Signatures of anisotropic sources in the trispectrum of the cosmic microwave background
aa r X i v : . [ a s t r o - ph . C O ] D ec UMN-TH-3317/13
Prepared for submission to JCAP
Signatures of anisotropic sources inthe trispectrum of the cosmicmicrowave background
Maresuke Shiraishi, a,b
Eiichiro Komatsu c,d and Marco Peloso e a Dipartimento di Fisica e Astronomia “G. Galilei”, Universit`a degli Studi di Padova, viaMarzolo 8, I-35131, Padova, Italy b INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy c Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany d Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Ad-vanced Study, the University of Tokyo, Kashiwa, Japan 277-8583 (Kavli IPMU, WPI) e School of Physics and Astronomy, University of Minnesota, Minneapolis 55455, USA
Abstract.
Soft limits of N -point correlation functions, in which one wavenumber is muchsmaller than the others, play a special role in constraining the physics of inflation. Anisotropicsources such as a vector field during inflation generate distinct angular dependence in all thesecorrelators. In this paper we focus on the four-point correlator (the trispectrum T ). We adopta parametrization motivated by models in which the inflaton φ is coupled to a vector fieldthrough a I ( φ ) F interaction, namely T ζ ( k , k , k , k ) ≡ P n d n [ P n (ˆ k · ˆ k ) + P n (ˆ k · ˆ k ) + P n (ˆ k · ˆ k )] P ζ ( k ) P ζ ( k ) P ζ ( k ) + (23 perm), where P n denotes the Legendre polynomials.This shape is enhanced when the wavenumbers of the diagonals of the quadrilateral are muchsmaller than the sides, k i . The coefficient of the isotropic part, d , is equal to τ NL / I ( φ ) F interaction generates d = 2 d which is, in turn, relatedto the quadrupole modulation parameter of the power spectrum, g ∗ , as d ≈ | g ∗ | N with N ≈
60. We show that d and d can be equally well-constrained: the expected 68% CL errorbars on these coefficients from a cosmic-variance-limited experiment measuring temperatureanisotropy of the cosmic microwave background up to ℓ max = 2000 are δd ≈ δd = 105.Therefore, we can reach | g ∗ | = 10 − by measuring the angle-dependent trispectrum. Thecurrent upper limit on τ NL from the Planck temperature maps yields | g ∗ | < .
02 (95% CL). ontents d and d
74 Expected error bar on g ∗ from the CMB trispectrum 95 Conclusion 12 Cosmic inflation [1–5] is thought to have occurred in nearly de Sitter spacetime. Recentconvincing detection of a small deviation from the exact scale invariance of primordial cur-vature perturbations [6, 7] shows that time-translation invariance is slightly broken duringinflation. This provides strong evidence for inflation, as the expansion rate during inflationmust be time-dependent in order for inflation to end eventually, and the time dependencemust be weak in order for inflation to occur. This then leads to a natural question: “Areother symmetries also broken?”Invariance under spatial rotation remains unbroken in the usual inflation models basedon scalar fields; however, it can be broken in the presence of vector fields (see ref. [8–10] forreviews). In such a case, the two-point correlation function in Fourier space (power spec-trum) of primordial curvature perturbations defined by h ζ k ζ k i = (2 π ) δ (3) ( k + k ) P ζ ( k )generically exhibits a direction dependence as [11] P ζ ( k ) = P ( k ) h g ∗ ( k )(ˆ k · ˆ E cl ) i , (1.1)where ˆ E cl is a preferred direction in space and P ( k ) is the isotropic power spectrum. Theamplitude, g ∗ ( k ), may depend on wavenumbers.Temperature anisotropy of the cosmic microwave background (CMB) offers a stringenttest of rotational invariance of correlation functions. Assuming that g ∗ is independent ofwavenumbers (which is a reasonable assumption for inflation models we mostly focus on inthis paper, up to a logarithmic correction), ref. [12] finds g ∗ = 0 . ± .
