PHYSICAL REVIEW D, VOLUME 63, 023504 Gravitational time delay effects on cosmic microwave background anisotropies Wayne Hu* Institute for Advanced Study, Princeton, New Jersey 08540 and Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637 Asantha Cooray† Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637 ͑Received 2 August 2000; published 18 December 2000͒ We study the effect of gravitational time delay on the power spectra and bispectra of the cosmic microwave background ͑CMB͒ temperature and polarization anisotropies. The time delay effect modulates the spatial surface at recombination on which temperature anisotropies are observed, typically by ϳ1 Mpc. While this is a relatively large shift, its observable effects in the temperature and polarization fields are suppressed by geometric considerations. The leading order effect is from its correlation with the closely related gravitational lensing effect. The change to the temperature-polarization cross power spectrum is of order 0.1% and is hence comparable to the cosmic variance for the power in the multipoles around lϳ1000. While unlikely to be extracted from the data in its own right, its omission in modeling would produce a systematic error comparable to this limiting statistical error and, in principle, is relevant for future high precision experiments. Contributions to the bispectra result mainly from correlations with the Sachs-Wolfe effect and may safely be neglected in a low density universe. DOI: 10.1103/PhysRevD.63.023504 PACS number͑s͒: 98.80.Es, 95.85.Nv I. INTRODUCTION small effects can accumulate along the path. Indeed it is well known that the gravitational lensing of CMB photons has a In order that the full potential of anisotropies in the cos- substantial effect on the power spectrum of the anisotropies mic microwave background ͑CMB͒ temperature and polar- ͓11,12͔. ization fields be realized, effects that have been previously In addition to the lensing effect, gravitational potentials of dismissed as negligible in their own right must now be re- large-scale structure contribute a time-delay ͓13͔ that accu- considered as potential sources of systematic error. Projec- mulates along the path—an effect familiar from studies of tions as to the ability of CMB experiments to measure fun- the light-curves of lensed quasars ͑e.g. ͓14͔͒. In the case of damental cosmological quantities ͓1͔ precisely and reveal the CMB, the time-delay warps the spatial surface at recom- information about the structure formation process from sec- bination from which the primary anisotropies arise ͓5͔. Be- ondary effects ͓2͔ rely on the fact that statistical errors from cause the lensing depends on the angular gradient of the the sampling of a finite sky rapidly decrease toward smaller projected potentials whereas the delay depends on the pro- angular scales. Statistical errors in the power spectra decline jected potential itself, the fractional delay is generically from ϳ1% at degree scales to ϳ0.1% at the several arc- smaller than lensing and has not been explicitly calculated in minute scale. To achieve this precision in practice, all physi- the literature. We shall see however that because of the an- cal, astrophysical and instrumental effects at this level must gular smoothness ͑or coherence͒ of the lensing, the reduction be included in the analysis to avoid generating systematic in amplitude is not in and of itself large. Furthermore, gravi- errors that are comparable to the statistical errors. tational time delays are strongly correlated with lensing, A host of physical effects contribute to the anisotropies at leading to additional effects in the power spectrum. Indeed, second order in perturbation theory ͓3–6͔. Since primary the typical perturbation in comoving units is on the order of anisotropies are formed at recombination when the cosmo- 1 Mpc. The effect of gravitational time-delay on the spectra logical density perturbations are at the 10Ϫ5 level, most sec- of temperature and polarization anisotropies therefore merits ond order effects are entirely negligible. There are two gen- further study. eral ways in which higher order effects can be important. In Sec. II we present the formalism required to understand First, the primordial perturbations responsible for the pri- these gravitational effects on the temperature and polariza- mary anisotropies grow into non-linear structures today by tion fields. We proceed in Secs. III and IV to evaluate the gravitational instability. Effects that take advantage of this delay and lensing-delay correlation effects on the power fact mainly involve scattering of CMB photons at low red- spectra of temperature and polarization anisotropies respec- shifts in large-scale structure and non-linear objects ͓7–10͔. tively. In Sec. V, we consider their effects on the bispectra Secondly, since