walls are considered in collision avoidance $R_{max}$ (length), the size of
the leg swing space $B_{l}$ (length), $k_{s}$ (eq. 63, inverse of time), and
the “goal noise” standard deviation $\sigma_{g}$ (eq. 64). In this work we
assumed, following [32] $k_{v_{p}}=1.52$ s-1, and fixed $R_{max}=2$ m, leaving
the 8 parameters $C$, $d_{int}$, $d_{ext}$, $\mu_{v}$, $\sigma_{v}$, $B_{l}$,
$k_{s}$ and $\sigma_{g}$ to be calibrated.

A further parameter is the integration step $\Delta t$, that was fixed to
$0.05$ s. The equations governing the behavioural dynamics (collision
avoidance) of pedestrians are written as differential equations but
implemented as difference equations, by discretising through an Euler scheme.
The integration step is common to all behavioural modules (including the short
range ones described below), and assumes that pedestrians take decisions at
regular time steps and act based on them. Using a Euler scheme with a time
step comparable to human reaction times is thus not an issue, as long as, as
described in Section 5.3, physical dynamics is resolved at a finer scale.

The “short range” module introduces 6 new parameters: $k^{v}_{\theta}$
(inverse of time), $k_{\omega}$ (inverse of time) and $k^{\omega}_{\theta}$
(inverse of squared time), $\gamma$ (dimensionless), $\tau_{1}$ (time) and
$\tau_{2}$ (time) (eqs. 2, 3, 4).

###  E.4 Constraints

All terms explained until now are considered to be the results of pedestrian
decisional processes, and as such they should provide velocities and
accelerations that are compatible with realistic pedestrian behaviour. For
this reason, in case after computing the discrete equations, the values of
$|\dot{\mathbf{v}}|$ and $\dot{\omega}$ are higher than thresholds $a_{max}=5$
m s-2 and $\dot{\omega}_{max}=2\pi$ s-2, the final acceleration and angular
accelerations are scaled (i.e., preserving their direction and sign) to such
maximum values. In an equivalent way, thresholds $v_{max}=3$ m/s and
$\omega_{max}=2\pi$ s-1 are applied to velocity and angular velocity.

##  Appendix F Statistical analysis of observables

In this appendix we provide a definition of some recurrent statistical terms
used in this work.

We assume that $O$ is an observable whose value is measured over $n$
repetitions, assumed statistically independent, of an experiment. Here, both
the term “observable” and “experiment” are used in a very broad sense.
“Observable” may refer not only to those defined in Section 3, but also to the
values assumed by the evaluation metrics defined in Appendix G. By
“experiment”, we may refer to the actual experiments with human pedestrians,
to the simulations using the same initial conditions of a given experiment
repetition, or even to an “experiment with pedestrians-simulation” pair in
which the simulation uses the same initial conditions as the experiment.

The “$n$ independent repetitions” are assumed to share some common
experimental condition (e.g., the initial density $\rho_{I}$) or a set of
common experimental conditions (e.g., initial density and model specification
used in simulation). Given these assumptions, we use some statistical tools to
understand what our experiments tell us about the expected behaviour under
these conditions and about its statistical variation. All these tools are
quite basic and are intended to provide some quantitative tool for comparison,
without any claim to “prove” or “disprove” anything (although terms like
“statistically significant” are sometimes used in our manuscript when the
evidence appears to be compelling, or on the other hand, when we want to
stress the lack of such evidence).

Given these premises, writing the value assumed by $O$ in the $k$th repetition
as $O_{k}$ we define the mean of the observable as

$<O>=\frac{\sum_{k=1}^{n}O_{k}}{n},$ (69)

its standard deviation as

$\sigma=\sqrt{\frac{\sum_{k=1}^{n}O_{k}^{2}}{n}-<O>^{2}},$ (70)

and its standard error as

$\varepsilon=\frac{\sigma}{\sqrt{n-1}}.$ (71)

When comparing results between two different conditions (e.g., when comparing
the evaluation metric values attained by different models), which we may call
condition $A$ and $B$, a possible rule of thumb is to say that $O$ assumes a
different value under the two conditions provided that

$|<O>^{A}-<O>^{B}|\gg\sqrt{2}\max(\varepsilon^{A},\varepsilon^{B}).$ (72)

This rule of thumb is obviously related to the ANOVA $p$ values reported in
the text. In general, when comparing results between 2 different conditions,
corresponding to $n^{A}$ and $n^{B}$ repetitions, we may define the overall
mean as

$<O>=\frac{<O>^{A}n^{A}+<O>^{B}n^{B}}{n^{A}+n^{B}}$ (73)

and then the deviations of each condition from this average as

$d_{A}=<O>-<O>^{A},\;d_{B}=<O>-<O>^{B}.$ (74)

The degrees of freedom are

$\gamma_{1}=1,\qquad\gamma_{2}=n^{A}+n^{B}-2.$ (75)

and the $F$ value is then defined as

$F=\frac{\left(d_{A}^{2}n^{A}+d_{B}^{2}n^{B}\right)}{\left((\sigma^{A})^{2}n^{A}+(\sigma^{B})^{2}n^{B}\right)}\frac{\gamma_{2}}{\gamma_{1}}.$
(76)

In our tables, we report the celebrated $p$-value, that provides the
probability, under the hypothesis of independence of data, that the difference
between the distributions is due to chance [62]

$p=1-\int_{0}^{F}f_{\gamma_{1},\gamma_{1}}(x)dx.$ (77)

The $F$ distribution has to be computed numerically [63], but a value $F\gg 1$
assures a small $p$-value.

