# New generalizations of cosmography inspired by the Padé approximant

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## Abstract

The current accelerated expansion of the universe has been one of the most important fields in physics and astronomy since 1998. Many cosmological models have been proposed in the literature to explain this mysterious phenomenon. Since the nature and cause of the cosmic acceleration are still unknown, model-independent approaches to study the evolution of the universe are welcome. One of the powerful model-independent approaches is the so-called cosmography. It only relies on the cosmological principle, without postulating any underlying theoretical model. However, there are several shortcomings in the usual cosmography. For instance, it is plagued with the problem of divergence (or an unacceptably large error), and it fails to predict the future evolution of the universe. In the present work, we try to overcome or at least alleviate these problems, and we propose two new generalizations of cosmography inspired by the Padé approximant. One is to directly parameterize the luminosity distance based on the Padé approximant, while the other is to generalize cosmography with respect to a so-called \(y_\beta \)-shift \(y_\beta =z/(1+\beta z)\), which is also inspired by the Padé approximant. Then we confront them with the observational data with the help of the Markov chain Monte Carlo (MCMC) code *emcee*, and find that they work fairly well.

### Keywords

Markov Chain Monte Carlo Taylor Series Cosmic Microwave Background Dark Energy Model Cosmic Acceleration## 1 Introduction

From the observation of distant type Ia supernovae (SNIa) [1, 2], it has been found in 1998 that the universe is experiencing an accelerated expansion. This amazing discovery was confirmed later by the observations of e.g. cosmic microwave background (CMB) [3, 4, 5, 6], and large scale structure (LSS) [7, 8]. In fact, this mysterious phenomenon has been one of the most important fields in physics and astronomy.

*z*is the redshift. Another powerful model-independent approach is cosmography [14, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87]. To the best of our knowledge, it was first discussed by Weinberg [27, 28] and extended by Visser [29, 30] recently. Using cosmography, one can analyze the evolution of the universe without assuming any underlying theoretical model. The only necessary assumption of cosmography is the cosmological principle, so that the spacetime metric is the one of the Friedmann–Robertson–Walker (FRW) universe,

*c*is the speed of light,

*a*is the scale factor, and \(k>,\,=,\,<0\) corresponds to a spatially close, flat, open universe, respectively. Introducing the so-called cosmographic parameters, namely the Hubble constant \(H_0\), the deceleration \(q_0\), the jerk \(j_0\), and the snap \(s_0\) (defined below), one can expand the scale factor

*a*in terms of a Taylor series with respect to cosmic time

*t*[27, 28, 29, 30],

*z*[14, 24, 25, 27, 28, 29, 30, 31, 32],

The best-fit model parameters with \(1\sigma \), \(2\sigma \), \(3\sigma \) uncertainties. The corresponding \(\chi ^2_{\mathrm{min}}\) and \(\chi ^2_{\mathrm{min}}/\mathrm{dof}\) are also given. These results are obtained by fitting the Padé parameterization (6) to SN and SN + CMB data, respectively. See the text for details

Dataset | SN | SN + CMB |
---|---|---|

\(\chi ^2_{\mathrm{min}}\) | 562.530 | 562.171 |

\(\chi ^2_{\mathrm{min}}/\mathrm{dof}\) | 0.980 | 0.978 |

| \(0.67389_{-0.05318}^{+0.18215}\,(1\sigma )_{-0.14960}^{+0.22007}\,(2\sigma )_{-0.17192}^{+0.22578}\,(3\sigma )\) | \(0.69994_{-0.04318}^{+0.07525}\,(1\sigma )_{-0.08990}^{+0.09644}\,(2\sigma )_{-0.09942}^{+0.09987}\,(3\sigma )\) |

\(\alpha _0\) | \(0.00043_{-0.00033}^{+0.00080}\,(1\sigma )_{-0.00080}^{+0.00153}\,(2\sigma )_{-0.00127}^{+0.00226}\,(3\sigma )\) | \(0.00029_{-0.00033}^{+0.00055}\,(1\sigma )_{-0.00071}^{+0.00102}\,(2\sigma )_{-0.00109}^{+0.00153}\,(3\sigma )\) |

