Angular momentum evolution can be predicted from cosmological initial conditions

 

With A. Pontzen & H. V. Peiris — arXiv:2012.02201

The origin of angular momentum

 

  • Galaxies         → origin of rotational support?
  • Physics           → origin of vector quantity irrotational field?
  • Cosmology    → origin of spin alignment signal?

 

New-Horizon simulation — Dubois+20

Tempel+13

Focus on closed systems: Lagrangian patch

Predict one value for \( \vec{L} \)

Focus on open systems: dark matter haloes

Predict \( p(\vec{L}) \)

Tidal Torque Theory

& semi-Analytical Models

$$ \vec{L} = a^2 \dot{D} \vec{L}_0 $$

White 84, Codis+15

Vitvitska+02

AM can be predicted

AM is stochastic/chaotic

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?

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?

How to change \(\color{blue}\vec{L}_0\) all other things being equal?

\(\Lambda\)CDM gives you \(\neq\color{blue} \vec{L}_0\) … all other things different.

*Only an illustration, ICs do not correspond to the halos below.

\(N\)-body numerical integration

\(N\)-body numerical integration

+

Original \(\vec{L}_0\)

Genetic modif.

\(N\)-body numerical integration

Contrast x50

Roth+15, Stopyra+20

Can compute

$$ P\big(\delta \big| \delta(x) = v\big) $$

or

$$ P\Big(\delta \Big| \sum_i \alpha_i \delta_i = v\Big) $$

or 

$$ P\Big(\delta \Big| L_x = v \Big) $$

 

+ minimization of \(\Delta \chi^2\)
⇒ unique solution
=  "genetically modified density field"

+

=

Original \(\vec{L}_0\)

Genetic modif.

New initial \(\vec{L}_0'\)

\(N\)-body numerical integration

\(N\)-body numerical integration

Same region & environment

 

Different \(\color{blue} \vec{L}_0\)

Contrast x50

Roth+15, Stopyra+20

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

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

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

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

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

AM genetic modifications

Initial \(\vec{L}\) given in initial conditions by

$$ L_{z,0} \propto \sum_{i,j,k}\left[ (q_{x}^{ijk} - \bar{q}_x) \nabla_y\phi^{ijk} - (q_{y}^{ijk} - \bar{q}_y) \nabla_x\phi^{ijk}\right]. $$

 

Genetic modifications: find \(\vec{u}\) such that

$$ \vec{u} \cdot \vec{\delta} = \vec{L}_0,$$

with \(\vec{\delta} = \{\delta^{i_0j_0k_0}, \dots, \delta^{i_nj_nk_n}\} \) and solve for

$$ \vec{u}\cdot \vec{\delta}' = \vec{L}_0'$$

with minimal change.

deck

By Corentin Cadiou