Xavier Siemens et al- Gravitational wave bursts from cosmic (super)strings: Quantitative analysis and constraints

8/3/2019 Xavier Siemens et al- Gravitational wave bursts from cosmic (super)strings: Quantitative analysis and constraints

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Gravitational wave bursts from cosmic (super)strings: Quantitative analysis andconstraints

Xavier Siemens1, Jolien Creighton1, Irit Maor2, Saikat Ray Majumder1, Kipp Cannon1, Jocelyn Read1

1Center for Gravitation and Cosmology, Department of Physics,University of Wisconsin Milwaukee, P.O. Box 413, Wisconsin, 53201, USA

2 CERCA, Department of Physics, Case Western Reserve University,

10900 Euclid Ave, Cleveland, OH 44106-7079 USA(Dated: January 24, 2006)

We discuss data analysis techniques that can be used in the search for gravitational wave burstsfrom cosmic strings. When data from multiple interferometers is available, we describe consistencychecks that can be used to greatly reduce the false alarm rates. We construct an expression for therate of bursts for arbitrary cosmic string loop distributions, and apply it to simple known solutions.The cosmology is solved exactly, including the effects of a late-time acceleration. We find burst ratesthat are substantially lower than previous estimates and explain the disagreement. Initial LIGOis unlikely to detect field theoretic cosmic strings with the usual loop sizes, though it may detectcosmic superstrings as well as cosmic strings and superstrings with non-standard loop sizes (whichmay be more realistic). In the absence of a detection, we show how to set upper limits based on theloudest event. Using Initial LIGO sensitivity curves, we show that these upper limits may resultin interesting constraints on the parameter space of theories that lead to the production of cosmicstrings.

PACS numbers: 11.27.+d,98.80.Cq, 11.25. -w

I. INTRODUCTION

Cosmic strings can form during phase transitions inthe early universe [1]. Although cosmic microwave back-ground data has ruled out cosmic strings as the primarysource of density perturbations, they are still potentialcandidates for the generation of a host of interesting as-trophysical phenomena including gravitational waves, ul-tra high energy cosmic rays, and gamma ray bursts. Fora review see [2].

Following formation, a string network in an expand-ing universe evolves toward an attractor solution calledthe scaling regime (see [2] and references therein). Inthis regime, the statistical properties of the system, suchas the correlation lengths of long strings, scale with thecosmic time, and the energy density of the string net-work becomes a small constant fraction of the radiationor matter density.

The attractor solution arises due to reconnections,which for field theoretic strings essentially always oc-cur when two string segments meet. Reconnections leadto the production of cosmic string loops, which in turnshrink and decay by radiating gravitationally. This takesenergy out of the string network, converting it to grav-itational waves. If the density of strings in the networkbecomes large, then strings will meet more often, therebyproducing extra loops. The loops then decay gravitation-ally, removing the surplus energy from the network. If,on the other hand, the density of strings becomes toolow, strings will not meet often enough to produce loops,and their density will start to grow. Thus, the networkis driven toward a stable equilibrium.

Recently, it was realised that cosmic strings could be

produced in string-theory inspired inflation scenarios [36]. These strings have been dubbed cosmic superstrings.The main differences between field theoretic strings andsuperstrings are that the latter reconnect when they meetwith probabilitiesp that can be less than 1, and that morethan one kind of string can form. The values suggestedfor p are in the range 103 1. The suppressed recon-nection probabilities arise from two separate effects. Thefirst is that fundamental strings interact probabilistically,and the second that strings moving in more than 3 di-

mensions may more readily avoid reconnections: Eventhough two strings appear to meet in 3 dimensions, theymiss each other in the extra dimensions [5].

The effect of a small reconnection probability is to in-crease the time it takes the network to reach equilibrium,and to increase the density of strings at equilibrium.The amount by which the density is increased is the sub-

ject of debate, with predictions ranging from p2 [6],to p0.6 [7]. In this work we will assume the densityscales like p1, and that only one kind of cosmicsuperstring is present [8].

Cosmic strings and superstrings can produce powerfulbursts of gravitational waves. These bursts may be de-

tectable by first generation interferometric earth-basedgravitational wave detectors such as LIGO and VIRGO[810]. Thus, the exciting possibility arises that a certainclass of string theories may have observable consequencesin the near future [11].

The bursts we are most likely to be able to detectare produced at cosmic string cusps. Cusps are regionsof string that acquire phenomenal Lorentz boosts, withgamma factors 1. The combination of the largemass per unit length of cosmic strings and the Lorentz


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boost results in a powerful burst of gravitational waves.The formation of cusps on cosmic string loops and longstrings is generic, and their gravitational waveforms aresimple and robust [12].

The three LIGO interferometers (IFOs) are operatingat design sensitivity. The VIRGO IFO is in the finalstages of commissioning. Thus, there arises a need for aquantitative and observationally driven approach to the

problem of bursts from cosmic strings. This paper at-tempts to provide such a framework.

We consider the data analysis infrastructure that canbe used in the search for gravitational wave bursts fromcosmic string cusps, as well as cosmological burst ratepredictions .

The optimal method to use in the search for signals ofknown form is match-filtering. In Sec. II we discuss thestatistical properties of the match-filter output, templatebank construction, an efficient algorithm to compute thematch-filter, and, when data from multiple interferome-ters is available, consistency checks that can be used togreatly reduce the false alarm rate. In Sec. III we discuss

the application of the loudest event method to set upperlimits on the rate, the related issue of sensitivity, andbackground estimation. In Sec. IV we derive an expres-sion for the rate of bursts for arbitrary cosmic string loopdistributions, and apply it to the distribution proposedin Refs. [810]. In Sec. V we show how to computethe relevant cosmological functions necessary to evaluatethe rate, including the effects of a late-time acceleration.In Sec. VI we compare our estimates for the rate withthose obtained previously (in [810]). We find substan-tially lower event rates, and explain the disagreement.We also compute the rate for simple loop distributions,using the results of Sec. IV and show how the detectabil-ity of bursts is sensitive to the loop distribution. In Sec.

VII we discuss how, in the absence of a detection, it ispossible to use the upper limits on the rate discussed inSec. III to place interesting constraints on the parameterspace of theories that lead to the production of cosmicstrings. We show an example of these constraints. Wesummarise our results and conclude in Sec. VIII.

II. DATA ANALYSIS

A. The signal

In the frequency domain, the waveforms for bursts ofgravitational radiation from cosmic string cusps are givenby [9],

h(f) = A|f|4/3(fh f) (f fl). (1)

The low frequency cutoff of the gravitational wave signal,fl, is determined by the size of the feature that producesthe cusp. Typically this scale is cosmological. In prac-tice, however, the low frequency cutoff of detectable ra-diation is determined by the low frequency behaviour of

the instrument (for the LIGO instruments, for instance,by the seismic wall). The high frequency cutoff, on theother hand, depends on the angle between the line ofsight and the direction of the cusp. It is given by,

fh 23L

, (2)

where L is the size of the feature that produces the cusp,

and in principle can be arbitrarily large. The amplitudeof the cusp A is,

A GL2/3

r, (3)

where G is Newtons constant, is the mass per unitlength of strings, and r is the distance between the cuspand the point of observation. We have taken the speedof light c = 1.

