Overlap reduction function for pulsar timing arrays in an expanding Universe

Abstract

Since it was confirmed two decades ago that the expansion of the Universe is accelerating, it would be of theoretical interests to figure out what is the influence from cosmological constant on detection of stochastic gravitational wave background. This paper studies the overlap reduction functions in de-Sitter space-time for a pair of one-way tracking gravitational wave detectors. It is shown to be non-trivial in an expanding Universe, because the propagation of light along line of sight also has effect on the response of GW detectors. It is found that the expansion of the Universe can enhance the value of magnitude of the overlap reduction functions, when the detector pairs are close to each other. For nanohertz gravitational waves, this effect can dominate the values of overlap reduction functions when the galactic pulsar pairs are separated by milliarcsecond.

1 Introduction

It was confirmed 2 decades ago that the expansion of the Universe is accelerating, and the current Universe is dominated by the dark energy which still remains mysterious [1, 2]. It would be of theoretical interests to figure out what is the influence of the accelerated expansion of the Universe on physical quantities or observables [3,4,5,6,7,8,9].

The first direct detection of gravitational waves (GWs) open up a new window for exploring the Universe [10]. It implies the existence of stochastic gravitational waves background (GWB), which can be originated from inflationary GW [11,12,13,14,15], produced from early-time phase transitions [16,17,18], sourced by cosmic string [19,20,21,22], or formed by superpositions of unresolved individual GW sources such as binary systems [23,24,25,26,27], core-collapse supernovae [28,29,30,31], and deformed rotating neutron stars [29, 32]. To date, the experiments for GW detections were built or designed in a broad frequency band [33,34,35,36,37,38]. In the $10\mathrm Hz$–$1\mathrm kHz$ frequency band, the ground-based GW detectors LIGO/Virgo network at current sensitivity did not find evidence of GWB, and therefore presented an upper limits of it [39, 40]. In $\mathrm nHz$ frequency band, the timing pulsar array projects, namely, NANOGrav [41], PPTA [42] and EPTA [43], found and confirmed a common spectrum process from the pulsar-timing data sets, and suggested that further evidence for GWB might rely on its angular correlation signature [44,45,46].

With the assumption of isotropic GWB, the cross-correlations of output of GW detectors depend on the angular separation of a pair of detectors. And the angular-dependence of it is completely described by overlap reduction functions (ORFs) for the detector pairs. For GW detector networks made by pulsar timing array, the ORF of GWs is known as Hellings-Downs curve for a pair of pulsars [47]. Theoretically, it is necessary to clarify possible physical causes that can lead to deviations of the Hellings-Downs curve. For instance, it might come from the GWB beyond isotropy approximation [48, 49], polarized GWB [50,51,52], non-tensor modes from modified gravity [53,54,55,56,57,58], non-linear contributions from higher order perturbation theory [59, 60]. Through a careful calculation on the pulsar terms, it also was found that the value for the magnitude of the ORFs can get larger for the pulsar pairs close to each other [61,62,63]. It was suggested an additional correlated phase changes between the close pulsar pairs. In the present study, instead of pulsar terms, we will show a similar behavior of the ORFs due to the cosmological constant.

Pulsar timing residuals can reflect GWs propagating in the Universe, thus the effects of the expanding Universe on PTAs was explored [7,8,9, 64], and could be utilized for local measurement of Hubble constant [7, 64, 65]. It is an independent way for determining the value of $H_0$, and distinguished from traditional approaches, e.g. via standard candles or standard sirens [9]. Differed from pioneers’ studies focusing on individual GW sources, in this paper, we consider GWB and investigate the theoretical ORFs in an expanding Universe (de-Sitter background). We present a rigorous formalism for calculating the ORFs. The calculation is shown to be non-trivial for de-Sitter background, because the propagation process of light along line of sight also has effect on the response of GW detectors. One can not account for the results in the expanding Universe with simply a redshift factor $1+z$. It is found that the cosmological constant leads to a much larger value of ORFs for close GW detector pairs, which is differed from Hellings-Downs curve.

The rest of the paper is organized as follows. In Sect. 2, we brief review the dark energy dominated epochs described by the de-Sitter space-time, and present conventions used in the following. In Sect. 3, we show how the cosmological perturbations freely propagate in the de-Sitter vacuum. In Sect. 4, we calculate the light ray affected by the GWs by solving perturbed geodesic equations in de-Sitter background. In Sect. 5, we calculate the ORFs for a pair of one-way tracking GW detectors, and present its deviations from the Hellings-Downs curve. In Sect. 6, the conclusions and discussions are summarized.

2 Dark energy dominated epochs and the conventions

Due to the accelerating expansion of our Universe, it would be interesting to extend the studies from the previously in Minkowski space-time into the Friedmann–Lemaitre–Robertson–Walker (FLRW) space-time, and figure out the influence from the expansion of the Universe on the observables. In this section, we will brief review the metric for describing the expanding Universe at late time, and present the conventions used in rest of the paper.

In cosmology, the spatially flat FLRW metric is given by

$$\begin{aligned} \textrm{d} s^2= & {} - \textrm{d} t^2 + a^2 (t) (\textrm{d} x^2 + \textrm{d} y^2 + \textrm{d} z^2)~, \end{aligned}$$

(1)

where the scale factor a(t) describes the evolution of the Universe. The expression of a(t) can be obtained by solving Einstein field equations sourced by perfect fluids, namely,

$$\begin{aligned} H^2= & {} \frac{8 \pi G}{3} \rho ~, \end{aligned}$$

(2)

where G is gravity constant, $H(\equiv {{\dot{a}}}/a)$ is Hubble parameter, and $\rho$ is matter density in the Universe. Based on the standard cosmology, the $\varLambda$CDM model, the matters in the Universe consist of relativistic matter $\rho _r$ (radiation or massless particles), non-relativistic matter $\rho _m$ (baryon, or dark matter), and dark energy $\rho _\varLambda$. Thus Eq. (2) can be rewritten in the form of [66]

$$\begin{aligned} H^2= & {} \frac{8 \pi G}{3} (\rho _r+\rho _m+\rho _\varLambda ) \nonumber \\=: & {} H_0^2 \sqrt{\frac{\varOmega _r}{a^4} + \frac{\varOmega _m}{a^3} + \varOmega _{\varLambda }} ~, \end{aligned}$$

(3)

where the $H_0 (\equiv {8 \pi G \rho _c}/{3})$ is Hubble constant, and the $\rho _c$ is critical density that describes the average density in the Universe at the present. In the second equality of Eq. (3), each components $i = r$, m or $\varLambda$ of matter density are given by $\rho _i = (\varOmega _i \rho _c) a^{- 3(1 + w_i)}$, which can be obtained by using the equations of state $\rho = w p$ and conservation of energy-momentum tensor. Here, the $\varOmega _i$ is density fraction, and $\varOmega _r+\varOmega _m+\varOmega _\varLambda =1$. For different compositions, $w_r=1/3$, $w_m=0$, and $w_\varLambda =-1$, which give different equation of state.

Though the constraints from observation on the cosmic microwave background [67], and local measurement of Hubble law in the late-time Universe [1], the density fractions $\varOmega _r$, $\varOmega _m$ and $\varOmega _{\varLambda }$ are determined, in which $\varOmega _r \ll \varOmega _m \lesssim \varOmega _{\varLambda }$. For the late time Universe, there exists an epoch $(\varOmega _m / \varOmega _{\varLambda })^{\frac{1}{3}}< a < 1$, in which the cosmological redshift is lower than 2.3, approximately. It is known as the epoch that the Universe is dominated by the dark energy or the cosmological constant. In this epochs, the scale factor in Eq. (2) takes the form of

$$\begin{aligned} a= & {} e^{H_0 t}~. \end{aligned}$$

(4)

And the metric in Eq. (1) with scalar factor in Eq. (4) is known as de-Sitter space-time. Here, we adopt the convention $t = 0$ for the present time of the Universe, and $t < 0$ for the history of the Universe.

