Dechiphering Precision Cosmic Microwave Background Data By MARIUS MILLEA M.S. (University of California, Davis) 2015 DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Physics in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Prof. Lloyd Knox, Chair Prof. Andreas Albrecht Prof. Chris Fassnacht 2015 i Pro Q ue st Num b e r: 3723682 All rig hts re se rve d INFO RMATIO N TO ALL USERS The q ua lity o f this re p ro d uc tio n is d e p e nd e nt up o n the q ua lity o f the c o p y sub m itte d . In the unlike ly e ve nt tha t the a utho r d id no t se nd a c o m p le te m a nusc rip t a nd the re a re m issing p a g e s, the se will b e no te d . Also , if m a te ria l ha d to b e re m o ve d , a no te will ind ic a te the d e le tio n. Pro Q ue st 3723682 Pub lishe d b y Pro Q ue st LLC (2015). Co p yrig ht o f the Disse rta tio n is he ld b y the Autho r. All rig hts re se rve d . This wo rk is p ro te c te d a g a inst una utho rize d c o p ying und e r Title 17, Unite d Sta te s Co d e Mic ro fo rm Ed itio n © Pro Q ue st LLC. Pro Q ue st LLC. 789 Ea st Eise nho we r Pa rkwa y P.O . Bo x 1346 Ann Arb o r, MI 48106 - 1346 Abstract This thesis centers around work which was performed in the lead up to and during analysis of high resolution cosmic microwave background (CMB) data coming from the South Pole Telescope and from the Planck satellite. Analysis of such data necessitates modeling of extra-galactic foregrounds, and here we develop a model for these foregrounds for use with current data. This model is shown to be both necessary and suﬃcient for unbiased estimation of cosmological parameters. Additionally, we develop a novel method for using auxiliary data in the form of galaxy number counts to put priors on certain components of this model. Finally, we perform an analysis of CMB data, taking into account these extra-galactic foregrounds, to calculate new constraints on hypothetical particles like axions or axion-like particles. We ﬁnd new data from Planck and from measurements of primordial deuterium newly rule out some theoretically interesting regions of parameter space. In addition, we show that in a more speculative scenario which includes the presence of both axions and other dark radiation, some of these regions are again allowed and could explain hints of high primordial helium fraction, or could allow one or two thermalized sterile neutrinos to exist. ii Acknowledgments Thanks to Lloyd Knox, who was the best advisor anyone could ask for. I couldn’t be where I am today without your help, and I hope to make you proud with where I am going. Thanks to my friends and family who supported me throughout this journey and made sacriﬁces so I could be here. Finally, thanks to the South Pole Telescope and Planck collaborations for allowing me to join and work with such cutting edge data, and for the chance to give many talks on these results. iii Contents Abstract ii Acknowledgments iii List of Figures vii List of Tables xii Chapter 1. 1.1. 1 Theoretical Background Chapter 2. 2.1. Introduction 3 Modeling Extragalactic Foregrounds Modeling 2.1.1. 6 9 Emission from External Galaxies 9 2.1.1.1. Radio Galaxies 10 2.1.1.2. Dusty Star-Forming Galaxies 10 2.1.1.3. Polarization 12 2.1.2. Thermal SZ Eﬀect 12 2.1.3. Kinetic SZ Eﬀect 16 2.1.3.1. Ostriker-Vishniac Eﬀect 16 2.1.3.2. Patchy Reionization 18 2.1.4. tSZ-DSFG Correlation 19 2.1.5. Galactic Foregrounds 20 2.1.6. CMB 20 2.2. Fiducial Model and Current Constraints 21 2.3. Survey Properties 22 2.4. Forecasting Methodology 24 2.4.1. 2.5. Non-Gaussianity 26 Compression to a CMB Power Spectrum Estimate iv 28 2.5.1. Splitting the power spectra into CMB–free and a CMB estimate 28 2.5.2. Modeling the foreground residuals with principal components 31 2.5.3. Discussion of linear combination analysis 31 2.6. Results 32 2.6.1. Importance of the Diﬀerent Components 33 2.6.2. Modeling Suﬃciency 35 2.6.3. Statistical Error Increase with and without Auxiliary Data 37 2.7. Conclusions Chapter 3. 38 Predicting Poisson Foreground Amplitudes 40 3.1. Methodology 40 3.2. Source Count Modeling 41 3.3. Results 43 3.4. Conclusions 44 Chapter 4. New Constriants on Axions and Axion-like Particles 48 4.1. The Scenario 49 4.2. Constraints 55 4.2.1. Cosmic Microwave Background 55 4.2.1.1. Frequency Spectrum 55 4.2.1.2. Angular Power Spectrum 60 4.2.2. Primordial Abundance Inferences 61 4.2.3. Laboratory 66 4.2.4. Globular Clusters and SN1987A 66 4.3. Discussion 67 4.3.1. The MeV-ALP Window 67 4.3.2. A Loophole in the Presence of Extra Radiation 70 4.3.3. A Simple Expression for Exclusion Bounds 71 4.3.4. Forecasts 72 Conclusion 73 4.4. Chapter 5. Conclusion 75 v Appendix A. Principal Component Analysis for Power Spectra 76 Appendix B. CMB Linear Combination Generalization to Oﬀ-Diagonal Correlations 77 Bibliography 79 vi List of Figures 2.1 Three model DSFG clustering auto-spectra at 217 GHz (black), and approximations to them with our parameterized model (solid, red), all normalized (with one exception) at = 3000. Our ﬁducial model is the thickest curve. Also plotted are estimates of the clustering power from Planck [Planck Collaboration, 2011r] and SPT [Hall et al., 2010]. For both sets of data points we have subtracted estimates of the Poisson power from the reported total CIB power. The lowest amplitude solid (red) curve is the result of a “by-hand” adjustment of our model parameters to ﬁt the Planck and SPT data. 13 2.2 Comparison between recent models and simulations of the tSZ eﬀect (black lines) and ﬁts of our PCA model to each (thin red lines). The thickest red line shows the ﬁducial tSZ power spectrum used in this work. All results are plotted at 146 GHz and are scaled to σ8 = 0.8. The blue arrow shows the SPT 95% conﬁdence upper limit on thermal SZ power at = 3000 [Shirokoﬀ et al., 2010]. 15 2.3 Comparison between recent simulations of the kSZ eﬀect (black lines) and our model. We consider contributions from the post-reionization kSZ eﬀect (solid red line) and from patchy reionization (dashed red line). Note that the simulations plotted here assume homogeneous reionzation and thus do not include a patchy contribution. The SPT 95% conﬁdence upper limit for the kSZ power at = 3000 is 6.5 μK2 . 16 2.4 All 36 power spectra which can be formed from Planck 70 GHz–217 GHz temperature and E-mode polarization, and the prediction of our ﬁducial model for the CMB and foreground power in each of them (with the exception of the tSZ-DSFG correlation which is shown at 30% instead of its ﬁducial value of 0%). The black dashed line shows the errors bars for -bins of width Δ = 256. Dotted lines indicate negative power. 2.5 Top: The -dependent weightings which form the CMB linear combination (Eq. 2.29). All possible auto/cross spectra from Planck channels in Table 2.2 were considered. Dashed lines indicate negative weight. Middle: The mean foreground contribution to the CMB vii 25 linear combination for our ﬁducial model. Note for example that tSZ (purple) is not present at high because only 217 GHz is used there. The dominant non-Poisson component for the -range where Planck is most sensitive is the DFSG clustering. Bottom: Principal components of foreground residuals (constrained by the CMB–free linear combinations) with amplitudes set to 1-σ. Note that we only need two principal component amplitudes to be accurate to > 1μK2 . (The errors in bin widths of Δ = 256 for both the CMB linear combination and for 217 GHz alone are plotted as dashed lines in the bottom two plots.) 29 2.6 68% (and 95% in the bottom panel) conﬁdence contours for a suite of test cases examining the eﬀect of neglecting to model diﬀerent foregrounds. Unless explicitly stated above, other parameters were included in the data at their ﬁducial values listed in Table 2.3 and were marginalized over in the analysis. N corresponds to the maximum number of -bins per power spectrum one could use and still detect the error in modeling at 3 sigma (see Sec. 2.6.1 for further discussion). 34 2.7 The eﬀect on cosmological parameters from trying to ﬁt our model to simulated data which includes (orange) the Battaglia et al. [2010] tSZ template and (black) the Amblard and Cooray [2007] clustering template. These two models are the most dissimilar to ours, and thus show our model can protect against biases of a few percent up to Planck sensitivity. The inclusion of Ground data necessitates more detailed modeling of only the clustering. 36 3.1 The blue points with error bars show the M13 source count data, and the line drawn through them is our best ﬁt model integrated down to individual frequencies. At 150 GHz we also show the dusty sources in green, as these are used in calculating the dusty Poisson contribution there (see text). The lines at the bottom of each plot show the fractional contribution from each logarithmic ﬂux bin to the Planck radio Poisson power. Dotted/dashed/dot-dashed lines are for the contribution at 100, 143, and 217 GHz. For example, the dotted line in the 220 GHz plot corresponds to the contribution to the 100 GHz Poisson power from sources with those particular 220 GHz ﬂuxes. The normalization is arbitrarily. The best-ﬁt model is used to calculate these curves. 3.2 Posteriors on radio Poisson amplitudes at 100 and 143 GHz from LCDM chains when the data used is (blue) Planck2013+WP (green) Planck2013+WP+highL. These posteriors are nearly unchanged for extended models. The prior from this procedure is in (black). To viii 45 arrive at the radio contribution in the 2013 chains, 143 GHz has had the expected dusty contribution of (8 ± 2)μK2 subtracted from the total Poisson amplitude (see text). 45 3.3 (Top row) Slices through the joint incompleteness function, I(S100 , S143 , S217 ). Each column shows the incompleteness for S143 vs. S217 , at a ﬁxed value of S100 , given in the label above the plot. (Bottom row) The same slices but approximating the joint incompleteness as independent, I(S100 , S143 , S217 ) ≈ I(S100 )I(S143 )(S217 ). Doing the integral in Eq. 3.2 with this approximation leads to about a 3% diﬀerence, thus we use the more accurate joint incompleteness. The top plots shows visible shot noise from the simulations used to compute the incompleteness function. We have split these simulations in two and using either half gives the same answer to within 1%. 46 3.4 Constraints on the six parameter source count model from (blue) integrating the model down to individual frequencies and ﬁtting the M13 diﬀerential source count measurements at 90, 150, and 220 GHz and (orange) also including the M13 distribution of spectral indices as a direct constraint on the spectral index and scatter parameters. 47 4.1 Key regions and contours in the mass-lifetime parameter space according to the analytic approximations in Sec. 4.1. As in other plots in this paper, dashed lines correspond to the temperature at neutrino decoupling, dot-dashed lines the start of BBN, and dotted lines the end of BBN. Blue lines show contours of constant Primakoﬀ freeze-out temperature, Tfo , and black lines show contours of constant two-photon re-equilibration temperature, Tre . The line Tre = mφ divides two regions A and B. Region A is vertically hatched and corresponds to out-of-equilibrium decays. Region B is cross hatched and corresponds to in-equilibrium decays. Constant decay-time contours in region A are Tre = const whereas they are mφ = const in region B. Region C has no hatching and corresponds to decays before neutrino decoupling, where ALPs leave no cosmologically observable traces. The line Tfo =QCD leaves a sharp feature on cosmological constraints as g∗ changes suddenly during this phase transition. 52 4.2 The evolution of the energy densities in the various components of the universe for diﬀerent scenarios which have similar decay time. The temperature of the photons today is held ﬁxed and the y-axis units are such that the ﬁnal value of the neutrino line is the value CMB . As in other plots in this paper, dashed lines correspond to the scale factor at of Neﬀ ix neutrino decoupling, dot-dashed lines the start of BBN, and dotted lines the end of BBN. The dashed red line is not actually a component, but is shown for illustrative purposes; it is the equilibrium ALP energy density (that is, the energy density APLs would have if they were in chemical and kinetic equilibrium with the photons). As per Eqn. 4.5, interactions serve to always drive the ALP energy density towards equilibrium. The plots labeled A and B correspond to the same regions in Fig. 4.1. 56 4.3 Exclusion regions in the ALP mass-lifetime parameter space. The dashed and dotted lines labeled “ν dec” (neutrino decoupling), and “BBN start/end” correspond to particles which decay at these particular times (with decay here arbitrarily deﬁned as when maximum energy injection occurs). The two thick dashed lines are the consistency relations for two particular axion models (see Sec. 4.3.1). The CMB, D/H, and Yp regions are excluded at 3σ, the Collider and Beam Dump regions are excluded at 2σ, and the SN1987a and HB Stars regions are less formal, rough bounds (see Sec. 4.2.4). 57 4.4 A comparison of exclusion regions from previous works (left panel) and those presented here (right panel). The right panel is identical to Fig. 4.3. 58 4.5 The colored contours show the prediction for each of the labeled quantities as a function of diﬀerent values of ALP mass and lifetime. The dotted/dashed/solid lines give 1/2/3 σ contours given the measurements for these quantities discussed in Sec. 4.2. No lines are visible on the lithium plot because the entire parameter space is excluded at > 3σ (our CMB scenario does not alleviate the lithium problem). We do not give contours for the Neﬀ plot because the CMB constraint is highly degenerate with Yp . For the D/H panel, the colored contours are calculated assuming a best-ﬁt η from the CMB, and uncertainties in η and nuclear reaction rates are taken into account in producing the σ contours (see Sec. 4.2.2 for discussion). 59 CMB and Y from 4.6 The contours show the 1- and 2-σ conﬁdence regions for Neﬀ p Planck +WP+highL. The dotted lines give the 1-σ constraint on Yp from Aver et al. [2013]. The dashed line is the relation if standard BBN is assumed, and the dot CMB = 3.046. Colored points show along this line corresponds to the standard value of Neﬀ CMB and Y arising from ALP masses and lifetimes sampled from a grid over values of Neﬀ p √ the entire region shown in Fig. 4.3. They are colored by mφ τφγ which is an important x quantity for the CMB constraint since it controls the fractional energy injected into the CMB ≈ 2.44 gives photons (Eqn. 4.10). The maximum value for in-equilibrium decays of Neﬀ the sharp cutoﬀ visible above. Points along the standard BBN consistency line arise from decays happening between neutrino decoupling and the beginning of BBN, and correspond to the island of low helium visible in Fig. 4.5. 61 4.7 Parameter constraints in the MeV-ALP region of parameter space when we also allow extra radiation present besides neutrinos and the ALP. Our likelihood includes all of the constraints shown in Fig. 4.3. In the ALP case (top panel) the lifetime is marginalized over whereas in the DFSZ-EN2 case (bottom panel) it is ﬁxed by the consistency relation (Eqn. 4.28). The vertical dashed line is a forecast for SUPER-KEKB, showing that it could close the remaining allowed parameter window or detect a particle there. We show Neﬀ evaluated both prior to BBN when neutrinos, extra radiation, and the ∼MeV ALP (which here adds 4/7 to Neﬀ ) contribute, and at the CMB epoch after the ALP has decayed. 69 CMB and Y from 4.8 (Top) The contours show the 1- and 2-σ conﬁdence regions for Neﬀ p Planck +WP+highL. The dotted lines give the 1-σ constraint on Yp from Aver et al. [2013]. The dashed line is the relation if standard BBN is assumed, and the dot along this line CMB = 3.046. Colored points show values of N CMB corresponds to the standard value of Neﬀ eﬀ and Yp arising from ALP masses and lifetimes taken from the ΛCDM+ΔNeﬀ +ALP chain described in Sec. 4.3.2. They are colored by mφ which controls decay time and can alter CMB . (Bottom) Same as the top panel, but with BBN but otherwise does not aﬀect Neﬀ 105 D/H shown on the x-axis. The vertical dotted lines give the 1-σ constraint from Cooke preBBN . Suﬃciently tight constraints around et al. [2014]. Points are colored instead by Neﬀ the standard value (black dot) could rule out the ALP scenario even in the presence of extra radiation. 69 xi List of Tables 2.1 SZ Cosmological Scaling 18 2.2 Survey Properties 23 2.3 Summary of Model Parameters 24 2.4 Statistical Error Degradation 37 4.1 Best-ﬁt parameters 71 4.2 Best-ﬁt χ2 71 xii CHAPTER 1 Introduction The era of precision cosmology is currently upon us, and observations of the cosmic microwave background (CMB) are at the forefront of driving current and future progress. The CMB is a remarkable cosmological probe because of the wealth of physics to which it sensitive; this ranges from the conditions at the very beginning of time during the inﬂationary era, to the formation of the light elements during big bang nucelosynthesis, to the evolution of structure across orders of magnitude of cosmic time. In addition, the properties of the CMB can be predicted accurately to ﬁrst order in linear perturbation theory, meaning precise theoretical predictions can be readily calculated and compared to observations. For these reasons, the CMB is a truly powerful probe of physics. It can be said that the precision era was ushered in by the results from the WMAP satellite [Bennett et al., 2003, Spergel et al., 2003]. By the time of the ﬁnal results from nine years of observation were published [Hinshaw et al., 2013], the temperature ﬂuctuations in the CMB at scales larger than about a third of a degree, or multipole moments smaller than about ∼ 180◦ /(1/3◦ ) ∼ 540, were now measured to the best possible theoretical limit, the cosmic variance limit. The picture of universe which emerged was a parametrically simple but physically rich “standard model of cosmology”, ΛCDM, which took the universe to be made up mostly of dark matter and dark energy, with a small but important contribution from known atoms, with stars which reionized the interstellar medium between redshifts z ∼ 7 − 14, and with initial conditions consistent with those predicted from the simplest models of inﬂation. Falsifying this model, or alternatively conﬁrming and reﬁning its predictions, has continued to drive the ﬁeld of cosmology. Further progress on the CMB side necessitated better measurements of the temperature ﬂuctuations at small scales, and better measurements of the CMB polarization at all scales. The latter is outside of the scope of this thesis, while the former is the main topic. The ﬁrst results following WMAP pertaining to small scale temperature ﬂuctuations were published by the ACT and SPT collaborations [Dunkley et al., 2011, Keisler et al., 2011, respectively], which were, at least in certain regions of the sky, now at the cosmic variance limit up to and above ∼ 2500. 1 This essentially completed the measurement of the CMB temperature ﬂuctuations in these regions since above this scale almost no cosmological information can be extracted due to irreducible foregrounds and confusion from gravitational lensing. While some hints of failures of ΛCDM did arise, [Dunkley et al., 2010, Hou et al., 2013, 2014], none were of convincing statistical signiﬁcance. Thus in the lead up to the results from the Planck satellite, the post-WMAP picture of the universe had remained qualitatively very similar [Calabrese et al., 2013], albeit with more and more alternatives to ΛCDM ruled out. The release of the ﬁrst Planck results [Collaboration et al., 2013a] marked the completion of measurements of the temperature ﬂuctuations up to ∼ 2500 to the cosmic variance, now across the entire sky. Measurements by Planck, ACT, and SPT, probe a qualitatively new regime of the CMB power spectrum called the damping tail. This region corresponds to scales small enough that they entered the horizon deep in the radiation dominated epoch of the universe, and additionally whose amplitude has been damped signiﬁcantly due to photon diﬀusion in the last scattering surface. This allows these measurements to probe the physics of known or hypothetical relativistic particles which contribute to the radiation energy density, or which can aﬀect this photon diﬀusion. Additionally, by virtue of being at small scales, extraction of cosmological information from these modes now necessitates taking into account new foreground components which are otherwise negligible at large scales. These foregrounds, called extra-galactic foregrounds, arise from emission at microwave frequencies or distortion of the CMB spectrum by sources outside of our own galaxy such as other galaxies and galaxy clusters. A large part of this thesis centers around these foregrounds, and contains work which was performed by the author in the lead up and during the analysis of SPT and Planck data. Ch. 2 follows from Millea et al. [2012] and develops a realistic model for such contamination. Since Millea et al. [2012] was written before the release of the Planck results, it uses a set of Planck -like simulations to show that subtracting this model is crucial for such an analysis. Ch. 3 follows from work soon to be published as part of a Planck publication, and gives a novel method for constraining particular components of this foreground model with auxiliary data. Finally, in Ch. 4, which follows from Millea et al. [2014], we use CMB observations to constrain a model of a hypothetical scalar particle called the axion, and a more general class of so called axion-like particles. 2 1.1. Theoretical Background We begin with a very brief theoretical introduction to the power spectrum of the temperature ﬂuctuations in the CMB, a quantity which will be central to this thesis. The temperature of the primary1 CMB as a function of position on the sky, T (n̂), is modeled as a homogeneous and isotropic Gaussian random ﬁeld. This means that the correlation between diﬀerent spots on the sky is independent of their location and orientation, and thus can only be dependent on the distance between them, (1.1) T (n̂)T (m̂) = ξ(|n̂ − m̂|) where ξ is known as the correlation function. Here, the brackets denote the “ensemble average”, taken over many hypothetical realizations of our universe. One can show that in terms of the spherical harmonic coeﬃcients of the same map, (1.2) am = dn̂ Ym (n̂)T (n̂) Eqn. 1.1 is equivalent to, (1.3) a∗m a m = δ δmm C where C is referred to as the power spectrum. Higher order correlations, i.e. ones with more than two am ’s appearing such as am a m a m can be calculated using Wick’s theorem. It states that such correlations can be written as the sum over all permutations of products of correlations of one and two am ’s. Using am = 0 and Eqn. 1.3, it is clear that all such higher order correlations can themselves only depend on C and various numerical factors, thus the entirety of the statistical information contained in the T (n̂) ﬁeld is contained in its power spectrum, C . For this reason, using the power spectrum is a form of loss-less data compression, one which is also very convenient as it is most directly predicted by theory. The ﬂuctuations in the temperature of the CMB are sourced by quantum ﬂuctuations which become classical perturbations during inﬂation, then causally evolve according to classical physics until today. Their power spectrum can be written as an integral over spatial Fourier modes k, taking the product of the initial perturbation set up during inﬂation, P(k), and a transfer function, Δ2 (k), 1We will use “primary” to refer to the CMB before gravitational lensing 3 which describes the subsequent causal evolution and projects each mode onto the last scattering surface, (1.4) C = d ln k P(k)Δ2 (k) Here P(k) is related to ﬂuctuations in the inﬂaton ﬁeld, P(k) = |φ(k)|2 (evaluated long after each mode has left the horizon). The transfer function, Δ2 (k), comes from solving the coupled set of Einstein and Boltzmann equations which track how perturbations in the various components of the universe and also in the background metric evolve. While constraining both P(k) and Δ2 (k) are of interest in cosmology, in this thesis we will focus only on the transfer function. In particular, in Ch. 4, we discuss how the addition of an axion or an axion-like particle to the coupled set of Boltzmann and Einstein equations aﬀect the transfer function and how CMB and other cosmological observations can place constraints on these types of hypothetical particles. Maps at microwave frequencies contain not just CMB ﬂuctuations, but also contamination from extra-galactic foregrounds. Here, objects along the line of sight to the CMB last-scattering surface emit microwaves, or distort the CMB spectrum. The observed temperature, T will then be T = T CMB + T FG with, (1.5) T FG (n̂) = f N (n̂) where we have assumed that each one of N objects along the line of sight in direction n̂ contributes an amount t to the temperature (we will shortly generalize to the realistic case where objects contribute diﬀerent amounts). The distribution of object locations is modeled as being randomly sampled from some smooth underlying distribution (which in our case is a biased traced of the total matter distribution, although in general it does not need to be). Thus the ﬁeld N (n̂) arises from a Cox Poisson process, and the actual number density in any direction (for our realization of the universe) will be drawn from a Poisson distribution, (1.6) N (n̂) ∼ P(N̄ (n̂)) ≈ N (N̄ (n̂), N̄ (n̂)) where N̄ is the underlying distribution and P(λ) represents the Poisson distribution with mean λ. As will become important in a moment, in the limit of large N , this is approximately equal to a Gaussian distribution N (μ, σ 2 ) with mean μ and variance σ 2 both equal to N̄ (n̂). The ﬁeld 4 N̄ is itself a Gaussian random ﬁeld, whose distribution is most cleanly expressed in terms of its harmonic coeﬃcients, N̄m ∼ N (0, N̄ ) (1.7) where N is the power spectrum of the smooth underlying distribution as predicted by theory. One observation which helps understand the resulting power spectrum of N (n̂) is that, under the Gaussian approximation to the Poisson distribution, the ﬁeld can be written as, (1.8) N (n̂) = N̄ (n̂) + ΔN (n̂) N̄ (n̂) with (1.9) ΔN (n̂) ∼ N (0, 1) where we have separated out the mean contribution, and written the remainder explicitly as a uniform white noise term multiplied by a term which spatially scales its variance. In this form, it is more obvious that the power spectrum N (n̂) will have three terms, one auto correlation for each of the two terms in Eqn. 1.8, and one cross correlation. The auto power of N̄ (n̂) is by deﬁnition N , which is the quantity predicted by theory. The auto power of the second term is the mean object density, N̄ (which is independent of direction), also as predicted by theory. To arrive at this result one can simply plug in Eqn. 1.8 into Eqns. 1.2 and 1.3. Using the same method, one can also see that the cross term is zero. Therefore, the power spectrum of the foregrounds is, (1.10) CFG = f 2 (N + N̄ ) The ﬁrst term is known as the “clustering” term, because it depends on how the objects trace the underlying distribution, and hence how they cluster. The second term is known as the Poisson term because it is independent of the underlying distribution and only depends the Poisson ﬂuctuations around the mean number of objects. Ch. 2 will give theoretical predictions for both of these terms for a number of diﬀerent objects. It will turn out that the clustering term is only important for dusty galaxies, whereas for radio galaxies, and for distortion of the CMB spectrum by galaxy clusters via the tSZ and kSZ eﬀect, only the Poisson term is important. 