016 (68% CL) fromthe temperature data obtained recently by the Planck satellite [13]. The 95% CL limit is − . < g ∗ < . Planck satellite [14] and emission from our own Galaxy [15].The three-point function (bispectrum) offers an additional test of rotational invariance ofcorrelation functions. As breaking of rotational invariance during inflation requires multiplefields (e.g., a scalar field driving inflation and a vector field), it also breaks the so-called single-field consistency relation of the bispectrum [16, 17]; namely, there can be a non-negligiblecorrelation in a “soft limit” of the three-point correlation function, in which one wavenumber,say k , is much smaller than the other two, i.e., k ≪ k ≈ k . Breaking of rotational– 1 –nvariance then introduces a dependence of the soft-limit bispectrum on angles between thewavenumbers. Defining the bispectrum as h ζ k ζ k ζ k i = (2 π ) δ (3) ( k + k + k ) B ζ ( k , k , k ),we write [18] B ζ ( k , k , k ) = X n c n P n (ˆ k · ˆ k ) P ζ ( k ) P ζ ( k ) + (2 perm) , (1.2)where P n ( x ) denotes the Legendre polynomials. Note that this form is valid for an isotropicmeasurement of the bispectrum, namely for the case in which we fix a triangular shape,and we then average over all possible orientations of this shape in Fourier space (this isequivalent to taking an average over all possible directions for the preferred direction ˆ E cl ).The Planck temperature data give constraints on the first three coefficients as c = 3 . ± . c = 11 . ± c = 3 . ± . g ∗ . For example, therelation is c = 2 c = 320 | g ∗ | ( N/
60) (with N ≈
60 being the number of e -folds counted fromthe end of inflation) for inflation models with a scalar field driving inflation, φ , coupled toa vector field in the form of I ( φ ) F where F is a vector-field strength tensor [18, 20–23]. We then obtain | g ∗ | < .
05 and 0 .
36 (95% CL) from c and c , respectively.The goal of this paper is to investigate the four-point function (trispectrum) definedby DQ i =1 ζ k i E c = (2 π ) T ζ ( k , k , k , k ) δ (3) (cid:16)P i =1 k i (cid:17) , where h . . . i c denotes the connectedpart of the trispectrum. The trispectrum is fully parametrized by six independent numbers,i.e., three wavenumbers and three angles between wavevectors, e.g., k , k , k , ˆ k · ˆ k , ˆ k · ˆ k ,and ˆ k · ˆ k , where k ≡ k + k . We then find a simple linear parametrization as T ζ ( k , k , k , k ) = X n h A n P n (ˆ k · ˆ k ) + B n P n (ˆ k · ˆ k ) + C n P n (ˆ k · ˆ k ) i × P ζ ( k ) P ζ ( k ) P ζ ( k ) + (23 perm) . (1.3)Symmetry under permutations of k i imposes B n = C n , while A n remains independent ingeneral. By construction, this trispectrum has the largest values in soft limits in whichdiagonals of a quadrilateral, k , etc., are much smaller than the sides, k i .Instead of studying the most general form, we shall study a simpler form motivated byinflation with I ( φ ) F coupling, which yields A n = B n = C n [18, 22, 23]. Our parametriza-tion is T ζ ( k , k , k , k ) = X n d n h P n (ˆ k · ˆ k ) + P n (ˆ k · ˆ k ) + P n (ˆ k · ˆ k ) i × P ζ ( k ) P ζ ( k ) P ζ ( k ) + (23 perm) . (1.4)The readers who are familiar with the primordial trispectrum would find that the first coeffi-cient, d , is equal to τ NL / E cl ). As we shall show later in sec-tion 4, I ( φ ) F inflation gives d = d / ≈ | g ∗ | N . While d is yet to be constrained by A bispectrum with a nontrivial angular dependence in the squeezed limit is also obtained in the model ofsolid inflation [24], which is a model characterized by three scalar fields with a nontrivial spatial profile. Inthis model, c ≫ c . With this parametrization, we divide the quadrilateral in the two triangles having sides k , k , k , and k , k , k , respectively. We then specify each triangle and their relative orientation. The odd terms in the expansion (1.4) may arise if the source of the anisotropic modulation breaks parity. – 2 –he data, the current upper limit on d = τ NL / <
470 (95% CL) from the
Planck data [19]yields | g ∗ | < .
