New observational constraints on cosmology from radio quasars
Abstract
Using a new recently compiled milliarcsecond compact radio data set of 120 intermediateluminosity quasars in the redshift range , whose statistical linear sizes show negligible dependence on redshifts and intrinsic luminosity and thus represent standard rulers in cosmology, we constrain three viable and most popular gravity models, where is the torsion scalar in teleparallel gravity. Our analysis reveals that constraining power of the quasars data (N=120) is comparable to the Union2.1 SN Ia data (N=580) for all three models. Together with other standard ruler probes such as Cosmic Microwave Background and Baryon Acoustic Oscillation distance measurements, the present value of the matter density parameter obtained by quasars is much lager than that derived from other observations. For two of the models considered (CDM and CDM) a small but noticeable deviation from CDM cosmology is present, while in the framework of CDM the effective equation of state may cross the phantom divide line at lower redshifts. These results indicate that intermediateluminosity quasars could provide an effective observational probe comparable to SN Ia at much higher redsifts, and gravity is a reasonable candidate for the modified gravity theory.
∎
e1corresponding author
email: caoshuo@bnu.edu.cn
1 Introduction
The current cosmic acceleration has been supported by many independent astrophysical observations, including type Ia supernovae (SN Ia) (1), large scale structure (2), cosmic microwave background (CMB) anisotropy (3), etc. A mysterious component with negative pressure, dubbed as dark energy, has been proposed to explain this phenomenon in the framework of Einstein’s general relativity, which gave birth to a large number of dark energy models including the cosmological constant (CDM), scalar field theory (4); (5); (6), and dynamical dark energy models (7); (8); (9); (10); (11). The other direction one could follow in search for solution of the accelerating cosmic expansion enigma is to construct modified theories of gravity instead of invoking exotic dark energy. Large majority of works in this direction concentrated on the braneworld DvaliGabadadzePorrati (DGP) model (12), gravity (13), and GaussBonnet gravity (14).
Equally well, one can also modify the gravity according to the scenario described by the socalled theory (15), which was proposedin the framework of the Teleparallel Equivalent of General Relativity (also known as Teleparallel Gravity). In this approach, the LeviCivita connection used in Einstein’s general relativity is replaced by the Weitzenböck connection with torsion, while the Lagrangian density of this theory is the torsion scalar . Compared with the theory leading to the fourth order equations, the field equations of the theory are in the form of second order differential equations, which provides an important advantage of this approach. In addition, if certain conditions are satisfied, the behavior of cosmologies is similar to several popular dark energy models, such as quintessence (16), phantom (17), DGP model (18) and transient acceleration (19). Due to the above mentioned property, theory and its cosmological applications has gained a lot interest in the literature. The detailed introduction to the theory could be found in (18); (20).
In this paper, we focus on using the currently released quasar data (21) to provide the constraints on various gravity models. Recently, the angular size of compact structure in radio quasars versus redshift data from the verylongbaseline interferometry (VLBI) observations have become an effective probe in cosmology. Reliable standard rulers and standard candles at cosmological scales are crucial for measuring cosmic distances at different redshifts. For instance, the type Ia supernovae are regarded as standard candles, while the BAO peak location is commonly recognized as a fixed comoving ruler. The increasing observational material concerning these two distance indicators has been widely used in various cosmological studies. In the past, there were controversial discussions about whether the compact radio sources could act as standard rulers (22); (23); (24); (25); (26). The difficulty lies in the fact that the linear sizes of compact radio sources might not be constant, i.e., its value is dependent on both redshifts and some intrinsic properties of the source (luminosity, for example). Based on a 2.29 GHz VLBI allsky survey of 613 milliarcsecond ultracompact radio sources (27); (28), Cao et al. (21) presented a method to divide the full sample into different subsamples, according to their optical counterparts and luminosity (lowluminosity quasars, intermediateluminosity quasars, and highluminosity quasars). The final results indicated that intermediateluminosity quasars show negligible dependence on both redshifts and intrinsic luminosity , which makes them a fixed comovinglength standard ruler. More recently, based on a cosmologicalmodelindependent method to calibrate the linear sizes of intermediateluminosity quasars, Cao et al. (29) investigated the cosmological application of this data set and obtained stringent constraints on both the matter density and the Hubble constant , which agree very with the recent Planck results. The advantage of this data set, compared with other standard rulers: BAO (30); (31); (32), clusters (33), strong lensing systems (34); (35); (36)), is that quasars are observed at much higher redshifts (). Therefore, it may be rewarding to test the theory with this newly revised quasar data. In this paper, we examine constraints on the viable cosmological models imposed by the quasars. We compare them with analogous results obtained with the newly revised Union2.1 set — the largest published and spectroscopically confirmed SN Ia sample to date. We expect that different systematics and sensitivities of these two different probes (rulers vs. candles) can give complementary results on the theory.
