Understanding the Lower Redshift Universe
Adi Nusser Physics Department Technion, Haifa
Collaborators:
❖ Enzo Branchini ❖ Marc Davis ❖ Martin Feix (postdoc at Technion) ❖ Wojtek Hellwing (postdoc at Durham) ❖ Ziv Mikulizky (Student at Technion) ❖ Jim Peebles ❖ Steven Phelps “Happy families are all alike, every unhappy family is unhappy in its own way”
–Tolstoy, Anna Karenina - J. Diamond, The Anna Karenina Principle
``Happy Community”: • All reliable large scale data tell the same story. • Very low level (but important) systematics. • LSS from Local Group to ~150Mpc - traditional and New probes
The LCDM is a ``Happy model”… but a little ``moody”
Therapy maybe required, perhaps by Dark sector physics
no z, just fluxes
Condon et al 1998 Theoretical tools Observational Support The Cosmological Principle Probing Super-Survey Scales Alternative probes of large scale motions The Cosmological Principle (Einstein 1917)
. Einstein 1931 . Allen Stellen des Universums sind gleichwertig; Im speziellen solo also such die ¨ortlich gemittelte Dichte der .Sternmaterie ¨uberall gleich seine . In English . All places in the Universe are equivalent. In particular, the local, averaged density of stellar material ought to be .the same everywhere. . Note . The second part of this statement is vague (not unusual for Einstein!) as the averaging process should refer to some physical scale. A more explicit Cosmological Principle is expressed by Milne (33,35). .The name (Cosmological Principle) was given by Milne. . In this talk, CP implies . average ρ exists < δδ >= ξ(r) approaches zero on the “largest . scales”. Basics: cosmological principle Approaching homogeneity in LCDM 1 10 H Cosmic Variance 0 Planck cosmology
0 10 km/s/Mpc 0
H −1 σ 10 Volume for local H0 measurements
<<2% accuracy in 1-2yrs −2 10 by Freedman et al
1 2 3 10 10 10 Basics: cosmological principle R [Mpc] 550
500
450 V Cosmic Variance 400 Planck cosmology 350 LSS motions Tully-Fisher 300 Dn-sigma [km/s]
V 250 σ 200
150 LSS motions
100 New/old probes
50
1 2 3 10 10 10 Basis: cosmological principle R [Mpc] Approaching homogeneity in observations Dipoles in the Sky
100 NVSS data, S>10mJy && Galactic latitude |b|>10o(only 10% of the data) no redshifts, just fluxes
50
0 Dec
-50
-100 -200 -150 -100 -50 0 50 100 150 200 RA
Condon et al 1998 Mean number as a function of flux at 1.4GHz
Angular dependence:
Tiwari et al 15 The NVSS observed dipole:
Solar motion in CMB frame: 369km/s, RA=168, DEC=-7
Tiwari et al 15 How to Resolve this ``Discrepancy’’? open big brackets ( (radial) peculiar velocities of galaxies
SFI++, Cosmic Flows II Springob et al, Tully et al
close big brackets ) B on 100Mpc/h is almost parallel to solar motion
There must be a dipole component of mass fluctuations at R>100Mpc/h to account for B.
This needs to considered when we compare NVSS dipole component of n(S) to theoretical predictions, e.g. LCDM ``NVSS” Dipole in 100 mocks conditioned to yield proper B
Entries 100 12 Mean 0.002925 RMS 0.000935
10
8
6
4
2
0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Dipole
This conditioning makes NVSS data consistent with LCDM at ~1sigma level Dipoles in the Sky: SDSS
Alternative V Advantage of SDSS: measured redshifts! 2 Nusser 2 Nusser 2 Nusser 2 Nusser 1. BASICS 1. BASICS 1. BASICS 1. BASICS 2 2 P = w⇢vc P = w⇢2vc P = w⇢vc 2 =0.55 + 0.05(1 + w) =0P .=55w +⇢ 0v.c05(1 + w) =0.55 + 0.05(1 + w) 2 2 P = w⇢ c =0.55P += 0.05(1w⇢vc + w) 2v P = w⇢vc 2 =0.55 + 0.05(1 + w) =0P .=55w +⇢ 0v.c05(1 + w) =0.55 + 0.05(1 + w) ⌦ =0.55 + 0.05(1⌦ + w) =⌦ = = b ⌦ b b = ⌦ /b ⌦ /b b ⌦ /b galaxies b mass ⌦ /b b ⇡ galaxies ⇡ mass galaxies↵ ↵ b mass ↵ V = H⇡¯ r + V galaxies↵ ¯ ↵ b mass ↵ gal B Vgal = H r + VB V ↵ = H¯ ↵ r + V ↵ ⇡ gal B ↵ ¯ ↵ ↵ Vgal = H r + VB f = ⌦ f = ⌦ f = ⌦ A reincarnation of an old idea = f/b Tammann, Yahil & Sandage 79 and long before f ==⌦ f/b = f/b galaxies = f mass 8 8 galaxies = f/b= f mass galaxies = f mass 8 8 8 8 galaxies = f mass cz = Hr + V 8 8 cz = Hr + V cz =czHr >+ HrV cz =czHr >+ HrV czcz >
< Hr Hr czL <(cz Hr) >L 0 t cz < Hr L0(cz) >Lt L L(cz()cz>L)
Galaxies as standard candles For a Schechter luminosity function (↵ = 1): < L >L>2L =2.77L , L>2L =0.81L ⇤ ⇤ ⇤ ⇤
< L >L>4L =4.85L , L>4L =0.74L ⇤ ⇤ ⇤ ⇤
How do we use this? (Tamman, Yahil & Sandage 79 ) take a very large redshift survey as estimate of magnitudes, compute M = m 5log(cz)=m 5log(Hr + V ) 0 true magnitudes are M = m 5log(Hr) t Simultaneously constrain a model for V and P(Mt ) by maximizing P[M = M 2.17log(1 V /cz)],assuming 0 t P(Mt ) does not depend on velocity.
Alternative V: luminosity modulation Problems: Re-visiting the basics Internal dynamics and masses of M31 and MW Luminosity variations Linear theory application to data • coherent photometric mis-calibration: affects all Gaia The Cosmological Principle Alternative probes of large scale motions clustering studies on large scales Degree of the e↵ect • environmental dependences of the luminosity Observations give M / 0.1 (this depends on the band and ⇤ ⇠ distribution: affects certainit is actuallyvelocity models zero for r-band).
Re-visiting the basics 0.02 Internal dynamics and masses of M31 and MW Luminosity variations Linear theory application to data Gaia The Cosmological Principle Alternative probes of large scale motions 0.015 env: 0.1σ d Degree of the e↵ect V-signal: 2.17σ v/cz, z =0.1
Observations give M / 0.1 (this* depends on the band and M 0.01
⇤ ⇠ it is actually zero for r-band).
0.02 0.005
0.015 env: 0.1σ d 0 V-signal: 2.17100σ /cz, z =0200.1 300 400 500 600 v R [Mpc] Alternative V: luminosity* modulation: caveats
M 0.01
0.005
0 100 200 300 400 500 600 R [Mpc] Theoretical tools Linear theory application to data Luminosity variations The Cosmological Principle Gaia Alternative probesTheoretical of large scale tools motions Linear theory applicationSDSS to data DR7 GalaxyLuminosity Catalog variations The Cosmological Principle real stuGaia↵ Alternative probes of large scale motions
Theoretical tools Linear theory application to data Luminosity variations SDSS DR7The Cosmological GalaxyNYU Principle Value-Added CatalogGaia Catalog Alternative probes of large scale motions realSDSS stu↵ DR7 Galaxy Catalog seereal Blanton stu↵ et al. (2005) Use extinction-corrected r-band magnitudesApplicationNYU Value-Added (Petrosian) Catalog to SDSS (z~0.1) NYU Value-Added Catalog 14.5 < seemgeneralr Blanton< 17. et6 velocity al. (2005) model Use extinction-corrected r-band 22.5 100 C C Angular incompleteness2 100 Adopt best-fitting1 WMAP cosmological 50 50 parameters (Calabresebothbins et al. 2013) low-z binonly 0 00.511.5200.511.52 0 0 0100200300400500 050σ8 100 150200250300σ8 ˜ ˜ 2 C1 [km/s] FIG. 9. Raw estimates of σ8 obtained from the real NYU-VAGCCi [km/s] galaxy data: shown is the derived ∆χ as a function of σ8 for both redshift bins (left panel) and the first redshift bin with 0.02 s 25 0.07 c dipole (dashed lines) also eliminates the bias, thus lead- param planck or choosing a different LF estimator has o ing to the same mean value of σ8 in both redshift bins. only a minor impact on the results. m f 20 Expressing the bias in numbers, the dipole contribution Again, one may ask whether fixing the linear evolution o r to galaxy magnitudes amounts to a systematic shift of as described in section IV A causes an additional bias in e b ∆σ8 ≈ 0.13 and ∆σ8 ≈ 0.52 in the first and second red- our measurements of σ8. To answer this question, we plot m shift bin, respectively. the derived values of σ8 for both redshift bins against the u 15 N Considering now the real SDSS galaxy sample, we per- estimate of the evolution parameter Q0 in figure 10, using form exactly the same analysis to obtain measurements the simple NYU-VAGC mocks with (black squares) and 10 of σ8 for the different LF estimators introduced in sec- without (red circles) the magnitude dipole. As before tion IV C. Our results are presented in figure 9 which (see section IV C), the linear correlation coefficients turn 2 < shows the derived ∆χ as a function of σ8, obtained after out ∼ 0.1, and there is no indication for a correlation between these quantities. 5 combining the information from both redshift bins (left panel) and using the first redshift bin only (right panel). Of course, one is not restricted to σ8, but also free to Similar to what we have discovered in our investigation look at other cosmological parameters or various com- of “bulk flows”, the values based on different LF mod- binations thereof. Considering the two parameters h 0 0 0.511.52els agree very well011 within their0.5 corresponding 1σ errors,.52and Ωb which, together, determine the baryonic matter and we get σ8 ∼ 1.0–1.1 in the low-z and σ8 ∼ 1.5–1.6 density, for instance, we have found that the respective σ8 in the high-z bin. Remarkably, the measuredσ8 values and constraints turn out weaker than before, and are also FIG. 8. Distribution of σ8 estimated from the simple NYU-VAGC mocks: shown are the recovered histograms (black lines) with (solid lines) and without (dashed lines) the inclusion of a systematic dipole in the galaxy magnitudes, using the information in both redshift bins (left) and the first redshift bin with 0.02 are found by directly fitting eq. (17) to the observed dis- Applying this procedure to the full suite of mock cata- tribution. As is customary, σ8 is inferred from discretely logs with and without the inclusion of a systematic mag- sampling the posterior probability and interpolating the nitude dipole, we obtain the histograms shown in figure corresponding result. In our calculations, we will choose 8. As described in section III B, the NYU-VAGC mocks a step size of 0.05. are based on the parameter set param mock and assume Re-visiting the basics Internal dynamics and masses of M31 and MW Luminosity variations Linear theory application to data Gaia The Cosmological Principle Alternative probes ofGrowth large scale motions Rate From SDSS Constraints on Growth Rate from SDSS As a velocity model, take V ( = f /b) from the SDSS galaxy distribution. Tune such as P(M0) is maximum. This is basically an elaborate alternative for minimizing the variance in M = m 5log(cz V ( ) with respect to . est 2 Nusser Martin Feix, AN, Enzo Branchini 15 2 Nusser 1. BASICS 27 Nusser 3.5 1. BASICS 2 6 3P = w⇢vc 1. BASICS = . ± . > = 0.37 ± 0.13 ( > 1) f σ8 0 56 0 25 (l 5) f σ8 2 l P = w⇢vc =0.55 + 0.05(1 + w) 5 P = w⇢ c2 2.5 v 2 =0.55 + 0.05(1 + w) P = w⇢vc 4 =0.55 + 0.05(1 + w) 2 2 2 2 =0.55 + 0.05(1 + w) P = w⇢vc2 ∆χ P = w⇢ c ∆χ v 1.5 3 =0.55 + 0.05(1 + w) ⌦ =0.55 + 0.05(1 + w) = 2 Nusser b 2 ⌦ 1 = ⌦ ⌦ /b 1.=BASICSb b 0.5 1 galaxies b mass ⌦ /b 2 ⌦P =/bw⇢vc ⇡ ↵ 0 ¯ ↵ ↵ 0 =0.55 + 0b. 05(1 + w) Vgal = H r + VB galaxiesgalaxies ⇡ b massmass 0.1 0.2 0.3⇡ 0.42 0.5 0.6 0.7 0.8 0.20.3 0.4 0.5 0.6 0.7 0.8 0.9 ↵↵ P↵ ↵ = w ⇢vc ↵↵ V = H¯¯ r + V Vgalgal = H r +f σVBB =0.55 + 0.05(18 + w) f = ⌦ f σ8 ⌦ = f/b ff == =⌦ b galaxies mass == ⌦f/b /b 8 = f 8 1. from (xxx) to V (xxx) - OK galaxiesgalaxies galaxies= fb massmass 88 =⇡f 88 V ↵ = H¯ ↵ r + V ↵ linear theory is enough for current LSS data 1. 1.fromfrom ( xxx()xxx)totoVV(xxx(xxx))--OKOKgal B • Peebles’ action method for future data and LG linear theory is enough for current LSS data• •linear theory is enoughf for= ⌦ current LSS data • 2. biasing: galaxies = dark but we know how to model that - OK Peebles’Peebles’ action action method method for= f/b future future data data and and LG LG 6 • • 3. redshift distortions: cz = Hr + V - OK & NOK 2. biasing: galaxies = dark butgalaxies we= knowf mass how to model that - OK 2. biasing: galaxies =6 dark but8 we know8 how to modelF.o.G: thatin general- OK a bad e↵ect 3. redshift1. from distortions: (xxx) 6 to V (xxxcz) -=OKHr + V - OK & NOK• 3. redshift distortions: cz = Hr + V - OK & NOKLarge scale compression in the radial direction: good e↵ect F.o.G:linearin general theory isa enough bad e↵ forect current LSS data• •F.o.G:•in general a bad e↵ect Kaiser’s rocket e↵ect: hopless at r > 150Mpc • Large scalePeebles’ compression action method in for the future radial data direction: and• LG good e↵ect • • ⇠ Large2. biasing: scale galaxies compression= dark but in we the know radial how to direction: model that -goodOK e↵ect • Kaiser’s rocket e6 ↵ect: hopless at r >1.150radialMpc component only - not a big deal > •Kaiser’s3. redshift rocket distortions: e↵ect:cz =hoplessHr + V - atOKr⇠2. & NOKsparseness150Mpc with < 104 galaxies 1. radial• component only - not a big deal F.o.G: in general a bad e↵ect ⇠ ⇠ 1. radial component• < only4 - not a big deal > 2. sparseness withLarge scale10 compressiongalaxies in the radial direction:limitsgood us e↵ect to scales 20 30Mpc • < 4 • ⇠ 2. sparseness with ⇠10 galaxies > < Kaiser’s rocket> e↵ect: hopless at r 3. 150largeMpc velocity errors 0.15H0r limits• us to⇠ scales 20 30Mpc ⇠ •limits1. radial us component to scales only>⇠ -20not a30 bigMpc deal ⇠ 3. large velocity errors < 0.15H r spatial Malmquist bias if galaxies are placed at rset • < ⇠4 0 2. sparseness with <⇠ 10 galaxies • 3. large velocity errors ⇠ 0.15H0r spatiallimits Malmquist us to⇠ scales bias> if20 galaxies30Mpc are placed at rset • • ⇠ spatial Malmquist bias< if galaxies are placed at r 3. large velocity errors 0.15H0r set • ⇠ spatial Malmquist bias if galaxies are placed at rset • The result is competitive and completely independent of z-distortions 2 Nusser 2 Nusser 1. BASICS 2 Nusser 1. BASICS 2 1. BASICS P = w⇢vc 2 P = w⇢vc =0.55 + 0.05(1 + w) P = w⇢ c2 2 Nusser v 2 =0.55 + 0.05(1 + w) P = w⇢vc =0.55 + 0.05(1 + w) 1. BASICS 2 =0.55 + 0.05(1 + w) P = w⇢vc2 P = w⇢vc 2 P = w⌦⇢ vc =0.55 + 0.05(1 + w) = =0.55 + 0.05(1 + w) =0.55 + 0.05(1 + w) 2 Nusser b ⌦ ⌦ 2 == P⌦= /bw⇢vc 1. BASICSb =0galaxies.55 + 0.b05(1 mass + w) ⌦ /b 2 ⌦P =/bw⇢vc ⇡ ↵ ¯ ↵ ⌦ ↵ =0.55 + 0b. 05(1 + w) Vgal = H = r + VB galaxiesgalaxies ⇡ b massmass b ⇡ 2 ↵↵ ¯¯P↵ ↵ = w ⇢vc ↵↵ ⌦ /b VVgalgal == HH r ++VVBB The value of =0.55 + 0.05(1 + w) f = ⌦ galaxies b mass ⌦ = f/b⇡ ff == =⌦ VVV = VVV 0 + H¯ b galaxies mass == ⌦f/b /b 1. from (xxx) to V (xxx ) - 8OK = f 8 1. from (xxx) to V (xxx) - OK galaxiesgalaxies galaxies= fb massmass linear theory is enough for current LSS data 88 =⇡f 88 • Planck ↵ ¯ ↵ ↵ Vgal = H r + VB linearPeebles’ theory action is enough method for for current future data LSS and data LG 1. 1.fromfrom ( xxx()xxx)totoVV(xxx(xxx))--OKOK 0.7 • • 2. 