CMB Angular Power Spectrum from Correlated Primordial Fluctuation
aa r X i v : . [ a s t r o - ph . C O ] F e b CMB Angular Power Spectrum from Correlated Primordial Fluctuation
B.Yu ∗ Department of Physics, Nanjing University, Nanjing 210093, China
T.Lu † Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China andJoint Center for Particle, Nuclear Physics and Cosmology,Nanjing University-Purple Mountain Observatory, Nanjing 210093, China (Dated: November 8, 2018.)The usual inflationary scenario predicts a Gaussian random primordial density fluctuation, differ-ent Fourier modes of which do not correlate with each other. In this paper we propose a correlationbetween these different modes. A simple case is that these different Fourier modes correlate witheach other following a Gaussian function. For such a primordial density fluctuation we calculate theCMB angular power spectrum and find that its amplitude decreases but the decrease is different fordifferent l . This feature can be used to constrain the the correlation strength from the real data. PACS numbers: 98.70.Vc, 98.80.-k
I. INTRODUCTION
In the inflationary scenario, the primordial fluctuationcomes from the quantum vacuum fluctuation of the infla-ton which drive the inflation [1, 2, 3, 4]. Then the quan-tum fluctuation becomes classical when the correspond-ing scale goes out of the horizon. The primordial fluctua-tion produced in this method is regarded to be Gaussianrandom and scale invariant, which are confirmed by manyastronomical observations [5, 6, 7, 8].This explanation of the origin of the primordial fluc-tuation seems reasonable and successful. However it isnot a rigorous theory and needs more experimental tests.Besides the inflation, there exist other ideas to producethe primordial fluctuations, for example, the string gasmodel [9, 10, 11, 12]. Even in the inflationary scenario,there are many kinds of modification to the primordialpower spectrum, for example, the case that translationalinvariance is broken [13] or rotational invariance is bro-ken [14, 15, 16, 17]. The quantum-classical transition inthe inflationary scenario is also criticized by [18]. Con-sidering these uncertainties, our paper is motivated bychecking one of the main features of the Gaussian randomfluctuation from the experiments: the different modes ofthe primordial fluctuation in the Fourier space are in-dependent from each other. As we know, the CosmicMicrowave Background (CMB) is the relics of early uni-verse, which can provide precise measurement on theprimordial fluctuation. So in this paper we introduce asmall correlation between different modes of the primor-dial fluctuation (in Fourier space) and then study how thecorrelation affects the CMB angular power spectrum.This paper is organized as follows. In section II weintroduce the above mentioned correlations of the pri- ∗ Electronic address: [email protected] † Electronic address: [email protected] mordial fluctuation in Fourier space. In section III wecalculate the CMB angular power spectrum. The numer-ical results are presented in section IV and the discussionsare given in section V.
