Recent Topics in Axion Cosmology -Isocurvature Perturbations and Alignment

Recent topics in axion cosmology -isocurvature perturbations and alignment-

30th June 2016 seminar@Osaka U. Fuminobu Takahashi (Tohoku)

Kawasaki, FT, Yamada, 1511.05030 Higaki, Jeong, Kitajima, FT, 1512.05295, 1603.02090, Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 The Strong CP Problem

g2 = ✓ s Gaµ⌫ G˜a L✓ 32⇡2 µ⌫ Experimental bound from neutron electric dipole moment reads 10 ✓ < 10 | | Why is ✓ so small is the strong CP problem. cf. More precisely, the physical strong CP phase is ✓¯ ✓ arg det (M M ) ⌘ u d which makes the problem even more puzzling. A plausible solution is the Peccei-Quinn mechanism where the CP phase is promoted to a dynamical variable: Peccei, Quinn `77, Weinberg `78, Wilczek `78 a g2 = ✓ + s Gaµ⌫ G˜a L✓ f 32⇡2 µ⌫ ✓ a ◆

T QCD

T QCD a

The QCD axion dynamically cancels the stronga CP phase. Accordingly, the axion DM is produced as coherent oscillations [misalignment mechanism].

f 1.19 ⇤ ⌦ h2 =0.18 ✓2 a QCD a i 1012 GeV 400 MeV ✓ ◆ ✓ ◆ If the axion exists during inﬂation, it acquires isocurvature ﬂuctuations. ⌦ ✓ a =2 i ⌦a ✓i

a = Hinf /2 T QCD

T QCD a a Axion isocurvature perturbations

Adiabatic perturbation Isocurvature perturbation ρ ρ photon photon

DM/baryon DM/baryon x x ⌦ ⌦ ⌦ 2✓ ⌦ H S = a a = a i = a inf ⌦CDM ⌦a ⌦CDM ✓i ⌦CDM ⇡✓ifa

S Planck 2015 iso = P < 0.038 (95% CL) (Planck TT, TE, EE + lowP) + S PR P CMB angular power spectrum

Adiabatic

Isocurvature

Planck CMB angular power spectrum

Adiabatic

Isocurvature

⌦ ✓ ⌦ H S =2 a i = a inf ⌦CDM ✓i ⌦CDM ⇡✓ifa (Taken from Kawasaki’s slide) Isocurvature constraint on Hinf

r = 0.01 9 10 r = 0.1 108 r = ⌦a/⌦DM 107 r = 1

106 7 8 105 Hinf 10 GeV 104 GeV / 103 inf 102 Anharmonic H 101 effects 100 - 10 1 - 10 2 - 10 3 109 1010 1011 1012 1013 1014 Kobayashi, Kurematsu, FT, 1304.0922 fa / GeV Axion DM is in severe tension w/ many inﬂation models! Solutions to isocuvature problem

1)Restoration of Peccei-Quinn symmetry during inﬂation. Linde and Lyth `90 Lyth and Stewart `92

Figure taken from M. Kawasaki’s slide Solutions to isocuvature problem

1)Restoration of Peccei-Quinn symmetry during inﬂation. Linde and Lyth `90 Lyth and Stewart `92 • Axions are produced from domain walls and 10 axion DM is possible for fa = 10 GeV.

Hiramatsu, Kawasaki, Saikawa and Sekiguchi, 1202.5851,1207.3166 Solutions to isocuvature problem

1)Restoration of Peccei-Quinn symmetry during inﬂation. Linde and Lyth `90 Lyth and Stewart `92 • Axions are produced from domain walls and 10 axion DM is possible for fa = 10 GeV.

Hiramatsu, Kawasaki, Saikawa and Sekiguchi, 1202.5851,1207.3166 2)(Super-)Planckian saxion ﬁeld value during

inﬂation. (Saxion could be the inﬂaton) Linde and Lyth `90 Linde, `91

Axion: phase component Saxion: radial component Solutions to isocuvature problem

3)Resonant mixing with another axion. Kitajima, FT 1411.2011 cf. Hill, Ross `88

2 2 4)Heavy axions during inﬂation ma & Hinf

• Stronger QCD during inﬂation cf. Dvali, `95, Jeong, FT 1304.8131 Choi et al, 1505.00306

• Extra explicit PQ breaking Dine, Anisimov hep-ph/0405256 Higaki, Jeong, FT, 1403.4186, Barr and J.E.Kim, 1407.4311 FT and Yamada 1507.06387

a The extra PQ breaking term must be suﬃciently suppressed at present. Solutions to isocuvature problem

In the following, I will talk about two solutions and their implications:

(1) The Witten effect Explicit PQ breaking (2) Alignment mechanism Restoration of PQ symmetry Witten effect Witten `79 Let us first consider the θ-term of hidden U(1): This θ is not the ✓e2 ✓e2 strong CP phase. = F F˜µ⌫ = E B L✓ 32⇡2 µ⌫ 8⇡2 · which usually has no effect as it is a total derivative. The term, however, has an interesting effect in the monopole background. B B +θ-term

Mono Dyon E pole

QM =4⇡/e QM =4⇡/e Q =0 E 6 Witten effect Witten `79 Consider a static EM field and a monopole sitting at the origin: due to Coleman Q rˆ E = grad = A0 B = A + M r r⇥ 4⇡ r2 Then, the Lagrangian is Monopole ✓e2 L = d3r E B ✓ 8⇡2 · Z ✓ ◆ ✓e2 Q rˆ = d3r A0 A + M 8⇡2 r · r⇥ 4⇡ r2 Z ✓ ◆ ✓ ◆ ✓e2Q rˆ = d3rA0 M 8⇡2 r 4⇡r2 Z ✓ ◆ ✓ ◆ ✓e2Q = d3rA0 M (3)(r) 8⇡2 Z ✓ Electric◆ charge at the origin Witten effect Witten `79

A monopole acquires an electric charge, B and becomes a dyon if ✓ =0 . 6 Dyon E e✓ Q = ne + n =0, 1, 2, E 2⇡ ± ± ···

A monopole magnetic charge Q M =4 ⇡ /e is used. QM =4⇡/e Q =0 E 6 The mass of dyons is heavier than the monopole mass: 1 m2 ✓ 2 M = M + W n D M 2 M 2⇡ M ✓ ◆ v Q2 + Q2 ' M E The non-zero value ofq θ costs more energy as it induces an electric ﬁeld around the monopole. Axion coupled to hidden monopoles

Kawasaki, FT, Yamada, 1511.05030 Consider the axion coupling to U(1)H:

2 2 e a ˜µ⌫ e a ✓ = 2 Fµ⌫ F = 2 E B L 32⇡ fa 8⇡ fa ·

Then, the QCD axion acquires an extra mass about a =0 by the Witten eﬀect in the presence of hidden monopoles:

2 ↵ nM ma,M 2 2 ' 16⇡ rcfa cf. Fischler and Preskill, `83 e2 nM :(anti-)monopole number density ↵ = 4⇡ 1 1 rc = : monopole radius ' ev mW •The extra axion mass naturally decreases as the monopole density is diluted by the cosmic expansion. Consistent w/ the neutron EDM constraint. Evolution of axion

•The axion starts to oscillate when m 2 H 2. a,M ' 2 ⌦ h2 f T 65 GeV↵2 M a . osc,1 ' 0.12 1012 GeV ✓ ◆✓ ◆ Witten eﬀect a •No extra axion coherent oscillations are induced during the QCD phase transition if m /H (10) . a,M & O ⌦ h2 0.55 f 1012 GeV↵1.1 M . a . 0.12 ✓ ◆ “Adiabatic suppression mechanism” Linde `96. +QCD instanton Witten eﬀect a Axion abundance

Kawasaki, FT, Yamada, 1511.05030 The ﬁnal axion abundance is given by

3 2 4 2 2 0.12 fa ⌦ h 3 10 ↵ ✓ a ' ⇥ i ⌦ h2 1012 GeV ✓ M ◆✓ ◆

The axion abundance is suppressed, and it is inversely proportional to the monopole abundance!

The Witten eﬀect also solves the axion domain wall problem when the PQ symmetry spontaneously broken after inﬂation. QCD axion and monopole DM

0.12 Monopole dominated DM 2 3 2 12 3 0.10 θinifa /α =(4×10 GeV)

0.08 Ω a h 2 + 2 Ω m 2h h ⌦M h 0.06 2 m = 0.12 Ω 0.04 2 3 2 12 3 θinifa /α =(2×10 GeV) Axion dominated 0.02 2 3 2 12 3 θinifa /α =(1×10 GeV) DM

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 2 ⌦ h 2 Kawasaki, FT, Yamada, 1511.05030 Ωaah Hidden monopole DM

The monopoles with mass M =(1 10) PeV account for M the observed DM. Murayama, Shu 0905.1720

In the minimal case of the ’t Hooft-Polyakov monopole w/o other charged particles, the massive gauge bosons give a

larger contribution. Baek, Ko, Park 1311.1035, Khoze, Ro 1406.2291 e.g. The observed DM is realized with v 102 TeV, ↵ = (0.1) ' O for which the monopole fraction is O(10)%.