02 (95% CL), which is already better than the limit from the power spectrumor the bispectrum. The limits on d and d from the next Planck data release should improvethe limit further.This paper is organized as follows. In section 2, we calculate the trispectrum of CMBtemperature anisotropy from eq. (1.4) both with the flat-sky approximation and the full-skyformalism. In section 3, we calculate the expected 68% CL error bars on d and d from acosmic-variance-limited CMB experiment. In section 4, we translate the error bars on d and d to that on g ∗ . We conclude in section 5. Let us rewrite the trispectrum given in eq. (1.4) as * Y i =1 ζ k i + c = (2 π ) Z d K δ (3) ( k + k + K ) δ (3) ( k + k − K ) X n d n t k k k k ( K , n )+(23 perm) , (2.1)with t k k k k ( K , n ) ≡ (cid:20) P n (ˆ k · ˆ k ) + 1 + ( − n P n (ˆ k · ˆ K ) + 1 + ( − n P n (ˆ k · ˆ K ) (cid:21) × P ζ ( k ) P ζ ( k ) P ζ ( K ) . (2.2)Here, a reduced curvature trispectrum, t k k k k ( K , n ), satisfies t k k k k ( K , n ) = t k k k k ( K , n ). Wethen write eq. (2.2) using spherical harmonics as t k k k k ( K , n ) = P ζ ( k ) P ζ ( k ) P ζ ( K ) 4 π n + 1 X µ × (cid:20) Y ∗ nµ (ˆ k ) Y nµ (ˆ k ) + 1 + ( − n (cid:16) Y ∗ nµ (ˆ k ) + Y ∗ nµ (ˆ k ) (cid:17) Y nµ ( ˆ K ) (cid:21) . (2.3) To gain analytical insights into the structure of the CMB trispectrum, we first derive theCMB trispectrum in the flat-sky approximation. The coefficients of the two-dimensionalFourier transform of temperature anisotropy in a small flat section of the sky are related tothe curvature perturbation as [26] a ( ℓ ) = Z τ dτ Z ∞−∞ dk z π ζ (cid:18) k k = − ℓ D , k z (cid:19) S I (cid:16) k = p k z + ( ℓ/D ) , τ (cid:17) D e − ik z D , (2.4)where D ≡ τ − τ denotes the conformal distance between a given conformal time, τ , andthe present time, τ ; k = ( k k , k z ) with k k = ( k x , k y ); and S I is the so-called source function.The flat-sky approximation is accurate for ℓ ≫ a ( ℓ ) in the limits of ℓ i ≫ k i D and L ≫ k z r is given by * Y i =1 a ( ℓ i ) + c = (2 π ) Z d L δ (2) ( ℓ + ℓ + L ) δ (2) ( ℓ + ℓ − L ) X n d n t ℓ ℓ ℓ ℓ ( L , n )+(23 perm) , (2.5)– 3 – l l l L l l l l Lcollinear triangleisosceles triangle
Figure 1 . Two shapes of the trispectrum in ℓ space in soft limits in which the diagonal, L , is muchsmaller than the sides, ℓ i . The red lines show ℓ and ℓ , while the blue solid and dashed lines show twoconfigurations of ℓ and ℓ . Specifically, the blue solid and dashed lines show configurations in whichˆ ℓ · ˆ ℓ ≈ − L . (Top) Collinear configurations:ˆ ℓ · ˆ L ≈ −
1, ˆ ℓ · ˆ L ≈ ∓
1, and ˆ ℓ · ˆ ℓ ≈ ±
1. (Bottom) Isosceles configurations: ˆ ℓ · ˆ L ≈ ˆ ℓ · ˆ L ≈
0, andˆ ℓ · ˆ ℓ ≈ ± where t ℓ ℓ ℓ ℓ ( L , n ) is the so-called CMB reduced trispectrum: t ℓ ℓ ℓ ℓ ( L , n ) ≡ Z ∞−∞ r dr " Y i =1 Z τ dτ i Z ∞ ℓ i /D i dk i π G ( ℓ i , k i , τ i , r ) P ζ ( k ) P ζ ( k ) P ζ (cid:18) L | r | (cid:19) × (cid:20) P n (ˆ ℓ · ˆ ℓ ) + 1 + ( − n P n (ˆ ℓ · ˆ L ) + 1 + ( − n P n (ˆ ℓ · ˆ L ) (cid:21) , (2.6)with G ( ℓ, k, τ, r ) = " − (cid:18) ℓkD (cid:19) − / S I ( k, τ ) 2 D cos s − (cid:18) ℓkD (cid:19) k ( r − D ) . (2.7)The flat-sky reduced CMB trispectrum directly reflects the angular dependence of the Leg-endre polynomials in the reduced curvature trispectrum, t k k k k ( K , n ), given in eq. (2.2).The isotropic term, n = 0, has the largest values when the diagonal, L , is much smallerthan the sides, ℓ i , i.e., ℓ ≈ ℓ ≫ L or ℓ ≈ ℓ ≫ L [27]. The amplitude of the trispectrumin this limit is modulated when n = 0. For example, in the “collinear configurations,”– 4 – ℓ · ˆ L ≈ −
