A Small-Scale Modification to the Lensing Kernel Boryana Hadzhiyska,1 David Spergel,1, 2 and Joanna Dunkley1, 3 1Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08540 USA 2Flatiron Institute, Simons Foundation, 162 Fifth Avenue, New York, NY 10010, USA 3Department of Physics, Princeton University, Princeton, NJ 08544 USA Calculations of the Cosmic Microwave Background lensing power implemented into the standard cosmological codes such as CAMB and CLASS usually treat the surface of last scatter as an in- finitely thin screen. However, since the CMB anisotropies are smoothed out on scales smaller than the diffusion length due to the effect of Silk damping, the photons which carry information about the small-scale density distribution come from slightly earlier times than the standard recombination time. The dominant effect is the scale dependence of the mean redshift associated with the fluctua- tions during recombination. We find that fluctuations at k = 0:01 Mpc−1 come from a characteristic redshift of z ≈ 1090, while fluctuations at k = 0:3 Mpc−1 come from a characteristic redshift of z ≈ 1130. We then estimate the corrections to the lensing kernel and the related power spectra due to this effect. We conclude that neglecting it would result in a deviation from the true value of the lensing kernel at the half percent level at small CMB scales. For an all-sky, noise-free experiment, this corresponds to a ∼ 0:1σ shift in the observed temperature power spectrum on small scales (2500 . l . 4000). I. INTRODUCTION information need to have come from slightly earlier times than the standard recombination time [17, 18]. The dom- Density fluctuations along the line of sight distort the inant effect is the scale dependence of the mean redshift images of observed galaxies. This effect is generally associated with the fluctuations during recombination. known as gravitational lensing. By analyzing such dis- The standard calculation of the CMB lensing power torted images, one can obtain a map of the lensing poten- implemented into numerical codes such as CAMB [19] tial, which can then be related to the matter power spec- and CLASS [20] treats the surface of last scatter as an in- trum at a given redshift. In the case of cosmic microwave finitely thin screen. In this paper, we provide a modified background (CMB) photons, these lensing distortions en- estimation of the distance to last scattering as a function code information about the density fluctuations between of scale. This correction takes into account the scale de- the early Universe at z ≈ 1100 and the present-day Uni- pendence of the recombination redshift in the calculation verse. We observe them in the CMB anisotropies as slight of the lensing kernel, which is needed to obtain the lens- modifications to their statistical properties [1{3]. Over ing power spectrum. We finally evaluate the percentage the past decade, cosmologists have measured the CMB difference in the lensing kernel resulting from this mod- lensing signal through both auto-correlations and cross- ification and discuss its significance given the expected correlations with other density probes (e.g., cosmic in- sensitivity of future experiments. frared background, galaxy lensing, galaxy counts, 21 cm probes) [3{8]. Lensing measurements can put constraints on the nature of dark energy and the expansion history II. EFFECT ON LENSING KERNEL of the Universe [9{11]. Lower-redshift information can be inferred from weak The weighted projection of the matter density contrast lensing studies, which measure the distortions of the δ, known as the convergence field, encodes information shapes of galaxies caused by lensing. In both cases, the about the density fluctuations in the Universe since the goal is to reconstruct the convergence field, which can period of recombination and is expressed as: be directly related to the projected matter density by Z 1 measuring the magnification and shear effects from ei- κ(^n) = dzW κ(z)δ(χ(z)^n; z); (1) arXiv:1711.03168v3 [astro-ph.CO] 25 Jan 2018 ther distribution [12]. Since the lensing reconstruction 0 information is encoded mostly in the smallest scales ob- where χ(z) is the conformal distance between us and served, high resolution and sensitivity are crucial for such some event at redshift z [21]. In a flat universe, the measurements [13]. lensing kernel W κ is given by: With the improvement in sensitivity expected in fu- Z 1 ture experiments [14{16], it is becoming increasingly κ 3 2 1 + z χ(z) χ(zs) − χ(z) W (z) = ΩmH0 dzsps(zs) ; more important to take into