Application of the cross-spectrum method to the timing of millisecond
Person in charge: François Vernotte, UTINAM, Observatory THETA, UBFC
Laboratory: UTINAM Institute, UMR UFC/CNRS 6213, Besançon (France)
- Enrico Rubiola, Femto-ST, Observatory THETA, UBFC
- Ismaël Cognard, LPC2E, Region Centre Observatory (OSUC), Orléans University (UO)
- Gilles Theureau, LPC2E, OSUC, UO
- Lucas Guillemot, LPC2E, OSUC, UO
The cross-spectrum method (CSM) has been developed by Rubiola for estimating the phase noise of an oscillator by using two measuring instruments simultaneously. Under certain assumptions, this method rejects the instrument background noise and converges to the phase power spectral density (PSD), even if it is significantly lower than the background. The aim of this project is to apply this method to the phase noise analysis of the millisecond pulsars (MsP) as it is performed at Nançay Observatory in the framework of the European Pulsar Timing Array (EPTA). This phase noise is composed of a high observational white noise due to the very poor signal to noise ratio of the MsP observations and very low frequency noise which could come either from the pulsar itself or from gravitational waves inducing fluctuations of the space-time metric between the pulsar and the Earth but have never been observed. The detection of the spectral signature of such gravitational waves is a major scientific issue since, firstly, it would be the first direct detection of gravitational waves in the frequency range from nHz to uHz and, secondly, it would provide tight constraints over cosmological theories. Finally, far from diminishing the interest of this project, the results of LIGO further strengthen the emulation of the international community to detect other types of sources with other types of experimentation.
A pulsar is a neutron star emitting a beam of electromagnetic waves in the manner of a lighthouse, so it is possible to receive a RF pulse at each turn of the pulsar. The rotation of the millisecond pulsars (MSP) exhibit extremely low time instabilities. The study of these instabilities is useful not only to understand the physics of the pulsar but also to potentially detect gravitational waves (GW) in the range of nanoHertz (nHz) propagating between the MSP and us. Various types of GW sources are expected from, firstly, supermassive binary black holes (or a background of binary black holes) and from, secondly, a background noise of cosmic origin (string cosmic or relics of inflation). In all cases, these are major issues for astrophysics, as well as for fundamental physics.
The main difficulty is the very low signal to noise ratio which alters the pulse receptions by the radio telescopes. This leads to a strong white noise hiding the expected gravitational instabilities. However, these instabilities corresponding to a very low frequency noise, the time series analysis of the time of arrival (TOA) of pulses over several years should be able to detect them. On the other hand, as GW affect all MsP, one can detect their passage by analyzing the correlations between the TOA of several pulsars. The timing of a set of very stable MsP to detect GW is called a pulsar timing array (PTA).
Today, three consortia (European Pulsar Timing Array — EPTA — in Europe, North America NanoGRAV, Parkes Pulsar Timing Array — PPTA — in Australia) are involved in a fruitful competition to achieve the first direct detection of gravitational waves by observing PTA. This competition presents:
- theoretical aspects, by developing GW signature models and by foreseeing the effects of the interstellar medium (ISM),
- instrumental aspects, by increasing the RF bandwidth of MsP observation and by improving the dedispersion systems in order to reduce the impact of ISM,
- observational aspects, by programming coordinated observation campaigns of the same MsP by several radio telescopes,
- and finally signal processing aspects, by optimizing the statistical methods of TOA extraction as well as the time series analysis of TOA.
In particular, we want to be involved in this last point by using a method derived from the cross-spectrum method (CSM) developed for the analysis of oscillator phase noise. This method is based on the use of two independent instruments simultaneously measuring the phase of a single oscillator. In the case of the CS-pulsar project, this method can be varied according to three approaches:
- First, a pulsar can be likened to the oscillator and two radio telescopes, observing simultaneously the same MsP, are likened to the independent measurement instruments; this method should reject the observational noise contaminating the TOA sequences obtained with each radio telescope.
- Secondly, two pulsars can be observed with one only radio telescope; in this case, the intrinsic noise of each pulsar is rejected and the GW affecting the position of the Earth is highlighted.
- Finally, a method inspired by the MSC can also be used to improve the detection of the individual pulses in the raw signals received from pulsars by two radio telescopes.
It is important to remember that the CSM is further enhanced by increasing the number of radio telescopes or of pulsars simultaneously used.
