#### Invisible ${{\mathit A}^{0}}$ (Axion) MASS LIMITS from Astrophysics and Cosmology

$\mathit v_{1}$ = $\mathit v_{2}$ is usually assumed ($\mathit v_{\mathit i}$ = vacuum expectation values). For a review of these limits, see RAFFELT 1991 and TURNER 1990 . In the comment lines below, D and K refer to DFSZ and KSVZ axion types, discussed in the above minireview.
VALUE (eV) CL% DOCUMENT ID TECN  COMMENT
• • We do not use the following data for averages, fits, limits, etc. • •
$>1.4 \times 10^{-21}$ 95 1
 2021
ASTR Fuzzy DM
$<1.9 \times 10^{4}$ 2
 2021
COSM warm dark matter
3
 2021
ASTR SN 1987A, axion-muon coupling
4
 2021
ASTR Microlensing
5
 2021
ASTR SN 1987A, ${{\mathit \Lambda}}$ decay
$\text{none } 1.3 - 2.7 \times 10^{-13}$ 6
 2021
$>2 \times 10^{-20}$ 95 7
 2021
COSM Lyman-$\alpha$
$\text{none } 0.8 - 6.5 \times 10^{-13}$ 95 8
 2021
$>2 \times 10^{-17}$ 9
 2020
COSM Isocurvature fluctuations
10
 2020
ASTR Compact binary systems
$>2.1 \times 10^{-21}$ 11
 2020
COSM Fuzzy DM
$\text{none } 6.4 - 8.0 \times 10^{-13}$ 95 12
 2020
$\text{none } 2.9 - 4.6 \times 10^{-21}$ 13
 2019
$\text{none } 10^{-21} - 6 \times 10^{-20}$ 14
 2019
ASTR Fuzzy DM
$\text{none } 1.1 - 4 \times 10^{-13}$ 95 15
 2019
$<0.06$ 16
 2018
ASTR K, SN 1987A
$<0.67$ 95 17
 2013 A
COSM K, hot dark matter
$\text{none } 0.7 - 3 \times 10^{5}$ 18
 2011
COSM ${}^{}\mathrm {D}$ abundance
$<105$ 90 19
 2011 A
CNTR D, solar axion
20
 2010
CAST K, solar axions
$<0.72$ 95 21
 2010
COSM K, hot dark matter
22
 2009
CAST K, solar axions
$<191$ 90 23
 2009 A
CNTR K, solar axions
$<334$ 95 24
 2009
HPGE K, solar axions
$<1.02$ 95 25
 2008
COSM K, hot dark matter
$<1.2$ 95 26
 2007
COSM K, hot dark matter
$<0.42$ 95 27
 2007 A
COSM K, hot dark matter
$<1.05$ 95 28
 2005 A
COSM K, hot dark matter
$3\text{ to }20$ 29
 1998
COSM K, hot dark matter
$<0.007$ 30
 1997
ASTR D, neutron star
$<4$ 31
 1997
ASTR D, neutron star cooling
$<(0.5 - 6){\times }\text{ 10}$$^{-3} 32  1997 ASTR SN 1987A <0.018 33  1995 ASTR D, red giant <0.010 34  1994 ASTR D, red giants, white dwarfs 35  1993 ASTR K, SN 1987A <0.01  1992 ASTR D, white dwarf <0.03  1992 C ASTR D, C-O burning \text{none 3 - 8} 36  1991 ASTR D, K, intergalactic light <10 37  1991 C COSM D, K, mass density of the universe, supersymmetry 38  1991 B ASTR D,K, SN 1987A <1 \times 10^{-3} 39  1991 ASTR K, intergalactic light \text{none } 10^{-3} - 3  1990 ASTR D,K, SN 1987A 40  1990 ASTR D,K, SN 1987A <0.02 41  1990 D ASTR D, red giant <1 \times 10^{-3} 42  1989 ASTR D,K, SN 1987A <(1.4 - 10){\times }\text{ 10}$$^{-3}$ 43
 1989
ASTR D,K, SN 1987A
$<3.6 \times 10^{-4}$ 44
 1989
ASTR D,K, SN 1987A
$<12$
 1988
ASTR D, Sun
$<1 \times 10^{-3}$
 1988
ASTR D,K, SN 1987A
45
 1988 B
ASTR red giant
$<0.07$
 1987
ASTR D, red giant