N.B. The massive gauge boson abundance can be reduced by adding matter ﬁelds. Discussion

• The monopoles (as well as W’) have self-interactions which may solve the small-scale tension of CDM. Baek, Ko, Park 1311.1035, Khoze, Ro 1406.2291 • The monopole DM may also have the SM electromagnetic mini-charges via the kinetic mixing:

a µ⌫ (EM) v µ⌫ (EM) Fa Fµ⌫ F3 Fµ⌫ a =(0, 0,v) M M h i 2 13 For v 10 TeV,M Mp the kinetic mixing is 10 ⇠ ⇠ (& magnetic) ⇠ and it is consistent with observations. cf. Jaeckel and Ringwald, 1002.0329

• If U(1)H is broken by another Higgs, monopoles and W’ become unstable and decay into the SM particles via the portal to the SM Higgs. The QCD axion abundance can be

suppressed for even larger f a . Kawasaki, FT, Yamada, 1511.05030 cf. Nomura, Rajendran, Sanches, 1511.06347 Aligned QCD axion

Higaki, Jeong, Kitajima, FT, 1512.05295, 1603.02090, Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 No axion isocurvature perturbations if the PQ symmetry is restored during or after inﬂation.

Is high TR or Hinf necessary? Classical axion window: TR,Hinf & Fa 9 12 10 GeV . Fa . 10 GeV Decay constant = PQ breaking scale?

In a simple set-up, V () F h i⇠ a : PQ scalar

However, this is not necessarily the case. If there are multiple PQ scalars, F h i⌧ a is possible. Alignment mechanism

The decay constant can be enhanced by the largest hierarchy among the PQ charges in the alignment mechanism,

See also Sikivie `86 Kim, Nilles, Peloso, hep-ph/0409138 Choi, Kim, Yun, 1404.6209, Higaki, FT, 1404.6923 Harigaya and Ibe, 1407.4893, Choi and Im, 1511.00132, Kaplan and Rattzzi, 1511.01827.

Clockwork axion model with N=2: 2 2 V = ( m2 2 + 4) 1 i | i| i| i| i=1 X 3 + ✏ 12 +h.c. fi + si iai/fi i = e p2

2 2 2 3f1a1 f2a2 fa = 3 f1 + f2 ,a= fa q Alignment with multiple axions

Choi, Kim, Yun, 1404.6209, Higaki, FT, 1404.6923 Harigaya and Ibe, 1407.4893, Choi and Im, 1511.00132, Kaplan and Rattzzi, 1511.01827.

N V = m2 2 + 4 i | i| i| i| i=1 N 1 X 3 + ✏ ii+1 +h.c. i=1 X f 3N f a ⇠ Aligned QCD axion

Higaki, Kitajima, FT, 1408.3936 , Higaki, Jeong, Kitajima, FT, 1512.05295. In general, if we have N 1 a a V (a )= ⇤4 cos i + n i+1 , i i f i f i=1 i i+1 X ✓ ◆ N-1 of the N axions becomes massive, leaving one massless mode, a. N N N N 1 i 1 2 2 2 a = ( 1) nj fiai, with fa = nj fi , fa 0 1 0 1 i=1 j=i i=1 j=i X Y X Y @ A @ A Adding a coupling to the PQ quarks = y QQ.¯ L q N N a becomes the QCD axion with Fa = fa 3 f for fi = f ⇠ There are many axions and saxions at f (e.g. at TeV scale) much lower than the conventional axion window! Aligned QCD axion

Higaki, Jeong, Kitajima, FT, 1512.05295. •The PQ symmetry is likely restored in the early Universe, avoiding the isocurvature bound.

•High quality of the PQ symmetry is partially explained.

•One of the axions or saxions may be responsible for the 750GeV diphoton excess. gg a ! 750 ! gg s or s a a 2() ! 750 ! 750 ! i i ! ✓Its coupling to gluons and raison d’être are due to the QCD axion (DM).

See also Chiang, Fukuda, Ibe and Yanagida 1602.07909, Gherghetta, Nagata, Shifman 1604.01127, Dimopoulos, Anson, Huang, Marques-Tavares, 1606.03097 for visible axion as the origin of the diphoton excess. ATLAS-CONF-2015-081, CMS-PAS-EXO-15-004 Quality of U(1)PQ

Higaki, Jeong, Kitajima, FT, 1512.05295, 1603.02090, In the conventional scenario, one needs to suppress PQ

breaking terms up to high order Carpenter, Dine, Festuccia, `09

n+4 e.g. n>10 for F 1012 GeV n h i⇠ a ⇠ Mp

In the aligned QCD axion models, the PQ symmetry breaking scale is much smaller, F h ii⌧ a which relaxes the required high quality of the PQ symmetry. Quality of U(1)PQ

Higaki, Jeong, Kitajima, FT, 1512.05295, 1603.02090, For instance, let us consider 6 6 1 V6 = 2 +h.c., 6 Mp which gives an extra contribution to the axion mass,

Assuming a CP phase of order unity, we expect

1 10 Fa/f1 ✓¯ 10 ⇠ 108 ✓ ◆ which satisﬁes the neutron EDM bound for enhancement greater than 108.