The time domain waveform can be computed by takingthe inverse Fourier transform of Eq. (1). The integral hasa solution in terms of incomplete Gamma functions. Fora cusp at t = 0 it can be written as

h(t) = 2A|t|1/3 { i(1/3, 2iflt) + c.c. (i(1/3, 2ifht) + c.c.)}. (4)

The duration of the burst is on the order of the inverseof the the low frequency cutoff fl, and the spike at t = 0is rounded on a timescale 1/fh.

Eq. (4) is useful to make a rough estimate the am-plitude of changes in the strain h(t) that a cusp wouldproduce. If we consider a time near t = 0, we can expandthe incomplete Gamma functions according to [13],

(, x) = () +

n=0

(

1)nx+n

n!( + n) . (5)

Keeping the first term of the sum we see that the ampli-tude in strain of the cusp is,

hcusp = 6A(f1/3l f1/3h ). (6)

B. Definitions and conventions

The optimal method to use in the search for signals ofknown form is match-filtering. For our templates we take

t(f) = f4/3

(fh f) (f fl), (7)so that a signal would have h(f) = A t(f). We define theusual inner product [18],

(x|y) 4

0

dfx(f)y(f)

Sh(f), (8)

where Sh(f) is the single-sided spectral density definedby n(f)n(f) = 12(f f)Sh(f), where n(f) is theFourier transform of the detector noise.


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The templates of Eq.( 7) can be normalised by theinner product of a template with itself,

2 = (t|t) (9)so that,

t = t/, (t|t) = 1. (10)

If the output of the instrument is s(t), the signal tonoise ratio (SNR) is defined as,

(s|t). (11)In general, the instrument output is a burst h(t) plussome noise n(t), s(t) = h(t) + n(t). When the signal isabsent, h(t) = 0, the SNR is Gaussian distributed withzero mean and unit variance,

= (n|t) = 0, 2 = (n|t)2 = 1. (12)When a signal is present, the average SNR is

= (h|t) + (n|t) = (At|t) = A, (13)and the fluctuations have variance

2 2 = (h + n|t)2 A22 = 1. (14)Thus, when a signal of amplitude A is present, the sig-nal to noise ratio measured, , is a Gaussian randomvariable, with mean A and unit variance,

= A 1 (15)

The amplitude that we assign the event A depends onthe template normalisation ,

A =

= A 1

. (16)

This means that in the presence of Gaussian noise therelative difference between measured and actual signalamplitudes will be proportional to the inverse of the SNR,

A

A= 1 . (17)

If an SNR threshold th is chosen for the search, on av-erage only events with amplitude

Ath

th

, (18)

will be detected.

C. Template bank

Cosmic string cusp waveforms have two free parame-ters, the amplitude and the high frequency cutoff. Sincethe amplitude is an overall scale, the template need notmatch the signal amplitude. However, the high frequency

cutoff does affect the signal morphology, so a bank oftemplates is needed to match possible signals.

The template bank, {ti}, where i = 1, 2..,N, is one-dimensional, and fully specified by a collection of high fre-quency cutoffs {fi}. The number of templates N shouldbe large enough to cover the parameter space finely. Wechoose the ordering of the templates so that fi > fi+1.

Although in principle the high frequency cutoff can

be infinite, in practice the largest distinguishable highfrequency cutoff is the Nyquist frequency of the time-series containing the data, fNyq = 1/(2t), where t isthe sampling time of that time-series. The normalisationfactor for the first template is given by

21 = (t1|t1) = 4fNyqfl

dff8/3

Sh(f). (19)

This template is the one with the largest , and thus thelargest possible SNR at fixed amplitude.

To determine the high frequency cutoffs of the remain-ing templates, we proceed as follows. The fitting factorbetween two adjacent templates ti and ti+1, with highfrequency cutoffs fi and fi+1, is defined as [19]

F =(ti|ti+1)

(ti|ti)(ti+1|ti+1)= 1 , (20)

where is the maximum mismatch. The maximum mis-match is twice the maximum fractional SNR loss due tomismatch between a template in the bank and a signal,and should be small. Since ti+1 has a lower frequencycutoff than ti,

(ti|ti+1) = (ti+1|ti+1). (21)Thus, the maximum mismatch is,

= 1

(ti+1|ti+1)(ti|ti) = 1

i+1i

, (22)

and once fixed, determines the high frequency cutofffi+1,given fi. Thus the template bank can be constructediteratively.

D. Single interferometer data analysis

A convenient way to apply the match filter is to use theinverse spectrum truncation procedure [20]. The usualimplementation of the procedure involves the creation ofan FIR (Finite Impulse Response) digital filter for thesingle-sided power spectral density, Sh(f), of some seg-ment of data. This filter, along with the template canthen be efficiently applied to the data via FFT (FastFourier Transform) convolution. Here we describe a vari-ation of this method in which the template as well asSh(f) are incorporated into the digital filter.

Digital filters may be constructed for each template jin the bank as follows. First we define a quantity which


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corresponds to the square root of our normalised tem-plate divided by the single-sided power spectral density,

Tj,1/2(f) =

tj(f)

Sh(f). (23)

This quantity is then inverse Fourier transformed to cre-ate,

Tj,1/2(t) =

df e2iftTj,1/2(f). (24)

At this point the duration of the filter is identical tothe length of data used to estimate the power spectrumSh(f). We then truncate the filter, setting to zero val-ues of the filter sufficiently far from the peak. This isdone to minimise the amount of data corrupted by filterinitialisation. After truncation, however, the filter stillneeds to be sufficiently long to adequately suppress linenoise in the data (such as power line harmonics). Thetruncated filter is then forward Fourier transformed, and

squared, creating the frequency domain FIR filter Tj(f).The SNR is the time-series given by,

j(t) = 4

0

df s(f)Tj(f)e2ift. (25)

where s(f) is a Fourier transform of the data.To generate a discrete set of events or triggers we can

take the SNR time series for each template j(t) andsearch for clusters of values above some threshold th.We identify these clusters as triggers, and determine thefollowing properties: 1) The SNR of the trigger which isthe maximum SNR of the cluster, 2) the amplitude of

the trigger, which is the SNR of the trigger divided bythe template normalisation j , 3) the peak time of thetrigger, which is the location in time of the maximumSNR, 4) the start time of the trigger, which is the timeof the first SNR value above threshold, 5) the duration ofthe trigger, which includes all values above the threshold,and 6) the high frequency cutoff of the template. Thefinal list of triggers for some segment of data can thenbe created by choosing the trigger with the largest SNRwhen several templates contain triggers at the same time.

The rate of events we expect can be estimated for thecase of Gaussian noise. In the absence of a signal the SNRis Gaussian distributed with unit variance. Therefore, atan SNR threshold of th, the rate of events we expectwith a single template is

R(th) t1Erfc(th/

2) (26)

where t is the sampling time of the SNR time series, andErfc is the complementary error function. For example,if the time series is sampled at 4096 Hz and an SNRthreshold of 4 is used we expect a false alarm rate ofR(th = 4) 0.26 Hz. The rate of false alarms risesexponentially with lower thresholds.

