# High repetition rate and coherent Free-Electron Laser in the tender X-rays
based on the Echo-Enabled Harmonic Generation of an Ultra-Violet Oscillator
pulse

N. S. Mirian1, M. Opromolla2,3, G. Rossi2, L. Serafini3 and V. Petrillo2,3
1Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany  2Universitï¿œ
degli Studi di Milano,Via Celoria,16 20133 Milano, Italy  3INFN - Sezione di
Milano, Via Celoria 16, 20133 Milano and LASA, Via F. Cervi 201, 20090 Segrate
(MI), Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fine time-resolved analysis of matter - i.e. spectroscopy and photon
scattering - in the linear response regime requires a fs-scale pulsed, high
repetition rate, fully coherent X-ray source. A seeded Free-Electron Laser
(FEL) driven by a Super-Conducting Linac, generating $10^{8}$-$10^{10}$
coherent photons at 2-5 keV with about 0.5 MHz of repetition rate, can address
this need. The seeding scheme proposed is the Echo-Enabled Harmonic
Generation, alimented by a FEL Oscillator working at 13.6 nm with a cavity
based on Mo-Si mirrors. The whole chain of the X-ray generation is here
described by means of start-to-end simulations. Comparisons with the Self
Amplified Spontaneus Emission and a fresh-bunch harmonic cascade performed
with similar electron beams show the validity of this scheme.

###### pacs:

41.60. Cr

##  I Introduction

Fine time-resolved analysis of matter is currently performed with synchrotron
radiation (SR) sources or with X-ray Free Electron Lasers (FELs), whose
extremely brilliant pulses are able to detect matter in highly excited states
dominated by non-linear response. Spectroscopic studies and photoemission
experiments require probes with fluxes smaller than 108 photons/pulse and MHz-
class repetition rates, for remaining below the linear response threshold and
collecting adequate statistics. Current FELs’ photon number exceeds this level
by 2-4 orders of magnitude, requiring severe attenuation with huge waste of
energy. On one hand, sources based on Warm Linacs, operating at 10/120 Hz, are
inadequate for collecting statistics for high resolution spectroscopy. On the
other hand, signals like the one by EuXFEL XFEL ; xfel2 , shaped in thousands
micropulses grouped in 10 macropulses per second, are also non ideal for
spectroscopy as both attenuation is needed and the high repetition rate of the
micropulses overruns detector and pump-probe set-up capability. SASE
fluctuations severely limit the use of FELs in X-ray spectroscopy and full
seeding, routinely implemented at FERMI@Elettra fermi2  and SXFEL selfmod ,
should be extended to tender/hard X-ray energies.

There is therefore scientific need and ample room for a novel type of source:
a source delivering to the sample $10^{7}-10^{8}$ photons in 10 fs coherent
pulses at 0.5-2 MHz in the tender/hard X-ray range, thus bridging the gap in
time resolution and average photon flux between the most advanced SR and the
current FELs. These requests are addressed by conceiving a tailored seeded FEL
driven by Linacs based on Super-Conducting cavities, providing
$10^{8}$-$10^{10}$ coherent photons at 2-5 keV, at about 1 MHz of repetition
rate.
In the seeded FEL configuration, an external coherent pulse imprints its
temporal phase on the electron beam at the undulator entrance. The direct
seeding directseeding  is not possible in the soft-hard X-ray range due to the
lack of high-power coherent seeds at these wavelengths, while self-seeding
processes self1 ; self2  only achieve partial coherence. High Gain Harmonic
Generation (HGHG) multistage cascades lee , seeded by the harmonics of an IR
laser generated in crystals dcls  or in gases HHG ; hhg2 ; Sparc_cascata ;
sparc in gas , have been demonstrated and applied up to few nm wavelengths
fermi2 . However, their implementation in the tender/hard X-ray range is
highly demanding, while the extention to higher repetition rates, obtained by
using oscillators NJP_Oscill  or lasers in cavity desy , has been studied
sofar only theoretically. As demonstrated at SXFEL selfmod , a step towards
high repetition rate seeded FELs is also foreseen by using an optical
klystron-like configuration, allowing to reduce the requirement for the peak
power of the seed laser. FEL oscillators oscill ; dattoli-1 ; Freund ; Xfelo2
; xfelo3 ; instru  or regenerative amplifiers reg ampl-1 ; mcneil ; lind ;
rafel_2019 ; Marcusconfronto  could directly produce coherent X-rays oscill
seed-1 ; NJP_Oscill , but the operational scenario proposed so far, with
electrons at several GeVs, impedes their realization in small/medium size
research laboratories. UV/soft X-ray coherent radiation has been generated
with the Echo-Enabled Harmonic Generation (EEHG) EEHG0 ; EEHG ; EEHG-1 ;
EEHG-2 ; EEHG-3 , a technique which requires two coherent radiation pulses,
usually delivered by optical lasers, seeding the electron beam in two
sequential modulators interspersed by a strong dispersive section. A second
dispersive section and the radiator are placed dowstream. As explained in
Ref.s EEHG0 ; EEHG , the combination of energy modulation and dispersion,
replicated twice, warps the electron longitudinal phase space, producing a
significant bunching on very high harmonics which drives the emission of a
short wavelength coherent pulse in the radiator. A combined EEHG-oscillator
scheme with two oscillators as modulators has also been proposed eehg-osc .

