Low-z BAOs:
proving acceleration and testing Neff
Will Sutherland (QMUL)
Talk overview
Cases for low-redshift BAO surveys :
1.
Smoking-gun test of cosmic acceleration - assumes
only homogeneity & isotropy, not GR.
2.
Testing fundamental assumptions from CMB era, in
particular the number of neutrino species.
2005: first observation of predicted BAO feature
by SDSS and 2dFGRS
(Eisenstein et al 2005)
BAO feature in BOSS DR9 data: ~ 6 sigma
(Anderson et al 2012)
BAO observables: transverse and radial
Spherical average gives rs / DV ,
BAOs : strengths and weaknesses
 BAO length scale calibrated by the CMB .
+ Uses well-understood linear physics (unlike SNe).
- CMB is very distant: hard to independently verify assumptions.
 BAO length scale is very large, ~ 153 Mpc:
+ Ruler is robust against non-linearity, details of galaxy formation
+ Observables very simple: galaxy redshifts and positions.
- Huge volumes must be surveyed to get a precise measurement.
- Can’t measure BAO scale at “ z ~ 0 ”
 + BAOs can probe both DA(z) and H(z); no differentiation
needed for H(z). More sensitive to “features” in H(z);
enables consistency tests for flatness, homogeneity.
“Cosmic speed trap:”
Proving cosmic acceleration with BAOs only
 Assuming homogeneity, evidence for accelerating
expansion is strong: SNe, CMB+low-z measurements .
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SNe require acceleration independent of GR (if no evolution, and
photon number conserved)
CMB + LSS : acceleration evidence very strong, but requires
assumption of GR.
 Possible loophole to allow non-accelerating model :
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Assume SNe are flawed by evolution and/or photon nonconserving processes (peculiar dust, photon/dark sector
scattering).
AND: GR not correct, so CMB inferences are misleading.
 This is contrived, but we should close this loophole
Cosmic expansion rate: da/dt
Cosmic expansion rate, relative to today
BOSS: Busca et al 2012
Caveat: assumed flatness and standard rs
Speed-trap: motivations
 Radial BAO scale directly measures rs H(z) / c
 Ratio of two such measurements will cancel rs , and detect
acceleration directly.
 BUT, there is a practical problem:
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very feeble acceleration at z > 0.3
Not enough volume to measure radial BAOs at z < 0.3 .
Can’t measure rs H0 at “z ~ 0”.
 Spherical-average BAOs can prove acceleration IF we
assume almost-flatness, but we don’t want to rely on this.
 Workaround: use radial BAO at z ~ 0.7, compare to
spherical-average BAO observable at z ~ 0.2 .
Limit relating DV(z1) and H(z2) for any non-accelerating model:
Comoving radial distance:
No acceleration requires :
therefore:
Assuming homogeneity, angular-diameter distance is :
No acceleration requires :
Therefore :
Open curvature:
()>1
Closed
curvature :
radial BAO
observable:
Spherical-average BAO
observable, at z1 :
Divide:
Use previous limit for DV :
Rearrange square-bracket onto LHS:
now RHS becomes 1 + O(z2),
depends very weakly on curvature.
Define XS as “excess speed” , ratio of BAO observables:
Flat models : RHS = 1 exactly .
Open models : RHS < 1 … limit gets stronger.
Closed models : RHS > 1 … need to constrain this.
But, closed models have a maximum angular diameter
distance < Rc / (1+z) , so z ~ 3 galaxy sizes
eliminate “super-closed” models.
(Sutherland, MN 2012,
arXiv:1105.3838)
Blue/green: predictions for LambdaCDM / wCDM
Red: upper limits for non-accelerating model, various (extreme)
curvatures.
Speed-trap result:
 If we assume
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Homogeneity and isotropy
Redshift due to cosmic expansion, and constant speed of light
BAO length conserved in comoving coordinates
No acceleration after redshift z2
 Then :
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Observable BAO ratio must be below red-lines above
 If observed XS > 3 sigma above red-line ,
at least one of four statements above is false.
