Probing Particle Physics from Top Down with Cosmic Strings

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by  Robert H. Brandenberger

Probing Particle Physics from Top Down with Cosmic Strings
Robert H. Brandenberger1,∗
1Physics Department, McGill University, 3600 University Street, Montreal, QC, H3A 2T8, Canada
Makinguseofthewealthofnewobservationaldatacomingfromtheskyitispossibletoconstrain
particlephysicstheoriesbeyondtheStandardModel. Onewaytodothisisillustratedinthisarticle:
a subset of theories admits cosmic string solutions, topologically stable matter field configurations.
Inthesemodels,anetworkofcosmicstringsinevitablyformsintheearlyuniverseandpersiststothe
presenttime. Thegravitationaleffectsofthesestringsleadstocosmologicalsignatureswhichcould
bevisibleincurrentandfuturedata. Themagnitudeofthesesignaturesincreasesastheenergyscale
of the new physics involved in cosmic string formation increases. Thus, searching for cosmological
4
1 signatures of strings is a way to probe particle physics model “from top down”, as opposed to
0 “from bottom up” as is done using data from accelerators such as the Large Hadron Collider.
2 Different ways of searching for cosmic strings are illustrated in this article. They include Cosmic
MicrowaveBackgroundtemperatureandpolarizationanisotropymaps,LargeScaleStructureoptical
n
and infrared surveys, and 21cm intensity maps.
a
J
PACSnumbers: 98.80.Cq
8
1
I. INTRODUCTION natures. Searching for these signatures is thus a way to
]
O probe particle physics beyond the Standard Model. The
magnitudeofthesignalsisproportionaltothetensionof
C Observationalcosmologyisinitsgoldenage. Improved
the strings, which in turn is proportional to the square
instrumentationhasallowedobserverstoexploretheuni-
.
h of the energy scale characteristic of the string. Hence,
verse to much greater depth than ever before. Optical
p looking for cosmic strings in the sky is a way to probe
and infrared telescopes yield maps of the distribution of
- particle physics “from top down”, i.e. with the tightest
o galaxiestolargerdistances, microwaveexperimentshave
constraints on high energy physics processes. Searching
r provided accurate cosmic microwave anisotropy (CMB)
t for cosmic strings in the sky is thus an approach to test
s temperature maps, and promise to give us CMB polar-
a particlephysicswhichiscomplementarytoterrestrialac-
izationmapsinthenearfuture. Therearealsoprototype
[ celerator experiments which test models “from bottom
21cmtelescopeswhichpromisetoyieldthree-dimensional
1 intensity maps of the distribution of neutral hydrogen. up”,i.e. wherethetightestconstraintsareonlowenergy
v This data is giving us precise information about the in- scales (see [6] for a more technical recent review of this
9 topic).
homogeneities in the universe.
1 In the following section, the basics of cosmic strings
As has been realized for some time, the seeds for the
6
will be reviewed. Section 3 introduces the main effects
4 current structure of the universe had to have been laid
by which strings lead to cosmological signatures. Sec-
. down in the very early universe. Inflationary cosmol-
1 tions 4 - 7 then focus on specific observational windows:
ogy [1] is one paradigm for generating structure in the
0 large-scalestructure,CMBtemperatureandpolarization
4 universe, and alternatives such as string gas cosmology
maps, and 21cm redshift surveys.
1 [2], the Ekpyrotic scenario [3] or the “matter bounce”
A word on notation: units in which the speed of light,
: scenario [4] exist. These scenarios all involve physics at
v Boltzmann’s constant and Planck’s constant are set to 1
energy scales much larger than those currently explored
Xi interrestrialacceleratorexperiments,scaleswhichinfact willbeused. Themetricofthebackgroundhomogeneous
and isotropic space-time is
r could approach thePlanck and superstringscales. Thus,
a
itisnaturaltoaskwhethercosmologicaldatacanbeused ds2 =dt2−a(t)2dx2, (1)
to explore particle physics beyond the Standard Model.
Thisarticlewillsurveyonewayofprobingtheconnec- where t is cosmic time, x are comoving spatial coordi-
tion between particle physics and cosmology. It is based nates(coordinatespaintedontotheexpandingspace,co-
onthefactthatinalargeclassofparticlephysicstheories ordinates which are constant for particles at rest in the
beyond the Standard Model there are topologically sta- expanding universe), and a(t) is the scale factor which
ble field configurations called cosmic strings which are describes the expansion of space. A crucial length scale
predicted to form in the early universe and which sur- when considering the formation of structure in the uni-
vive until the present time [5]. They correspond to lines verse is the Hubble radius, the inverse expansion rate.