2dFGRSbiasing:Peebles’ galaxies action= methoddark but for we future know how data to and model LG that - OK linear theory is enoughf for= ⌦ current LSS data•2SLAQ 6 •linear theory is enough for current 0.65 LSS3. dataredshift distortions: cz = Hr + V - OK & NOK • 2. biasing:VVDS galaxies = dark but we know how to model that - OK Peebles’ action method for= f/b future dataSDSS and LRG LG 6 •Peebles’ action method for future data and LGF.o.G: in general a bad e↵ect • 0.63. redshiftWiggleZ• distortions: cz = Hr + V - OK & NOK 2. biasing: galaxies = dark butgalaxies we knowmass how to modelBOSS that - OK 2. biasing: = but8 we= knowf 8 how to modelLarge that scale - OK compression in the radial direction: good e↵ect galaxies 6 dark 6dFGSF.o.G:• in general a bad e↵ect 3. redshift distortions:6 cz = Hr + V - OK & NOK > 3. redshift1. distortions:from (xxx) to V (czxxx) =- OKHr + V - OK 0.55 & NOKVIPERS• Kaiser’s rocket e↵ect: hopless at r 150Mpc Large• scale compression in the radial⇠ direction: good e↵ect F.o.G:linearin general theory isa enough bad e↵ forect current LSS1. dataradial• component only - not a big deal •F.o.G:•in general a bad e↵ect 0.5 Kaiser’s rocket e↵ect: hopless at r > 150Mpc Peebles’ action method for future8 data and LG < 4 • Large• scale compression in theσ radial direction:• good e↵ect f 2. sparseness with 10 galaxies • ⇠ Large2. biasing: scale galaxies compression= dark but in we the know radial how> to direction: model that -goodOK e↵⇠ect • Kaiser’s rocket e6 ↵ect: hopless at 0.45r 1.150radialMpc component only - not> a big deal > limits us to scales 20 30Mpc •Kaiser’s3. redshift rocket distortions: e↵ect:cz =hoplessHr + V - atOKr⇠2. & NOKsparseness150Mpc• with < 104 galaxies⇠ 1. radial• component only - not a big deal < F.o.G: in general a bad e↵ect 0.4⇠ 3. large velocity errors⇠ 0.15H0r 1. radial component• < only4 - not a big deal limits us to scales⇠ > 20 30Mpc 2. sparseness withLarge scale10 compressiongalaxies in the radial direction: goodspatial e↵ect Malmquist bias if galaxies are placed at r • < 4 0.35 • ⇠ set 2. sparseness with ⇠10 galaxies > • < limits usKaiser’s to scales rocket> e↵ect:20 hopless30Mpc at r 3. 150largeMpc velocity errors 0.15H0r • • ⇠ ⇠ limits1. radial us component to scales only>⇠ -20not a30 bigMpc deal 0.3 ⇠ 3. large velocity errors < 0.15H r spatial Malmquist bias if galaxies are placed at rset • < ⇠4 0 2. sparseness with <⇠ 10 galaxies • 3. large velocity errors ⇠ 0.15H0r spatial Malmquist bias> if galaxies 0.25 are placed at rset limits us to⇠ scales 20 30Mpc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 • • ⇠ spatial Malmquist bias< if galaxies are placed at rset 3. large velocity errors 0.15H0r z • ⇠ spatial Malmquist bias if galaxies are placed at rset • Fig. 19. A plot of f 8 versus redshift, showing VIPERS result contrasted with a compilation ofde recent la measurements. Torre et al The (VIPERS) previous results from 2dFGRS (Hawkins et al. 2003), 2SLAQ (Ross et al. 2007), VVDS (Guzzo et al. 2008), SDSS LRG (Cabré & Gaztañaga 2009; Samushia et al. V2mrs2012), vs WiggleZ VTF (Blake et al.WMAP9 2012), BOSS (Reid et al. 2012), and 6dFGS (Beutler et al. 2012) surveys are shown with the di↵erent symbols (see inset). The thick solid (dashed) curve corresponds to the prediction for General Relativity in a ⇤CDM model with WMAP9 (Planck) parameters, while the dotted, dot-dashed, and dot-dot-dashed curves are respectively Dvali-Gabadaze-Porrati (Dvali et al. 2000), coupled dark energy, and f (R) model expectations. For these models, the analytical growth rate predictions given in di Porto et al. (2012) have been used. the price of slightly larger systematic error. Therefore from this Mocks test we decided to use model B and a compromise value for the Likelihood data 1 minimum scale of smin = 6 h Mpc. 1 7.5. The VIPERS result for the growth rate These comprehensive tests of our methodology give us con- fidence that we can now proceed to the analysis of the real VIPERS data and expect to achieve results for the growth rate 68% c.l. that are robust, and which can be used as a trustworthy test of PDF the nature of gravity at high redshifts. As explained earlier, we assume a fixed shape of the mass power spectrum consistent with the cosmological parameters ob- tained from WMAP9 (Hinshaw et al. 2012) and perform a max- imum likelihood analysis on the data, considering variations in the parameters that are not well determined externally. The best- fitting models are shown in Fig. 17 when considering either a c.l. 95% Gaussian or a Lorentzian damping function. Although the mock samples tend to slightly prefer models with Lorentzian damping 0 as seen in Fig. 16, we find that the Gaussian damping provides 0 0.2 0.4 0.6 0.8 1 a much better fit to the real data and we decided to quote the f σ corresponding f 8 as our final measurement. 8 We measure a value of Fig. 18. Marginalized likelihood distribution of f 8 in the data (solid curve) and distribution of fitted values of f 8 for the 26 individual Mul- f (z = 0.8) 8(z = 0.8) = 0.47 0.08, (32) tiDark simulation mocks (histogram). These curves show a preferred ± value and a dispersion in the data that is consistent at the 1 level with which is consistent with the General Relativity prediction in a the distribution over the mocks. flat ⇤CDM Universe with cosmological parameters given by WMAP9, for which the expected value is f (0.8) 8(0.8) = 0.45. We find that our result is not significantly altered if we adopt as expected given the minimum scales we consider, although in a Planck cosmology (Planck Collaboration et al. 2013) for the the case of model B the change in f 8 is at most 5%. Includ- shape of the mass power spectrum, changing our best-fitting f 8 ing smaller scales in the fit reduces the statistical error but at by only 0.2%. This shows that given the volume probed by the Article number, page 16 of 19 Do galaxy motion follow gravity? 1991ApJ...379....6N Theoretical tools Hence, it is good to do a combined analysis the twoDynamics independent in datasets: an expanding background Linear theory application to data Initial value vs Boundary value problems The Cosmological Principle observations “Snapshot” solutions to the boundary value problem Alternative probes of large scale motions Applying LAP to recover large scale velocities Velocities are a direct probe of 3D mass distribution very long arrow theory matching z-surveys & Vpec How well can velocities be recovered? Theoretical tools Dynamics in an expanding background TheoreticalLinear theory tools application to data Initial value vs Boundary value problems LinearTheoreticalLinear theory theory application tools The to data CosmologicalObservational Principle Support Dynamics in an expanding background“Snapshot” solutions to the boundary value problem Linear theory applicationTheAlternative Cosmological to data probes Principle of largeProbing scale Super-Survey motions Scales Alternative probes of large scale motionsInitial value vs Boundary value problems Linear theoryThe Cosmological Principle 1 unsmoothed Linear theory “Snapshot” solutions to the boundary value problem Alternative probesAtrivialfact:localvelocitiesarea of large scale motions Velocity-Density↵ected by= mass relation distribution atV far a given away time 1 f (⌦)r · Velocity-Density = relation atV a given time f (⌦)r · 2002MNRAS.335...53B Theoretical tools Linearf ( theory⌦)= applicationd ln D ⌦ to data Observational Support Theoreticald ln t tools⇡ d ln D Linear theoryThe application Cosmological to data PrincipleObservational SupportProbing Super-Survey Scales 1991ApJ...379....6N f (⌦)= ⌦ AlternativeSolutiond ln t to theThe probes Fundamental Cosmological of large Principle Relation scale motionsProbing Super-Survey Scales Alternative⇡ probesSolution of large scale motions 1991ApJ...379....6N Zeldovich approximation etc From⌦Theoretical Distance tools 3 to Redshiftx0 x Zeldovich approximationLinear etcv theory= application to datad x0 Observational(x0) Support The Cosmological4⇡ Principle Probingx Super-Surveyx 3 Scales Alternative probes of large scaleZall spacemotions | 0 | = ( )+ ( ) · · Zsurvey Zexternal Hence Hence Definition In redshifts spacer 1 @ 2 radial = rradialVNusser et al 91. It is a Monge-Ampe´re equation (as pointed out by s Hrr 2+@r V U.Frisch) Nusser et al⌘ 91.1 It is a Monge-Ampe´re⇣ equation⌘ (as pointed out byradial Hards to= solve directly.rV [ See1VMohayaee,] @ 2 Frisch, Matarrese & U.Frisch)= f r · r · 2 radial r V Sobolovski recoveringr @r displacements...smoothed Hard to solve directly. See Mohayaee, Frisch, Matarrese & 5TH Sobolovski recovering displacements... r is isotropic but s is not (Kaiser1 87, Davis & Peebles⇣ 83) ⌘ = V [ V] f r · r · radial r is isotropic but s is not (Kaiser 87, Davis & Peebles 83) The Astrophysical Journal,788:1(12pp),2014??? Nusser, Davis, & Branchini Based on Millennium 2MRS mocks (De Lucia & Blaizot) This choice is justified by the fact that strong nonlinearities 8 inside 5 Mpc seem to be absent in the real universe, as in- 10Mpc/h 5Mpc/h 4 dicated by the fact that the flow is fairly quiet within that b=1.23 b=1.27 radius. Hereafter, we take Rlg 5Mpctobetheradiusof 2 = g the LG and treat the motion of the central sphere of that δ 1 radius as the motion of the LG. 1+ The mocks were drawn from the original computational 0.4 box, taking into account periodic boundary conditions, i.e., 0.2 wrapping around the distribution of objects when LG-like observer happened to be close enough to the edge of the 0.2 0.4 1 2 4 8 0.2 0.4 1 2 4 8 1+δ 1+δ box. These mocks are not fully independent in the sense dm dm that their volumes overlap. This could effectively result in an Figure 1.ScatterScatter is mostly plot (logarithmicshot-noise scale) of the galaxy vs. the dark matter over- underestimation of the error in the recover of the LG motion. densities in the simulation. For each of the 53 mocks, densities of 125 randomly 1 selected pointsAN, Davis with & Branchini a distance <70 h− Mpc from the LG candidate, are shown. However, the main point is the location of the mock LGs with 1 The left and right panels correspond to densities in cubic cells of 10 h− Mpc respect to the large-scale structure in the simulation. We expect 1 and 5 h− Mpc on the side, respectively. The thick solid curve in each panel the lack of full independence to play an insignificant role. is the mean of 1 + δg at a given 1 + δdm. The two thin solid curves are 1σ The mocks are taken from the z 0simulationoutputand, scatter-computed from points above and below the mean. Dashed curves are± the = expected 1σ Poisson (shot-noise) scatter. The nearly straight red lines show therefore, are free from any possible galaxy evolution. The two- ± δg bδdm + const, where b (indicated in the figure) are determined using linear point correlation function of the mock galaxies fits reasonably = regression from points in the range 0.5 < δdm < 4. well the observed one (Westover 2007), but less so does the − (A color version of this figure is available in the online journal.) K-band luminosity function, resulting in a discrepancy with the observed number of galaxies. To fix this problem, the original luminosity of mock galaxies was shifted to brighter values by Smith et al. 2007), confirmed by numerical experiments (e.g., 1.5magnitude.Weobtained,onaverage, 50,000 galaxies Kauffmann et al. 1997;Narayananetal.2000;Bensonetal. ∼ ∼ per mock, slightly larger than, but close to, the real Ks 11.75 2000;Huffetal.2007)andsupportedbyobservationsinvolving 2MRS catalog. Each of the 53 mock catalogs contains= galaxy galaxy samples dominated, like in the 2MRS case, by late type distances, peculiar velocities (and hence redshifts), angular galaxies (e.g., Tegmark et al. 2001;Verdeetal.2002;Westover positions, and Ks-band magnitudes. 2007). Here, we explore the bias of the distribution of the mock 4.1. The Selection Function galaxies with respect to the dark matter density field in the simulation. Gerard Lemson has kindly used the facilities of the In the application to a flux limited survey like 2MRS, each Millennium Simulation Database to produce for us the density galaxy in the summation of the relations (6) and (5) should field from all 21603 dark matter particles in the simulation box be weighted by the inverse of the selection function, ϕ,to on a cubic grid of 1 h 1 Mpc spacing. Density fields from the compensate for missing faint galaxies that fall below the flux − distribution of mock galaxies have also been directly computed limit. The selection function depends on the galaxy distances for all of the mocks. Figure 1 is a scatter plot of the over densities and it is physically determined by the distribution of galaxy Q4 computed from the mock galaxy distribution versus the dark luminosities. In the mocks, where galaxy distances and apparent matter density field. For δ 3, the scatter in the relation is magnitude are both known, we compute ϕ using a direct method dm ! mainly Poissonian. However, at higher densities, intrinsic scatter which avoids the explicit calculation of the luminosity function in the biasing relation dominates. The relation in small (right (Turner 1979;Kirshneretal.1979;Davis&Huchra1982). panel) and large (left) cells is fairly linear, δ bδ ,inthe The method provides discrete values of ϕ in distance bins, g dm moderate density ( 0.2 δ 4 in the two panels)= regions, which are then interpolated to the galaxy distances to yield ! dm ! with a weak dependence− on b scale: b 1.23 and 1.27 in the the weights to be assigned to individual galaxies. In realistic large and small cells, respectively. The values∼ change according applications, however, the distances to the galaxies are not to the density cut used in fits. Departure from linear bias in low known. Using redshifts rather than distances as arguments to the density regions is a well-known feature also detected in real selection function induces systematic errors, sometimes dubbed galaxy catalogs in the local universe (Branchini 2001)aswell the “Kaiser Rocket” effect (Kaiser 1987). The hazards of not as at moderate (z [0.5, 1.2]) redshift (Marinoni et al. 2005; explicitly accounting for the Kaiser rocket effect are given in Kovacetal.ˇ 2011).= Here, we shall continue to assume linear bias, Section 6. noticing that the value of the linear bias parameter for various 4.2. Bias of the Selected Galaxies densities yields 1.2 4 I will show next an excellent agreement between: Peculiar motions derived (using linear theory) from the distribution of galaxies in the Two Mass Redshift Survey (2MRS) and The observed peculiar motions from the SFI++ V2mrs vs VTF Mon. Not. R. Astron. Soc. 413, 2906–2922 (2011) doi:10.1111/j.1365-2966.2011.18362.x Local gravity versus local velocity: solutions for β and non-linear bias Marc Davis,1⋆ Adi Nusser,2 Karen L. Masters,3 Christopher Springob,4 John P. Huchra5 and Gerard Lemson6 21Departments of Astronomy & Physics, Nusser University of California, Berkeley, CA 94720, USA 2Physics Department and the Asher Space Science Institute-Technion, Haifa 32000, Israel 23Institute for Cosmology and Gravitation, University Nusser of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX 4Anglo-Australian Observatory, PO Box 296, Epping, NSW 1710, Australia 5Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 6 1. BASICS Downloaded from Max-Planck Institute of Astrophysics,1. Karl-Schwarzschild-Str.BASICS 1, 85741 Garching, Germany Accepted 2011 January 14. Received 2011 January 13; in original form 2010 November 12 2 P = w⇢vc 2 P = w⇢ c http://mnras.oxfordjournals.org/ ABSTRACT v We perform a reconstruction of the cosmological large-scale flows in the nearby Universe =0 .