OU-HEP-220401

On dark radiation from string moduli decay to ALPs
Howard Baer1111Email<EMAIL_ADDRESS>, Vernon Barger2222Email:
<EMAIL_ADDRESS>and Robert Wiley Deal1333Email<EMAIL_ADDRESS>
1Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma,
Norman, OK 73019, USA
2Department of Physics, University of Wisconsin, Madison, WI 53706 USA

We examine the issue of dark radiation (DR) from string moduli decay into
axion-like particles (ALPs). In KKLT-type models of moduli stabilization, the
axionlike phases of moduli fields are expected to decouple whilst in LVS-type
moduli stabilization some can remain light and may constitute dark radiation.
We evaluate modulus decay to Minimal Supersymmetric Standard Model (MSSM)
particles and dark radiation for more general compactifications. In spite of
tightening error bars on $\Delta N_{eff}$, we find only mild constraints on
modulus-ALP couplings due to the somewhat suppressed modulus branching
fraction to DR owing to the large number of MSSM decay modes. We anticipate
that future CMB experiments with greater precision on $\Delta N_{eff}$ may
still turn up evidence for DR if the ALP associated with the lightest modulus
field is indeed light.

##  1 Introduction

String theory[1, 2] offers a consistent and finite quantum theory of gravity
which can include the matter states and gauge symmetries that form the basis
for the Standard Model (SM). The price to pay is that string theory must be
formulated in 10 (or 11 for $M$-theory) spacetime dimensions. To gain our
observable four spacetime dimensions, the additional space dimensions are
assumed compactified into a tiny compact manifold such as a Calabi-Yau space
(which preserves some spacetime supersymmetry (SUSY) under compactification).
The resultant 4-d theory then consists minimally of the supersymmetric SM
(MSSM) plus at least an assortment of gravitationally coupled scalar fields
(the moduli) with no classical potential. The moduli fields, which parametrize
the size and shape of the compact space, must be stabilized and then their
vacuum expectation values (vevs) determine many features of the low energy 4-d
effective theory such as coupling constants and soft SUSY breaking terms.
Typical string compactifications then contain, in the 4-d limit, on order of
tens-to-hundreds of moduli fields in addition to visible and hidden sector
fields.

In II-B string theory, the moduli can be classified as complex structure
($U_{\alpha}$) and Kähler ($T_{\beta}$) along with the axio-dilaton ($S$).
Under flux compactifications[3], the $S$ and $U_{\alpha}$ moduli are
stabilized by flux and should gain ultra-high KK-scale masses. Under KKLT
stabilization[4], the $T_{\beta}$ are stabilized non-perturbatively while in
LVS stabilization[5], the $T_{\beta}$ are stabilized by a balancing of
perturbative and non-perturbative effects. Thus, the Kähler moduli may have
much lighter masses which may be as low as the soft SUSY breaking scale
$m_{soft}\sim 1$ TeV. The lightest of the moduli, $\phi$ (with mass labeled
here as $m_{\phi}$) may be cosmologically dangerous in that

  1. 1.

they may live long enough to decay after the onset of BBN thus destroying the
successful BBN predictions of light element abundances[6, 7, 8],

  2. 2.

they may overproduce neutralino dark matter (the moduli-induced LSP
problem)[9],

  3. 3.

they may overproduce gravitinos if $m_{\phi}>2m_{3/2}$ (the moduli-induced
gravitino problem)[10, 11, 12, 13, 14] where then the gravitinos could decay
after onset of BBN or overproduce dark matter and

  4. 4.

they may decay into relativistic particles such as the axion-like particles
(ALPs)[15, 16] which are endemic to 4-d string models[17], thus potentially
violating limits on dark radiation as parametrized by the parameter $N_{eff}$,
the effective number of light neutrinos which inhabit the cosmic soup.

Under KKLT stabilization, the shift symmetry $T\rightarrow T+i\alpha$ is
destroyed and the corresponding ALPs, which comprise the phase fields of the
$T_{\beta}$, are expected to obtain masses comparable to the corresponding
real components: $m_{\text{A}LP}\gtrsim m_{soft}$. However, for LVS
stabilization, the shift symmetry can survive, and the corresponding ALPs end
up with small but non-zero masses and thus may comprise a portion of the
measured dark radiation.

Indeed, years ago the measured value of $N_{eff}$ seemed somewhat displaced
from the SM value, motivating great excitement that string remnants may have
left a detectable imprint on the cosmic microwave background radiation (CMB).
In recent years, increasingly precise CMB measurements have brought $N_{eff}$
more into accord with SM expectations, so that $\Delta N_{eff}$ provides now
increasingly tight constraints on new physics models which include dark
radiation.

The 2018 Planck analysis of cosmological parameters[18] in relation to CMB
measurements is able to fit the amount of dark radiation at 95% CL as

$N_{eff}=2.99^{+0.34}_{-0.33}\ \ (TT,TE,EE+lowE+lensing+BAO)$ (1)

based upon joint fits to Planck CMB polarization, lensing and baryon acoustic
oscillations (BAO). From these fits, and using the Standard Model (SM) value
$N_{eff}(SM)=3.046$[19], we will require that at 95% CL

$\Delta N_{eff}<0.29\ \ \ (95\%\ CL,\ Planck,2018).$ (2)

In this paper, we continue our earlier investigation[20] of the cosmological
moduli problem (CMP) wherein we computed the various modulus decay rates into
the MSSM particles including all phase space and mixing effects which are
routinely ignored in the literature. Once these are known, then one may
compute the modulus decay temperature
$T_{D}\simeq\sqrt{\Gamma_{\phi}m_{P}}/(\pi g_{*}/90)^{1/4}$ and use these to
implement BBN constraints ($T_{D}>T_{BBN}\sim 3-5$ MeV). One may also compute
the modulus oscillation temperature $T_{osc}$, the modulus-radiation equality
temperature $T_{e}$, and the temperature $T_{3/2}$ of radiation at the time of
gravitino decay. Comparing to the inflaton reheat temperature $T_{R}$ and
neutralino freeze-out temperature $T_{fo}$, then one may compute the entropy
dilution factor $r=S_{f}/S_{0}=T_{e}/T_{D}$ and the ultimate non-thermal
neutralino abundance and constraints on relic gravitinos. Assuming a well-
motivated natural SUSY spectrum of MSSM particles, it was found that very
large modulus masses $m_{\phi}\gtrsim 5000$ TeV were needed (assuming an
initial modulus field strength $\phi_{0}\sim m_{P}$) to avoid the moduli-
induced LSP problem. Also, typically $m_{\phi}<2m_{3/2}$ was needed to avoid
the moduli-induced gravitino problem. For the well-motivated gravity-mediated
SUSY breaking model, wherein the soft SUSY breaking scale $m_{soft}\sim
m_{3/2}$, then the high mass modulus solution to the CMP would bring physics
into conflict with naturalness of SUSY models which requires sparticles (save
light higgsinos) typically in the several TeV range[20]111For natural SUSY
models, we require models to have low electroweak finetuning with
$\Delta_{EW}\lesssim 30$[21, 22].. An alternative solution which allows much
lighter values of $m_{\phi}\sim 30$ TeV is to find an anthropic selection on
modulus field strength $\phi_{0}\sim 10^{-7}m_{P}$ which allows for a more
comparable dark-matter-to-baryonic-matter ratio $\sim 1-10$[23].
Alternatively, $m_{\phi}\sim m_{weak}$ with $\phi_{0}\sim m_{P}$ leads to a
dark matter dominated universe wherein baryons might only occur as diffuse
hardly gravitating clouds and minimal structure for baryonic matter[24, 25,
26].