E. Multiple interferometer consistency checks

When data from multiple interferometers is available,a coincidence scheme can be used to greatly reduce thefalse alarm rate of events. For example, in the case ofthe three LIGO interferometers, using coincidence wouldresult in a coincident false alarm rate of

R = RH1RH2RL1(2HH)(2HL), (27)

where RH1 and RH2 are the rate of false triggers in the4 km and 2 km IFOs in Hanford, WA, and RL1 is therate of false triggers in the 4 km IFO in Livingston, LA.The two time windows HH, and HL are the max-imum allowed time differences between peak times forcoincident events in the H1 and H2 IFOs, and H1-L1 (orH2-L1), respectively. The time windows should allow forlight-travel time between sites, shifting of measured peaklocation due to noise, timing errors and calibration errors.For example, if we take HL = 12 ms and HH = 2 ms,

which allows for the light travel time of 10 ms between theHanford and Livingston sites as well as 2 ms for other un-certainties, and the single IFO false alarm rate estimatedin the previous subsection (0.26 Hz), we obtain a triplecoincident false alarm rate,

R 1.7 106 Hz, (28)

which is about 50 false events per year.

Additionally, when two instruments are co-aligned,such as the H1 and H2 LIGO IFOs, we can demand somedegree of consistency in the amplitudes recovered from

the match-filter (see, for example, the distance cut usedin [21]). If the instruments are not co-aligned then theamplitudes recovered by the match-filter could be differ-ent. This is due to the fact that the antenna patternof the instrument depends on the source location in thesky. If the event is due to a fluctuation of the noise or aninstrumental glitch we do not generally expect the am-plitudes in two co-aligned IFOs to agree. The effect ofGaussian noise to 1 on the estimate of the amplitudecan be read from Eq. (17). This equation can be used toconstruct an amplitude veto. For example, to veto eventsin H1 we can demand,

AH1 AH2AH1 <

H1+

, (29)

where is the number of standard deviations we allow,and is an additional fractional difference in the ampli-tude that accounts for other sources of uncertainty, suchas the calibration.

These tests can be used to construct the final triggerset, a list of survivors for each of our instruments whichhave passed all the consistency checks.


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III. UPPER LIMITS, SENSITIVITY AND

BACKGROUND

In this section we describe the application of the loud-est event method [22] to our problem, the related issueof sensitivity, and discuss background estimation.

We define the loudest event to be the survivor with thelargest amplitude. Suppose, in one of the instruments,

we find the loudest event to have an amplitude A

. Foran observation time T, on average, the number of eventswe expect to show up with an amplitude greater than A

is,

N>A = T , (30)

where the effective rate,

=

0

(A, A)dR(A)

dAdA. (31)

Here, (A, A) is the search efficiency, namely the fractionof events with an optimally oriented amplitude A found

at the end of the pipeline with an amplitude greater thanA. Aside from sky location effects, the measured am-plitude of events is changed as a result of the noise (seeEq. (17)). The efficiency can be measured by adding sim-ulated signals to detector noise, and then searching forthem. The efficiency accounts for the properties of thepopulation, such as the distribution of sources in the skyand the different high frequency cutoffs (as we will see,dR/df f5/3 for cusps), as well as possible inefficien-cies of the pipeline. The quantity

dR(A)

dAdA (32)

is the cosmological rate of events with amplitudes be-tween A and A + dA and will be the subject of the nextSection.

If the population produces Poisson distributed events,the probability that no events show up with a measuredamplitude greater than A, in an observation time T, is

P = eT. (33)

The probability p, that at least one event with amplitudegreater than A shows up is p = 1P, so that if p = 0.9,90% of the time we would have expected to see an eventwith amplitude greater than A.

From Eq. (33) we see that,

ln(1 p) = T , (34)Setting p = 0.9, say, we have

90% 2.303T . (35)

We say that the value of < 90% with 90% confidence,in the sense that if = 90%, 90% of the time we wouldhave observed at least one event with amplitude greater

than A. If we take a nominal observation time of oneyear, T = 3.2107 s, our upper limit statement becomes,

< 90% 2.303/year 7.3 108s1. (36)IfA50% is the amplitude at which half of the injections

are found with an amplitude greater than A, we can ap-proximate the sigmoid we expect for the efficiency curveby,

(A, A) (AA50%), (37)meaning we assume all events in the data with an am-plitude A > A50% survive the pipeline and are foundwith an amplitude greater than A. This means we canapproximate the rate integral of Eq. (30) as,

A50%

dR(A)

dAdA = R>A50%. (38)

We expect A50% to be proportional to A. It is at least

a factor of

5 larger, due to the averaging of source

locations on the sky [23], and potentially even largerthan this due inefficiencies of the data analysis pipeline.If there are no significant inefficiencies then we expectA50%

5A.

At design sensitivity the form of the LIGO noise curvemay be approximated by [24],

Sh(f) = 1.09 1041

30Hz

f

28

+ 1.44 1045

100Hz

f

4

+ 1.28

10461 + f

90Hz2

s. (39)The first term is the so-called seismic wall, the secondterm comes from thermal noise in the suspension of theoptics, and the third term from photon shot noise. In theabsence of a signal, this curve can be used to estimatethe amplitude of typical noise induced events.

A reasonable operating point for the pipeline involvesan SNR threshold of textth = 4. The value of the am-plitude is related to , the SNR, and , the templatenormalisation,

2 = 4

0

dft(f)2

Sh

(f)= 4

fh

fl

dff8/3

Sh

(f)(40)

by Eq.( 18). If the loudest surviving event has an SNRclose to the threshold, 4, a low frequency cutoff atfl = 40 Hz (the seismic wall is such that its not worthrunning the match filter below 40 Hz), and a typical highfrequency cutoff fh = 150 Hz then the amplitude of theloudest event will be A 4 1021s1/3. So we expectA50% 9 1021s1/3.

The question of sensitivity is subtly different. In thiscase we are interested in how many burst events we can


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detect at all, not just those with amplitudes greater thanA. The rate of events we expect to be able to detect isthus,

s =

0

(0, A)dR(A)

dAdA. (41)

The efficiency includes events of all amplitudes and if thecosmological rate dR/dA favors low amplitude events,

this rate could be larger than that given by Eq. (30).Furthermore this rate must be compared with a rate of1/T, rather than, say, the 90% rate of Eq. (35). The neteffect is that the parameter space of detectability can besubstantially larger than the parameter space over whichwe can set upper limits.

In the remainder of the paper we will use an amplitudeestimate of A50% = 10

20s1/3 for Initial LIGO upperlimit and sensitivity estimates. Advanced LIGO is ex-pected to be somewhat more than an order of magnitudemore sensitive then Initial LIGO, so for the AdvancedLIGO rate estimates we will use A50% = 10

21s1/3. Itshould be noted that the loudest event resulting from a

search on IFO data could be larger than this, for instanceif it was due to a real event or a surviving instrumentalglitch.

The rate of events that we expect in the absence ofa signal, called the background, can be estimated eitheranalytically (via Eq. (27), in which the single IFO ratesare measured), or by performing time-slides (see, for ex-ample [25]). In the latter procedure single IFO triggersare time-shifted relative to each other and the rate of ac-cidental coincidences is measured. Care must be takento ensure the time-shifts are longer than the duration ofreal events. If the rate of events is unknown, then onecan look for statistically significant excesses in the fore-

ground data set. In this case a statistically significantexcess could point to a detection. However, as we shallsee, there is enough freedom in models of string evolu-tion to lead to statistically insignificant excesses in anybackground. Therefore all surviving triggers in the fore-ground need to be followed up.

