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# The second data release from the European Pulsar Timing Array: VI. Challenging the ultralight dark matter paradigm

Clemente Smarra, Boris Goncharov, Enrico Barausse, J Antoniadis, S Babak, A.-S. Bak Nielsen, C.G Bassa, A Berthereau, M Bonetti, E Bortolas, et al.

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## ► To cite this version:

Clemente Smarra, Boris Goncharov, Enrico Barausse, J Antoniadis, S Babak, et al.. The second data release from the European Pulsar Timing Array: VI. Challenging the ultralight dark matter paradigm. Physical Review Letters, 2023, 131 (17), pp.171001. 10.1103/PhysRevLett.131.171001. hal-04158928

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https://hal.science/hal-04158928v1

Submitted on 15 Jul 2025

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# The second data release from the European Pulsar Timing Array
## VI. Challenging the ultralight dark matter paradigm

Clemente Smarra<sup>1,2,*</sup> Boris Goncharov<sup>3,4</sup> Enrico Barausse<sup>1,2</sup> J. Antoniadis<sup>5,6</sup> S. Babak<sup>7</sup>

A.-S. Bak Nielsen<sup>6,8</sup> C. G. Bassa<sup>9</sup> A. Berthereau,<sup>10,11</sup> M. Bonetti<sup>12,13,14</sup> E. Bortolas,<sup>12,13,14</sup>

P. R. Brook<sup>15</sup> M. Burgay<sup>16</sup> R. N. Caballero<sup>17</sup> A. Chalumeau<sup>12</sup> D. J. Champion<sup>6</sup> S. Chanlaridis<sup>5</sup>

S. Chen<sup>18</sup> I. Cognard<sup>10,11</sup> G. Desvignes<sup>6</sup> M. Falxa,<sup>10,7</sup> R. D. Ferdman,<sup>19</sup> A. Franchini<sup>12,13</sup>

J. R. Gair<sup>20</sup> E. Graikou<sup>6</sup> J.-M. Grießmeier<sup>10,11</sup> L. Guillemot<sup>10,11</sup> Y. J. Guo<sup>6</sup> H. Hu<sup>6</sup> F. Iraci,<sup>16,21</sup>

D. Izquierdo-Villalba<sup>12,13</sup> J. Jang<sup>6</sup> J. Jawor<sup>6</sup> G. H. Janssen<sup>9,22</sup> A. Jessner<sup>6</sup> R. Karuppusamy<sup>6</sup>

E. F. Keane<sup>23</sup> M. J. Keith<sup>24</sup> M. Kramer,<sup>6,24</sup> M. A. Krishnakumar<sup>6,8</sup> K. Lackeos<sup>6</sup> K. J. Lee,<sup>5,6,11</sup>

K. Liu,<sup>6</sup> Y. Liu<sup>8,25</sup> A. G. Lyne,<sup>24</sup> J. W. McKee<sup>26,27</sup> R. A. Main,<sup>6</sup> M. B. Mickaliger,<sup>24</sup> I. C. Nițu<sup>24</sup>

A. Parthasarathy<sup>6</sup> B. B. P. Perera<sup>28</sup> D. Perrodin<sup>16</sup> A. Petiteau<sup>29,7</sup> N. K. Porayko,<sup>6,12</sup> A. Possenti,<sup>16</sup>

H. Quelquejay Leclere<sup>7</sup> A. Samajdar<sup>30</sup> S. A. Sanidas,<sup>24</sup> A. Sesana,<sup>12,13,14</sup> G. Shaifullah<sup>12,13,16</sup>

L. Speri<sup>20</sup> R. Spiewak,<sup>24</sup> B. W. Stappers,<sup>24</sup> S. C. Susarla<sup>31</sup> G. Theureau<sup>10,11,32</sup> C. Tiburzi,<sup>16</sup>

E. van der Wateren<sup>22,9</sup> A. Vecchio<sup>15</sup> V. Venkatraman Krishnan<sup>6</sup> J. Wang<sup>8,33,34</sup> L. Wang,<sup>24</sup> and Z. Wu<sup>25</sup>

(The European Pulsar Timing Array)