Let us see how this relates to the rule of thumb for standard errors. Let us
assume, as usual in our comparisons

$n^{A}=n^{B}=n$ (78)

We have

$<O>=\frac{<O>^{A}+<O>^{B}}{2},$ (79)
$|d_{A}|=|d_{B}|=\frac{|<O>^{A}-<O>^{B}|}{2},$ (80)

and

$F=\frac{|<O>^{A}-<O>^{B}|^{2}}{(\sigma^{A})^{2}+(\sigma^{B})^{2}}(n-1).$ (81)

Using

$\frac{(\sigma^{A,B})^{2}}{n-1}=(\varepsilon^{A,B})^{2},$ (82)

we get the expression

$F\approx\frac{|<O>^{A}-<O>^{B}|^{2}}{(\varepsilon^{A})^{2}+(\varepsilon^{B})^{2}}>\frac{|<O>^{A}-<O>^{B}|^{2}}{2(\max(\varepsilon^{A},\varepsilon^{B}))^{2}},$
(83)

so that the rule of thumb eq. 72 corresponds to have a high $F$-value and thus
a low $p$-value.

The $F$ value is high if the $\sigma$s are smaller than the $d$s, i.e., if the
variations inside the categories are smaller than the ones outside the
categories, and if the total number of observations is high. It may be useful
to use also an estimator that does not depend on the number of observations.
In this work, we also use effect size, defined as

$\begin{split}\delta&=\frac{<O>^{A}-<O>^{B}}{\overline{\sigma}},\\\
\overline{\sigma}&=\sqrt{\frac{(n^{A}-1)(\sigma^{A})^{2}+(n^{B}-1)(\sigma^{B})^{2}}{n^{A}+n^{B}-2}}.\end{split}$
(84)

While a $p$-value tells us about the significance of the statistical
difference between two distributions, the difference may often be so small
that it can be verified only if a large amount of data are collected. But if
we have also $\delta\approx 1$, then the two distributions are different
enough to be distinguished also using a relatively reduced amount of data.

##  Appendix G Evaluation metrics

###  G.1 Earth Mover’s Distance

The Earth Mover’s Distance (EMD, [58]) may be used to compare probability
distributions.

The EMD process can be visualised as filling holes by moving piles of dirt.
Assume that $P$ and $Q$ denote two pdfs, and that a proper metric, named
ground distance is defined to measure the distance between the bins $i$ and
$j$ 191919It is common practice to use the Euclidean distance from $i$ to $j$
as the ground distance. Namely, in the 1D case this reduces to $|i-j|$..
Suppose also that a flow $f(i,j)$ is applied to morph $P$ to $Q$, namely a
(signed) quantity is subtracted from $P(i)$ and added to $Q(j)$ in the process
of making the $P$ and $Q$ distributions more similar.

In this contest, EMD can be formulated and solved essentially as a
transportation problem. Namely, EMD aims at finding the amount of flow $f$
that minimises the overall cost of morphing $P$ to $Q$. Explicitly, the work
required to morph $P$ to $Q$ (or vice versa) given an explicit flow $f$ is,

$\sum_{i}\sum_{j}f(i,j)d(i,j),$ (85)

where $f(i,j)$ and $d(i,j)$ are respectively, the flow and the ground distance
between $P(i)$ and $Q(j)$. By solving for the optimal flow and normalising it
with the total flow, EMD is described as,

$EMD(P,Q)=\frac{\min_{f}\sum_{i}\sum_{j}f(i,j)d(i,j)}{\sum_{i}\sum_{j}f(i,j)}.$
(86)

The normalisation operation in the above equation yields the average distance
travelled by unit weight under the optimal flow.

When computing the EMD between two discrete histograms defined on the same
array of bins $i=0,...,N-1$, the following algorithm may be used

$\displaystyle\begin{split}\text{EMD}_{0}&=0,\\\
\text{EMD}_{i+1}&=Q_{i}-P_{i}+\text{EMD}_{i},\\\
\text{EMD}&=\frac{\sum_{i=1}^{N}|\text{EMD}_{i}|}{N}.\end{split}$ (87)

In the last step, we normalise dividing by the number of bins $N$.

In this work, the comparison is performed between observables averaged over
independent repetitions of experiments according to eq. 20.

###  G.2 “Standard” metric

As shown by eq. 87, when the absolute difference between the two histograms is
small with respect to the sum over all bins, $|Q_{i}-P_{i}|\ll 1$, the
contribution of bin $i$ to the overall EMD will be $\ll 1$ (due to the final
$1/N$ normalisation), regardless of the ratio $|Q_{i}|/|P_{i}|$. Nevertheless,
such relative difference may be relevant to our study.

We were particularly interested in identifying simulated observable
distributions presenting relatively high values in regions in which the
empirical distribution presents relatively high values, and we defined the
following evaluation metric, that we name “Standard” since it is based on a
comparison using standard errors.

Let us assume that the empirical distribution of the observable under
consideration can be expressed through $\langle E_{i}\rangle\pm\epsilon_{i}$,
defined according to eqs. 20, 21. Furthermore, let us assume the corresponding
simulated distribution as given by $\langle S_{i}\rangle$.

Since we want to compare the difference between empirical and simulated
distributions with the uncertainty in the empirical data

$\delta_{i}=\frac{S_{i}-E_{i}}{\epsilon_{i}},$ (88)

the simulated distribution standard error does not play a role in the
definition.

