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Hideo Kodama Cosmophysics Group, KEK Theory Center @ Shanghai Asia-Pacific School and Workshop on Gravity 2011.2.10-14

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How to Probe the Ultimate Theory Test predictions or possible effects characteristic to effective 4D theories derived from String theory/M-theory – Mini-black holes, beyond SM phenomena ( ) LHC, ILC) – KK particles /relics ( ) DM) – Stringy cosmic strings/topological relics ( ) cosmology ) – Fields with non-standard kinetic terms ( ) inflation, DE ) DBI action, nonstandard Kahler potential, moduli-dependent gauge coupling. – Low energy phenomena caused by moduli Change of the fundamental constants in cosmological time scales. New forces in submm ranges

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– Axionic fields descendent from form fields ( ) Lab. experiments /astrophysics/cosmology) Birefringence of CMB polarisation Step structures in the cosmological power spectrum Black hole bomb/axion siren Circular polarisation of primordial GWs Penetration of the GZK type barrier of CMB for high energy gamma rays.

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Contents Lecture 1 Introduction Ultimate theory probe. Black holes – Basic Conepts – Bound states and scattering – Superradiance instability Lecture 2 Axiverse – String axions – Axiverse overview – G-atom and axion siren – Axions in cosmology – Axions in astrophysics

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Basic Concepts

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Concept of Infinity Conformal Embedding Minkowski Spacetime Static Einstein Universe

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Generalisation Conformal Infinity [Penrose R 1963] Weakly Asymptotically Simple – Infinity of vacuum spacetime is sensitive to . – In general, when a spacetime M has a neighborhood of infinity that is isomorphic to a neighborhood of infinity of either E n,1, dS n+1 or adS n+1, M is called to be weakly asymptotically simple.

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Definition of Black Hole Let M be an AF spacetime and be its conformal infinity ． Asymoptotically predictable Horizon Black hole region DOC （ Domain of outer communiation ）

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Killing Horizon Stationary spacetime – There exists a Killing vector » = t that is timelike around infinity ： – » ) a one-param trf group © a : t ! t+a Null hypersurface – A hypersurface whose tangent plane is null (i.e., has a degenerate metric). A null hypersurface is tangential to a light cone and shares a unique null geodesic ( k ¢ k=0 ) at each point. – Horizon is a null hypersurface. Killing horizon – A null hypersurface whose null geodesic generators coincide with a (stationary) Killing vector on the surface. – Rigidity Theorem: A horizon of a stationary spacetime is a Killing horizon.

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Static Black Holes Static spacetime Stationary and time-reversal invariant Schwarzschild spacetime Horizon=Killing horizon: f(r)=0 Horizon ¼ S n £ R The time translation Killing vector is null on the horizon and timelike in DOC.

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Stationary Rotating Black Hole Stationary rotating spaceti me The time-translation Killing vector » satisfies Exmaple ： Kerr solution – H + is a Killing horizon : ¢ (r)=0 ． – H + ¼ S 2 £ R – The time-translation Killing vector » is spacelike on H +. – There is an ergo region in DOC where » is spacelike. Ergo region

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AF Stationary Rotating Black Hole Global Structure Symmetries Rotation DOC Ergo region around the horizon in DOC

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Asymptotic Structure – At infinity where f(r)=1-2M/r n-1 and n=D-2. – Near horizon This metric can be written in a regular form in terms of the advanced/retarded time coordinate

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Bound states and scattering

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Particle Motion around a BH Schwarzschild BH Conservation laws Effective potential Massive particle Massless particle 9 stable bound orbits No stable bound orbits 4D

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Kerr BH Conservation laws Effective potential Massive particle Massless particle Potential depends on the sign of L No stable bound orbits 4D

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Massless Scalar Field around a Kerr BH Klein-Gordon product From the field equation the KG product defined by is independent of the choice of the Cauchy surface in DOC. Scattering problem No incoming wave from the black hole

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Flux Integral on a Null Surface t x Hyperboloid in Minkowski spacetime Tangent vector (dt,dx) Unit normal n Integral along the Hyperboloid n kds

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Flux Integral – At infinity – At horizon where * = – m h,.