¹SISSA — International School for Advanced Studies,
Via Bonomea 265, 34136, Trieste, Italy and INFN, Sezione di Trieste

²IFPU — Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

³Gran Sasso Science Institute (GSSI), I-67100 L'Aquila, Italy

⁴INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy

⁵Institute of Astrophysics, FORTH, N. Plastira 100, 70013, Heraklion, Greece

⁶Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany

⁷Université Paris Cité CNRS, Astroparticule et Cosmologie, 75013 Paris, France

⁸Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

⁹ASTRON, Netherlands Institute for Radio Astronomy,
Oude Hoogeveensedijk 4, 7991 PD, Dwingeloo, The Netherlands

¹⁰Laboratoire de Physique et Chimie de l'Environnement et de l'Espace,
Université d'Orléans / CNRS, 45071 Orléans Cedex 02, France

¹¹Observatoire Radioastronomique de Nançay, Observatoire de Paris,
Université PSL, Université d'Orléans, CNRS, 18330 Nançay, France

¹²Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

¹³INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

¹⁴INAF - Osservatorio Astronomico di Brera, via Brera 20, I-20121 Milano, Italy

¹⁵Institute for Gravitational Wave Astronomy and School of Physics and Astronomy,
University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

¹⁶INAF - Osservatorio Astronomico di Cagliari, via della Scienza 5, 09047 Selargius (CA), Italy

¹⁷Hellenic Open University, School of Science and Technology, 26335 Patras, Greece

¹⁸Kavli Institute for Astronomy and Astrophysics,
Peking University, Beijing 100871, P. R. China

¹⁹School of Physics, Faculty of Science, University of East Anglia, Norwich NR4 7TJ, UK

²⁰Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muühlenberg 1, 14476 Potsdam, Germany

²¹Universitá di Cagliari, Dipartimento di Fisica,
S.P. Monserrato-Sestu Km 0,700 - 09042 Monserrato (CA), Italy

²²Department of Astrophysics/IMAPP, Radboud University Nijmegen,
P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

²³School of Physics, Trinity College Dublin, College Green, Dublin 2, D02 PN40, Ireland

²⁴Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Manchester M13 9PL, UK

²⁵National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, P. R. China

²⁶E.A. Milne Centre for Astrophysics, University of Hull,
Cottingham Road, Kingston-upon-Hull, HU6 7RX, UK

²⁷Centre of Excellence for Data Science, Artificial Intelligence and Modelling (DAIM),
University of Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX, UK

²⁸Arecibo Observatory, HC3 Box 53995, Arecibo, PR 00612, USA

²⁹IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

³⁰Institut für Physik und Astronomie, Universität Potsdam,
Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany

³¹Ollscoil na Gaillimhe — University of Galway, University Road, Galway, H91 TK33, Ireland
Due to its high occupation number and non-relativistic nature, the ULDM scalar field can be thought as a classical wave [37]:

$$ \phi(\vec{x}, t) = \frac{\sqrt{2\rho_{\phi}}}{m_{\phi}} \hat{\phi}(\vec{x}) \cos(m_{\phi}t + \gamma(\vec{x})), \quad (2) $$

where $m_\phi$ is the mass of the scalar field, $\gamma(\vec{x})$ is a space-dependent phase and $\hat{\phi}(\vec{x})$ accounts for the pattern of interference in the proximity of $\vec{x}$ caused by the wave-like nature of ULDM. The scalar field density $\rho_\phi$ is conveniently normalized to the local DM density $\rho_{DM}$, which can be determined e.g. by fitting the MW rotation curve or, in a more refined way, by studying the vertical oscillations of disc stars [42–45]. In the following, we assume a fiducial value $\rho_{DM} \approx 0.4 \text{ GeV/cm}^3$. The oscillating nature of ULDM induces an oscillating gravitational potential leading to a periodic displacement $\delta t_{DM}$ in the TOAs of radio pulses emitted by pulsars, which can be written as [37, 39]:

$$ \delta t_{DM} = \frac{\Psi_c(\vec{x})}{2m_{\phi}} [\hat{\phi}_E^2 \sin(2m_{\phi} + \gamma_E) - \hat{\phi}_P^2 \sin(2m_{\phi} + \gamma_P)], \quad (3) $$