$<0.7$ 46
 1987
ASTR K, red giant
$\text{< 2-5}$
 1987
COSM K, thermal production
$<0.01$ 47
 1986
ASTR D, red giant
$<0.06$
 1986
ASTR D, red giant
$<0.7$ 48
 1986
ASTR K, red giant
$<0.03$
 1986 B
ASTR D, white dwarf
$<1$ 49
 1985
ASTR K, red giant
$\text{<0.003 - 0.02}$
 1984
ASTR D, K, neutron star
$>1 \times 10^{-5}$
 1983
COSM D,K, mass density of the universe
$>1 \times 10^{-5}$
 1983
COSM D,K, mass density of the universe
$<0.04$
 1983 B
ASTR D, red giant
$>1 \times 10^{-5}$
 1983
COSM D,K, mass density of the universe
$<0.1$
 1982
ASTR D, red giant
$<1$ 50
 1982
ASTR D, stellar cooling
$<0.07$
 1982 B
ASTR D, red giant
 1 BANIK 2021 use the subhalo mass function inferred from the analyses of the GD-1 and Pal 5 stellar streams. The limit is strengthened to $2.2 \times 10^{-21}$ eV when adding dwarf satellite counts.
 2 BAUMHOLZER 2021 study the freeze-in production of axion dark matter through couplings to photons, and set the limit using Lyman-$\alpha$ forest data and the observed number of Milky Way subhalos.
 3 CROON 2021 study the supernova cooling effect of the axion-muon coupling, taking account of semi-Compton scattering and muon-proton bremsstrahlung, as well as the loop-induced axion-photon coupling, and exclude the range of $\mathit g_{ {{\mathit A}} {{\mathit \mu}} {{\mathit \mu}} }$ $\simeq{}$ $7 \times 10^{-3} - 2 \times 10^{-10}$ for ${\mathit m}_{{{\mathit A}^{0}}}$ $<$ $0.5$ GeV. See their Fig. 8 for mass-dependent limits.
 4 FUJIKURA 2021 use the EROS-2 survey and the Subaru HSC observation to set limits on spherically symmetric axion clumps, taking account of the finite lens and source size effects. $\mathit f_{{{\mathit A}^{0}} }{ {}\gtrsim{} }$ $10^{12}$ GeV can be constrained depending on the fraction of the axion dark matter collapsed into clumps, and the clump densities. See their Figs. $7 - 10$ for the limits.
 5 MARTINCAMALICH 2021 considered axion emission from a supernova core through the ${{\mathit \Lambda}}$ hyperon decay, and set the limit on B( ${{\mathit \Lambda}}$ $\rightarrow$ ${{\mathit n}}{{\mathit A}^{0}}$ ) ${ {}\lesssim{} }$ $8 \times 10^{-9}$, or equivalently, $\mathit f_{{{\mathit A}^{0}} }/C_{sd}{ {}\gtrsim{} }$ $2.6 \times 10^{9}$ GeV in terms of the flavor-violating axion coupling to the down and strange quarks.
 6 NG 2021 use the binary black holes reported by LIGO and Virgo to determine the black hole spin distribution at formation and the scalar boson mass simultaneously, neglecting the boson self-interaction.
 7 ROGERS 2021 set the limit by using a framework involving Bayesian emulator optimization to accurately forward-model the Lyman-$\alpha$ flux power spectrum, and comparing this with small-scale data to constrain the predicted suppression of cosmic structure growth.
