Draft version September 15, 2021
Preprint typeset using LATEX style emulateapj v. 08/22/09
THE PECULIAR VELOCITIES OF LOCAL TYPE Ia SUPERNOVAE AND THEIR IMPACT ON COSMOLOGY
James D. Neill
California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125
Michael J. Hudson
University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, CANADA
Alex Conley
University of Toronto, 60 Saint George Street, Toronto, ON M5S 3H8, CANADA
Draft version September 15, 2021
ABSTRACT
We quantify the effect of supernova Type Ia peculiar velocities on the derivation of cosmological
parameters. The published distant and local Ia SNe used for the Supernova Legacy Survey first-year
cosmology report form the sample for this study. While previous work has assumed that the local SNe
are at rest in the CMB frame (the No Flow assumption), we test this assumption by applying peculiar
velocity corrections to the local SNe using three different flow models. The models are based on the
IRAS PSCz galaxy redshift survey, have varying β = Ω0.6m /b, and reproduce the Local Group motion
in the CMB frame. These datasets are then fit for w, Ωm, and ΩΛ using flatness or ΛCDM and a
BAO prior. The χ2 statistic is used to examine the effect of the velocity corrections on the quality
of the fits. The most favored model is the β = 0.5 model, which produces a fit significantly better
than the No Flow assumption, consistent with previous peculiar velocity studies. By comparing the
No Flow assumption with the favored models we derive the largest potential systematic error in w
caused by ignoring peculiar velocities to be ∆w = +0.04. For ΩΛ, the potential error is ∆ΩΛ = −0.04
and for Ωm, the potential error is ∆Ωm < +0.01. The favored flow model (β = 0.5) produces the
following cosmological parameters: w = −1.08+0.09
−0.08, Ωm = 0.27
+0.02
−0.02 assuming a flat cosmology, and
ΩΛ = 0.80
+0.08
−0.07 and Ωm = 0.27
+0.02
−0.02 for a w = −1 (ΛCDM) cosmology.
Subject headings: cosmology: large-scale structure of the universe – galaxies: distances and redshifts
– supernovae: general
1. INTRODUCTION
Dark Energy has challenged our knowledge of funda-
mental physics since the direct evidence for its existence
was discovered using Type Ia supernovae (Riess et al.
1998; Perlmutter et al. 1999). Because there are cur-
rently no compelling theoretical explanations for Dark
Energy, the correct emphasis, as pointed out by the Dark
Energy Task Force (DETF, Albrecht et al. 2006), is on
refining our observations of the accelerated expansion of
the universe. Recommendation V from the DETF Re-
port (Albrecht et al. 2006) calls for an exploration of the
systematic effects that could impair the needed observa-
tional refinements.
A couple of recent studies (Hui & Greene 2006;
Cooray & Caldwell 2006) point out that the redshift
lever arm needed to accurately measure the universal
expansion requires the use of a local sample, but that
coherent large-scale local (z < 0.2) peculiar velocities
add additional uncertainty to the Hubble diagram and
hence to the derived cosmological parameters.
Current analyses (e.g., Astier et al. 2006; Riess et al.
2007; Wood-Vasey et al. 2007) of the cosmological pa-
rameters do not attempt to correct for the effect of local
peculiar velocities. As briefly noted by Hui & Greene
Electronic address: neill@srl.caltech.edu
Electronic address: mjhudson@uwaterloo.ca
Electronic address: conley@astro.utoronto.ca
(2006) and Cooray & Caldwell (2006), it is possible
to use local data to measure the local velocity field
and hence limit the impact on the derived cosmo-
logical parameters. Measurements of the local ve-
locity field have improved to the point where there
is consistency among surveys and methods (Hudson
2003; Hudson et al. 2004; Radburn-Smith et al. 2004;
Pike & Hudson 2005; Sarkar et al. 2006). Type Ia
supernova peculiar velocities have been studied re-
cently by Radburn-Smith et al. (2004); Pike & Hudson
(2005); Jha et al. (2006); Haugboelle et al. (2006);
Watkins & Feldman (2007) and others. Their results
demonstrate that the local flows derived from SNe are
in agreement with those derived from other distance in-
dicators, such as the Tully-Fisher relation and the Funda-
mental Plane. Our aim is to use the current knowledge
of the local peculiar motions to correct local SNe and,
together with a homogeneous set of distant SNe, fit for
cosmological parameters and measure the effect of the
corrections on the cosmological fits.
To produce this measurement, we analyze the lo-
cal and distant SN Ia sample used in the first-year
cosmology results from the Supernova Legacy Survey
(SNLS, Astier et al. 2006, hence A06). This sam-
ple is composed of 44 local SNe (A06, Table 8:
Hamuy et al. 1996; Riess et al. 1999; Krisciunas et al.
