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CU-TP-767CAL-615 astro-ph/9609132 A Probe of Primordial Gravity Waves and Vorticity Marc Kamionkowski∗ arXiv:astro-ph/9609132v1 19 Sep 1996 Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027 Arthur Kosowsky† Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138 and Department of Physics, Lyman Laboratory, Harvard University, Cambridge, Massachusetts 02138 Albert Stebbins‡ NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500 (September 1996) Abstract A formalism for describing an all-sky map of the polarization of the cosmic microwave background is presented. The polarization pattern on the sky can be decomposed into two geometrically distinct components. One of these components is not coupled to density inhomogeneities. A non-zero amplitude for this component of polarization can only be caused by tensor or vector metric perturbations. This allows unambiguous identification of long-wavelength gravity waves or large-scale vortical flows at the time of last scattering. 98.70.V, 98.80.C Typeset using REVTEX ∗ [email protected] † [email protected] ‡ [email protected] 1 Calculations of this sort have been done with a small-angle approximation. Inflation damps out vector modes but will produce some tensor modes and also predicts a specific relationship between the spectrum of the scalar and tensor fluctuations [2]. vector. since rotational non-invariance disappears when considering only a small patch of the sky. which constitute a complete orthonormal set of rank-2 STF tensors on the sphere. topological defects. QU. (1) The “ab” are the tensor indices. For example. topological defects will produce a mixture of scalar. describing linear polarization.and cross-correlations between the Stokes parameters Q and U and the temperature T have been considered. COBE has already mapped the polarization pattern with an angular resolution of 7◦ (although the data has not been analyzed). can be described as a tensor field on the celestial sphere. or perhaps some other mechanism. results from numerous balloon-borne and ground-based experiments. while in the foreseeable future we can only expect to observe the consequences of tensor or vector modes through their effects on the CMB. any mechanism which produces temperature anisotropies will invariably lead to non-zero polarization as well [3–5]. in spherical polar coordinates (θ. However. Detection of a stochastic gravity-wave background (tensor modes) [1] or vortical motions in the primeval fluid (vector modes) would help to discriminate between these models.With the COBE detection of large-angle anisotropy in the cosmic microwave background (CMB). Scalar modes give rise to both the observed large-scale structure and CMB fluctuations. As we demonstrate in this Letter. this polarization signal can be used to discriminate between scalar and vector or tensor metric perturbations. the CMB is becoming an increasingly precise probe of the early Universe. this formalism is not optimal for describing the complete temperature and polarization correlations present in full-sky maps. Here we present a rotationally covariant formalism for describing the polarization pattern on a full sky.8]. UU. In prior work. there is no rotationally invariant way to lay down orthogonal basis vectors on a sphere. the contribution of scalar. vector. However. one of each for every 2 . are just given by the symmetric trace-free (STF) part of this tensor. The Q and U parameters. Without a model of primordial fluctuations. so the meaning of QQ. φ). sin2 θ). and we use standard tensor notation throughout. and tensor modes. and tensor modes to the CMB temperature anisotropy are indistinguishable. defined by the 2 × 2 correlation matrix of the electric field of incoming photons. the auto. and MAP [6] will measure the polarization with a resolution of around 0. Y(lm)ab and Y(lm)ab . QT . It is natural to decompose the linear-polarization pattern into STF tensor spherical harmonics [7. G C There are two types of harmonic STF tensors. CMB anisotropies will help determine whether density perturbations (scalar modes) are the result of inflation. and the advent of a new generation of satellite missions. The Stokes parameters. or UT will depend on absolute positions of the points being correlated rather than just the relative position. the polarization tensor is 1 Pab (ˆ n) = 2 Q(ˆ n) −U(ˆ n) sin θ −U(ˆ n) sin θ −Q(ˆ n) sin2 θ ! . where the spherical metric is gab = diag(1. While this formalism does provide a complete description of the polarization. In contrast. Here Q and U are defined with respect to particular orthogonal axes on the celestial sphere.3◦ . C∗ dˆ n Y(lm)ab (ˆ n)Y(lC′ mab′ ) (ˆ n) = δll′ δmm′ . Since Compton scattering can produce no net circular polarization. Two sets of tensor harmonics are required as there are two modes of linear polarization. (lm) Y(lm) (ˆ T0 lm i Pab (ˆ n) X h G G C a(lm) Y(lm)ab (ˆ n) + aC Y (ˆ n ) . 