Gravitational Wave Limits in the Presence of Red Noise

Authors: G. Hobbs, A. Lommen

Introduction

One of the main goals of pulsar timing array projects is to limit the existence of a background of gravitational waves (GWs). Jenet et al. (2006) developed a method to determine such a limit if (and only if) the pulsar timing residuals were statistically white. In the paper we used data from six pulsars from the PPTA project to obtain a limit. The remaining pulsars were rejected because their timing residuals were not white. It is not currently clear if the "noise" seen in the timing residuals of the other pulsars is due to calibration errors, to the interstellar medium or to intrinsic pulsar timing noise.

Andrea Lommen + collaborators would like to obtain a limit using their long B1855+09 data set. However, the timing residuals for this pulsar are also not white. We therefore need a technique to produce a limit in the presence of non-white noise.

Definition of the problem

We have timing residuals for one or more pulsars where

• the data are unevenly sampled (and different sampling for different pulsars)
• each pulsar data-set is of a different length
• each timing residual has a corresponding uncertainty which can vary widely between different residuals (even for the same pulsar)
• the timing residuals are not statistically white in most cases

We would like

• a limit on the amplitude of a specified GW background.

The noise in the timing residuals

We expect that the timing residuals are due to a white noise process (measurement error etc.) plus a red-noise process.

GW backgrounds

GW backgrounds are predicted to produce timing residuals with a steep power-law spectrum. A correlation will exist between the timing residuals of different pulsars (where the correlation coefficient is given by the angular separation of the pulsars).

Analysis for one pulsar

J. Verbiest et al., have considered placing a limit on the background by simulating GW backgrounds (using tempo2) of different amplitudes. For each amplitude a histogram of the rms timing residuals induced by the background is obtained. If the rms in the actual data is not consistent with the distribution from the simulation then the amplitude of the background is raised or lowered until it is just consistent. This is a limit on the GW background. However, arguments against this method include the fact that this is not taking in to account the statistics of the measured timing residuals (there is no "false-alarm" probability). It may happen that the timing noise and the GW signature cancel each other out leading to excellent rms timing residuals, but no stringent limit on the GW background.

In order to study this, and to attempt a completely general method for limiting the GW background, we have started to develop a large Monte-Carlo simulation that will simulate different red-noise data-sets in order to model the timing residuals. We use the "red-noise statistic" described in Jenet et al. (2006). 1000 red-noise data sets are simulated with a given red-noise amplitude (A) and spectral exponent (alpha) plus a specified white noise rms (wrms). The data sets are sampled in an identical manner tothe B1855+09 data-set and the post-fit residuals are studied after fitting for the standard pulsar parameters using tempo2. The statistic is calculated for the measured data set (S = 29.8) and for each realisation of the red-noise process. Histograms are plotted of "S":

Figure 1: distribution of "S" for alpha = -1, amp = 1e-6, wrms = 0

Figure 2: distribution of "S" for alpha = -2, amp = 1e-8, wrms = 0

However, the statistic "S" is normalised by the variance of the data-span. This means that the distribution of "S" only depends on the power law exponent and not on the amplitude. We have now decided to remove this normalisation and so the distribution of "S" is dependent upon the amplitude of the red-noise.

Figure 3: distribution of "S2" for alpha = -2, amp = 1e-9, wrms = 0

Some thoughts

• The statistic "S" is only one way of describing the data. We can also use other parameters such as the measured spectral exponent in the measured residuals and the rms of the timing residuals etc.
• For some amplitudes the noise is greater than the period of the pulsar and no coherent timing fit can occur. The post-fit timing residuals are then basically meaningless for this work.