=055 +.55using 0 +. two05(1 0 complementary.05(1 + +w observational)w) sets. The first, the SFI sample of Tully–Fisher ++ (TF) measurements of galaxies, provides a direct probe of the flows. The second, the whole sky distribution2 of2 galaxies in the 2MASS (Two Micron All Sky Survey) redshift survey P =P(2MRS),w=⇢vw yieldsc⇢v ac prediction of the flows given the cosmological density parameter, ", and a biasing relation between mass and galaxies. We aim at an unbiased comparison between the peculiar velocity fields extracted from the two data sets and its implication on the cosmological =0.55 + 0.05(1 + w) at Generalverwaltung der Max-Planck-Gesellschaft on June 3, 2014 =0.55 +parameters 0.05(1 and the + biasingw relation.) We expand the fields in a set of orthonormal basis functions, each representing a plausible realization of a cosmological velocity field smoothed in such a way as to give⌦ a nearly constant error on the derived SFI velocities. The statistical analysis ++ is done⌦= on the coefficients of the modal expansion of the fields by means of the basis functions. Our analysis completelyb avoids the strong error covariance in the smoothed TF velocities by =the use of orthonormal basis functions and employs elaborate mock data sets to extensively calibrateb the errors in 2MRS predicted velocities. We relate the 2MRS galaxy distribution to the mass density field by a linear bias factor, b,andincludealuminosity-dependent, Lα, b ∝ galaxiesgalaxy weighting.mass We assess the agreement between the fields as a function of α and β = V2mrs vs VTF: visual f (")/b,where⇡ f is the growth factor of linear perturbations. The agreement is excellent with galaxiesareasonableb massχ 2 per¯ degree of freedom. For α 0, we derive 0.28 < β < 0.37 and 0.24 < VVV = VVV 0 + H = β ⇡< 0.43, respectively, at the 68.3 per cent and 95.4 per cent confidence levels (CLs). For β = 0.33, we get¯α < 0.25 and α < 0.5, respectively, at the 68.3 per cent and 95.4 per cent CLs. 1. from (xxx) to V (xxxVVV) -=OKVVV We set+ a constraintH on the fluctuation normalization, finding σ 8 0.66 0.10, which is only 0 = ± 1.5σ deviant from Wilkinson Microwave Anisotropy Probe (WMAP)results.Itisremarkable linear theory is enoughthat σ 8 determined for current from this local LSS cosmological data test is close to the value derived from the 1. from (xxx) •to V (xxx) - OK cosmic microwave background, an indication of the precision of the standard model. Peebles’ action methodKey words: forcosmological future parameters data – dark and matter LG – large-scale structure of Universe. linear• theory is enough for current LSS data •2. biasing: galaxies = dark but we know how to model that - OK Peebles’ action method6 for future data and LG •3. redshift distortions: cz = Hr + V - OK &Willick NOK 1995; Zaroubi 2002). They are limited to small redshifts 1INTRODUCTION 1 ( 100 h− Mpc) at which distance indicators can reliably be used. 2. biasing: galaxiesF.o.G:For 15 yr,=in the problemdark general of large-scalebut a we bad flows of know e galaxies↵ect has seen howThese∼ to earlier model forays into the that subject led - toOK disagreements that few • little attention6 relative to other probes of the large-scale structure people wanted to sift through. But in the interval, the data have in the Universe. The data on peculiar velocities have been difficult improved dramatically, thus stirring recent activity in the subject. 3. redshift distortions:Largeto obtain, scale and thecz results compression= had contradictoryHr + conclusionsV in- the(StraussOK radial& & NOKPeculiar direction: velocities are uniquegood in that e they↵ect provide explicit in- • formation on the three-dimensional mass distribution, and mea- > 1 Kaiser’s rocket e↵ect: hopless at r sure150 mass onMpc scales of 20–50 h− Mpc, a scale untouched by al- F.o.G: in⋆E-mail: general [email protected] a bad e↵ect ternative methods. Local peculiar velocity data are, in principle, • • ⇠ C 1. radial component only - not a big deal ⃝ 2011 The Authors C Large scale compression in the radial direction:Monthly Notices of thegood Royal Astronomical e↵ Societyect⃝ 2011 RAS • < 4 2. sparseness with 10 galaxies > Kaiser’s rocket e⇠↵ect: hopless at r 150Mpc • limits us to scales > 20 30Mpc⇠ • ⇠ 1. radial component only - not< a big deal 3. large velocity errors 0.15H0r 2. sparseness with < 104 galaxies⇠ spatial Malmquist bias if galaxies are placed at rset • ⇠ limits us to scales > 20 30Mpc • ⇠ < 3. large velocity errors 0.15H0r ⇠ spatial Malmquist bias if galaxies are placed at rset • correlation of SFI-2MRS correlation of SFI (not to be compared with models) V2mrs vs VTF: quantitative Implications:2 Nusser 1. BASICS 2 P = w⇢vc • finally, we have an excellent match. =0.55 + 0.05(1 + w) - no cosmic variance uncertainty 2 P = w⇢vc - Great job by the observers.=0.55 + 0.05(1 + w) ⌦ • GI is confirmed with no indication= for deviations on 30-70 Mpc scales. b • no scale dependence of ⌦ /b b - likely to constraint alternativegalaxies ⇡ modelsmass VVV = VVV 0 + H¯ 1. from (xxx) to V (xxx) - OK linear theory is enough for current LSS data • Peebles’ action method for future data and LG • 2. biasing: galaxies = dark but we know how to model that - OK 6 3. redshift distortions: cz = Hr + V - OK & NOK F.o.G: in general a bad e↵ect • Large scale compression in the radial direction: good e↵ect • V2mrs vs VTF: whyKaiser’s do we rocket care e↵ect: hopless at r > 150Mpc • ⇠ 1. radial component only - not a big deal 2. sparseness with < 104 galaxies ⇠ limits us to scales > 20 30Mpc • ⇠ < 3. large velocity errors 0.15H0r ⇠ spatial Malmquist bias if galaxies are placed at rset • Still, standard paradigm might have some problems! The Curious Case of the Local Neighborhood Peebles & AN Note the impressive Local Void revealed by B. Tully and puzzled J. Peebles 337 galaxies with good distances 172 SDSS galaxies 53 HIPASS galaxies Nearby LSS 2002MNRAS.335..410C s>2 s<2 density density background background density density empty R R initial profile: Nearby: void The Challenge of Pure-Disk Galaxies 21 The problem of pure disk galaxies proves to depend on Table 2 lists the resulting 19 galaxies in order from pure disk environment – it is a puzzle in the field but not in rich clusters. to pure elliptical. Distances are a complicated problem; we Also, we need detailed observations to classify (pseudo)bulges. use averages (Column 3) of the most accurate determinations These considerations motivate us to restrict ourselves to a that we could find in the sources in Column (4). Column (5) nearby volume that contains small groups of galaxies like the gives the K-band absolute magnitude of the galaxy from the Local Group but not any denser environments that approach total magnitude in the 2MASS Large Galaxy Atlas (Jarrett et al. the conditions in the Virgo cluster. M 101 is the most distant 2003). Column (6) is the V-band total absolute magnitude. bulgeless disk discussed in §3, at D =7Mpc.Welookforall Column (7) gives the outer rotation velocity Vcirc from sources giant galaxies with D 8Mpc.Asourcutoffforgiantgalaxies, in Column (8). For the two ellipticals, we use Vcirc = √2σ, ≤ !1 we will be conservative and choose Vcirc > 150 km s or central where σ is an approximate velocity dispersion. Finally, !1 σ Vcirc/√2 > 106 km s .WeuseTully(1988),HyperLeda, classical-bulge-to-total and pseudobulge-to-total ratios B/T and∼ NED to construct a master list of nearby galaxies and then and PB/T,respectively,arelistedinColumns(9)and(10).We Aboutuse individual half papers thatof provide the accurate brightest measures of D, V20circ, nearbyaveraged the valuesgalaxies given by the sourcesare listedpure in Column (11). and σ to cull a sample that satisfies the above criteria. Bulge classifications are discussed in the Appendix. disks just like the MW (KormendyTABLE 2 et al 10) BULGE,PSEUDOBULGE, AND DISK INVENTORIES IN GIANT GALAXIES CLOSER THAN 8MPC DISTANCE Galaxy Type D S MK MV Vcirc S B/TPB/T S (Mpc) (km s!1) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) NGC 6946 Scd 5.9 a,b !23.61 !21.38 210 10 a,b 0 0.024 0.003 a ± ± NGC 5457 Scd 7.0 c,d,e !23.72 !21.60 210 15 c,d,e 0 0.027 0.008 a ± ± IC 342 Scd 3.28 f,w !23.23 !21.4 : 192 3a,f0 0.030 0.001 c,e,f ± ± NGC 4945 SBcd 3.36 g !23.21 !20.55 174 10 e 0 0.036 0.009 b ± ± NGC 5236 SABc 4.54 d,i,j !23.69 !21.018015 e,i 0: 0.074 0.016 c,e ± ± NGC 5194 Sbc 7.66 h !23.94 !21.54 240 20 a,i,j 0: 0.095 0.015 d,e ± ± NGC 253 SBc 3.62 g,k !24.03 !20.78 210 5a,f0:0.15 c ± Maffei 2 SBbc 3.34 l !23.0: !20.8 : 168 20 f 0: 0.16 0.04 b ± ± Galaxy SBbc 0.008 m,n,o !23.7 !20.8 : 220 20 k,l 0: 0.19 0.02 g,h ± ± Circinus SABb: 2.8 a !22.8 !19.81555o,p0:0.30 0.03 b,e ± ± NGC 4736 Sab 4.93 h,p !23.36 !20.66 181 10 e,q 0: 0.36 0.01 d,e ± ± NGC 2683 SABb 7.73 h !23.12 !19.80 152 5g,h0.05 0.01 0: b ± ± NGC 4826 Sab 6.38 h,u !23.71 !20.72 155 5 m,n 0.10 0.10 d,e,f,i,j ± NGC 2787 SB0/a 7.48 h !22.16 !19.19 220 10 r,s,t 0.11 0.28 0.02 d,k ± ± NGC 4258 SABbc 7.27 g,h,q !23.85 !20.95 208 6e,u0.12 0.02 0: b,d,e,l ± ± M31 Sb 0.77 c,h,r !23.48 !21.20 250 20 e 0.32 0.02 0 b,m,n ± ± M81 Sab 3.63 d,r,s !24.00 !21.13 240 10 e,v 0.34 0.02 0 d,e,f,i,o,p ± ± Maffei 1 E 2.85 l !23.1: !20.6 : (264 10) w 1 0 q ± NGC 5128 E 3.62 e,h,t,v !23.90 !21.34 (192 2) x 1 0: q ± NOTE.— Galaxies are ordered from pure disk to pure elliptical, i.e., by increasing pseudobulge-to-total luminosity ratio PB/T and then by increasing bulge-to-total luminosity ratio B/T. Column (2): Hubble types are from NED. Column (3): Adopted distance. Column (4): Distance sources are: (a) Karachentsev et al. 2004; (b) Karachentsev et al. 2000; (c) Sakai et al. 2004; (d) Saha et al.2006;(e)Rizzietal.2007;(f)Sahaetal.2002;(g)Mouhcineetal.2005;(h)Tonryetal.2001;(i)Thim et al. 2003; (j) Karachentsev et al. 2002; (k) Karachentsev et al. 2003d; (l) Fingerhut et al. 2007, which is also the sourceforGalacticextinctionsand,togetherwith Buta & McCall 1999, for VT ; (m) Paczynsky´ & Stanek 1998; (n) Stanek & Garnavich 1998; (o) Eisenhauer et al. 2003; (p) Karachentsev et al. 2003b; (q) Caputo et al. 2002; (r) Ferrarese et al. 2000; (s) Jensen et al. 2003; (t) Rejkuba 2004; (u) Mould & Sakai 2008; (v) Ferrarese et al. 2007; (w) Karachentsev et al. 2003a. Columns (5) and (6): Absolute magnitudes MK and MV are calculated from apparent integrated magnitudes (in K band, from Jarrett et al. 2003; in V band, preferably from HyperLeda, otherwise from NED) and colors (preferably (B!V)T from RC3, otherwise from HyperLeda). Galactic absorptions are from Schlegel et al. 1998. Column (7): Circular rotation velocity at large radii, Vcirc,correctedtoedge-oninclination.Valuesinparenthesesare √2σ. In many galaxies (e. g., M 31) error bars reflect variations with radius, not errors of measurement. Column (8): Source of Vcirc measurements: (a) Sofue 1996; (b) Tacconi & Young 1986; (c) Bosma et al. 1981; (d) Kenney et al. 1991; (e) Sofue 1997; (f) Kuno et al. 2007; (g) Casertano & van Gorkom 1991; (h) McGaugh 2005; (i) Bosma 1981; (j) Tilanus & Allen 1991; (k) Gunn et al. 1979; (l) McMillan & Binney 2010 – Caution: Vcirc may be more uncertain (although not smaller) than we commonly think; (m) Braun et al. 1994; (n) Rubin 1994a; (o) Jones et al. 1999; (p) Curran et al. 2008; (q) Bosma etal.1977;(r)Shostak1987;(s)Sarzietal.2001;(t)Erwinetal.2003;(u)vanAlbada1980;(v)Visser 1980; (w) Fingerhut et al. 2003; (x) Silge et al. 2005. Columns (9) and (10) are averages of measured classical-bulge-to-total and pseudobulge-to-total luminosity ratios. Quoted errors are from the variety of decompositions discussed in this paper or, when there are multiple sources, are the dispersions in the published values divided by the square root of the number of values averaged. In the latter case, the smallest values are unrealistically optimistic estimates of the true measurement errors and indicate fortuitously good agreement between published values (e. g., for IC 342). Colons indicate uncertainty in the sense that we know of no observational evidence that this component is present in the galaxy but we are also not aware of a rigorous proof that a small contribution by this component is impossible. Column (11): References for Columns (9) and (10): (a) This paper, §3: I band for NGC 6946; K band for NGC 5457; (b) This paper and Kormendy 2010; see Appendix for details on individual galaxies; (c) Simien & de Vaucouleurs 1986; (d) Fisher & Drory 2008; (e) Fisher & Drory 2010; (f) Baggett et al. 1998; (g) Kent et al. 1991; (h) Dwek et al. 1995; (i) Méndez-Abreu et al. 2008; (j) Möllenhoff&Heidt2001;(k)Erwinetal.2003;(l)Sánchez-Portaletal. 2004; (m) Seigar et al. 2008; (n) Tempel et al. 2010; (o) Möllenhoff 2004; (p) Laurikainen et al. 2004; (q) From assumed Hubble type. For NGC 4826, the five sources of photometric decompositions give a total (pseudo)bulge-to-total luminosity ratio of 0.20 0.05; we conservatively assign half of this to a classical bulgeandhalftoapseudobulge,forreasonsdiscussed ± in the Appendix. Note: Since we convert our bulge-pseudobulge-disk luminosity inventory into a stellar mass inventory using MK and K-band mass-to-light ratios, (P)B/T values were determined in the infrared (H to L bands) whenever possible, especially for spiral galaxies. Some sources that list (P)B/T determined in optical bandpasses are therefore not used here. Kormendy 2010 discusses the dependence of (P)B/T on bandpass in more detail. Merging in Simulation Wang, Peebles & AN Figure 3: 23 Andromeda has a huge bulge 18 pure disk Kormendygalaxy et al. FIG.14.—ColorimageofNGC6503takenwiththeHubble Space Telescope Advanced Camera for Surveys. Colors are bland because the wavelength range available is small. Blue corresponds to the F650N filter (Hα), red to F814W (I band) and green to their average. Brightness here is proportional to the square root of the brightness in the galaxy. North is up and east is at left.LikeNGC5457andNGC6946,thisisapure-diskgalaxy.ButNGC6503issmaller;ithasaflat !1 !1 outer rotation curve with Vcirc ≃ 115 km s compared with Vcirc ≃ 210 km s for the previous galaxies. Like those galaxies, its Hubble type is Scd. And like them, atiny,brightcentervisibleinthisimageprovestobeapseudobulge that makes up 0.11 % of the I-band light of the galaxy (see text). The nucleus that we use to constrain M• makes up only 0.040 % of the I-band light of the galaxy. It is completely invisible here butisillustratedinFigure15. 