in space with trapped energy density. The gravitational Onsub-Hubblescales,fluctuationstypicallyoscillatelike
effects of this energy leads to distinct cosmological sig- theydoinMinkowskispace-time,onsuper-Hubblescales
they are frozen in and squeezed (see e.g. [7] for a com-
prehensive review of the theory of cosmological pertur-
bations and [8] for an introductory overview). As a final
∗Electronicaddress: [e-mail] point, we remark that it is common in cosmology to la-
2
beltimetbycosmologicalredshiftz(t), ameasureofthe on that scale, then the field vector φ will circle the point
expansion of space between time t and the present time. φ = 0. By continuity, it then follows that there needs
The relation is to be at least one point on any disk bounded by C with
φ = 0. These points connect to a line, and this is the
a(t )
z(t)+1= 0 , (2) center of the cosmic string. The energy density µ of the
a(t)
stringisobtainedbyminimizingthesumofpotentialand
where t is the present time. gradient energies and is given by
0
µ(cid:39)η2, (4)
II. COSMIC STRING BASICS
wherethedependenceonthecouplingconstantλcancels
out. Astheobservantreaderwillhavenoticed,theabove
According to quantum field theory, our best descrip-
argumentisbasedoncausalityandholdsatalltempera-
tion of matter at high energies, matter is described in
turesbelowthecriticaltemperature,andthusatalltimes
terms of fields which live in space and evolve in time.
up to the present time. Thus, and this is the second as-
A particle corresponds to a localized point-like fluctua-
pectoftheKibblemechanism,inparticlephysicsmodels
tion of the corresponding field, in the same way that a
whichadmitcosmicstringsolutions,anetworkofstrings
phonon is a localized point-like excitation in a crystal.
will persist until the present time.
Somecrystalsadmitthepossibilityofstabledefectlines,
The detailed form of the distribution of the cosmic
lines at which the orientation of the atoms in the crys-
string network must be studied by means of numerical
tal changes abruptly. In a similar way, certain quantum
simulations (see e.g. [10] for a selection of references on
field theories allow the existence of stable field configu-
such simulations). However, the qualitative aspects can
rations with linear defects. Such defect lines are called
be derived using analytical considerations (see e.g. [11–
cosmic strings. In the same way that not all crystalline
13]forearlyreviewsonthecosmologyofcosmicstrings).
materials admit defect lines, not all field theories admit
First of all, note that cosmic strings cannot have ends.
cosmicstrings. Forexample, theStandardModelofpar-
Hence, they are either part of an infinite string network,
ticle physics does not admit string configurations which
or else there are loops. The infinite string network is de-
are stable in the vacuum. However, many extensions of
scribedbyacorrelationlengthξ(t)whichdependsoncos-
the Standard Model do admit such defects.
mic time t and gives the mean separation and mean cur-
In the same way that in condensed matter systems
vatureradiusofthelongstrings(inthe“onescale”model
which admit defect solutions, such solutions inevitably
of the string network, these two lengths are the same).
form during a fast crystallization process, in a quantum
Due to the tension, a curved string will typically have
field theory model which admits cosmic string solutions,
velocity in direction perpendicular to the string whose
anetworkofsuchdefectlinesinevitablyformsduringthe
magnitudev isasignificantfractionofthespeedoflight.
cooling process in the early universe. This is the famous
As a consequence of the induced motion, there will be
Kibble mechanism [9].
frequent string intersections which both produce string
In the simplest models which yield cosmic strings, the
loopsandleadtoanincreaseinξ(t). Ontheotherhand,
field configuration can be visualized as a vector of fixed
by causality ξ(t) is bounded from above by the horizon
length η which can point in any direction in a two-
distancet(weareusingunitsinwhichthespeedoflightis
dimensional plane. For example, consider a complex
setequalto1). Thedynamicscanbestudiedbymeansof
scalar field φ with potential energy given by
a Boltzmann equation (see e.g. [13]) which in fact yields
V(φ)=λ(cid:0)|φ|2−η2(cid:1)2, (3)
ξ(t)∼t, (5)
where η is the characteristic energy scale of the field
theory and λ is a positive number (the so-called “self- where the proportionality constant must be determined
coupling constant”). In the high temperature plasma of numerically. It is found that a horizon volume has on
theveryearlyuniverse,thescalarfieldwillhavesufficient theaverageN longstringspassingthroughit,where1<
thermal energy to cross the potential energy barrier at N <10 based on simulations by different groups.
φ=0. However,whenthetemperaturefallsbelowacrit- To summarize, it follows from analytical arguments
ical temperature T which is given by the energy scale η, that the cosmic string network will at times much later
c
thefieldnolongerhasenoughenergy. Wesaythatatthe than the time of the phase transition take on a “scaling
temperature T a “phase transition” takes place. Below solution” according to which the statistical properties of
c
T . then in order to minimize the energy, the field would thestringnetworkaretime-independentifalllengthsare
c
then like to be in a ground state with |φ|=η. However, scaledtothehorizondistancet(seeFigure1foranillus-
by causality the orientation of the φ vector in the com- tration of the scaling solution).
plex plane must be random on scales beyond which no The cosmic string loops surviving at late times are
information has had time to propagate since the time of theremnantsoftheinteractionsbetweenlongstringseg-
the Big Bang. Hence, on such scales there is a probabil- ments. Theirdistributionalsoscales,andischaracterized
ity of order one that if one considers a circle C in space by a maximal radius αt, where α is a positive number of
3
Since in theories which admit cosmic strings a net-
workofsuchstringssurvivesuntilthepresenttime,these
strings will have observational consequences which can
be searched for in observations (this was first realized in
[5, 18, 19]). This will be the topic of the rest of this
review.
