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タイトル Pulsar timing residual induced by ultralight vector dark 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

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 . 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 -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 ) 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-

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    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 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 . Thus, we can neglect At . During inflation, Higgs or the Stueckelberg mechanisms. only the longitudinal mode survives. Hence, the directions of Here, we follow the discussion in [29], where the case the vector at different points in a coherent region align. We of the scalar DM is considered. Taking into account that the take a coordinate system so that the direction of the oscillation − typical velocity in the galaxy is given by 10 3 times the speed is along z-axis. From Eq. (7), we obtain of light, we can estimate the occupation number of the vector DM as

123 Eur. Phys. J. C (2020) 80:419 Page 3 of 10 419

The second term in (15), the traceless part, is Az(t, x) = A(x) cos(mt + α(x)). (9) 1 k Tij − δijT k Here we neglected the spatial derivative when we solve the 3 ⎛ ⎞ equation. However, we left spatial dependence of the ampli- 10 0 1 tude and phase. Note that the scale of variation of these quan- =− m2 A2(x) cos(2mt + 2α(x)) ⎝01 0⎠ . (17) 3 tities is larger than λdB. 00−2 Let us see metric perturbations induced by the oscillat- ing DM. The energy-momentum tensor for matter fields is In the rest of this section, we will consider the trace part and defined by the traceless part of the energy-momentum tensor separately. First, let us focus on the trace part. We take the Newtonian −2 δS gauge and write the perturbed metric as Tμν = √ . (10) −g δgμν 2 2 i j ds =−(1 + 2(t, x))dt + (1 + 2(t, x))δijdx dx , For a free massive vector field (2), we have (18)   where  and  correspond to the gravitational potential. We 1 ρα σβ 1 2 ρσ Tμν = gμν − g g Fαβ Fρσ − m g Aρ Aσ can expect that time dependence of the potential is induced 4 2 by coherent oscillations of the DM field. For convenience, ρσ 2 + g Fμρ Fνσ + m Aμ Aν. (11) we write the potential as the sum of a time-independent part and an oscillating part with a frequency 2m: In a flat background, the components of the energy-momentum tensor (11) read (t, x) = 0(x) + osc(x) cos(2mt + 2α(x)), (19) (t, x) = 0(x) + osc(x) cos(2mt + 2α(x)). (20) 1 2 2 Ttt = m A (x), (12) 2 Given the energy-momentum tensor, the tt component of 1 2 2 linearized Einstein equations is Txx = Tyy =− m A (x) cos(2mt + 2α(x)), (13) 2 ∂2 =− π . 1 2 2 i 4 GTtt (21) Tzz = m A (x) cos(2mt + 2α(x)), (14) 2 From the result (12), the right-hand side does not depend on where we have neglected the spatial derivative of the field. time, so this relation determines the time-independent part Notice that the energy density of the DM Ttt is time- 0(x). In order to find time dependence of the gravitational independent. On the other hand, the anisotropic pressure is potential, we use the trace of the spatial component of Ein- time-dependent. When we average the pressure over cosmo- stein equations, logical time scales which are much longer than the oscillation − ¨ + ∂2( + ) = π k . period, the pressure vanishes. This tells us that a coherently 3 i 4 GT k (22) oscillating massive vector field behaves as a non-relativistic matter with zero-pressure on cosmological scales. As we will Now we substitute Eqs. (16), (19) and (20) into this see below, however, the oscillating pressure affects the space- expression and split terms into time-dependent and time- time metric on the time scale relevant to the PTAs. independent terms. Focusing on time-independent terms, we × ∂2( + ) =  =− In general, a symmetric 3 3 tensor Tij can be decomposed can see i 0 0 0, which implies 0 0.Onthe into a trace part and a traceless part as other hand, as for the time-dependent parts, one can neglect ∂2( + )     i term because the spatial gradients on or typically bring out k. Thus these terms are suppressed com- 1 k 1 k Tij = δijT k + Tij − δijT k . (15) ¨ 2/ 2 ∼ v2 3 3 pared with term because k m is tiny. Assuming the amplitude of the oscillating part of the gravitational poten-  The first term corresponds the trace part, which behaves as tial, osc, is sufficiently homogeneous over the length scale a scalar under three-dimensional rotations. From Eqs. (13) considered, we have and (14), we get

k 1 2 2 T k =− m A (x) cos(2mt + 2α(x)). (16) 2 123 419 Page 4 of 10 Eur. Phys. J. C (2020) 80:419

