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The cold spot as a large : Rees-Sciama effect on CMB power spectrum and bispectrum

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Received August 22, 2008 Revised December 18, 2008 Accepted January 18, 2009 Published February 13, 2009 Abstract. Isabella Masina and Alessio Notari The cold spot asRees-Sciama a effect large on void: CMBspectrum power and bispectrum J of an anomalously large spherical underdense region (with r JCAP02(2009)019 – 5 7 9 1 3 8 4 11 13 10 15 16 17 [ ies of ] has claimed e northern and 10 ] for a paper that that appears to be 11 ◦ ould be due to such an t 10 n good agreement with the been pointed out by several ling it through an inhomoge- tead be due to an anomalously n the line of sight between us Some of them are localized at tic fluctuations, fully described lactic Radio Sources (the NVSS uires an overdense compensating appear from Gaussian primordial ch objects is that an underdense ignment of the low multipoles (for 1 (see, however [ ], if we happen to live near its centre. ∼ 12 z ]. So, while this could still be a statistical 6 – ]. We may also remind that [ 4 8 – 1 – could be enough to give an acceptable fit to the , 7 8% [ /h . measurements NL f ]. Another anomaly is the presence of the so-called Cold Spot 3 ] experiment has measured with great accuracy the anisotrop 1 ] and references therein) and the power asymmetry between th 2 In this paper we also take the point of view that the Cold Spot c 4.1 Non-gaussianity4.2 from RS effect Contamination of 2.2 Decomposition into spherical harmonics 3.1 Rees-Sciama power spectrum 2.1 Temperature profile ], which is a large circular region on an angular scale of abou 6 A Vanishing of the P-P-RS contribution to the bispectrum 1 Introduction The recent WMAP [ 5 Other voids 6 Conclusions 3 Power spectrum 4 Bispectrum Contents 1 Introduction 2 A void in the line of sight southern hemispheres [ challenges this claim). Anotherregion motivation of for two studying or su three hundreds of Mpc underdense region, customarily denoted (LTB)neous as metric Lemaˆıtre-Tolman-Bondi (which a also “Void”. req By mode the Cosmic Backgroundexpectations (CMB), from whose inflation features of areby a i Gaussian a spectrum nearly ofauthors scale-invariant adiaba power that spectrum. the datavery large However, contain angular it various scales, such has unexpected asa the features. review low see quadrupole, [ the al fluke, some authors have putlarge forward the spherical idea underdense that it regionand could of the ins some Last unknownthat, Scattering origin, looking Surface at o the (LSS) directionsurvey), [ of an the underdense Cold region Spot is in the visible Extraga at redshift Supernova data and the CMB without [ anomalously cold: the probability thatfluctuations such a is pattern would estimated to be about 1 JCAP02(2009)019 e ]. |∼ 15 ] has 18 T/T ]. ∆ | 14 . For Void 3 ) 0 LH ( , which would be /h ], with the possibility ∝− he comoving radius of : this would be at odds 21 – ome blue-shift due to the sent . s, it is interesting to note /h ter. Hence, here we focus ] there are many localized d be significant only when T/T 19 00Mpc nd briefly comment on the ond physical effect due to a anomalous Voids could well onal quantities, focusing on mation of the cosmological it be detectable also in the 17 he issue of the primordial ori- , id, does the power spectrum wo-point correlation function ,∆ rature anisotropy of is much smaller than the total he photons: the lensing of the nventional structure formation pe [ ge Scale Structures: as [ 0 fere with the measurement of a that they are located along the sian primordial fluctuations (at aussian features in the primor- l temperature fluctuations [ 16 ppear as Voids in the sky today. (the bispectrum). This is inter- ture formation scenarios, the RS H probability of having large Voids. be an effect already at the linear in a companion paper [ nt in time — the so-called Rees- r example, one could consider cer- SW) effect, which however would volume and time) compared to the 300)Mpc rtant piece of information in order at odds with the usual structure for- − s could generate a coherent spherical region on imordial Gaussian spectrum, and which would (200 ∼ L – 2 – This is an important point because a Void at the LSS 1 ]. By RS effect we mean the one associated to the variation of th 13 ]). We may also mention that according to [ 8 [ σ times the present value of the Hubble parameter L It is well known that, passing through a Void, photons suffer s It is also known that the RS effect scales as the third power of t In the case of an explanation of the Voids via tunneling event , the Void should be of very large size, i.e. 5 It is possible, also, to imagine that some primordial proces 1 − the statistical properties(power of spectrum) the and CMB: theesting in three-point for correlation particular the function onget following the a reasons. t sizableparameters First, correction? is if affected? And, therebispectrum? if is Second, Third, such yes, if does a there the toprimordial presence is non-gaussianity, Vo what which of such constitutes such extent a a anto the very object Void, distinguish impo esti inter could between models of inflation? shell), we compute the impact of such a Void on some observati with the standard scenariomore of than structure 10 formation from Gaus recently stressed the existence ofmation these scenario. regions is Even already though wegin do of not such address large in objects,tain this we models paper mention of t some inflation, possibilities. such Fo as the ones of the “extended” ty In order to produce a signal comparable to the measured tempe regions in the sky (both underdense and overdense) of about 1 of tunneling events and nucleationAlternatively, of one bubbles, could which would also a dial imagine fluctuations which the would presence seedhistories of a in large non-G the Void late-time or Universe even which non-co could enhance the fact that the gravitationalSciama potential (RS) is effect notgravitational [ exactly potential from consta non-linearlevel, effects. usually There referred can tovanish in as a the matter Integrateda dominated Sachs-Wolfe Dark flat (I Energy Universe. component Theour becomes ISW attention dominant effect with only woul respect onextension to to the mat a RS ΛCDM effect,Void Universe on later which the on. is line Note alwaysprimordial of that perturbations, present, sight, there which in a is will addition a be to sec analyzed the in RS detail the blue-shift Void of t that if a nucleation processHubble has rate small probability (at (per some unit be time very during small, inflation) then suchIn the as addition, number having since of just the one LSSvolume is inside or a it, a rather if few thinline there of shell of are them whose sight, only in volume rather few the than Voids pre at it the is LSS. more likely 10 the LSS surface, with small density contrast, on top of the pr sizes compatible with theeffect expectations happens from to the be usual suppressed struc with respect to the primordia responsible for the correlations between the CMB and the Lar JCAP02(2009)019 . 4 6 the (2.2) (2.1) by: ]. As . 