III. MAIN EFFECTS
The network of cosmic strings at late times contains
stringloopsandnetworkofinfinitestringswithacorrela-
tionlengthξ(t)-wewillcallthelatterthe“longstrings”.
Both loops and long strings lead to gravitational effects.
Since, according to the current cosmic string evolution
simulationsmostoftheenergyofthestringnetworkisin
thestringloops,theloopswilldominatethecosmological
effects. However, the long strings lead to more charac-
teristic signatures and hence might be easier to detect in
the sky. An important message of the following sections
isthatitisbettertolookforthesignalofstringsinposi-
tion space rather than in Fourier space. The first reason
is that in Fourier space the specific signals of the strings
arewashedout. Thesecondreasonisthat(Fourierspace)
powerspectratypicallydependquadraticallyonthecon-
stant N which parametrizes our ignorance of the exact
natureofthescalingsolution,whereasthepositionspace
signals in the idealized case are independent on N.
A. Effects of string loops
FIG.1: Sketchofthecosmicstringscalingsolution. Whatis
shown is the distribution of strings in a fixed Hubble volume
A string loop will oscillate and slowly emit gravita-
(projected onto a plane).
tional radiation. Viewed from a distance larger than the
loop radius, the time-averaged gravitational effect of the
the order but somewhat smaller than 1. As a function of loop is like that of a point mass of magnitude
radius R, the number density n(R,t) of loops scales as
M(R)=βµR, (7)
R−5/2 for loops formed before the time of equal matter
and radiation (see e.g. [14]). String loops oscillate and
where the deviation of the number β from 2π describes
emit gravitational radiation [15]. This effect leads to the
the difference of the loop shape from being circular.
fact that the number density n(R,t) becomes constant
Hence, cosmic string loops will be the seeds of roughly
below a cutoff radius given by
spherical accretion (for simplicity we here neglect the
initial center of mass motion of the loops). Before the
R =γGµt, (6)
c
time t of equal matter and radiation, the accretion of
eq
whereµisthestringtension,GisNewton’sgravitational baryons and dark matter is suppressed. Thus, the main
constant and γ is a real number whose value is deter- growth starts at t .
eq
mined in numerical simulations to be about 100. As long as there is enough matter to feed the growth,
Since a cosmic string arises in a relativistic quantum thegrowthaboutstringloopswillproceedindependently.
field theory, its mass per unit length is equal in mag- Loops of different radii R will lead to objects of different
nitude to the tension. The gravitational effects of a final masses M (R)
f
string are hence proportional to µ. They are typically
z
parametrized by the dimensionless constant Gµ whose Mf(R)(cid:39)βµRzeq , (8)
valueisintherange10−8−10−6forcosmicstringsformed to
in“GrandUnified”modelsofparticlephysics,andwhich wherethelastfactordescribesthegravitationalaccretion
are predicted to be in the range 10−12−10−6 for cosmic from time t (with corresponding redshift z ) until the
eq eq
superstrings [16], fundamental strings which can be sta- redshift z when accretion stops, e.g. the present time
to
bilized as macroscopic objects in certain string models forsmallvaluesofGµandforloopssufficientlydisplaced
[17]. from any larger loop.
4
Theinitialworkoncosmicstringsandstructureforma- tributetotheformationofultra-compactmini-halosem-
tion[14]mostlyfocusedontheeffectsofstringloops,and bedded within a galaxy halo [33]. For a more general
made the hypothesis that cosmic strings were responsi- recent discussion of the role of cosmic string loops see
ble for all of the inhomogeneities and anisotropies in the e.g. [34]. However, in this article we will focus on sig-
universe, and thus would be an alternative to cosmic in- natures of long strings since their geometric patterns are
flationasanexplanationfortheoriginofstructure. Asa more distinctive.
consequence of the scaling distribution of cosmic strings,
thespectrumofdensityfluctuationsfromstringsisscale-
invariant [20]. For a value of Gµ (cid:39) 10−6 the spectrum B. Kaiser-Stebbins effect
would have the right overall amplitude to explain the
large-scale structure (LSS) data. Becauseofthefactthatacosmicstringhasrelativistic
However, thefluctuationsstartoutasentropyfluctua- tension (tension equal in magnitude to the energy den-
tions on super-Hubble scales, rather than adiabatic per- sity), the gravitational effects of long straight strings are
turbations in the case of inflation. They are incoherent very special. There is in fact no local gravitational force
andactive(i.e. theinducedadiabaticcomponentisgrow- exerted by the string. On the other hand, globally space
ing on super-Hubble scales). As a consequence of these is non-trivial - space perpendicular to a cosmic string
facts, the CMB anisotropies on small angular scales are is conical with a “deficit angle” α of magnitude propor-
predicted to be different from the results for primordial tional to the tension [35]
adiabatic fluctuations: there are no acoustic oscillations
α=8πGµ. (10)
in the CMB angular power spectrum [21], but only one
broad Doppler peak. The conical structure of space perpendicular to a long
Once the acoustic oscillations in the CMB angular string leads to lensing of light passing on different sides
power spectrum were discovered by the Boomerang [22] of the string. This, in turn, gives rise to a specific signa-
experimentandlaterconfirmedbydatafromtheWMAP ture of long straight strings in CMB temperature maps
satellite mission [23] it became clear that cosmic strings [36,37]: considerasourceoflightbehindastringmoving
couldnotaccountforallofthestructureintheuniverse. with velocity v in the plane perpendicular to the string,
s
This leads to an upper bound on the cosmic string ten- fromthepointofviewofanobserveratrestintheframe
sion. The most robust current bound on the string ten- of the cosmological background. Light from this source
sioncomesfromtherecentCMBangularpowerspectrum will reach the observer on two different paths which pass
measurements, in particular the South Pole Telescope on different sides of a string. There will be a relative
(SPT) [24], the Atacama Cosmology Telescope (ACT) Doppler shift in the frequency which the observer sees.
