Abstract

The merger of two dense stellar remnants including at least one neutron star is predicted to produce gravitational waves (GWs) and short-duration gamma ray bursts1,2. In the process, neutron-rich material is ejected from the system and heavy elements are synthesized by r-process nucleosynthesis1,3. The radioactive decay of these heavy elements produces additional transient radiation termed kilonova or macronova4,5,6,7,8,9,10. We report the detection of linear optical polarization, P = (0.50 ± 0.07)%, 1.46 days after detection of the GWs from GW 170817—a double neutron star merger associated with an optical macronova counterpart and a short gamma ray burst11,12,13,14. The optical emission from a macronova is expected to be characterized by a blue, rapidly decaying component and a red, more slowly evolving component due to material rich in heavy elements—the lanthanides15. The polarization measurement was made when the macronova was still in its blue phase, during which there was an important contribution from a lanthanide-free outflow. The low degree of polarization is consistent with intrinsically unpolarized emission scattered by galactic dust, suggesting a symmetric geometry of the emitting region and low inclination of the merger system. Stringent upper limits to the polarization degree from 2.45–9.48 days post-burst are consistent with the lanthanides-rich macronova interpretation.

The search for the optical counterpart to gravitational wave (GW) 17081711 quickly allowed the discovery of the possible counterpart, named AT2017gfo (also known as SSS17a), in the outskirts of the elliptical galaxy NGC4993 at about 40 Mpc14. Spectroscopic observations showed that this source was highly unusual and probably associated with the GW event and a short gamma ray burst (GRB)16. Only a few possible detections of macronova emission have been reported in the literature and in all of these studies the temporal and spectral evolution of the afterglows of on-axis short GRBs was analysed17,18,19,20. GRB 170817 A is intrinsically the weakest short GRB detected so far, which may be a regular short GRB but viewed from an off-axis orientation. The off-axis scenario is also helpful in reconciling the probability of GW–GRB association for this event21,22. Therefore SSS17a offers a unique opportunity to study a macronova emission plausibly not polluted by the GRB emission. The combination of a likely off-axis jet and the potential ability of Laser Interferometer Gravitational-Wave Observatory/Virgo data to constrain the merger geometry and orientation gave great urgency to a polarimetric measurement of the symmetry and orientation of the optical emission region(s) post-merger. Measuring the degree of polarization of the electromagnetic emission provides unique constraints on the geometry of the system and the properties of any incipient magnetic fields23,24.

Our linear polarimetry campaign consisted of a set of five observations carried out with the European Southern Observatory Very Large Telescope equipped with the FOcal Reducer and low dispersion Spectrograph (FORS2; http://www.eso.org/sci/facilities/paranal/instruments/fors.html) starting on 18 August 2017 and spanning almost ten days. After that, the transient was too faint for a reliable polarimetric analysis. Details of the observing setup, data reduction and analysis are reported in the Methods. The derived degree of linear polarization, position angle and optical brightness after instrumental corrections had been applied are given in Table 1. A complete observation log, including the dates of the observations, exposure times, filters and seeings, is reported in Table 2. The Stokes parameters for optical transient and nearby field stars for the first four epochs are shown in Fig. 1. Over the duration of our campaign, the transient showed a degree of linear polarization and a position angle fully consistent with that shown by stars in the field whose polarization is induced by dust in our Galaxy. This implies that the macronova emission is essentially unpolarized at a level driven by the photometric uncertainties and the spread of polarization shown by field stars (that is, 0.4–0.5%).

Table 1: Results of the polarimetric campaign
Table 2: Observation log
Fig. 1: Q/U Stokes parameter plot for the optical transient and several field stars near to the transient.
Fig. 1

The reported numbers in the plot indicate the observation run as in Table 2. The polarization of AT2017gfo (circles) is essentially indistinguishable from that shown by field stars (blue stars). Errors are at 1σ. The Stokes parameters are a set of four parameters that describe the full polarization state of electromagnetic radiation. Q measures the difference between radiation intensity in the horizontal and vertical direction of a given reference frame, whereas U measures the difference between directions inclined by 45° and 135° with respect to the reference frame. I is the total intensity of the radiation. Together, Q and U therefore give the amplitude and angle of the linearly polarized component of the received intensity.