For simplicity, it is more practical to transform the metric into the conformally flat one. Namely, by introducing the conformal time,

$$\begin{aligned} \eta= & {} \int ^t_{- \infty } \frac{\textrm{d} t}{a}\nonumber \\= & {} - \frac{1}{H_0} e^{- H_0 t}~, \end{aligned}$$

(5)

the metric in Eq. (1) reduces to

$$\begin{aligned} \textrm{d} s^2= & {} \frac{1}{(H_0 \eta )^2} (- \textrm{d} \eta ^2 + \textrm{d} x^2 + \textrm{d} y^2 + \textrm{d} z^2)~. \end{aligned}$$

(6)

For the present time of the Universe $t = 0$, the conformal time corresponds to $\eta = - \frac{1}{H_0}$. And $\eta < - \frac{1}{H_0}$ describes the history of the Universe. For $t\rightarrow 0^-$, the Eq. (5) can be expanded in the form of

$$\begin{aligned} \eta = -\frac{1}{H_0}+t+{\mathcal {O}}((H_0 t)^2)~. \end{aligned}$$

(7)

In the late time of the Universe, the $\eta +1/H_0$ is equal to the cosmic time t, approximately. We will utilize Eq. (7) in the following for identifying the freely-propagating GWs at $t\rightarrow 0^-$.

3 Cosmological perturbations propagation in vacuum

In de-Sitter background, the propagation of the GW is different from that in Minkowski space-time. Due to GW detectors constituted by co-moving periodic sources in the Universe, the GWs, in fact, propagate within the GW detector network. Thus, the cosmological constant might affect the response of GW detectors. In this section, we will show the evolutions of GWs to the first order with the assumption that the metric perturbations are freely-propagating in the de-Sitter background.

The perturbed de-Sitter metric to the first order is given based on helicity decomposition [66],

$$\begin{aligned} \textrm{d} s^2= & {} \frac{1}{(H_0 \eta )^2} \big (- \textrm{d} \eta ^2 + (\delta _{i j} (1 - 2 \psi ) + 2 \partial _i \partial _j E \nonumber \\{} & {} + \partial _i C_j + \partial _j C_i + h_{i j}) \textrm{d} x^i \textrm{d} x^j\big )~, \end{aligned}$$

(8)

where the $\psi$ and the E are scalar perturbations, the $C_j$ is vector perturbation providing $\delta ^{i j}\partial _j C^i=0$, and the $h_{ij}$ is tensor perturbation providing $\delta ^{i j}\partial _j h_{i j} = \delta ^{i j}h_{i j}=0$. Here we adopt the Synchronous gauge, because the GW detectors are set to be freely-falling in the Universe.

Evaluating the Einstein field equations, we obtain the evolution equations for the first order metric perturbations,

$$\begin{aligned} 0= & {} h_{i j}'' - \frac{2}{\eta } h_{i j}' - \varDelta h_{i j} - \frac{2}{\eta } \partial _i C_j' + \partial _i C_j'' - \frac{2}{\eta } \partial _j C_i' + \partial _j C_i'' \nonumber \\{} & {} + 2 \partial _i \partial _j \left( \psi + E'' - \frac{2}{\eta } E' \right) \nonumber \\{} & {} + 2 \delta _{i j} \left( - \frac{4}{\eta } \psi ' + 2 \psi '' + \frac{2}{\eta } \varDelta E' - \varDelta E'' - \varDelta \psi \right) ~. \end{aligned}$$

(9a)

By making of using helicity decomposition, we can split the equations in the form of [68]

$$\begin{aligned}{} & {} h_{i j}'' - \frac{2}{\eta } h_{i j}' - \varDelta h_{i j} = 0~,\end{aligned}$$

(10a)

$$\begin{aligned}{} & {} - \frac{2}{\eta } C_j' + C_j'' = 0~,\end{aligned}$$

(10b)

$$\begin{aligned}{} & {} \psi + E'' - \frac{2}{\eta } E' = 0~,\end{aligned}$$

(10c)

$$\begin{aligned}{} & {} - \frac{4}{\eta } \psi ' + 2 \psi '' + \frac{2}{\eta } \varDelta E' - \varDelta E'' - \varDelta \psi = 0~. \end{aligned}$$

(10d)

Simplifying and evaluating above equations in Fourier space, we obtain

$$\begin{aligned}{} & {} \psi '_{\varvec{k}} = 0~,\end{aligned}$$

(11a)

$$\begin{aligned}{} & {} \frac{2}{\eta } E_{\varvec{k}}' - E''_{\varvec{k}} = \psi _{\varvec{k}}~,\end{aligned}$$

(11b)

$$\begin{aligned}{} & {} C_{j, \varvec{k}}' = 0~,\end{aligned}$$

(11c)

$$\begin{aligned}{} & {} h_{i j, \varvec{k}}'' - \frac{2}{\eta } h_{i j, \varvec{k}}' + k^2 h_{i j, \varvec{k}} = 0~. \end{aligned}$$

(11d)

It shows that the evolution of tensor perturbation $h_{i j}$ is described by the wave equations in de-Sitter background, while rest of the metric perturbations are not. Since we only consider that the metric perturbations freely propagate in de-Sitter background, the solutions of above equations can be formally expressed as initial stochastic variables (${\bar{\psi }}_{\varvec{k}}, {\bar{E}}_{\varvec{k}}, {\bar{C}}_{j, \varvec{k}}, \text {and } {\bar{h}}_{i j, \varvec{k}}$) and temporal transfer functions $T_{\varvec{k}, *}$, namely,

$$\begin{aligned} \psi _{\varvec{k}}= & {} T_{\psi , \varvec{k}} (\eta ){\bar{\psi }}_{\varvec{k}}~,\end{aligned}$$

(12a)

$$\begin{aligned} E_{\varvec{k}}= & {} T_{E, \varvec{k}} (\eta ) {\bar{E}}_{\varvec{k}}~,\end{aligned}$$

(12b)

$$\begin{aligned} C_{j, \varvec{k}}= & {} T_{C, \varvec{k}} (\eta ) {\bar{C}}_{j, \varvec{k}}~,\end{aligned}$$

(12c)

$$\begin{aligned} h_{i j, \varvec{k}}= & {} T_{h, \varvec{k}} (\eta ) {\bar{h}}_{i j, \varvec{k}}~. \end{aligned}$$

(12d)

The initial stochastic variables contain physical information about how the perturbations are generated and propagates before its reaching the GW detectors. Because of its stochastic nature, the physical information should be extracted in statistics. The transfer functions describe the propagation of perturbations within the GW detectors, and thus can affect the response of GW detectors. In Sect. 4, we will explicitly show that the expression of $T_{h,\varvec{k}}$ can affect the response of GW detectors.

By making use of Eqs. (11) and (12), we obtain the expressions of the transfer functions $T_{*,\varvec{k}}$ in the form of

$$\begin{aligned} T_{\psi , \varvec{k}}= & {} T_{\psi ,0}~,\end{aligned}$$

(13a)

$$\begin{aligned} T_{E, \varvec{k}}= & {} T_{\psi ,0}\frac{\eta ^2}{2}+T_{E,0}~,\end{aligned}$$

(13b)

$$\begin{aligned} T_{C, \varvec{k}}= & {} T_{C,0}~,\end{aligned}$$

(13c)

$$\begin{aligned} T_{h, \varvec{k}}= & {} e^{- i k \left( \eta +\frac{1}{H_0}\right) } \left( \frac{H_0}{k} \left( \frac{H_0}{2 k} + i \right) (i k \eta + 1) \right. \nonumber \\ {}{} & {} \left. + \frac{1}{2} \left( \frac{H_0}{k} \right) ^2 (i k \eta - 1) e^{2 i k \left( \eta + \frac{1}{H_0} \right) } \right) ~, \end{aligned}$$

(13d)

where $T_{\psi ,0}$, $T_{E,0}$, and $T_{C,0}$ are integral constants from solving Eqs. (11), and $k\equiv |\varvec{k}|$. In order to obtain an expression of the $T_{h, \varvec{k}}$ that could reduce to the results in Minkowski space-time at $t\rightarrow 0^-$, we adopt the boundary conditions that $T_{h, \varvec{k}} \rightarrow e^{- i k (\eta +1/H_0)}$ as $\eta \rightarrow - {1}/{H_0}$. For the transfer function $T_{E, \varvec{k}}$ shown in Eq. (13b), there seems not a physical mechanism for a large amplitude of the $E_{\varvec{k}}$ at large $| \eta |$. Therefore, we let $T_{\psi , \varvec{k}} = 0$ and $T_{E, \varvec{k}} = \mathrm{{const}}.$.

