Effects of the image universe on cosmic strings

V o l u m e 242, number 1 PHYSICS LETTERS B 31 May 1990

E F F E C T S O F T H E I M A G E U N I V E R S E O N C O S M I C S T R I N G S

Tanmay VACHASPATI Institute of Cosmology and Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA

Craig J. H O G A N Steward Observatory, University of Arizona, Tucson. AZ, USA

and

Mart in REES

Institute of Astronomy, Madingley Road. University of Cambridge, Cambridge, UK

Received 16 January 1990

We investigate some of the cosmological effects of the gravitational attraction of straight cosmic strings that arises due to the conical geometry of the string. Although this effect is second order in Newton's gravitational constant, its effects in the early universe can be significant. We find that the image masses responsible for this second order attraction effectively "fill up" the volume deficit due to the conical geometry of a static straight string. A moving string also experiences a frictional force due to the images and this provides a mechanism for energy dissipation. The energy loss due to the image effect is comparable to the energy loss in gravitational radiation for strings on the size of the horizon scale but is probably not important when compared to the energy loss due to loop production. The image effect can also become important when a string comes close to a black hole. Our analysis of these effects is newtonian.

1. Introduction

Cosmic strings have been the subject of intensive research over the last decade. This research is fueled by the many interest ing and even pecul iar features of cosmic strings and of the whole network o f strings. One such pecul iar feature is the conical metr ic of a static straight infini te cosmic string [ 1 ]. it is gener- ally bel ieved that due to the flat nature of the conical metric, a static straight infini te cosmic string will not exert any force on neighbouring masses. This, however, is true only i f the neighbouring mass is a test mass. It has been shown [2-5 ] that, due to the global non- t r iv ia l i ty of the conical geometry, the string will repel a charge kept in its vicini ty and will a t t ract a neighbouring mass. This force has a newtonian fall off and, in the gravi ta t ional case we may write

G( 8;,rGlt)m 2 F = - f l r2 , (1)

where f l~ 1 for small G/z, G is Newton 's gravi ta t ional constant, # is the energy per unit length of the string, m is the mass o f the neighbouring part icle and r is the distance of the mass f rom the string. Note that r is not the spherical coordinate but the cylindrical distance. In part icular , i f the string is along the z-axis, then r = ~ 2 when the mass is located at (x, y) . A similar expression can be wri t ten for the repulsive electromagnet ic force exerted on neighbouring

charges .

The force in eq. ( 1 ) is very small since the d imen- sionless pa ramete r G# is typical ly taken to be 1 0 - 6

or even less. However, inspite of the small magni tude of the force exerted by the string on a single body o f mass m, the cosmological effects may still be interesting i f not impor tant . This is easily seen i f we realize that the force in eq. (1) is to be thought o f as the a t t ract ion between the mass and the " image" that the Another way of seeing this is to apply Gauss ' law:

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Volume 242, number 1 PHYSICS LETTERS B 31 May 1990

mass sees in the string due to the conical geometry. The effective mass of this image is Gltm upto some numerical factor. Now, in the cosmological scenario, not only will a mass see its own image in the string but it will also see the image of every other mass in the universe. Hence a small mass near the string will be attracted by the "image universe". Clearly the effect can be large. In fact, we shall show that the magnitude of this effect is exactly such that it will nullify the volume deficit due to the conical geometry. This is done in section 2, where we first demonstrate the effect in a cosmological setting and then show that a static straight infinite string does not accrete matter. Previously this result had been arrived at by using general relativistic arguments. Our analysis shows how to get the same result by using newtonian arguments.

The next setting we examine is a straight infinite string moving with uniform velocity through a homogeneous medium. It is well known that a wedge- shaped wake will form behind the string [6,7 ] as it moves through the medium due to the conical geometry of the string. In the presence of the wake, the matter is not distributed uniformly around the string. The result of this is that the wake exerts an excess gravitational pull on the string and slows it down. This, then, is a mechanism for energy loss from strings due to wake formation. For strings on the horizon scale, the energy loss due to the image effect is about equal to, if not greater than, the energy loss due to gravitational radiation.

Once we have realized that the wake exerts a drag on the string, an interesting quantity to calculate is the distance that the string will travel before it is brought to a halt by the wake behind it. The calculation of this "stopping distance" also leads us to the observation that the equivalence principle does not hold for cosmic strings: This is not a new observation [ 8 ]: even if we ignore the newtonian force, the string does not attract masses but surely there is a non-van- ishing inertial mass associated with the string. How- ever, we are not aware of any situation where it is necessary to differentiate between the inertial and gravitational masses of the string in the absence of the image effect.

We then compare the direct gravitational interaction of a string with a black hole to the interaction via the images. We find that the image effect can become

important for the motion of a long string in the vicin- ity of a black hole.

