Gravitational wave luminosity distance for Starobinsky gravity in viscous cosmological models
Authors: Chi Jia-nan , Wu Ya-Bo , Sang Yu , Zhang Xue , Hu Yi-liang , Liu Wen-zhong , Chen Bo-hai 
Published: January 2025

Abstract
In this paper, in the Starobinsky gravity (SG) model, the gravitational wave (GW) luminosity distance

$$d_L^{GW}(z)$$

d
L

GW

(
z
)

is analytically and numerically investigated by considering the dark matter as the two different kinds of shear viscous fluids in universe. Concretely, on the basis of the propagation equation of the metric perturbation derived by using transverse-traceless gauge for the linear perturbation of the metric in FRW background, we obtain the analytical expressions of the modified friction term

$$\delta (z)$$

δ
(
z
)

and the ratio

$$d_L^{GW}(z)/d_L^{EM}(z)$$

d
L

GW

(
z
)

/

d
L

EM

(
z
)

of GW to electromagnetic (EM) wave luminosity distance, and find that they only depend on the SG model parameter

$$\alpha $$

α

and the shear viscous coefficient

$$\eta $$

η

, regardless of the bulk viscous coefficient

$$\zeta $$

ζ

. Evidently, both of them are affected by

$$\alpha $$

α

and

$$\eta $$

η

. It worth stressing that the results given by us can reduce to ones in documents [13, 22, 30]. It follows that we can extend the previous results as the special cases in our model. Furthermore, by combining the latest observational data, we more tightly constrain the values of related parameters and numerically analyze the influence of both

$$\alpha $$

α

and

$$\eta $$

η

on

$$\delta (z)$$

δ
(
z
)

and

$$d_L^{GW}(z)/d_L^{EM}(z)$$

d
L

GW

(
z
)

/

d
L

EM

(
z
)

. We observe that, in the range of

$$0< z\le 8.2$$

0
<
z
≤
8.2

, the evolutionary curves of

$$\delta (z)$$

δ
(
z
)

and

$$d_L^{GW}(z)/d_L^{EM}(z)$$

d
L

GW

(
z
)

/

d
L

EM

(
z
)

exhibit that

$$\delta (z)$$

δ
(
z
)

(

$$d_L^{GW}(z)/d_L^{EM}(z)$$

d
L

GW

(
z
)

/

d
L

EM

(
z
)

) monotonically decrease (increase) as redshift rises, moreover there are always

$$\delta (z)<0$$

δ
(
z
)
<
0

and

$$d_L^{GW}(z)\ge d_L^{EM}(z)$$

d
L

GW

(
z
)

≥

d
L

EM

(
z
)

, specially

$$d_L^{GW}(0)=d_L^{EM}(0)$$

d
L

GW

(
0
)

=

d
L

EM

(
0
)

in the limit of

$$z\rightarrow 0$$

z
→
0

, which means that detecting GW signals might represent a more efficient tool than detecting EM signals to test modified gravity model on the given scale of cosmic distances. It is worth noting that the numerical analyses to

$$\delta (z)$$

δ
(
z
)

and

$$d_L^{GW}(z)/d_L^{EM}(z)$$

d
L

GW

(
z
)

/

d
L

EM

(
z
)

illustrate once again that the GW luminosity distance

$$d_L^{GW}(z)$$

d
L

GW

(
z
)

is indeed influenced by

$$\alpha $$

α

and

$$\eta $$

η

, in particular more sensitive to

$$\alpha $$

α

than

$$\eta $$

η

in our model. In addition, by comparing the GW and EM luminosity distances of SG model with

$$d_{L(GR)}$$

d

L
(
G
R
)

of Einstein GR, we find that the GW and EM signals in SG1 and SG2 change from stronger to weaker than the ones in GR as redshift increases and more difficultly to be detected in the same distance and detection sensitivity. Meanwhile, the EM signals change from strong to weak earlier than GW signals do. Finally, we confirm that

$$\delta (z)$$

δ
(
z
)

and

$$d_L^{GW}(z)$$

d
L

GW

(
z
)

of our model meet the constraints imposed by the standard siren GW170817 and its electromagnetic companion GRB170817A.