3.4. NGC 6503 NGC 6503 (Figures 14 and 15) is an Scd galaxy that is smaller than NGC 5457 and NGC 6946. It has a rising rotation curve over the inner 100′′,i.e.,roughlytheradiusrangeshown in Figure 14, and then a well known, flat outer rotation curve !1 with Vcirc 115 km s (van Moorsel & Wells 1985; Begeman 1987; Begeman≃ et al. 1991) out to r 800′′.Thisissimilar ≃ to Vcirc in M 33. NGC 6503 is another example of a pure-disk !1 galaxy; it is not in the § 4 sample because Vcirc < 150 km s . Two HST archive images include the nucleus, an F814W image that defines our I photometry bandpass and an F650N image that includes Hα emission. Color images of the galaxy and its nucleus plus pseudobulge are constructed from these images in Figures 14 and 15. The wavelength range is small, so colors look bland. But absorption and star-formation regions are recognizable, and the figures serve to emphasize how thoroughly this is a pure-disk galaxy. The tiny, bright center that is saturated in Figure 14 is resolved in Figure 15 into an elongated structure that resembles anuclearbar(seealsoGonzález-Delgadoetal.2008).The disk-like or bar-like morphology is sufficient to identify this as apseudobulge.Itsurroundsadistinct,high-surface-brightness nucleus. NGC 6503’s distance is only 5.27 Mpc (Karachentsev et al. 2003c; Karachentsev & Sharina 1997). So the nucleus ′′ ′′ FIG.15.—Colorimageofthecentral20. 5×20. 5ofNGC6503madeasin provides another opportunity to use ground-based spectroscopy Figure 14 but with a different square-root stretch to show thecentralbar-like to derive an M• limit in a pure-disk galaxy. pseudobulge and nuclear star cluster. Both together are saturated in Figure 14. Pure disk NGC4247 12 Kormendy et al. 3.3. NGC 6946 Like NGC 5457, NGC 6946 has no hint of a classical bulge. In photometry discussed below, the overexposed red center Globally, NGC 6946 is very similar to NGC 5457. It has the same Scd Hubble type. It has almost the same luminosity shownin Figure7 provestobea pseudobulge. As inNGC5457, it is easy to identify an engine for secular evolution: the spiral (MV !21.4versus!21.6forNGC5457),inclination- ≃ !1 structure and associated dust lanes reach the nucleus, so there corrected maximum rotation velocity (Vmax =210 10 km s versus 210 15 km s!1 for NGC 5457), and distance± (5.9 Mpc is no effective inner Lindblad resonance (see Kormendy & versus 7.0 Mpc± for NGC 5457; see Tables 1 and 2). It is less Norman 1979) that acts as a barrier to inflowing gas. However, we expect that secular evolution is slow in a barless Scd galaxy well known than NGC 5457 because it is heavily obscured by our Galactic disk. We adopt absorptions A =1.133, A =0.663, (Kormendy & Kennicutt 2004; Kormendy & Cornell 2004). So, V I as in NGC 5457, it is no surprise that the pseudobulge of NGC and A =0.125 (NED, following Schlegel et al. 1998). K 6946 is tiny. It adds up to 2.4 % of the I-band light of the galaxy. Figure 7 illustrates the similarity to NGC 5457. We tried to match the color scheme of Figure 4 but did not fully succeed: At the center of NGC 6946 is an even tinier nucleus (Fig. 8) that is seen in the V -band decomposition of Fisher & Drory the bandpasses are different, and the correction for foreground (2008) but that is still more obvious in I band. Large color reddening is not perfect. In fact, the galaxies have similar gradients in NGC 6946 imply (in contrast to NGC 5457) that dereddened total colors: (B !V) 0.46 for NGC 6946 and T0 the nucleus is dominated by young stars. To measure its mass, 0.44 for NGC 5457. Both disks are≃ dominated by ongoing star it is important that we measure its brightness profile at the same formation. A difference is that NGC 6946 has a compact central concentration of molecular gas and a nuclear starburst; we will wavelength that we used in our spectroscopy to measure σ. A pure disk withWe a therefore pseudo work in I band.-bulge detect this gas dynamically. We will not find a secure M• limit. FIG.7.—ColorimageofNGC6946takenwiththeLargeBinocularTelescope (http://medusa.as.arizona.edu/lbto/astronomical.htm). This !1 galaxy is very similar to NGC 5457: it is a giant galaxy (Vcirc ≃ 210 ± 10 km s :Table2),butitiscompletelydominatedbyitsdisk(Hubbletype Scd). As in NGC 5457, the tiny, bright center visible in this image proves to be a pseudobulge that makes up 2.4 % of the I-band light of the galaxy (see text). The nucleus whose dispersion we measure makes up only 0.12 % of the I-band light of the galaxy. It is completely invisible here butisillustratedinFigure8. Tweaking \w Peebles, Gubser & Keselman Some motivations: • Need for deeper voids: Mon. Not. R. Astron. Soc. 000,000–000(0000) Printed3December2013 (MNLATEXstylefilev2.2) Local Void is too deep PossibleISW is evidencetoo large in for big anvoids inverted (Granett temperature-density et al) relation in the intergalactic medium from the flux distributionHotter voids of in the the Ly α forestforest (Botlon et al) J.S.• Quieter, Bolton1,M.Viel more2 ,isolated3, T.-S. Kim galaxies4,M.G.Haehnelt (Kormendy5 &R.F.Carswell et al) 5 1 Max Planck Institut f¨ur Astrophysik, Karl-Schwarzschild Str. 1, 85748 Garching, Germany 2 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo11,I-34131Trieste,Italy 3 INFN/National Institute for Nuclear Physics, Via Valerio 2,I-34127Trieste,Italy •4 Astrophysikalischesfewer satellites Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany 5 Institute of Astronomy, University of Cambridge, MadingleyRoad,Cambridge,CB30HA • many LxCDM models include these modification 20 February 2008 ABSTRACT We compare the improved measurement of the Lyα forest flux probability distribution at 1.7 1INTRODUCTION Lidz et al. 2006; Becker et al. 2007) and wavelet analysis (Meiksin 2000; Theuns & Zaroubi 2000; Zaldarriaga 2002) Traditional analyses of the Lyα forest observed in the spec- have also been developed. These measures are simple and - tra of high redshift quasi-stellar objects (QSOs) decom- at least in principle - easy to compare to the same quan- arXiv:0711.2064v2 [astro-ph] 19 Feb 2008 pose the flux distribution into a series of discrete absorp- tities extracted from theoretical models of the Lyα for- tion profiles, generally characterised by a Voigt function est (Cen 1992; Miralda-Escud´eet al. 1996; Bi & Davidsen (Hu et al. 1995; Lu et al. 1996; Kirkman & Tytler 1997; 1997; Croft et al. 1999; Dav´eet al. 1999; Theuns et al. 1998; Kim et al. 1997, 2001). Voigt profiles provide an accurate Jena et al. 2005). In particular, the power spectrum of the description for an absorption line if the absorber is a lo- Lyα forest flux distribution has recently been successfully calised gas cloud with a Gaussian velocity dispersion (Rauch developed into a quantitative tool for measuring the matter 1998). In the now widely established paradigm for the power spectrum on scales of 0.5h−1 Mpc to 40h−1 Mpc (e.g. origin of the Lyα forest most of the absorption is, how- Croft et al. 2002; Viel et al. 2004b; McDonald et al. 2005). ever, caused by extended gas distributions broadened by the Hubble flow, removing much of the physical motiva- The simplest pixel statistic is the Lyα flux probability tion for the decomposition of the flux distribution into distribution function (PDF), which is sensitive to the density Voigt profiles. Such a decomposition is nevertheless use- distribution and thermal state of the IGM (e.g. Becker et al. ful, but it is non-unique and the line fitting process is 2007). Several attempts have been made to obtain joint con- time consuming and somewhat subjective (see Kim et al. straints on a variety of cosmological and astrophysical pa- 2007, hereafter K07, for an exhaustive discussion). Al- rameters using the flux PDF together with the flux power ternative characterisations of the flux distribution based spectrum (Choudhury et al. 2001; Desjacques & Nusser on pixel statistics (Jenkins & Ostriker 1991; Rauch et al. 2005; Lidz et al. 2006; Desjacques et al. 2007). However, the 1997; Gazta˜naga & Croft 1999; McDonald et al. 2000; PDF is also sensitive to a range of systematic uncertainties, Theuns et al. 2000; Meiksin et al. 2001; Viel et al. 2004c; most notably the assumed continuum level, the noise prop- ⃝c 0000 RAS Tweaking • bellow is a simplified version which incorporates much of effects of more complicated models Tweaking How do we do that? WARNING: ONLY BETWEEN DARK MATTER PARTICLES!!!! The Baryons do not feel the extra term Tweaking: 5th force Physical realization Lorentz factor factor relativistic kinetic term of Lagrangian of a the field particle Ansatz: massive relativistic Tweaking: 5th force: Lagrangian 2 bers of particles, compared to what can be done, and where the sum is over all the particles, and mα(φ)= should therefore be regarded as only a first exploration m yφ or ysφ depending on the species of particle α. of what the model has to offer. Nevertheless, as discussed During− structure formation the scalar field dynamics in section III C, we can conclude that the effects we do can be treated in a quasi-stationary approximation: establish are§ observationally attractive. In IV, we com- § 2φ = φ/r2 yn(r,t) , (3) ment on issues related to massive halos which we cannot ∇ s − analyze using our simulations, but which we hope will be where 2 is the spatial laplacian and explored in future work. ∇ 2 rs = ϵs/ys n¯s . (4) # II. DYNAMICS The last term in (3) accounts for the non-relativistic particles in a hydrodynamic approximation. The pre- 2 The form (1) is identical [37] to the standard vious term, φ/rs , follows by noting [2] that the source parametrization of hypothetical fifth force corrections to term for φ for a particle with speed v includes a factor Newton’s law of gravity in the visible sector (see for ex- ds/dt = √1 v2,andthatforquasi-staticconfigurations − 2 ample the review article [10]). The interesting regime for of φ the screening particle energy, ϵs = ms/√1 v = the strength parameter β is also the same as in discus- 2 − ysφ/√1 v ,isnearlyindependentofposition.Elimi- sions of a fifth force: β O(1). But the interesting scale − 2 ∼ nation of √1 v in favor of ϵs results in the screening for the screening length in our case is rs 1Mpc,vastly − ∼ length (4). The energy ϵs does change with time, scal- larger than the regime rs 100 µmofinterestinmodern −1 ing with the expansion of the universe as ϵs a(t) .2 fifth force experiments. And,∼ crucially, we assume that ∝ The screening length thus scales as rs a(t), that is, the the scalar force acts only in the dark sector: visible mat- ∝ bers of particles, compared to what can be done, and comovingwhere the length sum is is constant. over all the particles, and m (φ)= ter interacts with the dark matter only through ordinary The scalar field produced by a single dark matterα par- gravity.should therefore be regarded as only a first exploration m yφ or ysφ depending on the species of particle α. ticle− at distance r rs is φ = y/4πr.Theforcethisfield ofIt what is essential the model that has we to assume offer. Nevertheless, that the self-interaction as discussed During structure≪ formation the scalar field dynamics in section III C, we can conclude that the effects we do exertscan be on treated another in dark a quasi-stationary matter particle approximation: is the negative of potential V §(φ)forthescalarfieldcanbeneglected:a F establish are observationally′ attractive. In IV, we com- the gradient of the mass m yφ,thatis, = y φ, significant non-zero V (0) would drive φ away§ from 0 at 2 2− ∇ ment on issues related to′′ massive halos′ which we cannot so we see that theφ particles= φ/rs areyn( attractedr,t) , with force(3) late times, and non-zero V (0) (with V (0) = 0) amounts 2 2 Important:∇ − analyze using our simulations, but which we hope will be F = y /4πr at r rs.Thismeanstheratioofthe to a mass for φ which dominates over the screening ef- scalarwhere and2 gravitationalis the spatial≪ forces laplacian of attraction and of two dark explored in future work. ∇ fect of the relativistic particles at late times. Non-zero matter particles is 2 higher derivatives of V at φ =0mightendangerour • r s =~ a fewϵs/y Mpcs n¯s . comoving (4) story in subtler ways which have not been fully probed. #y2 II. DYNAMICS The last term in• (3)β = ~ accounts 1 . for the non-relativistic(5) These requirements on V (φ)areinconflictwithstandard 4πGm2 notions of field-theoretic naturalness. We nevertheless particles in a hydrodynamic approximation. The pre- 2 • careful with element production claimThe some form motivation (1) is for identical our model [37] from to string the standard theory Thevious relations term, φ (4)/rs and, follows (5) summarize by noting how [2] the that parameters the source andparametrization supersymmetry, of hypothetical as we briefly fifth review force in correctionsII B. to ofterm the forpotentialφ for a (1) particle emerge with from speed the dynamics v includes cf. Hamann, we a Hannestad, factorhave Raffelt & Wong 10 2 Dvorkin, Wyman, Rudd, Douglas & Hu 14 Newton’sOur further law discussion of gravity in in sections the visible II A sector and§ II (see B of for the ex- addedds/dt to= the√1 darkv ,andthatforquasi-staticconfigurations sector. − 2 physicsample the behind review the article force law[10]). (1) The includes interesting some regime recapit- for ofTheφ the effect screening of the scalar particle interaction energy, ϵ ons = them evolutions/√1 v of= the strength parameter β is also the same as in discus- 2 − ulation of earlier discussions [2, 8, 9], presented here in theysφ mass/√1 densityv ,isnearlyindependentofposition.Elimi- contrast δ = δρ/ρ in linear perturba- sions of a fifth force: β O(1). But the interesting scale − 2 the interest of a self-contained∼ presentation. tionnation theory of √ is1 simplyv in expressed favor of ϵs inresults terms in of the the screening Fourier for the screening length in our case is rs 1Mpc,vastly − ∼ amplitudeslength (4).δk The(t). In energy the approximationϵs does change that with all time, the mass scal- larger than the regime rs 100 µmofinterestinmodern −1 ising in the with dark the matter, expansion the of evolution the universe equation as ϵ iss a(t) . fifth force experiments. And,∼ crucially, we assume that ∝ A. The screening mechanism The screening length thus scales as rs a(t), that is, the the scalar force acts only in the dark sector: visible mat- a˙ β ∝ comovingδ¨k +2 lengthδ˙k is=4 constant.