Dim. 6 Planck-suppressed PQ symmetry breaking operators 8 are allowed if Fa / f > 10 . Topological defects

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 Cosmic strings appear when the PQ symmetry is broken, and their tension is dominated by the gradient energy outside the core. f = ei✓ in cylindrical coordinate p2 R 1 @ 2 R µ µ + 2⇡rdr ⇡f 2 ln , ⇠ core r @✓ ⇡ Z ✓ ◆ R : distance b/w strings

: core radius (credit: M. Hindmarsh)

In the aligned QCD axion, the tension of each string is of order f 2, much smaller than Fa2. Any correspondence between the two? Topological defects

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 There is an isolated string-wall solution, “string bundle”, where many strings are glued by domain walls:

string

wall N =2 N =3

The aligned structure in the ﬁeld space exhibits itself in the real space! Topological defects

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 There is an isolated string-wall solution, “string bundle”, where many strings are glued by domain walls: N =2

The aligned structure in the ﬁeld space exhibits itself in the real space! Topological defects

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 The tension of the isolated string-wall system is equal to that of a single PQ ﬁeld.

2(N 1) 2 2 2 2 R 2 R µe↵ ⇡(3 f1 + +3 fN 1 + fN )ln = ⇡Fa ln ' ··· ✓ ◆ ✓ ◆ Topological defects

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 However, such string bundles are probably not produced in the Universe, as they require exponentially large hierarchy in the cosmic string distribution.

N i #(S ) #(S¯ ) =3 i i

Initial condition, #(S ) #(S¯ ) = (1). or scaling law: i i O Numerical simulation

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552

N =2

Strings glued by walls remain, and domain walls disappear in the case of N=2. Numerical simulation

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552

N =3

Domain walls remain, stretching between strings, in the case of N > 2. They follow the scaling law. Gravitational waves from domain walls

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 The domain walls annihilate at the QCD phase transition, producing a signiﬁcant amount of gravitational waves.

Hiramatsu, Kawasaki, Saikawa, Sekiguchi, 1202.5851

Gravitational waves Gravitational waves from domain walls

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552

The tension of walls: =8m f 2 m p✏f aH aH ⇠

The energy density of walls: ⇢ H dw ⇠

M 2 (H 2)2 Amount of GWs: EGW G G 1 ⇠ R ⇠ H

The peak frequency is determined by the Hubble horizon at the QCD phase transition. Gravitational waves from domain walls

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 Frequency corresponding to the Hubble radius at the QCD phase transition. Gravitational waves from domain walls

Higaki, Jeong, Kitajima, Sekiguchi, FT, 1606.05552 GW density parameter:

4 4 6 2 11 g 3 Tann f ⌦ (⌫ )h 2 10 ✏ ⇤ GW peak ' ⇥ 80 1 GeV 100 TeV ⇣ ⌘ ✓ ◆ ✓ ◆ 3 with a frequency dependence, ⌦GW (⌫) ⌫ , for ⌫ < ⌫ . / peak Hiramatsu, Kawasaki, Saikawa, 1309.5001

Pulsar timing constraint:

2 10 8 ⌦GW h < 2.3 10 at ⌫1yr 3 10 Hz ⇥ ' ⇥ P. D. Lasky et al. 1511.05994

Constraint on the PQ breaking scale:

1/6 f 200 TeV ✏ . ⇥ Will be improved by a factor of 2 by future SKA.

credit: D. J. Champion Summary

The axion isocurvature problem point to either

(1)low-scale inﬂation, or

(2)various interesting (exotic) possibilities (symmetry restoration, axion string-wall system, hidden monopoles, many (s)axions, GWs), Summary

✓The Witten eﬀect suppresses the QCD axion B abundance, thereby relaxing the isocurvature E bound.

✓Hidden monopoles are self-interacting and mini- charged DM. Summary

✓The Witten eﬀect suppresses the QCD axion B abundance, thereby relaxing the isocurvature E bound.

✓Hidden monopoles are self-interacting and mini- charged DM.

✓In the aligned QCD axion, the PQ symmetry is likely

2 restored, avoiding the isocurvature bound. 1

✓Robust against Planck suppressed PQ symmetry breaking terms.

✓Domain walls remain until the QCD phase transition, producing gravitational waves thru annihilation.