1 Introduction

While there are various lines of evidence for the existence of dark matter in the universe, the nature of the dark matter particle remains a challenging dilemma at the interface of astrophysics, cosmology, and particle physics [1]. There is a large variety of dark matter candidates with masses spanning many orders of magnitude. Of particular interest here, fuzzy dark matter (FDM) is made up of non-interacting ultralight bosonic particles that exhibit coherent dynamics and a wave-like behaviour on galactic scales [2]. On sub-galactic length scales, FDM brings to light a distinctive phenomenology alternative to that of cold dark matter (CDM). However, FDM predictions are indistinguishable from those of CDM on large scales, and so benefits from the remarkable success of \(\Lambda \)CDM cosmology.

The main parameter regulating the two FDM regimes is the particle’s mass, with a range that spans 3 decades of energy, \(10^{-24} \lesssim m/\textrm{eV} \lesssim 10^{-22}~\textrm{eV}\). Particles of such tiny mass and with typical velocities v found in haloes hosting Milky Way-sized galaxies, acquire a very long de Broglie wavelength,

$$\begin{aligned} \lambda _\textrm{dB} \equiv \frac{2 \pi }{mv} = 4.8~\textrm{kpc} \left( \frac{10^{-23}~\textrm{eV} }{m}\right) \left( \frac{250~\mathrm{km/s}}{v}\right) \,, \end{aligned}$$
(1)

to deliver the wave-like behavior at galactic scales.

FDM can populate the galactic haloes with large occupation numbers and behave as self-gravitating dark matter waves. This engenders a pressure-like effect on macroscopic scales which catalyzes a flat core at the center of galaxies, with a relatively marked transition to a less dense outer region that follows the typical CDM-like distribution.

Before proceeding, we take note of a serious challenge for FDM models. The criticism centers on numerical simulations to accommodate Lyman-\(\alpha \) forest data which provide bounds on the fraction of FDM [3,4,5]. In response, it was noted that these bounds strongly depend on the modeling of the intergalactic medium [6]. More recently, new constraints for FDM models have emerged, e.g. (i) from inferences of the low-mass end of the subhalo mass function [7], (ii) from observations of ultra-faint dwarf galaxies [8], and (iii) from superradiance of FDM which would cause the supermassive black hole at the center of M87 to spin down excessively [9]. Constraints (i) and (ii) also depend on simulations and are subject to a completely different set of assumptions and systematic uncertainties. The Event Horizon Telescope measurement of the spin of M87\(^*\) excludes FDM masses in the range \(10^{-21} \lesssim m/\textrm{eV} \lesssim 10^{-20}\). Whichever point of view one may find more convincing, it seems most conservative at this point to depend on experiment (if possible) rather than numerical simulations to resolve the issue.

In this paper we show that the dark dimension scenario [10] embraces a well motivated FDM candidate. The layout is as follows: in Sect. 2 we outline the basic setting of the dark dimension scenario and we identify the radion as a FDM candidate, in Sect. 3 we discuss the process of radion production to estimate the corresponding relic abundance, and conclusions are given in Sect. 4.

2 The good, the bad, and the fuzzy

The Swampland program seeks to understand which are the “good” low-energy effective field theories (EFTs) that can couple to gravity consistently (e.g. the landscape of superstring theory vacua) and distinguish them from the “bad” ones that cannot [11]. In theory space, the frontier discerning the good theories from those downgraded to the swampland is drawn by a family of conjectures classifying the properties that an EFT should call for/avoid to enable a consistent completion into quantum gravity. These conjectures provide a bridge from quantum gravity to astrophysics, cosmology, and particle physics [12,13,14].

For example, the distance conjecture (DC) forecasts the appearance of infinite towers of states that become exponentially light and trigger the collapse of the EFT at infinite distance limits in moduli space [15]. Connected to the DC is the anti-de Sitter (AdS) distance conjecture, which correlates the dark energy density to the mass scale m characterizing the infinite tower of states, \(m \sim |\Lambda |^\alpha \), as the negative AdS vacuum energy \(\Lambda \rightarrow 0\), with \(\alpha \) a positive constant of \(\mathcal {O} (1)\) [16]. Besides, under the hypothesis that this scaling behavior holds in dS (or quasi dS) space, an unbounded number of massless modes also pop up in the limit \(\Lambda \rightarrow 0\).