23 plus K 73 . and that the that would be = 2 , while section ) 3 i NL ( f ¯ T e drawn in section i t completely homoge- ize would be strongly we extend some of the P in of the Void, relatively 5 eas, such as the idea that = tion of n section e physical characteristics of the temperature fluctuation ties. T ecause of the existence of one er size and less visible — see , cal Void located at comoving lar a cosmic texture [ m the same physical process be lensed by it, leading to an present on the LSS that will the line of sight. Our aim is similar fashion if the origin of d ) ction the CMB photons, whose fluc- ration, the observer detects one L ( r the following cosmological con- ee-point correlation functions, in act, it is conceivable that if there /T and Gaussian (as those generated T ) T P ∆ ( ) ted perturbations on the LSS i T ( + ¯ ) T we calculate the RS-induced temperature RS T − 2 ( ) i T ( T T ∆ – 3 – + ≡ ) ) i P ( ( T T T T ∆ ∆ , where (RS) stands for Rees-Sciama. As a second step, = ]. On the contrary, Voids on the line of sight are not subject /T ) 22 T T , RS ∆ ( 20 )-labeled fluctuation is given by . So, finally, the total temperature fluctuation will be given T [ i /T /h ) L ( T 4Mpc . fluctuation. However, we do not consider here such possibili L 5 from him. We assume that the object does not intersect the LSS − ] for some other candidates in the CMB sky. Our conclusions ar D 17 , As a first step we consider that the Universe and the LSS are jus The paper is organized as follows. In section Note also that other authors have put forward alternative id At this point it is important to be more precise about the orig 16 secondary effects due toto give this the anomalous theoretical structureorder prediction to located for compare in the them two-point with and the thr observations. observer is not insidetuations it. we The assume observer to receivese.g. be from by adiabatic, the the nearly usual LSS scale-invariant inflationaryparticular mechanism). realization of Given the this primordial configu inflation-genera done by neglecting the presence of such a Void. Finally, in se neous, and study what kindinhomogeneous of Void profile located we between should us seeobtained and in in the the LSS: sky, this b we way denote as ∆ produce a 10 deals with the impact on the bispectrum and on the overestima constrained: would have a huge impact on the CMB and, as a consequence, its s we consider that there are Primordial fluctuations ∆ we are going tothe discuss, Cold some Spot of our is considerations a apply texture, in rather a than a Void. the Cold Spot. The impact on the power spectrum is discussedconsiderations i to the caseis of several one objects such in structure,e.g. the there [ sky. may In be f other ones, maybe of small 2 A void inAs the anticipated in line the of introduction,figuration: sight we an would observer like looking todistance at conside the CMB through one spheri with the bar representing the angular average over the sky an profile of the CMB photons passing through a large Void with th to the latter constraint. where by definition each ( also be affected byadditional the effect, presence ∆ of the Void: in fact, they will the Cold Spot could be due to a , in particu to the Primordial fluctuations. If such a structure comes fro JCAP02(2009)019 ≃ are and ] as T ; the σ 23 A D mostly ] and the A and 9 , 8 A is close to 1. An e choice of the M e RS fluctuation , or equivalently ons coming from the centre mordial Gaussian . This is probably r term (RS effect) is A µK 1, the linear ISW would te the RS (and Lens- 80) location of the Void in such an inhomogeneous ≪ . As for the angular size 0 ± el, in addition to the RS the Void radius, but with δ A : the amplitude he parameters at there is some correlation 0 B metric, which is matched bsence of dark energy. The is large or if Ω lightly different from theirs, he Cold Spot area is ∆ ) and only two observational 0 0 leads to an estimation for the 2 y choosing some shape for the (190 δ = 1. Including a dark energy . e range of values of interest. In δ ] corresponds to the profile due en studied e.g. by [ ] the entire cold region extends 0 − mption since, in the presence of from a different process, such as son-Walker (FLRW) flat model. 6 δ ues of 23 ], a shape that fits well the Cold M D,L,δ 6 = a Void via the following quantities: d be generically enhanced. and , its present-day density contrast at , T 5 L L inates if of the Cold region. We also need some nsity contrast, , we rely on the values given by [ σ , while it is is not very sensitive to D 0 A δ – 4 – . As explained in the next paragraph, we choose the ] use a temperature profile for the secondary effect, add for the very cold part, but we also show in all plots the 4 L/D ◦ from us, its comoving radius and the density contrast D L : in fact as one can see in figure1 of [ ). ◦ dependence is different in the presence of Λ: the second-orde σ phenomenologically. Clearly this leaves a degeneracy in th 0 . In this paper we assume the latter range of values for δ σ 5 . The authors of [ = 18 − correlated at all with the Primordial temperature fluctuati and /T σ 10 ) , and some shape for the density profile. As we will discuss, th and A depends on the ratio P 0 ], but this value includes as well a contribution from the Pri × ( turns out to be described by two parameters: its amplitude at not δ 4 A [ σ T 3) /T ± ) µK In order to fix a range for the amplitude Then, we need to fix the properties of the Void in order to compu Having fixed the numerical values, we choose a metric to model RS ( Specifically, the 2 = (7 of the profile, we proceed as follows. According to [ T 400 not entirely accurate (thesince we shape consider of a compensated the Void, profile while that the profile we in use [ is s A suppressed, but there is a non-zero linear term. For small de dominate in a Λ-dominated cosmology. The quadratic term dom physical parameters, since we have three of them ( values for σ constraints ( diameter between the two.the However sky we is will assume, from now on, that the as the Primordial Gaussian spectrum, then this would mean th − its angular extension, characterized by the diameter value of the secondary effect at the centre, given by ∆ Spot has a diameterextreme of case about 10 up to such large size. component would lead toeffect that an we ISW consider, effecteffect of which already a is at cosmological present constant the in in linear an any analogous lev case, setup has even be in a the Gaussian fluctuation and fit the observational data: this profile, ∆ to a cosmic texture), butany it case, should our give results a can good be estimate for easily th rescaled for different val two effects turn outsome to difference be on similar, the dependence with on the the same density dependence contrast on follows. The minimal temperature in real space observed in t related to the physical parameters of the Void, namely shape for the temperaturedensity profile, profile which of will the be Void. determined b Let us discuss in more detail how t ing) contributions. Physically, one mayits characterize comoving such distance inflation. This is true,nucleation for of example, bubbles. if such Notea structures that correlation, come the this temperature is correlation a functions conservative woul assu depends on the radius the centre region. The choiceto that a we make homogeneous is andFor a isotropic simplicity spherically Friedman-Lemaˆıtre-Robert we symmetric consider LT a flat FLRW Universe with Ω ∆ JCAP02(2009)019 ) r 