# Resolving symmetron dark energy

### Table of contents

### Abstract

This page gives an introduction the (a)symmetron dark energy model, and summarises the work performed in our paper on the convergence of non-linear features of the field using our implementation introduced in the previous paper. In addition, we will introduce some illustrations and animations further visualising the simulations. The larger idea of this project is to validate the dynamic and non-linear solutions we obtain so that we can stand on firm ground when making statements about the impact of their inclusions and the stability of domain walls. We establish that the domain wall stability is very sensitive to the resoution of the simulation.

## Model

We start out defining the action of the asymmetron as $$S = \int \rm d ^{4} x \sqrt{-g} \, \Bigg[\frac{1}{2} M_{\text{pl}}^2 R-\frac{1}{2} \nabla_{\mu} \phi \nabla^{\mu} \phi-V(\phi)\Bigg]+ S_{m}\left(\psi_{i}, \tilde{g}_{\mu \nu}\right) ,\tag{1} $$ where \(R\) is the Ricci scalar, \(S_m\) is the matter action of the standard model of cosmology, and \(\phi\) is the asymmetron scalar field. The asymmetron reduces to the symmetron whenever we define its potential, \(V\), to be symmetric. The matter action, \(S_m\) is a function of the Jordan frame metric \(\tilde g\) which relates to the Einstein frame metric, in which the gravitational action is the Einstein-Hilbert action, as $$\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu}, \tag{2} $$ and the conformal coupling \(A\) is defined as $$ A(\phi) = 1 + \frac{1}{2}\left(\frac{\phi}{M}\right)^2 ,\tag{3} $$ where \(M\) is the conformal coupling scale. This allows us to find the effective potential that sources the Klein-Gordon field equation of the asymmetron field $$ V_{\rm {eff}} \equiv V - \ln A(\phi) T_{\rm m} = V_0 - \frac{1}{2}\mu^2\phi^2 - \frac{1}{3}\kappa \phi^3 + \frac{1}{4}\lambda \phi^4 - \ln A(\phi) T_{\rm m}, \tag{4} $$ in which we take note of the interaction with cold dark matter and baryons through the trace of the stress energy tensor of matter \(T_m\). In our paper, we studied the resultant behaviour of the field through our implementation of it in the fully cosmological and relativistic N-body code gevolution. We will now review some of the setups that we considered.## Ideal wall

At first, we considered a simple system of a single domain wall at the centre of the simulation volume, on a homogeneous matter background. We choose for simplicity symmetron parameters \(L_C,z_*,\beta_* = 1\) Mpc/h, \(2, 1\), but we will take the analysis to be representative for a larger parameter space. Here the cubic term is zero, and therefore we are considering the symmetron.

At first, we initialised the field in the symmetry-broken regime, at redshift \(z=1.8\), as a simple step-function across the domain wall, and equal to the respective vacuum field value at each side. We then ran some iterations of the Gauss-Seidel relaxation to allow the field to settle in to its lowest energy state. If we run enough iterations, we notice that the field is behaving statically, once we start evolving it.

The less iterations of the Gauss-Seidel relaxation we do, the more excitedly the field will behave once we let it go. At a relatively low amount of iterations, we see the behaviour below.

After a while, some numerical imprecision caused by the relaxation scheme break the hemispherical symmetry, and the fields starts oscillating in a more complicated way. Since our purposes here is to establish convergence on the dynamic part of the theory, we therefore need to initialise the field in a different configuration than its equilibrium. We do so, while keeping more control of the configuration across different box resolutions, by initialising the field with its analytic solution for a homogeneous matter background, found among other in (...Claudio..) as

$$ \chi(x) = \sqrt{1-\left(\frac{a_*}{a}\right)^3} \tanh\left( \frac{a x}{2L_{IC}}\sqrt{1-\left(\frac{a_*}{a}\right)^3} \right),\tag{5} $$where \(\chi\equiv\phi/v\) is the symmetron normalised to its vacuum expectation value. Initialising the wall with \(L_{IC}=L_C\), would leave it static as expected. Using instead \(L_{IC}=L_C/2\), we see the oscillations below.

