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BAO: Where We Are Now, What To Be Done, and Where We Are Going Eiichiro Komatsu The University of Texas at Austin UTAP Seminar, December 18, 2007 Dark Energy Everybody talks about it... What exactly do we need Dark Energy for?

  1. BAO: Where We Are Now, What To Be Done, and Where We Are Going Eiichiro Komatsu The University of Texas at Austin UTAP Seminar, December 18, 2007

  2. Dark Energy • Everybody talks about it... • What exactly do we need Dark Energy for? Baryon Dark Matter Dark Energy

  3. Need For Dark “Energy” • First of all, DE does not even need to be energy. • At present, anything that can explain the observed (1) Luminosity Distances (Type Ia supernovae) (2) Angular Diameter Distances (BAO, CMB) simultaneously is qualified for being called “Dark Energy.” • The candidates in the literature include: (a) energy, (b) modified gravity, and (c) extreme inhomogeneity.

  4. μ = 5Log 10 [D L (z)/Mpc] + 25 w(z) =P DE (z) / ρ DE (z) = w 0 +w a z/(1+z) Current Type Ia Supernova Samples Redshift, z Wood-Vasey et al. (2007)

  5. [residuals to this model] w(z)=w 0 +w a z/(1+z) Current Type Ia Supernova Samples Redshift, z Wood-Vasey et al. (2007)

  6. 2.0 • Within the standard ESSENCE+SNLS+gold ( ! M , ! " ) = (0.27,0.73) framework of ! Total =1 cosmology based on 1.5 General Relativity... • There is a clear ! " 1.0 indication that the matter density alone cannot explain the 0.5 supernova data. • Need Dark Energy. Current Type Ia Supernova Samples 0.0 0.0 0.5 1.0 1.5 2.0 ! M Wood-Vasey et al. (2007)

  7. • Within the standard 15 framework of ESSENCE+SNLS+gold cosmology based on (w 0 ,w a ) = ( − 1,0) General Relativity... 10 w(z) = P DE (z) / ρ DE (z) = w 0 + w a z/(1+z) • Dark Energy is Vacuum Energy consistent with 5 “vacuum energy,” w a a.k.a. cosmological 0 constant. • The uncertainty is − 5 still large. Goal: 10x Current Type Ia Supernova Samples reduction in the − 10 uncertainty. [StageIV] − 3 − 2 − 1 0 1 w 0 Wood-Vasey et al. (2007)

  8. D L (z) = (1+z) 2 D A (z) D L (z) Type 1a Supernovae D A (z) Galaxies (BAO) CMB 0.02 0.2 2 6 1090 Redshift, z • To measure D A (z), we need to know the intrinsic size. • What can we use as the standard ruler ?

  9. How Do We Measure D A (z)? d BAO θ Galaxies D A (galaxies)=d BAO / θ d CMB θ CMB D A (CMB)=d CMB / θ 0.02 0.2 2 6 1090 Redshift, z • If we know the intrinsic physical sizes, d, we can measure D A . What determines d?

  10. Just To Avoid Confusion... • When I say D L (z) and D A (z), I mean “physical distances.” The “comoving distances” are (1+z)D L (z) and (1+z)D A (z), respectively. • When I say d CMB and d BAO , I mean “physical sizes.” The “comoving sizes” are (1+z CMB )d CMB and (1+z BAO )d BAO , respectively. • Sometimes people use “r” for the comoving sizes. • E.g., r CMB = (1+z CMB )d CMB , and r BAO = (1+z BAO )d BAO .

  11. CMB as a Standard Ruler θ ~the typical size of hot/cold spots θ θ θ θ θ θ θ θ • The existence of typical spot size in image space yields oscillations in harmonic (Fourier) space. What determines the physical size of typical spots, d CMB ?

  12. Sound Horizon • The typical spot size, d CMB , is determined by the physical distance traveled by the sound wave from the Big Bang to the decoupling of photons at z CMB ~1090 (t CMB ~380,000 years). • The causal horizon (photon horizon) at t CMB is given by • d H (t CMB ) = a ( t CMB )*Integrate [ c dt/ a (t), {t,0,t CMB }]. • The sound horizon at t CMB is given by • d s (t CMB ) = a ( t CMB )* Integrate[ c s (t) dt/ a (t), {t,0,t CMB }], where c s (t) is the time-dependent speed of sound of photon-baryon fluid .

  13. l CMB =301.8 ± 1.2 Hinshaw et al. (2007) • The WMAP 3-year Number: • l CMB = π / θ = π D A (z CMB )/d s (z CMB ) = 301.8 ± 1.2 • CMB data constrain the ratio, D A (z CMB )/d s (z CMB ) .

  14. What D A (z CMB )/d s (z CMB ) Gives You • Color: constraint from l CMB =301.8 ± 1.2 l CMB = π D A (z CMB )/d s (z CMB ) with z EQ & Ω b h 2 . 1- Ω m - Ω Λ = • Black contours: Markov 0.3040 Ω m +0.4067 Ω Λ Chain from WMAP 3yr (Spergel et al. 2007)

  15. 2.0 ESSENCE+SNLS+gold ( ! M , ! " ) = (0.27,0.73) ! Total =1 1.5 ! " 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 ! M

  16. BAO as a Standard Ruler Okumura et al. (2007) Position Space Fourier Space Percival et al. (2006) (1+z)d BAO • The existence of a localized clustering scale in the 2-point function yields oscillations in Fourier space. What determines the physical size of clustering, d BAO ?

