Kleidis KostasKleidis Kostas

GRAVITATIONAL WAVESGRAVITATIONAL WAVESVSVS

COSMIC STRINGSCOSMIC STRINGS

kleidis@astro.auth.grkleidis@astro.auth.gr

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Cosmological Gravitational Waves (CGWs)

They represent small – scale perturbations to the Universe metric tensor.

Gravitational radiation may travel virtually unaffected since emission!

The metric perturbations penetrate through the e/m surface of last scattering, carrying a picture of the Universe as it was at very early stages.

This is due to their weak interaction with ordinary matter.

Which makes them difficult to observe.

Pioneering experiments have been proposed: LIGO (Advanced)LISAASTRODBBODECIGO

They could detect a stochastic background of CGWs, generated at the very early stages of the Universe!

(Relic Gravitational Waves – RGWs)

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Characteristic Spectra

In order to detect them we need to specify what we expect to see!

A stochastic GW background of cosmological origin is:

Isotropic: Same profile in all spatial directions.

Stationary: Remains unaltered during the time – scale of observation.

Gaussian: A multi–variable normal distribution.

All the information is included in its frequency spectrum!

The logarithmic spectrum:

εgw : the (averaged) energy – density k : the wave-number

The power spectrum:

2 21( )

ln 2gwd

k P kd k G

22 3( ) ( , )P k k k t

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Equations of Propagation

To evaluate these spectra we need to determine the time–dependent amplitude α(k, t)!

In other words…

To solve the equations of propagation in curved space time!

The far–field propagation of a weak gravitational wave in a curved and non–vacuum space-time is governed by the differential equations:

under the gauge condition:

Rαμνβ : is the Riemann tensor with respect to the background metric gαβ .

h : is the trace of hαβ and the semicolon denotes covariant derivative.

;α; 2 0h R h

;

1( ) 0

2h g h

5

RGWs in a Model Universe

In a spatially flat FRW background the cosmological-wave perturbations are defined by:

: the conformal time

The scale factor R(τ) is a solution to the Friedmann equation

With matter–content in the form of a perfect fluid: Tμν = diag (ρ, -p, -p, -p)

Obeying:

The conservation law

The equation of state

2 2 2( ) i jij ijds R d h dx dx

/ ( )dt R t

2

2

8

3

R΄ G

R

; 0T

3 ( ) 0R

pR

13

mp

6

Determining the Matter–Content

Which applies to :

1. Quantum vacuum m = 02. A network of domain walls m = 13. A gas of cosmic strings m = 24. Dust m = 35. Radiation m = 46. Zel’ dovich ultra – stiff matter m = 6

The conservation law implies:

For an one–component fluid:

Mm : a typical mass–density of the m-th component.

For a multi–component fluid:

Provided that:

The various components do not interact!

In this case the conservation law holds for every matter–component separately!

mm

M

R

mm

m

M

R

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RGWs in a Friedmann Model with k = 0

The equation of propagation reduces to:

We decompose it, by representing the metric perturbations as a linear super–position of plane-wave modes:

εik : the polarization tensor (solely a function of kj )

h(k,τ) : the time–dependent part of the modes

In this model, the temporal evolution of a RGW is governed by:

It can be treated as the Schrödinger equation in the presence of the effective potential:

,2 0΄΄ ΄ lmik ik ik lm

R΄h h h

R

( , ): the amplitude

( , )( , )

( )

jjik xj

ik ik

a k

h kh x e

R

2 0k k

R΄΄h ΄΄ k h

R

΄΄

eff

RV

R

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What Kind of Process could Produce RGWs?

Phenomena related to phase transitions Cosmic-string oscillations Low energy–density bubble collisions

Amplification of quantum fluctuations during inflation!

Some of them escape from the visible Universe once their physical wavelength [λ ~ R(τ)] becomes larger than the constant inflationary horizon

Outside the horizon they freeze out!

(Abbott & Wise 1984)

During the subsequent radiation epoch they may re–enter inside the Hubble sphere, since:

1H dSl H

2

23

16 1dsk

plm k

2ph Hl

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RGWs in the Radiation Era

During this epoch the general solution is

For kτ << 1: Outside the horizon:

α(k,τ) ~ αkThey remain frozen!

For kτ >> 1: Inside the horizon:

They oscillate with decreasing amplitude.

They produce a part of the spectrum we observe today.

The k – dependence of their amplitude implies a scale – invariant power spectrum:

sin( , ) k

ka k t a

k

2

2 16( ) ds

pl

HP k

m

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Constancy of the Effective Potential

The choice: M: a non–negative constant of units gr/cm3

reduces the equation which governs the temporal evolution of a RGW to:

Now, the condition:

discriminates the metric perturbations into:

Oscillating modes : k > kc

Non–oscillating modes : k < kc

8

3

΄΄R GM

R

2 80

3΄΄k k

Gh k M h

8

3c

Gk k M

2 2ci k k

kh e

2 2ck k

kh e

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Cosmological Models of Constant Effective Potential

Combine the constant Veff with the Friedmann equation!

Then :

Which admits the solution:

We distinguish cases with respect to M:

For M = 0 ~> A radiation–dominated Universe!

kc = 0 ~> Every mode may oscillate!