5 CHAPTER 2 Modeling Extragalactic Foregrounds At angular scales smaller than a tenth of a degree, extragalactic foregrounds1 become important for three reasons: 1) the CMB power spectrum is dropping in amplitude, 2) cosmic variance is smaller and 3) foregrounds are growing in amplitude. At suﬃciently small angular scales, foregrounds become the dominant signal at all CMB frequencies. Furthermore, unlike galactic foregrounds, they are statistically isotropic and thus cannot be avoided by masking regions of higher contamination. Their modeling is an unavoidable necessity. In this chapter we present a parameterized, physically-motivated, phenomenological model for the extragalactic foregrounds and consider it in the context of extracting cosmological parameters from the primary CMB anisotropy. We demonstrate that for an analysis of Planck data, such modeling is necessary to avoid signiﬁcant biases in cosmological parameter estimates, but that marginalization over even a very rich foreground model is essentially “for free”; the foregrounds are suﬃciently orthogonal to the primary CMB that the statistical errors on cosmological parameters are degraded by at most 20% for ns and less than 10% for other parameters. With the addition of higher resolution ground-based data or non-CMB Planck bands to clean the foregrounds, the degradation is reduced to a few percent for all parameters. The importance of extragalactic foregrounds for CMB analysis has been recognized for a long time [Tegmark and Efstathiou, 1996, Bouchet and Gispert, 1999, Knox, 1999, Tegmark et al., 2000, Leach et al., 2008, Cardoso et al., 2008, Dunkley et al., 2011]. Potential biases from extragalactic contaminants have been pointed out previously by Knox et al. [1998], Santos et al. [2003], Zahn et al. [2005], Serra et al. [2008] and Taburet et al. [2009]. Distinguishing our work is the simultaneous consideration of all foreground components necessary for an analysis of Planck data, and physical modeling of these components informed from recent measurements beyond the damping tail by SPT [Hall et al., 2010, Vieira et al., 2010, Shirokoﬀ et al., 2010] and ACT [Dunkley et al., 2011]. In this chapter, we will consider the foreground power contributions from shot noise due to radio galaxies 1We will henceforth refer to both extragalactic foreground contaminants and secondary anisotropies as just “fore- grounds” since they cannot be modeled from ﬁrst principles like the primary CMB. 6 and dusty star forming galaxies (DSFGs), the clustering of the DSFGs, the thermal and kinetic Sunyaev-Zeldovich eﬀects (tSZ and kSZ), and correlation between the tSZ and DSFG components. We now turn to summarizing recent developments in both modeling and measurements of these extragalactic foregrounds. Our understanding of the power spectrum due to DSFGs at frequencies relevant for CMB analysis has been rapidly improving. We demonstrate here that for analysis of Planck data, the eﬀects of DSFG clustering are the most important of the foregrounds to model. Although it is the most important eﬀect, it has been almost entirely ignored by previous cosmological parameter error forecasting work. To date, the only papers to consider the impact of DSFG clustering on cosmological parameter estimates are Dunkley et al. [2011] and Serra et al. [2008]. DSFG clustering power was ﬁrst detected at CMB frequencies by the SPT [Hall et al., 2010], with subsequent conﬁrmation and improved constraints from ACT [Dunkley et al., 2011] and SPT [Shirokoﬀ et al., 2010]. The recent suite of early Planck papers [Planck Collaboration, 2011r, in particular] have also provided signiﬁcant constraints on both the amplitude and shape of the clustering power. The Planck measurements rule out many otherwise viable models which generally predict higher power (on the scales relevant for analysis of the primary CMB power spectrum) than observed. Radio galaxy source counts from high-resolution ground-based data are particularly useful for Planck since they are sensitive to the decade in brightness below Planck ’s ﬂux cut. The radio sources in this brightness range create the dominant source of shot noise power in most of the Planck frequencies which contain signiﬁcant CMB information. SPT measurements of point source populations [Vieira et al., 2010] have oﬀered valuable information about the amplitude of Poisson power, as well as the coherence of these shot-noise ﬂuctuations from frequency to frequency. Recent data, as well as recent theoretical developments, inform our modeling of the power spectrum of the tSZ eﬀect—a spectral distortion that arises due to inverse Compton scattering of CMB photons oﬀ the hot electrons in groups and clusters. The magnitude of the tSZ signal is proportional to the thermal pressure of the intra-cluster medium (ICM) integrated along the line of sight. Upper limits on the amplitude of the tSZ power (set by Lueker et al. [2010], conﬁrmed by Dunkley et al. [2011] and further tightened by Shirokoﬀ et al. [2010]) were found to be surprisingly low compared to predictions from halo model calculations [Komatsu and Seljak, 2002] and nonradiative hydrodynamical simulations [White et al., 2002]. Recent work has demonstrated that the 7 inclusion of a signiﬁcant non-thermal contribution to the total gas pressure in groups and clusters in analytic models can signiﬁcantly reduce the predicted amplitude of the tSZ power spectrum [Shaw et al., 2010, Trac et al., 2010]. Non-thermal pressure, sourced by bulk gas motions and turbulence, reduces the thermal pressure required to support the ICM against gravitational collapse and thus the amplitude of the tSZ signal. Similarly, Battaglia et al. [2010] demonstrated that the inclusion of radiative cooling, star formation and AGN feedback in hydrodynamical simulations substantially lowers the tSZ power compared to simulations that omit these processes. Current predictions for tSZ power from models and simulations are consistent with the upper limits derived from observations. These recent modeling developments are supported by data from Planck ; when the models are used to extrapolate from X-ray measurements to a predicted tSZ signal, the predictions agree with Planck SZ observations. Agreement is seen both in observations of single galaxy clusters [Planck Collaboration, 2011h,i] and via a stacking analysis over a broad range in X-ray luminosity down to masses as small as M500 ∼ 5 × 1013 M [Planck Collaboration, 2011j]. Current data provide no direct lower limits to the amplitude of tSZ power due to a degeneracy with the kinetic SZ power spectrum [Lueker et al., 2010]. The kinetic SZ eﬀect arises due to the Doppler Thomson scattering of CMB photons oﬀ of regions of ionized gas with bulk peculiar velocities. Upper limits on kSZ power set by Lueker et al. [2010] and now substantially tightened by Shirokoﬀ et al. [2010], are ruling out some models of patchy reionization. It is useful to decompose the kSZ power into contributions arising from an inhomogeneous transition from a neutral to ionized inter-galactic medium, so called “patchy reionization,” and those from the post-reionization era, the “Ostriker-Vishniac” (OV) eﬀect. The former is much more uncertain than the latter, and our best knowledge of its amplitude comes directly from the upper limits in Shirokoﬀ et al. [2010]. The OV power level has a current theoretical uncertainty that we estimate to be about a factor of 2. Despite its low levels, kSZ power is a worrisome source of potential bias of cosmological parameters since its spectral dependence is the same as the primary CMB temperature anisotropies. We expect that the only potentially signiﬁcant extragalactic contributions to polarization anisotropy are Poisson power from radio sources and DSFGs. A polarization analog for DSFG clustering could only arise due to (unexpected) correlations between galaxies in the polarization orientations of their emission. Polarization signals arise from scattering oﬀ of electrons in clusters and groups [Sazonov and Sunyaev, 1999, Carlstrom et al., 2002, Amblard and White, 2005] and in reionized patches [Knox et al., 1998, Santos et al., 2003], but these are also expected to be negligibly small. 8 In addition to developing and exploring the implications of an extragalactic foreground model that takes into account recent developments, we introduce a new approach to analyzing the multifrequency data. We show how the complexities of our modeling can be reduced to a fairly simple description of the contamination of the estimates of CMB power spectra. The contamination can be described by just a few principal components whose amplitudes are constrained by CMB–free linear combinations of the auto and cross-frequency power spectra. The outline of this chapter is as follows. In Sec. 2.1 we describe our foreground models before describing our ﬁducial models and surveys in Sec. 2.2 and 2.3 respectively. In Sec. 2.4 we describe our general methodology before detailing our principal component approach in Sec. 2.5. We ﬁnally present our results in Sec. 2.6 and discuss them in Sec. 2.7. 2.1. Modeling 2.1.1. Emission from External Galaxies. In the frequency range in which Planck is most sensitive to the CMB (roughly 70 GHz to 217 GHz), external galaxies are well approximated by power-law intensities Iν ∝ ν α , and divide fairly cleanly into those with spectral indices α < 1 (radio galaxies) and those with α > 1 (DSFGs) [Vieira et al., 2010]. We assume all sources have no spatial extent, an approximation which might be worrisome for radio sources because of long relativistic jets. However, if we extrapolate from 1.4 GHz up to our frequency range, we ﬁnd 99% of sources have a major axis FWHM less than 30 arc-seconds [Hodge et al., 2011], too small to be detectable with the 1.5 arc-minute beams of typical ground-based experiments. External galaxies lead to anisotropy via their discreteness, usually modeled with a Poisson distribution, and also via correlations due to their tracing of the large-scale structure. The Poisson ﬂuctuations are important for both radio galaxies and DSFGs, while clustering is only signiﬁcant for the dusty galaxies [Hall et al., 2010]. The Poisson contribution depends on the brightness function, dN/dS, via (2.1) C = Sc 0 dS S 2 dN dS where Sc is the ﬂux cut; map pixels with sources with S > Sc are masked. 9 Clustering power, in contrast, scales approximately with the square of the mean intensity, Iν2 , with Iν = (2.2) Sc dS S 0 dN . dS Although radio sources do cluster, their mean intensity at the relevant frequencies is much smaller than for the DSFGs; suﬃciently smaller that their clustering power is negligible. 2.1.1.1. Radio Galaxies. From Vieira et al. [2010] we know the radio galaxies at 150 GHz and 220 GHz and at ﬂux densities below 100 mJy are described quite well by the de Zotti et al. [2005] model2. This model has a brightness function that is approximately a power-law SdN/dS ∝ S γR . This translates into Poisson power which depends on the ﬂux cut via C ∝ ScγR +2 . Due to the inhomogeneity of the Planck sky coverage, Sc will vary signiﬁcantly across the sky. So that these angular variations can be taken into account, we chose to model the radio galaxies in terms of dN/dS rather than C . For frequency dependence, we assume the spectral indices of the source population form a Gaussian distribution with mean α and width δα2 = σ 2 (uncorrelated from source to source). With these assumptions our power spectra from radio sources are given by3 2 γR + 2 αR + ln(νν /ν02 )σR /2 Sc νν (2.3) CR = C R,0 S0 ν02 where C R,0 is some overall normalization factor. 2.1.1.2. Dusty Star-Forming Galaxies. Due the shape of DSFG brightness function, the integrals in Equations 2.1 and 2.2 are nearly independent of the upper bound [Hall et al., 2010], thus dusty power is nearly independent of ﬂux cut and we choose to build our model in C rather than dN/dS. In that case, the DSFG Poisson contribution is given simply by (2.4) CD = C D,0 νν ν02 2 αD + ln(νν /ν02 )σD /2 and again C D,0 is an overall normalization. 2We also know from recent Planck results [Planck Collaboration, 2011m] that at brighter ﬂux densities the deZotti model signiﬁcantly over predicts the number counts. 3In deriving this form we have used the identity that for a zero-mean Gaussian random variable x, exp(−x) = exp(x2 /2). This identity and its applicability in this context, was pointed out to us by Challinor, Gratton and Migliaccio. 10 A number of authors have considered the clustering of the infrared background, starting with Bond et al. [1986, 1991]. Further theoretical investigation [Scott and White, 1999, Haiman and Knox, 2000] was stimulated by the detection of the infrared background in COBE data [Puget et al., 1996, Fixsen et al., 1998], and the detection of bright “sub-millimeter” galaxies in SCUBA data [Hughes et al., 1998]. Subsequently, the clustering has been detected at 160 microns [Lagache et al., 2007], at 250, 350 and 500 microns by the Balloon-borne Large Aperture Submillimeter Telescope [Viero et al., 2009, BLAST] and at 217 GHz [Hall et al., 2010, Dunkley et al., 2011]. Recent Planck measurements of the Cosmic Infrared Background [CIB; Planck Collaboration, 2011r] have extended to much larger angular scales than before at 217 GHz, 353 GHz, 545 GHz, and 857 GHz and recent Herschel measurements [Amblard et al., 2011] have tightened up the BLAST measurements and extended them to smaller angular scales. The ﬁeld is rapidly evolving. For the clustering, we assume the same model as in Hall et al. [2010], extended to phenomenologically include the consequences of non-linear clustering by including a multiplicative factor which is a power-law in for > 1500. This extension is able to ﬁt many diﬀerent models in the literature and allows us to explore the theoretical uncertainty in a statistical manner by marginalizing over the value of the multiplicative factor. We neglect one aspect of the Hall et al. [2010] model because it leads to corrections of only about 1% across the relevant frequency range; we ignore the -dependent spectral index. Thus, the DSFG clustering power spectra are given by (2.5) CC = C C,0 ΦH10 ⎧ α C ⎪ ⎨ 1 n C ⎪ ν02 ⎩ 1500 νν < 1500 > 1500 is the Hall et al. [2010] clustering template. where ΦH10 Though the same sources generate both the Poisson power and the clustering power, they are weighted diﬀerently, thus for our baseline model we conservatively assume no relationship between the clustering spectral index and the Poisson spectral index. To gain some idea of the range of possible shapes of the DSFG clustering power spectrum, we show a sampling of power spectra from models in the literature in Figure 2.1. They are all normalized at = 3000 to highlight similarities/diﬀerences in shape. The models are the ﬁducial model from Righi et al. [2008], the β = 0.6 model from Amblard and Cooray [2007] and a non-linear version of the model by Haiman and Knox [2000], hereafter HK00. Righi et al. [2008] associate 11 the sources of infrared light with starbursts triggered by mergers. Amblard and Cooray [2007] incorporate nonlinearities using a halo model. For the ‘HK00nonlin’ curve, we used the luminosity densities for the ﬁducial model of HK00, assumed light is a biased tracer of mass, and calculated the non-linear mass power spectrum using the prescription by Peacock and Dodds [1996]. Though these template arise from very diﬀerent modeling assumptions, they have similar shapes in the linear regime at large scales, then turn to a power-law behavior at small scales. It is this observation which informed our phenomenological model. We will also show our model to be suﬃcient for reproducing these shapes with enough accuracy for Planck cosmological parameter estimation. One result of the Planck measurements, available only after our calculations for this paper were completed, is that the CIB power spectrum uncertainty at < 2000 is now much smaller than before. At least two of the three models shown in Figure 2.1 that guided our understanding of the range of possible amplitudes have shapes that are inconsistent with the combined Planck and SPT data. That range of possible amplitudes is now given by the Planck CIB power spectrum measurement uncertainty. 2.1.1.3. Polarization. We expect polarized emission from the sources we consider to be very small and uncorrelated from source to source. For a collection of sources with polarization fraction f , contributing a Poisson temperature power spectrum of CT T,P , we have (2.6) CEE = CBB = f 2 CT T,P CT E = f CT T,P We parameterize both radio source and DSFG contributions with the above forms, with f = fD for DSFGs and f = fR for radio sources. 2.1.2. Thermal SZ Eﬀect. The thermal SZ eﬀect is a distortion of the CMB caused by inverse Compton scattering of CMB photons oﬀ electrons in the high temperature plasma within galaxy clusters. To ﬁrst order, the temperature change of the CMB at frequency ν is given by ΔT /TCMB (xν ) = f (xν )y, where f (xν ) = xν (coth(xν /2) − 4), xν = hν/kB TCMB , and y is the dimensionless Compton-y parameter (2.7) y= kB σT me c2 ne (l)Te (l)dl , 12 Figure 2.1 Three model DSFG clustering auto-spectra at 217 GHz (black), and approximations to them with our parameterized model (solid, red), all normalized (with one exception) at = 3000. Our ﬁducial model is the thickest curve. Also plotted are estimates of the clustering power from Planck [Planck Collaboration, 2011r] and SPT [Hall et al., 2010]. For both sets of data points we have subtracted estimates of the Poisson power from the reported total CIB power. The lowest amplitude solid (red) curve is the result of a “by-hand” adjustment of our model parameters to ﬁt the Planck and SPT data. where the integral is along the line of sight. TCMB is the CMB temperature, ne and Te are the number density and electron temperature of the ICM, respectively. The thermal SZ power spectrum can be calculated by simply summing up the squared, Fourierspace SZ proﬁles, ỹ, of all clusters: (2.8) CtSZ = f (xν ) 2 dV dz dz d ln M dn(M, z) 2 ỹ (M, z, ) d ln M where V(z) is the comoving volume per steradian and n(M, z) is the number density of objects of mass M at redshift z. For the latter we use the ﬁtting function of Tinker et al. [2008]. y(M, z, r) is the projected radial SZ proﬁle for a cluster of mass M and redshift z. Note that this calculation assumes that halos are not spatially correlated; Komatsu and Kitayama [1999] demonstrated that for > 1000 the two-halo (or clustered) contribution to the tSZ power spectrum is several orders of magnitude smaller than the Poisson contribution given by Eq. 2.8. 13 To calculate the thermal SZ signal we adopt the analytic intra-cluster gas model presented in Shaw et al. [2010]. This model provides a prescription for calculating the compton-y (or equivalently, thermal pressure) proﬁles of hot gas in groups and clusters. The model assumes that gas resides in hydrostatic equilibrium in the potential well of dark matter halos with a polytropic equation of state. The dark matter potential is modeled by a Navarro-Frenk-White proﬁle [Navarro et al., 1997] using the halo mass - concentration relation of Duﬀy et al. [2008]. The model includes parameters to account for gas heating via energy feedback (from AGN or supernovae) plus dynamical heating via mergers. The stellar component of the baryon fraction in groups/clusters is determined using the stellar mass fraction - total mass relation observed by [Giodini et al., 2009]. A radially-dependent non-thermal pressure component of the gas is incorporated by calibrating oﬀ the non-thermal pressure proﬁles measured in hydrodynamical simulations [Lau et al., 2009]. In total the model has four free parameters relating to astrophysical processes in groups and clusters. Shaw et al. [2010] explored the range in which these parameters reproduce radial proﬁles and scaling relations derived from X-ray observations of nearby groups and clusters. To allow the astrophysical uncertainty to be marginalized over quickly in our MCMC chains, we perform a principal component analysis (PCA) described in Appendix A. A suite of 10,000 simulated power spectra were created, each time randomly sampling from the input astrophysical parameter distribution (with the cosmological parameters ﬁxed to their ﬁducial values described in Sec. 2.2). We ﬁnd that two principal components are suﬃcient to achieve 1% accuracy out to = 10, 000 on the model power spectra. In Figure 2.2 we plot the thermal SZ power spectrum predicted by a number of recent simulations (black lines) as well as a ﬁt to each with our PCA model (red lines). The dotted line represents the thermal SZ power spectrum measured from the Mare-Nostrum simulation – a nonradiative simulation run using the smoothed-particle hydrodynamics code, Gadget-2. The black solid line shows the results of the non-radiative simulation of Battaglia et al. [2010] and the black dashed line the results of a rerun of this simulation including radiative cooling, star-formation and energy feedback. The dot-dashed line shows the ‘standard’ tSZ model from the simulations ofTrac et al. [2010]. The thickest red lines represents our ﬁducial thermal SZ model in this work. The blue point with errorbars show the recent SPT constraint on the amplitude of thermal SZ power at = 3000. All models are plotted at 146 Ghz and have been scaled to our ﬁducial cosmology. 14 16 14 M B B T t D [K 2 ] 12 1 8 6 4 2 2 4 6 8 1 Figure 2.2 Comparison between recent models and simulations of the tSZ eﬀect (black lines) and ﬁts of our PCA model to each (thin red lines). The thickest red line shows the ﬁducial tSZ power spectrum used in this work. All results are plotted at 146 GHz and are scaled to σ8 = 0.8. The blue arrow shows the SPT 95% conﬁdence upper limit on thermal SZ power at = 3000 [Shirokoﬀ et al., 2010]. Our PCA model can accurately reproduce all the simulations in Fig 2.2 other than the nonradiative simulation of Battaglia et al. [2010] which peaks at much smaller angular scales than the other simulations. We note that the Shaw et al. [2010] model inherently assumes that some fraction of cluster gas has been converted to stars, whereas this simulation did not include these processes. Turning oﬀ star formation produces a power spectrum that peaks at smaller scales. In Section 2.6.2 we investigate the bias on measured cosmological parameters when the Battaglia et al. [2010] non-radiative template is used for the tSZ signal. We ﬁnd that the PCA will adapt suﬃciently to prevent a bias in the measured cosmological parameters. The ﬁnal step is to determine the cosmological scaling of the power spectrum of our ﬁducial model so that the amplitude can be scaled accordingly in our analysis. We ﬁnd that the tSZ power spectrum is principally sensitive to Ωm , Ωb , ns , and σ8 , with a particularly strong dependence on the latter. To determine the scaling we simply evaluate the Shaw et al. [2010] model varying each cosmological parameter in the range ±25% of its ﬁducial value while holding the other three ﬁxed (at their ﬁducial value). We then ﬁt to the resulting power spectra, with our results summarized in Table 2.1. 15 T M B t t t 40 D [K 2 ] 4 30 3 20 2 10 1 0 2 4 6 8 1 Figure 2.3 Comparison between recent simulations of the kSZ eﬀect (black lines) and our model. We consider contributions from the post-reionization kSZ eﬀect (solid red line) and from patchy reionization (dashed red line). Note that the simulations plotted here assume homogeneous reionzation and thus do not include a patchy contribution. The SPT 95% conﬁdence upper limit for the kSZ power at = 3000 is 6.5 μK2 . 2.1.3. Kinetic SZ Eﬀect. The kinetic SZ eﬀect is a temperature anisotropy that arises from the Compton scattering of CMB photons oﬀ of electrons that have been given a line-of-sight peculiar velocity by density inhomogeneities in the matter ﬁeld. We break up the kSZ into contributions from the post-reionization period and from a period of inhomogeneous “patchy” reionization. 2.1.3.1. Ostriker-Vishniac Eﬀect. When the density ﬂuctuations which source electron velocities are in the linear regime the eﬀect is known as the Ostriker-Vishniac (OV) eﬀect, as derived in Ostriker and Vishniac [1986] and Vishniac [1987]. The post-reionization kSZ eﬀect can then be modeled as the nonlinear extension of the OV eﬀect as described below. We follow the analytic prescription given in Hu [2000] which describes the angular power spectrum of the linear Vishniac eﬀect as (2.9) π2 C = 5 2 3 dχDA Ġ g G 2 Δ4δb IV . where χ is the conformal time, G is the cosmological growth function, DA is the comoving angular diameter distance, g is the visibility function, Δ2δb is the linear theory baryon density power spectrum 16 and IV represents the mode coupling of the linear density and velocity ﬁelds: (2.10) IV = ∞ 0 dy1 1 −1 dμ 2 2 (1 − μ2 )(1 − 2μy1 ) Δδb (ky1 ) Δδb (ky2 ) , y13 y25 Δ2δb (k) Δ2δb (k) with μ = k̂ · k̂1 y1 = k1 /k (2.11) y2 = k2 /k = 1 − 2μy1 + y12 . Due to an incomplete treatment of the eﬀects of pressure feedback from baryons we slightly overpredict the power on very small scales. As described in Hu [2000], in this formulation we can consider the kSZ eﬀect to be the nonlinear extension of the linear Vishniac eﬀect. This approximation requires replacing the linear density power spectrum in Eq. 2.9 with its nonlinear extension while leaving the contribution from the velocity power spectrum unchanged: (2.12) CkSZ,OV π2 = 5 2 3 dχDA Ġ g G 2 2(N L) Δ δb Δ2δb IV . For the nonlinear power spectra (NL) we utilize the HALOFIT [Smith et al., 2003] model. In this calculation of the kSZ eﬀect we assume that the nonlinear density ﬂuctuations are uncorrelated with the bulk velocity ﬁeld in which they lie. Zhang et al. [2004] argue that this approximation may not hold in highly nonlinear regimes where contributions from the curl of the nonlinear velocity ﬁeld may become important however we neglect these corrections here. As in the previous section, we ﬁnd a power-law approximation for estimating the kSZ power as a function of cosmological parameters. The kSZ angular power spectrum was calculated under the full analytic formulation for a large suite of WMAP7-allowed LCDM cosmologies. An MCMC was then performed in the 6 dimensional ﬁtting-function parameter space and best-ﬁt marginalized values were found and are listed in Table 2.1 In Figure 2.3 we compare our calculation of the kSZ power spectrum (solid red line) with that measured from recent simulations (black lines). As in Figure 2.2, we plot the power spectrum predicted by the Mare-Nostrum simulation (dotted), the ‘standard’ model of Trac et al. [2010] (dot-dashed) and the non-radiative hydrodynamical simulations of Battaglia et al. [2010] (solid). 17 Table 2.1 SZ Cosmological Scaling AOV [μK2 ] 500 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1.18 1.81 2.64 3.06 3.33 3.53 3.67 3.78 3.87 3.94 4.00 AtSZ [μK2 ] 500 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 2.06 3.59 4.86 5.04 4.82 4.44 4.03 3.62 3.24 2.89 2.59 ns Ωb Ωc σ8 τ 0.96 0.045 0.22 0.8 0.09 -1.44 1.83 -1.06 4.36 0.25 -1.36 1.91 -1.13 4.82 0.24 -0.94 1.96 -1.12 5.26 0.22 -0.45 1.94 -1.03 5.38 0.20 -0.22 1.96 -1.04 5.54 0.17 -0.03 1.98 -1.05 5.66 0.15 0.13 2.00 -1.06 5.75 0.13 0.27 2.01 -1.07 5.83 0.12 0.38 2.02 -1.08 5.89 0.10 0.49 2.03 -1.09 5.94 0.09 0.58 2.04 -1.10 5.99 0.08 ns Ωb Ωm σ8 h 0.96 0.045 0.265 0.8 0.71 -1.01 2.41 0.69 8.57 1.30 -0.75 2.45 0.63 8.49 1.43 -0.36 2.52 0.53 8.40 1.65 -0.08 2.57 0.48 8.36 1.73 0.15 2.62 0.44 8.34 1.88 0.31 2.66 0.42 8.33 2.03 0.49 2.70 0.39 8.32 2.13 0.58 2.73 0.38 8.32 2.18 0.77 2.78 0.36 8.32 2.27 0.87 2.80 0.35 8.33 2.32 0.96 2.83 0.34 8.33 2.37 Note. — The -dependent power-law cosmological scalings for the Ostriker-Vishniac eﬀect and the thermal SZ eﬀect. The numbers immediately below the cosmological parameters are the pivot points for the power law, and the numbers in the table are the power-law indices. For example, the top row says that for the OV eﬀect, D500 = 1.18μK2 (ns /0.96)−1.44 (Ωb /0.045)1.83 ... The red dashed line shows our model for the contribution to the kSZ signal from inhomogeneous reionization (not included in the other lines), as described in the following section. 2.1.3.2. Patchy Reionization. We also consider the contribution to the kSZ power from inhomogeneous reionization [Gruzinov and Hu, 1998, Knox et al., 1998, Hu, 2000, Zahn et al., 2005, McQuinn et al., 2005, Iliev et al., 2007]. Simulations and analytic models of HII bubble formation both indicate that the ﬁrst galaxies and quasars were highly clustered and led to gradual reionization in “bubbles” that quickly grew to sizes of several Mpc [Zahn et al., 2010]. 18 In our estimates we use the analytic Monte-Carlo “FFRT” model of Zahn et al. [2010]. It has been shown to agree well with the most sophisticated radiative transfer simulations on scales of 100 comoving Mpc/h, while having the added advantage of allowing the modeling of arbitrarily large volumes (the analytic scheme is about 4 orders of magnitude faster, at a given dynamic range, than radiative transfer). This is especially important for kSZ, since large scale velocity streams lead to the bulk of the signal. Our particular template (dashed red line in Figure 2.3) was calculated in a 1.5 Gpc/h cosmological volume where x- and y-axes correspond to roughly 15 degrees on a side and z-axis corresponds to redshift, with a median redshift of 8. We shift this template left-right logarithmically by a “patchy shift” parameter RP ; that is, the power spectrum for a given shift is related to the ﬁducial RP = 1 spectrum by, f id C = CR P × (2.13) RP is to be thought of as scaling the size of the bubbles, and is, to good approximation, proportional to the duration of the patchy phase. The timing of reionization has a secondary small eﬀect on the shape and amplitude, which we neglect here. 2.1.4. tSZ-DSFG Correlation. It is reasonable to expect some correlation between the DSFG clustering and tSZ components since they both trace the same underlying dark matter distribution, with signiﬁcant overlap in redshift. Simulations which associate emission with individual cluster member galaxies predict anti-correlations—DSFGs ﬁll in SZ decrements at frequencies below 217 GHz considered here—on the order of tens of percent, with a correlation coeﬃcient nearly independent of scale [Sehgal et al., 2010]. This eﬀect was explored in Shirokoﬀ et al. [2010] which found correlation consistent with zero but with signiﬁcant uncertainty due to degeneracies with the tSZ and kSZ components. Assuming a ﬁxed correlation rtSZ,C , the total power spectrum isn’t simply the sum of the tSZ and DSFG clustering terms, but must also include a term given by, (2.14) C,νν = rtSZ,C C C tSZ + C,νν ,ν ν tSZ C C C,νν ,ν ν Note that this eﬀect can be larger than either component individually in cross spectra between frequencies in which each component is large. For example, even with only moderate levels of 19 correlation, the 217 × 70 GHz foreground contribution would be dominated by this correction because of the large DSFG power at 217 GHz and tSZ power at 70 GHz. 2.1.5. Galactic Foregrounds. Though galactic foreground cleaning represents a key challenge for Planck , the aim of this chapter is to understand the impact of the extragalactic foregrounds rather than to provide the most accurate Planck forecast possible. For treatments of galactic foregrounds see, for example, Tegmark et al. [2000], Gold et al. [2011]. The impact of ignoring the galactic foregrounds on the temperature power spectrum is minimal provided suﬃciently conservative masking and template cleaning. Our neglect of galactic foregrounds will be a bad approximation at low- polarization and leaves our forecasts for r and τ overly optimistic; we don’t report these forecasts. 