This paper is organized as follows: In section 2 we briefly introduce the gravity and its cosmological consequences. In section 3 we present the latest data sets for our analysis and perform a Markov chain Monte Carlo analysis using different data sets. Finally, we summarize the main conclusions in Section 4.
2 The theory
In this section we brief review the gravity in the framework of cosmology, and then present three specific models to be analyzed in this work.
2.1 The cosmology
We use the vierbein fields (), which is an orthonormal basis for the tangent space at each point of the manifold , and whose components are (here Latin indices stand for the tangent space and Greek indices refer to the manifold). Its dual vierbein gives the metric tensor . In theory, instead of the torsionless LeviCivita connection in Einstein’s General Relativity, the curvatureless Weitzenböck connection is considered, and hence the torsion tensor describing the gravitational field is
(1) 
The Lagrangian of teleparallel gravity is constructed by the torsion scalar as (15)
(2) 
where
(3) 
and the contorsion tensor is given by
(4) 
In the theory, the Lagrangian density is a function of (15), and the action reads
(5) 
where . The corresponding field equation is
(6) 
where , , , , and is the matter energymomentum tensor. Considering a flat homogeneous and isotropic FRW universe, we have
(7) 
where is the cosmological scale factor. By substituting Eqs. (2.1), (1), (3) and (4) into Eq. (2), one could obtain the torsion scalar as (15)
(8) 
where is the Hubble parameter . The dot represents the first derivative with respect to the cosmic time. Substituting Eq. (2.1) into (6), one can obtain the corresponding Friedmann equations
(9)  
(10) 
where and are the total energy density and pressure, respectively. By defining the effective energy density density , pressure and effective equation of state (EoS) parameter as
(11)  
(12)  
(13) 
The Friedmann equations could be rewrite as
(14)  
(15) 
Therefore, the cosmic acceleration could be driven by the torsion instead of dark energy. In this cosmological framework, the corresponding normalized Hubble parameter is
(16) 
where (the subscript “0” denotes the current value). Here we consider the matter and radiation in the Universe — the components whose energy density evolves with redshift as , , respectively. And then, Eq. (16) could be expressed as (37); (38)
(17) 
where , and . In this way, a specific form of is embodied in the function , whose expression is
(18) 
where p stands for the parameters in different forms of theory.
2.2 Specific models
In this subsection we briefly review three specific models,
which have passed basic observational tests (37)
and will be further investigated in this paper.
(1) The powerlaw model (15) (hereafter CDM) assumes that the Lagrangian density of the theory is the following:
(19) 
where and are two model parameters. The distortion parameter quantifies deviation from the CDM model, whereas the parameter can be expressed through the Hubble constant and density parameter by inserting Eq. (19) into Eq. (17) with the boundary condition :
(20) 
Now Eq. (18) may be rewritten as
(21) 
Depending on the choice of parameter , this model can be
connected with some popular dark energy models. For , it
reduces to the CDM, while it can mimic the
DvaliGabadadzePorrati (DGP) model when .
(2) The exponential model (39) (hereafter CDM) is characterized by
(22) 
where and are two dimensionless parameters. Similarly the expressions for and can also be obtained as
(23) 
(24) 
This model reduces to the CDM in the limit . By setting , Eq. (24) is rewritten as
(25) 
and CDM is recovered when .
(3) Motivated by the exponential gravity, the hyperbolictangent model (17) (hereafter CDM) arises from the ansatz
(26) 
where and are the two model parameters. We obtain the expressions for and as
(27) 
(28) 
respectively. Compared with two previous theories, this model cannot be reduced to the CDM for any value of its parameters. In addition, in order to have a positive value for , the parameter must be greater than (17).
