times a normalization factor.

Jocelyn: So the implication here is that they’ve found a way to make the ORF for v < one finite and correctly approaches that limiting value, which is what we expect from the underlying physics of subluminal wave propagation <ref:2407.21567#pg1>.

Subrahmanyan: This work connects directly to how we test modified gravity because it provides a mechanism—this cut-off—that allows us to use these PTA measurements in a way that aligns with established theoretical limits.

Conclusion: Vera: So, looking at the paper "Finite distance effects on the Hellings-Downs curve in modified gravity," it really boils down to this: when gravitational waves travel slower than light, v < one the standard way we calculate how two pulsars overlap gets messy at very close angles <ref:2407.21567#pg1,Finite distance effects on the Hellings-Downs curve in modified gravity>.

Jocelyn: The authors found that this messiness is actually caused by finite distance effects, and they quantify that by showing a cut-off at about sqrt one - v squared kL in the spherical harmonic decomposition <ref:2407.21567#pg1>.

Subrahmanyan: In simple terms, this means the shape of the overlap reduction function depends on both how far away your pulsars are and how fast the gravitational wave is moving relative to light.

Vera: And what this means for us as observers is that if we want to use pulsar timing arrays to test theories where gravity modifies propagation speed, we have a specific cutoff scale that matters for our data analysis.

Jocelyn: So, the main takeaway from this paper is that finite distance effects are not just noise; they introduce a physical structure into the signal at certain angular scales dictated by v and kL.

Subrahmanyan: It gives us a concrete way to interpret what we see in pulsar timing arrays, suggesting that if we measure deviations from General Relativity, those deviations will show up in how the overlap function behaves at these specific multipole numbers.

Vera: The authors’ work is important because it confirms that this function is finite and has a defined limit when you look at very small angular separations.

Jocelyn: It gives us a better roadmap for interpreting the data from pulsars, showing exactly how to account for these distance-dependent corrections when we are trying to probe physics beyond Einstein's gravity.

Guillem Domenech, Apostolos Tsabodimosa

Institute for Theoretical Physics, Leibniz University Hannover · Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut

gr-qc, astro-ph.CO

Submitted: 2024-07-31

Updated: 2024-08-09

The gist: The gist The overlap reduction function for subluminal gravitational wave propagation in modified gravity is sensitive to finite distance effects, introducing an effective cut-off at multipole

Key concepts

Overlap Reduction Function (ORF): The ORF, also known as the Hellings-Downs curve in General Relativity, measures the angular correlation between two pulsars. It is used to test gravitational wave phase velocity; deviations from the GR prediction indicate modifications to gravity.
Spherical Bessel Functions: These special mathematical functions are used to reformulate harmonic coefficients in the ORF calculation. By expressing the curve in terms of these functions, the authors can calculate complex integrals related to gravitational wave propagation more effectively.
Cut-off at $\ell \sim \sqrt{1 - v^2kL}$: This specific multipole number limit represents a physical boundary where approximations for the ORF become accurate. It arises because the angular separation corresponding to this multipole is smaller than the typical features in the pulsar antenna response pattern.

Terminology

Pulsar Timing Array — 22× in this paper
Overlap Reduction Function — 12× in this paper
pulsar timing arrays — 7× in this paper · explained in 2 episodes
Gravitational Wave Background — 4× in this paper · explained in 2 episodes
Total Angular Momentum — 4× in this paper
Polarization — 3× in this paper · explained in 2 episodes
In General Relativity — 2× in this paper
Inflation — 2× in this paper · explained in 2 episodes
Non-Gaussianity — 2× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Effective Field Theory — 1× in this paper
Effective Field Theory (EFT) — 1× in this paper · explained in 2 episodes
Frequency Dependence — 1× in this paper · explained in 2 episodes
Overlap Reduction Function (ORF) — 1× in this paper · explained in 2 episodes

Summary

The gist The overlap reduction function for subluminal gravitational wave propagation in modified gravity is sensitive to finite distance effects, introducing an effective cut-off at multipole numbers around l ∼ √1 − v2kL, which reconciles analytical approximations with autocorrelation values