Table 1: Electron beam for MariX FEL.  electron beam energy | GeV | 2-3.8
---|---|---
Charge | pC | 30-50
Current | kA | 1.6
rms normalized emittance | mm mrad | 0.3-0.5
rms relative energy spread | $10^{-4}$ | 3
electron beam duration | fs | 12
slice energy spread | $10^{-4}$ | 4-2
slice normalized emittance | mm mrad | 0.3

In this paper, we show the operation of a FEL in the tender X-ray range, based
on EEHG. We propose a FEL Oscillator in the far Ultra-Violet frequency range
as seeding source. The advantage of such a scheme is twofold: the oscillator
operation, combined with an electron beam accelerated in a Super-Conducting
Linac, increases the device repetition rate by many orders of magnitude (4-6).
Moreover, since the oscillator frequency and peak power are much higher than
the ones of a conventional laser, the soft-hard X-ray range can be more easily
reached by a lower harmonic number.

##  II Lay-out of the coherent source and simulations

The electron beam is supposed to be generated by an accelerator similar to the
project MariX’s (Multi-disciplinary Advanced Research Infrastructure for the
generation and application of X-rays) mariX1-1 ; bacciprab-1 , whose compact
footprint with a total length of about 500 m and contained costs should permit
its construction also in medium-size research infrastructures or within
university campuses. MariX is based on the innovative design of a two-pass
two-way Super-Conducting linear electron accelerator bacciprab-1 , equipped
with an arc compressor arco-1 ; arco2-1  to be operated in CW mode at 1 MHz.
The characteristics of the electron beam are listed in Table 1.

Figure 1: Scheme of the Echo-Enhanced Harmonic Generation FEL.
$\mathrm{b}_{1}$ and $\mathrm{b}_{2}$: electron bunches alternatively sent in
the oscillator and in the undulators. FEL Oscillator undulator:
$\lambda_{w}$=5 cm, $a_{w}$=3.71, total length 9 m. $m_{1}$, $m_{2}$, $m_{3}$,
$m_{4}$: oscillator cavity mirrors. D: beam dump. $m_{5}$, $m_{6}$, $m_{7}$,
$m_{8}$: mirrors of the optical line that splits and couples the seed to the
modulators. $m_{2}$ and $m_{5}$: beam splitters. First ($M_{1}$) and second
($M_{2}$) modulators: $\lambda_{w}$=5 cm, $a_{w}$=3.71, respectively 1.6 m and
1.5 m long. The two chicanes $C_{1,2}$ are both 0.45 m long. Radiator: about
20 m long, $\lambda_{w}$=1.5 cm, maximum magnetic field: about 1 T.

As studied and demonstrated in Ref. hom , the superconducting technology
allows to achieve low jitters and fluctuations of the electron beam. These
increased stability conditions, together with the seeded mode FEL operation,
give the possibility to produce a fully coherent, high repetition rate and
highly stable x-ray source.

Fig. 1 shows the scheme of the source. After the acceleration stage,
successive electron bunches are alternatively driven in the oscillator or
matched to the EEHG undulator device. The electron beam alimenting the
oscillator is extracted upstream the linac end at an energy of 2 GeV, while
the ones entering the EEHG device may have the same energy or be further
accelerated. The oscillator is constituted by a 9 m long undulator segment
with period $\lambda_{w}=5$ cm and produces 70 $\mu J$ of intracavity
radiation at $\lambda_{O}=$13.6 nm. It is embedded into a folded ring cavity
composed by 4 mirrors, two of which focusing, with optics heat loading
requiring an intelligent cooling system. For an oscillator repetition rate of
0.5 MHz, the round trip length $L_{c}$ is 600 m and the distance between two
mirrors is $L_{c}/4=150$ m. The oscillator supermodes oscill  are calculated
fully numerically mcneil ; NJP_Oscill  by extracting the radiation field
simulated by GENESIS 1.3 genesis-1  from the oscillator, driving it through
the optical line accounting for mirrors and propagation, and superimposing it
on the successive electron bunch. The microscopic distribution of the electron
beam is changed shot to shot in order to simulate the passage of a sequence of
different bunches. After the passage into the oscillator, the electron bunch
is deteriorated by the radiation process and driven to the dump.

Fig. 2 presents the intracavity pulse temporal and spectral densities of the
seed at saturation, whose Table 2 summarizes the characteristics of the
intracavity seed pulse at saturation. After an optical transport line that
splits it in two pulses, the seed is synchronized to the electron bunches at
the beginning of both modulators. Energy losses along the transport line have
been taken into account. The oscillator seed pointing stability and transverse
overlap with the electron bunches after the transport line will be checked and
adjusted with multipurpose stations and beam stabilization systems as the ones
described in end_stat1 ; end_stat2 .

