Weak Lensing and Redshift Space Data:
Tests of Gravity
Bhuvnesh Jain, University of Pennsylvania
Jake VanderPlas, Joseph Clampitt, Anna Cabre, Vinu Vikram
BJ & Khoury (2010) arXiv: 1004.3294
BJ (2011) arXiv: 0223977
BJ & VanderPlas (2011) arXiv: 1106.0065
Dark Energy Tests
• Lensing sensitive to geometry+growth: shear-shear and
galaxy-shear spectra
• Redshift space power spectra measure D(z) through BAO
peaks, and growth factor+bias from full 3D power spectra
• Joint constraints on Dark Energy are powerful due to
complementary dependence on parameters and bias
constraints.
See Gaztanaga, Bernstein, Kirk talks.
In this talk, I will focus on small-scale tests of gravity.
Caveat: Much of this work is preliminary, quantitative
connections to DESpec are yet to be worked out.
Recent progress in gravity theories
• Models that produce cosmic acceleration have been
proposed
• Mechanisms exist to recover GR in the solar system
• General features arise in the dynamics of galaxies
and large-scale structure
Modified Gravity I
how changing gravity affects galaxies
• Modified gravity (MG) theories generically involve scalar fields that
provide an attractive, fifth-force:
a = (ΨS + ΨN)

• This can enhance effective forces on galaxies by 10-100%!
• For large-scale structure, deviations from GR are measured through power
 or galaxy clustering (MG suppressed at high-z -> smaller
spectra of lensing
deviations accumulate in the growth factor).
• For low-z galaxies or clusters with dynamical timescales ~Gyr or less, the
effects can be larger.
Modified Gravity II
two potentials, not one
• Galaxies and Photons respond to different potentials: the mass
distribution inferred from dynamics is different from lensing.
• Conformal transformation of metric -> lensing masses are true masses!
• So a fairly generic signature of modified gravity:
Dynamical mass > Lensing masses
…on a variety of scales: kpc-Gpc.
Modified Gravity III
how the Milky Way protects GR
• Modified gravity theories generically involve large force enhancements.
• BUT…GR must be restored in the Milky Way - via ``natural’’ mechanisms
that work for massive/dense objects. Khoury & Weltman 2004; Vainshtein 72
• So small galaxies or the outer regions of big galaxy/cluster halos may show
deviations from GR.
• The best place to look for signatures of cosmic acceleration could be
through the dynamics and infall of modest-sized galaxies.
– A broad class of theories requires  < 10-6 for objects to feel the scalar force; dwarf
galaxies have  < 10-7 .
Gravity tests in nearby galaxies
• The infall velocities of small galaxies can be enhanced due to the fifth force of
the scalar field: small-scale redshift space distortions
• Enhanced forces alter the luminosities, colors and ages of stars in
``unscreened’’ galaxies.
- For realistic parameters, main sequence stars self-screen, but red giants
in dwarf galaxies will be hotter. Chang & Hui 2010
• Stars may be screened due to their own Newtonian potential: so in dwarf
galaxies they may move differently than dark matter and gas (which feel the
fifth/scalar force).
- Stars move slower than DM/gas
- Stars separate from gas component
Small Scale Tests: III
• Enhanced forces between dwarf galaxies can displace stellar disk from halo
center.
• The neutral Hydrogen gas disk observed in 21cm would track the dark matter
halo -> observable offsets between the disks, and distortions stellar disks.
BJ & VanderPlas, arXiv: 1106.0065
Small Scale Tests: III
• Enhanced forces between dwarf galaxies displace stellar disk from halo center
(and from HI disk) by up to 1kpc.
• Rotation curves of stars are displaced from HI gas, and are asymmetric
• Related effects may be seen in velocity dispersions of dwarf
ellipticals/spheroidals – to be studied
Designing Spectroscopic Surveys
•
Ultra low-z component with three goals:
– Map the gravitational field of the universe out to 100s of Mpc
– Obtain redshifts and velocity dispersions of field dwarf ellipsoids/spheroidals
– Obtain infall patterns around galaxy groups
• Medium z component: obtain lensing and dynamics of hosts with redshifts
z~0.1-0.5
– Sample a sufficient number of galaxy groups (0.1-few Mpc) more densely with
spectroscopy
– See Bernstein talk for advantage of estimating halo masses
Probes of metric potentials
bulk flows
Galaxies