πGρ¯ 1+ δk, (6) ter interacts with the dark matter only through ordinary a $ 1+(kr )−2 % Let one dark matter species have mass m yφ,while The scalar field produced by a singles dark matter par- gravity. ticle at distance r r is φ = y/4πr.Theforcethisfield asecondspecieshasmassy φ.Thecouplings− y and y s It is essential that we assumes that the self-interactions whereexertsk onand anotherrs are≪ constant dark matter in comoving particle is coordinates. the negative In of are positive dimensionless constants. Assume that the potential V (φ)forthescalarfieldcanbeneglected:a thethe Einstein-de gradient of Sitter the mass model,m modesyφ,thatis, with wavenumberF = y kφ, particlessignificant of mass non-zeroysφ haveV ′(0) a would much larger drive φ numberaway from density, 0 at grow as − ∇ ′′ ′ so we seeTweaking: that 5th the force particles are attracted with force n¯laten¯ times,s,sothatasthefieldmovestominimizetheenergy and non-zero V (0) (with V (0) = 0) amounts 2 2 ≪ F = y /4πr at r rs.Thismeanstheratioofthe1/2 into particle a mass masses for φ which it is dominates pulled to 0 over< φ the screeningm.That ef- p 1≪ 24β 1 ≪ scalark and gravitational forces of attraction of two dark makes the screening particles relativistic and gives the δ t ,p= 25 + −2 , (7) fect of the relativistic particles at late times. Non-zero matter∝ particles is 6$ 1+(krs) % − 6 darkhigher matter derivatives particles of massesV at φ that=0mightendangerour are close to m.The 2 particlestory in dynamics, subtler ways independent which have of spin not been or statistics, fully probed. can to be compared to the usual powery law, p =2/3, at krs β = . ≪(5) forThese present requirements purposes on beV modeled(φ)areinconflictwithstandard as gasses of classical 1. The scalar interaction thus4π causesGm2 earlier development point-likenotions of particles field-theoretic with the naturalness. action We nevertheless of small-scale structure. We comment on the possible claim some motivation for our model from string theory implicationsThe relations in (4)IV. and (5) summarize how the parameters and supersymmetry,4 1 as we2 briefly review in II B. ofThe the value potential of the§ (1) interaction emerge from cuto theff length dynamicsr is limited we have S = d x (∂φ) ds mα(φ) , (2) s Our further! discussion2 in− sections! II A and§ II B of the added to the dark sector. "α γα by the allowed energy density ρs = ϵsn¯s in the relativistic physics behind the force law (1) includes some recapit- The effect of the scalar interaction on the evolution of ulation of earlier discussions [2, 8, 9], presented here in the mass density contrast δ = δρ/ρ in linear perturba- the interest of a self-contained presentation. tion theory is simply expressed in terms of the Fourier amplitudes δk(t). In the approximation that all the mass is in the dark matter, the evolution equation is A. The screening mechanism a˙ β δ¨k +2 δ˙k =4πGρ¯ 1+ δk, (6) a $ 1+(kr )−2 % Let one dark matter species have mass m yφ,while s − asecondspecieshasmassysφ.Thecouplingsy and ys where k and rs are constant in comoving coordinates. In are positive dimensionless constants. Assume that the the Einstein-de Sitter model, modes with wavenumber k particles of mass ysφ have a much larger number density, grow as n¯ n¯s,sothatasthefieldmovestominimizetheenergy ≪ 1/2 in particle masses it is pulled to 0 < φ m.That p 1 24β 1 ≪ k makes the screening particles relativistic and gives the δ t ,p= 25 + −2 , (7) ∝ 6$ 1+(krs) % − 6 dark matter particles masses that are close to m.The particle dynamics, independent of spin or statistics, can to be compared to the usual power law, p =2/3, at krs for present purposes be modeled as gasses of classical 1. The scalar interaction thus causes earlier development≪ point-like particles with the action of small-scale structure. We comment on the possible implications in IV. 4 1 2 The value of the§ interaction cutoff length r is limited S = d x (∂φ) ds mα(φ) , (2) s ! 2 − ! "α γα by the allowed energy density ρs = ϵsn¯s in the relativistic Violation of the Equivalence Principle If B and E contain DM Tweaking: DM only LCDM AN, Gubser & Peebles 05 LCDM+5th Tweaking: halo assembly Baryonic Fraction as a function of mass Simulations with 5th Observations McGaugh et. al. 09 Tweaking: implications A test for ReBEL Galactic satellites/streams (Frieman & Gradwohl 93, Kesden & Kamionkowski 06) e.g. Sagittarrius stream is ok (Keselma, AN & Peebles in response to Kesden and Kamionkowski) Tweaking: falsify? The Sagittarius dwarf/stream disk 15kpc Ibata et al Gravity alone 3 component gravitational potential: halo+disk+``bulge” Gravity + weak 5th (beta=0.1) A demonstration of the Frieman & Gradwohl 93 and Kesden & Kamionkowski 2006 test Gravity + strong ReBEL (beta=1) Observations Simulation (at 3 Gyrs) Close to pericenter, Distance from Close to pericenter, center of galaxy at 16 Kpc (Law et al., 2004) at 18 Kpc Luminosity, 2 to 5.8e7 Assuming (Ibata 97, 8e7 M/L=2.5 solar Mateo 98) Line of sight 9.6 Km/s, Velocity and constant 9.8 Km/s dispersion (Bellazzini, 2008) and constant Observed core radius is about 1 Kpc (Majewski, 2003) – matched well by the simulation. The Astrophysical Journal,755:58(6pp),2012August10 Nusser, Branchini, & Davis The Astrophysical Journal,755:58(6pp),2012August10 Nusser, Branchini, & Davis possibly making Gaia’spropermotionsanexcellentprobeofthe thanks to the interesting property that anTheoretical irrotational tools (or poten- large-scale flows.possibly This making probeGaia of’spropermotionsanexcellentprobeofthe large-scale flows is completely tial) flowthanks in to real the space interesting remains propertyLinear irrotational theory that application an irrotational also to in data redshift (or poten-Luminosity space variations The Cosmological Principle Gaia independentlarge-scale of any assumption flows. This probe on the of large-scale intrinsic flows relations is completely of (Chodorowskitial) flow in & realThe Nusser space AlternativeGaia remains1999 prospect). probes irrotational of large scale also motions in redshift space galaxies. Further,independent the two-dimensional of any assumption (2D) on transverse the intrinsic motions relations of (Chodorowski & Nusser 1999). The Astrophysical Journal,755:58(6pp),2012August10 Nusser, Branchini, & Davis are orthogonalgalaxies. (in information Further, the two-dimensional content as well (2D) as in transverse geometry) motions agalaxywouldbedetectedinasingleresolutionelementof are orthogonal (in information content as well as in geometry) 2.1. From 2D Transverse Velocities to 3D Flows 700 Gaia.Wefindthatthemajorityofearly-andlate-typegalaxies to standard line-of-sight peculiar velocities. 2.1. From 2D Transverse Velocities to 3D Flows could be detected as point sources at G 20 if, respectively, to standard line-of-sight peculiar velocities. 600 G<16 placed at 500 h 1 Mpc and 250 h 1 Mpc.= Overall, it looks The outline of the paper is as follows. In Section 2,wepresent Here, we offer basic expressions for the derivation of the full G<15 70− 70− The outline of the paper is as follows. In Section 2,wepresent 500 like the overwhelming majority of early-type galaxies and more Here, we offer basic expressions for the derivation of the full G<14 the generalthe setup general and setup describe and describetheoretical theoretical tools for tools analyzing for analyzing peculiar velocity field vvv(sss)fromthesmoothed2Dtransverse than 50% of late types will have peculiar motions measured by peculiar velocity field vv(ss)fromthesmoothed2Dtransverse SFI++ future transversefuture transverse velocity data. velocity We data. present, We present, in Section in Section3,a3,a 400 Gaia with errors in transverse velocities given in the top panel velocityvelocity field, field,vvv (vvsss).(ss Assuming). Assuming a a potential potential flow flow vvvvv((sssss)) Φ [km/s] Φ(ss()sss) in Figure 2.Inaddition,asignificantfractionoftheiremitted ⊥ ⊥ ⊥ 300 = −∇ σ rough estimaterough of estimate the expected of the expected error in errorthe transverse in the transverse velocity velocity andand expanding expanding the the angular angular dependence dependence of of ΦΦ inin= spherical spherical−∇ light will be within Gaia’s detection window, which justifies the obtained by smoothing individual velocities. Expected errors on 200 simple relation between galaxy number density and luminosity obtained by smoothing individual velocities. Expected errors on harmonics,harmonics,(sssΦ) (ss) Φlm(s()sY)Ylm(sss(ss),), gives gives (Arfken & & Weber Weber Φ = lm lmΦlm lm ˆ function that we have adopted in Section 3.AGNswillbeeasily astrometry for Gaia’s galaxies are discussed in Section 4, and 2005) = ˆ 100 astrometry for Gaia’s galaxies are discussed in Section 4, and 2005) Lunched Dec 2013 & doing well detected by Gaia as bright, pointlike sources and possibly mis- amoregeneraldiscussiononastrometryofextendedobjects ! taken by galaxies. However, their contamination to a relatively amoregeneraldiscussiononastrometryofextendedobjects ! dΦlm is given in Section 5.Intheconcludingsection,Section6, v dΦlm Y (3) local sample of objects with measured redshift, like the one we is given in Section 5.Intheconcludingsection,Section6, Alternative V: Gaia lm v ∥ = − ds Ylm 300(3) consider here, should be negligible. we present a general assessment of the transverse velocity lm we present a general assessment of the transverse velocity ∥ = − " ds In fact, since we are interested in studying the velocity field data in comparison to other probes of large-scale motions. We lm 250 of the local ( 100 h 1 Mpc) universe, the situation is likely to " 70− data in comparisonalso discuss to other possible probes sources of for large-scale redshifts of motions. the population We of G<16 be even more favorable. Within this distance the typical galaxy 200 also discussgalaxies possible expected sources to for be observed redshifts by ofGaia the. population of Φlm G<15 will be resolved in high-SB substructures that, if brighter than vv !lm, (4) G 20, can be detected as individual sources and analyzed as a Φlm [km/s] 150 G<14 galaxies expectedUnless to beotherwise observed specified, by Gaia magnitudes. observed by Gaia ⊥ = − s B = v lm ! σ group. Examples of multiple high-SB sources are star-forming vv lm, (4) ΛCDM will refer to an aperture photometry of 0.65 arcsec. They are ⊥ = − " s 100 regions, globular clusters, and bulges with steep SB profiles that Unless otherwise specified, magnitudes observed by Gaia lm given in the G band (350–1000 nm). Transformation from the where ! r Y is the vector" spherical harmonic. Thanks to are more extended than Gaia’s window (for example, the SB will refer to an aperture photometry of 0.65 arcsec. They are lm lm 50 profile of M87 drops below 18.5magarcsec 2 at 700 h 1 pc = ∇Pros: free of biases, allows K K − 70− more familiar V and Ic bands is performed using constant colors the orthogonality conditions dΩ!lm !l m l(l +1)δ δ 1 ∼ given in the G band (350–1000 nm). Transformation from the ! ′ ′ ll′ mm′ from the center; if placed at 50 h70− Mpc, it will be detected where lm r Ytestslm is theof vectorpotential spherical flow· harmonic.ansatz= Thanks0 to ∼ V G 0.27 and V Ic 1forallgalaxies(Fukugita the potential= ∇ coefficients can be recovered by K K 2 4 6 8 10 12 14 as 10 individual sources by Gaia). Detecting multiple sources more familiar −V and=Ic bands is performed− = using constant colors the orthogonality conditions dΩ!lm !l m l(l +1)δ δ ∼ et al. 1995;Jordietal.2010). We also assume that Gaia will # ′ ′ ll′ mm′ cz [1000km/s] from the same objects significantly improves the astrometric V G 0.27 and V I 1forallgalaxies(Fukugita · = precision, as we shall show in the next section. identify all sourcesc with G<20 within 0.65 arcsec with the potential coefficients can1 be recovered by Figure 2. Expected errors (1σ) on two quantities computed from the Gaia et al.− 1995=;Jordietal.2010− ).= We also assume that Gaia will1 1 # astrometric galaxy data. Top: errors in the 2D transverse peculiar velocity 100% completeness. Finally, we use H0 70 km s− Mpc− Φlm(s) − dΩvv (ss) !lm(ss), field obtained(5) by filtering the data with a Gaussian window of width RG 1 = 5. ASTROMETRY WITH EXTENDED OBJECTS = = l(l +1) ⊥ · ˆ 1500 km s− .Forcomparison,thethinsolidmagentalineistheerrorinthe identify allto sources set the distance with G< scale and20 use withinh70 0.65H0/ arcsec70 to parameterize with 1 $ SFI++ line-of-sight peculiar velocities smoothed with the same window. Errors 1 1 3/2 The possibility of placing multiple constraints on the same = (s) d vvv (sss) ! (sss), scale like R (5). Bottom: errors in the bulk (dipole) motion of spherical shells 100% completeness.uncertainties. Finally, we use H0 70 km s− Mpc− Φlm − Ω lm G objects allows one, in principle, to improve the astrometric ac- ⊥ 1 1/2 = for l>0. This means= l(l that+1)Φ(ss)canberecoveredfromthe· ˆ of thicknessvv up∆cz 3000 km s− . Errors scale like (∆cz) . For reference, to set the distance scale and use h H /70 to parameterize = curacy. We discuss this possibility in a general context and with 70 0 to a monopole term that corresponds$ to a purely radial flowpredictions⊥ with from the WMAP7 ΛCDM for the dipole on shells are also plotted. 2. METHODOLOGY= In both panels, dash-dotted, solid, and dotted curves correspond to G 14, 15, aformalismthatcontemplateboththepossibilityofperform- uncertainties. zero transverse motions. That is not a serious drawbackand 16 since mag cuts, as indicated in the figure. = ing resolved photometry with high-resolution instruments like for l>0. This means that Φ(sss)canberecoveredfromthevvv up HST,10 JWST,LSST,orPan-STARRS(Saha&Monet2005; We will assume an all-sky catalog of redshifts and proper the monopole term can always be removed from the predictions(A color version⊥ of this figure is available in the online journal.) to a monopole term that corresponds to a purely radial flow with Chambers 2005)andthatofsplittinganextendedsourcein motions. We2. METHODOLOGY denote the physical peculiar velocity by vv and of any model to be compared with the data. individual sources, like in the case of Gaia. more conservative choice and assume that only sources with zero transverse motions. That is not a serious drawback since Suppose for simplicity we observe a galaxy at two different the real space comoving coordinate by rr,bothexpressedin µ < 18.5 mag arcsec 2 will be used for astrometric purposes. G − epochs, t and t .LetusdefineI (θ )astheSBoftheobject We will assume1 an all-sky catalog of redshifts and proper the monopole term can always be removed from the predictionsAsurveyoftheliteratureshowsthatthisconditionissatis- 1 2 i i km s− .Further,v vv rr and vv vv v rr are, respectively, at the epoch t measured at the angular position of a pixel ∥ = · ˆ ⊥ = − ∥ ˆ 2.2. Testing the Potential Flow Ansatz fied for the central region of a significant fraction of galaxies i motions. Wethe denote components the physical of vv parallel peculiar and perpendicular velocity by tovvv theand line of of any model to be compared with the data. θ .InthecaseoftraditionalphotometryI (θ )representsthe (e.g., Kormendy 1977;Allenetal.2006;Oohamaetal.2009; i i i SB profile of the object at θ ,whereasinthecaseofGaia the real spacesight, comoving where rr is coordinate a unit vector by inrrr,bothexpressedin the line-of-sight direction. Balcells et al. 2007;Smithetal.2009;Graham2011; Ferrarese i Initial conditions in the early universe might have been it represents the magnitude of the SB substructure measured 1 ˆ 1 et al. 1994;Carolloetal.1998;Laueretal.2007). For example, km s− .Further,We restrictv vvv therrr analysisand vvv to czvvv v15rrr,000are, km respectively, s− and neglect somewhat chaotic, so that the original peculiar velocity field within the detection window. In principle, the astrometric shift, 2.2. Testing the Potential Flow Ansatz this can be seen in Figure 3 in Oohama et al. (2009)show- cosmological∥ = geometric· ˆ ⊥ effects,= − so∥ thatˆ the redshift coordinate p,couldbedeterminedbyminimizing,withrespecttop, the components of vvv parallel and perpendicular to the line of was uncorrelated with the mass distribution or even containeding a scatter plot of the B-band effective SB versus half-light χ 2 [I (θ ) I (θ )]2/σ 2,wherethesummationisover is ss rr + v rr.Notess rr and cz r + v ss rr s.Proper radius for various galaxy types.9 More importantly, we have vi- i 1 i 2 i′ i sight, where rrr is a unit vector in the line-of-sight direction. vorticity (e.g., Christopherson et al. 2011). At late time, a all pixels,= θ −θ p,andσ here is the 1σ error in the = ∥ ˆ ˆ = ˆ = ∥ = · ˆ = Initial conditions in the early universe might havesually inspected been the observed V-band SB profiles of 200 out of ′ Ii motionsˆ transverse to the line of sight will be1 denoted by µ.The measurement! of= the− SB (since p is small, we assume that σ We restrict the analysis to cz 15,000 km s− and neglect cosmological velocity field should have a negligible rotational600 galaxies in the Carnegie-Irvine Galaxy Survey (Ho et al. Ii somewhat chaotic,rot so that the original peculiar velocity field in pixel i is the same for both images). We have assumed that transverse 2D space velocity of a galaxy at real-space distance component, vv on large scale, away from orbit mixing2011∼ regions.;Lietal.2011). Most of these galaxies are nearby (me- cosmological geometric effects, so that the redshift coordinate I and I differ only by a linear displacement. In principle, one r is was uncorrelated with the mass distributionrot or even contained h 1 B 1 2 The reason is that any circulation, Γ vv dss,isconservedbydian distance of 25 70− Mpc) and with mean -band absolute should take into account changes in the internal structure of is sss rrr + v rrr.Notesss rrr and cz r + v sss rrr s.Proper vorticity (e.g., Christopherson et al.