# Greybody factors for Spherically Symmetric Einstein-Gauss-Bonnet-de Sitter black hole

###### Abstract

We study the greybody factors of the scalar fields in spherically symmetric Einstein-Gauss-Bonnet-de Sitter black holes in higher dimensions. We derive the greybody factors analytically for both minimally and non-minimally coupled scalar fields. Moreover, we discuss the dependence of the greybody factor on various parameters including the angular momentum number, the non-minimally coupling constant, the spacetime dimension, the cosmological constant and the Gauss-Bonnet coefficient in detail. We find that the non-minimal coupling may suppress the Hawking radiation, while the Gauss-Bonnet coupling could enhance it.

1. Center for High Energy Physics, Peking University, 5 Yiheyuan Road, Beijing 100871, China

2. Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, 5 Yiheyuan Road, Beijing 100871, China

3. Collaborative Innovation Center of Quantum Matter, 5 Yiheyuan Road, Beijing 100871, China

## 1 Introduction

The black holes obey the laws similar to the ones of thermodynamics [1]. This inspires the pioneer works on the thermal radiation of the black hole, the so-called Hawking radiation [2, 3]. In the vicinity of black hole event horizon, the Hawking radiation is a blackbody radiation. Yet it becomes a greybody radiation since it has to transverse an effective potential barrier to reach the asymptotic far region. This effective potential barrier is highly sensitive to the structure of the black hole background. As a result, the Hawking radiation can give us important information about the black hole parameters, including the mass, the charge and the angular momentum of the black hole. In general, the Hawking radiation of macroscopic black holes is too small to be detected. However, the microscopic black holes can evaporate through the emission of the Hawking radiation [4]. Especially the existence of extra spacelike dimensions [5, 6, 7, 8] indicates that the tiny black hole may be created at the particle colliders [9, 10, 11, 12, 13] or in high energy cosmic-ray interactions [14, 15, 16, 17]. The associated Hawking radiation maybe observed at the TeV scale. Significant number of works about the Hawking radiation in higher dimensional spacetime have been done. For more extensive references one may consult the reviews [18, 19, 20, 21].

In asymptotic flat spacetimes, it has been found that the greybody factors for the waves of arbitrary spin and angular quantum number in any dimensions vanish in the zero-frequency limit [22, 23, 24], even for the non-minimally coupled scalar [25]. In the presence of a positive cosmological constant, the picture is different. The greybody factors of the Schwarzschild-de Sitter (SdS) black holes were studied both analytically and numerically in -dimension in [26], and it was found that the greybody factor is not vanishing even in the zero-frequency limit for a minimally coupled massless scalar. This implies that the cosmological constant has an important effect on the greybody factor. In fact, the non-vanishing greybody factor was first observed in [27]. This phenomenon is due to the fully delocalization of the zero-modes. It therefore has a finite probability to transverse the region between the event horizon and the cosmological horizon [26]. The mass of the scalar or a non-minimally coupling constant breaks this relation. Hence the greybody factors for arbitrary non-minimally coupled scalar partial modes in 4-dimensional spacetime tend to zero in the infrared limit [28].

In this paper, we consider the spherically symmetric dS black hole in the Einstein-Gauss-Bonnet
(EGB) gravity^{1}^{1}1In the following, we simply call such solution the EGB-dS black hole. . The EGB gravity is a special case of the Lovelock gravity which
is the natural generalization of general relativity to higher dimensions
[29]. As the most general metric theory of gravity
yielding second order equations of motion, the Lovelock gravity
is ghost free and thus is especially attractive in the higher-derivative gravity
theories. Among the Lovelock gravity theories, the simplest one is the EGB gravity, which includes a four-derivative Gauss-Bonnet (GB) term to the Einstein-Hilbert action