where

$$ \frac{\Psi_c(\vec{x})}{10^{-18}} \approx 6.52 \left( \frac{10^{-22} \text{ eV}}{m_{\phi}} \right)^2 \left( \frac{\rho_{\phi}}{0.4 \text{ GeV/cm}^3} \right), \quad (4) $$

and $\gamma_P \equiv 2\gamma(\vec{x}_P) - 2m_\phi d_P/c$ ($\gamma_E \equiv 2\gamma(\vec{x}_E)$) are related to the phases of Eq. (2) evaluated at the pulsar (Earth) location, with $d_P$ standing for the pulsar-Earth distance. The amplitude in Eq. (4) is computed assuming a constant DM density background and possible deviations caused by the wave-like nature of the ultralight scalar field are parametrized in terms of the pulsar (Earth) dependent phase factors $\hat{\phi}^2(\vec{x}_P) \equiv \hat{\phi}_P^2$ ($\hat{\phi}^2(\vec{x}_E) \equiv \hat{\phi}_E^2$). The approximation of constant DM density across pulsars is sufficient, as their distances from Earth are all ~kpc and subject to large measurement uncertainties [39]. Notice that accurate measurements of pulsar-Earth distances might help to reduce the number of free parameters in the limit in which $\gamma(\vec{x}_P) = \gamma(\vec{x}_E)$. Moreover, precise determination of pulsar positions could provide us with more information about the dark matter density in its surroundings. On scales smaller than the de Broglie wavelength, the ULDM scalar field oscillates coherently, with the same amplitude $\hat{\phi}$ (see Eq. (2)). Since the typical ULDM velocity is expected to be $v_\phi \sim 10^{-3}$, the coherence length is approximately

$$ l_c \approx \frac{2\pi}{m_\phi v_\phi} \approx 0.4 \text{ kpc} \left( \frac{10^{-22} \text{ eV}}{m_\phi} \right). \quad (5) $$

Therefore, $\hat{\phi}_E^2$ and $\hat{\phi}_P^2$ are:

* uncorrelated if the coherence length of ULDM is less than the average inter-pulsar and pulsar-Earth separation. In this case, $\hat{\phi}_E^2$ and $\hat{\phi}_P^2$ will thus be separate parameters;

* correlated if the coherence length of ULDM is larger than the inter-pulsar and pulsar-Earth separations and encloses the typical Galacto-centric region tested by the most precise MW rotation curves measurements (roughly the inner ~20 kpc [46]). In this case, $\hat{\phi}_E^2 = \hat{\phi}_P^2$ for all the pulsars. Moreover, rotation curves also sample from the same coherence patch, and thus measure the local abundance $\rho_{DM}$ of DM. Therefore, the stochastic parameter $\hat{\phi}^2$ can be safely absorbed in a redefinition of $\Psi_c$.

* pulsar-correlated if the coherence length of ULDM is larger than the inter-pulsar and pulsar-Earth separations, but smaller than the typical Galacto-centric radius sampled by rotation curves. In this case, $\hat{\phi}_E^2 = \hat{\phi}_P^2$ for all the pulsars. However, DM density estimates from rotation curves average over different patches. We therefore keep $\hat{\phi}^2$ as a free parameter and consistently marginalize over it. In this way, the limits on $\rho_{DM}$ obtained from pulsars will constrain the same quantity measured by rotation curves.

We perform the analysis in the three limits above, noting that the fully correlated limit has not been considered in the previous studies [47, 48]. From Eq. (5), recalling that the pulsar-Earth distance is $O(\text{kpc})$ for the observed systems, it follows that the correlated regime is an excellent approximation for masses lower than $m_\phi \sim 2 \times 10^{-24} \text{ eV}$; the pulsar-correlated regime holds for $2 \times 10^{-24} \text{ eV} \lesssim m_\phi \lesssim 5 \times 10^{-23} \text{ eV}$ and the uncorrelated regime is valid for $m_\phi \gtrsim 5 \times 10^{-23} \text{ eV}$.