II. PRIMORDIAL FLUCTUATION
The meaning of the Gaussian random primordial den-sity fluctuation is < ξ ( ~k ) ξ ∗ ( ~k ′ ) > = P s ( k ) δ ( ~k − ~k ′ ),where ξ ( ~k ) is the Fourier mode of the primordial den-sity fluctuation ξ ( ~x ). The delta function indicates thatdifferent mode does not correlate with each other. Onecan relax this property by allowing different modes tocorrelate. Generally speaking, one can rewrite the aboveequation as < ξ ( ~k ) ξ ∗ ( ~k ′ ) > = f ( ~k, ~k ′ ), where f ( ~k, ~k ′ ) isnot required to be a delta function any more. How-ever f ( ~k, ~k ′ ) is not arbitrary, it must satisfy some con-straints since ξ ( ~x ) is a real field. < ξ ( ~k ) ξ ∗ ( ~k ′ ) > ∗ = <ξ ( ~k ′ ) ξ ∗ ( ~k ) > gives f ∗ ( ~k, ~k ′ ) = f ( ~k ′ , ~k ), which meansthe real part of f ( ~k, ~k ′ ) is symmetric under the inter-change of ~k, ~k ′ , Ref ( ~k, ~k ′ ) = Ref ( ~k ′ , ~k ), and the imag-inary part of f ( ~k, ~k ′ ) is antisymmetric under the inter-change of ~k, ~k ′ , Imf ( ~k, ~k ′ ) = − Imf ( ~k ′ , ~k ). Besides this, ξ ∗ ( ~k ) = ξ ( − ~k ), which indicates that < ξ ∗ ( − ~k ) ξ ( − ~k ′ ) > = f ( − ~k ′ , − ~k ) = f ( ~k, ~k ′ ) = < ξ ( ~k ) ξ ∗ ( ~k ′ ) > . Thus oneobtains Ref ( − ~k, − ~k ′ ) = Ref ( − ~k ′ , − ~k ) = Ref ( ~k, ~k ′ )and Imf ( − ~k, − ~k ′ ) = − Imf ( − ~k ′ , − ~k ) = − Imf ( ~k, ~k ′ ). P s ( k ) δ ( ~k − ~k ′ ) is such a special function with a zeroimaginary part.However, these constraints provide no positive infor-mation on the function f ( ~k, ~k ′ ). We must construct aconcrete form of the above function. It is not strangeto require that delta function is a good approxima-tion to this function, and noting that when σ → √ πσ exp [ − ( x − y ) σ ] → δ ( x − y ), one can use a very simpleform ( 1 p πσ k o ) exp [ − ( ~k − ~k ′ ) σ k o ] (1)to replace the delta function. When σ →
0, it ap-proaches δ ( ~k − ~k ′ ). Here σ is a pure number and k is the comoving wave number. This is a Gaussian formfunction, which is completely determined by the param-eter σ ≡ σ k and its maximum is located at ~k = ~k ′ .The FWHM is 2 √ σ . Then in order to satisfy theabove constraints, one can generalize P s ( k ) to P s ( √ kk ′ )or P s ( k + k ′ ). Their difference can be ignored in the weakcorrelation case ( σ is small), but cannot be ignored inthe strong correlation case ( σ is great). However, it iseasy to note that the strong correlation case is excludedby the CMB observation, so in this paper we only take f ( ~k, ~k ′ ) = P s ( k + k ′ p πσ k o ) exp [ − ( ~k − ~k ′ ) σ k o ] (2)as an example to study the correlation effects on CMBangular power spectrum. In fact, the following discus-sions not only apply to primordial fluctuation with form(2) but also to such a form f ( ~k, ~k ′ ) = f ( k, k ′ , | ~k − ~k ′ | ). Itcan be rewritten as f ( k, k ′ , | ~k − ~k ′ | ) = Z d ~ ∆ f ( k, k ′ , | ~ ∆ | ) δ ( ~k − ~k ′ − ~ ∆) (3) III. CMB POWER SPECTRUMA. Two point correlator