The transfer functions of scalar and vector perturbations are constant, while the transfer function of the tensor perturbation $h_{i j, \varvec{k}}$ oscillates with conformal time. The latter one seems to be more interesting, and could interpret the GWB in the Universe. Thus, in the following, we would limit our study to tensor perturbations $h_{i j}$.

4 Propagation of light in the perturbed de-Sitter space-time

The GWB formulated by the metric perturbations $h_{i j}$ in the space can affect the propagation of light within the GW detectors. Thus, in this section, we will calculate the propagation of light rays in the perturbed de-Sitter space-time.

Expanding the geodesic equations to the first order, we have

$$\begin{aligned} 0= & {} p^{\mu } \nabla _{\mu } p^{\nu }~, \end{aligned}$$

(14)

$$\begin{aligned} 0= & {} \delta p^{\mu } \nabla _{\mu } p^{\nu } + p^{\mu } \nabla _{\mu } \delta p^{\nu } \nonumber \\{} & {} + g^{\nu \rho } \left( \nabla _{\mu } \delta g_{\lambda \rho } - \frac{1}{2} \nabla _{\rho } \delta g_{\mu \lambda } \right) p^{\mu } p^{\lambda }~, \end{aligned}$$

(15)

where $p^{\mu }$ and $\delta p^{\mu }$ are the background and the first order 4-velocity of the light, respectively, the $g_{\mu \nu }$ and $\delta g_{\mu \nu }$ are the background metric and the first order metric perturbation, respectively, and ${\nabla _\mu }$ is the covariant derivative with respect to the background metric. In Appendix B, the derivation of Eq. (15) is presented.

Using background metric in Eq. (6), the zeroth order geodesic equations in Eqs. (14) can be evaluated to be

$$\begin{aligned} \partial _0 p^0= & {} \frac{2}{\eta } p^0~,\end{aligned}$$

(16a)

$$\begin{aligned} \partial _0 p^i= & {} \frac{2}{\eta } p^i~, \end{aligned}$$

(16b)

where we have used the normalization condition $p_{\mu } p^{\mu } = 0$ for the null 4-velocities in Eqs. (16). By integration over the conformal time, we obtain the 4-velocities of backward-propagating light rays,

$$\begin{aligned} p^{\mu }= & {} w_0 \eta ^2 (1, - {\hat{n}}^i)~, \end{aligned}$$

(17)

where the normalized vector ${\hat{n}}_i (\equiv -{p^i}/{p^0})$ is a constant vector, and $w_0$ is an integral constant from the geodesic equations. In the case of $\eta \rightarrow -1/H_0$, the $w_0/H_0^2$ represents the frequency of a light ray. By solving Eq. (17), we obtain the trajectories of the light rays,

$$\begin{aligned} x^i (\eta ) - x^i_0 = -{\hat{n}}^i (\eta - \eta _0)~, \end{aligned}$$

(18)

where $(\eta _0, x_0^i)$ represents initial event.

For the events of distant objects $(\eta _{\mathrm{{emt}}}, d {\hat{n}}^i)$ and the event on the earth $\left( - {1}/{H_0}, 0 \right)$, the trajectories can be formulated as

$$\begin{aligned} d= & {} \frac{1}{H_0} + \eta _{\mathrm{{emt}}}~, \end{aligned}$$

(19)

where the d is co-moving distance. Since the redshift of co-moving objects can be given by $1 + z = {1}/{a} = - ({H_0 \eta _{\mathrm{{emt}}}})^{-1}$, it is not difficult to find the distance-redshift relation in de-Sitter space-time [66],

$$\begin{aligned} (1 + z) d= & {} \frac{z}{H_0}~, \end{aligned}$$

(20)

where one can also define the luminosity distance $d_L\equiv (1+z)d$.

Using the background 4-velocities $p^\mu$ in Eq. (17), we can further solve the perturbed geodesic equations in Eqs. (15). Since the GW detectors are set to be freely-falling in the Universe, we evaluate the perturbed geodesic in the Synchronous gauge,

$$\begin{aligned} 0&= g_{00} p^0 (\partial _0 - {\hat{n}}^i \partial _i) \delta p^0 + \frac{2}{\eta } p_a \delta p^a - \frac{1}{2} (p^0)^2 {\hat{n}}^a {\hat{n}}^b \partial _0 h_{a b}~, \nonumber \\ \end{aligned}$$

(21a)

$$\begin{aligned} 0&= p^{\mu } \partial _{\mu } \delta p^j - \frac{2}{\eta } p^0 \delta p^j + \left( p^a p^0 g^{j b} \partial _0 + p^a p^c g^{j b} \partial _c \right. \nonumber \\&\quad \left. - \frac{1}{2} p^a p^b g^{j c} \partial _c + \frac{2}{\eta } g^{j b} p^0 p^a \right) h_{a b}~, \end{aligned}$$

(21b)

where the Latin letters denote spatial indices, and we limit our study to the tensor perturbation $h_{i j}$ in above equations.

In order to obtain ORFs of GW detectors, one should solve $\delta p^0 / p^0$ from perturbed geodesic equations. In Minkowski space-time, the 0-component of Eqs. (15) is enough for obtaining the $\delta p^0 / p^0$, namely,

$$\begin{aligned} (\partial _0 - {\hat{n}}^i \partial _i) \left( \frac{\delta p^0}{p^0} \right)= & {} - \frac{1}{2} {\hat{n}}^a {\hat{n}}^b \partial _0 h_{a b}~. \end{aligned}$$

(22)

However, differed from the calculation in Minkowski space-time, the 0-component of the perturbed geodesic equations in Eq. (21a) can not be solved without knowing the $p_a \delta p^a$. It indicates that the ORFs in de-Sitter space-time is non-trivial, because the propagation of light along the direction of $p^a$ also has effect.

For solving the $\delta p^0/p^0$, we rewrite the Eq. (21b) by contracting a vector $p_j$, which leads to

$$\begin{aligned} 0= & {} \partial _0 \left( \frac{p_j \delta p^j}{(p^0)^2} \right) + \frac{2}{\eta } \frac{p_j \delta p^j}{(p^0)^2} - {\hat{n}}^c \partial _c \left( \frac{p_j \delta p^j}{(p^0)^2} \right) \nonumber \\ {}{} & {} + {\hat{n}}^a {\hat{n}}^b \left( \partial _0 - \frac{1}{2} {\hat{n}}^j \partial _j + \frac{2}{\eta } \right) h_{a b}~, \end{aligned}$$

(23)

Expressing Eqs. (21a) and (23) in Fourier space, we obtain

$$\begin{aligned} 0= & {} p_0 (\partial _0 - i {\hat{n}} \cdot k) \delta p^0_{\varvec{k}} + \frac{2}{\eta } p_a \delta p^a_{\varvec{k}} \nonumber \\ {}{} & {} - \frac{1}{2} (p^0)^2 {\hat{n}}^a {\hat{n}}^a \partial _0 h_{a b, \varvec{k}}~,\end{aligned}$$

(24a)

$$\begin{aligned} 0= & {} \left( \partial _0 + \frac{2}{\eta } - i {\hat{n}} \cdot k \right) \left( \frac{p_j \delta p^j_{\varvec{k}}}{(p^0)^2} \right) \nonumber \\ {}{} & {} + {\hat{n}}^a {\hat{n}}^b \left( \partial _0 - \frac{1}{2} i {\hat{n}} \cdot k + \frac{2}{\eta } \right) h_{a b, \varvec{k}}~, \end{aligned}$$