2.1.6. CMB. For the primary CMB signal itself, we use an 8-parameter model which includes the baryon density Ωb h2 , the density of cold dark matter Ωc h2 , the optical depth to recombination τ , the angular size of the sound horizon at last scattering Θ, the amplitude of the primordial density ﬂuctuations ln[1010 As ], the scalar spectral index ns , the dark energy equation of state parameter w, and the tensor-to-scalar ratio r. Freeing w opens up the “geometric degeneracy” which is typically broken by adding an external dataset such as supernovae data. Rather than do this, we simply put a ±0.3 Gaussian prior on w to reasonably constrain the chain while allowing it to explore the parameter space. Because we are interested in the simplest description of how the foregrounds aﬀect cosmological parameters, we do not consider extensions to our model such as a running spectral index, non-ﬂat universes, non-standard eﬀective number of neutrino species, or a diﬀerence in primordial helium from standard big bang nucleosynthesis. Due to the small angular scales where they aﬀect the CMB anisotropy, it is possible that such parameters could be even more degenerate with the foregrounds than the “vanilla” set we consider. For quick and highly accurate CMB calculations during our MCMC chains, we use a PICO [Fendt and Wandelt, 2007] interpolation of a training set generated by CAMB [Lewis et al., 2000]. PICO was trained using the June 2008 version of CAMB which uses a now-outdated recombination code. Though we use the older code, we do it in a self-consistent manner and don’t expect any impact on our forecasting. Additionally, the training set includes the option of a non-linear lensing contribution described in Challinor and Lewis [2005] which we use. 20 2.2. Fiducial Model and Current Constraints For our forecasting, we create simulated power spectra (henceforth the “simulated data”) using the model described in the previous sections. We pick one single set of model parameter values, called the “ﬁducial values” or the “ﬁducial model” in general, which is the baseline for the diﬀerent cases of simulated data which we consider. The model used to analyze the simulated data, which generally contains small changes relative to the ﬁducial model, will be called the “analysis model.” In Table 2.3 we summarize all of the parameters in our model, the naming convention, and their ﬁducial values. The ﬁducial values are chosen to be consistent with current cosmological constrains from WMAP7 [Komatsu et al., 2010], and with constraints on the foreground components from ground-based data such as SPT and ACT. In the following paragraphs, we describe the method used to arrive at our ﬁducial model. Because the expected SZ power depends strongly on cosmology, special care was taken so that our ﬁducial SZ power and cosmology agree. To achieve this, we use the constraint from Lueker et al. [2010] on the linear combination DtSZ + .46 × DkSZ = 4.2 ± 1.5 μK2 (at = 3000 and 153 GHz) along with the cosmological scalings in Table 2.1. We then importance sample the WMAP7 ΛCDM+TENS4 chain by calculating at each step the expected SZ (kinetic and thermal) power assuming no theory uncertainty, then applying the prior from Lueker et al. [2010]. The new best ﬁt point in the post-processed chain mainly shifts SZ power up relative to best ﬁt SPT value and σ8 down relative to the best ﬁt WMAP7 value. All other cosmological parameters are also aﬀected (at a smaller level), and their new mean values form our ﬁducial cosmology, which remains 1-σ consistent across all parameters with WMAP7. For the radio sources, the tightest constraints on the expected Planck power come from the Vieira et al. [2010] catalog which contains sources in the decade of brightness just below the Planck ﬂux cut. Fitting a de Zotti et al. [2005] model to the data yields the values listed in Table 2.3, notably radio Poisson power of 53 μK2 at 143 GHz assuming a 330 mJy ﬂux cut. Since the DSFG Poisson contribution is nearly independent of ﬂux cut, we expect the same Poisson power in Planck maps as in SPT maps, adjusting only for bandpass diﬀerences. We get our ﬁducial value for Planck Poisson power at 143 GHz by extrapolating in frequency from the best-ﬁt value of the SPT 150 GHz power as given in Shirokoﬀ et al. [2010]. Our ﬁducial values for αD and αC also come from the best-ﬁt values in Shirokoﬀ et al. [2010]. We set σD to 0.4 following 4Available at http://lambda.gsfc.nasa.gov/ 21 the arguments in Knox et al. [2004], although it is not yet well constrained by observations. We adopt a clustering tilt nC = 1 so that it (roughly) has the shape expected at small scales due to the the observed clustering properties of high-redshift galaxies. As argued by Scott and White [1999], the observed clustering properties of z ∼ 3 Lyman break galaxies, namely an angular correlation function proportional to θ−0.9 [Giavalisco et al., 1998], correspond to D ∝ 1.1 . Since we multiply the power-law by the linear theory template, this is similar (at > 1500) to the power-law only D ∝ 0.8 shape used as baseline models in both Dunkley et al. [2011] and Shirokoﬀ et al. [2010]. Although ruled out by the Planck data, our ﬁducial model is at least closer to the measurements than all of the other models plotted in Figure 2.1. The agreement is suﬃcient for our purposes here, though we will certainly be updating our CIB modeling in the near future. Following Battye et al. [2011] which found a mean fractional polarization of 4.5% at 86 GHz (and varying weakly with frequency) for the WMAP point source catalog [Wright et al., 2009], we adopt a ﬁducial value of fR = 0.05. For DSFGs we expect an even smaller level of average polarization fraction. Polarized dust emission arises due to alignment of grains in interstellar magnetic ﬁelds. We somewhat arbitrarily set fD = 0.01 for our ﬁducial model which is consistent with the ﬁnding that, in our own galaxy, the coherence length for magnetic ﬁelds is much smaller than the extent of the dust emission [Prunet et al., 1998]. Figure 2.4 shows the ﬁducial CMB and foreground contribution to Planck TT, TE, and EE power spectra (with the exception of tSZ-DSFG correlation which is plotted at 30% rather than its ﬁducial value of 0%). 2.3. Survey Properties We consider simulated Planck data in the four bands between 70 GHz and 217 GHz. These are chosen because they contain nearly all of the signiﬁcant CMB information. Though the neglected channels place little extra constraints on the CMB, they are crucial for understanding and cleaning the foregrounds. We consider their eﬀect implicitly by testing limits such as lowered Poisson power amplitudes, or ﬁxed DSFG clustering shapes. Additionally, we also consider the beneﬁt of higher resolution ground-based data, which we model after SPT 90 GHz, 150 GHz, and 220 GHz channels. We divide the data into two ﬁelds: a 100 deg2 “deep” ﬁeld and a 1000 deg2 “wide” ﬁeld. We henceforth refer to these two datasets as Ground-deep and Ground-wide. The depths, sky coverage, and ﬂux cuts used in our forecasting are summarized in Table 2.2. 22 Table 2.2 Survey Properties Band T (E/B) Beam Notes (μK-arcmin) (arcmin) (GHz) Planck 70 177 (253) 14 fsky = 70% 61 (98) 10 Sc = 330 mJy 100 42 (80) 7.1 143 217 64 (132) 5 Ground-Deep 90 53 1.6 fsky = 100 deg2 150 13 1.15 Sc = 6.4 mJy 35 1.05 220 Ground-Wide 90 53 1.6 fsky = 1000 deg2 18 1.15 Sc = 6.4 mJy 150 80 1.05 220 Note. — Instrument properties used to generate simulated power spectra. The beam width is given as a full-widthhalf-max. Sc refers to the ﬂux cut above which brighter sources are masked out. Our simulated data take the form of auto and cross spectra from as many bands as are present for a given patch of sky. The four Planck frequency channels form 10 TT, EE, and BB and 16 TE power spectra, with an additional 18 TT power spectra from the three extra frequencies in regions of Ground overlap5. We do not assume overlap between Ground deep and wide, nor do we form cross spectra between Ground temperature and Planck polarization as these are expected to be a very small contribution to the CMB and foreground information. Planck BB polarization is also ignored except in one test case where we ﬁnd its impact is minimal on our cosmological parameterization. We simulate power spectrum assuming a uniform masking threshold across the sky. The only exception is in the case of Planck and Ground overlap. For such patches of sky, we assume Planck maps can be masked using a point source mask from the higher resolution Ground data. Thus, for the overlap areas, even the Planck auto spectra will have greatly reduced radio Poisson power. The non-zero width of frequency bandpasses creates a diﬀerent eﬀective frequency for each component in each band. For components with uncertain spectral shapes, the variation in eﬀective frequency leads to percent level corrections which can be neglected. In this chapter, values quoted from ground-based experiments are normalized at, and explicitly cite, the corresponding eﬀective 5In general N frequency channels can be used to create N (N + 1)/2 power spectra of type TT, EE, and BB, and N 2 of type TE 23 Table 2.3 Summary of Model Parameters Parameter Cosmological Ω b h2 Ω c h2 Θ τ w ns ln(1010 As ) r Dusty Poisson αD σD DD fD Radio Poisson αR σR DR γR fR Dusty Clustered αC DC nC SZ Eﬀects DtSZ DkSZ,OV DkSZ,P RP Correlations rtSZ,C Fiducial Value Current Constraints (1σ) Deﬁnition .022565 .10709 .010376 .0799 -1 .9669 3.1462 .13 .00073 .0063 .000029 .015 .13 .014 .045 <.36 (95%) Baryon density Cold dark matter density Angular size of the sound horizon at last scattering Optical depth to reionization Dark energy equation of state parameter Scalar spectral index Scalar amplitude Tensor-to-scalar ratio 3.8 .4 5.9 μK2 .01 0.35 Spectral index Spectral index intrinsic spread Amplitude at = 3000, ν = 143 GHz Dusty polarization fraction -.5 .1 53 μK2 -.8 .05 0.1 < 0.6 (95%) 10 μK2 0.1 Spectral index Spectral index intrinsic spread Amplitude at = 3000, ν = 143 GHz, Sc = 330 mJy Brightness function power law index Polarization fraction 3.8 3.9 μK2 1 0.4 1.2 μK2 Spectral index Amplitude at = 3000, ν = 143 GHz Nonlinear tilt 4.3 μK2 2.7 μK2 1.5 μK2 1 < 6.8 μK2 (95%) < 6.5 μK2 (95%) < 6.5 μK2 (95%) tSZ amplitude at = 3000, ν = 143 GHz OV amplitude at = 3000 Patchy amplitude at = 3000 Patchy shift 0.8 Correlation between tSZ and DSFGs at = 3000 0 Note. — A summary of the parameters in our model. The ﬁducial values generate our simulated data. The current constraints column gives the 1σ constraints on our model given WMAP power spectra and radio source counts, SPT power spectra and radio/DSFG source counts, and ACT power spectra. Note that due to the process by which the ﬁducial values were chosen (Sec. 2.2) they are not necessarily the most likely values given current data; they are, however, totally consistent with the most likely value to within 1σ. frequency. For Planck , it is suﬃcient for our forecasting purposes to ignore this and use the nominal band centers for all components. 2.4. Forecasting Methodology The analysis of the simulated data assumes perfectly known Gaussian beams, no calibration uncertainty, isotropic noise, and ignores the eﬀects of mode-mode coupling on the cut sky. While these assumptions are not suﬃcient for modeling real data, we expect them to be adequate for our purpose of modeling the extragalactic foregrounds, and understanding their importance on cosmological parameter biases and statistical errors. 24 T 1000 z 100 H G 10 7 T 1000 z 100 H G m T l D z H 100 3 10 G 4 T z H 100 7 10 G 2 1000 000 000 > = < 6 1000 000 000 1000 000 000 1000 000 000 tSZCIB Rad io Poisson 1000 - kSZ u sty Poisson 1000 - tSZ u sty Cu ste red 10 K @ CMB ; : 9 8 " ! # $ % &' ()* + E V Q F N ?AF J ?A?F V F? O L U Q F N ?AF L ?A?F O ~ L W | { x v $ % ,'' ()* + E $ % ,./ ()* + E $ % 5,& ()* + E F? U } V w F? U F N ?AF W X ?A?F V F? O Q Y U F N ?AF W ?A?F O Q J [ F??? \??? ]??? ^_ `bc f g F??? \??? ]??? h__ `bc f g F??? \??? ]??? hjp `bc f g F??? \??? ]??? qh^ `bc f g Figure 2.4 All 36 power spectra which can be formed from Planck 70 GHz–217 GHz temperature and E-mode polarization, and the prediction of our ﬁducial model for the CMB and foreground power in each of them (with the exception of the tSZ-DSFG correlation which is shown at 30% instead of its ﬁducial value of 0%). The black dashed line shows the errors bars for -bins of width Δ = 256. Dotted lines indicate negative power. 25 Under these assumptions, the so called “pseudo power spectrum” which includes both signal and noise is given by, (2.15) S + δij w−1 exp(2 σb2 ) Cij, = Cij, Here i and j each label one of the maps, and the noise is parameterized by the weight per solid angle w, and the beam width in radians σb . Assuming the Gaussianity (the validity of which we discuss in Sec. 2.4.1), the covariance on our estimate of the signal power spectra is, Σ(ij)(kl) ≡ (2.16) = S S ij S − C S ) (C − Cij )(C kl kl 1 (Cil Cjk + Cik Cjl ) (2l + 1)fsky where we have suppressed the dependence for notational simplicity. We use the covariance to form the likelihood as a function of parameters θ, (2.17) S S S ij S (θ) − C (θ) − C Σ−1 C −2 ln L(θ) = Cij kl kl (ij)(kl) Note we have neglected the normalization term since it does not vary with θ. Our simulated data are the mean expected power spectra; i.e., they do not include a sample of the errors from the bandpower covariance matrix. Leaving out these ﬂuctuations has the beneﬁt of making the best ﬁt χ2 equal to exactly 0 (as long as our analysis model and simulation model are the same) and has no aﬀect on our forecasting abilities. Tests we performed showed that a Gaussian propagation of uncertainty from the C ’s to the model parameters can be insuﬃciently accurate due to the highly non-Gaussian shape of the foreground parameter posterior likelihoods. Instead, we run a full Markov-Chain Monte-Carlo (MCMC) analysis using a custom multi-frequency extension to CosmoMC [Lewis and Bridle, 2002]. 2.4.1. Non-Gaussianity. In the previous section, we make assumptions that the signal and noise are a Gaussian random ﬁeld (Equation 2.16) and that the likelihood itself is Gaussian (Equation 2.17). The latter is a good approximation for examining the foregrounds since they are relevant at high- where averaging over many alm ’s drives the likelihood to Gaussianity. The former, however, can be a very bad approximation for the radio/dusty Poisson and tSZ (which is just the one-halo term) since they are sourced by Poisson number ﬂuctuations. In general, we expect the tSZ to suﬀer most from non-Gaussian eﬀects since the ﬂuctuation power is most heavily weighted 26 towards a small number of bright objects; the dusty Poisson component by contrast receives most of the power from well below the ﬂux-cut where there are many sources. Non-Gaussianity can impact our likelihood both by producing long tails in the C probability distribution and by the C variance diﬀering from that expected from sample variance. The ﬁrst problem is alleviated by large sky coverage. We can use the tSZ as a worst-case scenario, where calculations by Zhang and Sheth [2007] and simulations by Peel et al. [2009] suggest for 1000 deg2 one would expect a skewness of less than 0.1 in the distribution of C ’s. The second problem, however, does not average away with fsky , and notably becomes worse when binning in . The full expression for the C covariance is, (2.18) ΔC ΔC = −1 fsky 2C 2 δ T + (2 + 1)Δ 4π where the ﬁrst term is the sample variance for a Gaussian ﬁeld, and the second is the non-Gaussian tri-spectrum contribution [Komatsu and Seljak, 2002]. To examine the impact of ignoring the trispectrum in our analysis, we use the following method. Since binning in brings out the eﬀect of a non-zero tri-spectrum, we consider making one giant -bin which is inverse variance weighted across the entire angular range. If the variance in that bin still receives a negligible contribution from the tri-spectrum, then we can safely ignore its eﬀect (including -to- correlations) in our more reasonably binned analysis. The primary contribution to the tSZ tri-spectrum comes from galaxy cluster shot noise [Komatsu and Seljak, 2002]. Shaw et al. [2009] calculate this contribution, and we use their tri-spectrum to apply the binning procedure described above. Using only Planck 143 GHz data and forming the single -bin, we ﬁnd the error bar is increased by 1%. For Ground 150 GHz, it is increased by 30%, although we expect this number to decrease for an analysis which does not ignore mode-coupling. To calculate the radio tri-spectrum, we create a set of 1000 full sky radio realizations by Poisson sampling our ﬁducial brightness function. We ﬁnd, somewhat surprisingly, that the radio power spectrum with a Planck ﬂux cut is even more non-Gaussian than the tSZ, dropping to a similar level for a Ground ﬂux cut. Applying the binning procedure yields an error increase of 6% for a Planck ﬂux cut and Planck 143 GHz data, and 4% for a Ground ﬂux cut and Ground 150 GHz. Given the small error increase in all cases, we conclude it is safe to ignore the foreground nonGaussianities in our analysis. We do note, however, the possibility that non-Gaussianities could 27 be relevant to other analyses, for example, component separation methods which produce heavily -binned foreground-only maps, or lensing reconstructions which rely on oﬀ-diagonal correlations. 2.5. Compression to a CMB Power Spectrum Estimate Before getting to our results, it is useful to explore the foreground contamination in a more model-independent manner, motivated by two drawbacks of our procedure. First, there is a large amount of data one must work with—our bandpower covariance matrix at each is 46×46 for Planck and 64×64 for Planck +Ground. Second, 17 foreground parameters must be marginalized over, and if one wanted to examine constraints on a new cosmological model, the whole procedure would have to be repeated. Here we present a procedure for compressing all the power spectra to 1) a single CMB estimate and 2) a low dimensional parametrization of the residual foregrounds in this estimate. We describe the procedure here for temperature-only power spectra with errors that are uncorrelated from multipole to multipole. The generalization to include to correlations and polarization is in Appendix B. Given N power spectra (for example the 10 TT spectra we consider for Planck ), we would like to split our data up into N − 1 linear combinations of power spectra that have no sensitivity to the CMB and then ﬁnd the remaining linear combination that contains CMB and whose errors are uncorrelated with those of the N − 1. With this split made, we can then derive our foreground constraints using the CMB–free linear combinations. Doing so means foreground constraints can be made independent of our modeling of the CMB (other than the assumed frequency dependence). We use the N − 1 CMB–free linear combinations to ﬁnd the constraints they place on our foreground model parameters via MCMC. For each point in the chain we can determine the contribution to the CMB linear combination. We sample over all these contributions to ﬁnd the mean contribution and ﬂuctuations about that mean. We ﬁnd a low-dimensional description of the ﬂuctuations via a principal component decomposition. 2.5.1. Splitting the power spectra into CMB–free and a CMB estimate. Let us begin by ﬁrst considering arbitrary linear combination of the power spectra, (2.19) C̃ μ = wiμ Ci i where Ci are the i = 1 to N power spectra. The weightings we consider will be -dependent; the lack of any labeling by is solely for notational simplicity. Here, μ is merely a label to distinguish 28 1.00 0.50 0.20 t h ig 0.10 e W 0.02 4 44 4 CMB usy Cusrd usy Posson Rado Posson 1000 4 0.05 SZ kSZ H op aL H 7 GzL 100 K m@ l D 10 1 CMB kSZ H op aL H 7 GzL Rsdua PC's 1000 100 K m@ D l 10 1 1000 1500 2000 2500 3000 3500 Figure 2.5 Top: The -dependent weightings which form the CMB linear combination (Eq. 2.29). All possible auto/cross spectra from Planck channels in Table 2.2 were considered. Dashed lines indicate negative weight. Middle: The mean foreground contribution to the CMB linear combination for our ﬁducial model. Note for example that tSZ (purple) is not present at high because only 217 GHz is used there. The dominant non-Poisson component for the -range where Planck is most sensitive is the DFSG clustering. Bottom: Principal components of foreground residuals (constrained by the CMB–free linear combinations) with amplitudes set to 1-σ. Note that we only need two principal component amplitudes to be accurate to > 1μK2 . (The errors in bin widths of Δ = 256 for both the CMB linear combination and for 217 GHz alone are plotted as dashed lines in the bottom two plots.) diﬀerent weightings; the C̃ μ are a linear combination of the old power spectra with weight wiμ . Note that if a weighting satisﬁes, (2.20) wiμ = 0 i it is not sensitive to the CMB. 29 We would ﬁrst like to ﬁnd the CMB weighting wCMB which will be statistically orthogonal to the N − 1 linear combinations which satisfy the CMB–free condition (Eq. 2.20). We would also like this weighting to be properly normalized so that, (2.21) wiCMB = 1. i To satisfy the orthogonality condition it helps to work in a primed space deﬁned by a linear transformation via wαμ = (2.22) Lαi wiμ i where L is the Cholesky decomposition of the bandpower error covariance matrix, Σ = LLT . The advantage of the primed space is that the basis vectors in the primed space correspond to power spectra whose errors are statistically orthogonal; i.e., with the weightings set so that wαμ = δμα (now setting μ = 1...N ) the errors in the corresponding power spectra satisfy δ C̃ μ δ C̃ ν = δμν . (2.23) The primed weights that satisfy the CMB–free condition satisfy (2.24) L−1 iα wα = 0. α,i Thus any power spectrum with primed weighting proportional to wαCMB = (2.25) L−1 iα i is perpendicular to any vector satisfying the CMB–free condition, as one can easily verify. To ﬁnd the CMB weighting in the unprimed space we perform the inverse transform and normalize to satisfy the normalization condition (Eq. 2.21) (2.26) wkCMB = ⎡ −1 ⎣ L−1 kα Li,α i,α (2.27) i 30 ⎤−1 −1 ⎦ L−1 kα Liα i,k,α = ⎡ ⎣ Σ−1 ik i,k ⎤−1 ⎦ Σ−1 ik . Note that this is the expression for inverse-variance weighting. Our remaining task is to construct the N − 1 CMB–free weightings in a manner that leaves them all statistically orthogonal to the CMB weighting. We do so by applying the Gramm-Schmidt procedure in the primed space. This gives us N − 1 orthogonal vectors that are all orthogonal to the CMB direction as well, that we will call vαμ for μ = 2, N . The weightings in the unprimed space are then given by μ viμ = L−1 iα vα . (2.28) We now deﬁne the matrix W so that (2.29) W1i = wiCMB Wμi = viμ {for μ = 2...N }. This matrix deﬁnes the linear combinations of the power spectra that have all the properties we desire. The ﬁrst row is the optimal CMB weighting and subsequent rows give the N − 1 CMB–free linear combinations. All the linear combinations are statistically orthogonal; i.e., the covariance matrix for the new power spectra, W T ΣW , is diagonal. Furthermore, W is non-singular so we have not lost any information through this re-weighting. 2.5.2. Modeling the foreground residuals with principal components. With the weight matrix W deﬁned, we can constrain the foreground power in the N − 1 CMB–free power spectra by running an MCMC chain. Despite the large number of parameters and power spectra, this analysis is fast in practice because the foreground model consists of simple analytic forms and precomputed templates, and does not depend on any costly Einstein-Boltzmann solver or lensing models. For the set of foreground parameters at each step in this chain, we calculate the corresponding foreground contribution to the CMB linear combination. These -dependent contributions form the columns of the Y matrix in a principal component analysis (see Appendix A). Following the PCA procedure, we have a few principal components and priors on their amplitudes which must be marginalized over in a separate chain which uses only the CMB linear combination. 2.5.3. Discussion of linear combination analysis. The weightings which make up the CMB and CMB–free linear combinations depend on the bandpower covariance matrix, and thus on the noise properties of the instrument, any ﬁltering which is performed, and on the true power 31 spectrum on the sky. The principal components for the foreground residuals also depend on the choice of foreground model. For a Planck temperature-only forecast and for our ﬁducial model, we present the results of a linear combination analysis. In the top panel of Figure 2.5 we plot the weights for the CMB linear combination as a function of . At high where the measurement is noise dominated, nearly all of the CMB information is contained in the 217 GHz map which is the least noisy. At lower where we become dominated by cosmic variance, the CMB information comes from the channels with the lowest foreground contamination. Given these weights, we plot in the middle panel of Figure 2.5 the foreground contribution to the CMB linear combination and the error bars on this new power spectrum. Also shown are the error bars for the 217 GHz channel alone for comparison. The maximum improvement is at = 2000 where the error bars tighten by a factor of 1.4. We also see that the dominant contribution to the foreground power in the range where Planck is most sensitive to the CMB is the radio Poisson, followed by the DSFG clustering. Finally, we perform a PCA on the foreground residuals in the CMB linear combination. The ﬁrst several principal components are shown in the bottom panel of Figure 2.5. We ﬁnd that all of the variation > 1 μK2 can be described by two amplitude parameters, as compared to the 14 parameters which govern these foregrounds. Another way to put this is that using the CMB–free combinations we can clean out almost 140 μK2 of foregrounds (at = 3000), leaving only tens of μK2 of residual uncertainty, modeled with the two principal components. 2.6. Results With the model and forecasting tools in place, we are ready to present the results of our main analysis. We want to ﬁnd which components can potentially cause large biases in an analysis of Planck data, so that we can model them with suﬃcient care. We would also like to know how much constraining power is reduced due to foreground confusion. Could signiﬁcant improvements in cosmological parameter constraints be achieved by using additional data and/or modeling? To answer these questions, we run a suite of forecasting analyses aimed at singling out the eﬀects of each foreground contribution. The next sub-sections are organized as follows. First we examine the importance of each component by turning it on or oﬀ in the analysis. For the components which we deem important, 32 we check whether our modeling is suﬃcient to protect the cosmological parameters from biases, both at the Planck and Planck +Ground sensitivity levels. We then examine the degradation in statistical errors from the need to marginalize over the foregrounds, and ﬁnally we explore the impact of adding in ground-based data. 2.6.1. Importance of the Diﬀerent Components. Figure 2.6 shows the eﬀect of removing four foreground components—the DSFG clustering, tSZ, kSZ, and tSZ-DSFG correlation—one at a time from the analysis model, while they are actually present in the simulated data at their ﬁducial value. We present the results by plotting likelihood contours in the ns and Ωc h2 plane, since changes in those two parameters aﬀect the primary CMB at the smallest scales and are the most susceptible to foreground biases. We also show the amplitudes of the clustering and SZ eﬀects as their -space shapes make them most degenerate with cosmological parameters. All of the chains in this section include only Planck power spectra in the simulated data. We expect the DFSG clustering to be extremely important to model since it is the second largest foreground contribution to the CMB linear combination in the -range where Planck is most sensitive. When marginalized over, this contribution is constrained to be 10.5 ± 0.6 μK2 at = 1500, so setting it to zero is about an 18σ change. The dot-dashed green contours in Figure 2.6 show that this is compensated by a systematic bias of 7σ is ns and 11σ in Ωc h2 , along with an increased kSZ power to about 30 μK2 . Using the middle panel of Figure 2.5 as a visual guide, we can examine how this happens. Though the kSZ increases to compensate for the missing clustering power at high-, its shape is ﬂatter than the DSFG contribution to the CMB linear combination, so ns decreases to roughly remove the extra power at a low-. At 217 Ghz, the kSZ power in our ﬁducial model is a factor of 20 times smaller than the DSFG clustering and is therefore (in terms of cosmological parameter estimation) less troublesome. However, due to its identical frequency dependence to the CMB, we do expect the amplitude of the kSZ signal to be degenerate with cosmological parameters. The solid black contours in Figure 2.6 demonstrate the bias introduced if the kSZ component is omitted from the foreground modeling. We ﬁnd a roughly 0.5σ bias in ns as it increases to try to ﬁll in the missing 4.2 μK2 of kSZ power. The DSFG and tSZ plane (bottom right panel of Figure 2.6) shows that the other foregrounds are largely unaﬀected. The thermal SZ component is neither frequency independent, nor does it contribute as much power as the DSFG clustering, so we do not expect a bias as large as in either previous case. It 33 0.98 0.97 0.96 0.95 Fiducial No kSZ No tSZ No tSZ-CIB correlation No DSFG Clustering 0.94 0.106 0.108 0.110 W h 0.112 0.114 0.116 D 3 D @ mK No No No No D D D @ mK D @ mK Data Analysis Δχ2 N tSZ DtSZ = 4.3 DtSZ = 0 43 6 DkSZ = 4.2 DkSZ = 0 2 1 kSZ DC = 3.9 DC = 0 791 1930 DSFG Clustering 3 1 tSZ-DSFG Corr. rtSZ,C = −.3 rtSZ,C = 0 Figure 2.6 68% (and 95% in the bottom panel) conﬁdence contours for a suite of test cases examining the eﬀect of neglecting to model diﬀerent foregrounds. Unless explicitly stated above, other parameters were included in the data at their ﬁducial values listed in Table 2.3 and were marginalized over in the analysis. N corresponds to the maximum number of -bins per power spectrum one could use and still detect the error in modeling at 3 sigma (see Sec. 2.6.1 for further discussion). 