As demonstrated in [10], the generalization of the AdS-DC to dS space could help elucidate the radiative stability of the cosmological hierarchy \(\Lambda _4/M_p^{4} \sim 10^{-120}\), because it connects the size of the compact space \(R_\perp \) to the dark energy scale \(\Lambda _4^{-1/4}\),

$$\begin{aligned} R_\perp \sim \lambda \ \Lambda _4^{-1/4} \,, \end{aligned}$$
(2)

where the proportionality factor is estimated to be within the range \(10^{-1}< \lambda < 10^{-4}\). Actually, (2) derives from constraints by theory and experiment. On the one hand, since the associated Kaluza–Klein (KK) tower contains massive spin-two bosons, the Higuchi bound [17] provides an absolute upper limit to \(\alpha \), whereas explicit string calculations of the vacuum energy (see e.g. [18,19,20,21]) yield a lower bound on \(\alpha \). All in all, the theoretical constraints lead to \(1/4 \le \alpha \le 1/2\); see [22] for a recent discussion. On the other hand, experimental arguments (e.g. constraints on deviations from Newton’s gravitational inverse-square law [23] and neutron star heating [24]) lead to the conclusion encapsulated in (2); namely, that there is one extra dimension of radius \(R_\perp \) in the micron range, and that the lower bound for \(\alpha = 1/4\) is basically saturated [10]. This in turn implies that the KK tower of the new (dark) dimension opens up at the mass scale \(m_\textrm{KK} \sim 1/R_\perp \). Within this set-up, the 5-dimensional Planck scale (or species scale where gravity becomes strong [25, 26]) is given by \(M_* \sim m_\textrm{KK}^{1/3} M_p^{2/3}\). Note that for \(m_\textrm{KK} \sim 1~\textrm{eV}\), we have \(M_* \sim 10^{9}~\textrm{GeV}\) and therefore the species scale is outside the reach of collider experiments [27].

The dark dimension stores a top-notch phenomenology [28,29,30,31,32,33,34,35,36,37,38]. For example, it was noted in [32] that the universal coupling of the SM fields to the massive spin-2 KK excitations of the graviton in the dark dimension provides a dark matter candidate. Complementary to the dark gravitons, it was observed in [29] that primordial black holes with Schwarzschild radius smaller than a micron could also be good dark matter candidates, possibly even with an interesting close relation to the dark gravitons [31]. Next, in line with our stated plan, we propose a new dark matter candidate within this framework.

It is unnatural to entertain that the size of the dark dimension would remain fixed during the evolution of the Universe right at the species scale. To accommodate this hierarchy we need to inflate the size of the dark dimension. To see how this works explicitly, we consider that the inflationary phase can described by a 5-dimensional dS (or approximate) solution of Einstein equations [33]. All dimensions (compact and non-compact) expand exponentially in terms of the 5-dimensional proper time. This implies that when inflation starts the radius R of the compact space is small and the 4-dimensional Planck mass is of order the 5-dimensional Planck scale \(M_*\). However, when inflation ends the radius of the compact space is on the micron-scale size and the 4-dimensional Planck scale is much bigger,

$$\begin{aligned} M_p^2 = 2 \pi \ M_*^3 \ R_\perp \, . \end{aligned}$$
(3)

A straightforward calculation shows that the compact space requires 42 e-folds to expand from the fundamental length \(1/M_*\) to the micron size. We can interpret the solution in terms of 4-dimensional fields using 4-dimensional Planck units from the relation (3), which amounts going to the 4-dimensional Einstein frame. Namely, the higher-dimensional metric in \(M_*\) units is given by

$$\begin{aligned} ds_5^2 = a_5^2 \ (-d\eta ^2 + d\vec x^2 + r^2_0 \ dy^2) \,, \end{aligned}$$
(4)

where \(\eta \) is the conformal time, \(a_5 = 1/(H\eta )\), H is the Hubble parameter, \(\vec x\) denotes the 3 uncompactified dimensions, and \(r_0 \sim 1\) is the radius of the dark dimension y at the beginning of the inflationary phase. The 4-dimensional decomposition in the Einstein frame is given by