consideration corrections to 2 H(z) c z χ(zs) the observed power spectra, which have until now been (2) negligible. One such effect comes from the fact that where ps(z) is the normalized distribution of sources as on scales smaller than the diffusion length, the density a function of redshift. anisotropies are smoothed out due to photon diffusion In the case of the CMB, it is standard to assume that damping. This means that photons carrying small-scale the photons come predominantly from the redshift of re- 2 1 ] 2 102 0.8 [Mpc 1 2 10 D 0.6 k 200 220 240 260 280 300 320 340 0.4 1.0 Visibility Function Visibility 0.2 ()0.5 0 v 600 800 1000 1200 1400 1600 Redshift 0.0 200 220 240 260 280 300 320 340 2 D 1.0 k / 2 k k = 0.01Mpc 1 e 0.5 k = 0.1Mpc 1 k = 0.2Mpc 1 , ) = k 1 ( k = 0.3Mpc D 0.0 200 220 240 260 280 300 320 340 1.0 large-scale modes intermediate modes small-scale modes ,) k ( 0.5 D () FIG. 1. The CMB anisotropies can be broken into small-scale, v intermediate-scale and large-scale. Due to diffusion damping, 0.0 200 220 240 260 280 300 320 340 the small-scale information is provided by photons which last [Mpc] scattered at redshifts larger than the redshift of recombina- tion. In contrast, at smaller redshifts, the information on small scales is lost, and large-scale information is provided by FIG. 2. Top two panels: Damping scale and visibility func- late-time photons. tion as a function of conformal time. Bottom two pan- els: Damping factor and the normalized product between the damping factor and the visibility function for different wavenumbers k. Smaller-scale modes are most likely to have combination z(η∗) ≡ z∗, so that one can approximate the scattered at earlier times. source distribution as ps ≈ δD(z − z∗) and thus obtain the kernel [12]: given mode last scattered: κ 3 2 1 + z χ(z) χ(z∗) − χ(z) W (z) = ΩmH0 : (3) 2 H(z) c χ(z ) 2 2 ∗ g(k; η) = e−k =kD (η) v(η): (4) We, thus, see that the usual approach for calculating the lensing kernel, also employed by the cosmological codes As seen in the lowest panel in Fig. 2, for small- CLASS and CAMB, treats the surface of last scatter scale modes, the product between the visibility func- as an infinitely thin screen. However, the CMB pho- tion and the damping factor peaks at earlier times than tons come from a range of redshifts which peaks at the the standard recombination time η∗, which shows that period of recombination. The photons which last scat- the CMB photons providing information on small-scale −1 tered at earlier times contain more small-scale informa- anisotropies (k & 0:1Mpc ) are more likely to have come tion than those coming from later times because as the from an earlier time than the mean recombination time. −1 diffusion damping scale increases with time, anisotropies On even smaller scales (k & 0:3Mpc ), where the pri- are smoothed out and information on small scales is lost mary anisotropies are washed out, the effective emission [17, 18]. The main effect is that the mean redshift associ- time is shifted to later times, as the motion of the photon- ated with the fluctuations during recombination becomes baryon fluid starts to be dominated by its infall into the scale-dependent. This effect is illustrated in Fig. 1. CDM potential wells during matter domination. In this This claim can be supported quantitatively by con- regime, the approximation which we are adopting breaks sidering the visibility function v(η), which expresses the down and more careful analysis is needed. However, since most probable time at which a CMB photon last scat- we are interested in the effect on the temperature and the 2 2 polarization power spectra for l 4000, we can neglect tered, and the damping factor exp[−k =kD(η) ], which . measures how much the growth of a given mode is sup- the baryon effect. pressed as a function of time [22]. Their product, com- A plot of conformal time η∗(k) [Mpc] versus wavenum- puted for each mode, informs us about the most likely ber k [Mpc−1] obtained by numerically computing the time at which the photons encoding information on the peak position for each mode is shown in Fig. 3. We fit a 3 cubic polynomial to this function, finding the form: 4 3 η∗(k) = −2:14[ln(k)] − 15:67[ln(k)] 2 0.014 z = 1 −42:46[ln(k)] − 50:77[ln(k)] + 257:76 (5) [%] 0.012 CMB , 0.010 W )/ 0.008 CMB , 0.006 ] 280 c p 0.004 M [ 279 W ) g k n 0.002 i ( r e W t 278 ( t 0.000 a c 0.00 0.05 0.10 0.15 0.20 0.25 0.30 S 277 t s a L f 276 o 0.06 z = 5 [%] e m i 275 0.05 T CMB Numerical answer , Polynomial fit 274 W 0.04 10 2 10 1 )/ 1 Wavenumber k [Mpc ] 0.03 CMB , FIG.