Gravitational wave background from rotating neutron stars

The background of gravitational waves produced by the ensemble of rotating neutron stars (which includes pulsars, magnetars, and gravitars) is investigated. A formula for $\Omega (f)$ (a function that is commonly used to quantify the background, and is directly related to its energy density) is derived, without making the usual assumption that each radiating system evolves on a short time scale compared to the Hubble time; the time evolution of the systems since their formation until the present day is properly taken into account. Moreover, the formula allows one to distinguish the different parts of the background: the unresolvable (which forms a stochastic background or confusion noise, since the waveforms composing it cannot be either individually observed or subtracted out of the data of a detector) and the resolvable. Several estimations of the background are obtained, for different assumptions on the parameters that characterize neutron stars and their population. In particular, different initial spin period distributions lead to very different results. For one of the models, with slow initial spins, the detection of the background by present or planned detectors can be rejected. However, other models do predict the detection of the background, that would be unresolvable, by the future ground-based gravitational wave detector ET. A robust upper limit for the background of rotating neutron stars is obtained; it does not exceed the detection threshold of two cross-correlated Advanced LIGO interferometers. If gravitars exist and constitute more than a few percent of the neutron star population, then they produce an unresolvable background that could be detected by ET. Under the most reasonable assumptions on the parameters characterizing a neutron star, the background is too faint to be detected. Previous papers have suggested neutron star models in which large magnetic fields (like the ones that characterize magnetars) induce big deformations in the star, which produce a stronger emission of gravitational radiation. Considering the most optimistic (in terms of the detection of gravitational waves) of these models, an upper limit for the background produced by magnetars is obtained; it could be detected by ET, but not by BBO or DECIGO. Simple approximate formulas to characterize both the total and the unresolvable backgrounds are given for the ensemble of rotating neutron stars, and, for completion, also for the ensemble of binary star systems.

Figure 1

Spectral strain sensitivity of aLIGO, two possible configurations of ET (named ETB and ETD), DECIGO and BBO. The sources of the curves are given in the text.

Figure 2

Redshift versus observed frequency of the gravitational waves produced by the ensemble of gravitars, assuming an ellipticity of $ϵ={10}^{-7}$. The vertical axis gives the redshift of the gravitational waves observed today (redshift at present). Those waves were emitted by gravitars that started radiating at the redshifts given by the gray scale (redshift of formation). The solid lines follow points of equal redshifts of formation (corresponding, from bottom to top, to redshifts 1, 2, 3, 4, and 5). The upper plot accounts for all signals in the universe, whereas the lower plot accounts only for unresolvable signals.

Figure 3

Same as Fig. 2, but for the ensemble of rotating neutron stars with magnetic field $B={10}^8\mathrm{T}$ and ellipticity $ϵ={10}^{-7}$.

Figure 4

Spectral function, versus observed frequency, of the total and unresolvable backgrounds produced under the assumption that all rotating neutron stars are gravitars, i.e., neutron stars which spin down primarily by emitting gravitational waves. The gravitar limit is also shown, as a robust upper limit of the background from rotating neutron stars. The background is calculated by using three different initial frequency distributions (see Sec. 4e). The unresolvable background is calculated with ${\mathcal{N}}_0=1$ and $\Delta f=1{\mathrm{yr}}^{-1}$.

Figure 5

Spectral function, versus observed frequency, of the total background produced by rotating neutron stars, assuming the magnetic field distribution of Eq. (88) (which is taken from Ref. 77) and the ellipticity distribution of Eq. (89) (from Ref. 15). Three initial frequency distributions are used (see Sec. 4e). None of the present or planned gravitational wave detectors is sensitive enough to observe such a background.

Figure 6

Spectral function, versus observed frequency, of the total and unresolvable backgrounds produced by magnetars. The rate of magnetars is assumed to be 10% of the one of neutron stars, the average magnetic field is $B={10}^{10}\mathrm{T}$, and the average ellipticity, $ϵ=-6.4\times {10}^{-4}$. This corresponds to the so-called TD model in Ref. 34, which is the most optimistic model (regarding the detection) considered in that paper. Other models predict levels of background several orders of magnitude lower. The total background plotted here can thus be considered an upper limit for the background of magnetars.

Figure 7

Spectral function, versus observed frequency, of the total background produced if all rotating neutron stars had the same magnetic field and ellipticity. Different line types correspond to different magnetic fields (as the legend of the upper plot describes). Each plot corresponds to a certain ellipticity.

Figure 8

Same as Fig. 7, but for the unresolvable part of the background.

Figure 9

Signal-to-noise ratio produced by the total background of rotating neutron stars, assuming that all of them have the same magnetic field and ellipticity. Each curve corresponds to a certain magnetic field (specified on top of each curve), and each point on a curve corresponds to a certain ellipticity (specified on the horizontal axis). These values of SNR are obtained by cross-correlating 1 year of data of two interferometers of ETB. Upper, middle, and lower plots are obtained using the initial frequency distribution called Distribution 0, 1, and 2, respectively, in Sec. 4e.

Figure 10

Same as Fig. 9, calculated in this case for ETD.

Figure 11

Same as Fig. 9, but calculated for two interferometers of BBO. The observation time assumed is also of 1 year.

Figure 12

Same as Fig. 9, but for two interferometers of DECIGO. The observation time assumed is also of 1 year.

Figure 13

Same as Fig. 9, but for the unresolvable part of the background (using ${\mathcal{N}}_0=1$ and $\Delta f=1{\mathrm{yr}}^{-1}$). These values of SNR are obtained by cross-correlating 1 year of data of two interferometers of ETB.

Figure 14

Same as Fig. 13, but for ETD.

Figure 15

Spectral function, versus observed frequency, of the total background produced by the ensemble of rotating neutron stars, using the three initial frequency distributions (Distribution 0, 1, and 2, of Sec. 4e). All systems are assumed to have $B={10}^8\mathrm{T}$, and $ϵ={10}^{-7}$. The light-gray curves are obtained with different star formation rates [67, 68, 69, 70, 71, 72, 73, 74], whereas the black dashed lines use the constant star formation rate of Eq. (106). The black solid line is obtained with the approximate formula of Eq. (a7).