In the present work, we extend our previous analyses to include the effects of
modulus decay into ALPs, which may occur for LVS moduli stabilization, but may
occur more generally in other (possibly unthought of) 4-d string models. This
addresses a fourth facet of the CMP: the moduli-induced ALP problem[15, 16,
27]. But first, we review several related works that precede our contribution,
and explain the new aspects of our own work. Then in Sec. 2, we explain our
calculation of modulus field coupling to ALPs and decay rate into dark
radiation. In Sec. 3, we present details of our calculation of $\Delta
N_{eff}$ in the sudden decay approximation and in Sec. 5 we present numerical
results which mainly restrict the values of $m_{\phi}$ and modulus-ALP
coupling $\lambda_{\text{A}LP}$. We find that in spite of the tightening error
bars from the experimental determination of $\Delta N_{eff}$, the moduli-
induced ALP problem is perhaps not overly constraining, even for values of
$\lambda_{\text{A}LP}\sim 1$, unless $m_{\phi}$ is very large $m_{\phi}\gtrsim
10^{4}$ TeV wherein the modulus field begins to oscillate before the end of
inflation (where $T_{osc}\gtrsim T_{R}$, assuming a reheat temperature of
$T_{R}\sim 10^{12}$ GeV). On the other hand, the anthropic solution to the
moduli-induced LSP problem does not help much with the modulus-induced ALP
problem since decreasing the field strength $\phi_{0}$ hardly affects the
$\phi\rightarrow ALPs$ branching fraction. A brief summary and conclusions are
given in Sec. 6.

###  1.1 Some previous work

Consequences of moduli stabilization for the QCD axion were addressed by
Conlon in Ref. [28]. The issue of dark radiation in string models was
addressed by Cicoli, Conlon and Quevedo[15] and Higaki and Takahashi[16] in
2012 in the context of LVS, the moduli stabilization scheme wherein the Kähler
moduli shift symmetry $T_{\alpha}\rightarrow T_{\alpha}+i\alpha$ is maintained
under perturbative stabilization resulting in (nearly) massless ALP partners
$a_{\alpha}$ of the fields $T_{\alpha}=\phi_{\alpha}+ia_{\alpha}$. These
papers pointed out that dark radiation should be generic in string models with
LVS stabilization whilst models with non-perturbative Kähler moduli
stabilization should yield $m_{a_{\alpha}}\sim m_{\phi_{\alpha}}$ so that
these models typically do not produce DR from ALPs. Much of this work was
inspired by CMB measurements which at the time seem to favor enhanced $\Delta
N_{eff}\gg 0$. Thus, other papers examined different dark radiation sources
from e.g. saxion decay to QCD axions $a$ in SUSY axion models[29, 30, 31, 32].
In Ref. [33], Conlon and Marsh investigated the effect of relic ALPs on BBN
and on dark matter production rates (axiogenesis). In Ref. [27], Higaki et al.
illustrated the severity of the moduli-induced axion problem as a fourth
aspect of the CMP and emphasized two solutions: decrease the partial width
$\Gamma_{\phi}(\phi\rightarrow ALPs)$ or increase the width
$\Gamma_{\phi}(\phi\rightarrow$ visible sector particles). In Ref. [34],
Cicoli presented a minireview on ALPs from string compactifications,
especially for sequestered LVS models. In Ref. [35], Allahverdi et al.
examined non-thermal dark matter production in sequestered LVS models
including dark radiation; in [36], they pointed out correlations between dark
matter and dark radiation production in sequestered LVS models. In Ref. [37],
Angus noted that dark radiation bounds can effectively rule out certain
extended LVS models where the bulk volume is stabilized by two rather than one
moduli fields. In Hebecker et al. Ref. [38], dark radiation predictions from
general LVS scenarios are examined. In Ref. [39], a general analysis of dark
radiation in sequestered string models is made: by including addition modulus
visible sector decay modes, the DR is significantly reduced. Acharya and
Pongkitvanichkul[40] considered general string compactifications giving rise
to an axiverse[17] and asked the question: given the general expectation of an
axiverse from string compactifications, then why is $\Delta N_{eff}$ so small?
Supersymmetric axions, dark radiation and inflation were examined in Ref.
[41]. Takahashi and Yamada[42] consider an anthropic bound on $\Delta N_{eff}$
in that if $\Delta N_{eff}$ is too big, then it suppresses the growth of
matter fluctuations in the early universe and hence suppresses structure
formation. In Ref. [43], Acharya et al. consider the case of multiple light
string moduli and find that the early-on produced dark radiation is
significantly diluted but generically WIMP dark matter is overproduced. They
present a scenario where the WIMP DM abundance is reduced due to annihilation
to dark radiation while the DR abundance is reduced by entropy dilution as
long as the DM annihilation is prompt. In Ref. [44], M. Reig considers a
stochastic axiverse wherein a low scale of inflation extending to $T\sim 100$
eV dilutes all relics while relic axions can still be produced via a maximal
misalignment mechanism. In Ref. [45], decay of multiple dark matter particles
to dark radiation in different epochs is shown to not alleviate the tension in
the Hubble constant determination. As we were completing this work, a
comprehensive overview of axions in string theory with implications for dark
radiation and inflationary models appeared in Ref. [46].

##  2 Modulus decay to ALPs

We begin discussion of modulus decays into ALPs by briefly reviewing how the
interaction arises within the Kähler potential. After describing how this
arises in concrete moduli stabilization scenarios, we then illustrate our
“stabilization-agnostic” approach.

