What if Planck's Universe isn't flat?
WWhat if Planck’s Universe isn’t flat?
Philip Bull
Department of Astrophysics, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK
Marc Kamionkowski
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Inflationary theory predicts that the observable Universe should be very close to flat, with aspatial-curvature parameter | Ω K | (cid:46) − . The WMAP satellite currently constrains | Ω K | (cid:46) . − . Suppose that Planck were to findΩ K (cid:54) = 0 at this level. Would this necessarily be a serious problem for inflation? We argue thatan apparent departure from flatness could be due either to a local (wavelength comparable tothe observable horizon) inhomogeneity, or a truly superhorizon departure from flatness. If thereis a local inhomogeneity, then secondary CMB anisotropies distort the CMB frequency spectrumat a level potentially detectable by a next-generation experiment. We discuss how these spectraldistortions would complement constraints on the Grishchuk-Zel’dovich effect from the low- (cid:96) CMBpower spectrum in discovering the source of the departure from flatness.
Inflation predicts that the observable Universe shouldbe very nearly flat, with a spatial-curvature parameter | Ω K | < − in most models [1]. WMAP data currentlyconstrain | Ω K | (cid:46) − (95% CL) [2], and Planck shouldbe sensitive to Ω K at around the 10 − level [3], improvingto ∼ − when combined with 21-cm intensity maps [4](which represents the limit of detectability [5]).Suppose that Planck were to find a nonzero value forΩ K . What might this mean for inflation? Such an ob-servation would nominally be evidence for a genuine de-parture from flatness on superhorizon scales, with wide-ranging implications for a broad class of inflationarymodels; for example, a measurement of Ω K < − − is sufficient to rule out the majority of eternal inflationscenarios with high confidence [6].Before jumping to such conclusions, though, one mightwonder whether the deviation could be explained simplyby a local inhomogeneity that biases our determinationsof cosmological parameters. This would allow us to pre-serve flatness (and thus some relatively natural sort ofinflation) by explaining the discrepancy as the result ofsystematic distortions of, e.g., the distance-redshift rela-tion due to lensing by the inhomogeneity [7]. Althougha local density fluctuation of a large enough amplitude(Φ (cid:38) − ) would be inconsistent with the simplest in-flationary models, it might conceivably arise if there issome strongly scale-dependent non-Gaussianity, or per-haps if some sort of semi-classical fluctuation arises atthe beginning or end of inflation [8].For a sufficiently large and smooth local inhomogene-ity, it would be difficult to definitively distinguish thesetwo situations using standard cosmological tests. Purelygeometric observables such as distance measures wouldbe inhibited by degeneracies with evolving-dark-energymodels [9], and the deviation from flatness would be toosmall to significantly affect the growth of structure.In this Letter, we show that a class of observables basedon spectral distortions of the CMB offer the prospect todisentangle the two scenarios. These observables exploitthe strong relationship between spatial homogeneity and the isotropy of spacetime; by using them to measure thedipole anisotropy of the CMB about distant points, itis possible to place stringent constraints on the possiblesize of a local inhomogeneity [10]. Furthermore, these ob-servables unambiguously distinguish between subhorizonand superhorizon effects, owing to a cancellation of thedipole induced by superhorizon perturbations [11, 12].We begin by calculating the bias in Ω K due to a localinhomogeneity. We take the form of this local inhomo-geneity throughout to be a spherically symmetric poten-tial perturbation Φ( r, t ) = D ( t ) a − ( t )Φ exp[ − ( r/r ) ],where the linear-theory growth factor is normalized to D =1 today. The presence of a large, local inhomogeneitymodifies the apparent distance to last scattering througha combination of lensing, integrated Sachs-Wolfe effect,gravitational redshift, and Doppler shift. Ref. [13] de-rived a full expression for the (subhorizon) luminosity-distance perturbation δd L up to linear order in perturba-tions. (The observed distance d L ( z ) = ¯ d L (1+ δd L ), wherethe overbar denotes a background quantity.) When con-sidering the CMB, it is useful to rewrite this as a pertur-bation δd A = δd L − δz/ (1 + z ) to the angular-diameterdistance.An observer sitting at the center of a spherically-symmetric inhomogeneity will measure a distance to lastscattering which deviates from the background quan-tity by a uniform amount over the whole sky. Thisintroduces a shift in angular scale of the entire CMBpower spectrum. The value of Ω K inferred from obser-vations depends primarily on the angular scale of thefirst few CMB acoustic peaks [14], and will therefore bebiased away from its background value. Fig. 1 showsthe distance perturbation as a function of the depthand width of a local inhomogeneity, compared with thechange in (background) distance between a flat model,and one with | Ω K | = 10 − . (For numerical work wetake [ h, Ω m , Ω Λ , σ ] = [0 . , . , . , .