2.5 he (2.5) (2.3) (2.4) (2.6) ) is small. r ( . From ( k α , ′ ) is small. The ) , r r ) is approximately 0 ( . As a function of ) r 6 k r ( / ( k 1 LH k − ( 6 , ure and determines the ) 2 / oid, with the underdense is the present value of the  1 π is the average density in τ ) j arge even if 0 3 1 ρ π / mes e.g. from a primordial ong as ) as follows: H hes to a flat FLRW universe 2 dx of the density profile at the ional dust, which describes a r i ) 3 ntial Φ given by the following = (6 ( egion. This is a requirement / 2(6 π 4 k um and bispectrum comes from act a bubble would have a thin 0 dx he condition that the Void does e choice of 3 ¯ τ r , = ture, the angular size of the entire ower energy contained in the true rd 10(2 L 2Φ) )¯ r r − ≡ (¯ ≤ k ) r r ( + (1 Z , since we have a hot region as well. In any 2 3 σ we now compute its shape in the following , the gravitational potential is given by: / r 2 3 dτ ) / ] that this metric can be treated, in some cases, 1 , B π /T – 5 – ) ) 24 , r r 9 3 ) ( RS 2 5 (2 r ( B (  T − α (1 + 2Φ) B ]: are dimensionless comoving coordinates. We have chosen  − 1+ i  24 L )= r x r ) − r τ  ( = Φ( 2 − a ¯ ρ 1 = −  ¯ ρ 2 ) 0 (0) = 0, where the prime denotes a derivative with respect to t ′ r k k ( ds ρ the density profiles corresponding to certain values of )= 1 ≡ r ( ) k r is the dimensionless radius of the LTB patch and ( δ L r ) = 0). We have chosen arbitrarily the function L 1, ( ) is the matter energy density inside the LTB patch, ¯ ) is an arbitrary function which represents the local curvat k r r is the conformal time and ( ( ρ k τ α> )= Another important point is that our profile is a compensated V Given these specifications for ∆ L ( ′ k given by the following expression [ only constraints oncentre this (which function dictates comes from the smoothness Hubble parameter. The density contrast that results from th where coordinate) and from the( requirement that the LTB patch matc where shape of the density profile. This approximation is valid as l as a perturbation of an(Newtonian FLRW gauge) metric expression with a gravitational pote the inner underdense region. case, we show that the main contribution to the power-spectr central region surroundeddictated by by a the matching thinnernot conditions, external which distort corresponds overdense the to r bubble outer t of FLRW true metric. vacuum,wall this with Physically, is localized if a gradient the consistentvacuum energy, requirement: in compensating Void the for co in interior the region. f LTB l Note patch that, that because we of consider this fea will be larger than where units such that the present value of the conformal time is subsections. 2.1 Temperature profile First of all, wecompensated model Void. the It profile has as been an shown LTB metric in with [ irrotat the dimensionless comoving radial coordinate where We show in figure the outer FLRW region. Note that the density contrast can be l JCAP02(2009)019 0 2. δ . 0 (2.8) (2.7) (2.9) − factor (2.10) . = ) prefactor A 0 0 0 δ δ , we may ] which is πδ − π 26 2 20 3(1 , 1. = . 25 0 φ < of the LTB patch, k , L L ≤ urse the integrand is ]: between the observer L D . . 26 is directly related to r/r D , ≡  , 0 ) = 4 0 k 1 L r , 24 and 0 (˜ θ α ′′ 1. The profile has no depen-  π LH ). The conformal time evolves 0 − f the Void): )Φ for tan ession given in [ Υ ≤ r : the radius (˜ ′ 0.8 for a photon which travels through θ 21 10 ], which are of about 0 Φ 9 , , (0) = , r = 4 − ≤ 8 f refers to the observer space-time point. 2  parameter) comes in the numerical factor and L L ′ α T/T 0 θ α + 2 ˜ Φ O 0.6 ses [ and the distance ), normalized for a central value of 2 3): the two effects are roughly compensating and 6 1 2 ≥ . ′ r . ) 0 ( 0 DH  θ θ<θ M δ δ r ≈ ) analytically along a radial trajectory, and it = 2 − Φ(˜ 0 = 2). 1 α δ dττ  0.4 – 6 – 2.7 at leading order in α ˜  ) if r E 3 τ θ d O 2 0 ( τ δ ¯ r L Z 3 r ) 0 if = 4 ( Z 0 0.2 A f α ¯ 2 r axis to point towards the centre of the Void. The = 2 ¯ ( r d z LH ) , which ranges between 0.05 and 1.85. This variability in the r ( L ) r RS )= Z ( ) is normalized so that 0 , where the subscript +27 θ ) we can compute the ∆ T − α O ( 0 r T τ θ, φ f 0.2 0.2 0.8 0.6 0.4 26624 51205 ∆ ( = +90 - ) 2 ) 0 = )+ α 4 r RS Υ O α ( ( A r δ T +105 104 3 T − α ∆ r +50 4 α ) = ( 8 ( r 7 ( because we choose the ˆ τ is the value of the conformal time at emission (although of co φ is a non-local quantity, which is given in terms of Φ as [ : Plot of the present day density contrast ) it follows that the value of the normalization constant E 0 τ 2.6 Given the potential Φ( Then, using the spherical coordinates angles, 0 The dependence on the shape of the density profile (the 3 this patch, by computing a line integral, following the expr parameterize the temperature profile as interesting case is the one for intermediate values ( where nonzero only in thesimply region as where the LTB structure is located valid at second order in perturbation theory in Φ: is a function of the physical parameters of the configuration the density contrast atand the the centre centre of of the the Void. Void The result is, Here Υ could account for the prefactors obtained by previous analy it is given by the calculation does not depend much on the value of Ω Figure 1 where the profile function (the value of the present day density contrast at the centre o The solid (dashed) line corresponds to and ( dence on can be computed performing the integral ( JCAP02(2009)019 ) ◦ θ we , ( = 4 axis as a f 2 α (2.11) (2.12) = 18 z = 4 = 2 /A 1 at the α α σ ) − 0 RS ℓ ( for for = a = 0 are non- , 12 , the linear ISW can m , 0 T/T x δ 12 spect to the ˆ x 601 . θ which shows, for com- 2 940 . 2 − 20° ompute the function e profile, while the texture 10 +2 x 10 x rovides the best fit to the Cold = 4 . 904 we show the ratio ) . order corrections). In figure α ˆ n 361 . For the reader’s convenience, we 3 coefficients with ( . α 15° ∗ +17 ℓm 19 8 ℓm Y − a x 8 ) ). The light solid line shows, for comparison, 1 at the centre) of the texture that gives the ◦ x ˆ n texture 190 ( − . )-th profile for the temperature anisotropy ) ] that this case does not lead to a hot ring. i i 8 T ( 959 = . 46 = 16 10° T is a weak correction, unless the patch happens − σ . The dark solid (dashed) line corresponds to ∆ ( – 7 – 6 ◦ D = 2 T/T +33 x ◦ ˆ n 6 α d x ]. for two values of 272 = 2), hence to a Cold Spot with diameter = 20 . = 18 Z θ 23 5° L 258 α σ . θ ≡ +58 17 ) 4 i ( ℓm − x ) along a non-radial trajectory (computed as an unperturbed ) for a 4 = 4 ( θ x ( 2.7 576 α f . The dark solid (dashed) curve corresponds to a temperature . ℓ 1 0 954 . 37 ] (its profile has also been normalized so that ∆ - 0.4 0.6 0.8 0.2 0.4 0.2 5 − and - - - - 23 − 2 ) ◦ 2 x θ x , however: in a Λ dominated cosmology and with small contrast ( ) as a function of ) is defined as: θ f 191 ( . 