What if the galaxy had formed here instead?

What if the galaxy had formed here instead?

or here?

The “splicing” technique

1. Generate ICs
2. Integrate (\(N\)-body)
3. Select region of interest
4. Trace back to ICs
5. “Splice” the region
6. Integrate again

Splicing: equivalent of constraining density field at all points in region

\(\longleftarrow  50\,\mathrm{Mpc} \longrightarrow\)

Splicing in 1D

Splicing in 1D

Most likely field \(f\) with

• same value in spliced region (\(a\)),
• as close as possible outside (\(b\))

Mathematically, \({\color{green}f}\) is the unique solution that satisfies:

• \( {\color{green}f(x)}= {\color{blue}a(x)},\qquad\qquad\qquad\forall x \in\Gamma\)
• minimizes \[\mathcal{Q} = ({\color{red}b}-{\color{green}f})^\dagger\mathbf{C}^{-1}({\color{red}b}-{\color{green}f}),\quad \forall x \not\in \Gamma \]

The causal origin of DM halo concentration

The causal origin of DM halo concentration

The causal origin of DM halo concentration

\(\log\left|\dfrac{{\color{blue}\rho_\mathrm{i}} - \color{red}\rho_\mathrm{i}}{\sigma_\rho}\right|\)

\(\log\left|\dfrac{{\color{blue}v_\mathrm{i}} - \color{red}v_\mathrm{i}}{\sigma_v}\right|\)

Issue : splicing the density does not fix the velocity & tides

Issue : splicing the density does not fix the velocity & tides
 

Solution: splice the potential

\(\log\left|\dfrac{{\color{blue}\rho_\mathrm{i}} - \color{red}\rho_\mathrm{i}}{\sigma_\rho}\right|\)

\(\log\left|\dfrac{{\color{blue}v_\mathrm{i}} - \color{red}v_\mathrm{i}}{\sigma_v}\right|\)

Remember \(\nabla^2\phi = 4\pi G\rho\) so

\[\rho_n\propto\frac{\phi_{n+1}-2\phi_n+\phi_{n-1}}{\Delta x^2}\]

\[v_n\propto\frac{\phi_{n+1}-\phi_{n-1}}{2\Delta x}\]

\(z=0\)

\( z = 100\)

\(z=0\)

\( z = 100\)

[White 84]

\(z=0\)

\( z = 100\)

[Genetic modifications: Roth+16, see also Rey&Pontzen 18, Stopyra+20]

Time

Angular
momentum

✅ AM of fixed DM regions responds ~linearly (so is not chaotic!)

However, particle membership hence DM halo does not

Angular
momentum

Time