Eq. 88 is not defined when we have $E_{i}=\epsilon_{i}=0$. To deal with such
situations, we compute

$\overline{\epsilon}=\frac{\min_{\epsilon_{i}\neq 0}\epsilon_{i}}{2}$ (89)

and define $\delta_{i}$ for the $E_{i}=\epsilon_{i}=0$ case as

$\delta_{i}=\frac{S_{i}}{\overline{\epsilon}}.$ (90)

Finally the Standard Metric is defined as

$S=\sqrt{\frac{\sum_{i=1}^{N}\delta_{i}^{2}}{N}}.$ (91)

##  Appendix H Evaluation using the Standard Metric

In Tables 5, 6, 7, 8 we evaluate the different models at $\rho_{I}=2.5$ ped/m2
and compare them using the standard metric.

Table 5: Standard Metric comparison between Full and No-short elliptical models, evaluation at $\rho_{I}=$ 2.5 ped/m2 |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
Total | $5.18\pm 0.55(2.6)$ | $10.8\pm 1.5(7)$ | 0.00084 | 1.1
$\rho$ | $3.35\pm 1.1(5.2)$ | $4.48\pm 0.88(4.2)$ | 0.43 | 0.24
$v$ | $3.71\pm 0.55(2.7)$ | $5.18\pm 0.65(3.1)$ | 0.092 | 0.51
$\theta^{v}$ | $7.49\pm 1(4.9)$ | $9.95\pm 1.5(7.2)$ | 0.18 | 0.4
$\delta^{s}$ | $7.52\pm 1.2(6)$ | $27.6\pm 5.2(25)$ | 0.00051 | 1.1
$\delta^{o}$ | $5.87\pm 2.1(9.9)$ | $28\pm 7.2(35)$ | 0.0049 | 0.87
$\phi^{s}$ | $2.27\pm 0.074(0.36)$ | $2.6\pm 0.09(0.43)$ | 0.0079 | 0.82
$\phi^{o}$ | $4.88\pm 0.3(1.4)$ | $5.16\pm 0.33(1.6)$ | 0.53 | 0.18
$\theta$ | $5.34\pm 0.33(1.6)$ | $5.19\pm 0.66(3.2)$ | 0.84 | 0.061
$\delta\theta$ | $6.16\pm 0.5(2.4)$ | $8.59\pm 1.2(5.6)$ | 0.061 | 0.57
Table 6: Standard Metric comparison between Full Elliptical and Full Circular models, evaluation at $\rho_{I}=$ 2.5 ped/m2. |  Elliptical $\langle\;\rangle\pm\varepsilon(\sigma)$ |  Circular $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $5.59\pm 0.91(4.4)$ | $7.05\pm 0.93(4.4)$ | 0.27 | 0.33
$\rho$ | $3.35\pm 1.1(5.2)$ | $5.65\pm 0.9(4.3)$ | 0.11 | 0.48
$v$ | $3.71\pm 0.55(2.7)$ | $3.69\pm 0.4(1.9)$ | 0.98 | 0.0068
$\theta^{v}$ | $7.49\pm 1(4.9)$ | $18.1\pm 3.1(15)$ | 0.0024 | 0.95
$\delta^{s}$ | $7.52\pm 1.2(6)$ | $5.46\pm 0.81(3.9)$ | 0.17 | 0.41
$\delta^{o}$ | $5.87\pm 2.1(9.9)$ | $2.41\pm 0.15(0.73)$ | 0.1 | 0.49
$\phi^{s}$ | $2.27\pm 0.074(0.36)$ | $2.19\pm 0.079(0.38)$ | 0.43 | 0.24
$\phi^{o}$ | $4.88\pm 0.3(1.4)$ | $4.67\pm 0.29(1.4)$ | 0.63 | 0.14
Table 7: Standard Metric comparison between Full Elliptical and No-Short Circular (CP) models, evaluation at $\rho_{I}=$ 2.5 ped/m2. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  CP $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $5.59\pm 0.91(4.4)$ | $6.39\pm 0.89(4.3)$ | 0.53 | 0.19
$\rho$ | $3.35\pm 1.1(5.2)$ | $4.79\pm 1.8(8.4)$ | 0.49 | 0.21
$v$ | $3.71\pm 0.55(2.7)$ | $5.82\pm 0.78(3.7)$ | 0.032 | 0.65
$\theta^{v}$ | $7.49\pm 1(4.9)$ | $12.5\pm 2(9.8)$ | 0.032 | 0.65
$\delta^{s}$ | $7.52\pm 1.2(6)$ | $5.72\pm 0.21(1)$ | 0.16 | 0.42
$\delta^{o}$ | $5.87\pm 2.1(9.9)$ | $3.09\pm 0.18(0.86)$ | 0.19 | 0.4
$\phi^{s}$ | $2.27\pm 0.074(0.36)$ | $2.52\pm 0.062(0.3)$ | 0.013 | 0.76
$\phi^{o}$ | $4.88\pm 0.3(1.4)$ | $4.61\pm 0.35(1.7)$ | 0.56 | 0.17
Table 8: Standard metric comparison between Full Circular and No-short Circular models, evaluation at $\rho_{I}=$ 2.5 ped/m2. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $7.05\pm 0.93(4.4)$ | $6.39\pm 0.89(4.3)$ | 0.61 | 0.15
$\rho$ | $5.65\pm 0.9(4.3)$ | $4.79\pm 1.8(8.4)$ | 0.67 | 0.13
$v$ | $3.69\pm 0.4(1.9)$ | $5.82\pm 0.78(3.7)$ | 0.019 | 0.72
$\theta^{v}$ | $18.1\pm 3.1(15)$ | $12.5\pm 2(9.8)$ | 0.15 | 0.44
$\delta^{s}$ | $5.46\pm 0.81(3.9)$ | $5.72\pm 0.21(1)$ | 0.75 | 0.094
$\delta^{o}$ | $2.41\pm 0.15(0.73)$ | $3.09\pm 0.18(0.86)$ | 0.0061 | 0.85
$\phi^{s}$ | $2.19\pm 0.079(0.38)$ | $2.52\pm 0.062(0.3)$ | 0.0015 | 0.99
$\phi^{o}$ | $4.67\pm 0.29(1.4)$ | $4.61\pm 0.35(1.7)$ | 0.88 | 0.044

##  Appendix I Comparison on low and medium density ranges

###  I.1 Full Elliptical vs No-short Elliptical

We compare the Full Elliptical and No-short Elliptical models on the
$\rho_{I}=0.25$ ped/m2 (see Tables 9 and 10) and $\rho_{I}=1.5$ ped/m2 (see
Tables 11 and 12) initial conditions on which they were calibrated.

