Based on:

• arXiv:1607.0076
• arXiv:1611.08456
• arXiv:1809.03384

With:

Mark Hindmarsh • Anna Kormu • Asier Lopez-Eiguren • Kari Rummukainen

Cosmic strings: motivation

• Extended, long lived, very tense pieces of 'string' formed during symmetry breaking in early universe.
• They produce a scaling network:
  • 📏 All lengths grow with the horizon size.
  • ↔️ Fraction of energy in the network is constant.
• Lose energy by gravitational + particle radiation.
• Loops of string are long-lived
  ☛ emit GWs over many wavelengths as they shrink.

Next page: video credit David Daverio, John Favre

Topological stability

In field theory, cosmic strings are topologically stable:

Can also be modelled by thin Nambu-Goto strings.

What about cosmic necklaces?

• Necklaces are networks of strings carrying monopoles.
• Models can be embedded in GUTs, including $\mathrm{SO}(10)$.
• May not behave quite like strings; role of monopoles?
• Observations:
  • 📡 Cosmic ray and $\gamma$-ray signals? astro-ph/9704257
  • 〰️ Gravitational waves? (or lack thereof)

The model

• "$\mathrm{SU}(2)$ Georgi-Glashow model, two adjoint Higgses" $$ \begin{multline} V(\Phi_1,\Phi_2) = - m_1^2\mathrm{Tr}\Phi_1^2 - m_2^2\mathrm{Tr}\Phi_2^2 \\ +\lambda (\mathrm{Tr}\Phi_1^2)^2 + \lambda (\mathrm{Tr}\Phi_2^2)^2 +\kappa(\mathrm{Tr}\Phi_1\Phi_2)^2. \end{multline} $$
• If $m_1^2 > m_2^2$: necklaces with monopoles, charge $\{\pm 1\}$.
• If $m_1^2 = m_2^2$, behaviour depends on $\kappa/2\lambda$:
  • If $\kappa = 2\lambda$, there's a global $\mathrm{U}(1)$ - superfluidity.
  • If $\kappa \neq 2\lambda$, monopoles split into two semipoles with complex charge $\{1,i,-1,-i\}$.

Necklaces

• When $m_1^2 > m_2^2$, $\mathrm{U}(1)$ breaks to $Z_2$
• Strings form, joining up the monopoles

Semipoles 1

• When $m_1^2 = m_2^2$ and $\kappa/2\lambda < 1$, get semipoles.
• Four types of pole; complex charge $\{1, i, -1, -i\}$.
• Linked by four types of string; complexified flux $\Phi_B^{(1)} + i \Phi_B^{(2)} \in \{1 + i, -1 + i, -1 - i, 1 - i\}$.

Semipoles 2

• When $m_1^2 = m_2^2$ and $\kappa/2\lambda > 1$, slight difference...
• Vacua rotated by 45° so string fields are rotated, too, $$\Phi_\pm = (\Phi_1 \pm \Phi_2)/\sqrt{2}.$$

Special case

• When $m_1^2 = m_2^2$ and $\kappa/2\lambda = 1$, there is a global $\mathrm{U}(1)$.

(We won't look at this any further today...)

Do the simulations work?

• ✅ Yes, we can obtain the residual $\mathrm{U}(1)$ field.
• ✅ Yes, we can measure the vortex winding associated with the other Higgs field.
• ✅ Yes, they're both gauge invariant.
• Necklaces: heavier field (our $\Phi_1$) forms monopoles.
• For semipoles, we have to make two magnetic fields,
  • either $\mathbf{B}^{(1)}$ and $\mathbf{B}^{(2)}$ from $\Phi_1$ and $\Phi_2$
  • or $\mathbf{B}^{(\pm)}$ from $\Phi^{(\pm)} = \frac{1}{\sqrt{2}}(\Phi_1 \pm \Phi_2)$.

String and pole position movie

Limitations of the simulations

(in addition to the usual caveats for field theory strings)

• Mass scales $m_1^2$ and $m_2^2$ will never be that different 😕
  • Largest we have is $m_2^2/m_1^2 = 0.04$.
  • Decrease $m_2$? Fatter strings reduce statistics.
  • Decrease $m_2$? Defect formation dynamics happens on time $1/m_2$ - need longer simulations.
  • Increase $m_1$? Small monopoles can get stuck on lattice.

Movie interlude

(We also have 3D and 360 movies!)

Necklaces

Semipoles 1 ($\kappa/2\lambda < 1$)

Semipoles 2 ($\kappa/2\lambda > 1$)

(note, isosurfaces of unrotated $\Phi_1$, $\Phi_2$)

Dynamics: key quantities

• Lattice projectors give the positions of monopoles (total $N$) and plaquettes with winding (total $L$).
• Average defect separations in volume $V$ $$\xi_\mathrm{m} = \left(\frac{V}{N}\right)^{1/3}; \qquad \xi_\mathrm{s} = \left(\frac{V}{L}\right)^{1/2}.$$
• Comoving pole density $$ n = \frac{\xi_\mathrm{s}^2}{\xi_\mathrm{m}^3}. $$

$\xi_\mathrm{s}$ and $\xi_\mathrm{m}$ - necklaces

• String separation: linear scaling in conformal time $\tau$.
• Monopole separation: $\xi_m \mathrm \propto{\tau}^{2/3}$.
• No real dependence on initial $\xi_\mathrm{m}$.

$\xi_\mathrm{s}$ and $\xi_\mathrm{m}$ - semipoles

• Same general story as for necklaces.
• Some hints of different $\kappa/2\lambda$ behaviour...

Monopole linear density - necklaces and semipoles

• Note 'pseudopoles': wrong complexified $\mathbf{B}$ fields.
• Everything physical goes to constant $n$ (more or less).