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This article briefly reviews our current understanding of the evolution of magnetic fields in neutron stars, which basically defines the evolutionary pathways between different observational classes of neutron stars. The emphasis here is on the evolution in binary systems and the newly emergent classes of millisecond pulsars.

Gravitational waves are expected to be radiated by supermassive black hole binaries formed during galaxy mergers. A stochastic superposition of gravitational waves from all such binary systems would modulate the arrival times of pulses from radio pulsars. Using observations of millisecond pulsars obtained with the Parkes radio telescope, we constrained the characteristic amplitude of this background, Ac,yr, to be <1.0 × 10–15 with 95% confidence. This limit excludes predicted ranges for Ac,yr from current models with 91 to 99.7% probability. We conclude that binary evolution is either stalled or dramatically accelerated by galactic-center environments and that higher-cadence and shorter-wavelength observations would be more sensitive to gravitational waves.

Pulsar timing array (PTA) collaborations in North America, Australia, and Europe, have been exploiting the exquisite timing precision of millisecond pulsars over decades of observations to search for correlated timing deviations induced by gravitational waves (GWs). PTAs are sensitive to the frequency band ranging just below 1 nanohertz to a few tens of microhertz. The discovery space of this band is potentially rich with populations of inspiraling supermassive black hole binaries, decaying cosmic string networks, relic post-inflation GWs, and even non-GW imprints of axionic dark matter. This article aims to provide an understanding of the exciting open science questions in cosmology, galaxy evolution, and fundamental physics that will be addressed by the detection and study of GWs through PTAs. The focus of the article is on providing an understanding of the mechanisms by which PTAs can address specific questions in these fields, and to outline some of the subtleties and difficulties in each case. The material included is weighted most heavily toward the questions which we expect will be answered in the near-term with PTAs; however, we have made efforts to include most currently anticipated applications of nanohertz GWs.

Spider pulsars are neutron stars that have a companion star in a close orbit. The companion star sheds material to the neutron star, spinning it up to millisecond rotation periods, while the orbit shortens to hours. The companion is eventually ablated and destroyed by the pulsar wind and radiation 1 , 2 . Spider pulsars are key for studying the evolutionary link between accreting X-ray pulsars and isolated millisecond pulsars, pulsar irradiation effects and the birth of massive neutron stars 3 – 6 . Black widow pulsars in extremely compact orbits (as short as 62 minutes 7 ) have companions with masses much smaller than 0.1  M ⊙ . They may have evolved from redback pulsars with companion masses of about 0.1–0.4  M ⊙ and orbital periods of less than 1 day 8 . If this is true, then there should be a population of millisecond pulsars with moderate-mass companions and very short orbital periods 9 , but, hitherto, no such system was known. Here we report radio observations of the binary millisecond pulsar PSR J1953+1844 (M71E) that show it to have an orbital period of 53.3 minutes and a companion with a mass of around 0.07  M ⊙ . It is a faint X-ray source and located 2.5 arcminutes from the centre of the globular cluster M71. PSR J1953+1844 (M71E) has an orbital period of 53.3 minutes and a companion with a mass of 0.07  M ⊙ , making it a bridging object between redbacks and black widows in the evolutionary track.

Observations of 12 X-ray binaries that contain black holes within the central parsec of the Galaxy suggest the existence of hundreds more, and even more isolated black holes. Many black holes in the Galactic Centre Simulations predict that the supermassive black holes near the centres of all large galaxies are surrounded by a concentration of stellar-mass black holes. Such black holes, however, have not previously been detected at the centre of our galaxy. Low-mass X-ray binary systems containing black holes are proxies for single black holes. Charles Hailey and collaborators report finding a dozen such binary systems in the central parsec of the Milky Way. By extrapolating these observations they conclude that the total population of such binary systems in the central parsec of the Galaxy is in the hundreds, with a much greater number of isolated black holes. They cannot, however, rule out the contribution of a sub-dominant population of rotating neutron stars that have become millisecond pulsars through the accretion of gas from close companion stars. The existence of a ‘density cusp’ 1 , 2 —a localized increase in number—of stellar-mass black holes near a supermassive black hole is a fundamental prediction of galactic stellar dynamics 3 . The best place to detect such a cusp is in the Galactic Centre, where the nearest supermassive black hole, Sagittarius A*, resides. As many as 20,000 black holes are predicted to settle into the central parsec of the Galaxy as a result of dynamical friction 3 , 4 , 5 ; however, so far no density cusp of black holes has been detected. Low-mass X-ray binary systems that contain a stellar-mass black hole are natural tracers of isolated black holes. Here we report observations of a dozen quiescent X-ray binaries in a density cusp within one parsec of Sagittarius A*. The lower-energy emission spectra that we observed in these binaries is distinct from the higher-energy spectra associated with the population of accreting white dwarfs that dominates the central eight parsecs of the Galaxy 6 . The properties of these X-ray binaries, in particular their spatial distribution and luminosity function, suggest the existence of hundreds of binary systems in the central parsec of the Galaxy and many more isolated black holes. We cannot rule out a contribution to the observed emission from a population (of up to about one-half the number of X-ray binaries) of rotationally powered, millisecond pulsars. The spatial distribution of the binary systems is a relic of their formation history, either in the stellar disk around Sagittarius A* (ref. 7 ) or through in-fall from globular clusters, and constrains the number density of sources in the modelling of gravitational waves from massive stellar remnants 8 , 9 , such as neutron stars and black holes.

