arXiv subjects

Andrew Melatos

Publications and source records attributed to Andrew Melatos.

At least 37 records · Page 2Linked to original sources

arXiv2025

Hierarchical Bayesian estimation of population-level torque law parameters from anomalous pulsar braking indices

Abridged. Stochastic fluctuations in the spin frequency $ν$ of a rotation-powered pulsar affect how accurately one measures the power-law braking index, $n_{\rm pl}$, defined through $\dotν=Kν^{n_{\rm pl}}$, and can lead to measurements of anomalous braking indices, with $\vert n \vert = \vert ν\ddotν/ \dotν^{2} \vert \gg1$, where the overdot symbolizes a derivative with respect to time. Previous studies show that the variance of the measured $n$ obeys the predictive, falsifiable formula $\langle n^{2} \rangle = n_{\rm pl}^{2}+σ^{2}_{\ddotν}ν^{2}γ_{\ddotν}^{-2}\dotν^{-4}T_{\rm obs}^{-1}$ for $\dot{K}=0$, where $σ_{\ddotν}$ is the timing noise amplitude, $γ_{\ddotν}^{-1}$ is a stellar damping time-scale, and $T_{\rm obs}$ is the total observing time. Here we combine this formula with a hierarchical Bayesian scheme to infer the population-level distribution of $n_{\rm pl}$ for a pulsar population of size $M$. The scheme is validated using synthetic data. For a plausible test population with $M=100$ and injected $n_{\rm pl}$ values drawn from a population-level Gaussian with mean $μ_{\rm pl}=4$ and standard deviation $σ_{\rm pl}=0.5$, intermediate between electromagnetic braking and mass quadrupole gravitational radiation reaction, the Bayesian scheme infers $μ_{\rm pl}=3.89^{+0.24}_{-0.23}$ and $σ_{\rm pl}=0.43^{+0.21}_{-0.14}$. The $M=100$ per-pulsar posteriors for $n_{\rm pl}$ and $σ^{2}_{\ddotν}γ_{\ddotν}^{-2}$ contain $87\%$ and $69\%$, respectively, of the injected values within their $90\%$ credible intervals. Comparable accuracy is achieved for (i) population sizes spanning the range $50 \leq M \leq 300$, and (ii) wide priors satisfying $μ_{\rm pl} \leq 10^{3}$ and $σ_{\rm pl} \leq 10^{2}$, which accommodate plausible spin-down mechanisms with $\dot{K}\neq0$.

astro-ph.HE↗

arXiv2025

State-space algorithm for detecting the nanohertz gravitational wave background

The stochastic gravitational wave background (SGWB) can be observed in the nanohertz band using a pulsar timing array (PTA). Here a computationally efficient state-space framework is developed for analysing SGWB data, in which the stochastic gravitational wave strain at Earth is tracked with a non-linear Kalman filter and separated simultaneously from intrinsic, achromatic pulsar spin wandering. The filter is combined with a nested sampler to estimate the parameters of the model, and to calculate a Bayes factor for selecting between models with and without a SGWB. The procedure extends previous state-space formulations of PTA data analysis applied to individually resolvable binary black hole sources. The performance of the new algorithm is tested on synthetic data from the first International PTA Mock Data Challenge. It is shown that the algorithm distinguishes a SGWB from pure noise for $A_{\rm gw} \geq 3 \times 10^{-14}$, where $A_{\rm gw}$ denotes the standard normalization factor for a power spectral density with power-law exponent $-13/3$. Additional, systematic validation tests are also performed with synthetic data generated independently by adjusting the injected parameters to cover astrophysically plausible ranges. Full posterior distributions are recovered and tested for accuracy. The state-space procedure is memory-light and evaluates the likelihood for a standard-sized PTA dataset in $\lesssim 10^{-1}$ s without optimization on a standard central processing unit.

astro-ph.IM↗

arXiv2025

Gravitational waves from r-mode oscillations of stochastically accreting neutron stars

