Irreducible background of gravitational waves from a cosmic defect
  network: update and comparison of numerical techniques
  Cosmological phase transitions in the early Universe may produce relics in
the form of a network of cosmic defects. Independently of the order of a phase
transition, topology of the defects, and their global or gauge nature, the
defects are expected to emit gravitational waves (GWs) as the network
energy-momentum tensor adapts itself to maintaining {scaling}. We show that the
evolution of any defect network (and for that matter any scaling source) emits
a GW background with spectrum $\Omega_{\rm GW} \propto f^3$ for $f \ll f_0$,
$\Omega_{\rm GW} \propto 1/f^2$ for $f_0 \lesssim f \lesssim f_{\rm eq}$, and
$\Omega_{\rm GW} \propto~const$ (i.e.~exactly scale-invariant) for $f \gg
f_{\rm eq}$, where $f_0$ and $ f_{\rm eq}$ denote respectively the frequencies
corresponding to the present and matter-radiation equality horizons. This
background represents an irreducible emission of GWs from any scaling network
of cosmic defects, with its amplitude characterized only by the symmetry
breaking scale and the nature of the defects. Using classical lattice
simulations we calculate the GW signal emitted by defects created after the
breaking of a global symmetry $O(N) \rightarrow O(N-1)$. We obtain the GW
spectrum for $N$ between 2 and 20 with two different techniques: integrating
over unequal time correlators of the energy momentum tensor, updating our
previous work on smaller lattices, and for the first time, comparing the result
with the real time evolution of the tensor perturbations sourced by the same
defects. Our results validate the equivalence of the two techniques. Using CMB
upper bounds on the defects' energy scale, we discuss the difficulty of
detecting this GW background in the case of global defects.