2001; Jha 2002; Strolger et al. 2002; Altavilla et al.
http://arxiv.org/abs/0704.1654v1
mailto:neill@srl.caltech.edu
mailto:mjhudson@uwaterloo.ca
mailto:conley@astro.utoronto.ca
2 Neill, Hudson, & Conley
2004; Krisciunas et al. 2004a,b) and 71 distant SNe
(A06, Table 9). The distant SNe are the largest homo-
geneous set currently in the literature. The local sample
span the redshift range 0.015 < z < 0.125 and were
selected to have good lightcurve sampling (A06, § 5.2).
Using three different models encompassing the range of
plausible local large-scale flow, we assign and correct for
the peculiar velocity of each local SN. We then re-fit the
entire sample for w, Ωm, and ΩΛ to assess the system-
atics due to the peculiar velocity field, and to asses the
change in the quality of the resulting fits.
2. PECULIAR VELOCITY MODELS
Peculiar velocities, v, arise due to inhomogeneities in
the mass density and hence in the expansion. Their ef-
fect is to perturb the observed redshifts from their cos-
mological values: czCMB = cz + v · r̂, where cz is the
cosmological redshift the SN would have in the absence
of peculiar velocities. With the advent of all-sky galaxy
redshift surveys, it is possible to predict peculiar veloc-
ities from the galaxy distribution provided one knows
β = f(Ω)/b, where b is a linear biasing parameter relating
fluctuations in the galaxy density, δ, to fluctuations in the
mass density. The peculiar velocity in the CMB frame is
then given by linear perturbation theory (Peebles 1980)
applied to the density field (see, e.g. Yahil et al. 1991;
Hudson 1993):
∫ Rmax
δ(r′)
(r′ − r)
|r′ − r|3
d3r′ +V. (1)
In this Letter, we use the density field of IRAS PSCz
galaxies (Branchini et al. 1999), which extends to a
depth Rmax = 20000 km s
−1. Contributions to the pe-
culiar velocity arising from masses on scales larger than
Rmax are modeled by a simple residual dipole, V. Thus,
given a density field, the parameters β and V describe
the velocity field within Rmax. For galaxies with dis-
tances greater than Rmax, the first term above is set to
zero.
The predicted peculiar velocities from the PSCz den-
sity field are subject to two sources of uncertainty: the
noisiness of the predictions due to the sparsely-sampled
density field, and the inapplicability of linear perturba-
tion theory on small scales. Typically these uncertainties
are accounted for by adding an additional “thermal” dis-
persion, which is assumed to be Gaussian. From a care-
ful analysis of predicted and observed peculiar velocities,
Willick & Strauss (1998) estimated these uncertainties
to be ∼ 100 km s−1, albeit with a dependence on den-
sity. Radburn-Smith et al. (2004) found reasonable χ2
values if 150 km s−1 was assumed in the field, with an
extra contribution to the small-scale dispersion added in
quadrature for SNe in clusters. Here we adopt a thermal
dispersion of 150 km s−1.
For this study, we explore the results of three different
models of large-scale flows and compare them to a case
where no flow model is used. These models have been
chosen to span the range of flow models permitted by
peculiar velocity data, and all of these models reproduce
the observed ∼ 600 km s−1 motion of the Local Group
with respect to the CMB. The first model assumes a
pure bulk flow (model PBF, hence β = 0) with V having
vector components (57,−540, 314) km s−1 in Galactic
Cartesian coordinates. The second model assumes β =
0.5 (model B05), with a dipole vector of (70,−194, 0) km
s−1. The third model adopts β = 0.7 (model B07) which
requires no residual dipole. We compare these models
to the no-correction scenario adopted by A06 and others
with β = 0, V = 0 which we call the “No Flow” or NF
scenario. Note that a recent comparison (Pike & Hudson
2005) of results from IRAS predictions versus peculiar
velocity data yields a mean value fit with β = 0.50±0.02
(stat), so the B05 model is strongly favored over the NF
scenario by independent peculiar velocity analyses.
3. COSMOLOGICAL FITS
Prior to the fitting procedure, the peculiar veloci-
ties for each model are used to correct the local SNe
(using a variation of Hui & Greene 2006, equations 11
and 13). We then fit our corrected SN data in two
ways using a χ2-gridding cosmology fitter1 (also used by
Wood-Vasey et al. 2007). The first fit uses a flat cos-
mology (Ω = 1) with the equation of state parameter w
and Ωm as free parameters. The second fit assumes a
ΛCDM (w = −1) cosmology with ΩΛ and Ωm as free
parameters. We used the same intrinsic SN photometric
scatter (σint = 0.13 mag, A06) for every fit. The result-
ing χ2 probability surfaces for both fits are then further
constrained using the BAO result from Eisenstein et al.
(2005). The final derived cosmological parameters are
then used to calculate the χ2 for each fit (see A06, § 5.4).