GW 170817 originated in the coalescence of two neutron stars11. Numerical simulations show that these events can eject a small part of the original system into the interstellar medium3,25,26 and also form a centrifugally supported disk that is quickly dispersed in space with a neutron-rich wind7. These two different ejection mechanisms are characterized by material of differing composition. The outflows from the disk are probably lanthanide free since the synthesis of heavier elements is suppressed by the high temperature8, while the surface material is the site of an intense r-process nucleosythesis, producing heavy elements. In both cases, the spectrum should be close to a black-body, peaking in the optical in the disk-wind case and in the infrared for the lanthanide-rich material due to its very large opacity8,9. Ejecta should flow out anisotropically around the orbital plane of the system and outflows in the polar region can be produced by a strong shock driven by the merger and by processes such as viscous heating and magnetic effects in the disk27. Anisotropies induced by electron scattering can then produce some polarization28. As pointed out by Kyutoku at al.28, in the case of high optical depth to electron scattering (1) and assuming spectral lines do not significantly depolarize the global emission, the linear polarization observed from the equatorial plane could be as high as a few per cent. As is the case of supernovae, this depends on the degree of asymmetry of the photosphere. However, with respect to the supernova case, if the ejecta are mainly composed of r-process elements, the ionization degree is not particularly high7,29 and the number density of free electrons is proportionally lower. With typical parameters, the electron scattering opacity is lower than the total opacity (κ 102 cm2 g−1) by three orders of magnitude and the expected linear polarization is reduced by a similar factor compared with the electron-scattering dominant case28.

The emission from the wind could instead have a different composition and be more similar to typical supernova ejecta. A blue component was identified in the spectra of AT2017gfo16 and it showed a more rapid evolution than the redder component. The emission from the lanthanide-rich material is not expected to display any detectable linear polarization, but the situation can be significantly different for the early emission phase that is dominated by the lanthanide-free material, for which the temperature is higher (as is the ionization degree) and the electron scattering optical depth is 1. Assuming that at our first epoch the blue component was at least as important as the lanthanide-rich emission, we can derive an upper limit of 1% on the polarization of the lanthanide-free component. The low gamma-ray luminosity of GRB 170817 A12,13, despite its location in the local Universe, seems to indicate that it is an off-axis event, although the possiblity of a peculiarly weak event cannot be ruled out. A natural prediction of the off-axis scenario is that, following the deceleration of the outflow, the afterglow emission will be visible for the observer at very late times2. Although the off-axis afterglow emission could be linearly polarized30, during our polarimetric observations there was no evidence for such a component in the optical bands16. The absence of polarization in our late-time optical data is therefore quite natural, while at earlier times the limits on the polarization of the lanthanide-free component are still within the allowed possibilities since the actual polarization fraction also depends on the degree of asymmetry of the outflows. This would be different in the case of a neutron star–black-hole merging, since we would expect the ejecta to be much more anisotropic than in the neutron star–neutron star case15,28.

Our non-detection of linear polarization, which was unambiguously due to a macronova emission up to very stringent limits, is thus consistent with the theoretical expectations28 for this category of sources. It also strengthens the identification of AT2017gfo with a macronova resulting from a neutron star–neutron star coalescence associated with a short GRB and a GW event, and indirectly confirms that these sources are the site of r-process production. If these events are fairly common even in the local Universe22, it is likely that in the near future more macronovae will be observed enabling the exploration of a variety of merging conditions and system parameters such as viewing angles, mass ratios, possible off-axis afterglows, and so on. A spectro-polarimetric coverage that tracks the evolution of the phenomenon will shed light on possible deviations from the main expectations that are inaccessible with other techniques.