1, ˆ ℓ · ˆ L ≈ ∓
1, and ˆ ℓ · ˆ ℓ ≈ ± t ℓ ℓ ℓ ℓ ( L ,
0) : t ℓ ℓ ℓ ℓ ( L ,
1) : t ℓ ℓ ℓ ℓ ( L , ≈ ±
13 : 1 . (2.8)In the “isosceles configurations,” ˆ ℓ · ˆ L ≈ ˆ ℓ · ˆ L ≈
0, and ˆ ℓ · ˆ ℓ ≈ ± n = 2 trispectrum vanishes: t ℓ ℓ ℓ ℓ ( L ,
0) : t ℓ ℓ ℓ ℓ ( L ,
1) : t ℓ ℓ ℓ ℓ ( L , ≈ ±
13 : 0 . (2.9)Note that the sign of the n = 1 trispectrum can change, as the Legendre polynomial with n = 1 is an odd function. These signatures will affect the expected error bars on d n asdiscussed in section 3. We shall move onto the full-sky formalism. The spherical harmonics coefficients of tempera-ture anisotropy are related to the curvature perturbation as a ℓm = 4 π ( − i ) ℓ Z k dk (2 π ) T ℓ ( k ) ζ ℓm ( k ) , (2.10)where ζ ℓm ( k ) is the curvature perturbation in spherical harmonics space: ζ ℓm ( k ) ≡ R d ˆ k ζ ( k ) Y ∗ ℓm (ˆ k ),and T ℓ ( k ) is the radiation transfer function, which is related to the source function as T ℓ ( k ) = R τ dτ S I ( k, τ ) j ℓ ( kD ). Using this and the computational technique developed inref. [28], the CMB trispectrum is given by * Y i =1 a ℓ i m i + c = " Y i =1 π ( − i ) ℓ i Z k i dk i (2 π ) T ℓ i ( k i ) Y i =1 ζ ℓ i m i ( k i ) + c , (2.11)where * Y i =1 ζ ℓ i m i ( k i ) + c = X LM ( − M (cid:18) ℓ ℓ Lm m − M (cid:19) (cid:18) ℓ ℓ Lm m M (cid:19) × (2 π ) X n d n t k k ℓ ℓ k k ℓ ℓ ( L, n ) + (23 perm) , (2.12)and t k k ℓ ℓ k k ℓ ℓ ( L, n ) = P ζ ( k ) P ζ ( k ) X L L L ′ ( − L ℓ L ℓ + ℓ + ℓ + ℓ + ℓ × Z ∞ r dr Z ∞ r ′ dr ′ j L ( k r ) j ℓ ( k r ) j L ( k r ′ ) j ℓ ( k r ′ ) × π h F L ′ L ′ ( r, r ′ ) I ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) + F L ′ L ( r, r ′ ) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L )+ F LL ′ ( r, r ′ ) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) i . (2.13) We have the same relationship between magnitudes for the other collinear configurations: ˆ ℓ · ˆ L ≈ ℓ · ˆ L ≈ ±
1, and ˆ ℓ · ˆ ℓ ≈ ± – 5 –he F function, defined by F LL ′ ( r, r ′ ) ≡ π Z K dKP ζ ( K ) j L ( Kr ) j L ′ ( Kr ′ ) , (2.14)projects the K dependence onto L . The I and J symbols are defined by I ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) ≡ π n + 1 ( − ℓ + ℓ + L ′ + n + L I L ℓ L ′ I L ℓ L ′ I ℓ L n I ℓ L n × (2 L + 1) (cid:26) ℓ ℓ LL ′ n L (cid:27) (cid:26) ℓ ℓ LL ′ n L (cid:27) , (2.15) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) ≡ π n + 1 1 + ( − n − ℓ + L + L ′ + L I L ′ Ln I L ℓ L ′ I ℓ ℓ L I ℓ L n × δ L ,ℓ (cid:26) ℓ ℓ LL ′ n L (cid:27) , (2.16)with I l l l ≡ q (2 l +1)(2 l +1)(2 l +1)4 π (cid:18) l l l (cid:19) , and they reflect characteristic ℓ dependenceimposed by P n (ˆ k · ˆ k ) and P n (ˆ k , · ˆ K ), respectively. The selection rules in these symbolsrestrict summation ranges of L , L and L ′ to the values close to ℓ , ℓ and L , respectively.They also guarantee parity invariance of the trispectrum; namely, ℓ + ℓ + ℓ + ℓ = evenalthough ℓ + ℓ or ℓ + ℓ can take on both even and odd numbers in the I function.Substituting eq. (2.13) into eq. (2.11) leads to the final expression of the CMB trispectrum: * Y i =1 a ℓ i m i + c = X LM ( − M (cid:18) ℓ ℓ Lm m − M (cid:19) (cid:18) ℓ ℓ Lm m M (cid:19) × X n d n t ℓ ℓ ℓ ℓ ( L, n ) + (23 perm) , (2.17)where the reduced form is given by t ℓ ℓ ℓ ℓ ( L, n ) ≡ X L L L ′ ( − L L ℓ ℓ + ℓ + ℓ × Z ∞ r dr Z ∞ r ′ dr ′ β ℓ L ( r ) α ℓ ( r ) β ℓ L ( r ′ ) α ℓ ( r ′ ) × h F L ′ L ′ ( r, r ′ ) I ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) + F L ′ L ( r, r ′ ) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L )+ F LL ′ ( r, r ′ ) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) i , (2.18)with α ℓ ( r ) = 2 π Z k dk T ℓ ( k ) j ℓ ( kr ) , (2.19) β ℓL ( r ) = 2 π Z k dkP ζ ( k ) T ℓ ( k ) j L ( kr ) . (2.20)When n = 0, we have I ℓ ℓ ℓ ℓ ( L , L , L ′ ; 0 , L ) = J ℓ ℓ ℓ ℓ ( L , L , L ′ ; 0 , L ) = J ℓ ℓ ℓ ℓ ( L , L , L ′ ; 0 , L )= I ℓ ℓ L I ℓ ℓ L δ ℓ ,L δ ℓ ,L δ L ′ L , (2.21)– 6 –hich agrees with the previous results [27, 29].The r and r ′ integrals in eq. (2.18) are dominated by contributions from r ≃ r ′ ≃ r ∗ ≡ τ − τ ∗ with τ ∗ being the recombination epoch, as α ℓ ( r ) and β ℓL ( r ) peak at r ≃ r ∗ . If F LL ′ ( r, r ′ ) varies slowly for r ≃ r ′ ≃ r ∗ , i.e., in the small- L limit, the r and r ′ integrals maybecome separable: t ℓ ℓ ℓ ℓ ( L, n ) ≈ X L L L ′ ( − L L ℓ ℓ + ℓ + ℓ R ( ℓ , L , ℓ ) R ( ℓ , L , ℓ ) × h F L ′ L ′ ( r ∗ , r ∗ ) I ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) + F L ′ L ( r ∗ , r ∗ ) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L )+ F LL ′ ( r ∗ , r ∗ ) J ℓ ℓ ℓ ℓ ( L , L , L ′ ; n, L ) i , (2.22)where R ( ℓ , L , ℓ ) ≡ Z ∞ r drβ ℓ L ( r ) α ℓ ( r ) . (2.23)This approximate formula enables us to calculate the trispectrum in the whole ℓ space withina reasonable computational time. This approximation is justified, as the signal-to-noise ofthe trispectrum is dominated by soft limits in which L is small [27]. Using a scale-invariantcurvature power spectrum, k P ζ ( k )2 π = A S , we obtain F LL ′ ( r ∗ , r ∗ ) as F LL ′ ( r ∗ , r ∗ ) = π A S Γ( L + L ′ )Γ( L − L ′ +32 )Γ( L ′ − L +32 )Γ( L + L ′ +42 ) . (2.24)Let us also derive eq. (2.22) from eq. (2.18) using the Sachs-Wolfe approximation. In theSachs-Wolfe limit, the transfer function is given by T ℓ ( k ) → − j ℓ ( kr ∗ ), and hence α ℓ ( r ) →− r ∗ δ ( r − r ∗ ). Performing the r and r ′ integrals, we recover eq. (2.22) with R ( ℓ , L , ℓ ) → F ℓ L ( r ∗ , r ∗ ). d and d In this section, we calculate the expected 68% CL error bars on d n using the full-sky formal-ism. Let us define a Fisher matrix element for d n as [30] F nn ′ ≡ X ℓ >ℓ >ℓ >ℓ X L T ℓ ℓ ℓ ℓ ( L, n ) T ℓ ℓ ℓ ℓ ( L, n ′ )(2 L + 1) C ℓ C ℓ C ℓ C ℓ , (3.1)where C ℓ is the temperature power spectrum. We shall consider an ideal, noise-free, cosmic-variance limited experiment measuring temperature anisotropy up to a maximum multipoleof ℓ max ; thus, C ℓ contains the CMB only.The trispectrum averaged over possible orientations of quadrilaterals, T ℓ ℓ ℓ ℓ ( L, n ), isgiven by [30] T ℓ ℓ ℓ ℓ ( L, n ) = P ℓ ℓ ℓ ℓ ( L, n ) + (2 L + 1) X L ′ (cid:20) ( − ℓ + ℓ (cid:26) ℓ ℓ Lℓ ℓ L ′ (cid:27) P ℓ ℓ ℓ ℓ ( L ′ , n )+( − L + L ′ (cid:26) ℓ ℓ Lℓ ℓ L ′ (cid:27) P ℓ ℓ ℓ ℓ ( L ′ , n ) (cid:21) , (3.2)– 7 – F nn l max n = 0: all Ln = 1: all LL ≤
5n = 2: all LL ≤ Figure 2 . Diagonal elements of the Fisher matrix, F , F , and F , computed using the Sachs-Wolfeapproximation. The lines with “all L” use all L in the summation of the Fisher matrix, while thelines with “ L ≤