**Dataset and methodology.**—The EPTA monitors 42 millisecond radio pulsars with five telescopes located in France, Germany, Italy, the Netherlands, and the United Kingdom. The second data release (DR2) of the EPTA contains 24.7 years of observations of pulse arrival times of 25 pulsars, surveyed with an approximate cadence of once every 3 weeks [49], which translates into a Nyquist frequency of approximately $3 \times 10^{-7} \text{ Hz}$. TOAs are measured at the position of the Solar System Barycenter, and are primarily described by pulsar-specific deterministic timing models accounting for the position of each pulsar in the sky, spin down, proper motion, the presence of a binary companion, etc. The timing models are provided with the data set. Deviations between the TOAs predicted by these models and the measured TOAs are referred to as the timing residuals, $\delta t$. The residuals contain contributions from various sources of noise, from variations in the dispersion measure to irregularities in pulsar rotation, but they may also contain signals of astrophysical interest. The sources of noise are identified as part of the noise analysis of the EPTA DR2 [50]. The DR2 data set also hints at growing evidence for the stochastic gravitational-wave background, which manifests itself as a temporally-correlated stochastic process with a hallmark inter-pulsar correlation signature of General Relativity [51]. Our work is complementary to the
EPTA-wide interpretation effort of the spatial and temporal correlations in DR2 [41].

We use Bayesian inference techniques to search for the ULDM signal while simultaneously fitting timing model parameters and all known sources of noise to the data, to correctly marginalize over the associated uncertainties. The likelihood of the timing residuals, $\mathcal{L}(\delta t|\theta)$, given the parameters of the models, $\theta$, is [52–56]

$$ \ln \mathcal{L}(\delta t|\theta) \propto -\frac{1}{2}(\delta t - \mu)^{\mathrm{T}} C^{-1} (\delta t - \mu). \quad (6) $$

This is a time-domain Gaussian likelihood, multivariate with respect to a number of observations, i.e., $\delta t$ has dimension equal to the number of observations. The contribution of ULDM from Eq. (3) is added to $\mu$, which also contains contributions from the timing model [49] and noise processes, according to the noise analysis of Ref. [50]. The diagonal part of the covariance matrix $C$ contains TOA measurement uncertainties that include temporally-uncorrelated “white” noise. Contributions from temporally correlated “red” noise may be added as off-diagonal elements in $C$. However, for computational efficiency, red noise contributions are modeled in $\mu$ [53, 54]. The priors $\pi(\theta)$ are set based on Table I, see Ref. [47] for further details. To obtain a sufficient amount of posterior samples across the mass-frequency parameter space, the search is effectively performed across equally-spaced segments of $\pi(m_{\phi})$, which we refer to as bins. The measurements of parameters are obtained as posterior distributions, $\mathcal{P}(\theta|d) \propto \mathcal{L}(\delta t|\theta)\pi(\theta)$. The posteriors are evaluated using the Parallel-Tempering-Markov-Chain Monte-Carlo sampler [57] implemented in ENTERPRISE [55] and ENTERPRISE_EXTENSIONS [56]. The conclusion about the presence or absence of the ULDM signal in the data is based on the Bayesian odds ratio. In our case, that is equal to the Bayes factor, because we assume the prior odds of both scenarios to be equal. We evaluate Bayes factors, $\mathcal{B}$, using the Savage-Dickey density ratio [58]. In particular, finding $\ln \mathcal{B} \gtrsim 5$ would indicate robust evidence for the ULDM signal.

There is strong evidence for a temporally-correlated red signal in EPTA DR2, characterized by the same Fourier spectrum of $\delta t$ in all pulsars [51]. This signal may contain contributions from the stochastic gravitational-wave background [59]. Because in the 24.7-yr data set this signal does not show significant evidence for inter-pulsar correlations (unlike in the 10.3-yr data set [41, 49–51]), we model it as a spatially uncorrelated red noise process. Individual Fourier components of this broadband signal may contaminate frequencies at which the presence of ULDM is evaluated. Thus, this common red noise signal is included in the null hypothesis, $\emptyset$, along with the pulsar-intrinsic noise. The signal hypothesis is based on adding ULDM to the null hypothesis.