4 . We have added also a light solid line in figure 663 = 20 2.1 f . L caveat L θ of ( θ/θ 1+6 1+11 − − ≡ : Plot of the profile /T ) x ˆ n ( )= )= For the (RS) component, since the profile is symmetric with re ) θ θ i ( ( ( There is one = 2) and a Cold Spot with diameter 4 f f T α vanishing (and they are real). In the left plot of figure pointing towards the centre of the Void, only the provide polynomial interpolations of the profiles: 2.2 Decomposition intoThe spherical spherical harmonics harmonic decomposition of the ( to be located at distances comparable to the horizon. We can c Figure 2 centre). Notice thatdoes the not compensated have Void it. has a hot ring in th ∆ by numerical integration of ( profile with straight line, since any deviation would lead only to higher parison, the temperature profileSpot, of as the cosmic discussed texture in that [ p Note that the dependence on the distance plot the profile where function of the multipole ( become the dominant effect; and it has been shown by [ the temperature profile (normalizedbest so fit that to ∆ the Cold Spot, as claimed in [ JCAP02(2009)019 ]. ℓ 23 80 (3.1) ) in our , so that 2 ◦ = 4 (solid rposes the and 1% for , where the equal to the α i 60 ); dashed lines ◦ ℓ ◦ = 11 that one would C 40. h L and (10 θ ◦ /A = 20 . coefficients. As an ◦ 0 ℓ ℓ ) L 40 a θ ll (see figure . = 18 0 RS tead ℓ ( = 20 σ a or non-compensated տ L coefficients, the two-point θ he compensated տ effect mainly comes from the rature profile only to the cold 20 tty similar. ℓm best fit to the Cold Spot [ ց a , the ratio with of the density profile only to the texture 20 and 15 ’s coefficients as nse compensating shell. One can ) ℓ axis aligned with the centre of the . C the cold region (Void). The light curve 0 ). Right: The solid line corresponds to RS z ℓ ( ℓ ◦ set of . a 0 (9 . 2 ◦ | 0.01 0.02 0.03 0.04 0.05 0.01 + 1 - - - - - ) ℓm ℓ , while the dashed line to the non-compensated a 0 RS = 16 A | 2 ◦ ( ℓ ℓ a σ a single − and, in order to face it with the experimentally ℓ ’s do not depend on the choice of the coordinate = . Left: The dark (light) solid line corresponds to a = 20 – 8 – ℓ X ]. ) ℓ ℓ coefficients one can compute the associated two- m L L C ( ℓm 23 θ 80 = 2 ≡ a = 4 ℓm ℓ = 2 and a α we focus on the α + that one would obtain from the temperature profile of the C ) α 3 /A RS ), namely a Cold Spot with diameter ℓm ( 0 = 4 and 60 ◦ ℓ solid: a a α ◦ dashed: + (11 ◦ ) as a function of P = 11 ( ℓm a L 40 ◦ = 20 /A θ ) = L , we have to estimate the theoretical prediction for θ 0 RS ℓ ℓ ( = 20 ℓm are now suppressed but still quite similar, so that for our pu a C a L ) θ 20 ). Notice that the medium amplitude of such a ratio is roughly 0 RS ◦ ℓ ( = 4 and a = 9 α σ : Plots of ); the light solid (dashed) curve is obtained by choosing ins , respectively concentrated at multipoles 10 ( ◦ ◦ 0 ◦ One may wonder whether the presence of a compensating hot she 0.01 0.02 0.03 0.04 0.05 0.01 ) - - - - - = 16 = 11 = 10 0 RS A ( ℓ L σ a profile with Figure 3 Void. In our case observed value of system on the sphere. Therefore we are free to keep our ˆ correspond to similar profiles but with profile has a significant impact on the magnitude and shape of t Notice that the texture and the non-compensated Void are3 pre Power spectrum Given a temperature profile with its ( example, in the right plot of figure point correlation function. In general, given Note that this definition ensures that the correlation function (power spectrum) is defined via the line), and show how theypart would change (dashed by line). truncating the Thisunderdense tempe approximately mimics region a (the truncation true Void),see disregarding that the the overde the full compensated profile with σ obtain from the temperature profile of the texture giving the result is not going tounderdense be region. very The different. same plot This also shows shows, that for the comparison RS texture giving the best fit to the Cold Spot [ θ fraction of the sky covered by our LTB patch, namely about 3% f profile, namely the previousshows, one for truncated so comparison, to the include ratio only JCAP02(2009)019 ) ], ∝ ± ) 23 2.1 (3.4) (3.3) (3.2) RS . The = (7 ( ℓ 2 C A re of [ erate Primor- 6= 0. Hence the RS , is just: of the fit, which gets ) m ameter estimation due 2 cosmic variance is the range, namely RS) component in ( ’s. χ RS ℓ he RS correction is of the ( Cold Spot, the correction ℓ σ uding the cosmic variance , and the P-L contribution ion functions are given by: C he 2 | =0 if ) + of possible realizations of the tatistical analysis of the CMB with respect to the Void case. , correlated observers. ts on the power spectrum due ) 50 and its magnitude is about ans that it can be factored out i ) i RS fferent ) ℓm ( in its 1- P . RS ( P ℓm ( a ℓ 1 | ( ℓ ℓ a A C C h h . , . 2 = 2 ), we get two types of non-zero contri- ensemble 2 m ) i | 1 ℓ 2 ) ℓ + 1). For comparison, we also show in the m RS σ ℓ C + 1 ( 2.12 ℓ δ 0 h RS 2 ℓ ( ℓ (2 C ℓ 2 a 1 / -direction, | ℓ can be extracted just by noticing that ℓ receives corrections; we have plotted respectively the quantities 2 ]. For the purpose of this paper, we may just z δ ≤ 2 i X A = – 9 – 27 p ) 4 = ) and ( i ) P ) = ( i ℓ P 2 ∗ 2 RS ( 2.1 C ℓ ( ) ℓ m h χ 2 C P C ℓ ( h ∆ = a ]. 1 ℓ = ) m C 14 1 , for the range of parameters mentioned in section P P ℓ ( ℓ 2 0 C a T h . The first effect is computed in the next subsection, while the π +1) i 2 ℓ ) ( L ℓ ( ℓm i a ℓ ∗ the correction to the power spectrum obtained from the textu C ) h P 4 ( ℓm are predicted by some mechanism (i.e. inflation) that can gen a h i : the RS-RS contribution proportional to ) and i ℓ P ( 2 ℓ 0 C h T C h is the variance of the two-point correlation function, incl . The results for different values of π +1) 2 ℓ 5 2 ℓ ( σ − ℓ 25% with respect to the Primordial signal. We also note that t brackets stand for a statistical average over an In order to see how large is the impact on the cosmological par As we have already stressed, we assume that the Rees-Sciama ( For a Primordial and Gaussian signal the two-point correlat ) 10 i − . As one can see, for a Void with a size which can account for the RS × (i) the inflationary prediction ( 2 ℓ (ii) there is also some non-diagonal correlation between di . . . contribution to the power spectrum coefficients, second in the companion paper [ where h and the instrumental noise. For an experiment like WMAP, the proportional to In the left and right panels of figure the RS effect from adata, large Void, which one we should perform postpone a for detailed s future work [ which turns out to be smaller and shifted at lower multipoles shaded regions are indeed obtained varying the amplitude C 3.1 Rees-Sciama powerAs spectrum we have seen, for a spherical Void in the ˆ estimate the impact thatincreased the by inclusion the amount: of the RS effect has on t butions for left panel of figure is uncorrelated with the Primordialof (P) the component, brackets. which Given me this fact, from ( Universe — or, equivalently, an average over many distant un where the 5% 3) A dial Gaussian fluctuations. Then,to there our secondary are effects: two types of effec order of the cosmic variance, ∆ to the power spectrum is non-zero only in the range 5 JCAP02(2009)019 ℓ esti- ), for (4.1) (3.5) gular ◦ 5050 (10 ◦ and the RS = 18 L σ 2020 asons: i) we deal . onal structures, i.e. ) 1010 RS ( ℓ C ]. In the right panel the RS . 50 and is of the same order 55 ents, also to check whether 23 not the case in the conven- ◦ ◦ − contribution to the two-point . rum, ures seeded by the Primordial , nsity of the Void itself, which is 3 = 10 = 18 um is the main tool used to make m exture [ 3 22 σ σ data points are also shown. ℓ ] the associated Lensing effect. ing to see whether the presence of a a the Cold Spot equal to ; ii) we are assuming no correlations m the usual ΛCDM concordance model; 14 3 2