The Kähler potential for the lightest geometrical moduli typically takes the
approximate form

$\displaystyle K$ $\displaystyle\supset-n_{i}\log(T_{i}+\overline{T}_{i})$ (3)

where $n_{i}$ is determined by the form of the compactification manifold
volume, $\mathcal{V}$. In minimal LVS models, the volume is assumed to be of
the form $\mathcal{V}\propto\tau_{b}^{3/2}$ (where $\tau_{b}$ is the “big”
volume modulus) so that $n\simeq 3$, which gives the required “no-scale
structure”[47, 48]. The volume may however take more complicated forms
depending on the compactification details. In e.g. fibred LVS models such as
those considered in [37], the volume of the compactification manifold instead
goes as $\mathcal{V}\propto\sqrt{\tau_{1}}\tau_{2}$ \- corresponding to a
Kähler potential
$K\supset-\log(T_{1}+\overline{T}_{1})-2\log(T_{2}+\overline{T}_{2})$ for the
light geometrical moduli.

The Kähler metric associated to Eq. 3 is diagonal with entries
$K_{ii}\simeq\frac{n_{i}}{4\tau_{i}^{2}}$, so that the (non-canonical) kinetic
term in the Lagrangian becomes

$\displaystyle\mathcal{L}$ $\displaystyle\supset
K_{i\overline{\jmath}}\partial
T^{i}\partial\overline{T}^{\overline{\jmath}}=\sum\limits_{i}\frac{n_{i}}{4\tau_{i}^{2}}\partial_{\mu}\tau_{i}\,\partial^{\mu}\tau_{i}+\frac{n_{i}}{4\tau_{i}^{2}}\partial_{\mu}c_{i}\,\partial^{\mu}c_{i}$
(4)

where the $T_{i}=\tau_{i}+ic_{i}$. Through the field redefinitions

$\displaystyle\tau_{i}$
$\displaystyle=\exp\left(\sqrt{\frac{2}{n_{i}}}\phi_{i}\right)\quad\text{and}\quad
c_{i}=\sqrt{\frac{2}{n_{i}}}a_{i}$ (5)

the canonical kinetic terms are recovered and after expanding the exponential,
we obtain the interaction terms:

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\frac{1}{m_{P}}\sqrt{\frac{2}{n_{i}}}\phi_{i}\,\partial_{\mu}a_{i}\,\partial^{\mu}a_{i}+\text{higher
order}$ (6)

where we have now explicitly restored $m_{P}$. The form of this interaction
term loosely matches the expectation for a $\phi_{i}a_{i}a_{i}$ interaction
where the $a_{i}$ terms possess a shift symmetry. The coupling here is then
set explicitly by the field space geometry. However, for models with more than
one light modulus, the $\phi_{i}$ fields may still need to get rotated into
the mass eigenbasis. While the specific form of the mass matrix is determined
by the moduli stabilization details and hence is model dependent, the generic
feature is that each modulus mass eigenstate may then decay to multiple ALP
types, $a_{i}$, with couplings determined by $n_{i}$ and the form of the mass
matrix.

In this work, we take the general form of Eq. 6 but make some minor
adjustments. We begin by first writing down the interaction term as

$\displaystyle\mathcal{L}$ $\displaystyle\supset-\frac{\lambda_{\rm
ALP}}{m_{P}}\phi\,\partial_{\mu}a\,\partial^{\mu}a.$ (7)

Here, we assume that the lightest modulus $\phi$ is written in the mass
eigenbasis. Since that may introduce couplings to multiple ALPs $a_{i}$ (which
we expect to be nearly massless if the shift symmetry is preserved and
kinematically inaccessible if the symmetry is broken), we choose to
parameterize this by a single ALP field $a$ with an effective coupling
$\lambda_{\rm ALP}$. The $\lambda_{\rm ALP}$ coupling then parameterizes the
“total” coupling between the lightest modulus and the (possibly many) ALPs it
may decay into, without relying on a specific stabilization model. In the
familiar case of minimal LVS, we have $\lambda_{\rm ALP}=\sqrt{2/3}\simeq
0.816$. Larger values of $\lambda_{\rm ALP}\gtrsim 0.816$ can then
parameterize decays to multiple ALPs, whereas $\lambda_{\rm ALP}\lesssim
0.816$ may correspond to some (possibly yet undiscovered) stabilization scheme
in which the shift symmetry is broken for some ALPs coupled to $\phi$, making
those decays kinematically forbidden and hence lowering the effective
coupling.

The matrix element squared for this decay, taking for now a small but non-zero
ALP mass, is

$\displaystyle|\mathcal{M}|^{2}$ $\displaystyle=\frac{\lambda_{\rm
ALP}^{2}}{m_{P}^{2}}m_{\phi}^{4}\left(1-2\frac{m_{\rm
ALP}^{2}}{m_{\phi}^{2}}\right)^{2}.$ (8)

The decay width can then be easily computed:

$\displaystyle\Gamma_{\phi\rightarrow aa}$ $\displaystyle=\frac{\lambda_{\rm
ALP}^{2}}{32\pi}\frac{m_{\phi}^{3}}{m_{P}^{2}}\left(1-2\frac{m_{\rm
ALP}^{2}}{m_{\phi}^{2}}\right)^{2}\lambda^{1/2}\left(1,\frac{m_{\rm
ALP}^{2}}{m_{\phi}^{2}},\frac{m_{\rm ALP}^{2}}{m_{\phi}^{2}}\right).$ (9)

Taking the massless limit for the ALPs $a$, the decay width simplifies to:

$\displaystyle\Gamma_{\phi\rightarrow aa}$ $\displaystyle=\frac{\lambda_{\rm
ALP}^{2}}{32\pi}\frac{m_{\phi}^{3}}{m_{P}^{2}}.$ (10)

In the case of e.g. minimal LVS with $\lambda_{\rm ALP}=\sqrt{2/3}$, we
recover the results in [15, 16].

##  3 Estimating $\Delta N_{eff}$

Our task now is to quantify the amount of dark radiation produced in the early
universe via modulus decay. As these light, relativistic degrees of freedom
should contribute to the total radiation energy density, $\rho_{R}^{tot}$, it
is useful to parameterize the radiation density as

$\displaystyle\rho_{R}^{tot}$
$\displaystyle=\left(1+N_{eff}\frac{7}{8}\left(\frac{T_{\nu}}{T}\right)^{4}\right)\rho_{\gamma}.$
(11)

Here, $T_{\nu}$ is the neutrino temperature and $T$ is the temperature of the
thermal bath, related by $T_{\nu}=(\frac{4}{11})^{1/3}T$ [49]. In the SM,
$N_{eff}^{SM}=3.046$ [19]. Any additional light degrees of freedom then
correspond to an increase in $N_{eff}$ from its SM value, $\Delta
N_{eff}=N_{eff}-N_{eff}^{SM}$. We can then estimate $\Delta N_{eff}$ from
modulus decay into ALPs by

$\displaystyle\Delta N_{eff}$
$\displaystyle=\frac{\rho_{\text{ALP}}(T)}{\rho_{\nu}}=\frac{120}{7\pi^{2}}\left(\frac{11}{4}\right)^{4/3}\frac{\rho_{\text{A}LP}(T)}{T^{4}}$
(12)

where $\rho_{\nu}$ is the energy density of a single neutrino species so that
$\rho_{\nu}=\frac{7}{8}\frac{\pi^{2}}{15}T_{\nu}^{4}$.