= 2011·). At latetotal time,magnitude a ∼ of 20.2, close to M .Weidentifiedgalaxies = ∥ ˆ ˆ = ˆ = ∥ = · ˆ = Kelvin’s theorem. Hence, any rotational component will decay − ∗ 2 the object. Those, however, will have little effect compared to µ reaching a central SB of 18.5magarcsec− and tabulated the motions transverse to thevv linerµ of sight will be denoted by .The cosmologicalas 1/a,where velocitya is the field scale should factor. have In% contrast, a negligible the irrotational rotational the overall observational accuracy. Since we will eventually be ⊥ = µ r rot corresponding radii (in arcsec). Since we did not have access to interested in the mean coherent displacement of an ensemble of transverse 2D space velocity of a galaxy at real-space distance component, vvv on large scale, away from orbit mixingthe regions. actual data, the minimal radius we could determine using a 677.22 h70, (1) component of the peculiar velocity will have a growing v √a. many galaxies, incoherent changes in the internal structure of 1 4 1 rot ruler∼ is 1–2 arcsec. About 70% of the galaxies we inspected were r is = 1 µas yr− 10 km s− Therefore, on large scales, away fromv collapseds objects, the galaxies will be insignificant. The reason is that any circulation, Γ vv dss,isconservedbybrighter than 18.5magarcsec 2,allowingthemtobedetected irrotational component is expected= to be negligible.· The absence − This procedure of minimizing the image differences exploits which corresponds to a transverse peculiar velocity of Kelvin’s theorem. Hence, any rotational component willby Gaia decay.SinceSBisadistance-independentquantity,wecan all information contained in both images, but it requires a possi- vvv rµ of any significant large-scale vorticity is, therefore,use a strongthis threshold to compute the maximum distance at which 1 1 as 1/a,wherea is the scale factor. In% contrast, the irrotational bly non-trivial interpolation of θ ′ on the observed pixel positions 474⊥ = km s− for 1 µasµ yr− at d r 100 Mpc. = prediction of the standard cosmological paradigm. To9 For assess old stellar populations, B V + 1 (Fukugita et al. 1995), and since However,677.22 the true distances, r,areunknown,and,therefore,h , (1) component of the peculiar velocity will have a growing v √a. ∼ 10 70 this prediction, the observed transverse motions can beG usedV +0.27, to the astrometric condition G 18.5translatestoB 19.7. http://www.stsci.edu/hst/ = 1 µas yr 1 104 km s 1 = ∼ we make the approximation− − Therefore,constrain on the large amplitude scales, of theaway irrotational from collapsed component. objects, This can the 4 irrotational component is expected to be negligible.vrot The absence which corresponds to a transversevv peculiarsµ. velocity of (2) be done by writing the transverse component of vv as (Arfken 1 1 ⊥ = of any& Weber significant2005) large-scale vorticity is, therefore, a strong 474 km s− for 1 µas yr− at d 100 Mpc. prediction of the standard cosmological paradigm. To assess This introduces a relative= error v /s in the determination of vv rot rot However, the true distances, r,areunknown,and,therefore,∥ ⊥ vv Vlm "lm, (6) 2 1/2 1 this prediction, the observed⊥ = transverse motions can be used to we make thewhere approximationv 200–300 km s− (Davis et al. 2011). Hence, lm ⟨ ∥ ⟩ ∼ the error is negligible as we go to s 2000 km s 1.Theerrorconstrain the amplitude of the" irrotational component. This can − be done by writing the transverse component of vvvrot as (Arfken is also random sincevvv vv svµ. 0. (2) where "lm ss Ylm belong to another class of vector spher- Therefore, the estimated⊥ =⟨ ⊥ ∥⟩ = velocity field will be given as a & Weberical harmonics2005)= that×∇ satisfy the same orthogonality conditions as function of the redshift space coordinate. To linear order, veloc- !.Hence,V rot is equal to the right-hand side of Equation (5)but This introduces a relative error v /s in the determination of vvv lm vvvrot V rot" , (6) 2 1ity/2 fields expressed in real∥1 and redshift spaces are equivalent.⊥ with " instead of ! .Further,lm dlmΩ" ! 0; hence, where v 200–300 km s (Davis et al. 2011). Hence, lm ⊥ lm= lm l′m′ In the quasilinear regime,− dynamical relations can be derived the recovery of the rotationallm mode is formally· independent= of ⟨ ∥ ⟩ ∼ 1 " the error isfor negligible the velocity as we field go in to redshifts 2000 space km (Nusser s− .Theerror & Davis 1994), the potential flow mode. # is also random since vvv v 0. where "lm sss Ylm belong to another class of vector spher- Therefore, the estimated⟨ ⊥ ∥⟩ = velocity field will be given as a ical2 harmonics= that×∇ satisfy the same orthogonality conditions as rot function of the redshift space coordinate. To linear order, veloc- !.Hence,Vlm is equal to the right-hand side of Equation (5)but ity fields expressed in real and redshift spaces are equivalent. with "lm instead of !lm.Further, dΩ"lm !l′m′ 0; hence, In the quasilinear regime, dynamical relations can be derived the recovery of the rotational mode is formally· independent= of for the velocity field in redshift space (Nusser & Davis 1994), the potential flow mode. # 2 Problem I: Alternative V: Gaia Theoretical tools Linear theory application to data Luminosity variations The Cosmological Principle Gaia Alternative probes of large scale motions Theoretical tools Linear theoryAstrometry applicationTheoretical to of data galaxies tools Luminosity with Gaia variations (2013-2018) TheLinear Cosmological theory application Principle to data GaiaLuminosity variations Alternative probes of largeThe Cosmological scale motions Principle Gaia Alternative probes of large scale motions AstrometryAstrometry of galaxies of galaxiesProblem with with Gaia Gaia II: (2013-2018) (2013-2018) Gaia’s onboard thresholding is optimized for point sources. But, a large number of galaxies have stellar light concentrated in compact regions, making them appear as point sources. Gaia’sFor onboard example, the thresholding nuclei of M87 and N5121 is optimized (both d=17.8Mpc) for point should be sources. detectable by Gaia with Gaia’s onboardan end ofthresholding mission accuracy of 600 iskm/ optimizeds in V . for point sources. But,Visual a large inspection number of SB profiles of galaxies of the Carnegie-Irvine have? stellar Galaxy Survey light (Ho concentrated et al 2011) shows that in compact70% of galaxies regions, in this survey making could be them detected appear by Gaia. asThe point majority of sources. those nearby galaxies But, a largewill number be detected if of placed galaxies at > 500Mpc have(early types) stellar and > 250 lightMpc (late concentrated type). For example, the nuclei of M87 and⇠ N5121 (both d=17.8Mpc) should⇠ be detectable by Gaia with in compactan regions, end of mission accuracy making of 600km them/s in V . appear as point sources. DistancesVisual inspection of galaxies of SB profiles are of the Carnegie-Irvineneeded? to Galaxy get SurveyV (Hobut et al 2011) not shows a big that error For example,70% the of nuclei galaxies of in M87 this survey and could N5121 be detected (both d=17.8Mpc) by Gaia. The majority should? of be those detectable nearby galaxies by Gaia with an endis of made missionwill be bydetected accuracy approximating if placed of 600 atkm> 500/sMpcin Vd(early=. types)cz/ andH >in250µ˙Mpcd. (late type). ⇠ ⇠ Visual inspection of SB profiles of the Carnegie-Irvine? Galaxy Survey (Ho et al 2011) shows that 70% ofDistances galaxies in this of galaxies survey could are be detected needed by to Gaia. get TheV majoritybut not of those a big nearby error galaxies will be detected if placed at > 500Mpc (early types) and >?250Mpc (late type). is made by approximating⇠ d = cz/H in µ˙⇠d. Distances of galaxies are needed to get V but not a big error ? is made by approximating d = cz/H in µ˙ d. Alternative V: Gaia The Astrophysical Journal,755:58(6pp),2012August10 Nusser, Branchini, & Davis The AstrophysicalThe Journal Astrophysical,755:58(6pp),2012August10 Journal,755:58(6pp),2012August10 Nusser,Nusser, Branchini, Branchini, & & Davis possibly making Gaia’spropermotionsanexcellentprobeofthe thanks to the interesting property that anTheoretical irrotational tools (or poten- The Astrophysical Journal,755:58(6pp),2012August10 Nusser, Branchini, & Davis large-scale flows.possibly This making probeGaia of’spropermotionsanexcellentprobeofthe large-scale flows is completely tial) flowthanks in to real the space interesting remains propertyLinear irrotational theory that application an irrotational also to in data redshift (or poten-Luminosity space variations possibly making Gaia’spropermotionsanexcellentprobeofthe thanks to the interesting property that anTheoretical irrotational tools (or poten- large-scale flows. This probe of large-scale flows is completely tial) flow in real space remains irrotationalThe Cosmological also Principle in redshiftGaia space large-scaleindependent flows.possibly This of making any probe assumptionGaia of’spropermotionsanexcellentprobeofthe large-scale on the flows intrinsic is completely relations of tial)(Chodorowski flowthanks in to real the space interesting & Nusser remainsAlternative property1999Linear irrotational). theoryprobes that applicationof an large irrotational also scale to in motions data redshift (or poten-Luminosity space variations galaxies.large-scale Further,independent flows. the two-dimensional This of any probe assumption of large-scale (2D) on transverse the flows intrinsic is completely motions relations of tial)(Chodorowski flow in real space & Nusser remains1999 irrotationalThe). Cosmological also Principle in redshiftTheGaia space Astrophysical Journal,755:58(6pp),2012August10 Nusser, Branchini, & Davis independent of anygalaxies. assumption Further, the on two-dimensional the intrinsic (2D) relations transverse of motions(Chodorowski & NusserAlternative1999). probes of large scale motions are orthogonalindependent (in of information any assumption content on as the well intrinsic as in geometry) relations of (Chodorowski & Nusser 1999). End-of-missionThe Astrophysical (2018) Journal expectations,755:58(6pp),2012August10 agalaxywouldbedetectedinasingleresolutionelementofNusser, Branchini, & Davis galaxies. Further,are the orthogonal two-dimensional (in information (2D) content transverse as well motions as in geometry) 2.1. From 2D Transverse Velocities to 3D Flows 700 Gaia.Wefindthatthemajorityofearly-andlate-typegalaxies to standardgalaxies. line-of-sight Further, the peculiar two-dimensional velocities. (2D) transverse motions 2.1. From 2D Transverse Velocities to 3D Flows agalaxywouldbedetectedinasingleresolutionelementof are orthogonal (in information content as well as in geometry) G<16 could be detected as point sources at G 20 if, respectively, to standard line-of-sight peculiar velocities. 700600 1 1 = are orthogonal (in information content as well as in geometry) 2.1. From 2D Transverse Velocities to 3D Flows placedGaia.Wefindthatthemajorityofearly-andlate-typegalaxies at 500 h− Mpc and 250 h− Mpc. Overall, it looks The outlineThe of the outline paper of is the as paper follows. is as follows. In Section In Section2,wepresent2,wepresent Here,2.1. we offer From basic 2D Transverse expressions Velocities for the to derivation 3D Flows of the full G<15 70 70 to standard line-of-sight peculiar velocities. Here, we offer basic expressions for the derivation of the full500 G<16 likecould the be overwhelming detected as point majority sources of early-type at G galaxies20 if, respectively, and more the generalto standard setup line-of-sight and describe peculiar theoretical velocities. tools for analyzing 600 G<14 1 1 = the general setup and describe theoretical tools for analyzing peculiar velocity field vv(ss)fromthesmoothed2Dtransverse thanplaced 50% at of500 lateh types70− Mpc will and have peculiar250 h70− motionsMpc. Overall, measured it looks by The outline of the paper is as follows. In Section 2,wepresent Here,peculiar we offer velocity basic expressions field vv(ss)fromthesmoothed2Dtransverse for the derivation of the full400 G<15SFI++ The outline of the paper is as follows. In Section 2,wepresent 500 Gaialike thewith overwhelming errors in transverse majority velocities of early-type given galaxies in the top and panel more future transversefuture transverse velocity data. velocity We data. present, We present, in Section in Section3,a3,a Here, we offer basic expressions for the derivation of the full G<14 the general setup and describe theoretical tools for analyzing velocityvelocity field, field,vv (vvss).