Finally, we can add one more ingredient and define a static inhomogeneous configuration of matter, in this case a \(4\times4\) grid of Guassian overdensities that have an amplitude of \(\delta_A = 10\).

In this case, we observe the same convergence behaviour in time as when we are evolving the field on a homogeneous background. In fact, as we show in the paper, the convergence seems to depend neither on the resolution, the scale of the initial conditions nor the Compton wavelength; As long as all these sizes are resolved, we are achieving convergence for Courant factors \(C_\phi \leq 1.4\).

In the above plot, 'fraction +' indicates the fraction of lattice where the field has a positive value (above some threshold \(\chi\gt 0.01\)). In the cases where the plot goes to zero, it is because the solution diverges and sets to zero, and once the field is completely zero, it has no source and will remain zero. We show below the field when it diverges.

We note the divergence seems to take place at first close to the domain walls at \(x=0,B/2\), where \(B\) is the length of the simulation box. This is a nice feature of using explicit methods, like fourth order Runge-Kutta, for these kinds of problems: When the solver starts diverging from the correct solution, it quickly becomes unstable and diverges completely. On the other hand, when it does not diverge, it seems to sufficiently resolve the oscillations in the box-averaged fraction of nodes in positive minimum that we are considering.

## Cosmological scenarios

The full set of differential equations includes that of cold dark matter too, which in the previous section has been left static or evolving according to the background density. We find that in the cosmological scenario, including cold dark matter's evolution, there is a dramatic decrease in the stability of the asymmetron field solver, so that we need a Courant factor of \(C_\phi\sim 0.2\) for the numerical scheme to remain stable throughout the simulations time. At least when we are spatially resolving all other relevant scales. We expect that the reason for this is not the inclusion of cold dark matter, but rather that we need to resolve the zero-point oscillations of the scalar field in the filaments where it attains a large mass due to the conformal coupling. Choosing the stable time resolution for the field, and a Courant factor for the cold dark matter of \(C_{\rm{cdm}} = 1\), as informed by our discussion in the paper, we find the evolution of the field shown below.

### Model Ⅰ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 2, 1, 0) \)

For the first choice of parameters, we have an early redshift symmetry breaking that coincides with the expected redshift parameter \(a_*\). This is presumably because the cold dark matter density field at the scale of the Compton wavelength still is rather homogeneous. We see an equation of state parameter that sets up the eventual domain walls before the time of the symmetry breaking, indicating important field evolution taking place, so that we need to initialise the simulations before this time. The domain walls are not very pinned to the filamentary cosmic structure, and sweep accross the box before eventually collapsing and releasing the excess energy as scalar waves.

### Model Ⅱ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 0.1, 1, 0) \)

For the remaining parameter choices, we pick small redshifts \(z_*=0.1\). For this choice, we see symmetry breaking happen locally within underdense voids, at approximately \(z_{\rm{SSB}}\sim 0.6\), due to the late-time, large density contrasts at Compton wavelength scales. Keeping the parameters otherwise similar to the above, we see a large suppression in the energy scale, and similarly in clustering enhancement.

### Model Ⅲ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 0.1, 8, 0) \)

We now increase the fifth force strength relative to gravity \(\beta_*=8\), so that the size of the conformal coupling \(\Delta A\) is similar to what it was in model Ⅰ. Apart from a larger energy scale, the field seems to behave and evolve similarly to in model Ⅱ.

### Model Ⅳ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(0.75\) Mpc/h\(\ , 0.1, 8, 0) \)

We choose a smaller Compton wavelength \(L_{C}=0.75\) Mpc/h and see finer structure in the symmetry broken field. Additionally, it collapses slightly earlier than in model Ⅲ, as expected from density contrasts being larger when smoothed over a smaller Compton scale. The resultant domains are completely different from the other late-time simulations, owing to the different phase velocity and symmetry-breaking time, giving different phases at the time of the symmetry breaking.

### Model Ⅴ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 0.1, 8, 10\%) \)

For the final model Ⅴ, we pick the same parameters as model Ⅲ, but add 10 % asymmetricity in the fifth force strengths of the two potential minimas. This reduces the domains' stability and causes the walls to collapse towards the true vacuum in the late-time when the domains have grown sufficiently close.