Exploring the Early Universe with the Cosmic Microwave Background

Ruth Durrer

Departement´ de Physique Theorique´ and CAP

PPC - ETHZ, August 2018

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 1 / 27 Contents

1 The CMB

2 Quantum ﬂuctuations from inﬂation

3 Polarisation of the CMB

4 CMB lensing

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 2 / 27 At T > 3000K 0.3eV hydrogen was ionised and the ’cosmic plasma’ of protons, electrons and photons∼ was coupled by Thomson scattering and in thermal equilibrium. At T 3000K protons and electrons combined to neutral hydrogen. The photons' became free and their distribution evolved simply by redshifting of the photon energies to a thermal distribution with T0 = (2.72548 0.00057)K today. ± This corresponds to about 400 photons per cm3 with mean energy of 4 Eγ = kT0 2.3 10− eV 150GHz( λ 0.2cm). This is the observed CMB. ' × ' ' At T > 8300K 0.8eV the Universe was ’radiation dominated’, i.e. its energy density was dominated' by the contribution from these photons (and 3 types of relativistic neutrinos which made up about 35%). Hence initial ﬂuctuations in the energy density of the Universe should be imprinted as ﬂuctuations in the CMB temperature.

The cosmic microwave background (CMB)

The Universe is expanding. In the past it was much denser and hotter.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 3 / 27 At T 3000K protons and electrons combined to neutral hydrogen. The photons' became free and their distribution evolved simply by redshifting of the photon energies to a thermal distribution with T0 = (2.72548 0.00057)K today. ± This corresponds to about 400 photons per cm3 with mean energy of 4 Eγ = kT0 2.3 10− eV 150GHz( λ 0.2cm). This is the observed CMB. ' × ' ' At T > 8300K 0.8eV the Universe was ’radiation dominated’, i.e. its energy density was dominated' by the contribution from these photons (and 3 types of relativistic neutrinos which made up about 35%). Hence initial ﬂuctuations in the energy density of the Universe should be imprinted as ﬂuctuations in the CMB temperature.

The cosmic microwave background (CMB)

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 3 / 27 This corresponds to about 400 photons per cm3 with mean energy of 4 Eγ = kT0 2.3 10− eV 150GHz( λ 0.2cm). This is the observed CMB. ' × ' ' At T > 8300K 0.8eV the Universe was ’radiation dominated’, i.e. its energy density was dominated' by the contribution from these photons (and 3 types of relativistic neutrinos which made up about 35%). Hence initial ﬂuctuations in the energy density of the Universe should be imprinted as ﬂuctuations in the CMB temperature.

The cosmic microwave background (CMB)

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 3 / 27 At T > 8300K 0.8eV the Universe was ’radiation dominated’, i.e. its energy density was dominated' by the contribution from these photons (and 3 types of relativistic neutrinos which made up about 35%). Hence initial ﬂuctuations in the energy density of the Universe should be imprinted as ﬂuctuations in the CMB temperature.

The cosmic microwave background (CMB)

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 3 / 27 The cosmic microwave background (CMB)

The Universe is expanding. In the past it was much denser and hotter. At T > 3000K 0.3eV hydrogen was ionised and the ’cosmic plasma’ of protons, electrons and photons∼ was coupled by Thomson scattering and in thermal equilibrium. At T 3000K protons and electrons combined to neutral hydrogen. The photons' became free and their distribution evolved simply by redshifting of the photon energies to a thermal distribution with T0 = (2.72548 0.00057)K today. ± This corresponds to about 400 photons per cm3 with mean energy of 4 Eγ = kT0 2.3 10− eV 150GHz( λ 0.2cm). This is the observed CMB. ' × ' ' At T > 8300K 0.8eV the Universe was ’radiation dominated’, i.e. its energy density was dominated' by the contribution from these photons (and 3 types of relativistic neutrinos which made up about 35%). Hence initial ﬂuctuations in the energy density of the Universe should be imprinted as ﬂuctuations in the CMB temperature.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 3 / 27 The cosmic microwave background (CMB)

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 4 / 27 Fluctuations in the CMB

T0 = 2.7363K P ∆T (n) = `m a`mY`m(n) 2 C` = a`m , h| | i D` = `(` + 1)C`/(2π)

Planck Collaboration: Cosmological parameters

6000

5000

] 4000 2 K µ [ 3000 TT D 2000 From the Planck Collaboration 1000 2015 0 600 60 300 30 TT 0 0 D

-300 -30 -600 -60 2 10 30 500 1000 1500 2000 2500

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 5 / 27 Fig. 1. The Planck 2015 temperature power spectrum. At multipoles ` 30 we show the maximum likelihood frequency averaged temperature spectrum computed from the Plik cross-half-mission likelihood with foreground and other nuisance parameters determined from the MCMC analysis of the base ⇤CDM cosmology. In the multipole range 2 ` 29, we plot the power spectrum estimates from the Commander component-separation algorithm computed over 94% of the sky. The best-fit base ⇤CDM theoretical spectrum fitted to the Planck TT+lowP likelihood is plotted in the upper panel. Residuals with respect to this model are shown in the lower panel. The error bars show 1 uncertainties. ± sults to the likelihood methodology by developing several in- the full “TT,TE,EE” likelihoods) would di↵er in any substantive dependent analysis pipelines. Some of these are described in way had we chosen to use the CamSpec likelihood in place of Planck Collaboration XI (2015). The most highly developed of Plik. The overall shifts of parameters between the Plik 2015 these are the CamSpec and revised Plik pipelines. For the likelihood and the published 2013 nominal mission parameters 2015 Planck papers, the Plik pipeline was chosen as the base- are summarized in column 7 of Table 1. These shifts are within 2⌧ line. Column 6 of Table 1 lists the cosmological parameters for 0.71 except for the parameters ⌧ and Ase which are sen- base ⇤CDM determined from the Plik cross-half-mission like- sitive to the low multipole polarization likelihood and absolute lihood, together with the lowP likelihood, applied to the 2015 calibration. full-mission data. The sky coverage used in this likelihood is In summary, the Planck 2013 cosmological parameters were identical to that used for the CamSpec 2015F(CHM) likelihood. pulled slightly towards lower H0 and ns by the ` 1800 4-K line However, the two likelihoods di↵er in the modelling of instru- systematic in the 217 217 cross-spectrum, but⇡ the net e↵ect of mental noise, Galactic dust, treatment of relative calibrations and this systematic is relatively⇥ small, leading to shifts of 0.5 or multipole limits applied to each spectrum. less in cosmological parameters. Changes to the low level data As summarized in column 8 of Table 1, the Plik and processing, beams, sky coverage, etc. and likelihood code also produce shifts of typically 0.5 or less. The combined e↵ect of CamSpec parameters agree to within 0.2 , except for ns, which these changes is to introduce parameter shifts relative to PCP13 di↵ers by nearly 0.5 . The di↵erence in ns is perhaps not sur- 2⌧ prising, since this parameter is sensitive to small di↵erences in of less than 0.71 , with the exception of ⌧ and Ase . The main scientific conclusions of PCP13 are therefore consistent with the the foreground modelling. Di↵erences in ns between Plik and CamSpec are systematic and persist throughout the grid of ex- 2015 Planck analysis. tended ⇤CDM models discussed in Sect. 6. We emphasise that Parameters for the base ⇤CDM cosmology derived from the CamSpec and Plik likelihoods have been written indepen- full-mission DetSet, cross-year, or cross-half-mission spectra are dently, though they are based on the same theoretical framework. in extremely good agreement, demonstrating that residual (i.e. None of the conclusions in this paper (including those based on uncorrected) cotemporal systematics are at low levels. This is