(ss Assuming). Assuming a a potential potential flow flow vvvv((ssss)) Φ [km/s] Φ(ss()ss) in Figure 2.Inaddition,asignificantfractionoftheiremitted the general setup and describe theoretical tools for analyzing peculiar velocity field⊥ ⊥ vv(svs)fromthesmoothed2Dtransverses ⊥ 300 than 50% of late types will have peculiar motions measured by rough estimaterough of estimate the expected of the expected error in errorthe transverse in the transverse velocity velocity peculiar velocity field vv(ss)fromthesmoothed2Dtransverse==−∇−∇σ 400 SFI++ future transversefuture transverse velocity data. velocity We data. present, We present, in Section in Section3,a3,a andand expanding expanding the the angular angular dependence dependence of of ΦΦ inin spherical spherical lightGaia willwith be errors within inGaia transverse’s detection velocities window, given which in justifies the top panel the obtained by smoothing individual velocities. Expected errorsvelocity on velocity field, field,vv (vvss).(ss Assuming). Assuming a a potential potential flow flow vvvv((ssss)) Φ [km/s] Φ(ss200()ss) simplein Figure relation2.Inaddition,asignificantfractionoftheiremitted between galaxy number density and luminosity obtained by smoothing individual velocities. Expected errors on harmonics,harmonics,⊥(ssΦ⊥) (ss) lm Φlm(s()sY)Ylm(ss(ss),), gives gives (Arfken & & Weber⊥ Weber300 Φ Φlm lm = −∇ σ rough estimaterough ofastrometry estimate the expected of for theGaia expected error’s galaxies in errorthe transverse are in discussed the transverse velocity in Section velocity4, andandand expanding expanding the the angular= angularlm dependence dependenceˆ of of ΦΦ inin= spherical spherical−∇ functionlight will that be within we haveGaia adopted’s detection in Section window,3.AGNswillbeeasily which justifies the astrometry for Gaia’s galaxies are discussed in Section 4, and 2005) = ˆ 100 detected by Gaia as bright, pointlike sources and possibly mis- obtained by smoothing individual velocities. Expected errors on 2005) 200 simple relation between galaxy number density and luminosity obtained by smoothingamoregeneraldiscussiononastrometryofextendedobjects individual velocities. Expected errors on harmonics,harmonics,Φ(ssΦ) (ss) lmΦ!lmΦlm(s()sY)Ylmlm(ss(ss),), gives gives (Arfken & & Weber Weber taken by galaxies. However, their contamination to a relatively amoregeneraldiscussiononastrometryofextendedobjects = lm! dˆΦlm function that we have adopted in Section 3.AGNswillbeeasily astrometry forastrometryGaiais given’s for galaxies inGaia Section’s are galaxies5 discussed.Intheconcludingsection,Section are discussed in Section in Section4, and4, and 6, 2005) = dˆ 100 local sample of objects with measured redshift, like the one we is given in Section 5.Intheconcludingsection,Section6, 2005) v Φlm Ylm (3) detected by Gaia as bright, pointlike sources and possibly mis- amoregeneraldiscussiononastrometryofextendedobjects !v ∥ = − ds Ylm 300(3) considertaken by here, galaxies. should However, be negligible. their contamination to a relatively amoregeneraldiscussiononastrometryofextendedobjectswe present a general assessment of the transverse velocity ! lmd we presentis given a in general Section assessment5.Intheconcludingsection,Section of the transverse velocity6, ∥ = − Φdslm localIn fact, sample since of we objects are interested with measured in studying redshift, the like velocity the one field we data in comparison to other probes of large-scale motions. We v lm"dΦlm Ylm (3)250 1 is given in Section 5.Intheconcludingsection,Section6, of the local ( 100 h70− Mpc) universe, the situation is likely to data inwe comparison present a generalto other assessment probes of large-scale of the transverse motions. velocity We v ∥ = − " ds Ylm 300(3) consider here, should be negligible. also discuss possible sources for redshifts of the population of ∥ = − lm ds G<16 beIn even fact, more since favorable. we are interestedWithin this in distance studying the the typical velocity galaxy field we present a general assessment of the transverse velocity " Φlm 200 will be resolved in high-SB substructures that, if brighter than also discussdata ingalaxies possible comparison expected sources to other to for be probes observed redshifts of by large-scale ofGaia the. population motions. of We lm 250 G<15 of the local ( 100 h 1 Mpc) universe, the situation is likely to vv !lm, (4) G 20, can be detected70− as individual sources and analyzed as a " [km/s] data in comparison to other probes of large-scale motions. We Φlm 150 G<14 galaxiesalso expected discussUnless possible to beotherwise observed sources specified, by forGaia redshifts magnitudes. of the observed population by ofGaia ⊥ = − s B G<16 be= even more favorable. Within this distance the typical galaxy vv lm !lm, σ 200 (4) group. Examples of multiple high-SB sources are star-forming also discussgalaxies possiblewill expected refer sources to to an for be aperture observed redshifts photometry by ofGaia the. population of 0.65 arcsec. of They are ⊥ "Φlm G<15ΛCDM regions,will be resolved globular clusters, in high-SB and substructures bulges with steep that, SB if profiles brighter that than Unless otherwise specified, magnitudes observed by Gaia vv = − s !lm, (4)100 G 20, can be detected as individual sources and analyzed as a lmΦlm [km/s] 150 G<14 are more extended than Gaia’s window (for example, the SB galaxies expectedUnlessgiven to beotherwise in observed the G band specified, by (350–1000Gaia magnitudes. nm). Transformation observed by Gaia from the ⊥ = − s B = where !lm rv Ylm is the vector" spherical harmonic. Thanksσ to group. Examples of multiple high-SB sources2 are star-forming1 will refer to an aperture photometry of 0.65 arcsec. They are vv freelm of biases,!lm, allows (4)50 profile of M87 drops below 18.5magarcsec at 700 h− pc = ∇Pros: " K K ΛCDM − 70 Unless otherwisewill refermore to specified, familiar an apertureV and magnitudes photometryIc bands is performed observed of 0.65 arcsec. using by Gaia constant They are colors the orthogonality⊥ conditions= − sdΩ!lm !l m l(l +1)δ 100δ regions, globular clusters, and bulges1 with steep SB∼ profiles that given in the G band (350–1000 nm). Transformation from the where ! r Y is the vectorlm spherical harmonic.′ ′ Thanksll′ mm′ to from the center; if placed at 50 h70− Mpc, it will be detected V G 0.27 and V I 1forallgalaxies(Fukugita lm testslm of potential flow· ansatz= 0 are more extended than Gaia∼’s window (for example, the SB will refer togiven an aperture in the G band photometry (350–1000 of nm).0.65c Transformation arcsec. They are from the wherethe! potentiallm = r∇ coefficientsYlm is the vector" can be spherical recovered harmonic. by ThanksK toK 2 4 6 8 10 12 14 as 10 individual sources by Gaia). Detecting multiple2 sources1 more familiar V and Ic bands is performed using constant colors Pros: free of biases, allows 50 profile∼ of M87 drops below 18.5magarcsec− at 700 h− pc − = − = the orthogonality= ∇ conditions #dΩ!lm !l′m′ l(l +1)K δllK δmm cz [1000km/s] from the same objects significantly improves the astrometric70 more familiaret al. 1995V and;Jordietal.Ic bands is2010 performed). We also using assume constant that colorsGaia will the orthogonality conditions dΩ!lm !l m l(l +1)δ δ ′ ′ 1 ∼ givenV in theGG band0.27 (350–1000 and V I nm). Transformation1forallgalaxies(Fukugita from the ! · ′ ′ = ll′ mm′ precision,from the center; as we shall if placed show at in the50 nexth70− section.Mpc, it will be detected c wherethe potentiallm r coefficientsYtestslm is theof vectorpotential can be spherical recovered flow· harmonic.ansatz by= ThanksFigure 2. 0Expected to errors (1σ) on two quantities computed from the Gaia ∼ V Gidentify0.27 all and sourcesV withIc G<1forallgalaxies(Fukugita20 within 0.65 arcsec with the potential coefficients can1 be recovered by 2 4 6 8 10 12 14 as 10 individual sources by Gaia). Detecting multiple sources − = − = 1 1 = ∇ astrometricK K galaxy data. Top: errors in the 2D transverse peculiar velocity more familiaret al. 1995−V and;Jordietal.=Ic bands is2010 performed).− We= also using assume constant that colorsGaia will the orthogonality conditions(s) d#Ω!dlm vv !(ssl )m ! (lss()l,+1)fieldδ obtainedδ(5) by filtering the data with a Gaussian window of width R ∼ et al. 1995100%;Jordietal. completeness.2010 Finally,). We we also use assumeH0 that70 kmGaia s− willMpc− ΦlmNew probes:− Gaia# Ω ′ ′ lm ll′ mm′ cz [1000km/s] G from5. the ASTROMETRY same objects significantly WITH EXTENDED improves OBJECTS the astrometric = l(l +1) ⊥ · ˆ 1500 km s 1.Forcomparison,thethinsolidmagentalineistheerrorinthe= V Gidentify0.27 all and sourcesV withI G<1forallgalaxies(Fukugita20 within 0.65= arcsec with · = Figure 2. Expected− errors (1σ) on two quantities computed from the Gaia precision, as we shall show in the next section. identifyto set all the sources distancec with scaleG< and use20 withinh70 H 0.650/70 arcsec to parameterize with the potential coefficients can1 be recovered$ by SFI++ line-of-sight peculiar velocities smoothed with the same window. Errors − = − = 1 1 1 astrometric galaxy3/2 data. Top: errors in the 2D transverse peculiar velocity The possibility of placing multiple constraints on the same = 1 1 Φlm(s) − # dΩvv (ss) !lm(ss), scale like R (5). Bottom: errors in the bulk (dipole) motion of spherical shells et al. 100%1995;Jordietal. completeness.uncertainties.2010 Finally,). We we also use assumeH0 that70 kmGaia s− willMpc− Φlm(s) − dΩvv (ss) !lm(ss), field obtained(5)G by filtering the data with a Gaussian window of width RG objects allows one, in principle, to improve the astrometric ac- 100% completeness. Finally, we use H0 70 km s− Mpc− ⊥ 1 1/2 5. ASTROMETRY WITH EXTENDED OBJECTS = for l>0. This means= ll((ll that+1)+1)Φ(ss)canberecoveredfromthe⊥ · ˆ ˆ 1500of thickness kmvv s 1.Forcomparison,thethinsolidmagentalineistheerrorintheup∆cz 3000 km s− . Errors scale like (∆cz) . For reference,= identifyto set all the sources distance with scaleG< and use20h withinH 0.65/70= arcsec to parameterize with = · − = curacy. We discuss this possibility in a general context and with to set the distance scale and use70 h70 0 H0/70 to parameterize to a monopole term that1 corresponds$$ to a purely radialSFI++ flowpredictions line-of-sight⊥ with from the peculiar WMAP7 velocitiesΛCDM smoothed for the dipole with theon shells same arewindow. also plotted. Errors 1 1 In both panels,3/2 dash-dotted, solid, and dotted curves correspond to G 14, 15, aformalismthatcontemplateboththepossibilityofperform-The possibility of placing multiple constraints on the same 2. METHODOLOGY= = (s) d vv (ss) ! (ss), scale like R (5). Bottom: errors in the bulk (dipole) motion of spherical shells 100%uncertainties. completeness.uncertainties. Finally, we use H0 70 km s− Mpc− Φlm − Ω lm and 16 magG cuts, as indicated in the figure. = ingobjects resolved allows photometry one, in principle, with high-resolution to improve the instruments astrometric like ac- zero transverse motions. That is not⊥ a serious drawback since 1 1/2 = forforl>l>0.0. This This means means= l(l that that+1)ΦΦ(s(sss)canberecoveredfromthe)canberecoveredfromthe· ˆ of thicknessvv vvup∆czup3000 km s− . Errors scale like (∆cz) . For reference, HST,10 JWST,LSST,orPan-STARRS(Saha&Monet2005; to set the distance scale and use h H /70 to parameterize (A color⊥ version= of this figure is available in the online journal.) curacy. We discuss this possibility in a general context and with We will assume an70 all-sky0 catalog of redshifts and proper the monopole term can always$ be removed from the predictions from⊥ the WMAP7 ΛCDM for the dipole on shells are also plotted. Chambers 2005)andthatofsplittinganextendedsourcein = toto a a monopole monopole term term that that corresponds corresponds to to a apurely purely radial radial flowIn both flow panels, with with dash-dotted, solid, and dotted curves correspond to G 14, 15, aformalismthatcontemplateboththepossibilityofperform- uncertainties. motions. We2. METHODOLOGY denote2. METHODOLOGY the physical peculiar velocity by vv and of any model to be compared with the data. = individual sources, like in the case of Gaia. zero transverse motions. That is not a serious drawbackandmore 16 since mag conservative cuts, as indicated choice in the and figure. assume that only sources with ing resolved photometry with high-resolution instruments like for zerol>0. transverse This means motions. that ( Thatss)canberecoveredfromthe is not a serious drawbackvv sinceup Suppose10 for simplicity we observe a galaxy at two different the real space comoving coordinate by rr,bothexpressedin Φ (Aµ color< version18.5 magof this arcsec figure is2 availablewill be in used the online for astrometric journal.) purposes. HST, JWST,LSST,orPan-STARRS(Saha&Monet2005; We will assume an all-sky catalog of redshifts and proper the monopole term can always be removed from the predictionsG ⊥ − epochs, t and t .LetusdefineI (θ )astheSBoftheobject We will assume1 an all-sky catalog of redshifts and proper to athe monopole monopole term term that can corresponds always be removed to a purely from radial the predictions flowAsurveyoftheliteratureshowsthatthisconditionissatis- with Chambers1 20052)andthatofsplittinganextendedsourceini i km s− .Further,v vv rr and vv vv v rr are, respectively, at the epoch t measured at the angular position of a pixel motions. We2. METHODOLOGY denote the∥ = physical· ˆ peculiar⊥ = − velocity∥ ˆ by vv and of any model to2.2. be compared Testing the with Potential the data. Flow Ansatz fied for the central region of a significant fraction of galaxies individual sources,i like in the case of Gaia. motions. Wethe denote components the physical of vv parallel peculiar and perpendicular velocity by tovv theand linezero of of transverse any model motions. to be compared That is with not the a serious data. drawbackmore conservative since choice and assume that only sources with θ .InthecaseoftraditionalphotometryI (θ )representsthe the real space comoving coordinate by rr,bothexpressedin (e.g., Kormendy 1977;Allenetal.2 2006;Oohamaetal.2009; i Suppose for simplicity we observe a galaxyi i at two different µG < 18.5 mag arcsec− will be used for astrometric purposes. SB profile of the object at θ i ,whereasinthecaseofGaia the real space1sight, comoving where rr is coordinate a unit vector by inrr,bothexpressedin the line-of-sight direction. Balcells et al. 2007;Smithetal.2009;Graham2011; Ferrarese epochs, t1 and t2.LetusdefineIi (θ i )astheSBoftheobject We will assume an all-sky catalogv r ofv redshiftsv r and proper the monopoleInitial term conditions can always in the be early removed universe from might the predictionsAsurveyoftheliteratureshowsthatthisconditionissatis- have been it represents the magnitude of the SB substructure measured 1km s− .Further,v ˆ vv rr and vv vv v rr are, respectively,1 et al. 1994;Carolloetal.1998;Laueretal.2007). For example, at the epoch t measured at the angular position of a pixel km s− .Further,We restrictv vv the∥ =rr analysisand· ˆ vv to ⊥czvv= v15−rr,000are,∥ ˆ km respectively, s− and neglect somewhat2.2. chaotic, Testing so the that Potential the original Flow peculiar Ansatz velocityfied for the field central region of a significant fraction of galaxies within the detectioni window. In principle, the