34 does, however, project into the CMB linear combination to an -shape very similar to the CMB itself (see the middle panel of Figure 2.5), making it more likely to be degenerate with cosmological parameters. From the results, we see about 0.3σ biases in each of ns , Ωc h2 , and Ωb h2 . Finally, we consider neglecting a 30% tSZ-DSFG correlation, a value on the high end of expected correlation, but still consistent with Shirokoﬀ et al. [2010]. We expect this to have the smallest eﬀect on the cosmological parameters since the power contribution is sub-dominant to all of the other foreground components at all frequencies which appear in the CMB linear combination at > 1%. While the measured tSZ amplitude is biased at a few sigma as it raises to compensate for the missing power, the eﬀect is not large enough to signiﬁcantly impact any of the cosmological parameters. One question is whether any of these analysis errors would be caught by a goodness-of-ﬁt test. To address this question we present Δχ2 values in the table in Fig. 2.7. We can expect rms √ ﬂuctuations in χ2 to be Nb where Nb is the total number of bandpowers which is roughly equal to the number of degrees of freedom. If one is searching for signs of a contaminant that is very slowly varying in , then one would bin coarsely to reduce the statistical ﬂuctuations in χ2 , to make a more stringent goodness-of-ﬁt test. We deﬁne N to be the number of -space bins such that the absolute Δχ2 from the ﬁt is 99.7% inconsistent with random ﬂuctuations. Thus we have N = (Δχ2 )2 /(9 × Nspec ) where Nspec is the number of power spectra (36 here). We see that binning would not have to be coarse at all to detect the poor ﬁt caused by neglecting clustering. We also see that for the other entries in the table, binning would have to be extremely coarse for the ﬁts to be noticeably poor. Indeed, the binning would have to be coarser than is practical since the signals of interest, as well as the contaminants, would vary signiﬁcantly across a bin. We conclude that only the “no clustering” case would produce a noticeably bad ﬁt for Planck only. 2.6.2. Modeling Suﬃciency. Given the demonstrated importance of the foreground components, we would now like to see if our modeling is suﬃcient to protect the cosmological parameters from biases if we have modeled the components, but modeled them incorrectly. In this section we consider the DSFG clustering and the tSZ eﬀect. For the DSFG clustering, we turn to the models plotted in Figure 2.1. Our parameterization should have the most trouble reproducing the Amblard and Cooray [2007] model, which switches to a power-law (as a consequence of non-linear clustering) at ≈ 2500 rather than at = 1500 as 35 0.970 0.965 0.960 0.955 Fiducial Battaglia Battaglia Amblard Amblard 0.106 tSZ H P nk L tSZ H P nk +Groud L Clusterig H P nk L Clusterig H P nk +Groud L 0.107 W h 0.108 2 0.109 Figure 2.7 The eﬀect on cosmological parameters from trying to ﬁt our model to simulated data which includes (orange) the Battaglia et al. [2010] tSZ template and (black) the Amblard and Cooray [2007] clustering template. These two models are the most dissimilar to ours, and thus show our model can protect against biases of a few percent up to Planck sensitivity. The inclusion of Ground data necessitates more detailed modeling of only the clustering. in our ﬁducial model. The orange contours in Figure 2.7 show the results obtained when ﬁtting our model to simulated power spectra that assume the Amblard and Cooray [2007] clustering template. As we had hoped, for the case of Planck only (solid lines), there is no signiﬁcant biasing. We also explore the ability of our tSZ principal component model [based on the analytic model of Shaw et al., 2010] to encompass the variations in the tSZ models shown in Figure 2.2. We elect the Battaglia et al. [2010] one as the most dissimilar, since it lacks the eﬀects of radiative cooling, and should be the most diﬃcult for the Shaw et al. [2010] model to reproduce. Despite these diﬀerences, Figure 2.7 shows that for Planck the model is suﬃcient to encompass the shape uncertainty and protect cosmological parameters. When we add in Ground (dashed lines), the requirements on the modeling accuracy are more stringent. For the clustering case, we see an almost 1σ bias from using our ﬁducial model when the true model is the Amblard and Cooray [2007] clustering template. Analyses with current data can tolerate much more discrepant clustering shapes [Dunkley et al., 2011]. For future Planck +Ground analyses, the clustering shape will need to be modeled more accurately. For tSZ the modeling appears to be more robust; tSZ-induced biases are small even in the Planck +Ground case. 36 Table 2.4 Statistical Error Degradation Planck (fgs ﬁxed) Planck (fgs marginalized) Planck +Ground Planck (Clean DSFG) 104 Ωb h2 1.1 1.2 1.1 1.1 103 Ωc h2 1.0 1.0 1.0 1.0 104 Θ 2.6 2.6 2.6 2.6 103 ns 3.0 3.6 3.3 3.3 ln(1010 As ) 1.3 1.4 1.3 1.3 DD – 3.4 0.3 2.0 DR – 6.0 3.0 6.0 DC – 1.3 0.5 DtSZ – 1.0 0.6 1.0 DkSZ – 4.4 2.6 4.4 Note. — Entries are 1-σ constraints. Dashes indicate the parameter was ﬁxed, while blanks mean the parameter is not applicable to that case. The normalization parameters Dx are in units of μK2 . The diﬀerent cases correspond to: (fgs ﬁxed) Fixing all of the foregrounds at their ﬁducial values. (fgs marginalized) Marginalization over our full foreground model. (+Ground) Also including Ground auto and cross spectra in the simulated data. (Clean DSFG) Assuming 90% reduced clustering power due to cleaning from higher frequencies. 2.6.3. Statistical Error Increase with and without Auxiliary Data. We have demonstrated the possibility of σ-level biases in cosmological parameters arising from failure to model foregrounds. To prevent these biases, the foregrounds need to be jointly estimated or marginalized over. We now turn to two questions: 1) How much do the cosmological parameter statistical errors degrade due to foreground uncertainty? and 2) How much can be gained from using other data to constrain foregrounds and thereby reduce that degradation? The top two rows of Table 2.4 show the eﬀect of marginalizing over our entire foreground model as opposed to ﬁxing it at ﬁducial values. In each row, the diﬀerence from 100% is the percent degradation due to foreground marginalization. The second row shows the degradation is limited to 20% for ns and 10% for As and Ωb h2 . We see no degradation in τ and r since they are mainly constrained by large scales where the extragalactic foregrounds we consider are negligible. The dark energy equation of state w is unaﬀected because it is mainly constrained by our ±0.3 prior. Ground data can help reduce this degradation by better constraining the foregrounds using auto and cross spectra that are more sensitive at small scales. The improvement from adding these to the simulated data is shown in the row labeled Planck +Ground. The measurement of DSFG shot noise is improved ten-fold, with the clustering and SZ eﬀects also tightened by a factor of two. The radio amplitude is improved through constraints on the spectral dependence, and could be further improved though a prior on γR from Ground source counts. The eﬀect on the cosmological parameters is to remove essentially all of the degradation we incurred from marginalizing over the foreground model. Above about 300 GHz, the DSFGs are the dominant source of anisotropy power on all scales. Correlations with maps at these higher frequencies, for example maps from Planck bands above 37 217 GHz or Herschel, can be used to place tight constraints on the DSFG components, at the price of requiring more sophisticated modeling for the spectral dependence and shape. Even with such modeling, the correlations are no longer fully coherent across frequencies so there is a limit to how much of the DSFG power can be “cleaned out” of the lower frequency maps. Following results in Knox et al. [2001], which assumes a redshift dependent grey-body emissivity density tracing the linear matter power spectrum, we assume that we could clean out 90% of DSFG clustering power at the lower frequencies. As in the previous case of adding in Ground data, this again is enough to eliminate nearly all of the degradation on cosmological parameters. 2.7. Conclusions To make full use of Planck ’s very small statistical error on CMB power spectra out to ∼ 2500, without introducing signiﬁcant bias in the cosmological parameters, we must include contributions from extragalactic foregrounds and secondaries in our model of the data. Here we have presented a model of these contaminants, based on the latest data and modeling developments, and demonstrated its ability to remove biases in an 8-parameter cosmological model. The foreground model has 17 parameters – many more than any extragalactic foreground model used in analysis of CMB data to date. Despite the large number of nuisance parameters, marginalizing over all of them only increases statistical uncertainties in the cosmological parameters by, at most 10 to 20%. Almost all of this degradation can be avoided by inclusion of ground-based data or higher frequency Planck bands. Our model includes Poisson components from both radio galaxies and DSFGs, a clustering component due to DSFGs, contributions to kSZ power from patchy reionization, as well as after reionization is complete, and tSZ power. If kSZ power and tSZ power are at our ﬁducial values (slightly higher than the preferred values given current high-resolution ground-based data) then ignoring them in an analysis of Planck data would produce small, almost negligible biases, to cosmological parameter estimates. On the other hand, ignoring the clustering of DSFGs, would lead to a very large bias in cosmological parameters. To avoid having to marginalize over these 17 parameters every time a new cosmological model is analyzed, we broke our procedure up into a two-step process, with the ﬁrst step independent of the model of the primary CMB power spectra. The second step is an analysis of the CMB power spectra estimated in the ﬁrst step, with a small number of foreground template amplitude 38 parameters to marginalize over. The shapes of these templates, and priors on their amplitudes, are also outputs of the ﬁrst step. Only the second step needs to be repeated in order to get constraints on the parameters of a new model of the primary CMB power spectra. Looking toward the near future, the model will deﬁnitely evolve, increasing the faithfulness with which it represents reality, as we gain more information from the CMB-dominated channels in Planck , higher-frequency Planck channels, higher-resolution ground-based data (SPT, SPTpol and ACTpol) and higher-resolution, higher-frequency space-based data (Herschel). One could easily use our foreground model to study potential biases in extensions of the primary cosmological model, to include, for example, departure of the Helium mass fraction from the predicition of Big Bang Nucleosynthesis, or a diﬀerence in the number of eﬀective neutrino species from the standard model value. 39 CHAPTER 3 Predicting Poisson Foreground Amplitudes In the previous chapter, we showed that a realistic model of extra-galactic foregrouds must include a term due to Poisson ﬂuctuations of dusty and radio galaxies. Including and marginalizing over such a term degrades constraints on cosmological parameters, so it is of interst to try and put priors on it. Alternatively, we would like to check that the resulting constraints on the Poisson amplitude are in general agreement with other sources of knowledge about these galaxies. In this section, we show how we can use auxilliary data in the form of number counts of such galaxies to put tight priors on the expected Poisson amplitudes for Planck. The method will center around using source catalogs from SPT, which resolves many more such galaxies than Planck due to its better angular resolution. Additionally, because of the non-trivial way in which sources are masked in the Planck analysis, we develop a novel method based on simulations for calculating the Poisson amplitude. 3.1. Methodology In the previous chapter, and in general in the literature, the Poisson power is calculated as, C = (3.1) Scut 0 dS S 2 dN , dS where dN/dS are the diﬀerential number counts, Scut is some eﬀective ﬂux cut above which which sources are masked, and the integral is performed independently at each frequency of interest. Although satisfactory for rough consistency checks, Eq. 3.1 ignores that Planck point source mask are built from a union of sources detected at diﬀerent frequencies, that the Planck ﬂux cut varies across the sky, and the eﬀect of Eddington bias. In order to accurately account for all of these eﬀects, the method we have developed calculates the Poisson power as (3.2) Cij = ∞ 0 dS1 ...dSn Si Sj dN (S1 , ..., Sn ) I(S1 , ..., Sn ), dS1 ...dSn 40 where 1...n are a set of frequencies, the diﬀerential source count model, dN/dS, is now jointly a function of the ﬂux at each frequency, and I(S1 , ..., Sn ) is the joint “incompleteness” of our catalogue for the particular cut that was used to build the point source mask. The joint incompleteness was determined by injecting simulated point sources into the Planck sky maps, using the procedure described in Planck Collaboration et al. [2015]. The same point source detection pipelines that were used to produce the second Planck catalogue of compact sources (PCCS2) were run on the injected maps, producing an ensemble of simulated Planck sky catalogues with realistic detection characteristics. The joint incompleteness is deﬁned as the probability that a source would not be included in the mask as a function of the source ﬂux, given the speciﬁc masking thresholds being considered. The raw incompleteness is a function of sky location, as the Planck noise varies across the sky. The incompleteness which appears in Eq. 3.2 is integrated over the region of the sky used in the analysis; the injection pipeline estimates exactly this quantity by injecting sources only in these regions. Because computing the N-dimensional incompleteness I(S1 , ..., Sn ) can be costly, we have explored whether it can be approximated as independent, i.e., (3.3) I(S1 , ..., Sn ) = I(S1 )...I(Sn ) For a set of realistic model parameters, performing the integral in Eq. 3.2 with this approximation given the same answer to within about 3%. This is to say that the chance to detect a source at each frequency is (nearly) independent; the appearance of some small correlation comes the fact that part of the noise that contributes to point source detection comes from CMB and galaxy ﬂuctuations, which are correlated frequency-to-frequency. However, as 3% is on the order of the statistical uncertainty, we use the more accurate joint incompleteness in our calculations. 3.2. Source Count Modeling The auxilliary data we use comes from [Mocanu et al., 2013, hereafter M13], which gives a catalog of detected sources at 90 GHz, 150 GHz, and 220 GHz. These source counts are shown in Fig. 3.1. To model these source counts, we take a simple phenomenological ﬁt to the M13 data which works well in the range relevant for Planck Poisson power. That is, we take the scaling of the diﬀerential number counts with ﬂux to be power-law, and we take the frequency dependence of individual sources to be power-law with a spectral index which is scattered around some mean, 41 with a break in this mean and scatter at 150 GHz. With a convenient choice of normalization, this is, (3.4) (α(S2 , S3 ) − ᾱ32 )2 A(S1 S2 S3 )γ−1 (α(S1 , S2 ) − ᾱ12 )2 dN (S1 , S2 , S3 ) exp − = exp − 2 2 dS1 dS2 dS3 2πσ12 σ23 2σ12 2σ32 with, ln(Sj /Si ) ln(νj /νi ) α(Si , Sj ) = (3.5) and 1, 2, and 3 representing the SPT 90, 150, and 220 GHz bands. This form conveniently integrates down to single frequencies analytically, which we give here for reference, (3.6) dN (S1 ) = dS1 ∞ ∞ 0 0 AS13γ−1 (3.7) = (3.8) dN (S2 ) = AS23γ−1 dS2 (3.9) dN (S3 ) = AS33γ−1 dS3 dN (S1 , S2 , S3 ) dS1 dS2 dS3 2 ln ν3 γα23 + 1 γ 2 σ23 dS2 dS3 ν3 ν2 ν2 ν1 ν2 ν1 2 ν2 2 ln ν2 −γα12 + 1 γ 2 σ12 2 ν1 2 ln ν2 −γα12 + 1 γ 2 σ12 2 ν1 ν2 1 2 2 ν2 2γα12 + 2 (2γ) σ12 ln ν1 ν3 ν2 ln ln ν1 ν2 ν1 2 ln ν3 γα32 + 1 γ 2 σ32 2 ν2 ν3 ν3 ν2 ln ln ν2 ν2 ν1 2 ln ν3 −2γα32 + 1 (2γ)2 σ32 2 ν2 ν3 ν3 ν2 ln ln ν2 ν2 ν1 Finally, we will need to evaluate this model at Planck frequencies. Under our assumption that sources have power-law frequency dependence (with a break at 150 GHz), it is straight forward to write down the relation between Planck and SPT ﬂuxes for a given source, (3.10) S100 = S100 (S90 , S150 ) (3.11) S143 = S143 (S90 , S150 ) (3.12) S217 = S217 (S150 , S220 ) Then we simply need to invert this set of equations, substitute it into Eq. 3.4 and apply the Jacobian determinant factor. The computation is straight-forward, albeit messy, so we give the expression 42 with the Planck and SPT frequencies numerically evaluated, (3.13) S10.203 dN (S1 , S2 , S3 ) A(S11.203 S20.898 S30.899 )γ−1 =1.506 0.101 dS1 dS2 dS3 2πσ12 σ23 S2 S30.102 (α(S2 , S3 ) − ᾱ32 )2 (α(S1 , S2 ) − ᾱ12 )2 exp − × exp − 2 2 2σ12 2σ32 where now 1, 2, and 3 refer to Planck 100, 143, and 217 GHz. 3.3. Results We ﬁt the source count model to the M13 data using an MCMC chain. In constructing the likelihood, if M13 had provided a three-dimensional measured dN/dS in a grid of ﬂux bins, as well a joint covariance between all these bins, we could just construct a Gaussian likelihood directly. As is standard, however, only single frequency dN/dS and covariance are provided, with no crosscorrelation between diﬀerent frequencies. M13 do, however, provide estimates of the spectral index of each source, and error bars on this quantity. Thus, to ﬁt model we perform two steps. First we integrate the multifrequency model down the model to individual frequencies (Eqs. 3.6) and construct a Gaussian likelihood with the single frequency covariances, ignoring cross correlations. These correlation should be small and dominated by calibration uncertainties which are taken into account. Note that some information is lost by not using the full three-dimensional source count data, however this is inevitable without further processing of the source count data. We then use the distribution of spectral indices to constrain {ᾱ12 , ᾱ23 , σ12 , σ23 } directly. M13 give asymmetric error bars on the spectral index of each source, which we use in the likelihood by “gluing” together two half-normal distribution in such a way that the mean and 68% intervals correspond to those quoted in M13. After running the chain, we can also apply Eq. 3.2 to calculate the predicted Poisson power in each Planck band and post process this quantity into the chain. The posteriors on these parameters are shown in Fig. 3.2 along with posteriors from Planck chains. At 143 GHz we must also take into account a sub-dominant contribution to the Poisson amplitude from dusty sources. The dusty Poisson amplitude, unlike the radio, is largely insensitive to increasing the ﬂux-cut from SPT to Planck levels. The constraint from George et al. [2015] scaled (only in frequency) to 143 GHz gives (6.3 ± 0.3) μK2 and will be a large part of the total Planck dusty Poisson. We can estimate the remaining contribution by integrating a power-law ﬁt to the 43 measured dusty sources in between ﬂux cuts, shown in the middle panel of Fig. 3.1. A careful treatment of this contribution à la Eq. 3.2 is not necessary as the contribution is very small. We ﬁnd 0.3 μK2 , and if we ﬂatten the best-ﬁt power-law index from −2.5 to −1.5, since we expect lensed dusty sources at higher ﬂuxes to boost the number counts, the power increases to only 1.3 μK2 . We will take (8 ± 2) μK2 , having artiﬁcially inﬂated the error bar to account for the possibility of lensed sources and our rough treatment of the integration. However we stress that the uncertainty in this number is still sub-dominant in our analysis. This would not be the case at 217 GHz, hence why we do not attempt to give a prior there. 3.4. Conclusions As shown in Fig. 3.2, our main conclusion is that we ﬁnd excellent agreement between the Poisson priors and the 100 and 143 GHz posteriors in all cases except chains with highL. The ﬁts from the non-highL chains give us physically very believable Poisson amplitudes. The tension with highL will have to be examined further. We note the highL constraint on Poisson amplitudes is indirect, via constrains on SZ and CIB clustering, and interaction with data and cosmological model. We ﬁnd that our priors lead to smaller amplitudes than what was calculated in the Collaboration et al. [2013b]. Using our procedure we conclude that those numbers have overestimated the power by neglecting the eﬀect of the union mask; e.g., a signiﬁcant number of sources at 100 GHz are masked because they were detected only with 143 and 217 GHz and this can be shown directly by setting the joint incompleteness I(S1 , S2 , S3 ) to be independent of S2 and S3 . We see marginal improvement in constraints on extended parameters, about 10% on Yp , Neﬀ , mν , and nrun . The prior improves the constraint in Planck +highL chains more than in Planck -only chains, suggesting that once the tension is resolved between the two datasets, they could be used to jointly break a degeneracy that neither alone can. 44 90 GHz 150 GHz 220 GHz 1.0 S dN/dS [deg−2 ] 0.8 0.6 0.4 0.2 10 -2 10 -1 S [Jy] 10 0 10 -2 10 -1 S [Jy] 10 0 10 -2 10 -1 10 0 0.0 S [Jy] Figure 3.1 The blue points with error bars show the M13 source count data, and the line drawn through them is our best ﬁt model integrated down to individual frequencies. At 150 GHz we also show the dusty sources in green, as these are used in calculating the dusty Poisson contribution there (see text). The lines at the bottom of each plot show the fractional contribution from each logarithmic ﬂux bin to the Planck radio Poisson power. Dotted/dashed/dot-dashed lines are for the contribution at 100, 143, and 217 GHz. For example, the dotted line in the 220 GHz plot corresponds to the contribution to the 100 GHz Poisson power from sources with those particular 220 GHz ﬂuxes. The normalization is arbitrarily. The best-ﬁt model is used to calculate these curves. Figure 3.2 Posteriors on radio Poisson amplitudes at 100 and 143 GHz from LCDM chains when the data used is (blue) Planck2013+WP (green) Planck2013+WP+highL. These posteriors are nearly unchanged for extended models. The prior from this procedure is in (black). To arrive at the radio contribution in the 2013 chains, 143 GHz has had the expected dusty contribution of (8 ± 2)μK2 subtracted from the total Poisson amplitude (see text). 45 A γ -0.35 -0.375 -0.4 α12 0 -0.5 -1 α23 0 -0.5 -1 σ12 0.6 0.4 0.2 σ23 0.6 0.4 σ23 0.6 0.4 0.2 σ12 0.6 0.4 0.2 α23 0 -0.5 -1 α12 0 -1 -0.5 γ -0.35 -0.375 -0.4 A 0.04 0.02 0 0.2 Figure 3.4 Constraints on the six parameter source count model from (blue) integrating the model down to individual frequencies and ﬁtting the M13 diﬀerential source count measurements at 90, 150, and 220 GHz and (orange) also including the M13 distribution of spectral indices as a direct constraint on the spectral index and scatter parameters. 47 CHAPTER 4 New Constriants on Axions and Axion-like Particles With a number of analysis methods laid out in the previous two chapters, we now turn to using CMB data to constrain intersting models. Recent improvements in both CMB measurements as well as inferences of primordial elemental abundances formed during Big Bang Nucleosynthesis (BBN) motivate us to reconsider bounds on hypothetical scenarios which alter the expansion rate or inject energy into the plasma around these two epochs. One of the simplest scenarios involves a radiatively decaying particle, for which axions or so called “axion-like particles” (ALPs) provide a theoretically well motivated candidate. Axions arise from perhaps the most elegant solution to the strong CP problem as the pseudo Nambu-Goldstone boson of a new spontaneously broken symmetry [Peccei and Quinn, 1977a,b, Wilczek, 1978, Weinberg, 1978]. Axions have a mass and standard model couplings controlled by a single parameter: the energy scale of the symmetry breaking. ALPs form a more general class of particles where the mass and couplings are independent. Such models can arise from other new symmetries which are spontaneously broken [Chikashige et al., 1981, Gelmini and Roncadelli, 1981], and in string theory [Arvanitaki et al., 2009]. In the parameter space of interest here, ALPs are weakly interacting, making them diﬃcult to detect in the laboratory. Cosmological constraints serve as a natural complement as the weak coupling generally leads to later decays allowing the particles to become non-relativistic and pick up energy compared to the plasma, leading to observable consequences upon their decay. Astrophysical bounds are also important as ALPs provide a new method for energy release from stars and supernovae. Some early calculations and compilation of cosmological, astrophysical, and laboratory bounds on ALPs include those from Massó and Toldrà [1995, 1997]. Cosmological bounds were recently updated by Cadamuro et al. [2011] which considered only axions, and Cadamuro and Redondo [2012] who extended this more generally to ALPs. Among other signiﬁcant advances, Cadamuro and Redondo [2012] used newer data, treated out-of-equilibrium decays more carefully, and performed precise calculations of the implications for BBN. These and other known ALP bounds are tabulated in Hewett et al. [2012], Essig et al. [2013], Olive and Group [2014]. Our work updates these by 1) using the latest inferences of primordial element abundances, 2) using the 48 latest measurements of the CMB power spectrum measurements from Planck, 3) having improved calculations of CMB spectral distortions. We also highlight the importance of a region in parameter space near m = 1 MeV and τ = 100 ms which we term the MeV-ALP window. This region is interesting because it previously has evaded all known constraints (also noted in Hewett et al. [2012], Mimasu and Sanz [2014]) and, as we show, can correspond to a particular axion model we will call the DFSZ-EN2. Additionally, in this mass window the symmetry breaking scale is much lower than the often-considered “invisible” axion models. A main conclusion of this paper is that the possibility of the DFSZ-EN2 or any other ALP hiding in plain sight in the MeV-ALP window is ruled out by the newer bounds presented here. We show, however, that these bounds are model dependent and the MeV-ALP window can be reopened if there is other exotic radiation present. The paper is structured as follows. In Sec. 4.1 we discuss in more detail the scenario and its implications for cosmology. Sec. 4.2 describes the new and tabulated constraints which we use. In Sec. 4.3.1 we further discuss the MeV-ALP window and in Sec. 4.3.4 we give forecasts for future probes. 