**Results.**—We carry out the search for ULDM in the correlated, pulsar-correlated and uncorrelated limit for $\hat{\phi}_{\mathrm{E}}^{2}$ and $\hat{\phi}_{\mathrm{P}}^{2}$ with the parameters in Table I. The factors $\hat{\phi}_{\mathrm{E}}^{2}$ and $\hat{\phi}_{\mathrm{P}}^{2}$ are drawn from an exponential prior, to correctly

model the stochastic nature of the ULDM field [60, 61]. We find no evidence for a signal in the mass range $m_{\phi} \sim [10^{-24} \text{ eV}, 10^{-22} \text{ eV}]$. The largest $\ln \mathcal{B}$ we find across frequency-mass bins is $< 1$, i.e. the null hypothesis is favored. Therefore, we calculate the 95% upper limits on the signal amplitude $\Psi_{\mathrm{c}}$ and, through Eq. (4), on the scalar field density $\rho_{\phi}$. The results are shown in Fig. 1.

A highlight of our study is that not only does EPTA DR2 yield more stringent constraints than previous results [38, 39], but it also rules out that particles with masses $m_{\phi} \sim [10^{-24} \text{ eV}, 10^{-23.3} \text{ eV}]$ can be 100% of the observed local DM density. In particular, the scalar field density is $\rho_{\phi} \lesssim 0.15 \text{ GeV/cm}^3$ in the mass range $m_{\phi} \sim [10^{-24} \text{ eV}, 10^{-23.7} \text{ eV}]$, while it is constrained to $\rho_{\phi} \lesssim 0.30 \text{ GeV/cm}^3$ between $m_{\phi} \sim [10^{-23.7} \text{ eV}, 10^{-23.4} \text{ eV}]$. Furthermore, the correlated limit in Fig. 1 confirms Lyman-$\alpha$ bounds, which exclude ULDM in this mass range unless it constitutes less than 30% of DM [30]. It is worth noticing that the low-frequency end of Fig. 1 extends below the näive expectation $f = 1.3 \text{ nHz}$ corresponding to the inverse of the observation time $T_{\mathrm{obs}} = 24.7 \text{ yr}$. In fact, while an ULDM candidate in this mass region does not complete an oscillation cycle during the observation timescale, the signal can still be approximated by a polynomial expansion in $(m_{\phi}t)$ [47]. The sensitivity in this region is limited by the simultaneous fitting to pulsar spin frequency derivatives [62, 63]. PTAs are only sensitive to the $(m_{\phi}t)^3$ term, as the first terms in the expansion are degenerate with the timing model [64]. However, since the expected amplitude $\Psi_{\mathrm{c}}$ of an ULDM candidate increases as its mass decreases, we can still obtain competitive constraints at low frequency. While, in principle, our analysis could be pushed to even lower masses [65], we choose to focus on the region $m_{\phi} \gtrsim 10^{-24} \text{ eV}$ to comply with the aforementioned CMB bounds. We find that the significant improvement in sensitivity to ULDM at low frequencies arises thanks to the larger data span of EPTA DR2, in accordance with the theoretical sensitivity scaling proposed in Eq. (13) of Ref. [65]. In particular, because of the longer data span, we expect EPTA DR2 limits to be better than NANOGrav [66] ones by a factor of roughly $\sim 3.6$, which is in agreement with what observed. At high frequencies, we find that the advantage of the long timing baseline compared to NANOGrav diminishes, also in accordance with the scaling, as pulsar white noise levels become more important. We also performed an identical analysis of the 10-year subset of EPTA DR2 [49, 51], as well as of the MeerTime data [67], which yield less stringent upper limits in agreement with the scaling. For comparison, the bounds in both the correlated and uncorrelated limit for the 10-year subset of the EPTA, shown in Fig. 22 in Ref. [41], appear at the level of the pulsar-correlated limit in Fig. 1 of this manuscript.

