Probing violation of the Copernican principle via the integrated Sachs-Wolfe effect

arXiv:0903.1541v1 [astro-ph.CO] 9 Mar 2009 Probing violation of the Copernican principle via the integrated Sachs-Wolfe effect Kenji Tomita Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Kaiki Taro Inoue Department of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502, Japan (Dated: November 5, 2018) Recent observational data of supernovae indicate that we may live in an underdense region, which challenges the Copernican principle. We show that the integrated Sachs-Wolfe (ISW) effect is an excellent discriminator between anti-Copernican inhomogeneous models and the standard Copernican models. As a reference model, we consider an anti-Copernican inhomogeneous model that consists of two inner negatively curved underdense regions and an outer flat Einstein-de Sitter region. We assume that these regions are connected by two thin-walls at redshifts z = 0.067 and z = 0.45. In the inner two regions, the first-order ISW effect is dominant and comparable to that in the concordant flat-Λ models. In the outer Einstein-de Sitter region, the first-order ISW effect vanishes but the second-order ISW effect plays a dominant role, while the first-order ISW effect is dominant in the flat-Λ models at moderate redshifts. This difference can discrimate the anti- Copernican models from the concordant flat-Λ model. At high redshits, the second-order ISW effect is dominant both in our inhomogeneous model and the concordant model. In the outer region, moreover, the ISW effect due to large-scale density perturbations with a present matter density contrast ǫm0 ≪0.37 is negligible, while the effect due to small-scale density perturbations (such as clusters of galaxies, superclusters and voids) with ǫm0 ≫0.37 would generate anisotropies which are larger than those generated by the ISW effect in the concordant model. PACS numbers: 98.80.-k, 98.70.Vc, 04.25.Nx I. INTRODUCTION Assuming a uniform distribution of matter on large scales, the observed data of high-redshift type Ia supernovae(SNIa)[1, 2, 3, 4] point to Λ-dominated flat Friedmann-Robertson-Walker (FRW) models. The darkness of the SNIa is reduced to accelerating expansion of the universe due to a positive Λ term. These Λ-dominated FRW models are consistent also with the observed data of temperature anisotropy in the Cosmic Microwave Background (CMB) radiation[5, 6], except for the low-multipole components[7, 8]. Moreover the observed correlation between the CMB and large-scale structure supports these Λ-dominated models, which can generate anisotropies due to the first-order(linear) ISW effect[9, 10, 11]. On the other hand, alternative inhomogeneous models that can explain the SNIa data without introducing a cosmological constant Λ have been independently proposed by C´el´erier[12], Goodwin et al.[13] and Tomita[14, 15, 16, 17] and subsequently studied by several authors[18, 19, 20, 21]. It turned out that some inhomogeneous cosmological models with an inner large-scale underdense region (which we called a local void in our previous works) with a small Hubble constant (h ≈0.5) in the outer flat region can also explain the CMB data[18, 20, 22, 23] as well as the SNIa data. In these models, the cosmological Copernican principle is violated since we need to live near the center of an underdense region. However, recent observational studies such as the baryon acoustic oscillations (BAO)[24, 25, 26, 27, 28, 29, 30], the kinematic Sunyaev-Zeldovich effect either from clusters [31] or reionized regions[32] put stringent constraints on these anti-Copernican models. As a result, models with a local void on 300 Mpc scales seem to be ruled out. At the moment, we need to consider inhomogeneous models with a local void on Gpc scales so that the constraints from BAO at epochs of z ≤0.45 may be avoided. Recently several Gpc void models have been studied by Clifton et al.[33] and Garc´ıa-Bellido and Haugbølle[34]. In this paper we study the ISW effect1 in flat FRW models with an inner underdense region on Gpc scales based on previous results[35, 36, 37, 38, 39, 40]. Then we compare it with the ISW effect in the concordant flat FRW model with a cosmological constant Λ. As we shall show, the ISW effect will be an excellent discriminator 1 In this paper, “the ISW effect” means redshift/blueshift of the CMB photons due to time-evolving first-order or second-order metric perturbations. 2 between our anti-Copernican models and the standard concordant Copernican model. In §2, we present our inhomogeneous cosmological model with inner underdense regions and in §3 we derive analytic formulae for calculating the ISW effect in the inner and outer regions and we discuss the property of temperature anisotropy due to the ISW effect in our models and the concordant model. §4 is dedicated to concluding remarks. In what follows, we use the units of 8πG = c = 1. For spatial coordinates, we use Latin subscripts running from 1 to 3. II. A COSMOLOGICAL MODEL WITH INNER UNDERDENSE REGIONS Our in

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