3 Observational data and fitting method
In order to measure the angular diameter distance, we always turn to objects of known comoving size acting as “standard rulers”. In this paper, we will consider a combination of three types of standard rulers using the most recent and significantly improved observations, i.e., the compact radio quasars data from VLBI, baryonic acoustic oscillations (BAO) from the largescale structure, and the cosmic microwave background (CMB) measurements.
3.1 Quasars data
It is well known that the baryon acoustic oscillations (BAO) peak location is commonly recognized as a fixed comoving ruler of about 100 Mpc. Therefore it has already been used in cosmological studies (30); (31); (32). In the similar spirit, as extensively discussed in the literature, compact radio sources (quasars, in particular) constitute another possible class of standard rulers of about pc comoving length. Following the analysis of Gurvits (28), luminosity and redshift dependence of the linear sizes of quasars can be parametrized as
(29) 
where and are two parameters quantifying the “angular size  redshift” and “angular size  luminosity” relations, respectively. The parameter is the linear size scaling factor representing the apparent distribution of radio brightness within the core. The data used in this paper were derived from an old 2.29 GHz VLBI survey undertaken by Preston et al.(1985), which contains 613 milliarcsecond ultracompact radio sources covering the redshift range . More recently, Cao et al. (29) presented a method to identify a subsample which can serve as a certain class of individual standard rulers in the Universe. According to the optical counterparts and luminosities, the full sample could be divided into three subsamples: lowluminosity quasars ( W/Hz), intermediateluminosity quasars ( W/Hz W/Hz) and highluminosity quasars (W/Hz). The final results showed that only intermediateluminosity quasars show negligible dependence (, ), and thus they could be a population of rulers once the characteristic length is fixed. In our analysis, we will use the observations of 120 intermediateluminosity quasars covering the redshift range , while the linear size of this standard ruler is calibrated to pc through a new cosmologyindependent technique (29).
The observable quantity in this dataset is the angular size of the compact structure in intermediateluminosity radio quasars, whose theoretical (i.e. determined by the cosmological model) counterpart is
(30) 
where is the angular diameter distance at redshift and the model parameters p directly enter the angular diameter distance through
(31) 
where is the dimensionless Hubble parameter and is the dimensionless Hubble constant. We estimate the parameters by minimizing the corresponding defined as
(32) 
where is the observed value of the angular size and is the corresponding uncertainty for the th data point in the sample. In order to properly account for the intrinsic spread in linear sizes and systematics we have added in quadrature uncertainties to the .
3.2 CMB and BAO data
In order to diminish the degeneracy between model parameters we also used the accurate measurements of BAO and CMB.
The CMB experiments measure the temperature and polarization anisotropy of the cosmic radiation in the early epoch. In general, they are a very important tool for the inference of cosmological model parameters. In particular, the shift parameter defined as:
(33) 
where denotes the decoupling redshift, is a convenient quantity for a quick fitting of cosmological model parameters. The firstyear data release of Planck reported its value of (41). We estimate the model parameters by minimizing the corresponding
(34) 
The measurements of Baryon acoustic oscillation (BAO) in the largescale structure power spectrum and CMB angular power spectrum have also been widely used for cosmological applications. In this work we consider the measurements of , where is the decoupling time, is the comoving angulardiameter distance, and the dilation scale is given by
(35) 
The BAO data are shown in Table 1. Similarly, the corresponding for the BAO probes is defined as
(36) 
where and is the inverse covariance matrix given by Ref. (40).
4 Observational constraints
In this section, we determine the model parameters of three cosmologies through the maximum likelihood method based on introduced in previous section using the Markov Chain Monte Carlo (MCMC) method. Our code is based on CosmoMC (42) and we generated eight chains after setting to guarantee the accuracy of this work.
4.1 CDM model:
In the case of the theory based on , different data sets and their combinations led to the marginalized 2D confidence contours presented in Fig. 12. The corresponding marginal error bars can also be seen in Table 2.