Table 9: EMD metric comparison between Full and No-short elliptical models, $\rho_{I}=0.25$ ped/m2 initial condition. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
Total | $0.00885\pm 0.00026(0.0013)$ | $0.00862\pm 0.0002(0.00098)$ | 0.49 | 0.21
$\rho$ | $0.00458\pm 0.00013(0.00062)$ | $0.00458\pm 0.00012(0.00056)$ | 1 | 0
$v$ | $0.0189\pm 0.0015(0.0073)$ | $0.0166\pm 0.00083(0.004)$ | 0.2 | 0.39
$\theta^{v}$ | $0.00459\pm 0.0003(0.0014)$ | $0.00474\pm 0.00023(0.0011)$ | 0.7 | 0.12
$\delta^{s}$ | $0.00969\pm 0.00081(0.0039)$ | $0.00761\pm 0.0007(0.0033)$ | 0.057 | 0.58
$\delta^{o}$ | $0.00957\pm 0.00089(0.0043)$ | $0.0113\pm 0.00068(0.0033)$ | 0.13 | 0.45
$\phi^{s}$ | $0.0103\pm 0.00082(0.0039)$ | $0.00896\pm 0.00096(0.0046)$ | 0.29 | 0.31
$\phi^{o}$ | $0.00845\pm 0.00072(0.0035)$ | $0.00862\pm 0.00063(0.003)$ | 0.86 | 0.053
$\theta$ | $0.00721\pm 0.00012(0.00058)$ | $0.00742\pm 0.0001(0.0005)$ | 0.2 | 0.39
$\delta\theta$ | $0.00743\pm 0.00025(0.0012)$ | $0.00808\pm 0.00021(0.00099)$ | 0.055 | 0.58
Table 10: Standard Metric comparison between Full and No-short elliptical models, $\rho_{I}=0.25$ ped/m2 initial condition. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
Total | $3.63\pm 0.12(0.57)$ | $3.79\pm 0.12(0.56)$ | 0.37 | 0.27
$\rho$ | $0.898\pm 0.034(0.16)$ | $0.901\pm 0.024(0.11)$ | 0.94 | 0.023
$v$ | $4.46\pm 0.33(1.6)$ | $3.67\pm 0.21(1)$ | 0.05 | 0.59
$\theta^{v}$ | $6.04\pm 0.45(2.2)$ | $6.65\pm 0.42(2)$ | 0.33 | 0.29
$\delta^{s}$ | $2.78\pm 0.22(1)$ | $3.34\pm 0.38(1.8)$ | 0.21 | 0.37
$\delta^{o}$ | $3.16\pm 0.26(1.2)$ | $3.72\pm 0.13(0.65)$ | 0.057 | 0.58
$\phi^{s}$ | $1.41\pm 0.043(0.2)$ | $1.33\pm 0.038(0.18)$ | 0.19 | 0.39
$\phi^{o}$ | $1.84\pm 0.06(0.29)$ | $1.93\pm 0.049(0.24)$ | 0.24 | 0.35
$\theta$ | $3.89\pm 0.18(0.87)$ | $3.99\pm 0.28(1.3)$ | 0.78 | 0.085
$\delta\theta$ | $4.21\pm 0.27(1.3)$ | $4.24\pm 0.17(0.81)$ | 0.92 | 0.031
Table 11: EMD metric comparison between Full and No-short elliptical models, $\rho_{I}=1.5$ ped/m2 initial condition. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
Total | $0.0128\pm 0.00056(0.0027)$ | $0.0162\pm 0.001(0.005)$ | 0.0062 | 0.85
$\rho$ | $0.00975\pm 0.00094(0.0045)$ | $0.011\pm 0.001(0.0049)$ | 0.35 | 0.28
$v$ | $0.0211\pm 0.0025(0.012)$ | $0.0295\pm 0.0041(0.02)$ | 0.087 | 0.52
$\theta^{v}$ | $0.00813\pm 0.00056(0.0027)$ | $0.00892\pm 0.0006(0.0029)$ | 0.34 | 0.28
$\delta^{s}$ | $0.0202\pm 0.0009(0.0043)$ | $0.0274\pm 0.00095(0.0046)$ | 1.3e-06 | 1.6
$\delta^{o}$ | $0.0114\pm 0.0012(0.0058)$ | $0.0139\pm 0.003(0.014)$ | 0.44 | 0.23
$\phi^{s}$ | $0.0153\pm 0.0015(0.0072)$ | $0.0174\pm 0.0021(0.01)$ | 0.41 | 0.25
$\phi^{o}$ | $0.0312\pm 0.0026(0.012)$ | $0.0328\pm 0.0025(0.012)$ | 0.67 | 0.13
$\theta$ | $0.00978\pm 0.00076(0.0037)$ | $0.00969\pm 0.0019(0.0092)$ | 0.97 | 0.012
$\delta\theta$ | $0.00916\pm 0.00071(0.0034)$ | $0.0127\pm 0.00081(0.0039)$ | 0.0021 | 0.96
Table 12: Standard Metric comparison between Full and No-short elliptical models, $\rho_{I}=1.5$ ped/m2 initial condition. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
Total | $5.85\pm 0.36(1.7)$ | $11.3\pm 2(9.7)$ | 0.01 | 0.79
$\rho$ | $2.35\pm 0.37(1.8)$ | $2.61\pm 0.29(1.4)$ | 0.58 | 0.16
$v$ | $8.73\pm 1(5)$ | $11\pm 1.4(6.6)$ | 0.19 | 0.39
$\theta^{v}$ | $6.29\pm 0.37(1.8)$ | $7.41\pm 0.72(3.4)$ | 0.17 | 0.41
$\delta^{s}$ | $9.65\pm 0.99(4.8)$ | $35.5\pm 8.9(43)$ | 0.006 | 0.85
$\delta^{o}$ | $3.55\pm 0.55(2.7)$ | $11.3\pm 3.2(15)$ | 0.022 | 0.7
$\phi^{s}$ | $1.73\pm 0.069(0.33)$ | $1.88\pm 0.075(0.36)$ | 0.15 | 0.43
$\phi^{o}$ | $3.98\pm 0.26(1.2)$ | $4.57\pm 0.24(1.2)$ | 0.099 | 0.5
$\theta$ | $5.17\pm 0.4(1.9)$ | $4.69\pm 0.81(3.9)$ | 0.6 | 0.16
$\delta\theta$ | $5.26\pm 0.56(2.7)$ | $6.83\pm 1.2(6)$ | 0.26 | 0.34