$r$-mode oscillations in rotating neutron stars are a source of continuous gravitational radiation. We investigate the excitation of $r$-modes by the mechanical impact on the neutron star surface of stochastically accreted clumps of matter, assuming that the Chandrasekhar-Friedman-Schutz instability is not triggered. The star is idealised as a slowly-rotating, unmagnetised, one-component fluid with a barotropic equation of state in Newtonian gravity. It is found that the $r$-mode amplitude depends weakly on the equation of state but sensitively on the rotation frequency $ν_{\rm s}$. The gravitational wave strain implicitly depends on the equation of state through the damping timescale. The root-mean-square strain is $h_{\rm rms} \approx 10^{-35} (ν_{\rm s}/ 10 {\rm Hz})^{2} (R_*/10 {\rm km})^2 (Δt_{\rm acc}/1 {\rm yr})^{1/2} (f_{\rm acc}/1 {\rm kHz})^{-1/2} (\dot{M}/10^{-8} \text{M}_{\odot} \text{yr}^{-1}) (v/0.4c) (d/1 {\rm kpc})^{-1}$, which is comparable to the strain from $g$-, $p$- and $f$-modes excited by stochastic accretion, where $R_*$ is the radius of the star, $Δt_{\rm acc}$ is the uninterrupted duration of an accretion episode, $f_{\rm acc}$ is the mean clump impact frequency, $\dot{M}$ is the accretion rate, $v$ is the impact speed, and $d$ is the distance of the star from the Earth. An observational test is proposed, based on the temporal autocorrelation function of the gravitational wave signal, to discern whether the Chandrasekhar-Friedman-Schutz instability switches on and coexists with impact-excited $r$-modes before or during a gravitational wave observation.

astro-ph.HE↗

arXiv2024

Quadrupole Moment of a Magnetically Confined Mountain on an Accreting Neutron Star in General Relativity

General relativistic corrections are calculated for the quadrupole moment of a magnetically confined mountain on an accreting neutron star. The hydromagnetic structure of the mountain satisfies the general relativistic Grad-Shafranov equation supplemented by the flux-freezing condition of ideal magnetohydrodynamics, as in previous calculations of the magnetic dipole moment. It is found that the ellipticity and hence the gravitational wave strain are up to $12\%$ greater than in the analogous Newtonian system. The direct contribution of the magnetic field to the nonaxisymmetric component of the stress-energy tensor is shown to be negligible in accreting systems such as low-mass X-ray binaries.

astro-ph.HE↗

arXiv2024

Axion sourcing in dense stellar matter via CP-violating couplings

Compact objects such as neutron stars and white dwarfs can source axion-like particles and QCD axions due to CP-violating axion-fermion couplings. The magnitude of the axion field depends on the stellar density and on the strength of the axion-fermion couplings. We show that even CP-violating couplings one order of magnitude smaller than existing constraints source extended axion field configurations. For axion-like particles, the axion energy is comparable to the magnetic energy in neutron stars with inferred magnetic fields of the order of $10^{13}$ G, and exceeds by more than one order of magnitude the magnetic energy content of white dwarfs with inferred fields of the order of $10^{4}$ G. On the other hand, the energy stored in the QCD axion field is orders of magnitude lower due to the smallness of the predicted CP-violating couplings. It is shown that the sourced axion field can polarize the photons emitted from the stellar surface, and stimulate the production of photons with energies in the radio band.

hep-ph↗

arXiv2024

Adaptive cancellation of mains power interference in continuous gravitational wave searches with a hidden Markov model

Continuous gravitational wave searches with terrestrial, long-baseline interferometers are hampered by long-lived, narrowband features in the power spectral density of the detector noise, known as lines. Candidate GW signals which overlap spectrally with known lines are typically vetoed. Here we demonstrate a line subtraction method based on adaptive noise cancellation, using a recursive least squares algorithm, a common approach in electrical engineering applications such as audio and biomedical signal processing. We validate the line subtraction method by combining it with a hidden Markov model (HMM), a standard continuous wave search tool, to detect an injected continuous wave signal with an unknown and randomly wandering frequency, which overlaps with the mains power line at $60 \, {\rm Hz}$ in the Laser Interferometer Gravitational Wave Observatory (LIGO). The performance of the line subtraction method is tested on an injected continuous wave signal obscured by (a) synthetic noise data with both Gaussian and non-Gaussian components, and (b) real noise data obtained from the LIGO Livingston detector. In both cases, before applying the line subtraction method the HMM does not detect the injected continuous wave signal. After applying the line subtraction method the mains power line is suppressed by 20--40 dB, and the HMM detects the underlying signal, with a time-averaged root-mean-square error in the frequency estimate of $\sim 0.05 $ Hz. The performance of the line subtraction method with respect to the characteristics of the 60 Hz line and the control parameters of the recursive least squares algorithm is quantified in terms of receiver operating characteristic curves.