Left panel of the Fig. 1 shows the contours obtained from the quasars only and in combination with CMB and BAO. We remark that the quasar data only can not tightly constrain the model parameters. In order to clearly illustrate the constraint comparison between different data sets, a prior is applied to the likelihood contours obtained from the quasar data. Quantitatively, the value of the distortion parameter , which quantifies the deviation from the CDM model varies over the interval [3, 0.56] within confidence level. As it is well known, the main evidence for cosmic acceleration came from the other type of distance indicators in cosmology, those probing the luminosity distance, by observing the flux of type Ia supernovae (SN Ia). In order to compare our fits with the results obtained using SN Ia, likelihood contours obtained with the latest Union2.1 compilation (43) consisting of 580 SN Ia data points are also plotted in the right panel of the Fig. 1. It is clear that the quasar data could give more stringent constraints than SN Ia, and its constraining power becomes obvious when the large size difference between the samples is taken into consideration. This may happen due to the wider redshift range of the quasars data () compared with SN Ia (). Moreover, one can clearly see from Fig. 1 that principal axes of confidence regions obtained with SN and quasars are inclined at higher angles, which sustains the hope that careful choice of the quasar sample would eventually provide a complementary probe breaking the degeneracy in the model parameters. Finally, our method based on the observations of intermediateluminosity quasars may also contribute to testing the consistency between luminosity and angular diameter distances (44); (45); (46).
With the combined standard ruler data sets of quasars, BAO and CMB, the bestfit value for the parameters are and within 68.3% confidence level. For comparison, fitting results from SN+BAO+CMB are also given in Fig. 1. The bestfit value is and , which is in good agreement with that of the Quasar+BAO+CMB data. It is obvious that the quasar data, when combined to CMB and BAO observations, can give more stringent constraints on this cosmology, which demonstrates the strong constraining power of BAO and CMB on the cosmological parameters. This situation has also been extensively discussed in the previous works investigating dark energy scenarios with other astrophysical observations ((47); (48); (49); (50); (46)). Again, the constraining power of 120 quasar data is comparable to that of 580 SN Ia. On the one hand, the present value of the matter density parameter given by quasars is much lager than that derived from other observations. This has been noted by our previous analysis Cao et al. (29) and the firstyear Planck results, in the framework of CDM cosmology. Such a result indicates that quasars data at high redshifts may provide us a different understanding of the parameters describing the components of the Universe. On the other hand, the parameter , which captures the deviation of cosmology from the CDM scenario, seems to be vanishing or slightly larger than 0 with the combined Quasar+BAO+CMB data. It is interesting to note that CDM is not included at confidence level (), this slight deviation from CDM is also consistent with a similar conclusion obtained in Ref. (37) for this CDM model. This tendency can be more clearly seen from Fig. 3, which illustrates the comparison between the effective equation of state for and the EoS for CDM model at , with the bestfitted value as well as the 1 and 2 uncertainties derived from the joint data of Quasars, BAO and CMB.
The contours constrained with the total combination of Quasars+SN Ia+BAO+CMB are presented in Fig. 2, and the bestfit value is and . The combined data give no stronger constraint, which indicates the constraint ability of quasars data is already very strong, while SN Ia do not play a leading role in the joint constraint. From the results above, we can see the CDM model which corresponds to () is still included within 1 range. For comparison, in Table 2 we also list alternative constraints obtained by the others using different probes.
Data  Ref.  
Quasars+BAO+CMB  This paper  
SN Ia+BAO+CMB  This paper  
Quasars+SN Ia+BAO+CMB  This paper  
OHD+SN Ia+BAO+CMB  (37)  
SN Ia+BAO+CMB+dynamical growth data  (38)  
SN Ia+BAO+varying fundamental constants  (51) 
4.2 CDM model:
Performing a similar analysis as before, this time with the other model in which CDM is also nested, namely, , we made the same comparison as CDM discussed above, i.e. Quasars vs. SN Ia and Quasars+BAO+CMB vs. SN Ia+BAO+CMB. The results are presented in Fig. 4 and the estimated cosmic parameters are briefly summarized in Table 3. It is apparent that the quasars data exhibit similar constraining power as in the case of CDM model, which implies that the constraint ability of 120 quasar data can be comparable to that of 580 SN Ia. By fitting the CDM model to Quasars+BAO+CMB, we obtain and (let us recall that here we introduced ).