4.1. The Scenario We begin by discussing the cosmological impact of ALPs, which we deﬁne as any particle with a mass mφ and two-photo coupling gφγ . Following standard conventions in the ALP literature, the eﬀective Lagrangian is (4.1) gφγ 1 1 φFμν F̃ μν , L = (∂μ φ)(∂ μ φ) − m2φ φ2 − 2 2 4 where F is the electromagnetic ﬁeld strength tensor, F̃ its dual, and φ the ALP ﬁeld. We often describe the two dimensional parameter space with mφ and, in place of gφγ , the lifetime for decay into photons (4.2) τφγ ≡ Γ−1 φγ = 64π 2 . m3φ gφγ Two processes drive the cosmological evolution of the ALP energy density. The ﬁrst is the Primakoﬀ interaction which allows for conversion between photons and ALPs in the presence of a charged particle q, via γq ↔ φq. Because it is a four-point interaction, the scattering rate for the Primakoﬀ process will depend on the density of scatterers (in this case charged particles). At early 49 times, this density grows faster than the Hubble rate, meaning the Primakoﬀ process will always begin in equilibrium and freeze out at later times. The second process is the direct two-photon interaction, γγ ↔ φ. Conversely, this three-point interaction has no dependence on scatterers, and will always begin out of equilibrium then re-equilibrate at later times. Qualitatively, the ALP scenario depends strongly on the time ordering in which ALPs 1) freezeout from the Primakoﬀ interaction, 2) become non-relativistic, and 3) recouple via the two-photon interaction. The details of how these events are controlled by the two free parameters forbids certain orderings, and in fact the vast majority of solutions fall into one of just two cases. If there is a gap between freeze-out and recoupling during which the ALPs become non-relativistic, meaning Tfo > mφ > Tre , there is an out-of-equilibrium decay. In this case, upon becoming non-relativistic ALPs cease to track their equilibrium abundance and instead increase in energy density relative to the plasma. Decay happens when the two-photon interaction becomes eﬀective, which here is controlled by only the ALP lifetime, independent of the mass. On the other hand, if ALPs become nonrelativistic only after recoupling, Tre > mφ , they will track their equilibrium abundance throughout decay, a scenario which might better be called a “Boltzmann suppression.” This suppression occurs when the temperature reaches the mass, independent of the lifetime. Although some subtleties can occur if other events reheat the plasma while the ALPs are decoupled, to a large degree only these two in- and out-of-equilibrium scenarios are important. In discussing the cosmological impact of ALPs, it is useful to deﬁne two quantities. The ﬁrst is the eﬀective number of relativistic species, Neﬀ . As usual, this is taken so that in the limit of complete neutrino decoupling prior to electron-positron annihilation, each neutrino species (with antineutrinos) contributes 1 to Neﬀ , making the total relativistic energy density, (4.3) ρrel = ργ 7 1 + Neﬀ 8 4 11 4 3 . The other quantity is the baryon-to-photon ratio η, or equivalently the energy density in baryons Ω b h2 , (4.4) η= nb = 2.74 × 10−8 Ωb h2 . nγ We sometimes superscript these quantities with BBN or CMB depending on the epoch at which they are evaluated, although we note there is no exact deﬁnition as they can change with time in this scenario. 50 CMB . If the decays occur ALP decays can only reduce or leave unchanged the value of Neﬀ CMB does not change. Later decays increase the while the neutrinos are still fully coupled then Neft CMB . The value of N BBN temperature of the plasma relative to the neutrinos and thus decrease Neﬀ eﬀ can also be decreased in the same manner if decay happens before BBN, although this region in parameter space is fairly small. A much larger region of parameter space corresponds to ALPs BBN by simply contributing to the relativistic energy density decaying after BBN and increasing Neﬀ during BBN. ALP decays aﬀect η similarly to Neﬀ by causing its value to be less after the decay. The diﬀerence is that with η CMB held constant, η before the decay is now increased. Assuming BBN serve that the physics of BBN is unchanged, both an increase in ηBBN and an increase in Neﬀ to increase the amount of primordial helium produced. For deuterium, however, there is partial cancellation, leaving the primordial abundance only slightly reduced. This cancellation is key to allowing the ALP scenario at all in light of the very tight constraints on primordial deuterium (see CMB moves along a Sec. 4.2.2). The increase in primordial helium coupled with the decrease in Neﬀ degeneracy direction for the CMB constraints, and is hence also generally allowed. More quantitatively, we must track the evolution of the phase space distribution function fφ , which is governed by the Boltzmann equation dfφ = (Cq + Cγ )(fφeq − fφ ), dt (4.5) where the f ’s are comoving, fφeq is a Bose-Einstein distribution, and a dependence on the comoving momentum p is omitted for brevity. When electrons and positrons are the only charged particles present, the scattering rate for the Primakoﬀ interaction, Cq , is given by (4.6) Cq ≈ 2 nep gφγ α [4E(me + 3T )]2 log 1 + 2 2 16 mγ [me + (me + 3T )2 ] ! , where nep is the number density of electrons and positrons, E is the ALP energy, and mγ = eT /3 is the plasmon mass in an electron-positron plasma [Bolz et al., 2001]. As shown by Cadamuro and Redondo [2012], an accounting of other charged particles gives an approximation for the Primakoﬀ freeze-out temperature of (4.7) " 2 g∗ (Tfo ) 10−9 GeV−1 GeV, Tfo ≈ 123 gq (Tfo ) gφγ 51 In practice, we begin evolving the Boltzmann equations at a temperature T0 when electrons and positrons are the only remaining charged particles and we can thus use Eqn. 4.6 for the Primakoﬀ scattering rate. If the Primakoﬀ process has yet to freeze out at T0 , as given by Eqn. 4.7, we take take as the initial conditions for fφ a Bose-Einstein distribution at temperature T0 . If freeze-out is earlier, we use the conservation of comoving entropy to calculate the increase in photon temperature after Primamkoﬀ freeze-out. The result is that ALPs are now at a reduced relative temperature of T0 (g∗ (T0 )/g∗ (Tfo ))1/3 . We now describe key regions in the ALP parameter space using Fig. 4.1 as a guide. An important quantity governing the ALP evolution is the temperature, Tre , at which the two-photon interaction re-equilibrates. Eqn. 4.5 shows that the ALP distribution fφ at a given momentum p becomes equal to its equilibrium value fφeq when the scattering rate, Cγ (p), is on the order of the Hubble constant. We consider only those momenta which contribute dominantly to the total energy density, since this is the quantity we are interested in. For mφ T , these are p ≈ 0. In this case, Cγ vastly simpliﬁes to 1/τφγ and thus recoupling occurs when the Hubble time reaches the lifetime. For mφ T , the important momenta are instead E ≈ p ≈ T . Ignoring the term in brackets in Eqn. 4.8 which can be shown to depend only logarithmically on temperature for these √ momenta, and using H = 1.66 g∗ T 2 , we arrive at (in Planck units), (4.9) ⎧ 1/2 ⎪ 1 ⎪ ⎪ ⎪ ⎨ τφγ 1.66√g∗ Tre = 1/3 ⎪ ⎪ mφ 1 ⎪ ⎪ √ ⎩ τφγ 1.66 g∗ mφ > Tre mφ < Tre Fig. 4.1 shows constant Tre contours in black for three values of Tre corresponding to neutrino decoupling, and the start and end of BBN. Here and throughout this paper, we take these temperatures to be 1 MeV, 200 KeV, and 20 KeV respectively. Fig. 4.1 also shows constant mass contours at these three values. The line implicitly formed by Tre = mφ divides regions A and B where the ALP becomes non-relativistic before and after the two-photon interaction re-equilibrates, respectively. Note that in region A we have mφ > Tre , which is only part of the requirement for an out-of-equilibrium decay. The other is that Tfo > mφ , however it turns out that this is always satisﬁed for any combination of mass and lifetime in region A; that is to say, ALPs can never decay by the Primakoﬀ process alone. 53 The evolution of the energy densities in the relevant components of the plasma for a typical inequilibrium decay (region B) is shown in the third panel of Fig. 4.2. If ALPs decay after neutrinos are decoupled, conservation of comoving entropy implies that the temperature of the neutrinos relative to the photons after the ALP decay and after electrons and positrons have annihilated CMB will be is (4/13)1/3 , as compared to (4/11)1/3 in the standard scenario. This means that Neﬀ reduced by (13/11)4/3 and, at ﬁxed η CMB , the value of η prior to the decay is increased by a factor of CMB ≈ 2.44. Because the exact timing with respect 13/11. Assuming three neutrinos, this gives Neﬀ to electron-positron annihilation is unimportant for the ﬁnal temperature, the entirety of region B CMB . The BBN data, however, are sensitive to the exact time of decay shares this same value for Neﬀ BBN and η via sensitivity to Neﬀ BBN . Since decay time when in equilibrium is controlled only by mφ , we ﬁnd characteristic constant-mass contours in this region in the BBN constraints in Figs. 4.3, 4.4, and 4.5. The second panel of Fig. 4.2 shows instead a typical out-of-equilibrium decay (region A). In this region an important quantity is the fractional increase in the energy of the photons (or equivalently the decrease in Neﬀ ) once the ALP decays at t ∼ τφγ . Because the ALP here is non-relativistic, this is, (4.10) # # ρφ ## mφ a−3 ## 1 √ ∼ ∼ ∼ mφ τφγ # # −4 Neﬀ ργ t=τφγ a t=τφγ where we have assumed that the ALP does not remain non-relativistic for too long before decay √ hence the universe is radiation dominated and a(t) ∼ t. This assumption is true for models right √ on the edge of the allowed region, thus we ﬁnd characteristic contours of constant mφ τφγ for cosmological constraints in region A as seen in Figs. 4.3, 4.4, and 4.5. In both regions A and B there are characteristic contours arising from phenomena which depend on the fractional energy injection after a certain reaction has frozen out, i.e. after a certain temperature. For example, CMB and BBN constraints are only sensitive to energy injected after neutrino decoupling, as any earlier injection is “invisible” because it is rethermalized among all components. We will also consider bounds from CMB spectral distortions, which depend on the energy injected only after the freeze-out of reactions which can bring the CMB spectrum back into chemical equilibrium. In region B, fractional energy injection is independent of either mass or lifetime, and the amount after a certain temperature is controlled only by the mass. On the non-equilibrium side the story 54 is slightly more subtle. The fractional energy increase at a certain time or scale factor depends on the mass, as per Eqn. 4.10. However, we cannot simply assume a ∼ 1/T because the ALP decay alters this relation. In particular, the Friedmann acceleration equation shows that it does so in a way which exactly cancels the mφ dependence, leaving only sensitivity to τφγ . Essentially, more massive ALPs lead to more total energy injection, but delay the time it takes to reach a certain temperature, leaving the same amount of energy injected after that temperature. Thus, contours of constant fractional energy injection are m = const in region B and τ = const in region A. We note that this is the same as a contour of constant decay time. The ﬁnal key region in Fig. 4.1, region C, is delineated by such a contour, and corresponds to zero energy injection after neutrino decoupling and thus no cosmologically observable imprints. The Primakoﬀ freeze-out temperature plays a smaller role in the ALP evolution than the recoupling temperature, although we brieﬂy mention two eﬀects stemming from events happening in the gap between freeze-out and recoupling (in regions of parameter space where the gap exists). We ﬁrst note that if no reheating of the plasma occurs in this gap and ALPs are relativistic, they still track their equilibrium abundance even though they are decoupled. One possible reheating occurring during the gap is the QCD phase transition which imprints a sharp feature our constraints. During this transition the temperature of the ALPs roughly doubles along with the rest of the plasma if they are still coupled, leading to an energy density roughly twenty times larger. Another possibility is reheating from electron-positron annihilation, a case in which the ALPs can re-equilibrate relativistically thereafter. Such a scenario happens near the top-left part of Fig. 4.1 and is shown in the fourth panel of Fig.4.2. For deuterium and helium constraints shown in Fig. 4.5, near this region we can ﬁnd both lines of constant Tre and Tfo . 4.2. Constraints 4.2.1. Cosmic Microwave Background. 4.2.1.1. Frequency Spectrum. The measurement of the CMB frequency spectrum by COBE/FIRAS places very tight bounds on spectral distortions away from a black-body spectrum [Fixsen et al., 1996], limiting possible energy injection into the plasma [Wright et al., 1994]. The eﬀects of energy injection depend crucially on when it occurs, with the time-line roughly divided into three eras. In the earliest era, reactions that change photon number are fast and any injection of photon energy is quickly rethermalized. This leads to only an adjustment of the 55 [2012] under a further approximation of Eqn. 4.11 where DC and BR are taken to be inﬁnitely fast until they instantaneously freeze-out at T ≈ 750 eV. In the “previous constraints” panel of Fig. 4.4 we reproduce their result, showing that qualitatively this is a very good approximation. The use of Eqn. 4.11 becomes more important, however, for constraining scenarios with even smaller chemical potentials generated deeper into what is currently called the T -era. Two future missions which are predicted to reach such sensitivity are PIXIE and COrE [Kogut et al., 2011, Collaboration et al., 2011], for which we give forecasts in Sec. 4.3.4. 4.2.1.2. Angular Power Spectrum. Measurements of CMB anisotropies have been recently improving, both from the ground [Keisler et al., 2011, Das et al., 2011, Story et al., 2013, Das et al., 2014] and from space [Collaboration et al., 2013b, collaboration et al., 2013]. Better angular resolution and lower noise have tightened up small-scale constraints where the CMB is most sensitive to changes in Neﬀ and Yp , both of which are altered by the decay of ALPs. A fully general treatment would include ALPs in the set of Boltzmann equations for calculating the CMB power spectrum, but it turns out this is not necessary for the scales that are currently well measured. For these scales, all of the physical eﬀects of ALPs are in fact identical to changes in Neﬀ and Yp , as long as we assume adiabatic initial conditions. This is essentially because decays must happen early enough, as enforced by the spectral distortion bound discussed in the previous section, which requires the decay happen by T ≈ 750 eV or equivalently z ≈ 3 × 106 . The angular scales well constrained by CMB measurements, roughly 3000, correspond to physical scales which do not begin to enter the horizon until about z ≈ 3 × 105 . At this point two things are diﬀerent in the ALP scenario as opposed to the standard case 1) the amplitude of neutrino density perturbations upon horizon entry is reduced relative to the photons and 2) the expansion rate is diﬀerent. However, both are exactly captured in the standard scenario by changing Neﬀ . No other scale-dependent changes to spatial perturbations are possible because the relevant scales are still outside of the horizon by the time of the decay. Finally, we note that the altered helium abundance is taken as an input to the CMB spectrum calculation, and in the ALP scenario it is now just at a diﬀerent value. In their work, Cadamuro and Redondo [2012] took as a CMB constraint a lower bound on Neﬀ as given by WMAP7. The Planck data tighten this constraint and are also sensitive to Yp . We use the joint constraint on these two parameters given by the combination of Planck+WP+highL from Collaboration et al. [2013b], approximating the likelihood as Gaussian and taking just the mean 60 element abundances thus probe new physics at play during this epoch [e.g., Pospelov and Pradler, 2010]. For our case of an ALP, light element production is aﬀected by changes to the cosmic expansion rate during BBN and to the extrapolation of η back from the CMB epoch. For the majority of the parameter space where ALPs decay after neutrino decoupling, they are still present BBN and the expansion rate. Additionally, because the decay during BBN and hence increase Neﬀ decreases η, ﬁxing ηCMB to the observed value generally leads to an increased ηBBN . Both eﬀects BBN increases D/H serve to increase Yp , but partially cancel for D/H and 7 Li. The increase in Neﬀ while the increase in η has the opposite eﬀect, with the latter about twice as large, leaving an overall reduced D/H. For 7 Li it is instead the former which wins out. The light-element trends in the mass-lifetime planes of Fig. 4.5 bear out these expectations, as we now see in detail. In practice we have modiﬁed the AlterBBN code of Arbey [2012] to include changes to the expansion history and η due to ALPs. Our code assumes the photon spectrum is instantaneously rethermalized, in eﬀect ignoring the possibility that high energy photons from the decay can break apart already-formed nuclei. Bounds due to this phenomenon constitute so called “photo-erosion” bounds, discussed in e.g. Cyburt et al. [2009] and references therein. We will consider them separately at the end of this section. Deuterium is observable at z ∼ 3 in QSO absorption systems, via the ∼ 82 km/s isotope shift between D and H Lyman absorption lines. Recent D/H measurements have been reported Cooke et al. [2014], (4.16) D = (2.53 ± 0.04) × 105 , H which represents a factor ∼ 3 improvement in precision. These bounds are now so tight as to place the measurement errors on level footing with uncertainties associated with nuclear reaction rates and with a determination of η from the CMB. To account for these uncertainties, we ﬁrst consider the joint likelihood for the CMB, D/H, and nuclear reaction rate measurements, which can be written as DH(mφ , τφγ , η, αi ) − DH − log L = 2 2σMEAS + log LCMB (η, Neﬀ , Ω) (4.17) + log LNUCL (αi , Ω ) 62 2 where DH ± σMEAS = (2.53 ± 0.04) × 105 as per Eqn. 4.16, η and Neﬀ are evaluated at the CMB epoch but we omit the label for brevity, Neﬀ = Neﬀ (mφ , τφγ ) is uniquely set by the mass and lifetime, αi are parameters describing the nuclear reaction rates, and Ω and Ω are any remaining cosmological and nuisance parameters. Neﬀ is important and appears explicitly because it is both dependent on the ALP parameters and its measurement from the CMB is signiﬁcantly degenerate with η. We next analytically marginalize over all parameters other than mφ and τφγ under the assumption that these other parameters have Gaussian posterior likelihoods and that D/H depends linearly on η. This gives (4.18) −Neﬀ DH(mφ , τφγ , η̄ + rση Neﬀ σNeﬀ , αi ) − DH 2 − log L = 2 2 2 σMEAS + σNUCL + σETA 2 with (4.19) 2 σETA = dDH ση dη 2 (1 − r2 ) where η̄ ± ση and Neﬀ ± σNeﬀ are the mean and standard deviation of the posterior likelihoods from the CMB with all other parameters marginalized over, and r is the correlation coeﬃcient between η and Neﬀ . The presence of r in this equation can be understood by considering the r = 1 case, which would imply that CMB measurements could turn a ﬁxed Neﬀ into a perfect determination of η; the quantity above at which the D/H prediction is evaluated, η̄ + rση (Neﬀ − Neﬀ )/σNeﬀ , is the mean of this determination. Because in our case Neﬀ is ﬁxed by the mass and lifetime, it would mean η is also ﬁxed, leading to no extra uncertainty in D/H. In reality, we ﬁnd r ≈ 0.4 from the Planck measurements. We make one further approximation which is that neither the D/H derivative nor the nuclear reaction rate uncertainty depends on the values of mass and lifetime or the fact that η evolves with time in the ALP scenario, which we have checked is suﬃcient. We ﬁnd σNUCL = 4.5 × 10−7 using AlterBBN which takes the αi to be principal components in the nuclear reaction rate parameter space [Fiorentini et al., 1998]. Numerically evaluating the D/H derivative and taking posterior likelihoods from Planck +WP+highL, we ﬁnd σETA = 6.9×10−7 . When added in quadrature these lead to an eﬀective deuterium constraint of (4.20) D = (2.53 ± 0.091) × 105 H 63 which is meant to be compared to a theoretical prediction calculated for the particular values of η and αi given in Eqn. 4.18. The eﬀects of these D/H constraints on the (mφ , τφγ ) plane appear in Fig. 4.5. We see that the eﬀect of an ALP is always to decrease D/H due to the ALP’s eﬀective increase of ηBBN winning BBN . Moreover, we see that the high precision of the D/H measurements out over the increase in Neﬀ leads to a tight constraint on the ALP space in all regions where the decays occur after neutrino decoupling. Indeed, D/H is now a very powerful probe of ALPs. The primordial 4 He abundance is inferred astronomically from observations of emission spectra of highly ionized gas in primitive nearby dwarf galaxies, i.e., in low-metallicity extragalactic HII regions. The primordial abundance is traditionally inferred by extrapolation to zero metallicity. To derive helium and metal abundances from the observed spectra requires characterization of the thermodynamic properties of the emitting gas (i.e., temperature, density). The analysis of [Aver et al., 2013] derives these quantities simultaneously in a self-consistent manner, and ﬁnds a primordial abundance (4.21) Yp = 0.2465 ± 0.0097 where the uncertainty is quantiﬁed with an MCMC analysis. We adopt this as our ﬁducial primordial 4 He constraint. Using a similar data set, Izotov and Thuan [2010] give a helium constraint of, Yp = 0.254 ± 0.003 where the errors are derived in a less conservative manner. In Fig. 4.5 we see that, as expected, the eﬀect of an ALP is to increase Yp for almost all of the parameter space where the decays occur after neutrinos decouple. However, the constraints are not as strong as those of D/H. There is an island of parameter space around mφ > 100 MeV and τφγ ∼ 1 sec where Yp decreases. This region corresponds to decays happening between neutrino decoupling and the start of BBN. Here we have ηBBN = ηCMB and Neﬀ < 3, the latter of which serves to decrease Yp . It is interesting to note this low helium region does not extend along the entire constant decay-time contour, cutting oﬀ once we enter the in-equilibrium decay side. This occurs because in-equilibrium decays reduce the ALP energy density more slowly than do out-of-equilibrium ones, and thus the entire decay cannot ﬁt in the short time between neutrino decoupling and BBN. Finally, the primordial 7 Li abundance is inferred from observations of the atmospheres of lowmetallicity (extreme Population II) stars in the Galactic stellar halo. Down to a metallicity of (Fe/H) ∼ 10−2.8 (Fe/H) , these stars have lithium abundances that are the same to within a small 64 scatter consistent with observational errors. The independence of Li with Fe in this “Spite plateau” indicates that lithium is primordial [Spite and Spite, 1982], and implies a primordial abundance (4.22) Li = (1.6 ± 0.3) × 10−10 H [Sbordone et al., 2010]. At lower metallicity, however, the Li/H abundance scatter increases dramatically, but always below the Spite plateau value. This suggests that in these very metal-poor stars some lithium destruction has occurred; the reason for this remains unclear. The astronomically-inferred lithium abundance in Eqn. 4.22 is inconsistent with the primordial value expected from standard BBN theory combined with CMB determinations of η. The observed Li/H value is low at the ∼ 5σ level. This is the “lithium problem” [reviewed in, e.g., Fields, 2011]. Stellar astrophysics uncertainties may be the origin of the problem, but solutions to date require ﬁne tuning and do not explain the observed Li/H “meltdown” at very low metallicities. A more radical and intriguing solution is the presence of new physics during or after BBN. The challenge for such scenarios is to reduce lithium substantially without drawing other light elements–particularly deuterium–from their concordant primordial abundances. The ALP scenario tends to aggravate the problem by increasing lithium slightly, as seen in Fig. 4.5, and as expected due to the ALP eﬀect on η. Thus, awaiting a resolution to the lithium problem, we do not consider lithium bounds. In closing we note that our calculations have neglected the eﬀects of photoerosion of the light elements. This occurs 1) when the ALP mass exceeds light-element binding energies mφ > ∼ B ∼ 10 MeV, and 2) for decay time scales long enough so that the decay photons interact with light elements before thermalization. This leads to some deuterium destruction via γd → np, but a net production due to e.g., γ 4 He → dd. Thus constraints arise from D/H, Yp , and 3 He/D [Ellis et al., 1984, Kawasaki and Moroi, 1995, Cyburt et al., 2003]. These were recently computed for purely electromagnetic decays by Cyburt et al. [2009], assuming that the decay photons provide a negligible contribution to the energy density and thus expansion rate. In this case, the constraints are only 7 4 important for τX > ∼ 10 sec and mX > ∼ 10 eV, with X the decaying particle. This regime shows a Yp drop due to photoerosion and a corresponding D/H increase. These trends could potentially bring Yp and D/H predictions back into agreement with observations, but would require a more detailed calculation. Since the regions of parameter where this can happen are already ruled out by CMB observations, we ignore the eﬀects of photo-erosion. 65 4.2.3. Laboratory. Laboratory bounds on ALPs come from a variety of diﬀerent experimental setups. At lower masses, roughly m eV, some examples include photo-regeneration experiments (“shining light through walls”), microwave cavities, and helioscopes [for a review, see Hewett et al., 2012, Essig et al., 2013, Olive and Group, 2014]. For the larger masses considered here, the best constraints come from electron-positron colliders and beam dumps. The presence of the ALP-photon interaction allows for the possibility of single-photon ﬁnal states at electron-positron colliders. Early interpretation of searches for these events in terms of constraints on the ALP coupling gφγ was done by Massó and Toldrà [1995]. Both Kleban and Rabadan [2005] and Mimasu and Sanz [2014] have further shown the ability of current and future colliders to improve these bounds. Here we reproduce the constraint from LEP given by Mimasu and Sanz [2014] of, (4.23) gφγ < 4.5 × 10−4 GeV−1 , valid in the entire lifetime range considered here. The excluded region is labeled “Collider” and shown in magenta in Fig. 4.3. Additional constraints come from beam dump experiments, where ALPs would be produced in the beam dump, penetrate through shielding, then decay to photons which can be detected by a downstream detector. We use the constraints from Bjorken et al. [1988] which ﬁnd that at 95% conﬁdence, (4.24) mφ τφγ > 1.4 keV sec. This is labeled “Beam Dump” and shown in red in Fig. 4.3. 4.2.4. Globular Clusters and SN1987A. ALPs oﬀer a new means for energy loss from stars if they can both be produced in stellar interiors and have suﬃciently weak interaction strengths to subsequently escape. In the case of SN1987A, the energy loss can aﬀect the duration of the neutrino pulse from the handful of neutrinos which were detected, placing constraints on the ALP interaction strength [Massó and Toldrà, 1995, 1997]. These bounds are reproduced in Fig. 4.3. We note that they assume the ALP has only a two-photon coupling, although constraints based on other couplings exist. Energy loss can also aﬀect the duration of the red giant phase and of the horizontal branch, leading to a diﬀerent observed ratio of such stars in globular clusters. We 66 use the bounds from Cadamuro and Redondo [2012] based on arguments of Raﬀelt and Dearborn [1988] and Raﬀelt [1996]. These are also reproduced in Fig. 4.3. 4.3. Discussion 4.3.1. The MeV-ALP Window. An interesting feature of the exclusion regions prior to this work is the allowed window bounded on all sides near m ∼ 1 MeV and τ ∼ 100 ms corresponding to an ALP decay during BBN. We will henceforth call this the MeV-ALP window. It can be seen in the left panel of Fig. 4.4 as well as in Hewett et al. [2012], Mimasu and Sanz [2014]. Further interest is driven by the fact that a particle in this window could actually be a DFSZ axion, in which case its symmetry breaking scale is close to the electroweak scale. One of the main conclusions of this work is to show that this region is now, in fact, ruled out by the combination of CMB+D/H measurements. We ﬁrst brieﬂy review two relevant generic axion models, referred to as the KSVZ and DFSZ models. Both models introduce a new global U (1) symmetry which is approximately broken at some energy scale fφ giving rise to an axion with mass mφ . The symmetry breaking scale is related to the axion mass by non-perturbative eﬀects and given by √ mφ = (4.25) z mπ f π 1 + z fφ where mπ is the pion mass, fπ its decay constant, z = mu /md the ratio of up to down quark masses. Axion models diﬀer in what other new ﬁelds are introduced to implement the symmetry breaking and how these, as well as standard model ﬁelds, transform under the new U (1). The KSVZ model [Kim, 1979, Shifman et al., 1980] has the standard model fermions neutral, whereas in the DFSZ model [Zhitnitskij, 1980, Dine et al., 1981] they can carry U (1) charge. These model dependent choices in turn aﬀect the axion’s eﬀective photon coupling which arises from fermion loops, ultimately leading to a consistency relation between axion mass and photon lifetime which can be written as, (4.26) τφγ = 2(4 + z) E − N 3(1 + z) −1 √ z 16π 3/2 fπ mπ 1+z α 2 m−5 φ with the model dependence captured by the E/N factor. In evaluating this relation, we will adopt ﬁxed values of mπ = 135 MeV, fπ = 92 MeV, and z = 0.56, ignoring small uncertainties that lead to roughly a 10% uncertainty in the axion mass [Cadamuro et al., 2011, Beringer et al., 2012]. 