\(\alpha _1\) | \(0.94784_{-0.09201}^{+0.23285}\,(1\sigma )_{-0.22181}^{+0.30391}\,(2\sigma )_{-0.26775}^{+0.34522}\,(3\sigma )\) | \(0.98786_{-0.06788}^{+0.09886}\,(1\sigma )_{-0.13506}^{+0.14137}\,(2\sigma )_{-0.16705}^{+0.16747}\,(3\sigma )\) |

\(\alpha _2\) | \(1.74351_{-0.10466}^{+3.11237}\,(1\sigma )_{-0.87546}^{+5.24045}\,(2\sigma )_{-1.47114}^{+6.14252}\,(3\sigma )\) | \(1.42179_{-0.14728}^{+0.44075}\,(1\sigma )_{-0.42018}^{+0.86413}\,(2\sigma )_{-0.63735}^{+1.34014}\,(3\sigma )\) |

\(\beta _1\) | \(0.87449_{-0.14387}^{+2.28839}\,(1\sigma )_{-0.76494}^{+3.72218}\,(2\sigma )_{-1.24755}^{+4.09612}\,(3\sigma )\) | \(0.54097_{-0.10055}^{+0.24989}\,(1\sigma )_{-0.25922}^{+0.49820}\,(2\sigma )_{-0.39612}^{+0.78251}\,(3\sigma )\) |

\(\beta _2\) | \(-0.08709_{-0.57889}^{+0.04635}\,(1\sigma )_{-0.97493}^{+0.19247}\,(2\sigma )_{-1.18410}^{+0.29763}\,(3\sigma )\) | \(-0.00008_{-0.00011}^{+0.00007}\,(1\sigma )_{-0.00024}^{+0.00014}\,(2\sigma )_{-0.00038}^{+0.00020}\,(3\sigma )\) |

It is easy to see that the key of cosmography is to expand the quantities under consideration as a Taylor series with respect to redshift *z*. However, it is well known that the Taylor series converges only for small *z* around 0, and it might diverge at high redshift (especially when \(z>1\)). A possible remedy is to replace the redshift *z* with the so-called *y*-shift, \(y\equiv z/(1+z)\) [26, 33, 34, 35, 80, 81]. In this case, \(y<1\) holds in the whole cosmic history \(0\le z<\infty \), and hence the Taylor series with respect to *y* converges. However, there still exist several serious problems in the case of \(y=z/(1+z)\). The first is that the error of a Taylor approximation throwing away the higher order terms will become unacceptably large when *y* is close to 1 (say, when \(z>9\)). The second is that the cosmography in terms of \(y=z/(1+z)\) cannot work well in the cosmic future \(-1<z<0\). The Taylor series with respect to \(y=z/(1+z)\) does not converge when \(y<-1\) (namely \(z<-1/2\)), and it drastically diverges when \(z\rightarrow -1\) (it is easy to see that \(y\rightarrow -\infty \) in this case). So, this *y*-shift cosmography fails to predict the future evolution of the universe. Note that there are other *y*-shifts considered in the literature, for instance, \(y_1\equiv \arctan \left( z/(1+z)\right) \), \(y_2\equiv z/(1+z^2)\), and \(y_3\equiv \arctan z\) [36]. However, they are purely written by hand, without solid motivation. On the other hand, \(|y|>1\) at suitable redshift *z*, since the function \(\arctan x\in \left( -\pi /2,\,+\pi /2\right) \), and hence the Taylor series does not converge.

In the present work, we try to overcome the problems of cosmography mentioned above. We are mainly interested in the cosmography of the luminosity distance \(d_L\), since it can be confronted with the observational data directly. In fact, the new generalizations considered in the present work are inspired by the so-called Padé approximant. In Sect. 2, we briefly review the key points of the Padé approximant, and then we parameterize the luminosity distance \(d_L\) based on the Padé approximant. We confront this parameterization of the luminosity distance \(d_L\) with the observational data and see whether it works well. Note that in this work we use the Markov chain Monte Carlo (MCMC) code *emcee* [37, 38] in the data fitting. In Sect. 3, inspired by the Padé approximant, we propose a new \(y_\beta \)-shift and then derive the cosmography of the luminosity distance \(d_L\) by expanding it as a Taylor series with respect to the new \(y_\beta \)-shift. This cosmography is completely free from the problems mentioned above. We also confront it with the observational data. Finally, some brief concluding remarks are given in Sect. 4.