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Superradiance Flux conservation Superradiance condition This condition is equivalent to Cf. Penrose process in the ergo region [Penrose 1969]

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Penrose Process Energy conservation law – P : the 4-momentum of a free partilce » : the time-translation Killing vector. Then, the energy defined by E= - » ¢ p is conserve: Ergo region of a rotating bh – Because » is spacelie, a physical partical can have a negative energy E=- » ¢ p <0. – We can extract energy from a black hole through reactions in the ergo region. »p Ergo Region E E1E1 E2E2 E 1 =E - E 2 >E

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Superradiance Instability

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Black Hole Bombs Black hole in a mirror box [Zel’dovich 1971; Press, Teukolsky 1972; Cardoso, Dias, Lemos, Yoshida 2004] If a rotating black hole is put inside a box with reflective boundary, superradiance provokes infinite growth of massless bosonic field inside the box. Cf. No instability for fermions. Massive bosonic fields around a black hole [Damour, Deruelle, Ruffini 1976;] A similar instability occurs for massive bosonic fields!

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Massive Scalar Wave Equation Klein-Gordon equation Separation of variables Angular modes

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Radial modes Boundary Condition At horizon: At infinity

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WKB wavefunction for a bound state

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Instability Condition Mode expansion Energy integral

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Regularity/boundary condition Flux conditions All are positive!

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Instability criterion – The mode is bounded. – R is peaked far outside the ergo region => A>0. – is nearly real: | I |<< R – satisfies the superradiance condition: R < m h [Zouros TJM, Eardley DM 1979]

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Large Mass Case WKB estimate Infalling flux to BH: Zouros TJM, Eardley DM 1979

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Instability growth rate Generic features (a) Smallest permissible l (b) Largest permissibJe m (m = 1) (c) Largest permissible a (a/M = 1) (d) Largest permissible real frequency ω R ( ～ 0.98μ · m h )

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Small Mass Case A bound state solution with 2 = 2 - 2 >0 Schroedinger eq. for a hydrogen atom! This solution behaves in the region M << x <<1 as Detwiler SL (1980)

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The solution ingoing at horizon This solution behaves in the overlapping region 1 << z << l/( M) as

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Matching the two approximate solutions in the overlapping region, we obtain This determines the instability growth rate: where

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Numerical Estimation Continued Fraction Method – Bounded series expression – Convergence condition – Equation for omega

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Dolan SR(2007)PRD76,084001 Numerical Results

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General Features The growth rate is greatest for M <0.5. The mode with l=m=1 is most unstable: The maximum growth rate at a=0.99 i s

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BH SR instability: Summary Massive bosonic fields around a black hole [Damour, Deruelle, Ruffini 1976;] For light axions around an astrophysical black hole, an instability occurs. Its growth rate is [Zouros, Eardley 1979; Detweiler 1980] Numerical calculations show that the maximum instability is : [Furuhashi, Nambu 2004; Dolan 2007], Here note that Cf. AdS-Kerr black holes [Hawking, Reall 1999; Cardoso, Dias 2004; Cardoso, Dias, Yoshida 2006] Magnetic Penrose process and relativistic cosmic jets in GRB [van Putten 2000; Aguirre 2000; Nagataki, Takahashi, Mizuta, Tachiwaki 2007]

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Axiverse

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String axions

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What Is Axion? Originally, a psued-Goldstone boson for the Peccei-Quinn chiral symmetry to resolve the strong CP problem. CP problem in QCD – Winding number and topological vacuum – Instanton – Theta vacuum and CP violation a q

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Peccei-Quinn symmetry – L Y = -h(R * * L + cc) is invariant under the chiral trf – Due to the chiral anomaly, this trf shifts : (r=2) – When the SU(2)xU(1) symmetry is spontaneous broken, if we put 0 by the PQ transformation, this term becomes – Thus, CP is still violated in general. – Theory has a shift symmetry + in the tree level.

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Non-perturbative effect. – In vacuum, the SU(2) chiral symmetry for (u,d) is explicitly broken by non-perturbative effects: – This chiral symmetry produces a potential for axion and violate where f a is the scale of PQ symmetry breaking. – Thus, the shift symmetry of the axion is broken, and CP invariance is dynamically restored: Axion interactions – Fermions: – Gauge fields:

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Basic features of the invisible QCD axion – Weak coupling (chiral) : g aq ¼ m q /f a ; f a & 10 8 GeV – Small mass by the QCD instaton effect: m a ¼ 10 -3 eV (10 10 GeV/ f a ) – Dark matter candidate a. 0.01 (f a /10 10 GeV) 1.175 – Coupling to gauge fields via anomaly: g a F Æ F General Definition (ALP) – A pseudo scalar with tree-level shift symmetry and P/CP violation a q

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Form Fields in HUnTs Form fields have gauge invariance and Chern-Simons couplings: – M-theory : G 4 =dC 3 This action is invariant under the gauge transformation – 10D SST (NS sector): H 3 =dB 2 d B 2 =d l For type II theories, For the Heterotic/type I theories, H 3 should be modified as for anomaly cancelation. The Chern-Simons term breaks C and CP.