With the combined data set of Quasars+SN Ia+BAO+CMB, we also get the marginalized 1 constraints of the parameters as and . The marginalized 1 and 2 contours of each parameter are presented in Fig. 6. In Table 3, the bestfit parameters and their 1 uncertainties for three data sets are displayed. As previously the results from the others using different probes are shown for comparison. Obviously, the present matter density parameter fitted by quasars is lager than given by other observations. The parameter quantifying the deviation from the CDM scenario, tends to be zero for all of observations listed in Table 3, which results in that the exponential gravity is practically undistinguishable from CDM. As can be seen from the results presented in Fig. 5, even at 2 confidence level, the effective EoS of CDM model from the joint analysis of Quasars, BAO and CMB agrees very well with that of CDM at , which strongly indicates the consistency between the two types of cosmological models at much higher redshifts.
Data  Ref.  
Quasars+BAO+CMB  This paper  
SN Ia+BAO+CMB  This paper  
Quasars+SN Ia+BAO+CMB  This paper  
OHD+SN Ia+BAO+CMB  (37)  
SN Ia+BAO+CMB+dynamical growth data  (38)  
SN Ia+BAO+varying fundamental constants  (51) 
4.3 CDM model:
Now we will discuss the third cosmology which is truly an alternative to the CDM since the concordance cosmological model cannot be recovered as a limiting case of CDM model. Consequently, the parameter does not characterize the deviation from CDM.
In Fig. 7 we presented contour plots of CDM model parameters fitted to four different probes, namely Quasars, SN Ia, Quasars+BAO+CMB, and SN Ia+BAO+CMB. As we can see, the quasar data provide more stringent constraints than SN Ia, which indicates that the constraining ability of quasar data can be comparable to or better than that of SN Ia at least in this particular model. In Fig. 8 we show the contour plots for the combination of all data sets Quasars+SN Ia+BAO+CMB. Additionally, in Table 4 we summarize the bestfit values for the three combined data sets respectively. The table 4 also includes the bestfit values and their 68% confidence levels for the previous results from the literature. Similar to the cases of CDM model and CDM model, the present matter density parameter implied by quasars is lager than that given by other observations. Concerning the value of the parameter , its the value constrained by all of the current observations satisfies the condition , which is necessary to achieve the cosmic acceleration in the framework of CDM.
In Fig. 9 we show the evolution of the effective equation of state for CDM model as a function of redshift, concerning the bestfitted value with the 1 and 2 uncertainties from the joint data of Quasars, BAO and CMB. In particular, we find that the value of obtained with quasars suggests that the effective equation of state crosses the phantom divide line at lower redshifts (17).
Data  Ref.  
Quasars+BAO+CMB  This paper  
SN Ia+BAO+CMB  This paper  
Quasars+SN Ia+BAO+CMB  This paper  
GRB+OHD+SN Ia+BAO+CMB  (52) 
4.4 Model selection
In order to to make a good comparison between different models or decide which model is preferred by the observational data, we will use two standard information criteria, namely the Akaike Information Criterion (AIC) ((53)) and the Bayesian Information Criterion (BIC) ((54)) to study competing models. The above two information criteria are respectively defined as
(37) 
and
(38) 
where , represents the number of free parameters in the model and is the sample size used in the statistical analysis. In addition, we introduce the ratio of to the degrees of freedom (d.o.f), , to judge the quality of observational data set.
In Table 5, we list the values of AIC, BIC and for different models from the joint analysis Quasar+BAO+CMB and SN Ia+BAO+CMB. It is obvious that both of AIC and BIC criteria support CDM to be the best cosmological model consistent with the available observations, since the IC value it yields is the smallest. Concerning the ranking of the three models, AIC and BIC criteria tend to provide the same conclusions as follows. The CDM model performs the best in explaining the current data, which can be clearly seen from the similarity between CDM and CDM shown in Fig. 5. Then next after CDM is the CDM model, which can also reduce to the CDM model and its bestfit parameters indeed do so. The worst model according to the AIC and BIC criteria is CDM, which is unable to provide a good fit to the data and can not nest CDM.