 8 TSUKADA 2021 look for a stochastic GW background produced by extragalactic BH-hidden photon cloud systems through the superradiant instability. They assume a uniform spin distribution at birth of isolated BHs from 0 to 1.
 9 IRSIC 2020 used the Lyman-$\alpha$ forest constraint on small-scale isocurvature perturbation to derive limits on the axion mass and decay constant, assuming that the axion makes up all dark matter in the post-inflationary scenario. See their Fig. 1 for other astrophysical limits as well as the limits on the case of the temperature-dependent axion mass.
 10 PODDAR 2020 used the observed decay in orbital period of four compact binary systems to derive a limit on the emission of axions with ${\mathit m}_{{{\mathit A}^{0}}}$ $<$ $1 \times 10^{-19}$ eV, assuming they couple to nucleons and the strong $\mathit CP$ phase vanishes at the potential minimum. They exclude $\mathit f_{{{\mathit A}^{0}} }{ {}\lesssim{} }$ $10^{11}$ GeV for such axions.
 11 SCHUTZ 2020 set a limit on fuzzy dark matter based on the existing limits for warm dark matter derived from the inferred subhalo mass function.
 12 SUN 2020 look for quasimonochromatic gravitational waves emitted from boson clouds around the Cygnus X-1 black hole. The quoted limit assume the black hole age of $5 \times 10^{6}$ years. A mass range of $9.6 - 15.5 \times 10^{-13}$ eV is disfavored when repeated induction of bosenova for string axions with decay constant $\mathit f_{{{\mathit A}^{0}} }$ $\simeq{}$ $10^{15}$ GeV prevents the superradiance from being saturated.
 13 DAVOUDIASL 2019 used the observed data of M87* by the Event Horizon Telescope to set the limit. A mass range of $0.85 - 4.6 \times 10^{-21}$ eV is disfavored for a spin-1 boson.
 14 MARSH 2019 considered heating of star clusters due to the stochastic oscillations of the core and granular quasiparticles in the outer halo. The limit was derived by requiring the survival of the old star cluster in Eridanus II, where the lower end is set by the validity of diffusion approximation. The effect of tidal stripping is also discussed for lower masses.
 15 PALOMBA 2019 used the LIGO O2 dataset to derive limits on nearly monochromatic gravitational waves emitted by boson clouds formed around a stellar-mass black hole. They exclude boson masses in a range of $1.1 \times 10^{-13}$ and $4 \times 10^{-13}$ eV for high initial black hole spin, and $1.2 \times 10^{-13}$ and $1.8 \times 10^{-13}$ eV for moderate spin. See their Figs. 2 and 3 for limits based on various values of black hole initial spin, boson cloud age, and distance.
 16 CHANG 2018 update axion bremsstrahlung emission rates in nucleon-nucleon collisions, shifting the excluded mass range to higher values. They rule out the hadronic axion with mass up to a few hundred eV, closing the hadronic axion window. See their Fig. 11 for results based on several different choices of the temperature and density profile of the proto-neutron star.
 17 ARCHIDIACONO 2013A is analogous to HANNESTAD 2005A. The limit is based on the CMB temperature power spectrum of the Planck data, the CMB polarization from the WMAP 9-yr data, the matter power spectrum from SDSS-DR7, and the local Hubble parameter measurement by the Carnegie Hubble program.
 18 CADAMURO 2011 use the deuterium abundance to show that the ${\mathit m}_{{{\mathit A}^{0}}}$ range 0.7$~$eV -- 300$~$keV is excluded for axions, complementing HANNESTAD 2010 .
 19 DERBIN 2011A look for solar axions produced by Compton and bremsstrahlung processes, in the resonant excitation of ${}^{169}\mathrm {Tm}$, constraining the axion-electron ${\times }$ axion nucleon couplings.