Even if the first direct detection of GW has already been performed, let us remind that the GW of cosmological origin have not yet been identified. The use of the CSM to improve this detection could be decisive.
The pulsar TOA residuals are the difference between the observed TOA and the computed TOA. This computation is based upon a perfectly stable pulsar frequency relative to TAI (the pulsar frequency is one of the parameters estimated from TOA sequences). Therefore, these residuals are time error samples x(t) exactly like those obtained by comparing two oscillators. The methods conventionally used in time and frequency metrology are then applicable, with the notable difference that the residuals of TOA are very far from being equally spaced. Thus, the CSP can still be used but the FFT-like algorithms are no longer available to compute the spectrum of the residuals.
The principle consists then in processing two sequences of residuals obtained from the same pulsar, at the same time interval, with two different radio telescopes. We consider that each of the spectra of these residuals is composed of a white noise coming from the in-scope radio telescope and of a common low-frequency signal characteristic of the GW spectral signature. The principle of the CSP amounts to compute the square of the modulus of the product of the two spectra. It is shown that in this case, the level of white noise is zero-mean while the averaged low frequency noise converges to the GW noise level. The figure below is a simulation of the PSD of two signals coming from two radio telescopes (Sx1(f) and Sx2(f)) each consisting of a white noise due to each radio telescope and a very low frequency noise, with a low level, common to both signals (SG(f)). The resulting CSP (S12(f)) clearly shows that the white noises are strongly rejected and that the low frequency noise is clearly detected at the lowest frequencies.
The PSD of the gravitational wave noise of this pulsar is depicted by the curve Sg(f) and S12(f) is the resulting CSP.
Once optimized, this method can be adapted to process the signals coming from several pulsars and observed with several radio telescopes to better target the gravitational waves lying on both lines of sight. Finally, this method will be further enhanced by using 3, 4, ... , N different radio telescopes (and possibly pulsars).
Similarly, determining the TOA could be improved by using a method inspired by the CSM. For each channel (each RF frequency band) a TOA time series is obtained by computing the correlation function between the received signal and a pulse shape model which is specific to each pulsar. Each TOA is then a maximum of the correlation function. By jointly using the signals obtained during the same time interval but with sufficiently distant channels for assuming their noises are uncorrelated, we can better reject the noise from the correlation function and, therefore, we can date more precisely the TOA. This method also applies to the case of raw signals from two radio telescopes, on the express condition that these signals are well synchronized, at the nanosecond level of absolute date. It’s one of the purposes of the ERC European Large Array for Pulsars (LEAP) that collects, for 2 years, a large number of simultaneous observations of MSP by 3 or 4 radio telescopes. These data can also be used with this new method.
We wish to recruit a young PhD who, ideally, would have expertise in metrology and statistics, including the Bayesian approach as well as a good knowledge of calculation tools. But he could also come from the pulsar community and have focused his thesis on the analysis of the timings of pulsars. Indeed, as part of this post-doctoral fellowship, the researcher will have to make the link between time and frequency metrology and radio astronomy, his good knowledge of statistics providing such a link.
This project comes as the entire international pulsar community tracks all sources of disturbances affecting the TOA of MSP in order to characterize and, ideally, to overcome it. The arrival of a metrologist in the European collaboration EPTA, bringing moreover a method not yet used in this field, will be a bonus.
At the national level, this study will deepen the collaboration between researchers in time-frequency (UTINAM and Femto-ST) and the astronomers of the Nançay observatory in order to exchange methods and know-how, especially in time series analysis. The post-doctoral fellow will also be involved in the European collaboration EPTA, ideally by training with the team responsible for the statistical analysis of data MsP in Cambridge (UK) and with the working group in continuity of ERC LEAP (Bonn, Germany) responsible for the collection of multi-radio-telescope data. In addition, an extended stay (1 month or more) within one of two teams from other countries (NanoGRAV or PPTA) will allow him to have a clear idea of the state of the art of this issue while providing him strong international collaborations.
Date and duration
This contract will start January 1st, 2017. Its duration is 18 months (1 year, renewable 6 months).
The salary will be determined by the candidate’s professional experience and should be at least 2000 € monthly (gross salary plus social security including contribution for retirement and medical insurance).
The application deadline is February 5, 2017. The candidate should hold a PhD before January 1st, 2017.
The applications can be sent to: firstname.lastname@example.org