8] and red-shifts of reionization and last scattering to be z re = 10and z ∗ = 1090 .
79 respectively.) Based on the distance tolast scattering alone, an inhomogeneity with Φ ∼ − a r X i v : . [ a s t r o - ph . C O ] J u l would induce an apparent shift in Ω K of order 10 − fora wide range of widths.The inhomogeneity will also cause the observed red-shift z ∗ of the surface of last scattering, to differ from itsbackground value, ¯ z ∗ = z ∗ − δz ∗ , where [13] δz = (1+ z s ) (cid:20) Φ s − Φ o + ( v o − v s ) · n + 2 (cid:90) η o η s dη n · ∇ Φ (cid:21) , (1)and n is a unit vector along the line of sight. For the cen-tral observer, the effect of the redshift perturbation is tochange the inferred conformal time (and thus expansionrate) of last scattering, which will bias the estimation ofparameters such as Ω m . For an observer who is off-center in the inhomogeneity, however, an additional anisotropywill also be induced in the CMB. This is because the red-shift perturbation, Eq. (1), depends on direction; a lineof sight looking towards the center of the inhomogene-ity will experience a different change in redshift to onelooking away from it, and thus there will be a direction-dependent change in temperature.In general, anisotropies will be induced over a rangeof angular scales, but at least for observers close to thecenter of a large (wide) inhomogeneity, the dipole, β , willdominate. While there is also a dipole contribution dueto the peculiar velocity of the observer, velocity perturba-tions due to the matter distribution on smaller scales areexpected to be Gaussian random distributed with meanzero, whereas the dipole due to a large inhomogeneity willgenerally present a systematic trend in redshift and an-gle on the sky. This allows us to distinguish between thetwo contributions. For a spherical inhomogeneity, axialsymmetry dictates that the dipole will be aligned in theradial direction, and that all spherical-harmonic modesof the induced anisotropy with m (cid:54) = 0 on the sky of theobserver will be zero, so that β ∝ (cid:82) δz ∗ ( θ ) cos θ sin θdθ .We now discuss spectral distortions due to a local in-homogeneity. There is a close relationship between ho-mogeneity and the isotropy of spacetime. A number ofobservational tests that are sensitive to CMB anisotropiesabout distant points can be used to exploit this link anddetect local inhomogeneities of the kind that would causea systematic bias in measurements of Ω K .The strength of the connection between homogeneityand isotropy is most clearly demonstrated by the Ehlers-Geren-Sachs (EGS) theorem [15]. According to EGS,if all comoving observers in a patch of spacetime see anisotropic CMB radiation field, then that patch is uniquelyFLRW (i.e., it is necessarily homogeneous and isotropic).Generalizations of this result show that it is perturba-tively stable, in the sense that small departures fromperfect isotropy imply only small departures from homo-geneity (up to some assumptions; see [16] for a critique).A corollary to the EGS theorem is that observers in aninhomogeneous region of spacetime will in general see ananisotropic CMB sky. We can therefore use measure-ments of the anisotropy of the CMB about a collectionof spacetime points to constrain the degree of inhomo- FIG. 1: The change δd A in the distance to last scattering as afunction of the width and depth of the inhomogeneity. Blacklines denote δd A . The thick blue line plots the difference indistance (in background) between models with | Ω K | = 10 − and Ω K = 0 (with identical h and Ω m ), equivalent to δd A =2 . × − . geneity inside our Hubble volume [17].Compton scattering of CMB radiation by ionized gasprovides a way to detect anisotropy about remote points.The scattered radiation spectrum consists of a weightedsuperposition of spectra from all directions on the scat-terer’s sky, I (cid:48) ν ∼ (cid:82) τ (1 + cos θ ) I ν ( θ, φ ) d Ω. If the scat-terer’s sky is a perfectly isotropic blackbody of uniformtemperature, the scattered spectrum is simply a black-body of the same temperature, plus spectral distortionsdue to the random thermal motions of the electrons inthe scattering medium (the thermal Sunyaev-Zel’dovicheffect, TSZ [18]). If its sky is anisotropic, however, theresulting spectrum is a combination of blackbodies ofdifferent temperatures. This induces additional black-body spectral distortions, and shifts the