AXIVERSE AND BLACK HOLE Hideo Kodama Cosmophysics Group

AXIVERSE AND BLACK HOLE Hideo Kodama Cosmophysics Group, KEK Theory Center @ Shanghai Asia-Pacific School and Workshop on Gravity 2011. 2. 10 -14

Introduction

How to Probe the Ultimate Theory • Test predictions or possible effects characteristic to effective 4 D 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

– 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.

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

Lecture 1 Black Holes

Black Holes Basic Concepts

Concept of Infinity Conformal Embedding Minkowski Spacetime Static Einstein Universe

Generalisation • Conformal Infinity [Penrose R 1963] • Weakly Asymptotically Simple – Infinity of vacuum spacetime is sensitive to L. – In general, when a spacetime M has a neighborhood of infinity that is isomorphic to a neighborhood of infinity of either En, 1, d. Sn+1 or ad. Sn+1, M is called to be weakly asymptotically simple.

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

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.

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

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

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

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

Black Holes Bound states and scattering

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

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

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

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

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

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

Penrose Process p • 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. E 2 E 1 Ergo Region E 1=E - E 2>E E

Black Holes Superradiance Instability

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!

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

Radial modes • Boundary Condition At horizon: At infinity

WKB wavefunction for a bound state

Instability Condition Mode expansion Energy integral

• Regularity/boundary condition • Flux conditions All are positive!

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]

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

• Instability growth rate • Generic features (a) Smallest permissible l (b) Largest permissib. Je m (m = 1) (c) Largest permissible a (a/M = 1) (d) Largest permissible real frequency ωR (～ 0. 98μ · m h )

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

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

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

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

Numerical Results Dolan SR(2007)PRD 76, 084001

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 is

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. Ad. S-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]

L ecture 2 Axiverse

Axiverse String axions

What Is Axion? a 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 q g 5 g g

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

• 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 fa 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:

• Basic features of the invisible QCD axion – Weak coupling (chiral) : gaq ¼ mq/fa ; fa &108 Ge. V – Small mass by the QCD instaton effect: m a ¼ 10 -3 e. V (1010 Ge. V/ fa) – Dark matter candidate a q g 5 a. 0. 01 (fa/1010 Ge. V)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 g g

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

– 10 D SST (type II) RR fields In the democratic formulation, in terms of the polyforms C= p C p and G= p Gp , 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.

Coupling to Gauge Fields Shift symmetry ) Axion • Model-independent axion – B 2 field in 4 D 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, JEHP 06 (2006) 051:

• Model-dependent axions – B 2 field and C p fields in the internal space: b =s B 2 , c =s Cp – 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 Dbrane Chern-Simons term Svrcek P, Witten E: Axions in string theory, JEHP 06 (2006) 051:

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 extradimensions, 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.

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

Axiverse Overview

Characteristic Mass Scales • 3/m= Horizon size (=1/H) – – Present t=t 0: m=m 0=4. 5£ 10 -33 e. V CMB last scattering t=tls: m=0. 7£ 10 -28 e. V H recombination t=trec: m=mrec=1. 2£ 10 -28 e. V Equidensity time t=teq : m=meq =0. 9£ 10 -27 e. V • 1/m = BH size (=Mpl 2/M ) – Supermassive BH M=1010 M¯: m=mbh, max=1. 3£ 10 -20 e. V – Solar mass BH M=1 M¯: m=mbh, min=1. 3£ 10 -10 e. V • QCD axion m ¼ LQCD 2/fa – fa=1016 Ge. V: m » 10 -9 e. V – fa= 1012 Ge. V: m » 10 -5 e. V

Probing the Ultimate Theory by Axion Cosmophysics String theories ) superlight axionic fields + QCD axion Superlight axionic moduli ) new cosmophysical phenomena. Arvanitaki A, Dimopoulos S, Dubovsky S, Kaloper N, March. Russell, J: “String Axiverse” ar. Xiv: 0905. 4720

Axiverse Black Hole Instability

Superradiance Instability • The growth rate becomes maximum at g= Rg » 1 (Rg=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.

SR instability strip in the M- plane Arvanitaki A, Dubovsky S: ar. Xiv: 1004. 3558

Black Hole Spin Arvanitaki A, Dubovsky S: ar. Xiv: 1004. 3558

G-Atom

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

Gravitational Wave Emission Arvanitaki A, Dubovsky S: ar. Xiv: 1004. 3558 • Estimation by the quadruple formula • Quantum level transition • Axion annilation: 2 a G

Axion annihilation emision

Level transition emission

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

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

• 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 nonsuperradiant if

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

BH Regge Trajectories

Axiverse Axions in Cosmology

Behavior of a light coherent field Field equation Basic behavior – H & m/3: DE/¤ – H. m/3: dust-like matter m/H 0 =103, L =0

• Simple case where º = (3 g-1)/2, and Á ! Á0 as t! 0. For large t, where • Energy density The effective particle number a 3 nÁ = a 3 h m Á 2 i is conserved for m & 3 H. Hence, the energy density at m ¿ H is given by where

Time Evolution of rf and the expansion rate Axion field behaves as a cosmological constant in the early stage and as non-relativistic matter in the late stage. Marsh DJE, Ferreira PG: ar. Xiv: 1009. 3501

Density Parameter Dark Energy Dark Matter

Present Abundance For 10 -26 e. V < m < 10 -28 e. V, the density parameter of the axion is (1 -0. 1) m, if fa » mpl. QCD axion f

Finite Temperature Effect The QCD axion mass at finite T by the dilute gas approximation Turner MS: prd 33(1986) 889.