5” for n = 1 and 2 use only L ≤ with P ℓ ℓ ℓ ℓ ( L, n ) = 2 t ℓ ℓ ℓ ℓ ( L, n ) + 2( − ℓ + ℓ + L t ℓ ℓ ℓ ℓ ( L, n )+2( − ℓ + ℓ + L t ℓ ℓ ℓ ℓ ( L, n ) + 2( − ℓ + ℓ + ℓ + ℓ t ℓ ℓ ℓ ℓ ( L, n ) . (3.3)Figure 2 shows the diagonal elements of the Fisher matrix, F , F , and F , computedusing the Sachs-Wolfe approximation. We show the results from summation over all possiblediagonals, L , as well as those from summation over only soft limits, L ≤
5. We find that F grows as ℓ in agreement with the previous work [27], and F also grows as ℓ ; however, F is smaller than F by two orders magnitude. Most of information of the trispectrumwith n = 2 is contained in the soft limit, L ≤
5, just like that with n = 0. On the other hand, F grows more slowly as ℓ , implying that the error bar on d would be too large to beuseful. We thus do not consider d any further in this paper. Information of the trispectrumwith n = 1 is not completely contained in the soft limit, and sizable contributions come from L > d and d when they are estimated jointly.We use (2) F ij = (cid:18) F F F F (cid:19) , (3.4)to obtain ( δd , δd ) = (cid:18)q (2) F − , q (2) F − (cid:19) . (3.5)– 8 – δ d / δ d l max actualSW limit Figure 3 . Ratio of the expected error bars, δd /δd . The solid line shows the results from eq. (2.22)with the full radiation transfer function, while the dashed line shows the Sachs-Wolfe approximation. In figure 3, we show the ratio of δd to δd as a function of ℓ max . The error bar on δd improves slightly faster than that on δd as ℓ max increases. We find δd /δd = 4 for ℓ max = 1000. We also find that these two parameters are not correlated very much: thecross-correlation coefficient, F / √ F F , is as small as 0.2. For ℓ max = 1000, we find( δd , δd ) = (105 , F ∝ F ∝ F ∝ ℓ holds for ℓ max > d and d would become ( δd , δd ) = (26 , ℓ max = 2000.Recalling d = τ NL /
6, the error bar on d we obtain here agrees with that given in ref. [27]. g ∗ from the CMB trispectrum The parameters of the power spectrum ( g ∗ ), the bispectrum ( c n ), and the trispectrum ( d n )can be related to each other once a model of inflation is specified. Such a relation is a powerfulprobe of the physics of inflation. In this section, we use inflation models with a particularcoupling between a scalar field driving inflation and a vector field given by I ( φ ) F to relatethe trispectrum parameters with g ∗ . The trispectrum averaged over all possible orientationsof quadrilaterals is given by [18] T I F ζ ≈ N | g ∗ | h (ˆ k · ˆ k ) + (ˆ k · ˆ k ) + (ˆ k · ˆ k ) − (ˆ k · ˆ k )(ˆ k · ˆ k )(ˆ k · ˆ k ) i × P ζ ( k ) P ζ ( k ) P ζ ( k ) + (23 perm) , (4.1)where N ≈
60 is the number of e -folds counted from the end of inflation. The shape ofthis trispectrum is 99% correlated with the trispectrum without (ˆ k · ˆ k )(ˆ k · ˆ k )(ˆ k · ˆ k ).Adjusting the amplitude, we find that the following trispectrum is an excellent approximation– 9 – δ g * l max d extrapolate ( ∝ l -2max )d extrapolate ( ∝ l -2max )c c Figure 4 . Expected 68% CL error bars on g ∗ from the bispectrum parameters ( c and c in eq. (1.4))and the trispectrum parameters ( d and d ), for N = 60. The lines for the trispectrum in ℓ max > to eq. (4.1): T I F ζ ≈ . × N | g ∗ | h (ˆ k · ˆ k ) + (ˆ k · ˆ k ) + (ˆ k · ˆ k ) i × P ζ ( k ) P ζ ( k ) P ζ ( k ) + (23 perm) . (4.2)The 99% correlation means that eqs. (4.1) and (4.2) have nearly identical shapes. The pre-factor 0.89 in eq. (4.2) is the ratio of the overall averages of trispectra computed numerically.One can understand this ratio by angular-averaging the trispectra in soft limits, using [23]:(ˆ k · ˆ k ) | av = 1 /