Introduction and Motivation

The unique nature of pulsars allows for their utilization in applications ranging from Galactic maps to the exploration of Gravitational Waves (GWs) in the frequency range of 1–100 nHz Of particular interest is the Overlap Reduction Function (ORF), commonly denoted as Γab(ξab), where ξab is the angle between pulsars “a” and “b”. In General Relativity (GR), the ORF is known as the Hellings-Downs curve. It has been shown that the ORF can also be used to test the phase velocity of GWs, as a phase velocity of v ≠ 1 is a signature of modifications of gravity For example, it is possible to have v 1 in scalar-tensor theories of gravity. PTAs provide a complementary probe to modified gravity because they can measure the ORF Γab(kL, ξab), where k is the wavenumber of the GW and L the distance to the pulsars.

Reformulation and Analytical Approximations

The paper investigates finite distance effects by reformulating harmonic coefficients in terms of spherical Bessel functions The starting point involves expressing the Hellings-Downs curve in a spherical harmonic expansion Γ(kL, ξ) = C X∞l=2 alPl(cos ξ). An alternative formulation connects the Legendre functions to spherical Bessel functions via an integral of spherical Bessel functions This leads to the main formula for cl given by Eq. (2.9). The calculation of Il(kL, v), which is central to determining cl, is reduced to calculating a definite integral of a spherical Bessel function.

Finite Distance Effects and Cut-offs

The analysis reveals that the approximations derived in Refs. [23, 25] are valid up to a cut-off at l ∼ √1 − v2kL This cut-off is expected because the angular separation corresponding to the multipole number (roughly ξ ∼ 1/l) is smaller than the typical features in the antenna response pattern (which is of the order of 1/kL). The paper shows that this cut-off reconciles the ORF of Refs. [23, 24] with the autocorrelation value of Ref. [26] In Sec. III, an exponential suppression in the latter case is shown for l > √1 − v2kL.

Results and Conclusions

The ORF (2.1) is computed by focusing on the coincident pulsar limit, where previous approximations gave a divergent result For v < 1, the ORF (4.3) is well approximated by summing only up to the cut-off at lcut = √1 − v2kL The asymptotic value for ξ → 0 matches the analytical approximation given by Eq. (4.7), which is Γ(kL, ξ → 0) ≈ 3π/8 kL v2(1 − v2)2 This confirms that the ORF for v < 1 is finite and correctly approaches the limiting value for ξ → 0 The effect of √1 − v2 is slightly more apparent in the case of v = 1/2.

Acknowledgments

G.D. would like to thank B. Allen, M. Sasaki, K. Schmitz and J. Tr¨ankle for their helpful discussions. We also thank R. C. Bernardo and Kin-Wang Ng for useful correspondence. G.D. is supported by the DFG under the Emmy-Noether program grant no DO 2574/1-1, project number 496592360.

Appendix A

In this appendix, we compare our formulas with those derived in the total angular momentum formalism [59] We show that Eq. (A8) can be written in terms of Eq. (2.10) as I(2l)(kL, v) = −1/l(l + 2)(l − 1) v2(l + 2) − l I∗l(kL, v) − 2ivI∗n(kL, v) - eikLv jn(kL)/jn(kL−l)! We conclude that Eq. (A10) identifies the functions defined in (A4) and (2.10) via I∗l(kL, v) = I(0l)(kL, v).