8 It is well known that coupling a quantum field to a time dependent external classical field can lead to particle creation. The energy of the modes which can be generated is determined by the kinetic energy in the time dependence of the external field. In an expanding universe, this is determined by the Hubble parameter H(t) which is nearly constant during inflation, 8πG 8πG H2 = φ˙2 + V (φ) V , φ˙2 V (φ) (slow roll) 3 ' 3

Let us consider small ﬂuctuations of the metric in ADM form, 2 2 2 i i j j ds = N dt + hij (dx + N dt)(dx + N dt) − There is one scalar and two tensor degrees of freedom.

Quantum ﬂuctuations from inﬂaton I

Our present understanding is that these fluctuations stem from quantum fluctuations imprinted on the space-time metric during a phase of very rapid expansion called ’inflation’.

We consider a homogeneous and isotropic Universe with vanishing spatial curvature. The metric is 2 2 2 i j ds = dt + a (t)δij dx dx . − a(t) is the scale factor, H(t) = a˙ /a is the Hubble parameter.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 6 / 27 Let us consider small ﬂuctuations of the metric in ADM form, 2 2 2 i i j j ds = N dt + hij (dx + N dt)(dx + N dt) − There is one scalar and two tensor degrees of freedom.

Quantum ﬂuctuations from inﬂaton I

Our present understanding is that these fluctuations stem from quantum fluctuations imprinted on the space-time metric during a phase of very rapid expansion called ’inflation’.

It is well known that coupling a quantum field to a time dependent external classical field can lead to particle creation. The energy of the modes which can be generated is determined by the kinetic energy in the time dependence of the external field. In an expanding universe, this is determined by the Hubble parameter H(t) which is nearly constant during inflation, 8πG 8πG H2 = φ˙2 + V (φ) V , φ˙2 V (φ) (slow roll) 3 ' 3

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 6 / 27 Quantum ﬂuctuations from inﬂaton I

Our present understanding is that these fluctuations stem from quantum fluctuations imprinted on the space-time metric during a phase of very rapid expansion called ’inflation’.

Let us consider small ﬂuctuations of the metric in ADM form, 2 2 2 i i j j ds = N dt + hij (dx + N dt)(dx + N dt) − There is one scalar and two tensor degrees of freedom.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 6 / 27 Quantum ﬂuctuations from inﬂaton II

In comoving gauge

φ(t, x) = φ0(t), 2 2ζ(t,x)ˆ ˆ i i hij (t, x) = a (t)e hij , hij = exp [γij ] γi = ∂i γj = 0

i where γj is (transverse traceless) graviton, and ζ is the comoving curvature perturbation. On a quasi de Sitter background,

φ˙2 H˙ ˙ := 0 = 1 = i 1 2 2 2 , i+1 , ≡ 2H Mpl − H Hi the second order action is Z 2 4 3 ˙2 1 2 1 1 S2 = M d x a ζ (∂ζ) + γ˙ ij γ˙ ij ∂k γij ∂k γij pl − a2 8 − a2

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 7 / 27 Quantum ﬂuctuations from inﬂaton III

Quantizing ζ and γij , one ﬁnds that the coupling of the vacuum ﬁelds to the expansion of the Universe generates the following spectra of scalar and tensor modes out of an initial (Bunch Davis) vacuum state.

3 ζ(k, t)ζ∗(k0, t) = δ(k k0)(2π) Pζ (k) h i −

ns 1 H2 k − Pζ (k) = , ns 1 = 21 2 , (Mukhanov & Chibisov, 1982) 8π2 M2 k? 1 pl − − − n 2H2 k T Pγ (k) = , nT = 21 , (Starobinskii, 1979) π2M2 k? pl − Pγ (k∗) r = = 161 = 8nT Pζ (k∗) − for k aH .

One can show that the long duration of inflation leads to squeezed states so that denoting the canonically normalized field operator by y(k, t) and its canonically conjugate momenta by p(k, t), we obtain p α(k, t)y, where α is a function, so that → [p, y] 0 and the field becomes a classical stochastic field (Kiefer & Polarski, 2009). →

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 8 / 27 After inﬂation, the Universe ’reheats’ into a thermal bath of relativistic particles. In the radiation phase, 1 λ a √t while H− t, the Hubble scale catches∝ ∝ up. ∝ Due to the relativistic pressure, P = ρ/3, scalar perturbations undergo acoustic oscillations, ζ(k) = A(k) cos(cskτ), at ﬁxed wave number k they are all in phase. These are the acoustic peaks we see in the CMB.

Quantum ﬂuctuations from inﬂaton: resume´

The physical wavelength λ (inverse of frequency) of a mode is growing faster than the Hubble scale 1 H− during inﬂation.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 9 / 27 Quantum ﬂuctuations from inﬂaton: resume´

The physical wavelength λ (inverse of frequency) of a mode is growing faster than the Hubble scale 1 H− during inflation. After inflation, the Universe ’reheats’ into a thermal bath of relativistic particles. In the radiation phase, 1 λ a √t while H− t, the Hubble scale catches∝ ∝ up. ∝ Due to the relativistic pressure, P = ρ/3, scalar perturbations undergo acoustic oscillations, ζ(k) = A(k) cos(cskτ), at fixed wave number k they are all in phase. These are the acoustic peaks we see in the CMB.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 9 / 27 Acoustic peaks in the CMB Planck Collaboration: Cosmological parameters

