Did m 2 φ 2 bite the dust? A closer look at ns and r Raphael Flauger Cosmology with the CMB and its Polarization, University of Minnesota, January 15, 2015
The Cosmic Microwave Background COBE EBEX WMAP SPTPol POLARBEAR BICEP2 ACTPol ABS Planck SPIDER
The Cosmic Microwave Background Observations of the CMB have taught us that the primordial perturbations existed before the hot big bang are nearly scale invariant are very close to Gaussian are adiabatic What generated them?
Generating Primordial Perturbations The system of equations describing the early universe contains two important scales k/a and H. H k/a To generate the perturbations causally, they cannot have been outside the horizon very early on, requiring a phase with d dt k a H < 0 a rec (inflation or bounce) a
Inflation The simplest system leading to a phase of inflation (that ends) is S = 1 16πG d 4 x gr d 4 x g 1 2 gµν µ φ ν φ + 1 2 m2 φ 2 If the scalar field is nearly homogeneous, and at a position in field space such that the potential energy dominates its energy density, this leads to nearly exponential expansion.
Inflation The perturbations are generated as quantum fluctuations deep inside the horizon, and eventually exit the horizon. H k/a Outside the horizon, a quantity is conserved. R a rec a This sets the initial conditions for the equations describing the universe from a few kev to the present. We observe the density perturbations in the plasma at recombination that were seeded by the inflationary perturbations.
Inflation For standard single field slow-roll inflation, the primordial spectrum of scalar perturbations is 2 R(k) = H2 (t k ) 8π 2 (t k ) 2 R k k ns 1 with and n s = 1 4 2δ = Ḣ H 2 δ = Ḧ 2HḢ in agreement with observations.
Inflation Assuming inflation took place, what can we learn about it beyond and? n s 2 R What is the energy scale of inflation? How far did the field travel? Are there additional light degrees of freedom? What is the propagation speed of the inflaton quanta? tensor modes non-gaussianity
Energy Scale of Inflation In addition to the scalar modes, inflation also predicts a nearly scale invariant spectrum of gravitational waves 2 h(k) = 2H2 (t k ) π 2 A measurement of the tensor contribution would provide a direct measurement of the expansion rate of the universe during inflation, as well as the energy scale V 1/4 inf = 1.06 10 16 GeV r 1/4 0.01
The Field Range For r>0.01 the inflaton must have moved over a super-planckian distance in field space. (Lyth, Turner) Motion of the scalar field over super-planckian distances is hard to control in an effective field theory V (φ) =V 0 + 1 2 m2 φ 2 + 1 3 µφ3 + 1 4 λφ4 + φ 4 n=1 c n (φ/λ) n (Λ <M p )
Possible Solution: The Field Range Use a field with a shift symmetry and break the shift symmetry in a controlled way. e.g. Linde s chaotic inflation with V (φ) = 1 2 m2 φ 2 with m M p natural inflation Freese, Frieman, Olinto, PRL 65 (1990) V (φ) =Λ 4 1 + cos φ f with f M p
The Field Range In field theory we may simply postulate such a symmetry, but it is far from obvious that such shift symmetries exist in a theory of quantum gravity. In fact, the most naive implementation of an axion with f M p seems hard to realize string theory. Banks, Dine, Fox, Gorbatov hep-th/0303252 (However, see Kim, Nilles, Peloso hep-ph/0409138, Bachlechner, Long, McAllister 1412.1093) This motivates a systematic study of large field models of inflation in quantum gravity/string theory
Axions arise from integrating gauge potentials over non-trivial cycles in the compactification manifold. b I (x) = B where Axion Monodromy Inflation Consider string theory on M X Σ (p) c α (x) = Σ (2) I Σ (p) α C (p) α is an element of an integral basis of H p (X, Z) Inspection of the vertex operator for b I (x) reveals that these fields possess a shift symmetry to all orders in string perturbation theory.