IV. THE RATE OF BURSTS

In this section we will derive the expression for thecosmological rate as a function of the amplitude neededto evaluate the upper limit and sensitivity integrals(Eqs. (30) and (41)). To make the account self-containedwe will re-derive a number of the results presented in [810]. The main differences are that we derive an expres-sion for a general cosmic string loop distribution that canbe used when such a distribution becomes known, we ab-sorb some uncertainties of the model, as well as factorsof order 1, into a set of ignorance constants, and we con-sider a generic cosmology with a view to including theeffects of a late time acceleration.

Numerical simulations show that at any cosmic timet, the network consists of a few long strings that stretch

across the horizon and a large collection of small loops.In early simulations [14] the size of loops, as well asthe substructure on the long strings, was at the size ofthe simulation resolution. At the time, the consensusreached was that the size of loops and sub-structure onthe long strings was at the size given by gravitationalback-reaction. Very recent simulations [1517] suggestthat the size of loops is given by the large scale dynamics

of the network, and is thus un-related to the gravita-tional back-reaction scale. Further work is necessary toestablish which of the two possibilities is correct.

In either case, the size of loops can be characterised bya probability distribution,

n(l, t)dl, (42)

the number density of loops with sizes between l and l+dlat a cosmic time t. This distribution is unknown, thoughanalytic solutions can be found in simple cases (see [2]9.3.3).

The period of oscillation of a loop of length l is T = l/2.If, on average, loops have c cusps per oscillation, the

number of cusps produced per unit space-time volumeby loops with lengths in the interval dl at time t is [30],

(l, t)dl =2c

ln(l, t)dl. (43)

We can write the cosmic time as a function of the redshift,

t = H10 t(z), (44)

where H0 is current value of the Hubble parameter andt(z) is a function of the redshift z which depends onthe cosmology. In [810], an approximate interpolatingfunction for t(z) was used for a universe which contains

matter and radiation. For the moment we will leave itunspecified. Later we will compute it numerically fora more realistic cosmology which includes the late-timeacceleration. We can therefore write the number of cuspsper unit space-time volume produced by loops with sizesbetween l and l + dl at a redshift z,

(l, z)dl =2c

ln(l, z)dl. (45)

Now we would like to write the length of loops in termsof the amplitude of the event an optimally oriented cuspfrom such a loop would produce at an earth-based de-tector. The amplitude we expect from a cusp produced

at a redshift z, from a loop of length l can be read offEq. (4.10) of [10]. It is,

A = g1Gl2/3H0

(1 + z)1/3r(z). (46)

Here g1 is an ignorance constant that absorbs the uncer-tainty on exactly how much of the length l is involved inthe production of the cusp, as well as factors of O(1) thathave been dropped from the calculation (see the deriva-tion leading up to Eq. (3.12) in [10]). We have expressed


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r(z), the amplitude distance [10], as an unspecified cos-mology dependent function of the redshift r(z),

r(z) = H10 r(z). (47)

The amplitude distance is a factor of (1+ z) smaller thanthe luminosity distance. Since,

l(A, z) =

Ar(z)(1 + z)

1/3

g1GH03/2

, dldA = 32A l, (48)

we can write the number of cusps per unit space-timevolume which produce events with amplitudes betweenA and A + dA from a redshift z as,

(A, z)dA =3c

An(l(A, z), z)dA. (49)

As discussed in [9], we can observe only a fraction ofall the cusps that occur in a Hubble volume. Supposewe look, via match-filtering, for signals of the form ofEq. (1). The highest frequency we observe fh is relatedto the angle that the line of sight makes to the directionof the cusp. Iff is the lowest high frequency cutoff thatour instrument can detect with confidence, the maximumangle the line of sight and the direction of a detectablecusp can subtend is,

m = (g2fl)1/3. (50)

The ignorance constant g2 absorbs a factor of about 2.31[10], other factors of O(1), as well as the fraction of theloop length l that actually contributes to the cusp (seethe derivation leading up to Eq. (3.22) of [10]). For burstsoriginating at cosmological distances, red-shifting of thehigh frequency cutoff must be taken into account. So we

write,

m(z, f, l) = (g2(1 + z)fl)1/3, (51)

The waveform in Eq. (1), was derived in the limit 1, so that cusps with 1 will not have the form ofEq. (1) (indeed, they are not bursts at all). Thereforecusps with 1 should not contribute to the rate. Thus,the subset of observable cusps, , is

(z, f, l) 2m(z, f, l)

4(1 m(z, f, l)). (52)

The first term on the right hand side is the beaming

fraction for the angle m. At fixed l, the beaming fractionis the fraction of cusp events with high frequency cutoffsgreater than f. The function cuts off cusp eventsthat have m 1; it ensures we only count cusps whosewaveform is indeed given by Eq. (1).

The rate of cusp events we expect to see from a volumedV(z) (the proper volume in the redshift interval dz), inan amplitude interval dA is [31]

dR

dV(z)dA= (1 + z)1(A, z)(z, f, A), (53)

where (z, f, A) = (z, f, l(A, z)). The factor of(1 + z)1 comes from the relation between the observedburst rate and the cosmic time: Bursts coming from largeredshifts are spaced further apart in time. We write theproper volume element as,

dV(z) = H30 V(z)dz, (54)

where V(z) is a cosmology dependent function of theredshift. Thus, for arbitrary loop distributions, we canwrite the rate of events in the amplitude interval dA,needed to evaluate the upper limit integral Eq. (30), andsensitivity integral Eq. (41) as,

dR

dA= H30

0

dz V(z)(1 + z)1(A, z)(z, f, A).

(55)Damour and Vilenkin [810] took the loop distribution

to be,

n(l, t) = (pG)1t3(l t), (56)where p is the reconnection probability, which can besmaller than 1 for cosmic superstrings, and 50. Inthis loop distribution all loops present at a cosmic timet, are of the same size t.

This distribution is consistent with the assumptionusually made in the literature that the size of loops isgiven by gravitational back-reaction, and that G.Recently it was realised that damping of perturbationspropagating on strings due to gravitational wave emissionis not as efficient as previously thought [27]. As a conse-quence, the size of the small-scale structure is sensitiveto the spectrum of perturbations present on the strings,and can be much smaller than the canonical value Gt[28].

If the value of is given by gravitational back-reactionwe can write it as

= (G)n . (57)

We use two parameters , introduced in [8], as well as nthat can be used to vary the size of loops. The parametern arises naturally from gravitational back-reaction mod-els and is related to the power spectrum of perturbationson long strings [28]. For example, if the spectrum is in-versely proportional to the mode number, then n = 3/2.This is the spectrum of perturbations we expect if, say,the shape of the string is dominated by the largest kink[28].

As we have mentioned, recent simulations suggest thatloops are produced at sizes unrelated to the gravitationalback-reaction scale, i.e. 1. If this is the case, thenthe loops produced survive for a long time, and the formof the distribution can be computed analytically in thematter and radiation eras using some simple assumptions(see [2] 9.3.3). For these analytic solutions it turns out[8] that the dominant contribution to the energy densityand the rate of bursts comes from loops of a size l Gt, and (in the matter era) we can still use Eq. (56),


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8

and Eq. (57) with = n = 1. We will see explicitly inSec. VI that this is indeed a good approximation.

