Axiverse Cosmology and the Energy Scale of Inflation David J. E. Marsh Wits, 25 th March 2013 arxiv:1303.3008
Collaborators Pedro G. Ferreira (Oxford) Daniel Grin (IAS) Renée Hlozek (Princeton) ULA = ultralight axion
The Principle of Plenitude: This best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature s perfection. Gottfried Leibniz (1646-1716), in Theodicee Quoted in Arvanitaki et al (2009) Pangloss sometimes said to Candide: There is a concatenation of events in this best of all possible worlds: for if you had not been kicked out of a magnificent castle for love of Miss Cunégonde if you had not come under the Inquisition if you had not walked over America if you had not stabbed the Baron if you had not lost all your sheep from the fine country of El Dorado why, then, you would not be here, eating preserved citrons and pistachio-nuts. All that is very well, answered Candide, but let us cultivate our garden. Voltaire (1694-1778), in Candide
The String Axiverse Arvanitaki et al (2009) Svrcek and Witten (2006) Topological complexity à many axions CMB Polarization Matter Power Spectrum Anthropically Constrained Black Hole Super-radiance Decays Inflated Away 10-33 4! 10-28 3! 10-18 10 8 f a M pl S 2! 10-20 Axion Mass in ev 3! 10-10 QCD axion 4 a = µ 4 e S Figure: Arvanitaki et al
The Axiverse: how? Svrcek and Witten (2006) v String theory has extra dimensions: compactify. v Axions are KK zero-modes of antisymmetric tensor fields compactified on closed cycles. http://www.hep.ph.ic.ac.uk/cms/physics/higgs.html v Potentials from non-perturbative physics (D-branes, instantons etc.). many pseudo-goldstone bosons Isocurvature and the String Axiverse, David J. E. Marsh, Oxford 10/1/13
Axions as Fuzzy DM Hu et al (2000) Amendola and Barbieri (2005) DJEM and Ferreira (2010) 10 5 P(k)[h 3 Mpc 3 ] 10 4 10 3 10 2 decreasing m a 10 1 decreasing f ax 10 0 10 2 10 1 10 0 k [h 1 Mpc] v Quantum pressure in the sound speed leads to structure suppression: cosmological Compton wavelength.
I.C. Modes v Adiabatic (curvature): v Isocurvature (entropy): n 1 n 1 n 2 n 2 =0 T 00 / X i i i =0 v Eigenmodes of Einstein.
Inflationary axion fluctuations v Stable DM axions are massless during inflation. q v Power spectrum: h 2 ai 4 h 2 1 i = H I 2 * 1 0 quantum fluctuations in ds space 2 + = (H I/M pl ) 2 2 ( 0,i /M pl ) 2 v These fluctuations are created and survive to late times if: f a > Max{T GH,T max } f a & 10 12 10 16 GeV String Axiverse
Vacuum misalignment and relic density v T-dependence of mass effects relic density for QCD axions. v Negligible for light axions that begin oscillations while: 1) at zero T mass AND 2) energetically sub-dominant v 1) true for ULAs, 2) true by observation if DM. ) 0,i M pl 2 6 104 a h 2 m 2 aa 3 osc Fn of mass and cosmology à Different scaling with cosmological parameters than a QCD axion
Axion Isocurvature v In single field thermal cosmologies all fluctuations are seeded by inflaton decay and are adiabatic: (1 + w j ) i =(1+w i ) j v ULAs have w=-1 and no initial perts in adiabatic mode v Isocurvature (S) parameterised by ratio to inflaton (R): Isocurvature fraction (k 0 ) 1 (k 0 ) P S(k 0 ) P R (k 0 ) k 3 2 2 P R(k 0 )=A s = 1 2 HI /M pl 2 2 =2.41 10 9 WMAP 9
ULAs versus CDM axions v Normalisation effects parameter dependence. v CDM axions are indistinguishable from WIMPs etc à Single fluid normalisation d =1) P S =( a / d ) 2 P a v ULAs are distinct in their clustering Commonly used à Separate fluid normalisation a =1) P S = P a ) a 1 a = 8 ( 0,i /M pl ) 2 = d a Necessary for light DM 2 CDM 1 CDM
Power Spectrum and Correlations v Defining the spectral index by k 3 k 2 2 P S(k) =A i k 0 ni 1 ) n i =1 2 = n T +1 v Adiabatic mode initially zero for an axion à Totally uncorrelated isocurvature: simply add spectra using α à c.f. curvaton is totally anti-correlated v We will fix ε by our choice of α and Ω a (phenomeneological) à Tensors and index fixed by consistency
Inflation and Tensors v All massless modes fluctuate in ds. r = P h P R = 16 < 0.13, (95%C.L.) WMAP 9 v The adiabatic amplitude and ε together give H I. v Observable r has large H I and large inflaton motion. Lyth (1997) r 0.01 ) V 1/4 10 16 GeV v Ambiguous when other sources possible. Senatore et al (2011)
WMAP Bounds for CDM v QCD relic abundance depends on f a, θ i and γ à So do isocurvature constraints CDM < 0.047, (95%C.L.) WMAP 9 v Gives prediction for tensors ( r =(1.6 10 12 Ωc h 2 ) γ )( Ωc Ω a ( ) )( ) 5/6 f a α 0 10 12. GeV 1 α 0 CDM ) V 1/4. 4.2 10 12 GeV Without tuning, In the desert Observation of primordial tensors would rule out a string QCD axion as DM But in everything inflationary cosmology. changes for Fox generalised, et (2004), WMAP, ultralight Mack and axions! Steinhardt (2009)
The ULA Case: Predictions 1.0 a d 0.8 0.6 0.4 a osc >a eq a osc <a eq iso. ) r! 0 (high ma, lowhi) 0.2 0.0 tensors )! 0 (Low axion density) 32 30 28 26 24 22 log 10 (m a /ev)
The ULA Case: Predictions Credit: Macaulay N eff L 20 10 0 3.5 3 2.5 0.04 WEAVE Euclid Planck WEAVE+Planck Euclid+Planck f ax 0.02 0 0 0.1 m ν [ev] 2.5 3 3.5 N eff 0 0.02 0.04 f ax
The ULA Case: Methodology v Direct solution limited to small range & too slow for MCMC. 0 +2H o + m 2 a 2 0 =0 1 +2H 1 +(m 2 a 2 + k 2 ) 1 + 1 2 0ḣ =0 v Effective fluid formalism copes with this. v Pre-oscillations field and fluid are equivalent re-writing. P c2 ad = a = 3H(1 + w a ) a 1+ 2m aa 3H r 1 wa 1+w a Computed exactly from φ 0 e.o.m.