In the following, we clarify some specific aspects of our results. First, we notice that a similar analysis has been done by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration [66].
<table><caption>TABLE I: Parameters employed for the search along with their respective priors. In the correlated limit, the parameters $\hat{\phi}_E^2, \hat{\phi}_P^2$ are accounted for by a redefinition of $\Psi_c$, while in the pulsar-correlated regime $\hat{\phi}_E^2 = \hat{\phi}_P^2 = \hat{\phi}^2$ is a free parameter.</caption><thead><tr><th>Parameter</th><th>Description</th><th>Prior</th><th>Occurrence</th></tr></thead><tbody><tr><td colspan="4">White Noise (σ = E<sub>f</sub><sup>2</sup>σ<sub>TOA</sub><sup>2</sup> + E<sub>q</sub><sup>2</sup>)</td></tr><tr><td>E<sub>f</sub></td><td>EFAC per receiver-backend system</td><td>Uniform [0, 10]</td><td>1 per pulsar</td></tr><tr><td>E<sub>q</sub></td><td>EQUAD per receiver-backend system</td><td>Log<sub>10</sub>-Uniform [-10, -5]</td><td>1 per pulsar</td></tr><tr><td colspan="4">Red Noise</td></tr><tr><td>A<sub>red</sub></td><td>Red noise power-law amplitude</td><td>Log<sub>10</sub>-Uniform [-20, -6]</td><td>1 per pulsar</td></tr><tr><td>γ<sub>red</sub></td><td>Red noise power-law spectral index</td><td>Uniform [0, 10]</td><td>1 per pulsar</td></tr><tr><td colspan="4">ULDM</td></tr><tr><td>Ψ<sub>c</sub></td><td>ULDM signal amplitude</td><td>Log<sub>10</sub>-Uniform [-20, -12]</td><td>1 per PTA</td></tr><tr><td>m<sub>φ</sub> [eV]</td><td>ULDM mass</td><td>Log<sub>10</sub>-Uniform [-24, -22]</td><td>1 per PTA</td></tr><tr><td>φ̂<sub>E</sub><sup>2</sup></td><td>Earth factor</td><td>e<sup>-x</sup></td><td>1 per PTA</td></tr><tr><td>φ̂<sub>P</sub><sup>2</sup></td><td>Pulsar factor</td><td>e<sup>-x</sup></td><td>1 per pulsar</td></tr><tr><td>γ<sub>E</sub></td><td>Earth signal phase</td><td>Uniform [0, 2π]</td><td>1 per PTA</td></tr><tr><td>γ<sub>P</sub></td><td>Pulsar signal phase</td><td>Uniform [0, 2π]</td><td>1 per pulsar</td></tr><tr><td colspan="4">Common spatially Uncorrelated Red Noise (CURN)</td></tr><tr><td>A<sub>GWB</sub></td><td>Common process strain amplitude</td><td>Log<sub>10</sub>-Uniform [-20, -6]</td><td>1 per PTA</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>FIG. 1: Upper limits on ULDM, and namely on the dimensionless amplitude ($\Psi_c$, left panel) and the ULDM fraction of the local DM density $\rho_{DM} = 0.4\text{GeV/cm}^3$ ($\rho_\phi/\rho_{DM}$, right panel), at 95% credibility. The bottom horizontal axes show the ULDM particle mass, whereas the top horizontal axes show the equivalent oscillation frequency of the scalar field. The upper limits from previous searches [38, 39] are shown for comparison. As a reference, we highlight the frequency $T_{\text{obs}}^{-1}$. In the right panel, we zoom in on the excluded ULDM masses. The horizontal dotted line represents the value of $\rho_\phi$ that would saturate the local DM density. Notice that based on our results ULDM particles with mass $-24.0 < \log_{10}(m_\phi/\text{eV}) < -23.7$ can only make up at most 30–40 % of the total DM energy density, while particles with mass $-23.7 < \log_{10}(m_\phi/\text{eV}) < -23.3$ can contribute at most up to $\sim 70\%$.</figcaption></figure>

There, the upper limits provided in the correlated and uncorrelated scenarios differ at low frequency. This can be understood by noticing that the correlated limit of NANOGrav corresponds to our pulsar-correlated limit. However, in the low mass limit of Fig. 1, the pulsars, the