3, (ˆ k · ˆ k ) | av = (ˆ k · ˆ k ) | av = 1 /
2, and (ˆ k · ˆ k )(ˆ k · ˆ k )(ˆ k · ˆ k ) | av = 1 / d , d and g ∗ as d = 12 d ≈ . × | g ∗ | . (cid:18) N (cid:19) . (4.3)In figure 4, we show the expected error bars on g ∗ computed from those on d and d using eq. (4.3). We show the results of the direct calculation of δd and δd up to ℓ max = 1000,and use the extrapolation for 1000 < ℓ max ≤ g ∗ from the bispectrum parameters using [18] c = 2 c = 32 | g ∗ | . N . (4.4)The trispectrum parameters are proportional to | g ∗ | N , whereas the bispectrum param-eters are proportional to | g ∗ | N . More generally, we have h ζ n i ∝ | g ∗ | N n − , (4.5)– 10 – igure 5 . Diagrams for the 2- (left), 3- (middle), and 4-point (right) functions of ζ ∝ δφ in the I ( φ ) F model. The external dashed lines are δφ lines, while the internal propagators are δ E lines.The labels denote the momentum of the external lines, which is taken to flow inside the diagram. Inthe power spectrum, k + k = 0, while in the other two diagrams we are interested in the soft-limitconfigurations k + k →
0. A bullet denotes a mass insertion, namely a quadratic δ E δφ couplingproportional to the vector vacuum expectation value E cl . where ζ ∝ − H ˙ φ δφ in uniform density gauge. To understand this scaling, consider the diagramsshown in figure 5, which represent the dominant contributions to h ζ , , i arising from thisinteraction. By Taylor-expanding the I ( φ ) F coupling in the inflaton perturbations δφ ∝ ζ ,and by retaining only the linear terms, we have the two interactions H ∝ R d xa E cl · δ E ζ and H ∝ R d xa δ E · δ E ζ . In this expression H i denotes a contribution to the interactionHamiltonian, and a is the scale factor ( a = √− g in conformal time τ ). For each value of n , the diagram shown in the figure corresponds to the following terms in the in-in formalismcomputation h ζ n ( τ ) i ∝ " n − Y i =1 Z dτ i h [[ . . . [ ζ n ( τ ) , H ( τ )] , . . . ] , H ( τ n − )] i , (4.6)where ζ denotes the (“unperturbed”) curvature perturbation in the absence of the I ( φ ) F term. We are interested in the correlators h ζ n i in the super-horizon regime. The integrals ineq. (4.6) are dominated by the regions in which also the fields arising from the vertices arein the super-horizon regime [21]. Each interaction contains one ζ ( τ i ) ∝ δφ ( τ i ) field which,once commuted with one of the external fields, gives [ ζ ( τ ) , ζ ( τ i )] ∝ τ − τ i [21]. Thesecommutators, and the measure a ( τ i ) ∝ τ i in each vertex, are the only time-dependentcontributions to the integrand in eq. (4.6), leading to [21] h ζ n ( τ ) i ∝ n − Y i Z τ dτ i τ i (cid:0) τ − τ i (cid:1) ∝ N n − . (4.7)We thus see that the contribution to h ζ n i from the corresponding diagram in figure 5 is ∝ E N n − . The diagram shown for the power spectrum ( n = 2 in this expression) adds upwith the vacuum one, and provides the subdominant quadrupole modulation ∝ | g ∗ | ∝ E N .Therefore, h ζ n i ∝ E N n − ∝ | g ∗ | N n − , as indicated in eq. (4.5). It is also worth noting thateach internal line in the diagram produces in the final expression for h ζ n i a power spectrumwhich