Axion Monodromy Inflation The corresponding canonically normalized scalar fields are periodic with period 2πf, with typically. f 0.1M p
Axion Monodromy Inflation The corresponding canonically normalized scalar fields are periodic with period 2πf, with typically. One way to achieve super-planckian excursions is monodromy 1. 10 10 f 0.1M p 8. 10 11 VΦMp 4 6. 10 11 4. 10 11 2. 10 11 0 0.6 0.4 0.2 0.0 0.2 0.4 0.6 ΦM p
Axion Monodromy Inflation Monodromy occurs in various contexts in non-abelian gauge theories in string theory in the presence of branes in the presence of fluxes
Axion Monodromy Inflation For a D5-brane wrapping a two-cycle of size L. Σ (2) α 1 S DBI = (2π) 5 α 3 g s (2π) 5 α 2 g s d 6 ξ det( ϕ (G + B)) d 4 x (4) g L 4 + b 2 For large field values in terms of the canonically normalized fields the potential then becomes V (φ) µ 3 φ
Axion Monodromy Inflation Basic setup Type IIB orientifolds with O3/O7 Stabilize the moduli a la KKLT NS5 5B C (2) = c anti 5B anti-ns5
Axion Monodromy Inflation This is just one example of a larger class of models with potentials V (φ) =µ 4 p φ p so far with p = 2 3, 1, 4 3, 2, 3 (see Alexander s talk) Even though one can arrange p=2, would we believe any of this if we found that the data was consistent with m 2 φ 2?
Axion Monodromy Inflation Indeed the models make additional predictions ED1 5B NS5 C (2) = c anti 5B anti-ns5 Instanton corrections may lead to oscillatory contributions to the potential. V (φ) =µ 3 φ + Λ 4 cos φ f These lead to oscillations in the power spectrum that can be searched for.
Axion Monodromy Inflation In the larger class of models they are of the form V (φ) =µ 4 p φ p + Λ(φ) 4 1+pf φ cos 0 φ f 0 φ 0 + ϕ This can be shown to lead to a power spectrum of the form 2 R (k) = 2 R ns 1 k φ k 1+δn s cos 0 f φk φ 0 pf +1 + ϕ δn s =3b 2π α 1/2 with α =(1+p f ) φ 2pN0 1+pf 0 2fN 0 φ 0
Axion Monodromy Inflation Search for oscillations with drifting period in Planck nominal mission data
Axion Monodromy Inflation Improvement of the fit over ΛCDM: χ 2 = 18 Expectation based on simulations in the absence of a signal: χ 2 = 16.5 ± 3.5 One should keep in mind that not the entire parameter space was searched and more work is required, but as of now there is no evidence for oscillations in the primordial power spectrum. The amplitude is very model dependent, and a nondetection does not rule out these models, but it means for now* we are stuck with and r. n s (*) LSS may some day dramatically improve the constraints
A closer look at ns and r Using Planck temperature anisotropies 0.05 0.04 B2xB2 B2xB1c B2xKeck (preliminary) BICEP2 BB power spectrum l(l+1)c l BB /2π [µk 2 ] 0.03 0.02 0.01 0 0.01 0 50 100 150 200 250 300 Multipole Planck 353 GHz BB power spectrum
