Gravitational wave signals of extra dimensions

Ruth GregoryDurham, UK

COSMO12, Beijing10/9/12

Tasos Avgoustidis, Sarah Chadburn, Ghazal Geshnizjani, Eimear O’Callaghan, and Ivonne ZavalaPRL 105:081602 (2010) [1003.4395], JCAP 09 (2010) 013 [1005.3220], & JCAP 03 (2011) 004

Outline

Brief overview Gravitational waves and strings String motion with extra dimensions Cusp and Kink events Prospects for detection

Brane Inflation

Brane Anti-Brane

Cosmic Superstring

G~ 10-6 10-12

In brane inflation, we imagine an (anti-)brane moving on the internal extra dimensions and extended along the noncompact directions.

The higher dimensional information is encoded in a scalar describing the brane position: The Inflaton.

Inflation ends when the anti-brane meets the brane and annihilates.

Cosmic Superstrings

Cosmic superstrings form as a remnant of this decay, and can persist as a cosmological network. The network scales with expansion, roughly, the length of a typical loop is proportional to cosmological time, and the density inversely proportional to T3. There are two main inputs to this picture: string motion and intercommutation, both of which are affected by extra dimensions.

String motionMotion of the string is governed by its effective action, obtained by integrating out over the detailed core structure. This gives the Nambu action:

Area of worldsheet

Essentially the same action as string theory. The loops have left moving and right moving waves, which can interact in a curved background.€

S =− μ dτ dσ γ∫

IntercommutationThe other key feature of string motion is that strings tend to intercommute, or break off when they self-intersect:

This is related to a standard “90º scattering” property of solitons.

Gravitational wave observationsStrings are detected by gravitational waves. The string emits characteristic signals from two important events: cusps and kinks.

LIGO f ~ 150 Hz LISA f ~ 4 mHz

Cusps and KinksThe string can have a sharp profile for two reasons:• KINKS: occur when a string self-intersects and cuts off; the loop or string has a kink in it.

• CUSPS: occur when the left and right moving waves constructively interfere to allow the string to (instantaneously) move at the speed of light.

Linearized gravity

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hμν ≈4G

re−iω (t −r )

ω

∑ Tμν (k,ω)

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I±μ = dσ ± lμ +σ ±

˙ ̇ X ±μ

( )∫ exp[− 2πimL lμ X±

μ ] ≈˙ ̇ X ±

μ

| ˙ ̇ X ± |4 / 3

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f

⎛

⎝ ⎜

⎞

⎠ ⎟

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In the far field the metric is well approximated by:

For a cusp or kink, the Fourier integral is dominated when the momentum aligns with the cusp vector. This gives a characteristic high frequency power law tail, giving a distinct signal.

Damour and Vilenkin

Cusp Signal

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h cusp ( f ,θ ) ≈GμL2 / 3

r f1/ 3 H θ m −θ[ ]

The left and right moving modes each contribute to h so when these align, a strong signal is produced – the CUSP

Cusp beams out gravity waves in a tight cone around the cusp vector Opening angle defined by saddle point of I±

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Lf

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⎝ ⎜

⎞

⎠ ⎟

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Damour-Vilenkin

Kink Signal

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h cusp ( f ,θ ) ≈GμL1/ 3

r f2 / 3 H θ m −θ[ ]

The discontinuity dominates the integral for the left mover, but the right mover still has the saddle around its wave vector, signal a combination of these. Kink beams out gravitational waves along the cts wave-vector, localised transverse – a FAN Opening angle defined by saddle point of I+

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Lf

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⎝ ⎜

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⎠ ⎟

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Cosmological SignalThe expanding universe modifies the signal from an individual cusp/kink

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d ˙ N ≈ν (z)

(1+ z)

πθ m2(z)DA (z)2

(1+ z)H(z)dz

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CnL

PTL

≈2C

Pα 2t 4

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f →(1+ z) f

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r →a0r = (1+ z)DA (z)

Then must estimate the total signal from the network

C – number cusps/kinks per loop

DV Ligo result

Cusps (c=1, 0.1)

kinks

Ligo

AdvLigo

Ln

Damour and Vilenkin use analytic approximations to variables, and take a desired event rate of 1 per year. Signal dominated by maximal redshift, use this to obtain amplitude at desired fiducial frequency.

The effect of extra dims?

Extra dimensions reduce the probability of intercommutation, but also the kinematics of strings will be different. The strings can move in the extra dimensions as classical solitons provided their width is sufficiently small.