Figure 2: Signal (a) and spectrum (b) of the intracavity power of the oscillator at saturation. Table 2: Characteristics of the seed, generated by the FEL oscillator with undulator length $L_{w}$=9 m. The repetition rate of the source is 0.5 MHz. $=Photons/s/$mm^{2}$/$mrad^{2}$/bw(‰). $\lambda_{O}$(nm) | 13.6 | E($\mu$J) | 70
---|---|---|---
Photons/shot | 4.5$\times 10^{12}$ | Photons/s | 2.27$\times 10^{18}$
relative bandwidth | 1.7$\times 10^{-3}$ | rms length($\mu m$) | 2
div($\mu rad$) | 50 | size($\mu$m) | 80
peak brilliance($) | 2.4$\times 10^{31}$ | average brilliance($) | 8$\times 1$$0^{22}$
Figure 3: Radiation energy E(J) vs z(m). In blue, radiation at
$\lambda_{o}=$13.6 nm in the modulators. In violet, radiation at
$\lambda=$13.6/25 nm=5.44 Å in the radiator. Point (a) is the $1^{st}$
modulator end, (b) $1^{st}$ chicane end, (c) $2^{nd}$ modulator end and (d)
$2^{nd}$ chicane end. Inner figures: (i) bunching on the $25^{th}$ harmonics
at the end of the $2^{nd}$ chicane, (ii) power density, (iii) spectral
amplitude (SASE at 40 m and EEHG at the radiator exit). Windows (a), (b), (c),
(d): electron phase space, electron Lorentz factor $\gamma$ vs seed phase
$\varphi$, in the corresponding points. (e) and (f): SASE (red) and EEHG
(blue) coherence degree for one pulse $\Gamma$ and for different pulses
$\Gamma_{ij}$ vs $\varsigma$=c$\tau$. Dotted segments: FWHM coherence lengths.

To obtain radiation in the range $\lambda=$5-2 Å, namely 2-5 keV, with the
MariX’s moderate energy electron bunch (2.5-3.8 GeV), a short period radiator
must be considered. From the resonance relation
$\lambda=\frac{\lambda_{w}}{2\gamma^{2}}(1+a_{w}^{2})$ ($a_{w}$ is the
undulator parameter and $\gamma$ the electron Lorentz factor), taking a
maximum magnetic field B=1T corresponding to $a_{w}$=0.98, we can deduce that
an undulator period of $\lambda_{w}$=1.5 cm is suitable. Besides, starting
from a seed at $\lambda_{o}=$ 13.6 nm, a significant electron bunching on
harmonics’ order n in the range from 25 to 70 is required. By following the
uni-dimensional model based on plane waves exposed in Ref.s EEHG0 ; EEHG , the
electron bunching is expressed in terms of four free parameters, namely the
dispersion strengths of the two chicanes $R_{56,1}$ and $R_{56,2}$ and the
normalized electric fields of the seeds:

$A_{1,2}=\frac{ea_{w}\lambda_{O}}{mc^{2}\pi\sigma_{1,2}}\sqrt{\frac{P_{1,2}}{c\varepsilon_{0}\pi}},$

$P_{1,2}$ and $\sigma_{1,2}$ being the peak power and the rms transverse
dimension of the seeds, e electron charge, m electron mass, c speed of light
and $\varepsilon_{0}$ the vacuum dielectric constant. Assuming
$P_{1,2}=100$,200 MW respectively (0.86 and 1.72 $\mu J$ of total energy), and
keeping the longitudinal lengths of the two chicanes constant, $R_{56,1}$ and
$R_{56,2}$ have been optimized to maximize the electron bunching on the
desired harmonics of the seed. Using an electron energy of 2.66 GeV and a
slice relative energy spread of $\Delta E/E=3\times 10^{-4}$, proper bunching
at the second chicane end is obtained for n=25 , corresponding to
$\lambda=5.44$ Å with $R_{56,1}$=132 $\mu m$ and $R_{56,2}$=4.72 $\mu m$. The
power growth in this case is shown in Fig. 3, its central windows showing the
electron phase spaces after the first modulator (a), after the first chicane
(b), after the second modulator (c) and at the radiator entrance (d). The
initial bunching on $\lambda=$5.44 Å (in violet) reaches a peak of 4.5% on the
bunch, while the rms energy spread increases from 0.03% to 0.045% in the
modulators and in the chicanes. These parameters are sufficient to trigger a
consistent FEL emission in the radiator, and the radiation is extracted at the
minimum bandwidth position, occurring after 12 m of radiator (about 16 m of
the total device). The neat single spike structures in power and spectral
amplitude at the end of the undulator, are shown in the inner windows in blue
and red respectively, compared with the corresponding SASE profiles extracted
after 40 m of undulator. Source coherence and stability are evaluated through
the modulus of the complex coherence degree:

