The Flow Around a Cosmic String, Part I: Hydrodynamic Solution

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by  Andrey Beresnyak

Draftversion January19,2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE FLOW AROUND A COSMIC STRING, PART I: HYDRODYNAMIC SOLUTION
Andrey Beresnyak
LosAlamosNationalLaboratory,LosAlamos,NM,87545and
Nordita,KTHRoyalInstitute ofTechnology andStockholm University,SE-10691
Draft version January 19, 2015
ABSTRACT
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Cosmicstringsarelineartopologicaldefectswhicharehypothesizedtobeproducedduringinflation.
1
Mostsearchesforstrings havebeen relying onthe string’slensing of backgroundgalaxiesorCMB. In
0
2 thispaperIderiveasolutionforthe supersonicflowofthecollisionalgaspastthecosmicstringwhich
hastwo planarshockswith shockcompressionratiothatdepend onthe angledefect ofthe stringand
n
its speed. The shocks result in compression and heating of the gas and, given favorable condition,
a
particle acceleration. The gas heating and overdensity in an unusual wedge shape can be detected
J
by observing HI line at high redshifts. The particle acceleration can occur in present-day Universe
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when the string crosses the hot gas contained in galaxy clusters and, since the consequences of such
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collision persist for cosmological timescales, could be located by looking at the unusual large-scale

radio sources situated on a single spatial plane.
]
O Subject headings: cosmology: theory—hydrodynamics—shock waves: acceleration of particles—radio
continuum: general
C
.
h
1. INTRODUCTION an exact hydrodynamic solution of a homogeneous flow
p
- Cosmic strings are hypothetical objects generically past a linear angle defect. Section 3 estimates over-
o densitiesandheatingproducedbytheshocksbehindthe
predicted by most modern inflationary models and are
r string in the early Universe and briefly discusses obser-
t expected to survive till present time as large over-
s vational possibilities compared with previously reported
horizon kinked linear objects and smaller loops with
a dark matter wakes. Section 4 estimates particle accel-
[ about 10 horizon-scale strings in the observable por-
tion of the Universe (Polchinski & Rocha 2007). The eration on the shocks and their potential observability.
1 velocities of strings and loops are expected to be Section 5 contains the discussion.
v trans-relativistic such with the rms velocity v ≈
4 s 2. SUPERSONIC FLOW PAST THE LINEAR
0.7c. The main parameter of the string is the sym-
1 ANGLE DEFECT
metry breaking scale η which determines the mass
8 per unit length µ = η2 and the angle defect of the Ordinary matter in a form of either neutral or ionized
3
straight string θ = 8πGµ/c2. For a general intro- gas is normally considered within a fluid framework due
0
duction of cosmic strings see, e.g., Vilenkin & Shellard to its relatively high collisionality. In the case of atomic
.
1 (1994); Hindmarsh & Kibble (1995); Copeland et al. gas the mean free path in hydrogen right after recombi-
0 (2011). Cosmic strings are expected to lens back- nationisaround4×10−5pc,whichistinycomparedwith
5 groundsourcesoflight (Vilenkin 1984; Morganson et al. cosmologicalscales or the scales of the string. The ordi-
1 2010; Sazhina et al. 2011) and cosmic microwave back- nary matter should, therefore, be considered collisional
v: ground (Vilenkin 1986). The loops and kinks will for the purpose of considering a large-scale solution of a
i emit gravitational waves (Spergel et al. 1987) and flow around the string.
X leave wakes behind themselves (Sornborger et al. 1997; BelowI willdescribe asupersonicflowofordinarycol-
r Duplessis & Brandenberger2013). Thiswillaffectstruc- lisionalfluidpastthestring(fortheestimateofcollision-
a ture formation in the early Universe (Khatri & Wandelt ality see Section 3). The straight string segment has no
2008;Shlaer et al.2012). Kinkstendtostraightenthem- gravity of its own, but is manifested by the presence of
selves by emitting gravitationalwavesand loops tend to an angle defect. It is convenient to consider the flow in
evaporate for the same reason. thestringrestframeandtomapthespacearounditonto
The current upper limits on the angle defect θ come the Euclidean space. As the perpendicular cross-section
from the lensing of background galaxies, θ < 6×10−5 ofthestringisacone,theprojectioninvolvesanangular
(Morgansonet al. 2010), and the CMB lensing, θ < cut in flat space with the sides of the cut mapped onto
7×10−6 (Wyman et al. 2005). The pulsar timing ex- each other. Fig. 1 shows perpendicular cross-section of
periments(Damour & Vilenkin2005)givetighterlimits, the space around the string, where I have chosen to use
butaremoremodel-dependent. Itispotentiallyinterest- the cut with sides, which are parallel to the fluid veloc-
ing to lookfordirectinteractionofstringswith ordinary ity. Such a cut ensures that the flow pattern is symmet-
collisionalmatter. Thissubjectwasmostlyoverlookedin ric with respect to the cut direction. The flow changes
the literature. The straight segment of the string could its velocity from v1 to v2 at two oblique shocks having
produce interaction signatures that are peculiar in that angle β with the tail direction x. I will also designate
they lay on a single spatial plane. the deflection angle α = β +θ/2 and the ratio of spe-
The paper is organized as follows. Section 2 describes cific heats γ and I assume γ = 5/3, as I mostly deal
with either monoatomic gas or very cold hydrogen. I
2
will also introduce the β = v /c and the Mach number
s 1
of the inflow M = v /c , where c is the sound speed.