Earth and the stellar and gaseous tracers used for rotation curves estimates lie well within the area spanned by the coherence length; thus, one can only measure the combination $\Psi_c^0 = \Psi_c \hat{\phi}^2$, which represents the realization of DM in our Galaxy. Therefore, we remove the
$\hat{\phi}_E^2 = \hat{\phi}_P^2 \equiv \hat{\phi}^2$ parameter in the correlated limit, as it
can be accounted for by a redefinition of $\Psi_c$. Fitting for
$\Psi_c$ and $\hat{\phi}^2$ separately, instead, introduces an additional
uncertainty, which leads our pulsar-correlated analysis to
produce a similar mismatch as the one found in Ref. [66],
as shown in Fig. 1. Second, Fig. 1 hints at a steep in-
crease in the upper limits at $m_\phi \gtrsim 10^{-23.2}$ eV. In fact,
we report the presence of excess signal power on top of
the common red noise process, corresponding to a mass
of $m_\phi \simeq 10^{-23}$ eV and an amplitude of $\Psi_c \simeq 6 \times 10^{-14}$
or, equivalently, a density of $\rho_\phi = 90$ GeV/cm$^3$. At face
value, this excess is not compatible with an ULDM can-
didate, as the corresponding density is outside the local
DM measurement uncertainties [42–45]. Moreover, such
a mass would be in tension with astrophysical bounds, as
extensively discussed in the introduction [27–34]. Any-
way, the Bayesian odds ratio suggests that it is still con-
sistent with noise ($\ln \mathcal{B} \sim 0.1$). We find a similar excess
in the analysis of 10-yr subset of the EPTA DR2 [41].
Moreover, the boson mass corresponding to the excess
also matches the frequency of the continuous gravita-
tional wave (CGW) candidate studied in [68]. This mo-
tivates further investigations as part of the International
Pulsar Timing Array [69].

Conclusions.— ULDM is a theoretically motivated
paradigm that may alleviate the small-scale crisis of
structure formation. Here, we focused on the most robust
scenario, in which ULDM features only gravitational in-
teractions. These interactions produce a periodic oscilla-
tion in the TOAs of the radio beams emitted by pulsars,
which can then be collected in PTA telescopes. PTAs
stand out as excellent laboratories to test the effects of
ULDM in the mass range $m_\phi \sim [10^{-24}$ eV, $10^{-22}$ eV]. In
this work, we showed that PTAs constrain the presence
of ULDM below a few tenths of the observed DM abun-
dance in the mass range $m_\phi \sim [10^{-24}$ eV, $10^{-23.3}$ eV].
Therefore, in this range, ULDM cannot constitute 100%
of the observed DM.

We wish to thank Diego Blas, Alessio Zicoschi and Paolo Salucci for useful discussions and insights. Moreover, we thank the anonymous referees for improving the quality of our manuscript. C.S. and E.B. acknowledge support from the European Union's H2020 ERC Consolidator Grant "GRavity from Astrophysical to Microscopic Scales" (Grant No. GRAMS-815673) and the EU Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement No. 101007855. B.G. is supported by the Italian Ministry of Education, University and Research within the PRIN 2017 Research Program Framework, n. 2017SYRTCN. The European Pulsar Timing Array (EPTA) is a collaboration between European and partner institutes, namely ASTRON (NL), INAF/Osservatorio di Cagliari (IT), Max-Planck-Institut für Radioastronomie (GER), Nançay/Paris Observatory (FRA), the University of Manchester (UK), the University of Birmingham (UK), the University of East Anglia (UK), the University of Bielefeld (GER), the University of Paris (FRA),