The fitting procedure employed here differs in imple-
mentation from that used in A06. Three additional pa-
rameters, often called nuisance parameters, must be fit
along with the two cosmological parameters. These pa-
rameters are the constant of proportionality for the SN
lightcurve shape, αs, the correction for the SN observed
color, βc, and a SN brightness normalization,M. We dis-
tinguish βc from the β used to describe the flow models
above. A06 used analytic marginalization of the nuisance
parameters αs and βc in their fits. Here these parame-
ters are fully gridded like the cosmological parameters.
This avoids a bias in the nuisance parameters that re-
sults because, in the analytic method, their values must
be held fixed to compute the errors. The result is that
our fits using the NF scenario produces slightly different
cosmological parameters than quoted in A06.
4. RESULTS
The results of the cosmological fits for each model are
listed in Table 1 and plotted in Figure 1 and Figure 2.
They demonstrate two effects of the peculiar velocity
corrections: a change in the values of the cosmological
parameters, and a change in the quality of the fits as
measured by the χ2 statistic.
We expect, if a given model is correct, to improve the
fitting since our corrected data should more closely re-
semble the homogeneous universe described by a few cos-
mological parameters. The χ2 of the fits for each flow
model can be compared to the χ2 for the NF scenario
(shown by the dashed line in the figures) as a test of this
hypothesis. Using ∆χ2 = −2 lnL/LNF , where L is the
likelihood, we find that the pure bulk flow is over 103
times less likely than the NF scenario, while the B05 and
1 http://qold.astro.utoronto.ca/conley/simple cosfitter/
http://qold.astro.utoronto.ca/conley/simple_cosfitter/
Peculiar Velocities of SNe 3
TABLE 1
Peculiar Velocity Model Parameters and Results
Ω = 1 + BAO prior w = −1 + BAO prior
Model β V (km s−1) w Ωm χ
ΩΛ Ωm χ
ΩΛ,Ωm
A06a 0.0 · · · −1.023± 0.090 0.271± 0.021 · · · 0.751± 0.082 0.271 ± 0.020 · · ·
NF 0.0 · · · −1.054
+0.086
−0.084
0.270
+0.024
−0.018
115.5 0.770
+0.083
−0.071
0.269
+0.033
−0.017
115.4
PBF 0.0 57,-540,314 −1.026+0.085
−0.083
0.273+0.024
−0.019
129.4 0.741+0.084
−0.073
0.273+0.034
−0.017
129.2
B05 0.5b 70,-194,0 −1.081
+0.087
−0.085
0.268
+0.024
−0.018
110.3 0.796
+0.081
−0.070
0.267
+0.032
−0.017
110.1
B07 0.7 · · · −1.094
+0.087
−0.085
0.267
+0.024
−0.018
111.2 0.809
+0.082
−0.069
0.265
+0.032
−0.017
111.1
a results quoted in A06 marginalizing analytically over αs and βc (see § 3)
b best fit value from Pike & Hudson (2005)
Fig. 1.— Parameter values for the w, Ωm fit (Ω = 1 + BAO prior)
for each of the four peculiar velocity models in Table 1. The values
for the NF scenario are indicted by the dashed lines. The largest
systematic error in w compared with the NF fit is +0.040 for the
B07 model, which demonstrates the amplitude of the systematic
error if peculiar velocity is not accounted for. The offsets for Ωm
are all within ±0.003 showing that this parameter is not sensitive
to the peculiar velocity corrections due to the BAO prior. The χ2
of the fits improve when using the two β models (B05, B07), while
the PBF model provides a significantly worse fit.
B07 models are 13.5 and 8.6 times more likely, respec-
tively.
We also use these data to assess the systematic er-
rors made in the parameters if no peculiar velocities are
accounted for. The largest of these are obtained by com-
paring the B07 model with the NF scenario. This com-
parison yields ∆wB07 = +0.040 and ∆ΩΛ,B07 = −0.039.
The same comparison for the B05 model, which is only
slightly preferred by the χ2 statistic over model B07, pro-
duces ∆wB05 = +0.027 and ∆ΩΛ,B05 = −0.026. The
systematic offsets for Ωm are all 0.004 or less, demon-
strating the insensitivity of this parameter to peculiar
velocities. This is due to the BAO prior which is insensi-
tive to local flow and provides a much stronger constraint
for Ωm than for w or ΩΛ (see A06, Figures 5 and 6).
5. DISCUSSION AND SUMMARY
The systematic effect of different flow models is at the
level of ±0.04 in w. This is smaller than the present level
of random error in w, which is largely due to the small
numbers of high- and low-redshift SNe. However, com-
pared to other systematics discussed in A06, which total
Fig. 2.— Parameter values for the ΩΛ, Ωm fit (w = −1 + BAO
prior) for each of the four peculiar velocity models as in Figure 1.
Again, comparing the NF fits to the B07 model produces the largest
systematic in ΩΛ of −0.039. We also find Ωm insensitive to the
corrections, having all offsets within ±0.004. The χ2 values show
the same pattern as in Figure 1, favoring the β models over no
correction (NF), and over pure bulk flow.