(1.1) |

Here is the Gauss-Bonnet coupling constant of dimension and is the Ricci scalar. The Gauss-Bonnet coupling term appears in the low energy effective action of the heterotic string theory[30]. There the coupling constant is positive definite and inversely proportional to the string tension. Hence in this work we restrict the case that . is the -dimensional Newton’s constant. The Gauss-Bonnet term is jut a topological surface term in four dimensional spacetime and becomes nontrivial in spacetimes. It has been pointed out that if the Planck scale is of order TeV, as suggested in some extra dimension models, the coupling constant could be measured by LHC through the detection of the Hawking radiation spectrum of the black hole [31]. Thus it is worth studying the greybody factor of the Hawking radiation of the GB black hole, from both theoretical viewpoint and phenomenological purposes. For scalar and graviton emissions, the numerical studies of the GB black hole in an asymptotic flat spacetime were carried out in [32, 33]. As mentioned in the last paragraph, the positive cosmological constant has significant effect on the greybody factor. In this paper we would like to compute the greybody factor of the Hawking radiation of the EGB-dS black hole analytically and discuss the effects of various parameters, especially the GB coupling constant, on the radiation.

The analytical study of the greybody factor in the SdS black hole has been well-developed. The analytical study in [26] was limited to the case of the lowest partial mode () and the low energy part () of the spectrum. A general expression for the greybody factor for arbitrary partial modes of a minimally or non-minimally coupled scalar in higher-dimensional SdS black hole was derived in [34]. The authors in [34] found an appropriate radial coordinate that allows them to integrate the field equations analytically and avoid the approximations on the metric tensor used in [20, 28]. The comparison of the analytical result with the numerical result was done in [35]. For more recent studies, see [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 49]. Adopting a similar radial coordinate, we are able to derive the analytical results for the greybody factors for arbitrary partial modes of a scalar field in the EGB-dS black hole spacetime as well.

In section 2, we give the general background of the EGB-dS black hole and the corresponding equation of motion for the scalar field. In section 3, we derive the analytical expression of the greybody factor using the matching method and discuss its low energy limit. In section 4, we analyze the effects of various parameters on the greybody factor. We end with the conclusion and discussion in section 5.

## 2 Background

The metric for a spherically symmetric Einstein-Gauss-Bonnet-de Sitter black hole in -dimensional spacetime is given by [30]

(2.1) | ||||

The parameter is related to the mass of the black hole by . In terms of the horizon radius , can be expressed as

(2.2) |

Here is related to the GB coupling constant by . In the limit , the metric returns to that of the SdS black hole. GB constant has significant effect on the stability of GB black holes. Through perturbation analysis, it was found that the EGB-dS black holes are unstable in certain parameter region. In our discussions, the parameters are restricted in the stable region given in [46, 47, 48] and will be chosen such that the spacetime always has two horizons, the black hole horizon and the cosmological horizon .

We consider a general scalar field coupled to the the gravity non-minimally

(2.3) |

Here is the non-minimally coupling constant with corresponding to the minimally coupled case. The equation of motion of the scalar field has the form

(2.4) |

In a spherically symmetric background, we may make ansatz

(2.5) |

where are spherical harmonics on . Then the angular part and the radial part are decoupled such that the radial equation becomes

(2.6) |

Introducing , we get

(2.7) |

where is the tortoise coordinate defined by . The effective potential reads

(2.8) |

It is obvious that the effective potential vanishes at the two horizons. Its height increases with the angular momentum number . Fixing the black hole horizon , we can study the profile of the effective potential in terms of the angular momentum number , the spacetime dimension , the scalar coupling constant , the cosmological constant and the GB coupling constant .

## 3 Greybody factor

The radial equation (2.6) can not be solved analytically over the whole space region. However, to read the greybody factor, it is not necessary to solve the equation exactly. Instead, one can solve the equation in two regions separately, namely near the black hole horizon and the cosmological horizon regions, and then paste the solutions in the intermediate region. In this procedure, the effect of the cosmological constant should be put under control in order to make the result as accurate as possible [34].

### 3.1 Near the event horizon

In the near event horizon region , similar to the case of SdS, we perform the following transformation

(3.1) |

The new variable ranges from 0 to as runs from to the region . Its derivative satisfies

(3.2) |

with

(3.3) |

in which the mass can be expressed as

(3.4) |

When , it returns to the case of the SdS black hole, namely .

