Focus on closed systems: Lagrangian patch

Predict one value for $\vec{L}$

Tidal Torque Theory

& semi-Analytical Models

TTT

Focus on closed systems: Lagrangian patch

Predict one value for $\vec{L}$

sAMs

Focus on open systems: dark matter haloes

Predict $p(\vec{L})$

TTT & semi-analytical models

AM can be predicted from the initial conditions

AM is stochastic/chaotic

Is AM chaotic or not?

Not chaotic if for a given region,

$$\vec{L}(t) = {\color{red}f(t)} \times \color{blue}\vec{L}_0,\phantom{(1+\epsilon),}$$

with $\color{red} f$ indep. of $\color{blue}\vec{L}_0$.

Is AM chaotic or not?

Not chaotic if for a given region,

$$\vec{L}(t) = {\color{red}f(t)} \times \color{blue}\vec{L}_0\phantom{,}{(1+\epsilon)},$$

with $\color{red} f$ indep. of $\color{blue}\vec{L}_0$.

Is AM chaotic or not?

Original $\vec{L}_0$

$N$-body numerical integration

Original $\vec{L}_0$

$N$-body numerical integration

Original $\vec{L}_0$

New initial $\vec{L}_0'$

Original $\vec{L}$

New initial $\vec{L}'$

We can predict/control AM down to $z = 0$.*

How does it compare to predictions from linear theory?

*AM of regions, not halos

Prediction from linear theory:

$$\vec{L}_\mathrm{pred.} = {\color{blue}\underbrace{\quad a^2\dot{D} \quad}_{\color{blue}\mathrm{time}}} \times {\color{red}\underbrace{\quad\vec{L}_0\quad}_{\mathrm{space}}}.$$

Simulations with genetically modified initial conditions

$$\vec{L}_\mathrm{pred.} = {\color{blue}\underbrace{\quad f(t) \quad}_{\color{blue}\mathrm{time}}} \times {\color{red}\underbrace{\quad\vec{L}_0\quad}_{\mathrm{space}}},$$where $\color{blue}f(t)$ is the measured growth rate of a given region in the ref. simu.

Comparison with tidal torque theory

Comparison with tidal torque theory

Median of deviations

Prediction from linear theory:

$$\vec{L}_\mathrm{pred.} = {\color{blue}\underbrace{\quad a^2\dot{D} \quad}_{\color{blue}\mathrm{time}}} \times {\color{red}\underbrace{\quad\vec{L}_0\quad}_{\mathrm{space}}}.$$

Simulations with genetically modified initial conditions

$$\vec{L}_\mathrm{pred.} = {\color{blue}\underbrace{\quad f(t) \quad}_{\color{blue}\mathrm{time}}} \times {\color{red}\underbrace{\quad\vec{L}_0\quad}_{\mathrm{space}}},$$where $\color{blue}f(t)$ is the measured growth rate of a given region in the ref. simu.

Comparison with tidal torque theory

Median of deviations

Scatter of deviations

Prediction from linear theory:

$$\vec{L}_\mathrm{pred.} = {\color{blue}\underbrace{\quad a^2\dot{D} \quad}_{\color{blue}\mathrm{time}}} \times {\color{red}\underbrace{\quad\vec{L}_0\quad}_{\mathrm{space}}}.$$

Simulations with genetically modified initial conditions

$$\vec{L}_\mathrm{pred.} = {\color{blue}\underbrace{\quad f(t) \quad}_{\color{blue}\mathrm{time}}} \times {\color{red}\underbrace{\quad\vec{L}_0\quad}_{\mathrm{space}}},$$where $\color{blue}f(t)$ is the measured growth rate of a given region in the ref. simu.

Comparison with tidal torque theory

Median of deviations

Scatter of deviations

• Much more accurate pred. than linear theory
• Departure happen at later time
• The smaller the modif., the smaller the scatter
  ⇒ linear response

Conclusions

What's next?

• apply to baryons instead of DM only?
• extend analysis to halos?
• predict $\color{red}f(t)$ rather than measure it?

Read more in Cadiou, Pontzen & Peiris arXiv:2012.02201.

Explored how $\vec{L}$ of given fixed regions changes with changed $\color{blue}\vec{L}_0$

• angular momentum can be predicted accurately from initial conditions (not chaotic)
• possible to improve linear theory by predicting growth rate $\color{red} f(t)$
• apparent stochasticity of AM of halos ⇒ result of particle membership