gr-qc↗

arXiv2024

Analysing one- and two-bit data to reduce memory requirements for F-statistic-based gravitational wave searches

Searches for continuous-wave gravitational radiation in data collected by modern long-baseline interferometers, such as the Laser Interferometer Gravitational-wave Observatory (LIGO), the Virgo interferometer and the Kamioka Gravitational Wave Detector (KAGRA), can be memory intensive. A digitisation scheme is described that reduces the 64-bit interferometer output to a one- or two-bit data stream while minimising distortion and achieving considerable reduction in storage and input/output cost. For the representative example of the coherent, maximum-likelihood matched filter known as the F-statistic, it is found using Monte-Carlo simulations that the injected signal only needs to be ~24 per cent stronger (for one-bit data) and ~6.4 per cent stronger (for two bit data with optimal thresholds) than a 64-bit signal in order to be detected with 90 per cent probability in Gaussian noise. The foregoing percentages do not change significantly when the signal frequency decreases secularly, or when the noise statistics are not Gaussian, as verified with LIGO Science Run 6 data.

gr-qc↗

arXiv2024

State-space analysis of a continuous gravitational wave source with a pulsar timing array: inclusion of the pulsar terms

Pulsar timing arrays can detect continuous nanohertz gravitational waves emitted by individual supermassive black hole binaries. The data analysis procedure can be formulated within a time-domain, state-space framework, in which the radio timing observations are related to a temporal sequence of latent states, namely the intrinsic pulsar spin frequency. The achromatic wandering of the pulsar spin frequency is tracked using a Kalman filter concurrently with the pulse frequency modulation induced by a gravitational wave from a single source. The modulation is the sum of terms proportional to the gravitational wave strain at the Earth and at every pulsar in the array. Here we generalize previous state-space formulations of the pulsar timing array problem to include the pulsar terms; that is, we copy the pulsar terms from traditional, non-state-space analyses over to the state-space framework. The performance of the generalized Kalman filter is tested using astrophysically representative software injections in Gaussian measurement noise. It is shown that including the pulsar terms corrects for previously identified biases in the parameter estimates (especially the sky position of the source) which also arise in traditional matched-filter analyses that exclude the pulsar terms. Additionally, including the pulsar terms decreases the minimum detectable strain by $14\%$. Overall, the study verifies that the pulsar terms do not raise any special extra impediments for the state-space framework, beyond those studied in traditional analyses. The inspiral-driven evolution of the wave frequency at the Earth and at the retarded time at every pulsar in the array is also investigated.

astro-ph.HE↗

arXiv2024

Stochastic and secular anomalies in pulsar braking indices

Stochastic and secular variations in the spin frequency $ν$ of a rotation-powered pulsar complicate the interpretation of the measured braking index, $n$, in terms of a power-law spin-down torque $\propto ν^{n_{\rm pl}}$. Both categories of variation can lead to anomalous braking indices, with $\vert n \vert = \vert ν\ddotν / \dotν^2 \vert \gg 1$, where the overdot symbolizes a derivative with respect to time. Here we quantify the combined effect of stochastic and secular deviations from pure power-law spin down on measurements of $n$. Through analytic calculations, Monte Carlo simulations involving synthetic data, and modern Bayesian timing techniques, it is shown that the variance of $n$ satisfies the predictive, falsifiable formula $\langle n^{2} \rangle = (n_{\rm pl}+\dot{K}_{\rm dim})^{2}+σ_{\rm dim}^{2}$, where $\dot{K}_{\rm dim}$ is inversely proportional to the time-scale $τ_K$ over which the proportionality constant of the power-law spin-down torque varies, $σ_{\rm dim}$ is proportional to the timing noise amplitude and inversely proportional to the square root of the total observing time, and the average is over an ensemble of random realizations of the timing noise process. The anomalous regime $\langle n^2 \rangle \gg 1$ occurs for $\dot{K}_{\rm dim} \gg 1$, $σ_{\rm dim} \gg 1$, or both. The sign of $n$ depends in part on the sign of $\dot{K}_{\rm dim}$, so it is possible to measure unequal numbers of positive and negative $n$ values in a large sample of pulsars. The distinguishable impact of stochastic and secular anomalies on phase residuals is quantified to prepare for extending the analysis of synthetic data to real pulsars.