Since the ALPs dilute as radiation, we can relate their energy density between
$T_{D}$ and $T$ (with $T<T_{D}$) as

$\displaystyle\rho_{\text{ALP}}(T)$
$\displaystyle\simeq\left(\frac{g_{*S}(T)^{1/3}\,T}{g_{*S}(T_{D})^{1/3}\,T_{D}}\right)^{4}\rho_{\text{ALP}}(T_{D})$

where we assume conservation of entropy density,
$s=\frac{2\pi^{2}}{45}g_{*S}T^{3}$, between $T_{D}$ and $T$.

What remains is to calculate $\rho_{\text{ALP}}(T_{D})$. Here, to gain the
overall big picture, we adopt the sudden decay approximation. Non-sudden
effects will be included in forthcoming coupled Boltzmann calculations. Using
conservation of energy density, we can then express the energy density of the
ALPs in terms of the modulus:

$\displaystyle\rho_{\text{ALP}}(T_{D})=\mathcal{B}(\phi\rightarrow
aa)\rho_{\phi}(T_{D})/r$ (13)

where $r\equiv S_{f}/S_{i}=T_{e}/T_{D}$ is the entropy dilution factor[50,
51].

As the modulus behaves as non-relativistic matter, to estimate
$\rho_{\phi}(T_{D})$ we first need the number density at $T_{D}$, which is
given by

$\displaystyle n_{\phi}(T_{D})$
$\displaystyle=\frac{1}{2}m_{\phi}\phi_{0}^{2}\left(\frac{g_{*S}(T_{D})T_{D}^{3}}{g_{*S}(T_{\text{osc}})T_{\text{osc}}^{3}}\right).$
(14)

Putting all of this together, we arrive at the expression for $\Delta N_{eff}$

$\displaystyle\Delta N_{eff}$
$\displaystyle\simeq\frac{60}{7\pi^{2}}\left(\frac{11}{4}\right)^{4/3}\left(\frac{g_{*S}(T)^{4/3}}{g_{*S}(T_{D})^{1/3}g_{*S}(T_{\text{osc}})}\right)\mathcal{B}\left(\phi\rightarrow
aa\right)\frac{m_{\phi}^{2}\phi_{0}^{2}}{T_{e}T_{\text{osc}}^{3}}$ (15)

where for convenience we show $g_{*S}$ as a function of $T=m_{\phi}$ in Fig.
1.

Figure 1: The value of $g_{*S}$ vs. $m_{\phi}$ for a natural SUSY BM point.

Furthermore, we note that the temperature of radiation/modulus energy density
equality, determined by requiring $\rho_{R}=\rho_{\phi}$, is found to be[51]

$T_{e}\equiv\begin{cases}\left(15/\pi^{2}g_{*}(T_{e})\right)^{1/4}\sqrt{m_{\phi}\phi_{0}}&(T_{osc}<T_{e}<T_{R})\\\
(3/2m_{P}^{2})\phi_{0}^{2}\sqrt{m_{P}m_{\phi}}\left(10/\pi^{2}g_{*}(T_{e})\right)^{1/4}&(T_{e}<T_{osc}<T_{R})\\\
(3/2m_{P}^{2})\phi_{0}^{2}T_{R}&(T_{e}<T_{R}<T_{osc})\end{cases}$ (16)

and where the oscillation temperature is found to be

$T_{osc}\simeq\begin{cases}\left(10/\pi^{2}g_{*}(T_{osc})\right)^{1/4}\sqrt{m_{P}m_{\phi}}&(T_{osc}\leq
T_{R})\\\
\left(10g_{*}(T_{R})/\pi^{2}g_{*}^{2}(T_{osc})\right)^{1/8}\left(T_{R}^{2}m_{P}m_{\phi}\right)^{1/4}&(T_{osc}>T_{R}),\end{cases}$
(17)

where $m_{P}\equiv M_{Pl}/\sqrt{8\pi}$ is the reduced Planck mass.

###  3.1 Approximate analytic expression for $\Delta N_{eff}$

We can now make some simple assumptions that should give us a reasonable
estimate for $\Delta N_{eff}$. Since the leading decay modes for the modulus
are to the massless gauge bosons (assuming all couplings as given in [20] are
$\lambda_{i}\sim 1$), the total width is well approximated by simply

$\displaystyle\Gamma_{\phi}^{\text{tot}}$
$\displaystyle\sim\Gamma_{\phi\rightarrow\gamma\gamma}+\Gamma_{\phi\rightarrow
gg}\sim\frac{9}{8\pi}\lambda_{G}^{2}\frac{m_{\phi}^{3}}{m_{P}^{2}}$ (18)

where we make the additional simplifying assumption that
$\lambda_{U(1)}\simeq\lambda_{SU(2)}\simeq\lambda_{SU(3)}\equiv\lambda_{G}$.
The branching ratio can then be approximated as

$\displaystyle\mathcal{B}\left(\phi\rightarrow aa\right)$
$\displaystyle\simeq\frac{\lambda_{\text{A}LP}^{2}}{36\lambda_{G}^{2}}.$ (19)

giving us the estimate

$\displaystyle\Delta N_{eff}$
$\displaystyle\simeq\frac{5}{21\pi^{2}}\left(\frac{11}{4}\right)^{4/3}\frac{\lambda_{\text{A}LP}^{2}}{\lambda_{G}^{2}}\left(\frac{g_{*S}(T)^{4/3}}{g_{*S}(T_{D})^{1/3}g_{*S}(T_{\text{osc}})}\right)\frac{m_{\phi}^{2}\phi_{0}^{2}}{T_{e}T_{\text{osc}}^{3}}$
(20)

or, plugging in values

$\displaystyle\Delta N_{eff}$ $\displaystyle\simeq
0.11\frac{\lambda_{\text{A}LP}^{2}}{\lambda_{G}^{2}}$
$\displaystyle\left(\frac{224}{g_{*S}(T_{\text{osc}})}\right)\left(\frac{g_{*S}(T)}{3.9}\right)^{4/3}\left(\frac{63.7}{g_{*}(T_{D})}\right)^{1/3}\left(\frac{m_{\phi}}{10^{6}\text{
GeV}}\right)^{2}$ $\displaystyle\times\left(\frac{\phi_{0}}{1.94\times
10^{18}\text{ GeV}}\right)^{2}\left(\frac{3.85\times 10^{11}\text{
GeV}}{T_{\text{osc}}}\right)^{3}\left(\frac{4.0\times 10^{11}\text{
GeV}}{T_{e}}\right).$ (21)

##  4 Modulus-to-ALPs branching fraction numerics

In Eq. 15, we found that the amount of dark radiation depends directly on the
modulus field branching fraction to ALPs. From Eq. 19, we expect the branching
fraction ${\cal B}(\phi\rightarrow ALPs)$ to be of order $\sim 0.01-0.1$,
although this approximate expression assumes that the modulus decay to SM
modes includes only massless gauge bosons. The actual result can be very
model-dependent in that various MSSM decay modes may be suppressed or not, and
these details can seriously affect the ultimate modulus branching fraction
into DR.