$$\begin{aligned} ds_5^2 = \frac{1}{R} ds_4^2 + R^2 dy^2 \,, \end{aligned}$$
(5)

where \(ds_4^2 = a_4^2 (-d\eta ^2 + d^2\vec x)\). Comparing (4) and (5) we arrive at \(a_4/\sqrt{R} = R\). After inflation of N e-folds, where the scale factor was expanded by \(a_5 = e^N\), the radius becomes \(R=e^N\). This implies that if R expands N e-folds, then the 3-dimensional space would expand 3N/2 e-folds as a result of a uniform 5-dimensional inflation [33]. We want \(r_0\) to grow fast up to the micron scale. Altogether, the 3-dimensional space has expanded by about 60 e-folds to solve the horizon problem, while connecting this particular solution to the generation of a mesoscopic size dimension. A consistent model requires the size of the dark dimension to be stabilized at the end of inflation; an investigation along this line is already presented in [39].

The 5-dimensional action of uniform dS (or approximate) inflation

$$\begin{aligned} S_5=\int [d^4x][dy] \left( {1\over 2} M_*^3 \mathcal{R}_{(5)}-\Lambda _{5}\right) \,, \end{aligned}$$
(6)

leads to a runaway potential for the radion R coming from the 5-dimensional cosmological constant, where \(\mathcal {R}_{(5)}\) is the higher dimensional curvature scalar and \(\Lambda _5\) is the 5-dimensional cosmological constant at the end of inflation. The quintessence-like potential of the radion is seen explicitly upon dimensional reduction to 4 dimensions. The resulting 4-dimensional action in the Einstein frame is found to be,

$$\begin{aligned} S_{4}= & {} \int [d^4x] \left\{ {1\over 2} \ M_p^{2} \ \mathcal{R}_{(4)} - {3\over 4} \ M_p^2 \ \Bigg ({\partial R\over R}\right) ^2 \nonumber \\{} & {} - (2\pi \langle R \rangle )^{2} {\Lambda _{5}\over (2\pi R)}\Bigg \}, \end{aligned}$$
(7)

where \(\mathcal {R}_{(4)}\) is the Ricci scalar and \(\langle R \rangle \) is the vacuum expectation value (vev) of R after the end of inflation. Because the radion field R is not canonically normalized, we define \(\phi = \sqrt{3/2} \ \ln \left( R/\langle R \rangle \right) \). In terms of the normalized field \(\phi \) the scalar potential takes the advertised quintessence-like form

$$\begin{aligned} V(\phi ) = 2 \pi \ \Lambda _5 \ \langle R \rangle \ e^{- \sqrt{2/3} \phi } \, . \end{aligned}$$
(8)

Exponential potentials of the form \(e^{-\alpha \phi }\) are constrained by cosmological and astrophysical observations. The existing data lead to an upper bound \(\alpha \lesssim 0.8\) [40]. Curiously, the upper limit on the allowed value of \(\alpha \) is the one predicted by (8).Footnote 1 Even though the potential in (8) could be used to explain the current acceleration of the universe, herein we consider the possibility that the radion is stabilized by additional terms in the potential. The mass of the radion,

$$\begin{aligned} m \sim \sqrt{V^\textrm{tot}_{\phi \phi } (0)}/M_p \, , \end{aligned}$$
(9)

depends on the functional form of the various additional terms \(V_i(\phi )\) that allow minimization of the potential, i.e.

$$\begin{aligned} V^\textrm{tot} (\phi ) = V(\phi ) + \sum _i V_i(\phi ) + \Lambda _4 \end{aligned}$$
(10)

with \(V_\phi ^\textrm{tot} (0) = 0\) and where \(V_\phi \equiv dV/d\phi \). Adding only the term originating from the Casimir energy [42] leads to a lower bound on \(m \sim \sqrt{\Lambda _4}/M_p \sim 10^{-30}~\textrm{eV}\) [39].

An important aspect of this model is that the coupling of the radion to SM fields must be suppressed to avoid conflicts with limits on long range forces. Herein, we assume that the radion has a localized kinetic term through (e.g. an expectation value of a brane field) that suppresses the coupling to matter. Alternatively, in the absence of a scalar potential, the 5-dimensional radion is equivalent to a Brans–Dicke scalar with a parameter \(\omega = -4/3\). It has been argued that an appropriate modification of such theories due to bulk quantum corrections can lead to a logarithmic scale (time) dependence of \(\omega \) that suppresses the radion coupling to matter, consistently with the experimental limits [43]. An investigation along these lines is obviously important to be done.

