1. Adaptive gravitational softening in GADGET
Iannuzzi & Dolag (2011) and more

2. What softening is
*Monaghan & Lattanzio 1985, A&A, 149, 135

3. What we need it for
Depends on the kind of simulation one is performing:
Collisional simulation - need softening to avoid divergences and reduce
the computational time
Collisionless simulation - need softening to reduce computational time
and moderate the noise induced by the under-representation of the
particles’ DF

4. Noise vs. Bias and the optimal h
(Merritt 1996, AJ, 111, 2462)
NOISEr = (Fr − Fr )2
BIASr = Fr −Fr,true
ASEr = BIAS2
r +NOISEr = (Fr − Fr,true)2

5. Optimal h: does it always exist?

6. Adaptive softening
(Price & Monaghan 2007, MNRAS, 374, 1347)
Definition 4
3πh3ρ = mNngbs =⇒ h ∝ ρ−1/3
New Lagrangian Lgrav = N
i=1 mi
1
2v2
i − Φ (ri , h(ri ))
Equation of Motion
dvj
dt
= − G
N
i=1
mi
φij (hj ) + φij (hi )
2
rj − ri
|rj − ri |
−
G
2
N
i=1
mi
ζj
Ωj
∂Wij (hj )
∂rj
+
ζi
Ωi
∂Wij (hi )
∂rj

7. Adaptive softening
(Price & Monaghan 2007, MNRAS, 374, 1347)
Definition 4
3πh3ρ = mNngbs =⇒ h ∝ ρ−1/3
New Lagrangian Lgrav = N
i=1 mi
1
2v2
i − Φ (ri , h(ri ))
Equation of Motion
dvj
dt
= − G
N
i=1
mi
φij (hj ) + φij (hi )
2
rj − ri
|rj − ri |
−
G
2
N
i=1
mi
ζj
Ωj
∂Wij (hj )
∂rj
+
ζi
Ωi
∂Wij (hi )
∂rj

8. Adaptive softening
(Price & Monaghan 2007, MNRAS, 374, 1347)
Definition 4
3πh3ρ = mNngbs =⇒ h ∝ ρ−1/3
New Lagrangian Lgrav = N
i=1 mi
1
2v2
i − Φ (ri , h(ri ))
Equation of Motion
dvj
dt
= − G
N
i=1
mi
φij (hj ) + φij (hi )
2
rj − ri
|rj − ri |
−
G
2
N
i=1
mi
ζj
Ωj
∂Wij (hj )
∂rj
+
ζi
Ωi
∂Wij (hi )
∂rj

9. Adaptive softening
(Price & Monaghan 2007, MNRAS, 374, 1347)
Definition 4
3πh3ρ = mNngbs =⇒ h ∝ ρ−1/3
New Lagrangian Lgrav = N
i=1 mi
1
2v2
i − Φ (ri , h(ri ))
Equation of Motion
dvj
dt
= − G
N
i=1
mi
φij (hj ) + φij (hi )
2
rj − ri
|rj − ri |
−
G
2
N
i=1
mi
ζj
Ωj
∂Wij (hj )
∂rj
+
ζi
Ωi
∂Wij (hi )
∂rj

10. Adaptive softening
(Price & Monaghan 2007, MNRAS, 374, 1347)
Zeta
ζi ≡
∂hi
∂ρi
N
k=0
mk
∂φik(hi )
∂hi
Omega
Ωi ≡ 1 −
∂hi
∂ρi
N
k=0
mk
∂Wik(hi )
∂hi

11. Optimal h: fixed vs. adaptive softening

12. Correction term and energy conservation

13. Effects in a cosmological environment
No analytical solution
Keep the same power spectrum and increase the resolution
Fixed 643
Fixed 1283
Fixed 2563
Adapt 643
Adapt 1283
Adapt+corr 643
Adapt+corr 1283

14. Mass functions

15. Correlation functions

16. The most massive halo
M200 1015
M ; r200 2.4 h−1
Mpc; 3000, 21000, 160000 particles

17. The most massive halo
M200 1015
M ; r200 2.4 h−1
Mpc; 3000, 21000, 160000 particles

18. The most massive halo:
substructures

19. An “adaptive” mini-MillenniumII
low-resolution version of the MillenniumII (Boylan-Kolchin et al. 2009)
same power-spectrum, 125 times less particles
mII mini-mII
adaptive
mini-mII

20. An “adaptive” mini-MillenniumII:
mass function

21. An “adaptive” mini-MillenniumII:
correlation function

22. Conclusions (I)
adaptive softening lengths provide near-optimal softening with little
dependance on Nngbs
in order to retain energy conservation the equation of motion needs to
be modified
the use of adaptive softening in a cosmological simulation enhances
the clustering of particles at small scales, anticipating the results
obtained in higher resolution simulations

23. Hybrid simulations
Fields sampled by particles of different masses:
Collisionless species - to start with
Hydrodynamical simulations - eventually
Do we have an impact on two-body effects?

24. Hybrid collisionless simulation:
evaporation of the light component from halos

25. Hybrid collisionless simulation:
mass functions

26. Hybrid collisionless simulation:
evaporation of the light component from halos

27. Hybrid collisionless simulation:
evaporation of the light component from halos

28. Conclusions (II)
simulations with equal-mass particles are well behaved
in simulations with particles of different masses the effect of varying
Nngbs on the correction term is only partially understood
possible solutions may require non-immediate modifications of the
method
the use of adaptive softening (without correction) moderates the
impact of two-body effects in hybrid simulations

30. Distribution of the softening lengths

31. Time bins

32. Dependence on the number of neighbours
Inner density profile of a Hernquist sphere

33. Passive evolution of a bulge+halo system