Recently, in Ref. [20], all MSSM decay modes of light moduli fields were
computed assuming Moroi-Randall operators[52]. The decay widths were evaluated
including all mixing and phase space effects. Even so, some of these decay
modes are still model dependent. In particular, modulus decay to gauginos may
or may not be helicity suppressed. The helicity suppression is displayed in
the decay widths by whether or not the width numerator contains a factor
$m_{\phi}^{3}$ (unsuppressed, case A) or $m_{\phi}m_{\lambda}^{2}$ (helicity-
suppressed, case B). The suppression factor depends on details of the gauge
kinetic function[11, 14]. Likewise, modulus decay to gravitinos may not be
(case 1) or may be (case 2) helicity-suppressed depending on details of the
Kähler function $K$[11, 14].

In addition, the moduli branching fractions depend on the specific details of
the assumed SUSY particle mass spectrum. In Ref. [20], a natural SUSY
benchmark (BM) spectrum was adopted with low finetuning measure
$\Delta_{EW}\sim 20$. The BM point came from the NUHM3 model with parameters
$m_{0}(1,2)=10$ TeV, $m_{0}(3)=5$ TeV, $m_{1/2}=1.2$ TeV, $A_{0}=-8$ TeV and
$\tan\beta=10$. It also had $\mu=200$ GeV and $m_{A}=2$ TeV. Using the
Isasugra spectrum calculator[53], it is found to have $m_{h}=125.3$ GeV and
$m_{\tilde{g}}=2.9$ TeV– in accord with LHC Higgs mass measurement and
sparticle mass limits. Such a point is expected to emerge with a high
probability (relative to finetuned models) from string landscape selection[54,
55].

In Fig. 2, we plot the various modulus to MSSM particle partial widths as in
Ref. [20] for the BM point in scenario B1, except here we also include the
partial width for decay to ALPs (brown curve). As $m_{\phi}$ increases, most
of the partial widths increase as $\Gamma_{\phi}\sim m_{\phi}^{3}$ as commonly
assumed in the literature. However, here we see the partial widths to gauginos
(and also SM fermion pairs) increases only as $\Gamma_{\phi}\sim m_{\phi}^{1}$
due to the assumed helicity suppression. Furthermore, the partial widths into
MSSM sfermion pairs suffers additional suppression and so
$\Gamma_{\phi}(\phi\rightarrow sfermions)\sim
m_{\tilde{f}}^{4}/(m_{\phi}m_{P}^{2})$ and thus these partial widths decrease
with increasing $m_{\phi}$. The important result is that there are very many
MSSM decay modes, making the ultimate modulus BF into DR very model dependent.
In much of the literature, it is assumed instead just that
$\Gamma_{\phi}(\phi\rightarrow ALPs)\sim\Gamma_{\phi}(\phi\rightarrow MSSM)$.

Figure 2: Decay widths of lightest modulus field $\phi$ into ALPs plus various
other MSSM particles in the B1 scenario with helicity suppressed decay to
gauginos but not gravitinos. We adopt the natural SUSY BM point from Ref.
[20].

In Fig. 3, we show the resultant branching fraction ${\cal B}(\phi\rightarrow\
ALPs)$ for the same BM case and scenario B1 (blue curve) as in Fig. 2. From
the plot, we see that initially at low $m_{\phi}$ the BF oscillates somewhat
due to turn on of various MSSM decay modes. Ultimately, the ${\cal
B}(\phi\rightarrow ALPs)$ settles down to $\sim 1.4\times 10^{-2}$. The rather
low $BF$ result, compared to other estimates in the literature, is due to our
assumption in taking all couplings $\sim 1$, so that the gauge boson modes
(and not the Higgs modes) are dominant - whereas in the explicit sequestered
models usually studied for DR, the gauge bosons typically possess a loop
suppression factor. The inclusion of the very many possible MSSM modes
entering the modulus decay width then suppresses this branching ratio further
to the percent level.

Figure 3: Lightest modulus field $\phi$ branching fraction into ALPS vs.
$m_{\phi}$ for $\lambda_{\rm ALP}=0.816$ and all other moduli couplings
$\lambda_{i}=1$ for the case B1 (blue, case with suppressed decay to gauginos
but unsuppressed decays to gravitinos) and B2 (orange, suppressed decays to
gauginos and gravitinos with decays to matter and Higgs superfields turned
off) with all other MR couplings $\lambda_{i}=1$ for a natural SUSY BM point
from Ref. [20].

Also in Fig. 3, we show the modulus branching fraction ${\cal
B}(\phi\rightarrow ALPs)$ versus $m_{\phi}$ for the same SUSY BM point but in
decay scenario B2 (orange curve). This time we assume in addition that the
$\phi$ decay modes to Higgs and matter superfields are turned off, as might be
expected if these fields carry Peccei-Quinn (PQ) charges and are involved in a
solution to the strong CP problem with a Kim-Nilles[56] solution to the SUSY
$\mu$ problem[57]. Although the B2 scenario is quite different from scenario
B1, the branching fractions are rather similar since the decays into MSSM
particles are dominated by decays to gauge bosons.

##  5 Results for $\Delta N_{eff}$

Now that the modulus branching fraction to ALPs is estimated, we can proceed
to using Eq. 15 to evaluate $\Delta N_{eff}$ numerically, including all MSSM
decay modes. Our main result is shown in Fig. 4 where we plot $\Delta N_{eff}$
vs. $m_{\phi}$ for the case of our SUSY BM point and in scenario B2. The
horizontal dashed red line denotes the Planck 2018 95% CL bound on $\Delta
N_{eff}$, i.e. one is constrained to live below the red dashed line. We also
show the projected limit of the Stage 4 CMB experiment[58] where $\Delta
N_{eff}$ as low as 0.06 may be detected (dot-dashed line). We also show the
value of $\Delta N_{eff}$ computed from Eq. 15 assuming three values of
$\lambda_{\text{A}LP}=1,\ 2$ and the minimal LVS value of $0.816$. From the
plot, we see that much of the green curve for $\lambda_{\rm ALP}=2$ would be
excluded, while the remaining curves are excluded at very low $m_{\phi}\sim
1-2$ TeV. As $m_{\phi}$ increases, one might naively expect $\Delta N_{eff}$
to increase as $m_{\phi}^{2}$ according to Eq. 15. But for most of the range,
the denominator $T_{e}T_{osc}^{3}\sim m_{\phi}^{2}$ as well, so that the
$m_{\phi}$ dependence roughly cancels out. Instead, the bulk of $m_{\phi}$
dependence comes from the $g_{*}$ degrees of freedom parameters: namely, as
$m_{\phi}$ increases, then $T_{D}$ and $T_{osc}$ also change and consequently
the various values of $g_{*}$ change. For convenience, we plot in Fig. 1 the
value of $g_{*S}(T)$ vs. $T=m_{\phi}$. Here, we see that as various thresholds
are passed, then the effective number of MSSM degrees of freedom can increase
significantly leading to the various slope discontinuities in $\Delta N_{eff}$
of Fig. 4.

