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9.1 Linearized waves 321

The angular functions f

in the metric (9.48) depend on the axial parameter m. We list

these functions in the order m =±2, ±1, 0, with the functions corresponding to the upper

sign displayed on top of those corresponding to the lower sign:

= sin



cos 2φ

sin 2φ



, 2sinθ cos θ



cos φ

sin φ



, 2 − 3sin

θ,

rθ

= sin θ cos θ



cos 2φ

sin 2φ



, (cos

θ −sin

θ)



cos φ

sin φ



, − 3sinθ cos θ,

rφ

= sin θ



−sin 2φ

cos 2φ



, cos θ



−sin φ

cos φ



, 0,

(1)

θθ

= (1 + cos

θ)



cos 2φ

sin 2φ



, 2sinθ cos θ



−cos φ

−sin φ



, 3sin

θ,

(2)

θθ



−cos 2φ

−sin 2φ



, 0, − 1,

θφ

= cos θ



sin 2φ

−cos 2φ



, sin θ



−sin φ

cos φ



, 0,

(1)

φφ

=−f

(1)

θθ

(2)

φφ

= cos



cos 2φ

sin 2φ



, 2sinθ cos θ



−cos φ

−sin φ



, 3sin

θ − 1.

(9.54)

Exercise 9.4 Consider a superposition of ingoing and outgoing waves with

(x) =−F

(x). (9.55)

(a) Show that if F

(x) is an odd function in x,thent = 0 corresponds to a moment

of time symmetry (i.e., show that K

= 0att = 0).

(b) Now consider the speciﬁc choice

(x) =−F

(x) = Axe

−(x /λ)

, (9.56)

where

A is an amplitude and λ measures the spatial extent of the wave packet.

Perform an expansion around r = 0 to show that the waveform is regular (i.e.,

nonsingular) at the origin.

Hint: It may be helpful to use an algebraic package like Mathematica, or else to

restrict the analysis to t = 0.

To translate to the h

and h

polarizations, we can introduce a local orthonormal

coordinate system and identify

= h

= Cf

(1)

θθ

(9.57)

and

= h

=−2Cf

θφ

. (9.58)

Here we have kept only those terms that fall off with 1/r, dropping higher-order terms.

322 Chapter 9 Gravitational waves

Exercise 9.5 Use equation (9.30), without averaging over wave cycles, to show that

the total energy emitted by an even-parity l = 2, m = 0waveis

=−



(5)



, (9.59)

provided the F

(4)

/r term constitutes the dominant contribution to C at large r .

Similarly, the odd-parity or axial metric takes the form

=−dt

+ dr

+ (2Kd

rθ

) rdrdθ + (2Kd

rφ

) r sin θ drdφ + (1 + Ld

θθ

) r

dθ

+(2Ld

θφ

) r

sin θdθdφ + (1 + Ld

φφ

) r

sin

θdφ

. (9.60)

Here we construct the coefﬁcients K and L from a function

G = G

(t − r ) + G

(t + r ) (9.61)

and its derivatives

(n)

≡

(x)

x=t−r

+ (−1)

(x)

x=t+r

(9.62)

according to

K =

(2)

(1)

, (9.63)

L =

(3)

(2)

(1)

. (9.64)

The angular functions d

are again listed in the order m =±2, ±1, 0, yielding

rθ

= 4sinθ



cos 2φ

sin 2φ



, −2cosθ



cos φ

sin φ



, 0,

rφ

=−4sinθ cos θ



sin 2φ

−cos 2φ



, −2(cos

θ −sin

θ)



sin φ

−cos φ



, −4cosθ sin θ,

θθ

=−2cosθ



cos 2φ

sin 2φ



, −sin θ



cos φ

sin φ



, 0,

θφ

= (2 −sin

θ)



sin 2φ

−cos 2φ



, cos θ sin θ



sin φ

−cos φ



, −sin

θ,

φφ

= 2cosθ



cos 2φ

sin 2φ



, sin θ



cos φ

sin φ



, 0.

(9.65)

As we have seen in exercise 9.4, we can construct superpositions of ingoing and outgo-

ing waves in vacuum that at t = 0 are time symmetric with K

= 0. If the amplitude A

of the wave is small (A  1), the metric for the complete spacetime is given by the lin-

earized solutions (9.48)or(9.60)toequation(9.44). For a nonlinear wave, the 3 + 1

9.2 Sources 323

equations must be solved numerically to determine the spacetime. However, a time-

symmetric waveform at t = 0 automatically satisﬁes the momentum constraint (2.133).

