Probing the integrated Sachs-Wolfe effect using embedded lens models

B. Chen and R. Kantowski

Phys. Rev. D 91, 083014 – Published 29 April 2015

Abstract

The photometry profile of the integrated Sachs-Wolfe (ISW) effect, recently obtained by the Planck consortium by stacking patches of cosmic microwave background (CMB) sky maps around a large number of cosmic voids, contains a cold ring at about half the void’s effective radius surrounded by a hot ring near the void’s boundary. The source of the temperature structure is assumed to be the ISW effect but the exact cause of the ringed structure is not currently well understood, particularly the outer hot ring. Numerical simulations have suggested that hot/cold ring structures can be produced by motions associated with nonlinear growths of cosmic structures whose gravitational potentials produce the ISW effect. We have recently developed the embedded lens theory and the Fermat potential formalism which can be used to model the ISW effect caused by intervening individual lens inhomogeneities evolving arbitrarily. This theory only requires knowledge of the void’s projected mass profile as a function of the passing CMB photons’ impact radius and the rate of change of that mass distribution at passage. We present two simple embedded void lens models with evolving mass densities and investigate the ISW effect caused by these lenses. Both models possess expanding mass shells which produce hot rings around central cold regions, consistent with the recent observations. By adding a small overdensity at the void’s center we can produce the slight positive temperature excess hinted at in Planck’s photometric results. We conclude that the embedded lens theory and the Fermat potential formalism is well suited for modeling the ISW effect.

Received 8 March 2015

DOI:https://doi.org/10.1103/PhysRevD.91.083014

Authors & Affiliations

B. Chen^1,* and R. Kantowski^2,†

¹Research Computing Center, Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306, USA
²Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, 440 West Brooks, Norman, Oklahoma 73019, USA

^*bchen3@fsu.edu
^†kantowski@ou.edu

Time delay in Swiss cheese gravitational lensing

B. Chen, R. Kantowski, and X. Dai

Phys. Rev. D 82, 043005 (2010)

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Vol. 91, Iss. 8 — 15 April 2015

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Images

Figure 1
ISW temperature profiles across embedded void lens models of type I are shown in the left panel. The right panel shows the signals averaged by applying a compensated top-hat filter; see Eq. (11). The void is at redshift $z_{d} = 0.5$ with physical radius $r_{d} = 50 Mpc$ and density contrast $- 0.5$ . The respective solid, long-dashed, and dashed curves show results for shells which are coexpanding with (blue), expanding relative to (magenta), and contracting relative to (red) the FRLW background. The relative velocities shown are $\pm 1000 k m s^{- 1}$ and $\pm 500 k m s^{- 1}$ . The cyan dotted-dashed curve is for the case when the shell is comoving but the density contrast $δ$ is evolving according to linear growth theory. A central cold spot is produced for all cases (left panel) but linear growth produces the smallest ISW effect. Near the void boundary the time-delay contribution diminishes and the ISW signal is dominated by the movement of the compensating shell, which results in a hot ring for expansion. On the right the ISW filtered cold spot is seen to be most significant at radius $R \approx 0.6$ .
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Figure 2
ISW temperature profiles for embedded model II voids without a central compact object are shown in the left column and the corresponding filtered signals are shown on the right. The void is at redshift $z_{d} = 0.5$ with physical radius $r_{d} = 50 Mpc$ and density contrast $δ = - 0.9$ . A compensating shell is located at $y (t_{d}) = 0.5$ or at 0.8 (top or bottom row respectively). For the upper row the lens behaves like a cluster lens producing a hot spot toward the center. For the lower row most of the lens mass is located near the void boundary and a central cold spot is produced except for the case of an expanding shell with $v = + 500 k m s^{- 1}$ where $Δ T > 0$ across the entire lens. A diverging/converging flow increases/decreases the CMB temperature but only within the shell [i.e., the solid, dashed, and long-dashed curves merge beyond $x = y (t_{d})$ ]. A hot ring near the void boundary can be produced by this lens model. The filtered ISW cold spot is again most significant at a filter radius $R \approx 0.6$ ; see the bottom right panel.
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Figure 3
ISW temperature and photometry profiles (left and right panels respectively) for embedded model II voids possessing a central compact object. The void is at redshift $z_{d} = 0.5$ with physical radius $r_{d} = 50 Mpc$ . The overdense shell is located at $y (t_{d}) = 0.8$ and contains 90% of the lens mass ( $ξ = - 0.9$ ). The central compact object contains 1% of the lens mass (about $7.2 \times 1 0^{14} M_{⊙}$ ). Hot rings again appear near the boundary of these large void models and the central cold spot is again most significant at filter radius $R \sim 0.6$ . However, now a small central positive excess is present for filter radii $R ≲ 0.2$ due to the point mass at the void’s center.
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