Methods

Our target, AT2017gfo, is the optical counterpart of the first GW signal detected by the Advanced Laser Interferometer Gravitational-Wave Observatory and Virgo network31,32 from the merger of a binary system of neutron stars, GW 170817. Data were in general of excellent quality and were reduced following standard methods (that is, frames were bias-corrected and flat fielded, and bad pixels were flagged). Photometric analysis was carried out with standard aperture photometry. An additional complication arose since the target is located in the outskirts (at 10 arcsec) of the bright galaxy NGC 4993 and for the latest epochs the galaxy light was comparable or more intense than the SSS17a brightness. We therefore modelled the external part of the galaxy with an analytical profile and effectively removed its contribution at the transient position. Photometry for the target was obtained by the acquisition frames calibrating with objects in the same field of view with the Pan-STARRS1 data archive (https://panstarrs.stsci.edu). Polarimetric observations were carried out using a Wollaston prism to split the image of each object in the field into two orthogonal polarization components that appeared in adjacent areas of the detector. A mask was used to avoid overlap of the two component images. For each position angle ϕ/2 of the half-wave plate rotator, we obtained two simultaneous images of cross-polarization at angles ϕ and ϕ + 90°. We obtained observations at position angles 0, 22.5, 45 and 67.5° of the half-wave plate. This technique allowed us to remove any differences between the two optical paths (ordinary and extraordinary ray), including the effects of seeing and airmass changes. It was also possible to estimate the polarization introduced by galactic interstellar grains along the line of sight by studying the polarization of a large number of stars in the same detector area of the target to avoid a possible dependence of the instrumental polarization on the field of view position. The optical extinction in our Galaxy is AV 0.3 mag and there is negligible extinction in the host galaxy16. This implies that dust-induced polarization up to almost 1% would be possible33, although these are only statistical estimates and large variations on the main trend may be expected.The weighted average polarization shown by a set of stars near the transient turned out to be P 0.35% with a position angle PA 56°. Polarization is a positive quantity. At a low signal-to-noise ratio it suffers from a statistical bias, which was properly corrected34. The last observation in the z-band did not allow us to derive constraining results because of the increasing difficulties due to the fading of the counterpart and the rapidly reducing visibility window. With the same setup we also observed polarized and unpolarized standard stars to convert position angles from the telescope to the celestial reference frame, and to correct for the small instrumental polarization introduced by the telescope. More details on the imaging polarimetric data analysis can be found in ref. 35.

Data availability

The data that support the plots within this paper andother findings of this study are available from the corresponding author upon reasonable request.

Additional Information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Change history

  • Correction 30 October 2017

    In the version of this Letter originally published, in the third paragraph of the text Kyutoku  et al. were not correctly cited and the sentence should have read: “As pointed out by Kyutoku at al.28, in the case of high optical depth to electron scattering (~1) and assuming spectral lines do not significantly depolarize the global emission, the linear polarization observed from the equatorial plane could be as high as a few per cent.” Also, in the Author contributions section, the final sentence should have read: “C.G.M. contributed to the writing of the paper.”

References

  1. 1.

    Eichler, D., Livio, M., Piran, T. & Schramm, D. N. Nucleosynthesis, neutrino bursts and gamma-rays from coalescing neutron stars. Nature 340,126–128 (1989).

  2. 2.

    Berger, E. Short-duration gamma-ray bursts. Annu. Rev. Astron. Astrophys. 52, 43–105 (2014).

  3. 3.

    Rosswog, S. et al. Mass ejection in neutron star mergers. Astron. Astrophys. 341, 499–526 (1999).

  4. 4.

    Li, L.-X. & Paczyński, B. Transient events from neutron star mergers. Astrophys. J. 507, L59–L62 (1998).

  5. 5.

    Rosswog, S. Mergers of neutron star-black hole binaries with small mass ratios: nucleosynthesis, gamma-ray bursts, and electromagnetic transients. Astrophys. J. 634, 1202–1213 (2005).

  6. 6.

    Metzger, B. D. et al. Electromagnetic counterparts of compact object mergers powered by the radioactive decay of r-process nuclei. Mon. Not. R. Astron. Soc. 406, 2650–2662 (2010).

  7. 7.

    Kasen, D., Badnell, N. R. & Barnes, J. Opacities and spectra of the r-process ejecta from neutron star mergers. Astrophys. J. 774, 25 (2013).