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– 10D SST (type II) RR fields In the democratic formulation, in terms of the polyforms C= p C p and G= p G p, we have and the action In contrast to its appearance, P and CP are violated in this sector because the duality condition should be imposed on the flux in the field equations: After compactification, these form fields produce axionic fields that violate P and CP.

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Coupling to Gauge Fields Shift symmetry ) Axion Model-independent axion – B 2 field in 4D spacetime: – The invariance under the gauge transformation B= d ¸ guarantees the shift symmetry. – The coupling of B 2 to gauge fields can be derived from the anomaly cancelation condition Svrcek P, Witten E: Axions in string theory, JEHP06 (2006) 051:

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Model-dependent axions – B 2 field and C p fields in the internal space: b = s B 2, c = s C p – The coupling of b to gauge fields can be derived from the Green-Schwarz counter-term – The coupling of c to gauge fields comes from the D- brane Chern-Simons term Svrcek P, Witten E: Axions in string theory, JEHP06 (2006) 051:

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Stringy Axion Mass If the shift symmetry is not violated at the tree level by flux, branes and compactification (i.e., by moduli stabilisation), it can be preserved by perturbative quantum corrections (for supersymmetric states). Then, axions acquire mass only by non-perturbative effects (possibly associated with SUSY breaking), such as instanton effects as in the case of the QCD axion. If a light QCD axion really exists, it is natural that there survive lots of other light axions coming from the large number of non-trivial cycles in extra-dimensions, which can be order of several hundreds or more to allow for the tuning of the cosmological constant. Thus, it is expected that there are lots of superlight axions whose mass spectrum is homogeneous in log m, producing the axiverse.

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Rough Estimates Lagrangian of an axion where ¤ is the energy scale of the axion potential, which can be represented as ¤ 4 ¼ F susy m pl 2 e -S, in many cases, in terms of the SUSY breaking scale F susy and the intanton action S. From the relations We have f a » m pl /S. Hence, If we require that this stringy effect is less than the QCD instanton effect for the QCD axion, we

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Overview

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Characteristic Mass Scales 3/m= Horizon size (=1/H) – Present t=t 0 : m=m 0 =4.5 £ 10 -33 eV – CMB last scattering t=t ls : m=0.7 £ 10 -28 eV – H recombination t=t rec : m=m rec =1.2 £ 10 -28 eV – Equidensity time t=t eq : m=m eq =0.9 £ 10 -27 eV 1/m = BH size (=M pl 2 /M ) – Supermassive BH M=10 10 M ¯ : m=m bh,max =1.3 £ 10 -20 eV – Solar mass BH M=1 M ¯ : m=m bh,min =1.3 £ 10 -10 eV QCD axion m ¼ QCD 2 /f a – f a =10 16 GeV: m » 10 -9 eV – f a = 10 12 GeV: m » 10 -5 eV

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Probing the Ultimate Theory by Axion Cosmophysics String theories ) superlight axionic fields + QCD axion Superlight axionic moduli ) new cosmophysical phenomena.

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Black Hole Instability

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Superradiance Instability The growth rate becomes maximum at g = R g » 1 (R g =GM). Instability occurs for gravitational bound states with where n is a positive integer such that n>l. In general, for the most unstable mode Hence, the mode is peaked outside the ergo region and far from the horizon: This implies that states are quantum for near extremal cases.

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SR instability strip in the M- plane Arvanitaki A, Dubovsky S: arXiv:1004.3558

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Black Hole Spin Arvanitaki A, Dubovsky S: arXiv:1004.3558

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G-Atom

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Fate of G-Atom? Kerr BH Accretion Disk Axion Cloud Gravitational Waves 2a G axions photons SR mode Bosenova??

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Gravitational Wave Emission Estimation by the quadruple formula Quantum level transition Axion annilation: 2a G Arvanitaki A, Dubovsky S: arXiv:1004.3558

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Axion annihilation emision

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Level transition emission

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Non-linear Effects Axion Action Non-relativisitc effective action Averaging S over a time scale >> 1/ Attractive interaction

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Non-linear Effects Direct axion emission: 3 bounded axions 1 free axion Self-force dominance Bosenova collapse

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Superradiance suppression Due to level mixing, axions in a margially superradiant mode (l 0 =m 0 = h ) make the level l=m=m 0 +1 non- superradiant if

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Expected Evolution of a BH system 1.The most ustable mode with n 0 >l 0 =m 0 grows due to the SR instability. 2.BH angular momentum reduces to the margially superradiant value: h0 = l 0 3.BH angular momentum remains at this value until the axions at the level n 0 >l 0 =m 0 are transferred the next unstable mode with n 1 >l 1 =m 1. 4.Bosenova happens serveral times until the next marginally superradiant value is achieved: h1 = l 1 5.Repeats 3-4 until SR modes disappear.