2 Nl c Y(lm):ac ǫ b + Y(lm):bc ǫc a . = 2 q (5) Here Nl = 2(l − 2)!/(l + 2)! is a normalization factor. and “:” indicates a covariant derivative on the sphere. and the V Stokes parameter will not be considered further. = T0 (3) which can be derived from the orthonormality properties Z Z Z Z ∗ dˆ n Y(lm) (ˆ n)Y(l′ m′ ) (ˆ n) = δll′ δmm′ . the CMB is expected to have V = 0. T0 Z 1 G ab ∗ = dˆ n Pab (ˆ n) Y(lm) (ˆ n). (4) Here T0 is the cosmological mean CMB temperature and we are assuming Q and U are measured in brightness temperature units rather than flux units. hence we use the notation G for “gradient” and C for “curl”. This decomposition is quite similar to the decomposition of a vector field into a part which is the gradient of a scalar field and a part which is the curl of a vector field.one of the usual spherical harmonics Y(lm) with l ≥ 2. Q and U. ǫab is the completely antisymmetric tensor. the Y(lm)ab ’s and Y(lm)ab ’s provide a complete basis for G-type and C-type STF 3 . T0 Z 1 C ab ∗ dˆ n Pab (ˆ n) Y(lm) (ˆ n). Since the Y(lm) ’s provide a complete basis for scalar functions on G C the sphere. G∗ dˆ n Y(lm)ab (ˆ n)Y(lC′ mab′ ) (ˆ n) = 0. The harmonic expansion of an all-sky map of the CMB temperature and polarization can be written X T (ˆ n) =1+ aT n). In two dimensions. any STF tensor can be uniquely decomposed into a part of the form A:ab − (1/2)gabA:c c and another part of the form B:ac ǫc b +B:bc ǫc a where A and B are two scalar functions. The two geometrically distinct tensor harmonics are G Y(lm)ab C Y(lm)ab 1 = Nl Y(lm):ab − gab Y(lm):c c . = (lm) (lm)ab T0 lm (2) The mode amplitudes are given by aT (lm) aG (lm) aC (lm) Z 1 ∗ = dˆ n T (ˆ n) Y(lm) (ˆ n). G∗ dˆ n Y(lm)ab (ˆ n)Y(lG′ mab′ ) (ˆ n) = δll′ δmm′ . and thus the Fourier vector does not completely define the direction of the polarization orientation. Another way to state this argument is that scalar perturbations have no handedness so they cannot produce any “curl”. If the cosmological inhomogeneities are Gaussian random noise. i. (6) with the definitions W(lm) ≡ X(lm) ! ∂ m2 ∂2 − cot θ + Y(lm) . In (θ. consider a scalar perturbation with single Fourier mode k in the ˆ z direction. respectively. in linear theory. in a given region of the sky all of the orientations are parallel and thus the polarization pattern has no curl. This is in contrast to tensor or vector metric perturbations which will produce a mixture of both types. U} ↔ {U.tensors. ∂θ2 ∂θ sin2 θ 2im ≡ sin θ ! ∂ − cot θ Y(lm) . or α = π/2 if the orientation is perpendicular to the direction of k. the harmonics are given explicitly by G Y(lm)ab (ˆ n) Nl = 2 W(lm) X(lm) sin θ X(lm) sin θ −W(lm) sin2 θ C Y(lm)ab (ˆ n) Nl = 2 −X(lm) W(lm) sin θ W(lm) sin θ X(lm) sin2 θ ! ! . then to the extent linear theory is valid. Similarly only G-type polarization contributes to P ab :ab . We now turn to statistics of CMB polarization. φ) coordinates. (1) holds. To understand why scalar metric perturbations do not produce a C-type polarization pattern. scalar perturbations can produce only G-type polarization and not C-type polarization.e. In either case. when vector or tensor perturbations are present. the orientation of the polarization can be determined only by the direction of k: thus α = 0 if the polarization orientation is along the direction of k. the azimuthal symmetry in the scalar case is explicitly broken. −Q} as G↔C indicates that Y(lm)ab and Y(lm)ab ◦ represent polarizations rotated by 45 . A most useful property of the G/C decomposition is that. Given a polarization map of even a small part of the sky one could in principle test for vector or tensor contribution by computing the combination of derivatives of the polarization given by P ab :bc ǫc a which will be non-zero only for C-type polarization. more robust measures are given below. the CMB fluctuations 4 . For