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ma = 5.70µeV

1012GeVfa

⎛

⎝⎜⎞

⎠⎟ fa

ma ≈

µ2

fae−# moduli/2

5

m ≈10−27 eV m ≈10

3eV m ≈103GeV m ≈10−33eV m ≈10

13GeV

10−20eV ≤ m ≤10−10eV

Kitajima’stalk

m ≈10−6eV

m−1 ≈ GHz m−1 ≈10−18 Hz

ω + i∂z( )− ma

2

2ω⎡

⎣⎢

⎤

⎦⎥a = −

12

gaγγ BT A!

ω + i∂z( )A! = − 12 gaγγ BT a

ω + i∂z( )A⊥ = 0

MM⊙

⎛

⎝⎜⎞

⎠⎟≈

RS3km

⎛⎝⎜

⎞⎠⎟≈ m

10−10eV⎛⎝⎜

⎞⎠⎟

−1

10−20eV ≤ m ≤10−10eV

1M⊙ ≤ M ≤1010 M⊙

ρDM =

12!a2 + 1

2m2a2 ≈ 1

2m2a0

2

a = a0 cos mt

pDM =

12!a2 − 1

2m2a2 ≈ − 1

2m2a0

2 cos 2mt( )

!4 !2 2 4

2

4

6

8

m10−23eV

⎛⎝⎜

⎞⎠⎟≈ f

10−8 Hz⎛⎝⎜

⎞⎠⎟

H0 ≈10−33eV

β ≡gaγγ Bm

, κ ≡ km, ε ≡

gaγγ k 2ρm2

γ

γa γ

γa

a

B

hij = d2n̂ hij t − n̂ ⋅x( )∫

z ≡ ν0 −ν (te )ν0

= 12

d 2n̂ p̂i p̂ j

1+ n̂ ⋅ p̂Δhij n̂( )∫ == d 2n̂ z te, n̂( )∫Δt ≡ dte z te, n̂( )0

t

∫

n̂ :direction of wave propagation

7

PSR B1855+09 [21],

h20Ωgwstring < 4 × 10−9 (6)

improving over the previously best limit [16] bymore than one order of magnitude.

In addition to this truly cosmological stochas-tic GW background, a contribution is also ex-pected from astrophysical processes, in partic-ular from the coalescence of massive BH bina-ries during early galaxy evolution [28,15]. WhileLISA may detect individual mergers of the moremassive BH binaries, these sources also form astochastic background. The spectrum of this“foreground” contribution follows hc(f) ∝ f−2/3

or h20Ωgw(f) ∝ f+2/3 [26]. Whilst the exact am-plitude of this signal depends on the mass func-tion of the massive BHs and their merger rate[15] (see shaded area in Fig. 2), massive BH bi-naries are the among primary sources expectedfor the background in the frequency range de-tectable with LISA and the PTA [15,40,9]: thenHz frequency background is thought to be dom-inated by BH binaries at redshifts z ∼ 7. Acomplementary detection of this background anda measurement of its amplitude from nHz to mHzwith LISA and the PTA would allow to discrimi-nate between this foreground signal from the cos-mological sources and would provide unique in-formation about the physics and history of BHgrowth in galaxies.

3.2. Experimental StrategyThe sensitivity of the PTA scales according to

h20Ωgw ∝ RMS2f4 (7)

producing a wedge-like sensitivity curve as shownin Fig. 2 (e.g. [15]). For timing precision thatis only limited by radiometer noise, the RMS isexpected to scale with the collecting area of theobserving telescope. In reality, the precision isalso affected by pulse phase jitter, propagationeffects in the interstellar medium and gain andpolarization calibration (see pulsar science for de-tails). While we discuss the resulting technicalrequirements such as multi-frequency capabilities