recombination, CMB photons propagate ͑three point correlations͒. We conclude in Sec. VI with a across essentially the whole horizon volume. Intrinsically discussion of our results. To illustrate our calculations, we assume a cold dark mat- ter model ͑CDM͒ with a cosmological constant (⌳) with ⍀ ϭ ⍀ ϭ *Email address: [email protected] parameters c 0.30 for the CDM density, b 0.05 for the † Email address: [email protected] baryon density, ⍀⌳ϭ0.65 for the vacuum density, hϭ0.65 0556-2821/2000/63͑2͒/023504͑11͒/$15.0063 023504-1 ©2000 The American Physical Society WAYNE HU AND ASANTHA COORAY PHYSICAL REVIEW D 63 023504 FIG. 1. Gravitational lensing vs time delay. Lensing introduces an angular perturbation in the mapping of a plane-wave source field at recombination onto anisotropies today. Time delay introduces a radial modulation. When the wavevector is perpendicular to the line-of-sight, features in the angular spectrum—such as the acoustic FIG. 2. Power spectra for the lensing deflection angles (␾␾), peaks—are created, geometrically distinguishing the otherwise time-delay (dd) and deflection-delay cross correlation (␾d). The ␾␾ ␾d similar lensing and delay effects. underlying lensing potential spectrum Cl and cross spectrum Cl are weighted by l(lϩ1) and ͓l(lϩ1)͔1/2 respectively to reflect the for the dimensionless Hubble constant and a scale invar- angular gradients in the deflection angles. iant spectrum of primordial fluctuations, normalized to COBE ͓15͔. used the Born approximation to evaluate the lensing poten- tial on the unperturbed path; it has been shown to be an ͓ ͔ II. FORMALISM excellent approximation for lensing 18 . We have also used a Newtonian approximation where the time-time perturba- As the CMB photons propagate to the observer from the tion ⌿ϭϪ⌽. It is also an excellent approximation since recombination epoch (zϳ103) through large-scale structure anisotropic stresses from the radiation are negligible at the in the universe, they suffer the effects of gravitational lensing low redshifts involved ͓19͔. and time delay. These effects are both formally second order The projected potential itself is a field on the sky and may in perturbation theory because they would leave a homoge- be decomposed into multipole moments neous and isotropic CMB unperturbed ͑cf. ͓16͔͒. There are a host of other second order effects ͓3–5͔. To the extent that ␾͑ ˆ ͒ϭ ␾ m͑ ˆ ͒ ͑ ͒ n ͚ lmY l n , 3 they are uncorrelated with each other, they may be viewed as lm independent effects. The lensing and time delay effects are strongly correlated because both arise from the gravitational and described by its power spectrum potentials of large scale structure and must be considered together. We therefore begin with a review of lensing effects. ͗␾ ␾ ͘ϭ␦ ␦ ␾␾ ͑ ͒ lm* lЈmЈ l,lЈ m,mЈCl . 4 A. Lensing This power spectrum is shown in Fig. 2. In the Newtonian Lensing involves a deflection that remaps the temperature approximation its shape simply reflects the underlying den- sity power spectrum tilted according to the Poisson and Lim- and polarization fields according to the angular gradient of ␾␾ ber equations: P(k)ϰkn implies C ϰlnϪ4 with lϳk␩ /2. the lensing-weighted projected potential nˆ!nˆϩٌ␾(nˆ) ͑see l 0 Fig. 1͒ A useful measure of the amplitude of the lensing effects is ␪2 the rms deflection angle rms as defined by ␾͑ ˆ ͒ϭϪ ͵ ␩ ͑␩͒⌽͑ ˆ ␩͒ ͑ ͒ ϱ n 2 d g␾ rn, , 1 2lϩ1 ␪2 ϭ ͑ ϩ ͒ ␾␾ ͑ ͒ rms ͚ l l 1 Cl . 5 lϭ1 4␲ where The factor l(lϩ1) reflects the angular gradient in the defini- ␩ ЈϪ 1 Ϫ␶ r r tion of the deflection. Note that the lϭ0 monopole does not g␾͑␩͒ϭ ͵ d␩Ј ˙␶e . ͑2͒ r 0 rЈ contribute since its angular gradient vanishes. In the fiducial ⌳ ␪ ϭ ϫ Ϫ4 CDM model rms 7.5 10 or 2.6Ј. Note that this is in Here overdots represent derivatives with respect to confor- sharp contrast with the angular coherence of the deflection ␪2 len mal time ␩ϭ͐dt/a, ⌽(x,␩) is the Newtonian gravitational angle. The variance rms reaches half its total value by l1/2 ϭ ϭ ␪lenϵ ␲ len potential and c 1 throughout. In an open universe, the con- 30 in the fiducial model or an angle of 1/2 2 /l1/2 ␩ ϭ␩ Ϫ␩ formal distance traveled by a photon r( ) 0 should be Ϸ0.2 (10°). The smaller scale potential fluctuations tend to replaced by angular diameter distances. We have implicitly produce deflections that cancel out along the line of sight. 023504-2 GRAVITATIONAL TIME DELAY EFFECTS ON COSMIC . PHYSICAL REVIEW D 63 023504 B. Time delay C. Temperature field Now consider the time-delay effect. Photons follow null The CMB temperature field on the sky may be written geodesics in the perturbed metric so that in the fixed time implicitly as the projection of sources S which contribute in interval since last scattering, the distance traveled by the the optically thin regime and are so weighted by eϪ␶ where ␶ photons is perturbed as r!r͓1ϩd(nˆ)͔ ͑see Fig.