tensor and vector perturbations. where tan 2α = U/Q. This G/C decomposition is also known as the scalar/pseudo-scalar decomposition [8]. Since the curl is a linear operator. taking derivatives of noisy data is problematic. For scalar perturbations. summing over Fourier modes does not alter this conclusion. choose θ). The polarization in a given direction can be represented by a magnitude P = (Q2 + U 2 )1/2 ˆ and an orientation angle α from the axis defining the Stokes parameters (here. Of course. whereas vector and tensor perturbations do have a handedness and therefore can. where Eq. ∂θ (7) G C The exchange symmetry {Q. Finding a non-zero component of C-type polarization in the CMB would provide compelling evidence for significant contribution of either vector or tensor perturbations at the time of last scattering. indicating a handedness to the inhomogeneities in our universe. this statistic provides a powerful and unambiguous model-independent probe of tensor and vector perturbations. rotational invariance requires that the 2-point correlations be of the form E D T ∗ T aT (lm) a(l′ m′ ) = Cl δll′ δmm′ . 5 . for X =T. D D D D E ∗ G G aG (lm) a(l′ m′ ) = Cl δll′ δmm′ . Measuring a non-zero Cl and/or Cl would be quite interesting.will also be Gaussian random noise. The mean square polarization is Q2 + U 2 = 2P ab Pab = PG2 + PC2 (11) where ∞ X PG2 2l + 1 d = ClG . Given an all-sky temperature/polarization map.e. then G C ClTC = ClGC = 0 since the Y(lm) and the Y(lm)ab have parity (−1)l while the Y(lm)ab have parity l+1 TC GC (−1) . 2 T0 8π l=2 (12) Since scalar modes do not contribute to PC2 . m=−l 2l + 1 2 l X |aC (lm) | d C Cl = . are related to various quantities in previous work (see Ref. one can determine the a(lm) ’s using Eq. 2 |aT (lm) | . vector. the same techniques developed to analyze anisotropy with incomplete sky coverage [10] may be applied to polarization to construct other estimators of the various Cl ’s. (3). (8) If we also require that the distribution of inhomogeneities be invariant under parity. The first is the well-known angular power spectrum of temperature anisotropies while the last three. We have argued that ClC scalar = 0. E GC ∗ C aG (lm) a(l′ m′ ) = Cl δll′ δmm′ . and tensor contribution to each of the Cl ’s adds in quadrature. 2l + 1 m=−l d G C l = l X (10) If only part of the sky is mapped. ClG . and then construct estimators for the Cl ’s in the usual way. ClTG }. Regardless of whether the distribution is Gaussian. new to this paper. m=−l 2l + 1 d T C l = l X 2 |aG (lm) | . i. ClC . Note that the scalar.e. E E TG ∗ G aT (lm) a(l′ m′ ) = Cl δll′ δmm′ . (9) and this is true whether or not the fluctuations are Gaussian. i. m=−l 2l + 1 ∗ G l X aT (lm) a(lm) d TG Cl = . C.. [9]). However we do not expect this and will henceforth only consider the four angular power spectra {ClT . E TC ∗ C aT (lm) a(l′ m′ ) = Cl δll′ δmm′ . TG ClX = ClXscalar + ClXvector + ClXtensor . 2 T0 8π l=2 ∞ X PC2 2l + 1 dC = Cl . D ∗ C C aC (lm) a(l′ m′ ) = Cl δll′ δmm′ . G. along with undetermined cosmological parameters. Although correlation functions of Stokes parameters which appear in the previous literature depend on the positions of the points being correlated. the most information can be extracted from the map by comparing the entire set of predicted moments. but the ClC ’s are not contaminated in this way. ClG . {ClT . This small-angle formalism is completely analogous to that developed above and will provide an accurate description of a region of sky small enough to be approximated by a flat surface. 0)]. then the vector or tensor signal in {ClT . Note that only the ClC ’s potentially allow detection of a small vector or tensor signal. define Stokes parameters Qr and Ur with respect to axes which are parallel and perpendicular to the great arc (or geodesic) which connects the two points being correlated. we can write the two-point temperature and polarization correlation functions [3–5. The two-point correlation functions are C T (θ) = hT (ˆ n1 )T (ˆ n2 )inˆ1 ·ˆn2 =cos θ X 2l + 1 = T02 ClT Pl (cos θ). ClG . 0) + iClG X(l2) (θ. 4π l (13) where θ is the angle separating the two points and Pl2 is the m = 2 associated Legendre function.13–16] can be reproduced by replacing the Y(lm) (ˆ n)’s in our formalism with Fourier modes. ClTG } may be swamped by the cosmic variance in the scalar signal. so the pixel-noise variance to these moments will be substantially reduced. ClC .13–16] in terms of multipole moments [9]. Furthermore. which differ for scalar. 