To completely specify the initial data for such a nonlinear wave, we only need to solve

the Hamiltonian constraint (2.132).

Linearized waves provide useful analytic solutions

for testing initial value routines and 3 + 1 vacuum evolution codes designed to handle

nonlinear problems.

9.2 Sources

The gravitational waves that we hope to observe on Earth have exceedingly small ampli-

tudes.

For a binary, for example, we can evaluate equation (9.27) to estimate the typical

gravitational wave strain h. For equal-mass binaries we have µ = M/4, so

h 

µM

 5 × 10

−20



1Mpc









. (9.66)

For a binary of stellar-mass black holes with M = 10M



, for example, located in the

Virgo cluster at a distance of about 20 Mpc, at a binary separation of about

R = 6M,

we see that the strain h is smaller than about 10

−20

. Similar estimates hold for other

stellar sources of gravitational radiation. Clearly it is a formidable challenge to observe

this radiation, and we will return to this issue in Section 9.3. In the meantime we recognize

that for an astrophysical object of a given mass M to emit strong gravitational radiation,

its compaction M/R needs to be large. This requirement singles out compact objects as

the most promising stellar sources of gravitational radiation.

Of course, there may also be other astrophysical sources of gravitational radiation that are

not related to stars. Processes in the early Universe, for example, may generate gravitational

radiation that may still be observable today, not unlike the cosmic microwave background.

Potential origins of such a radiation include primordial ﬂuctuations in the Universe’s

geometry, ampliﬁed during inﬂation, and phase transitions. String theory also predicts

possible mechanisms for the generation of gravitational radiation, including vibrating

cosmic strings and the condensation of branes. Finally, one might suspect that there are

sources of detectable gravitational radiation of which we are unaware at the time of the

writing of this book. Every time another form of radiation has opened up a new window

for us to view the Universe, we have been surprised by some unexpected observations.

These observations, in turn, have led to a new and deeper understanding of the Universe.

It is entirely possible that nature will reward us with a similar surprise when we are able to

observe and measure gravitational radiation at sufﬁciently high sensitivities.

Time-symmetric, nonlinear wave solutions constructed in this way in axisymmetry are often refered to as Brill waves

(see exercise 3.5).

Our discussion in this section is drawn from Flanagan and Hughes (2005), who provide a more detailed overview as

well as numerous references.

As we will see in Chapters 12 and 13, this binary separation is close to the “innermost stable circular orbit”, at which

the binary inspiral becomes increasingly rapid and decreasingly circular.

324 Chapter 9 Gravitational waves

Different sources of gravitational radiation emit waves at vastly different frequencies. It is

useful to divide this large spectrum of gravitational radiation into different frequency bands.

The high frequency band includes frequencies in the range 1 Hz

∼

Hz, the low

frequency band covers the range 10

−5

∼

1Hz,thevery low frequency band spans

−9

∼

−7

Hz, and the ultra low frequency band covers 10

−18

∼

−13

Hz.

The partition into these different bands is motivated in part by the means by which

we detect them, as we will discuss below. It is also motivated by the different classes of

potential sources. To distinguish the later, we shall estimate the characteristic gravitational

wave frequency from a stellar object of mass M,radiusR and compaction M/R by

f 





3/2

 2 × 10









3/2

. (9.67)

The highest frequency sources are compact objects with large compactions (black holes or

neutron stars) and small masses; stellar-mass compact objects fall into this category. These

sources fall into the high frequency band. Objects with either larger masses (supermassive

black holes) or smaller compactions (white dwarfs, or binaries with large binary separation)

radiate in the low frequency band. Other stellar sources may radiate with even lower

frequencies; however, the strain of their radiation is so weak that this radiation is not

interesting from an observational perspective. That leaves nonstellar sources as the most

interesting sources of gravitational radiation in the very low and ultra low frequency bands.