-15 -10 -5 0 5 10

-16

-14

-12

-10

-8

-6

LIGO I

LIGO II/VIRGO

LISA

PULSARS

COBE

Slow-roll inflation - Upper Bound

0.9KBlackbodySpectrum

Inflation

Inflation

Localstrings

Globalstrings

1st OrderEW PhaseTransition

ExtendedInflation

EW PhaseTransition

1st Order COBE

CMB−POL

LISA AdvancedLIGO

SKA−PTA

MBH−MBH

Current PTA

Binaries

Figure 2. Summary of the potential cosmologicalsources of a stochastic gravitational background,including inflationary models, first-oder phasetransitions and cosmic strings and a primordial0.9K black-body graviton spectrum as presentedBattye & Shellard (1996). We also overlay boundsfrom COBE, from current millisecond pulsar tim-ing and the goals from CMB polarization, LISAand Advanced LIGO (see e.g. Maggiore 2000).The PTA provided by the SKA will improve onthe current MSP limit by about four orders ofmagnitudes. The gray area indicates the spec-trum of an additional astrophysical backgroundcaused by the merger of massive black holes(MBHs) in early galaxy formation (e.g. Rajagopal& Romani 1995, Jaffe & Backer 2003). For thisbackground, Ωgw ∝ f2/3, whilst its amplitudedepends on the MBH mass function and mergerrate. The uncertainty is indicated by the size ofthe shaded area.

Δ t = − Ω t '( )−Ω0Ω00

t

∫ dt '

ds2 = − 1+ 2Φ( )dt2 + 1− 2Φ( )δ ijdxidx j

Φ =

ρDM8m2

cos 2mt( )

Ω t( )−Ω0Ω0

= Φ x ,t( )−Φ x p ,t '( )

Ba = −

gaeee

∇a

fa =

ω a2π

=ma c

2

h= 0.24

ma1.0µeV

⎛⎝⎜

⎞⎠⎟

GHz

Lint = −igaeeaψ γ 5ψ ! −

gaee"2me

σ̂ ⋅∇a = 2µBŜ ⋅ −gaeee

∇a⎛⎝⎜

⎞⎠⎟

µB =e!

2me

! 4.4×10−8 gaee

ρDM0.45GeV/cm3

⎛⎝⎜

⎞⎠⎟

1/2v

300km/s⎛⎝⎜

⎞⎠⎟

[T]

Ba < 4.1×10−14[T]

gaee

H ! −µB 2δ ij + hij( )S i B j

geff <

3.5×10−12eV3.1×10−11eV

⎧⎨⎪

⎩⎪

Sh <7.6×10−22 Hz−1/2 at 14 GHz

1.2×10−20 Hz−1/2 at 8.2 GHz

⎧⎨⎪

⎩⎪

geff =

1

4 2µB Bz sinθ N cos

2θ h(+ )( )2 + h(× )( )2 + 2cosθ sinθh(+ )h(× )⎡⎣⎢⎤⎦⎥

1/2

!!A + k2 1+ ε λρksinmt

⎡

⎣⎢

⎤

⎦⎥A = 0 , ε = ±1

fr = 1.2 ×104 m10−10 eV

⎛⎝⎜

⎞⎠⎟Hz

ρaxionρDM

≤10−12

30

1012GeV( )−1λ

0.3GeV/cm3

ρpc

Δ f = 2.6×10−14 λ

1016GeV( )−1ρ

0.3GeV/cm3Hz

hij = hReijR n( ) + hLeijL n( )

!!hA +

mεAδ cosmtm + εAkδ cosmt

k !hA + k2hA = 0

kres =m2= 1.2 ×104 m

10−10 eV⎛⎝⎜

⎞⎠⎟Hz

S =

M p2

2d 4∫ x −g R +

M p8ℓ2 d 4x −g a(x)ε µνλρRαβµν R

αβλρ∫ − d 4∫ x −g

12

∂a( )2 + 12 m2a2

⎡

⎣⎢

⎤

⎦⎥

a = a0 cos mt

δ = m2ℓ2

a0M p

εA =1 for R−1 for L

⎧⎨⎪

⎩⎪

ℓ ≤108km

101012

ℓ ≤1km

10−8 10

−10eVm

⎛⎝⎜

⎞⎠⎟

108 kmℓ

⎛⎝⎜

⎞⎠⎟

20.3GeV/cm3

ρpc

100Mpc

ℓ = 100km

.

1.0× 10

−10eVm

⎛⎝⎜

⎞⎠⎟

100kmℓ

⎛⎝⎜

⎞⎠⎟

210−29 g/cm3

ρMpc

Ωa = 0.01

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