Back to Fig. 4, we see that as $m_{\phi}$ increases past $\sim 2$ TeV, then
$\Delta N_{eff}$ drops well below the Planck 95% CL bound for
$\lambda_{\text{ALP}}\lesssim 1$ and stays in the allowed region all the way
until $m_{\phi}\gtrsim 20,000-30,000$ TeV whence $\Delta N_{eff}$ begins a
sharp increase into forbidden territory. The sharp increase is due to the fact
that $m_{\phi}$ is so large that $\phi$ starts oscillating earlier than our
assumed value of reheat temperature $T_{R}\sim 10^{12}$ GeV, i.e. that the
field $\phi$ begins oscillating before the reheating period of inflation has
finished. In this case, the dependence of $T_{e}$ and $T_{osc}$ changes as in
Eq’s 16 and 17 and the value of $\Delta N_{eff}$ consequently increases with
increasing $m_{\phi}$.

Figure 4: The value $\Delta N_{eff}$ vs. $m_{\phi}$ for three cases of
$\lambda_{\rm ALP}$ along with upper bound from Planck 2018 results (red
dashed line) and projected reach of CMS-S4 (dot-dashed line).

Thus, the bulk of the region for $m_{\phi}\sim 2-30,000$ TeV is allowed by
present CMB data if $\lambda_{\text{ALP}}\lesssim 1$. Furthermore, the $\Delta
N_{eff}$ distribution reaches a minimal value in $\Delta N_{eff}\sim
0.04-0.28$ for $m_{\phi}\sim 700-6000$ TeV. This is the same value of
$m_{\phi}$ for which the dark matter abundance $\Omega_{\chi}h^{2}$ drops into
the allowed region $\lesssim 0.1$ in Ref. [20]. Thus, for these very heavy
values of $m_{\phi}$, both the modulus induced BBN, DM and DR problems are all
solved. A remaining outlier problem would be the moduli-induced gravitino
problem for $m_{\phi}\gtrsim 2m_{3/2}$. Assuming $m_{3/2}\sim m_{soft}$ as in
gravity mediation, then one is faced with antagonism between SUSY naturalness
requiring TeV-scale soft terms and a solution to the various CMP problems
which may favor much heavier moduli masses[20].

Figure 5: Upper bound on $\lambda_{\rm ALP}$ vs. $\phi_{0}/m_{P}$ for various
values of $m_{\phi}$. The dashed line denotes the LVS value of $\lambda_{\rm
ALP}$.

One way to reconcile the four CMPs with naturalness is to assume the lightest
modulus $\phi$ is stabilized by SUSY breaking so that $m_{\phi}\sim
m_{soft}\sim m_{3/2}$ so the gravitino decay mode is closed but with
$m_{\phi}\sim 1-10$ TeV. The anthropic solution[23] to the CMPs then selects
low $\phi_{0}$ values $\sim 10^{-7}m_{P}$ in order to gain a comparable DM-to-
baryons abundance. Notice that the tiny $\phi_{0}$ value does not suppress the
${\cal B}(\phi\rightarrow ALPs)$ since $\phi_{0}$ tends to cancel in Eq. 15.
To show this more clearly, we plot in Fig. 5 the upper bound on
$\lambda_{\text{A}LP}$ versus $\phi_{0}/m_{P}$ which is required to stay below
the Planck CMB limit on $\Delta N_{eff}$ for various values of $m_{\phi}$. For
most $m_{\phi}$ values, the upper bound on $\lambda_{\text{A}LP}$ exceeds the
naive LVS limit although for $m_{\phi}\sim 10^{5}$ TeV the upper limit drops
to $\lambda_{\text{A}LP}\lesssim 0.31$. Thus, for much of the range of
$m_{\phi}$, assuming the bulk of $\phi\rightarrow MSSM$ decay modes are open,
the DR aspect of the CMP does not seem too severe, at least at present. On the
other hand, at least for the LVS stabilization scheme, one might expect a good
chance for a future deviation in the measured value of $\Delta N_{eff}$
compared to the SM value.

###  5.1 Scatter plots

To gain more perspective on $\lambda_{\text{A}LP}$ constraints, we plot in
Fig. 6 the allowed and disallowed regions as scatter plots in the
$\lambda_{\text{A}LP}$ vs. $m_{\phi}$ plane assuming in the first three frames
that all MSSM decay modes are allowed with values
$\lambda_{G}=\lambda_{M}=\lambda_{H}=$ a) 0.1, b) 1 and c) 10. In frame d), we
adopt the sequestered LVS model where modulus decays to the gauge sector are
suppressed (with $\lambda_{G}=0.01$) but with modulus decay to Higgs and
matter at full strength ($\lambda_{M}=\lambda_{H}=1$). The color-coding
corresponds to the computed value of $\Delta N_{eff}$ where red has $\Delta
N_{eff}\gtrsim 0.29$ (excluded) and purple has $\Delta N_{eff}<0.06$ (not even
seeable by CMB S4).

Figure 6: Color-coded value of $\Delta N_{eff}$ in the $\lambda_{\text{A}LP}$
vs. $m_{\phi}$ plane for a) $\lambda_{i}=0.1$, b) $\lambda_{i}=1$, c)
$\lambda_{i}=10$ and d) sequestered model with $\lambda_{G}=0.01$ but with
$\lambda_{M}=\lambda_{H}=1$. The dashed line denotes the value of
$\lambda_{\text{A}LP}$ in the LVS scenario.

For frame a), with decays to MSSM particles suppressed, then ${\cal
B}(\phi\rightarrow aa)$ is large and all $m_{\phi}$ values with
$\lambda_{\text{A}LP}\gtrsim 0.1$ are already excluded. As $\lambda_{i}$ for
$i=G,\ M$ and $H$ increases (thus increasing decays to MSSM particles and
decreasing ${\cal B}(\phi\rightarrow aa)$) as in frames b) $\lambda_{i}=1$ and
c) $\lambda_{i}=10$, then more and more of parameter space becomes allowed
(non-red colors). For the sequestered LVS prediction where decays to gauge
bosons and gauginos are suppressed, then again the ${\cal B}(\phi\rightarrow
aa)$ becomes large and predicts a value of $\Delta N_{eff}$ that is beyond
Planck 2018 bounds.