s shifts from previous results 0.2 1.0 60 1.50 0.975 1.75 1.00 0.8 s P/Pmax AL 1.25 3.2 1.50 4.0 1.75 AL 4.8 3.2 4.0 Neff Neff 100 4.8 0.0 pend on data cutaand on l range closer look at dn /dn lnsk 0.00 0.4 0.8 Σmν [ev] dns /d ln k 1.2 [ev] Spec = 1200 for 143,1.0 217 0.8 0.20 0.25 0.30 CamSpec YP Σmν [ev] 0.20 0.25 0.30 0.35 min = 1200 for 143, 217 YP= 2000 max Plik No 217 217 max = 20000.8 P/Pmax 1.2 P/Pmax 0.4 Plik 100 GHz 1.00 P/Pmax max No 2 Plik 0.8 No 217 217 0.6 0.4 0.0 1.25 0.960 ns 1.50 0.975 1.00 1.75 3.2 AL 1.0 1.25 3.2 4.0 4.8 Neff 0.4 1.75 0.975 3.2 1.0 AL 4.0 Neff 4.8 1.00 1.25 1.50 AL Neff 0.8 1.75 0.6 1.50 0.960 ns 1430.8 GHz 4.0 4.8 217 GHz 0.6 0.4 0.2 0.2 0.0 0.0 0.00 max = 2000 0.945 0.945 AL -100 0.0 0.4 Wednesday, May 22, 13 1.50-200 P 0.35 1.0 0.2 0.975 1. 0.2 0.6 No 217 217 VI -300 CamSpec 0.20 0.25(Planck 0.30 XVI) 0.35 min = 1200Y for 143, 217 1.2 0.8 Σmν [ev] 1-400 0.04 0.02 0.00 0.4 10 s 0.4 0.0 8 2 0.00 0.2 d ln k 60 s 0.04 0.02 P/Pmax 2 0.6 0.4 0.04 0.02 0.8 dns /d 1.2 0.00 ln k 0.4 0.20 0.25 0.8 1.2 0.30 Σmν0.35 [ev] 0.04 0.02 0.20 0.00 0.25 0.30 dns /d ln k YP 0.35 0.4 0.8 1. Σmν [ev]
A closer look at ns Is 217x217 consistent with the other frequencies? Use Planck covariance matrix to compute P (C 217 217 {C 100 100,C 143 143,C 143 217 }) and C 217 217 {C 100 100,C 143 143,C 143 217 } to predict the spectrum
2.0 10 9 1.8 10 9 A closer look at ns Predicted versus measured 217x217 spectrum 217x217 predicted from 100x100, 143x143, and 143x217 2 1 2 C2ΠΜK 2 1.6 10 9 1.4 10 9 1.2 10 9 1.0 10 9 8.0 10 8 6.0 10 8 500 1000 1500 2000 2500
In our reanalysis A closer look at ns We use survey cross spectra rather than detector set spectra We have used various cleaning procedures and a range of masks to test for stability We have performed the analysis with the same treatment of foregrounds as Planck (but for survey cross-spectra) We do not find strong dependence on mask or treatment of foregrounds
A closer look at ns 0.975 0.972 0.315 0.969 0.966 0.305 ns Ωm 0.963 0.960 0.957 Hybrid cleaning - SA47 Hybrid cleaning - SA50 Hybrid cleaning SA47 - no 217x217 GHz 353 cleaning - SA47 545 cleaning - SA47 0.295 0.285 Planck+WP Planck+WP no 217x217 GHz Uncleaned Survey Crosses Hybrid cleaning - SA24 Hybrid cleaning - SA38 Hybrid cleaning - SA44 0.285 0.295 0.305 0.315 Ω m 65 66 67 68 69 70 H 0
A closer look at ns 0.975 0.972 0.315 0.969 0.966 0.305 ns Ωm 0.963 0.960 0.957 Hybrid cleaning - SA47 Hybrid cleaning - SA50 Hybrid cleaning SA47 - no 217x217 GHz 353 cleaning - SA47 545 cleaning - SA47 0.295 0.285 Planck 2013 CYE Planck+WP Planck+WP no 217x217 GHz Uncleaned Survey Crosses Hybrid cleaning - SA24 Hybrid cleaning - SA38 Hybrid cleaning - SA44 0.285 0.295 0.305 0.315 Ω m 65 66 67 68 69 70 H 0 With the new prior on the optical depth we find, e.g. n s =0.9651 ± 0.0066 H 0 = 67.45 ± 1.07
A closer look at r l(l+1)c l BB /2! [µk 2 ] 0.02 0.015 0.01 0.005 lens+r=0.2 BSS LSA FDS PSM DDM1 DDM2 0 0.005 0 50 100 150 200 250 300 Multipole
A closer look at r Foreground models made in collaboration with David Spergel, Colin Hill, and Aurelien Fraisse BICEP2 BICEP2xKeck BICEP2 BICEP2xKeck BICEP2 BICEP2xKeck