(24b)

where $\delta p_{\varvec{k}}^\mu$ and $\delta h_{a b,\varvec{k}}$ are the Fourier modes of $\delta p^\mu$ and $\delta h_{a b}$, respectively, and ${\hat{n}}\cdot k \equiv {\hat{n}}_j k^j$. Because the $p^\mu$ and ${\hat{n}}_i$ are free of spatial coordinates in de-Sitter background, they thus have no relevance with the $\varvec{k}$ in Fourier space. It should be clarified that the ${\hat{n}}_i$ represents the directions of the backward-propagating light rays, and the ${\hat{k}}^i\equiv k^i/{|\varvec{k}|}$ represents the directions of propagation of the gravitational waves $h_{i j}$. For simplification, we introduce the $\pi _{\varvec{k}}$ and $\chi _{\varvec{k}}$ in the form of

$$\begin{aligned} \pi _{\varvec{k}}\equiv & {} \frac{\delta p_{\varvec{k}}^0}{{\hat{n}}^a {\hat{n}}^b {\bar{h}}_{a b , \varvec{k}}}~,\end{aligned}$$

(25a)

$$\begin{aligned} \chi _{\varvec{k}}\equiv & {} \frac{\frac{1}{(p^0)^2} p_j \delta p_{\varvec{k}}^j}{{\hat{n}}^a {\hat{n}}^b {\bar{h}}_{a b , \varvec{k}}}~. \end{aligned}$$

(25b)

Substituting Eqs. (12d) and (25) into Eqs. (24), we obtain

$$\begin{aligned} \pi _{\varvec{k}}' - i ({\hat{n}} \cdot k) \pi _{\varvec{k}}= & {} - \frac{p^0}{g_{00}} \left( \frac{2}{\eta } \chi _{\varvec{k}} - \frac{1}{2} T'_{h,\varvec{k}} \right) ~,\end{aligned}$$

(26a)

$$\begin{aligned} \chi _{\varvec{k}}' - i ({\hat{n}} \cdot k) \chi _{\varvec{k}} + \frac{2}{\eta } \chi _{\varvec{k}}= & {} - T'_{h,\varvec{k}} + \frac{1}{2} i ({\hat{n}} \cdot k) T_{h,\varvec{k}} - \frac{2}{\eta } T_{h,\varvec{k}}~.\nonumber \\{} & {} \end{aligned}$$

(26b)

Here, the $\pi _{\varvec{k}}$ depends on the transfer functions $T_{h,\varvec{k}}(\eta )$ within the GW detectors. With the expression of $T_{h,\varvec{k}}(\eta )$ in Eq. (13d), solutions of Eq. (26) can be obtained,

$$\begin{aligned} \frac{\pi _{\varvec{k}}}{p^0}= & {} e^{\frac{i k \left( \eta H_0+1\right) }{{H_0}}} \left( i {\left( \frac{H_0}{k} \right) } \frac{ \eta ^3 H_0^3 }{4 ({{\hat{n}}\cdot {\hat{k}}}-1)} + i {\left( \frac{H_0}{k} \right) }^3 \left( \frac{\eta H_0}{2 ({{\hat{n}}\cdot {\hat{k}}}-1)}\right. \right. \nonumber \\{} & {} \left. \left. -\frac{\eta H_0}{({{\hat{n}}\cdot {\hat{k}}}-1)^2}-\frac{3 \eta H_0}{2 ({{\hat{n}}\cdot {\hat{k}}}-1)^3}\right) \right. \nonumber \\{} & {} \left. -{\left( \frac{H_0}{k} \right) }^2 \left( -\frac{\eta ^2 H_0^2}{2 ({{\hat{n}}\cdot {\hat{k}}}-1)}-\frac{3 \eta ^2 H_0^2}{4 ({{\hat{n}}\cdot {\hat{k}}}-1)^2}\right) \right. \nonumber \\{} & {} \left. +{\left( \frac{H_0}{k} \right) }^4 \left( \frac{1}{2 ({{\hat{n}}\cdot {\hat{k}}}-1)^2}-\frac{1}{({{\hat{n}}\cdot {\hat{k}}}-1)^3}-\frac{3}{2 ({{\hat{n}}\cdot {\hat{k}}}-1)^4}\right) \right) \nonumber \\{} & {} +e^{-\frac{i k \left( \eta H_0+1\right) }{{H_0}}} \left( \frac{\eta ^3 H_0^3}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)} -i {\left( \frac{H_0}{k} \right) } \left( \frac{\eta ^3 H_0^3}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)}\right. \right. \nonumber \\{} & {} \left. \left. -\frac{\eta ^2 H_0^2}{{{\hat{n}}\cdot {\hat{k}}}+1}+\frac{3 \eta ^2 H_0^2}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}\right) \right. \nonumber \\{} & {} \left. +i {\left( \frac{H_0}{k} \right) }^3 \left( -\frac{\eta H_0}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)}-\frac{\eta H_0}{({{\hat{n}}\cdot {\hat{k}}}+1)^2}+\frac{3 \eta H_0}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)^3}\right. \right. \nonumber \\{} & {} \left. \left. -\frac{1}{({{\hat{n}}\cdot {\hat{k}}}+1)^2}-\frac{2}{({{\hat{n}}\cdot {\hat{k}}}+1)^3}+\frac{3}{({{\hat{n}}\cdot {\hat{k}}}+1)^4}\right) \right. \nonumber \\{} & {} \left. -{\left( \frac{H_0}{k} \right) }^2 \left( -\frac{\eta ^2 H_0^2}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)}+\frac{3 \eta ^2 H_0^2}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}-\frac{\eta H_0}{{{\hat{n}}\cdot {\hat{k}}}+1}\right. \right. \nonumber \\{} & {} \left. \left. -\frac{2 \eta H_0}{({{\hat{n}}\cdot {\hat{k}}}+1)^2}+\frac{3 \eta H_0}{({{\hat{n}}\cdot {\hat{k}}}+1)^3}\right) +{\left( \frac{H_0}{k} \right) }^4 \right. \nonumber \\{} & {} \left. \left( -\frac{1}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}-\frac{1}{({{\hat{n}}\cdot {\hat{k}}}+1)^3}+\frac{3}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)^4}\right) \right) ,\nonumber \\ \end{aligned}$$

(27)

$$\begin{aligned} \chi _{\varvec{k}}= & {} e^{\frac{i k \left( \eta H_0+1\right) }{{H_0}}} \left( {\left( \frac{H_0}{k} \right) }^4 \left( \frac{1}{2 \eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}-1)^2}-\frac{1}{\eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}-1)^3}\right. \right. \nonumber \\{} & {} \left. \left. -\frac{3}{2 \eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}-1)^4}\right) +i {\left( \frac{H_0}{k} \right) }^3\right. \nonumber \\{} & {} \left. \left( \frac{1}{2 \eta H_0 ({{\hat{n}}\cdot {\hat{k}}}-1)}-\frac{1}{\eta H_0 ({{\hat{n}}\cdot {\hat{k}}}-1)^2}-\frac{3}{2 \eta H_0 ({{\hat{n}}\cdot {\hat{k}}}-1)^3}\right) \right. \nonumber \\{} & {} \left. -i {\left( \frac{H_0}{k} \right) } \left( \frac{\eta H_0}{4}-\frac{\eta H_0}{4 ({{\hat{n}}\cdot {\hat{k}}}-1)}\right) \right. \nonumber \\{} & {} \left. -{\left( \frac{H_0}{k} \right) }^2 \left( -\frac{1}{2 ({{\hat{n}}\cdot {\hat{k}}}-1)}-\frac{3}{4 ({{\hat{n}}\cdot {\hat{k}}}-1)^2}-\frac{1}{4}\right) \right) \nonumber \\{} & {} +e^{-\frac{i k \left( \eta H_0+1\right) }{{H_0}}} \left( \frac{\eta H_0}{2}+{\left( \frac{H_0}{k} \right) }^4 \left( -\frac{1}{2 \eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}\right. \right. \nonumber \\{} & {} \left. \left. \frac{1}{\eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}+1)^3}+\frac{3}{2 \eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}+1)^4}\right) \right. \nonumber \\{} & {} \left. +i {\left( \frac{H_0}{k} \right) }^3 \left( -\frac{1}{\eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}-\frac{2}{\eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}+1)^3}\right. \right. \nonumber \\{} & {} \left. \left. +\frac{3}{\eta ^2 H_0^2 ({{\hat{n}}\cdot {\hat{k}}}+1)^4}-\frac{1}{2 \eta H_0 ({{\hat{n}}\cdot {\hat{k}}}+1)}\right. \right. \nonumber \\{} & {} \left. \left. -\frac{1}{\eta H_0 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}+\frac{3}{2 \eta H_0 ({{\hat{n}}\cdot {\hat{k}}}+1)^3}\right) \right. \nonumber \\{} & {} \left. -{\left( \frac{H_0}{k} \right) }^2 \left( -\frac{1}{\eta H_0 ({{\hat{n}}\cdot {\hat{k}}}+1)}-\frac{2}{\eta H_0 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}\right. \right. \nonumber \\{} & {} \left. \left. +\frac{3}{\eta H_0 ({{\hat{n}}\cdot {\hat{k}}}+1)^3}-\frac{1}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)}+\frac{3}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}+\frac{1}{4}\right) \right. \nonumber \\{} & {} \left. -i {\left( \frac{H_0}{k} \right) } \left( \frac{\eta H_0}{4}+\frac{\eta H_0}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)}-\frac{1}{{{\hat{n}}\cdot {\hat{k}}}+1}+\frac{3}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}\right. \right. \nonumber \\{} & {} \left. \left. +\frac{1}{2}\right) +\frac{\eta H_0}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)}\right) . \end{aligned}$$

