Baryon oscillations
Theory
Martin White
UC Berkeley
Outline
• Linear theory
– Stability
– Physics
– Weird stuff at z~103
• Beyond linear theory
– Dark matter
– Redshift space distortions
– Galaxy bias
Linear perturbation theory
• Key to the BAO method is the fact that the (linear) theory
of perturbations is well understood and the sound horizon
can be inferred from z~103 physics.
• However the physics is not completely trivial - no analytic
model exists.
Seljak, Sugiyama, White & Zaldarriaga (2003)
Numerical stability
Recombination
Non-standard scenarios
• Our method hinges on being able to predict the
sound horizon, s:
– Recombination (atomic physics) is very robust.
– Remaining dependence is on ρB/ργ and zeq.
• We can get ρB/ργ from CMB (peaks & damping)
• The CMB also fixes zeq very well (from high l)
– Potential envelope depends on zeq
– s is relatively insensitive to zeq.
(Hu & White 1997)
• Decreasing zeq by 500 decreases s by 5%
Extra radiation?
• For 3 relativistic ν species, knowing ργ (from Tγ)
gives ωrad=Ωradh2.
• Knowing zeq gives ωm.
• What if ωrad was different?
• As long as zeq is still known reasonably well it
doesn’t matter! Misestimate ωm
–
–
–
–
Comparing rulers at z~103 and z~1.
Same ωm prefactor enters H, dA as s.
All DE inferences go through unchanged!
Misestimate H0.
Eisenstein & White (2004)
Decaying “X” ?
A non-relativistic (massive) particle which undergoes a momentum
conserving decay into massless neutrinos with lifetime τ leads to excess
small-scale power and a shift in sound horizon.
Beyond linear theory …
• Unfortunately we don’t measure the linear theory
matter power spectrum in real space.
• We measure:
– the non-linear
– galaxy power spectrum
– in redshift space
• How do we handle this and what does it mean for
the method?
BAO surveys are always in the sample variance dominated regime.
Cannot afford to take a large “hit” due to theoretical uncertainties!
Numerical simulations
• Our ability to simulate structure formation has
increased tremendously in the last decade.
• Simulating the dark matter for BAO:
– Meiksin, White & Peacock (1999)
• 106 particles, 102 dynamic range, ~1Gpc3
– Springel et al. (2005)
• 1010 particles, 104 dynamic range, 0.1Gpc3
– Huff, Schulz, Schlegel, Warren & White (in prep)
• Many runs of 109 particles, 104 dynamic range, several Gpc3
• Our understanding of galaxy formation has also
increased dramatically.
Non-linearities (easy part)
White (2005)
δP(k)/P(k)
Updated from Heitmann et al. (2005)
0.1
1.0
Current accuracy is a few percent among the better codes.
Galaxy bias
• The hardest issue is galaxy bias.
– Galaxies don’t faithfully trace the mass
• Here we use large numerical simulations with adhoc galaxy recipes.
– Rather than try to predict the unique “right” answer for
galaxy formation we want to explore a range of
plausible alternatives.
– We do this by assigning galaxies to the halos found in
dark matter simulations using phenomenological rules.
– The resulting catalogs exhibit scale-dependent,
stochastic, non-linear bias of the galaxies wrt the dark
matter.
Huff, Schulz, Schlegel, Warren, White.
Eisenstein, Seo, White.
An example
A slice, 10Mpc
thick, through a
1Gpc3 simulation.
Each panel zooms in
on the previous 1 by
a factor of 4.
The color scale is
logarithmic, from
just below mean
density to 102x
mean density.
Points mark galaxy
positions.
An example
A slice, 10h-1 Mpc
thick, through a
1h-3Gpc3 simulation.
Each panel zooms in
on the previous 1 by
a factor of 8.
The color scale is
logarithmic, from
just below mean
density to 102x
mean density.
Points mark galaxy
positions.
Insight vs Numbers
• Trying to learn from these simulations
– What range of behaviors do we see?
– Which D/A algorithms work best?
– How do we parameterize the effects?
• Can we gain an analytic understanding of the
issues?
• Are there shortcuts for describing the
complexities?
– Bias on large scales, excess power on small scales.
Toy model I
• We can understand the main features with a
simple “toy” model: halo model.
• There are two contributions to the 2-point function
of objects:
1-halo
2-halo
2-halo
1-halo
Toy model II
• If the halos form a biased tracer of the linear
theory density field, with a bias depending on their
mass, then
• Definite predictions for Pgal(k) which depend on
the number of galaxies in halos of mass M, N(M),
and how they are spatially distributed.
– However on the scales of interest only N(M) matters.
satellite
N(M)
central
M
Toy model III
If we work on scales much larger than the virial radius of a
typical halo, the halo profile is sub-dominant. Then
With a similar expression for the dark matter with the
replacement of Ngal with Mhalo.
The tradeoff between the 1- and 2-halo terms occurs at
different k for the galaxies and DM, leading to a scaledependent bias.
Schulz & White (2005)
Scale-dependent bias
Power ~ k3 P(k)
Schulz & White (2005)
In our model scale
dependence of bias is
enhanced when:
At fixed ngal, bias
increases.
At fixed bias, ngal
decreases.
Scale dependence
increases faster with b
for rarer objects.
Wavenumber
Perhaps a real space
description is better!
Bias + shot-noise decomposition
Real space
k (h/Mpc)
Huff et al.
Real space description
But the 1-halo term is confined to small-r in real space!
Measuring ξ( r) in periodic boxes is problematic -- instead measure
Huff et al.
which is insensitive to
low-k modes, mean
density estimate etc.
Look at residual scale
dependence and any
systematic shifts in the
peaks.
Conclusions
• Baryon oscillations are a firm prediction of CDM models
relying (mostly) on linear physics.
• For DE inferences method is surprisingly robust to
uncertainties in physics at z~103
• Both precision and systematic mitigation are dramatically
improved with Planck data.
• Understanding structure and galaxy formation to the level
required to maximize our return on investment will be an
exciting and difficult challenge for theorists!
• We need a “turn-key” method for extracting DE science from
mock data to evaluate the effects of various choices a realworld survey needs to make.