In ref. [ 5 ], some other questions regarding the interaction of masses with cosmic strings were studied. However, we believe that the estimate for the frictional force given in ref. [ 5 ] is mistaken because the non-linear nature of the image effect was not taken into account. In particular, a particle scatters off the string due to its interaction with the image universe and not just due to its own image.

2. The image effect and accretion

Consider a static string along the z-axis of a carte- sian coordinate system. The conical metric of the string then implies that the xy-cross section is conical. This can be visualized by removing a wedge from the xy-plane and by identifying the two edges where the wedge has been cut. This is shown in fig. 1.

Next consider a uniform fluid of density p all around the string. We wish to find the acceleration, g, that a test particle of this fluid will experience due to the presence of the string. Clearly, there is cylindrical symmetry around the string. Hence, g has to point in the radial direction everywhere: g=g(r)f. Now consider the matter within a cylinder of radius r around the string in the wedge picture of fig. 1. The uniform distribution of matter within the cylinder of radius r can only produce a radial g if there is an image distribution of matter that "fills up" the wedge.

A ' J

X

;TRING

Fig. 1. The conical geometry of a cosmic string along the z-axis can be visualized by cutting out the wedge-shaped region between the lines A and A' and then identifying the two edges. The angular deficit 5 is the angle of the wedge that has been removed.

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~ g .dS=AL, (2) C

where C denotes the surface of the cylinder, L ~ c e is the length and 2 is the mass per unit length of the cylinder. This gives

n--,~/2 t l

rg(r) | d 0 = 2 . (3) - n + d / 2

Here f = 8riG# is the deficit angle of the cone (fig. 1 ) and then

1 2 - ~ . (4) g=g(r)~= 2 n - f r

So we may define an effective mass per unit length of the matter in the cylinder by 2 ' = 2 / a , where a - 1 - f / 2 n . Now we know that the mass per unit length in the wedge is a fraction f / 2 n of the mass per unit length in the wedge plus the physical (conical) region. Therefore, if we denote the mass per unit length in the wedge by A2, then A2= (2+AZ)f/2n. This immediately gives 2+ A 2 = 2 / a = Z ' . Therefore the effective mass per unit length of the cylindrical distribution of matter on the cone is that of a cylindrical distribution of matter on a plane. And so it makes no difference, as far as the acceleration g is concerned, if we have a cone (any deficit angle) or a plane (zero deficit angle). Another way of stating this result is that the image of the (conical) cylinder acts to exactly cancel the volume deficit due to the re- moval of the wedge. Hence, in order to calculate the acceleration of a particle at a distance r from the string, we should forget about the presence of the string and, in particular, about the volume deficit due to the conical geometry. Instead we should find the acceleration of the particle as if it had been placed in (globally) flat space with a homogeneous distribution of matter all around it.

An immediate corollary to this result is that the rate of expansion of the universe in the x- and the y-direc- tions is unchanged due to the presence of the string and hence, a static straight string does not accrete matter.

3. Wake formation, energy loss and other issues

We now consider a string moving with constant speed through a homogeneous medium. The motion of the string produces a wake behind the string [ 6,7 ]. The cosmological implications of such a wake have been studied before [ 9,10 ]. Here we focus on the en- ergetics of the wake.

We shall take the string to lie along the z-direction and to be moving with uniform velocity v in the x- direction. This causes the particles that fall in the wake to have a velocity in the y-direction given by vy~ G#v. (We ignore relativistic factors.) The density in the wake is twice the background density p (t) (ignoring accretion in the matter-dominated era) and the mass per unit length in the wake is

R

Mw ~ 2 f j p(t)R dR, (5) o

where f = 8riG# is the deficit angle which is equal to the opening angle of the wedge-like wake region and R is the extent to which the wake has formed. If we assume that the background matter density is approximately constant, we get

M w ~ @ R 2 . (6)

This can be expected to be a reasonable approximation if the duration over which eq. (5) is applied is much less than the typical time over which the density changes. If the density is changing due to the Hubble expansion, eq. (6) is a good approximation provided we are considering times that are less than the Hubble time.

The force per unit length that the wake exerts on the string is equal to the attractive gravitational force that the mass of the wake exerts on the total mass of the images that the wake will see in the string. This total image mass is the mass of the "image universe". If we imagine that the wake's center of gravity is at a distance R from the string, the image universe is the image of all matter within a cylinder of radius R cen- tered about the string. Therefore the force between the wake and the image universe is

f= - G MwMim Gp2f2R 3 (7) R

w h e r e M i m = fMuniverse (r < R) = Bin R 2p is the mass per

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unit length of the image universe. The length R is the length of the wake that is trailing behind the string and exerting the drag. We would expect R to be given by the horizon distance, or, for curved strings, by their radius of curvature.