π ρ( ) sensitive region of PTAs. In the next section, we will evalu- 1 2 G x osc(x) =− πGA (x) =− 6 3m2 ate the pulsar timing signals from the vector DM discussed    −23 2 above. − ρ(x) 10 eV =−2.2 × 10 16 . 0.4GeV/cm3 m (23) 3 Effects on pulsar timing arrays

In the second equality, we used the energy density of the DM First of all, we would like to clarify the picture we envisage. ρ given by the tt component of the energy-momentum tensor During inflation, the vector field was frozen in a coherent (12). Moreover, the oscillation frequency is given by direction, the vector field started oscillating at some point and acting as a dark matter. It is legitimate to assume that the 2m − m f = = 4.8 × 10 9 Hz . (24) coherence of the vector field over the region we are observing π −23 2 10 eV survives even in the present universe as is always assumed Next, we consider the effect of the traceless part of the in the axion dark matter. In fact, the de Broglie wavelength energy-momentum tensor. We denote traceless metric per- of the vector dark matter is several kpc, within which the direction of the vector field is coherent. In the Milky Way, turbation as hij: there are many domains of the vector condensations. The 2 2 i j direction of a domain is different from another domain. We ds =−dt + (δij + hij(t, x))dx dx , assumed that we are in one of them where there are many i = . h i 0 (25) pulsars. The aim of this work is to determine the direction of the vector field in our vicinity with the pulsar timing arrays. ¨ In the linearized Einstein equations, a combination hij − Let us investigate how time-dependent metric perturba- ∂2 k hij appears. By the same reasoning as the above discussion tions due to the coherently oscillating vector DM field affect 2 for the trace part, the contribution from ∂ hij term can be the observed periodic electromagnetic fields from pulsars. ¨ k neglected compared to hij. Thus we have We choose a coordinate system so that the observation point   is at the spatial origin, and the direction of the vector DM ¨ 1 k oscillation is along z-axis as is done in (9). A unit vector hij = 16πG Tij − δijT k . (26) 3 pointing from the observer to a pulsar is written as

By using (17), the traceless part of the metric perturbation n = (sin θ cos φ,sin θ sin φ,cos θ). (29) can be obtained: ⎛ ⎞ Then the pulsar is located at xp(=|xp|n). The rotational 10 0 period of the pulsar is T0 = 2π/ω0, where we have intro- ( , ) = 4π 2( ) ( + α( )) ⎝ ⎠ . hij t x GA x cos 2mt 2 x 01 0 duced the angular frequency ω0. Moreover, the observed 3 − 00 2 angular frequency of the pulses is denoted by ωobs(t). Then, (27) the redshift of electromagnetic fields propagating from the pulsar to the observer is defined by For later convenience, we define the amplitude of the oscil- ω − ω (t) lation as z(t) ≡ 0 obs . (30) ω0 π ρ( ) 4 2 8 G x hosc(x) ≡ πGA (x) = ( ) =−(ω ( ) − ω )/ω  3 3m2 Since we have a relation z t obs t 0 0    −23 2 (Tobs(t) − T0)/T0, where Tobs(t) ≡ 2π/ωobs(t), z(t) stands − ρ(x) 10 eV = 1.7 × 10 15 . for the relative variation of the observed pulsar timing. Then, 0.4GeV/cm3 m conventionally, the timing residual with respect to a reference (28) time t = 0 is defined as  From Eq. (27), we find that anisotropic metric perturbations t ( ) =  ( ). appear. This effect comes from the anisotropy of the vector R t dt z t (31) 0 DM field. In the case of the scalar DM such as axion-like particles, this kind of anisotropy does not occur. Therefore, First, we focus on the effect of scalar perturbations on the it is possible to distinguish whether the DM is scalar or not pulsar timing, in particular , which is induced by the trace from the presence or absence of anisotropy of the metric part of the energy-momentum tensor of the DM field. In this perturbations. Notice that the frequency given by (24)isthe case, the redshift is given by (see Appendix A.1) 123 Eur. Phys. J. C (2020) 80:419 Page 5 of 10 419