###  I.2 Full Elliptical vs Full Circular

We compare the Full Elliptical and Full Circular models on the $\rho_{I}=0.25$
ped/m2 (see Tables 13 and 14) and $\rho_{I}=1.5$ ped/m2 (see Tables 15 and 16)
initial conditions on which they were calibrated.

Table 13: EMD metric comparison between Full Elliptical and Full Circular models, $\rho_{I}=0.25$ ped/m2 initial condition. |  Elliptical $\langle\;\rangle\pm\varepsilon(\sigma)$ |  Circular $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $0.00947\pm 0.00036(0.0018)$ | $0.00996\pm 0.00037(0.0018)$ | 0.35 | 0.28
$\rho$ | $0.00458\pm 0.00013(0.00062)$ | $0.00515\pm 0.00016(0.00077)$ | 0.0082 | 0.82
$v$ | $0.0189\pm 0.0015(0.0073)$ | $0.02\pm 0.0013(0.006)$ | 0.58 | 0.16
$\theta^{v}$ | $0.00459\pm 0.0003(0.0014)$ | $0.00385\pm 0.00029(0.0014)$ | 0.079 | 0.53
$\delta^{s}$ | $0.00969\pm 0.00081(0.0039)$ | $0.00915\pm 0.00056(0.0027)$ | 0.59 | 0.16
$\delta^{o}$ | $0.00957\pm 0.00089(0.0043)$ | $0.0117\pm 0.00095(0.0046)$ | 0.12 | 0.47
$\phi^{s}$ | $0.0103\pm 0.00082(0.0039)$ | $0.0101\pm 0.00074(0.0036)$ | 0.89 | 0.043
$\phi^{o}$ | $0.00845\pm 0.00072(0.0035)$ | $0.0111\pm 0.00093(0.0045)$ | 0.032 | 0.65
Table 14: Standard Metric comparison between Full Elliptical and Full Circular models, $\rho_{I}=0.25$ ped/m2 initial condition. |  Elliptical $\langle\;\rangle\pm\varepsilon(\sigma)$ |  Circular $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $3.47\pm 0.14(0.67)$ | $3.51\pm 0.17(0.8)$ | 0.84 | 0.058
$\rho$ | $0.898\pm 0.034(0.16)$ | $0.949\pm 0.025(0.12)$ | 0.23 | 0.36
$v$ | $4.46\pm 0.33(1.6)$ | $4.98\pm 0.49(2.3)$ | 0.38 | 0.26
$\theta^{v}$ | $6.04\pm 0.45(2.2)$ | $4.92\pm 0.4(1.9)$ | 0.069 | 0.55
$\delta^{s}$ | $2.78\pm 0.22(1)$ | $3.13\pm 0.3(1.4)$ | 0.35 | 0.28
$\delta^{o}$ | $3.16\pm 0.26(1.2)$ | $3.57\pm 0.22(1.1)$ | 0.23 | 0.36
$\phi^{s}$ | $1.41\pm 0.043(0.2)$ | $1.36\pm 0.044(0.21)$ | 0.44 | 0.23
$\phi^{o}$ | $1.84\pm 0.06(0.29)$ | $2.14\pm 0.094(0.45)$ | 0.0091 | 0.8
Table 15: EMD metric comparison between Full Elliptical and Full Circular models, $\rho_{I}=1.5$ ped/m2 initial condition. |  Elliptical $\langle\;\rangle\pm\varepsilon(\sigma)$ |  Circular $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $0.0141\pm 0.00077(0.0037)$ | $0.015\pm 0.00067(0.0032)$ | 0.38 | 0.26
$\rho$ | $0.00975\pm 0.00094(0.0045)$ | $0.0101\pm 0.00069(0.0033)$ | 0.74 | 0.097
$v$ | $0.0211\pm 0.0025(0.012)$ | $0.0248\pm 0.0019(0.009)$ | 0.24 | 0.35
$\theta^{v}$ | $0.00813\pm 0.00056(0.0027)$ | $0.00694\pm 0.0005(0.0024)$ | 0.12 | 0.47
$\delta^{s}$ | $0.0202\pm 0.0009(0.0043)$ | $0.0224\pm 0.0012(0.0058)$ | 0.15 | 0.43
$\delta^{o}$ | $0.0114\pm 0.0012(0.0058)$ | $0.0108\pm 0.0018(0.0088)$ | 0.77 | 0.086
$\phi^{s}$ | $0.0153\pm 0.0015(0.0072)$ | $0.0192\pm 0.0023(0.011)$ | 0.16 | 0.43
$\phi^{o}$ | $0.0312\pm 0.0026(0.012)$ | $0.0259\pm 0.0024(0.012)$ | 0.14 | 0.44
Table 16: Standard Metric comparison between Full Elliptical and Full Circular models, $\rho_{I}=1.5$ ped/m2 initial condition. |  Elliptical $\langle\;\rangle\pm\varepsilon(\sigma)$ |  Circular $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $6.11\pm 0.43(2.1)$ | $6.28\pm 0.4(1.9)$ | 0.78 | 0.083
$\rho$ | $2.35\pm 0.37(1.8)$ | $2.62\pm 0.23(1.1)$ | 0.53 | 0.19
$v$ | $8.73\pm 1(5)$ | $6.69\pm 0.86(4.1)$ | 0.14 | 0.44
$\theta^{v}$ | $6.29\pm 0.37(1.8)$ | $8.96\pm 1.1(5.2)$ | 0.025 | 0.68
$\delta^{s}$ | $9.65\pm 0.99(4.8)$ | $10.1\pm 1.6(7.5)$ | 0.82 | 0.069
$\delta^{o}$ | $3.55\pm 0.55(2.7)$ | $3.03\pm 0.3(1.5)$ | 0.42 | 0.24
$\phi^{s}$ | $1.73\pm 0.069(0.33)$ | $1.91\pm 0.076(0.37)$ | 0.093 | 0.51
$\phi^{o}$ | $3.98\pm 0.26(1.2)$ | $3.84\pm 0.24(1.1)$ | 0.69 | 0.12