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 5 D 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

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

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

• The rotation angle of the CMB polarisation is given by • if 1/m is larger than the width of the last scattering surface dtls ' 10 kpc. The effect becomes maximum for the mass range • The value of Db for this range is independent of both the axion energy scale fa and the inflation energy scale: When there exist N axions, Db is multiplied by N • Observational Limits: – Current limit: Db < 2 o =3. 5 ¢ 10 -2. – Planck: accuracy < 0. 1 o – CMBPol: accuracy < 0. 005 o 1/2.

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

• 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

Fuzzy Cold Dark Matter Wayne Hu, R Barkana, A Gruzinov: PRL 85, 1158 (2000) For k ¿ m, non-relativistic fluid with

Resolution of the Cusp/Substructure Problem JEANS LENGTH 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 e. V.

Influences on LSS H>m frozen H < m & k < k. J the same as the standard CDM H < m & k > k. J damped oscillation k < km = a. H at H=m k > k. J, 0 = k. J (t 0) S=1 S=const <1 Step-like deformation!

The characteristic scales depend on m

Numerical Results Marsh DJE, Ferreira PG: ar. Xiv: 1009. 3501 • 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.

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

Cf. Axiverse Paper • The suppression factor S is given by where For m» 10 -22 e. V for which k. J » 1 Mpc-1, • Observations – Current limits: Ly-® lines ) a/ m. 0. 1 for k=(0. 1 -10)Mpc-1, zo=2» 4. – Future observations: BOSS (SDSSIII) and the 21 cm line measurement will give much stronger limits/detections.

Axiverse Axions in Astrophysics

Axions in Astrophysics • Key point Axions are converted to and from photos by mixing: q a g 5 • Solar axions due to the Primakov effect: – CAST experiment at CERN(2007, 2008) Cast Collaboration (2008) ar. Xiv: 0810. 4482 g B, E

CAST Bounds CAST Collaboration (2008) ar. Xiv: 0810. 4482

J. Jaeckel, A. Ringwald: ar. Xiv: 1002. 0329 The Low-Energy Frontier of Particle Physics

-a Conversion by Magnetic Fields • Propagation equation where with pl 2=4 ne/me 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: PLB 543: 23(2002)] • Resonant conversion

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*=102 Ge. V » 1 Te. V if ma ¼ 10 -6» 10 -8 e. V at the CAST bound on ga and – Intergalactic fields: Ldom» 1 Mpc, B=(1 -5)¢ 10 -9 G for D=200» 500 Mpc – Intracluster fields: Ldom» 10 kpc, B=10 -6 G, ne' 10 -3 cm-3 for D= 1 Mpc – Galactic fields: Ldom» 10 kpc, B=(2 -4)¢ 10 -6 G, ne' 10 -3 cm-3 for D= 1 Mpc De Angelis A, Mansutti O, Roncadelli M: ar. Xiv: 0707. 2695 [astro-ph]

• Strong mixing can occur between cosmic gray and axions by cosmic magnetic fields Fairbairn, Rashba, Troitsky 2009 ar. Xiv 0910. 4085

– UHE gammas from QSOs and Blazers can penetrate the CMB/CIRB barrier to explain the observed flux. Expected flux from 3 C 279 Optical depth against CMB Fairbairn, Rashba, Troitsky 2009; Roncadelli, de Angelis, Mansutti 2009

• Deformation of the g-ray spectra from AGN, GRB and other sources may be detected by Fermi if ma < 10 -8 e. V and gag is close to the CAST bound. M=109 M⊙, B=0. 5 G over 200 pc, =10 -3 pc, g 11=1, ma=1 e. V. Hochmuth, Sigl 2007; de Angelis et al 2008 Cf. Burrage, Davis, Shaw 2009

Summary

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.

References

Axiverse • Pverview – “String Axiverse” Arvanitaki A, Dimopoulos S, Dubovsky S, Kaloper N, March-Russell, J (2010) PRD 81: 123530 [ar. Xiv: 0905. 4720] • String axions – “Axions in string theory” Svrcek P, Witten E (2006) JHEP 0606: 051 [ar. Xiv: ]

Superradiance Instability • massive scalar – “Instabilities of massive scalar perturbations of a rotating black hole” Zouros TJM, Eardley DM(1979)Ann. P 118: 139. – “Klein-Gordon equation and rotating black holes Detwiler SL(1980)PRD 22: 2323 – “Instability of massive scalar fields in Kerr-Newman space-time” Furuhashi H, Nambu Y (2004)PTP 112: 983 – “Instability of the massive Klein-Gordon field on the Kerr spacetime” Dolan SR (2007) PRD 76: 084001 [ar. Xiv: . 0705. 2880]

• Recurrence relation method for SR instability – Leaver EW(1985)PRSA 402, 285 : 3 -term rec. rel for QNMs – Konoplya R, Zhidenko A (2006) PRD 73, 124040: QNM of massive scalar – Cardoso V, Yoshida S(2005) JHEP 0507, 009: 5 term rec. rel. for bound states of massive scalar – Dolan SR(2007)PRD 76, 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: ar. Xiv: 1004. 3558 – “Black Hole Portal into Hidden Valleys” Dubovsky S, Gorbenko R: ar. Xiv: 1012. 2893

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