astrometric shift, motions. Wethe denote components the physical of vv parallel peculiar and perpendicular velocity by tovv theand line of of any model to2.2. be compared Testing the with Potential the data. Flow Ansatz this can be seen in Figure 3 in Oohama et al. (2009)show- θ .InthecaseoftraditionalphotometryI (θ )representsthe cosmological∥ = geometric· ˆ ⊥ effects,= − so∥ thatˆ the redshift coordinate (e.g., Kormendy 1977;Allenetal.2006;Oohamaetal.2009; p,couldbedeterminedbyminimizing,withrespecttoi i i p, the components of vv parallel and perpendicular to the line of was uncorrelated with the mass distribution or even containeding a scatter plot of the B-band effective SB versus half-light SB profile of the object at θ ,whereasinthecaseofGaia the real spacesight, comoving where rr is coordinate a unit vector by inrr,bothexpressedin the line-of-sight direction. Balcells et al. 2007;Smithetal.2009;Graham2011; Ferrarese χ 2 [I (θ ) I (θ )]2/σ 2,wherethesummationisoveri is ss rr + v rr.Notess rr and cz r + v ss rr s.Proper Initial conditions in the early universe might haveradius been for various galaxy types.9 More importantly, we have vi- it representsi 1 thei magnitude2 i′ ofi the SB substructure measured sight,1 where rr is aˆ unit vector in the line-of-sight1 direction. vorticity (e.g., Christopherson et al. 2011). At lateet al. time,1994;Carolloetal. a 1998;Laueretal.2007). For example, all pixels,= θ −θ p,andσ here is the 1σ error in the km s− .Further,We restrictv =vv therr analysis∥andˆ vv toˆ =czvvˆ v15rr,000are,= km respectively,∥ s−= and· ˆ = neglect somewhatInitial conditions chaotic, so in that the the early original universe peculiar might velocity havesually field inspected been the observed V-band SB profiles of 200 out of within the detection′ window. InIi principle, the astrometric shift, motionsˆ transverse to the line of sight will be1 denoted by µ.The measurement! of= the− SB (since p is small, we assume that σ We restrict the∥ = analysis· ˆ to ⊥cz= 15−,000∥ ˆ km s− and neglect cosmological2.2. Testing velocity the field Potential should Flow have a Ansatz negligiblethis rotational600 can galaxies be seen in in the Figure Carnegie-Irvine 3 in Oohama Galaxy et al. Survey (2009 (Ho)show- et al. Ii the componentscosmological of vv parallel geometric and effects, perpendicular so that the to redshift the line coordinate of somewhatwas uncorrelated chaotic,rot with so the that mass the distribution original peculiar or even velocity contained field inp,couldbedeterminedbyminimizing,withrespectto pixel i is the same for both images). We have assumed thatp, transverse 2D space velocity of a galaxy at real-space distance component, vv on large scale, away from orbit mixinging2011∼ regions.a scatter;Lietal. plot2011 of the). MostB-band of these effective galaxies SB versus are nearby half-light (me- 2 2 2 cosmologicals r geometricr effects,s r so that the redshifts r coordinate 9 χ i [I1(θ i ) I2(θ i′ )] /σi ,wherethesummationisover sight, whereis srsr isrrris a+ unitv rr.Note vectorss inrr and thecz line-of-sightr + v ss direction.rr s.Proper wasvorticity uncorrelated (e.g., Christopherson with the mass et distribution al. 2011rot). Ator even lateradius time, contained for various a galaxy types.1 More importantly, we have vi- I1 and= I2 differ only− by a linear displacement. In principle, one = ∥ ˆ ˆ = ˆ = ∥ = · ˆ = InitialThe conditions reason is that in any the circulation, early universeΓ vv mightdss,isconservedby havedian distance been of 25 h70− Mpc) and with mean B-band absolute all pixels, θ ′ θ p,andσIi here is the 1σ error in the is ss motionsrr ˆ+ v rr.Note transversess torr theand linecz of sightr + v will be1 ss denotedrr s.Proper by µ.The sually inspected the∼ observed V-band SB profiles of 200 out of should take into= account− changes in the internal structure of vorticitycosmological (e.g., velocity Christopherson field should et have al.= 2011 a negligible·). At late rotationaltotal time,magnitude a of 20.2, close to M .Weidentifiedgalaxies measurement! of the SB (since p is small, we assume that σIi We restrict= the analysis∥ ˆ toˆ =czˆ 15,000= km∥ s−= and· ˆ = neglect somewhatKelvin’s chaotic, theorem. so that Hence, the any original rotational peculiar component velocity will600 galaxies decay field in the− Carnegie-Irvine Galaxy∗ 2 Survey (Ho et al. the object. Those, however, will have little effect compared to rot reaching a central SB of 18.5magarcsec− and tabulated the in pixel i is the same for both images). We have assumed that motionstransverse transverse 2D to space thevv line velocityrµ of sight of a will galaxy be at denoted real-space by µ distance.The cosmologicalcomponent, vv velocityon large field scale, should away fromhave% orbit a negligible mixing2011∼ regions. rotational;Lietal.2011). Most of these galaxies are nearby (me- the overall observational accuracy. Since we will eventually be cosmological geometric effects,⊥ = so that the redshift coordinate as 1/a,wherea is the scale factor. In contrast, the irrotationalcorresponding radii (in arcsec). Since we did not have access to I and I differ only by a linear displacement. In principle, one transverser is 2D space velocity of a galaxyµ at real-spacer distance was uncorrelatedrot with the mass distributionrot or even contained h 1 B interested1 2 in the mean coherent displacement of an ensemble of component,Thecomponent reason isvv that ofon the any large peculiar circulation, scale, velocity awayΓ will fromvv have orbitdss a,isconservedby growing mixingdianthev distance regions. actual√a data,. of the25 minimal70− Mpc) radius and with we could mean determine-band absolute using a should take into account changes in the internal structure of is ss rr + v rr.Notess rr and cz677.22r + v ss rr s.Properh70, (1) = · total magnitude∼ of 20.2, close to M .Weidentifiedgalaxies many galaxies, incoherent changes in the internal structure of r is ∥ = 1 µas∥ yr 1 104 km s 1 vorticityKelvin’s (e.g., theorem. Christopherson Hence, any rotational et al. 2011 componentrot). At late willruler time, decay∼ is 1–2 arcsec. a About 70% of the galaxies we inspected were the object. Those, however, will have little effect compared to = ˆ ˆ = ˆ = =− · ˆ = − The reasonTherefore, is that on any large circulation, scales, awayΓ fromvv collapseddss,isconservedby objects,reaching a the central SB− of 18.5magarcsec2 ∗ 2 and tabulated the galaxies will be insignificant. motions transverse to thevv linerµ of sight will be denoted by µ.The cosmological velocity field should have% a negligible rotationalbrighter than 18.5magarcsec− ,allowingthemtobedetected− theThis overall procedure observational of minimizing accuracy. the Sinceimage we differences will eventually exploits be ⊥ = as 1irrotational/a,where componenta is the scale is expected factor. In= to contrast, be negligible.· the irrotational Thecorrespondingby absenceGaia.SinceSBisadistance-independentquantity,wecan radii (in arcsec). Since we did not have access to transverse 2D spacewhich velocity corresponds of a galaxy toµ a at transverse real-spacer peculiar distance velocity of Kelvin’s theorem.rot Hence, any rotational component will decay allinterested information in the contained mean coherent in both displacement images, but it requires of an ensemble a possi- of vv rµ component,componentof anyvv significant ofon the large peculiar large-scale scale, velocity away vorticity will from have orbit is, a growing therefore, mixingthevuse regions.a actual strongthis√a data,threshold. the minimal to compute radius the we maximum could determine distance at using which a 1 677.22 1 h70, (1) as 1/a,wherea is the scale factor. In% contrast, the irrotational blymany non-trivial galaxies, interpolation incoherent of changesθ ′ on the in observed the internal pixel structure positions of r is 474⊥ = km s=− for 1 µ1asµ yras− yrat 1d104100km Mpc. s 1 rot ruler∼ is 1–2 arcsec. About 70% of the galaxies we inspected were µ − r − The reasonTherefore,prediction is that on ofany large the circulation, scales, standard away cosmologicalΓ fromvv collapsedds paradigm.s,isconservedby objects, To9 assess the 2 galaxies will be insignificant. = component of the peculiar velocity will have a growingbrightervFor old than stellar√ 18a populations,..5magarcsecB V− +,allowingthemtobedetected 1 (Fukugita et al. 1995), and since However,677.22 the true distances, r,areunknown,and,therefore,h70, (1) = · ∼ 10 This procedure of minimizing the image differences exploits 1 4 1 Kelvin’sirrotationalthis theorem. prediction, component Hence, the observed is any expected rotational transverse to be negligible. component motions can The willby be absenceGGaia usedV decay∼.SinceSBisadistance-independentquantity,wecan+0.27, to the astrometric condition G 18.5translatestoB 19.7. http://www.stsci.edu/hst/ whichwe corresponds make= the approximation1 µ toas a yr− transverse10 km s peculiar− velocity of Therefore, on large scales, away from collapsed objects,= the all information contained in both images, but it requires a possi- vv rµ of anyconstrain significant the amplitude large-scale of the vorticity irrotational is, therefore, component.use a This strongthis threshold can to compute the maximum distance at which 1 1 as 1/a,wherea is the scale factor. In% contrast, the irrotational 4 bly non-trivial interpolation of θ ′ on the observed pixel positions 474⊥ = km s− for 1 µasµ yr− at d r 100 Mpc. irrotationalpredictionbe done componentof by the writing standard the is expected transverse cosmological to component be negligible. paradigm. of vvrot The Toas (Arfken assess absence which corresponds to a transversev=v peculiarsµ. velocity of component(2) of the peculiar velocity will have a growing9 vFor old stellara populations,. B V + 1 (Fukugita et al. 1995), and since However,677.22 the true distances, r,areunknown,and,therefore,h , (1) of any significant large-scale vorticity is, therefore, a strong√ ∼ 10 1 1 ⊥ = 70 this& prediction, Weber 2005 the) observed transverse motions can beG usedV +0.27, to the astrometric condition G 18.5translatestoB 19.7. http://www.stsci.edu/hst/ 474 km s= for 1 µas1 µ yras yrat 1d104100km Mpc. s 1 = ∼ we make− the approximation− − − Therefore,predictionconstrain on the of large amplitude the scales,standard of theaway cosmological irrotational from collapsed component. paradigm. objects, This To can assess the This introduces a relative= error v /s in the determination of vv rot rot 4 However, the true distances, r,areunknown,and,therefore,∥ irrotational⊥ component is expectedvv toV belm " negligible.lm, rot The absence(6) 2 1/2 1 thisbe done prediction, by writing the the observed transverse⊥ = transverse component motions of vv canas (Arfken be used to whichwe corresponds make thewhere approximation tov a transverse200–300vv km peculiarsµ. s− (Davis velocity et al. 2011 of ). Hence,(2) lm ⟨ ∥ ⟩ ∼ of anyconstrain significant the amplitude large-scale of the vorticity irrotational is, therefore, component. a This strong can 474 km s 1 for 1theµas error yr is1 negligibleat d 100 as⊥ Mpc.=we go to s 2000 km s 1.Theerror & Weber 2005) " − − − predictionbe done of by the writing standard the transverse cosmological component paradigm. of vvrot Toas (Arfken assess However,This the introduces trueis also distances, random a relative sincevv=r,areunknown,and,therefore, errorvv svµv. /s0.in the determination of(2)vv where "lm ss rotYlm belong torot another class of vector spher- ⊥ ∥ ∥ ⊥ = ×∇vv Vlm "lm, (6) Therefore,2 1/2 the estimated⊥ =⟨ ⟩ =1 velocity field will be given asthis a & prediction, Weberical harmonics2005 the) observed that satisfy⊥ = transverse the same orthogonality motions can conditions be used as to we make thewhere approximationv 200–300 km s− (Davis et al. 2011). Hence, lm ⟨ ∥ ⟩ ∼ constrain the amplituderot of the irrotational component. This can This introducesthe errorfunction is a negligible relative of the redshift error as wev space go/s toin coordinate.s the2000 determination km To linear s 1.Theerror order, of vv veloc- !.Hence,Vlm is equalrot to" the right-handrot side of Equation (5)but − vv V "lm, rot (6) 2 1ity/2 fields expressed in real∥1 and redshift spaces are equivalent.⊥ be donewith by writing"lm instead the of transverse!lm.Further, componentlm dΩ"lm of!vvl m as0; (Arfken hence, whereisv also random200–300 sincevv vv kmsvµ. s− 0.(Davis et al. 2011). Hence,(2) where "lm ss Ylm⊥belong= to another class of′ vector′ spher- In the quasilinear⟨ ⊥ regime,∥⟩ = dynamical relations can be derived the recovery= ×∇ of the rotationallm mode is formally· independent= of ⟨ Therefore,∥ ⟩ ∼ the estimated⊥ = velocity field will1 be given as a & Weberical harmonics2005) that satisfy the" same orthogonality conditions as the error isfor negligible the velocity as we field go in to redshifts 2000 space km (Nusser s− .Theerror & Davis 1994), the potentialrot flow mode. # function of the redshiftv space/s coordinate. To linear order,vv veloc- !.Hence,Vlm is equalrot to the right-handrot side of Equation (5)but This introducesis also random a relative since errorvv v 0.in the determination of where "lm ss vvYlm belongV to another" , class of vector spher-(6) 2 1ity/2 fields expressed in real∥1 and redshift spaces are equivalent.⊥ with " instead of ! .Further,lm dlmΩ" ! 0; hence, where v 200–300⟨ km⊥ ∥ s⟩ =(Davis et al. 2011). Hence, ical2 harmonicslm = that×∇ satisfy⊥ lm= the same orthogonalitylm l′m′ conditions as Therefore,In the quasilinear the estimated regime,− velocity dynamical field relations will be can given be derived as a the recovery of the rotationallm mode is formally· independent= of ⟨ ∥ ⟩ ∼ 1 !.Hence,V rot is equal to the" right-hand side of Equation (5)but the errorfunction isfor negligible of the the velocity redshift as we field space go in to redshift coordinate.s 2000 space km To (Nusser linear s− .Theerror & order, Davis veloc-1994), the potentiallm flow mode. # is alsoity random fields expressedsince vv v in real0. and redshift spaces are equivalent. wherewith""lmlm insteadss Y oflm!belonglm.Further, to anotherdΩ" classlm ! ofl vectorm 0; spher- hence, · ′ ′ = Therefore,In the quasilinear the estimated⟨ ⊥ regime,∥⟩ = velocity dynamical field relations will be cangiven be as derived a ical2 the harmonics recovery= that×∇ of the satisfy rotational the same mode orthogonality is formally independent conditions as of for the velocity field in redshift space (Nusser & Davis 1994), the potentialrot flow mode. # function of the redshift space coordinate. To linear order, veloc- !.Hence,Vlm is equal to the right-hand side of Equation (5)but ity fields expressed in real and redshift spaces are equivalent. with "lm instead of !lm.Further, dΩ"lm !l′m′ 0; hence, In the quasilinear regime, dynamical relations can be derived 2the recovery of the rotational mode is formally· independent= of for the velocity field in redshift space (Nusser & Davis 1994), the potential flow mode. # 2