###  I.3 Full Elliptical vs No-short Circular (CP)

We compare the Full Elliptical and No-short Circular models on the
$\rho_{I}=0.25$ ped/m2 (see Tables 17 and 18) and $\rho_{I}=1.5$ ped/m2 (see
Tables 19 and 20) initial conditions on which they were calibrated.

Table 17: EMD metric comparison between Full Elliptical and No-Short Circular (CP) models, $\rho_{I}=0.25$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  CP $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $0.00947\pm 0.00036(0.0018)$ | $0.0108\pm 0.00046(0.0022)$ | 0.031 | 0.66
$\rho$ | $0.00458\pm 0.00013(0.00062)$ | $0.00479\pm 0.00013(0.0006)$ | 0.24 | 0.35
$v$ | $0.0189\pm 0.0015(0.0073)$ | $0.0198\pm 0.0009(0.0043)$ | 0.62 | 0.15
$\theta^{v}$ | $0.00459\pm 0.0003(0.0014)$ | $0.00418\pm 0.00023(0.0011)$ | 0.28 | 0.32
$\delta^{s}$ | $0.00969\pm 0.00081(0.0039)$ | $0.0116\pm 0.001(0.005)$ | 0.15 | 0.43
$\delta^{o}$ | $0.00957\pm 0.00089(0.0043)$ | $0.0135\pm 0.0014(0.0066)$ | 0.02 | 0.71
$\phi^{s}$ | $0.0103\pm 0.00082(0.0039)$ | $0.00959\pm 0.00096(0.0046)$ | 0.57 | 0.17
$\phi^{o}$ | $0.00845\pm 0.00072(0.0035)$ | $0.00992\pm 0.00045(0.0021)$ | 0.09 | 0.51
Table 18: Standard Metric comparison between Full Elliptical and No-Short Circular (CP) models, $\rho_{I}=0.25$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  CP $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $3.47\pm 0.14(0.67)$ | $4.41\pm 0.22(1.1)$ | 0.00079 | 1.1
$\rho$ | $0.898\pm 0.034(0.16)$ | $0.908\pm 0.022(0.11)$ | 0.8 | 0.077
$v$ | $4.46\pm 0.33(1.6)$ | $5.3\pm 0.43(2.1)$ | 0.13 | 0.46
$\theta^{v}$ | $6.04\pm 0.45(2.2)$ | $5.74\pm 0.37(1.8)$ | 0.61 | 0.15
$\delta^{s}$ | $2.78\pm 0.22(1)$ | $5.6\pm 0.57(2.7)$ | 3.1e-05 | 1.4
$\delta^{o}$ | $3.16\pm 0.26(1.2)$ | $4.47\pm 0.31(1.5)$ | 0.0021 | 0.96
$\phi^{s}$ | $1.41\pm 0.043(0.2)$ | $1.29\pm 0.035(0.17)$ | 0.045 | 0.61
$\phi^{o}$ | $1.84\pm 0.06(0.29)$ | $2.22\pm 0.058(0.28)$ | 4.2e-05 | 1.3
Table 19: EMD metric comparison between Full Elliptical and No-Short Circular (CP) models, $\rho_{I}=1.5$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  CP $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $0.0141\pm 0.00077(0.0037)$ | $0.0199\pm 0.0012(0.0057)$ | 0.00019 | 1.2
$\rho$ | $0.00975\pm 0.00094(0.0045)$ | $0.0125\pm 0.0011(0.0052)$ | 0.062 | 0.56
$v$ | $0.0211\pm 0.0025(0.012)$ | $0.0377\pm 0.004(0.019)$ | 0.001 | 1
$\theta^{v}$ | $0.00813\pm 0.00056(0.0027)$ | $0.00765\pm 0.00045(0.0022)$ | 0.51 | 0.2
$\delta^{s}$ | $0.0202\pm 0.0009(0.0043)$ | $0.0268\pm 0.00096(0.0046)$ | 7.6e-06 | 1.5
$\delta^{o}$ | $0.0114\pm 0.0012(0.0058)$ | $0.0147\pm 0.0013(0.006)$ | 0.066 | 0.55
$\phi^{s}$ | $0.0153\pm 0.0015(0.0072)$ | $0.0177\pm 0.0014(0.0066)$ | 0.23 | 0.36
$\phi^{o}$ | $0.0312\pm 0.0026(0.012)$ | $0.0254\pm 0.0024(0.012)$ | 0.11 | 0.48
Table 20: Standard Metric comparison between Full Elliptical and No-Short Circular (CP) models, $\rho_{I}=1.5$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  CP $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $6.11\pm 0.43(2.1)$ | $8.28\pm 0.48(2.3)$ | 0.0015 | 1
$\rho$ | $2.35\pm 0.37(1.8)$ | $2.77\pm 0.28(1.3)$ | 0.36 | 0.27
$v$ | $8.73\pm 1(5)$ | $12.3\pm 1.7(8.2)$ | 0.084 | 0.52
$\theta^{v}$ | $6.29\pm 0.37(1.8)$ | $8.94\pm 0.88(4.2)$ | 0.0079 | 0.82
$\delta^{s}$ | $9.65\pm 0.99(4.8)$ | $12.5\pm 0.9(4.3)$ | 0.039 | 0.63
$\delta^{o}$ | $3.55\pm 0.55(2.7)$ | $4.95\pm 0.42(2)$ | 0.05 | 0.59
$\phi^{s}$ | $1.73\pm 0.069(0.33)$ | $1.92\pm 0.081(0.39)$ | 0.091 | 0.51
$\phi^{o}$ | $3.98\pm 0.26(1.2)$ | $4.03\pm 0.28(1.3)$ | 0.89 | 0.041