[25]andthePlanckmission[26]. Theboundonthestring Specifically, this effect applies to the CMB background
tension from a combination of these experiments is [27] radiation. Hence, a string will produce a line in the sky
(see [28] for earlier limits) acrosswhichthereisarelativetemperaturejumpofmag-
nitude
Gµ<1.5×10−7, (9)
δT
= 8πγ(v )v Gµ, (11)
where the coefficient in fact depends on the constant N T s s
describing the string scaling solution (which has been whereγ istherelativisticgammafactorassociatedwith
s
mentioned earlier). This implies that cosmic strings can thevelocityv . ForanillustrationofthiseffectseeFigure
s
be responsible for at most about 5% of the power of 2.
density fluctuations in the current universe. The dom- For cosmic strings formed during a phase transition in
inant source of inhomogeneities must be a different pro- the early universe, the conical structure of space extends
cess such as inflation [29] or one of its alternatives (see only a finite distance away from the string. As derived
e.g. [30] for a recent comparison between inflation and in [38], this distance is the time that light can travel
some of its alternatives). between string formation and the time t which is being
Fromtheparticlephysicspointofview,however,there considered. Beyondthatdistance,spaceperpendicularto
are good reasons to expect cosmic strings to be present, the string rapidly converges to being Euclidean. Hence,
even if the universe underwent a period of inflation. In a string segment leads to a temperature surplus on one
many supergravity models of inflation, strings are pro- side of the string and a deficit on the other extending a
duced at the end of inflation [31], and in brane inflation distance corresponding to the comoving horizon at the
models motivated by superstring theory, similarly a net- time the light is passing by the string.
workofstringsistheresultoftheterminationofinflation
[32]. In string gas cosmology, a network of cosmic su-
perstrings may survive. Hence, looking for cosmological C. Cosmic string wakes
signals of strings provides a good way to probe particle
physics beyond the Standard Model. The conical structure of space perpendicular to a long
Cosmic string loops will accrete matter in a roughly straight string is also responsible for the second impor-
spherical manner. String loops could, for example, con- tant effect, the “cosmic string wake”. We consider the
5
c t
1 i
v
4!Gµtivs"s
tivs"s
FIG. 3: The geometry of a cosmic string wake.
rections following the expansion of space. A long string
presentattimet (takentobelargerthant )willcreate
i eq
a wake whose physical dimensions at a later time t will
be
a(t) a(t) a(t)
c t ×v γ t ×4πGµγ v t [ ]2, (14)
t ia(t ) s s a(t ) s s i a(t )
i i i
where the extra factor of a(t) in the third term comes
a(ti)
fromthegrowthofthewidthviagravitationalaccretion.
The magnitude of this factor is easy to understand: it
represents the linear cosmological perturbation theory
growth of fluctuations. An improved derivation is based
on the Zel’dovich approximation [40]. The thickness is
definedasthephysicalradiusofthemassshellwhichhas
FIG. 2: Geometry of space perpendicular to a long straight
string segment. Flattened out, there is a deficit angle whose just “turned around” and is beginning to collapse onto
magnitude is proportional to Gµ. A string moving with ve- thecentralplaneofthewake(initiallyeverymassshellof
locity v in transverse direction leads to a relative Doppler particles above the wake is moving away from the wake
s
shiftbetweenphotonspassingondifferentsidesofthestring. because of cosmological expansion. The Zel’dovich ap-
proximation has been applied to cosmic string wakes in
[41].
samestringsegmentmovingwithvelocityv indirection
s A key point is that cosmic string wakes correspond
perpendiculartoitstangentvector. Itismovingthrough
to nonlinear structures present already at the earliest
the cosmic gas. From the point of view of an observer
times. This is in contrast to what happens in theories
behind the string, a velocity perturbation of the gas to-
withGaussianfluctuationssuchasthoseproducedbyin-
wards the ”world plane” of the moving string (the plane
flation. In those theories fluctuations remain in the lin-
spanned by the tangent vector to the string and the ve-
ear regime until times close to the epoch of reionization
locity vector v ) is induced whose magnitude is
s at redshifts of about 10. Hence, observational windows
which probe nonlinear structures at high redshifts are
δv =4πGµv . (12)
s
more likely to find signatures of cosmic strings than sur-
Hence, a wedge-shape region of overdensity 2 behind the veys which probe cosmological structures at the present
string is induced, the “wake” [39]. The wedge is a three- time.
dimensionalobject. Astringsegmentoflengthc t,where
1
c is a constant, induces a wake of dimensions
1
IV. COSMIC STRINGS AND LARGE-SCALE
c t×v γ t×4πGµγ v t, (13)
1 s s s s STRUCTURE
wheretheseconddimensionisthedepthandthethirdis
themeanwidth, thewidthbeingzeroatthepointwhere In this section we will discuss signatures of cosmic
thestringislocated,andtwicetheabovevalueatthefar strings in the large-scale structure observed through op-
end (see Fig. 3). ticalandinfraredtelescopes. Wewillfocusontheroleof
Once formed, a wake will grow in thickness by grav- theinfinitestringnetworksincethelongstringsgiverise
itational accretion, and it will expand in the planar di- to more distinctive geometrical patterns.