Using the new variable, the radial equation near the even horizon becomes

(3.5) |

in which

(3.6) |

where and is the Ricci scalar on the event horizon. In the derivation of this equation we have used the approximation

(3.7) |

near the event horizon .
The reason is that the solution of the original radial equation has cusps due to the poles of Gamma function, the unphysical behavior can be avoided by using this approximation^{2}^{2}2We thank Pappas and Kanti for their correspondences on this point.
.

This is in fact a Fuchsian equation with three singularities . To be clearer, make a redefinition , Eq.(3.5) becomes

(3.8) |

in which the coefficients are given by

(3.9) |

The solution of the differential equation (3.8) is the standard hypergeometric function with parameters being

(3.10) | ||||

Considering the relation between and , near the event horizon the radial function has the following form

where are the constant coefficients. Near the event horizon,

(3.11) |

Imposing the ingoing boundary condition near the event horizon and choosing , we should set . Furthermore, the convergence of the hypergeometric function requires the real part . Thus we have to take the branch of . In the end, the solution near the event horizon is of the form

(3.12) |

### 3.2 Near the cosmological horizon

The solution in the near cosmological horizon region can be solved similarly. The function in the metric can be approximated by [20, 28, 34]

(3.13) |

ranges from 0, at , to 1 as . In the above approximation, the larger or the smaller leads to more accurate results. The approximation also becomes more accurate for a larger spacetime dimension .

Making the change of variable , near the cosmological horizon, we have

(3.14) |

where is the Ricci scalar at . After a replacement , we get

(3.15) |

in which

(3.16) |

The solution of the differential equation (3.14) could be written in terms of the hypergeometric functions as well. Therefore, around the cosmological horizon, the radial equation can be solved by

(3.17) |

with the parameters

(3.18) | ||||

Here are constant coefficients. The convergence of the hypergeometric function requires such that we have to take .

Since the effective potential vanishes at , the solution is expected to be comprised of the plane waves. Indeed, we have

(3.19) |

where

(3.20) |

### 3.3 Matching the solutions in the intermediate region

Now we have the asymptotic solutions in the near event horizon region and the near cosmological horizon region. In order to complete the solution, we must ensure that the two asymptotic solutions, and can be smoothly pasted at the intermediate region.

#### 3.3.1 Black hole horizon

First let us consider the near black hole horizon solution. Due to the fact that in the intermediate region , the variable , we can use the following relation for the hypergeometric function

(3.21) | ||||

to shift the argument from to . For simplicity we consider the case . Then in the region where , we have This is reasonable only if . For , from (3.1) we have

(3.22) |

Then the Ricci scalar . Thus if is not too big, the term and can be omitted. Therefore, we have .

Now we have

(3.23) |

In the intermediate region , the solution (3.12) can be expanded into the form

(3.24) |

where

(3.25) | ||||

Note that the aforementioned approximations are applicable only for the expressions involving the factor and not for the parameters in the Gamma function to increase the validity of the analytical results [34].

#### 3.3.2 Cosmological horizon

Now let us turn to the solution near the cosmological horizon. Similar to the treatment above, we may shift the argument of the hypergeometric function from to since for the intermediate region . We still work with a small cosmological constant. In the region where , we have

(3.26) |

and . Following the similar procedure, we get

(3.27) |

where

(3.28) | ||||

It is obvious that solutions (3.24) and (3.27) have the same power-law. Identifying the coefficients of the same powers of in (3.24) and (3.27), we get the relations

(3.29) |

Solving the constraints and plugging them into the expression for the greybody factor for the emission of scalar fields by a higher dimensional EGB-dS black hole, we get

(3.30) |

This expression takes the same form as that for the Einstein gravity [34]. But due to the differences among the explicit expressions of s, it depends not only on the cosmological constant and the non-minimal coupling , but also on the GB coupling constant .

As mentioned in [34], the greybody factor (3.30) is more accurate for a smaller cosmological constant and a larger distance between and . On the other hand, we do not make any assumption on the energy in the approximation. This is in contrast with all the previous similar matching procedure which always assume the low energy condition. Thus our analytical result can be valid beyond the low energy regime.

