Production mechanisms for cold dark matter axions
Ken’ichi Saikawa (Institute for Theoretical Physics, Kanazawa University)
25 January 2021, Axions Beyond Gen 2, University of Washington (Online) 1/25 Axion dark matter mass?
[GeV] 1017 1016 1015 1014 1013 1012 1011 1010 109 108 107 106
Possibility of dark matter Excluded (Astrophysics)
10-4 10-3 10-2 10-1 1 10 102 103 104 105 106 107 [ eV] • Relic axion abundance depends on the Peccei-Quinn scale, and hence on the axion mass.
• One can guess the axion DM mass from its relic density.
2/25 Misalignment mechanism
[Preskill, Wise and Wilczek (1983); Abbott and Sikivie (1983); Dine and Fischler (1983)] Equation of motion for • Important input: Topological susceptibility
[Wantz and Shellard, 0910.1066] ×1012
a 25 102 1.5 10-1 Na 20 ma < 3H 0 10 -2 axion is frozen 15 10 1 -2 10 -3 axion number Na 10 10 a a ]
N -4 0.5 is conserved 10 5 -4 10-4
[fm -6 100 150 200 250 χ 10 0 0 10-8 -5 -0.5 10-10 -10 0 0.5 1 1.5 2 2.5 -12 10 10 100 200 500 1000 2000 1 [Borsanyi et al., 1606.07494] T[MeV] 10-1 10-2 10-3 Late times, axion comoving number -4 • 10 -5 m 3H 10 a ≈ is conserved. 10-6 axion starts rolling, 10-7 turns into pressureless matter. 10-8 -9 10 0 ma / ma 10-10 -11 0 10 3 H / ma 0 0.5 1 1.5 2 2.5 1/T (Gev-1) 3/25 Initial condition (I): Pre-inflationary scenario
• Assume that Peccei-Quinn (PQ) symmetry is never restored after inflation. • Relic axion abundance depends on the initial misalignment angle.
[Borsanyi et al., 1606.07494] [Armengaud et al., 1904.09155] 4/25 Initial condition (II): Post-inflationary scenario
[Armengaud et al. 1904.09155] • Naive calculation: Use the misalignment formula with angle-average. • But topological defects could have an important role. [Davis (1986)] 5/25 Production mechanisms and prediction for dark matter mass
1016 1015 1014 1013 1012 1011 1010 109 108
Misalignment mechanism [pre-inflationary] tuned tuned dominant / subdominant subdominant
Strings and domain walls (NDW = 1) [post-inflationary] dominant overclosure subdominant (uncertainty?)
Long-lived domain walls (NDW > 1, e.g. DFSZ models) [post-inflationary] [Kawasaki, KS and Sekiguchi, 1412.0789; Ringwald and KS, 1512.06436]
overclosure dominant / subdominant
Kinetic misalignment mechanism [pre-inflationary] [Co and Harigaya, 1910.02080; Co, Hall and Harigaya, 1910.14152]
reduced to conventional misalignment mechanism dominant / subdominant
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
6/25 Production mechanisms and prediction for dark matter mass
1016 1015 1014 1013 1012 1011 1010 109 108
Misalignment mechanism [pre-inflationary] tuned tuned dominant / subdominant subdominant
Strings and domain walls (NDW = 1) [post-inflationary] This talk dominant overclosure subdominant (uncertainty?)
Long-lived domain walls (NDW > 1, e.g. DFSZ models) [post-inflationary] [Kawasaki, KS and Sekiguchi, 1412.0789; Ringwald and KS, 1512.06436]
overclosure dominant / subdominant
Kinetic misalignment mechanism [pre-inflationary] [Co and Harigaya, 1910.02080; Co, Hall and Harigaya, 1910.14152]
reduced to conventional misalignment mechanism dominant / subdominant
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
6/25 Axionic strings
Position space
• Form when U(1)PQ symmetry is spontaneously broken. • Disappear around the epoch of the QCD phase transition. 7/25 Axionic strings
Position space
• Form when U(1)PQ symmetry is spontaneously broken. • Disappear around the epoch of the QCD phase transition. 7/25 8/25 Simulations of axion production in the post-inflationary scenario
• Kawasaki, KS and Sekiguchi (2014) [arXiv:1412.0789] lattice
• Fleury and Moore (2015) [arXiv:1509.00026]
• Klaer and Moore (2017) [arXiv:1707.05566, 1708.07521]
• Gorghetto, Hardy and Villadoro (2018) [arXiv:1806.04677]
• Buschmann, Foster and Safdi (2019) [arXiv:1906.00967]
• Gorghetto, Hardy and Villadoro (2020) [arXiv:2007.04990]
• Redondo, KS and Vaquero (2021?) (to be confirmed, RAVEN at MPCDF)
9/25 Difficulty in string dynamics
• Two extremely different length scales. • String core radius : mass scale of UV completion • Hubble radius • String tension acquires a logarithmic correction.
• Realistic value
• Difficult to reach the large-log regime for simulations with limited dynamical range ( ).
10/25 Scaling solution
• strings per horizon volume:
• The net energy density of radiated axions should be the same order.