6
The long strings form a network of infinite extent.
However, for analytical studies it is convenient to break
up this network into a set of segments, each of length
ξ(t). As the infinite strings self-intersect and split off
loops, thesetofstringsegmentschanges. Thetimescale
forstringintersectionsistheHubbleexpansiontimescale
t. Hence, string segments can be viewed as statistically
independentontimescaleslargerthant. Thisideagives
risetoatoymodel[42]forthedistributionoflongstrings:
we divide the time period under consideration (usually
starting either at t or else at the time t of recom-
eq rec
bination and ending at the present time t ) into Hubble
0
time steps. In each time step (with initial time t , where
i
i is the index of the time step), we take a fixed number
N of straight string segments of length ξ(t), and moving
with velocity v . The centers, orientations, and velocity
s
unit vectors are chosen at random.
Each string segment present at time t will produce a
i
wakewhosedimensionsatalatertimetaregivenby(14).
The wakes will thus imprint a planar topology to the
distribution of dark matter in the universe. In the “old
days” of cosmic string cosmology the effects of wakes on
the large-scale structure of visible matter was analyzed
in many works, e.g. in [43]. Since there is only loga-
rithmic growth of the density fluctuations before t , the
eq
time interval to consider for studies of large-scale struc-
ture is the interval between t and the present time. As
eq
a consequence of the scaling distribution of the string
network, the overall power spectrum of induced density
fluctuations is scale-invariant [14]. The specific signa-
tures of cosmic strings cannot be seen in the two point
FIG. 4: The temperature pattern on the sky induced by a
correlation function, but only in non-Gaussian statistics.
single string segment.
A better approach than naively computing three- and
four-point correlation functions is to look at topological
statistics such as Minkowski functionals [44]. This has
Infact, duetothefinitedepthoftheconicalgeometry
been done in some early studies in [45].
about the string, a string segment whose world sheet is
For the small values of Gµ which are now allowed by
crossed by our past light cone at time t leads to a rect-
i
the bounds on the cosmic string tension, it was recently
angular pattern in the sky with δT (cid:54)=0. In the direction
shown that halos which are induced by the accretion of
on the sky corresponding to the string motion, there is a
matterontowakesaretoosmallandhavetoolowatem-
rectangle of depth
perature to induce star formation. Hence, they will be
difficulttodetectdirectly[46],inspiteofthefactthatat 1
θ v γ (15)
higher redshifts the power spectrum of nonlinear matter 2 i s s
becomesdominatedbythestringwakes. Amoredetailed
study of this issue is in preparation [47]. in front of the string with a positive temperature fluc-
tuations δT (given by (11), whereas behind the string
there isa rectangle ofsimilar size thea negative temper-
aturefluctuationofequalmagnitude. Thewidthofthese
V. CMB TEMPERATURE ANISOTROPIES
rectangles in direction tangent to the string is given by
FROM STRINGS
c θ , (16)
1 i
Thecosmicmicrowavebackground(CMB)providesan
image of the distribution of matter at the time trec of where θi is the angle corresponding to the comoving Hu-
recombination, a time when the fluctuations are in the ubble radius at the time t , and is given by
i
linear regime and precise analytical calculations are pos-
sible. AsalreadymentionedinSection3,longstringseg- θ =90o[z(t )+1]−1/2. (17)
i i
mentsleadtoacleargeometricalsignatureinCMBtem-
perature anisotropy maps: a line in space across which A sketch of the pattern of an individual string segment
the temperature jumps by (11). is given in Figure 4.
7
obtainable. InitialstudiesofCMBmapsusingMinkowski
functionals have been done in [49], and an application of
the Canny edge detection algorithm [50] to constructed
maps including the contributions from string wakes has
shown [48, 51] that an improvement of the limit by one
orderofmagnitudemightbeachievablewithcurrentSPT
data.
VI. CMB POLARIZATION PATTERNS FROM A
COSMIC STRING WAKE
CMB radiation can be polarized. The polarization
data of CMB anisotropies carries interesting cosmolog-
ical information. Polarization can be decomposed into
two independent components which are called E-mode
andB-modepolarization. TheprimordialE-modepolar-
ization has already been discovered [52], but only upper
bounds are available on the B-mode. In the near future,
experiments such as the SPT [53], ACT [54] and Planck
willprovideuswithgreatlyimprovedpolarizationmaps.
In this context it is interesting that cosmic strings lead
to a specific pattern of CMB polarization, in particular
B-mode polarization.
