Red noise models

A time-correlated stochastic process whose power spectral density is a decreasing function of the conjugate frequency (not to be confused with observing frequency) is known as red noise. Depending on the pulsar, different types of red noise processes will be present in the TOAs, including spin noise, dispersion noise, and the stochastic gravitational wave background.

Such a process is usually modeled as truncated Fourier series.

$\Delta(t) = (\frac{\nu_{\text{ref}}}{\nu})^\alpha \sum_{j=1}^N \{ a_j\cos(2\pi j f_1(t-t_0)) + b_j\sin(2\pi j f_1(t-t_0)) \}\,.$

where $\nu$ is the observing frequency, $\nu_{\text{ref}}$ is a reference frequency, $\alpha$ is the chromatic index, $N$ is the number of harmonics, $f_1$ is the fundamental frequency, $t_0$ is a fiducial time, and $a_j$ and $b_j$ are Fourier coefficients.

Two types of red noise models are currently available in Vela.jl.

The WaveX family of models treat the Fourier coefficients as unconstrained free parameters (e.g., WXSIN_ and WXCOS_).

The PowerlawRedNoiseGP family of models treat these coefficients as Gaussian random variables whose variances, interpreted as power spectral densities, follow a power law spectrum.

$\left\langle a_j \right\rangle = \left\langle b_j \right\rangle = 0$

$\left\langle a_j a_k \right\rangle = \left\langle b_j b_k \right\rangle = \sigma_j^2 \delta_{jk}$

$\left\langle a_j b_k \right\rangle = 0$

$\sigma_j = P(f_j) = \frac{A^2}{12\pi^2 f_{\text{yr}}^3} f_1 \left(\frac{f_{\text{yr}}}{f}\right)^\gamma$

where $A$ is the powerlaw amplitude $\gamma$ is the powerlaw index, and $f_{\text{yr}}=1 \text{yr}^{-1}$. The prior parameters $A$ and $\gamma$ are also treated as free parameters and sampled over. However, in this case, the geometry of the parameter space exhibits Neil's funnel-like geometry, which is hard for MCMC samplers to deal with. To avoid this, we use $\bar{a}_j=a_j/\sigma_j$ and $\bar{b}_j=b_j/\sigma_j$ as free parameters where

$\left\langle\bar{a}_j^2\right\rangle = \left\langle\bar{b}_j^2\right\rangle = 1\,.$

In PowerlawRedNoiseGP, this is represented by the parameters PLREDSIN_ and PLREDCOS_. The powerlaw parameters are PLREDAMP and PNREDGAM. Please note that the PINT-format par files do not support PLREDSIN_ and PLREDCOS_. Hence, while they are included in the posterior samples, they will not be included in the output par file.

Similar representations also exist for dispersion noise ($\alpha=2$) and chromatic noise (see Dispersion delays and Chromatic delays).

Other types of spectral models such as free spectrum, t-process, broken powerlaw, running power law, etc are not yet implemented.

By default, the noise amplitudes described above are treated as free parameters are sampled. However, this can sometimes lead to inefficient sampling due to Neil's funnel effect. One way of avoiding this is to analytically marginalize the posterior distribution over the noise amplitudes. This can be achieved in Vela.jl by moving the Gaussian noise components (PowerlawRedNoiseGP, PowerlawDispersionNoiseGP, PowerlawChromaticNoiseGP) from the TimingModel components tuple into a WoodburyKernel. The is_gp_noise() function identifies the components that support this operation.

is_gp_noise(::Component)::Bool