  17. Sound Horizon Again • The clustering scale, d BAO , is given by the physical distance traveled by the sound wave from the Big Bang to the decoupling of baryons at z BAO ~1080 (c.f., z CMB ~1090). • The baryons decoupled slightly later than CMB. • By the way, this is not universal in cosmology, but accidentally happens to be the case for our Universe. • If 3 ρ baryon /(4 ρ photon ) =0.64( Ω b h 2 /0.022)(1090/(1+z CMB )) is greater than unity, z BAO >z CMB . Since our Universe happens to have Ω b h 2 =0.022, z BAO <z CMB . (ie, d BAO >d CMB )

  18. The Latest BAO Measurements • 2dFGRS and SDSS main samples at z=0.2 z=0.2 • SDSS LRG samples at z=0.35 z=0.35 • These measurements constrain the ratio, D A (z)/d s (z BAO ) . Percival et al. (2007)

  19. Not Just D A (z)... • A really nice thing about BAO at a given redshift is that it can be used to measure not only D A (z), but also the expansion rate, H(z), directly, at that redshift. • BAO perpendicular to l.o.s => D A (z) = d s (z BAO )/ θ • BAO parallel to l.o.s => H(z) = c Δ z/[(1+z)d s (z BAO )]

  20. Measuring D A (z) & H(z) Data Linear Theory c Δ z/(1+z) = d s (z BAO ) H(z) 2D 2-pt function from the SDSS LRG samples (Okumura et al. 2007) (1+z)d s (z BAO ) θ = d s (z BAO )/ D A (z)

  21. D V (z) = {(1+z) 2 D A2 (z)[cz/H(z)]} 1/3 Since the current data are not good enough to constrain D A (z) and H(z) separately, a combination distance, D V (z) , has been constrained. (1+z)d s ( t BAO )/D V (z) 2dFGRS and SDSS main samples SDSS LRG samples Ω m =1, Ω Λ =1 Ω m =0.3, Ω Λ =0 Ω m =0.25, Ω Λ =0.75 Redshift, z Percival et al. (2007)

  22. CMB + BAO => Curvature • Both CMB and BAO are absolute distance indicators. • Type Ia supernovae only measure relative distances. • CMB+BAO is the winner for measuring spatial curvature.

  23. BAO: Current Status • It’s been measured from SDSS main/LRG and 2dFGRS. • The successful extraction of distances demonstrated. (Eisenstein et al. 2005; Percival et al. 2007) • CMB and BAO have constrained curvature to 2% level. (Spergel et al. 2007) • BAO, CMB, and SN1a have been used to constrain various properties of DE successfully. (Many authors)

  24. BAO: Challenges • Non-linearity, Non-linearity, and Non-linearity! Data 1. Non-linear clustering Linear Theory 2. Non-linear galaxy bias Model 3. Non-linear peculiar vel. Is our theory ready for the Do we trust this theory? future precision data?

  25. Toward Modeling Non-linearities • Conventional approaches: • Use fitting functions to the numerical simulations • Use empirical “halo model” approaches • Our approach: • The linear (1st-order) perturbation theory works beautifully. (Look at WMAP!) Let’s go beyond that. • The 3rd-order Perturbation Theory (PT)

  26. Is 3rd-order PT New? • No, it’s actually quite old. (25+ years) • A lot of progress made in 1990s (Bernardeau et al. 2002 for a comprehensive review published in Phys. Report) • However, it has never been applied to the real data, and it was almost forgotten. Why? • Non-linearities at z=0, for which the galaxy survey data are available today, are too strong to model by PT at any orders. PT had been practically useless.

  27. Why 3rd-order PT Now? • Now, the situation has changed, dramatically. • The technology available today is ready to push the galaxy surveys to higher redshifts , i.e., z>1. • Serious needs for such surveys exist: Dark Energy Task Force recommended BAO as the “cleanest” method for constraining the nature of Dark Energy. • Proposal: At z>1, non-linearities are much weaker. We should be able to use PT.

  28. Perturbation Theory “Reloaded” • My message to those who have worked on the cosmological perturbation theory in the past but left the field thinking that there was no future in that direction... Come Back Now! Time Has Come!

  29. Three Equations To Solve • Focus on the clustering on large scales, where baryonic pressure is completely negligible. • Ignore the shell-crossing of matter particles, which means that the velocity field is curl-free: rot V =0. • We just have simple Newtonian fluid equations:

  30. In Fourier Space ˙ δ ( k , τ ) + θ ( k , τ ) d 3 k 1 d 3 k 2 δ D ( k 1 + k 2 − k ) k · k 1 � � = δ ( k 2 , τ ) θ ( k 1 , τ ) , − k 2 (2 π ) 3 1 a 2 θ ( k , τ ) + ˙ a θ ( k , τ ) + 3˙ a ˙ 2 a 2 Ω m ( τ ) δ ( k , τ ) d 3 k 1 d 3 k 2 δ D ( k 1 + k 2 − k ) k 2 ( k 1 · k 2 ) � � = θ ( k 1 , τ ) θ ( k 2 , τ ) , − 2 k 2 1 k 2 (2 π ) 3 2 • Here, is the “velocity divergence.”

  31. Taylor Expanding in δ 1 • δ 1 is the linear perturbation. n ∞ (2 π ) 3 · · · d 3 q n − 1 d 3 q 1 � � � � a n ( τ ) d 3 q n δ D ( δ ( k , τ ) = q i − k ) F n ( q 1 , q 2 , · · · , q n ) δ 1 ( q 1 ) · · · δ 1 ( q n ) , (2 π ) 3 n =1 i =1 n ∞ (2 π ) 3 · · · d 3 q n − 1 d 3 q 1 � � � � a ( τ ) a n − 1 ( τ ) d 3 q n δ D ( θ ( k , τ ) = − ˙ q i − k ) G n ( q 1 , q 2 , · · · , q n ) δ 1 ( q 1 ) · · · δ 1 ( q n ) (2 π ) 3 n =1 i =1

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