Joe Polchinski- The Cosmic String Inverse Problem

8/3/2019 Joe Polchinski- The Cosmic String Inverse Problem

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The Cosmic String Inverse ProblemThe Cosmic String Inverse Problem

JP & Jorge Rocha, hep-ph/0606205JP & Jorge Rocha, gr-qc/0702055Florian Dubath & Jorge Rocha, gr-qc/0703109JP, arXiv:0707.0888

Florian Dubath, JP & Jorge Rocha, work in progress

Joe PolchinskiJoe PolchinskiKITP, UCSBKITP, UCSB

Berkeley Center for Theoretical PhysicsBerkeley Center for Theoretical PhysicsOpening Symposium, 10/20/07Opening Symposium, 10/20/07


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There are many potential cosmic strings fromThere are many potential cosmic strings fromstring compactifications:string compactifications:

The fundamental string themselves D-strings Higher-dimensional D-branes, with all but one

direction wrapped. Solitonic strings and branes in ten dimensions Solitons involving compactification moduli Magnetic flux tubes (classical solitons) in the effective

4-d theory: the classic cosmic strings.

Electric flux tubes in the 4-d theory. A network of any of these might form in an appropriate phasetransition in the early universe, and then expand with the universe.


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What are the current bounds, and prospects for improvement?

To what extent can we distinguish different kinds of cosmicstring?


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The cosmic string inverse problem:The cosmic string inverse problem:



ObservationsMicroscopicmodels


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The cosmic string inverse problem:The cosmic string inverse problem:



ObservationsMicroscopicmodels

There is an intermediate step:

Macroscopicparameters


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Macroscopic parameters:Macroscopic parameters:

Tension Q Reconnection probability P :

P P

Light degrees of freedom: just the oscillations in3+1, or additional bosonic or fermionic modes? Long-range interactions: gravitational only, or

axionic or gauge as well? One kind of string, or many? Multistring junctions?

F

D F+D


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Vanilla Cosmic Strings:Vanilla Cosmic Strings:

P = 1 No extra light degrees of freedom

No long-range interactions besides gravity

One kind of string

No junctions

Even for these, there are major uncertainties.


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A simulation of vanillastrings (radiation era,box size ~ .5 t ).

Simple argumentssuggest that t ~ Hubblelength ~ Horizon length isthe only relevant scale.

If so, simulations( A lbrecht & Turok, Bennett & Bouchet, A llen & Shellard, ~ 1989 )

would have readily given a quantitative understanding.However, one sees kinks and loops on shorter scales(BB). Limitations: U V cutoff, expansion time. A nalyticapproachs limited by nonlinearities, fairly crude.


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Estimates of the sizes at whichloops are produced range over more than fifty orders of magnitude, in a completely well-posed, classical problem.

Since the problem is a large ratio of scales, shouldntsome approach like the RG work?

Not exactly like the RG: since the comoving scaleincreases more slowly than t , structure flows from longdistance to short.However, with the aid of recent simulations, we haveperhaps understood what the relevant scales are, andwhy.


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Outline:Outline:

Review of network evolution*

Signatures of vanilla strings* A model of short distance structure

The current picture

*Good references:V ilenkin & Shellard, Cosm ic String s and Other T opolo gica l Defect s ;Hindmarsh & Kibble, hep-ph/9411342.


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Processes:

I. Review of Vanilla Network EvolutionI. Review of Vanilla Network Evolution

1. Formation of initial network in a phase transition.2. Stretching of the network by expansion of the

universe.3. Long string intercommutation.4. Long string smoothing by gravitational radiation.5. Loop formation by long string self-

intercommutation.6. Loop decay by gravitational radiation.


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String solitons exist whenever a U (1) is broken, and they

are actually produced whenever aU (1) becomes broken duringthe evolution of the universe (Kibble):

Phase is uncorrelated over distances >horizon. O (50% ) of string is in infinite

random walks.

1. Network creation1. Network creation

(Dual story for other strings).


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1.5 Stability1.5 Stability

We must assume that the strings are essentiallystable against breakage and axion domain wallconfinement (this is model dependent).


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2. Expansion2. Expansion

FRW metric:

String EOM:

L/R form: define unit vectors:

Then:

gauge

(Comoving expansion above horizon scale, oscillationand redshifting below horizon scale.


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3. Long string intercommutation3. Long string intercommutation

Produces L- and R-movingkinks. Expansion of theuniverse straightens theseslowly, but more enter thehorizon (BB).


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4. Long string gravitational radiation4. Long string gravitational radiation

This smooths the long strings at distances less thansome scale l G.

Simple estimate gives l G = +G Qt , with + b

Subtle suppression when L- and R-moving wave-lengths are very different,

so in fact l G = + G Q G , with Gto be explainedlater (Siemens & Olum; & V ilenkin; JP & Rocha).


