RESCEU SYMPOSIUM ON

GENERAL RELATIVITY AND GRAVITATION

JGRG 22

November 12-16 2012

Koshiba Hall, The University of Tokyo, Hongo, Tokyo, Japan

Tatsuya Narikawa, JGRG 22(2012)111418

“Testing gravity with galaxy cluster”

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Based on TN, Yamamoto, JCAP (2012) TN, Kobayashi, Yamauchi, Saito, work in progress

Outline �Modified gravity �Screening mechanism �Cluster lensing �Summary

758*9:8';)"<*#& �O+3A,93A+:M%PA<=+,-0C%+Q%=R00-:3%=+<.A=%9==-;-093A+:% [Riess, Schmidt et al. ‘98, Perlmutter et al. ‘99] %%�S0-9DK+E:%+Q%"#%93%;90T-N<=9;-<�%%�U<%9:%9;3-0:93A,-%3+%K90D%-:-0TC%%�O+KAV-K%T09,A3C%EA34%-W309%V-;K<%X%'=9;90N3-:<+0%T09,A3C5%QF#H%T09,A3C5%P"Y%.+K-;5 %"9;A;-+:%.+K-;5%P-=+R>;A:TN;A.A3%+Q%"4+<3NQ0--%.9<<A,-%%T09,A3C5%-3=Z% [Mukohyama-san’s talk]

a > 0

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No. 3] Weak Lensing Study of 30 Clusters 827

Fig. 8. Left panel: The mean distortion profile with 1! statistical uncertainties as a function of the projected radius, which is obtained by stacking thedistortion signals for 10 clusters that are selected with the virial masses Mvir ! 6 " 1014 h#1 Mˇ from 19 clusters in figure 6. Note that the distortionprofile is plotted in units of the projected mass density, and the projected radius is computed from the weighted mean redshift of clusters. The dashed andsolid curves are the best-fit SIS and NFW models, respectively. Right panel: A similar plot, but for 9 halos with Mvir > 6 " 1014 h#1 Mˇ. For bothresults, the SIS model is strongly disfavored: !"2 $ "2

SIS;min # "2NFW;min ' 41 and 127 for the low- and high-mass cluster samples, respectively.

The combined results also show 2!-level evidence that the NFW concentration is greater for more massive halos, which is exactly consistent withthe result in figure 6.

status of the clusters: more relaxed clusters tend to have highconcentrations, and vice versa (also see Smith & Taylor 2008).It will therefore be important to further explore the concentra-tion distribution by enlarging the sample size of clusters.

5.5. Stacked Lensing Signal

In this section we study the stacked weak-lensing signal of19 clusters in figure 6. This approach has several importantadvantages. First, the average distortion profile is less sensitiveto substructures within and asphericity of the individual clustermass distributions and also to uncorrelated large-scale structurealong the same line-of-sight. This is because these “contam-inating signals” are averaged out via the stacking, under theassumption that the universe is statistically homogeneous andisotropic. Second, stacking should boost the signal-to-noiseratio of the distortion signal at very small and large radii. Thesignal-to-noise ratio at small radii is limited for a single clusterbecause the solid angle subtended by a radial bin shrinks atsmall radii, thus reducing the number of galaxies over whichthe mean distortion signal is calculated. Hence, the signal-to-noise ratio suffers despite the signal peaking in these regions.On the other hand, at large radii, the binned solid-angle is muchlarger, helping to maintain the signal-to-noise; however, thesignal becomes very small, and thus the signal-to-noise ratiodeclines. As discussed in subsection 5.3, the signal-to-noiseratio is optimized at intermediate radii. Therefore, stackinghelps to improve the signal-to-noise as a function of the radius,thus enabling a clearer investigation of (i) the curvature of the

density profile, which is a characteristic signature of the NFWprediction, helping us potentially to address the nature of darkmatter (e.g., Yoshida et al. 2000), and (ii) the distribution ofmass outside the virial radius to address whether or not theouter slope of the NFW profile, # / r#3, continues outsidethe virial radius (e.g., Bertschinger 1985; Busha et al. 2003).

To study the stacked lensing signal, we divide the 19 clus-ters into two mass bins, based on whether the NFW model fitsto individual clusters yielded a virial mass estimate of greaterthan or less than Mvir = 6 " 1014 h#1 Mˇ. This results intwo sub-samples of 10 low-mass and 9 high-mass clusters.Figure 8 shows the average distortion profile as a functionof the projected radius in the physical length scale. Notethat the effect of different cluster redshifts was taken intoaccount by using the weighting method in terms of the lensingefficiency functions of averaging clusters (Mandelbaum et al.2006; also see Sheldon et al. 2009a), and the projected radiusis computed from the weighted mean redshift of the sampledclusters. However, we checked that, even if we use the singlelensing efficiency for the mean cluster redshift, the resultsremains almost unchanged due to the narrow redshift coverageof our cluster samples. Note that the mean lens redshifts arehzli = 0.251 and 0.236 for the low- and high-mass samples,respectively.

First, unsurprisingly, the stacked profiles yield very signifi-cant detections: the total signal-to-noise ratios are S=N = 24and 30 for the low- and high-mass samples, respectively.Second, the lensing distortion signals are recovered over a wide

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[Okabe, Takada, Umetsu, Futamase, Smith ’10]

[Itoh-san’s talk]

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2

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is equivalent to the generalized galileon [Deffayet et al. ‘11, Kobayashi, Yamaguchi , Yokoyama ‘11]

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+G4(φ, X)R+G4X × (field derivatives)

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−1

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20

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