In general the rate integral needs to be computed, pre-sumably numerically, through Eq. (55). In the case ofthe more simple loop distribution of Eq. (56), it is conve-nient to proceed slightly differently. In this case all loopsat a given redshift are of the same size, so we can directlyassociate amplitudes with redshifts.

We can write the rate of cusp events we expect to seefrom the redshift interval dz, from loops of length in theinterval dl as

dR

dzdl= H30 V(z)(1 + z)

1(l, z)(z, f, l). (58)

Using Eq. (45), Eq. (56) with t expressed in terms of theredshift, and integrating over l we find,

dR

dz= H0

c(g2fH10 )

2/3

25/3pG14/3t (z)V(z)(1 + z)

5/3

(1 m(z, f, H10 t(z)) (59)

At a redshift of z, all cusps produce bursts of thesame amplitude, given by replacing l with H10 t(z)in Eq. (46). Therefore the solution of,

2/3t (z)

(1 + z)1/3r(z)=

AH1/30

g1G2/3(60)

for z gives the redshift from which a burst of amplitude Aoriginates. This means we can perform the rate integral,say Eq. (38), over redshifts rather than over amplitudes,

z50%0

dR

dzdz, (61)

where z50% is the solution of Eq. (60) for A = A50%.

V. COSMOLOGICAL FUNCTIONS

To derive exact expressions for t(z), r(z), andV(z), we begin with the evolution of the Hubble func-tion. The Hubble function is given by

H(z) = H0h(z), (62)

with, for a flat universe,

h(z) =

m(1 + z)

3

+ r(1 + z)

4

+ 1/2

. (63)Here, i = i(z = 0)/c(z = 0) is the present energydensity of the ith component, relative to the critical den-sity. The subscripts m, r, and stand for matter (darkand baryonic), radiation, and the cosmological constant,respectively. Since we assume the universe flat, i = 1.

We will use the set of cosmological parameters in[29], which provide a good fit to recent cosmologicaldata. The precise values of the parameters are not crit-ical, but we include them here for clarity. They are,

H0 = 73 km/s/Mpc = 2.4 1018 s1, m = 0.25,r = 4.6 105, and = 1 mr. In the remain-der of the paper we will refer to this cosmological modelas the universe.

To compute the relation between the time and the red-shift, we use the fact that dz/dt = (1 + z)H, and write

t = t

0

dt =

z

dz

(1 + z

)H(z

). (64)

The dimensionless function t(z) of Eq. (44) is thus,

t(z) =

z

dz

(1 + z)h(z). (65)

In order to compute the amplitude distance [10] as afunction of the redshift, we consider null geodesics in anFRW universe. We can take the polar and azimuthalcoordinates to be constant and let the radial coordinate,r, vary. In this case we have that dr/dt = (1 + z), andthus dr/dz = 1/H. So we write

r =

r

0

dr =

z

0

dz

H(z). (66)

We can therefore express the dimensionless function ofEq. (47), r(z), as

r(z) =

z0

dz

h(z). (67)

Finally, we would like to derive an expression for thedifferential volume as a function of the redshift, dV/dz.The differential volume element is given by

dV = a

3

(t)r

2

dr sin d d ,

where a(t) is the scale factor. Integrating over the polarand azimuthal coordinates and using dr/dz gives

dV = 4a3(t)r2dr =4r2

(1 + z)3H(z)dz . (68)

Using Eqs. (62) and (66) gives the dimensionless functionof Eq. (54)

V(z) =42r(z)

(1 + z)3h(z). (69)

VI. RESULTS I: DETECTABILITY

A. Comparison with previous estimates

In [810] Damour and Vilenkin introduced a set of in-terpolating functions for t(z), r(z), and V(z). Theywere

t(z) = t0H0(1 + z)3/2(1 + z/zeq)

1/2, (70)


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1012

1010

108

106

104

1018

1016

1014

1012

1010

108

106

G

Rate[Hz]

Cosmology A50%

=1020

s1/3

Cosmology A50%

=1021

s1/3

DV Cosmology A50%

=1020

s1/3

DV Cosmology A50%

=1021

s1/3

1/year

FIG. 1: Rate of gravitational wave bursts as a function ofG.The dash-dot and dashed curves show the rate computed withthe Damour-Vilenkin cosmological functions Eqs. (70), (71)

and (72), with A50% = 1021 s1/3, and A50% = 10

20 s1/3

respectively. The thick and thin solid curves show the ratecomputed in a universe with a cosmological constant with

amplitudes A50%

= 10

20

s

1/3

, and A50%

= 10

21

s

1/3

respectively.

r(z) = t0H0z

1 + z, (71)

and [26],

V(z) = (t0H0)3102z2(1 + z)13/2(1 + z/zeq)

1/2. (72)

Damour and Vilenkin used zeq 103.9 as the redshift ofradiation matter equality and set the age of the universet0 = 1017.5 s. Notice that if we substitute Eqs. (70) and(72) into Eq. (59) and set = G and g2 = p = 1

we recover Eq. (5.17) of [10] (aside from a factor of zrelated to their use of a logarithmic derivative). We willuse Eqs. (70), (71) and (72) to compare their results withthose of a cosmology solved exactly including the effectsof a late time acceleration.

Figure 1 shows the rate of burst events as a functionof G for two different models. For all four curves wehave set = G, = 50, f = 75 Hz, c = p = 1, andthe ignorance constants g1 = g2 = 1. We will refer tostring models with these parameters as classic, whichis appropriate for field theoretic strings with loops of sizel = Gt. The dashed-dot and dashed curves of Fig. 1show the rates computed using the Damour-Vilenkin cos-mological functions namely, Eqs. (70), (71) and (72). Forthe dashed-dot curve we have used an amplitude estimateofA50% = 10

21 s1/3. This estimate is obtained by set-ting the SNR threshold to 1 and assuming all cusp eventsare optimally oriented: It corresponds to the estimatesmade in [810] for Initial LIGO (and also our estimatefor the amplitude in the case of Advanced LIGO). Thedashed curve shows the rate computed with the ampli-tude A50% = 10

20 s1/3, which we feel is more appro-priate for Initial LIGO. The thick and thin solid curvesshow the rates computed by evaluating the cosmological

1012

1010

108

106

104

1020

1015

1010

105

G

Rate[Hz]

Cosmology A50%

=1020

s1/3

, p=103

Cosmology A50%

=1020

s1/3

, n=3/2

Cosmology A50%

=1020

s1/3

, n=3/2, p=103

1/year

FIG. 2: Comparison of the rate of gravitational wave burstsas a function of G. For all three curves an Initial LIGOamplitude estimate of A50% = 10

20 s1/3 has been used,and the cosmological functions have been evaluated in the universe. The model parameters are identical to those ofFig. 1 except where indicated. The solid curve shows the ratecomputed with a reconnection probability of p = 103. Thedashed curve shows rate computed with a size of loops giveby Eq. (57) with = 1, and n = 3/2. The dot-dash curveshows the combined effect of a low reconnection probability,p = 103, as well as a size of loops given by Eq. (57) with = 1, and n = 3/2.

functions (Eqs. (65), (67) and (69)) numerically for the universe. The thick solid curve corresponds to our ampli-tude estimate for Initial LIGO, and the thin solid curvecorresponds to our estimate for Advanced LIGO.

The functional dependence of the rate of gravitationalwave bursts on G is discussed in detail in the Appendix.