 “Signal” ~ 10 percent: need < 3% (ideally 2%) precision
on ratio of two BAO observables. Challenging, but
definitely achievable.
Measuring the absolute scale of BAOs :
 BAO length scale is essentially the sound horizon at “drag
redshift” zd ~ 1020.
 If we assume
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Standard GR
Standard neutrino content
Standard recombination history
Nearly pure adiabatic fluctuations
Negligible early dark energy
Negligible variation in fundamental constants
 Then BAO length depends on just two numbers, ωm and
ωb ; both well determined by WMAP and Planck.
 WMAP results give rs(zd) = 153 ± 2 Mpc (1.3 percent).
Planck gives rs(zd) = 151.7 ± 0.5 Mpc (0.33 percent).
Measuring the absolute scale of BAOs (2):
 Above assumptions are (mostly) testable from CMB
acoustic peaks structure.
 But there’s a risk of circular argument… a wrong
assumption may be “masked” by fitting biased values of
cosmological parameters – especially H0 ; also Ωm, w etc.
 Highly desirable to actually measure the BAO length with a
CMB-independent method.
 “Obvious” way: measure transverse BAOs and DL(z) at
same redshift; distance duality gives DA(z) and absolute
BAO scale.
 Would like to work at lower z , and use DV(z)
 Snag: DV(z) is not directly measurable with standard
distance indicators.
Effect of non-standard radiation density
Definition of Neff :
Matter density:
Sound horizon in
terms of rad. density
and zeq :
Define
and use
base parameter set :
(WMAP7: Komatsu et al 2010)
WMAP7 likelihood contours:
Strong degeneracy between Neff and ωm ;
but zeq is basically unaffected.
WMAP7 likelihood contours:
Not exact, but accurate summary :
 If we drop the assumption of standard Neff, then:
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WMAP still tells us redshift of matter-radiation equality ~ 3200,
(Planck ~ 3350) , but the physical matter and radiation densities
are much less precise.
Keeping CMB acoustic angle constant requires physical dark
energy density to scale in proportion to matter & radiation.
best-fit inferred H0 scales as √(Xrad)
Sound horizon rs scales as 1/ √(Xrad) .
The BAO observables don’t change: inferred Ωm , w are nearly
unbiased (Eisenstein & White 2004).
 If a 4th neutrino species, equivalent to 13.4% increase in
densities, 6.5% increase in H (e.g. 70 to 74.5) and 6.1%
reduction in cosmic distances/ages. Substantial effect !
Neff affects all dimensionful parameters :
 Nearly all our WMAP + SNe + BAO observables are actually
dimensionless (apart from photon+baryon densities) :
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redshift of matter-radiation equality
CMB acoustic angle
SNe give us distance ratios or H0 DL /c .
BAOs also give distance ratios.
 All these can give us robust values for Ω’s , w, E(z) etc ; almost
independent of Neff .
 But: there are 3 dimensionful quantities in FRW cosmology ;
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Distances, times, densities.
Two inter-relations : distance/time via c ,and Friedmann equation relates
density + timescale, via G.
This leaves one short, i.e. any number of dimensionless distance ratios
can’t determine overall scale.
Usually, scales are (implicitly) anchored to the standard radiation density,
Neff ~ 3.04 . But if we drop this, then there is one overall unknown scale
factor.
Neff , continued…
 Photon and baryon densities are determined in absolute units…
but these don’t appear separately in Friedmann eq., only as
partial sums.
 Rescaling total radiation, total matter and dark energy densities
by a common factor leaves WMAP, BAO and SNe observables
(almost) unchanged; but changes dimensionful quantities e.g. H.
 Potential source of confusion: use of h and ω’s. These are
unitless but they are not really dimensionless, since they involve
arbitrary choice of H = 100 km/s/Mpc , and corresponding
density.