3 Production of the fuzzy radion and its relic abundance

The issue that remains to be assessed is whether there is a mechanism which allows enough radion production to accommodate the relic dark matter density. An interesting possibility emerges if the inflaton has roughly equal couplings to brane and bulk fields, such that on decay will produce the SM fields while also populating the KK towers.

We begin by considering a tower of equally spaced dark gravitons, indexed by an integer l, and mass \(m_l = l m_\textrm{KK}\). We assume that the cosmic evolution of the dark sector is mostly driven by “dark-to-dark” decay processes that regulate the decay of KK gravitons within the dark tower. The proposed decay model then provides a particular realization of the dynamical dark matter model [44]. The intra-KK decays in the bulk require a spontaneous breakdown of the translational invariance in the compact space such that the 5-dimensional momenta are not conserved. An explicit realization of this idea, in which the KK modes acquire a nonzero vev \(\langle \varphi _l \rangle \) has been given in [45]. Following [46], we further assume transitions by instanton-induced tunneling dynamics associated with such vacuum towers. The effect of the instanton processes is to accelerate the cascade dynamics to collapse into the radion.

Bearing this in mind, we calculate the decay of a given massive KK graviton into the radion \(\phi \) and a lighter KK graviton in the presence of an expectation value for the bulk scalar \(\varphi \) that breaks momentum conservation. Following [45] we postulate the existence of a coupling of the form

$$\begin{aligned} \mathscr {L}_I \supset \ \lambda \ \frac{1}{M_*} \ \Phi \ h_{AB} \ h_{CD} \ h_{EF} \ \mathcal {C}^{ABCDEF} \,, \end{aligned}$$
(11)

coming from

$$\begin{aligned} \int d^4x \ dy \ \sqrt{g} \ T (x,y) \,, \end{aligned}$$
(12)

where T(x, y) is the trace of the energy momentum tensor of the bulk theory, \(h_{AB}\) is the 5-dimensional graviton which comes from expansion of the 5-dimensional metric around flat space, \(g_{AB} = \eta _{AB} +M_*^{-3/2} h_{AB}\), and where in (11) \(\lambda \) is a dimensionless coupling and \(\mathcal {C}^{ABCDEF}\) is a constant tensor. Next, we expand all 5-dimensional fields in terms of their 4-dimensional KK modes, for instance

$$\begin{aligned} \Phi (x,y) = \sum _{l} \frac{1}{\sqrt{2 \pi R_\perp }} \varphi _{l} (x) \ e^{i l y/R_\perp } \, . \end{aligned}$$
(13)

After integration over dy, (11) can be recast as

$$\begin{aligned} \mathscr {L}_I \supset \sum _{l,l'} \lambda \ \left( \frac{M_*}{M_p} \right) ^2 \ \varphi _{(l-l')} \ \phi \ h_{l} \ h_{l'} \ \mathcal {C} \, , \end{aligned}$$
(14)

where we have made use of (3) and considered a factor of \(1/\sqrt{2\pi R_\perp }\) for each KK decomposition of the four fields and a factor of \(2 \pi R_\perp \) from the integration over y. Now, following [45] we assume that \(\varphi _l\) takes a vev which is independent of l. The total decay width of a KK graviton of mass \(m_l = l m_\textrm{KK}\) is found to be [27]

$$\begin{aligned} \Gamma ^l_\textrm{tot}= & {} \frac{\lambda ^2}{8\pi } \ \frac{1}{m_l^2} \sum _{l' < l} \left( \frac{M_*}{M_p}\right) ^4 \frac{m_l^2-m_{l'}^2}{2 m_l} \langle \varphi _{l-l'} \rangle ^2 \nonumber \\= & {} \frac{\lambda ^2}{8 \pi } \ \frac{1}{m_l^2} \ \left( \frac{M_*}{M_p}\right) ^4 \ \frac{(l-1) (4l+1)}{12} m_\textrm{KK} \ \langle \varphi \rangle ^2 \nonumber \\= & {} \frac{\lambda ^2}{96 \pi ^3} \left( m_\textrm{KK} - \frac{3 m_\textrm{KK}^2}{4m_l} - \frac{m_\textrm{KK}^3}{4 m_l^2} \right) \left( \frac{\langle \varphi \rangle }{M_*} \right) ^2 \!\!\!\!, \end{aligned}$$
(15)