∆w = ±0.054, the systematic effect of large-scale flows
is important. Wood-Vasey et al. (2007, Table 5) list 16
sources of systematic error which total ∆w = ±0.13.
Aside from three method-dependent systematics and the
photometric zero-point error, they are all smaller than
the flow systematic. As the number of SNe continues to
increase, and understanding of other systematics (e.g.
photometric zero-points) improves, it is possible that
large-scale flows will become one of the dominant sources
of systematic uncertainty.
The peculiar velocities of SN host galaxies arise from
large-scale structures over a range of scales. The compo-
nent arising from small-scale, local structure is the least
important: it is essentially a random variable which is
reduced by
N . More problematic is the large-scale co-
herent component. Such a large-scale component can
take several forms: an overdensity or underdensity; a
large-scale dipole, or “bulk” flow.
The existence of a large-scale, but local (< 7400
km s−1) underdensity, or “Hubble Bubble” was first
discussed by Zehavi et al. (1998). Recently Jha et al.
(2006) have re-enforced this claim with a larger SN data
set: they find that the difference in the Hubble constant
inside the Bubble and outside is ∆H/H = 6.5 ± 1.8%.
4 Neill, Hudson, & Conley
If correct, this could have a dramatic effect on the de-
rived cosmological parameters (Jha et al. 2006, Fig 17),
especially for those studies that extend their local sample
down below z < 0.015. However, the “Hubble Bubble”
was not confirmed by Giovanelli et al. (1999) who found
∆H/H = 1.0 ± 2.2% using the Tully-Fisher (TF) pe-
culiar velocities, nor by Hudson et al. (2004) who found
∆H/H = 2.3± 1.9% using the Fundamental Plane (FP)
distances.
According to equation 1, a mean underdensity of IRAS
galaxies of order ∼ 40% within 7400 km s−1 would
be needed to generate the “Hubble Bubble” quoted by
Jha et al. (2006). However, we find that the IRAS
PSCz density field of Branchini et al. (1999) is not un-
derdense in this distance range; instead it is mildly over-
dense (by a few percent) within 7400 km s−1 (see also
Branchini et al. 1999, Figure 2). As a further cross-
check, when we refit the Jha et al. (2006) data after hav-
ing subtracted the predictions of the B05 flow model, the
“Bubble” remains in the Jha et al. (2006) data. Thus,
the Jha et al “Bubble” cannot be explained by local
structure, unless that structure is not traced by IRAS
galaxies. Moreover, when we analyze the 99 SNe within
15000 km s−1 from Tonry et al. (2003) in the same way,
we find no evidence of a significant “Hubble Bubble”
(∆H/H = 1.5 ± 2.0%), in agreement with the results
from TF and FP surveys. The Tonry et al. (2003) sam-
ple and that of Jha et al. (2006) have 67 SNe in common.
The high degree of overlap suggests that the difference
lies in the different methods for converting the photom-
etry into SN distance moduli.
A local large-scale flow can also introduce systematic
errors if the low-z sample is biased in its sky coverage:
in this case, an uncorrected dipole term can corrupt the
monopole term, which then biases the cosmological pa-
rameters. For the large-scale flow directions considered
here, this does not appear to affect the A06 sample: we
note that the PBF-corrected case has similar cosmolog-
ical parameters to the “No Flow” case. However, if co-
herent flows exist on large scales, this may affect surveys
with unbalanced sky coverage, such as the SN Factory
(Aldering et al. 2002) or the SDSS SN survey2.
The most promising approach to treating the effect
of large-scale flows is a more sophisticated version of
the analysis presented here: combine low-redshift SNe
with other low-redshift peculiar velocity tracers, such as
Tully-Fisher SFI++ survey (Masters et al. 2006) and the
NOAO Fundamental Plane Survey (Smith et al. 2004),
and use these data to constrain the parameters of the
flow model (β and the residual large-scale flow V) di-
rectly. One can then marginalize over the parameters of
the flow model while fitting the cosmological parameters
to the low- and high-z SNe.
2 http://sdssdp47.fnal.gov/sdsssn/sdsssn.html
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  We quantify the effect of supernova Type Ia peculiar velocities on the
derivation of cosmological parameters. The published distant and local Ia SNe
used for the Supernova Legacy Survey first-year cosmology report form the
sample for this study. While previous work has assumed that the local SNe are
at rest in the CMB frame (the No Flow assumption), we test this assumption by
applying peculiar velocity corrections to the local SNe using three different
flow models. The models are based on the IRAS PSCz galaxy redshift survey, have
varying beta = Omega_m^0.6/b, and reproduce the Local Group motion in the CMB
frame. These datasets are then fit for w, Omega_m, and Omega_Lambda using
flatness or LambdaCDM and a BAO prior. The chi^2 statistic is used to examine
the effect of the velocity corrections on the quality of the fits. The most
favored model is the beta=0.5 model, which produces a fit significantly better
than the No Flow assumption, consistent with previous peculiar velocity
studies. By comparing the No Flow assumption with the favored models we derive
the largest potential systematic error in w caused by ignoring peculiar
velocities to be Delta w = +0.04. For Omega_Lambda, the potential error is
Delta Omega_Lambda = -0.04 and for Omega_m, the potential error is Delta
Omega_m < +0.01. The favored flow model (beta=0.5) produces the following
cosmological parameters: w = -1.08 (+0.09,-0.08), Omega_m = 0.27 (+0.02,-0.02)
assuming a flat cosmology, and Omega_Lambda = 0.80 (+0.08,-0.07) and Omega_m =
0.27 (+0.02,-0.02) for a w = -1 (LambdaCDM) cosmology.

