then impose that

\[ \int _ { f _ { \operatorname* { m i n } } } ^ { f _ { \operatorname* { m a x } } } \Omega _ { \mathrm { G W } } ( f ) \, d \ln f < 1 . \eqno ( 1 5 ) \]

The lower frequency limit of this integral is unimpor-tant since the integral is dominated by the high frequency behavior. Performing the integral results in a limit on A of the form

\[ A ^ { 2 } < f _ { y } ^ { - 4 / 3 } f _ { \mathrm { m a x } } ^ { - 2 / 3 } \frac { H _ { 0 } ^ { 2 } } { 2 \pi ^ { 2 } } . \eqno ( 1 6 ) \]

For \( f _ { \mathrm { m a x } } ~ = ~ 2 . 0 \, \times \, 1 0 ^ { - 6 } \) Hz, this implies that \( A ~ < \)\( 4 . 4 \times 1 0 ^ { - 1 2 } \), which is the upper limit we use when em-ploying a uniform prior in A throughout our analyses. We take the uniform prior on the logarithm of the amplitude to share the same upper limit derived for the uniform am-plitude prior, but there is no natural value for the lower limit. This prevents us from using this prior to set up-per limits, but by choosing a lower limit several orders of magnitude below the noise level, it is possible to use the uniform logarithmic prior for parameter estimation once a detection has been made.

The other parameters that enter into our model for the GW spectrum are \( f _ { b } \) and \( \kappa \), as discussed above. The prior we choose to employ on \( f _ { b } \) is uniform in \( \ln ( f _ { b } ) \), indicating our lack of knowledge about even the order of magnitude of the frequency at which the GW slope may bend, and has a lower limit of \( f _ { b } = 2 . 5 \times 1 0 ^ { - 9 } \)\( \mathrm { H z } \) (which is below the lower edge of the sensitivity band for a 10 year data span), and an upper limit of \( f _ { b } = 1 0 ^ { - 7 } \) Hz. This upper limit is somewhat arbitrary, but it does indicate our belief that GWs must, at some high enough frequency, dominate the evolution of SMBH binaries. For \( \kappa \), we use a prior that is uniform in \( \kappa \), with a range of 0 to 23/3 (the non-integer value for the upper end of the range is due to us changing our conventions for the parameterization of the spectrum late in the project).

We note that we have chosen to study our ability to measure the low-frequency slope, bend frequency, and amplitude for a set of simulated data, rather than try-ing to assess how efficiently or frequently the last parsec problem is solved, or what the correct correlation is be-tween the black hole mass and a particular property of the host galaxy. Our principle motivation for this choice is that the astrophysically-motivated distributions only differ by a factor of ∼ 2–3 in predicted signal amplitude between the most conservative and the most optimistic estimates, despite making different assumptions about the solution to the last parsec problem and the correla-tion between black-hole mass and host-galaxy properties. Furthermore, there are multiple elements that contribute to this level of amplitude uncertainty, including the over-all galaxy merger rate and its dependence on mass and environment, in addition to the last parsec solution and the choice of black-hole mass - host-galaxy-property cor- relation. Given our comparative ignorance of the low-frequency signal slope and the transition frequency be-tween the dynamical process dominating the final parsec and GW-driven evolution, we choose to focus our study on constraining these parameters for a given amplitude.

Another approach we could have taken with the am-plitude prior is to separate the uncertainties into an ob-servational part (from factors such as the rate of galaxy mergers and the \( M _ { - } \sigma \) relation) and a theoretical part (from factors such as the efficiency of dynamical friction in hardening the binary and the fraction of systems where the last parsec problem is overcome). This could be done by writing the prior as a Gaussian distribution in ln A, centered at some value \( \ln { \bar { A } } \), with width \( \sigma _ { \mathrm { o b s } } \) to account for the observational uncertainties. The central value, \( \ln { \bar { A } } \), would then be a hyper-parameter to be determined by the data. If we quantify the uncertainty on the cen-tral value \( \bar { A } \) as \( \bar { A } \doteq \eta A _ { * } \), where \( A _ { * } \) is some reference value and \( \eta \) encodes the theoretical uncertainty in the merger efficiency, then assigning a Gaussian prior on the hyper-parameter \( \ln \eta \) of width \( \sigma _ { \mathrm { t h } } \) centered on \( \eta _ { * } \), and marginalizing over the hyper parameter, leads to Gaus-sian distributions of the form used for Models A and B with width \( \sigma \, = \, \sqrt { \sigma _ { \mathrm { o b s } } ^ { 2 } + \sigma _ { \mathrm { t h } } ^ { 2 } } \). Alternatively, a uniform prior on \( \ln \eta \) over some range leads to a roughly uniform prior on ln A between \( \ln ( \eta _ { \mathrm { m i n } } A _ { * } ) \) and \( \ln ( \eta _ { \mathrm { m a x } } A _ { * } ) \) with rounded edges of width \( \sigma _ { \mathrm { o n s } } \). Thus, the analyses we con-sider are equivalent to a hierarchical Bayesian analysis that separates out the theoretical and observational un-certainties for certain choices of the hyper-prior.

# III. SIMULATED DATA AND ANALYSIS

Our simulated data set consists of the timing residuals from 20 pulsars, randomly distributed on the sky, and observed for 10 years with a two-week cadence. The data from each pulsar is generated including white noise at a level of 200 ns, and no red noise. We then recover this simulated signal using a model that includes the three parameters that describe the GW spectrum, \( A , f _ { b } , \kappa \), and the noise parameters for each pulsar. These include a white noise level, red noise level, and red noise slope. The white noise is fully described by the amplitude of its power spectral density (PSD), which we label \( S _ { n } \). The prior on \( S _ { n } \) was uniform in \( \ln ( S _ { n } ) \), and ranged from \( S _ { n } = \)\( 4 \times 1 0 ^ { - 1 8 } \, \mathrm { H z ^ { - 1 / 2 } } \) to \( S _ { n } \, = \, 1 0 ^ { - 2 } \, \mathrm { H z } ^ { - 1 / 2 } \). The red noise is parameterized by its PSD amplitude, \( S _ { r } \), and by a slope, r, as \( S _ { r } ( f ) = S _ { r } ( f / f _ { \mathrm { y e a r } } ) ^ { r } \). The prior on the red noise amplitude was also uniform in \( \ln ( S _ { r } ) \), with the same range as \( S _ { n } \), and the prior on the slope r was uniform from −2 to −6.

Finally, we include two parameters for each pulsar that encode the effects of the timing model on the timing residuals. As discussed briefly in the introduction, the timing model used to predict the TOA of radio pulses from a given pulsar includes a large set of parameters [8–10]. In this analysis we only consider the two timing

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.