6000

5000

] 4000 2 K µ [ 3000 TT

D 2000

1000

0 600 60 300 30 TT 0 0 D

-300 -30 -600 -60 2 10 30 500 1000 1500 2000 2500

Fig. 1. The Planck 2015 temperature power spectrum. At multipoles ` 30 we show the maximum likelihood frequency averaged Ruthtemperature Durrer (Universit spectrume´ de computed Geneve)` from the Plik cross-half-missionCMB & Early Universe likelihood with foreground and other nuisanceAugust parameters 20, 2018 deter- 10 / 27 mined from the MCMC analysis of the base ⇤CDM cosmology. In the multipole range 2 ` 29, we plot the power spectrum estimates from the Commander component-separation algorithm computed over 94% of the sky. The best-ﬁt base ⇤CDM theoretical spectrum ﬁtted to the Planck TT+lowP likelihood is plotted in the upper panel. Residuals with respect to this model are shown in the lower panel. The error bars show 1 uncertainties. ±

sults to the likelihood methodology by developing several in- the full “TT,TE,EE” likelihoods) would di↵er in any substantive dependent analysis pipelines. Some of these are described in way had we chosen to use the CamSpec likelihood in place of Planck Collaboration XI (2015). The most highly developed of Plik. The overall shifts of parameters between the Plik 2015 these are the CamSpec and revised Plik pipelines. For the likelihood and the published 2013 nominal mission parameters 2015 Planck papers, the Plik pipeline was chosen as the base- are summarized in column 7 of Table 1. These shifts are within 2⌧ line. Column 6 of Table 1 lists the cosmological parameters for 0.71 except for the parameters ⌧ and Ase which are sen- base ⇤CDM determined from the Plik cross-half-mission like- sitive to the low multipole polarization likelihood and absolute lihood, together with the lowP likelihood, applied to the 2015 calibration. full-mission data. The sky coverage used in this likelihood is In summary, the Planck 2013 cosmological parameters were identical to that used for the CamSpec 2015F(CHM) likelihood. pulled slightly towards lower H0 and ns by the ` 1800 4-K line However, the two likelihoods di↵er in the modelling of instru- systematic in the 217 217 cross-spectrum, but⇡ the net e↵ect of mental noise, Galactic dust, treatment of relative calibrations and this systematic is relatively⇥ small, leading to shifts of 0.5 or multipole limits applied to each spectrum. less in cosmological parameters. Changes to the low level data As summarized in column 8 of Table 1, the Plik and processing, beams, sky coverage, etc. and likelihood code also produce shifts of typically 0.5 or less. The combined e↵ect of CamSpec parameters agree to within 0.2 , except for ns, which these changes is to introduce parameter shifts relative to PCP13 di↵ers by nearly 0.5 . The di↵erence in ns is perhaps not sur- 2⌧ prising, since this parameter is sensitive to small di↵erences in of less than 0.71 , with the exception of ⌧ and Ase . The main scientiﬁc conclusions of PCP13 are therefore consistent with the the foreground modelling. Di↵erences in ns between Plik and CamSpec are systematic and persist throughout the grid of ex- 2015 Planck analysis. tended ⇤CDM models discussed in Sect. 6. We emphasise that Parameters for the base ⇤CDM cosmology derived from the CamSpec and Plik likelihoods have been written indepen- full-mission DetSet, cross-year, or cross-half-mission spectra are dently, though they are based on the same theoretical framework. in extremely good agreement, demonstrating that residual (i.e. None of the conclusions in this paper (including those based on uncorrected) cotemporal systematics are at low levels. This is

8 A non inﬂationary toy model

Is inﬂation really needed to generate such a ﬂuctuation spectrum?

To test this we considered a model of rapidly expanding thin spherical shells of energy (rapid explosions) in a radiation dominated universe.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 11 / 27 A non inﬂationary toy model

(from Scodeller, Kunz & RD, ’09) The best fit CMB anisotropies from a causal model of expanding shells (fat black line) compared with the data from WMAP and ACBAR, and to the best fit ΛCDM model (thin green line). Left: the rise to the first peak, ` 200 which can not be fitted satisfactorily ≤ by this model. Right: the spectrum up to ` = 2500. The secondary peaks are well fitted.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 12 / 27 A non inﬂationary toy model

(from Scodeller, Kunz & RD, ’09) The best fit T-E correlation spectrum from a causal model of expanding shells (fat black line) compared with the data from WMAP and with a standard ΛCDM model (thin green line). Left: the first acausal anti-correlation peak, at ` 150 which is absent in the causal model (as predicted by Spergel & Zaldarriaga,' ’97). Right: the spectrum up to ` = 1500. The secondary peaks are very similar to the inflationary case. Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 13 / 27 (from Scodeller, Kunz & RD, ’09) The best fit CMB anisotropies from an a-causal seed model (v1 2, v2 1.5, fat black line) are compared with the data from WMAP and ACBAR and to' a standard' ΛCDM model (thin green line).

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 14 / 27 (from Scodeller, Kunz & RD, ’09) The best ﬁt CMB temperature–polarization cross correlation from an acausal seed model (v1 2, v2 1.5, fat black line) are compared with the data from and to a standard Λ'CDM model' (thin green line).

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 15 / 27 Quantum ﬂuctuations of the graviton

Planck Collaboration: Cosmological parameters

0.25 0.25 Planck TT+lowP+lensing (Ne = 0.39) Planck TT+lowP CDM Planck TT+lowP Planck TT+lowP+BKP N=60 N=60 +lensing+ext +lensing+ext N=50 N=50 0.20 0.20 2D marginalized limits (68% and 95% CL) for the tensor-to-scalar ra- 2 2 tio r, and the scalar spectral in- 0.15 0.15 Convex Convex dex ns, for Planck data. The pre-

0.002 Concave 0.002 Concave dicted values for a chaotic inﬂa- r r tionary model with inﬂaton poten- 0.10 0.10 tial V (φ) φp with 60 e-folds are ∝ shown for p = 2, 1; p > 1 is disfavored at > 80% CL. 0.05 0.05 The Planck Collaboration 2015.