## 2 Padé parameterization of the luminosity distance

*f*(

*x*), its Padé approximant of order \((m,\,n)\) is given by the rational function [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]

*m*and

*n*are both non-negative integers, and \(\alpha _i\), \(\beta _i\) are all constants. Obviously, it reduces to the Taylor series when all \(\beta _i=0\). Actually in mathematics, a Padé approximant is the best approximation of a function by a rational function of given order [42]. In fact, the Padé approximant often gives a better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge [42]. So, considering the Padé approximant in cosmology is well motivated.

*c*and Hubble constant \(H_0\) are introduced from dimensional point of view. How to choose the order \((m,\,n)\) of the Padé approximant is important. If the order is too low, the error of the Padé approximant deviating from the real luminosity distance \(d_L\) will be unacceptably large. If the order is too high, the number of free coefficients are too many and the uncertainties will be large. So, we choose a moderate order \((2,\,2)\) in this work, and then Eq. (5) becomes

*z*for suitable \(\beta _1\) and \(\beta _2\). Thus, this parameterization based on the Padé approximant can easily avoid the problems of the usual cosmography mentioned above. It is worth noting that this parameterization is a generalization of cosmography, since the Padé approximant is a generalization of the Taylor series in fact.

*h*is the Hubble constant \(H_0\) in units of \(100\,\mathrm{km/s/Mpc}\). In our case, \(D_L\) has been given in Eq. (6). Correspondingly, the \(\chi ^2\) from 580 Union2.1 SNIa is given by

*emcee*[37, 38] to find the best fits and the corresponding 68.3 %, 95.4 %, and 99.7 % confidence levels. We present the best-fit parameters with \(1\sigma \), \(2\sigma \), \(3\sigma \) uncertainties and the corresponding \(\chi ^2_{\mathrm{min}}\) in Table 1. The 1D marginalized distribution, and \(1\sigma \), \(2\sigma \), \(3\sigma \) contours in the 2D model parameter spaces are also given in Fig. 1.

*R*[64, 65] from CMB, which has been used extensively in the literature (including the works of the Planck and the WMAP Collaborations). It is argued in e.g. [66, 67, 68] that the shift parameter

*R*is model-independent and contains the main information of the observation of CMB. As is well known, the shift parameter

*R*is defined by [62, 63, 64, 65, 66, 67, 68, 69]

*z*, which can be related to the luminosity distance \(d_L\) through (see e.g. the textbooks in Refs. [27, 28])

*R*has also been determined to be \(R_{\mathrm{obs}}=1.7382\pm 0.0088\) by the latest Planck 2015 data [63]. So, the \(\chi ^2\) from CMB is given by \(\chi ^2_{\mathrm{CMB}}=(R-R_{\mathrm{obs}})^2/\sigma _R^2\), and then the total \(\chi ^2\) from the combined SN + CMB data reads

*emcee*[37, 38] to find the best fits and the corresponding 68.3 %, 95.4 %, and 99.7 % confidence levels to the combined SN + CMB data. We present the best-fit parameters with \(1\sigma \), \(2\sigma \), \(3\sigma \) uncertainties and the corresponding \(\chi ^2_{\mathrm{min}}\) in the last column of Table 1. The 1D marginalized distribution, and \(1\sigma \), \(2\sigma \), \(3\sigma \) contours in the 2D model parameter spaces are also given in Fig. 2. Thanks to CMB data, it is easy to see that the constraints on all parameters are significantly tightened (nb. Table 1). By confronting the Padé parameterization (6) with the observational data, we see that this generalized cosmography works well.

## 3 \(y_\beta \)-shift cosmography

In this section, we propose another generalization of cosmography inspired by the Padé approximant. We stress that it is completely independent of the one proposed in the previous section. At first, we propose a new \(y_\beta \)-shift inspired by the Padé approximant, and then derive the cosmography of the luminosity distance \(d_L\) by expanding it as a Taylor series with respect to this new \(y_\beta \)-shift. We will also confront it with the observational data and see whether it works well.