temperature ofthe ‘base’ blackbody spectrum as seen by an observer[10, 19]. If the dipole anisotropy dominates, we call thesethe Compton- y distortion and the kinematic Sunyaev-Zel’dovich (KSZ) effect [20], respectively.By measuring the Compton- y distortion and KSZ ef-fects for many scattering regions on our own sky, we canbuild up a picture of the degree of anisotropy, and thusinhomogeneity, within our past lightcone. We will nowoutline three observational tests based on these effects,and estimate their sensitivities to a local inhomogeneity.We begin with the KSZ effect from galaxy clusters.Galaxy clusters contain a significant amount of ionizedgas. Since they are effectively individual collapsed ob-jects, they can be used to sample the dipole anisotropyinduced by a local inhomogeneity at discrete points inspace. This is useful to reconstruct the systematic trendin dipole anisotropy as a function of redshift that a localinhomogeneity produces. Each cluster has a character- FIG. 2: The KSZ power D (cid:96) = (cid:96) ( (cid:96) + 1) C (cid:96) T / π at (cid:96) = 3000,in µ K . The thick red line is the SPT upper limit of D < . µ K (95% CL). istic integrated optical depth of τ ∼ − − − . TheKSZ signal due to a single galaxy cluster at redshift z is∆ T /T = − β ( z ) τ , and can be extracted from CMB skymaps given a sufficiently accurate component separationmethod and low-noise data.The KSZ effect from individual clusters is difficult tomeasure owing to the smallness of the signal, confusionwith primary CMB anisotropies, and other dominant sys-tematic errors. Currently, only upper limits are available,but this is likely to change as data from Planck and small-scale CMB experiments such as ACT and SPT becomeavailable. Current data have nevertheless been used toconstrain inhomogeneous relativistic cosmological mod-els for dark energy [21].We now consider the KSZ angular power spectrumfrom gas in the intergalactic medium. This angular powerspectrum is easier to measure than the KSZ effect fromindividual clusters because it is an integrated quantityand has additional contributions from the diffuse in-tergalactic medium that is not associated with clusters(sometimes called the Ostriker-Vishniac effect [22]). TheKSZ power spectrum from a large inhomogeneity is [23] C (cid:96) ≈ π (cid:90) r re dr r − [ β ( z )( dτ /dr )] P ( k ( r ) , z ( r )) . (2)The Limber approximation has been used, giving k ( r ) =(2 (cid:96) +1) / r ( z ). We model the distribution of scatterers inthe late Universe with dτ /dz ∝ σ T f b ρ ( z ) /H ( z ) [23], andtake reionization to be an abrupt transition at z re . Athigh (cid:96) , the KSZ signal is strongly dependent on the non-linear matter power spectrum, P ( k, z ), which we modelusing HaloFit/CLASS [24]. Results for our toy model areshown in Fig. 2.At a characteristic angular scale of (cid:96) ∼ FIG. 3: The Compton- y distortion induced by the inhomo-geneity. Also plotted is the projected upper limit from PIXIE(thick blue line). ter inhomogeneities at lower redshifts, where the inducedanisotropy is mostly dipolar. These scales are accessi-ble to CMB experiments such as ACT and SPT, whichhave recently put stringent upper limits on the combinedTSZ+KSZ power [25]. Accessing the bare KSZ signal iscomplicated by difficulties in modeling the distributionof extragalactic point sources [26], and contains a the-oretical uncertainty due to the unknown ‘patchiness’ ofreionization, which also contributes a KSZ effect [27].We now turn to the Compton- y distortion inducedby the inhomogeneity. Spectral distortions arising fromthe Compton scattering of an anisotropic CMB canbe parametrized as a Compton- y blackbody distortion.When the dipole dominates, the observed Compton- y dis-tortion is a monopole [23], y = (7 / (cid:90) r re dr ( dτ /dr ) β ( r ) . (3)Results for our model are shown in Fig. 3.Measurement of the Compton- y distortion requires aninstrument for which an absolute calibration of the spec-tral response can be obtained. This excludes most recentCMB experiments, and so the best current constraintscome from COBE/FIRAS [28]. The planned PIXIE mis-sion [29] could improve the determination of y by somefour orders of magnitude.Our toy-model calculations give some sense of the effec-tiveness of the different spectral-distortion tests in con-straining the size of a local inhomogeneity. A depth ofΦ ∼ × − is sufficient to induce a