  8. 8.

    Barnes, J. & Kasen, D. Effect of a high opacity on the light curves of radioactively powered transients from compact object mergers. Astrophys. J. 775, 18 (2013).

  9. 9.

    Tanaka, M. & Hotokezaka, K. Radiative transfer simulations of neutron star merger ejecta. Astrophys. J. 775, 113 (2013).

  10. 10.

    Baiotti, L. & Rezzolla, L. Binary neutron star mergers: a review of Einstein’s richest laboratory. Rep. Prog. Phys. 80, 096901 (2017).

  11. 11.

    Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration) GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. https://doi.org/10.1103/PhysRevLett.119.161101 (2017).

  12. 12.

    Savchenko, V. et al. INTEGRAL detection of the first prompt gamma-ray signal coincident with the gravitational event GW170817. Astrophys. J. Lett. https://doi.org/10.3847/2041-8213/aa8f94 (2017).

  13. 13.

    Goldstein et al. An ordinary short gamma-ray burst with extraordinary implications: Fermi-GBM detection of GRB 170817A. Astrophys. J. Lett. https://doi.org/10.3847/2041-8213/aa8f41 (2017).

  14. 14.

    Coulter, D. et al. Swope Supernova Survey 2017a (SSS17a), the optical counterpart to a gravitational wave source. Science https://doi.org/10.1126/science.aap9811 (2017).

  15. 15.

    Kasen, D., Fernàndez, R. & Metzger, B. D. Kilonova light curves from the disc wind outflows of compact object mergers. Mon. Not. R. Astron. Soc. 450, 1777–1786 (2015).

  16. 16.

    Pian, E. et al. Spectroscopic identification of r-process nucleosynthesis in a double neutron star merger. Nature https://doi.org/10.1038/nature24298 (2017).

  17. 17.

    Tanvir, N. R. et al. A 'kilonova' associated with the short-duration γ-ray burst GRB 130603B. Nature 500, 547–549 (2013).

  18. 18.

    Yang, B. et al. A possible macronova in the late afterglow of the ‘long-short’ burst GRB 060614. Nat. Commun. 6, 7323 (2015).

  19. 19.

    Jin, Z.-P. et al. The light curve of the macronova associated with the long-short burst GRB 060614. Astrophys. J. 811, L22 (2015).

  20. 20.

    Jin, Z.-P. et al. The macronova in GRB 050709 and the GRB–macronova connection. Nat. Commun. 7, 12898 (2016).

  21. 21.

    Ghirlanda, G. et al. Short gamma-ray bursts at the dawn of the gravitational wave era. Astron. Astrophys. 594, A84 (2016).

  22. 22.

    Jin, Z.-P. et al. Short GRBs with small opening angles: implications on local neutron star merger rate and GRB/GW association. Preprint athttps://arxiv.org/abs/1708.07008 (2017).

  23. 23.

    Mundell, C. G. et al. Highly polarized light from stable ordered magnetic fields in GRB 120308A. Nature 504, 119–121 (2013).

  24. 24.

    Wiersema, K. et al. Circular polarization in the optical afterglow of GRB 121024A. Nature 509, 201–204 (2014).

  25. 25.

    Goriely, S., Bauswein, A. & Janka, H.-T. r-process nucleosynthesis in dynamically ejected matter of neutron star mergers. Astrophys. J. 738, L32 (2011).

  26. 26.

    Hotokezaka, K. et al. Remnant massive neutron stars of binary neutron star mergers: evolution process and gravitational waveform. Phys. Rev. D 88, 044026 (2013).

  27. 27.

    Kiuchi, K., Cerdà-Duràn, P., Kyutoku, K., Sekiguchi, Y. & Shibata, M. Efficient magnetic- field amplification due to the Kelvin–Helmholtz instability in binary neutron star mergers. Phys. Rev. D 92, 124034 (2015).

  28. 28.

    Kyutoku, K., Ioka, K., Okawa, H., Shibata, M. & Taniguchi, K. Dynamical mass ejection from black hole-neutron star binaries. Phys. Rev. D 92, 044028 (2015).

  29. 29.

