## Abstract

The Copernican principle remains largely unproven at Gpc radial scale and above. Generally, violations of this type cause a first-order anisotropic kinetic Sunyaev Zel'dovich (kSZ) effect. Here we show that, if large-scale radial inhomogeneities have amplitude large enough to explain the ‘dark energy’ phenomena, the induced kSZ power spectrum will be orders of magnitude larger than the Atacama cosmology telescope/South Pole telescope upper limit. This single test rules out the void model as the cause of the apparent cosmic acceleration, confirms the Copernican principle on Gpc radial scale and above, and closes a loophole in the standard cosmology.

## 1. Introduction

The unusual dimming of distant supernovae type 1a (SN1a) [1,2] requires new understandings of the Universe. An incomplete list of possible explanations includes

—

*new physics*, such as the cosmological constant, dark energy and modified gravity;—

*new calculation of old physics*; the cosmological back-reaction arising from the coupling between the nonlinear field equation of general relativity and the inhomogeneity of the Universe belongs to this catalogue; and—

*new geometry*; a popular model is the void model in which we live or near the centre of a large void as described by a spherically symmetric Lematre–Tolman–Bondi (LTB) space–time [3].

The last possibility provokes us to re-investigate a fundamental tenet of modern science, the Copernican principle. The Copernican principle states that there should be no special regions in the Universe and hence our Universe should be homogeneous at sufficiently large scales. This is the foundation of the cosmological principle. So, testing the Copernican principle is of crucial importance for modern cosmology and, in particular, for cosmological physics. For example, the void model explicitly violates the Copernican principle and challenges the existence of cosmic acceleration [4–9].

Cosmic microwave background (CMB) observations verify the statistical homogeneity in angle on our celestial sphere [10]. Galaxy surveys verify the radial homogeneity up to the Gpc scale [11]. However, radial homogeneity at larger scales remains unproven. Strong evidence in favour of this large-scale radial inhomogeneity and tests based on it includes:

— a strong kinetic Sunyaev Zel'dovich (kSZ) effect of galaxy clusters [12,14];

— varying spatial curvature [15];

— features in the SN1a Hubble diagram [16]; and

— features in the time drift of cosmological redshift [17] and abnormal cosmic parallax [18].

Joint analysis has also been performed [19]. These tests disfavour the void model and support the Copernican principle. Recently, we proposed a new powerful test [20]. It is by far the most stringent single test and it confirms the Copernican principle at Gpc scales and above. We summarize the major results here.

## 2. The kinetic Sunyaev Zel'dovich test

A generic consequence of violating the Copernican principle is that some regions will expand faster or slower than others and, as photons transit between these regions, there will be a relative motion between the matter frame and CMB. When relative motions between free electrons and photons exist, the inverse Compton scattering will induce a shift of the brightness temperature of CMB photons via the kSZ effect [21,22]. This temperature shift will be anisotropic on our sky tracing the anisotropy of the projected free electron surface density. This test of the Copernican principle has been applied to cluster kSZ observations [12,14,23], where the electron surface density is high. However, this effect applies to all free electrons that exist in great abundance everywhere in the Universe up to the reionization epoch at redshift *z*∼10, whereas clusters are rare above *z*∼1. So, one can expect a more sensitive test from blank-field CMB anisotropy power spectrum measurements than from cluster measurements, as has been demonstrated for the ‘dark flow’-induced small-scale kSZ effect [24]. The amplitude of the effect is much larger for the proposed LTB models and is in conflict with recent observations. Furthermore, this power spectrum test limits flows on a much larger range of redshifts than cluster measurements can.

When the Copernican principle is violated, the electron peculiar motion ** v** has two components.

*v*_{H}is the relative motion between the matter co-moving frame and CMB and

*v*_{L}is the local motion of electrons with respect to the co-moving frame. Correspondingly, the induced kSZ temperature fluctuation has two contributions 2.1 The first term on the right-hand side is the conventional kSZ effect, 2.2 Here, is the radial direction on the sky.

*τ*

_{e}is the mean Thomson optical depth to the corresponding redshift and

*δ*

_{e}is the fractional fluctuation in the free electron number density. Both

*v*_{L}and

*δ*

_{e}fluctuate about zero, and cancellations along the line of sight cause the small-scale anisotropy power spectrum to be dominated by terms cubic and higher in the amplitude of the inhomogeneities [25,26]. The last term in equation (2.1) is new and does not vanish in a non-Copernican universe [24], 2.3 The last expression neglects the term, which has no direction dependence in LTB models in which we live at the centre, and is therefore not observable.