(28)

In the zeroth order with $H_0 / k \rightarrow 0$ and $H_0 \eta \rightarrow -1$, the Eq. (27) reduces to

$$\begin{aligned} \frac{\pi _{\varvec{k}}}{p^0} \Bigg |_{\tiny \begin{array}{l} H_0 / k \rightarrow 0 \\ H_0 \eta \rightarrow -1 \end{array}} = - \frac{1}{2 (1 + {\hat{n}} \cdot {\hat{k}})} e^{- i k (\eta + 1/H_0)}~, \end{aligned}$$

(29)

which is consistent with the results in Minkowski space-time [47, 69].

Fig. 1

The time evolution of the relative frequency shift for light rays with GW frequency $k=H_0/500$. The comparison with the results in Minkowski space-time is also presented

In Fig. 1, we show the time evolution of the $\pi _{\varvec{k}}/p^0$ in Eq. (27). Because of expansion of the Universe, there are phase shift and decay of the amplitude for the $\pi _{\varvec{k}}/p^0$.

Finally, by making use of Eqs. (25a) and (27), the frequency shift of the light in configuration space is shown to be

$$\begin{aligned} \frac{\delta p^0}{p^0}= & {} \int \frac{\textrm{d}^3 k}{(2 \pi )^3} \left\{ {\bar{h}}_{a b, \varvec{k}} {\hat{n}}^a {\hat{n}}^b \frac{\pi _{\varvec{k}} (\eta )}{p^0} e^{i k \cdot x} \right\} \nonumber \\= & {} \int \frac{\textrm{d}^3 k}{(2 \pi )^3} \left\{ {\bar{h}}_{a b, \varvec{k}} {\hat{n}}^a {\hat{n}}^b \left( e^{-\frac{i k}{H_0} \left( H_0(\eta - {\hat{k}}\cdot x) + 1 \right) } \left( \frac{\eta ^3 H_0^3}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)} \right. \right. \right. \nonumber \\{} & {} \left. \left. \left. -i {\left( \frac{H_0}{k} \right) } \left( \frac{\eta ^3 H_0^3}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)}-\frac{\eta ^2 H_0^2}{{{\hat{n}}\cdot {\hat{k}}}+1}+\frac{3 \eta ^2 H_0^2}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}\right) \right. \right. \right. \nonumber \\{} & {} \left. \left. \left. -{\left( \frac{H_0}{k} \right) }^2 \left( -\frac{\eta ^2 H_0^2}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)}+\frac{3 \eta ^2 H_0^2}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}-\frac{\eta H_0}{{{\hat{n}}\cdot {\hat{k}}}+1}\right. \right. \right. \right. \nonumber \\{} & {} \left. \left. \left. \left. -\frac{2 \eta H_0}{({{\hat{n}}\cdot {\hat{k}}}+1)^2}+\frac{3 \eta H_0}{({{\hat{n}}\cdot {\hat{k}}}+1)^3}\right) \right) \right. \right. \nonumber \\{} & {} \left. \left. + e^{\frac{i k}{H_0} \left( H_0(\eta - {\hat{k}}\cdot x) + 1 \right) } \left( -{\left( \frac{H_0}{k} \right) }^2 \left( \frac{\eta ^2 H_0^2}{2 ({{\hat{n}}\cdot {\hat{k}}}+1)}\right. \right. \right. \right. \nonumber \\{} & {} \left. \left. \left. \left. -\frac{3 \eta ^2 H_0^2}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)^2}\right) -\frac{i \eta ^3 H_0^3 {\left( \frac{H_0}{k} \right) }}{4 ({{\hat{n}}\cdot {\hat{k}}}+1)}\right) +{\mathcal {O}}\left( \left( \frac{H_0}{k} \right) ^3 \right) \right) \right\} .\nonumber \\ \end{aligned}$$

(30)

It is found that the $\delta p^0/p^0$ depends non-linearly on the factor $1/(1+{\hat{n}}\cdot {\hat{k}})$, which is even different from the higher order corrections of $\delta p^0/p^0$ [59]. This might give rise to difficulties in analytical calculations for the ORFs.

In the following, we will calculate the ORFs numerically to the leading order of $H_0/k$ based on the expression of $\delta p^0/p^0$ shown in Eq. (30).

5 Gravitational wave detector and overlap reduction function

Providing the isotropic GWB, the ORFs describe the angular correlations of the output of a pair of GW detectors. In this section, we will calculate the ORFs for one-way tracking of light with the assumption that the distant clock and receiver both co-moving with the expansion of the Universe.

In de-Sitter background, the time is dilated due to the expansion of the Universe. It can be described by the cosmological redshift,

$$\begin{aligned} 1 + z= & {} \frac{ u_{\mu } p^{\mu } \big |_{\mathrm{{rec}}}}{u_{\nu } p^{\nu } \big |_{\mathrm{{emt}}}}~, \end{aligned}$$

(31)

where $u_{\mu }$ is the 4-velocities of co-moving objects, and $p^{\mu }$ is the background 4-velocities of light rays. In Synchronous gauge, the 4-velocities of co-moving objects are $u_{\mu } = (- a, 0, 0, 0)$, the subscripts ‘rec’ and ‘emt’ represent the events of receivers, and the event of emitted light from the distant objects, respectively. In this case, the redshift in Eq. (31) reduces to $1+z=p^0_{\textrm{obs}}/(a_{\textrm{src}} p^0_\textrm{src})$. For a pulsar as distant clock, its distance from the earth is around $\text {kpc}$. One can estimate its redshift $z \sim 10^{-6}$ based on Eq. (20).

The redshift fluctuations from a distant clock can reflect the space-time fluctuations. Here, it can be derived from the fluctuations of the cosmological redshift,

$$\begin{aligned} \varDelta z\equiv & {} \frac{\delta (1 + z)}{1 + z}\nonumber \\= & {} \frac{\delta p_{\mathrm{{rec}}}^0}{p^0_{\mathrm{{rec}}}} - \frac{\delta p_{\mathrm{{emt}}}^0}{p^0_{\mathrm{{emt}}}}~. \end{aligned}$$

(32)

In principle, the $\varDelta z$ contains the contributions from the perturbed 4-velocities $\delta u_\mu$. In Synchronous gauges, it turns out to be zero. From Eq. (32), the redshift fluctuation $\varDelta z$ depends linearly on the perturbed frequency of light $\delta p^0 / p^0$. The fluctuation of the distant clock timing can be formulated by the redshift fluctuation of distant objects, because $\varDelta t/t=-\varDelta f/f$ is independent of specific timing mechanism. Namely, a clock timing by the rotation frequency of a pulsar, or characteristic frequency of an atom must give the same $\varDelta t/t$. We shall clarify that the $\varDelta z$ here is redshift fluctuation with respect to the cosmological redshift, while $\varDelta z$ was simply called redshift in the calculation of Hellings-Downs curves [47, 69]. It is because latter one was considered in Minkowski space-time, the background redshifts between distant objects are zero, and the leading order redshift comes from the $\varDelta z$. Note that they are different physical quantities. The cosmological redshift z can indicate the luminosity distance of co-moving objects, while the redshift fluctuation $\varDelta z$ here have no relevance with the distance.