astro-ph.HE↗

arXiv2024

Kalman tracking and parameter estimation of continuous gravitational waves with a pulsar timing array

Continuous nanohertz gravitational waves from individual supermassive black hole binaries may be detectable with pulsar timing arrays. A novel search strategy is developed, wherein intrinsic achromatic spin wandering is tracked simultaneously with the modulation induced by a single gravitational wave source in the pulse times of arrival. A two-step inference procedure is applied within a state-space framework, such that the modulation is tracked with a Kalman filter, which then provides a likelihood for nested sampling. The procedure estimates the static parameters in the problem, such as the sky position of the source, without fitting for ensemble-averaged statistics such as the power spectral density of the timing noise, and therefore complements traditional parameter estimation methods. It also returns the Bayes factor relating a model with a single gravitational wave source to one without, complementing traditional detection methods. It is shown via astrophysically representative software injections in Gaussian measurement noise that the procedure distinguishes a gravitational wave from pure noise down to a characteristic wave strain of $h_0 \approx 2 \times 10^{-15}$. Full posterior distributions of model parameters are recovered and tested for accuracy. There is a bias of $\approx 0.3$ rad in the marginalised one-dimensional posterior for the orbital inclination $ι$, introduced by dropping the so-called `pulsar terms'. Smaller biases $\lesssim 10 \%$ are also observed in other static parameters.

astro-ph.HE↗

arXiv2024

Measuring the magnetic dipole moment and magnetospheric fluctuations of accretion-powered pulsars in the Small Magellanic Cloud with an unscented Kalman filter

Many accretion-powered pulsars rotate in magnetocentrifugal disequilibrium, spinning up or down secularly over multi-year intervals. The magnetic dipole moment $μ$ of such systems cannot be inferred uniquely from the time-averaged aperiodic X-ray flux $\langle L(t) \rangle$ and pulse period $\langle P(t) \rangle$, because the radiative efficiency of the accretion is unknown and degenerate with the mass accretion rate. Here we circumvent the degeneracy by tracking the fluctuations in the unaveraged time series $L(t)$ and $P(t)$ using an unscented Kalman filter, whereupon $μ$ can be estimated uniquely, up to the uncertainties in the mass, radius and distance of the star. The analysis is performed on Rossi X-ray Timing Explorer observations for $24$ X-ray transients in the Small Magellanic Cloud, which have been monitored regularly for $\sim 16$ years. As well as independent estimates of $μ$, the analysis yields time-resolved histories of the mass accretion rate and the Maxwell stress at the disk-magnetosphere boundary for each star, and hence auto- and cross-correlations involving the latter two state variables. The inferred fluctuation statistics convey important information about the complex accretion physics at the disk-magnetosphere boundary.

astro-ph.HE↗

arXiv2024

Gravitational waves from nonradial oscillations of stochastically accreting neutron stars

Oscillating neutron stars are sources of continuous gravitational waves. We study analytically the excitation of stellar oscillations by the mechanical impact on the stellar surface of ''clumps'' of stochastically accreted matter. We calculate the waveform and spectrum of the gravitational wave signal emitted by the accretion-driven pulsations. Results are generated for an idealised model of a nonrotating, unmagnetised, one-component star with uniform polytropic index $n_{\rm poly}$ assuming Newtonian gravity and the Cowling approximation. We find that the excited mode amplitudes grow with increasing $n_{\rm poly}$ and mode order $n$. The gravitational wave signal forms a sequence of amplitude-modulated packets for $n_{\rm poly}=1$, lasting $\sim 10^{-3}$s after each impact. The gravitational wave strain increases with increasing $n_{\rm poly}$, but decreases with increasing $n$ and increasing multipole order $l$ for $n_{\rm poly}=1$. In the observing band of current long-baseline interferometers, $g$-modes emit higher, narrower peaks in the amplitude spectral density than $f$- and $p$-modes, with the highest peaks reaching $\sim 10^{-26}$Hz$^{-1/2}$ for modes with damping time $τ_{nl} \sim 10^{8}$yr. The root-mean-square strain $h_{\text{rms}}$, calculated by summing over modes with $2\leq l\leq4$ and $τ_{nl} \leq 10^{8}$yr, spans the range $10^{-33} \leq h_{\text{rms}} \leq 10^{-32}$ for $1\leq n_{\text{poly}}\leq 2$.