Quasar+BAO+CMB  SN Ia+BAO+CMB  

Model  AIC  AIC  BIC  BIC  AIC  AIC  BIC  BIC  
CDM  
CDM  
CDM  
CDM 
5 Conclusions and discussions
As an interesting approach to modify gravity, theory based on the concept of teleparallel gravity, was proposed to explain the accelerated expansion of the Universe without the need of dark energy. In this paper, we have used the recentlyreleased sample of VLBI observations of the compact structure in 120 intermediateluminosity quasars () to get the constraints on the viable and most popular gravity models. The statistical linear sizes of these quasars observed at 2.29 GHz show negligible dependence on redshifts and intrinsic luminosity, and thus represent a fixed comovinglength of the standard ruler. Therefore, the other motivation of this work was to investigate the constraining ability of quasar data in the context of models. In particular, we have considered three models with two parameters, out of which two could nest the concordance CDM model and we quantifed their deviation from CDM cosmology through a single parameter . For the third cosmology which can not be directly reduced to CDM, we discussed the possibility for the effective equation of state to cross the phantom divide line.
In our investigation we have used (i) the very recently released “angular size  redshift” data sets of 120 intermediateluminosity quasars in the redshift range , (ii) the cosmic microwave background and baryon acoustic oscillation data points. Meanwhile, in order to compare our fits obtained with 120 quasars (standard rulers), to the similar constraints obtained with the Union 2.1 compilation consisting of 580 SN Ia data points (standard candles) we also carried out respective analysis based on SNIa data. Here we summarize our main conclusions in more detail:

For all of the three the models, all of the fitting results show that the quasar data (N=120) could provide more stringent constraints than the Union2.1 SN Ia data (N=580). This may be associated with the wider redshift range covered by the quasar data () compared with SN Ia (). The constraining power of the former becomes obvious when the large size difference between the samples is taken into consideration. Moreover, one can clearly see that principal axes of confidence regions obtained with SN and quasars are inclined at higher angles, which sustains the hope that careful choice of the quasar sample would eventually provide a complementary probe breaking the degeneracy in the model parameters. Our method based on the observations of intermediateluminosity quasars may also contribute to testing the consistency between luminosity and angular diameter distances.

The present value of the matter density parameter implied by quasars is much lager than that derived from other observations, which has been noted by our previous analysis and the firstyear Planck results, in the framework of CDM cosmology. Such result indicates that quasar data at high redshifts may provide us a different understanding of the components in the Universe.

For CDM and CDM models, deviation from CDM cosmology is also allowed in the obtained confidence level, although the bestfit value is very close to its CDM one. It is interesting in the present work to note that CDM is not included at confidence level for the powerlaw model CDM model, this slight deviation from CDM is also consistent with a similar conclusion obtained in the previous observational studies on gravity. In the framework of CDM, the value of constrained by all of the current observations satisfies the limit of , which is necessary to achieve the cosmic acceleration. Moreover, we find that the value of obtained with quasars suggests that the effective equation of state can cross the phantom divide line at lower redshifts .

The information criteria (AIC and BIC) demonstrate that, compared with other three scenarios considered in this paper, the cosmological constant model is still the best cosmological model consistent with the available observations. Concerning the ranking of the cosmologies, the CDM model performs the best in explaining the current data, while the CDM model gets the smallest support and can not nest the concordance CDM model.

In summary, using for the recently released quasar data acting as a new source of standard rulers, we were able to set more stringent limits on the viable and most used gravity models. Our results highlight the importance of quasar measurements to provide additional information of various candidates for modified gravity, especially the possible deviation from CDM cosmology. More importantly, given the usefulness of this angular size data in pinning down parameter values, we also anticipate that nearfuture quasar observations will provide significantly more restrictive constraints on other torsional modified gravity theories ((55); (56); (11)).
Acknowledgements.
This work was supported by the National Key Research and Development Program of China under Grants No. 2017YFA0402603; the Ministry of Science and Technology National Basic Science Program (Project 973) under Grants No. 2014CB845806; the National Natural Science Foundation of China under Grants Nos. 11503001, 11373014, and 11690023; the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University; China Postdoctoral Science Foundation under grant No. 2015T80052; and the Opening Project of Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences. This research was also partly supported by the PolandChina Scientific & Technological Cooperation Committee Project No. 354. M.B. was supported by Foreign Talent Introducing Project and Special Fund Support of Foreign Knowledge Introducing Project in China.Footnotes
 journal: Eur. Phys. J. C
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