Fig. 1 Angular dependence of z (t) = (t, 0) − (t −|xp|, xp). (32) the redshift due to the oscillation of the vector DM is shown. The Paying attention only to the oscillating part, which is mea- blue line and red line represent the contribution of the trace part surable with PTAs, we have z (33) and the traceless part zh (36), respectively. Actually, we only observe the summation of z (t)  z and zh, which is depected by the green line. The angle θ is = osc cos(2mt + 2α(0)) − cos(2mt − 2m|xp|+2α(xp)) measured from the direction of =−  ( | |+α( ) − α( )) 2 osc sin m xp 0 xp the oscillation chosen as the z × sin(2mt − m|xp|+α(0) + α(xp)), (33) axis. A gray dashed line shows the magnitude of the redshift when the DM is a scalar field where we assumed that the oscillation amplitude at the obser- vation point and that at the pulsar are approximately equal, and used the same symbol osc for them. Integrating z(t) over time according to (31), we can evaluate the timing resid- ual induced by the trace part of the metric perturbations as

osc R = sin(m|x |+α(0) − α(x )) Recall that the timing residual due to the coherent oscil- m p p lation of an ultralight scalar DM is [29], × cos(2mt − m|xp|+α(0) + α(xp)). (34) πGρ R = sin(m|x |+α(0) − α(x )) It depends on the distance to the pulsar. scalar m3 p p Next, we consider the effect of the traceless part of metric × cos(2mt − m|xp|+α(0) + α(xp)). (38) perturbations hij on the pulsar timing, which is induced by the traceless part of the energy-momentum tensor of the DM The main difference is that the timing residual of the vector field. For this part, the redshift is expressed as (see Appendix. DM has a nontrivial direction dependence as shown in Fig. A.2) 1, in contrast to that of the scalar DM. Thus, it is possible to discriminate the vector DM from the scalar DM. Moreover,  1 i j from Eqs. (23), (28), (34) and (37), we can see that the mag- zh(t) = n n hij(t, 0) − hij(t −|xp|, xp) . (35) 2 nitude of the maximal timing residual of the vector DM is three times larger than that of the scalar DM. Substituting the result (27) in the previous section and the Finally, we assess the detectability of the vector DM by definition (29) into this expression, we obtain means of PTAs. The maximal amplitude of the timing resid- ual due to the vector DM oscillation is given by

( ) =−1 ( + θ) zh t 1 3cos2 hosc max|R + Rh| 4      × cos(2mt + 2α(0)) − cos(2mt − 2m|xp|+2α(xp))  osc 1 hosc  = max  − (1 + 3 cos 2θ)  = 1 ( + θ) ( | |+α( ) − α( ))  m 4 m  1 3cos2 hosc sin m xp 0 xp   2  πGρ 1 8πGρ  × ( − | |+α( ) + α( )). = max − − (1 + 3 cos 2θ)  sin 2mt m xp 0 xp (36) 3m3 4 3m3 π ρ = 3 G Correspondingly, the timing residual reads m3    −23 3 − ρ 10 eV = 1.3 × 10 7 sec . 1 hosc 0.4GeV/cm3 m Rh =− (1 + 3 cos 2θ) sin(m|xp|+α(0) − α(xp)) 4 m (39) × cos(2mt − m|xp|+α(0) + α(xp)). (37) In Fig. 2, the amplitudes of the timing residual estimated Thus, the timing residual depends on the distance to the pulsar above are shown together with observational thresholds. The and its angular position with respect to the direction of the threshold line at 100 ns comes from the best timing precision vector DM oscillation. on the existing PTAs. For example, PSR J0437-4715 has a 123 419 Page 6 of 10 Eur. Phys. J. C (2020) 80:419

dark matter mass / eV vature perturbations induced by the vector [11] and the pre- 10-24 10-23 10-22 10-3 ferred direction detected by the PTAs. We assumed the vector DM has no couplings to the Stan- 10-4 -5 dard Model. Then, the constraints on the mass of the vec- 10 Vector DM tor DM come from the superradiance of astrophysical black -6 10 holes and the structure formation. Since the superradiance 100 ns 10-7 reduces the rotation of a black hole when the Compton 10 ns 10-8 length of the vector DM has the same order of the grav- 10-9 itational radius of the black hole, the existence of rotat- Scalar DM 10-10 ing stellar-mass black holes excludes vector particles with × −14 × −11 -11 masses, 5 10 –2 10 eV. Also, vectors with lighter 10 × −20 × −17