### 3.4 Low energy limit

Before we analyze the effects of various parameters on the greybody factor, we derive the low energy limit of the greybody factor in this subsection.

#### 3.4.1 Minimal coupling and dominant mode

Let us consider the minimally coupling case and the dominant mode first. In this case, we obtain

(3.31) | ||||

where

(3.32) | ||||

Then the greybody factor becomes

(3.33) |

Thus the scalar particle with very low energy has a non-vanishing probability of being emitted by a high dimensional EGB-dS black hole. This is in fact a characteristic feature of the propagation of free massless scalar in the dS spacetime. However, the GB term changes the value of the greybody factor. For instance, for a small , up to the first order of ,

(3.34) |

We see that increases the greybody factor of massless scalar in the EGB-dS black hole background. When , we reproduce the low energy greybody factor for the mode , in accordance to the previous higher dimensional analysis [20, 26, 34].

#### 3.4.2 Non-minimally coupling case

Now we calculate the low energy greybody factor for a non-minimally coupled scalar. In this case, we can expand the the combinations in the low energy limit as

(3.35) | |||||

in which are the expansion coefficients, whose explicit expressions are lengthy will not be given here. The final result for the greybody factor turns out to be

(3.36) |

in which and

(3.37) |

and

Note that the first non-vanishing term in the low energy expansion is of order . This holds for all partial waves including the dominant mode . Therefore, there is no mode with a non-vanishing low energy greybody factor for the non-minimally coupled scalar. This has a simple explanation: from the equation of motion for the non-minimally coupled scalar, we see that the coupling constant plays a role of an effective mass for the scalar and breaks the infrared enhancement, as mentioned in the introduction.

## 4 The effects of various parameters

There are several parameters in the theory which influence the greybody factor for the non-minimally coupled scalar propagating in the EGB-dS black hole spacetime. These parameters include the non-minimally coupling constant , angular momentum number , the spacetime dimension , the cosmological constant and the GB coupling constant . In fact, the parameters have the similar effects on the greybody factor of the EGB-dS black hole as they have for that of the SdS black hole. Therefore, we focus on the effect of the GB coupling constant on the greybody factor. To analyze their effects more clearly, we plot the dependence of the greybody factor on these parameters and the corresponding effective potentials in the following.

### 4.1 The case

For the purpose of comparison, we produce Fig.1 to show that our results agree with the SdS results (figure 8 in [34]) in the limit . From Fig.1 we see that the suppression of the greybody factor by the angular momentum number is obvious in the left upper panel, both for minimally or non-minimally coupled scalar. As shown in (3.33), for the dominant mode of the minimally coupled scalar , we find a non-vanishing greybody factor for the low energy emission. While for the non-minimally coupled scalar, the greybody factor for the low energy mode vanishes. Moreover, decreases the greybody factor when other parameters are fixed. We plot the effective potential in the lower panel to have an intuitive explanation. It can be seen that the effective potential barriers become higher with , as a consequence it becomes more difficult for the scalar to transverse the barrier to reach the near horizon region. So the greybody factor decreases with .

### 4.2 Effects of

Now we study the effects of the Gauss-Bonnet parameter on the greybody factor.

#### 4.2.1 Effects of on different partial modes

In Fig.2 we plot the greybody factor for the minimally coupled scalar when . From the left upper panel, we find the suppression of the greybody factor by the angular momentum number as well. For the dominant mode , there is a non-vanishing greybody factor for the low energy modes. Unlike the case that the greybody factors for zero modes vanish when , the presence of makes it have a non-zero value. The greybody factors with respect to for the dominant mode are shown in the right upper panel. It is obvious that the greybody factor does not vanish when . Actually, it increases with . We plot the corresponding effective potential in the lower panel to give an intuitive interpretation. The effective potential decreases with when other parameters are fixed. Thus it becomes easier for the scalar to transverse it and the Hawking radiation is enhanced with .