E.o’C, SC, GG, RG, IZ 1003.4395, 1005.3220 [hep-th]

Motion with extra dims:

With flat extra dimensions, the picture is similar to 4D, with left and right moving waves, however, the string appears to slow in our noncompact dimensions. This has important consequences for cusps and kinks.With warped extra dimensions, things are more complex, the warping induces coupling between left and right movers, and the width of the string relative to the internal dimension size tells us if the string can sample the extra dimensions.

Warped throat Geometry

ds2 = h-1/2 dx2 + h1/2 dy2

FRW cosmology throat

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M p2 >

1

κ102 hdV6 ≈

L4rUV2

′ α 4∫

r0~2/3 ruv

L – curvature radiush0~L4/r0

4

Baumann-McAllister

String Width

This allows an estimate of the string width, w6~w4h-1/2, relative to r0.

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w62

r02 <w4

2M p2 ′ α

L2

⎛

⎝ ⎜

⎞

⎠ ⎟4

r0

rUV

⎛

⎝ ⎜

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⎠ ⎟

2

<10−14 w42M p

2

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1

Gμ~107 −1012

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r0

rUV

~10−3

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′L2 <10−2

Approximate SolutionsExpanding eqns around tip shows that motion is close to 4D, and modelling with an exact throat gives estimates of the slowdown of 4D motion. Most importantly, string motion in internal dimensions persists. Can use simple numerical solutions together with analytic approximations to estimate departure from the 4D motion.

Avgoustidis, Chadburn, RG 1204.0973

Cusp Rounding

The effect of internal velocity is that the wave vectors of 4D left & right movers no longer need be null.This rounds off the cusp, narrowing the beaming cone.

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a′ − b′ = 2Δ <<1

NEAR CUSP EVENT

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θm →θ m − Δ

Measure reductionIn 3d, cusps are generic, but in higher dimensions the Kibble-Turok sphere allows lines not to cross – have to estimate probability of near cusp event.

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C →C(Δ)

From cusp constraint, co-dimensionality of solns with exact cusps is n. Test trajectories give

N≈n

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C(Δ) ≈ nΔn −1

Total Effect:

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d ˙ N NCE

dz≈

2θ mn +2(z)

(n +1)(n + 2)

nL (z)

PTL (z)

πDA (z)2

(1+ z)2 H(z)

General network has range of NCE’s, up to critical where cone closes off. Must integrate cusp event rate over this range:

Can then use interpolating functions, as DV, or calculate DA(z) and H(z) exactly for CDM.

CaveatsAll tests with Nambu strings back up this picture, but strings have finite width. This will give more self-intersection as size of extra dims decreases, and possibly restrict parameter space. Model this empirically by changing measure on solution space to peak around 3d.

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∈ 0,Δ0[ ]

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C(Δ) =nΔn −1

Δ0n

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2θ mn +2

(n +1)(n + 2)→

2θ m (z)n +2

(n +1)(n + 2)

H[Δ0 −θ m ]

Δ0n + θ m

2(z) −2nΔ0θ m (z)

(n +1)+

nΔ02

(n + 2)

⎛

⎝ ⎜

⎞

⎠ ⎟H[θ m − Δ0]

On integration gives new rate dependence on θ

With Measure

= 10-4, 10-3, 10-2,0.1

n = 1, n = 3, n = 6

FIX

FIX n = 3

Kink waveformFor the kink, the right mover has a saddle as with the cusp, but the discontinuity contributes from its end points:

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I−∝v μ

f

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v μ =n1

μ

ˆ k ⋅ n1

−n2

μ

ˆ k ⋅ n2

Kibble-Turok sphere

fan

Kink amplitude for range of discontinuity angles

E.o’C & RG 1010.3942 [hep-th]

Kinks:

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d ˙ N Kdz

≈ KnL (z)θ m

2(z)

PTL (z)

πDA (z)2

(1+ z)2 H(z)

For the kink, the only factor from extra dimensions is the thinning of the fan to θwhich must be integrated over:

NB: This is independent of the number of extra dimensions

Plots show amplitudes are suppressed from DV result, but not more than cusp with n=3

If kinks proliferate by more than 6 orders of magnitude, they will be easily detectable.

Rate Plot

Perhaps more useful is an event rate plot at h ~ 10-21

Summary Kinematics of extra dimensions can have a strong effect on the gravitational wave signal Cusp signal is particularly sensitive to number of extra dimensions Warped extra dimensions and cosmological expansion seem not to change the qualitative result Lower frequency bands have less signal differentiation Kinks offer better bet for detection Differentiation between cusp/kink signals indicative of number of (eff) extra dimensions.