astro-ph.HE↗

arXiv2024

Search for continuous gravitational waves directed at sub-threshold radiometer candidates in O3 LIGO data

We present results of a follow-up search for continuous gravitational waves (CWs) associated with sub-threshold candidates from the LIGO-Virgo-KAGRA (LVK) All-Sky All-Frequency (ASAF) directed radiometer analysis, using Advanced LIGO data from the third observing run (O3). Each ASAF candidate corresponds to a $1/32\,\,\mathrm{Hz}$ frequency band and ${\sim}13\,\,\mathrm{deg}^2$ sky pixel. Assuming they represent possible CW sources, we analyze all $515$ ASAF candidates using a semi-coherent, $\mathcal{F}$-statistic-based matched filter search. The search algorithm incorporates a hidden Markov model (HMM), expanding the signal model to allow frequency spin-wandering, as well as unmodeled frequency evolution of less than $10^{-5}$ Hz per day that is not captured by the searched range of $\pm10^{-9}\,\,\mathrm{Hz/s}$ in frequency derivative. Significance thresholds with a $5\%$ probability of false alarm per ASAF candidate are determined empirically by searching detector noise at various off-target sky positions. We obtain $14$ outliers surviving a set of vetoes designed to eliminate instrumental artifacts. Upon further investigation, these outliers are deemed unlikely to represent astrophysical signals. We estimate the sensitivity of our search to both isolated and binary sources with orbital period greater than one year by recovering simulated signals added to detector data. The minimum detectable strain amplitude at $95\%$ confidence for isolated (long-period binary) sources is $h_0^{95\%} = 8.8\times 10^{-26}$ ($9.4\times 10^{-26}$) at a frequency of $222.6\,\,\mathrm{Hz}$. While this study focuses on ASAF sub-threshold candidates, the method presented could be applied to follow up candidates from future all-sky CW searches, complementing currently existing methods.

gr-qc↗

arXiv2024

Analysing radio pulsar timing noise with a Kalman filter: a demonstration involving PSR J1359$-$6038

In the standard two-component crust-superfluid model of a neutron star, timing noise can arise when the two components are perturbed by stochastic torques. Here it is demonstrated how to analyse fluctuations in radio pulse times of arrival with a Kalman filter to measure physical properties of the two-component model, including the crust-superfluid coupling time-scale and the variances of the crust and superfluid torques. The analysis technique, validated previously on synthetic data, is applied to observations with the Molonglo Observatory Synthesis Telescope of the representative pulsar PSR J1359$-$6038. It is shown that the two-component model is preferred to a one-component model, with log Bayes factor $6.81 \pm 0.02$. The coupling time-scale and the torque variances on the crust and superfluid are measured with $90\%$ confidence to be $10^{7.1^{+0.8}_{-0.5}}$ $\rm{s}$ and $10^{-24.0^{+0.4}_{-5.6}}$ $\rm{rad^2~s^{-3}}$ and $10^{-21.7^{+3.5}_{-0.9}}$ $\rm{rad^2~s^{-3}}$ respectively.

astro-ph.HE↗

arXiv2024

Impact of noise transients on gravitational-wave burst detection efficiency of the BayesWave pipeline with multi-detector networks

Detection confidence of the source-agnostic gravitational-wave burst search pipeline BayesWave is quantified by the log signal-versus-glitch Bayes factor, $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$. A recent study shows that $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ increases with the number of detectors. However, the increasing frequency of non-Gaussian noise transients (glitches) in expanded detector networks is not accounted for in the study. Glitches can mimic or mask burst signals resulting in false alarm detections, consequently reducing detection confidence. This paper an empirical study on the impact of false alarms on the overall performance of BayesWave, with expanded detector networks. The noise background of BayesWave for the Hanford-Livingston (HL, two-detector) and Hanford-Livingston-Virgo (HLV, three-detector) networks are measured using a set of non-astrophysical background triggers from the first half of Advanced LIGO and Advanced Virgo's Third Observing Run (O3a). Efficiency curves are constructed by combining $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ of simulated binary black hole signals with the background measurements, to characterize BayesWave's detection efficiency as a function of the per-trigger false alarm probability. The HL and HLV network efficiency curves are shown to be similar. A separate analysis finds that detection significance of O3 gravitational-wave candidates as measured by BayesWave are also comparable for the HL and HLV networks. Consistent results from the two independent analyses suggests that the overall burst detection performance of BayesWave does not improve with the addition of Virgo at O3a sensitivity, because the increased false alarm probability offsets the advantage of higher $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$.