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BH Regge Trajectories

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Axions in Cosmology

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Field equation Basic behavior – H & m/3: – H. m/3: Behavior of a light coherent field DE/ ¤ m/H 0 =10 3, =0 dust-like matter

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Simple case where º = and Á ! Á 0 as t ! 0. For large t, where Energy density The effective particle number a 3 n Á = a 3 h m Á 2 i is conserved for m & 3H. Hence, the energy density at m ¿ H is given by where

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Time Evolution of and the expansion rate Marsh DJE, Ferreira PG: arXiv:1009.3501 Axion field behaves as a cosmological constant in the early stage and as non-relativistic matter in the late stage.

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Density Parameter

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Present Abundance For 10 -26 eV < m < 10 -28 eV, the density parameter of the axion is (1-0.1) m, if f a » m pl. QCD axion

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Finite Temperature Effect The QCD axion mass at finite T by the dilute gas approximation Turner MS: prd33(1986) 889.

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Rotation of the CMB Polarisation Let us consider axionic fields that couple to the EM field, but not to the QCD sector. – This requirement can be realised in some orbifold SU(5) GUT models in 5D braneworld. – It is expected that there are only a few such fields: most axionic fields are localised on cycles that do not intersect our brane. Lagrangian density

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Field equation For Á = Á (t), under the gauge the field equation in the spatially flat universe with dt=a d reads Plane wave solution When the wave is propagating in the z-direction In the WKB approximation (large k), its solution is where

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Stokes Parameters for the Linear Polarisation Polarisation Tensor Cosmological Birefringence Á F Æ F term induces the rotation of the CMP polarisation when d Á /dt 0. [Ni W-T 1977; Carrol, Field, Jackiw 1990] ) Generates B-modes from E-modes after recombination. ) non-zero TB, EB correlations. Lue, Wang, Kamionkowski 1999 Pure B-mode Pure E-mode E cos + B sin

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The rotation angle of the CMB polarisation is given by if 1/m is larger than the width of the last scattering surface t ls ' 10kpc. The effect becomes maximum for the mass range The value of for this range is independent of both the axion energy scale f a and the inflation energy scale: When there exist N axions, is multiplied by N 1/2. Observational Limits: – Current limit: ¢ – Planck: accuracy < 0.1 – CMBPol: accuracy < 0.005

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Gravitational CS Term RÆR term produces circular polarisation to the primordial GWs generated during inflation if a is the inflaton. For ® =f( © ), ) Direct detections by BBO and DECIGO in the future. ) non-zero TB, EB correlations of CMB at recombination. Lue, Wang, Kamionkowski 1999; Satoh, Kanno, Soda 2008

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Subtle points – The evolution equation for the left-polarisation mode becomes singular at horizon crossing. – The final result appears to be sensitive to the initial condition for quantum gravitational fluctuatutions during inflation. Cf. Generation of baryon/lepton asymmetry through the gravitational anomaly: [Alexander, Peskin, Sheikh-Jabbari 2006] Lyth, Quimbay, Rodriguez 2005

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Fuzzy Cold Dark Matter For k ¿ m, non-relativistic fluid with Wayne Hu, R Barkana, A Gruzinov: PRL85, 1158 (2000)

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Resolution of the Cusp/Substructure Problem Suppression of inhomogeneities on scales smaller than The cusp/substructure problem in CDM can be resolved if the dark matter consists of scalar field of mass » 10 -22 eV.

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Influences on LSS frozen the same as the standard CDM damped oscillation S=1 S=const <1 Step-like deformation!

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The characteristic scales depend on m

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They derived an empirical formula for the suppression factor S that coincides with that in the axiverse paper for small deformation They also obtained an empirical formula for the transition wave number consistent with the analytic estimate. Numerical Results Marsh DJE, Ferreira PG: arXiv:1009.3501

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Transfer function: However, no oscillatory behavior appears in the transfer function!