0) + iClC X(l2) (θ. The G/C decomposition in the small-angle formalism can be used to detect non-scalar perturbations on small scales. To make contact with previous work.To test a given spectrum of tensor modes against a polarization map. vector. and using regular derivatives rather than covariant ones. comparing the d C complete set of predicted ClC with the estimators C l is more powerful than considering only PC2 . Therefore. however. = T02 4π l C T Q (θ) = hT (ˆ n1 )Qr (ˆ n2 )inˆ1 ·ˆn2 =cos θ X 2l + 1 = T02 Nl ClTG Pl2 (cos θ). the temperature-polarization cross correlation may be measured with some precision. [3–5. with the measured estimators [11.12]. eil·ˆn. If scalar perturbations dominate. ClTG }. 0)]. Since Qr and T are invariant under reflection along the great arc connecting the 6 . the noise in the temperature and polarization maps will be to a large extent uncorrelated. if the detection has sufficient signal-to-noise. 4π l Q C (θ) = hQr (ˆ n1 )Qr (ˆ n2 )inˆ1 ·ˆn2 =cos θ X 2l + 1 [ClG W(l2) (θ. will be larger than the polarization autocorrelation moments. The cross-correlation moments ClTG . Usually. If so. = T02 4π l C U (θ) = hUr (ˆ n1 )Ur (ˆ n2 )inˆ1 ·ˆn2 =cos θ X 2l + 1 [ClC W(l2) (θ. Much of the small-angle formalism of Refs. and tensor perturbations [13]. To construct them. though the tensor and vector signal are liable to drop off rapidly at angular scales smaller than a few degrees. the theory being tested has scalar as well as non-scalar modes. rotationally-invariant correlation functions exist which are closely related to those discussed above. isocurvature.E. NASA AST94-19400 at FNAL.K. Polarized emission from foreground sources is a relatively unknown factor at this time. and the Alfred P. C T Q (θ)} are a different way of representing {ClG . Seljak for helpful discussions.two points while Ur changes sign. Foreground emission and any Faraday rotation will certainly contribute to the C-type polarization. acknowledges the hospitality of the NASA/Fermilab Astrophysics Center and the CERN Theory Group. Polarization measurements will be difficult. although it may be larger in reionized [15]. but the promise of using them to detect gravity waves or vorticity. Eqs. Jungman and U. Sloan Foundation at Columbia. NASA NAG5-3091. hQr Ur i = hUr T i = 0 if statistical invariance under parity holds. and the Harvard Society of Fellows. C Q (θ). or topological-defect models. Thus the polarization signal is at least an order of magnitude below current experimental sensitivities. A. C U (θ).S. will likely attain the raw sensitivity necessary for detailed polarization investigations. M. In adiabatic models with standard recombination the polarization is only a few percent of the anisotropy. any measurable C-type polarization is a likely indicator of contamination. On sub-degree scales. In Gaussian models either set provides a complete statistical description of the temperature and polarization patterns. however. (13) reduce to the correct small-angle formulae [16] when θ ≪ 1. This work was supported by D. ClC . ClTG } and vice versa. acknowledges the hospitality of the Aspen Center for Physics and the support of the NASA grant NAG 5-2788. and hence to discriminate between cosmological models. ClT . contract DEFG02-92-ER 40699. Experiments planned or envisioned over the coming decade. where the signal from vector and tensor modes are liable to be negligible. The largest hurdles to detecting and characterizing CMB polarization are sensitivity and foregrounds. 7 . We thank G. The functions {C T (θ). but these contaminants can be subtracted using multi-frequency observations. makes these measurement potentially very valuable.O. A. 464. K. Coulson. D. Phys. Phys. J. [4] A. Kosowsky. Lett. J. 413 (1996). G.nasa. 521 (1995). 464. 69.-W. Turok. and A. Lett. Thorne. Phys. L. M. 76. Kosowsky. Astrophys. D. Phys. J. Davis et al. Lett. [3] J. [13] R. R. Stebbins. Wise. Lett. Crittenden. G.REFERENCES [1] L. Astrophys. Lett. [12] G. 2203 (1970). Spergel. Seljak. Mon. R. Nucl. L47 (1984). Wright et al. Spergel. [2] A. Phys. Tegmark. in preparation. R. Zaldarriaga. 3276 (1995). M. Math. R. and N. R. 96 (1993). Astrophys. F. Kamionkowski. Lett. D 51. 426.-W.gov. B 319. 1332 (1996). Phys. 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