maximal amplitude of timing residual / sec masses, 6 10 –2 10 eV, have been excluded from 10-12 10-9 10-8 10-7 measurements of supermassive black holes, although they frequency / Hz have less reliability [52]. We should also keep in mind that the superradiance constraints are given under the assumption Fig. 2 The maximum value of the amplitude of the pulsar timing resid- ual due to the vector DM is indicated by the red line. The gray line that the self-interaction is sufficiently weak [18]. Next, since corresponds to that of the scalar DM. Here we took the energy density the ultralight DM would suppress the structure formation on of the DM as ρ = 0.4GeV/cm3 small scales, CMB data constrained masses of the axion-like ultralight scalar DM in the range 10−33 ≤ m ≤ 10−24 eV [53]. Recently, the Lyman-α power spectrum in the ultra- . μ weighted root-mean-square of the timing residual 0 11 s light scalar DM model was calculated using hydrodynamical [36]. In the near future, the Square Kilometre Array (SKA) simulations and compared with the observed data. It gave a project with suitable 250 pulsars may be able to measure with lower limit on the mass of the scalar DM as m  10−21 eV an accuracy of the order of 10 ns [37]. [54,55]. Thus, the masses detectable by future PTAs are in tension with the above constraints at least in the case of the scalar DM. Whether those can be directly applicable to the 4 Discussion and conclusion vector DM is an issue to be considered. In any case, it is true that the PTA experiments can independently constrain the We have studied the pulsar timing signal of the ultralight vec- energy density of the ultralight DMs and determine whether tor DM. The vector DM in a oscillates coher- the dominant DM is vector or not. ently and monochromatically with a specific frequency deter- As a future work, it is interesting to consider the detectabil- mined by its mass (24). The oscillation induces the time- ity of the vector DM with gravitational wave interferome- dependent metric perturbations. The metric perturbations ters [56]. Our discussion would be applicable to the astromet- yield a redshift of propagating electromagnetic fields. Since ric effects in a similar manner [57]. Moreover, it is important the frequency of perturbations is typically in the nHz range, to evaluate gravitational waves from the vector DM during the redshift due to the DM is detectable by PTA experiments. cosmological evolution [58,59]. We have extended the analy- The signal is monochromatic unlike the stochastic gravita- sis of the scalar DM to the vector DM. In this line of thought, tional wave background e.g. generated from the very early it is intriguing to study ultralight higher spin fields as the universe [38–42], the population of massive black hole bina- DM [60–62]. ries [43–46] and the cosmic string network [47]. Remark- ably, we have shown that the pulsar timing residual due to Acknowledgements The authors thank R. Kato for useful discussions. the vector DM has a nontrivial angular dependence. Espe- A.I. was supported by Grant-in-Aid for JSPS Research Fellow and JSPS KAKENHI Grant No.JP17J00216. J.S. was in part supported by JSPS cially, when the direction of the vector oscillation and the KAKENHI Grant Numbers JP17H02894, JP17K18778, JP15H05895, line of sight to the pulsar are parallel, the magnitude of the JP17H06359, JP18H04589. J.S. was also supported by JSPS Bilat- signal becomes maximum. If the direction dependence of eral Joint Research Projects (JSPS-NRF collaboration) “String Axion the pulsar timing residual is found, it would be an evidence Cosmology.” Discussions during the YITP workshop YITP-T-19-02 on “Resonant instabilities in cosmology” were useful for this work. that the dominant component of the galactic DM is a vector field. Figure 2 suggests that when the DM halo is dominated Data Availability Statement This manuscript has no associated data by the vector DM, that is, ρ = 0.4GeV/cm3, and its mass or the data will not be deposited. [Authors’ comment: There are no experimental data associated with the letter]. m  2 × 10−23 eV, the amplitude of the timing residual reaches 10 ns at the maximum, which is the expected preci- Open Access This article is licensed under a Creative Commons Attri- sion of the SKA project. It is intriguing to observe the cor- bution 4.0 International License, which permits use, sharing, adaptation, relation between the statistical anisotropy [48–51] of isocur- distribution and reproduction in any medium or format, as long as you 123 Eur. Phys. J. C (2020) 80:419 Page 7 of 10 419 give appropriate credit to the original author(s) and the source, pro- A.1 Scalar perturbations vide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article In the Newtonian gauge, the line element with scalar pertur- are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not bations is written as included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- ds2 =−(1 + 2)dt2 + (1 + 2)δ dxi dx j . (A6) ted use, you will need to obtain permission directly from the copy- ij right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. The scalar perturbations  and  are induced by the trace 3 Funded by SCOAP . part of the energy-momentum tensor. Linearized Christoffel symbols are calculated as