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5. Loop formation by long string self 5. Loop formation by long string self- -intersectionintersection


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Review:

1. Formation of initial network in a phase transition.2. Stretching of the network by expansion of the

universe.3. Long string intercommutation .

4. Long string smoothing by gravitational radiation .5. Loop formation by long string self-

intercommutation .6. Loop decay by gravitational radiation .

(Simulations replace grav. rad. with a rule thatremoves loops after a while)


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Scaling hypothesis:Scaling hypothesis:

All s tati s tica l pr op ertie s of the netw ork are c ons tant when viewed on s ca l e t (K ibb l e ).

If only expansion were operating, the long stringseparation would grow as a (t ). With scaling, it

grows more rapidly, as t , so the various processesmust eliminate string at the maximum rate allowedby causality.

Simulations, models, indicate that the scaling

solution is an attractor under broad conditions (m & r)(more string p more intercommutions p more kinksp more loops p less string). Washes out initialconditions.


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Estimates of loop formation sizeEstimates of loop formation size

0. 1 t : original expectation, and some recent work(Vanchurin, Olum & V ilenkin)

t : other recent work (Martins & Shellard, Ringeval,Sakellariadou & Bouchet)

+G Qt : still scales, but dependent on gravitationalwave smoothing (Bennett & Bouchet)

+ G Q G t : corrected gravitational wavesmoothing (Siemens, Olum & V ilenkin; JP & Rocha)

Xstring : the string thickness - a fixed scale, not w t (V incent, Hindmarsh & Sakellariadou)


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Vanilla strings have only gravitational long-rangeinteractions, so we look for gravitational signatures:

II. Gravitational SignaturesII. Gravitational Signatures

1. Dark matter.2. Effect on CMB and galaxy formation.

3. Lensing.4. Gravitational wave emission.

Key parameter: G Q This is the typical gravitationalperturbation produced by string. In brane inflationmodels,

10 12 < G Q< 1 0 6

Normalized by HT T . (Jones, Stoica, Tye)


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1. Dark Matter 1. Dark Matter

Scaling implies that the density of string is a constanttimes Qt 2. This is the same time-dependence as thedominant (matter or radiation) energy density, sothese are proportional, with a factor of G Q

Simulations:

Vstring Vmatter = 7 0 G Q Vstring Vradiation = 4 00 G Q

Too small to be the dark matter.


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Bound from power spectrum: G Q< x(Pogosian, Wasserman & Wyman; Seljak & Slosar).

Bound from nongaussianity:G Q< x (Jeong &Smoot) .

Bound from Doppler distortion of black body:G Q< x

(Jeong & Smoot) .

Improved future bounds from polarization, non-gaussianity.


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3. Lensing:3. Lensing:

A string with tension G Q x has a deficitangle 1 arc-sec; lens angle depends on geometryand velocity, can be a bit larger or smaller. Only a

tiny fraction of the sky is lensed, so one needs al arge survey - radio might reach Mack, Wesley& King)

Network question: on the relevant scales is thestring straight (simple double images, and objectsin a line) or highly kinked (complex multipleimages, objects not aligned)?

(By long strings or loops)


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There is interesting radiation both from the lowharmonics of the loop and the high harmonics ; willconsider the low harmonics first.

4. Gravitational radiation4. Gravitational radiation

Primarily from loops (they have higher frequencies).

period = l 2 R = 2 n l

Most of the energy goes into the low harmonics


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Long strings

Follow the energy:

Loops

Stochastic gravitational radiation

density known from simulations

rate set by scaling

red-shift like matter

decay at t d t f = l +G Q

If l t f

| E " + G Qthen energy density is enhanced by(E +G Q)1/2 during the radiation era (relevant to LIGO,LIS A , and to pulsars down to G Q~ ).


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Pulsar bounds: the observed regularity of pulsar signalslimits the extent to which the spacetime through which

they pass can be fluctuating. Significant recentimprovement (PPT A , Jenet, et al. ):

< x

Energy balance gives:

Bound:K= initial boost of loop,

~1 for large loop.

(Will improve substantially.)

E.g. E = 0. 1, K= 1 gives G Q x but muchsmaller E gives a weak bound.

*

*factor of due to vacuum energy, = 16 in G Q


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High harmonics: kinks give [ spectrum, cusps give[

Cusp (Turok) : in conformal gauge, x (u,v ) = a (u ) + b (v),with a and b unit vectors.

a

b

When these intersect

the string has a cusp

QuickTime and a A nimation decompressor

are needed to see this picture.


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Initial calculations (Damour & V ilenkin) suggested thatthese might be visible at LIGO I, and likely at A dvanced LIGO. More careful analysis (Siemens,Creighton, Maor, Majumder, Cannon, Read) suggests thatwe may have to wait until LIS A . Large E helps, butprobably not enough.