Now we can estimate the work done per unit length in time d t = d R / v by the wake on the string. This is given by

d W = - f dR,~ -Gp2fi2R3 dR . (8)

The work done on the string must result in a loss of kinetic energy by the string. If we assume, for the time being, that the matter density is constant in time, this gives the stopping distance of the string to be

(~pp) :/2 Rs~op~ ~-1/4. (9)

The time taken to stop is

Rsto p ~ - - 1 / 4 v - 1 / 2

/stop "~ ~ ( 10 ) V ( @ ) 1 / 2

There are a number of interesting things to note about these results. The first is that the dependence on fi is quite weak. The GUT scale value for fi=8zcG.u is about 10- 5. Hence, for GUT strings, c~- 1/4 ~,~ 10. The second point to note is that in the case of the straight string moving through a medium, the image effect is the dominant effect in stopping the string. This is because the only other relevant effect is the direct gravitational pull of the wake on the string. To understand this direct pull we recall that although the string does not attract masses (if we ignore the image effect), external masses do exert forces on the string and influence the motion of the string. (A familiar example of this is that of strings moving in an expanding background. ) This force on the string is what we call the direct force and it is simply given by the gravitational potential produced by the external body times the mass per unit length of the string:

G •Mw fdirect ~ -- ( 1 1 )

R

This leads to an estimate of the stopping distance by the direct gravitational force: /~top ~ (v2 / @) 1/2~ -

1/2. Clearly, Rstop > Rstop and the image effect is dominant over the direct gravitational effect. (For c ~ 10 -5, however, the two effects are roughly equal. )

The estimates in eqs. (9) and (10) are not valid in the cosmological context because, if we put Gp ~ t - 2

then Rstop > t and the time dependence ofp cannot be ignored. This means that the Hubble damping is always dominant over the drag due to the image effect. In this case, however, the relevant calculation is the rate of energy loss of the string due to wake formation. This should be compared to the energy lost in gravitational radiation and also to the energy lost in loop production. The energy lost per unit length per unit time due to the image effect follows immediately from eq. (8):

dW dt ~ -Gp2c~2R3v" (12)

The rate of energy lost in gravitational radiation per unit length for strings on the horizon scale is

dEg ~ls G/A 2 (13) d t ~ t

Here ~:s is a numerical factor. For loops that are much smaller than the horizon, the corresponding numerical factor has been estimated to be about 50 [1 ]. For the long strings, we can expect that the factor will be smaller since the motion of long strings is damped due to Hubble expansion. Hence we will take 7:s ~ 1. A comparison of eq. (12) and eq. (13) shows that the two energy losses are about equal for strings on the horizon scale, that is, R ~ t.

Note that the above analysis ignores accretion due to the wake. This would occur in the matter-dominated era and would imply that the energy loss due to the image effect is greater than the loss due to gravitational radiation.

The energy lost by the network of long strings is presently being investigated by various workers [ 11 ]. A useful estimate can still be obtained by assuming that the energy lost by the network is the same as if a string produces one horizon size loop per Hubble time. (Simulations [ 11 ] indicate that the loops produced are very much smaller than the horizon. We will, for the sake of convenience, view this as the rapid fragmentation of the horizon size loop. ) The the energy lost per unit length per unit time in loop production is

dE:oOp # (14) dt t"

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Volume 242, number l PHYSICS LETTERS B 31 May 1990

A comparison of eqs. ( 12 ) and (14) shows that the image effect is never as efficient as loop production in dissipating the energy of long strings. This is strictly true if we ignore the accretion onto wakes and take p ~ 1 / G t z. However, we find that even if we take accretion onto the wakes into account, the energy loss due to the image effect does not dominate over the loop production energy loss till the present age of the universe.

The calculation in eq. (9) works because we had assumed that the density is approximately constant in time, eq. (6). I f the density is not constant in time, we will have to find R as a function of t by solving the equation of mot ion with the force being an integral over the wake extent. We know the gravitational force between the string and the wake, eq. (7). Newton 's second law then gives

R

0

where ~inertial is the inertial mass associated with the string, Pu is the energy density of the universe and pw(R) is the energy density of the wake at a distance R from the string. In eq. (15) we have assumed that the mass of the image universe (without the wake) is much larger than the mass of the image wake.

An interesting feature of eq. ( 15 ) is that the inertial mass per unit length of the string is not equal to the gravitational mass o f the string. The equivalence principle does not hold for the string even if we ignore the image effect. In that case, the gravitational force exerted due to the string vanishes but the inertial mass of the string is non-zero. However, in our knowledge, no calculation with cosmic strings so far was directly confronted with the inequivalent inertial and gravitational masses of the string. Eq. (15), on the other hand, inescapably forces us to make the distinction.