67 The KSVZ model has E/N = 0, so that we have, (4.27) τ φγ sec = 6.20 × 104 m −5 φ eV The DFSZ model we consider here has E/N = 2, hence we refer to it as the “DFSZ-EN2” model, with consistency relation, (4.28) τ φγ sec = 2.66 × 105 m −5 φ eV These two consistency relations are shown as the dashed lines in Fig. 4.3. Other values for E/N are possible, but the DFSZ-EN2 has the distinction of having a particularly weak coupling because E/N ≈ 2(4 + z)/3(1 + z) and so these terms nearly cancel in Eqn. 4.26 [Kaplan, 1985, Cheng et al., 1995]. This weaker coupling means that the DFSZ-EN2 is consistent with the collider bounds over a larger range of masses as compared to the KSVZ. Ultimately it is that the consistency relation passes through the MeV-ALP window which motivates our interest in this model. The lower mass limit for the DFSZ-EN2 in the MeV-ALP window is around mφ ∼ 200 keV, corresponding to fφ ∼ 30 GeV, less than an order of magnitude from the electroweak scale vweak ∼ 246 GeV where the axion was initially thought to lie. While it is interesting that this mass range for the DFSZ-EN2 was previously allowed, this part of parameter space for the DFSZ-EN2, and more generally the entire MeV-ALP window, is now ruled out by the combination of CMB+D/H data. This region corresponds to in-equilibrium decays CMB = 2.44. The decay happens essentially in the middle of BBN, increasing N BBN hence it gives Neﬀ eﬀ CMB and ηBBN , which, as discussed previously, increases Yp and decreases D/H. The decrease in Neﬀ and increase in Yp moves along the degeneracy direction for CMB measurements, and is allowed even by our updated CMB constraints coming from Planck (see Fig. 4.6). It is in combination with the D/H constraints that the MeV-ALP window is closed, with the best ﬁtting model within the window ruled out at about 3.5σ. If we replace the CMB constraint from Planck with previous measurements from the combination of WMAP, ACT, and SPT, the window is ruled out at a similar signiﬁcance. This is despite the 20% tighter η constraint from Planck because the central value also shifts lower, increasing D/H back towards the measured value. Conversely, replacing the D/H measurement with previous bounds does open the MeV-ALP window again, as seen in the left panel of Fig. 4.4. It is thus the new bounds from Cooke et al. [2014] that are the key improvement. 68 4.3.2. A Loophole in the Presence of Extra Radiation. While the MeV-ALP region is now excluded by the CMB+D/H measurements, this result depends on the assumption of having no extra radiation besides neutrinos and the ALP. In some scenarios, for example as predicted by the string axiverse [Arvanitaki et al., 2009], it is natural to have many ALPs, some of which could also contribute to Neﬀ but be light enough to remain otherwise invisible. Motivated by this possibility, we explore constraints when in addition to the ALP mass and lifetime, we also allow an extra arbitrary addition to Neﬀ . The MeV-ALP region is ruled out largely because it predicts too low an abundance of primorBBN increases D/H and can bring it back into agreement with dial deuterium. An addition to Neﬀ measurements. The penalty is a further increase in Yp , but because the helium constraints are not as tight as D/H, an allowed window now opens up again. We explore this window with Markov Chain Monte Carlo (MCMC) 2. We run two MCMC chains, one for the ALP case where both mass and lifetime are free parameters, and another for the DFSZ-EN2 case with only the mass free and the lifetime given by Eqn. 4.28. In both cases we also leave free the quantity we call ΔNeﬀ which controls any extra relativistic energy density preBBN = 3 + 4/7 + ΔNeﬀ . The ALP contributes at some early time before BBN, meaning that Neﬀ 4/7 because it is one bosonic degree of freedom and is fully thermalized in all regions of parameter space explored by the chain. The likelihood includes all of the bounds in Fig. 4.3. In either the ALP of DFSZ-EN2 cases, we ﬁnd that the MeV-ALP window is again allowed and the best-ﬁtting model consistent with all of the data at 1σ. The mass posterior distributions for the ALP and DFSZ-EN2 chains are shown in Fig. 4.7. In both cases masses below 200 keV are excluded. Masses above 1 MeV in DFSZ-EN2 case are excluded by the collider bound, but are allowed in the ALP case because models can evade this constraint by having a smaller photon coupling. We also show a forecast for a next generation electron-positron collider SUPER-KEKB after two years of integration (discussed in Sec. 4.3.4) which can probe down to almost exactly the 200 keV minimum. The corresponding likelihoods for Neﬀ are given in the right panel. These show that the data preBBN ∼ 4.7 and diluting this down to accommodate the ALP scenario by initially having Neﬀ CMB ∼ 3.4 via the ALP decay. Constraints on the extra radiation are ΔN Neﬀ eﬀ = 1.13 ± 0.30. Thus, one or (marginally) two extra neutrino-like particles allow for an ALP in the MeV-ALP window. 2https://github.com/marius311/cosmoslik 70 Table 4.1 Best-ﬁt parameters preBBN CMB 105 D/H m [keV] τ [ms] Neﬀ Neﬀ Yp 1010 Li7 ΛCDM 3 3.046 2.56 0.247 4.58 1936 4.6 4.73 3.52 2.46 0.255 5.02 ΛCDM+ΔNeﬀ +ALP ΛCDM+ΔNeﬀ +DFSZ-EN2 734 6.2 4.61 3.30 2.45 0.258 5.15 Table 4.2 Best-ﬁt χ2 ΛCDM ΛCDM+ΔNeﬀ +ALP ΛCDM+ΔNeﬀ +DFSZ-EN2 Planck(2) Cooke(1) Aver(1) Izotov(1) Planck+Cooke+Aver(4) Planck+Cooke+Izotov(4) 0.96 0.10 0.00 5.62 1.06 6.68 0.22 0.57 0.74 0.08 1.53 0.87 0.02 0.91 1.43 1.86 2.35 2.79 Alternatively, the decay of such an ALP can hide the existence of one or two additional neutrino-like particles from the tight CMB constraints which would otherwise rule them out. A similar loophole allowing for extra radiation has been proposed by Ho and Scherrer [2013]. We next test the extent to which the data prefer these extended models. We perform a simple test using best-ﬁt χ2 values given in Tab. 4.2. If the χ2 for the extended model decreases signiﬁcantly as compared to ΛCDM, then roughly that model is preferred. Although all bounds from Fig. 4.3 are included in the ﬁt, we only give χ2 for those which are not hard cutoﬀs. When using the combination of Planck +D/H+Yp , we ﬁnd the baseline data choice slightly disfavors both the DFSZ-EN2 and ALP models. The only case where there is a preference for the extend model is in the ALP case when using the helium constraint from Izotov and Thuan [2010]. Here we ﬁnd an improvement in χ2 of 5.81 when we have added 3 new free parameters, something we expect to happen by chance only 12% of the time. If the high helium value inferred by Izotov and Thuan [2010] is conﬁrmed, then this scenario is a natural explanation as it can increase helium compared to the standard value CMB roughly unchanged. while keeping the deuterium abundance and Neﬀ 4.3.3. A Simple Expression for Exclusion Bounds. Given the improved constraints from CMB+D/H measurements, we suggest a simple expression for ALP bounds which can be adopted by those who prefer a simpler picture than the many probes shown in Fig. 4.3. The CMB+D/H data alone now essentially rule out any energy injection after neutrino decoupling, giving allowed parameters of (4.29) τφγ mφ > 107 and < 10−2 . eV sec 71 This assumes no extra radiation besides ALPs, and is valid roughly until masses become small enough or lifetimes long enough that decays happen after CMB last scattering. These late decays are analyzed in more detail by Cadamuro and Redondo [2012], who ﬁnd approximately (4.30) τφγ mφ < 101 or > 1024 , eV sec are once again allowed. 4.3.4. Forecasts. Measurements relevant for placing bounds on ALP parameters have been recently improving and will continue to do so in the near future. It is expected that several probes will soon have the sensitivity to further test the MeV-ALP window. In this section we compute forecasts for some of them. Currently CMB anisotropies alone are not enough to rule out the MeV-ALP window where there is a maximum of Neﬀ = 2.44 and an increase in Yp , but which lie along the CMB degeneracy direction and are thus allowed. Abazajian et al. [2013] show that a Stage-IV CMB experiment could CMB to within 0.02 at 1-σ. Given such tight constraints, the arguments of Sec. 4.2.1.2 measure Neﬀ may need to be revisited; while it is true that the ALP decays before any modes relevant for the CMB enter the horizon, the diﬀerence is only an order of magnitude in scale factor. Assuming any eﬀ such corrections do not provide loopholes, if a Stage-IV CMB measurement found a value of Neﬀ consistent with the standard value of 3.046, the MeV-ALP window would be strongly ruled out 3. For the CMB constraints, however, there will always be the possibility of extra radiation exactly canceling the dilution due to the ALP decay (as in Sec. 4.3.2). Helium and deuterium measurements will continue to improve as more systems are discovered and systematic errors are better understood. Additionally, D/H measurements can be signiﬁcantly improved by better measuring nuclear reaction rates in the laboratory. Fig. 4.8 shows that in the D/H-Yp plane, the ΛCDM+ΔNeﬀ +ALP scenario is not continuous with the standard model. This allows for suﬃciently tight Yp and D/H constraints around the standard values to rule out the presence of ALPs, independent of assumptions about extra radiation. We ﬁnd that the minimum requirement for all points in the ΛCDM+ΔNeﬀ +ALP chain to be ruled out at > 3σ by the combination of Yp and D/H measurements is a factor of two improvement in the D/H error and a 3Their forecast assumes standard BBN, however we believe it unlikely that freeing Y could degrade it so much as to p allow 2.44 72 factor of three improvement in the Yp error bar. We note the former is possible by eliminating the uncertainty due to nuclear reaction rates alone. On the laboratory side, in Sec. 4.2.3 we used the constraint from LEP which limited gφγ 4.5 × 10−4 GeV−1 . If a search for single-photon events were performed using the entire 1000 fb−1 of currently existing KEKB data and the standard model background was found, forecasts from Kleban and Rabadan [2005] show that the constraints could improve to gφγ 10−6 GeV−1 . Similar improvement could come from reinterpreting the constraints on dark photons from 500 fb−1 of BABAR data given in Collaboration [2014] in terms of ALPs. Attempting either of these is outside of the scope of this paper, but could make signiﬁcant improvements in the mass bounds shown in Fig. 4.7. SUPER-KEKB, an ongoing upgrade to KEKB, plans to improve on the integrated luminosity of KEKB by a factor of ten with two years of integration, and by a factor of ﬁfty with ten years [Abe et al., 2010]. Taking the constraint on gφγ to scale with the square root of the integrated luminosity, we ﬁnd that within the ﬁrst two years, SUPER-KEKB can rule out the last remaining part of the MeV-ALP window through which the DFSZ-EN2 passes (the mass limit forecast is shown in Fig. 4.7). The full ten year forecast is exactly enough to close the MeV-ALP window entirely, bringing the collider bound up to gφγ 10−6 GeV−1 where the SN1987a constraint begins. This simple forecast is in broad agreement with a more sophisticated calculation given by Mimasu and Sanz [2014]. Finally, PIXIE is a proposed mission which would greatly improve constraints on CMB spectral distortions [Kogut et al., 2011]. Expected bounds on the μ parameter are, (4.31) |μ| < 5 × 10−8 . Bounds due to spectral distortion constitute constant decay-time boundaries (Sec. 4.1), and we ﬁnd that the PIXIE forecast moves up the exclusion region by only a factor of ﬁve in decay time as compared to FIRAS. This is much less than needed to reach the MeV-ALP window. Evidently it is very diﬃcult to constrain decays happening very much into the T era using spectral distortions. 4.4. Conclusion We have shown how cosmological, astrophysical, and laboratory bounds can provide complementary constraints in the mass-lifetime parameter space of axions and ALPs. We have updated 73 the work of Cadamuro and Redondo [2012] with constraints from the Planck satellite and the latest inferences of primordial D/H and helium abundances, and provided a more detailed calculation of spectral distortions. The most important change is that CMB+D/H constraints now rule out the entire region corresponding to decays happening after neutrino decoupling but before CMB last scattering. This includes closing the MeV-ALP region of parameter space which we have also shown can correspond to a type of DFSZ axion. The presence of additional radiation can relax the exclusion regions and once again allow the MeV-ALP window. Although it is allowed in this case, including such a particle slightly degrades the overall ﬁt to the CMB+BBN data if our most robust data combination is used. Alternatively, this model can provide a natural explanation for a high value of helium such as found by Izotov and Thuan [2010]. Forecasts for future primordial abundance measurements and for SUPER-KEKB are promising; both have the ability to test the MeV-ALP window even in the presence of extra radiation. A detection by either would be very exciting. Even the null result, however, would signify a new level of precision in our understanding of the contents of the primordial plasma. 74 CHAPTER 5 Conclusion In this thesis we have examined the challenges involved in analyzing precision data from the CMB damping tail, such as that from Planck or SPT. One such challenge involves the removal of extra-galactic foregrounds, and we have developed a realistic model for extra-galactic foregrounds, and shown that such a model is necessary to extract unbiased cosmological information from the CMB. Additionally, we have given a new method to put priors on this model with auxiliary data in the form of source count data from SPT, and shown that the results from Planck generally agree very well with these priors. The exception is when including highL data, which we leave for further examination. We have also used Planck data, whose analysis involved some of the methods discussed in this thesis, to constrain a model of axions and axion-like particles. We found that the interesting region in parameter space which we termed the MeV-ALP window is newly ruled out by the combination of Planck and D/H data. If additional radiation is present aside from the axion or ALP, then the bounds relax. Depending on the position one takes, this can be interpreted as either a way to allow one or two extra thermalized neutrino-like species, or a way to allow an MeV axion, both interesting possibilities. The future of CMB science continues to be exciting. Forecasts for next generation experiments promise signiﬁcant improvements in constraints and even guaranteed detection of new physics [Abazajian et al., 2013]. The precision era has forced us to improve and scrutinize our analyses, some examples of which we have demonstrated here. We expect this will be a continued necessity and look forward to it eagerly. 75 APPENDIX A Principal Component Analysis for Power Spectra In Ch. 2 we use a principal component analysis (PCA) to reduce the dimensionality of the tSZ astrophysical parameter space (Sec. 2.1.2) and of the entire foreground contribution to the CMB linear combination (Sec. 2.5.2). Here we present in more detail the procedure used in those sections. Given nsim realizations of an n -length power spectrum, drawn from a statistically signiﬁcant sample of parameter space, we ﬁrst form the [n × nsim ] matrix Y . In each column of Y we place the deviation from the mean power spectrum for that realization. This matrix is then subject to a singular value decomposition, (A.1) Y = U SV T , where the columns of U contain the orthogonal basis vectors, S is a diagonal matrix of the singular values and the columns of V are the principle component weights. The i-th realization can be written as (A.2) (i) y = Φμ wμ(i) , μ where the singular value-weighted orthogonal basis vectors are (A.3) Φμ = √ 1 Uμ Sμμ . nsim and the wμ are the weights, (A.4) wμ(i) = √ nsim Viμ . Because the singular values are in decreasing order, we can truncate the sum in Eq. A.2 at some small value of μ and still accurately describe each realization. Furthermore, the distribution of weights P (wμ ) sampled over nsim realizations provides a prior on our principal component amplitudes equivalent to the parameter space which was sampled to produce the Y matrix. 76 APPENDIX B CMB Linear Combination Generalization to Oﬀ-Diagonal Correlations The method for constructing a best estimate of the CMB presented in Section 2.5 assumes only temperature power spectra, and a covariance which is diagonal in . The generalization to include polarization and mode-mode coupling induced by sky masking is presented here. The math is, infact, indentical for the two scenarios, so in this appendix we’ll refer to polarization types with the understanding that we could just as well be talking about diﬀerent values of . The added diﬃculty in dealing with diﬀerent power spectrum types (for simplicity here just TT and EE) comes from the fact that we cannot arbitrarily create linear combinations which sum TT EE them. For example, C = C100 GHz − C100 GHz neither preserves CMB normalization, nor can we be sure it is CMB–free independent of model. To remedy this, we make sure that in our construction, any linear combination we consider must have the CMB signal cancel out for all but one type. For TT EE EE example, C = C100 GHz − C100 GHz + C143 GHz is a valid linear combination. We start by considering the covariance matrix for the TT and EE spectra. ⎡ ⎣ (B.1) ΣT T ··· ··· ΣEE ⎤ ⎦ By creating the single-type weight matrix (Eq. 2.29) for each of the diagonal blocks, we can cancel the CMB out of all but two weightings. The new covariance will look like, (B.2) ⎡ ⎣ ⎡ ⎛ ⎤⎡ WTTT 0 0 T WEE ⎦⎣ ΣT T ··· ··· ΣEE ⎤⎡ ⎦⎣ 0 0 WEE 77 ⎤ ⎢ ⎝ σT T 0 ⎠ ⎥ ··· ⎢ ⎥ .. ⎢ ⎥ . 0 ⎦=⎢ ⎞ ⎥ ⎛ ⎢ ⎥ ⎢ ⎥ 0 σ ⎢ ⎠ ⎥ ⎝ EE ⎦ ⎣ ··· .. . 0 ⎤ WT T ⎞ Under a permutation to place the two CMB weightings at the front, the covariance becomes, ⎡ ⎛ ⎢ ⎝ ⎢ ⎢ ··· ⎣ (B.3) ⎞ ··· σT T σEE ··· ⎤ ⎤ ⎡ ⎠ ··· ⎥ T Σ Σ cross ⎥ ⎣ cmb ⎦ ⎥≡ ⎦ Σ Σ cross dif f .. . where we’ve labeled Σcmb as the covariance between TT and EE estimates, Σdif f as the covariance of the CMB–free diﬀerenced spectra, and Σcross as the cross-correlation between the two. We now would like to do one ﬁnal reweighting in an attempt to zero out the cross-correlation. The reweighting should leave the diﬀerenced spectra unchanged, should not add TT and EE together, but will add CMB and CMB–free power spectra. Note that this will continue to satisfy our earlier condition that all but one CMB type canceling out. The reweighting matrix will look like, ⎤ ⎡ ⎣ (B.4) I 0 W I ⎦ The new covariance must satisfy, ⎡ (B.5) ⎣ I 0 W T I ⎤⎡ ⎦⎣ Σcmb Σcross ΣTcross ⎤ ⎤⎡ ⎦⎣ Σdif f I 0 W I Solving for W yeilds, (B.6) W = −Σ−1 dif f Σcross 78 ⎡ ⎦=⎣ ⎤ Σcmb 0 0 Σdif f ⎦ Bibliography K. N. Abazajian, K. Arnold, J. Austermann, B. A. Benson, C. Bischoﬀ, J. Bock, J. R. Bond, J. Borrill, E. Calabrese, J. E. Carlstrom, C. S. Carvalho, C. L. Chang, H. C. Chiang, S. Church, A. Cooray, T. M. Crawford, K. S. Dawson, S. Das, M. J. Devlin, M. Dobbs, S. Dodelson, O. Dore, J. Dunkley, J. Errard, A. Fraisse, J. Gallicchio, N. W. Halverson, S. Hanany, S. R. Hildebrandt, A. Hincks, R. Hlozek, G. Holder, W. L. Holzapfel, K. Honscheid, W. Hu, J. Hubmayr, K. Irwin, W. C. Jones, M. Kamionkowski, B. Keating, R. Keisler, L. Knox, E. Komatsu, J. Kovac, C.-L. Kuo, C. Lawrence, A. T. Lee, E. Leitch, E. Linder, P. Lubin, J. McMahon, A. Miller, L. Newburgh, M. D. Niemack, H. Nguyen, H. T. Nguyen, L. Page, C. Pryke, C. L. Reichardt, J. E. Ruhl, N. Sehgal, U. Seljak, J. Sievers, E. Silverstein, A. Slosar, K. M. Smith, D. Spergel, S. T. Staggs, A. Stark, R. Stompor, A. G. Vieregg, G. Wang, S. Watson, E. J. Wollack, W. L. K. Wu, K. W. Yoon, and O. Zahn. Neutrino Physics from the Cosmic Microwave Background and Large Scale Structure. ArXiv e-prints, 1309:5383, September 2013. URL http://adsabs.harvard.edu/abs/2013arXiv1309.5383A. T. Abe, I. Adachi, K. Adamczyk, S. Ahn, H. Aihara, K. Akai, M. Aloi, L. Andricek, K. Aoki, Y. Arai, A. Areﬁev, K. Arinstein, Y. Arita, D. M. Asner, V. Aulchenko, T. Aushev, T. Aziz, A. M. Bakich, V. Balagura, Y. Ban, E. Barberio, T. Barvich, K. Belous, T. Bergauer, V. Bhardwaj, B. Bhuyan, S. Blyth, A. Bondar, G. Bonvicini, A. Bozek, M. Bracko, J. Brodzicka, O. Brovchenko, T. E. Browder, G. Cao, M.-C. Chang, P. Chang, Y. Chao, V. Chekelian, A. Chen, K.-F. Chen, P. Chen, B. G. Cheon, C.-C. Chiang, R. Chistov, K. Cho, S.-K. Choi, K. Chung, A. Comerma, M. Cooney, D. E. Cowley, T. Critchlow, J. Dalseno, M. Danilov, A. Dieguez, A. Dierlamm, M. Dillon, J. Dingfelder, R. Dolenec, Z. Dolezal, Z. Drasal, A. Drutskoy, W. Dungel, D. Dutta, S. Eidelman, A. Enomoto, D. Epifanov, S. Esen, J. E. Fast, M. Feindt, M. Fernandez Garcia, T. Fiﬁeld, P. Fischer, J. Flanagan, S. Fourletov, J. Fourletova, L. Freixas, A. Frey, M. Friedl, R. Fruehwirth, H. Fujii, M. Fujikawa, Y. Fukuma, Y. Funakoshi, K. Furukawa, J. Fuster, N. Gabyshev, A. Gaspar de Valenzuela Cueto, A. Garmash, L. Garrido, Ch Geisler, I. Gfall, Y. M. Goh, B. Golob, I. Gorton, R. Grzymkowski, H. Guo, H. Ha, J. Haba, K. Hara, T. Hara, T. Haruyama, K. Hayasaka, K. Hayashi, H. Hayashii, M. Heck, S. Heindl, C. Heller, T. Hemperek, T. Higuchi, Y. Horii, W.-S. Hou, Y. B. Hsiung, C.-H. Huang, S. Hwang, H. J. Hyun, Y. Igarashi, C. Iglesias, Y. Iida, T. Iijima, M. Imamura, K. Inami, C. Irmler, M. Ishizuka, K. Itagaki, R. Itoh, M. Iwabuchi, G. Iwai, M. Iwai, M. Iwasaki, M. Iwasaki, Y. Iwasaki, T. Iwashita, S. Iwata, H. Jang, X. Ji, T. Jinno, M. Jones, T. Julius, T. Kageyama, D. H. Kah, H. Kakuno, T. Kamitani, K. Kanazawa, P. Kapusta, S. U. Kataoka, N. Katayama, M. Kawai, Y. Kawai, T. Kawasaki, J. Kennedy, H. Kichimi, M. Kikuchi, C. Kiesling, B. K. Kim, G. N. Kim, H. J. Kim, H. O. Kim, J.-B. Kim, J. H. Kim, M. J. Kim, S. K. Kim, K. T. Kim, T. Y. Kim, K. Kinoshita, K. Kishi, B. Kisielewski, K. Kleese van Dam, J. Knopf, B. R. Ko, M. Koch, P. Kodys, C. Koﬀmane, Y. Koga, T. Kohriki, S. Koike, H. Koiso, Y. Kondo, S. Korpar, R. T. Kouzes, Ch Kreidl, M. Kreps, P. Krizan, P. Krokovny, H. Krueger, A. Kruth, W. Kuhn, T. Kuhr, R. Kumar, T. Kumita, S. Kupper, A. Kuzmin, P. Kvasnicka, Y.-J. Kwon, C. Lacasta, J. S. Lange, I.-S. Lee, M. J. Lee, M. W. Lee, S.-H. Lee, M. Lemarenko, J. Li, W. D. Li, Y. Li, J. Libby, A. Limosani, C. Liu, H. Liu, Y. Liu, Z. Liu, D. Liventsev, A. Lopez Virto, Y. Makida, Z. P. Mao, C. Marinas, M. Masuzawa, D. Matvienko, W. Mitaroﬀ, K. Miyabayashi, H. Miyata, Y. Miyazaki, T. Miyoshi, R. Mizuk, G. B. Mohanty, D. Mohapatra, 79 A. Moll, T. Mori, A. Morita, Y. Morita, H.-G. Moser, D. Moya Martin, T. Mueller, D. Muenchow, J. Murakami, S. S. Myung, T. Nagamine, I. Nakamura, T. T. Nakamura, E. Nakano, H. Nakano, M. Nakao, H. Nakazawa, S.-H. Nam, Z. Natkaniec, E. Nedelkovska, K. Negishi, S. Neubauer, C. Ng, J. Ninkovic, S. Nishida, K. Nishimura, E. Novikov, T. Nozaki, S. Ogawa, K. Ohmi, Y. Ohnishi, T. Ohshima, N. Ohuchi, K. Oide, S. L. Olsen, M. Ono, Y. Ono, Y. Onuki, W. Ostrowicz, H. Ozaki, P. Pakhlov, G. Pakhlova, H. Palka, H. Park, H. K. Park, L. S. Peak, T. Peng, I. Peric, M. Pernicka, R. Pestotnik, M. Petric, L. E. Piilonen, A. Poluektov, M. Prim, K. Prothmann, K. Regimbal, B. Reisert, R. H. Richter, J. Riera-Babures, A. Ritter, A. Ritter, M. Ritter, M. Roehrken, J. Rorie, M. Rosen, M. Rozanska, L. Ruckman, S. Rummel, V. Rusinov, R. M. Russell, S. Ryu, H. Sahoo, K. Sakai, Y. Sakai, L. Santelj, T. Sasaki, N. Sato, Y. Sato, J. Scheirich, J. Schieck, C. Schwanda, A. J. Schwartz, B. Schwenker, A. Seljak, K. Senyo, O.-S. Seon, M. E. Sevior, M. Shapkin, V. Shebalin, C. P. Shen, H. Shibuya, S. Shiizuka, J.-G. Shiu, B. Shwartz, F. Simon, H. J. Simonis, J. B. Singh, R. Sinha, M. Sitarz, P. Smerkol, A. Sokolov, E. Solovieva, S. Stanic, M. Staric, J. Stypula, Y. Suetsugu, S. Sugihara, T. Sugimura, K. Sumisawa, T. Sumiyoshi, K. Suzuki, S. Y. Suzuki, H. Takagaki, F. Takasaki, H. Takeichi, Y. Takubo, M. Tanaka, S. Tanaka, N. Taniguchi, E. Tarkovsky, G. Tatishvili, M. Tawada, G. N. Taylor, Y. Teramoto, I. Tikhomirov, K. Trabelsi, T. Tsuboyama, K. Tsunada, Y.-C. Tu, T. Uchida, S. Uehara, K. Ueno, T. Uglov, Y. Unno, S. Uno, P. Urquijo, Y. Ushiroda, Y. Usov, S. Vahsen, M. Valentan, P. Vanhoefer, G. Varner, K. E. Varvell, P. Vazquez, I. Vila, E. Vilella, A. Vinokurova, J. Visniakov, M. Vos, C. H. Wang, J. Wang, M.-Z. Wang, P. Wang, A. Wassatch, M. Watanabe, Y. Watase, T. Weiler, N. Wermes, R. E. Wescott, E. White, J. Wicht, L. Widhalm, K. M. Williams, E. Won, H. Xu, B. D. Yabsley, H. Yamamoto, H. Yamaoka, Y. Yamaoka, M. Yamauchi, Y. Yin, H. Yoon, J. Yu, C. Z. Yuan, Y. Yusa, D. Zander, M. Zdybal, Z. P. Zhang, J. Zhao, L. Zhao, Z. Zhao, V. Zhilich, P. Zhou, V. Zhulanov, T. Zivko, A. Zupanc, and O. Zyukova. Belle II Technical Design Report. arXiv:1011.0352 [hep-ex, physics:physics], November 2010. URL http://arxiv.org/abs/1011.0352. arXiv: 1011.0352. A. Amblard and A. Cooray. Anisotropy Studies of the Unresolved Far-Infrared Background. ApJ, 670:903–911, December 2007. doi: 10.1086/522688. A. Amblard and M. White. Sunyaev-Zeldovich polarization simulation. ArXiv e-prints, 10:417–423, April 2005. doi: 10.1016/j.newast.2005.02.004. A. Amblard, A. Cooray, P. Serra, B. Altieri, V. Arumugam, H. Aussel, A. Blain, J. Bock, A. Boselli, V. Buat, N. Castro-Rodrı́guez, A. Cava, P. Chanial, E. Chapin, D. L. Clements, A. Conley, L. Conversi, C. D. Dowell, E. Dwek, S. Eales, D. Elbaz, D. Farrah, A. Franceschini, W. Gear, J. Glenn, M. Griﬃn, M. Halpern, E. Hatziminaoglou, E. Ibar, K. Isaak, R. J. Ivison, A. A. Khostovan, G. Lagache, L. Levenson, N. Lu, S. Madden, B. Maﬀei, G. Mainetti, L. Marchetti, G. Marsden, K. Mitchell-Wynne, H. T. Nguyen, B. O’Halloran, S. J. Oliver, A. Omont, M. J. Page, P. Panuzzo, A. Papageorgiou, C. P. Pearson, I. Pérez-Fournon, M. Pohlen, N. Rangwala, I. G. Roseboom, M. Rowan-Robinson, M. S. Portal, B. Schulz, D. Scott, N. Seymour, D. L. Shupe, A. J. Smith, J. A. Stevens, M. Symeonidis, M. Trichas, K. Tugwell, M. Vaccari, E. Valiante, I. Valtchanov, J. D. Vieira, L. Vigroux, L. Wang, R. Ward, G. Wright, C. K. Xu, and M. Zemcov. Submillimetre galaxies reside in dark matter haloes with masses greater than 3×1011 solar masses. Nature, 470:510–512, February 2011. doi: 10.1038/nature09771. Alexandre Arbey. AlterBBN: A program for calculating the BBN abundances of the elements in alternative cosmologies. Computer Physics Communications, 183(8):1822–1831, August 2012. ISSN 00104655. doi: 10.1016/j.cpc.2012.03.018. URL http://arxiv.org/abs/1106.1363. arXiv: 1106.1363. Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Nemanja Kaloper, and John MarchRussell. String Axiverse. arXiv:0905.4720 [astro-ph, physics:gr-qc, physics:hep-ph, physics:hepth], May 2009. URL http://arxiv.org/abs/0905.4720. arXiv: 0905.4720. 80 Erik Aver, Keith A. Olive, R. L. Porter, and Evan D. Skillman. The primordial helium abundance from updated emissivities. Journal of Cosmology and Astro-Particle Physics, 11:017, November 2013. ISSN 1475-7516. doi: 10.1088/1475-7516/2013/11/017. URL http://adsabs.harvard. edu/abs/2013JCAP...11..017A. N. Battaglia, J. R. Bond, C. Pfrommer, J. L. Sievers, and D. Sijacki. Simulations of the SunyaevZel’dovich Power Spectrum with Active Galactic Nucleus Feedback. ApJ, 725:91–99, December 2010. doi: 10.1088/0004-637X/725/1/91. R. A. Battye, I. W. A. Browne, M. W. Peel, N. J. Jackson, and C. Dickinson. Statistical properties of polarized radio sources at high frequency and their impact on cosmic microwave background polarization measurements. Mon.Not.Roy.As.Soc. , 413:132–148, May 2011. doi: 10.1111/j. 1365-2966.2010.18115.x. C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, L. Page, D. N. Spergel, G. S. Tucker, E. Wollack, E. L. Wright, C. Barnes, M. R. Greason, R. S. Hill, E. Komatsu, M. R. Nolta, N. Odegard, H. V. Peiris, L. Verde, and J. L. Weiland. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results. Astrophys. J. Supp. , 148:1–27, September 2003. doi: 10.1086/377253. J. Beringer, J.-F. Arguin, R. M. Barnett, K. Copic, O. Dahl, D. E. Groom, C.-J. Lin, J. Lys, H. Murayama, C. G. Wohl, W.-M. Yao, P. A. Zyla, C. Amsler, M. Antonelli, D. M. Asner, H. Baer, H. R. Band, T. Basaglia, C. W. Bauer, J. J. Beatty, V. I. Belousov, E. Bergren, G. Bernardi, W. Bertl, S. Bethke, H. Bichsel, O. Biebel, E. Blucher, S. Blusk, G. Brooijmans, O. Buchmueller, R. N. Cahn, M. Carena, A. Ceccucci, D. Chakraborty, M.-C. Chen, R. S. Chivukula, G. Cowan, G. D’Ambrosio, T. Damour, D. de Florian, A. de Gouvêa, T. DeGrand, P. de Jong, G. Dissertori, B. Dobrescu, M. Doser, M. Drees, D. A. Edwards, S. Eidelman, J. Erler, V. V. Ezhela, W. Fetscher, B. D. Fields, B. Foster, T. K. Gaisser, L. Garren, H.-J. Gerber, G. Gerbier, T. Gherghetta, S. Golwala, M. Goodman, C. Grab, A. V. Gritsan, J.-F. Grivaz, M. Grünewald, A. Gurtu, T. Gutsche, H. E. Haber, K. Hagiwara, C. Hagmann, C. Hanhart, S. Hashimoto, K. G. Hayes, M. Heﬀner, B. Heltsley, J. J. Hernández-Rey, K. Hikasa, A. Höcker, J. Holder, A. Holtkamp, J. Huston, J. D. Jackson, K. F. Johnson, T. Junk, D. Karlen, D. Kirkby, S. R. Klein, E. Klempt, R. V. Kowalewski, F. Krauss, M. Kreps, B. Krusche, Y. V. Kuyanov, Y. Kwon, O. Lahav, J. Laiho, P. Langacker, A. Liddle, Z. Ligeti, T. M. Liss, L. Littenberg, K. S. Lugovsky, S. B. Lugovsky, T. Mannel, A. V. Manohar, W. J. Marciano, A. D. Martin, A. Masoni, J. Matthews, D. Milstead, R. Miquel, K. Mönig, F. Moortgat, K. Nakamura, M. Narain, P. Nason, S. Navas, M. Neubert, P. Nevski, Y. Nir, K. A. Olive, L. Pape, J. Parsons, C. Patrignani, J. A. Peacock, S. T. Petcov, A. Piepke, A. Pomarol, G. Punzi, A. Quadt, S. Raby, G. Raﬀelt, B. N. Ratcliﬀ, P. Richardson, S. Roesler, S. Rolli, A. Romaniouk, L. J. Rosenberg, J. L. Rosner, C. T. Sachrajda, Y. Sakai, G. P. Salam, S. Sarkar, F. Sauli, O. Schneider, K. Scholberg, D. Scott, W. G. Seligman, M. H. Shaevitz, S. R. Sharpe, M. Silari, T. Sjöstrand, P. Skands, J. G. Smith, G. F. Smoot, S. Spanier, H. Spieler, A. Stahl, T. Stanev, S. L. Stone, T. Sumiyoshi, M. J. Syphers, F. Takahashi, M. Tanabashi, J. Terning, M. Titov, N. P. Tkachenko, N. A. Törnqvist, D. Tovey, G. Valencia, K. van Bibber, G. Venanzoni, M. G. Vincter, P. Vogel, A. Vogt, W. Walkowiak, C. W. Walter, D. R. Ward, T. Watari, G. Weiglein, E. J. Weinberg, L. R. Wiencke, L. Wolfenstein, J. Womersley, C. L. Woody, R. L. Workman, A. Yamamoto, G. P. Zeller, O. V. Zenin, J. Zhang, R.