##  6 Summary and conclusions

In this paper, we have continued our exploration of the various aspects of the
CMP, this time addressing the issue of dark radiation coming from the lightest
modulus $\phi$ decay to its associated ALPs. In this work, we have neglected a
so-called PQ sector QCD axion which may behave differently from a light
stringy ALP in that its interactions would be suppressed by the PQ scale
$f_{a}$ instead of $m_{P}$ and its mass and couplings would be related as in
the PQ solution to the strong CP problem[59]. For the SUSY PQ model, then one
also expects the presence of saxions and axinos which can all affect the
various particle abundances. In future work, we intend to include light moduli
fields into previous eight-coupled Boltzmann solutions to the dark matter
abundance from mixed axion-neutralino dark matter[60, 31, 61].

Our present work extends previous work[23, 20] by including modulus field
decay to its associated ALP particle assuming the ALP mass is quite light as
should occur in the LVS moduli stabilization scheme wherein moduli are
stabilized by a combination of perturbative and nonperturbative effects. In
other models such as KKLT with purely non-perturbative Kähler moduli
stabilization, then the ALPs tend to get masses comparable to the moduli, and
there is no ALP dark radiation problem. Our present analysis has been more
phenomenological, not restricting ourselves to a particular stabilization
scheme since for more realistic CY manifolds, one expects far more Kähler
moduli than the few that are assumed in simple toy moduli stabilization
schemes. In the more general case, where the full panoply of MSSM decay modes
of the lightest modulus may be allowed and the gauge boson decay modes may be
allowed at tree level, then the modulus branching fraction to dark radiation
is typically $\sim 0.01$, lower than is usually assumed. In such cases, where
$m_{\phi}>2m_{\rm ALP}$, then typically the $\Delta N_{eff}$ is below present
limits. Future experiments such as Stage 4 CMB (CMB-S4)[58] are expected to
probe much smaller $\Delta N_{eff}\sim 0.06$. If so, then a discrepancy could
still appear for light enough ALP particles coming from string
compactifications.

Acknowledgements:

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics under Award Number DE-
SC-0009956 and U.S. Department of Energy (DoE) Grant DE-SC-0017647.

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# A remark on Fremlin–Miller theorem concerning the Menger property and
Michael concentrated sets

J. Chaber, R. Pol

(10.10.2002)

###### Abstract

The theorem we prove is a slight strengthening of some results by Just,
Miller, Scheepers and Szeptycki [JMSS]. We use the Michael technique instead
of the combinatorial approach in the literature.

We consider only separable metrizable spaces. A space $M$ has the Menger
property if for any sequence ${\mathcal{U}}_{1},{\mathcal{U}}_{2},\ldots$ of
open covers of $M$ there are finite collections
${\mathcal{V}}_{i}\subset{\mathcal{U}}_{i}$ with
$\bigcup_{i}{\mathcal{V}}_{i}$ covering $M$. Let ${\rm{\bf d}}$ be the minimal
cardinality of a dominating set in ${{\mathbb{N}}^{\mathbb{N}}}$ (we recall
the definition below).

Fremlin and Miller [MF] proved that any space of cardinality less than
${\rm{\bf d}}$ has the Menger property. In particular, upon an identification
of ${{\mathbb{N}}^{\mathbb{N}}}$ with the irrationals, if $S$ is a scale in
${{\mathbb{N}}^{\mathbb{N}}}$ of cardinality ${\rm{\bf d}}$, and
${\mathbb{Q}}$ are the rationals, the union $S\cup{\mathbb{Q}}$ has the Menger
property (notice also that $S$ is concentrated about ${\mathbb{Q}}$). In
effect, no matter where ${\rm{\bf d}}$ is situated on the alephs scale, the
Fremlin–Miller theorem yields a non-$\sigma$-compact space with the Menger
property.

We shall add to this important result an observation that the Michael
technique of constructing concentrated sets in product spaces, introduced in
[Mi], allows one to use the Fremlin–Miller theorem to the following effect.

Theorem. In every completely metrizable, separable space $X$ without any
compact set with nonempty interior there is an $M\subset X$ whose all finite
powers have the Menger property, but $M$ is not contained in any
$\sigma$-compact subset of $X$.

Some closely related results were proved by Just, Miller, Scheepers and
Szeptycki with the use of a different, combinatorial approach, cf. Theorems
3.9 and 5.1 in [JMSS].

Before passing to a proof of the theorem, let us recall some facts about the
order $\leq^{*}$ in ${{\mathbb{N}}^{\mathbb{N}}}$, where $s\leq^{*}t$ means
that $s(n)\leq t(n)$ for all but finitely many $n$. The minimal cardinality of
a set in ${{\mathbb{N}}^{\mathbb{N}}}$ unbounded with respect to $\leq^{*}$ is
denoted by ${\rm{\bf b}}$, and ${\rm{\bf d}}$ is the minimal cardinality of a
set dominating in ${{\mathbb{N}}^{\mathbb{N}}}$, i.e., a set
$D\subset{{\mathbb{N}}^{\mathbb{N}}}$ such that for any
$s\in{{\mathbb{N}}^{\mathbb{N}}}$ there is a $t\in D$ with $s\leq^{*}t$, cf.
[vD]. Always ${\rm{\bf b}}\leq{\rm{\bf d}}$, and the equality ${\rm{\bf
b}}={\rm{\bf d}}$ yields a dominating set of cardinality ${\rm{\bf d}}$ well-
ordered by $\leq^{*}$, i.e., a scale in ${{\mathbb{N}}^{\mathbb{N}}}$.

Proof of Theorem. To begin with, let us recall that each
$t\in{{\mathbb{N}}^{\mathbb{N}}}$ determines a $\sigma$-compact set
$F(t)=\\{s\in{{\mathbb{N}}^{\mathbb{N}}}:s\leq^{*}t\\}$ and each
$\sigma$-compact set in ${{\mathbb{N}}^{\mathbb{N}}}$ is contained in some
$F(t)$.

Let $X$ satisfy the assumptions and let us fix a countable set $Q$ dense in
$X$.

Let

$f:{{\mathbb{N}}^{\mathbb{N}}}\rightarrow X$ (1)

be a perfect surjection.

Assume ${\rm{\bf b}}<{\rm{\bf d}}$, let $B\subset{{\mathbb{N}}^{\mathbb{N}}}$
be an unbounded set of cardinality ${\rm{\bf b}}$ and let $M=f(B)$. Since the
cardinality of $M$ is less than ${\rm{\bf d}}$, any finite power of $M$ has
the Menger property by Fremlin–Miller theorem and $M$ is not contained in any
$\sigma$-compact set in $X$.