the University of Milan-Bicocca (IT), the Foundation for
Research and Technology, Hellas (GR), and Peking Uni-
versity (CHN), with the aim to provide high-precision
pulsar timing to work towards the direct detection of
low-frequency gravitational waves. An Advanced Grant
of the European Research Council allowed to implement
the Large European Array for Pulsars (LEAP) under
Grant Agreement Number 227947 (PI M. Kramer). The
EPTA is part of the International Pulsar Timing Array
(IPTA); we thank our IPTA colleagues for their sup-
port and help with this paper and the external Detec-
tion Committee members for their work on the Detec-
tion Checklist. The work presented here is a culmination
of many years of data analysis as well as software and
instrument development. In particular, we thank Drs.
N. D'Amico, P. C. C. Freire, R. van Haasteren, C. Jor-
dan, K. Lazaridis, P. Lazarus, L. Lentati, O. Löhmer and
R. Smits for their past contributions. We also thank Dr.
N. Wex for supporting the calculations of the galactic ac-
celeration as well as the related discussions. The EPTA
is also grateful to staff at its observatories and telescopes
who have made the continued observations possible. Part
of this work is based on observations with the 100-m tele-
scope of the Max-Planck-Institut für Radioastronomie
(MPIfR) at Effelsberg in Germany. Pulsar research at
the Jodrell Bank Centre for Astrophysics and the ob-
servations using the Lovell Telescope are supported by
a Consolidated Grant (ST/T000414/1) from the UK's
Science and Technology Facilities Council (STFC). ICN
is also supported by the STFC doctoral training grant
ST/T506291/1. The Nançay radio Observatory is oper-
ated by the Paris Observatory, associated with the French
Centre National de la Recherche Scientifique (CNRS),
and partially supported by the Region Centre in France.
We acknowledge financial support from "Programme Na-
tional de Cosmologie and Galaxies" (PNCG), and "Pro-
gramme National Hautes Energies" (PNHE) funded by
CNRS/INSU-IN2P3-INP, CEA and CNES, France. We
acknowledge financial support from Agence Nationale de
la Recherche (ANR-18-CE31-0015), France. The West-
erbork Synthesis Radio Telescope is operated by the
Netherlands Institute for Radio Astronomy (ASTRON)
with support from the Netherlands Foundation for Sci-
entific Research (NWO). The Sardinia Radio Telescope
(SRT) is funded by the Department of University and
Research (MIUR), the Italian Space Agency (ASI), and
the Autonomous Region of Sardinia (RAS) and is op-
erated as a National Facility by the National Institute
for Astrophysics (INAF). The work is supported by the
National SKA programme of China (2020SKA0120100),
Max-Planck Partner Group, NSFC 11690024, CAS Cul-
tivation Project for FAST Scientific. This work is also
supported as part of the "LEGACY" MPG-CAS collab-
oration on low-frequency gravitational wave astronomy.
J.A. acknowledges support from the European Commis-
sion (Grant Agreement number: 101094354). J.A. and S.
Chanlaridis were partially supported by the Stavros Niar-
chos Foundation (SNF) and the Hellenic Foundation for
Research and Innovation (H.F.R.I.) under the 2nd Call of the “Science and Society – Action Always strive for excellence – Theodoros Papazoglou” (Project Number: 01431). A.C. acknowledges support from the Paris Île-de-France Region. A.C., A.F., A. Sesana, A. Samajdar, E.B., D.I., G.M.S., M. Bonetti acknowledge financial support provided under the European Union’s H2020 ERC Consolidator Grant “Binary Massive Black Hole Astrophysics” (B Massive, Grant Agreement: 818691). G.D., K. Liu, R.K. and M.K. acknowledge support from European Research Council (ERC) Synergy Grant “Black-HoleCam”, Grant Agreement Number 610058. This work is supported by the ERC Advanced Grant “LEAP”, Grant Agreement Number 227947 (PI M. Kramer). A.V. and P.R.B. are supported by the UK’s Science and Technology Facilities Council (STFC; grant ST/W000946/1). A.V. also acknowledges the support of the Royal Society and Wolfson Foundation.

N.K.P. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer PO 2758/1–1, through the Walter–Benjamin programme. A. Samajdar thanks the Alexander von Humboldt foundation in Germany for a Humboldt fellowship for postdoctoral researchers. A. Possenti, D.P. and M. Burgay acknowledge support from the research grant “iPeska” (P.I. Andrea Possenti) funded under the INAF national call Prin-SKA/CTA approved with the Presidential Decree 70/2016 (Italy). R.N.C. acknowledges financial support from the Special Account for Research Funds of the Helenic Open University (ELKE-HOU) under the research programme “GRAVPUL” (K.E.-80383/grant agreement 319/10-10-2022: PI N. A. B. Gizani). E. vd. W., C.G.B. and G.H.J. acknowledge support from the Dutch National Science Agenda, NWA Startimpuls – 400.17.608. L.S. acknowledges the use of the HPC system Cobra at the Max Planck Computing and Data Facility.

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