Galaxy Clusters

Linear regime LSS

Dynamical probes (blue) measure Newtonian potential 
Lensing and ISW (red) measures 
Constraints from current data are at 20-50% level
Linear Regime Growth Factors
ds2  (1 2)dt 2  (1 2 )a2 (t)dx 2
2 (  )  8Geff a2

Metric
Poisson
   /

8Geff 2
 2H
a   0
 1 
 and
Geff can be scale and time
dependent in modified gravity
Different growth factors for density and metric potentials:

– Density growth factor: D(z,k)
– Lensing growth factor: D+ Geff D,
– Dynamical growth factor D = /(1+ D+
This description is valid on scales of 10s-100s Mpc.
Lensing: what we assume about gravity
• Deflection angle formula
Generalize
GR
  2  2d
from Geodesic eqn
     (   ) 2d

• How the observable convergence  is related to mass fluctuations:

  12 12  22 (  )2d  G   dz W (z,zs )(z)
GR
Poisson eqn
   dz GeffectiveW (z,zs) (z)
Generalize


• For scalar-tensor gravity theories, lensing by a given mass
distribution is identical to GR.
How does lensing test gravity?
• By itself, lensing measures the sum of metric potentials
- Lensing power spectrum can only test specific models
• Lensing tomography  how D+ evolves with redshift
- This is the primary test for dark energy models as well
• Relation of lensing observables to matter correlations 
- Provided there is a tracer of the mass with known bias
• Cross-correlations: galaxy-lensing plus galaxy-dynamics
- Can give a model-independent measure of /
Robust Test
B. Galaxy-galaxy lensing
•Projected mass profile in three luminosity bins Mandelbaum et al 2005
•Statistical errors on lensing/dynamical comparison at 100-1000 kpc: ~20%
•Systematic errors are comparable or larger.
Redshift space power spectra
Pgv(k)
Tegmark et al 2006
Pgv can be combined with the lensing cross-spectrum Pg
Zhang et al 2007
Current Tests on Large Scales
<gγ>
<gg>
r
Reyes et al 2010
•SDSS data: 20% test of gravity (GR passed!) at 10-30 Mpc scale
•Other large-scale tests combine power spectra to constrain specific models.
The Future: Lensing and Redshift Space
Power Spectra
Lensing spectra
Redshift space spectra
Expected measurements from DES and BOSS surveys. Guzik, Jain, Takada 2009
See more recent work of Zhao et al; Gaztanaga et al; Kirk, Lahav, Bridle.
Forecasts for Geff and 

g


Forecasts for G, : time dependent
Results are sensitive to fiducial model and to time dependence of parameters!
Mpc-scales
C. Group/Cluster Masses: Dynamical
•Stack velocity differences of satellite galaxies around BCG
•Richer clusters  wider velocity histograms  higher mass
Velocity histogram within virial radius: modeling systematic errors
Main galaxies, fitting to 1 gaussian and 2 gaussians
1-d velocity disperion -> 3-d mass profiles
Velocity fields around SDSS
galaxies
Anna Cabre et al, in prep.:
• Measure velocity dispersion
and infall as a function of
radius and host luminosity
• Go out to 10 virial radius
• Compare to halo model
Theoretical models
Halo model and N-body predictions: Preliminary: Tsz-Yan Lam, M. Takada,
F. Schmidt
•
Spare Slides
Three regimes
•
Linear regime: >100 Mpc, z>0.5
•
Intermediate z, Mpc scales
•
Local universe, dwarf galaxies: within 100s Mpc
•
Can some fraction of fibers be used for the latter two regimes?