-Y. Zhu, G. Harper, V. S. Lugovsky, and P. Schaﬀner. Review of Particle Physics. Phys. Rev. D, 86(1):010001, July 2012. doi: 10.1103/PhysRevD.86.010001. J. D. Bjorken, S. Ecklund, W. R. Nelson, A. Abashian, C. Church, B. Lu, L. W. Mo, T. A. Nunamaker, and P. Rassmann. Search for Neutral Metastable Penetrating Particles Produced in the SLAC Beam Dump. Phys.Rev., D38:3375, 1988. doi: 10.1103/PhysRevD.38.3375. M. Bolz, A. Brandenburg, and W. Buchmüller. Thermal production of gravitinos. Nuclear Physics B, 606(1–2):518–544, July 2001. ISSN 0550-3213. doi: 10.1016/S0550-3213(01)00132-8. URL http://www.sciencedirect.com/science/article/pii/S0550321301001328. 81 J. R. Bond, B. J. Carr, and C. J. Hogan. Spectrum and anisotropy of the cosmic infrared background. ApJ, 306:428–450, July 1986. doi: 10.1086/164355. J. R. Bond, B. J. Carr, and C. J. Hogan. Cosmic backgrounds from primeval dust. ApJ, 367: 420–454, February 1991. doi: 10.1086/169640. Francois R. Bouchet and Richard Gispert. Foregrounds and CMB Experiments: I. Semi-analytical estimates of contamination. New Astron., 4:443–479, 1999. doi: 10.1016/S1384-1076(99)00027-5. D. Cadamuro and J. Redondo. Cosmological bounds on pseudo Nambu-Goldstone bosons. JCAP , 2:32, February 2012. doi: 10.1088/1475-7516/2012/02/032. Davide Cadamuro, Steen Hannestad, Georg Raﬀelt, and Javier Redondo. Cosmological bounds on sub-MeV mass axions. Journal of Cosmology and Astroparticle Physics, 2011(02):003, February 2011. ISSN 1475-7516. doi: 10.1088/1475-7516/2011/02/003. URL http://iopscience.iop. org/1475-7516/2011/02/003. Erminia Calabrese, Renée A. Hlozek, Nick Battaglia, Elia S. Battistelli, J. Richard Bond, Jens Chluba, Devin Crichton, Sudeep Das, Mark J. Devlin, Joanna Dunkley, Rolando Dünner, Marzieh Farhang, Megan B. Gralla, Amir Hajian, Mark Halpern, Matthew Hasselﬁeld, Adam D. Hincks, Kent D. Irwin, Arthur Kosowsky, Thibaut Louis, Tobias A. Marriage, Kavilan Moodley, Laura Newburgh, Michael D. Niemack, Michael R. Nolta, Lyman A. Page, Neelima Sehgal, Blake D. Sherwin, Jonathan L. Sievers, Cristóbal Sifón, David N. Spergel, Suzanne T. Staggs, Eric R. Switzer, and Edward J. Wollack. Cosmological parameters from pre-planck cosmic microwave background measurements. Physical Review D, 87:103012, May 2013. ISSN 0556-2821. doi: 10.1103/PhysRevD.87.103012. URL http://adsabs.harvard.edu/abs/2013PhRvD..87j3012C. J.-F. Cardoso, M. Martin, J. Delabrouille, M. Betoule, and G. Patanchon. Component separation with ﬂexible models. Application to the separation of astrophysical emissions. ArXiv e-prints, March 2008. J. E. Carlstrom, G. P. Holder, and E. D. Reese. Cosmology with the Sunyaev-Zel’dovich Eﬀect. Annu. Rev. Astron. Astrophys. , 40:643–680, 2002. doi: 10.1146/annurev.astro.40.060401.093803. Anthony Challinor and Antony Lewis. Lensed cmb power spectra from all-sky correlation functions. Phys. Rev., D71:103010, 2005. S. L. Cheng, C. Q. Geng, and W.-T. Ni. Axion-photon couplings in invisible axion models. Phys. Rev. D, 52:3132–3135, September 1995. doi: 10.1103/PhysRevD.52.3132. Y. Chikashige, R. N. Mohapatra, and R. D. Peccei. Are there real goldstone bosons associated with broken lepton number? Physics Letters B, 98(4):265–268, January 1981. ISSN 0370-2693. doi: 10.1016/0370-2693(81)90011-3. URL http://www.sciencedirect.com/science/article/ pii/0370269381900113. Planck Collaboration, P. a. R. Ade, N. Aghanim, M. I. R. Alves, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, H. Aussel, C. Baccigalupi, A. J. Banday, R. B. Barreiro, R. Barrena, M. Bartelmann, J. G. Bartlett, N. Bartolo, S. Basak, E. Battaner, R. Battye, K. Benabed, A. Benoı̂t, A. Benoit-Lévy, J.-P. Bernard, M. Bersanelli, B. Bertincourt, M. Bethermin, P. Bielewicz, I. Bikmaev, A. Blanchard, J. Bobin, J. J. Bock, H. Böhringer, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, H. Bourdin, J. W. Bowyer, M. Bridges, M. L. Brown, M. Bucher, R. Burenin, C. Burigana, R. C. Butler, E. Calabrese, B. Cappellini, J.-F. Cardoso, R. Carr, P. Carvalho, M. Casale, G. Castex, A. Catalano, A. Challinor, A. Chamballu, R.-R. Chary, X. Chen, H. C. Chiang, L.-Y. Chiang, G. Chon, P. R. Christensen, E. Churazov, S. Church, M. Clemens, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, B. Comis, F. Couchot, A. Coulais, B. P. Crill, M. Cruz, A. Curto, F. Cuttaia, A. Da Silva, H. Dahle, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, T. Déchelette, J. Delabrouille, J.-M. Delouis, J. Démoclès, F.-X. Désert, J. Dick, C. Dickinson, J. M. Diego, K. Dolag, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, O. Fabre, E. Falgarone, M. C. Falvella, Y. Fantaye, J. Fergusson, C. Filliard, F. Finelli, I. Flores-Cacho, S. Foley, 82 O. Forni, P. Fosalba, M. Frailis, A. A. Fraisse, E. Franceschi, M. Freschi, S. Fromenteau, M. Frommert, T. C. Gaier, S. Galeotta, J. Gallegos, S. Galli, B. Gandolfo, K. Ganga, C. Gauthier, R. T. Génova-Santos, T. Ghosh, M. Giard, G. Giardino, M. Gilfanov, D. Girard, Y. GiraudHéraud, E. Gjerløw, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, J. Haissinski, J. Hamann, F. K. Hansen, M. Hansen, D. Hanson, D. L. Harrison, A. Heavens, G. Helou, A. Hempel, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, S. Ho, M. Hobson, W. A. Holmes, A. Hornstrup, Z. Hou, W. Hovest, G. Huey, K. M. Huﬀenberger, G. Hurier, S. Ilić, A. H. Jaﬀe, T. R. Jaﬀe, J. Jasche, J. Jewell, W. C. Jones, M. Juvela, P. Kalberla, P. Kangaslahti, E. Keihänen, J. Kerp, R. Keskitalo, I. Khamitov, K. Kiiveri, J. Kim, T. S. Kisner, R. Kneissl, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, F. Lacasa, G. Lagache, A. Lähteenmäki, J.-M. Lamarre, M. Langer, A. Lasenby, M. Lattanzi, R. J. Laureijs, A. Lavabre, C. R. Lawrence, M. Le Jeune, S. Leach, J. P. Leahy, R. Leonardi, J. León-Tavares, C. Leroy, J. Lesgourgues, A. Lewis, C. Li, A. Liddle, M. Liguori, P. B. Lilje, M. Linden-Vørnle, V. Lindholm, M. López-Caniego, S. Lowe, P. M. Lubin, J. F. Macı́as-Pérez, C. J. MacTavish, B. Maﬀei, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero, D. Marinucci, M. Maris, F. Marleau, D. J. Marshall, P. G. Martin, E. Martı́nez-González, S. Masi, M. Massardi, S. Matarrese, T. Matsumura, F. Matthai, L. Maurin, P. Mazzotta, A. McDonald, J. D. McEwen, P. McGehee, S. Mei, P. R. Meinhold, A. Melchiorri, J.-B. Melin, L. Mendes, E. Menegoni, A. Mennella, M. Migliaccio, K. Mikkelsen, M. Millea, R. Miniscalco, S. Mitra, M.-A. Miville-Deschênes, D. Molinari, A. Moneti, L. Montier, G. Morgante, N. Morisset, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, M. Negrello, N. P. H. Nesvadba, C. B. Netterﬁeld, H. U. NørgaardNielsen, C. North, F. Noviello, D. Novikov, I. Novikov, I. J. O’Dwyer, F. Orieux, S. Osborne, C. O’Sullivan, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, R. Paladini, S. Pandolﬁ, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, P. Paykari, D. Pearson, T. J. Pearson, M. Peel, H. V. Peiris, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, P. Platania, D. Pogosyan, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, T. Poutanen, G. W. Pratt, G. Prézeau, S. Prunet, J.-L. Puget, A. R. Pullen, J. P. Rachen, B. Racine, A. Rahlin, C. Räth, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, A. Riazuelo, S. Ricciardi, T. Riller, C. Ringeval, I. Ristorcelli, G. Robbers, G. Rocha, M. Roman, C. Rosset, M. Rossetti, G. Roudier, M. Rowan-Robinson, J. A. RubiñoMartı́n, B. Ruiz-Granados, B. Rusholme, E. Salerno, M. Sandri, L. Sanselme, D. Santos, M. Savelainen, G. Savini, B. M. Schaefer, F. Schiavon, D. Scott, M. D. Seiﬀert, P. Serra, E. P. S. Shellard, K. Smith, G. F. Smoot, T. Souradeep, L. D. Spencer, J.-L. Starck, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, F. Sureau, P. Sutter, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, D. Tavagnacco, D. Taylor, L. Terenzi, D. Texier, L. Toﬀolatti, M. Tomasi, J.-P. Torre, M. Tristram, M. Tucci, J. Tuovinen, M. Türler, M. Tuttlebee, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, J. Varis, L. Vibert, M. Viel, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, C. Watson, R. Watson, I. K. Wehus, N. Welikala, J. Weller, M. White, S. D. M. White, A. Wilkinson, B. Winkel, J.-Q. Xia, D. Yvon, A. Zacchei, J. P. Zibin, and A. Zonca. Planck 2013 results. i. overview of products and scientiﬁc results. eprint arXiv:1303.5062, March 2013a. URL http://adsabs.harvard.edu/abs/2013arXiv1303.5062P. Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, E. Battaner, K. Benabed, A. Benoı̂t, A. Benoit-Lévy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, A. Bonaldi, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bridges, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, B. Cappellini, J.-F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, R.-R. Chary, X. Chen, H. C. Chiang, L.-Y Chiang, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, F. Couchot, A. Coulais, B. P. Crill, A. Curto, 83 F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J.-M. Delouis, F.-X. Désert, C. Dickinson, J. M. Diego, K. Dolag, H. Dole, S. Donzelli, O. Doré, M. Douspis, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, T. C. Gaier, S. Galeotta, S. Galli, K. Ganga, M. Giard, G. Giardino, Y. Giraud-Héraud, E. Gjerløw, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, J. Haissinski, J. Hamann, F. K. Hansen, D. Hanson, D. Harrison, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, Z. Hou, W. Hovest, K. M. Huﬀenberger, A. H. Jaﬀe, T. R. Jaﬀe, J. Jewell, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, T. S. Kisner, R. Kneissl, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lähteenmäki, J.-M. Lamarre, A. Lasenby, M. Lattanzi, R. J. Laureijs, C. R. Lawrence, S. Leach, J. P. Leahy, R. Leonardi, J. León-Tavares, J. Lesgourgues, A. Lewis, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. López-Caniego, P. M. Lubin, J. F. Macı́as-Pérez, B. Maffei, D. Maino, N. Mandolesi, M. Maris, D. J. Marshall, P. G. Martin, E. Martı́nez-González, S. Masi, M. Massardi, S. Matarrese, F. Matthai, P. Mazzotta, P. R. Meinhold, A. Melchiorri, J.-B. Melin, L. Mendes, E. Menegoni, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M.-A. Miville-Deschênes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterﬁeld, H. U. Nørgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, I. J. O’Dwyer, S. Osborne, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, D. Pearson, T. J. Pearson, H. V. Peiris, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, P. Platania, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, T. Poutanen, G. W. Pratt, G. Prézeau, S. Prunet, J.-L. Puget, J. P. Rachen, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, S. Ricciardi, T. Riller, I. Ristorcelli, G. Rocha, C. Rosset, G. Roudier, M. Rowan-Robinson, J. A. Rubiño Martı́n, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, D. Scott, M. D. Seiﬀert, E. P. S. Shellard, L. D. Spencer, J.-L. Starck, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, F. Sureau, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, D. Tavagnacco, L. Terenzi, L. Toﬀolatti, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, M. Türler, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Wilkinson, D. Yvon, A. Zacchei, and A. Zonca. Planck 2013 results. XVI. Cosmological parameters. ArXiv e-prints, 1303:5076, March 2013b. URL http://adsabs.harvard.edu/abs/2013arXiv1303.5076P. Planck collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, E. Battaner, K. Benabed, A. Benoit, A. Benoit-Levy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bridges, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J.-F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, L.-Y. Chiang, H. C. Chiang, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J.-M. Delouis, F.-X. Desert, C. Dickinson, J. M. Diego, H. Dole, S. Donzelli, O. Dore, M. Douspis, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Ensslin, H. K. Eriksen, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, T. C. Gaier, S. Galeotta, S. Galli, K. Ganga, M. Giard, G. Giardino, Y. Giraud-Heraud, E. Gjerlow, J. Gonzalez-Nuevo, K. M. Gorski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, F. K. Hansen, D. Hanson, D. Harrison, G. Helou, S. Henrot-Versille, C. Hernandez-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, K. M. Huﬀenberger, G. Hurier, T. R. Jaﬀe, A. H. Jaﬀe, J. Jewell, W. C. Jones, M. Juvela, E. Keihanen, R. Keskitalo, 84 K. Kiiveri, T. S. Kisner, R. Kneissl, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lahteenmaki, J.-M. Lamarre, A. Lasenby, M. Lattanzi, R. J. Laureijs, C. R. Lawrence, M. Le Jeune, S. Leach, J. P. Leahy, R. Leonardi, J. Leon-Tavares, J. Lesgourgues, M. Liguori, P. B. Lilje, V. Lindholm, M. Linden-Vornle, M. Lopez-Caniego, P. M. Lubin, J. F. Macias-Perez, B. Maﬀei, D. Maino, N. Mandolesi, D. Marinucci, M. Maris, D. J. Marshall, P. G. Martin, E. Martinez-Gonzalez, S. Masi, S. Matarrese, F. Matthai, P. Mazzotta, P. R. Meinhold, A. Melchiorri, L. Mendes, E. Menegoni, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M.-A. MivilleDeschenes, D. Molinari, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, P. Naselsky, F. Nati, P. Natoli, C. B. Netterﬁeld, H. U. Norgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, I. J. O’Dwyer, F. Orieux, S. Osborne, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, R. Paladini, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, P. Paykari, O. Perdereau, L. Perotto, F. Perrotta, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, T. Poutanen, G. W. Pratt, G. Prezeau, S. Prunet, J.-L. Puget, J. P. Rachen, A. Rahlin, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, S. Ricciardi, T. Riller, C. Ringeval, I. Ristorcelli, G. Rocha, C. Rosset, G. Roudier, M. RowanRobinson, J. A. Rubino-Martin, B. Rusholme, M. Sandri, L. Sanselme, D. Santos, G. Savini, D. Scott, M. D. Seiﬀert, E. P. S. Shellard, L. D. Spencer, J.-L. Starck, V. Stolyarov, R. Stompor, R. Sudiwala, F. Sureau, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, D. Tavagnacco, L. Terenzi, L. Toﬀolatti, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, M. Turler, L. Valenziano, J. Valiviita, B. Van Tent, J. Varis, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, D. Yvon, A. Zacchei, and A. Zonca. Planck 2013 results. XV. CMB power spectra and likelihood. arXiv:1303.5075 [astro-ph], March 2013. URL http://arxiv.org/abs/1303.5075. The BABAR Collaboration. Search for a dark photon in e+e- collisions at BABAR. Physical Review Letters, 113(20), November 2014. ISSN 0031-9007, 1079-7114. doi: 10.1103/PhysRevLett.113. 201801. URL http://arxiv.org/abs/1406.2980. arXiv: 1406.2980. The COrE Collaboration, C. Armitage-Caplan, M. Avillez, D. Barbosa, A. Banday, N. Bartolo, R. Battye, JP. Bernard, P. de Bernardis, S. Basak, M. Bersanelli, P. Bielewicz, A. Bonaldi, M. Bucher, F. Bouchet, F. Boulanger, C. Burigana, P. Camus, A. Challinor, S. Chongchitnan, D. Clements, S. Colafrancesco, J. Delabrouille, M. De Petris, G. De Zotti, C. Dickinson, J. Dunkley, T. Ensslin, J. Fergusson, P. Ferreira, K. Ferriere, F. Finelli, S. Galli, J. GarciaBellido, C. Gauthier, M. Haverkorn, M. Hindmarsh, A. Jaﬀe, M. Kunz, J. Lesgourgues, A. Liddle, M. Liguori, M. Lopez-Caniego, B. Maﬀei, P. Marchegiani, E. Martinez-Gonzalez, S. Masi, P. Mauskopf, S. Matarrese, A. Melchiorri, P. Mukherjee, F. Nati, P. Natoli, M. Negrello, L. Pagano, D. Paoletti, T. Peacocke, H. Peiris, L. Perroto, F. Piacentini, M. Piat, L. Piccirillo, G. Pisano, N. Ponthieu, C. Rath, S. Ricciardi, J. Rubino Martin, M. Salatino, P. Shellard, R. Stompor, L. Toﬀolatti J. Urrestilla, B. Van Tent, L. Verde, B. Wandelt, and S. Withington. COrE (Cosmic Origins Explorer) A White Paper. ArXiv e-prints, 1102:2181, February 2011. URL http://adsabs.harvard.edu/abs/2011arXiv1102.2181T. R. J. Cooke, M. Pettini, R. A. Jorgenson, M. T. Murphy, and C. C. Steidel. Precision Measures of the Primordial Abundance of Deuterium. ApJ, 781:31, January 2014. doi: 10.1088/0004-637X/ 781/1/31. R. H. Cyburt, J. Ellis, B. D. Fields, and K. A. Olive. Updated nucleosynthesis constraints on unstable relic particles. Phys. Rev. D, 67(10):103521, May 2003. doi: 10.1103/PhysRevD.67. 103521. Richard H. Cyburt, John Ellis, Brian D. Fields, Feng Luo, Keith A. Olive, and Vassilis C. Spanos. Nucleosynthesis constraints on a massive gravitino in neutralino dark matter scenarios. Journal of Cosmology and Astro-Particle Physics, 10:021, October 2009. ISSN 1475-7516. doi: 10.1088/ 1475-7516/2009/10/021. URL http://adsabs.harvard.edu/abs/2009JCAP...10..021C. 85 S. Das, T. A. Marriage, P. A. R. Ade, P. Aguirre, M. Amiri, J. W. Appel, L. F. Barrientos, E. S. Battistelli, J. R. Bond, B. Brown, B. Burger, J. Chervenak, M. J. Devlin, S. R. Dicker, W. Bertrand Doriese, J. Dunkley, R. Dünner, T. Essinger-Hileman, R. P. Fisher, J. W. Fowler, A. Hajian, M. Halpern, M. Hasselﬁeld, C. Hernández-Monteagudo, G. C. Hilton, M. Hilton, A. D. Hincks, R. Hlozek, K. M. Huﬀenberger, D. H. Hughes, J. P. Hughes, L. Infante, K. D. Irwin, J. Baptiste Juin, M. Kaul, J. Klein, A. Kosowsky, J. M. Lau, M. Limon, Y.-T. Lin, R. H. Lupton, D. Marsden, K. Martocci, P. Mauskopf, F. Menanteau, K. Moodley, H. Moseley, C. B. Netterﬁeld, M. D. Niemack, M. R. Nolta, L. A. Page, L. Parker, B. Partridge, B. Reid, N. Sehgal, B. D. Sherwin, J. Sievers, D. N. Spergel, S. T. Staggs, D. S. Swetz, E. R. Switzer, R. Thornton, H. Trac, C. Tucker, R. Warne, E. Wollack, and Y. Zhao. The Atacama Cosmology Telescope: A Measurement of the Cosmic Microwave Background Power Spectrum at 148 and 218 GHz from the 2008 Southern Survey. ApJ, 729:62–+, March 2011. doi: 10.1088/0004-637X/729/1/62. Sudeep Das, Thibaut Louis, Michael R. Nolta, Graeme E. Addison, Elia S. Battistelli, J. Richard Bond, Erminia Calabrese, Devin Crichton, Mark J. Devlin, Simon Dicker, Joanna Dunkley, Rolando Dünner, Joseph W. Fowler, Megan Gralla, Amir Hajian, Mark Halpern, Matthew Hasselﬁeld, Matt Hilton, Adam D. Hincks, Renée Hlozek, Kevin M. Huﬀenberger, John P. Hughes, Kent D. Irwin, Arthur Kosowsky, Robert H. Lupton, Tobias A. Marriage, Danica Marsden, Felipe Menanteau, Kavilan Moodley, Michael D. Niemack, Lyman A. Page, Bruce Partridge, Erik D. Reese, Benjamin L. Schmitt, Neelima Sehgal, Blake D. Sherwin, Jonathan L. Sievers, David N. Spergel, Suzanne T. Staggs, Daniel S. Swetz, Eric R. Switzer, Robert Thornton, Hy Trac, and Ed Wollack. The Atacama Cosmology Telescope: temperature and gravitational lensing power spectrum measurements from three seasons of data. Journal of Cosmology and Astro-Particle Physics, 04:014, April 2014. ISSN 1475-7516. doi: 10.1088/1475-7516/2014/04/014. URL http://adsabs.harvard.edu/abs/2014JCAP...04..014D. G. de Zotti, R. Ricci, D. Mesa, L. Silva, P. Mazzotta, L. Toﬀolatti, and J. González-Nuevo. Predictions for high-frequency radio surveys of extragalactic sources. Astron. & Astrophys., 431: 893–903, March 2005. doi: 10.1051/0004-6361:20042108. Michael Dine, Willy Fischler, and Mark Srednicki. A simple solution to the strong CP problem with a harmless axion. Physics Letters B, 104(3):199–202, August 1981. ISSN 0370-2693. doi: 10.1016/0370-2693(81)90590-6. URL http://www.sciencedirect.com/science/article/ pii/0370269381905906. A. R. Duﬀy, R. A. Battye, R. D. Davies, A. Moss, and P. N. Wilkinson. Galaxy redshift surveys selected by neutral hydrogen using the Five-hundred metre Aperture Spherical Telescope. Mon.Not.Roy.As.Soc. , 383:150–160, January 2008. doi: 10.1111/j.1365-2966.2007.12537.x. J. Dunkley, R. Hlozek, J. Sievers, V. Acquaviva, P. A. R. Ade, P. Aguirre, M. Amiri, J. W. Appel, L. F. Barrientos, E. S. Battistelli, J. R. Bond, B. Brown, B. Burger, J. Chervenak, S. Das, M. J. Devlin, S. R. Dicker, W. Bertrand Doriese, R. Dunner, T. Essinger-Hileman, R. P. Fisher, J. W. Fowler, A. Hajian, M. Halpern, M. Hasselﬁeld, C. Hernandez-Monteagudo, G. C. Hilton, M. Hilton, A. D. Hincks, K. M. Huﬀenberger, D. H. Hughes, J. P. Hughes, L. Infante, K. D. Irwin, J. B. Juin, M. Kaul, J. Klein, A. Kosowsky, J. M Lau, M. Limon, Y. Lin, R. H. Lupton, T. A. Marriage, D. Marsden, P. Mauskopf, F. Menanteau, K. Moodley, H. Moseley, C. B Netterﬁeld, M. D. Niemack, M. R. Nolta, L. A. Page, L. Parker, B. Partridge, B. Reid, N. Sehgal, B. Sherwin, D. N. Spergel, S. T. Staggs, D. S. Swetz, E. R. Switzer, R. Thornton, H. Trac, C. Tucker, R. Warne, E. Wollack, and Y. Zhao. The Atacama Cosmology Telescope: Cosmological Parameters from the 2008 Power Spectra. ArXiv:1009.0866, September 2010. J. Dunkley, R. Hlozek, J. Sievers, V. Acquaviva, P. A. R. Ade, P. Aguirre, M. Amiri, J. W. Appel, L. F. Barrientos, E. S. Battistelli, J. R. Bond, B. Brown, B. Burger, J. Chervenak, S. Das, M. J. Devlin, S. R. Dicker, W. Bertrand Doriese, R. Dünner, T. Essinger-Hileman, R. P. Fisher, J. W. Fowler, A. Hajian, M. Halpern, M. Hasselﬁeld, C. Hernández-Monteagudo, G. C. Hilton, M. Hilton, A. D. Hincks, K. M. Huﬀenberger, D. H. Hughes, J. P. Hughes, L. Infante, K. D. 86 Irwin, J. B. Juin, M. Kaul, J. Klein, A. Kosowsky, J. M. Lau, M. Limon, Y.-T. Lin, R. H. Lupton, T. A. Marriage, D. Marsden, P. Mauskopf, F. Menanteau, K. Moodley, H. Moseley, C. B. Netterﬁeld, M. D. Niemack, M. R. Nolta, L. A. Page, L. Parker, B. Partridge, B. Reid, N. Sehgal, B. Sherwin, D. N. Spergel, S. T. Staggs, D. S. Swetz, E. R. Switzer, R. Thornton, H. Trac, C. Tucker, R. Warne, E. Wollack, and Y. Zhao. The Atacama Cosmology Telescope: Cosmological Parameters from the 2008 Power Spectrum. ApJ, 739:52, September 2011. doi: 10.1088/0004-637X/739/1/52. J. Ellis, J. E. Kim, and D. V. Nanopoulos. Cosmological gravitino regeneration and decay. Physics Letters B, 145:181–186, September 1984. doi: 10.1016/0370-2693(84)90334-4. R. Essig, J. A. Jaros, W. Wester, P. Hansson Adrian, S. Andreas, T. Averett, O. Baker, B. Batell, M. Battaglieri, J. Beacham, T. Beranek, J. D. Bjorken, F. Bossi, J. R. Boyce, G. D. Cates, A. Celentano, A. S. Chou, R. Cowan, F. Curciarello, H. Davoudiasl, P. deNiverville, R. De Vita, A. Denig, R. Dharmapalan, B. Dongwi, B. Döbrich, B. Echenard, D. Espriu, S. Fegan, P. Fisher, G. B. Franklin, A. Gasparian, Y. Gershtein, M. Graham, P. W. Graham, A. Haas, A. Hatzikoutelis, M. Holtrop, I. Irastorza, E. Izaguirre, J. Jaeckel, Y. Kahn, N. Kalantarians, M. Kohl, G. Krnjaic, V. Kubarovsky, H. Lee, A. Lindner, A. Lobanov, W. J. Marciano, D. J. E. Marsh, T. Maruyama, D. McKeen, H. Merkel, K. Moﬀeit, P. Monaghan, G. Mueller, T. K. Nelson, G. R. Neil, M. Oriunno, Z. Pavlovic, S. K. Phillips, M. J. Pivovaroﬀ, R. Poltis, M. Pospelov, S. Rajendran, J. Redondo, A. Ringwald, A. Ritz, J. Ruz, K. Saenboonruang, P. Schuster, M. Shinn, T. R. Slatyer, J. H. Steﬀen, S. Stepanyan, D. B. Tanner, J. Thaler, M. E. Tobar, N. Toro, A. Upadye, R. Van de Water, B. Vlahovic, J. K. Vogel, D. Walker, A. Weltman, B. Wojtsekhowski, S. Zhang, and K. Zioutas. Dark Sectors and New, Light, Weakly-Coupled Particles. ArXiv e-prints, October 2013. W. A. Fendt and B. D. Wandelt. Computing High Accuracy Power Spectra with Pico. ArXiv e-prints, December 2007. Brian D. Fields. The Primordial Lithium Problem. Annual Review of Nuclear and Particle Science, 61(1):47–68, November 2011. ISSN 0163-8998, 1545-4134. doi: 10.1146/ annurev-nucl-102010-130445. URL http://arxiv.org/abs/1203.3551. arXiv: 1203.3551. G. Fiorentini, E. Lisi, Subir Sarkar, and F. L. Villante. Quantifying uncertainties in primordial nucleosynthesis without Monte Carlo simulations. Physical Review D, 58(6), August 1998. ISSN 0556-2821, 1089-4918. doi: 10.1103/PhysRevD.58.063506. URL http://arxiv.org/abs/ astro-ph/9803177. arXiv: astro-ph/9803177. D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer, and E. L. Wright. The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set. ApJ, 473:576, December 1996. doi: 10.1086/178173. D. J. Fixsen, E. Dwek, J. C. Mather, C. L. Bennett, and R. A. Shafer. The Spectrum of the Extragalactic Far-Infrared Background from the COBE FIRAS Observations. ApJ, 508:123–128, November 1998. doi: 10.1086/306383. G. B. Gelmini and M. Roncadelli. Left-handed neutrino mass scale and spontaneously broken lepton number. Physics Letters B, 99(5):411–415, March 1981. ISSN 0370-2693. doi: 10. 1016/0370-2693(81)90559-1. URL http://www.sciencedirect.com/science/article/pii/ 0370269381905591. E. M. George, C. L. Reichardt, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, N. W. Halverson, N. L. Harrington, G. P. Holder, W. L. Holzapfel, Z. Hou, J. D. Hrubes, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Millea, L. M. Mocanu, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, Z. Staniszewski, A. A. Stark, K. T. Story, A. van Engelen, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. A measurement of secondary cosmic microwave background anisotropies from the 2500 square-degree SPT-SZ 87 survey. The Astrophysical Journal, 799:177, February 2015. ISSN 0004-637X. doi: 10.1088/ 0004-637X/799/2/177. URL http://adsabs.harvard.edu/abs/2015ApJ...799..177G. M. Giavalisco, C. C. Steidel, K. L. Adelberger, M. E. Dickinson, M. Pettini, and M. Kellogg. The Angular Clustering of Lyman-Break Galaxies at Redshift Z approximately 3. ApJ, 503:543–+, August 1998. doi: 10.1086/306027. S. Giodini, D. Pierini, A. Finoguenov, G. W. Pratt, H. Boehringer, A. Leauthaud, L. Guzzo, H. Aussel, M. Bolzonella, P. Capak, M. Elvis, G. Hasinger, O. Ilbert, J. S. Kartaltepe, A. M. Koekemoer, S. J. Lilly, R. Massey, H. J. McCracken, J. Rhodes, M. Salvato, D. B. Sanders, N. Z. Scoville, S. Sasaki, V. Smolcic, Y. Taniguchi, D. Thompson, and the COSMOS Collaboration. Stellar and Total Baryon Mass Fractions in Groups and Clusters Since Redshift 1. ApJ, 703: 982–993, September 2009. doi: 10.1088/0004-637X/703/1/982. B. Gold, N. Odegard, J. L. Weiland, R. S. Hill, A. Kogut, C. L. Bennett, G. Hinshaw, X. Chen, J. Dunkley, M. Halpern, N. Jarosik, E. Komatsu, D. Larson, M. Limon, S. S. Meyer, M. R. Nolta, L. Page, K. M. Smith, D. N. Spergel, G. S. Tucker, E. Wollack, and E. L. Wright. Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission. Astrophys. J. Supp. , 192:15–+, February 2011. doi: 10.1088/0067-0049/192/2/15. A. Gruzinov and W. Hu. Secondary Cosmic Microwave Background Anisotropies in a Universe Reionized in Patches. ApJ, 508:435–439, December 1998. doi: 10.1086/306432. Z. Haiman and L. Knox. Correlations in the Far-Infrared Background. ApJ, 530:124–132, February 2000. doi: 10.1086/308374. N. R. Hall, R. Keisler, L. Knox, C. L. Reichardt, P. A. R. Ade, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, J. D. Hrubes, M. Joy, A. T. Lee, E. M. Leitch, M. Lueker, J. J. McMahon, J. Mehl, S. S. Meyer, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, B. Stalder, Z. Staniszewski, A. A. Stark, E. R. Switzer, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. Angular Power Spectra of the Millimeter-wavelength Background Light from Dusty Star-forming Galaxies with the South Pole Telescope. ApJ, 718:632–646, August 2010. doi: 10.1088/0004-637X/718/2/632. J. L. Hewett, H. Weerts, R. Brock, J. N. Butler, B. C. K. Casey, J. Collar, A. de Gouvea, R. Essig, Y. Grossman, W. Haxton, J. A. Jaros, C. K. Jung, Z. T. Lu, K. Pitts, Z. Ligeti, J. R. Patterson, M. Ramsey-Musolf, J. L. Ritchie, A. Roodman, K. Scholberg, C. E. M. Wagner, G. P. Zeller, S. Aefsky, A. Afanasev, K. Agashe, C. Albright, J. Alonso, C. Ankenbrandt, M. Aoki, C. A. Arguelles, N. Arkani-Hamed, J. R. Armendariz, C. Armendariz-Picon, E. Arrieta Diaz, J. Asaadi, D. M. Asner, K. S. Babu, K. Bailey, O. Baker, B. Balantekin, B. Baller, M. Bass, B. Batell, J. Beacham, J. Behr, N. Berger, M. Bergevin, E. Berman, R. Bernstein, A. J. Bevan, M. Bishai, M. Blanke, S. Blessing, A. Blondel, T. Blum, G. Bock, A. Bodek, G. Bonvicini, F. Bossi, J. Boyce, R. Breedon, M. Breidenbach, S. J. Brice, R. A. Briere, S. Brodsky, C. Bromberg, A. Bross, T. E. Browder, D. A. Bryman, M. Buckley, R. Burnstein, E. Caden, P. Campana, R. Carlini, G. Carosi, C. Castromonte, R. Cenci, I. Chakaberia, M. C. Chen, C. H. Cheng, B. Choudhary, N. H. Christ, E. Christensen, M. E. Christy, T. E. Chupp, E. Church, D. B. Cline, T. E. Coan, P. Coloma, J. Comfort, L. Coney, J. Cooper, R. J. Cooper, R. Cowan, D. F. Cowen, D. Cronin-Hennessy, A. Datta, G. S. Davies, M. Demarteau, D. P. DeMille, A. Denig, R. Dermisek, A. Deshpande, M. S. Dewey, R. Dharmapalan, J. Dhooghe, M. R. Dietrich, M. Diwan, Z. Djurcic, S. Dobbs, M. Duraisamy, B. Dutta, H. Duyang, D. A. Dwyer, M. Eads, B. Echenard, S. R. Elliott, C. Escobar, J. Fajans, S. Farooq, C. Faroughy, J. E. Fast, B. Feinberg, J. Felde, G. Feldman, P. Fierlinger, P. Fileviez Perez, B. Filippone, P. Fisher, B. T. Flemming, K. T. Flood, R. Forty, M. J. Frank, A. Freyberger, A. Friedland, R. Gandhi, K. S. Ganezer, A. Garcia, F. G. Garcia, S. Gardner, L. Garrison, A. Gasparian, S. Geer, V. M. Gehman, T. Gershon, 88 M. Gilchriese, C. Ginsberg, I. Gogoladze, M. Gonderinger, M. Goodman, H. Gould, M. Graham, P. W. Graham, R. Gran, J. Grange, G. Gratta, J. P. Green, H. Greenlee, R. C. Group, E. Guardincerri, V. Gudkov, R. Guenette, A. Haas, A. Hahn, T. Han, T. Handler, J. C. Hardy, R. Harnik, D. A. Harris, F. A. Harris, P. G. Harris, J. Hartnett, B. He, B. R. Heckel, K. M. Heeger, S. Henderson, D. Hertzog, R. Hill, E. A. Hinds, D. G. Hitlin, R. J. Holt, N. Holtkamp, G. Horton-Smith, P. Huber, W. Huelsnitz, J. Imber, I. Irastorza, J. Jaeckel, I. Jaegle, C. James, A. Jawahery, D. Jensen, C. P. Jessop, B. Jones, H. Jostlein, T. Junk, A. L. Kagan, M. Kalita, Y. Kamyshkov, D. M. Kaplan, G. Karagiorgi, A. Karle, T. Katori, B. Kayser, R. Kephart, S. Kettell, Y. K. Kim, M. Kirby, K. Kirch, J. Klein, J. Kneller, A. Kobach, M. Kohl, J. Kopp, M. Kordosky, W. Korsch, I. Kourbanis, A. D. Krisch, P. Krizan, A. S. Kronfeld, S. Kulkarni, K. S. Kumar, Y. Kuno, T. Kutter, T. Lachenmaier, M. Lamm, J. Lancaster, M. Lancaster, C. Lane, K. Lang, P. Langacker, S. Lazarevic, T. Le, K. Lee, K. T. Lesko, Y. Li, M. Lindgren, A. Lindner, J. Link, D. Lissauer, L. S. Littenberg, B. Littlejohn, C. Y. Liu, W. Loinaz, W. Lorenzon, W. C. Louis, J. Lozier, L. Ludovici, L. Lueking, C. Lunardini, D. B. MacFarlane, P. A. N. Machado, P. B. Mackenzie, J. Maloney, W. J. Marciano, W. Marsh, M. Marshak, J. W. Martin, C. Mauger, K. S. McFarland, C. McGrew, G. McLaughlin, D. McKeen, R. McKeown, B. T. Meadows, R. Mehdiyev, D. Melconian, H. Merkel, M. Messier, J. P. Miller, G. Mills, U. K. Minamisono, S. R. Mishra, I. Mocioiu, S. Moed Sher, R. N. Mohapatra, B. Monreal, C. D. Moore, J. G. Morﬁn, J. Mousseau, L. A. Moustakas, G. Mueller, P. Mueller, M. Muether, H. P. Mumm, C. Munger, H. Murayama, P. Nath, O. Naviliat-Cuncin, J. K. Nelson, D. Neuﬀer, J. S. Nico, A. Norman, D. Nygren, Y. Obayashi, T. P. O’Connor, Y. Okada, J. Olsen, L. Orozco, J. L. Orrell, J. Osta, B. Pahlka, J. Paley, V. Papadimitriou, M. Papucci, S. Parke, R. H. Parker, Z. Parsa, K. Partyka, A. Patch, J. C. Pati, R. B. Patterson, Z. Pavlovic, G. Paz, G. N. Perdue, D. Perevalov, G. Perez, R. Petti, W. Pettus, A. Piepke, M. Pivovaroﬀ, R. Plunkett, C. C. Polly, M. Pospelov, R. Povey, A. Prakesh, M. V. Purohit, S. Raby, J. L. Raaf, R. Rajendran, S. Rajendran, G. Rameika, R. Ramsey, A. Rashed, B. N. Ratcliﬀ, B. Rebel, J. Redondo, P. Reimer, D. Reitzner, F. Ringer, A. Ringwald, S. Riordan, B. L. Roberts, D. A. Roberts, R. Robertson, F. Robicheaux, M. Rominsky, R. Roser, J. L. Rosner, C. Rott, P. Rubin, N. Saito, M. Sanchez, S. Sarkar, H. Schellman, B. Schmidt, M. Schmitt, D. W. Schmitz, J. Schneps, A. Schopper, P. Schuster, A. J. Schwartz, M. Schwarz, J. Seeman, Y. K. Semertzidis, K. K. Seth, Q. Shaﬁ, P. Shanahan, R. Sharma, S. R. Sharpe, M. Shiozawa, V. Shiltsev, K. Sigurdson, P. Sikivie, J. Singh, D. Sivers, T. Skwarnicki, N. Smith, J. Sobczyk, H. Sobel, M. Soderberg, Y. H. Song, A. Soni, P. Souder, A. Sousa, J. Spitz, M. Stancari, G. C. Stavenga, J. H. Steﬀen, S. Stepanyan, D. Stoeckinger, S. Stone, J. Strait, M. Strassler, I. A. Sulai, R. Sundrum, R. Svoboda, B. Szczerbinska, A. Szelc, T. Takeuchi, P. Tanedo, S. Taneja, J. Tang, D. B. Tanner, R. Tayloe, I. Taylor, J. Thomas, C. Thorn, X. Tian, B. G. Tice, M. Tobar, N. Tolich, N. Toro, I. S. Towner, Y. Tsai, R. Tschirhart, C. D. Tunnell, M. Tzanov, A. Upadhye, J. Urheim, S. Vahsen, A. Vainshtein, E. Valencia, R. G. Van de Water, R. S. Van de Water, M. Velasco, J. Vogel, P. Vogel, W. Vogelsang, Y. W. Wah, D. Walker, N. Weiner, A. Weltman, R. Wendell, W. Wester, M. Wetstein, C. White, L. Whitehead, J. Whitmore, E. Widmann, G. Wiedemann, J. Wilkerson, G. Wilkinson, P. Wilson, R. J. Wilson, W. Winter, M. B. Wise, J. Wodin, S. Wojcicki, B. Wojtsekhowski, T. Wongjirad, E. Worcester, J. Wurtele, T. Xin, J. Xu, T. Yamanaka, Y. Yamazaki, I. Yavin, J. Yeck, M. Yeh, M. Yokoyama, J. Yoo, A. Young, E. Zimmerman, K. Zioutas, M. Zisman, J. Zupan, and R. Zwaska. Fundamental Physics at the Intensity Frontier. arXiv:1205.2671 [hep-ex, physics:hep-ph], May 2012. URL http://arxiv.org/abs/1205.2671. arXiv: 1205.2671. G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L. Wright. Nine-year wilkinson microwave anisotropy probe (WMAP) observations: Cosmological parameter results. The Astrophysical Journal Supplement Series, 208:19, October 2013. ISSN 0067-0049. doi: 89 10.1088/0067-0049/208/2/19. URL http://adsabs.harvard.edu/abs/2013ApJS..208...19H. C. M. Ho and R. J. Scherrer. Sterile neutrinos and light dark matter save each other. Phys. Rev. D, 87(6):065016, March 2013. doi: 10.1103/PhysRevD.87.065016. J. A. Hodge, R. H. Becker, R. L. White, G. T. Richards, and G. R. Zeimann. High-resolution Very Large Array Imaging of Sloan Digital Sky Survey Stripe 82 at 1.4 GHz. Astron. J. , 142:3–+, July 2011. doi: 10.1088/0004-6256/142/1/3. Z. Hou, C. L. Reichardt, K. T. Story, B. Follin, R. Keisler, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, R. de Putter, M. A. Dobbs, S. Dodelson, J. Dudley, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, S. Hoover, J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, J. T. Sayre, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, Z. Staniszewski, A. A. Stark, A. van Engelen, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. Constraints on Cosmology from the Cosmic Microwave Background Power Spectrum of the 2500 deg2 SPT-SZ Survey. The Astrophysical Journal, 782(2):74, February 2014. ISSN 0004-637X. doi: 10.1088/0004-637X/782/2/74. URL http: //iopscience.iop.org/0004-637X/782/2/74. Zhen Hou, Ryan Keisler, Lloyd Knox, Marius Millea, and Christian Reichardt. How massless neutrinos aﬀect the cosmic microwave background damping tail. Physical Review D, 87:83008, April 2013. ISSN 0556-2821. doi: 10.1103/PhysRevD.87.083008. URL http://adsabs.harvard. edu/abs/2013PhRvD..87h3008H. W. Hu. Reionization Revisited: Secondary Cosmic Microwave Background Anisotropies and Polarization. ApJ, 529:12–25, January 2000. doi: 10.1086/308279. W. Hu and J. Silk. Thermalization and spectral distortions of the cosmic background radiation. Phys. Rev. D, 48:485–502, July 1993. doi: 10.1103/PhysRevD.48.485. D. H. Hughes, S. Serjeant, J. Dunlop, M. Rowan-Robinson, A. Blain, R. G. Mann, R. Ivison, J. Peacock, A. Efstathiou, W. Gear, S. Oliver, A. Lawrence, M. Longair, P. Goldschmidt, and T. Jenness. High-redshift star formation in the Hubble Deep Field revealed by a submillimetrewavelength survey. Nature, 394:241–247, July 1998. doi: 10.1038/28328. I. T. Iliev, U.-L. Pen, J. R. Bond, G. Mellema, and P. R. Shapiro. The Kinetic Sunyaev-Zel’dovich Eﬀect from Radiative Transfer Simulations of Patchy Reionization. ApJ, 660:933–944, May 2007. doi: 10.1086/513687. F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P. D. Serpico. Primordial nucleosynthesis: From precision cosmology to fundamental physics. Physics Reports , 472:1–76, March 2009. doi: 10.1016/j.physrep.2009.02.002. Yuri I. Izotov and Trinh X. Thuan. The Primordial Abundance of 4He: Evidence for NonStandard Big Bang Nucleosynthesis. The Astrophysical Journal Letters, 710(1):L67, February 2010. ISSN 2041-8205. doi: 10.1088/2041-8205/710/1/L67. URL http://iopscience.iop. org/2041-8205/710/1/L67. David B. Kaplan. Opening the axion window. Nuclear Physics B, 260(1):215–226, October 1985. ISSN 0550-3213. doi: 10.1016/0550-3213(85)90319-0. URL http://www.sciencedirect.com/ science/article/pii/0550321385903190. M. Kawasaki and T. Moroi. Gravitino Production in the Inﬂationary Universe and the Eﬀects on Big-Bang Nucleosynthesis. Progress of Theoretical Physics, 93:879–899, May 1995. doi: 10.1143/PTP.93.879. R. Keisler, C. L. Reichardt, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H. M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, S. Hoover, Z. Hou, J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy, T. Natoli, S. Padin, T. Plagge, C. Pryke, J. E. 90 Ruhl, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, Z. Staniszewski, A. A. Stark, K. Story, A. van Engelen, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. A Measurement of the Damping Tail of the Cosmic Microwave Background Power Spectrum with the South Pole Telescope. ArXiv e-prints, May 2011. R. Keisler, C. L. Reichardt, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H. M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, S. Hoover, Z. Hou, J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy, T. Natoli, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, Z. Staniszewski, A. A. Stark, K. Story, A. van Engelen, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. A Measurement of the Damping Tail of the Cosmic Microwave Background Power Spectrum with the South Pole Telescope. The Astrophysical Journal, 743:28, December 2011. ISSN 0004-637X. doi: 10.1088/0004-637X/743/1/28. URL http://adsabs.harvard.edu/abs/2011ApJ...743...28K. Jihn E. Kim. Weak-Interaction Singlet and Strong $\mathrmCP$ Invariance. Phys. Rev. Lett., 43(2):103–107, July 1979. doi: 10.1103/PhysRevLett.43.103. URL http://link.aps.org/doi/ 10.1103/PhysRevLett.43.103. M. Kleban and R. Rabadan. Collider Bounds on Pseudoscalars Coupling to Gauge Bosons. ArXiv High Energy Physics - Phenomenology e-prints, October 2005. L. Knox. Forecasting foreground impact on cosmic microwave background measurements. Mon.Not.Roy.As.Soc. , 307:977–983, August 1999. doi: 10.1046/j.1365-8711.1999.02687.x. L. Knox, R. Scoccimarro, and S. Dodelson. Impact of Inhomogeneous Reionization on Cosmic Microwave Background Anisotropy. Physical Review Letters, 81:2004–2007, September 1998. doi: 10.1103/PhysRevLett.81.2004. L. Knox, A. Cooray, D. Eisenstein, and Z. Haiman. Probing Early Structure Formation with Far-Infrared Background Correlations. ApJ, 550:7–20, March 2001. doi: 10.1086/319732. L. Knox, G. P. Holder, and S. E. Church. Eﬀects of Submillimeter and Radio Point Sources on the Recovery of Sunyaev-Zel’dovich Galaxy Cluster Parameters. ApJ, 612:96–107, September 2004. doi: 10.1086/422447. A. Kogut, D. J. Fixsen, D. T. Chuss, J. Dotson, E. Dwek, M. Halpern, G. F. Hinshaw, S. M. Meyer, S. H. Moseley, M. D. Seiﬀert, D. N. Spergel, and E. J. Wollack. The Primordial Inﬂation Explorer (PIXIE): a nulling polarimeter for cosmic microwave background observations. Journal of Cosmology and Astro-Particle Physics, 07:025, July 2011. ISSN 1475-7516. doi: 10.1088/ 1475-7516/2011/07/025. URL http://adsabs.harvard.edu/abs/2011JCAP...07..025K. E. Komatsu and T. Kitayama. Sunyaev-Zeldovich Fluctuations from Spatial Correlations between Clusters of Galaxies. Astrophys. J. Lett. , 526:L1–L4, November 1999. doi: 10.1086/312364. E. Komatsu and U. Seljak. The Sunyaev-Zel’dovich angular power spectrum as a probe of cosmological parameters. Mon.Not.Roy.As.Soc. , 336:1256–1270, November 2002. doi: 10.1046/j. 1365-8711.2002.05889.x. E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. ArXiv eprints, January 2010. G. Lagache, N. Bavouzet, N. Fernandez-Conde, N. Ponthieu, T. Rodet, H. Dole, M.-A. MivilleDeschênes, and J.-L. Puget. Correlated Anisotropies in the Cosmic Far-Infrared Background Detected by the Multiband Imaging Photometer for Spitzer: Constraint on the Bias. Astrophys. J. Lett. , 665:L89–L92, August 2007. doi: 10.1086/521301. E. T. Lau, A. V. Kravtsov, and D. Nagai. Residual Gas Motions in the Intracluster Medium and Bias in Hydrostatic Measurements of Mass Proﬁles of Clusters. ApJ, 705:1129–1138, November 91 2009. doi: 10.1088/0004-637X/705/2/1129. S. M. Leach et al. Component separation methods for the Planck mission. Astron. Astrophys., 491: 597–615, 2008. doi: 10.1051/0004-6361:200810116. Antony Lewis and Sarah Bridle. Cosmological parameters from CMB and other data: a MonteCarlo approach. Phys. Rev., D66:103511, 2002. Antony Lewis, Anthony Challinor, and Anthony Lasenby. Eﬃcient computation of CMB anisotropies in closed FRW models. Astrophys. J., 538:473–476, 2000. M. Lueker, C. L. Reichardt, K. K. Schaﬀer, O. Zahn, P. A. R. Ade, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, E. M. George, N. R. Hall, N. W. Halverson, G. P. Holder, W. L. Holzapfel, J. D. Hrubes, M. Joy, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, J. J. McMahon, J. Mehl, S. S. Meyer, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, L. Shaw, E. Shirokoﬀ, H. G. Spieler, B. Stalder, Z. Staniszewski, A. A. Stark, K. Vanderlinde, J. D. Vieira, and R. Williamson. Measurements of Secondary Cosmic Microwave Background Anisotropies with the South Pole Telescope. ApJ, 719:1045–1066, August 2010. doi: 10.1088/0004-637X/719/2/1045. Eduard Massó and Ramon Toldrà. Light spinless particle coupled to photons. Physical Review D, 52(4):1755–1763, August 1995. doi: 10.1103/PhysRevD.52.1755. URL http://link.aps.org/ doi/10.1103/PhysRevD.52.1755. Eduard Massó and Ramon Toldrà. New Constraints on a Light Spinless Particle Coupled to Photons. Physical Review D, 55(12):7967–7969, June 1997. ISSN 0556-2821, 1089-4918. doi: 10.1103/PhysRevD.55.7967. URL http://arxiv.org/abs/hep-ph/9702275. arXiv: hepph/9702275. M. McQuinn, S. R. Furlanetto, L. Hernquist, O. Zahn, and M. Zaldarriaga. The Kinetic SunyaevZel’dovich Eﬀect from Reionization. ApJ, 630:643–656, September 2005. doi: 10.1086/432049. M. Millea, O. Doré, J. Dudley, G. Holder, L. Knox, L. Shaw, Y.-S. Song, and O. Zahn. Modeling Extragalactic Foregrounds and Secondaries for Unbiased Estimation of Cosmological Parameters from Primary Cosmic Microwave Background Anisotropy. The Astrophysical Journal, 746:4, February 2012. ISSN 0004-637X. doi: 10.1088/0004-637X/746/1/4. URL http://adsabs. harvard.edu/abs/2012ApJ...746....4M. Marius Millea, Brian Fields, and Lloyd Knox. New Bounds for Axions and Axion-Like Particles with keV-GeV Masses. In Preparation, November 2014. Ken Mimasu and Verónica Sanz. ALPs at Colliders. ArXiv e-prints, 1409:4792, September 2014. URL http://adsabs.harvard.edu/abs/2014arXiv1409.4792M. L. M. Mocanu, T. M. Crawford, J. D. Vieira, K. A. Aird, M. Aravena, J. E. Austermann, B. A. Benson, M. Béthermin, L. E. Bleem, M. Bothwell, J. E. Carlstrom, C. L. Chang, S. Chapman, H.-M. Cho, A. T. Crites, T. de Haan, M. A. Dobbs, W. B. Everett, E. M. George, N. W. Halverson, N. Harrington, Y. Hezaveh, G. P. Holder, W. L. Holzapfel, S. Hoover, J. D. Hrubes, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, D. P. Marrone, J. J. McMahon, J. Mehl, S. S. Meyer, J. J. Mohr, T. E. Montroy, T. Natoli, S. Padin, T. Plagge, C. Pryke, A. Rest, C. L. Reichardt, J. E. Ruhl, J. T. Sayre, K. K. Schaﬀer, E. Shirokoﬀ, H. G. Spieler, J. S. Spilker, B. Stalder, Z. Staniszewski, A. A. Stark, K. T. Story, E. R. Switzer, K. Vanderlinde, and R. Williamson. Extragalactic millimeter-wave point-source catalog, number counts and statistics from 771 deg2 of the SPT-SZ survey. The Astrophysical Journal, 779:61, December 2013. ISSN 0004-637X. doi: 10.1088/0004-637X/779/1/61. URL http://adsabs. harvard.edu/abs/2013ApJ...779...61M. J. F. Navarro, C. S. Frenk, and S. D. M. White. A Universal Density Proﬁle from Hierarchical Clustering. ApJ, 490:493–+, December 1997. doi: 10.1086/304888. K. A. Olive and Particle Data Group. Review of Particle Physics. Chinese Physics C, 38(9): 090001, August 2014. ISSN 1674-1137. doi: 10.1088/1674-1137/38/9/090001. URL http: //iopscience.iop.org/1674-1137/38/9/090001. 92 J. P. Ostriker and E. T. Vishniac. Generation of microwave background ﬂuctuations from nonlinear perturbations at the ERA of galaxy formation. Astrophys. J. Lett. , 306:L51–L54, July 1986. doi: 10.1086/184704. J. A. Peacock and S. J. Dodds. Non-linear evolution of cosmological power spectra. Mon.Not.Roy.As.Soc. , 280:L19–L26, June 1996. R. D. Peccei and Helen R. Quinn. CP Conservation in the Presence of Pseudoparticles. Physical Review Letters, 38(25):1440–1443, June 1977a. doi: 10.1103/PhysRevLett.38.1440. URL http: //link.aps.org/doi/10.1103/PhysRevLett.38.1440. R. D. Peccei and Helen R. Quinn. Constraints imposed by CP conservation in the presence of pseudoparticles. Physical Review D, 16(6):1791–1797, September 1977b. doi: 10.1103/PhysRevD. 16.1791. URL http://link.aps.org/doi/10.1103/PhysRevD.16.1791. M. W. Peel, R. A. Battye, and S. T. Kay. Statistics of the Sunyaev-Zel’dovich eﬀect power spectrum. Mon.Not.Roy.As.Soc. , 397:2189–2207, August 2009. doi: 10.1111/j.1365-2966.2009.15121.x. Planck Collaboration. Planck Early Results: The all-sky Early Sunyaev-Zeldovich cluster sample. ArXiv e-prints, January 2011h. Planck Collaboration. Planck early results: XMM-Newton follow-up for validation of Planck cluster candidates. ArXiv e-prints, January 2011i. Planck Collaboration. Planck early results: Statistical analysis of Sunyaev-Zeldovich scaling relations for X-ray galaxy clusters. ArXiv e-prints, January 2011j. Planck Collaboration. Planck Early Results: Statistical properties of extragalactic radio sources in the Planck Early Release Compact Source Catalogue. ArXiv e-prints, January 2011m. Planck Collaboration. Planck Early Results: The Power Spectrum Of Cosmic Infrared Background Anisotropies. ArXiv e-prints, January 2011r. Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, R. Barrena, J. G. Bartlett, N. Bartolo, E. Battaner, R. Battye, K. Benabed, A. Benoı̂t, A. Benoit-Lévy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, I. Bikmaev, H. Böhringer, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bucher, R. Burenin, C. Burigana, R. C. Butler, E. Calabrese, J.-F. Cardoso, P. Carvalho, A. Catalano, A. Challinor, A. Chamballu, R.-R. Chary, H. C. Chiang, G. Chon, P. R. Christensen, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, B. Comis, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, H. Dahle, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, F.-X. Désert, C. Dickinson, J. M. Diego, K. Dolag, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, P. R. M. Eisenhardt, F. Elsner, T. A. Enßlin, H. K. Eriksen, E. Falgarone, J. Fergusson, F. Feroz, A. Ferragamo, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli, K. Ganga, R. T. Génova-Santos, M. Giard, Y. Giraud-Héraud, E. Gjerløw, J. González-Nuevo, K. M. Górski, K. J. B. Grainge, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, F. K. Hansen, D. Hanson, D. L. Harrison, A. Hempel, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, K. M. Huﬀenberger, G. Hurier, A. H. Jaﬀe, T. R. Jaﬀe, T. Jin, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, I. Khamitov, T. S. Kisner, R. Kneissl, J. Knoche, M. Kunz, H. Kurki-Suonio, G. Lagache, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, R. Leonardi, J. Lesgourgues, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. LópezCaniego, P. M. Lubin, J. F. Macı́as-Pérez, G. Maggio, D. Maino, D. S. Y. Mak, N. Mandolesi, A. Mangilli, P. G. Martin, E. Martı́nez-González, S. Masi, S. Matarrese, P. Mazzotta, P. McGehee, S. Mei, A. Melchiorri, J.-B. Melin, L. Mendes, A. Mennella, M. Migliaccio, S. Mitra, M.-A. Miville-Deschênes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, A. Nastasi, F. Nati, P. Natoli, C. B. Netterﬁeld, H. U. Nørgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, M. Olamaie, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, G. Patanchon, T. J. Pearson, O. Perdereau, L. Perotto, Y. C. Perrott, 93 F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, G. W. Pratt, G. Prézeau, S. Prunet, J.-L. Puget, J. P. Rachen, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha, C. Rosset, M. Rossetti, G. Roudier, E. Rozo, J. A. Rubiño-Martı́n, C. Rumsey, B. Rusholme, E. S. Rykoﬀ, M. Sandri, D. Santos, R. D. E. Saunders, M. Savelainen, G. Savini, M. P. Schammel, D. Scott, M. D. Seiﬀert, E. P. S. Shellard, T. W. Shimwell, L. D. Spencer, S. A. Stanford, D. Stern, V. Stolyarov, R. Stompor, A. Streblyanska, R. Sudiwala, R. Sunyaev, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, L. Terenzi, L. Toﬀolatti, M. Tomasi, D. Tramonte, M. Tristram, M. Tucci, J. Tuovinen, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, I. K. Wehus, S. D. M. White, E. L. Wright, D. Yvon, A. Zacchei, and A. Zonca. Planck 2015 results. XXVII. the second planck catalogue of sunyaev-zeldovich sources. ArXiv e-prints, 1502:1598, February 2015. URL http://adsabs.harvard.edu/abs/2015arXiv150201598P. M. Pospelov and J. Pradler. Big Bang Nucleosynthesis as a Probe of New Physics. Annual Review of Nuclear and Particle Science, 60:539–568, November 2010. doi: 10.1146/annurev.nucl.012809. 104521. S. Prunet, S. K. Sethi, F. R. Bouchet, and M.-A. Miville-Deschenes. Galactic dust polarized emission at high latitudes and CMB polarization. Astron. & Astrophys., 339:187–193, November 1998. J.-L. Puget, A. Abergel, J.-P. Bernard, F. Boulanger, W. B. Burton, F.-X. Desert, and D. Hartmann. Tentative detection of a cosmic far-infrared background with COBE. Astron. & Astrophys., 308:L5+, April 1996. Georg G. Raﬀelt. Stars as Laboratories for Fundamental Physics: The Astrophysics of Neutrinos, Axions, and Other Weakly Interacting Particles. University of Chicago Press, Chicago, May 1996. ISBN 9780226702728. Georg G. Raﬀelt and David S. P. Dearborn. Bounds on light, weakly interacting particles from observational lifetimes of helium-burning stars. Physical Review D, 37(2):549–551, January 1988. doi: 10.1103/PhysRevD.37.549. URL http://link.aps.org/doi/10.1103/PhysRevD.37.549. M. Righi, C. Hernández-Monteagudo, and R. A. Sunyaev. The clustering of merging star-forming haloes: dust emission as high frequency arcminute CMB foreground. Astron. & Astrophys., 478: 685–700, February 2008. doi: 10.1051/0004-6361:20078207. M. G. Santos, A. Cooray, Z. Haiman, L. Knox, and C.-P. Ma. Small-Scale Cosmic Microwave Background Temperature and Polarization Anisotropies Due to Patchy Reionization. ApJ, 598: 756–766, December 2003. doi: 10.1086/378772. S. Y. Sazonov and R. A. Sunyaev. Microwave polarization in the direction of galaxy clusters induced by the CMB quadrupole anisotropy. Mon.Not.Roy.As.Soc. , 310:765–772, December 1999. doi: 10.1046/j.1365-8711.1999.02981.x. L. Sbordone, P. Bonifacio, E. Caﬀau, H.-G. Ludwig, N. T. Behara, J. I. González Hernández, M. Steﬀen, R. Cayrel, B. Freytag, C. van’t Veer, P. Molaro, B. Plez, T. Sivarani, M. Spite, F. Spite, T. C. Beers, N. Christlieb, P. Francois, and V. Hill. The metal-poor end of the Spite plateau. I. Stellar parameters, metallicities, and lithium abundances. Astron. & Astrophys., 522: A26, November 2010. doi: 10.1051/0004-6361/200913282. D. Scott and M. White. Implications of SCUBA observations for the Planck Surveyor. Astron. & Astrophys., 346:1–6, June 1999. N. Sehgal, P. Bode, S. Das, C. Hernandez-Monteagudo, K. Huﬀenberger, Y.-T. Lin, J. P. Ostriker, and H. Trac. Simulations of the Microwave Sky. ApJ, 709:920–936, February 2010. doi: 10.1088/ 0004-637X/709/2/920. Paolo Serra, Asantha Cooray, Alexandre Amblard, Luca Pagano, and Alessandro Melchiorri. Impact of Point Source Clustering on Cosmological Parameters with CMB Anisotropies. Phys. Rev., D78: 043004, 2008. doi: 10.1103/PhysRevD.78.043004. 94 L. D. Shaw, O. Zahn, G. P. Holder, and O. Doré. Sharpening the Precision of the Sunyaev-Zel’dovich Power Spectrum. ApJ, 702:368–376, September 2009. doi: 10.1088/0004-637X/702/1/368. L. D. Shaw, D. Nagai, S. Bhattacharya, and E. T. Lau. Impact of Cluster Physics on the SunyaevZel’dovich Power Spectrum. ApJ, 725:1452–1465, December 2010. doi: 10.1088/0004-637X/725/ 2/1452. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov. Can conﬁnement ensure natural {CP} invariance of strong interactions? Nuclear Physics B, 166(3):493 – 506, 1980. ISSN 0550-3213. doi: http://dx.doi.org/10.1016/0550-3213(80)90209-6. URL http://www.sciencedirect.com/ science/article/pii/0550321380902096. E. Shirokoﬀ, C. L. Reichardt, L. Shaw, M. Millea, P. A. R. Ade, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H. M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, E. M. George, N. W. Halverson, G. P. Holder, W. L. HOlzapfel, J. D. Hrubes, M. Joy, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, K. K. Schaﬀer, H. G. Spieler, Z. Staniszewski, A. A. Stark, K. Story, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. Improved constraints on cosmic microwave background secondary anisotropies from the complete 2008 South Pole Telescope data. ArXiv e-prints, December 2010. R. E. Smith, J. A. Peacock, A. Jenkins, S. D. M. White, C. S. Frenk, F. R. Pearce, P. A. Thomas, G. Efstathiou, and H. M. P. Couchman. Stable clustering, the halo model and nonlinear cosmological power spectra. Mon.Not.Roy.As.Soc. , 341:1311–1332, June 2003. doi: 10.1046/j.1365-8711.2003.06503.x. D. N. Spergel, L. Verde, H. V. Peiris, E. Komatsu, M. R. Nolta, C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, L. Page, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophys. J. Supp. , 148:175–194, September 2003. doi: 10.1086/377226. F. Spite and M. Spite. Abundance of lithium in unevolved halo stars and old disk stars - Interpretation and consequences. Astron. & Astrophys., 115:357–366, November 1982. G. Steigman. Primordial Nucleosynthesis in the Precision Cosmology Era. Annual Review of Nuclear and Particle Science, 57:463–491, November 2007. doi: 10.1146/annurev.nucl.56.080805. 140437. K. T. Story, C. L. Reichardt, Z. Hou, R. Keisler, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, B. Follin, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, S. Hoover, J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, J. T. Sayre, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, Z. Staniszewski, A. A. Stark, A. van Engelen, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. A Measurement of the Cosmic Microwave Background Damping Tail from the 2500-Square-Degree SPT-SZ Survey. The Astrophysical Journal, 779:86, December 2013. ISSN 0004-637X. doi: 10.1088/0004-637X/779/1/86. URL http://adsabs.harvard.edu/abs/2013ApJ...779...86S. N. Taburet, N. Aghanim, M. Douspis, and M. Langer. Biases on the cosmological parameters and thermal Sunyaev-Zel’dovich residuals. Mon.Not.Roy.As.Soc. , 392:1153–1158, January 2009. doi: 10.1111/j.1365-2966.2008.14105.x. M. Tegmark and G. Efstathiou. A method for subtracting foregrounds from multifrequency CMB sky maps**. Mon.Not.Roy.As.Soc. , 281:1297–1314, August 1996. M. Tegmark, D. J. Eisenstein, W. Hu, and A. de Oliveira-Costa. Foregrounds and Forecasts for the Cosmic Microwave Background. ApJ, 530:133–165, February 2000. doi: 10.1086/308348. Max Tegmark, Daniel J. Eisenstein, Wayne Hu, and Angelica de Oliveira-Costa. Foregrounds and Forecasts for the Cosmic Microwave Background. Astrophys. J., 530:133–165, 2000. doi: 95 10.1086/308348. J. Tinker, A. V. Kravtsov, A. Klypin, K. Abazajian, M. Warren, G. Yepes, S. Gottlöber, and D. E. Holz. Toward a Halo Mass Function for Precision Cosmology: The Limits of Universality. ApJ, 688:709–728, December 2008. doi: 10.1086/591439. H. Trac, P. Bode, and J. P. Ostriker. Templates for the Sunyaev-Zel’dovich Angular Power Spectrum. ArXiv e-prints, June 2010. J. D. Vieira, T. M. Crawford, E. R. Switzer, P. A. R. Ade, K. A. Aird, M. L. N. Ashby, B. A. Benson, L. E. Bleem, M. Brodwin, J. E. Carlstrom, C. L. Chang, H.-M. Cho, A. T. Crites, T. de Haan, M. A. Dobbs, W. Everett, E. M. George, M. Gladders, N. R. Hall, N. W. Halverson, F. W. High, G. P. Holder, W. L. Holzapfel, J. D. Hrubes, M. Joy, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. P. Marrone, V. McIntyre, J. J. McMahon, J. Mehl, S. S. Meyer, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, C. L. Reichardt, J. E. Ruhl, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G. Spieler, B. Stalder, Z. Staniszewski, A. A. Stark, K. Vanderlinde, W. Walsh, R. Williamson, Y. Yang, O. Zahn, and A. Zenteno. Extragalactic Millimeter-wave Sources in South Pole Telescope Survey Data: Source Counts, Catalog, and Statistics for an 87 Square-degree Field. ApJ, 719:763–783, August 2010. doi: 10.1088/0004-637X/719/1/763. M. P. Viero, P. A. R. Ade, J. J. Bock, E. L. Chapin, M. J. Devlin, M. Griﬃn, J. O. Gundersen, M. Halpern, P. C. Hargrave, D. H. Hughes, J. Klein, C. J. MacTavish, G. Marsden, P. G. Martin, P. Mauskopf, L. Moncelsi, M. Negrello, C. B. Netterﬁeld, L. Olmi, E. Pascale, G. Patanchon, M. Rex, D. Scott, C. Semisch, N. Thomas, M. D. P. Truch, C. Tucker, G. S. Tucker, and D. V. Wiebe. BLAST: Correlations in the Cosmic Far-Infrared Background at 250, 350, and 500 μm Reveal Clustering of Star-forming Galaxies. ApJ, 707:1766–1778, December 2009. doi: 10.1088/0004-637X/707/2/1766. E. T. Vishniac. Reionization and small-scale ﬂuctuations in the microwave background. ApJ, 322: 597–604, November 1987. doi: 10.1086/165755. Steven Weinberg. A New Light Boson? Physical Review Letters, 40(4):223–226, January 1978. doi: 10.1103/PhysRevLett.40.223. URL http://link.aps.org/doi/10.1103/PhysRevLett. 40.223. M. White, L. Hernquist, and V. Springel. Simulating the Sunyaev-Zeldovich Eﬀect(s): Including Radiative Cooling and Energy Injection by Galactic Winds. ApJ, 579:16–22, November 2002. doi: 10.1086/342756. F. Wilczek. Problem of Strong P and T Invariance in the Presence of Instantons. Physical Review Letters, 40(5):279–282, January 1978. doi: 10.1103/PhysRevLett.40.279. URL http://link. aps.org/doi/10.1103/PhysRevLett.40.279. E. L. Wright, J. C. Mather, D. J. Fixsen, A. Kogut, R. A. Shafer, C. L. Bennett, N. W. Boggess, E. S. Cheng, R. F. Silverberg, G. F. Smoot, and R. Weiss. Interpretation of the COBE FIRAS CMBR spectrum. The Astrophysical Journal, 420:450–456, January 1994. ISSN 0004-637X. doi: 10.1086/173576. URL http://adsabs.harvard.edu/abs/1994ApJ...420..450W. E. L. Wright, X. Chen, N. Odegard, C. L. Bennett, R. S. Hill, G. Hinshaw, N. Jarosik, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, J. L. Weiland, E. Wollack, J. Dunkley, B. Gold, M. Halpern, A. Kogut, D. Larson, M. Limon, S. S. Meyer, and G. S. Tucker. Five-Year Wilkinson Microwave Anisotropy Probe Observations: Source Catalog. Astrophys. J. Supp. , 180:283–295, February 2009. doi: 10.1088/0067-0049/180/2/283. O. Zahn, M. Zaldarriaga, L. Hernquist, and M. McQuinn. The Inﬂuence of Nonuniform Reionization on the CMB. ApJ, 630:657–666, September 2005. doi: 10.1086/431947. O. Zahn, A. Mesinger, M. McQuinn, H. Trac, R. Cen, and L. E. Hernquist. Comparison Of Reionization Models: Radiative Transfer Simulations And Approximate, Semi-Numeric Models. ArXiv e-prints, March 2010. P. Zhang and R. K. Sheth. The Probability Distribution Function of the Sunyaev-Zel’dovich Power Spectrum: An Analytical Approach. ApJ, 671:14–26, December 2007. doi: 10.1086/522913. 96 P. Zhang, U.-L. Pen, and H. Trac. Precision era of the kinetic Sunyaev-Zel’dovich eﬀect: simulations, analytical models and observations and the power to constrain reionization. Mon.Not.Roy.As.Soc. , 347:1224–1233, February 2004. doi: 10.1111/j.1365-2966.2004.07298.x. A. R. Zhitnitskij. On possible suppression of the axion-hadron interactions. 1980. URL http: //inis.iaea.org/Search/search.aspx?orig_q=RN:11563932. 97

1/--страниц