Draft version September 15, 2021
Preprint typeset using LATEX style emulateapj v. 08/22/09
THE PECULIAR VELOCITIES OF LOCAL TYPE Ia SUPERNOVAE AND THEIR IMPACT ON COSMOLOGY
James D. Neill
California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125
Michael J. Hudson
University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, CANADA
Alex Conley
University of Toronto, 60 Saint George Street, Toronto, ON M5S 3H8, CANADA
Draft version September 15, 2021
ABSTRACT
We quantify the effect of supernova Type Ia peculiar velocities on the derivation of cosmological
parameters. The published distant and local Ia SNe used for the Supernova Legacy Survey first-year
cosmology report form the sample for this study. While previous work has assumed that the local SNe
are at rest in the CMB frame (the No Flow assumption), we test this assumption by applying peculiar
velocity corrections to the local SNe using three different flow models. The models are based on the
IRAS PSCz galaxy redshift survey, have varying β = Ω0.6m /b, and reproduce the Local Group motion
in the CMB frame. These datasets are then fit for w, Ωm, and ΩΛ using flatness or ΛCDM and a
BAO prior. The χ2 statistic is used to examine the effect of the velocity corrections on the quality
of the fits. The most favored model is the β = 0.5 model, which produces a fit significantly better
than the No Flow assumption, consistent with previous peculiar velocity studies. By comparing the
No Flow assumption with the favored models we derive the largest potential systematic error in w
caused by ignoring peculiar velocities to be ∆w = +0.04. For ΩΛ, the potential error is ∆ΩΛ = −0.04
and for Ωm, the potential error is ∆Ωm < +0.01. The favored flow model (β = 0.5) produces the
following cosmological parameters: w = −1.08+0.09
−0.08, Ωm = 0.27
+0.02
−0.02 assuming a flat cosmology, and
ΩΛ = 0.80
+0.08
−0.07 and Ωm = 0.27
+0.02
−0.02 for a w = −1 (ΛCDM) cosmology.
Subject headings: cosmology: large-scale structure of the universe – galaxies: distances and redshifts
– supernovae: general
1. INTRODUCTION
Dark Energy has challenged our knowledge of funda-
mental physics since the direct evidence for its existence
was discovered using Type Ia supernovae (Riess et al.
1998; Perlmutter et al. 1999). Because there are cur-
rently no compelling theoretical explanations for Dark
Energy, the correct emphasis, as pointed out by the Dark
Energy Task Force (DETF, Albrecht et al. 2006), is on
refining our observations of the accelerated expansion of
the universe. Recommendation V from the DETF Re-
port (Albrecht et al. 2006) calls for an exploration of the
systematic effects that could impair the needed observa-
tional refinements.
A couple of recent studies (Hui & Greene 2006;
Cooray & Caldwell 2006) point out that the redshift
lever arm needed to accurately measure the universal
expansion requires the use of a local sample, but that
coherent large-scale local (z < 0.2) peculiar velocities
add additional uncertainty to the Hubble diagram and
hence to the derived cosmological parameters.
Current analyses (e.g., Astier et al. 2006; Riess et al.
2007; Wood-Vasey et al. 2007) of the cosmological pa-
rameters do not attempt to correct for the effect of local
peculiar velocities. As briefly noted by Hui & Greene
Electronic address: neill@srl.caltech.edu
Electronic address: mjhudson@uwaterloo.ca
Electronic address: conley@astro.utoronto.ca
(2006) and Cooray & Caldwell (2006), it is possible
to use local data to measure the local velocity field
and hence limit the impact on the derived cosmo-
logical parameters. Measurements of the local ve-
locity field have improved to the point where there
is consistency among surveys and methods (Hudson
2003; Hudson et al. 2004; Radburn-Smith et al. 2004;
Pike & Hudson 2005; Sarkar et al. 2006). Type Ia
supernova peculiar velocities have been studied re-
cently by Radburn-Smith et al. (2004); Pike & Hudson
(2005); Jha et al. (2006); Haugboelle et al. (2006);
Watkins & Feldman (2007) and others. Their results
demonstrate that the local flows derived from SNe are
in agreement with those derived from other distance in-
dicators, such as the Tully-Fisher relation and the Funda-
mental Plane. Our aim is to use the current knowledge
of the local peculiar motions to correct local SNe and,
together with a homogeneous set of distant SNe, fit for
cosmological parameters and measure the effect of the
corrections on the cosmological fits.