*v*_{H}varies slowly along the radial direction and does not suffer the cancellation of

*v*_{L}in the conventional kSZ effect. The small-scale anisotropy power spectrum will be quadratic in the amplitude of

*δ*

_{e}(which does fluctuate about zero), so we can say that Δ

*T*/

*T*is first order in the density fluctuations. Throughout this paper, unless otherwise specified, we will focus on this

*linear*kSZ effect. We restrict ourselves to adiabatic voids in which the initial matter, radiation and baryon densities track each other. This is what one would expect if baryogenesis and dark matter decoupling occur after the process which generates the void inhomogeneity. Non-adiabatic voids can in some cases suppress

*v*

_{H}[27]. We also restrict ourselves to small voids outside of which both matter and radiation are homogeneous. Additional inhomogeneities will generically lead to larger values of

*v*

_{H}.

To explain the dimming of SN1a and hence the apparent cosmic acceleration without dark energy and modifications of general relativity, we shall live in an underdense region (void) of size Gpc, with a typical outward velocity km s^{−1} [14]. Given the baryon density, *Ω*_{b}*h*^{2}=0.02±0.002 from the Big Bang nucleosynthesis [28], *τ*_{e}>10^{−3}. Scaling the observed weak lensing r.m.s. convergence *κ*∼10^{−2} at approximately 10 arcminute scale [29], the r.m.s. fluctuation in *δ*_{e} projected over Gpc length is at the same angular scale. Hence, such a void generates a kSZ power spectrum μk^{2} at these angular scales.

This large amplitude of the kSZ power spectrum is in sharp conflict with the recent kSZ observations. The South Pole telescope (SPT) collaboration [30] found Δ*T*^{2}<13 μK^{2} (95% upper limit) at multi-pole ℓ=3000 (approx. 7 arcminutes). The Atacama cosmology telescope (ACT) collaboration [31,32] found Δ*T*^{2}<8 μK^{2} (95% upper limit) at the same angular scale. SPT is able to further improve the 95% upper limit to 6.5 μK^{2} [33]. This simple order of magnitude estimation demonstrates the discriminating power of the kSZ power spectrum measurement. It implies that a wide range of void models capable of replacing dark energy are ruled out. This also demonstrates how purely empirical measurements of CMB anisotropies and the large-scale structure (e.g. weak lensing) can in principle be combined to limit non-Copernican models without any assumptions of how the inhomogeneities vary with distance.

We perform quantitative calculations for a popular void model, namely the Hubble bubble model ([13] and references therein). In this model, we live at the centre of a Hubble bubble of constant matter density *Ω*_{0}<1 embedded in a flat Einstein–de Sitter universe (*Ω*_{m}=1). The void extends to redshift *z*_{edge}, surrounded by a compensating shell (*z*_{edge}<*z*<*z*_{out}) and then the flat Einstein–de Sitter universe (*z*>*z*_{out}). The kSZ effect in this universe has two components—(i) the linear kSZ arising from the large angular scale anisotropies generated by matter (a) inside the void, (b) in the compensating shell, and (c) outside the void, and (ii) the conventional kSZ effect quadratic in density fluctuation [26] and the kSZ effect from patchy reionization [34]. The contributions of each of these to the anisotropy power spectrum are uncorrelated. Hence, the ACT/SPT measurements place an upper limit on the total. The latter contributes at least a few μK^{2} [35], so what is left for the first component is less than ∼3 μK^{2}. However, we will test the Copernican principle in a conservative way, by requiring the power spectrum of the first component generated by matter *inside the void* to be below the ACT/SPT upper limit 6.5 μK^{2} at ℓ=3000.

For a general Hubble bubble, *v*_{H} is determined by both Doppler and Sachs–Wolfe anisotropies generated by the void and depends qualitatively on the size of the void [13]. It is only small Hubble bubbles (technically *z*_{edge}<5/4) which are consistent with both the supernova data and the spectrum of the CMB [13] and for these small Hubble bubbles a simple Doppler formula can be used [36],
2.4
where *H*_{i}(*z*) is the Hubble expansion rate inside the void as a function of redshift, *H*_{e} gives the Hubble expansion rate exterior to the void at the same cosmological time, and *D*_{A,co}(*z*) is the co-moving angular diameter distance to redshift *z*. The above expression is valid in the limit of |*v*_{H}|≪*c*. Later, we will see that void models that pass the proposed kSZ test satisfy this condition.

