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= Gravitational Wave Background = From Wikipedia, the free encyclopedia

The gravitational wave background (also GWB and stochastic background) is a random gravitational-wave signal potentially detectable by gravitational wave detection experiments. Since the background is random, it is completely determined by its statistical properties such as mean, variance, and higher correlation functions.

Sources of a Stochastic Background[edit]
Several potential sources for the background are hypothesized across various frequency bands of interest, with each source producing a background with different statistical properties. The sources of the stochastic background can be broadly divided into two categories: cosmological sources, and astrophysical sources.

Cosmological Sources[edit]
Cosmological backgrounds may arise from several early universe sources. Some examples of these sources include time-varying scalar (classical) fields in the early universe, "preheating" mechanisms after inflation involving energy transfer from inflaton particles to regular matter, phase transitions in the early universe (such as the electroweak phase transition), cosmic strings, etc. While these sources are more hypothetical, a detection of a background from them would be a major discovery of new physics. The detection of such an inflationary background would have a profound impact on early-universe cosmology and on high-energy physics.

Astrophysical Sources[edit]
An astrophysical background may be seen as confusion noise in a detector. Confusion noise is the stochastic fluctuations of the background, caused by many weak and independent sources, below which individual sources cannot be resolved. For instance, the astrophysical background from stellar mass binary black-hole mergers is expected to be a key source of the stochastic background for the current generation of ground based gravitational-wave detectors. LIGO and Virgo detectors have already detected individual gravitational-wave events from such black-hole mergers. However, there would be a large population of such mergers which would not be individually resolvable which would produce a hum of random looking noise in the detectors. Other astrophysical sources which are not individually resolvable can also form a background. For instance, a sufficiently massive star at the final stage of its evolution will collapse to form either a black hole or a neutron star – in the rapid collapse during the final moments of an explosive supernova event, which can lead to such formations, gravitational waves may be produced. Also, in rapidly rotating neutron stars there is a whole class of instabilities driven by the emission of gravitational waves.

The nature of the source also depends on the sensitive frequency band of the signal. The current generation of ground based experiments like LIGO and Virgo are sensitive to gravitational-waves in the audio frequency band between approximately 10 Hz to 1000 Hz. In this band the most likely source of the stochastic background will be an astrophysical background from binary neutron-star and stellar mass binary black-hole mergers.

The background created by astrophysical sources can be described mathematically in terms of parameters relevant to the specific source one is considering, as well as other more general parameters such as cosmological redshift. Since the background is stochastic, many important quantities of interest such as the expected power spectrum, energy density of the background gravitational waves, and characteristic amplitude of the background can be obtained by examining averages and correlation functions of the relevant variables. If we have a Gaussian background, the average amplitude taken over the whole sky $$\langle h_{ij}^{TT}\rangle = 0$$, but the two-point correlation function $$\langle h_{ij}^{TT}h_{ij}^{TT}\rangle$$ is, in general, nonzero. Here, $$h_{ij}^{TT}$$ (which describes the gravitational wave) refers to the spatial components of the metric perturbation on a Minkowski background ($$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$) taken in the transverse-traceless gauge. The two-point correlation function can be taken as an ensemble average over all frequencies, and be written in terms of a characteristic amplitude $$h_c(f)$$ of some given frequency, as follows :

$$\langle h_{ij}^{TT}h_{ij}^{TT}\rangle = 2\int^{f=\infty}_{f=0}d\operatorname{log}\!f\ h_{c}^2(f)$$

The characteristic amplitude can further be specified in terms of the energy spectrum of gravitational waves $$dE^{(r)}_{GW}(f_r)/d\operatorname{log}\!f_r$$ (referring to the energy and frequency in the source's frame, which will be shifted in an observer's frame), the redshift $$z$$, and the number density of sources at a given redshift $$n(z)$$ as :

$$h_c^2(f) = \frac{4G}{\pi c^2f^2} \int_0^\infty \frac{dz}{1+z}n(z) \left. \frac{dE^{(r)}_{GW}(f_r)}{d\operatorname{log}\!f_r} \right|_{f_r=(1+z)f}$$