What BAOs really measure :
 Standard rule-of-thumb is “CMB measures ωm , and the sound
horizon; then BAOs measure h ” : only true assuming standard
radiation density.
 Really, CMB measures zeq ; adding a low-redshift BAO ratio
measures (almost) Ωm. These two tell us H0 / √(Xrad) , but not
an absolute scale.
 Thus, measuring the absolute BAO length provides a strong
test of standard early-universe cosmology, especially the
radiation content (Neff).
 Measuring just H0 is less good, since it mixes Neff, w and
curvature. The absolute BAO scale probes only the early
universe.
Measuring the absolute BAO scale (3) :
 Need two observations: a relative BAO ratio at some redshift,
and an absolute distance measurement to a matching redshift.
 It is generally easier to measure cosmic distances at lower z ~
0.25, which favours BAOs at moderate redshift.
 For SNe, the issue is evolution, so shorter time lever arm is
favourable.
 SNe are better in near-IR (Barone-Nugent et al 2012); sweet
spot at z~0.3 where rest-frame J, H appear in observed H,K.
 For lens time delays, degeneracy with cosmology: zl << zs is
favourable for absolute distances.
 The “ideal” distance indicators long-term may be gravitational
wave standard sirens; precision limited by SNR , favours lower
z.
 It is feasible to reach 1.5% precision on BAO ratio at z ~ 0.25 ;
this is probably better than medium-term distance indicators.
Measuring the absolute scale of BAOs (4):
 Most robust quantity from a BAO survey is rs / DV(z) ;
this is (almost) theory-independent.
 DV is related to comoving volume per unit redshift…
 Could measure DV exactly if we had a population of
“standard counters” of known comoving number density.
But prospects don’t look good – galaxy evolution.
 At very low z, DV ≈ c z / H0 . But error is 6% at z ~ 0.2 :
much too inaccurate.
 Next we’ll find much better approximations for DV(z)…
Pretty good approximations (< 0.5 percent at z < 0.4):
Suitable choice of z’s can eliminate H and gives :
Relative accuracy of approximation :
1 percent
Relative accuracy of approximation :
1 percent
Better
approximation:
1 percent
Accuracy < 0.2 percent at z < 0.5
An easy route to Ωm
h becomes a
derived parameter:
Define ε as error in
approximation :
BAO ratio is :
This is exact (apart from non-linear shifts in rs )
and fully dimensionless: all H and ω’s cancelled.
An easy route to Ωm
For WMAP baryon density,
the above simplifies to the following , to 0.4 percent :
This is all dimensionless, and nicely splits z-dependent effects:
• Zeroth-order term is just Ωm-0.5 (strictly Ωcb , without neutrinos)
• Leading order z-dependence is E(2z/3)
• The εV is second-order in z, usually ~ z2 / 25 and almost negligible
at z < 0.5
An easy route to Ωm
Repeat approximation from previous slide :
Substituting in the WMAP range for zeq ,
and the BAO measurement at z = 0.35
from Padmanabhan et al (2012),
and discarding the sub-percent εV , this gives
And just square and rearrange to :
Why DV approximation is good:
post-hoc explanation using Taylor series
Deceleration and
Jerk parameters:
For “reasonable” models, abs [ ] < 4 … leading order error < z2 / 27
Conclusions :
 BAOs are a gold standard for cosmological standard rulers.
Very well understood; observations huge in scope but clean.
 Most planned BAO surveys are targeting z > 0.7, to exploit the
huge available volume and sensitivity to dark energy w.
 However, there are still two good cases for optimal low-z
BAO surveys at z ~ 0.25 (e.g. extending BOSS to South and
lower galactic latitude) :
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A third direct test of cosmic acceleration, without GR
assumption. (arXiv:1105.3838)
In conjunction with precision distance measurements, can provide
a test of the CMB prediction rs ~ 151 Mpc, and/or a clean test for
extra “dark radiation”, independent of DE and curvature.
(arXiv:1205.0715)
Thank you !