    Tanaka, M. et al. Radioactively powered emission from black hole-neutron star mergers. Astrophys. J. 780, 31 (2014).

  30. 30.

    Granot, J., Panaitescu, A., Kumar, P. & Woosley, S. E. Off-axis afterglow emission from jetted gamma-ray bursts. Astrophys. J. 570, L61–L64 (2002).

  31. 31.

    Aasi, J. et al. Characterization of the LIGO detectors during their sixth science run. Classical and Quantum Gravity 32, 074001 (2015).

  32. 32.

    Acernese, F. et al. Advanced Virgo: a second-generation interferometric gravitational wave detector. Class. Quantum Grav. 32, 024001 (2015).

  33. 33.

    Serkowski, K., Mathewson, D. S. & Ford, V. L. Wavelength dependence of interstellar polarization and ratio of total to selective extinction. Astrophys. J. 196, 261–290 (1975).

  34. 34.

    Plaszczynski, S., Montier, L., Levrier, F. & Tristram, M. A novel estimator of the polarization amplitude from normally distributed Stokes parameters. Mon. Not. R. Astron. Soc. 439, 4048–4056 (2014).

  35. 35.

    Wiersema, K. et al. Detailed optical and near-infrared polarimetry, spectroscopy and broad-band photometry of the afterglow of GRB 091018: polarization evolution. Mon. Not. R. Astron. Soc. 426, 2–22 (2012).

Download references

Acknowledgements

This study was based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under European Southern Observatory programme 099.D-0116. We thank the European Southern Observatory—Paranal staff for carrying out excellent observations under difficult conditions during a hectic period. We also acknowledge partial funding from Agenzia Spaziale Italiana-Istituto Nazionale di Astrofisica grant I/004/11/3. K.W., A.B.H., R.L.C.S. and N.R.T. acknowledge funding from the Science and Technology Facilities Council. J.H. was supported by a VILLUM FONDEN Investigator grant (project number 16599). Y.Z.F. was supported by the National Natural Science Foundation of China under grant 11525313. C.G.M. acknowledges support from the UK Science and Technology Facilities Council. K.T. was supported by Japan Society for the Promotion of Science grant 15H05437 and a Japan Science and Technology Consortia grant. J.M. acknowledges the National Natural Science Foundation of China 11673062 and the Major Program of the Chinese Academy of Sciences(KJZD-EW-M06).

Author information

Affiliations

  1. Istituto Nazionale di Astrofisica / Brera Astronomical Observatory, via Bianchi 46, 23807, Merate (LC), Italy

    • S. Covino
    • , A. Melandri
    • , P. D’Avanzo
    • , M. G. Bernardini
    • , S. Campana
    • , W. Gao
    •  & G. Tagliaferri
  2. Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK

    • K. Wiersema
    • , A. B. Higgins
    • , N. R. Tanvir
    •  & R. L. C. Starling
  3. Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China

    • Y. Z. Fan
    •  & Z. P. Jin
  4. Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai, 980-8578, Japan

    • K. Toma
  5. Astronomical Institute, Tohoku University, Sendai, 980-8578, Japan

    • K. Toma
  6. Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    • C. G. Mundell
  7. Istituto Nazionale di Astrofisica / Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna, Via Gobetti 101, 40129, Bologna, Italy

    • E. Palazzi
    • , E. Pian
    •  & A. Rossi
  8. Laboratoire Univers et Particules de Montpellier, Université de Montpellier, Centre National de la Recherche Scientifique / IN2P3, Montpellier, 34095, France

    • M. G. Bernardini
  9. Gran Sasso Science Institute, 67100, L’Aquila, Italy

    • M. Branchesi
  10. Istituto Nazionale di Fisica Nucleare / Laboratori Nazionali del Gran Sasso, 67100, L’Aquila, Italy

    • M. Branchesi
  11. Istituto Nazionale di Astrofisica / Osservatorio Astronomico di Roma, Via di Frascati, 33, 00078, Monteporzio Catone, Italy

    • E. Brocato
  12. Istituto Nazionale di Astrofisica / Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy

