**5.3. CMB bispectrum from strings**

To calculate the sources of perturbations we have used an updated version of the cosmic string model first introduced by Albrecht et al. [1997] and further developed in [Pogosian & Vachaspati, 1999], where the wiggly nature of strings was taken into account. In these previous works the model was tailored to the computation of the two-point statistics (matter and CMB power spectra). When dealing with higher-order statistics, such as the bispectrum, a different strategy needs to be employed.

In the model, the string network is represented by a collection of uncorrelated straight string segments produced at some early epoch and moving with random uncorrelated velocities. At every subsequent epoch, a certain fraction of the number of segments decays in a way that maintains network scaling. The length of each segment at any time is taken to be equal to the correlation length of the network. This and the root mean square velocity of segments are computed from the velocity-dependent one-scale model of Martins & Shellard [1996]. The positions of segments are drawn from a uniform distribution in space, and their orientations are chosen from a uniform distribution on a two-sphere.

The total energy of the string network in a volume *V* at any time is
*E* = *N**µ**L*, where *N* is the total
number of string segments at that
time, *µ* is the mass per unit length, and *L* is the
length of one
segment. If *L* is the correlation length of the string network then,
according to the one-scale model, the energy density is
= *E* /
*V* = *µ*/*L*^{2}, where *V* =
*V*_{0} *a*^{3}, the expansion factor *a*
is normalized so that *a* = 1 today, and *V*_{0} is a
constant simulation volume. It follows that *N* = *V* /
*L*^{3} = *V*_{0} /
^{3}, where
= *L* / *a* is
the comoving correlation length. In the scaling regime
*l* is approximately proportional to the conformal time
and
so the number of strings
*N*() within the
simulation volume *V*_{0} falls as
^{-3}.

To calculate the CMB anisotropy one
needs to evolve the string network over at least four orders of
magnitude in cosmic expansion. Hence, one would have to start with
*N*
10^{12}
string segments in order to have one segment left at the present time.
Keeping track of such a huge number of segments is numerically
infeasible.
A way around this difficulty was suggested in
Ref.[3], where the idea was to consolidate all string segments
that decay at the same epoch. The number of segments that decay by the
(discretized) conformal time
_{i} is

(104) |

where *n*() =
[()]^{-3} is the number
density of strings at time
.
The energy-momentum tensor in Fourier space,
^{i}_{µ},
of these *N*_{d}(_{i}) segments is a sum

(105) |

where ^{im}_{µ} is the Fourier transform of the energy-momentum
of the *m*-th segment. If segments are uncorrelated, then

(106) |

and

(107) |

Here the angular brackets < ... > denote the
ensemble average, which in our case means averaging over many
realizations
of the string network. If we are calculating power spectra, then the
relevant quantities are the two-point functions of
^{i}_{µ}, namely

(108) |

Eq. (106) allows us to write

(109) |

where ^{i1}_{µ} is of the energy-momentum
of one of the segments that decay by the time
_{i}. The last
step in
Eq. (109) is possible because the segments are statistically
equivalent. Thus, if we only want to reproduce the correct power
spectra in
the limit of a large number of realizations, we can replace the sum in
Eq. (105) by

(110) |

The total energy-momentum tensor of the network in Fourier space is a sum over the consolidated segments:

(111) |

So, instead of summing over
_{i=1}^{K}
*N*_{d}(_{i})
10^{12}
segments we now sum over only *K* segments, making *K* a
parameter.

For the three-point functions we extend the above procedure. Instead of Eqs. (108) and (109) we now write

(112) |

Therefore, for the purpose of calculation of three-point functions, the sum in Eq. (105) should now be replaced by

(113) |

Both expressions in Eqs. (110) and (113), depend
on the simulation volume, *V*_{0}, contained in the
definition of
*N*_{d}(_{i}) given in Eq. (104). This is to be expected
and is consistent with our calculations, since this volume cancels in
expressions for observable quantities.

Note also that the simulation model in its present form does not allow
computation of CMB sky maps. This is because the method of finding the
two- and three-point functions as we described involves
"consolidated" quantities
^{i}_{µ} which do not
correspond to the energy-momentum tensor of a real string
network. These quantities are auxiliary and specially prepared to give
the correct two- or three-point functions after ensemble averaging.

In Fig. 1.16 we show the results for
*G*_{lll}^{1/3}
[cf. Eq. (99)]. It was calculated using the string model
with 800 consolidated segments in a flat universe with cold dark
matter and a cosmological constant. Only the scalar contribution to
the anisotropy has been included. Vector and tensor contributions are
known to be relatively insignificant for local cosmic strings and can
safely be ignored in this model [3, 131]
^{(18)}.
The plots are produced using a single realization of the
string network by averaging over 720 directions of
**k**_{i}. The comparison of
*G*_{lll}^{1/3} (or equivalently
*C*_{lll}^{1/3}) with its cosmic variance
[cf. Eq. (101)] clearly shows that the bispectrum (as
computed from the present cosmic string model) lies hidden in the
theoretical
noise and is therefore undetectable for any given value of *l*.

Let us note, however, that in its present stage the string code employed in these computations describes Brownian, wiggly long strings in spite of the fact that long strings are very likely not Brownian on the smallest scales, as recent field-theory simulations indicate. In addition, the presence of small string loops [Wu, et al., 1998] and gravitational radiation into which they decay were not yet included in this model. These are important effects that could, in principle, change the above predictions for the string-generated CMB bispectrum on very small angular scales.

The imprint of cosmic strings on the CMB is a combination of different effects. Prior to the time of recombination strings induce density and velocity fluctuations on the surrounding matter. During the period of last scattering these fluctuations are imprinted on the CMB through the Sachs-Wolfe effect, namely, temperature fluctuations arise because relic photons encounter a gravitational potential with spatially dependent depth. In addition to the Sachs-Wolfe effect, moving long strings drag the surrounding plasma and produce velocity fields that cause temperature anisotropies due to Doppler shifts. While a string segment by itself is a highly non-Gaussian object, fluctuations induced by string segments before recombination are a superposition of effects of many random strings stirring the primordial plasma. These fluctuations are thus expected to be Gaussian as a result of the central limit theorem.

As the universe becomes transparent, strings continue to leave their
imprint on the CMB mainly due to the
Kaiser & Stebbins
[1984]
effect. As we mentioned in previous sections, this effect results in line
discontinuities in the temperature field of photons passing on
opposite sides of a moving long string.
^{(19)}
However, this effect can result in non-Gaussian
perturbations only on sufficiently small scales. This is because on
scales larger than the characteristic inter-string separation at the
time of the radiation-matter equality, the CMB temperature
perturbations result from superposition of effects of many strings and
are likely to be Gaussian.
Avelino et al. [1998]
applied several non-Gaussian tests to the perturbations seeded by
cosmic strings. They found the density field distribution to be close
to Gaussian on scales larger than 1.5
(_{M}
*h*^{2})^{-1} Mpc,
where _{M}
is the fraction of cosmological matter density in
baryons and CDM combined. Scales this small correspond to the
multipole index of order *l* ~ 10^{4}.

^{18} The
contribution of vector and tensor modes is large in the case of global
strings
[Turok, Pen & Seljak,
1998;
Durrer, Gangui &
Sakellariadou, 1996].
Back.

^{19} The extension of
the Kaiser-Stebbins effect to polarization will be treated
below. In fact,
Benabed and Bernardeau
[2000]
have recently considered the generation of a B-type polarization field out
of E-type polarization, through gravitational lensing on a cosmic
string.
Back.