Substituting the expression of $\delta p^0 / p^0$ in Eq. (30) into Eq. (32), we obtain the redshift fluctuation in the form of

$$\begin{aligned} \varDelta z= & {} \int \frac{\textrm{d}^3 k}{(2 \pi )^3} \{ {\bar{h}}_{a b, \varvec{k}} {\hat{n}}^a {\hat{n}}^b F({\hat{n}},{\varvec{k}},\eta ) \}~, \end{aligned}$$

(33)

where

$$\begin{aligned} F({\hat{n}},{\varvec{k}},\eta )= & {} \int \frac{\textrm{d}^3 k}{(2 \pi )^3} \Bigg \{ {\bar{h}}_{a b, \varvec{k}} {\hat{n}}^a {\hat{n}}^b \left( \frac{1}{ (1 + {\hat{n}} \cdot {\hat{k}})} \left( -\frac{1}{2}{\mathcal {K}}_1\right. \right. \nonumber \\{} & {} \left. \left. + \frac{3i}{2} \left( \frac{H_0}{k}\right) {\mathcal {K}}_2 - \left( \frac{H_0}{k}\right) ^2 {\mathcal {K}}_4 \right) \right. \nonumber \\{} & {} \left. - \frac{i}{(1 + {\hat{n}} \cdot {\hat{k}})^2} \left( \frac{H_0}{k} \right) \left( \frac{3}{2}{\mathcal {K}}_3 -2i \left( \frac{H_0}{k}\right) {\mathcal {K}}_5 \right) \right. \nonumber \\{} & {} \left. +\left( \frac{H_0}{k}\right) ^2 \frac{3{\mathcal {K}}_6}{{(1 + {\hat{n}} \cdot {\hat{k}})^3}} +{\mathcal {O}}\left( \left( \frac{H_0}{k} \right) ^3 \right) \right) \Bigg \}, \nonumber \\ \end{aligned}$$

(34)

and

$$\begin{aligned} {\mathcal {K}}_1\equiv & {} 1+ (H_0 \eta )^3 e^{-\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})}~,\end{aligned}$$

(35a)

$$\begin{aligned} {\mathcal {K}}_2\equiv & {} 1+ \frac{(H_0 \eta -4)(H_0 \eta )^2}{6} e^{-\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})}\nonumber \\{} & {} + \frac{(H_0\eta )^3}{6} e^{\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})}~,\end{aligned}$$

(35b)

$$\begin{aligned} {\mathcal {K}}_3\equiv & {} 1 - (H_0 \eta )^2 e^{-\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})} ~,\end{aligned}$$

(35c)

$$\begin{aligned} {\mathcal {K}}_4\equiv & {} 1 + \frac{H_0\eta (H_0\eta +2)}{2} e^{-\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})}\nonumber \\{} & {} - \frac{1}{2} (H_0 \eta )^2 e^{\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})} ~, \end{aligned}$$

(35d)

$$\begin{aligned} {\mathcal {K}}_5\equiv & {} 1 +\frac{H_0\eta (8-3H_0\eta )}{8} e^{-\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})}\nonumber \\{} & {} + \frac{3}{8}(H_0\eta )^2 e^{\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})}~,\end{aligned}$$

(35e)

$$\begin{aligned} {\mathcal {K}}_6\equiv & {} 1 + H_0\eta e^{-\frac{i k}{H_0}(1+H_0\eta )(1-{\hat{n}}\cdot {\hat{k}})} \end{aligned}$$

(35f)

Here, the redshifts fluctuation $\varDelta z$ is proportional to the tensor perturbation ${\bar{h}}_{ab,\varvec{k}}$. As shown in Eq. (33), we also limit the calculation to the leading order effects of $H_0/k$.

The physical information of stochastic signals of GWBs can be extracted by using cross-correlation functions for the redshift fluctuation from distant clocks $\alpha$ and $\beta$, which ares given by

$$\begin{aligned}{} & {} \langle \varDelta z_{\alpha } \varDelta z_{\beta } \rangle = \int \frac{\textrm{d}^3k}{(2 \pi )^3} \int \frac{\textrm{d}^3 k'}{(2 \pi )^3}\nonumber \\{} & {} \qquad \times \left\{ \left\langle {\bar{h}}_{a b, \varvec{k}}^{*} {\bar{h}}_{c d,\varvec{k}'} \right\rangle {\hat{n}}^a_{\alpha } {\hat{n}}^b_{\alpha }{\hat{n}}^c_{\beta } {\hat{n}}^c_{\beta } F({\hat{n}}_\alpha ,{\varvec{k}},\eta )F({\hat{n}}_\beta ,{\varvec{k}}',\eta ) \right\} \nonumber \\{} & {} \quad = \int 4 \pi f^2 \textrm{d} f P_h (f) \int \frac{\textrm{d} \varOmega }{4 \pi } \left\{ \varLambda _{a b, c d} ({\hat{f}}) {\hat{n}}^a_{\alpha } {\hat{n}}^b_{\alpha } {\hat{n}}^c_{\beta } {\hat{n}}^c_{\beta } \right. \nonumber \\{} & {} \qquad \times \Bigg ( \frac{{\mathcal {K}}_{1, \alpha }^* {\mathcal {K}}_{1, \beta }}{4(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})} \nonumber \\{} & {} \left. \left. \qquad + \left( \frac{H_0}{k} \right) ^2 \left( \frac{\frac{9}{4} {\mathcal {K}}_{2, \alpha }^* {\mathcal {K}}_{2, \beta } + \frac{1}{2} {\mathcal {K}}_{1, \alpha }^* {\mathcal {K}}_{4, \beta } + \frac{1}{2} {\mathcal {K}}_{4, \alpha }^* {\mathcal {K}}_{1, \beta }}{(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})}\right. \right. \right. \nonumber \\{} & {} \left. \left. \left. \qquad + \frac{{\mathcal {K}}_{1, \alpha }^* {\mathcal {K}}_{5, \beta } - \frac{9}{4} {\mathcal {K}}_{2, \alpha }^* {\mathcal {K}}_{3, \beta }}{(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})^2} \right. \right. \right. \nonumber \\{} & {} \left. \left. \left. \qquad + \frac{{\mathcal {K}}_{5, \alpha }^* {\mathcal {K}}_{1, \beta } - \frac{9}{4} {\mathcal {K}}_{3, \alpha }^* {\mathcal {K}}_{2, \beta }}{(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}})^2 (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})} + \frac{9{\mathcal {K}}_{3, \alpha }^* {\mathcal {K}}_{3, \beta }}{4 (1 + {\hat{n}}_{\alpha } \cdot {\hat{k}})^2 (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})^2} \right. \right. \right. \nonumber \\{} & {} \left. \left. \left. \qquad - \frac{3{\mathcal {K}}_{1, \alpha }^* {\mathcal {K}}_{6, \beta }}{2 (1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})^3} - \frac{3{\mathcal {K}}_{6, \alpha }^* {\mathcal {K}}_{1, \beta }}{2 (1 + {\hat{n}}_{\alpha } \cdot {\hat{k}})^3 (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})} \right) \right. \right. \nonumber \\{} & {} \left. \left. \qquad +{\mathcal {O}}\left( \left( \frac{H_0}{k} \right) ^3 \right) \right) \right\} , \end{aligned}$$