$\Gamma_{ij}(\tau)=\left|\frac{\int dtE_{i}(t)E_{j}(t-\tau)}{\sqrt{\int
dt\left|E_{i}\right|^{2}}\sqrt{\int dt\left|E_{j}\right|^{2}}}\right|,$

between two different generic pulses i and j generated by different electron
beams ($\Gamma_{ij}$) or for one single pulse ($\Gamma=\Gamma_{i,i}$), where
$E_{i}$ and $E_{j}$ are the electric fields of the pulses as function of time.
These two quantities are shown vs $\varsigma=c\tau$ for n=25 in the right
windows of Fig. 3, together with those of the SASE case at saturation. The
FWHM coherence length of the EEHG pulse is $\mathfrak{L}_{c}=0.67\,\mu m$,
about 4 times the corresponding SASE’s. The equal time coherence between
different pulses $\Gamma_{ij}(0)$ is close to 1, differently from the SASE’s
of $10^{-1}$. The FWHM mutual coherence length, quantifying the shot-to-shot
stability, is 0.1 $\mu m$, compared to 0.015 $\mu m$ of the SASE one. The EEHG
coherent source provides an ultrashort pulse with 5 $\mu J$ of energy per shot
at $\lambda=$5.44 Å, corresponding to $1.4\mathrm{\times}10^{10}$ photon/shot
at 0.5 MHz or $0.7\times 10^{16}$ photons/s, while the SASE source reaches
$14.7\times 10^{16}$ photons/s at 1 MHz. Even if characterized by a smaller
number of photons, bandwidth, coherence and collimation of the EEHG give an
average brilliance of $1.7\times 10^{24}$ photons/s/$mm^{2}$/$mrad^{2}$/bw(‰),
being only a factor 5 smaller than the corresponding SASE’s.

Figure 4: SASE (black) and EEHG (red) power and spectral radiation profile for n=35 ($R_{56,1}$=184.7$\mu m$, $R_{56,2}$=4.86$\mu m$), 45 ($R_{56,1}$=281.6 $\mu m$, $R_{56,2}$=6.58$\mu m$) and 50 ($R_{56,1}$=293.2$\mu m$,$R_{56,2}$=6,07 $\mu m$). SASE power vs s=ct is on the left axis, EEHG power on the right. Spectra are in arbitrary units and not in scale. Table 3: SASE (extracted at saturation), EEHG (extracted at minimum bandwidth) and fresh-bunch HGHG cascaded FEL characteristics, $=Photons/s/$mm^{2}$/$mrad^{2}$/bw(‰). Mode |  | SASE | EEHG | HGHG | SASE | EEHG | HGHG | SASE | EEHG | SASE | EEHG
---|---|---|---|---|---|---|---|---|---|---|---
Harmonic number |  |  | 25 | 5$\times$5 |  | 35 | 7$\times$5 |  | 45 |  | 50
Wavelength | Å | 5.44 | 5.44 | 5.44 | 3.88 | 3.88 | 3.88 | 3.02 | 3.02 | 2.72 | 2.72
$\gamma$ |  | 5216 | 5216 | 5216 | 6151 | 6151 | 6151 | 6991 | 6991 | 7352 | 7352
Undulator length | m | 40 | 16 | 32 | 40 | 20 | 32 | 40 | 22 | 40 | 24
Energy | $\mu J$ | 58 | 5.1 | 7 | 58 | 0.9 | 3.2 | 46 | 0.9 | 42 | 0.18
Photon/shot | $10^{9}$ | 158 | 13.9 | 19 | 112 | 1.74 | 6.2 | 69 | 1.36 | 57 | 0.245
Bandwidth | 0.1% | 1 | 0.37 | 0.17 | 0.9 | 0.4 | 0.32 | 0.6 | 0.3 | 0.5 | 0.35
Length | fs | 10 | 1.5 | 3 | 10 | 0.8 | 4.5 | 10 | 1 | 10 | 1
Divergence | $\mathrm{\mu rad}$ | 3.7 | 3.9 | 3.6 | 2.7 | 3.2 | 5.1 | 2.2 | 2.5 | 2.05 | 2.3
Pulse size | $\mathrm{\mu m}$ | 35 | 27 | 24 | 32 | 25 | 29 | 27 | 25 | 27 | 30
Photon/s | $10^{15}$ | 158 | 7 | 4.7 | 112 | 0.87 | 3.1 | 69 | 0.67 | 57 | 0.09
Average brilliance |  $10^{23}$ $ | 94 | 17 | 37 | 151 | 3.39 | 5.51 | 325 | 5.71 | 251 | 0.51
Coherence length | $\mu m$ | 0.17 | 0.67 | 1.3 | 0.11 | 0.52 | 1 | 0.1 | 0.5 | 0.1 | 0.3

