The late-time integrated Sachs Wolfe effect

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The Late-time Integrated Sachs-Wolfe Effect











The Integrated Sachs-Wolfe (ISW) effect is due to the decays of graviational potential wells.  It can be split into two parts: the early ISW and the late-time ISW effects.   This page focusses on the latter.


While the early ISW effect occurs shortly after fecombination during the radiation era, the late-time ISW effect occurs much later, and is due to the gravitational potentials of Large Scale Structure (LSS).  It can also be a result of the potential of Dark Energy if one considers it is a fluid with perturbations.


The Origin of the late-time ISW effect


The gravitational potential of the LSS will distort space-time so that a photon travelling through it will be subject to a graviational blueshift on entry of the potential well, and redshift on exit. If the potential does not change during the travel time of the photon, then the net effect of the gravitational redshift will be null: the photon will emerge unaffected by LSS. This is always the case on linear scales in an Einstein-de Sitter Universe. 


If, however, these large scale potential wells vary with time, as they would in the presence of dark energy or curvature, the photon will emerge from the LSS gravitational field, either red- or blue-shifted depending on whether the potentials grow or decay respectively. This is illustrated in the image above. Photons travel (from the left of the image) towards the observer (right) through LSS potential wells. The photons at the top of the image travel through large scale structure in an Einstein-de Sitter universe and emerge from the LSS potential well with the same redshift as on entry. The photons at the bottom of the illustration travel through the LSS in an accelerating universe in which large scale potential wells decay and exit bluer than on entry.


For photons travelling from the surface of last scattering (CMB), the varying gravitational potential of LSS will create secondary temperature anisotropies which will add power to the temperature-temperature angular power spectrum Formula. The power added on large scales is:




where T is the temperature of the CMB, η is the conformal time, defined by  Formulaand Formula and Formula are the conformal times today and at the surface of last scattering respectively;

Formula is the unit vector along the line of sight; Formula is the gravitational potential at position

x and at conformal time Formula, and Formula.


The relative amplitude of the ISW signal to primary temperature anisotropies make it difficult to measure from the temperature-temperature power spectrum alone. However, Crittenden & Turok (1996) proposed using the cross-correlation between LSS and the CMB to detect the ISW effect independently from the intrinsic CMB fluctuations. A significant decay in the gravitational potentials will produce large scale hot spots in the CMB. The gravitational potentials will tend to host an overdensity of galaxies, so a positive correlation between the CMB and the galaxy distribution is expected.  This positive correlation is also expected in open universes (Kamionkowski 1996; Kinkhabwala & Kamionkowski 1999), whereas a negative correlation will occur in closed universes where the gravitational potentials of LSS grow. The Sunyaev-Zel’dovich effect also produces a negative correlation, but this is expected on smaller scales.


In the currently favoured cosmological model, it is believed that the Universe has recently ( z < 1) become dominated by dark energy, and since the ISW effect depends

directly on the growth of structure, this makes the ISW effect an evident probe of the cosmological model. Alternative models of gravity can predict ISW signatures (Carroll

et al. 2005, Song, Sawicki & Hu 2006), but these will also affect distance measurements (e.g., supernovæ). Measuring the ISW effect can therefore help break the degeneracy.



Cross-Correlation of Galaxy and Temperature Anisotropy Fields



Redshift Evolution of the ISW Signal



Cosmological Parameter Dependence


Galaxy Bias


Dark Energy vs. Curvature


Lambda vs. Dark Energy



Effect of Cosmic Magnification


Effect of Non-Trivial Topologies