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The suppression factor S is given by where For m » 10 -22 eV for which k J » 1Mpc -1, Observations – Current limits: Ly- ® lines ) a / m. 0.1 for k=( 0.1-10)Mpc -1, z o =2 » 4. – Future observations: BOSS (SDSSIII) and the 21cm line measurement will give much stronger limits/detections. Cf. Axiverse Paper

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Axions in Astrophysics

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Key point Axions are converted to and from photos by mixing: Solar axions due to the Primakov effect: – CAST experiment at CERN(2007, 2008) Cast Collaboration (2008) arXiv: 0810.4482 a q B, E

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CAST Bounds CAST Collaboration (2008) arXiv:0810.4482

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J. Jaeckel, A. Ringwald: arXiv:1002.0329 The Low-Energy Frontier of Particle Physics

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a Conversion by Magnetic Fields Propagation equation where with pl 2 =4 n e /m e being the plasma frequency, and R and CM represents the Faraday rotation effect and the vacuum Cotton-Mouton effect, respectively. Non-resonant conversion For homogeneous magnetic fields, where For a random sequence of N coherent domains [Grossman Y, Roy S, Zupan J: PLB543:23(2002)] Resonant conversion

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Spectral Deformation of Cosmic -rays by Galactic and Intergalactic Magnetic Fields Photon-ALP conversion rate where Estimations Can be observed by GLAST(10% deformation) and E * =10 2 GeV » 1 TeV if m a ¼ 10 -6 » 10 -8 eV at the CAST bound on g a and – Intergalactic fields: L dom » 1Mpc, B=(1-5) ¢ 10 -9 G for D=200 » 500 Mpc – Intracluster fields: L dom » 10kpc, B=10 -6 G, n e ' 10 -3 cm -3 for D= 1Mpc – Galactic fields: L dom » 10kpc, B=(2-4) ¢ 10 -6 G, n e ' 10 -3 cm -3 for D= 1Mpc De Angelis A, Mansutti O, Roncadelli M: arXiv:0707.2695 [astro-ph]

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Strong mixing can occur between cosmic - ray and axions by cosmic magnetic fields Fairbairn, Rashba, Troitsky 2009 arXiv0910.4085

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– UHE gammas from QSOs and Blazers can penetrate the CMB/CIRB barrier to explain the observed flux. Fairbairn, Rashba, Troitsky 2009; Roncadelli, de Angelis, Mansutti 2009 Optical depth against CMB Expected flux from 3C279

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Deformation of the -ray spectra from AGN, GRB and other sources may be detected by Fermi if m a < 10 -8 eV and g a is close to the CAST bound. Hochmuth, Sigl 2007; de Angelis et al 2008 Cf. Burrage, Davis, Shaw 2009 M=10 9 M ⊙, B=0.5G over 200pc, =10 -3 pc, g 11 =1, m a =1 eV.

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Summary Super-light axions as legacy of the string theory can produce quite rich cosmophysical phenomena. Future observational confirmation of these phenomena can provide valuable information on the hidden extra-dimensions and the ultimate theory behind them, complimentary to the inflation probe. Most of such cosmophysical phenomena have not been explored fully yet and can become a fruitful new research field, axion cosmophysics.

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Axiverse Pverview – “ – “String Axiverse” Arvanitaki A, Dimopoulos S, Dubovsky S, Kaloper N, March-Russell, J (2010) PRD81:123530 [arXiv: 0905.4720] String axions – “Axions in string theory” Svrcek P, Witten E (2006) JHEP0606:051 [arXiv:]

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Superradiance Instability massive scalar – “Instabilities of massive scalar perturbations of a rotating black hole” Zouros TJM, Eardley DM(1979)AnnP118:139. – “Klein-Gordon equation and rotating black holes Detwiler SL(1980)PRD22:2323 – “Instability of massive scalar fields in Kerr-Newman space-time” Furuhashi H, Nambu Y (2004)PTP112:983 – “Instability of the massive Klein-Gordon field on the Kerr spacetime” Dolan SR (2007) PRD76:084001 [arXiv:.0705.2880]

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Recurrence relation method for SR instability – Leaver EW(1985)PRSA402,285 : 3-term rec. rel for QNMs – Konoplya R, Zhidenko A (2006) PRD73,124040: QNM of massive scalar – Cardoso V, Yoshida S(2005) JHEP0507, 009: 5-term rec. rel. for bound states of massive scalar – Dolan SR(2007)PRD76,084001: 3 3-term rec. rel for bound states of massive scalar Astrophysics of superradiance instability of black holes – “Exploring the String Axiverse with Precision Black Hole Physics” Arvanitaki A, Dubovsky S: arXiv:1004.3558 – “Black Hole Portal into Hidden Valleys” Dubovsky S, Gorbenko R: arXiv:1012. 2893

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