removal of the plitude for a given dust total intensity. Choices of the cl model (Planck areas of the polarized sky cannot be made accurately scribed by the the A Planck closer totallook intensityat maps r alone. ived in Planck Component separation, or template cleaning, can b 1.59 andmeasurement T d = done of BB atin present the BICEP2 with the region Planck at HFI 353 353 GHz GHz data, b rescaled to 150 accuracy GHz of such cleaning is limited by Planck noise in o polarization not consistent 2 high Galactic neral properties endence. of the 353 GHz stematic errors ng conclusions ed at detection ctra computed ctic latitude to D BB = fields. Ground-based or balloon-borne experiments include dust channels at high frequency. Alternatively, intend to rely on the Planck data to remove the dust sion, they should optimize the integration time and a BICEP2 as to have a similar signal-to-noise level for the CM BICEP2xKeck dust power spectra. Turning specifically to the part of the sky mapped BICEP2 experiment, our analysis of the M B2 region ind the following results. Over the multipole range 40 < < 120, the Planck 35 D BB power spectrum extrapolated to 150 GHz yields a 1.32 10 2 µk 2 CMB, with statistical error ±0.29 10 2 µ and a further uncertainty (+0.28, 0.24) 10 2 µk 2 CMB
Joint Likelihood TT and BB is essentially independent so that the likelihoods factorize. BICEP2 and Planck 353 GHz are not independent and we need a joint likelihood. L(CCMB,C th dust C th (B) i ) exp 1 2 (P ) σ 2 P eff 2 L BICEP (C th CMB,C th dust C (B) i ) with (P ) =(C (P ) C dust ) (B) i C 1 ij C j (B) i = C (B) i C dust i C CMB i σ 2 P eff = C P C i C 1 ij C i
Frequency dependence The Planck 353 GHz measurement must be rescaled to 150 GHz, and the uncertainty in the rescaling must be included. L(β dust,c th CMB,C th dust C (B) i,c (P ) ) 1.0 0.8 0.6 PΒ 0.4 0.2 0.0 0 2 4 6 8 10 Β
A closer look at ns and r Assuming only frequency information in the patch CAMspec r<0.10 at 95% CL hybrid cleaning r<0.12 at 95% CL
Experimental Progress With the current data, we can constrain r by the tensor contribution to the temperature anisotropies on large angular scales the B-mode polarization generated by tensors. The two likelihood are essentially independent L(r TT,r BB )=L TT (r TT )L BB (r BB ) Typically we talk about L(r, r)
Experimental Progress L(r TT,r BB ) before March Planck+BICEP1 Constraint dominated by temperature data
Experimental Progress L(r TT,r BB ) after March Constraint from polarization data comparable to constraint from temperature and will soon be significantly stronger
Conclusions The BICEP2 BB power spectrum combined with Planck 353 GHz BB power spectrum in the BICEP region, and the Planck power spectrum of temperature anisotropies provide no evidence for primordial gravitational waves The simplest model of inflation is currently disfavored at 2-3 σ depending on the likelihood used for the temperature data, suggesting the existence of a second parameter in the inflationary model. Inclusion of cross-spectra between Planck and BICEP as well as the Keck Array 95 GHz data will soon tell us if the simplest model of inflation is relevant to our universe.
Conclusions Over the course of the decade we will find out if we are lucky enough that the CMB contains valuable information about quantum gravity or string theory Until then, we (theorists) should work harder to understand our theories at the relevant energy scales
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