800800 600600 400400 200200

14001400 12001200 10001000

π 2 L

0

ℓ m

] µK [ T i C

h 2

2 2 +1) ℓ ( ℓ ℓ 7 for 6 for . . a , has already been studied by several au- coefficients, defined as 0 3 1 ’s in the range 5 3 /h ( ℓ m 3 m 1 ℓ 2 ℓ ≃ 2 a ’s from the Rees-Sciama effect. The dark blue (light m – 10 – ℓ 2 ) 1 χ 1 m ℓ RS ≡ ℓ ( ℓ ∆ ◦ B 3 3 4 C ℓ . For comparison, we also show (light orange) the analogous m ’s, because we consider larger values of 5  2 2 − ℓ 5 = 10 5050 m ℓm ’s, so that we can just neglect the instrumental noise and ◦ 1 − 1 a 10 σ ℓ ℓ m 10 × A B = 18 × 3) σ 7 coefficients and the two-point correlation functions, we now ± 2020  ℓm . The sum gives roughly the following result, for different an a = (7 1010 A +1 2 ℓ 2 2 i ) 55 P

(

ℓ texture ]. However, the calculation here is quite different for two re of the Void: C h 29 σ , : In the left panel we plot the = 22 28 2 ℓ The basic quantities are now the Finally, as already mentioned, there is also a non-diagonal Note that the bispectrum from the the RS effect due to conventi

5050

σ

250250 200200 150150 100100 300300

π 2 ℓ

0

C T µK [ ]

2 2 +1) ℓ ( ℓ ) RS ( effect on the temperature profile is proportional to with much larger values for the between the RS andtional the calculation Primordial where fluctuations, thefluctuations themselves. RS while In effect this addition, arises is we from compute the in struct [ green) shaded region corresponds to an angular diameter for Figure 4 correction is added tothe the cosmic variance best-fit (gray Primordial band) spectrum, and fro the experimental binned dominant source of error at low- diameters correlation function, which correlates different a range of amplitudes of magnitude as the diagonal contribution to the power spect correction obtained in the case that the Cold Spot is due to a t use 4 Bispectrum Having computed the mate the impact thatthis a observable large could Void be has useda on to highly the constrain non-Gaussian bispectrum the object. size coeffici detections and Moreover, of since the the a de bispectr primordiallarge non-gaussianity, Void it can is contaminate interest this detection and towith what extent non-linearities at scales of order 10 Mpc thors [ JCAP02(2009)019 . , ) i 4 ) 0 RS 10 ℓ ( (4.2) (4.5) (4.6) (4.4) (4.3) RS tions a ( terms ∼ a i ) 3-j sym- 2 L . ) ( refore, for ) NL -axis along a L f ) z ( P a ( ), one realizes ( ]. a ) h 14 4.2 . , 2000 respectively, RS : i ( 3 ’s coefficients, one i 3 ℓ ℓ a 3 ∼ ) 2 h ℓ coefficients are those ℓ ) C 2 ℓ 1 , the potentially very ℓ ) ℓ tatistical average. As Void in the sky is not 1 b RS A ℓ ( ) and ( , RS his assumption, the four a B ℓm ( 3 (  h 3 that the coefficients a h ℓ 2.1 m 3 , . 2 m is carried over all possible ℓ + (5 permutations) 2 -axis, so that these quantities on. i ℓ e second in [ 0 ) 3 800 and z m i 2 with the 3 sponds roughly to 1 ) 1 3 ℓ RS ℓ m ng: ℓ evertheless keep our ˆ ∼ 3 RS m a m e non-gaussianity could be much ℓ ( 1 ons coming from inflation (this is RS 3 0 0 0 ℓ ℓ l ( a B 0 : the equivalent amplitude is very 2 ) 2 a RS ℓ 5  2  m a ) via the following: 2 RS 3 m 3 ℓ ( L 2 0 ℓ ( l 3 and, using ( a 1 m ℓ RS ℓ a ) 1 2 + 1) i 1 2 a ℓ , we can neglect the 2 m 3 ) ) 1 3 m 1 ℓ ℓ RS ℓ m ℓ ℓ ( 1 P P 2 b  ( ℓ ( ℓ a in figure 1 1 a a h 3 1 ℓ h ℓ ℓ ) m  B 2  h ≪ – 11 – 3  RS ℓ 3 3 + 1)(2 π ℓℓℓ ( 3 ) 3 ℓ b ℓ 4 m 1 2 m L 0 0 0 ℓ ℓ ( ,m 2 2 2 2 a 2  ℓ ℓ m m X ,m terms. 1 = 1 1 1 1 i m ℓ 3 + 1)(2 . However, as shown in appendix ) ℓ m ) ℓ m i 2 1 = 2 ℓ   ℓ RS ) RS 1 ( 3 ( 3 3 ) ℓ ℓ (2 a P 2 ) B ,m ,m ( ℓ are actually exactly zero (because of the absence of correla 2 2 L 1 a r ( ℓ term is very simple. The only non-zero ( i a X X ,m ,m ) 2 i B ) = 1 1 ) 3 P ) ) ℓ 3 m m RS ( 2 ℓ ( P ℓ a 2 ( a RS 1 ℓ h = = ( h a ℓ 1 ( ℓ i i ) 60 the RS signal due to the large Void goes rapidly to zero. The B ’s. One can indeed show, by using the properties of the Wigner ) 3 i h B ) ℓ 2 & RS 3 m and ℓ ( ℓ RS 1 2 ℓ a ( ℓ i ℓ h PLRS 1 2 ( B ℓ ) h ) B L h stochastic quantities, which means that the location of the ( = 0, so that: ], that this combination does not depend on the chosen ˆ a ( ) 30 m We are thus interested in evaluating We plot the diagonal contribution Summarizing, we are left with two types of contributions to not RS ( a bols [ are more suitable to make predictions. For convenience, we n the direction of the centre of the Void. correlated at all witha the conservative Primordial assumption: temperature fluctuati ifmore there important, is since sometypes some correlation, of th terms terms would that be potentially non-zero). survive are Under the t ones involvi h experiments like WMAP or Planck, which go up to about are which are coordinate-dependent quantities. So, in analogy introduces the angularly averaged bispectrum However, for high compared to a typical primordial signal, since it corre 4.1 Non-gaussianity fromTo RS compute effect the that it correspondsalready to mentioned, a a sum crucial of assumption 27 that terms, we of make which here 23 is have zero s large terms where the matrix representsvalues the for Wigner 3-j the symbols and the su with It is customary to define a reduced bispectrum between RS and P). Moreover, since only a small subset of the data is affected by the RS contributi We compute the first one in the following subsection, while th with respect to the JCAP02(2009)019 , or /h and 1 in (4.7) (4.8) . ◦ i ) t for the 1: for a P ( ℓ = 18 C > 200Mpc S/N > σ ≥ ≃ h RS ) , this is defined ℓ L i C , for ℓ S/N , instance, for a Void the region of physical 2 ) ℓ 3 we show the result for 6 3 ℓ l s detectable or not. The 2 2 ◦ 6 ℓ ) l Spot might be subject to i 1 1 ( 2 l ℓ e σ B the noise becomes dominant, ( = 10 50 , namely ( d. For instance, from the left for such a signal in the WMAP ] would give rise to , σ teresting constraints on the size 3 e: so we have to sum over all the max max ℓ l ◦ 23 ℓ 2 ≤ ℓ 3 1 l ℓ ≤ = 18 2 ∆ X l i σ ≤ 3 ℓ 1 20 l ≤ 2 ihC 2 ℓ = ) ratio. For a signal labeled by as a function of the multipole texture ii ihC – 12 – ) 1 ’s are different, if only two of them are equal or if 10 ℓ ℓ S/N RS ) ℓℓℓ ( 3 b , F ∼ hC 2 . ) 3 5 . ℓ 1 π 5 = 10 . For comparison, we also plot in red the prediction for the 2 − 5 − ℓ ↑ 5 ii − (2 1 1 − 2 / ℓ 10 F NL 2 . As one can see, the signal is detectable for a large part of 10 σ f 10 primordial × q , this happens for all the points in the light shaded region. F ( . The light (orange) shadow shows, for comparison, the resul × ◦ 7 3 × max 4 + 1) = ℓ ’s represent the sum of the CMB power spectrum plus the power 10 0 ℓ & i ℓ 3) 50%, a signal in the bispectrum would appear only if ) ( 30 40 50 60 10 20 ≥ C 2 = 10 ≥