where \(\langle \varphi _{(l-l')} \rangle = \langle \varphi \rangle \). Substituting in (15) our fiducial value for \(m_\textrm{KK} \sim 10~\textrm{eV}\) while considering as entertained in [45] \(\langle \varphi \rangle \sim M_*\) we obtain a total decay width of \(\Gamma ^l_\textrm{tot} \sim 5 \times 10^{12}~\textrm{s}^{-1}\), where we have taken \(\lambda \sim 1\) and \(m_l\gg m_\textrm{KK}\). If we instead adopt \(\langle \varphi \rangle \sim 5 \times 10^{-4} M_*\) we obtain \(\Gamma ^l_\textrm{tot} \sim 10^{6}~\textrm{s}^{-1}\), which implies that the energy the inflaton deposited in the KK tower ends up into the radion well before the QCD phase transition (with characteristic temperature \(\sim 150~\textrm{MeV}\) and age \(\sim 20~\mu \textrm{s}\)). Altogether, we conclude that even for \(\langle \varphi \rangle \ll M_*\) the energy the inflaton deposited in the KK tower collapses all into the radion well before the earliest observational verified landmark (viz., big bang nucleosynthesis with starting age of roughly 180 s).

Fig. 1
figure 1

Schematic representation of the evolution of \(\phi \) (left) and \(w_\phi \) (right) with the scale factor \(a_4\). The dimensionful quantities have arbitrary normalization. The vertical dashed lines indicate the condition defining \(m \sim 3\,H\)

Full size image

Qualitatively, radion cosmology reassembles that of ultralight axion-like particles [47]. Namely, the radion equation of motion is given by

$$\begin{aligned} \ddot{\phi }+ 3 H \dot{\phi }+ V^\textrm{tot}_\phi = 0 \,, \end{aligned}$$
(16)

where H is the Hubble parameter. We assume that \(\phi \) is around the minimum of the potential at the origin such that the total potential can be expanded around its minimum as \(V^\textrm{tot} \sim (m \phi )^2/2 + \Lambda _4\), and so (16) can be rewritten as

$$\begin{aligned} \ddot{\phi }+ 3 H \dot{\phi }+ m^2 \phi = 0 \, . \end{aligned}$$
(17)

At very early times, when \(m<3H\), the radion field is overdamped and frozen at its initial value by Hubble friction. During this epoch the equation of state is \(w_\phi = -1\) and the radion behaves as a sub-dominant cosmological constant. Once the Universe expands to the point where \(m \sim 3H\), the driving force overcomes the friction and the field begins to slowly roll. Finally, when \(m> 3H\), the field executes undamped oscillations. The equation of state oscillates around \(w_\phi = 0\) and the energy density scales as CDM. A visual representation of the evolution of \(\phi \) and \(w_\phi \) is shown in Fig. 1.

We now turn the estimate the required energy density of the radion field \(\rho _\phi \) to accommodate the observed relic density if the CDM evolution should duplicate that in \(\Lambda \)CDM after the epoch of matter-radiation equality. To this end we first reexamine the evolution of the radiation energy density, which can be conveniently expressed as

$$\begin{aligned} \rho _R = \left( \sum _B g_B + \frac{7}{8} \sum _F g_F \right) \frac{\pi ^2}{30} \ T^4 \equiv \frac{\pi ^2}{30} \ N(T) \ T^4 \,, \end{aligned}$$
(18)

where \(g_{B(F)}\) is the total number of boson (fermion) degrees of freedom and the sum runs over all boson (fermion) states with \(m_{B(F)} \ll T\), and where N(T) is the number of effective degrees of freedom (the factor of 7/8 is due to the difference between the Fermi and Bose integrals).