### 3.1 The formalism of \(y_\beta \)-shift cosmography

*z*is given by [14, 24, 25, 27, 28, 29, 30, 31, 32] [nb. Eq. (3)]

*z*. This inspires us to propose a so-called \(y_\beta \)-shift,

*z*[14, 24, 25, 27, 28, 29, 30, 31, 32] and

*y*-shift \(y=z/(1+z)\) [26, 33, 34, 35] are extensively used in the usual cosmography, our \(y_\beta \) can be regarded as their natural generalization. So, the cosmography with respect to \(y_\beta \)-shift is also a natural generalization of the usual cosmography in fact. We stress that the above discussions are only arguments to justify \(y_\beta \), rather than strict derivations. Now, let us see how it might overcome the problems of the usual cosmography mentioned in Sect. 1. First, \(y_\beta \) is inspired by the Padé approximant, and hence it is well motivated, not purely written by hand. Second, if \(1-\beta \ll 1\), we have \(y_\beta <1\) even for \(z\gg 1\). So, the \(y_\beta \)-shift cosmography can converge safely. Third, it can describe the future evolution of the universe, if \(1+\beta z\not =0\) in the redshift range \(-1<z<0\). In fact, to ensure \(y_\beta \) remains regular in not only the past but also in the future of the universe (\(-1<z<\infty \)), it is required that

*a*up to 5th order around the time \(t_0\) reads

*t*and absorbed at the current epoch \(t_0\) is given by

*D*as a Taylor series up to 5th order with respect to \(y_\beta \),

*r*-coordinate \(r=0\) at the time

*t*, and absorbed at

*r*-coordinate \(r=r_0\) at the time \(t_0\). Then the luminosity distance \(d_L\) is given by (see e.g. the textbooks in Refs. [27, 28])

The best-fit model parameters with \(1\sigma \), \(2\sigma \), \(3\sigma \) uncertainties. The corresponding \(\chi ^2_{\mathrm{min}}\) and \(\chi ^2_{\mathrm{min}}/\mathrm{dof}\) are also given. These results are obtained by fitting the \(y_\beta \)-shift cosmography (48) to SN data with the priors \(0<\beta <1\) and \(0<\beta <2\), respectively. See the text for details

Dataset | SN (\(0< \beta <1\)) | SN (\(0< \beta <2\)) |
---|---|---|

\(\chi ^2_{\mathrm{min}}\) | 565.317 | 564.668 |

\(\chi ^2_{\mathrm{min}}/\mathrm{dof}\) | 0.985 | 0.984 |

\(\beta \) | \(<1\) | \(0.94516_{-0.42111}^{+0.21990}\,(1\sigma )_{-0.75642}^{+0.44858}\,(2\sigma )_{-0.92425}^{+0.63060}\,(3\sigma )\) |

| \(0.69932_{-0.006}^{+0.0589}\,(1\sigma )_{-0.01170}^{+0.01221}\,(2\sigma )_{-0.01709}^{+0.01819}\,(3\sigma )\) | \(0.69818_{-0.00581}^{+0.00682}\,(1\sigma )_{-0.01208}^{+0.01338}\,(2\sigma )_{-0.01847}^{+0.01938}\,(3\sigma )\) |

\(q_0\) | \(-0.55618_{-0.26980}^{+0.28353}\,(1\sigma )_{-0.57862}^{+0.50908}\,(2\sigma )_{-0.85393}^{+0.68637}\,(3\sigma )\) | \(-0.52943_{-0.32969}^{+0.34475}\,(1\sigma )_{-0.67578}^{+0.65817}\,(2\sigma )_{-0.91769}^{+1.01555}\,(3\sigma )\) |

\(j_0\) | \(0.55909_{-4.21374}^{+4.59544}\,(1\sigma )_{-7.76870}^{+9.48256}\,(2\sigma )_{-11.10038}^{+14.08874}\,(3\sigma )\) | \(0.39037_{-5.56827}^{+6.47105}\,(1\sigma )_{-11.19167}^{+13.46629}\,(2\sigma )_{-16.72432}^{+17.82005}\,(3\sigma )\) |

\(s_0\) | \(1.90756_{-39.1773}^{+38.9127}\,(1\sigma )_{-87.1962}^{+75.4962}\,(2\sigma )_{-151.8190}^{+116.3544}\,(3\sigma )\) | \(3.97831_{-52.4932}^{+63.2142}\,(1\sigma )_{-127.5164}^{+145.1346}\,(2\sigma )_{-191.6690}^{+191.0794}\,(3\sigma )\) |