To produce this measurement, we analyze the lo-
cal and distant SN Ia sample used in the first-year
cosmology results from the Supernova Legacy Survey
(SNLS, Astier et al. 2006, hence A06). This sam-
ple is composed of 44 local SNe (A06, Table 8:
Hamuy et al. 1996; Riess et al. 1999; Krisciunas et al.
2001; Jha 2002; Strolger et al. 2002; Altavilla et al.
http://arxiv.org/abs/0704.1654v1
mailto:neill@srl.caltech.edu
mailto:mjhudson@uwaterloo.ca
mailto:conley@astro.utoronto.ca
2 Neill, Hudson, & Conley
2004; Krisciunas et al. 2004a,b) and 71 distant SNe
(A06, Table 9). The distant SNe are the largest homo-
geneous set currently in the literature. The local sample
span the redshift range 0.015 < z < 0.125 and were
selected to have good lightcurve sampling (A06, § 5.2).
Using three different models encompassing the range of
plausible local large-scale flow, we assign and correct for
the peculiar velocity of each local SN. We then re-fit the
entire sample for w, Ωm, and ΩΛ to assess the system-
atics due to the peculiar velocity field, and to asses the
change in the quality of the resulting fits.
2. PECULIAR VELOCITY MODELS
Peculiar velocities, v, arise due to inhomogeneities in
the mass density and hence in the expansion. Their ef-
fect is to perturb the observed redshifts from their cos-
mological values: czCMB = cz + v · r̂, where cz is the
cosmological redshift the SN would have in the absence
of peculiar velocities. With the advent of all-sky galaxy
redshift surveys, it is possible to predict peculiar veloc-
ities from the galaxy distribution provided one knows
β = f(Ω)/b, where b is a linear biasing parameter relating
fluctuations in the galaxy density, δ, to fluctuations in the
mass density. The peculiar velocity in the CMB frame is
then given by linear perturbation theory (Peebles 1980)
applied to the density field (see, e.g. Yahil et al. 1991;
Hudson 1993):
∫ Rmax
δ(r′)
(r′ − r)
|r′ − r|3
d3r′ +V. (1)
In this Letter, we use the density field of IRAS PSCz
galaxies (Branchini et al. 1999), which extends to a
depth Rmax = 20000 km s
−1. Contributions to the pe-
culiar velocity arising from masses on scales larger than
Rmax are modeled by a simple residual dipole, V. Thus,
given a density field, the parameters β and V describe
the velocity field within Rmax. For galaxies with dis-
tances greater than Rmax, the first term above is set to
zero.
The predicted peculiar velocities from the PSCz den-
sity field are subject to two sources of uncertainty: the
noisiness of the predictions due to the sparsely-sampled
density field, and the inapplicability of linear perturba-
tion theory on small scales. Typically these uncertainties
are accounted for by adding an additional “thermal” dis-
persion, which is assumed to be Gaussian. From a care-
ful analysis of predicted and observed peculiar velocities,
Willick & Strauss (1998) estimated these uncertainties
to be ∼ 100 km s−1, albeit with a dependence on den-
sity. Radburn-Smith et al. (2004) found reasonable χ2
values if 150 km s−1 was assumed in the field, with an
extra contribution to the small-scale dispersion added in
quadrature for SNe in clusters. Here we adopt a thermal
dispersion of 150 km s−1.
For this study, we explore the results of three different
models of large-scale flows and compare them to a case
where no flow model is used. These models have been
chosen to span the range of flow models permitted by
peculiar velocity data, and all of these models reproduce
the observed ∼ 600 km s−1 motion of the Local Group
with respect to the CMB. The first model assumes a
pure bulk flow (model PBF, hence β = 0) with V having
vector components (57,−540, 314) km s−1 in Galactic
Cartesian coordinates. The second model assumes β =
0.5 (model B05), with a dipole vector of (70,−194, 0) km
s−1. The third model adopts β = 0.7 (model B07) which
requires no residual dipole. We compare these models
to the no-correction scenario adopted by A06 and others
with β = 0, V = 0 which we call the “No Flow” or NF
scenario. Note that a recent comparison (Pike & Hudson
2005) of results from IRAS predictions versus peculiar
velocity data yields a mean value fit with β = 0.50±0.02
(stat), so the B05 model is strongly favored over the NF
scenario by independent peculiar velocity analyses.
