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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:
What are the current bounds, and prospects for improvement?
To what extent can we distinguish different kinds of cosmicstring?
ObservationsMicroscopicmodels
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The cosmic string inverse problem:The cosmic string inverse problem:
What are the current bounds, and prospects for improvement?
To what extent can we distinguish different kinds of cosmicstring?
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.