1 1 s s
For electron-protonplasma M can be approximated by
1
1.68×104β (T/1eV)−1/2,whereT istheplasmatemper-
s
ature.
Applying conservation of matter, momentum and en-
ergy to the flow depicted on Fig. 1, and excluding most
variables, I arrive at the oblique shock relation see, e.g.,
Landau & Lifshitz(1959),wherethedeflectionangleand
the shock angle are related by the angle defect of the
string:
θ (γ+1)M2
cot =tanα 1 −1 , (1)
2 (cid:20)2(M2sin2α−1) (cid:21)
1
The Equation 1 can be solved for α and has two
branches of solutions, one of which, giving larger α
is only realized in confined geometries, while in open
boundary flows the solution with smaller α is realized
Fig.1.— Flow around a linear topological angle defect of θ. I
(Landau & Lifshitz 1959). The expected values for the
am using Euclidean 2D plane with a dashed area representing an
angle defect θ for astrophysical strings are fairly small, angularcuttheta,whichisparalleltotheflowvelocity.
<10−4,see §1,andonthe secondbranchofthe solution
of Equation 1 this means that α,β ≪1 as well. Assum-
Brandenberger et al. 2010; Tashiro 2013; Brandenberger
ing that α,β,θ ≪ 1, but M θ, M β are not necessarily
1 1 2014). The gas will subsequently accrete on the dark
small,the secondbranchwillgivethe followingequation
matter to produce features in HI. This subsequent will
for α:
happen on much later times, however. In this Section I
will neglect the self-gravitational effect of the wake and
4M2α2−αM2θ(γ+1)−4=0. (2)
1 1 concentrate on the direct hydrodynamic interaction be-
If θ ≫ 1/M ≈ 10−4β−1T1/2, the shocks are strong tween the gas and the string.
1 s I will assume that M ≫ 1, γ = 5/3, which should be
and solving (2) gives β = θ(γ −1)/4. In the opposite
thecaseformolecularhydrogenwithT <70K.Theover-
limit, θ ≪10−4β−1T1/2 the shock is weak and the solu-
s density and temperature of such supersonic trail right
tionisβ =1/M , realizingthe “Machcone”. Inthe case
1 after the interaction with the string could be expressed
ofweakshockthe effective Machnumber inthe frameof
as:
the shockis M =1+θ(γ+1)M /8 andthe compression
1
ratio is ρ2/ρ1 =1+θM1/2,while in the case of a strong ρ /ρ =4/(1+3M−2)≈4, (3)
shockit is M =θ(γ+1)M /4andthe compressionratio 2 1
1
approaches (γ+1)/(γ−1).
It is potentially interesting to look for the flow solu- T /T ≈ 5 M2. (4)
tions inmagnetizedmedia, i.e. to considerthe magneto- 2 1 16
hydrodynamic(MHD) problem. The generalorientation
Assuming that the hydrogen temperature scales adia-
of the field will break the symmetry of the flow that I batically as (1+z)3(γ−1) = (1+z)2 after z = 500, the
have used to derive its structure on Fig. 1, so such a
effective Mach number will be
flow will be more complex. Two special MHD cases can
be treated relatively easily, however. If the magnetic
field is parallel to the string, both shocks will be per- 2 θ βs 1
M = θM =120 , (5)
pendicular shocks and the shock condition is the same, 3 1 (cid:18)10−5(cid:19)(cid:18)0.5(cid:19)(cid:18)1+z(cid:19)
except for adding magnetic pressure to the plasma pres-
sure (Landau & Lifshitz 1960). If the magnetic field is that is, I expect the shocks from the string to become
perpendicular to both the string and the inflow velocity strong starting from (1+ z) ≈ 120 and at later times
and θ is small, the solution will be similar to hydrody- before the re-ionization, and produce heating and com-
namicsolution. ThegeneralMHDcasewillbeconsidered pressionresulting in the excess of 21 cm emission due to
elsewhere. both higher temperature and high density.