A ------− ]. ± ℓ

A

) π (2 σ ℓℓℓ

2 16 10 b × 23

NL

=

S/N

16 f +1) ℓ ( ℓ ) RS

( ( or 6 respectively if all 2 2 0 the variance is dominated by the Primordial one, namely , δ = (7 of a large Void. We show in the right panel of figure one can see that the texture proposed in [ 2 , 0 6 A ]): δ is the variance of the bispectrum: max and ℓ 30 = 1 3 ◦ ℓ 3 2 as a function of ℓ ℓ : Plot of 10 such region gets larger and includes the dark shaded one. For 2 1 , with ℓ ℓ ◦ 1 = 18 ◦ σ ℓ RS ) Focusing on the RS signal, we now turn to estimate whether it i Other candidates for a structure that could explain the Cold Neglecting the instrumental noise, in the left panel of figur σ 18 = 10 and density ≥ ’s to find a bispectrum Signal-to-Noise ( S/N Figure 5 as (see e.g. [ σ ( parameter space which would give rise to a detectable signal signal on a single multipoleℓ is lower than the cosmic varianc primordial signal for the bispectrum only if the parameter space: so it should already be possible to look data. Conversely, the absenceL of any detection would give in texture considered in [ where cold region diameter and ∆ σ they are all equal. The constraints analogous to thosepanel of discussed figure above for a big Voi spectrum of the noise of the detector. In general, at some with which corresponds to while below JCAP02(2009)019 , ] , ◦ . 30 NL ) f (4.11) (4.10) = 10 RS r levels ( NL ]). The σ f 30 500500 e radiation and gous quantity 1 ◦ ), but in the for multipoles ] due to the RS ◦ k > 400400 i ( /h 10 sical model (like 3 ℓ NL RS = 18 NL 2 ≥ ) f ℓ f σ 1 σ ր ℓ [Mpc 300300 ) as follows: ◦ ’s, since it couples the B S/N . L for x ℓ for ( h 18 ( 3 ] is much smaller at low φ ℓ uld have on the measure- ≥ 2 is introduced (see e.g. [ 2 200200 max ℓ ) 14 σ ℓ 1 3 3 3 prim ]: ℓ ) (4.9) ℓ ℓ ℓ ˜ i 2 2 constant number. Note that NL 2 B ℓ ℓ from the data in order to get ) ℓ 33 for 3 f 1 1 1 ℓ 2 2 x ) ℓ ℓ at large prim ℓ , 2 100100 ( 11 σ σ ˜ ℓ B n, 0.80.8 0.60.6 0.40.4 0.20.2 RS 1 2 L ( ( 30 ℓ | s predict very small values for φ urbations 0 B ]. The impact on δ , max | , in terms of which it is customary to ℓ − h 3 , all the points of the light shaded (green) l 14 max ◦ ≤ 2 ) ℓ l NL 3 x 1 ℓ prim f ≤ l ( 3 ≤ ˜ ℓ 2 L B max 2 = 10 ℓ ℓ ≤ φ 2 ( ≤ σ ℓ NL 1 ℓ f ≤ NL 1 ≤ this region gets larger and includes the dark shaded 5050 ℓ f – 13 – 2 = ≤ ◦ 2 3 ). For P l | 2 )+ l 0 P x 1 δ texture prim l | ( = 18 4040 L B . L, σ φ 5 )= − measurements ◦ 10 max )= ℓ x 3030 ( × ]. Right: The black lines, from bottom to top, show the contou ( ) 1. For NL ], but other models may predict larger values (see e.g. [ φ = 10 for the RS as a function of the multipole f 23 ’s of the primordial signal [ > RS 32 σ ℓ =4 ( NL , thus avoiding to overestimate the latter by the amount ∆ is generically a function of the momenta, i.e. , f 50. On the contrary, the Lensing effect [ A ◦ RS in the plane ( . The light (orange) shadow shows, for comparison, the analo S/N 2020 NL ) ∆ 31 5 5 f ≤ NL − − f ℓ = 18 1) [ 10 S/N . 10 have a specific form in terms of the primordial spectrum and th σ ≤ × (0 × 3 1010 l 2 O 3) l 1 prim 10) l ± ) is the linear Gaussian part of the perturbation. Given a phy = , ˜ 22 11 11 B 7 x -- -- 1010 , ( ’s with the high : Left: Plot of NL L 1010 1010 ℓ f = (7 φ RS = (4 As we have seen, the RS effect leads to a large contribution to ) A A ’s, but could contaminate the primordial bispectrum signal slow-roll inflation) quantitative data analyses itsingle field is minimally usually coupled assumed slow-roll inflationary to model be just a effect can be computed by estimating the following ratio [ In other words, ifthe a correct large value Void for exists, one should subtract it low RS- primordial bispectrum coefficients can be written as: for details) parameterizing the primordial curvature pert ℓ for where in the range 10 where the for the texture considered in [ region would give ( Figure 6 transfer function. that is one, which goes down to 4.2 Contamination of with We now turn to thement impact of that the a primordial huge non-gaussianity Void parameter in the line of sight wo parametrize a primordial non-Gaussian signal. By definitio S/N ( JCAP02(2009)019 : l 7 ’s, ℓ the ’s is max ℓ ℓ 2000. an be (4.13) (4.12) − ]. 60 nal: ), plotted as experiment 23 800 σ are excluded ∼ is roughly the max 5 50 ℓ − give a negligible ( at 1- ) . NL max 10 f ) plotted in figure ℓ RS NL i from WMAP1 ( max NL f ) 40 × f ], so that one obtains: . Left: the log-log scale P 3 5 ( ℓ 60, while the search for ) for smaller values of 30 − C ւ ℓ>ℓ 7(8) . 10 30 ih corrections are localized at ) is well approximated by max he texture case [ vanishes if any of the ) ) turns out to be equal to 3 ℓ ≥ . × P appearing in the numerator i 1 ( ( ge Void does not affect much ,ℓ ℓ ) 3 reported by the WMAP 1-year ) A 3) ℓ 2 3 2000) [ NL 4.11 C proximation to get an estimate 2 ℓ 4.11 20 f h ℓ RS 2 . ± ,ℓ ( RS 1 ℓ NL NL he impact on ≈ 1 ( ℓ 1 f f + a-points up to prim ℓ ℓ B i ˜ B h ) = (7 rimental sensitivities. Right: the linear 10 max l P 3 ( ℓ A 0 C 500 500