is function of the momentum carried on that line. For each given n , the diagram The higher order terms can be shown to give subdominant contributions [21]. More in general, see ref. [21]for the detailed computation of the power spectrum and bispectrum. The computation of the trispectrum isperformed analogously [18]. – 11 –hown in the figure needs to be summed over with the diagrams obtained by permuting theposition of the external lines. The diagrams shown in the figure factor out a P ζ ( k ), andare enhanced in the soft limit k → c n and d n are equal, the trispectrum ismore sensitive to g ∗ than the bispectrum by a factor of N ≈
60. In addition, we find that theerror bars on g ∗ from the trispectrum decrease as δg ∗ ∝ ℓ − , while those from the bispectrumdecrease more slowly as δg ∗ ∝ ℓ − . In reality, the error bars on the trispectrum parametersare much larger than those on the bispectrum parameters for smaller ℓ max ; thus, we find thatthe trispectrum yields smaller error bars on g ∗ than the bispectrum for ℓ max & ℓ max = 2000, we find δg ∗ = 1 . × − and 2 . × − from the d and d measure-ments, respectively. These error bars are an order of magnitude better than those expectedfrom the bispectrum measurements, and are comparable to that expected from the powerspectrum measurement for the same ℓ max (in the absence of systematic errors such as ellip-ticity of beams) [31]. Inflation models with anisotropic sources can create the perturbations with a preferred direc-tion, and yield distinct angular dependence not only in the power spectrum and bispectrum,but also in the trispectrum of the CMB. Motivated by inflation models with I ( φ ) F coupling,we have studied the observational consequence of the parametrized form of the trispectrumgiven by eq. (1.4). The expected 68% CL error bars on the trispectrum parameters are δd = 26 and δd = 105 for a cosmic-variance-limited experiment measuring temperatureanisotropy up to ℓ max = 2000. The error bar on d is too large to be useful.Using the prediction of inflation models with I ( φ ) F coupling, we derive the relation-ship between the trispectrum parameters and the power spectrum parameter, g ∗ . We thenfind that the trispectrum measurements can give competitive limit on g ∗ reaching δg ∗ = 10 − for ℓ max = 2000, which is an order of magnitude better than the expected limit from the bis-pectrum for the same ℓ max . This is owing to two effects: the trispectrum parameters areproportional to | g ∗ | N whereas the bispectrum parameters are proportional to | g ∗ | N ; andthe error bar on g ∗ from the trispectrum decreases as ℓ − whereas that from the bispectrumdecreases as ℓ − .The signatures of broken rotational invariance in the power spectrum [12] and thebispectrum [19] have been constrained by the temperature data of the Planck satellite. Theyhave yielded the limit on g ∗ of order 10 − . This limit can be improved further by using thetrispectrum. The current limit on d from the Planck data, d = τ NL / <
470 (95% CL) [19],implies | g ∗ | < .
02 (95% CL), which is indeed competitive. The other parameter, d , has notbeen constrained by the data yet. Measurements of d and d from the full data of Planck should yield the best limit on g ∗ within the context of inflation with I ( φ ) F coupling. Acknowledgments
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