9.2.1 The high frequency band

The high frequency band includes frequencies in the approximate range 1 Hz

∼

Hz. This frequency band is observable with the new generation of ground-based gravita-

tional wave interferometers, which we will discuss in Section 9.3. The upper limit of this

band corresponds to the highest frequency of gravitational radiation that we may expect

from stellar sources. As discussed above, we can estimate this limit from equation (9.67)

for stellar-mass compact objects with large compactions, i.e., neutron stars or stellar-mass

black holes. The lower limit of this band is set by our ability to dectect gravitational radia-

tion with ground-based interferometers. For smaller frequencies, it becomes increasingly

difﬁculty to decouple the signal from the noise arising from low-frequency vibrations, both

mechanical and gravitational, on the ground. Curiously, the high frequency band coincides

more or less with the audible range of human hearing; if converted to sound, we would be

able to hear the gravitational radiation in this band.

Promising sources in this frequency band include compact, stellar-mass binaries, stellar

core collapse, rotating neutron stars, and stochastic backgrounds.

For binaries this relation follows directly from Kepler’s third law. For other objects we can estimate their characteristic

frequency from their dynamical time scale: f  1/τ

dyn



√

ρ, which again results in equation (9.67).

We refer to Cutler and Thorne (2002), both for a more detailed discussion of these and other sources and a perspective

on the possibility of their detection.

9.2 Sources 325

Compact binaries

Compact binaries – binary neutron stars, binary black holes, and binary black hole-neutron

stars – are probably the most promising sources of gravitational radiation for ground-

based gravitational wave detectors. Very simple estimates of the emitted gravitational

wave signals can be obtained from exercises 9.2, 9.3 and 12.3. More accurate predictions

applicable for close, mildly relativistic orbits require post-Newtonian calculations; see

Appendix E for a brief summary of these calculations. The highly relativistic late inspiral,

plunge and merger phases of the orbit require full numerical simulations for the evolution

and emitted gravitational radiation. Such numerical simulations will be described in detail

in Chapters 12–13 and 15–17.

Compact binaries are known to exist, at least in the case of binary neutron stars. Since

the discovery by Hulse and Taylor (1975) of PSR 1913+16, a binary neutron star system

containing a radio pulsar, a number of similar systems have been identiﬁed.

While several

of these binaries will coalesce within a Hubble time

none have small enough binary

separation to be observable with the current generation of gravitational wave detectors.

However, for some binaries the gravitational wave back-reaction is strong enough for us to

measure its effect on the binary orbit. By monitoring the orbit of PSR 1913+16 by radio

pulsar techniques, Hulse and Taylor were able to conﬁrm for the ﬁrst time the validity of

equation (9.37) and, hence, the rate at which gravitational radiation carries off energy as

predicted by general relativity in the weak-ﬁeld, slow-velocity limit.

We can use a statistical analysis based on the known sample of observed binary neutron

stars to estimate the rate of binary neutron star coalescence per Milky Way-type galaxy.

Since this sample is rather small, the estimates are not very rigorous, but presumably they

improve with each new discovery of a binary neutron star system.

To date, no binary

black hole or black hole–neutron star system has been discovered, so we cannot perform

such an analysis to estimate their merger rates.

An alternative way of estimating compact binary coalescence rates is to model the

evolution of stellar populations. These “population synthesis” calculations rely on theoret-

ical models for the late stages of stellar evolution and the formation of compact binaries

and have signiﬁcant uncertainties. Observational constraints can sharpen these population

synthesis estimates, yielding

merger rates of binary neutron stars of approximately 10

−5

–

−4

per year per Milky Way-type galaxy, 10

−6

–10

−5

for binary black hole-neutron stars,

and 10

−7

–10

−6

for binary black holes. Thus, to have an appreciable detection rate, gravi-

tational wave detectors must be able to observe mergers out to sufﬁciently large distances

so that the volume they survey contains many galaxies. We shall discuss the sensitivities

of the current generation of detectors in Section 9.3.

See, e.g., Stairs (2004)forareview.

At least seven such neutron star binaries have been discovered via radio pulsar observations at this time. They are

J0737−3039 (a double pulsar), J1518+4904, B1534+12, J1811−1736, J1829+2456, B1913+16 (the Hulse–Taylor

binary), and B2127+11C; see Stairs (2004).

See Burgay et al. (2003).

Kalogera et al. (2007).