0.00 0.00 0.95 0.96 0.97 0.98 0.99 1.00 0.95 0.96 0.97 0.98 0.99 1.00 ns ns

Fig. 21. Left: Constraints on the tensor-to-scalar ratio r0.002 in the ⇤CDM model, using Planck TT+lowP and Planck + + + + + TT lowP lensing BAO JLA H0 (red and blue, respectively) assumingRuth negligible Durrer (Universit runninge´ de Gen andeve)` the inflationaryCMB consistency & Early Universe rela- August 20, 2018 16 / 27 tion. The result is model-dependent; for example, the grey contours show how the results change if there were additional relativistic degrees of freedom with Ne↵ = 0.39 (disfavoured, but not excluded, by Planck). Dotted lines show loci of approximately con- p stant e-folding number N, assuming simple V (/mPl) single-field inflation. Solid lines show the approximate ns–r relation for quadratic and linear potentials to first order in slow/ roll; red lines show the approximate allowed range assuming 50 < N < 60 and a power-law potential for the duration of inflation. The solid black line (corresponding to a linear potential) separates concave and convex potentials. Right: Equivalent constraints in the ⇤CDM model when adding B-mode polarization results corresponding to the default configuration of the BICEP2/Keck Array+Planck (BKP) likelihood. These exclude the quadratic potential at a higher level of significance compared to the Planck-alone constraints. limited by cosmic variance of the dominant scalar anisotropies, from tensor modes. The constraints shown by the blue contours and it is also model dependent. In polarization, in addition to B- in Fig. 21, which add Planck lensing, BAO, and other astrophys- modes, the EE and TE spectra also contain a signal from tensor ical data, are therefore tighter in the ns direction and shifted to modes coming from reionization and last scattering. However, slightly higher values, but marginally weaker in the r-direction. in this release the addition of Planck polarization constraints at The 95 % limits on r0.002 are ` 30 do not significantly change the results from temperature and low-` polarization (see Table 5). r0.002 < 0.10, Planck TT+lowP, (39a) Figure 21 shows the 2015 Planck constraint in the ns–r plane, r0.002 < 0.11, Planck TT+lowP+lensing+ext, (39b) adding r as a one-parameter extension to base ⇤CDM. Note that for base ⇤CDM (r = 0), the value of ns is consistent with the results reported in PCP13. Note that we assume the second-order slow-roll consistency relation for the ten- n = 0.9655 0.0062, Planck TT+lowP. (38) sor spectral index. The result in Eqs. (39a) and (39b) are mildly s ± scale dependent, with equivalent limits on r0.05 being weaker by We highlight this number here since ns, a key parameter for in- about 5 %. flationary cosmology, shows one of the largest shifts of any pa- PCP13 noted a mismatch between the best-fit base ⇤CDM rameter in base ⇤CDM between the Planck 2013 and Planck model and the temperature power spectrum at multipoles ` < 40, 2015 analyses (about 0.7 ). As explained in Sect. 3.1, part of partly driven by the dip in the multipole range 20 < ` < 30.⇠ If this shift was caused by the ` 1800 systematic in the nominal- this mismatch is simply a statistical fluctuation of⇠ the ⇠⇤CDM mission 217 217 spectrum used⇡ in PCP13. model (and there is no compelling evidence to think otherwise), The red contours⇥ in Fig. 21 show the constraints from Planck the strong Planck limit (compared to forecasts) is the result of TT+lowP. These are similar to the constraints shown in Fig. 23 chance low levels of scalar mode confusion. On the other hand if of PCP13, but with ns shifted to slightly higher values. The ad- the dip represents a failure of the ⇤CDM model, the 95 % limits dition of BAO or the Planck lensing data to Planck TT+lowP of Eqs. (39a) and (39b) may be underestimates. These issues are 2 lowers the value of ⌦ch , which at fixed ✓ increases the small- considered at greater length in Planck Collaboration XX (2015) scale CMB power. To maintain the fit to⇤ the Planck tempera- and will not be discussed further in this paper. ture power spectrum for models with r = 0, these parameter As mentioned above, the Planck temperature constraints on shifts are compensated by a change in amplitude As and the tilt r are model-dependent and extensions to ⇤CDM can give sig- ns (by about 0.4 ). The increase in ns to match the observed nificantly di↵erent results. For example, extra relativistic de- power on small scales leads to a decrease in the scalar power grees of freedom increase the small-scale damping of the CMB on large scales, allowing room for a slightly larger contribution anisotropies at a fixed angular scale, which can be compensated

34 Polarisation of the CMB

Thomson scattering depends on polarisation. A local quadrupole induces linear polarisation, Q = 0 and U = 0, V = 0. 6 6

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 17 / 27 Polarisation of the CMB

Left: E-polarisation is pure gradient or pure curl, (n) = φ(n) or (n) = ψ(n) ∇ ∇ ∧ Right: B-polarisation is a mixture , (n) = φ(n) + ψ(n). ∇ ∇ ∧ A rotation by π/4 turns pure E-modes into pure B-modes and vice versa. Scalar perturbations only generate E-polarisation.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 18 / 27 Planck Collaboration: Cosmological parameters The CMBPlanck polarization Collaboration: Cosmological parameters spectra 140 140 ] 70 2 K ]

µ 70 2 [ 0 K µ [ TE ` 0

D -70 TE ` T-E correlation

D -70 -140 TE `(`+1) TE ` = C` -14010 D 2π TE ` 100 D TE ` -100 D

-10 30 500 1000 1500 2000 30 500 1000` 1500 2000 `

100 (The Planck Collaboration 2015) ] 2 10080 K ] µ 2 80 5 K 60 µ

5 60 [10 40

[10 40 EE 20

C E-E spectrum EE 200 C 04 EE 40 C EE -40 C

-4 30 500 1000 1500 2000 Ruth30 Durrer (Universit500e´ de Geneve)` 1000 1500 CMB2000 & Early Universe August 20, 2018 19 / 27 Fig. 3. Frequency-averaged TE and EE spectra (without fitting for T-P leakage). The theoretical TE and EE spectra plotted in the upper panel of each plot are computed from the Planck TT+lowP best-fit model of Fig. 1. Residuals with respect to this theoretical modelFig. 3. areFrequency-averaged shown in the lowerTE paneland EE in eachspectra plot. (without The error fitting bars for showT-P leakage).1 errors. The The theoretical green linesTE and in theEE lowerspectra panels plotted show in the the best-fitupper panel temperature-to-polarization of each plot are computed leakage from model the Planck of Eqs.TT (+11alowP) and best-fit (11b±), model fitted of separately Fig. 1. Residuals to the TE withand respectEE spectra. to this theoretical model are shown in the lower panel in each plot. The error bars show 1 errors. The green lines in the lower panels show the best-fit temperature-to-polarization leakage model of Eqs. (11a) and (11b±), fitted separately to the TE and EE spectra.