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III. A Model of ShortIII. A Model of ShortDistance StructureDistance Structure

Strategy: consider the evolution of a small (right- or left-moving) segment on a long string.

1. Small scale structure on short strings1. Small scale structure on short strings


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Evolution of a short segment, length l .Possible effects:

1. Evolution via Nambu-FRW equation

2. Long-string intercommutation

3. Incorporation in a larger loop

4. Emission of a loop of size l or smaller?

5. Gravitational radiation.


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Evolution of a short segment,length l . Possible effects:

1. Evolution via Nambu-FRW equation2. Long-string intercommutation

very small probability, w l

3. Incorporation in a larger loopcontrolled by longer-scale configuration, willnot change mean ensemble at length l*

4. Emission of a loop of size l or smaller ignore? not self-consistent, but againcontrolled by longer-scale physics

5. Gravitational radiationignore until we get to small scales


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Nambu-FRW equations

Separate segment into mean and (small) fluctuation:

where

Then

just precession over Hubble times, encounter many opposite-movingsegments, so average.

P+ P = 2Dv 2 w+, w a 2Dv 2


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Compare with simulations (Martins & Shellard) :

Random walk at longdistance. Discrepancyat short distance - butexpansion factor is only3.

radiation era matter era


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Compare with simulations (Martins & Shellard) :

Random walk at long

distance. Discrepancyat short distance - butexpansion factor is only3.

radiation era

matter era

Hindmarsh:

radiation era


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Lensing: fractal dimension is O ([l t ]2G).

Gives 1% difference between images.


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Loops form whenever string self-intersects. Thisoccurs when ( x = x = 0 on some segment, i.e.

2. Loop formation2. Loop formation

Rate per unit u, v, l :

Components of L+ L are of order l , l , l 1+2 G

Columns of J are of order l G l G l 2G

Rate ~ l 3+2G


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Rate of loop emission ~ l 3+2G

Rate of s tring emission ~ l 2+2 G

Rate per world-sheet area = d l l 2+2 G this diverges atthe lower end for G , even though the string isbecoming smoother there.

Total string conservation saturates at l ~ 0. 1t , butrapid loop formation occurs internally to the loops -this suggests a complicated fragmentation process.


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p +

p

Resolving the divergence: separate the motion into along-distance `classical piece plus short-distancefluctating piece:

Loops form near the cusps of the long-distance piece. All s ize s f o r m at the s a me ti me . Get loop productionfunction l 2+2 G, but with cutoff at gravitational radiationscale, and reduced normalization.


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Recent simulations (Vanchurin, Olum, V ilenkin) use volume-expansion trick to reach larger expansion factors. Result:

Tw o peak s , one near the horizon and one near the U V cutoff. V OV interpret the latter as a transient, but this isthe one we found.

radiation era

What about the large loops?


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Scorecard on loop formation sizeScorecard on loop formation size

0. 1 t : original expectation, and some recent work(Vanchurin, Olum & V ilenkin)

t : other recent work (Martins & Shellard, Ringeval,Sakellariadou & Bouchet)

+G Qt : still scales, but dependent on gravitationalwave smoothing (Bennett & Bouchet)

+ G Q G t : corrected gravitational wavesmoothing (Siemens, Olum & V ilenkin; JP & Rocha)

Xstring : the string thickness - a fixed scale, not w t (V incent, Hindmarsh & Sakellariadou)

80-90%

10-20%


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TwoTwo--peak distributionpeak distribution - - effect on bounds:effect on bounds:

Largeloops:

Smallloops:

Low harmonics High harmonics

Current (pulsar): 2 x 10 -7

PPT A : 10 -9 A dvanced LIGO: 10 -10

SK A , LIS A : 10 -11

A dvanced LIGO: ??LIS A : 10 -13

A dvanced LIGO: ??LIS A : 10 -10


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The inverse problem:The inverse problem:

Observation of low harmonics of large loopsprobably allows measurement of G Qonly (throughabsolute normalization) - if the netw ork s areunder s t oo d perfect l y .

Slightly less vanilla strings: P 1: Normalization w P # P # P # : degenerate with G Qfor lowharmonics (in pulsar range, slope of spectrum mayhave independent dependence on Q).

Observation of high harmonics gives severalindependent measurements: measure G Q, P , lookfor less vanilla strings.


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ConclusionsConclusions Long-standing problem perhaps nearing solution

Observations will probe most or all of braneinflation range

If so, there is prospect to distinguish differentstring models, maybe not until LIS A .

Precise understanding of string networks willrequire a careful meshing of analytic and numericalmethods.

This has been a lot of fun.

Joe Polchinski- The Cosmic String Inverse Problem

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