326 Chapter 9 Gravitational waves

One payoff of measuring gravitational waves from compact binaries is that we can use

the wave data to perform “stellar spectroscopy”. Speciﬁcally, once the data are analyzed

using theoretical templates and “matched-ﬁltering” techniques (see Section 9.3), they can

provide direct measurements of the masses and spins of the inspiraling companions. These

gravitational wave measurements are the only way we can obtain these parameters for

binary black holes, since they emit no other form of radiation. Unlike other means used

to estimate the masses and spins of black holes in gaseous environments, gravitational

wave measurements from binary inspiral do not have the complications and uncertainties

associated with the modeling of nonvacuum physics near black holes, such as turbulent

MHD accretion or electromagnetic radiation transport. The accuracy to which binary

parameters can be determined from a gravitational wave detection depends on several

factors, including the masses of the binary companions themselves.

For a “typical”

observation of a binary black hole system consisting of two 10 M



black holes with

ground-based gravitational wave detectors (see Section 9.3.1), an analysis of the the lowest-

order quadrupole radiation emitted during the inspiral phase

is sufﬁcient to determine

the so-called chirp mass (m

)

3/5

+ m

)

−1/5

to within a fraction of a percent.

Determining the individual masses m

and m

requires additional information, for example

the reduced mass, which appears at the next post-Newtonian order. Unfortunately, the

effects of black hole spin also appear in these post-Newtonian terms. Measuring these

quantities individually therefore requires some way of breaking the degeneracy. At least in

principle, one approach may involve observations of the binary merger. As we will see in

Chapter 13, the black hole spins do have an important effect on the dynamics of the binary

black hole mergers; matching observed gravitational wave signals from the merger phase

to numerical simulations may therefore hold the key to determining the individual black

hole parameters.

The expectation that compact binaries are signiﬁcant sources of detectable gravitational

radiation, together with their unique role as fundamental probes of general relativity, make

compact binaries of crucial importance for gravitational wave physics and astronomy. This

is why we devote so much attention to these objects in this book.

Stellar core collapse

Stellar core collapse, which occurs in a Type II supernova, for example, may also emit

a strong gravitational wave signal. Core collapse will lead to the formation of a neutron

star or, for the most massive progenitors, a black hole. Such an event involves a large

amount of mass moving very rapidly in a very small volume. Just how much gravitational

The mass of the binary affects the frequency range of the emitted gravitational wave pattern (equation 9.67), and hence

it determines whether or not the signal is received in a part of spectrum at which gravitational wave detectors are

sensitive; compare the gravitational wave “noise-curves” in Section 9.3.

See Chapter 12.1, including Figure 12.1 and exercise 12.3.

See, e.g., Cutler and Flanagan (1994); Cutler and Thorne (2002).

9.2 Sources 327

radiation is emitted nevertheless depends on how much the collapse deviates from spherical

symmetry. If the progenitor star has very little rotation, then the collapse will proceed almost

spherically, and only a very small amount of gravitational radiation will be emitted. If the

progenitor star rotates more rapidly, however, then the collapsing core will be deformed and

this will induce a time-varying quadrupole moment and, hence, a potentially signiﬁcant

burst of gravitational radiation.

A rotating star in equilibrium is oblate – even the Earth has a slightly larger radius

at the equator than at the poles. The degree of oblateness increases as the ratio of the

rotational kinetic energy T = I 

/2 = J

/(2I ) to the gravitational binding energy |W |

increases. Here I is the moment of inertia, J = I  the angular momentum, and W is the

gravitational potential energy of the star.

During the collapse the mean radius of the core,

R, decreases. Since I ∝ R

and |W |∝1/R,weﬁnd

|W |

∝

. (9.68)

The above relationship implies that, assuming conservation of angular momentum, the ratio

T/|W | increases during the collapse. Even a modestly oblate progenitor star will therefore

become increasingly oblate following collapse. Such a collapse leads to a gravitational

wave burst signal. It would be particularly exciting to observe such a signal in conjunction

with an electromagnetic or a neutrino counterpart.

It is also possible that the collapsed core does not remain axisymmetric. If the ratio

T/|W | becomes sufﬁciently large, nonaxisymmetric instabilities may be triggered. Best

understood is the so-called bar-mode instability (see Chapters 14.2.2 and 16.3), which

deforms the star into a triaxial ellipsoid, i.e., the shape of an American football. Typically,

the critical value of T/|W | for this to happen is about 0.14, which is quite large. However,

other instabilities – including a single-arm m = 1 spiral mode or the r-mode – may develop

at smaller values of T /|W |. If a bar-mode instability develops in a neutron star remnant

following core collapse, the remnant would emit a periodic gravitational wave signal in

addition to the initial burst. Depending on its strength, the periodic signal may be much

easier to identify via gravitational wave observations, since the periodicity can be exploited

to build up a higher signal-to-noise ratio in the output of a gravitational wave detector.