Regular adiabatic fluctuations such as those produced
during inflation do not produce any direct B-mode po-
larization. B-mode polarization in such models can be
generated from gravitational waves, or from the lensing
of E-mode polarization. In contrast, cosmic strings lead
todirectB-modepolarizationaswasfirstrealizedin[55].
The position space signal of B-mode polarization from a
FIG.5: CMBanisotropymapfora10o×10o patchofthesky
string was studied in detail in [56].
(cid:48)
at 1.5 resolution (the specifications are chosen with applica-
CMB polarization is induced by the CMB photons
tionstotheSPTandACTtelescopedatainmind)inamodel
from last scattering crossing a cosmic string wake. Only
in which the fluctuations are given by a scaling distribution
thequadrupolecomponentQoftheCMBinducespolar-
of cosmic strings. The color coding indicates the amplitude
ization. The wake is an overdensity of matter, including
of the temperature anisotropy.
baryonicmatter. Hence,italsorepresentsanoverdensity
offreeelectrons. TheCMBphotonsComptonscatteroff
ofthefreeelectrons,andthisprocessinducesCMBpolar-
A scaling network of string segments produces a CMB
ization. Each wake which is intersected by our past light
temperature anisotropy map which can be constructed
cone leads to a rectangle in the sky with extra CMB po-
as follows [48]: given a patch of the sky which is be-
larization. The angular dimensions of this wake-induced
ingmodelled,wecomputethenumberofstringsegments
rectangle for a wake created at time t are
present at each Hubble expansion time t which are in- i
i
tersected by the portion of the past light cone which the [c ×v γ ]θ2 (18)
1 s s i
patch of the sky corresponds to. We then choose ran-
(as long as the time t when the past light cone intersects
domcenters,directionsandvelocityvectorsforeachseg-
the wake is significantly smaller than t - for the gen-
ment and construct the resulting temperature rectangle 0
eral formula the reader is referred in [57]). Across the
intheskygivenbytheabove-mentioneddimensions. The
rectangleinthesky, thepolarizationdirectionisroughly
CMB map is a superposition of the rectangles given by
constant. The magnitude P is proportional to Q, to the
theindividualstringsegment. Figure5(takenfrom[48])
Thompsoncrosssection,tothewidthofthewake(which
presents an image corresponding to a 10o×10o patch of
in turn depends on Gµ and on the times t and t), to
thesky. Notethelargenumberofsmallrectangles(early i
the free electron fraction f(t) at the time t, and to the
t ) compared to the larger ones (late t ).
i i overall baryon energy fraction Ω . Inserting numbers,
The best current limits on the cosmic string tension B
the result is [56]
come from comparison of the angular power spectrum
of CMB temperature anisotropies [27]. By making use P(t,t )∼f(t)Gµv γ Ω (cid:0)z(t)+1(cid:1)2(cid:0)z(ti)+1(cid:1)1/2107.
of topological statistics such as Minkowski functionals Q i s s B 103 103
or edge detection algorithms, improved limits should be (19)
8
VII. IMPRINTS OF COSMIC STRINGS IN
21CM SURVEYS
Thefinalwindowtoprobeforthepossibleexistenceof
cosmicstringswhichwediscussinthisarticleisthewin-
dow which 21cm redshift surveys provide. These surveys
provide, in addition to many other purposes, a means of
probing the distribution of baryonic matter during the
dark ages, i.e. before the time of star formation. Since
21cm redshift maps are three-dimensional (two dimen-
sions the angles in the sky, the third dimension the red-
shift), they provide potentially much more information
than two dimensional CMB maps.
Beforestarformation(butafterrecombination), bary-
onic matter is almost entirely in the form of neutral hy-
drogen. Neutral hydrogen has a hyperfine transition line
with a wavelength of 21cm. If photons of the cosmic
microwave background travel through a cloud of neutral
FIG.6: Sketchofthepolarizationpatterninducedbyacosmic hydrogen, there will be either absorption or emission of
string wake. Shown is a portion of the sky, e.g. 10o×10o, 21cm radiation - absorption if the CMB temperature is
and the polarization which the wake induces. The direction larger than the gas temperature in the cloud, emission if
of the arrows indicates the polarization direction, the length the gas temperature is larger. Since cosmic string wakes
of the arrow gives the magnitude. are overdensities of neutral hydrogen, they will lead to a
larger 21cm signal.
The 21cm signal of a single string wake was discussed
The above is the mean amplitude across the rectangle. in[60](see[61]forinitialstudiesoftheimprintsofcosmic
The amplitude increases linearly moving away from the stringsinthe21cmbackground,andsee[62]forageneral
position of the string. The pattern is sketched in Fig. 6 overview of the physics of 21cm surveys). The geometry
(taken from [56]). of the string-induced pattern is very special: extended
The important point is that, as discussed in detail in in the two angular directions and thin in redshift direc-
[57], the mean amplitude of the induced E and B mode tion. The geometry is sketched in Figure 7 (taken from
polarizations are the same. Cosmic string wakes thus in- [60]). Ontheleftsideofthefigureisaspace-timesketch
duce directly B mode polarization, unlike standard adi- showing the location of the string wake as a function of
abatic fluctuations. Hence, the search for B mode polar- time, andthepastlightconeofourobserverintersecting
izationisaveryimportantcurrentgoalforobservational the wake. On the right side of the figure is the corre-
cosmology. Thediscoveryofsuchpolarizationcouldgive sponding signal in redshift space. The horizonal axes on
us a lot of very important information about the early both sides of the figure are the same. The vertical axis
universe. Itcouldconfirmtheexistenceofcosmicstrings is time on the left graph and redshift on the right graph.