The ULA Case: Methodology v Subtlety: frozen field w=-1 gives (apparent) divergences. v Perturbations: continuity+euler, with entropy perts. Hu 1998 w a a =(1 c 2 ad)( a +3Hu a /k) c 2 s P a a Eliminate: rest frame solution, boosts Solve and average oscillations: WKB EARLY TIMES LATE TIMES 1, k 2m a a; k 2 4m 2 aa 2, k 2m aa
The ULA Case: Methodology v CAMB/cosmomc modification. Lewis and Challinor v Piecewise generalised fluid + background solver: fast. a = ku a (1 + w a )ḣ/2 3H(1 w a) a 9H 2 (1 c 2 ad)u a /k u a =2Hu a + k a +3H(w a c 2 ad)u a hw a i!0 P a = ku a ḣ/2 3Hc 2 s a u a = Hu a + c 2 sk a
The ULA Case: Priors 33 < log 10 (m a /ev) < 20 Jeffrys prior: uniform log. 0.001 < a h 2 < 0.15 Uniform: treat as matter. P ( a h 2 ) / 1 a h 2 (*) Alternative from misalignment? Hertzberg et al 2008 0 < a < 1 Uniform: follow WMAP. v Question: r is derived à priors on inflation? v (*) does not take mass prior into account. Extend: Easther and McAllister (2006) N-flation
The ULA Case: Priors 1 0.9 0.8 0.7 Likelihood 0.6 0.5 0.4 Pr(Ω a h 2 )=c Pr(φ/M pl )=c 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 Ω a h 2
Isocurvature TT: norm. and suppression 10 4 `(` + 1)C`/2p [µ K 2 ] 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 4 =(0.01) 2 10 1 10 2 10 3 Multipole ` 100% CDM 1% ~10-20 ev 1% 10-29 ev 1% 10-32 ev
The ULA Case: (Preliminary) Results Ω a/ Ω m log(m a )
The ULA Case: (Preliminary) Results still preliminary runs/checks going Ω b h 2 0.021 0.022 0.023 0.024 0.025 τ WMAP9 SDSS P(k) Ω c h 2 0 0.05 0.1 0.15 n s 100θ A 1.035 1.04 1.045 Ω ax h 2 isocurvature modes specify tensor to scalar ratio r à tighter constraints on allowed amplitude 0.02 0.04 0.06 0.08 0.1 0.12 0.14 log(m a ) 30 25 20 Ω Λ 0.4 0.5 0.6 0.7 0.8 z re 0.94 0.96 0.98 1 1.02 α ax 0 0.1 0.2 0.3 0.4 0.5 Age/Gyr 13.5 14 14.5 r val 0 0.05 0.1 log[10 10 A s ] 3.05 3.1 3.15 3.2 3.25 3.3 Ω m 0.2 0.3 0.4 0.5 0.6 H 0 8 10 12 14 0 0.05 0.1 0.15 55 60 65 70 75
Summary 1.0 P(k)[h 3 Mpc 3 ] 10 5 10 4 10 3 10 2 decreasing m a a d 0.8 0.6 0.4 a osc >a eq a osc <a eq iso. ) r! 0 (high ma, lowhi) 10 1 decreasing f ax 10 0 10 2 10 1 10 0 k [h 1 Mpc] 10 4 0.2 0.0 tensors )! 0 (Low axion density) 32 30 28 26 24 22 log 10 (m a /ev) `(` + 1)C`/2p [µ K 2 ] 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 10 4 =(0.01) 2 10 5 10 6 10 1 10 2 10 3 Multipole `
Adiabatic Deflection 2.5 2.0 1.5 10 7 f ax = 0.001 f ax = 0.999 ACT Deflection m a = 10 28 ev m a = 10 26 ev m a = 10 21 ev m a = 10 20 ev C kk ` 1.0 0.5 0.0 10 1 10 2 10 3 Multipole ` 10 1 10 2 10 3 Multipole `