\(l_0\) | \(-6.32885_{-342.262}^{+344.384}\,(1\sigma )_{-554.510}^{+563.219}\,(2\sigma )_{-591.835}^{+604.266}\,(3\sigma )\) | \(0.02741_{-360.915}^{+357.489}\,(1\sigma )_{-562.618}^{+562.259}\,(2\sigma )_{-597.821}^{+597.730}\,(3\sigma )\) |

*z*(e.g. [14, 29, 30]) or

*y*-shift \(y=z/(1+z)\) (e.g. [14, 26, 33, 34, 35, 80, 81]), respectively. The cosmography with respect to \(y_\beta \)-shift \(y_\beta =z/(1+\beta z)\) obtained here is more general.

### 3.2 Cosmological constraints from the observational data

*emcee*[37, 38] to find the best fits and the corresponding 68.3 %, 95.4 % and 99.7 % confidence levels to SN data. Note that the prior \(0<\beta <1\) is required in Eq. (19). We present the best-fit parameters with \(1\sigma \), \(2\sigma \), \(3\sigma \) uncertainties and the corresponding \(\chi ^2_{\mathrm{min}}\) in Table 2. The 1D marginalized distribution, and \(1\sigma \), \(2\sigma \), \(3\sigma \) contours in the 2D model parameter spaces are also given in Fig. 3. It is easy to see that the parameter \(\beta \) cannot be well constrained, and its 1D marginalized distribution is not Gaussian. So, we temporarily relax the prior to \(0<\beta <2\). In this case, the best-fit parameters with \(1\sigma \), \(2\sigma \), \(3\sigma \) uncertainties and the corresponding \(\chi ^2_{\mathrm{min}}\) are given in the last column of Table 2, while the 1D marginalized distribution, and \(1\sigma \), \(2\sigma \), \(3\sigma \) contours in the 2D model parameter spaces are also given in Fig. 4. Now, the constraint on \(\beta \) looks well. The following discussions hold for both cases with \(0<\beta <1\) and \(0<\beta <2\) in fact. It is easy to see that SN data favor a best-fit \(\beta \) close to 1, and hence \(y_\beta \) is close to the usual

*y*-shift \(y=z/(1+z)\). However, \(\beta \) can still significantly deviate from 1 in the wide \(1\sigma \), \(2\sigma \), \(3\sigma \) regions. In particular, \(\beta \) can even be close to 0 in the \(3\sigma \) region. On the other hand, we find that \(q_0<0\) at \(1\sigma \) confidence level, which indicates the cosmic expansion is accelerating [nb. the definition in Eq. (21)]. Of course, it is not surprising that the constraints on \(l_0\) and \(s_0\) are fairly loose, since we use only SN data here.

*z*. Noting that SN data favors a best-fit \(\beta \) close to 1 as mentioned above, this indicates that there is tension between the SN and the CMB data. However, we stress that the constraints from SN and SN + CMB data are still consistent within the \(3\sigma \) confidence level. On the other hand, we see that \(q_0<0\), \(j_0>0\) beyond \(3\sigma \) confidence level, which strongly indicates the cosmic expansion is accelerating, and the acceleration is increasing [nb. the definitions in Eqs. (21) and (22)].

The same as in Table 2, except for SN + CMB data and the prior \(0<\beta <1\) only

Dataset | SN + CMB |
---|---|

\(\chi ^2_{\mathrm{min}}\) | 562.654 |

\(\chi ^2_{\mathrm{min}}/\mathrm{dof}\) | 0.979 |

\(\beta \) | \(0.13891_{-0.01202}^{+0.01050}\,(1\sigma )_{-0.03229}^{+0.02126}\,(2\sigma )_{-0.05907}^{+0.03294}\,(3\sigma )\) |

| \(0.70236_{-0.00556}^{+0.00611}\,(1\sigma )_{-0.01938}^{+0.01844}\,(2\sigma )_{-0.07461}^{+0.07016}\,(3\sigma )\) |