Here we summarise those findings. From left to right,the first steep rise in the rates as a function of G ofthe dashed and dashed-dot curves of Fig. 1 comes fromevents produced at small redshifts (z 1). The peak andsubsequent decrease in the rate starting around G 109 comes from events produced at larger redshifts butstill in the matter era (1 z zeq). The final rise comesfrom events produced in the radiation era (z zeq).

For classic cosmic strings (p = = n = 1), our es-timate for the rate of detectable events is substantiallylower than that in [810]. The bulk of the difference arisesfrom our estimate of the amplitude, which we believe ismore realistic. This is illustrated by the dash-dot anddashed curves of Fig. 1, which use the same cosmologi-cal functions, and two estimates for the amplitude. Theeffect of the amplitude estimate on the rate is discussedin detail in the Appendix. The remaining discrepancyarises from differences in the cosmology, as well as fac-tors ofO(1) that were dropped in the previous estimates.The net effect is that the chances of seeing an event fromclassic strings using Initial LIGO data have droppedfrom order unity to about 103 at the matter era peak.

Cosmic superstrings, however, may still be detectableby Initial LIGO. Furthermore, if the size of the small-


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10

scale structure is given by gravitational back-reaction,reasonable estimates for what the size of loops might bealso lead to an enhanced rate of bursts. Figure 2 illus-trates this point. All three curves use the Initial LIGOamplitude estimate of A50% = 10

20 s1/3, and the cos-mological functions Eqs. (65), (67) and (69) computed inthe universe. The solid curve shows the rate computedwith a reconnection probability of p = 103. The dashed

curve shows the rate computed for loops with a size givenby Eq. (57) with = 1, and n = 3/2. This is the value ofn we expect when the spectrum of perturbations on longstrings is inversely proportional to the mode number, andthe result we expect if the spectrum of perturbations onlong strings is dominated by the largest kink [28]. Thedashed-dot curve shows the combined effect of a low re-connection probability, p = 103, as well as a size ofloops given by Eq. (57) with = 1, and n = 3/2. Theremaining parameters for all three curves are identical tothose of Fig. 1.

To summarise, we find the chances of detecting clas-sic strings to be significantly smaller than previous es-timates. Even Advanced LIGO only has a few percentchance of detecting them at the matter era peak (aroundG 109 in Fig. 1). Initial LIGO requires the smallreconnection probability of cosmic superstrings and/orsmall loops to attain a reasonable chance of detection. Itshould be pointed out that the classic string model maywell be incorrect. If loop sizes are given by gravitationalback reaction then a reasonable guess for the spectrumof perturbations on long strings leads to n = 3/2. Thisvalue, as shown on Fig. 2 has a substantially larger chanceof detection by Initial LIGO. The size of loops, however,may not determined by gravitational back-reaction afterall.

B. Loop distributions

Recent simulations [1517] suggest that loops areproduced at sizes unrelated to the gravitational back-reaction scale, i.e. 1. In this case the loops producedby the network survive for a long time.

If all loops have the same size l = t at formation, thedistribution can be estimated for the one-scale model(see [2], 9.3.3 and 10.1.2). In the radiation era it is

n(l, t) = t3/2(l + Gt)5/2, l < t, (73)

where 1, and in the matter era,n(l, t) = t2(l + Gt)2, l < t. (74)

We can approximate the loop distribution using

n(l, t) =

t3/2(l + Gt)5/2 t < teq

t2(l + Gt)2 t > teq, l < t,

(75)where teq is the time of matter-radiation equality. Thisapproximation is not quite right because loops formed at

1012

1010

108

106

104

1018

1016

1014

1012

1010

108

106

G

Rate[Hz]

n(l,t) = (G )1

(lt)

n(l,t) ={t2

(lG t)2

t3/2

(lG t)5/2 ttEQ

n(l,t) = t2

(lG t)2

n(l,t) = t3/2 (lG t)5/2

1/year

n(l,t) ={t1

(lG t)3 ttEQ

FIG. 3: Plot of the rate of bursts as a function ofG for var-ious loops distributions. For all curves we have set = 50,f = 75 Hz, c = p = 1, = 1 (where appropriate) and theignorance constants to 1. As a reference we again show therate computed using the loop distribution from Eq. (56), ac-cording to Eq. (59), in the -universe with = G using thethick solid curve. The remaining curves have been computedthrough Eqs. (55) and (38) with A50% = 10

20 s1/3. Thethick dashed curve shows the rate computed using the loopdistribution of Eq. (74) with = 1. The thick dashed-dotcurve shows the rate computed with Eq. (73) with = 1.The thin dashed line shows the rate computed with Eq. (75),and the thin solid line the rate computed with Eq. (76).

the end of the radiation era survive into the matter era.The effect of this approximation is to underestimate therate of events that occur after (but near) the transition.Since at present it is unclear whether this distribution iscorrect, this seems a relatively minor problem. Recent re-

sults of one simulation group [15] (see also the discussionat end of [17]), instead suggest a distribution,

n(l, t) =

t1(l + Gt)3 t < teq

t3/2(l + Gt)5/2 t > teq, l < t,

(76)which favors smaller loops.

We can use the results of Sec. IV to compute the rateof bursts for these loop distributions. Figure 3 shows theburst rate as a function of G for all the above distribu-tions. For all curves we have set = 50, f = 75 Hz,c = p = 1, = 1 (where appropriate), and the igno-rance constants g1 = g2 = 1. We have evaluated all ratesin the universe. As a reference, we again show therate computed using the loop distribution from Eq. (56),according to Eq. (59), with = G using the thicksolid curve. The remaining curves have been computedthrough Eqs. (55) and (38) with A50% = 10

20 s1/3.The thick dashed curve shows the rate computed usingthe loop distribution of Eq. (74) with = 1. The thickdashed-dot curve shows the rate computed with Eq. (73)with = 1. The thin dashed line shows the rate com-puted with Eq. (75), and the thin solid line the rate com-puted with Eq. (76).


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11

There is remarkable agreement between the thick solidcurve and the thick dashed curve for matter era bursts.This shows that in the matter era we can indeed ap-proximate the loop distribution Eq. (74), using insteadEq. (56), and Eq (57) with = n = 1, as suggested in[8].

Distributions that favor small loop sizes are more likelyto produce a detectable signal. Indeed, for Initial LIGO,

the distribution given by Eq. (76) produces an average of10 events per year around G 5 109. Note that wehave not included the enhancement in the rate of burstsfor the case of cosmic superstrings p < 1.

Given this range of results, it is important to deter-mine whether the loop sizes at formation are determinedby gravitational back-reaction, or whether they are largeand we require the knowledge of a loop distribution.

VII. RESULTS II: CONSTRAINING THE

THEORY

If no events can be positively identified in a search, wecan use the results of Sec. III to constrain the parameterspace of theories that lead to the production of cosmicstrings. Unfortunately, as we have mentioned, consider-able uncertainties remain in models of cosmic string evo-lution. Nevertheless, we can place constraints that arecorrect in the context of a particular string model. Herewe will illustrate the procedure for the loop distribution,Eq. (56), and the resulting rate Eq. (59).