3. COSMOLOGICAL FITS
Prior to the fitting procedure, the peculiar veloci-
ties for each model are used to correct the local SNe
(using a variation of Hui & Greene 2006, equations 11
and 13). We then fit our corrected SN data in two
ways using a χ2-gridding cosmology fitter1 (also used by
Wood-Vasey et al. 2007). The first fit uses a flat cos-
mology (Ω = 1) with the equation of state parameter w
and Ωm as free parameters. The second fit assumes a
ΛCDM (w = −1) cosmology with ΩΛ and Ωm as free
parameters. We used the same intrinsic SN photometric
scatter (σint = 0.13 mag, A06) for every fit. The result-
ing χ2 probability surfaces for both fits are then further
constrained using the BAO result from Eisenstein et al.
(2005). The final derived cosmological parameters are
then used to calculate the χ2 for each fit (see A06, § 5.4).
The fitting procedure employed here differs in imple-
mentation from that used in A06. Three additional pa-
rameters, often called nuisance parameters, must be fit
along with the two cosmological parameters. These pa-
rameters are the constant of proportionality for the SN
lightcurve shape, αs, the correction for the SN observed
color, βc, and a SN brightness normalization,M. We dis-
tinguish βc from the β used to describe the flow models
above. A06 used analytic marginalization of the nuisance
parameters αs and βc in their fits. Here these parame-
ters are fully gridded like the cosmological parameters.
This avoids a bias in the nuisance parameters that re-
sults because, in the analytic method, their values must
be held fixed to compute the errors. The result is that
our fits using the NF scenario produces slightly different
cosmological parameters than quoted in A06.
4. RESULTS
The results of the cosmological fits for each model are
listed in Table 1 and plotted in Figure 1 and Figure 2.
They demonstrate two effects of the peculiar velocity
corrections: a change in the values of the cosmological
parameters, and a change in the quality of the fits as
measured by the χ2 statistic.
We expect, if a given model is correct, to improve the
fitting since our corrected data should more closely re-
semble the homogeneous universe described by a few cos-
mological parameters. The χ2 of the fits for each flow
model can be compared to the χ2 for the NF scenario
(shown by the dashed line in the figures) as a test of this
hypothesis. Using ∆χ2 = −2 lnL/LNF , where L is the
likelihood, we find that the pure bulk flow is over 103
times less likely than the NF scenario, while the B05 and
1 http://qold.astro.utoronto.ca/conley/simple cosfitter/
http://qold.astro.utoronto.ca/conley/simple_cosfitter/
Peculiar Velocities of SNe 3
TABLE 1
Peculiar Velocity Model Parameters and Results
Ω = 1 + BAO prior w = −1 + BAO prior
Model β V (km s−1) w Ωm χ
ΩΛ Ωm χ
ΩΛ,Ωm
A06a 0.0 · · · −1.023± 0.090 0.271± 0.021 · · · 0.751± 0.082 0.271 ± 0.020 · · ·
NF 0.0 · · · −1.054
+0.086
−0.084
0.270
+0.024
−0.018
115.5 0.770
+0.083
−0.071
0.269
+0.033
−0.017
115.4
PBF 0.0 57,-540,314 −1.026+0.085
−0.083
0.273+0.024
−0.019
129.4 0.741+0.084
−0.073
0.273+0.034
−0.017
129.2
B05 0.5b 70,-194,0 −1.081
+0.087
−0.085
0.268
+0.024
−0.018
110.3 0.796
+0.081
−0.070
0.267
+0.032
−0.017
110.1
B07 0.7 · · · −1.094
+0.087
−0.085
0.267
+0.024
−0.018
111.2 0.809
+0.082
−0.069
0.265
+0.032
−0.017
111.1
a results quoted in A06 marginalizing analytically over αs and βc (see § 3)
b best fit value from Pike & Hudson (2005)
Fig. 1.— Parameter values for the w, Ωm fit (Ω = 1 + BAO prior)
for each of the four peculiar velocity models in Table 1. The values
for the NF scenario are indicted by the dashed lines. The largest
systematic error in w compared with the NF fit is +0.040 for the
B07 model, which demonstrates the amplitude of the systematic
error if peculiar velocity is not accounted for. The offsets for Ωm
are all within ±0.003 showing that this parameter is not sensitive
to the peculiar velocity corrections due to the BAO prior. The χ2
of the fits improve when using the two β models (B05, B07), while
the PBF model provides a significantly worse fit.
B07 models are 13.5 and 8.6 times more likely, respec-
tively.
We also use these data to assess the systematic er-
rors made in the parameters if no peculiar velocities are
accounted for. The largest of these are obtained by com-
paring the B07 model with the NF scenario. This com-
parison yields ∆wB07 = +0.040 and ∆ΩΛ,B07 = −0.039.
The same comparison for the B05 model, which is only
slightly preferred by the χ2 statistic over model B07, pro-
duces ∆wB05 = +0.027 and ∆ΩΛ,B05 = −0.026. The
systematic offsets for Ωm are all 0.004 or less, demon-
strating the insensitivity of this parameter to peculiar
velocities. This is due to the BAO prior which is insensi-
tive to local flow and provides a much stronger constraint
for Ωm than for w or ΩΛ (see A06, Figures 5 and 6).