(scripts '$$r$$' refer to that quantity in the source's rest frame, whereas unscripted quantities are in the observer's frame of reference). One last complication arises in the fact that the energy spectrum will not be the same for every given individual source, even if they are of the same type. To account for this, we can add in a dependence on a set of parameters $$\xi = \{\xi_1,...\xi_m \}$$ which are relevant to a particular source, and modify the above expression to obtain :

$$h_c^2(f) = \frac{4G}{\pi c^2f^2}\int_0^\infty \left. \frac{dz}{1+z}\int d\xi\frac{dn(z;\xi)}{d\xi} \frac{dE^{(r)}_{GW}(f_r;\xi)}{d\operatorname{log}\!f_r} \right|_{f_r=(1+z)f}$$, and similarly for the energy density $$\rho_{GW}$$ of the background,

$$\frac{d\rho_{GW}}{d\operatorname{log}\!f}(f) =\int_0^\infty \left. \frac{dz}{1+z}\int d\xi\frac{dn(z;\xi)}{d\xi} \frac{dE^{(r)}_{GW}(f_r;\xi)}{d\operatorname{log}\!f_r} \right|_{f_r=(1+z)f}$$.

These expressions are general, and can be used to compute the stochastic background generated by the superposition of many different signals from an ensemble of individual sources. The expressions of course depend on the types of sources we are interested in, and can be applied to black hole binaries, neutron star binaries, etc., so long as we correctly specify the relevant parameters $$\xi$$, the energy spectrum, and the number density.

Example: Background Produced by Supermassive Black Hole (SMBH) Binaries
As stated in the previous section, black hole binary mergers are important astrophysical sources for stochastic gravitational wave backgrounds. The expected background generated from these sources can be derived from the general formulas given in the above section. The inspiral process can be divided into separate regimes, each with particular assumptions which allow us to calculate the quantities of interest more easily. The major regimes of the motion largely depend on how close the black holes are to merging, since different effects become dominant at different scales of the orbit's semi-major axis $$a$$. Some regimes of particular interest are the dynamical regime dominated by gravitational wave back-reaction (when the black holes are close together, and their mutual gravity dominates other effects), the dynamical regime dominated by three-body interactions (such as gas accretions), and the regime dominated by other effects like dynamical friction. Each of these regimes depend on different parameters, and each correspond separately to different frequency ranges. As an example, the region dominated by gravitational wave back reaction assuming a circular orbit can be characterized by the Chirp mass of the binary $$M_c$$. Using the Newtonian approximation in this regime yields the following expression for the energy spectrum :

$$\frac{dE^{(r)}_{GW}}{d\operatorname{log}\!f_r}=\frac{1}{3G}(GM_c)^{5/3}(\pi f_r)^{2/3}$$.

With this expression, we can then compute quantities of interest like the characteristic amplitude and energy density of the gravitational wave background due to the gravitational wave back-reaction regime of SMBH binaries, as :

$$h_c^2(f) = \frac{4G^{5/3}}{3\pi^{1/3} c^2}f^{-4/3}\int_0^\infty \frac{dz}{(1+z)^{1/3}}\int dM_c \frac{dn(z;M_c)}{dM_c}M_c^{5/3} $$,

$$\frac{d\rho_{GW}}{d\operatorname{log}\!f}(f)=\frac{(G\pi f)^{2/3}}{3}\int_0^\infty \frac{dz}{(1+z)^{1/3}}\int dM_c \frac{dn(z;M_c)}{dM_c}M_c^{5/3}$$.

Note that these expressions now only depend on the redshift and the Chirp mass. If we are interested in other regimes with more detailed dynamics, we can perform similar manipulations and choose parameters associated with the different regimes. In the three-body interaction regime for instance, the Chirp mass dependence is replaced with a more detailed dependence on the individual masses $$\{m_1, m_2\}$$. We also assumed that the orbits were circular in these expressions. In principle, the orbits could have eccentricity, and these more detailed effects could also be considered. To actually compute these quantities, estimates for the number density need to be found in each case. There are theoretical estimates one can find by simulations, and other techniques. More recently, observational estimates have been deduced via large-scale sky surveys. With all of these pieces, one can calculate the energy spectrum, characteristic amplitude, etc., and test the predictions by detecting the stochastic background using gravitational wave measurements.

Detection[edit]
On 11 February 2016, the LIGO and Virgo collaborations announced the first direct detection and observation of gravitational waves, which took place in September 2015. In this case, two black holes had collided to produce detectable gravitational waves. This is the first step to discovery of the gravitational wave background.

See also[edit]

 * Cosmic microwave background
 * Cosmic neutrino background