Working points covering the range between 5.44 and 2.7 Å, namely n=25, 35, 45
and 50, are presented in Table 3 and compared to similar SASE cases after 40 m
of undulator. Electron beams with progressively larger energies allow to reach
shorter wavelengths. Fig. 4 presents powers and spectral amplitudes for both
SASE (black) and EEHG (red) cases for n=35, 45, 50. The case with n= 35 is
performed with an electron beam of about 3 GeV and delivers 1.7$\times 10^{9}$
photons/shot at 3.88 Å, extracted after 16 m of radiator with an average
brilliance of $3.4\times 10^{23}$ photons/s/$mm^{2}$/$mrad^{2}$/bw(‰). With an
energy of 3.6 GeV, 1.36$\times 10^{9}$ photons/shot at 3.02 Å (n=45) can be
generated with an average brilliance of 5.7$\times 10^{23}$
photons/s/$mm^{2}$/$mrad^{2}$/bw(‰). Pushing the electron energy to the
maximum value foreseen for MariX, 3.8 GeV, the system produces 2.4$\times
10^{8}$ photons /shot at 2.7 Å (n=50) with a brilliance of 5.7$\times 10^{23}$
photons/s/$mm^{2}$/$mrad^{2}$/bw(‰). The coherence length is from 5 (n=35) to
3 (n=50) times the corresponding SASE’s, while the stability is larger by a
factor from 10 (n=35) to 20 (n=50). Since these estimations widely exceed the
target values set by the scientific case, the EEHG source will be capable to
satisfy the conditions requested by the envisaged experiments, considering a
safety margin compensating all the degradations due to errors, misalignment,
jitters and the losses dealing with the transport of the photon beams to the
experimental hutch. The comparison between the results of the EEHG technique
and those of a fresh-bunch HGHG cascade, performed with a similar electron
beam and with the same oscillator as seeding source NJP_Oscill , shows a
substantial equivalence in terms of number of emitted photons for n=5$\times$5
and a better performance of the HGHG cascade for n=7$\times$5, when the
induced energy spread limits the EEHG radiation. However, the HGHG efficiency
decreases very rapidly at higher harmonics (n>35) and radiation levels
comparable with the EEHG’s cannot be achieved. A total of four different
electron bunches per shot are needed for the fresh-bunch technique, while for
the EEHG only one bunch for the oscillator and one for the FEL are required,
doubling, in proportion, the radiation repetition rate. Regarding coherence, a
better performance of the HGHG cascade is observed, due to the more direct
transfer of the coherence properties from the seed to the radiation. The
advantage of the EEHG is a larger tunability and versatility of the source,
that permits to generate intense ($10^{10}-10^{8}$ photons/shot), ultra-short
(down to 1 fs) pulses at all the harmonics n of the seed up to n=50 and not
only at those corresponding to the product of two odd integer numbers
(n=5$\times$5, 3$\times$9, 7$\times$5…).

##  III Conclusions

A new generation accelerator complex is at the core of this coherent and
compact facility dedicated and optimized to ultra-fast coherent X-ray
spectroscopy and inelastic photon scattering, and to highly penetrating X-ray
imaging of mesoscopic and macroscopic samples. The X-ray generation scheme
here studied relies on a conventional EEHG device seeded by a far-UV FEL
Oscillator and reaching high harmonic orders up to n=50. Its comparison with a
SASE FEL performed with a similar electron beam proves the much higher
stability and coherence of the produced pulses. The major advantage with
respect to a fresh-bunch three-stage FEL cascade seeded by the same oscillator
is given by the tunability and simpler set-up of the EEHG scheme. Such
facility will be intrinsically multi-user and multidisciplinary as of the
research performed and science output.

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# AXION MASS IN THE CASE OF POST-INFLATIONARY PECCEI-QUINN SYMMETRY BREAKING

A. RINGWALD  Deutsches Elektronen-Synchritron DESY, Notkestr. 85, D-22607
Hamburg, Germany

###### Abstract

The axion not only solves the strong $CP$ puzzle, but it also may be the main
constituent of cold dark matter. We review the axion dark matter predictions
for the case that the Peccei-Quinn symmetry is restored after inflation.

##  1 Axion solution of the strong $CP$ puzzle and the axion mass

Already in the early days of Quantum Chromodynamics (QCD) it was realised that
the most generic Lagrangian of QCD contained also a term of the form
${\mathcal{L}}_{\rm
QCD}\supset-\frac{\alpha_{s}}{8\pi}\,\bar{\theta}\,G_{\mu\nu}^{b}\tilde{G}^{b,\mu\nu}$,
where $\alpha_{s}$ is the strong coupling, $G_{\mu\nu}^{b}$ is the gluonic
field strength, $\tilde{G}^{b,\mu\nu}$ its dual, and
$\bar{\theta}\in[-\pi,\pi]$ an angular parameter. This term violates parity
($P$) and time-reversal ($T$) invariances and, due to the $CPT$ conservation
theorem, also $CP$ invariance. Consequently, it induces $CP$ violation in
flavor-diagonal strong interactions, notably non-zero electric dipole moments
of nuclei. However, none have been detected up to date. The best constraint
currently comes from the electric dipole moment of the neutron, which is
bounded by $|d_{n}|<2.9\times 10^{-26}e$ cm. A comparison with the prediction,
$d_{n}\sim e\bar{\theta}m^{\ast}_{q}/m_{n}\sim 6\times 10^{-17}e$ cm, where
$m^{\ast}_{q}\equiv m_{u}m_{d}/(m_{u}+m_{d})$ is the reduced quark and $m_{n}$
the neutron mass, leads to the conclusion that $|\bar{\theta}|<10^{-9}$. This
is the strong $CP$ puzzle.