References

[1] M. V. Sazhin, Opportunities for detecting ultralong gravitational waves, Sov. Astron. 22, 36 (1978)

[2] S. L. Detweiler, Pulsar timing measurements and the search for gravitational waves, Astrophys. J. 234, 1100 (1979)

[3] R. w. Hellings and G. s Downs, Upper limits on the isotropic gravitational radiation background from pulsar timing analysis, Astrophys. J. Lett. 265, L39 (1983)

[4] S Chen, A Sesana, and C J Conselice, Constraining astrophysical observables of Galaxy and Supermassive Black Hole Binary Mergers using Pulsar Timing Arrays, Mon Not Roy Astron Soc. 488, 401 (2019), arXiv:1810.04184 [astro-ph.GA]

[5] D J Reardon et al., Timing analysis for 20 millisecond pulsars in the Parkes Pulsar Timing Array, Mon Not Roy Astron Soc. 455, 1751 (2016), arXiv:1510.04434 [astro-ph.HE]

[6] S R Taylor, R van Haasteren, and A Sesana, From Bright Binaries To Bumpy Backgrounds: Mapping Realistic Gravitational Wave Skies With Pulsar-Timing Arrays, Phys Rev. D 102, 084039 (2020), arXiv:2006.04810 [astro-ph.IM]

[7] Y.-M Wu, Z.-C Chen, Y.-C Bi, and Q.-G Huang, Constraining the graviton mass with the NANOGrav 15 year data set, Class. Quant. Grav. 41, 075002 (2024), arXiv:2310.07469 [astro-ph.CO]

[8] L S Finn and P J Sutton, Bounding the mass of the graviton using binary pulsar observations, Phys Rev. D 65, 044022 (2002), arXiv:gr-qc/0109049

[9] R C Bernardo and K W Ng, Constraining gravitational wave propagation using pulsar timing array correlations, Phys Rev. D 107, L101502 (2023), arXiv:2302.11796 [gr-qc]

[10] C de Rham and A J Tolley, Speed of gravity, Phys Rev. D 101, 063518 (2020), arXiv:1909.00881 [hep-th]

[11] R van Haasteren and M Vallisneri, New advances in the Gaussian-process approach to pulsar-timing data analysis, Phys Rev. D 90, 104012 (2014), arXiv:1407.1838 [gr-qc]

[12] L Lentati, M P Hobson, and P Alexander, Bayesian Estimation of Non-Gaussianity in Pulsar Timing Analysis, Mon Not Roy Astron Soc. 444, 3863 (2014), arXiv:1405.2460 [astro-ph.IM]

[13] J Ellis and N Cornish, Transdimensional Bayesian approach to pulsar timing noise analysis, Phys Rev. D 93, 084048 (2016), arXiv:1601.00650 [astro-ph.IM]

[14] J A Ellis, X Siemens, and R van Haasteren, An Efficient Approximation to the Likelihood for Gravitational Wave Stochastic Background Detection Using Pulsar Timing Data, Astrophys. J.

Improvements for AI systems

Improve gravitational wave background detection by incorporating finite distance effects into overlap reduction function approximations, specifically by using an effective cut-off in the spherical harmonics decomposition at l ∼ √ 1 − v2 kL. This allows AI systems to predict and correct divergences in the Hellings-Downs curve approximations for subluminal gravitational waves where current analytical methods fail.
Develop a more accurate model for the overlap reduction function by utilizing an alternative formulation of harmonic coefficients in terms of one integral of spherical Bessel functions, as detailed in Eq. (2.10). This allows AI to handle the complex integration required for calculating the ORF, which is currently intractable analytically due to the lack of known solutions for Il(kL, v).
Enhance the ability to test modified gravity theories by providing precise analytical templates for the Overlap Reduction Function (ORF) that correctly approaches the limiting value for ξ → 0 in PTAs. This enables AI systems to distinguish between General Relativity and modified gravity by comparing numerical ORF results against these high-precision limits, such as Eq. (4.7).
Improve parameter estimation in pulsar timing arrays by using the derived cut-off condition, lcut ∼ √ 1 − v2kL, to determine the relevant multipole range for gravitational wave detection. This allows AI to focus computational resources on multipoles that actually contribute significantly to the observable ORF, leading to more robust statistical significance claims.
Refine waveform modeling in massive gravity and scalar-tensor theories by incorporating frequency-dependent phase velocity signatures into predictive models, as suggested by the paper's discussion of massive gravity one finds that the phase velocity is frequency dependent. This allows AI to better constrain parameters like graviton mass or scalar field couplings based on PTA correlations.