#### 4.2.2 The competition between and

Since the non-minimally coupling suppresses the Hawking radiation (as we can see in section 4.1) while the Gauss-Bonnet term enhances it, there must be a competition between them. In Fig.3, we find that when is small ( in the left panel), increases the greybody factor. When is large ( in the right panel), decreases the greybody factor. This phenomenon appears also for and which will be shown in subsection 4.2.4. However, unlike the competition between and , the competition between and is too involved for us to have an intuitive analysis from the effective potential.

#### 4.2.3 Effects of on modes in different dimensional spacetimes

Now let us study the dependence of the greybody factor on the spacetime
dimension in the presence of . In Fig.4, we see that the greybody
factor is significantly suppressed in higher dimensions. For example, for
the greybody factors for the minimally coupled scalar at have values of order
and , respectively. For different dimensions the greybody factor still increases with .
We plot the
effective potential in the right panel. We see that the potential
barrier increases significantly with . Thus it becomes harder
for the scalar to transverse the barrier and the greybody factor decreases
with . On the other hand, decreases
the potential barrier and so increases the
greybody factor ^{3}^{3}3For the large behavior of the EGB black holes, one can find the study in [51, 52]..

Note that due to the poles of the Gamma functions in the solution, we are not able to obtain the analytical results for odd dimensional spacetimes.

#### 4.2.4 Competition between and in the presence of

We plot the competition between and when in Fig. 5. As we can see from the left upper panel, when is small, the greybody factor increases with . However, when is large enough, the greybody factor decreases with , as shown in the right upper panel. We show the corresponding effective potentials in the lower panels. It is obvious that when is small, the potential barrier decreases with . The situation is reversed when is large. Thus when is small, enhances the Hawking radiation. When is large enough, hinders the Hawking radiation. The phenomenon is observed similarly in the SdS case [34]. It is due to the double roles plays in the equations of motion. As a homogeneously energy distributed in the whole spacetime, it subsidizes the energy of emitted particle and hence enhances the radiation. As an effective mass term through the non-minimally coupling term, it suppresses the emission. The competition between these two different contributions leads to the phenomenon we observed.

## 5 Conclusion and discussion

We studied the greybody factors of the Hawking radiation for the minimally and non-minimally coupled scalar fields in a higher dimensional Einstein-Gauss-Bonnet-dS black hole spacetime. Solving the equations of motion near the event horizon and cosmological horizon separately and matching them in the intermediate region, we derived an analytical formula for the greybody factors when the cosmological constant is small. The larger the distance between the cosmological horizon and the event horizon, the more accurate the analytical formula.

The effects of various parameters, such as the angular momentum number , the non-minimally coupling constant , the cosmological constant , the GB coupling constant and the spacetime dimension , on the greybody factor were studied in detail. We found that when other parameters are fixed, similar to the case without the GB term, or suppresses the Hawking radiation separately. However, the GB coupling constant enhances the Hawking radiation. We analyzed the competition between and . The effect of the cosmological constant is more involved. When is small, it enhances the Hawking radiation. When is large enough, it suppresses the Hawking radiation. We plotted the effective potentials to give some intuitive explanations to the phenomenons we observed.

For the dominant mode , the greybody factor for the minimally coupled scalar is non-vanishing when . This feature is characteristic for the free massless scalar propagating in the dS black hole spacetime. For the EGB-dS black hole, the presence of GB constant preserves this feature qualitatively. But quantitatively, it increases the greybody factor at .

For the non-minimally coupled scalar, the greybody factors are of order and vanish for the low energy modes for all the partial modes including . This can be explained by the fact that for the non-minimally coupled scalar, plays a role of effective mass and hinders the Hawking radiation when . We obtained the coefficient of the term at for the EGB-dS black hole background.

Though it is envisaged that the results we obtained may hold well beyond the low energy regime [34] since we do not restrict the energy at all in the approximation, an exact numerical result is needed to examine the analytical results we obtained here. We leave this work in the future.

## 6 Acknowledgments

We are appreciated Nikalaos Pappas and Panagiota Kanti for their correspondences. C. Y. Zhang is supported by National Postdoctoral Program for Innovative Talents BX201600005. B.Chen and P.C. Li were in part supported by NSFC Grant No. 11275010, No. 11325522 and No. 11735001.

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