We would like to absorb the ignorance constants g1and g2 into the parameters of the model. The two igno-rance constants enter the expression for the rate Eq. (59)in three ways. First, they affect the upper limit of theintegral Eq. (61) through Eq. (60), which is,

2/3t (z)

(1 + z)1/3r(z)=

AH1/30

g1G2/3. (77)

Secondly, they enter through m in the theta functioncutoff of the rate,

m = (g2(1 + z)fH10 t(z))

1/3, (78)

and finally, the rate itself (Eq. (59)) is proportional to

g2/32 .If we write = (G)n, and substitute into Eqs. (78)

and (77), we can simultaneously absorb g1 and g2 intonew variables (, g1, g2, n) and X(G,g1, g2). Lookingat Eq. (78) we can write an identity for the ignoranceconstants and the variables we want to absorb them into,

g2(G)n = (X)n. (79)

Similarly, looking at the denominator of the right handside of Eq. (77) we write,

g1G((G)n)2/3 = X((X)n)2/3 (80)

g11

g1/32

p/c (=1/)

68%

95%

99.7

%

g1 1

g2/3

2

G

(=X)

105

104

103

102

101

100

1010

109

108

107

106

Allowed Region

Region ruled outat the 99.7% level

FIG. 4: Contour of excluded regions at various confidences.In the cosmic string model we have set = 1, and n = 3/2,and allowed X and to vary. For the loudest event we haveused our Initial LIGO estimate for the amplitude A50% =1020 s1/3.

These equations can be simultaneously solved to give,

G = g11 g2/32 X, (81)

and

= gn1 g2n/312 . (82)

If we replace G by Eq. (81) and by Eq. (82), in Eq. (78)we obtain,

m = ((1 + z)f(X)nH10 t(z))

1/3, (83)

and if we make the same replacements into Eq. (77),

2/3t (z)

(1 + z)1/3r(z)=

AH1/30

X((X)n)2/3, (84)

namely, we obtain functions of and X only.Finally, if we replace G by Eq. (81) and by Eq. (82),

in our expression for the rate, Eq. (59), we see that wecan absorb the remaining factors by defining a quantity,

= g1/32 g1

c

p. (85)

This yields a rate,

dRdz

= H0(fH10 )2/3

2((X)n)5/3X14/3t (z)V(z)(1 + z)

5/3

(1 m), (86)where m given by Eq. (83). The parameters that we canplace upper limits on are now X, n, , and .

Figure 4 illustrates the application of this procedure.We show a contour plot constructed by computing therate and comparing it to 68% 1.14/T, 95% 3.00/Tand 99.7% 5.81/T. In the cosmic string model we have


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12

set = 1, and n = 3/2, and allowed X and to vary. Inplace of the loudest event we have used our Initial LIGOestimate for the amplitude A50% = 10

20 s1/3. If weset the ignorance constants to unity, and c = 1, we seethat for G in the range 4 109 106, values of thereconnection probability around p = 103 are ruled outat the 99.7% level.

It is interesting to note that for cosmic superstrings a

bound on the reconnection probability might be turnedinto a bound on the string coupling and/or a bound onthe size of extra dimensions.

VIII. SUMMARY AND CONCLUSIONS

We begun this work by considering the data analysisinfrastructure necessary to perform a search for gravita-tional waves from cosmic (super)string cusps. The op-timal method to use in such a search is match-filtering.We have described the statistical properties of the match-filter output, a method for template bank construction,

and an efficient algorithm to compute the match-filter.When data from multiple interferometers is available, wediscussed consistency checks that can be used to greatlyreduce the false alarm rate.

The relevant output of the match-filter is the estimatedsignal amplitude. We have shown that in a search, theevent with the largest amplitude can be used to to set up-per limits on the rate via the loudest event method. Theupper limit depends on the cosmological rate of eventsand the efficiency, which can be computed using simu-lated signal injections into the data stream. The injec-tions should account for certain properties of the popula-tion, such as the frequency distribution (dR/df f5/3

for cusps), and source sky locations. We have briefly dis-cussed the related issue of sensitivity, which also dependson the search detection efficiency, as well as backgroundestimation. We have made a single estimate for the am-plitude of detectable and loudest events using the InitialLIGO design curve and an SNR threshold th = 4, whichwe believe is a reasonable operating point for a pipeline.Assuming that there are no significant inefficiencies in thepipeline, that the rate of false alarms is sufficiently low,and that no signals or instrumental glitches are presentin the final trigger set, we expect this amplitude to beA50% 1020s1/3. Advanced LIGO is expected to bean order of magnitude more sensitive then Initial LIGO,so for Advanced LIGO we expect A50%

1021s1/3.

We have computed the rate of bursts in the case of anarbitrary cosmic string loop distribution, using a genericcosmology. Motivated by the data analysis, we cast therate as a function of the amplitude. We have shownhow to calculate the relevant cosmological functions inthe case of a flat universe with matter, radiation and acosmological constant. We have applied this formalismto the distribution used in the estimates of [810]. Forthe classic (field theoretic strings with loops of sizel = Gt) model of cosmic strings considered in [9, 10],

we find substantially lower rates. The bulk of the differ-ence arises from our estimate of a detectable amplitude,which we believe is more realistic. The effect of the am-plitude estimate on the rate is discussed in detail in theAppendix. The remaining discrepancy arises from differ-ences in the cosmology (the cosmological model in [810]included only matter and radiation), as well as factors ofO(1) that were dropped in the previous estimates. The

net effect is that the chances of seeing an event fromclassic strings using Initial LIGO data have droppedfrom order unity to about 103 at the local maximumof the rate (which comes from bursts produced in thematter era). This result is illustrated in Fig. 1.

The classic string model may well be incorrect sothese results are not necessarily discouraging. Two newresults indicate that this is the case.

Recently it was realised that damping of perturbationspropagating on strings due to gravitational wave emissionis not as efficient as previously thought [27]. As a conse-quence, the size of the small-scale structure is sensitivethe spectrum of perturbations present on the strings, andcan be much smaller than the canonical value Gt [28].If the size of loops is given by gravitational back-reaction,a reasonable guess for the spectrum of perturbations onlong strings leads to

(G)3/2 , (87)which in turn could lead to detectable events by Ad-vanced LIGO, or Initial LIGO if we are dealing withcosmic superstrings. This is illustrated in Fig. 2.

Even more recently, simulations [1517] suggest thatloops are produced at sizes unrelated to the gravitationalback-reaction scale, 1. In this case loops of manydifferent sizes are present at any given time (because they

are very long-lived), and the use a loop distribution be-comes necessary. The particulars of the distribution arecurrently under debate. If the results of one simulationteam [15] are correct, then a wide range of values for thestring tension leads to events detectable by Initial LIGOeven from field theoretic strings. Results for the rate forvarious loop distributions are shown in Fig. 3.

Finally, we have have shown how the parameter spaceof theories that lead to the production of cosmic stringsmight be constrained in the absence of a detection. Re-sults for a model of cosmic superstrings, where the loopsize is given by Eq. (87), are shown in Fig. 4. InitialLIGO may yield interesting constraints on the reconnec-tion probability for some range of string tensions. In-

triguingly, a bound on the reconnection probability mightbe turned into a bound on the string coupling and/or abound on the size of extra dimensions, in the case of cos-mic superstrings.

Wish list

Unfortunately, there remain considerable uncertaintiesin models of cosmic string evolution. While we can es-


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13

timate the detectability, and place constraints, that arecorrect in the context of a particular string model, thecurrent parameter space allows for a wide spectrum ofburst rates in the interesting range of string tensions.