The expansion rate as a function of the temperature in the plasma is given by

$$\begin{aligned} H (T) = \left[ \frac{\pi ^2}{90} N(T) \right] ^{1/2} \frac{T^2}{M_p} \sim 0.33 \sqrt{N(T)} \, \frac{T^2}{M_p} \, . \end{aligned}$$
(19)

By inspection of (19) we can immediately see that for \(m \sim 3\,H\), only photons and neutrinos contribute to the sum in (18), yielding \(N = 7.25\). This corresponds to a temperature \(T_\textrm{osc} \sim 86~\textrm{eV}\), for which the total energy density of radiation (18) is \(\rho _R (T_\textrm{osc}) \sim 10^8~\textrm{eV}^4\). Now, let \(\rho _\phi (T_\textrm{osc})\) be the background energy density of the radion field at \(T_\textrm{osc}\). As the universe expands, the ratio of dark matter to radiation grows as 1/T, and in \(\Lambda \)CDM cosmology they are supposed to become equal at the temperature \(T_\textrm{MR} \sim 1~\textrm{eV}\) of matter-radiation equality. This implies that

$$\begin{aligned} \frac{\rho _\phi (T_\textrm{osc})}{\rho _R (T_\textrm{osc})} \ \frac{T_\textrm{osc}}{T_\textrm{MR}} \sim 1 \,, \end{aligned}$$
(20)

which leads to \(\rho _\phi (T_\textrm{osc}) \sim 10^6~\textrm{eV}^4\) [6]. In other words, if the density of the radion field were about \(10^6~\mathrm{eV^4}\), then today’s radion abundance would easily accommodate the observed dark matter density [48], i.e., \(\rho _{\phi , \textrm{today}} \sim \rho _\textrm{DM} \sim 1.26~\mathrm{keV/cm^{3}}\). We note that \(\rho _{\phi } (T_\textrm{osc})\) should be equal to the value of the potential (above \(\Lambda _4\)) at the constant value \(\phi _i\) that is the initial condition, i.e. \(V^\textrm{tot} (\phi _i) \sim 10^6~\textrm{eV}^4\). We note that although the initial value of radion field is a free parameter of the model it is subject to the constraint \(\phi _i/M_p \ll 1\), so that \(V^\textrm{tot}_\phi \sim m^2 \phi \) and the expansion in (17) is valid.

In closing, we note that when oscillations start, before the matter-radiation equality, the radion is non-relativistic and therefore \(\Delta N_\textrm{eff}\) (the number of “equivalent” light neutrino species in units of the density of a single Weyl neutrino [49]) stays unaffected at the earliest observationally verified landmarks (viz. big bang nucleosynthesis and cosmic microwave background). As a consequence, our model remains consistent with the bounds derived in [50, 51].

4 Conclusions

We have introduced a new dark matter contender within the context of the dark dimension. The dramatis personae is the radion, a bulk scalar field whose quintessence-like potential drives an inflationary phase described by a 5-dimensional de Sitter (or approximate) solution of Einstein equations. We have shown that within this set up the radion could be ultralight and thereby serve as a fuzzy dark matter candidate. We have put forward a simple cosmological production mechanism bringing into play unstable KK graviton towers which are fueled via inflaton decay.

We end with an observation. The coherent oscillation of the fuzzy radion in galactic haloes leads to pressure perturbations oscillating at twice its Compton frequency, \(\omega = 2\,m\) [52]. These oscillations induce fluctuations of the gravitational potential at frequency

$$\begin{aligned} f\equiv & {} \omega /(2\pi ) \nonumber \\\simeq & {} 4.8 \times 10^{-9}~\textrm{Hz} \ \left( \frac{m}{10^{-23}~\textrm{eV}}\right) \end{aligned}$$
(21)

and can give rise to distinctive profiles in the travel time of radio beams emitted from pulsars, which have been monitored for decades in Pulsar Timing Array (PTA) experiments [53].

Recently, several PTA experiments reported \(4\sigma \) evidence for a stochastic signal with quadrupolar angular correlations consistent with the expected isotropic background of gravitational waves radiated by inspiraling supermassive black hole binaries [54,55,56,57]. (A plethora of models have also been explored to explain the Hellings-Downs-correlated signal [58,59,60,61,62,63,64].) In addition, the NANOGrav Collaboration reported a monopole excess around 4 nHz [65]. The fuzzy radion can accommodate the PTA monopole anomaly if \(m \sim 8 \times 10^{-24}~\textrm{eV}\).