3000 2500 2000 1500 1000

ih

-

NL max ) ) ( ∆ ℓ f

) RS ( P 2 ( plateau in the power spectrum, while the full ℓ ℓ C 1 for Planck h . 10 for WMAP – 14 – + max ( i = 30. Of course, such an analysis would require a 19 for Planck ( ) ℓ . , together with ℓ P 2 ≈ 60 are strongly affected, they only represent a small ◦ ( ℓ ’s. However, since ) ℓ C (dependent on the experiment) the experimental noise in . 33 : values for the amplitude RS ih ◦ ( = 10 ℓ NL ) 1010 f P 1 σ max ( ℓ ℓ : the reason is indeed that a smaller Void affects higher ∆ arising from the inclusion of the RS effect, ∆ C ◦ = 18 800) and 0 allows a direct comparison with the WMAP-1year experimenta h and 7 σ NL 6( ≈ ◦ f ). This fact also allows to neglect the experimental noise in − = 10 get a sizable RS correction only if max = i σ 22 l = 18 ) with the Sachs-Wolfe approximation for the primordial sig ]. We can already see that some region of the parameter space c 3 4.11 3 ℓ 1010 ℓ σ 2 35 2 ℓ 4.6 [ ℓ ◦ . Instead the denominator is a quantity very sensitive to the 1 and 1 ℓ 3 prim ℓ ℓ b ◦ ˜ 2 B , for NL ◦ ℓ ], until the experimental noise is negligible. The result is h = 10 f 1 2 ℓ σ 34 max σ [ ℓ COBE WMAP Planck = 18 = 18 4 of the denominator becomes so large that the multipoles ]. The light (orange) region shows the analogous results in t σ for WMAP ( by the error bar localized at σ − 3 : The correction to 2 35 ℓ measurements for high resolution experiments. In fact, the 22 44 22 σ 11 ) by using ( 2 ]). In fact, at some high 10 − -- ℓ

1010 1010

1

1010 NL 2 ℓ It is also interesting to study the dependency of ∆ We may very easily give an approximation of the The right plot in figure 30 max ’s because × ∆ ) ( f ℓ 10 NL

. This can be easily done because the denominator in (

ℓ σ f ) RS ( 4.11 × 8 . max max constraints on numerator’s (see [ ℓ contribution to the sum. Accordingly, the denominator of ( from the left panel itthe turns out that the RS effect due to the lar excluded for the Void with 5 ℓ Figure 7 the at 1(2)- a function of allows a better understandingscale of is the more effect useful for for different expe a comparison with the constraints on of ( low analysis [ same for a primordial non-gaussianity uses all the experimental dat which are more relevant for the extraction of the primordial fraction of all the bispectrum data-points. Note also that t Therefore, even if the multipoles larger than about 60, weof consider the it to numerator be of a ( sufficiently fair ap expression should be used for higher This is a good approximation only for the low- JCAP02(2009)019 A N NL f and tion − ), by ] give ◦ is not 8 2 17 that the terms of N NL shows the 6 f N 8 ith respect to the wer spectrum and ue to the RS effect , for which only the A not be due to the RS are directly the observed ], whose authors claim a K); their density contrast . The authors of [ location, the amplitude at covers about 20% of the he angular size of the Voids erently, leading only to an µ σ recent WMAP 5-year data, 36 ld Spot, their measured y surveys; the knowledge of 11 pace, a full analysis should be P data. In their analysis they ith other cosmological sources: several Void regions are present Anyway, looking for ed to the Lensing effect will be and he standard structure formation ween the galaxy surveys and the he two-point correlation function, , which turns out to be of about e amplitude iderations to these Voids as well. ≈ − A ction, it would contain L . The left panel of figure T ◦ y not among them. ), we have to compute also 10 ] has interestingly pointed out that the 3.3 & 18 which have a mean radius of about 5 . Since we have shown in figure σ i 5 3 ) ℓ 2 ℓ RS 1 is smaller (about 1) for WMAP and, moreover, ( ℓ – 15 – B h — which we do not address in the present paper. NL f 3 ℓ interference terms. For a random distribution of Voids 2 ℓ (or, equivalently, ∆ 1 prim ℓ N 6 ] have claimed detection of the ISW effect because it could B − − 17 3 10 , ] (their location is sketched in the right panel of figure N 16 × 17 6 . 3 of its temperature profile. Note that the most of the contribu terms of the same type as ( σ ≈ ), plus , and there are 19 Voids with ¯ N A ◦ 4.5 ]. . As stressed in the introduction, [ is provided. to 14 14 /h ¯ ◦ A Voids are present in the sky, however, there are differences w N ] has catalogued about 50 of such Voids, 17 100Mpc If Finally, we may also give a comment on the results by [ For a precise computation of the effect of these Voids on the po We can nevertheless give an estimate of the impact on the CMB d The Cold Spot, located in the southern hemisphere, is clearl − 5 A full analysis should also include the correlation matrix w SZ-lensing effect, point sources and the primordial signal. the same kind as ( interference terms. As for the three-point correlation fun goes up byeffect, an because amount the of RS 7. contribution to In the light of our analysis, this can has opposite sign. The possibility that this could be ascrib discussed in [ for the luminous mattertheir redshift can also be allows directly to observed estimate in their the physical galax size 5 Other voids We briefly comment in this section about the possibility that sky-cuts, the full expression for in the sky. The authors of [ more refined technical treatment — for example including the the best way tobispectrum constrain data with a the large prediction Void: for one should rather comp a mean amplitude 50 existence of such large Voids is unlikely to be explained by t explain the correlations thatCMB they data. were By able exploring tosky, [ a identify region bet of the northern hemisphere th scenario. It could be relevant, therefore, to apply our cons these informations for the 50 Voids identified, except for th assigning to each of themspans an from amplitude 3 equal to the mean one. T RS Signal-to-Noise is aboveable unity to for set most interesting of constraints. the parameter s detection of primordial non-gaussianityalso already find in that, the masking WMA the region which corresponds to the Co in the sky, one expects the interference terms to add up incoh oscillatory modulation of the correlation functions. is expected to come from the few Voids with the largest analysis developed in the previousin sections. addition If we to focus the on t bispectrum, it wouldand be the necessary angular to size know, for each Void, its mean value of the Voids catalogued in [ JCAP02(2009)019 ¯ ), s) A . 2 K µ ipoles µK . These 1 (5 6 - 80) − O ± 10 = 0.5 × 2 - 0 ce terms. Right: 6 T . (190 and amplitude 0 ), assuming again for one Void with − π 40 for most of the +1) 2 ◦ 2 = 3 ℓ 3 1 ( − = . For the RS effect ℓ & A studied in this paper ) 0.5 10 t is non-negligible: we olution of the Einstein 0.5 ℓ T 0 its impact on statistical RS × ( ℓ 0.5 , due to the Rees-Sciama oefficients obtained from the 1 - C (5 ]). 1 11 00 - -- O 11 0.5 0.5 14 0.50.5 nity at - g the line of sight. Indeed, the ) are irrelevant, the latter value = by the following easy argument. le in the available data. Through e two and three point correlation oid of radius 5 o show that the interference terms RS ) for the WMAP fits of order unity. . Assuming that the interference terms 2 S/N 2 K χ ℓ µ 1 . . As done before, the final result is obtained 0 4 60 = 50, thus obtaining ≈ – 16 – 2 0 . We may estimate the contribution to the three the results obtained in section N 50 8 T ¯ ]. A π +1) this structure to be uncorrelated with the Primordial 17 2 ℓ 40 ( ℓ ) ], we consider the angular size of such a Void to be about : the dashed curve is obtained by selecting (among the 50 Void RS 23 ◦ ( ℓ 30 14 C − assuming ◦ . The dotted thin curve is the full result without interferen , as shown in section 25% to be added to the Primordial spectrum, localized at mult ◦ ], for all of which we assigned the mean amplitude 3 − 20 − 10 17 & , one gets: σ ◦ in the range 3 σ and its temperature at the centre to be characterized by ∆ : In the left panel we plot (solid line) the RS power spectrum c ◦ 50. This should lead to a variation in the 10 18 ≤ First, we have computed the temperature profile using an LTB s For the power spectrum the results are as follows. The RS effec Then we have studied the impact on the bispectrum coefficients