13 ACTPol spectrum and parameters 13

0.005

0.000 C / -0.005 TT C

-0.010 theory -0.015 simulation 0.020

0.010 C /

EE 0.000 C

-0.010

-0.020 2 1000 2000 3000 4000 Multipole Fig. 14.— E↵ect of aberration on the TT and EE CMB power 2 The Bspectra polarisation due to our proper motion spectrum with respect to the CMB. Our Fig. 13.— Distribution of the for the null tests described in aberrated simulations agree with the analytical estimate of the Section 3.7. The smooth line represents the expected distribution expected e↵ect. if the null tests were uncorrelated. The dashed black histogram shows our null test distribution after rescaling the errors by 3%. 3 We interpret this as an estimate of the uncertainty on our errors. 10 ACTPol 2 POLARBEAR The distribution for this set of null tests is shown in SPTpol Figure 13. The distribution is close to expectation, but BICEP2/Keck 2

we ﬁnd that the measured and predicted distribution ] 2

ﬁt best if we reduce the error bars by 3%. We interpret K this as an estimate of the uncertainty⇡ on our errors. µ

)[ Atacama telescope array

4 10 (2

3.8. E↵ect of aberration / ACTPOL Collaboration, 2017 BB The observed power spectra are a↵ected by aberration C due to our proper motion with respect to the CMB last scattering surface. We move at a speed of 369 km/s along the direction d =(l, b) = (264, 48) (e.g. Planck Collab- oration et al. (2014b)). This motion induces a kinematic 5 dipole of the form cos ✓ =(d n), where n is the vector 10 position of each pixel. Aberration· results in an angle- 0 500 1000 1500 2000 2500 11 dependent rescaling of the multipole moments ` and its Multipole e↵ect on the power spectrum can be approximated as Fig. 15.— Unblinded ACTPol BB power spectra compared to measurements from POLARBEAR (The Polarbear Collaboration: 7 13 1 1 C` d ln C` P. A. R. Ade et al. 2014), SPTpol (Keisler et al. 2015) and BI- BK+P = cos ✓ (2) CEP2/Keck array (BICEP2 Collaboration et al. 2016). The solid B+P 6

C` d ln ` h i ] 1 1 line is the Planck best ﬁt ⇤CDM model. The ACTPol data are K+P 2 BKxBK 0.8 0.8 K 0.05consistent with expectation and deviate from zero at 2. µ (Jeong et al. 2014), where = v/c and cos ✓ = 0.82 5 0.8 0.8 h i (BKxBK−αBKxP)/(1−α) in D56, 0.97 in D5 and 0.65 in D6, where the aver- 0.04et al. 2016). We ﬁt for an amplitude in the multipole 0.6 0.6 ] age is taken over the solid angle of each ACTPol patch.2 0.6 0.6 4 K

µ range 500 < ` < 2500, where Galactic and extragalac- [ 0.03 peak Bicep2-Keck-Planckpeak We generate0.4 a set of 120 aberrated0.4 simulations, compute π

/2 tic contamination is minimal, using the lensed B mode l their power spectra and compare it to the power spec- L/L L/L 3 0.2 0.2 0.02⇤CDM prediction. We ﬁnd A =2.03 1.01.0.4 This ampli- BICEP20.4 and Planck Col- tra of non-aberrated maps. The result is presented in tude is consistent with expectation, but± the signiﬁcance

Figure 140 together with the analytical0 estimate. We use BB l(l+1)C 0.01 2 @ l=80 & 353 GHz [

0 0.2 0.4 0 5 10 15 of the fit is not high enough to be interpreted as a detec- laborations, 2016 d 2 this set of simulationsr to correctA our @ l=80 power & 353GHz spectra [µK ] for 0.2 0.2 A d 0tion. the aberration e↵ect, such that Cˆ = C C . In earlier 1 ` ` ` releasesFIG. the 10. e↵ect Likelihoods was negligible for r and andAd,usingBICEP2/ we did notKeck correct −0.01 4. LIKELIHOOD and Planck,asplottedinFig.6, overplotted on constraints 0 50 100 150 200 250 300 0 0 0 for it. Section 5.2 discusses the impact of this correction Multipole 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 obtained from realizations of a lensed-⇤CDM+noise+dust We first construct a likelihood function to describe the 2 A @ l=80 & 353 GHz [µK ] on cosmologicalmodel with dustparameters. power similar to that favored by the real CMB and foreground emission present in the 149 GHz r d r 2 data (Ad =3.6 µK ). Half of the r curves peak at zero as power spectrum. To improve the estimation of the CMB expected. 3.9. Unblinded BB spectra part,1 we then add intensity power spectra estimated at Fiducial analysis FIG. 6. Likelihood results from a basic lensed-⇤CDM+r+dust model, fitting BB auto- and cross-spectra taken between maps We unblind the B mode power spectrum at the end of both 150r and< 2200.11 GHzCleaning at by analysis 90% the previous confidence ACT receiver, (in the minimal model with tensors). the analysis. The spectrum is shown in Figure 15 along MBAC.0.8 at 150GHz, 217, and 353GHz. The 217 and 353GHz maps come from Planck. The primary results (heavy black) use the with B mode measurements from The Polarbear Collab-Ruth DurrerUsing (Universit thesee´ multi-frequency de Geneve)` data150 GHz we estimate combinedCMB the & Early maps Universe from BICEP2/Keck. Alternate curvesAugust 20, (light 2018 blue 20 and / 27 red) show how the results vary when the oration:1 P. A. R. Ade et al. (2014),1 SPTpol (Keisler et al. foreground-marginalized0.6 CMB power spectrum in TT,

peak BICEP2 and Keck Array only maps are used. In all cases a Gaussian prior is placed on the dust frequency spectrum parameter

2015) and BICEP2/Keck array (BICEP2 Collaboration TE,L/L EE for ACT, for both the MBAC and ACTPol 0.8 0.8 0.4 d =1.59 0.11. In the right panel the two dimensional contours enclose 68% and 95% of the total likelihood. 0.6 0.6 ± 0.2 0.4 0.4