In Peccei-Quinn (PQ) extensions [1] of the Standard Model (SM), the symmetries
of the latter are extended by a global $U(1)_{\rm PQ}$ symmetry which is
spontanously broken by the vacuum expectation value (VEV) of a new complex
singlet scalar field, $\langle{|\sigma|^{2}}\rangle=v_{\rm PQ}^{2}/2$, which
is assumed to be much larger than the Higgs VEV. SM quarks or new exotic
quarks are supposed to carry PQ charges such that $U(1)_{\rm PQ}$ is also
broken by the gluonic triangle anomaly, $\partial_{\mu}J_{U(1)_{\rm
PQ}}^{\mu}\supset-\frac{\alpha_{s}}{8\pi}\,N_{\rm
DW}\,G_{\mu\nu}^{a}\tilde{G}^{a\,\mu\nu}$, where $N_{\rm DW}$ is a model-
dependent integer. Under these circumstances and at energies above the
confinement scale $\Lambda_{\rm QCD}$ of QCD, but far below $v_{\rm PQ}$, the
PQSM reduces to the SM plus a pseudo Nambu-Goldstone boson [2, 3] – the axion
$A$ – whose field, $\theta(x)\equiv A(x)/f_{A}\in[-\pi,\pi]$, corresponding to
the angular degree of freedom of $\sigma$, acts as a space-time dependent
$\bar{\theta}$ parameter,
${\mathcal{L}}_{\theta}\supset\frac{f_{A}^{2}}{2}\,\partial_{\mu}\theta\partial^{\mu}\theta-\frac{\alpha_{s}}{8\pi}\,\theta(x)\,G_{\mu\nu}^{c}{\tilde{G}}^{c,\mu\nu}$,
with $f_{A}\equiv v_{\rm PQ}/N_{\rm DW}$. Therefore, the
$\overline{\theta}$-angle can be eliminated by a shift
$\theta(x)\to\theta(x)-\overline{\theta}$. At energies below $\Lambda_{\rm
QCD}$, the effective potential of the shifted field, which for convenience we
again denote by $\theta(x)$, will then coincide with the vacuum energy of QCD
as a function of $\overline{\theta}$, which, on general grounds, has an
absolute minimum at $\theta=0$, implying that there is no strong $CP$
violation: $\langle\theta\rangle=0$. In particular,
$V(\theta)=\frac{1}{2}\chi\theta^{2}+{\mathcal{O}}(\theta^{4})$, where
$\chi\equiv\int d^{4}x\,\langle q(x)\,q(0)\rangle$, with
$q(x)\equiv\frac{\alpha_{s}}{8\pi}\,G_{\mu\nu}^{c}(x){\tilde{G}}^{c,\mu\nu}(x)$,
is the topological susceptibility. A recent lattice determination found [4]
$\chi=[75.6(1.8)(0.9){\rm MeV}]^{4}$, which agrees well with the result from
NLO chiral perturbation theory [5], $\chi=[75.5(5){\rm MeV}]^{4}$, leading to
the following prediction of the axion mass in terms of the axion decay
constant $f_{A}$,

$m_{A}\equiv\frac{1}{f_{A}}\sqrt{\frac{d^{2}V}{d\theta^{2}}}{|_{\theta=0}}=\frac{\sqrt{\chi}}{f_{A}}=57.0(7)\,\left(\frac{10^{11}\rm
GeV}{f_{A}}\right)\mu\textrm{eV}.$ (1)

##  2 Axion cold dark matter in the case of post-inflationary PQ symmetry
breaking

In a certain range of its decay constant, the axion not only solves the strong
$CP$ puzzle, but is also a cold dark matter candidate [6, 7, 8]. The extension
of this range depends critically on the cosmological history. It is
particularly constrained in the case on which we concentrate here: post-
inflationary PQ symmetry restoration and subsequent breaking.111Remarkably,
this case is strongly favored in the case of saxion (modulus of $\sigma$) or
saxion/Higgs inflation [9].

In the early universe, after the PQ phase transition, the axion field takes on
random initial values in domains of the size of the causal horizon. Within
each domain, the axion field evolves according to

$\ddot{\theta}+3H(T)\dot{\theta}+\frac{\chi(T)}{f_{A}^{2}}\sin\theta=0,$ (2)

with temperature dependent Hubble expansion rate $H(T)\sim T^{2}/M_{P}$ and
topological susceptibility [10] $\chi(T)\propto T^{-(7+3/n_{f})}$, for
temperatures far above the QCD quark-hadron crossover, $T_{c}^{\rm QCD}\simeq
150$ MeV ($n_{f}$ is the number of active quark flavors). At very high
temperatures, $v_{\rm PQ}>T\gg T_{c}^{\rm QCD}$, the Hubble friction term is
much larger than the potential term in (2), $3H(T)\gg\sqrt{\chi(T)}/f_{A}$,
and the axion field is frozen at its initial value. At temperatures around a
GeV, however, when $\sqrt{\chi(T)}/f_{A}\simeq 3H(T)$, the field starts to
evolve towards the minimum of the potential and to oscillate around the $CP$
conserving ground state. Such a spatially coherent oscillation has an equation
of state like cold dark matter, $w_{A}\equiv p_{A}/\rho_{A}\simeq 0$ (here
$p_{A}$ and $\rho_{A}$ are the pressure and the energy density of the axion
field, respectively). Averaging over the initial values of the axion field in
the many domains filling our universe – at temperatures around a GeV the size
of a domain is around a mpc – one obtains [4, 9]
$\Omega_{A}^{\rm(VR)}h^{2}=(3.8\pm 0.6)\times
10^{-3}\,\left(f_{A}\over{10^{10}\,{\rm GeV}}\right)^{1.165}$, for the
fractional contribution of axion cold dark matter to the energy density of the
universe from this so-called vacuum realignment (VR) mechanism [6, 7, 8].
Here, the exponent, $1.165$, arises from the temperature dependence of
$\chi(T)$ at $T\sim$ GeV, which has recently been determined quite precisely
from lattice QCD [4]. Requiring, that the axion dark matter abundance should
not exceed the observed one, this result implies a lower limit on the axion
mass [4]:

$m_{A}>28(2)\,\mu{\rm eV}\,.$ (3)

However, so far we have neglected that the domain-like structure discussed
above comes along with a network of one and two dimensional topological
defects – strings [11] and domain walls [12] – which are formed at the
boundaries of the domains. Their collapse will also produce axions.

Axion strings are formed at the same time when the domain-like structure
appears, cf. at the PQ phase transition. In the string cores, of typical
radius $1/m_{\rho}$, where $m_{\rho}\equiv\sqrt{2\lambda_{\sigma}}v_{\rm PQ}$
is the mass of the saxion (the particle excitation of the saxion field),
topology hinders the breaking of the PQ symmetry and a huge energy density is
stored. As the network evolves, the overall string length decreases by
straightening and collapsing loops. Moreover, some energy is radiated in the
form of low-momentum axions. The energy density in the network of global
strings is expected to reach a scaling behaviour, $\rho_{\rm
S}=\zeta\frac{\mu_{\rm S}}{t^{2}}$, with string tension $\mu_{\rm S}\equiv\pi
v_{\rm PQ}^{2}\ln\left(\frac{m_{\rho}t}{\sqrt{\zeta}}\right)$, where $\zeta$
is independent of time. This scaling behavior implies that the number density
of axions radiated from strings (S) can be estimated as

${n_{A}^{\rm(S)}(t)}\simeq\frac{\zeta}{\epsilon}\frac{v_{\rm
PQ}^{2}}{t}\left[\ln\left(\frac{m_{\rho}t}{\sqrt{\zeta}}\right)-3\right],$ (4)

where the dimensional parameter $\epsilon$ gives a measure of the average
energy of the radiated axions in units of the Hubble scale,
$\epsilon\equiv\langle E_{A}\rangle/(2\pi/t)$. A number of field theory
simulations have indicated that the network of strings evolves indeed toward
the scaling solution with $\zeta={\mathcal{O}}(1)$ and
$\epsilon={\mathcal{O}}(1)$. The latter value implies that most of the axions
produced from strings become non-relativistic during the radiation-dominated
era and contribute to the cold dark matter abundance. Adopting the values [13]
$\zeta=1.0\pm 0.5$ and $\epsilon=4.02\pm 0.70$, one finds from (4) for the
contribution of strings to today’s dark matter abundance [9]
$\Omega_{A}^{\rm(S)}h^{2}\approx 7.8^{+6.3}_{-4.5}\times 10^{-3}\times N_{\rm
DW}^{2}\left(\frac{f_{A}}{10^{10}\ {\rm GeV}}\right)^{1.165}$, where the upper
and lower end correspond to the maximum and minimum values obtained by using
the above error bars on $\zeta$ and $\epsilon$. They do not take into account
a possible large theoretical error due to the fact that the field theory
simulations can only be performed at values of $\ln(m_{\rho}t)\sim$ a few,
much smaller than the realistic value, $\sim 50$, and thus require an
extrapolation.

Domain walls appear at temperatures of the order of a GeV, when the axion
field, in any of the causally connected domains at this epoch, relaxes into
one of the $N_{\rm DW}$ distinct but degenerate minima of the effective
potential effective potential, $V(A,T)=\chi(T)\left[1-\cos(N_{\rm DW}A/v_{\rm
PQ})\right]$, in the interval $-\pi v_{\rm PQ}\leq A\leq+\pi v_{\rm PQ}$.
Between the domains, there appear two dimensional topological defects dubbed
domain walls whose thickness and stored energy density is controlled by
$\chi(T)$. Importantly, strings are always attached by $N_{\rm DW}$ domain
walls, due to the fact that the value of the phase of the PQ field $\sigma$
must vary from $-\pi$ to $\pi$ around the string core. Therefore, hybrid
networks of strings and domain walls, so-called string-wall systems, are
formed at $T={\mathcal{O}}(1)$ GeV. Their evolution strongly depends on the
model-dependent value of $N_{\rm DW}$.

