Kobe University Repository : Kernel タイトル Pulsar timing residual induced by ultralight vector dark matter Title 著者 Nomura, Kimihiro / Ito, Asuka / Soda, Jiro Author(s) 掲載誌・巻号・ページ European Physical Journal C,80(5):419 Citation 刊行日 2020-05-14 Issue date 資源タイプ Journal Article / 学術雑誌論文 Resource Type 版区分 publisher Resource Version © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party 権利 material in this article are included in the article’s Creative Commons Rights licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funded by SCOAP3 DOI 10.1140/epjc/s10052-020-7990-y JaLCDOI URL http://www.lib.kobe-u.ac.jp/handle_kernel/90007118 PDF issue: 2021-09-30 Eur. Phys. J. C (2020) 80:419 https://doi.org/10.1140/epjc/s10052-020-7990-y Regular Article - Theoretical Physics Pulsar timing residual induced by ultralight vector dark matter Kimihiro Nomuraa, Asuka Itob, Jiro Sodac Department of Physics, Kobe University, Kobe 657-8501, Japan Received: 12 March 2020 / Accepted: 28 April 2020 © The Author(s) 2020 Abstract We study the ultralight vector dark matter with for a free field to condense homogeneously during inflation. a mass around 10−23 eV. The vector field oscillating coher- Recently, however, a consistent ultralight vector model has ently on galactic scales induces oscillations of the spacetime been proposed [11] where the ultralight vector field is homo- metric with a frequency around nHz, which is detectable by geneously condensed during inflation when its mass is less pulsar timing arrays. We find that the pulsar timing signal due than the Hubble scale, and starts to oscillate coherently at to the vector dark matter has nontrivial angular dependence some epoch after inflation. The coherently oscillating vec- unlike the scalar dark matter and the maximal amplitude is tor field behaves as a non-relativistic matter and can be a three times larger than that of the scalar dark matter. candidate for the DM as well as the scalar field. Historically, there have been many works on the vector DM. Evolution of cosmological perturbations based on the 1 Introduction model of ultralight coherent vector DM field has been stud- iedin[12]. There, it is demonstrated that perturbations on Observations of the galactic rotation curves [1], structure the scales smaller than the de Broglie wave length of the formation [2], and gravitational lensing [3] suggest that the vector field have a specific feature compared to the scalar invisible matter, the so-called dark matter (DM) exists in the case. However, the magnitude of the feature is far below the Universe. Searching for the DM has been a long-standing sensitivity of present and future detectors. The approaches to challenge in cosmology and astrophysics. Recent observa- search for the vector DM or to constrain its couplings to parti- tional results show that the DM is accounting for 27% of cles in the Standard Model have been proposed, e.g. by using the energy density in the Universe [4]. The most promising ground-based gravitational-wave interferometers [13,14]or candidate for the DM has been the weak interacting massive taking into account the cosmological plasma effects with a particles (WIMPs) which are motivated by supersymmetric photon [15]. Some phenomenologies on the vector DM are theories of particle physics. However, inspite of the efforts of discussed together with its production mechanism [11,16– many researchers, no signal of the WIMPs has been detected. 23]. The effects of gravitational interaction between the vec- Recently, as an alternative candidate for the DM, an ultra- tor DM field and binary pulsar systems on the dynamics of light axion-like scalar with a mass ∼ 10−23 eV, often called the system have been considered in [24]. the fuzzy DM, has been intensively studied [5,6]. The fuzzy In this paper, we investigate the purely gravitational effects DM has a possibility to resolve small-scale problems of the of the vector DM on the pulsar timing. A coherently oscil- standard Cold DM model such as the galactic core-cusp prob- lating vector field behaves as a non-relativistic pressureless lem [7–9]. matter on cosmological scales. Hence it does not induce the Given the success of