\(q_0\) | \(-0.70925_{-0.07971}^{+0.09912}\,(1\sigma )_{-0.19356}^{+0.27603}\,(2\sigma )_{-0.27210}^{+0.55496}\,(3\sigma )\) |

\(j_0\) | \(1.98747_{-0.35293}^{+0.31463}\,(1\sigma )_{-0.92251}^{+1.03983}\,(2\sigma )_{-1.50048}^{+2.07407}\,(3\sigma )\) |

\(s_0\) | \(0.69908_{-1.11125}^{+0.99476}\,(1\sigma )_{-3.04250}^{+3.71175}\,(2\sigma )_{-6.00546}^{+7.70008}\,(3\sigma )\) |

\(l_0\) | \(-1.31401_{-3.56927}^{+3.29455}\,(1\sigma )_{-12.98544}^{+17.06251}\,(2\sigma )_{-35.29680}^{+38.88782}\,(3\sigma )\) |

## 4 Concluding remarks

The current accelerated expansion of the universe has been one of the most important fields in physics and astronomy since 1998. Many cosmological models have been proposed in the literature to explain this mysterious phenomenon. Since the nature and cause of the cosmic acceleration are still unknown, model-independent approaches to study the evolution of the universe are welcome. One of the powerful model-independent approaches is the so-called cosmography. It only relies on the cosmological principle, without postulating any underlying theoretical model. However, there are several shortcomings in the usual cosmography. In the present work, we try to overcome or at least alleviate these problems, and propose two new generalizations of cosmography inspired by the Padé approximant. We also confront them with the observational data with the help of the Markov chain Monte Carlo (MCMC) code *emcee* [37, 38], and find that they work fairly well.

In the literature (e.g. [14, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]), there exist a number of works on cosmography. They focused on various issues in cosmology and made much significant progress. However, the previous works used ordinary cosmography mainly with respect to the redshift *z* or the so-called *y*-shift \(y=z/(1+z)\). We stress that they are all plagued with the problem of divergence (or the unacceptably large error when \(y\sim 1\)), and all fail to predict the future evolution of the universe (especially when \(z\sim -1\) or \(y<-1\)), as mentioned in Sect. 1. In the present work, the two new generalizations of cosmography proposed in Sects. 2 and 3 can instead avoid or at least alleviate these problems of ordinary cosmography. In addition, our new generalizations of cosmography are well motivated by the Padé approximant, rather than written purely by hand. In fact, it has been solidly proved by mathematicians that any (even unknown) function can be well approximated by a Padé approximant [39, 40, 41, 42, 43, 44], and the Padé approximant often gives a better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge [42]. Noting that ordinary cosmography is actually based on a Taylor series, the advantages of the Padé approximant mentioned above make our new generalizations better than the ordinary cosmography used in the literature. Although the Padé approximant has been considered in cosmology (see e.g. [45, 46, 47, 48, 49, 70, 71, 72, 73, 74, 75]), here we instead use it in a fairly different issue. For example, in e.g. [48, 49] the Padé approximant was used to study the issues of an analytical approximation of the luminosity distance in the XCDM model, EoS parameterizations, and gamma-ray burst cosmology, while we use it to generalize cosmography in this work. By all the above arguments, we stress that the present work is significantly different from the previous works on both cosmography and the Padé approximant in the literature.

The key to avoid or at least alleviate the problems of the ordinary cosmography is the denominator of the Padé approximant. Considering Eqs. (5) or (4), if the order *n* of the denominator is larger than or equal to the order *m* of the numerator, this Padé approximant with \(n\ge m\) will not diverge even for \(z\gg 1\). In fact, this is just the case of our two generalizations [nb. Eqs. (6) and (18)]. On the other hand, for suitable parameters \(\beta _i\), it is easy to ensure the denominator not to equal zero for the very wide redshift range \(-1<z<\infty \), and hence the Padé approximant can avoid divergence. The shortcomings of ordinary cosmography mainly have their roots in the Taylor series. So, generalizing a Taylor series to the Padé approximant brings about a possible way out. If all \(\beta _i=0\) in the denominator of the Padé approximant, it reduces to the usual Taylor series. However, a denominator not equal to 1 and 0 makes a big difference.