We would like to finish by posing a set of questions tothe various cosmic string simulation groups which, fromthe perspective of gravitational wave detection, wouldgreatly improve predictability.

They are:

What is the size of cosmic string loops? Are theylarge when they are formed, so we need to considera loop distribution? If so, what is that distribu-tion? Is their size instead given by gravitationalback-reaction? If so, what is the spectrum of per-turbations on long strings?

What is the number of cusps per loop oscillation?Is it independent of the loop size?

What is the size of cusps? In particular, what frac-tion of the loop length l is involved in the cusp?

What are the effects of low reconnection probabil-ity? In particular, what is the enhancement in theloop density that results?

Acknowledgments

We would like to thank Alex Vilenkin, Benjamin Wan-delt, and Marialessandra Papa, for useful discussions. Wewould also like to thank Daniel Sigg for providing theanalytic form for the LIGO design curve, Eq. (39). Thework of XS, JC, SRM, JR, and KC was supported by

NSF grants PHY 0200852 and PHY 0421416. The workof IM was supported by the DOE and NASA.

APPENDIX: APPROXIMATE ANALYTIC

EXPRESSION FOR THE RATE

In this appendix we compute an approximate expres-sion for the rate as a function of the amplitude can beobtained using the interpolating functions introduced in[810], Eqs. (70), (71), and (72). The result is useful tounderstand the qualitative behaviour of the rate curvesshown in Fig.s 1, 2, and 3.

Using Eqs. (46) and (71) we can write the amplitudeof an burst from a loop of length l at a redshift z as

A Gl2/3(1 + z)2/3

t0z. (A.1)

We take the size of the feature that produces the cuspto be the typical size of loops, l(z) = H10 t(z), witht(z) given by Eq. (70), so that,

A G2/3t1/30 z1(1 + z)1/3(1 + z/zeq)1/3. (A.2)

We define a dimensionless amplitude a,

a AG2/3t

1/30

z1(1+z)1/3(1+z/zeq)1/3 (A.3)

Depending on the value ofz, the function a(z) has threedifferent asymptotic behaviours,

a(z) z

1

z 1z4/3 1 z zeqz1/3eq z5/3 z zeq

(A.4)

The regime where z 1 corresponds to cusp eventsoccurring nearby, the regime where 1 z zeq cor-responds to matter-era events, and the regime wherez zeq corresponds to radiation era events.

At z = 1, we define

a1 a(z = 1) 21/3 1,

and at z = zeq,

aeq a(z = zeq) z4/3eq 21/3 z4/3eq a1 4 106.

This means we can write,

zeq a3/4eq . (A.5)

The regime where z 1, corresponds to a 1; theregime where 1 z zeq, corresponds to aeq a 1;and the regime where z zeq, corresponds to a aeq.This means that we can write,

z(a)

a3/20eq a3/5 a aeq

a3/4 aeq

a

1

a1 a 1(A.6)

We can write this in terms of the interpolating function,

z(a) a3/20eq a3/5

1 +a

aeq

3/20

(1 + a)1/4 . (A.7)

Using Eqs. (70) and (72), we can write the rate ofevents as a function of the redshift, Eq. (59), as,

dR

dz bz2(1 + z)7/6(1 + z/zeq)11/6, (A.8)

where b is defined as

b 102c5/3(pG)1t10 (ft0)2/3, (A.9)

and the theta-function cutoff has been ignored for con-venience. In the three regimes considered above the rateis given by,

dR b

z2dz z 1z5/6dz 1 z zeq

z11/6eq z8/3dz z zeq

(A.10)


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14

In the regime where z 1, corresponds to a 1, andz a1. This means,

dR bz2dz = ba2dzda

da. (A.11)

Since,

dz

da a2

, (A.12)

we have that

dR ba4da, for a 1. (A.13)The regime where 1 z zeq, corresponds to aeq

a 1, and z a3/4. This means,

dR bz5/6dz = ba5/8 dzda

da. (A.14)

Since,

dz

da 3

4 a

7/4, (A.15)

we have that

dR 34ba19/8da, for aeq a 1. (A.16)Finally, the regime where z zeq, corresponds to a

aeq, and z (aeq/a1)3/20a3/5. Thus, using Eq. (A.5),

dR bz11/6eq z8/3dz = ba39/40eq a8/5dz

dada. (A.17)

Since,

dzda

35

a3/20eq a8/5, (A.18)

we have that

dR 35

ba33/40eq a16/5da, for a aeq. (A.19)Summarising,

dR(a)

da b

a33/40eq a16/5 a aeqa19/8 aeq a 1

a4 a 1(A.20)

Eq. (A.20) can be integrated, to give the rate of events

with reduced amplitude greater than a,

R>a b

a33/40eq a11/5 a aeq

a11/8 aeq a 1a3 a 1

(A.21)

We can determine the functional dependence of therate for the simple case when = G, which we can

then compare with the dashed and dashed-dot curves ofFig. 1. The factor ofb, as defined by Eq. (A.9), contains afactor of (G)1 as well as a factor of (G)5/3 throughits dependence on . So we take b (G)8/3. Thedimensionless amplitude a, as defined by Eq. (A.3) alsocontains a factor of (G)1 as well as a factor of (G)2/3

through its dependence on , so that a (G)5/3.In Eq. (A.21), (as we have mentioned) the regime

where a aeq maps into the radiation era, and in thiscase the rate,

R G. (A.22)

The regime where aeq a 1 maps into the matter erawhen the redshift z 1, and the rate in this case is,

R (G)31/24. (A.23)

The final regime when a

1 corresponds to bursts that

are coming from close by (z 1), and the rate,

R (G)7/3. (A.24)

So we immediately see that the first steep rise in the rateas a function ofG (from left to right) of the dashed anddashed-dot curves of Fig. 1 corresponds to bursts thatare coming from small redshifts, i.e. from z 1. Theslight decrease in the rate comes from bursts produced atlarge redshifts, but still in the matter era, i.e. 1 z zeq, and the final increase in the rate comes from burstsproduced in the radiation era, z zeq.

Eq. (A.21) also makes it easy to understand the lowerburst event rates we find relative to the previous esti-mates of Damour and Vilenkin [810] (see the the dashedand dashed-dot curves of Fig. 1). They compared thestrain produced by cosmic string burst events at a rateof 1 per year to a noise induced SNR 1 event, given theInitial LIGO design noise curve. In our treatment, theamplitude that corresponds to is A 1021s1/3. Onthe other hand, we have chosen an SNR of 4, which wethink is a more realistic operating point for a data analy-sis pipeline, and have included the effects of the antennapattern of the instrument, which averages to a factor of

5. The combined effect is to increase the amplitude esti-

mate by about a factor of 10, to make it A 1020

s1/3

.Looking at Eq. (A.21), we see that an increase in the

amplitude of a factor of 10, results in a decrease of afactor of 103 in the rate of nearby bursts (z 1), adecrease by a factor of about 24 in the rate of burstsproduced in the matter era (1 z zeq), and a decreaseby about a factor of 160 in the rate of bursts producedin the radiation era (z zeq).


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[31] In [810], the cutoff was placed on the strain of cusps (ata fixed rate) rather than their rate, as we do here. Thisis un-important, the purpose of the theta function is toensure we do not overestimate the rate.

Documents

Xavier Siemens et al- Gravitational wave bursts from cosmic (super)strings: Quantitative analysis and constraints