1. Simulations of Gravitational Lensing on High z Supernovae
Premana W. Premadi
Institut Teknologi Bandung
Collaborator: Hugo Martel (Université Laval, Canada)
Second EANAM, Daejeon, 1-3 Nov 2006

2. Distance-redshift relation depends on cosmological model
Distance to supernovae at given redshifts:
Different model ® different distance ® different apparent magnitude
W0 = 0.27, l0 = 0.73
W0 = 0.3, l0 = 0
W0 = 1, l0 = 0
z = 2
z = 1
z = 0

3. Effect of lensing: modifying the apparent magnitude
The effect of lensing is to produce an amplification or a deamplification.
® difficult to distinguish models
W0 = 0.27, l0 = 0.73
W0 = 0.3, l0 = 0
W0 = 1, l0 = 0
z = 2
z = 1
z = 0

4. Gravitational lensing due to large scale structure
Numerical simulation: reproduce the large scale structure of the universe
Multiple lens-plane algorithm: permits the calculation of lensing effect
Fobs = mF0 m
z = 0
z = 5

5. Numerical Simulation of the Large Scale Structure
Reproduce the large scale
structure of the universe
Locate galaxies using the
density of the underlying
dark matter distribution

6. Morphology – Density Relation
Use morphology-density relation to assign morphology of each galaxy (cf. Dressler 1980)

7. Dynamical Range of the Simulated LSS:
128 Mpc to 400 pc

8. Calculating the lensing effect
Project the mass density onto planes perpendicular to line of sights
Calculate lens effect and build statistics of magnification:
Vary source redshift
Vary cosmological models m
z = 0
z = 5

9. Magnification Distribution
Empirical “fit” for standard deviation:
sm = az / (1 + bz)
The values of a and b depend on cosmological model

10. Calculation of lensing effect
log (dLH0) = a ± da (a : measurement;
da : ‘intrinsic’ uncertainty)
F = L / 4pdL2 ; H0 ( L / 4p )1/2 = 10a 10±da F1/2
Lensing effect : F ® F ± DF
H0 ( L / 4p )1/2 = 10a 10±da ( F ± DF )1/2
Taylor expansion to 1st order in da and DF:
H0 ( L / 4p )1/2 = 10a 10±daF1/2 ( 1 ± DF / 2F )
log (dLH0) = a ± da ± DF / 2F ln 10
For a single supernova : DF / F = m - 1
problem : impossible to measure m.
Statistically : DF / F » sm
log ( dLH0) = a ± da ± sm / 2 ln 10

11. Tonry et al. 2003 supernovae sample
0 < z < 1.8
D(m - M) º (m - M) - (m - M)empty
Observations support LCDM model
Correction due to lensing effect is negligible: 10% for the higher bin

12. Lensing effect on a hypothetical population of supernovae at very high redshifts. The generated mock catalogue of supernovae are at redshifts 1.8 < z < 8.
Lensing effect is important overall in LCDM model.
At z > 3, it is difficult to distinguish between open and LCDM.
Einstein-de Sitter model is eliminated with high level of confidence.

13. Conclusion
We have used a series of numerical simulations to determine the distribution of magnification at various redshifts in various cosmological models.
Using a simple analysis, we have estimated the effect of lensing on the determination of distances of supernovae Type Ia.
The effect of lensing is negligible for currently actually observed supernovae (z < 1.8), but becoming important for supernovae at very high redshift (1.8 < z < 8).
Once supernovae (or other standard candles) at very high redshift are discovered, we need to understand in detail the lensing effect if we want to use those supernovae to constrain cosmological models.