For scalar type of primordial density fluctuation, theCMB temperature anisotropy at direction ~n can be ex-pressed as [19, 20] T s ( ~n ) = Z d ~k ∆ sT ( η , k, ˆ k · ˆ n ) ξ ( ~k )= Z d ~k Z η dηS sT ( η, k ) e − ix ˆ k · ˆ n ξ ( ~k ) (4)where S sT ( η, k ) is the transfer function, x ≡ k ( η − η ), η is the conformal time and η is the conformal time oftoday. Following the standard methods, the temperatureanisotropy can be expanded by the spherical harmoniccoefficients a lm = Z d Ω Y ∗ lm ( ~n ) T s ( ~n ) . (5)An important property of the modified primordial powerspectrum < ξ ( ~k ) ξ ∗ ( ~k ′ ) > = f ( k, k ′ , | ~k − ~k ′ | ) is that theCMB anisotropy produced by such a form of primor-dial density fluctuation is statistically rotational invari-ant which means that for any rotation R one always have < T s ( R~n ′ ) T s ( R~n ) > = < T s ( ~n ′ ) T s ( ~n ) > . (6) The definite proof of Eq.(6) is postponed to Appendix.With this property, one can obtain the following conclu-sion [21] < a lm a ∗ l ′ m ′ > = δ ll ′ δ mm ′ C l . (7)So the angular power spectrum C l is sufficient to reflectall the two point correlators of CMB anisotropy. B. Power spectrum
The CMB angular power spectrum can be calculatedby C l = 12 l + 1 X m < a ∗ lm a lm > = 12 l + 1 X m < ( Z d Ω Y ∗ lm ( ~n ′ ) T ( ~n ′ )) ∗ ( Z d Ω Y ∗ lm ( ~n ) T ( ~n )) > (8)Substituting Eq.(4) into Eq.(8), one obtains C l = (4 π ) l + 1 X m Z d ~k ′ d ~kG l ( k ′ ) G l ( k ) Y lm (ˆ k ′ ) Y ∗ lm (ˆ k ) < ξ ( ~k ) ξ ∗ ( ~k ′ ) > = Z d ~ ∆ F ( ~ ∆) , (9)where the formula e i~k · ~x = 4 π P lm i l j l ( kx ) Y ∗ lm (ˆ k ) Y lm (ˆ x )has been used and F ( ~ ∆) is defined by F ( ~ ∆) ≡ (4 π ) l + 1 X m Z d ~k ′ d ~kG l ( k ′ ) G l ( k ) Y lm (ˆ k ′ ) Y ∗ lm (ˆ k ) f ( k, k ′ , | ~ ∆ | ) δ ( ~k − ~k ′ − ~ ∆) , (10)where G l ( k ) ≡ R η dηS sT ( η, k ) j l ( x ). In the Appendix wehave proved F ( R~ ∆) = F ( ~ ∆) . (11)So, F ( ~ ∆) is only a function of ∆ ≡ | ~ ∆ | . Thus C l = 4 π Z ∞ ∆ d ∆ F (∆) . (12)Introducing new variables ~p = ~k + ~k ′ , ~p = ~k − ~k ′ , and | ∂ ( ~k, ~k ′ ) ∂ ( ~p, ~p ′ ) | = , one obtains F ( ~ ∆) = (4 π ) l + 1) X m Z d ~pd ~p ′ G l ( k ′ ) G l ( k ) Y lm (ˆ k ′ ) Y ∗ lm (ˆ k ) f ( k, k ′ , | ~ ∆ | ) δ ( ~p ′ − ~ ∆)= (4 π ) l + 1) X m Z d ~pG l ( k ′ ) G l ( k ) Y lm (ˆ k ′ ) Y ∗ lm (ˆ k ) f ( k, k ′ , ∆)= π Z d ~pG l ( k ′ ) G l ( k ) P l ( ˆ k ′ · ˆ k ) f ( k, k ′ , ∆) , (13)where ~k = ~p + ~p ′ , ~k = ~p − ~p ′ F ( ~ ∆) is onlya function of ∆, one can safely set ~ ∆ in the z di-rection ~ ∆ = ∆ ~e z , then k = q p x + p y + ( p z + ∆) k ′ = q p x + p y + ( p z − ∆) k ′ · ˆ k = p x + p y + p z − ∆ kk ′ .Noting that the integrand is only a function of p x + p y ,and using Z + ∞−∞ dp x Z + ∞−∞ dp y f ( p x + p y )= Z ∞ ¯ pd ¯ p Z π dθf (¯ p )= 2 π Z ∞ ¯ pd ¯ pf (¯ p ) , (14)one obtains C l = 4 π Z ∞ ∆ d ∆ Z + ∞−∞ dp z Z ∞ ¯ pd ¯ pG l ( k ′ ) G l ( k ) P l ( ˆ k ′ · ˆ k ) f ( k, k ′ , ∆) . (15)Finally noting that the integrand is an even function of p z , one obtains a simple form, C l = π π ) Z ∞ ∆ d ∆ Z ∞ dp z Z ∞ ¯ pd ¯ pG l ( k ′ ) G l ( k ) P l ( ˆ k ′ · ˆ k ) f ( k, k ′ , ∆) , (16)where G l ( k ) ≡ R η dηS sT ( η, k ) j l ( x ), k = p ¯ p + ( p z + ∆) k ′ = p ¯ p + ( p z − ∆) k ′ · ˆ k = ¯ p + p z − ∆ kk ′ . IV. NUMERICAL RESULTS
