model parameters that have the greatest effect on the low-frequency sensitivity. These are the quadratic and linear terms in the spin down model for each pulsar, which take the form

\[ P ( t ) = P _ { 0 } + { \dot { P } } _ { 0 } t + { \ddot { P } } _ { 0 } t ^ { 2 } + \dots \qquad \qquad ( 1 7 ) \]

Here, \( P _ { 0 } \) is the initial spin period of the pulsar, and \( \dot { P } _ { 0 } \) and \( \ddot { P } _ { 0 } \) encode the way that this spin evolves in time.

In order to speed up our analysis, we choose to perform our calculations in the Fourier domain. We therefore need to understand how the quadratic and linear terms in the timing model translate to effects on the timing residuals as a function of frequency. The Fourier trans-form of the timing model is given by

\[ \tilde { P } ( f _ { k } ) = \int _ { 0 } ^ { T } P ( t ) e ^ { i 2 \pi f _ { k } t } d t , \eqno ( 1 8 ) \]

where \( f _ { k } = k / T \) for integer k, and T is the observation time. This integral evaluates to

\[ \tilde { P } ( f _ { k } ) = T P _ { 0 } \delta _ { k 0 } - \frac { i \dot { P } _ { 0 } T ^ { 2 } } { 2 \pi k } - \frac { i \ddot { P } _ { 0 } T ^ { 3 } } { 2 \pi k } + \frac { \ddot { P } _ { 0 } T ^ { 3 } } { 2 \pi ^ { 2 } k ^ { 2 } } + \dots ~ ( 1 9 ) \]

The \( k \) = 0 term is simply a constant offset that we can ignore. Writing \( a \; = \; \bar { P } _ { 0 } \bar { T } ^ { 3 } / ( 2 \pi ^ { 2 } ) \) and \( b \ = \ - ( \dot { P } _ { 0 } T ^ { 2 } \ + \)\( \ddot { P } _ { 0 } T ^ { 3 } ) / ( 2 \pi ) \)), the timing model for each pulsar can be written as

\[ \tilde { P } _ { k } = \frac { a } { k ^ { 2 } } + \frac { i b } { k } . \eqno ( 2 0 ) \]

A model of this form, with independent a and b for each pulsar, is then subtracted from the TOAs.

The full set of parameters in our model thus con-sists of the five noise/timing parameters for each pulsar, and three parameters to describe the GW background -\( A , f _ { b } , \kappa \). With this set of parameters, the likelihood is defined by [43]

\[ p ( d | s , n ) = \frac { 1 } { \sqrt { ( 2 \pi ) ^ { L } \operatorname* { d e t } C } } \exp \left( - \frac { 1 } { 2 } \sum _ { a b } \sum _ { i j } r _ { a i } C _ { ( a i ) ( b j ) } ^ { - 1 } r _ { b j } \right) , \]

where C is the covariance matrix, which depends on both the noise in the individual pulsars and on the GW background, and \( r = d - s \) denotes the timing residuals after the subtraction of the timing model s from the data d. The indices a and b label individual pulsars, and run from 1 to the number of pulsars, \( N _ { p } \). The indices i and j label the data samples, i.e. individual frequency bins. Since our simulated data is stationary, the correlation matrix is diagonal in \( i , j \) and \( C _ { ( a i ) ( b j ) } \to C _ { a b } ( f ) \).

The timing model parameters for each pulsar enter the likelihood in the timing residuals; they are subtracted from the TOAs before the likelihood is evaluated. The red and white noise contributions for each pulsar enter along the diagonal of the covariance matrix. Finally, the GW signal enters via the Hellings and Downs [11] corre-lation matrix, which has the form

\[ H _ { a b } = \frac { 1 } { 2 } + \frac { 3 ( 1 - \cos \theta _ { a b } ) } { 4 } \ln \bigg ( \frac { 1 - \cos \theta _ { a b } } { 2 } \bigg ) - \frac { 1 - \cos \theta _ { a b } } { 8 } + \frac { 1 } { 2 } \delta _ { a b } . \]

The full covariance matrix is then given by

\[ C _ { a b } ( f ) = S _ { h } ( f ) \frac { H _ { a b } } { 3 } + \delta _ { a b } \left\{ S _ { n _ { a } } + S _ { r _ { a } } ( f / f _ { y } ) ^ { r _ { a } } \right\} , \quad ( 2 3 ) \]

where \( S _ { h } ( f ) \) is the PSD of the GW background, \( S _ { n _ { \alpha } } \) is the PSD of the white noise, \( S _ { r _ { a } } \) is the amplitude of the PSD of the red noise, and \( r _ { a } \) is the slope of the red noise.

# A. Methods

Our analysis is carried out within the framework of Bayesian inference, using the technique of Markov chain Monte Carlo (MCMC). To calculate Bayes factors1, we must calculate the evidence for each model, which ne-cessitates performing an integral over the full parameter space. For this integral, we use the technique of thermo-dynamic integration [44–46]. This technique necessitates the use of parallel tempering, in which multiple chains are run at different ‘temperatures,’ which are defined by changing the likelihood to

\[ p ( d | s , n , \beta ) = p ( d | s , n ) ^ { \beta } , \eqno ( 2 4 ) \]

where \( \beta \) is analogous to an inverse temperature. This effectively ‘softens’ the likelihood, allowing the chains to effectively sample the full posterior. Chains with differ-ent temperatures are allowed to swap parameters with a probability given by the Hastings ratio

\[ H _ { i \leftrightarrow j } = \operatorname* { m i n } \left( \frac { p ( d | s _ { i } , n _ { i } , \beta _ { j } ) p ( d | s _ { j } , n _ { j } , \beta _ { i } ) } { p ( d | s _ { i } , n _ { i } , \beta _ { i } ) p ( d | s _ { j } , n _ { j } , \beta _ { i } ) } , 1 \right) . \quad \quad ( 2 5 ) \]

The evidence for each model is then given by

\[ \ln p ( d ) = \int _ { 0 } ^ { i } E _ { \beta } [ \ln p ( d | \vec { x } ) ] d \beta , \eqno ( 2 6 ) \]

where \( E _ { \beta } \) denotes the expectation at inverse tempera-ture \( \beta \). Given equal prior belief in two models, the Bayes factor is then simply the evidence ratio between them.

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1 Given equal prior belief in the validity of two models, A and B, the Bayes factor, \( B _ { A B } \), between models A and B, given the observed data, is the ‘betting odds’ that model A is the better theory, rather than model B.