From now on we assume ${\rm{\bf b}}={\rm{\bf d}}$. Then we have a scale
$\\{t_{\alpha}:\alpha<{\rm{\bf d}}\\}$ in ${{\mathbb{N}}^{\mathbb{N}}}$,
$t_{\alpha}\leq^{*}t_{\beta}$ for $\alpha<\beta$. Let, cf. (1),

$F_{\alpha}=f(F(t_{\alpha})).$ (2)

Each $\sigma$-compact set in $X$ is contained in some $F_{\alpha}$. Let $Z$ be
a compactification of $X$ and let

$g_{m}:{{\mathbb{N}}^{\mathbb{N}}}\rightarrow Z^{m}\setminus Q^{m}$

be perfect surjections, $H_{\alpha}=X^{m}\setminus g_{m}(F(t_{\alpha}))$. Then
$H_{\alpha}\supset Q^{m}$ and, for any collection ${\mathcal{H}}$ of
$G_{\delta}$ sets in $X^{m}$ containing $Q^{m}$, if $|{\mathcal{H}}|<{\rm{\bf
d}}$ then there is $\alpha$ with $H_{\alpha}\subset\bigcap{\mathcal{H}}$.

In effect, changing the enumeration of the sets $H_{\alpha}$, we get
$G_{\delta}$-sets $G_{\xi}$ with

$Q^{m(\xi)}\subset G_{\xi}\subset X^{m(\xi)},\quad\xi<{\rm{\bf d}},$ (3)

such that for any collection ${\mathcal{H}}$ of $G_{\delta}$-sets in $X^{m}$
containing $Q^{m}$,

$\mbox{ if }|{\mathcal{H}}|<{\rm{\bf d}}\mbox{ then
}G_{\xi}\subset\bigcap{\mathcal{H}}\mbox{ for some }\xi\mbox{ with }m(\xi)=m.$
(4)

We shall define

$B=\\{x_{\alpha}:\alpha<{\rm{\bf d}}\\}\subset X\setminus
Q,\quad\mbox{and}\quad M=Q\cup B.$ (5)

such that the following concentration property holds true (cf. [PPR, Lemma
2.2.2]): for any $m$ and each collection ${\mathcal{H}}$ of $G_{\delta}$ sets
in $X^{m}$ containing $Q^{m}$, if $|{\mathcal{H}}|<{\rm{\bf d}}$ then
$M^{m}\setminus\bigcap{\mathcal{H}}$ is contained in the union of less than
${\rm{\bf d}}$ hyperplanes obtained by fixing one of the coordinates in
$M^{m}$.

A straightforward induction on $m$ shows that for such $M$ every product
$L\times M^{m}$, where $|L|<{\rm{\bf d}}$, has the Menger property (for $m=0$
use the Fremlin–Miller theorem). Therefore it is enough to define the set $B$
in (5), ensuring also that $B$ is not contained in any $\sigma$-compact set in
$X$.

To this end, we shall modify slightly a reasoning in [PPR, Comment 9.5], which
is a variation of the celebrated Michael technique [Mi].

We shall start with some notation. Let $S\subset\\{1,2,...,m\\}$, $u\in
X^{S}$, $v\in X^{\\{1,2,...,m\\}\setminus S}$. Then $(u,v)$ is the point $w\in
X^{m}$ with $w(i)=u(i)$ for $i\in S$ and $w(i)=v(i)$ for $i\notin S$. Further,
given a $G_{\delta}$-set $G$ in $X^{m}$ containing $Q^{m}$, we let

$G(S)=\\{u\in X^{S}:(u,v)\in G\mbox{ for all }v\in Q^{\\{1,2,...,m\\}\setminus
S}\\},$

and, for $u\in G(S)$, $j\in{\\{1,2,...,m\\}\setminus S}$, we let

$G(u,j)=\\{x\in X:(u,x)\in G(S\cup\\{j\\})\\}.$

Then $G(S)$, $G(u,j)$ are $G_{\delta}$-sets with

$Q^{S}\subset G(S)\subset X^{S},\quad Q\subset G(u,j)\subset X$ (6)

We shall pick distinct points $x_{\alpha}$ of the set $B$ in (5) by
transfinite induction, such that, with $\Delta(S)=\\{u\in X^{S}:\mbox{ at
least two coordinates of }u\mbox{ are equal}\\},$ we shall have for any
$\alpha<{\rm{\bf d}}$, $\xi<\alpha$, $S\subset\\{1,2,...,m(\xi)\\}$,

$\\{x_{\eta}:\xi<\eta\leq\alpha\\}^{S}\setminus\Delta(S)\subset G_{\xi}(S)$

We begin with an arbitrary $x_{0}\notin Q$. At each stage $\alpha>0$,
$x_{\alpha}$ must be chosen from the intersection of a collection
${\mathcal{H}}$ consisting of sets of the form $G_{\xi}(u,j)$, $\xi<\alpha$,
$u\in G_{\xi}(S)\cap\\{x_{\eta}:\xi<\eta<\alpha\\}^{S}\setminus\Delta(S)$,
$S\cup\\{j\\}\subset\\{1,2,...,m(\xi)\\}$, and $j\notin S$. But these are
$G_{\delta}$-sets in $X$ containing $Q$, cf. (6), (3), and
$|{\mathcal{H}}|<{\rm{\bf d}}$, hence by (4), the intersection
$\bigcap{\mathcal{H}}$ contains a $G_{\delta}$-set in $X$ containing $Q$. This
shows that a choice of $x_{\alpha}$ is always possible and, moreover, one can
pick $x_{\alpha}$ outside of the set $F_{\alpha}$, meager in $X$, cf. (2),
which will also guarantee that the resulting set $B$ is not contained in any
$\sigma$-compact set in $X$. A verification that the space $M=Q\cup B$ has the
concentration property described below (5) is similar to a simple reasoning in
[PPR, Comment 9.5].

## References

  * [vD] E.K. van Douwen, The Integers and Topology, Handbook of Set-Theoretic Topology, K. Kunen and J.E.Vaughan, editors, Elsevier Science Publishers, 1984, 111–167.
  * [JMSS] W. Just, A.W. Miller, M. Scheepers, P.J. Szeptycki, The combinatorics of open covers II, Topology and Appl. 73(1996), 241–266.
  * [Mi] E.A. Michael, Paracomactness and the Lindelöf property in finite and countable cartesian products, Compositio Math. 23(1971), 199–244.
  * [MF] A.W. Miller, D.H. Fremlin On some properties of Hurewicz, Menger, and Rothberger, Fund. Math. 129(1988), 17–33.
  * [PPR] E. Pol, R. Pol, M. Reńska, On countable-dimensional spaces with the Menger property, rational dimension and a question of S.D. Iliadis, Mh. Math. 128(1999), 331–348.