It should be noted that equations (3-5) describe over-
3. DETECTION IN THE EARLY UNIVERSE BY 21 densityandheatingonlyasafunctionofredshiftandthe
CM LINE angle defect. This makes them distinctly different from
expressionsobtainedforgravitationalaccretiononwakes
As I demonstrated in Section 2, strings will leave be-
which has an extra unknown, which is the time allowed
hind a wake of compressed and heated material, which
for accretion. Our expression will, therefore, be easier
has a well-defined shape, over-density and dimensions,
to use for direct estimation of θ, given the redshift and
dependingontheMachnumberoftheflowandtheangle
theexcess21cmemission,howeverspecialcareshouldbe
defectofthestring. Sofar,cosmicstringwakeshasbeen
given for not confusing the two effects.
considered primarily in the collisionless medium where
they produce the wedge-shaped wake with angle θ/2
4. DETECTION BY RADIO EMISSION
and over-density of two, see, e.g. (Silk & Vilenkin 1984;
3
Shocks propagating through magnetized plasma tend The new generation of radio telescopes, such as SKA,
toaccelerateparticlesandproduceradioemissionbysyn- should be able to detect such low surface brightness ob-
chrotron mechanism and γ-ray emission through inverse jects, e.g., 50% of the SKA will detect 5 mK surface
Compton mechanisms. I will consider string propagat- brightness at the 3σ level with beam of 8” in 1 hour
ing through the present-day well-ionized intergalactic or (Feretti et al. 2004), however the problem of confusion
intraclustermediumandestimatetheeffectsofshockac- withotherobjectsanddealinggalacticbackgroundisin-
celeration. Using the effective Mach number for oblique deed quite challenging. Several morphological features
shocks derived in Section 2, the change in enthalpy of specific to the remnants of string activity can help in
the gas per unit time per unit area – the power, in prin- differentiating these objects, however. Speaking of the
ciple available for acceleration, on both shocks can be collisionofthestringwiththegalaxycluster,otherlarge-
estimated as scale(>1Mpc)lowsurfacebrightnessobjectsexpectedto
be detected with the new generation of radio telescopes
include intergalactic shocks Keshet et al. (2004a,b), ac-
P =3nm c2cβ θ(γ+1)/4=
s p s s cretion shocks, currently detected only in some clusters
erg n(cm−3) T θ astheso-calledradiorelics(van Weeren et al.2010),and
3.2×10−6cm2s 10−3 1keVβs10−5, (6) diffuse radiohalos(Carilli & Taylor2002;Cassano et al.
2010), currently detected only in merging clusters but
for θ ≪10−4β−1T1/2, or thought to be present universally. Out of these three,
s
allhavedifferentmorphologicaland/orradiativefeatures
compared to remnants of string activity. Intergalactic
Ps =nmpc3βs3θ3((γ+1)/4)3 = and accretion shocks are expected to be detected on
erg n(cm−3) θ 3 the outskirts of the cluster, where the surface bright-
1.3×10−5 β3 , (7) ness is being enhanced due to the projection effect. For
cm2s 10−3 s(cid:18)10−3(cid:19) the string shocks, the projection factor (sini)−1 is, ba-
for θ ≫10−4β−1T1/2. sically, a constant, while the surface brightness should
s strongly increase with higher density, towards the cen-
Radiation efficiencies of the shocks are fairly uncer-
ter of the cluster. Both types of objects are expected to
tain, for several reasons. The first-principle calcula-
emit significantly polarized radio emission. Comparing
tion of acceleration efficiencies are still not available
cluster halos and string trails, the former are unpolar-
due to the complex nature of the acceleration pro-
ized and rather spherical, mimicking the shape of the
cess (Malkov & O’C Drury 2001), however some results
cluster,whilethelatterwillbepolarizedand,ingeneral,
basedon phenomenologicalmodel for particle scattering
rather elliptic, as they cross the cluster at some angle
is available (Kang et al. 2012). For the same reason,the
with respect to the field of view. In fact, due to the se-
injection process is not fully understood and the elec-
lectioneffect,thetrailswithhigherprojectionfactorwill
tron/proton ratio is not exactly known. While in super-
be much more likely observed, while their morphology
novashocks,giventypicaldensitiesandshockMachnum-
will be the most unusual – basically a thin bright stripe
bers,theamplifiedmagneticfieldattheshockdominates
across the cluster.