 20 ANDRIAMONJE 2010 search for solar axions produced from ${}^{7}\mathrm {Li}$ (478 keV) and ${}^{}\mathrm {D}({{\mathit p}}$ ,${{\mathit \gamma}}$ )${}^{3}\mathrm {He}$ (5.5 MeV) nuclear transitions. They show limits on the axion-photon coupling for two reference values of the axion-nucleon coupling for ${\mathit m}_{{{\mathit A}}}<$ 100 eV.
 21 This is an update of HANNESTAD 2008 including 7 years of WMAP data.
 22 ANDRIAMONJE 2009 look for solar axions produced from the thermally excited 14.4 keV level of ${}^{57}\mathrm {Fe}$. They show limits on the axion-nucleon ${\times }$ axion-photon coupling assuming ${\mathit m}_{{{\mathit A}}}<$ 0.03 eV.
 23 DERBIN 2009A look for Primakoff-produced solar axions in the resonant excitation of ${}^{169}\mathrm {Tm}$, constraining the axion-photon ${\times }$ axion-nucleon couplings.
 24 KEKEZ 2009 look at axio-electric effect of solar axions in HPGe detectors. The one-loop axion-electron coupling for hadronic axions is used.
 25 This is an update of HANNESTAD 2007 including 5 years of WMAP data.
 26 This is an update of HANNESTAD 2005A with new cosmological data, notably WMAP (3 years) and baryon acoustic oscillations (BAO). Lyman-$\alpha$ data are left out, in contrast to HANNESTAD 2005A and MELCHIORRI 2007A, because it is argued that systematic errors are large. It uses Bayesian statistics and marginalizes over a possible neutrino hot dark matter component.
 27 MELCHIORRI 2007A is analogous to HANNESTAD 2005A, with updated cosmological data, notably WMAP (3 years). Uses Bayesian statistics and marginalizes over a possible neutrino hot dark matter component. Leaving out Lyman-$\alpha$ data, a conservative limit is 1.4 eV.
 28 HANNESTAD 2005A puts an upper limit on the mass of hadronic axion because in this mass range it would have been thermalized and contribute to the hot dark matter component of the universe. The limit is based on the CMB anisotropy from WMAP, SDSS large scale structure, Lyman $\alpha$, and the prior Hubble parameter from HST Key Project. A ${{\mathit \chi}^{2}}$ statistic is used. Neutrinos are assumed not to contribute to hot dark matter.
 29 MOROI 1998 points out that a KSVZ axion of this mass range (see CHANG 1993 ) can be a viable hot dark matter of Universe, as long as the model-dependent $\mathit g_{ {{\mathit A}} {{\mathit \gamma}} }$ is accidentally small enough as originally emphasized by KAPLAN 1985 ; see Fig.$~$1.
 30 BORISOV 1997 bound is on the axion-electron coupling $\mathit g_{\mathit ae}<1 \times 10^{-13}$ from the photo-production of axions off of magnetic fields in the outer layers of neutron stars.
 31 KACHELRIESS 1997 bound is on the axion-electron coupling $\mathit g_{\mathit ae}<1 \times 10^{-10}$ from the production of axions in strongly magnetized neutron stars. The authors also quote a stronger limit, $\mathit g_{\mathit ae}<9 \times 10^{-13}$ which is strongly dependent on the strength of the magnetic field in white dwarfs.
 32 KEIL 1997 uses new measurements of the axial-vector coupling strength of nucleons, as well as a reanalysis of many-body effects and pion-emission processes in the core of the neutron star, to update limits on the invisible-axion mass.
 33 RAFFELT 1995 reexamined the constraints on axion emission from red giants due to the axion-electron coupling. They improve on DEARBORN 1986 by taking into proper account degeneracy effects in the bremsstrahlung rate. The limit comes from requiring the red giant core mass at helium ignition not to exceed its standard value by more than 5$\%$ ($0.025$ solar masses).