In this paper we use the Boltzmann code CMB-FAST ([22]) to do numerical calculation and adopt aflat Λ
CDM model with cosmological parameters: Ω b =0 . c = 0 . Λ = 0 . H = 70, τ = 0 . n s = 1and ∆ ξ = 1. Where τ is the optical depth, n s is spec-tral index and ∆ ξ is the primordial scalar perturbationamplitude. In CMB power spectrum calculation ∆ ξ isonly a normalization factor, the value of which is deter-mined by observational data. Thus we can safely set itto be unit (usually the value of ∆ ξ is taken to be about2 . × − ), which will not affect the following discus-sion, since we mainly care about the relative change ofthe power spectrum.First we calculate ( G l ( k )) = ( R η dηS sT ( η, k ) j l ( x )) for l = 5 , , , k .When there is no correlation between the differentmodes, the CMB angular power spectrum is C l = − ) FIG. 1: Dependence of ( G l ( k )) on k , from left to rightthese curves correspond to ( G ( k )) , ∗ ( G ( k )) , ∗ ( G ( k )) , ∗ ( G ( k )) respectively. -0.03-0.02-0.0100.010.021e-05 0.0001 0.001 0.01 0.1 1 G ( k ) k (Mpc − ) FIG. 2: Dependence of G ( k ) on k . (4 π ) R ∞ k dkP s ( k )( G l ( k )) , which reduces to C l =(4 π ) ∆ ξ R ∞ d ln k ( G l ( k )) when the spectrum is scale in-variant. From Fig.(1) it is easy to note that for different l the main contribution to C l comes from different regionof k . The larger is l , the main contributions to C l comesfrom larger k . Then we plot G l ( k ) = R η dηS sT ( η, k ) j l ( x )for l = 5 , , l = 5 G l ( k ) is almost positive exceptfor small k where there exist a negative peak. One alsonote that for l = 2000 the average period is longer thanthat for l = 1000.The CMB angular power spectra are plotted in Fig.(5)and the relative angular power spectra changes and therelative cosmic variance are plotted in Fig.(6). It is easyto note that the amplitude of angular power spectrumdecreases when correlation is introduced. The greateris σ , the lower is the amplitude. From Fig.(6) it canbe seen that the angular power spectrum decreases moresignificantly at the neighborhood of l = 1000 than atsmaller or larger l . This is because of the phenomenon wementioned in the last paragraph. Numerical calculationtells us that the contribution of F (∆) to C l (Eq.(12)) -0.0004-0.0003-0.0002-0.000100.00010.00020.00030.0004 0.075 0.08 0.085 0.09 0.095 0.1 0.105 G ( k ) k (Mpc − ) FIG. 3: Dependence of G ( k ) on k . -0.00015-0.0001-5e-0505e-050.00010.00015 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 G ( k ) k (Mpc − ) FIG. 4: Dependence of G ( k ) on k . concentrate at the neighbourhood of ∆ = √ σ andˆ k ′ · ˆ k ≈ σ is small (so P l ( ˆ k ′ · ˆ k ) ≈ C l ≡ (4 π ) R ∞ d ln kG l ( k ) ∗ G l ( k + δk ) (refer to Eq.