overradiationfieldandsynchrotronlossesdominateover
Finally, the surface of past interaction of the string
inverse Compton losses, in the tenuous ICM and IGM,
withdensematteroveraHubbletimewillbeverylarge1
theoppositecouldbetrue. Theconventionalapproachto
and it is likely to have patches where the shocks will
deal with these uncertainties is to introduce parameters
be amplified, when propagating down the density gra-
suchasaccelerationefficiencyandthemagneticfieldam-
dients (Ostriker & McKee 1988). Also, the acceleration
plification efficiency and make educated guess based on
efficiency of the twin shock is higher than that of a sin-
available theory studies as well as observations, see, e.g.
gle shock, as the downstream particle traveling from
Keshet et al. (2004a). I will further simplify the above
one shock could diffuse to the upstream of the other
approach,introducingtheradioemissionefficiencyofη ,
r (Melrose & Pope 1993).
keeping in mind that it depends on the gas density and
Anothermethodtodifferentiatestringtrailsandother
the Machnumberinafairlycomplexway. We wouldex-
pect radiation efficiency in the range 10−2−10−6. The objects can rely on the strings themselves and their
trajectories being fairly straight on sub-horizon scales.
radio spectrum is fairly uncertain for the same reasons
This means that the straight segments of the string will
asabove. I canestimatethe spectralbrightnessnearthe
leave behind relics which lay on the single spatial plane.
peak of the emission using the total emitted power as
Searchingforspatialplanesthatcontainsignificantnum-
ν I (ν ) ≈ P η /4πsini, where i is an angle between
m ν m s r ber of large-scale (>1Mpc) radio sources could be an-
the line of sight and the velocity of the string, provided
other viable method which will allow to avoid confusion
that it is not much smaller than 1/M . I obtain the fol-
1 with accretion/intergalacticshocks.
lowingexpressionfor the surface brightnesstemperature
Tb =Iνc2/2kν2: 5. DISCUSSION
In this paper I presented the solution of the flow of
T =83mK(sini)−1 ηr νm −3 collisional matter around the cosmic string for the first
b 10−4 (cid:16)1GHz(cid:17) 1 Also, cluster sound crossing times are cosmological, while for
n T θ bigger objects, such as filaments, sound crossing times are much
× β , (8)
10−3cm−31keV s10−5 larger than the age of the Universe. I also can ignore the shock
propagationspeedsasfarasthecoincidencedetectionmethodde-
where I used the weak shock case. scribedbelowisconcerned.
4
time. Aside from shocks in collisional medium, strings tended sources at least 1 Mpc in physical size. The
can also produce wakes in dark matter (Silk & Vilenkin cluster radio halos could be confused with the string
1984; Brandenberger et al. 2010), which also has wedge trails, but they are associated with turbulent acceler-
shape,butwithaconstantangleofθ/2andtheconstant ation in clusters (see, e.g., Brunetti & Lazarian 2007;
dark matter compression ratio of two, independent on Beresnyak et al. 2013) that underwent recent merger, so
θ. The subsequent self-gravitational contraction of such surveying smaller and quieter clusters is more advanta-
wakes will also draw in ordinary matter, possibly result- geous. Also, as I pointed out above should have differ-
ing in secondary shocks (Hern´andez & Brandenberger ent polarization properties and morphology. The colli-
2012) and various observational effect of the entrained sion of the string with giant molecular clouds (GMC)
hydrogen, e.g. the enhanced 21 cm emission (Tashiro could in principle produce much stronger signal, e.g. for
2013; Brandenberger et al. 2013; Brandenberger 2014). θ = 10−5, T = 10K the shocks will have an effective
Asthewakegravitationallycontracts,entrainsandheats Mach number of 3, and assuming density of 103cm−3,
hydrogen, it no longer presents such a clear and well- β =0.5 and the accelerationefficiency of 10−2, the sur-
s
definedangularshape. Incontrast,thetrailincollisional face brightness temperature is around 4 K. Given the
matter described in this paper heats the gas momentar- small volume fraction of GMCs in the Universe, such
ily. The relativeimportance ofthe trails consideredhere collision is fairly unlikely, however.
andthetrailsproducedbycollapseofdarkmatterwakes
to the HI structures inthe earlyUniversewill be consid- 6. ACKNOWLEDGMENTS
ered in a future publication.
I am grateful to Avi Loeb for a discussion. This work
Given that the typical string segment length as well
was supported by the LANL/LDRD program and the
as their distance between each other is of order of 1
DoE/Office of Fusion Energy Sciences. I am grateful for
Gpc, the radio searches for strings should survey large-
the hospitality of Nordita.
scale distant objects, such as clusters, and focus on ex-
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