(36)

where the $\textrm{d} \varOmega$ is surface element with respect to $\varvec{k}$, the $\theta _{\alpha \beta }$ is angular distance between the distant clocks $\alpha$ and $\beta$, the f is present-day physical frequency defined with $f \equiv (2 \pi )^{- 1} k$ and ${\hat{f}} \equiv {\hat{k}}$ [70], the transverse-traceless operator is given by

$$\begin{aligned} \varLambda _{a b, c d}\equiv & {} {\mathcal {T}}_{a c} {\mathcal {T}}_{b d} -{\mathcal {T}}_{a b} {\mathcal {T}}_{c d} +{\mathcal {T}}_{a d} {\mathcal {T}}_{b c}~, \end{aligned}$$

(37)

and above transverse operator ${\mathcal {T}}_{a c}$ is defined with

$$\begin{aligned} {\mathcal {T}}_{a b} ({\hat{f}})= & {} \delta _{a b} - {\hat{f}}_a {\hat{f}}_b~. \end{aligned}$$

(38)

Here, we have adopted homogeneous, isotropic and unpolarized GWs. And the two-point correlation functions for ${\bar{h}}_{a b,\varvec{k}}$ can be evaluated to be

$$\begin{aligned} \left\langle {\bar{h}}_{a b, \varvec{k}}^{*} {\bar{h}}_{c d, \varvec{k}'} \right\rangle= & {} e^{\lambda }_{a b} \left( \varvec{k} \right) e^{\lambda '}_{c d} \left( \varvec{k}' \right) \left\langle {\bar{h}}_{\lambda , \varvec{k}}^{*} {\bar{h}}_{\lambda ', \varvec{k}} \right\rangle \nonumber \\= & {} e^{\lambda }_{a b} \left( \varvec{k} \right) e^{\lambda '}_{c d} \left( \varvec{k}' \right) (2 \pi )^3 \delta \left( \varvec{k} - \varvec{k}' \right) \delta _{\lambda \lambda '} P_h (k)\nonumber \\= & {} (2 \pi )^3 \varLambda _{a b, c d} ({\hat{k}}) \delta \left( \varvec{k} - \varvec{k}' \right) P_h (k)~. \end{aligned}$$

(39)

In the regime $k(1/H_0+ \eta ) > 10$, one can take oscillation average that gives ${\mathcal {K}}^*_{i,\alpha } {\mathcal {K}}_{i,\beta }\rightarrow 1$. We thus neglect the “pulsar terms” from the oscillated parts of ${\mathcal {K}}_{i,\alpha }$ in Eq. (36). Therefore, we can read the ORFs in the form of

$$\begin{aligned}&\varGamma (\theta _{\alpha \beta },f) = \int \frac{\textrm{d} \varOmega }{4 \pi } \left\{ \varLambda _{a b, c d} ({\hat{f}}) {\hat{n}}^a_{\alpha } {\hat{n}}^b_{\alpha } {\hat{n}}^c_{\beta } {\hat{n}}^c_{\beta } \right. \Bigg ( \frac{1}{4(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})} \nonumber \\&\left. \left. \quad + \left( \frac{H_0}{k} \right) ^2 \left( \frac{13}{4(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})}\right. \right. \right. \nonumber \\&\quad \left. \left. \left. - \frac{5}{4(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})^2} - \frac{5}{4(1 + {\hat{n}}_{\alpha } \cdot {\hat{k}})^2 (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})} \right. \right. \right. \nonumber \\&\left. \left. \left. \quad + \frac{9}{4 (1 + {\hat{n}}_{\alpha } \cdot {\hat{k}})^2 (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})^2} - \frac{3}{2 (1 + {\hat{n}}_{\alpha } \cdot {\hat{k}}) (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})^3} \right. \right. \right. \nonumber \\&\quad \left. \left. \left. \quad - \frac{3}{2 (1 + {\hat{n}}_{\alpha } \cdot {\hat{k}})^3 (1 + {\hat{n}}_{\beta } \cdot {\hat{k}})} \right) \right) \right\} ~. \end{aligned}$$

(40)

In order to obtain $\varGamma (\theta _{\alpha \beta })$, we can let direction of propagation of GWs as

$$\begin{aligned} {\hat{f}}= & {} (\sin \theta \cos \phi , \sin \theta \sin \phi , \cos \theta )~, \end{aligned}$$

(41)

where the angular coordinate $\theta$, $\phi$ is defined with respect to $\varvec{k}$, and $\textrm{d} \varOmega \equiv \sin \theta \textrm{d} \theta \textrm{d} \phi$. Since the angular $\theta _{\alpha \beta }$ can be formulated by ${\hat{n}}_{\alpha }\cdot {\hat{n}}_{\beta } = \cos \theta _{\alpha \beta }$, the locations of the distant clocks $\alpha$, and $\beta$ can be

$$\begin{aligned} {\hat{n}}_{\alpha }= & {} (0, 0, 1)~,\end{aligned}$$

(42a)

$$\begin{aligned} {\hat{n}}_{\beta }= & {} (\sin \theta _{\alpha \beta }, 0, \cos \theta _{\alpha \beta })~. \end{aligned}$$

(42b)

From Eq. (40), the leading order effect from the expansion of the Universe is proportional to $(H_0/f)^2$ for the ORFs. By making use of the expression of the ${\hat{f}}$ in Eq. (41), the transverse-traceless operator acting on ${\hat{n}}_{\alpha }$ and ${\hat{n}}_{\beta }$ in Eq. (40) can be evaluated,

$$\begin{aligned} \varLambda _{a b, c d} {\hat{n}}^a_{\alpha } {\hat{n}}^b_{\alpha } {\hat{n}}^c_{\beta } {\hat{n}}_{\beta }^d= & {} \frac{1}{4} \sin ^2 \theta \Big ((3 + \cos (2 \theta )) \cos (2 \phi ) \sin ^2 \theta _{\alpha \beta } \nonumber \\{} & {} + \sin \theta \big (\sin \theta + 3 \cos (2 \theta _{\alpha \beta }) \sin \theta \nonumber \\ {}{} & {} - 4 \cos \theta \cos \phi \sin (2 \theta _{\alpha \beta })\big )\Big )~, \end{aligned}$$

(43)

and

$$\begin{aligned} {\hat{n}}_{\alpha } \cdot {\hat{f}}= & {} \cos \theta ~,\end{aligned}$$

(44a)

$$\begin{aligned} {\hat{n}}_{\beta } \cdot {\hat{f}}= & {} \cos \theta \cos \phi \sin \theta _{\alpha \beta } + \cos \theta \cos \theta _{\alpha \beta }~. \end{aligned}$$

(44b)

In Fig. 2, we present the ORFs for different values of $H_0 / (2\pi f)$. It shows that the cosmological constant could enhance the value of the magnitude of ORFs in the case of $\theta _{\alpha \beta } \rightarrow 0$. Similar behavior was also found from a careful calculation on the pulsar terms [61, 63]. In Fig. 3, we zoom in the angular correlation curves for small angle $\theta _{\alpha \beta }$. For $n\text {Hz}$ GWB in the PTA band, the $(H_0/k)^2$ is estimated to be $10^{-19}$. In this case, the enhanced values of the ORFs are shown to be dominated for the pulsars pairs that are separated by angular distance less than mas. This conclusions can be numerically presented by $\varGamma \left( \theta _{\alpha \beta } \rightarrow 0 \right) \propto \theta _{\alpha \beta }^{-2}$.

Fig. 2

ORFs of GW for different values of $H_0 / (2\pi f)$. The solid curve represents the Hellings–Downs curve

Fig. 3

ORFs of GW for different values of $H_0 / (2\pi f)$ in small $\theta _{\alpha \beta }$

6 Conclusions and discussions

We investigated the ORFs in de-Sitter background for one-way tracking GW detectors. It was found that the cosmological constant leads to a much larger value of magnitude of the ORFs, when GW detector pairs are close to each other. For nanohertz gravitational waves, this effect can dominate of value of ORFs when the galactic pulsar pairs are separated by milliarcsecond. We presented the ORFs in de-Sitter background for the first time. For GW detections in low frequency band in the future, it is inevitable to be confront with the effect from the expansion of the Universe, such as the gravitational wave timing array constituted by distant binaries [71].