(13)) andcompare it with C l = (4 π ) R ∞ d ln k ( G l ( k )) to under-stand the above mentioned phenomenon, where δk is asmall fixed value. From Fig.(2,3,4) one know that G l ( k )oscillate with k , so one can further consider such a func-tion, f ( x ) = x < − x + 1 − ≤ x ≤ − x + 1 0 ≤ x ≤ x > . The quantity D ≡ R f ( x ) dx = , while another quan-tity D ( δ ) ≡ R f ( x ) f ( x + δ ) dx = − δ (1 − δ ) (here δ ≪ D < D . As the same reason,the angular power spectrum decrease when the primor-dial density fluctuation is correlated in Fourier space. Atthe same time, when 0 < δ < δ , D ( δ ) > D ( δ ). Dueto this, for a fix δk , ˜ C l decrease less in the case that G l ( k )oscillates with a longer average period than the decreasein the case that G l ( k ) oscillates with a shorter averageperiod. Thus the angular power spectrum at large l de- l ( l + ) C l l FIG. 5: Unnormalized CMB angular power spectrum C l asa function of l . The solid line corresponds to the Gaussianrandom case, while the long dashed line, the short dashed lineand the dotted line corresponds to σ = 10 − , ∗ − and10 − respectively. -0.6-0.5-0.4-0.3-0.2-0.10 0 500 1000 1500 2000 2500 ∆ C l / C l l FIG. 6: Dependence of ∆ C l C l on l . ∆ C l is defined as the CMBangular power spectrum with correlation minus the one with-out correlation. The solid line, long dashed line and shortdashed line correspond to σ = 10 − , ∗ − and 10 − respectively. The dotted line is the cosmic variance of theangular power spectrum ∆ C l /C l = p / (2 l + 1) in the un-correlated case (here − p / (2 l + 1) is plotted). crease less than that at about l ∼ G ( k )oscillates with a shorter average period compared with G ( k ). At last one note that when G l ( k ) is alwayspositive, the integrand G l ( k ) ∗ G l ( k + δk ) is also positive,while when G l ( k ) is positive and negative alternate, theintegrand G l ( k ) ∗ G l ( k + δk ) is not always positive. Sowhen G l ( k ) is always positive ˜ C l tends to decrease lessthan the case that G l ( k ) takes positive value and nega-tive value alternately. This is the reason that the angu-lar power spectrum decrease less at small l than that at l = 1000. These features can be used to constrain thecorrelation strength parameter σ from the real data. V. DISCUSSION
In this paper we only discussed a special type of thecorrelation of the primordial density fluctuation in theFourier space. There exist other forms of correlation.From the point of experimental view, the correlation dis-cussed in our paper is typical because the correlation be-tween two modes is large only when they are almost thesame. However, it is possible that the correlation in theFourier space is scale dependent. Such possibilities cannot be excluded. A better method is to divide these pos-sible forms into several types, and then consider theireffects on CMB. Besides CMB temperature, the effectson CMB polarization and large scale structure also needa further study.
Acknowledgments
We would like to thank Dr Sun Weimin for improvingthe manuscript.