gr-qc↗

arXiv2024

Persistent gravitational radiation from glitching pulsars. II. Updated scaling with vortex number

Superfluid vortices pinned to nuclear lattice sites or magnetic flux tubes in a neutron star evolve abruptly through a sequence of metastable spatial configurations, punctuated by unpinning avalanches associated with rotational glitches, as the stellar crust spins down electromagnetically. The metastable configurations are approximately but not exactly axisymmetric, causing the emission of persistent, quasimonochromatic, current quadrupole gravitational radiation. The characteristic gravitational wave strain $h_0$ as a function of the spin frequency $f$ and distance $D$ from the Earth is bounded above by $h_0 = 1.2\substack{+1.3 \\ -0.9} \times 10^{-32} (f/30\;{\rm Hz})^{2.5} (D/1\;{\rm kpc})^{-1}$, corresponding to a Poissonian spatial configuration (equal probability per unit area, i.e. zero inter-vortex repulsion), and bounded below by $h_0 = 1.8\substack{+2.0 \\ -1.5} \times 10^{-50} (f/30\;{\rm Hz})^{1.5} (D/1\;{\rm kpc})^{-1}$, corresponding to a regular array (periodic separation, i.e.\ maximum inter-vortex repulsion). N-body point vortex simulations predict an intermediate scaling, $h_0 = 7.3\substack{+7.9 \\ -5.4} \times 10^{-42} (f/30\;{\rm Hz})^{1.9} (D/1\;{\rm kpc})^{-1}$, which reflects a balance between the randomizing but spatially correlated action of superfluid vortex avalanches and the regularizing action of inter-vortex repulsion. The scaling is calibrated by conducting simulations with ${N_{\rm v}} \leq 5\times10^3$ vortices and extrapolated to the astrophysical regime ${N_{\rm v}} \sim 10^{17} (f/30\;{\rm Hz})$. The scaling is provisional, pending future computational advances to raise ${N_{\rm v}}$ and include three-dimensional effects such as vortex tension and turbulence.

astro-ph.HE↗

arXiv2023

Search for gravitational waves from Scorpius X-1 with a hidden Markov model in O3 LIGO data with a corrected orbital ephemeris

Results are presented for a semi-coherent search for gravitational waves from the low-mass X-ray binary Scorpius X-1 in Observing Run 3 (O3) data from the Laser Interferometer Gravitational Wave Observatory, using an updated orbital parameter ephemeris and a hidden Markov model (HMM) to allow for spin wandering. The new orbital ephemeris corrects errors in previously published orbital measurements and implies a new search domain. This search domain does not overlap with the one used in the original Scorpius X-1 HMM O3 search. The corrected domain is approximately three times smaller by area in the $T_{\rm asc}$--$P$ plane than the original domain, where $T_{\rm asc}$ and $P$ denote the time of passage through the ascending node and the orbital period respectively, reducing the trials factor and computing time. No evidence is found for gravitational radiation in the search band from 60 Hz to 500 Hz. Upper limits are computed for the characteristic gravitational wave strain. They are consistent with the values from the original Scorpius X-1 HMM O3 search.

gr-qc↗

arXiv2023

Magnetically Confined Mountains on Accreting Neutron Stars in General Relativity

The general relativistic formulation of the problem of magnetically confined mountains on neutron stars is presented, and the resulting equations are solved numerically, generalising previous Newtonian calculations. The hydromagnetic structure of the accreted matter and the subsequent magnetic burial of the star's magnetic dipole moment are computed. Overall, it is observed that relativistic corrections reduce the hydromagnetic deformation associated with the mountain. The magnetic field lines are curved more gently than in previous calculations, and the screening of the dipole moment is reduced. Quantitatively, it is found that the dimensionless dipole moment ($m_{\rm d}$) depends on the accreted mass ($M_{\rm a}$) as $m_{\rm d} = -3.2\times10^{3}M_{\rm a}/M_\odot + 1.0$, implying approximately three times less screening compared to the Newtonian theory. Additionally, the characteristic scale height of the mountain, governing the gradients of quantities like pressure, density, and magnetic field strength, reduces by approximately $40\%$ for an isothermal equation of state.

astro-ph.HE↗