0.2 0.2 0 0 0.1 0.2 0.3 variation at high latitude, as explained in Sec. VA. though it does cause any given realization to shift r 0 0 0 0.2 0.4 0 5 10 15 Nevertheless, it is important to appreciate that the slightly—the change for the real data case is shown 2 r A @ l=80 & 353GHz [µK ] d FIG. 12. Upper: BB spectrum of the BICEP2/Keck maps r constraint curves shown in Fig. 6 shift left (right) in Fig. 7. before and after subtraction of the dust contribution, esti- FIG. 11. Constraints obtained when adding dust realizations mated from the cross-spectrum with Planck 353 GHz. The when assuming a lower (higher) value of d. For from the Planck Sky Model version 1.7.8 to the base lensed- error bars are the standard deviations of simulations, which, ⇤CDM+noise simulations. (Curves for 139 regions with peak in the latter case, have been scaled and combined in the same d =1.3 0.11 the peak is at r =0.021 and for Adding synchrotron: BK-I took the WMAP K - 2 way. The inner error bars are from lensed-⇤CDM+noise sim- ± Ad < 20 µK are plotted). We see that the results for r =1.9 0.11 the peak is at r =0.073. • are unbiased in the presence of dust realizations which do ulations as in the previous plots, while the outer error bars d band (23 GHz) map, extrapolated it to 150 GHz ac- 0.42 ± not necessarily follow the ` power law or have Gaussian are from the lensed-⇤CDM+noise+dust simulations. Lower: 3.3 fluctuations about it. constraint on r derived from the cleaned spectrum compared cording to ⌫ (mean value within the BICEP2 to the fiducial analysis shown in Fig. 6. Varying the dust power spectrum shape: in • the fiducial analysis the dust spatial power spec- field of the MCMC “Model f” spectral index map provided by WMAP [2]), and found a negligible B. Subtraction of scaled spectra trum is assumed to be a power law with ` 0.42 D / predicted contribution (rsync,150 =0.0008 0.0041). ` . Marginalizing over spectral indices in the as the level of Ad increases, and we should therefore not Figure 3 does not o↵er strong motivation± to reex- be surprised if the fraction of realizations peaking at a As previously mentioned, the modified blackbody range 0.8 to 0 we find little change in the r con- value higher than the real data is increased compared to model predicts that dust emission is 4% as bright in the 2 amine this finding—the only significant detections the simulations with mean Ad =3.6 µK . However we BICEP2 band as it is in the Planck 353 GHz band. There- straint (see also Sec. IV B for an alternate relax- still expect that on average 50% will peak above zero and fore, taking the auto- and cross-spectra of the combined of correlated BB power are in the BK150 P353 approximately 8% will have an L /L ratio less than BICEP2/Keck maps and the Planck 353 GHz maps, as ation of the assumptions regarding the spatial prop- ⇥ 0 peak and, to a lesser extent, BK150 P217 spectra. How- the 0.38 observed in the real data. In fact we find 57% shown in the bottom row of Fig. 2, and evaluating erties of the dust pattern). (BK BK ↵BK P)/(1 ↵), at ↵ = ↵ cleans out the ⇥ and 7%, respectively, consistent with the expected val- ⇥ ⇥ fid ever, here we proceed to a fit including all the ues. There is one realization which has a nominal (false) dust contribution (where ↵fid =0.04). The upper panel detection of non-zero r of 3.3, although this turns out to of Fig. 12 shows the result. Using Gaussian determinant likelihood: the polarized bands of Planck (as shown in Fig. 3) also have one of the lowest L0/Lpeak ratios in the Gaus- As an alternative to the full likelihood analysis pre-• and adding a synchrotron component to the base sian simulations shown in Fig. 10 (with which it shares sented in Sec. III B, we can instead work with the dif- fiducial analysis uses the HL likelihood approx- the CMB and noise components), so this is apparently ferenced spectra from above, a method we denote the imation, as described in Sec. III A. An alterna- lensed-⇤CDM+noise+r+dust model. We take syn- just a relatively unlikely fluctuation. “cleaning” approach. If ↵fid were the true value, the ex- tive is to recompute the covariance matrix C at chrotron to have a power law spectrum ` 0.6 D / each point in parameter space and take L = ` [23], with free amplitude Async,whereAsync is 1/2 T 1 det (C) exp ( (d C d)/2), where d is the de- the amplitude at ` = 80 and at 150 GHz, and scal- 3.3 viation of the observed bandpowers from the model ing with frequency according to ⌫ .Insucha expectation values. This results in an r constraint scenario we can vary the degree of correlation that which peaks slightly lower, as shown in Fig. 7.Run- is assumed between the dust and synchrotron sky ning both methods on the simulated realizations patterns. Figure 8 shows results for the uncorre- described in Sec. IV A, indicates that such a dif- lated and fully correlated cases. Marginalizing over 2 ference is not unexpected and that there may be r and Ad we find Async < 0.0003 µK at 95% con- a small systematic downward bias in the Gaussian fidence for the uncorrelated case, and many times determinant method. smaller for the correlated. This last is because once one has a detection of dust it e↵ectively becomes Varying the HL fiducial model: as mentioned a template for the synchrotron. This synchrotron • in Sec. III A the HL likelihood formulation requires limit is driven by the Planck 30 GHz band—we ob- that the expectation values and bandpower co- tain almost identical results when adding only this variance matrix be provided for a single “fiducial band, and a much softer limit when not including it. 3.0 model” (not to be confused with the “fiducial anal- If we instead assume synchrotron scaling of ⌫ ysis” of Sec. III B). Normally we use the lensed- the limit on Async is approximately doubled for the ⇤CDM+dust simulations described in Sec. IV A be- uncorrelated case and reduced for the correlated. low. Switching this to lensed-⇤CDM+r=0.2 pro- (Because the DS1 DS2 data-split is not available duces no change on average in the simulations, al- for the Planck LFI⇥ bands we switch to Y1 Y2 for ⇥ CMB lensing induces B-polarisation

The presence of foreground structure deﬂects photon geodesics from a straight line.

x x + δx 7→ n = (θ1, θ2) n + α 7→ The Jacobian of this lens map is given by

Aab = δab + bαa ∇ 1 κ γ1 γ2 + ω = − − − γ2 ω 1 κ + γ1 − − − κ describes magniﬁcation, γ is the shear and ω a rotation. (Figure by M. Vanvasselaer) In a ’quasi Newtonian’ situation the deﬂection angle is the gradient of the lensing potential,

Z λ∗ λ λ ψ(n, t ) = 2 dλ ∗ − Φ(λn, t0 λ) Aab = δab + b aψ ω = 0 . ∗ − 0 λ λ − ∇ ∇ ⇒ ∗

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 21 / 27 The lensing spectrum

Z r∗ (r r) ψ(n) = 2 dr ∗ − Φ(rn, t0 r) − 0 r r − ∗

Planck Collaboration: Gravitational lensing by large-scale structures with Planck 2 Planck (2015) SPT

Planck Collaboration: Gravitational lensing by large-scale structures with Planck ]

7 Planck (2013) ACT 1.5 10

Planck at the expected level. In Sect. 3.3, we cross-correlate the [

reconstructed lensing potential with the large-angle temperature T 2 1 / anisotropies to measure the CL correlation sourced by the ISW L

e↵ect. Finally, the power spectrum of the lensing potential is pre- C sented in Sect. 3.4. We use the associated likelihood alone, and 2 0.5 in combination with that constructed from the Planck temper- 1)] +