APPENDIX A: ROTATIONAL INVARIANCE
In this Appendix we shall give a proof for the statisti-cally rotational invariance of T s ( ~n ′ ) T s ( ~n ) and rotationalinvariance of F ( ~ ∆).From Eq.(4), < T s ( ~n ′ ) T s ( ~n ) > can be expressed as < T s ( ~n ′ ) T s ( ~n ) > = Z d ~k ′ Z d ~k Z η dη ′ S sT ( η, k ′ ) e ix ′ ˆ k ′ · ˆ n ′ Z η dηS sT ( η, k ) e − ix ˆ k · ˆ n < ξ ( ~k ) ξ ∗ ( ~k ′ ) > = Z d ~k ′ Z d ~k Z η dη ′ S sT ( η, k ′ ) e ix ′ ˆ k ′ · ˆ n ′ Z η dηS sT ( η, k ) e − ix ˆ k · ˆ n f ( k, k ′ , | ~k − ~k ′ | ) . (A1)Then the two point ensemble statistical average after ro-tation R is < T s ( R~n ′ ) T s ( R~n ) > = Z d ~k ′ Z d ~k Z η dη ′ S sT ( η, k ′ ) e ix ′ ˆ k ′ · R ˆ n ′ Z η dηS sT ( η, k ) e − ix ˆ k · R ˆ n f ( k, k ′ , | ~k − ~k ′ | ) . (A2) Replace the dummy variables ~k, ~k ′ by R~k, R~k ′ and notethat d ( R~k ) = d ~k , | R~k | = | ~k | , d ( R~k ′ ) = d ~k ′ , | R~k ′ | = | ~k ′ | and | R ( ~k − ~k ′ ) | = | ~k − ~k ′ | , one obtains < T s ( R~n ′ ) T s ( R~n ) > = Z d ~k ′ Z d ~k Z η dη ′ S sT ( η, k ′ ) e ix ′ R ˆ k ′ · R ˆ n ′ Z η dηS sT ( η, k ) e − ixR ˆ k · R ˆ n f ( k, k ′ , | ~k − ~k ′ | ) . (A3)Since R ˆ k ′ · R ˆ n ′ = ˆ k ′ · ˆ n ′ and R ˆ k · R ˆ n = ˆ k · ˆ n , one finallyobtain < T s ( R~n ′ ) T s ( R~n ) > = < T s ( ~n ′ ) T s ( ~n ) > . (A4)The same reasoning can be applied to F ( ~ ∆). FromEq.(10), one obtains F ( R~ ∆) = (4 π ) l + 1 X m Z d ~k ′ d ~kG l ( k ′ ) G l ( k ) Y lm (ˆ k ′ ) Y ∗ lm (ˆ k ) f ( k, k ′ , | ~ ∆ | ) δ ( ~k − ~k ′ − R~ ∆) . (A5)Replace the dummy variables ~k, ~k ′ by R~k, R~k ′ asbefore and noted that d ( R~k ) = d ~k , | R~k | = | ~k | , d ( R~k ′ ) = d ~k ′ , | R~k ′ | = | ~k ′ | , | R~ ∆ | = | ~ ∆ | , Y lm ( R~k ′ ) = P m ′ D lmm ′ ( R ) Y lm ( ~k ′ ), Y ∗ lm ( R~k ) = P m ′ D l ∗ mm ′ ( R ) Y ∗ lm ( ~k ) and δ [ R ( ~k − ~k ′ − ~ ∆)] = δ ( ~k − ~k ′ − ~ ∆) | R | = δ ( ~k − ~k ′ − ~ ∆), one gets F ( R~ ∆) = F ( ~ ∆) . (A6). [1] V. F. Mukhanov and G. V. Chibisov, JETP Lett. , 532(1981).[2] A. A. Starobinsky, Phys. Lett. B117 , 175 (1982).[3] A. H. Guth and S. Y. Pi, Phys. Rev. Lett. , 1110 (1982).[4] S. W. Hawking, Phys. Lett. B115 , 295 (1982).[5] G. F. Smoot et al., Astrophys. J. , L1 (1992).[6] C. L. Bennett et al., Astrophys. J. , L1 (1996), astro- ph/9601067.[7] D. N. Spergel et al. (WMAP), Astrophys. J. Suppl. ,377 (2007), astro-ph/0603449.[8] G. Hinshaw et al. (WMAP), Astrophys. J. Suppl. ,288 (2007), astro-ph/0603451.[9] A. Nayeri, R. H. Brandenberger, and C. Vafa, Phys. Rev.Lett. , 021302 (2006), hep-th/0511140.[10] R. H. Brandenberger, A. Nayeri, S. P. Patil, and C. Vafa,Phys. Rev. Lett. , 231302 (2007), hep-th/0604126.[11] R. H. Brandenberger, A. Nayeri, S. P. Patil, and C. Vafa,Int. J. Mod. Phys. A22 , 3621 (2007), hep-th/0608121.[12] R. H. Brandenberger et al., JCAP , 009 (2006), hep-th/0608186.[13] S. M. Carroll, C.-Y. Tseng, and M. B. Wise (2008),0811.1086.[14] L. Ackerman, S. M. Carroll, and M. B. Wise, Phys. Rev.
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