The document discusses methods for reconstructing gravitational fields using peculiar velocities derived from galaxy distributions, focusing on improving velocity maps at higher redshifts. It outlines a methodology to create observed velocity fields, and the integration of theoretical models and observational data is emphasized for accurate reconstructions. Key statistical principles and regularization techniques are highlighted to manage errors in inferred distances and enhance the quality of the reconstructed fields.

1. Reconstructing Gravity with
Peculiar Velocities
Russell Johnston
University of the Western Cape
South Africa
~collaborators~
David Bacon (ICG, Portsmouth)
Luis Teodoro (NASA, AMES)
Bob Nichol (ICG, Portsmouth)
Mike Warren (Los Alamos)
Catherine Cress (CHPC, Cape Town)
~in association with~

2. gravity mapspeculiar velocities weak lensing
( + )
ds2
= (1 2 )dt2
+ a2
(t)(1 + 2 )dx2
Newtonian potentials
temporal perturbations
spatial perturbations
THE CONCEPT
We are interested in extending velocity maps to higher redshifts,
concentrating on test.( )

3. vpec = czobs H0d
inferred distance
measured from e.g. Tully-Fisher or Fundamental Plane
typical error on d ~ 20%
THE CONCEPT

4. V true
pec
RECONSTRUCTING THE PECULIAR
VELOCITY FIELD
binning in observed redshift
384 Mpc

5. Creating an observed velocity field
vpec = czobs H0d
propagate errors on inferred
distance into the peculiar
velocities.
d = 0.3
low-z “high”-z

6. regularising the reconstruction
Pr(Vhyp|Vdata) / Pr(Vdata|Vhyp) Pr(Vhyp| v, Pk)
Velocity field derived from a
redshift galaxy survey
distribution.

7. The prior
gravitational potential
@ gal
z = i gal
(k)kz
a new velocity field
v
gal
(k) =
3⌦mH2
0 z(k)
2a(t)(k2 + k2
z)
Create a velocity field derived from the galaxy distribution:

8. The priorv
The field@ z The fieldcompared to V true
z
v =
Npix
X
i=1
( ˆV rec
@ z)2
v
regularisation
parameter

9. The prior
Uses a theoretical velocity power spectrum to regularise ˆV rec
Pk
Pk =
ktotX
i=1
[ ˆPv(k)rec
Pv(k)th
]2
[ v
P (k)]2
regularisation
parameter

10. Putting it altogether
Pr(Vhyp|Vdata) / Pr(Vdata|Vhyp) Pr(Vhyp| v, Pk)
The reconstructed field can be V or
The data could be PV+WL and jointly reconstruct
&
Pr( hyp|Vdata) / Pr(Vdata| hyp) Pr( hyp| , Pk )

11. 0.174 < z < 0.180
True V-field
Reconstructed
V-field
observed vs. truereconstructed
vs. true

12. r[Vnoisy, Vtrue]
r[Vrec, Vtrue]

13. v =
Npix
X
i=1
( ˆV rec
@ z)2
v
final reconstruction.
Vtrue
@
Embed artificial mass in the field
@Propagate through to field

14. an sdss type survey...

15. RECONSTRUCTING GRAVITY
r[ rec, gal]
Pr( hyp|Vdata) / Pr(Vdata| hyp) Pr( hyp| , Pk )