The auto-power spectrum at multi-pole ℓ generated by the linear kSZ effect inside the Hubble bubble, using the Limber approximation, is
2.5
Here, is the electron number overdensity power spectrum (variance) at wavenumber *k* and redshift *z*. Henceforth, we assume that , where is the matter power spectrum (variance), which is a sufficiently good approximation at the scales of interest.

It is non-trivial to calculate Δ^{2}_{m} in general LTB models, even at linear scales, as locally the expansion rate is anisotropic and so the inhomogeneities will have an anisotropic power spectrum (refer to the study of Clarkson *et al*. [37] for a linear perturbation treatment). We take a minimalist's approach to circumvent this obstacle. The measured matter clustering and its evolution agree with the standard Lambda cold dark matter (*Λ*CDM) [29,38–40] to *z*∼1, so do the galaxy clustering and evolution [41–43]. Hence the density inhomogeneities in any viable LTB models must be consistent with the *Λ*CDM prediction, within a factor of approximately 2 observational uncertainties. This allows us to approximate in the LTB models to be that of the standard *Λ*CDM model. We adopt *Ω*_{m}=0.27, *Ω*_{Λ}=1−*Ω*_{m}, *Ω*_{b}=0.044, *σ*_{8}=0.84 and *h*=0.71 and calculate the linear density clustering using the CMBFAST package [44], the nonlinear clustering from the halofit formula [45] and the kSZ power spectrum from equation (2.5). In this procedure, we have assumed that these LTB models agree with the existing matter clustering measurements. Certainly, they may not. So, the proposed kSZ test is a conservative test of the Copernican principle.

## 3. Constraints on the void model

The ACT/SPT upper limit rules out large voids with low density (figure 1). Only those voids with either () or (, corresponding to void radius h^{−1} Gpc) survive this test (figure 1). This is by far the most stringent *single* test of the void models and the Copernican principle at Gpc scale and above.

The kSZ test is highly complementary to other tests such as the supernova test. We have improved the SN1a constraints in the study of Caldwell & Stebbins [13] by using the UNION2 data with 557 SN1a [46]. The minimum *χ*^{2} is 605.4. Hubble bubble models within the 3*σ* contour have typical μK^{2} at ℓ=3000, two orders of magnitude larger than the ACT/SPT upper limit 6.5 μK^{2}, so they are robustly ruled out. On the other hand, Hubble bubble models consistent with the ACT/SPT results have Δ*χ*^{2}>195 (*χ*^{2}>800) for the SN1a test and hence fail too. Thus, in combination with SN1a observations, all Hubble bubble models are ruled out.

The above numerical calculation performed for the Hubble bubble model justifies our previous order of magnitude estimation on void models in general. This leads to a general conclusion that adiabatic void models capable of explaining the supernova Hubble diagram generate too much power in the kSZ sky to be consistent with the ACT/SPT upper limit. This conclusion is robust against various uncertainties in the kSZ modelling and measurement. Hence, we rule out the possibility to explain the apparent cosmic acceleration by these voids. This confirms that the observed apparent cosmic acceleration is indeed real.^{1}

## Footnotes

One contribution of 16 to a Theo Murphy Meeting Issue ‘Testing general relativity with cosmology’.

↵1 A general void model has three sets of free functions.

*M*(*r*) specifies the matter distribution and*E*(*r*) specifies the curvature variation along the radial direction.*t*_{B}(*r*) specifies the Big Bang time. Although one can tune*M*(*r*) or*E*(*r*) to fit SN1a data, these models are in general ruled out by our kSZ test. However, an inhomogeneous distribution of*t*_{B}(*r*) can cause CMB observed by distant electrons to have an intrinsic dipole. Such an intrinsic CMB dipole can be tuned to cancel the kSZ effect caused by the relative motion between CMB and matter frame. We thank Pedro Ferreira for pointing out this possibility. This kind of void model can then pass both the SN1a test and the kSZ test. The price to pay is to violate the Copernican principle not only at late time, but also in the early Universe.

- This journal is © 2011 The Royal Society