**2.4. String loops and scaling**

We saw before the reasons why gauge monopoles and domain walls were a
bit of a problem for cosmology. Essentially, the problem was that
their energy density decreases more slowly than the critical density
with the expansion of the universe. This fact resulted in their
contribution to
_{def} (the
density in defects normalized
by the critical density) being largely in excess compared to 1, hence
in blatant conflict with modern observations. The question now arises
as to whether the same might happened with cosmic strings. Are strings
dominating the energy density of the universe? Fortunately, the answer
to this question is *no*; strings evolve in such a way to make
their density
_{strings}
^{2}
*t*^{-2}. Hence, one
gets the same temporal behavior as for the critical density. The
result is that
_{strings}
~ *G**µ* ~
( /
*m*_{P})^{2} ~ 10^{-6} for GUT strings,
*i.e.*, we get an interestingly small enough,
constant fraction of the critical density of the universe and strings
never upset standard observational cosmology.

Now, why this is so? The answer is simply the efficient way in which a
network of strings looses energy. The evolution of the string network
is highly nontrivial and loops are continuously chopped off from the
main infinite strings as the result of (self) intersections within
the infinite-string network. Once they are produced, loops
oscillate due to their huge tension and slowly decay by emitting
gravitational radiation. Thus, energy is transferred from the cosmic
string network to radiation.
^{(11)}

It turns out from simulations that most of the energy in the string network (roughly a 80%) is in the form of infinite strings. Soon after formation one would expect long strings to have the form of random-walk with characteristic step given by the correlation length . Also, the typical distance between long string segments should also be of order . Monte Carlo simulations show that these strings are Brownian on sufficiently large scales, which means that the length of a string is related to the end-to-end distance d of two given points along the string (with d >> ) in the form

(8) |

What remains of the energy is given in the form of closed loops with
no preferred length scale (a scale invariant distribution) which
implies that the number density of loops having sizes between *R* and
*R* + *dR* follows just from dimensional analysis

(9) |

which is just another way of saying that *n*_{loops}
1/*R*^{3},
loops behave like normal nonrelativistic matter.
The actual coefficient, as usual, comes from string simulations.

There are both analytical and numerical indications in favor of the existence of a stable "scaling solution" for the cosmic string network. After generation, the network quickly evolves in a self similar manner with just a few infinite string segments per Hubble volume and Hubble time. A heuristic argument for the scaling solution due to Vilenkin [1985] is as follows.

If we take (*t*) to be
the mean number of infinite string segments
per Hubble volume, then the energy density in infinite strings
_{strings}
= _{s} is

(10) |

Now, strings will typically
have intersections, and so
the number of loops *n*_{loops}(*t*) =
*n*_{l}(*t*) produced per
unit volume will be proportional to
^{2}. We find

(11) |

Hence, recalling now that the loop sizes grow with the expansion like
*R* *t* we
have

(12) |

where *p* is the probability of loop formation per intersection, a
quantity related to the intercommuting probability, both roughly of
order 1.
We are now in a position to write an energy conservation equation for
strings plus loops in the expanding universe. Here it is

(13) |

where *m*_{l} = *µ**t* is just the
loop mass and where the second on the left hand side is the dilution term
3 *H*
_{s}
for an expanding radiation-dominated universe.
The term on the right hand side amounts to the loss of energy from
the long string network by the generation of small closed loops.
Plugging Eqs. (10) and (12) into (13)
Vilenkin finds the following kinetic equation for
(*t*)

(14) |

with *p* ~ 1. Thus if
>> 1 then *d*
/ *dt* < 0 and
tends to decrease in time, while if
<< 1 then
*d* / *dt* > 0 and
increases. Hence, there will
be a stable solution with
~ a few.

^{11} High-resolution cosmic string
simulations can be found in the Cambridge cosmology page at
http://www.damtp.cam.ac.uk/user/gr/public/cs_evol.html
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