    • S. di Serego Alighieri
  13. Commissariat à l’Énergie Atomique Saclay—Direction de la Recherche Fondamentale/Irfu/Dèpartement d’Astrophysique, 91191, Gif-sur-Yvette, France

    • D. Götz
  14. Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100, Copenhagen, Denmark

    • J. P. U. Fynbo
    • , J. Hjorth
    •  & D. Malesani
  15. Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210046, China

    • W. Gao
  16. Centre for Astrophysics and Cosmology, University of Nova Gorica, Vipavska 11c, 5270, Ajdovščina, Slovenia

    • A. Gomboc
  17. Space Telescope Science Institute, Baltimore, MD, 21218, USA

    • B. Gompertz
  18. Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, 85748, Garching, Germany

    • J. Greiner
  19. Anton Pannekoek Institute, University of Amsterdam, Science Park 904, 1098XH, Amsterdam, The Netherlands

    • L. Kaper
    •  & R. A. M. J. Wijers
  20. Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778, Tautenburg, Germany

    • S. Klose
  21. Astrophysics Research Institute, Liverpool John Moores University, ic2, Liverpool Science Park, 146 Brownlow Hill, Liverpool, L3 5RF, UK

    • S. Kobayashi
    •  & I. Steele
  22. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    • D. Kopac
  23. Department of Physics, University of Warwick, Coventry, CV4 7AL, UK

    • A. J. Levan
  24. Yunnan Observatories, Chinese Academy of Sciences, 650011, Kunming, Yunnan Province, China

    • J. Mao
  25. Istituto Nazionale di Astrofisica / Istituto di Astrofisica Spaziale e Fisica Cosmica Milano, via E. Bassini 15, 20133, Milano, Italy

    • R. Salvaterra
  26. Department of Astronomy, University of Maryland, College Park, MD, 20742, USA

    • E. Troja
  27. Department of Physics, The George Washington University, 725 21st Street NW, Washington, DC, 20052, USA

    • C. Kouveliotou
    •  & A. J. van der Horst
  28. Astronomy, Physics, and Statistics Institute of Sciences, Washington, DC, 20052, USA

    • A. J. van der Horst

Authors

  1. Search for S. Covino in:

  2. Search for K. Wiersema in:

  3. Search for Y. Z. Fan in:

  4. Search for K. Toma in:

  5. Search for A. B. Higgins in:

  6. Search for A. Melandri in:

  7. Search for P. D’Avanzo in:

  8. Search for C. G. Mundell in:

  9. Search for E. Palazzi in:

  10. Search for N. R. Tanvir in:

  11. Search for M. G. Bernardini in:

  12. Search for M. Branchesi in:

  13. Search for E. Brocato in:

  14. Search for S. Campana in:

  15. Search for S. di Serego Alighieri in:

  16. Search for D. Götz in:

  17. Search for J. P. U. Fynbo in:

  18. Search for W. Gao in:

  19. Search for A. Gomboc in:

  20. Search for B. Gompertz in:

  21. Search for J. Greiner in:

  22. Search for J. Hjorth in:

  23. Search for Z. P. Jin in:

  24. Search for L. Kaper in:

  25. Search for S. Klose in:

  26. Search for S. Kobayashi in:

  27. Search for D. Kopac in:

  28. Search for C. Kouveliotou in:

  29. Search for A. J. Levan in:

  30. Search for J. Mao in:

  31. Search for D. Malesani in:

  32. Search for E. Pian in:

  33. Search for A. Rossi in:

  34. Search for R. Salvaterra in:

  35. Search for R. L. C. Starling in:

  36. Search for I. Steele in:

  37. Search for G. Tagliaferri in:

  38. Search for E. Troja in:

  39. Search for A. J. van der Horst in:

  40. Search for R. A. M. J. Wijers in:

Contributions

All authors contributed to the work presented in this paper. S.C. and K.W. coordinated the data acquisition, analysed the data and wrote the paper. A.B.H., A.M., P.D., E.P. and N.T. contributed to the data analysis. Y.F. and K.T. contributed to the theoretical discussion. C.G.M. contributed to the writing of the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to S. Covino.