For M ≠ 0 ~> A two–component fluid consisting of: Radiation (R-4) and Cosmic strings (R-2)

kc ≠ 0 ~>

RGW modes of k< kc do not oscillate!

Important notice: Veff = constant only when Μ = Μ2: the mass-density attributed to the

gas of linear defects!

34 2

R΄ R΄΄ M

R R

4 2( )

C M

R R

44

( )M

R

4 24 2

( )M M

R R

2 2 0΄΄k c kh k k h

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A Tale of Cosmic Strings

Linear topological defects formed at a symmetry breaking phase transition during the (early) radiation epoch (τcr).

Initially, their motion is heavily damped due to string – particle scattering.

At this friction becomes subdominant to expansion damping.

For τ ≥ τ* the strings are essentially independent of anything else in the Universe!

A two – component fluid (the constituents of which do not interact) consisting of:

Radiation (dominant)

Cosmic strings (sub-dominant)

Radiation – plus – strings (RPS) stage!

*

1cr

G

2

8( ) sinh

3

GR M

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The Scaling of a Cosmic–String Network

When long cosmic strings intersect:

They are being “chopped off”

And / or “inter-commute”

Intersection of a string to itself leads to the production of small loops!

They oscillate and radiate gravitationally!

The network is led to a self–similar configuration:The number of long cosmic strings in a horizon volume remains fixed!

Numerical results: Albrecht & Turok 1989Allen & Shellerd 1990Bennett & Bouchet 1990

revealed the so – called “scaling” of a cosmic–string network!

Eventually:

and the Universe (re) enters in the (late) radiation era!

4

1str R

14

No matter how small a radiation–plus–strings stage may be,

it would have a measurable effect on the RGWs spectrum!

*20 50 sct t

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Evolution of the Non-oscillatory Modes Through a RPS – Stage

The condition of fitting inside the horizon (λph ≤ lH) during the RPS – stage reads:

RGW modes of k < kc remain outside the horizon during the whole RPS - stage!

Although they are outside the horizon they do not freeze out!

The RGW spectrum is no longer scale-invariant!

In terms of

Therefore, at present–time frequencies

2

8coth

3c

k GM

k

0 1c c

k fx

k f

22 1 1

223 2

c

c

x k

rps radgw gw

k

e

e

2/31/ 21 1

22pr pr cr rec

crec pr cr

t tf f

t t t

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The relic (inflationary) GW spectrum is distorted, departing from scale-invariance!

For GUT- scale strings: tcr ~ 10-31 sec

fcpr ~ 105 Hz

Too large!

For electroweak - scale strings: tcr ~ 10-13 sec

fcpr ~ 1,5 x 10-4 Hz

• close to the band–width of LISA

• within the band–width of ASTROD

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Evolution of the Oscillating Modes Through a RPS–Stage

For

The temporal evolution of the metric perturbations is given by:

The gravitational–perturbation field is in an adiabatic–vacuum state:

A conformally invariant field propagating in a conformally flat space-time!

For

The discontinuous change of R΄΄ at τ*

reflects

a discontinuous change in the scalar curvature

Graviton Production!

* 0R΄΄

R

2 20΄΄ in ik

k k kh k h h ek

*

8 0

3sc

R΄΄ GM

R

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The general solution:

Is a linear combination of positive and negative frequency modes.

: measures the departure from the pure–radiation case.

For

Once again, the temporal evolution of a RGW is governed by (adiabatic vacuum)

However, due to the discontinuity of the scalar curvature at τsc the general solution:

(αk and βk : Bogolubov coefficients)

Wronskian condition:

As long as βk ≠ 0 the final state (out–vacuum) differs from the initial state (in–vacuum).

Gravitons are present!

1 2

2 i ki kkh c e c e

2

0 1 1ck

k

0sc

R΄΄

R

2 0k kh k h

2 ikout ikk k kh a e e

k

2 2

1k ka

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The expectation value of the number operator is:

Δτ = 0: Nk = 0 No gravitons are produced!

k = kc : Well – defined and finite!

The distribution of the gravitational energy-density:

is the physical frequency

In terms of the logarithmic spectrum reads:

where, is the energy – density of the gravitons created in the

state denoted by kc!

22 2

42

2 2 2

sin

4

cc

k

c

c k kkN

k k k

21

4ck cN k c

22

2 3( ) 2

2g kd d dc

( )

ck

R

1c

kx

k

22

2

2

sin 1( ) 4 ( )

ln 1

cgc

c

k c xdk k x

d k k c x

3

1( ) 2

c ph

ph

c k cc

N ck

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Consequences

We observe that:

At high frequencies – f ≥ fcpr :

The RGW spectrum after being “filtered” by a cosmic-string network has a periodic profile!

The period depends solely on the duration of the RPS–stage!

What’s most important:

The relic GW signal is amplified by (almost) two orders of magnitude!

This is compatible to the absence of higher-harmonics on an infinite (long) string.

It reflects a resonant interaction between string–constituents and RGWs.

To the best of our knowledge, this is the first time that an amplification mechanism of a stochastic GW signal is proposed and discussed!