(this is the point emphasized in this article). If any dis- The signal of the string wake is a thin wedge, roughly
covered B mode polarization could be shown to be due but not quite perpendicular to the redshift axis. Within
to gravitational waves, it could shed light on the origin the region of the wedge there is extra 21cm absorption
of primordial gravitational waves. A roughly scale in- (or emission if Gµ were to be large). The mean thick-
variant spectrum of such waves is predicted not only by ness of the wedge depends on Gµ since it is determined
inflationary cosmology, but also by a scaling solution of by the width of the wake. On the other hand, it turns
cosmicstringloops, andinfactwithanamplitudelarger outthetheamplitudeofthesignalisindependentofGµ.
than what is predicted in the simplest inflationary mod- Thus,ifatelescopewithexcellentredshiftresolutionwere
els. A discovery of a gravitational wave spectrum with available, the 21cm window would be an excellent one to
a slight blue tilt would rule out all inflationary models search for lower tension strings.
basedGeneralRelativitywithmattersatisfyingtheusual The amplitude of the effect of the string wake is given
energy conditions, and it would [58] verify a key predic- by
tion of string gas cosmology (see [59] for a more detailed T (ν) = T (cid:0)1−e−τν(cid:1)+T (ν)e−τν , (20)
discussion of these points). b S γ
Theangularpowerspectrumofstringwake-inducedB- where T is the “spin temperature” of the hydrogen
S
mode CMB polarization was computed recently in [57]. atoms in the gas cloud, and T is the CMB tempera-
γ
The spectrum is almost degenerate with that of B-mode ture, τ is the dimensionless optical depth, and ν is the
ν
polarizationinducedbylensing. Thisisanotherexample frequency. The second term describes the absorption of
ofthemessagethatinordertodetectsignalsfromstring the primordial CMB photons, the first term yields the
wakes, one needs to analyze data in position space. emission.
9
t zi￿3000Redto1000Blue
t ! ∆Tbze￿20 K
0
# 0.01
￿ ￿￿ ￿
s
1
!2 "! 0.05 0.10 0.15 0.20 0.25 GΜ6
s
2
2ti !1 ￿0.01
t
i
xc x 2 x 1 x2 x1 x c ￿0.02
￿0.03
FIG.7: Geometryofthe21cmsignalofacosmicstringwake.
The left side shows a space-time sketch with the horizontal
FIG. 8: Pixel amplitude of the string wake signal (vertical
axis being comoving spatial coordinate and the vertical axis
axis) as a function of Gµ (horizontal axis) for various values
conformal time) of a wake created at time t . The string
i ofthewakeformationredshiftz ,forthepastlightconecross-
i
segment “lives” until the time 2t . Its initial and final po-
i ing the wake at redshift 20, just before rionization. For very
sitions are x and x , respectively. The wake extends from
1 2 large values of Gµ the kinetic temperature of the baryons in
x to x . Its thickness increases linearly from x to x . The
1 2 1 2 the wake is larger than the CMB temperature and the string
past light cone (indicated by the line labelled γ) intersects
signalisinemission. ForsmallervaluesofGµthesignalisin
the string wake. In the example shown, the past light cone
absorption. The string signal would asymptote to a constant
intersects the leading (thin) edge of the wake earlier than
amplitude were it not for the effect of diffusion. This leads
the trailing edge. Thus, 21cm photons from the leading edge
to a decrease of the string wake signal for very low values of
are redshifted more than those from the trailing edge. This
the tension. As the figure shows, the string signal remains
generates a characteristic wedge of extra 21cm absorption /
larger than the noise (the almost horizontal lines) to values
emission due to the string wake in redshift maps. This is
ofGµoneorderofmagnitudesmallerthanthecurrentupper
sketchedontheleftofthefigure. Here,thehorizontalaxisis
bound.
the same comoving spatial coordinate as in the left side, and
the vertical axis is the detected 21cm frequency.
Minkowskifunctionalsforthispurposehasbeenexplored
in [63]. One could also apply edge detection algorithms.
The spin temperature determines the excitation level
Thecalculationssummarizedaboveweredoneneglect-
of the hydrogen atoms and is related to the gas temper-
ingthethermalvelocitiesofthebaryonsbeforetheystart
ature T in the wake via a collision coefficient x (see
K c
falling into the wake. At early redshifts and for low val-
[62] for the exact equation). We are interested in the
ues of Gµ this is clearly not justified. Including these
difference δT between (20) and the CMB temperature.
b
initial thermal velocities will lead to the wake being dif-
redshifted to the present time:
fuse. This will increase the width of the wake (we speak
T (ν)−T (ν) ofa“diffusestringwake”), butwilldecreasethepixelby
δT (ν) = b γ , (21)
b 1+z pixel amplitude. The effect at a point in the sky inte-
grated over redshift remains the same, as was studied in
which yields [64].
