aa r X i v : . [ a s t r o - ph ] M a y Cosmic Flows on 100 h − Mpc scales
Hume A. Feldman ⋆, , Michael J. Hudson † , & Richard Watkins ‡ , ⋆ Department of Physics & Astronomy, University of Kansas, Lawrence, KS 66045, USA. † Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada. ‡ Department of Physics, Willamette University, Salem, OR 97301, USA.emails: [email protected]; [email protected]; [email protected] To study galactic motions on the largest available scales, we require bulk flow moments whosewindow functions have as narrow a peak as possible and having as small an amplitude aspossible outside the peak. Typically the moments found using the maximum likelihood esti-mate weights do not meet these criteria. We present a new method for calculating weightsfor moments that essentially allow us to ”design” the moment’s window function, subject, ofcourse, to the distribution and uncertainties of the available data.
On scales that are small compared to the Hubble radius, galaxy motions are manifest indeviations from the idealized isotropic cosmological expansion cz = H r + ˆ r · [ v ( r ) − v (0)] . (1)Redshift-distance samples, obtained from peculiar velocity surveys, allow us to determine theradial (line-of-sight) component of the peculiar velocity of each galaxy: v ( r ) = ˆ r · v ( r ) = cz − H r . (2)Galaxies trace the large-scale linear velocity field v ( r ) which is described by a Gaussianrandom field that is completely defined, in Fourier space, by its velocity power spectrum P v ( k ).The Fourier Transform of the line-of-sight velocity isˆ r · v ( r ) = 1(2 π ) Z d k ˆ r · ˆ k v ( k )e i k · r , (3)which provides us with the velocity power spectrum < v ( k ) v ∗ ( k ′ ) > = (2 π ) P v ( k ) δ D ( k − k ′ ) , (4)where δ D is the Dirac delta function. In linear theory, the velocity power spectrum is related tothe density power spectrum P v ( k ) = H k f (Ω m, Ω Λ ) P ( k ) , (5)where H is the Hubble constant and f is the rate of perturbation growth at the present. Ω m andΩ Λ are the matter and dark energy density parameters today. The power spectrum provides acomplete statistical description of the linear peculiar velocity field.For our power spectrum model, we follow Eisenstein and Hu (1998) in writing P ( k ) ∝ σ Ω . m k n T ( k/ Γ) , (6)where T is the transfer function, Γ is the power spectrum “shape” parameter and as usual,we normalize the power spectrum using the parameter σ , the amplitude of matter densityperturbations on the scale of 8 h − Mpc.To find the maximum likelihood likelihood parameters from peculiar velocity surveys westart with catalog of peculiar velocities galaxies, labeled by an index n , positions r n , estimatesf the line-of-sight peculiar velocities S n and uncertainties σ n . We assume that the observa-tional errors are Gaussian distributed. We then model the velocity field as a uniform streamingmotion, or bulk flow (BF), denoted by U , about which there are random motions drawn froma Gaussian distribution with a 1-D velocity dispersion σ ∗ . Then the likelihood function for theBF components is L ( U i ) = Y n p σ n + σ ∗ exp −
12 ( S n − ˆ r n,i U i ) σ n + σ ∗ ! . (7)where i go from 1 to 3 to specify the BF components. While this has been done previously formaximum likelihood estimates (MLE) of the bulk flow moments, the advantage of these newmoments is that they have been designed to be sensitive only to scales of order 100 h − Mpc.Thus we will be able to probe these scales without having to worry about the influence ofsmaller scales. Further, by isolating the very large scale motions we will be able to put strongerconstraints on power spectrum parameters.Maximum likelihood solution for BF moments is U i = A − ij X n ˆ r n,j S n σ n + σ ∗ , (8)where A ij = X n ˆ r n,i ˆ r n,j σ n + σ ∗ . (9)The measured peculiar velocity of galaxy nS n = ˆ r n,i v i ( r n ) + ǫ n , (10)where ǫ n is a Gaussian with zero mean and variance σ n + σ ∗ . The theoretical covariance matrixfor the BF components R ab = R ( v ) ab + R ( ǫ ) ab . (11)The first term is given as an integral over the matter fluctuation power spectrum, P ( k ), R ( v ) ab = Ω . m π Z ∞ dk W ab ( k ) P ( k ) , (12)where the angle-averaged tensor window function is W ab ( k ) = X n,m w a,n w b,m Z d ˆ k π (cid:16) ˆr n · ˆk ˆr m · ˆk (cid:17) exp ( i k · ( r n − r m )) . (13)In order to do better than the MLE solutions, we redesign the window functions by requiringthat they have narrow peaks and small amplitude outside the peak. We start with the BF MLEweights w i,n = A − ij X n x j · r n σ n + σ ∗ , (14)which depends on the spatial distribution and errors. Now consider an ideal survey with verylarge number of points, isotropic distribution and a Gaussian falloff n ( r ) ∝ exp( − r / R I ) where R I is the depth of the survey.The moments are specified by the weights u i = X n w i,n S n (15) Fig. 1:
The window functions of the bulk flow com-ponent for R I = 50 h − Mpc for the all the catalogs weconsidered. The thick (thin) lines are the window func-tions for the MV (MLE) bulk flow components. thex,y,z–component are dash-dot, short dash, long dashlines respectively. The Thick solid line is the ideal win-dow function.