###  I.4 Full Circular vs No-short Circular (CP)

We compare the Full Circular and No-short Circular models on the
$\rho_{I}=0.25$ ped/m2 (see Tables 21 and 22) and $\rho_{I}=1.5$ ped/m2 (see
Tables 23 and 24) initial conditions on which they were calibrated.

Table 21: EMD metric comparison between Full Circular and No-short Circular models, $\rho_{I}=0.25$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $0.00996\pm 0.00037(0.0018)$ | $0.0108\pm 0.00046(0.0022)$ | 0.17 | 0.41
$\rho$ | $0.00515\pm 0.00016(0.00077)$ | $0.00479\pm 0.00013(0.0006)$ | 0.088 | 0.51
$v$ | $0.02\pm 0.0013(0.006)$ | $0.0198\pm 0.0009(0.0043)$ | 0.89 | 0.042
$\theta^{v}$ | $0.00385\pm 0.00029(0.0014)$ | $0.00418\pm 0.00023(0.0011)$ | 0.36 | 0.27
$\delta^{s}$ | $0.00915\pm 0.00056(0.0027)$ | $0.0116\pm 0.001(0.005)$ | 0.041 | 0.62
$\delta^{o}$ | $0.0117\pm 0.00095(0.0046)$ | $0.0135\pm 0.0014(0.0066)$ | 0.28 | 0.33
$\phi^{s}$ | $0.0101\pm 0.00074(0.0036)$ | $0.00959\pm 0.00096(0.0046)$ | 0.65 | 0.14
$\phi^{o}$ | $0.0111\pm 0.00093(0.0045)$ | $0.00992\pm 0.00045(0.0021)$ | 0.28 | 0.33
Table 22: Standard metric comparison between Full Circular and No-short Circular models, $\rho_{I}=0.25$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $3.51\pm 0.17(0.8)$ | $4.41\pm 0.22(1.1)$ | 0.0023 | 0.95
$\rho$ | $0.949\pm 0.025(0.12)$ | $0.908\pm 0.022(0.11)$ | 0.22 | 0.36
$v$ | $4.98\pm 0.49(2.3)$ | $5.3\pm 0.43(2.1)$ | 0.63 | 0.14
$\theta^{v}$ | $4.92\pm 0.4(1.9)$ | $5.74\pm 0.37(1.8)$ | 0.14 | 0.45
$\delta^{s}$ | $3.13\pm 0.3(1.4)$ | $5.6\pm 0.57(2.7)$ | 0.00037 | 1.1
$\delta^{o}$ | $3.57\pm 0.22(1.1)$ | $4.47\pm 0.31(1.5)$ | 0.023 | 0.7
$\phi^{s}$ | $1.36\pm 0.044(0.21)$ | $1.29\pm 0.035(0.17)$ | 0.25 | 0.35
$\phi^{o}$ | $2.14\pm 0.094(0.45)$ | $2.22\pm 0.058(0.28)$ | 0.49 | 0.2
Table 23: EMD metric comparison between Full Circular and No-short Circular models, $\rho_{I}=1.5$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $0.015\pm 0.00067(0.0032)$ | $0.0199\pm 0.0012(0.0057)$ | 0.0009 | 1
$\rho$ | $0.0101\pm 0.00069(0.0033)$ | $0.0125\pm 0.0011(0.0052)$ | 0.072 | 0.54
$v$ | $0.0248\pm 0.0019(0.009)$ | $0.0377\pm 0.004(0.019)$ | 0.0059 | 0.85
$\theta^{v}$ | $0.00694\pm 0.0005(0.0024)$ | $0.00765\pm 0.00045(0.0022)$ | 0.3 | 0.31
$\delta^{s}$ | $0.0224\pm 0.0012(0.0058)$ | $0.0268\pm 0.00096(0.0046)$ | 0.006 | 0.85
$\delta^{o}$ | $0.0108\pm 0.0018(0.0088)$ | $0.0147\pm 0.0013(0.006)$ | 0.086 | 0.52
$\phi^{s}$ | $0.0192\pm 0.0023(0.011)$ | $0.0177\pm 0.0014(0.0066)$ | 0.58 | 0.16
$\phi^{o}$ | $0.0259\pm 0.0024(0.012)$ | $0.0254\pm 0.0024(0.012)$ | 0.89 | 0.041
Table 24: Standard metric comparison between Full Circular and No-short Circular models, $\rho_{I}=1.5$ ped/m2 initial conditions. |  Full $\langle\;\rangle\pm\varepsilon(\sigma)$ |  No-short $\langle\;\rangle\pm\varepsilon(\sigma)$ | $p$ | effect size
---|---|---|---|---
No body | $6.28\pm 0.4(1.9)$ | $8.28\pm 0.48(2.3)$ | 0.0024 | 0.95
$\rho$ | $2.62\pm 0.23(1.1)$ | $2.77\pm 0.28(1.3)$ | 0.68 | 0.12
$v$ | $6.69\pm 0.86(4.1)$ | $12.3\pm 1.7(8.2)$ | 0.0054 | 0.86
$\theta^{v}$ | $8.96\pm 1.1(5.2)$ | $8.94\pm 0.88(4.2)$ | 0.99 | 0.0047
$\delta^{s}$ | $10.1\pm 1.6(7.5)$ | $12.5\pm 0.9(4.3)$ | 0.19 | 0.39
$\delta^{o}$ | $3.03\pm 0.3(1.5)$ | $4.95\pm 0.42(2)$ | 0.00062 | 1.1
$\phi^{s}$ | $1.91\pm 0.076(0.37)$ | $1.92\pm 0.081(0.39)$ | 0.95 | 0.02
$\phi^{o}$ | $3.84\pm 0.24(1.1)$ | $4.03\pm 0.28(1.3)$ | 0.6 | 0.15