 34 ALTHERR 1994 bound is on the axion-electron coupling $\mathit g_{\mathit ae}<1.5 \times 10^{-13}$, from energy loss via axion emission.
 35 CHANG 1993 updates ENGEL 1990 bound with the Kaplan-Manohar ambiguity in $\mathit z={\mathit m}_{{{\mathit u}}}/{\mathit m}_{{{\mathit d}}}$ (see the Note on the Quark Masses in the Quark Particle Listings). It leaves the window $\mathit f_{\mathit A}=3 \times 10^{5}-3 \times 10^{6}$ GeV open. The constraint from Big-Bang Nucleosynthesis is satisfied in this window as well.
 36 BERSHADY 1991 searched for a line at wave length from $3100 - 8300$ $Å$ expected from 2${{\mathit \gamma}}$ decays of relic thermal axions in intergalactic light of three rich clusters of galaxies.
 37 KIM 1991C argues that the bound from the mass density of the universe will change drastically for the supersymmetric models due to the entropy production of saxion (scalar component in the axionic chiral multiplet) decay. Note that it is an $\mathit upperbound$ rather than a lowerbound.
 38 RAFFELT 1991B argue that previous SN$~$1987A bounds must be relaxed due to corrections to nucleon bremsstrahlung processes.
 39 RESSELL 1991 uses absence of any intracluster line emission to set limit.
 40 ENGEL 1990 rule out $10^{-10}~{ {}\lesssim{} }$ $\mathit g_{\mathit AN}{ {}\lesssim{} }~10^{-3}$, which for a hadronic axion with EMC motivated axion-nucleon couplings corresponds to $2.5 \times 10^{-3}~$eV ${ {}\lesssim{} }{\mathit m}_{{{\mathit A}^{0}}}{ {}\lesssim{} }$ $2.5 \times 10^{4}~$eV. The constraint is loose in the middle of the range, i.e. for ${\mathit g}_{\mathit AN}$ $\sim{}~10^{-6}$.
 41 RAFFELT 1990D is a re-analysis of DEARBORN 1986 .
 42 The region ${\mathit m}_{{{\mathit A}^{0}}}{ {}\gtrsim{} }$ 2 eV is also allowed.
 43 ERICSON 1989 considered various nuclear corrections to axion emission in a supernova core, and found a reduction of the previous limit (MAYLE 1988 ) by a large factor.
 44 MAYLE 1989 limit based on naive quark model couplings of axion to nucleons. Limit based on couplings motivated by EMC measurements is 2$-$4 times weaker. The limit from axion-electron coupling is weak: see HATSUDA 1988B.
 45 RAFFELT 1988B derives a limit for the energy generation rate by exotic processes in helium-burning stars $\epsilon$ $<$ 100 erg g${}^{−1}$ s${}^{-1}$, which gives a firmer basis for the axion limits based on red giant cooling.
 46 RAFFELT 1987 also gives a limit ${\mathit g}_{\mathit A{{\mathit \gamma}} }$ $<$ $1 \times 10^{-10}$ GeV${}^{-1}$.
 47 DEARBORN 1986 also gives a limit ${\mathit g}_{\mathit A{{\mathit \gamma}} }$ $<$ $1.4 \times 10^{-11}$ GeV${}^{-1}$.
 48 RAFFELT 1986 gives a limit ${\mathit g}_{\mathit A{{\mathit \gamma}} }$ $<$ $1.1 \times 10^{-10}$ GeV${}^{-1}$ from red giants and $<2.4 \times 10^{-9}$ GeV${}^{-1}$ from the sun.
 49 KAPLAN 1985 says ${\mathit m}_{{{\mathit A}^{0}} }$ $<$ 23 eV is allowed for a special choice of model parameters.
 50 FUKUGITA 1982 gives a limit ${\mathit g}_{\mathit A{{\mathit \gamma}} }$ $<$ $2.3 \times 10^{-10}$ GeV${}^{-1}$.
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 Also
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 Also
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