From the comparison between Eqs. (21) and (22), calculation on the ORFs in de-Sitter background is shown to be non-trivial, because spatial components of perturbed 4-velocities of light also have effect on the frequency fluctuation of light rays from the GW detectors. Thus, one can not account for the difference of the ORFs between those in an expansion Universe and Minkowski space-time from simply a redshift factor $1+z$.

The GW detectors are set to be co-moving with the expansion of the Universe, which is not suited for describing PTA, because the motion of a pulsar is dominated by gravitational field in the galaxy. Fortunately, in our formalism, one can take the local gravitational fields into considerations by giving physical 4-velocities in Eq. (31). It seems obvious that the geometric factors of PTAs should contain physical information about the motions of the composed pulsars.

It would be confusing that the values for the magnitude of the ORFs in de-Sitter background is shown to be divergent as $\theta _{\alpha \beta } \rightarrow 0$. It indicates that the auto-correlations of the output of GW detectors should be divergent. In this sense, difficulties might exist in estimation of the sensitivity of GW detectors.

Appendices

Appendix A: Polarization tensor

In principle, there is an additional the degree of freedom in polarization plane, which should be averaged [59, 69]. In the present paper, we present a consistent derivation by using the relation $\varLambda _{a b, c d} = e^{\lambda }_{a b} e_{c d}^{\lambda }$ in Eq. (39). Here, we will show that this relation can be reproduced from a non-specific choice of polarization vectors. Firstly, we express the transverse operator ${\mathcal {T}}_{a b}$ in the form of

$$\begin{aligned} {\mathcal {T}}_{a b}= & {} ({\hat{f}}_a {\hat{f}}_b + e_a e_b + {\bar{e}}_a {\bar{e}}_b) - {\hat{f}}_a {\hat{f}}_b\end{aligned}$$

(A.1)

$$\begin{aligned}= & {} e_a e_b + {\bar{e}}_a {\bar{e}}_b \end{aligned}$$

(A.2)

where the $e_a$ and ${\bar{e}}_a$ are the two unit polarization vector with respect to the ${\hat{f}}$. And the the transverse-traceless tensor reduces to be

$$\begin{aligned} \varLambda _{a b, c d}= & {} {\mathcal {T}}_{a c} {\mathcal {T}}_{b d} -{\mathcal {T}}_{a b} {\mathcal {T}}_{c d} +{\mathcal {T}}_{a d} {\mathcal {T}}_{b c} \nonumber \\= & {} (e_a e_c + {\bar{e}}_a {\bar{e}}_c) (e_b e_d + {\bar{e}}_b {\bar{e}}_d) \nonumber \\ {}{} & {} - (e_a e_b + {\bar{e}}_a {\bar{e}}_b) (e_c e_b + {\bar{e}}_c {\bar{e}}_b) \nonumber \\ {}{} & {} + (e_a e_d + {\bar{e}}_a {\bar{e}}_d) (e_b e_c + {\bar{e}}_b {\bar{e}}_c)\nonumber \\= & {} (e_a e_b - {\bar{e}}_a {\bar{e}}_b) (e_c e_d - {\bar{e}}_c {\bar{e}}_d) \nonumber \\ {}{} & {} + (e_a {\bar{e}}_b + {\bar{e}}_a e_b) (e_c {\bar{e}}_d + {\bar{e}}_c e_d) \end{aligned}$$

(A.3)

By introducing

$$\begin{aligned} e^+_{a b}\equiv & {} e_a e_b - {\bar{e}}_a {\bar{e}}_b\end{aligned}$$

(A.4)

$$\begin{aligned} e^{\times }_{a b}\equiv & {} e_a {\bar{e}}_b + {\bar{e}}_a e_b \end{aligned}$$

(A.5)

One can reproduce the relation $\varLambda _{a b, c d} = e^{\lambda }_{a b} e_{c d}^{\lambda }$ used in Eq. (39).

Appendix B: Perturbed geodesic equations

The geodesic equations are given by

$$\begin{aligned} \nabla _p p^{\mu }= & {} 0~, \end{aligned}$$

(B.6)

Based on the expansion $p^{\mu } \rightarrow {p}^{\mu } + \delta p^{\mu }$, and ${g}_{\mu \nu }\rightarrow g_{\mu \nu } + \delta g_{\mu \nu }$, we have

$$\begin{aligned} p^{\mu } \nabla _{\mu } p^{\nu }= & {} ( {p}^{\mu } + \delta p^{\mu }) (\partial _{\mu } ( {p}^{\nu } + \delta p^{\nu }) + \varGamma _{\mu \lambda }^{\nu } ( {p}^{\lambda } + \delta p^{\lambda }))\nonumber \\= & {} {p}^{\mu } {\nabla }_{\mu } {p}^{\nu } + \delta p^{\mu } {\nabla }_{\mu } {p}^{\nu } + {p}^{\mu } {\nabla }_{\mu } \delta p^{\nu } \nonumber \\ {}{} & {} + g^{\nu \rho } \left( 2 \nabla _{\mu } \delta g_{\lambda \rho } - \frac{1}{2} \nabla _{\rho } \delta g_{\mu \lambda } \right) {p}^{\mu } {p}^{\lambda }~, \end{aligned}$$

(B.7)

in which, we have used the expansion of Christoffel symbols,

$$\begin{aligned} \varGamma ^{\nu }_{\mu \lambda }= & {} \frac{1}{2} g^{\nu \rho } (\partial _{\mu } g_{\lambda \rho } + \partial _{\lambda } g_{\mu \rho } - \partial _{\rho } g_{\mu \lambda })\nonumber \\= & {} \varGamma ^{\nu }_{\mu \lambda } + \frac{1}{2} h^{\nu \rho } (\partial _{\mu } g_{\lambda \rho } + \partial _{\lambda } g_{\mu \rho } - \partial _{\rho } g_{\mu \lambda }) \nonumber \\{} & {} + \frac{1}{2} g^{\nu \rho } (\nabla _{\mu } \delta g_{\lambda \rho } + \varGamma _{\mu \lambda }^{\kappa } \delta g_{\kappa \rho } + \varGamma ^{\kappa }_{\mu \rho } \delta g_{\lambda \kappa } \nonumber \\ {}{} & {} + \nabla _{\lambda } \delta g_{\mu \rho } + \varGamma ^{\kappa }_{\lambda \mu } \delta g_{\kappa \rho } + \varGamma ^{\kappa }_{\lambda \rho } \delta g_{\mu \kappa } \nonumber \\ {}{} & {} - \nabla _{\rho } \delta g_{\mu \lambda } - \varGamma _{\rho \mu }^{\kappa } \delta g_{\kappa \lambda } - \varGamma ^{\kappa }_{\rho \lambda } \delta g_{\mu \kappa })\nonumber \\= & {} \varGamma ^{\nu }_{\lambda } + \frac{1}{2} g^{\nu \rho } (\nabla _{\mu } \delta g_{\lambda \rho } + \nabla _{\lambda } \delta g_{\mu \rho } - \nabla _{\rho } \delta g_{\mu \lambda })~, \nonumber \\ \end{aligned}$$

(B.8)

The $\nabla _\mu$ is covariant derivative with respect to background metric $g_{\mu \nu }$.

From Eq. (B.7), we obtain the first order geodesic equations,

$$\begin{aligned} {p}^{\mu } {\nabla }_{\mu } {p}^{\nu }= & {} 0~, \end{aligned}$$

(B.9)

and the second order geodesic equations,

$$\begin{aligned}{} & {} \delta p^{\mu } {\nabla }_{\mu } {p}^{\nu } + {p}^{\mu } {\nabla }_{\mu } \delta p^{\nu } \nonumber \\ {}{} & {} \quad + g^{\nu \rho } \left( \nabla _{\mu } \delta g_{\lambda \rho } - \frac{1}{2} \nabla _{\rho } \delta g_{\mu \lambda } \right) {p}^{\mu } {p}^{\lambda } = 0~. \end{aligned}$$

(B.10)