Appendix A: Photon redshift due to perturbations on a 0 = ,˙ 0 = 0 = ∂ , 0 = δ .˙ (A7) flat spacetime 00 i0 0i i ij ij

In this appendix, we calculate the photon redshift due to met- We use the photon geodesic equation ric perturbations on a flat spacetime. This section is based μ on [57], where the effect of gravitational waves on a photon dk μ ν ρ =−νρ k k . (A8) trajectory is focused on. dλ

Let the spatial origin 0 be the position of the observer. μ From (A2), k is independent of λ, so the left-hand side The photon is observed by the observer at the time t0.We μ0 reduces to dk /dλ. On the other hand, since Christoffel sym- write the unperturbed world line of a photon traveling from 1 μ a source to an observer as bols are already the first order of the perturbations, only k0 appears in the right-hand side. Considering μ = 0 compo- μ(λ) = ω (λ, −λ ) + ( , ), nent, we have x0 0 n t0 0 (A1)

ω λ 0 where 0 is a frequency, is an affine parameter, and n is a dk1 0 ν ρ =−νρ k k unit vector from the observer to the source (so that −n is the dλ 0 0 propagation direction of the photon). The trajectory (A1)is =−0 0 0 − 0 0 i − 0 i j μ 00k0k0 2 0i k0k0 ijk0k chosen so that the photon reaches x = (t , 0) at λ = λ = 0 0 0 obs =−ω˙ 2 − ∂ ω2(− i ) − δ˙ ω2 i j 0. The unperturbed four-momentum of the photon is 0 2 i 0 n ij 0n n ˙ 2 2 i ˙ 2 =−ω + 2 ∂i ω n − ω . (A9) μ(λ) 0 0 0 μ dx0 k = = ω0(1, −n). (A2) 0 λ i j d In the last line, we used δijn n = 1. We integrate (A9)to obtain a four-momentum: If the distance between the source and the observer is |xs| , the affine parameter value λs at which the photon is emitted  λ by the source is given by 0(λ) = ω2 λ ( ∂  i − ˙ − )˙ + , k1 0 d 2 i n C (A10) 0 |xs| λs =− . (A3) ω0 where C is a constant of integration. We determine C by con- sidering the initial condition that the source emits a photon of Let us find the expression for the photon redshift up to the frequency ω0 at the affine parameter value λs. In terms of μ the first order of perturbations. We write the world line of a the four-velocity of the source us , the condition is expressed photon as a sum of an unperturbed part and a perturbed part as as μ ν ω =−gμν(x )k (λ )u (λ ). (A11) μ(λ) = μ(λ) + μ(λ). 0 s s s s x x0 x1 (A4) Note that in the rest frame of the source, we obtain Similarly, the four-momentum is written as

− / μ(λ) = μ(λ) + μ(λ). u0 = (1 + 2) 1 2  1 − , (A12) k k0 k1 (A5) s

We will consider the contributions from the scalar perturba- to the first order of perturbations. Thus, from Eq. (A11), we tions and the traceless part of the metric, separately. have 123 419 Page 8 of 10 Eur. Phys. J. C (2020) 80:419

ω =− ( ) 0(λ ) 0(λ ) where hij corresponds to the traceless perturbations. From 0 g00 xs k s us s  this metric, linearized Christoffel symbols are calculated as λs = ω + ω2 λ ( ∂  i − ˙ − )˙ + + ω ( ). 0 0 d 2 i n C 0 xs 0 (A13) 0 1 ˙  = hij, ij 2 As a result, the constant C must be i = i = 1 ˙i , 0 j j0 h j  λ 2 s  1 C =−ω2 dλ ( ∂  ni − ˙ − )˙ − ω (x ). i = (∂ i + ∂ i − ∂i ). 0 2 i 0 s (A14) jk k h j j h k h jk (A19) 0 2