Finally, we mention the possibility of a neutron star remnant ultimately undergoing

collapse to a black hole (see Chapter 14.2.1). A number of different scenarios suggest how

such a “delayed collapse” could be triggered – by phase transitions as a newly-formed

neutron star cools, by accretion fall-back of the stellar mantle, or by changes in the angular

velocity proﬁle of the remnant induced by magnetic braking or viscosity. If the remnant

is rotating, delayed collapse will result in a delayed burst of gravitational waves, and this

For an incompressible, rotating Newtonian star T /|W |∼e

for small e

 1, where e

≡ 1 − R

pol

, R

pol

is the

polar radius and R

is the equatorial radius. See Shapiro and Teukolsky (1983), equation (7.3.24), for the exact

expression, as well as Chapters 7.3 and 16.8 in that text for further discussion.

328 Chapter 9 Gravitational waves

burst will be followed by emission from the quasinormal ringing of the black hole as it

settles into stationary equilibrium.

Rotating neutron stars

Rotating neutron stars are also promising sources of gravitational radiation in the high

frequency band. We have observed many rotating neutron stars as radio and X-ray pulsars,

so we know not only that they exist, but also where they are located. Also, such a star emits

a periodic gravitational wave signal at a nearly constant, known frequency, which serves

to increase the signal-to-noise ratio as mentioned above.

A perfectly axisymmetric, rotating neutron star in stationary equilibrium emits no grav-

itational radiation. For the star to radiate gravitational waves, its axisymmetry has to be

broken. A number of different mechanisms have been suggested that could accomplish

this: an ultra-strong internal magnetic ﬁeld that is not aligned with the axis of rotation,

accretion pile-up from a companion star, or a small “mountain” that emerges on the crust

from a starquake.

It is difﬁcult to predict how large a deformation to expect from these

mechanisms, making it somewhat uncertain how strong we may expect the gravitational

wave signal to be. Current gravitational wave detectors already provide useful upper limits

on the gravitational wave emission from a few rapidly rotating pulsars, and hence, on their

nonaxisymmetric deformation.

Stochastic backgrounds

Random, independent and uncorrelated sources that we cannot resolve observationally

give rise to so-called “stochastic background” radiation. Stochastic backgrounds include

the gravitational radiation that may have been produced in the early Universe, for example

by phase transitions, from quantum ﬂuctuations arising during inﬂation or from cosmic

strings.

Stochastic background radiation could exist at all frequencies, and thus spans

all the frequency bands considered here. Physically useful upper limits on stochastic

background radiation in the high frequency regime already can be established with current

gravitational wave detectors.

9.2.2 The low frequency band

The low frequency band between 10

−5

∼

1 Hz cannot be observed with ground-

based detectors, since their coupling – both mechanically and gravitationally – to a host of

See, e.g., Shapiro and Teukolsky (1983), Chapter 10.5 and 10.11 for discussion and references.

See The LIGO Scientiﬁc Collaboration: B. Abbott et al. (2007).

See, e.g., Peacock (1999)andFlanagan and Hughes (2005) for further discussion and references.

See The LIGO Scientiﬁc Collaboration: B. Abbott (2007).

9.2 Sources 329

Earth-related sources of noise is too strong. This leaves space-based observatories as the

best means of detecting such radiation, and we will describe one such planned instrument,

LISA, in Section 9.3.

As we can infer from equation (9.67), promising stellar sources in the low frequency

regime include both high compaction, supermassive objects, such as a supermassive black

hole binaries, and low compaction, stellar-mass objects, such as white dwarf binaries. As

discussed above, there also may be stochastic background radiation present in the low

frequency band, but here we will focus only on stellar sources.

Supermassive black holes

Supermassive black holes (M ∼ 10

–10



) are believed to be the engines that power

quasars and active galactic nuclei. Most, if not all, nearby bulge galaxies host a supermas-

sive black hole at their centers. Black holes can grow to supermassive scale in the early

Universe by a combination of accretion and binary mergers.