Fig. 2:
The Γ likelihood function for the compositecatalogs. that minimize the variance < ( u i − U i ) > . We expand the variance < ( u i − U i ) > = X n,m w i,n w i,m < S n S m > + < U i > − X n w i,n < U i S n > , (16)since the measurement error included in S n is uncorrelated with the bulk flow U i . For BFmoments we can impose the constraint thatlim k → W ii ( k ) = X n,m w i,n w i,m Z d ˆ k π (cid:16) ˆr n · ˆk ˆr m · ˆk (cid:17) = 1 / . (17)We now minimize this expression with respect to w i,n subject to the constraint which weenforce using a Lagrange multiplier to get X m ( h S n S m i + λP nm ) w i,m = h S n U i i . (18)We solve this to get the minimum variance (MV) weights.To apply this formalism, we used both cluster and galaxy peculiar velocity surveys. Specif-ically, we used the Willick cluster survey4, the SC1 5 and SC2 6 (Tully-Fisher clusters); TheEFAR survey 7 (a Fundamental Plane cluster survey); the SN are Type Ia supernovae from thecompilation of Tonry et al.
8; SMAC 9 ,
10 (a Fundamental Plane cluster survey); SFI++11 ,
12 (aTully-Fisher based survey of ∼ ∼
730 groups).The minimum variance, or MV, weights were calculated for the bulk flow component mo-ments using the method described above for each of our catalogs, as well as for three compositecatalogs, SFI++, consisting of both fields and groups from the catalog, OTHER, consisting ofall the surveys except for the SFI++ field galaxy and groups catalogs, and Composite, consistingof all of the catalogs combined. The OTHER catalog was included due to the fact that, giventheir large size, the SFI++ catalogs tend to dominate any composite catalog in which they areincluded. Here we will show results from a deep survey, R I = 50 h − Mpc.
Fig. 3:
The bulk flow of the large catalogs. The triangles (points) are the BF moments with R I = 20 h − Mpc ( R I = 50 h − Mpc ). There is a consistent and robust flow exhibited in allcatalogs to the negative Galactic y direction (upper right panel) for the large scale analysis whichis reflected in the BF magnitude (lower right panel)
In Figure 1 we show the window functions of the MV bulk flow component moments. Forcomparison, we also include the ideal window functions as well as those for the MLE momentsfor each survey. As expected, the match between the window functions for the MV momentsand the ideal is best for the large surveys and those with small measurement error and similardistribution to the ideal survey. For the sparse, noisy surveys, the window functions for theMV moments are not very different than those of the MLE moments, differing mostly in theamplitude of the tail of the window function for large k .In Figure 2. we show the likelihood function of the “shape” parameter Γ for the largecatalogs. We see clearly that Γ is smaller for large R I than for smaller R I . In Figure 3. weshow the BF velocity for the composite surveys. We see that for the shallow window functions,we get small flows, ∼
150 km/s. On large scales we get very large flows ∼
400 km/s. Thisunexpected flow which is exhibited in all the catalogs we analyzed, require a very steep powerspectrum which leads to small Γ (see figure 2.)
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