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Including millisecond pulsars inside the core of globular clusters in pulsar timing arrays

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Abstract

We suggest the possibility of including millisecond pulsars inside the core of globular clusters in pulsar timing array experiments. Since they are very close to each other, their gravitational wave-induced timing residuals are expected to be almost the same, because both the Earth and the pulsar terms are correlated. We simulate the expected timing residuals, due to the gravitational wave signal emitted by a uniform supermassive black-hole binary population, on the millisecond pulsars inside a globular cluster core. In this respect, Terzan 5 has been adopted as a globular cluster prototype and, in our simulations, we adopted similar distance, core radius, and number of millisecond pulsars contained in it. Our results show that the presence of a strong correlation between the timing residuals of the globular cluster core millisecond pulsars can provide a remarkable gravitational wave signature. This result can be therefore exploited for the detection of gravitational waves through pulsar timing, especially in conjunction with the standard cross-correlation search carried out by the pulsar timing array collaborations.

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Only simulated data have been used in the paper.

Notes

  1. In some rare cases an MSP can also have a planetary companion (or even more than one) which may be responsible for low-frequency timing residuals. The first exoplanet was indeed discovered just thanks to that effect [16].

  2. GC MSPs might also be important for the gravitational bursts with memory (BWMs) detection. Indeed, since GCs are often characterized by a high stellar density, this makes them suitable for the occurrence of BWM. Such events would induce impact parameter-dependent timing residuals on all GC MSPs (see ref. [19] for a more detailed discussion)

  3. Note that the MSP radio emission is actually observed from the Earth, which is not an inertial reference frame. Therefore, the ToA data are transformed to the SSB reference frame.

  4. For an exhaustive derivation see ref. [22].

  5. In this paper, geometrical units c=G=1 have been adopted.

  6. In this paper, the Einstein notation, which indicates the sum over repeated indices, has been adopted.

  7. Interestingly enough, \(B1821-24A\) is also characterized by a large value of the second-order spin-frequency derivative (i.e., \({\ddot{\nu }}=29.42\pm 15.75\times 10^{-27}\,\mathrm{s}^{-3}\)) but seems to be affected only slightly, probably at a level not much larger than \(15\%\), by the GC gravitational potential [37].

  8. In accordance with what is described by the GC core mass-density distribution [40].

  9. The distribution parameters have been arbitrarily chosen with the aim to simulate the GW emission from a possible SMBHB population observable by PTAs. The results in refs. [41,42,43] have been used as reference.

  10. Note that the Pearson correlation matrix coefficients are in the interval [− 1,1], where 1 means perfect correlation, 0 means non-correlation and − 1 means perfect anti-correlation.

  11. The mean value has been calculated by ignoring the diagonal elements of the matrix.

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Acknowledgements

We warmly acknowledge Andrea Possenti, of the Istituto Nazionale di Astrofisica (INAF), for many useful discussions. We also acknowledge the support of the Theoretical Astroparticle Physics (TAsP) and Euclid projects of the Istituto Nazionale di Fisica Nucleare (INFN). We thank the Referee for the useful comments.

Funding

No funding was received.

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Authors and Affiliations

  1. Department of Mathematics and Physics “Ennio De Giorgi”, University of Salento, Via per Arnesano, I-73100, Lecce, Italy

    Michele Maiorano, Francesco de Paolis & Achille Nucita

  2. Istituto Nazionale di Fisica Nucleare INFN, Sezione di Lecce, Via per Arnesano, I-73100, Lecce, Italy

    Michele Maiorano, Francesco de Paolis & Achille Nucita

  3. Istituto Nazionale di Astrofisica INAF, Sezione di Lecce, Via per Arnesano, I-73100, Lecce, Italy

    Francesco de Paolis & Achille Nucita

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  1. Michele Maiorano
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All authors equally contributed to the paper. The first draft of the manuscript was written by Michele Maiorano and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Michele Maiorano.

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The numerical codes are available on request.

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Appendix A: The Polarization Tensor Components

Appendix A: The Polarization Tensor Components

The GW polarization tensor components of the \(+\) polarization state and the \(\times \) polarization state can be determined from Eq. (6). Using Eqs. (1) and (2) one obtains:

$$\begin{aligned} \begin{aligned} e_{11}^+&= (\sin ^2\phi )-(\cos ^2\theta )(\cos ^2\phi )\\ e_{22}^+&= (\cos ^2\phi )-(\cos ^2\theta )(\sin ^2\phi )\\ e_{33}^+&= -(\sin ^2\theta )\\ e_{12}^+&= -\sin \phi \cos \phi (\cos ^2\theta +1)\\ e_{23}^+&= \sin \theta \cos \theta \cos \phi \\ e_{13}^+&= \sin \theta \cos \theta \sin \phi \\ e_{11}^\times&= 2\cos \theta \sin \phi \cos \phi \\ e_{22}^\times&= -2\cos \theta \sin \phi \cos \phi \\ e_{33}^\times&= 0\\ e_{12}^\times&= \cos \theta (\sin ^2\phi -\cos ^2\phi )\\ e_{23}^\times&= \sin \theta \cos \phi \\ e_{13}^\times&= -\sin \theta \sin \phi \\ \end{aligned} \end{aligned}$$
(14)

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Maiorano, M., de Paolis, F. & Nucita, A. Including millisecond pulsars inside the core of globular clusters in pulsar timing arrays. Eur. Phys. J. Plus 136, 1087 (2021). https://doi.org/10.1140/epjp/s13360-021-02098-0

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