5. DISCUSSION AND SUMMARY
The systematic effect of different flow models is at the
level of ±0.04 in w. This is smaller than the present level
of random error in w, which is largely due to the small
numbers of high- and low-redshift SNe. However, com-
pared to other systematics discussed in A06, which total
Fig. 2.— Parameter values for the ΩΛ, Ωm fit (w = −1 + BAO
prior) for each of the four peculiar velocity models as in Figure 1.
Again, comparing the NF fits to the B07 model produces the largest
systematic in ΩΛ of −0.039. We also find Ωm insensitive to the
corrections, having all offsets within ±0.004. The χ2 values show
the same pattern as in Figure 1, favoring the β models over no
correction (NF), and over pure bulk flow.
∆w = ±0.054, the systematic effect of large-scale flows
is important. Wood-Vasey et al. (2007, Table 5) list 16
sources of systematic error which total ∆w = ±0.13.
Aside from three method-dependent systematics and the
photometric zero-point error, they are all smaller than
the flow systematic. As the number of SNe continues to
increase, and understanding of other systematics (e.g.
photometric zero-points) improves, it is possible that
large-scale flows will become one of the dominant sources
of systematic uncertainty.
The peculiar velocities of SN host galaxies arise from
large-scale structures over a range of scales. The compo-
nent arising from small-scale, local structure is the least
important: it is essentially a random variable which is
reduced by
N . More problematic is the large-scale co-
herent component. Such a large-scale component can
take several forms: an overdensity or underdensity; a
large-scale dipole, or “bulk” flow.
The existence of a large-scale, but local (< 7400
km s−1) underdensity, or “Hubble Bubble” was first
discussed by Zehavi et al. (1998). Recently Jha et al.
(2006) have re-enforced this claim with a larger SN data
set: they find that the difference in the Hubble constant
inside the Bubble and outside is ∆H/H = 6.5 ± 1.8%.
4 Neill, Hudson, & Conley
If correct, this could have a dramatic effect on the de-
rived cosmological parameters (Jha et al. 2006, Fig 17),
especially for those studies that extend their local sample
down below z < 0.015. However, the “Hubble Bubble”
was not confirmed by Giovanelli et al. (1999) who found
∆H/H = 1.0 ± 2.2% using the Tully-Fisher (TF) pe-
culiar velocities, nor by Hudson et al. (2004) who found
∆H/H = 2.3± 1.9% using the Fundamental Plane (FP)
distances.
According to equation 1, a mean underdensity of IRAS
galaxies of order ∼ 40% within 7400 km s−1 would
be needed to generate the “Hubble Bubble” quoted by
Jha et al. (2006). However, we find that the IRAS
PSCz density field of Branchini et al. (1999) is not un-
derdense in this distance range; instead it is mildly over-
dense (by a few percent) within 7400 km s−1 (see also
Branchini et al. 1999, Figure 2). As a further cross-
check, when we refit the Jha et al. (2006) data after hav-
ing subtracted the predictions of the B05 flow model, the
“Bubble” remains in the Jha et al. (2006) data. Thus,
the Jha et al “Bubble” cannot be explained by local
structure, unless that structure is not traced by IRAS
galaxies. Moreover, when we analyze the 99 SNe within
15000 km s−1 from Tonry et al. (2003) in the same way,
we find no evidence of a significant “Hubble Bubble”
(∆H/H = 1.5 ± 2.0%), in agreement with the results
from TF and FP surveys. The Tonry et al. (2003) sam-
ple and that of Jha et al. (2006) have 67 SNe in common.
The high degree of overlap suggests that the difference
lies in the different methods for converting the photom-
etry into SN distance moduli.
A local large-scale flow can also introduce systematic
errors if the low-z sample is biased in its sky coverage:
in this case, an uncorrected dipole term can corrupt the
monopole term, which then biases the cosmological pa-
rameters. For the large-scale flow directions considered
here, this does not appear to affect the A06 sample: we
note that the PBF-corrected case has similar cosmolog-
ical parameters to the “No Flow” case. However, if co-
herent flows exist on large scales, this may affect surveys
with unbalanced sky coverage, such as the SN Factory
(Aldering et al. 2002) or the SDSS SN survey2.
The most promising approach to treating the effect
of large-scale flows is a more sophisticated version of
the analysis presented here: combine low-redshift SNe
with other low-redshift peculiar velocity tracers, such as
Tully-Fisher SFI++ survey (Masters et al. 2006) and the
NOAO Fundamental Plane Survey (Smith et al. 2004),
and use these data to constrain the parameters of the
flow model (β and the residual large-scale flow V) di-
rectly. One can then marginalize over the parameters of
the flow model while fitting the cosmological parameters
to the low- and high-z SNe.
2 http://sdssdp47.fnal.gov/sdsssn/sdsssn.html
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