the ultralight scalar DM, it is natu- anisotropic expansion of the Universe [25]. Actually, how- ral to ask if the ultralight vector can be the DM. In fact, the ever, there exists an oscillating anisotropic pressure on time possibility of the fuzzy DM being a massive vector boson scales corresponding to the oscillation of the vector field, (sometimes called a dark photon) has been proposed [10,11]. which induces nontrivial oscillations of the metric. The fre- In the case of a vector boson, it is known that it is difficult quency is determined by the mass of the DM considered. In the case of m ∼ 10−23 eV, it is on the order of nHz, which a e-mail: [email protected] (corresponding author) is in the range where the Pulsar Timing Arrays (PTAs) are b e-mail: [email protected] sensitive. PTAs can detect gravitational effects such as grav- c e-mail: [email protected] itational waves [26–28]. Furthermore, a method for detect- 0123456789().: V,-vol 123 419 Page 2 of 10 Eur. Phys. J. C (2020) 80:419 ing the axion-like scalar DM using PTAs has been proposed − 4 − 3 ρ ρ 10 23 eV 10 3 in [29], and actually the energy density of the DM is con- 1095 , m · (mv)3 0.4 GeV/cm3 m v strained by using observational data [30–32]. The detection method is applicable to the case of the vector DM and it is (3) expected that a specific signal depending on the vector prop- ρ erty appears. This is what we will discuss in this paper. where the energy density of the DM is normalized by the This paper is organized as follows. In Sect. 2, we show the local value [33–35]. Since the occupation number is so huge, oscillation of the vector DM field induces time-dependent we can treat the vector field as a classical wave. The de spacetime metric perturbations. In Sect. 3, we see the effect Broglie wavelength for the ultralight vector DM particles of the vector DM on the pulsar timing. We also discuss the withamassm reads detectability of the vector DM with PTAs. In Sect. 4,we 2π 10−23 eV 10−3 summarize the results. Appendix A is devoted to derivation λ = . dB v 4 kpc v (4) of the formula for the redshift of photons induced by metric m m perturbations. Due to the wave nature of the DM field, all inhomogeneities in the DM distribution on the scale smaller than λdB are smoothed out. So we can express the vector DM field as 2 Effects on metric perturbations a superposition of plane waves with a typical wave num- ber k = mv = 2π/λdB. In the present case, the energy In this section, we show that the spacetime metric fluctuates reads E m + mv2/2 m, because the velocity is non- due to the presence of the vector DM oscillating coherently relativistic. Thus, the vector DM field is coherently oscillat- on galactic scales. ing with a monotonic frequency determined by the mass m Since we work on the galactic scales, the cosmic expansion on the scale given by λdB. is negligible. Hence, we consider the metric On galactic scales, we can neglect the cosmic expansion. The equations of motion of the vector field in a flat back- μ ν ds2 = ημνdx dx ground read 2 i j i j − 2(t, x)dt + 2(t, x)δijdx dx + hij(t, x)dx dx , μν 2 ν (1) ∂μ F − m A = 0, (5) where ημν is the Minkowski metric ημν = diag(− 1, + 1, which is derived by taking the variation of the action (2) μν + 1, + 1). The metric perturbations induced by the DM are with respect to Aμ. Using the identity ∂ν∂μ F = 0, we can described by , , and hij. Weshall discuss the perturbations rewrite Eq. ((5)) as a set of Proca equations in the standard separately in Eqs. (18) and (25). form: The vector DM field Aμ with a mass m is expected to μ have little interaction with particles in the Standard Model. ∂μ A = 0, (6) We treat the vector DM field as a free field. Its action is given (−∂2 + ∇2 − 2) μ = . 0 m A 0 (7) by √ Taking into account that the field typically has a frequency 4 1 μν ρσ 1 2 μν S = d x −g − g g Fμρ Fνσ − m g Aμ Aν , m and a momentum k,Eq.((6)) gives 4 2 (2) k A ∼ A . (8) t m i where g is the determinant of the metric gμν and Fμν = ∂ − ∂ −3 μ Aν ν Aμ. We also consider the ultralight mass of the Hence, At gets suppressed by the order of k/m = v ∼ 10 ∼ −23 vector DM as m 10 eV, which can arise through the compared to Ai .