bias in the in-ferred spatial curvature of ∆Ω K ≈ − for a wide rangeof r (Fig. 1). Existing upper limits on the KSZ powerat (cid:96) = 3000 from SPT are sufficient to rule out an inho-mogeneity of this depth with a width less than around 5Gpc, although larger r are still allowed (Fig. 2). TheCompton- y distortion, on the other hand, provides muchweaker constraints even with the great increase in preci-sion that would be possible with PIXIE (Fig. 3). Part ofthe reason for the relative effectiveness of the KSZ powerspectrum is its density weighting, which enhances the sig-nal at the high (cid:96) probed by precision CMB experiments.The above calculations are only intended to be illus-trative, and more detailed modeling would be required toproduce firmer constraints. For example, the KSZ angu-lar power spectrum is sensitive to the non-linear contribu-tions to P ( k ) [23], and the form of β ( z ) to the shape of thepotential, Φ( r ), so uncertainties in these functions shouldbe treated carefully. For wider inhomogeneities, there isalso a (relatively minor) dependence on the details ofreionization. Finally, the assumption that the inhomo-geneity is perfectly spherically symmetric, and that weare exactly at its center, should also be relaxed. A realis-tic inhomogeneity cannot be too asymmetric, or place ustoo far from the center, however, without violating limitson isotropy, moderating a CMB dipole that is observedlocally [30], or inducing a CMB statistical anisotropy [8].Why should we expect to find ourselves near to thecenter of a large inhomogeneity in the first place? Al-though such a situation may seem unlikely [30], there areinflationary mechanisms known in the literature whichpreferentially place observers near the center of large un-derdensities [31]. Furthermore, Ellis [32] has argued thatit would be inconsistent to rule out such inhomogeneitieson strictly a priori probabilistic grounds, since we cur-rently accept features in our cosmological models thatare substantially less probable anyway. As such, obser-vations should be the final arbiter in deciding whether alarge inhomogeneity exists or not.Wouldn’t its presence have already been discoveredthrough other observational probes? Inhomogeneities ofthe kind considered here modify the low- (cid:96) CMB, causingalignment of low- (cid:96) multipoles [33], and changes in theISW signal, temperature-polarization cross-spectrum,and associated modifications to the reionization history[34]. Unfortunately, the induced effects tend either tobe smaller than cosmic variance at the relevant scales, orstrongly dependent on the details of the model, rendering these tests inconclusive.Superhorizon perturbations also produce fluctuationsin the low- (cid:96)
CMB through the Grishchuk-Zel’dovich ef-fect [11]. In a number of cosmological models (includingΛCDM), it has been shown that there is a cancellationbetween the anisotropy and peculiar velocity induced bysuch perturbations, resulting in no net dipole to first or-der [12]. Constraints from the low- (cid:96)
CMB are thereforecomplementary to spectral-distortion tests of the sortoutlined above: A deviation from spatial flatness causedby a local inhomogeneity results in a net dipole aboutmany locations within our horizon, which can be mea-sured using, e.g., the KSZ effect, whereas a superhori-zon deviation from flatness will produce no such signal,instead causing an enhancement of the quadrupole andhigher moments of our local CMB.In conclusion, an observation of | Ω K | (cid:38) − wouldhave considerable implications for inflation but wouldnot, on its own, be sufficient to rule out eternal inflation.It would also have to be shown that the inferred devia-tion from flatness was not caused by the effects of a localinhomogeneity instead. Observations of CMB spectraldistortions such as the KSZ effect and Compton- y distor-tion, taken with constraints on the size of the Grishchuk-Zel’dovich effect from the low- (cid:96) CMB power spectrum,present a viable method to constrain the source of a seem-ing departure from flatness.We thank P. G. Ferreira for useful discussions. PBacknowledges the support of the STFC. MK was sup-ported by DoE SC-0008108 and NASA NNX12AE86G.The computer code used in this paper is available onlineat , and makes useof the CosmoloPy package ( http://roban.github.com/CosmoloPy ). Note added:
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