The photon frequency measured by the observer with the In this case, the timelike component of the four-momentum μ ω =− ( ) μ four-velocity uobs can be found as obs gμν xobs k of the photon obeys (λ ) ν (λ ) μ = μ(λ ) obs uobs obs , where xobs x obs . Using the expres- μ = ( − ( ), , , ) sion uobs 1 xobs 0 0 0 in the rest frame of the 0 dk1 0 i j 1 ˙ 2 i j observer, we obtain =− k k =− hijω n n . (A20) dλ ij 0 0 2 0 0 0 ωobs =−g00(xobs)k (λobs)u (λobs)  obs 0 Integrating this expression yields = ω + ω2 λ ( ∂  i − ˙ − )˙ 0 0 d 2 i n λ  λ s 1    k0(λ) =− ω2ni n j dλ h˙ (λ ) + C , (A21) − ω0(xs) + ω0(xobs). (A15) 1 0 ij 2 0 We can use the relation t = ω λ + t along the unperturbed 0 0 where C is a constant of integration. We use the condition photon geodesic, t = t0 at λ = 0, and t = t0 −|xs| at λ =  λ (A11) to determine C . In the rest frame of the source, its s to rewrite this equation. In addition, the total derivative ν = ( , , , ) i four-velocity is us 1 0 0 0 .Sowehave with respect to λ is given by d/dλ = ω0∂t − ω0n ∂i . Then, eq. (A15) can be rewritten as ω =− ( ) μ(λ ) ν(λ )  0 gμν xs k s us s t0     λs ω = ω + ω [∂ ( ) i − ∂ ( ) i ] 1    obs 0 0 dt i t n i t n = ω − ω2ni n j dλ h˙ (λ ) + C . (A22) t −|x | 0 0 ij 0 s 2 0 − ω0(xobs) + ω0(xs). (A16) Thus, Wenow estimate the magnitude of the signal in PTAmeasure-  ments. The distance to a pulsar is typically |xs |  100 pc [36], λs − −  1 2 i j  ˙  which is much longer than m 1 = 0.6pc× (10 23 eV/m). C = ω n n dλ hij(λ ). (A23) 2 0 Therefore, the integrand of the second term in (A16)is 0 rapidly oscillating, and hence becomes small after the inte- The frequency of a photon measured by an observer is cal- gration. Moreover, since the spatial derivative gives k = μ ν − culated as ωobs =−gμν(xobs)k (λobs)u (λobs). Since 2π/λ and m 1 is factored out from the integral, the second μ obs dB u = (1, 0, 0, 0) in the rest frame of the observer, we find term in (A16) is suppressed by a factor k/m = v ∼ 10−3 obs compared to the last two terms (see the discussion in Sect. μ ν ωobs =−gμν(xobs)k (λobs)u (λobs) 2). Thus, we can neglect the second term. As a result, the  obs λs redshift of the photon is given by = ω + 1ω2 i j λ ˙ (λ). 0 0n n d hij (A24) 2 0 ω0 − ωobs z = = (xobs) − (xs) ω0 To the first order of perturbations, hij(λ) means hij(x0(λ)) = = (t , 0) − (t −|x |, x ). (A17) 0 0 s s hij(ω0λ + t0, −ω0λn). Hence, we obtain

A.2 Traceless part of metric perturbations dh (λ) d d ij = h˙ (ω λ + t ) + ∂ h (−ω λnk) dλ ij dλ 0 0 k ij dλ 0 We apply the above discussion to the traceless part of metric = ω ˙ − ω k∂ . perturbations. We write the line element as 0hij 0n k hij (A25)

2 2 i j ds =−dt + (δij + hij)dx dx , (A18) Using this relation, we can rewrite (A24)as 123 Eur. Phys. J. C (2020) 80:419 Page 9 of 10 419  λs ω = ω + 1ω2 i j λ ˙ (λ) 12. J. A. R. Cembranos, A. L. Maroto, and S. J. Núñez Jareño, Pertur- obs 0 0n n d hij bations of ultralight vector field dark matter. JHEP 02, 064 (2016). 2 0  λ    arXiv:1611.03793 [astro-ph.CO] s dh (λ ) = ω + 1ω i j λ ij + ω k∂ (λ) 13. A. Pierce, K. Riles, Y.Zhao, Searching for dark photon dark matter 0 0n n d  0n k hij 2 0 dλ with gravitational wave detectors. Phys. Rev. Lett. 121, 061102  (2018). arXiv:1801.10161 [hep-ph] 1 i j = ω0 + ω0n n hij(t0 −|xs|, xs) − hij(t0, 0) 14. H.-K. Guo, K. Riles, F.-W. Yang, Y. Zhao, Searching for dark 2 photon dark matter in LIGO O1 data, (2019). arXiv:1905.04316  −| | t0 xs [hep-ph] 1 i j  k   + ω0n n dt n ∂k hij(t , x(t )). (A26) 15. S. D. McDermott, S. J. Witte, The cosmological evolution of light 2 t0 dark photon dark matter, (2019). arXiv:1911.05086 [hep-ph] 16. P. Arias, D. 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