ature and polarization power spectra (Planck Collaboration XI L ( 0 L

2015), to constrain cosmological parameters in Sect. 3.5. [

0.5 3.1. Lensing potential 1 10 100 500 1000 2000 WF In Fig. 2 we plot the Wiener-filtered minimum-variance lensing ˆ (Data) L estimate, given by Fig. 6 Planck 2015 full-mission MV lensing potential power spectrum measurement, as well as earlier measurements using the SMICA Planck 2013 nominal-mission temperature data (Planck Collaboration XVII 2014), the South Pole Telescope (SPT, van Engelen Fig. 2 Lensing potential estimated from the full-mission et al. 2012), and the Atacama Cosmology Telescope (ACT, Das et al. 2014). The fiducial ⇤CDM theory power spectrum based on C, fid CMB maps using the MV estimator. The power spectrum of the parameters given in Sect. 2 is plotted as the black solid line. ˆWF L ˆMV LM = LM , (5) this map forms the basis of our lensing likelihood.(Planck The estimate Collaboration 2015) , fid + CL NL has been Wiener filtered following Eq. (5), and band-limited to In addition to the priors above, we adopt the same sampling Appendix E; see also Pan et al. (2014). Here, we aim to use priors and methodology as Planck Collaboration XIII (2015),† simple physical arguments to understand the parameter degen- , fid 8 L 2048. using CosmoMC and camb for sampling and theoretical predic- eracies of the lensing-only constraints. In the flat ⇤CDM model, where C is the lensing potential power spectrum in our fidu-   tions (Lewis & Bridle 2002; Lewis et al. 2000). In the ⇤CDM the bulk of the lensing signal comes from high redshift (z > 0.5) L 2 2 model, as well as ⌦bh and ns, we sample As, ⌦ch , and the where the Universe is mostly matter-dominated (so potentials are cial model and NL is the noise power spectrum of the recon- (approximate) acoustic-scale parameter ✓MC. Alternatively, we nearly constant), and from lenses that are still nearly linear. For can think of our lensing-only results as constraining the sub- fixed CMB (monopole) temperature, baryon density, and ns, in struction. As we shall discuss in Sect. 4.5, the lensing potential space of ⌦m, H0, and 8. Figure 7 shows the corresponding the ⇤CDM model the broad shape of the matter power spectrum 2 estimate is unstable for L < 8, and so we have excluded those constraints from CMB lensing, along with tighter constraints is determined mostly by one parameter, keq aeqHeq ⌦mh . Ruth Durrer (Universite´ de Geneve)` CMB & Earlyfrom combining Universe with additional external baryon acoustic oscil- The matter powerAugust spectrum 20, also 2018 scales with the⌘ 22 primordial / 27/ am- modes for all analyses in this paper, as well as in the MV lensing lation (BAO) data, compared to the constraints from the Planck plitude As; keeping As fixed, but increasing keq, means that the CMB power spectra. The contours overlap in a region of accept- entire spectrum shifts sideways so that lenses of the same typ- map. able Hubble constant values, and hence are compatible. To show ical potential depth lens become smaller. Theoretical ⇤CDM the multi-dimensional overlap region more clearly, the red con- models that keep èq keq fixed will therefore have the same As a visual illustration of the signal-to-noise level in the lens- ⌘ ⇤ tours show the lensing constraint when restricted to a reduced- number (proportional to keq ) of lenses of each depth along ⇤ ing potential estimate, in Fig. 3 we plot a simulation of the MV dimensionality space with ✓MC fixed to the value accurately mea- the line of sight, and distant lenses of the same depth will also sured by the CMB power spectra; the intersection of the red and maintain the same angular correlation on the sky, so that the reconstruction, as well as the input realization used. The re- black contours gives a clearer visual indication of the consis- shape of the spectrum remains roughly constant. There is there- tency region in the ⌦ – plane. fore a shape and amplitude degeneracy where ` constant, construction and input are clearly correlated, although the recon- m 8 eq ⇡ The lensing-only constraint defines a band in the ⌦m–8 As constant, up to corrections from sub-dominant changes in struction has considerable additional power due to noise. As can plane, with the well-constrained direction corresponding ap- the⇡ detailed lensing geometry, changes from late-time potential proximately to the constraint decay once dark energy becomes important, and nonlinear ef- WF be seen in Fig. 1, even the MV reconstruction only has S/N 1 ˆ (Sim.) 0.25 fects. In terms of standard ⇤CDM parameters around the best-fit 8⌦m = 0.591 0.021 (lensing only; 68 %). (13) model, ` ⌦0.6h, with the power-law dependence on ⌦ only ⇡ ± eq / m m for a few modes around L 50. varying slowly with ⌦ ; the constraint ` ⌦0.6h = constant ⇡ This parameter combination is measured with approximately m eq / m The MV lensing estimate in Fig. 2 forms the basis for a 3.5% precision. defines the main dependence of H0 on ⌦m seen in Fig. 7. The dependence of the lensing potential power spectrum on public lensing map that we provide to the community (Planck the parameters of the ⇤CDM model is discussed in detail in The argument above for the parameter dependence of the lensing power spectrum ignores the e↵ect of baryon suppres- Collaboration I 2015). The raw lensing potential estimate has a † For example, we split the neutrino component into approximately sion on the small-scale amplitude of the matter power spectrum very red power spectrum, with most of its power on large angular two massless neutrinos and one with m⌫ = 0.06 eV, by default. (e.g., Eisenstein & Hu 1998). As discussed in Appendix E, this P scales. This can cause leakage issues when cutting the map (for 8 example to cross-correlate with an additional mass tracer over a small portion of the sky). The lensing convergence  defined by L(L + 1)  = , (6) LM 2 LM has a much whiter power spectrum, particularly on large angular Input (Sim.) scales. The reconstruction noise on  is approximately white as well (Bucher et al. 2012). For this reason, we provide a map Fig. 3 Simulation of a Wiener-filtered MV lensing reconstruc- of the estimated lensing convergence  rather than the lensing tion (upper) and the input realization (lower), filtered in the potential . same way as the MV lensing estimate. The reconstruction and input are clearly correlated, although the reconstruction has considerable additional power due to noise. 3.2. Lensing B-mode power spectrum The odd-parity B-mode component of the CMB polarization is of great importance for early-universe cosmology. At first order in perturbation theory it is not sourced by the scalar fluctuations that dominate the temperature and polarization anisotropies, and so the observation of primordial B-modes can be used as a uniquely powerful probe of tensor (gravitational wave) or vec- analysis gives no statistically-significant evidence for primor- tor perturbations in the early Universe. A detection of B-mode dial gravitational waves, and establishes a 95 % upper limit fluctuations on degree angular scales, where the signal from r0.05 < 0.12. This still represents an important milestone for gravitational waves is expected to peak, has recently been re- B-mode measurements, since the direct constraint from the B- ported at 150 GHz by the BICEP2 collaboration (Ade et al. mode power spectrum is now as constraining as indirect, and 2014). Following the joint analysis of BICEP2 and Keck Array model-dependent, constraints from the TT spectrum (Planck data (also at 150 GHz) and the Planck polarization data, primar- Collaboration XIII 2015). ily at 353 GHz (BICEP2/Keck Array and Planck Collaborations In addition to primordial sources, the e↵ect of gravitational 2015), it is now understood that the B-mode signal detected lensing also generates B-mode polarization. The displacement of by BICEP2 is dominated by Galactic dust emission. The joint lensing mixes E-mode polarization into B-mode as (Smith et al.