ℓ

1 4 3 2 5

− π 2 ℓ

0

C T ] µK

[ ◦ 2 2 +1) ℓ ( ℓ ≤ ) RS ( a sketch of the location of the 50 Voids [ 50 Voids identified in [ Figure 8 Voids have consistently with the left plot of figure that the interference terms are negligible. 6 Conclusions Motivated by the so-calledthe impact Cold on Spot statisticalfunctions, in CMB of predictions, the in the WMAP particular presenceexistence data, th of of we an such have anomalously aeffect large (the Void Void Lensing alon could effect be is at analyzed the in the origin companion of paper the [ Cold Spot point function similarly. The signal-to-noise ratio for a V 10 by multiplying by the number of Voids, leading to ( equations, matched to anpredictions FLRW metric. for the Then, CMB, we have computed result for the two-point correlation function (where we als are negligible), whose order ofRescaling magnitude to can the be case understood radius of of an about amplitude 5 the 19 ones having fluctuations. As suggested by [ (which depend on the relative locationfor in a the single sky Void of has the to Voids be multiplied by would be of about 10 5 predict a bump of 5% we have found that the Signal-to-Noise ratio is larger than u parameter space, and therefore this should already be visib JCAP02(2009)019 i 3 ℓ . 2 ℓ 2) 2) 1 , (A.2) ℓ , uncor- 1 1 , B , ) ]. These h 3 i 3 ( which can 17 a e Universe → 2) (A.1) → , 3 e that the bis- 1 3 , , , 2 , for a Cold Spot 3 2 , , 5 → − }  3 10 , 2 claimed in [ 1)+(1 2 1)+(1 2) = 0 2 , m × , Bob McElrath, Oystein , ℓ , in which the Cold Spot 3 3 1 dered to explain the Cold − , , , is zero, for any 2 cture on the determination 2 2 3 2 i ) ℓ symbol and the fact that by m i → 0 l is large but localized at low ( → → 1) + (1 1 1 a j ℓ 3 be above unity for some part of , orrelation function leads also to ions. 0 not been studied in detail but it ne. ) 3 ℓ 3 δ {z , arge Void, through the RS effect. 3 , , P = 2 , − ( 2  2 , due to the RS effect turns out to be 2 , , a 2 ) m P in the bispectrum coefficients larger than 8 → ) NL ( +(1 i +(1 f a i 3 − ) 3 and average temperature at the centre h 3 ( , m 2 m 1) + (1 RS ◦ , T/T ( m , − − , , X | a 3 2 3 ) , 3 P ℓ m 2 P ∗ ,ℓ ( δ + (1 2 a 3 ℓ a i ℓ – 17 – ) δ → 3 2 2 i ℓ P 1 for Planck. Using the already existing WMAP1- ), one finds 2 m m ( 3 . δ ℓ 3 2 , a 0 2 P ℓ P ℓ ℓ h 2 C 4.2 a a , one can exclude extreme values of the temperature ] rather than a large Void: the effect on the power h , h C ≃ ℓ 0 2 3 3 1 23 ) RS ℓ m m m 2 a +(1 ) ) P ℓ RS ( . The effect on the two-point correlation function is about NL − − a = ( ( f 0 0 0 at low ) 1 1 1 µK RS ℓ RS ℓ RS ℓ a a a ). h 11 NL 0 0 0 f 1 1 1 PPRS 2.2 ( m m m i δ δ δ ≈ − ’s according to ( 3 50), so it has a small impact on high resolution experiments, i ℓ are likely to be excluded, via a full analysis. So, we conclud ) over a statistical ensemble of possible realisation for th 2 ] on ℓ = = = T m ◦ ≤ 1 ℓ 35 ) 4.1 ℓ . ) B h ≤ P = 0, see ( ( a PPRS ( RS 00 i 1 for WMAP and ∆ a 3 3 ℓ m ≃ 2 2 ) ℓ m We have also considered the 50 Voids whose detection has been Averaging ( Finally, we have also applied our considerations to the case 1 1 ℓ RS m ( NL f 5%, hence much smaller than the one due to the large Void consi B . h contrast of the Void. For example, values for ∆ go up to very large multipoles: the overestimation of of the primordial non-gaussianity.multipoles (10 The RS bispectrum signa the bispectrum, we have studied also the impact of such a stru ∆ year constraints [ Summing over the one gets: spectrum is similar but somewhat smaller; the three-point c Voids have meancharacterized angular by diameter ∆ of about 10 a smaller Signal-to-Noise ratio,the which parameter however space. turns out to Acknowledgments We would like toRudjord thank and Subir Tirtho Sarkar Biswas, for Marcos useful discussions Cruz, and Paul suggest A Hunt, Vanishing of theWe show P-P-RS here contribution that the to term containing the bispectrum definition 0 Spot. The effect onis the expected three-point to correlation lead function to has a Signal-to-Noise ratiois smaller explained than o by a cosmic texture [ is vanishing. More generally, any term of the kind where we have used a known property of the Wigner 3 related with with diameter of 18 pectrum is a valuable tool for constraining an anomalously l JCAP02(2009)019 , ]. ]; ]. uctures ]; SPIRES [ , , (2006) 868 ckground ]. ]. Detection of , 50 (2006) 029901] ]; SPIRES SPIRES SPIRES , [ ] [ ] [ SPIRES SPIRES ]. ] [ D 73 ] [ Local voids as the (2004) 1198] SPIRES Hemispherical power Asymmetries in the [ ]; ]. es on CMB anisotropies and 609 arXiv:0803.0732 SPIRES , Lilje, The non-Gaussian cold spot in ] [ lts e with directional spherical SPIRES , New. Astron. Rev. SPIRES .B. Lilje, y Probe sky maps n ]. , ]. Jin, ] [ ] [ es: the effect of a cosmological arXiv:0712.1118 Erratum ibid. lez and A.N. Lasenby, [ , ]. ]. Detection of a non-Gaussian Spot in astro-ph/0605704 The non-Gaussian Cold Spot in astro-ph/0602398 [ [ astro-ph/0603859 astro-ph/0405341 [ [ On the Rees-Sciama effect: maps and ]. SPIRES Erratum ibid. 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