What range of values of Gµ might be visible in 21cm
δT (ν) = T xc (cid:0)1− Tγ (cid:1)τ (1+z)−1. (22) surveys? A conservative criterion for visibility is that
b S1+x T ν
c K the pixel signal due to a string wake be larger than the
expected pixel noise. The study of [64] indicates that
The optical depth is proportional to the width of the
values of Gµ one order of magnitude smaller than the
wake and to the line profile, the broadening of the line
current limit might be visible. This is depicted in Figure
due to the varying time of travel of photons emitted at
8 (taken from [64]) which shows the string wave signal
different points in the wake. The line profile is inverse
as a function of Gµ for various values in the redshift z
proportionaltothewidth. Hence, theeffectofthewidth i
whenthewakeislaiddown,andforafixedreshiftz =20
(andhenceofGµ)cancelsoutoftheamplitude. Inserting
when our past light cone intersects the wake, compared
numbers, one obtains
to the noise (the almost horizontal lines to the top and
δT (ν) ∼200mK for , z+1=30, (23) bottom of the horizontal axis). Note that the noise is a
b
increasingfunctionasthepixelsizedecreases. Lowvalues
a very large effect (for details the reader is referred to ofGµrequiresmallerpixelsizesandhencethenoiselevel
[60]). will be higher. Making use of algorithms sensitive to the
Since string wakes produce such a distinctive geomet- particular geometrical pattern of the string wake signal
ricalpattern, oneshouldsearchforstringsmakinguseof will allow us to probe significantly smaller values of Gµ.
statistics which are sensitive to this pattern. The use of We close this section with a couple of final comments.
10
First, the amplitude of the 21cm signal from a cosmic are topologically stable and non-superconducting. Two
string loop is, as was studied in [65] larger by a factor interesting avenues of extension are to consider super-
of 16 compared to the amplitude of the wake signal, the conducting strings [67] for which electromagnetic inter-
reasonbeingthattheoverdensityislargerbythatfactor actions are important and will lead to a rather different
(contraction occurs in three instead of just one dimen- cosmologicalscenario(seee.g. [68]forareview)andnon-
sion). On the other hand, there is no distincitive geo- topological strings.
metricalsignatureoftheloopsignal,andhenceitwillbe
Non-topologicalstrings(see[69]forareview)arestring
more difficult to tease apart the loop signal from point
solutions which are unstable in the vacuum. However,
sourcenoise. Second,thetwo-dimensionalangularpower
as first discussed in [70] and more recently in [71] such
spectrum of the string wake signal has also been calcu-
strings can be stabilized in a plasma. In fact, even
lated [66]. However, the result does not show specific
the Standard Model of particle physics admits strings
stringy features.
(namely the electroweak Z-string and the pion string of
the low-energy sigma model of QCD) which can be sta-
bilized by the electromagnetic plasma present before the
VIII. CONCLUSIONS
timeofrecombination. Inthiscase,theywillforminthe
early universe like topological strings, and they can play
Since the seeds of the cosmological fluctuations which
a similar cosmological role.
we observe today have been laid down in the very early
universe, these fluctuations may carry imprints of the Strings are not the only type of defect solutions. We
physics at the earliest times, and this means the physics have focused on string solutions since they are the cos-
athighestenergyscales. Asanexample,wehavestudied mologically most interesting ones. Models with domain
how some observational windows can be used to probe wallsareruledoutiftheenergyscaleisbeyondthescale
for the possible existence of cosmic strings produced by of the LHC since a single domain wall would overclose
particle physics theories beyond the Standard Model. the universe [72]. Gauge monopoles are also ruled out
One of the important conclusions is that the signals by overclosure considerations for high energy scale walls
of strings are more prominent in position space than in [73].
Fourier space. Hence, to find or constrain strings it is
Finally, it is important to mention that all our studies
important to use good position space statistics (some
ofcosmicstringsignalshavebeeninthecontextofaone-
attempts were described in the text). For CMB tem-
scale toy model for the distribution of strings. It would
perature anisotropy maps the distinctive signal of string
beinterestingtostudythestringsignalsusingtheinputs
wakes is line discontinuities, for CMB polarization maps
of actual string evolution simulations. There are clearly
it is rectangles on the sky with a polarization amplitude
lots of interesting projects to be undertaken.
which increases linearly in one direction, and whose po-
larization angle is roughly constant. In particular, there
Acknowledgments
is direct B-mode polarization. In the case of 21cm red-
shift maps, string wakes lead to wedges which are ex- IwishtothankProf. PauchyHwangfortheinvitation
tended across the angular directions but thin along the to contribute to one of the first issues of “Universe”. I
redshift axis. The amplitude of the position space sig- wish to thank Rebecca Danos for permission to use vari-
nalsistypicallyindependentoftheunknownconstantN ousfiguresdrawnfrom[48,56,60],andtoOscarHernan-
which describes the string scaling solution, except when dez for permission to use Figure 8 which is taken from
overlapsontheskyofthesignalsofdifferentstringwakes [64]. This research has been supported in part by an
are important. NSERC Discovery Grant and by funds from the Canada
We have assumed here, for simplicity, that the strings Research Chair program.
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