For $N_{\rm DW}=1$, strings are pulled by one domain wall, which causes the
disintegration into smaller pieces of a wall bounded by a string [14]. String-
wall systems are short-lived in this case, and their collapse (C) contributes
an amount [13] $\Omega_{A}^{\rm(C)}h^{2}\approx 3.9^{+2.3}_{-2.1}\times
10^{-3}\times\left(\frac{f_{A}}{10^{10}\ {\rm GeV}}\right)^{1.165}$ to dark
matter, resulting in a total abundance

$\Omega_{A}h^{2}\approx\left(\Omega_{A}^{\rm{(VR)}}+\Omega_{A}^{\rm{(S)}}+\Omega_{A}^{\rm{(C)}}\right)h^{2}\approx
1.6^{+1.0}_{-0.7}\times
10^{-2}\times\left(\frac{f_{A}}{10^{10}\,\mathrm{GeV}}\right)^{1.165}.$ (5)

Therefore, in post-inflationary PQ symmetry breaking models with $N_{\rm
DW}=1$, the axion may explain all of cold dark matter in the universe if its
decay constant and mass are in the range

$f_{A}\approx(3.8-9.9)\times 10^{10}\,{\rm GeV}\hskip
12.91663pt\Leftrightarrow\hskip 12.91663ptm_{A}\approx(58-150)\ \mu{\rm
eV}\,.$ (6)

This prediction, however, has recently been challenged by the results from a
new field theory simulation technique designed to work directly at high string
tension with $\ln(m_{\rho}t)\sim 50$ and to treat vacuum realignment, string,
and string-wall contributions in a unified way [15]. The reported dark matter
axion mass,

$m_{A}=(26.2\pm 3.4)\,\mu{\rm eV}\,,$ (7)

where the error now only includes the uncertainty from $\chi(T)$, is
significantly lower than (6). It indicates that axions from strings and walls
are negligible, despite of the fact that the string networks appear to have a
higher energy density ($\zeta\sim 4$) than those observed in conventional
field theoretic simulations ($\zeta\sim 1$). This implies that the produced
axions have a larger energy, $\epsilon\sim 40$, and that dynamics at smaller
scales – outside the range of applicability of the new simulation method [15]
– can be relevant for the determination of the axion DM abundance. Further
studies on the dynamics of string-wall systems are required to include precise
modelling of physics at smaller distance scales.

Fortunately, there are new axion dark matter direct detection experiments
aiming to probe the mass region of interest for $N_{\rm DW}=1$ models with
post-inflationary PQ symmetry breaking, notably CULTASK [16], HAYSTAC [17],
and MADMAX [18].

For $N_{\rm DW}>1$, the string-wall systems are stable, since the strings are
pulled in $N_{\rm DW}$ different directions. The existence of such stable
domain walls is firmly excluded by standard cosmology [19]. Stability can be
avoided if there exist further interactions which explicitly break the PQ
symmetry, e.g. ${\mathcal{L}}\supset
gM_{P}^{4}\left(\frac{\sigma}{M_{P}}\right)^{N}+h.c.$, where $g$ is a complex
dimensionless coupling, $M_{P}$ is the reduced Planck mass, and $N$ is an
integer ($>4$). The appearance of such terms is motivated by the fact that
global symmetries are not protected from effects of quantum gravity. They give
rise to an additional contribution in the low energy effective potential of
the axion field, which lifts the degeneracy of the minima of the QCD induced
potential by an amount [20] $\Delta V\simeq-2|g|M_{P}^{4}\left(\frac{v_{\rm
PQ}}{\sqrt{2}M_{P}}\right)^{N}\left[\cos\left(\frac{2\pi N}{N_{\rm
DW}}+\Delta_{D}\right)-\cos\Delta_{D}\right]$, where
$\Delta_{D}=\arg(g)-N\overline{\theta}$, and acts like a volume pressure on
domain walls. If $\Delta V$ is small, domain walls live for a long time and
emit a lot of axions, potentially overclosing the universe. On the other hand,
if $\Delta V$ is large, it shifts the location of the minimum of the axion
effective potential and leads to large $CP$ violation, spoiling the axionic
solution of the strong $CP$ problem. A detailed investigation of the parameter
space exploiting the results of field theory simulations [13] showed [20] that
there exists a valid region in parameter space if $N=9$ or $10$.222The absence
of PQ symmetry breaking operators with $4<N<9$ can be naturally explained if
the PQ symmetry arises accidentally as a low energy remnant from a more
fundamental discrete symmetry [20, 21]. In the case of $N_{\rm DW}=6$ and
$N=9\,(10)$, and allowing a mild tuning of $|g|$, the axion can explain the
observed dark matter abundance for

$4.4\times 10^{7}\,(1.3\times 10^{9})\,{\rm GeV}<f_{A}<1\times 10^{10}\,{\rm
GeV}\ \Leftrightarrow\ 0.56\,{\rm meV}<m_{A}<130\,(4.5)\,{\rm meV}\,.$ (8)

Intriguingly, a DFSZ axion ($N_{\rm DW}=6$) in such a mass range can explain
the accumulating hints of excessive energy losses of stars in various stages
of their evolution [22]. In this range, axion dark matter direct detection may
be difficult, but not impossible [23, 24]. Fortunately, it is aimed to be
probed by the fifth force experiment ARIADNE [25] and the helioscope IAXO
[26].

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