Because supermassive black

hole masses are well-correlated with the velocity dispersions of stars in the central bulges

of their host galaxies, as well as with the total bulge masses, it appears that supermassive

black hole formation and growth are intimately connected with galaxy formation.

Binaries containing supermassive black holes are promising sources of quasiperiodic,

low frequency gravitational radiation. They can be divided into two broad categories. One

category consists of two supermassive black holes of comparable mass in binary orbit.

The other category consists of a supermassive black hole with a stellar-mass companion,

sometimes referred to as an “extreme mass ratio inspiral”, or EMRI, binary.

Binaries containing two supermassive black holes of comparable mass can form when

two galaxies collide and merge. Galaxy mergers are believed to provide an important

mechanism for the hierachical build-up of large-scale structure in the early Universe. Such

mergers continue to be observed in the present epoch. Thus, the formation of massive and

supermassive binary black holes may not be uncommon.

From a computational point of view, the inspiral and coalescence of two supermassive

black holes is identical to that of two stellar-mass black holes. We will discuss general

relativistic simulations of binary black hole coalescence in detail in Chapter 13.LISA,or

a detector like it, may be able to observe supermassive black hole mergers up to fairly high

redshift (possibly z ∼ 5–10), whereby we can use gravitational waves to probe structure

formation in the early Universe and provide insight into how supermassive black holes

form in the ﬁrst place.

One quasar, QSO SDSS 1148+5251, observed at redshift z = 6.43 (Fan, X. et al. (2003)), is believed to host a

supermassive black hole that has grown to a mass of about 10



within 0.9 Gyr after the Big Bang; see Shapiro

(2005) and references therein.

See articles in Ho (2004) for an overview and references.

See Sesana et al. (2007) for the tentative implications for LISA.

330 Chapter 9 Gravitational waves

EMRI binaries form when a supermassive black hole captures a smaller, stellar-mass

object. These binaries are promising sources of gravitational radiation if the binary is

compact, i.e., if the stellar-mass object is a black hole, neutron star or white dwarf. For

these compact EMRI binaries, tidal effects on the low-mass companion are small and

the stars are completely intact when they are swallowed by the supermassive black hole.

The challenge in modeling these binaries lies in correctly accounting for the gravitational

radiation reaction force, which causes a gradual deviation of the object’s orbit from a

pure geodesic. Observing such a binary may provide our best opportunity to map out the

strong-ﬁeld geometry of a Kerr black hole. Tentative estimates suggest that LISA may be

able to observe at least several of these systems per year.

We also mention the possibility that LISA may detect intermediate-mass black holes with

masses somewhere between stellar-mass and supermassive black holes (i.e., M ∼ 10

–



). There have been some tentative observational suggestions that such black holes

may exist.

Since binaries involving intermediate-mass black holes would also radiate in

the low frequency band, gravitational wave observations of, say, a binary containing one or

more intermediate-mass black hole could provide unambiguous evidence of their existence.

White dwarf binaries

White dwarf binaries constitute a well established and understood group of weak-ﬁeld

sources in the low frequency band. These systems emit gravitational radiation that is

strong enough to be detected by proposed detectors like LISA, but weak enough that the

radiation reaction force on the binary orbit is very small, making the signal almost perfectly

periodic over typical observation timescales. In fact, these sources are so well understood

that they can be used to calibrate LISA once the instrument is operational.

From a “new discovery” point of view, these sources are not so exciting as some of the

strong-ﬁeld sources we have described. Since we already understand these binaries fairly

well, there is not much we can learn about them from gravitational radiation. We may nev-

ertheless discover many more of these binaries, and, in addition, there are some aspects of

known binaries that we may be able to probe better with the help of gravitational waves. For

example, the orbit’s inclination angle, which is difﬁcult to determine from other observa-

tions, can be deduced from the relative amplitudes of the polarization waveforms h

and h

It is expected that LISA will be able to detect so many periodic binary sources that

they will form a stochastic background noise. This background is likely to exceed the

instrument’s intrinsic noise at low frequencies f

∼

−3

Hz.

9.2.3 The very low and ultra low frequency bands

The timescales associated with the very low frequency band, 10

−9

∼

−7

Hz,

correspond to a few months to a few decades. These limits are set by the observational

See van der Marel (2004) and references therein.