Here, we would like to clarify the main difference between the two new generalizations proposed in Sects. 2 and 3. Noting that ordinary cosmography is based on a Taylor series, the first one generalizes cosmography by directly generalizing the Taylor series to the Padé approximant when we expand the luminosity distance \(d_L\) [nb. Eq. (5)]. On the other hand, the second one instead generalizes cosmography by generalizing the redshift *z* or the *y*-shift \(y=z/(1+z)\) to the so-called \(y_\beta \)-shift \(y_\beta =z/(1+\beta z)\) (which is inspired by the Padé approximant), but we still expand the luminosity distance \(d_L\) in a Taylor series [nb. Eq. (48)], rather than the Padé approximant itself. This is the main difference. Noting that \(y_\beta =z\) and \(z/(1+z)\) when \(\beta =0\) and 1, respectively, we call \(y_\beta =z/(1+\beta z)\) the “\(y_\beta \)-shift” in analogy to the well-known terminology of the “*y*-shift” \(y=z/(1+z)\) used in the literature, while the terminology “*y*-shift” comes from the terminology “red-shift” in fact.

It is of interest to quantitatively compare our two new generalizations of cosmography, and also compare them with the ordinary cosmography (we thank the referee for pointing out this issue). Let us compare our \(y_\beta \)-shift cosmography with the ordinary cosmography first. The observational constraints on our \(y_\beta \)-shift cosmography from the SN + CMB data are presented in Table 3 and Fig. 5. It is easy to see that \(\beta =0\) and \(\beta =1\) deviate from the best fit far beyond the \(3\sigma \) confidence level; see especially the leftmost column of Fig. 5. Note that our \(y_\beta \)-shift reduces to the ordinary redshift *z* and the *y*-shift \(y=z/(1+z)\) when \(\beta =0\) and 1, respectively. Therefore, our \(y_\beta \)-shift cosmography can fit the SN + CMB data significantly better than ordinary cosmography. On the other hand, from the last column of Table 1 and Table 3, \(\chi ^2_{\mathrm{min}}=562.171\), and \(\chi ^2_{\mathrm{min}}/\mathrm{dof}=0.978\) by fitting the Padé parameterization to the SN + CMB data, while \(\chi ^2_{\mathrm{min}}=562.654\) and \(\chi ^2_{\mathrm{min}}/\mathrm{dof}=0.979\) by fitting the \(y_\beta \)-shift cosmography to the SN + CMB data. Therefore, our first generalization of cosmography is slightly better than the second one.

Other remarks are in order. First, the Padé approximant has been used previously in cosmology, e.g. in slow-roll inflation, the reconstruction of the scalar field potential, data fitting and analytical approximation of the luminosity distance, EoS parameterizations, gamma-ray burst cosmology, and cosmological perturbations in LSS (see e.g. [45, 46, 47, 48, 49, 70, 71, 72, 73, 74, 75]). We advocate further uses of the Padé approximant in cosmology. Second, in the present work we only derive the generalized cosmography of the luminosity distance \(d_L\). In fact, one can easily obtain the corresponding cosmography of other observable quantities, such as the angular diameter distance \(d_A\), the photon flux distance \(d_F\), the photon count distance \(d_P\), and the deceleration distance \(d_Q\), since they can be readily related to the luminosity distance \(d_L\) (see e.g. [26]). Third, we confront the generalized cosmography with the observational data only for the spatially flat FRW universe (\(k=0\)). In fact, one can do this for the \(k\not =0\) cases easily, and hence we do not present them here. Finally, the usual cosmography with respect to the redshift *z* and *y*-shift \(y=z/(1+z)\) has been extensively applied to various cosmological issues, for instance, the EoS of dark energy, modified gravity theories like *f*(*R*) and *f*(*T*) theories, gamma-ray burst cosmology, and so on (see e.g. [14, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]). The new generalizations of cosmography proposed in the present work can also be used in these cosmological issues, and we leave it to the future works.

## Notes

### Acknowledgments

We thank the anonymous referee for quite useful comments and suggestions, which helped us to improve this work. We are grateful to Zu-Cheng Chen, Jing Liu, Xiao-Peng Yan, Shoulong Li, Hong-Yu Li, and Dong-Ze Xue for kind help and discussions. This work was supported in part by NSFC under Grants No. 11575022 and No. 11175016.

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