Abstract
A low-frequency gravitational-wave background (GWB) from the cosmic merger history of supermassive black holes is expected to be detected in the next few years by pulsar timing arrays. A GWB induces distinctive correlations in the pulsar residuals—the expected arrival time of the pulse less its actual arrival time. Simplifying assumptions are made in order to write an analytic expression for this correlation function, called the Hellings and Downs curve for an isotropic GWB, which depends on the angular separation of the pulsar pairs, the gravitational-wave frequency considered, and the distance to the pulsars. This is called the short-wavelength approximation, which we prove here rigorously and analytically for the first time.
Highlights
Gravitational waves (GWs) are ripples in the fabric of space-time, originating from some of the most violent events in the Universe, including the mergers of supermassive black holes
High frequency GWs from the merger of stellar-mass black holes were first detected by the Laser Interferometer Gravitational-wave Observatory (LIGO) in September 2015 [1], hailing the dawn of gravitational-wave astronomy
With a Pulsar Timing Array (PTA), one can detect GWs from inspiraling supermassive black holes (SMBHs) binaries (SMBHBs), see e.g. [10, 11], but the GW background (GWB) from the cosmic merger history of SMBHBs [12,13,14]. This GWB is expected to be detected in the few years [15, 16], with the details depending on the underlying astrophysics of the SMBH
Summary
Gravitational waves (GWs) are ripples in the fabric of space-time, originating from some of the most violent events in the Universe, including the mergers of supermassive black holes. At the very low-frequency end of the GW spectrum, one expects to find nanohertz GWs from very massive inspiraling SMBHs, in the 108 − 109 M range These can be detected by timing millisecond pulsars, called a Pulsar Timing Array (PTA) [3,4,5,6]. For the first time, that the Hellings and Downs curve can be extracted from the cross-correlated pulsar residuals, without making assumptions that the pulsars are all at the same distance L from the Earth Part of this proof is a consequence of the application of the Riemann-Lebesgue Lemma and the Lebesgue Dominated Convergence Theorem – wellknown in the mathematics community, but somewhat obscure in the field of GWs. We emphasize that no previous work has been able to do this analytically, though computer-aided integration has been used to verify one’s intuition numerically
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