We derive a new interior solution for stellar compact objects in f(R) gravity assuming a differential relation to constrain the Ricci curvature scalar. To this aim, we consider specific forms for the radial component of the metric and the first derivative of f(R). After, the time component of the metric potential and the form of f(R) function are derived. From these results, it is possible to obtain the radial and tangential components of pressure and the density. The resulting interior solution represents a physically motivated anisotropic neutron star model. It is possible to match it with a boundary exterior solution. From this matching, the components of metric potentials can be rewritten in terms of a compactness parameter C which has to be C=2GM/Rc2<<0.5 for physical consistency. Other physical conditions for real stellar objects are taken into account according to the solution. We show that the model accurately bypasses conditions like the finiteness of radial and tangential pressures, and energy density at the center of the star, the positivity of these components through the stellar structure, and the negativity of the gradients. These conditions are satisfied if the energy-conditions hold. Moreover, we study the stability of the model by showing that Tolman–Oppenheimer–Volkoff equation is at hydrostatic equilibrium. The solution is matched with observational data of millisecond pulsars with a withe dwarf companion and pulsars presenting thermonuclear bursts.

Spider pulsars are millisecond pulsars in short-period (≲12-h) orbits with low-mass (~0.01–0.4  M ⊙ ) companion stars. The pulsars ablate plasma from the companion star, causing time delays and eclipses of the radio emission from the pulsar. The magnetic field of the companion has been proposed to strongly influence both the evolution of the binary system 1 and the eclipse properties of the pulsar emission 2 . Changes in the rotation measure (RM) have been seen in a spider system, implying that there is an increase in the magnetic field near the eclipse 3 . Here we report a diverse range of evidence for a highly magnetized environment in the spider system PSR B1744 – 24A 4 , located in the globular cluster Terzan 5. We observe semi-regular profile changes to the circular polarization, V , when the pulsar emission passes close to the companion. This suggests that there is Faraday conversion where the radio wave tracks a reversal in the parallel magnetic field and constrains the companion magnetic field, B   (> 10 G). We also see irregular, fast changes in the RM at random orbital phases, implying that the magnetic strength of the stellar wind, B , is greater than 10 mG. There are similarities between the unusual polarization behaviour of PSR B1744 – 24A and some repeating fast radio bursts (FRBs) 5 – 7 . Together with the possible binary-produced long-term periodicity of two active repeating FRBs 8 , 9 , and the discovery of a nearby FRB in a globular cluster 10 , where pulsar binaries are common, these similarities suggest that a proportion of FRBs have binary companions. The observation of pulsar emission at various orbital phases of a companion star probes the diverse magnetic structure in a binary system, and exhibits varying polarization behavior, akin to that observed in certain fast radio bursts.

Pulsars are famed for their rotational clocklike stability and their highly repeatable pulse shapes. However, it has long been known that there are unexplained deviations (often termed timing noise) from the rate at which we predict these clocks should run. We show that timing behavior often results from two different spin-down rates. Pulsars switch abruptly between these states, often quasi-periodically, leading to the observed spin-down patterns. We show that for six pulsars the timing noise is correlated with changes in the pulse shape. Many pulsar phenomena, including mode changing, nulling, intermittency, pulse-shape variability, and timing noise, are therefore linked and are caused by changes in the pulsar's magnetosphere. We consider the possibility that high-precision monitoring of pulse profiles could lead to the formation of highly stable pulsar clocks.

Over a dozen millisecond pulsars are ablating low-mass companions in close binary systems. In the original ‘black widow’, the eight-hour orbital period eclipsing pulsar PSR J1959+2048 (PSR B1957+20) 1 , high-energy emission originating from the pulsar 2 is irradiating and may eventually destroy 3 a low-mass companion. These systems are not only physical laboratories that reveal the interesting results of exposing a close companion star to the relativistic energy output of a pulsar, but are also believed to harbour some of the most massive neutron stars 4 , allowing for robust tests of the neutron star equation of state. Here we report observations of ZTF J1406+1222, a wide hierarchical triple hosting a 62-minute orbital period black widow candidate, the optical flux of which varies by a factor of more than ten. ZTF J1406+1222 pushes the boundaries of evolutionary models 5 , falling below the 80-minute minimum orbital period of hydrogen-rich systems. The wide tertiary companion is a rare low-metallicity cool subdwarf star, and the system has a Galactic halo orbit consistent with passing near the Galactic Centre, making it a probe of formation channels, neutron star kick physics 6 and binary evolution. ZTF J1406+1222 is a wide hierarchical triple system that hosts a low-metallicity subdwarf star and a ‘black widow’ millisecond pulsar that has a highly varying optical flux and a 62-minute period.

The gravitational wave radiation will release the energy momentum, which will dissipate the rotation of neutron star while in the accretion process. If the deformation of star is known, then we can estimate the maximum spin frequency of pulsar, based on which we can interpret why the spin periods of all millisecond pulsars cannot be less than one millisecond.