4 Planck Collaboration: Cosmological parameters

2000 1.2 0.000 0.000 1.0 1500 ] ] 0.8 2 2 K -0.005 K -0.005 µ µ [ [ 1000 0.6 EE TT D D 0.4

500 -0.010 -0.010 0.2

0 0.0 10 20 30 40 50 60 50 100 150 200 Cosmological parameters

Fig. 25. Power spectra drawn from the Planck TT+lowP posterior for the correlated matter isocurvature model, colour-coded by the value of the isocurvature amplitude parameter ↵, compared to the Planck data points. The left-hand figure shows how the negatively- correlated modes lower the large-scale temperature spectrum, slightly improving the fit at low multipoles. Including polarization, the negatively-correlated modes are ruled out, as illustrated at the first acoustic peak in EE on the right-hand plot. Data points at ` < 30 are not shown for polarization, as they are included with both the default temperature and polarization likelihood combinations.

The “geometric degeneracy” (Bond et al. 1997; +TE+EE 0.75 68 Zaldarriaga et al. 1997) allows for the small-scale linear +lensing CMB spectrum to remain almost unchanged if changes in ⌦K +lensing+BAO 64 are compensated by changes in H0 to obtain the same angular diameter distance to last scattering. The Planck constraint is 60 0.60 therefore mainly determined by the (wide) priors on H0, and the e↵ect of lensing smoothing on the power spectra. As discussed

56 H

0 in Sect. 5.1, the Planck temperature power spectra show a slight 52 preference for more lensing than expected in the base ⇤CDM 0.45 cosmology, and since positive curvature increases the amplitude 48 of the lensing signal, this preference also drives ⌦K towards negative values. 44 Taken at face value, Eq. (47) represents a detection of posi- 0.30 40 tive curvature at just over 2 , largely via the impact of lensing on the power spectra. One might wonder whether this is mainly 0.30 0.45 0.60 0.75 a parameter volume e↵ect, but that is not the case, since the best 2 fit closed model has 6 relative to base ⇤CDM, and the m fit is improved over almost⇡ all the posterior volume, with the mean chi-squared improving by 2 5 (very similar to the (Planck Collaboration, 2015)(Hutererh eti⇡ al., 2017) Fig. 26. Constraints in the ⌦m–⌦⇤ plane from the Planck phenomenological case of ⇤CDM+AL). Addition of the Planck TT+lowP data (samples; colour-coded by the value of H0) and polarization spectra shifts ⌦K towards zero by ⌦K 0.015: Planck TT,TE,EE+lowP (solid contours). The geometric degen- ⇡ +0.038 eracy between ⌦m and ⌦⇤ is partially broken because of the ef- ⌦K = 0.040 0.041 (95%, Planck TT,TE,EE+lowP), (48) fect of lensing on the temperature and polarization power spectra. These limits are improved significantly by the inclusion but ⌦K remains negative at just over 2 . of the Planck lensing reconstruction (blue contours) and BAO However the lensing reconstruction from Planck measures (solid red contours). The red contours tightly constrain the ge- the lensing amplitude directly and, as discussed in Sect. 5.1, this ometry of our Universe to be nearly flat. does not prefer more lensing than base ⇤CDM. The combined Ruth Durrer (Universite´ de Geneve)` CMB & Earlyconstraint Universe shows impressive consistency withAugust a flat 20, universe: 2018 23 / 27

+0.016 more speculatively, there has been interest recently in “multi- ⌦K = 0.005 0.017 (95%, Planck TT+lowP+lensing). (49) verse” models, in which topologically-open “pocket universes” form by bubble nucleation (e.g., Coleman & De Luccia 1980; The dramatic improvement in the error bar is another illustration Gott 1982) between di↵erent vacua of a “string landscape” (e.g., of the power of the lensing reconstruction from Planck. Freivogel et al. 2006; Bousso et al. 2013). Clearly, the detection The constraint can be sharpened further by adding external of a significant deviation from ⌦K = 0 would have profound data that break the main geometric degeneracy. Combining the consequences for inflation theory and fundamental physics. Planck data with BAO, we find The Planck power spectra give the constraint ⌦K = 0.000 0.005 (95%, Planck TT+lowP+lensing+BAO). +0.049 ± ⌦K = 0.052 0.055 (95%, Planck TT+lowP). (47) (50)

38 Second order lensing and B-polarisation

��

�� ℬ ( � ) ℓ 

/ � �� ℬ ℓ

Δ � ��

��

� �� ℓ (Figure from Marozzi, Fanizza, Di Dio, RD, 2016).

Different contributions to the higher order lensing B-mode spectrum. The contribution from the rotation ω is green, the sum in black.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 24 / 27 Why should we care

Not correctly subtracted B-modes from lensing (incorrect ’delensing’) can mimic primordial B-modes from inﬂation.

��-� ] �

��-� /( � π )[ μ � ℬ ℓ ��-� ℓ ( + � ) ��-�

� �� ℓ (Figure from Marozzi, Fanizza, Di Dio, RD, 2016).

Blue: ﬁrst order lensing B-modes. Red: B-modes from higher order lensing (rotation). 2 3 4 5 6 Grey-tones: primordial B-modes from inﬂation, r = (10− , 10− , 10− , 10− , 10− ).

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 25 / 27 If we define inflation as a cosmic era where correlations on scales larger than the present Hubble scale were generated, then we may nearly say the fluctuations of the CMB are a ’proof’ that inflation happened. Measurements of primordial B-polarisation would allow us to determine the energy scale of inflation. It is very important to correctly take into account the effect of lensing beyond leading order. In order to distinguish from lensing-induced and primordial B-modes. Thank You

Conclusions

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 26 / 27 Measurements of primordial B-polarisation would allow us to determine the energy scale of inﬂation. It is very important to correctly take into account the effect of lensing beyond leading order. In order to distinguish from lensing-induced and primordial B-modes. Thank You

Conclusions

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 26 / 27 It is very important to correctly take into account the effect of lensing beyond leading order. In order to distinguish from lensing-induced and primordial B-modes. Thank You

Conclusions

The CMB fluctuations are the most precious cosmological dataset. They not only let us determine cosmological parameters to % accuracy but they also contain information about the very early Universe: inflation. If we define inflation as a cosmic era where correlations on scales larger than the present Hubble scale were generated, then we may nearly say the fluctuations of the CMB are a ’proof’ that inflation happened. Measurements of primordial B-polarisation would allow us to determine the energy scale of inflation.

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 26 / 27 Thank You

Conclusions

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 26 / 27 Conclusions

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 26 / 27 Second order lensing and B-polarisation

�

��

ℬ ℓ �� / σ ℬ ℓ

Δ � ��

��-�

��-� � �� ℓ (ﬁgure from Marozzi, Fanizza, Di Dio, RD, 2017)

Ruth Durrer (Universite´ de Geneve)` CMB & Early Universe August 20, 2018 27 / 27