It would be worthwhile to point out that while the image effect may be the dominant frictional mechanism, it does not dominate the dynamics of the string under normal circumstances. The dominant force in the late cosmological evolution o f strings within the horizon scale is their tension. This is seen by compar- ing the drag on a string due to the image effect with the force of tension. If the string is curved on a scale 2, the force due to the curvature is £ ~ # / 2 . To esti-

mate the drag force due to the image, we use eq. (7): fiirn ~ Gp 2652R 3. Therefore, for the image force to dominate the curvature force we need 2 > ~ - 1 (Gp) - 2R - 3

I f we write Gp=1-2 this gives 2 > 14/R3~. On the cosmological scale ( l~ t, R ~ t), this inequality is never satisfied since 2 ~ t, and so the curvature of the strings always dominates over the image drag force. I f we are considering some other situation, say, when the string is smooth on the horizon scale (2~ t) but is passing through a region with unusually high density where we may take R ~ l, then, it is possible for the image force to dominate over the curvature force. This, however, may be a temporary effect as the string can become highly curved during its evolution and then, once again, become curvature dominated.

Another situation o f interest is that of a string in- teracting with a dense massive body like the sun or a black hole [ 12 ]. Although the string does not attract the massive body to first order in G, the gravitational field of the body does influence the dynamics of the string. The direct interaction force between the body and the string is given by the force per unit length on the string

G :O fai .... ~ - - - , (16)

g

where 3 ) is the mass per unit length of the body and r is the distance between the string and the body. can be estimated to be the total mass o f the body M divided b y / = m a x ( r , R) where R is the size of the body. The image force acts on a segment of the string whose length is roughly L Over this segment, the image force per unit length is

G 2~I2 ~ fimage ~-~ - - ( 1 7 )

Therefore fmage>faireot only if ~l~c~>/t. This leads to the condition G M / l > 1. Hence, the image interaction is important for the dynamics o f a cosmic string when it gets close to a black hole.

4. Conclusion

We have shown that although the image effect for cosmic strings is second order in Newton's gravitational constant G, its magnitude can be significant when we consider a cosmic string immersed in a cos-

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mological medium. Our analysis of a variety of cosmological situations shows that under some circumstances the effects of the image universe on cosmic strings can b e the dominan t effect. In particular, in studying the dynamics of cosmic strings near black

holes, the image effect must be taken into account. Some other cosmological situations that we have studied are a static straight string and a uniformly moving straight string in the universe. The analysis of the straight string led to the interesting observa- t ion that the volume deficit around a string is exactly compensated by the excess gravitational attraction due to the image masses. For the uniformly moving string, the image effect leads to a loss of kinetic energy of the string, For strings on the horizon scale, this energy loss is greater than the energy loss in gravitational radiation but less than the energy loss in loop production.

Acknowledgement

We would like to thank Alex Vilenkin for his com- ments. C.J.H. and T.V. would like to thank the Insti- tute of Astronomy, Cambridge for its hospitality. C.J.H. was supported by a NASA grant (NAGW-

1703 ) and the Alfred P. Sloan Foundat ion. T.V. was supported by the SERC at DAMTP, University of Cambridge where this work was started and by the NSF.

References

[ 1 ] See A. Vilenkin, Phys. Rep. 121 (1985) 263. [ 2 ] G. Smith, in: Proc. 1989 Cambridge Symp. on The formation

and evolution of cosmic strings, eds. G.W. Gibbons, S.W. Hawking and T. Vachaspati (Cambridge U.P., Cambridge), to be published.

[3] J.S. Dowker, J. Math. Phys. 30 (1989) 770; Phys. Rev. D 36 (1987) 3095.

[4] B. Liuet, Phys. Rev. D 33 (1986) 1833. [5] D.V. Gal'tsov, Are cosmic strings gravitationally sterile?,

preprint ITP-SB-SB-89-25. [ 6 ] J. Silk and A. Vilenkin, Phys. Rev. Lett. 53 (1984) 1700. [7 ] G.W. Gibbons, F. Ruiz-Ruiz and T. Vachaspati, Commun.

Math. Phys. 127 (1990) 127. [ 8 ] A. Vilenkin, private communication. [9] T. Vachaspati, Phys. Rev. Lett. 57 (1986) 1655.

[ 10] A. Stebbins et al., Astrophys. J. 322 (1987) 1. [ 11 ] See the contributions by D. Bennett, F. Bouchet, P. Shellard

and N. Turok, in: Proe. 1989 Cambridge Symp. on The formation and evolution of cosmic strings, eds. G.W. Gibbons, S.W. Hawking and T. Vachaspati (Cambridge U.P., Cambridge), to be published.

[12] S. Lonsdale andI. Moss, Nucl. Phys. B 298 (1988) 693.

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Effects of the image universe on cosmic strings