11/25 Recent simulations observe a logarithmic growth of string density
[Gorghetto, Hardy and Villadoro, 2007.04990]
1.0
0.8
0.6
0.4
0.2 physical 0.0 3 4 5 6 7 8 9 log(mr/H) [Fleury and Moore, 1509.00026]
1.4 [Redondo, KS and Vaquero, work in progress]
[Kawasaki et al., 1806.05566] 2 1.2 -3 ζ=9.5 (v/M*=5x10 ) -3 ζ=23.9 (v/M*=2x10 ) -3 ζ=47.7 (v/M*=1x10 ) previous result w/ ζ=9.5 (v/M =5x10-3) 1.5 * 1.0
ξ ª
0.8 1 string parameter 0.6 0.5
Previous dynamic range Preliminary
No loop removal 0.4 0 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 10 100 1000 physical time t/d log(ms/H)
(However, full consensus has not been reached, cf. [Hindmarsh et al., 1908.03522].) 12/25 Issue of large log • “Scaling” solution suggests that the energy density of the system is of order
If , this leads to an enhancement by a factor of than typical density at QCD temperatures. • Does it imply an enhancement of axion abundance (and DM mass)? • Studying the large-log regime is a challenge. • Two possible approaches: 1. [Direct approach] use some effective description. 2. [Indirect approach] rely on a careful extrapolation.
13/25 Effective theory approach
• Integrate out (unresolvable) logarithmically distributed field and only describe string “core” and low-momentum axions.
• Kalb-Ramond (KR) effective action [Kalb and Ramond (1974)]
Nambu-Goto string axion field string-axion coupling used in the literature to study axion radiation analytically. [Davis and Shellard (1989); Dabholkar and Quashnock (1990); Battye and Shellard (1994); Battye and Shellard (1996)] 14/25 Numerical implementation of effective description
• 3D simulations of KR strings have not been performed. (But worked in 2D, [Fleury and Moore, 1602.04818].)
• Alternative method (trick): Theory with 1 vector field + 2 complex scalars. [Klaer and Moore, 1707.05566]
(q1,q2) = (4,3)
• Two phases, one is eaten by vector field and the other is identified as axion with a decay constant
15/25 Tunable effective tension [Klaer and Moore, 1707.05566]
• Tension is dominated by the local string core.
• It becomes relatively high compared to :
Pure global string Hybrid string
16/25 Simulations at high effective tension [Klaer and Moore, 1708.07521]
• String density increases.
• Axion production becomes less efficient than the estimate based on the angle-average misalignment calculation.
• Prediction for axion DM mass:
17/25 Spectrum of radiated axions: a controversy?
[Gorghetto, Hardy and Villadoro, 1806.04677] • Differential energy transfer rate
• Slope matters.
(IR modes dominate) Many soft axions. Enhancement of DM abundance.
(UV modes dominate) Few hard axions. Suppression of DM abundance.
• Results of effective tension approach point to a hard spectrum ( ). (unphysical) UV modes may not have decoupled, breakdown of effective theory?
18/25 Characterizing axion spectrum in the scaling regime
[Gorghetto, Hardy and Villadoro, 2007.04990]
1.1 physical
- 2 (mr/H)= 10 log 1.0 6
F 7 10-3 q 0.9 7.9 0.8 10-4 mr/2H physical 0.7 1 5 10 50 100 500 1000 6.0 6.5 7.0 7.5 8.0 8.5 9.0
x log(mr/H)
• Simulations with 1 complex scalar (not relying on the effective description), but limitation on the value of string tension, . • Evidence of growing spectral index. • When extrapolated, the spectrum will turn IR dominated ( ) at large log.
19/25 Extrapolation to large log
[Gorghetto, Hardy and Villadoro, 2007.04990]
Assume at large log (spectrum is IR-dominated). If the typical gradient of the IR field is set by ,
Interpreted as large field amplitude. space
20/25 Nonlinear effect
[Gorghetto, Hardy and Villadoro, 2007.04990] • Energy density of radiated axions at the epoch of QCD crossover
• Axion number is fixed only after (later than ). • This delay alleviates the log-enhancement of the axion number.
103
str 2 na 10 But it is still enhanced by , mis, = 1 • na 0 x = 5 implying higher axion DM mass: 0 10 10
30 1 10 102 103 104
log(mr/H )
21/25 Skepticism about an enhancement of the axion abundance
[Dine, Fernandez, Ghalsasi and Patel, 2012.13065]
• The axion is a compact field: Any enhancement of the energy density should come from higher gradient or higher .
• Enhancement of the string density should shift the IR cutoff to higher momentum, .
22/25 Revisiting axion radiation spectrum
[Redondo, KS and Vaquero, work in progress]
1.00 log(ms/H)=5.5
log(ms/H)=6.5 0.95 log(ms/H)=7.5 2 10°
0.90
F (x) q Preliminary 0.85
0.80 3 10°
0.75 Preliminary
100 101 102 103 0.70 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50
x = k/H log(ms/H)
23/25 Summary of axion dark matter mass prediction
• Current best guesses • Direct calculation with a high effective tension [Klaer and Moore, 1708.07521]
(less efficient than the angle-average misalignment) • Extrapolation with a scaling solution [Gorghetto, Hardy and Villadoro, 2007.04990]
(more efficient than the angle-average misalignment) • The discrepancy should be attributed to our limited knowledge on the spectrum of axions around the QCD epoch.
24/25 Conclusions • Prediction for axion dark matter mass is strongly related to physics at the early universe. • Uncertainty in the calculation for post-inflationary scenario: • Results of effective theory approach and extrapolation from the scaling solution appear to disagree. • Careful study of the low momentum axion distribution with improved dynamical ranges would be highly motivated (project for Beyond Gen 2 simulations).
1014 1013 1012 1011 1010 109 108
Misalignment mechanism [pre-inflationary] tuned tuned dominant / subdominant subdominant
Strings and domain walls [post-inflationary]
overclosure ? subdominant
Klaer 2017 Gorghetto 2020
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
25/25