##  Appendix J Model pdf comparison

In this section, we compare the observable pdf obtained by each model on the
$\rho_{I}=2.5$ ped/m2 initial condition with the experimental data (See Figs.
25-32). Experimental data are reported as average over all experiment
repetitions, while the model standard errors are computed by averaging over
all GA solutions (i.e., for each solution we first compute the average over
experiment repetitions, and then use the variation of such averages on GA
solutions to compute the standard errors in the figures).

Figure 25:  $\rho$ standard error intervals in different models (dashed lines)
compared to the experimental values (squares) for the $\rho_{I}=2.5$ ped/m2
initial condition. (a): linear plot. (b): logarithmic plot. Figure 26:  $v$
standard error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(a): linear plot. (b): logarithmic plot.  Figure 27:  $\theta^{v}$ standard
error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(a): linear plot. (b): logarithmic plot.  Figure 28:  $\delta^{o}$ standard
error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(a): linear plot. (b): logarithmic plot.  Figure 29:  $\delta^{s}$ standard
error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(a): linear plot. (b): logarithmic plot.  Figure 30: (a): $\phi^{s}$ standard
error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(b): $\phi^{o}$ standard error intervals in different models (dashed lines)
compared to the experimental values (squares).  Figure 31:  $\theta$ standard
error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(a): linear plot. (b): logarithmic plot.  Figure 32:  $\delta\theta$ standard
error intervals in different models (dashed lines) compared to the
experimental values (squares) for the $\rho_{I}=2.5$ ped/m2 initial condition.
(a): linear plot. (b): logarithmic plot.

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# Quasar X-ray and UV flux, baryon acoustic oscillation, and Hubble parameter
measurement constraints on cosmological model parameters

Narayan Khadka,1 and Bharat Ratra,1
1Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan,
KS 66502, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We use the Risaliti $\&$ Lusso (2015) compilation of 808 X-ray and UV flux
measurements of quasars (QSOs) in the redshift range $0.061\leq z\leq 6.28$,
alone and in conjunction with baryon acoustic oscillation (BAO) and Hubble
parameter [$H(z)$] measurements, to constrain cosmological parameters in six
cosmological models. The QSO data constraints are significantly weaker than,
but consistent with, those from the $H(z)$ \+ BAO data. A joint analysis of
the QSO + $H(z)$ \+ BAO data is consistent with the current standard model,
spatially-flat $\Lambda$CDM, but mildly favors closed spatial hypersurfaces
and dynamical dark energy.

###### keywords:

(cosmology:) cosmological parameters – (cosmology:) observations –
(cosmology:) dark energy

††pubyear: 2019††pagerange: Quasar X-ray and UV flux, baryon acoustic
oscillation, and Hubble parameter measurement constraints on cosmological
model parameters–Quasar X-ray and UV flux, baryon acoustic oscillation, and
Hubble parameter measurement constraints on cosmological model parameters

##  1 Introduction

Type Ia supernova (SNIa) apparent magnitude measurements provided the first
convincing evidence for accelerated cosmological expansion (see Scolnic et al.
(2018) for a recent discussion). Supporting evidence soon came from other
cosmological probes, the most significant being cosmic microwave background
(CMB) anisotropy data (Plank Collaboration, 2018), baryon acoustic oscillation
(BAO) distance measurements (Alam et al., 2017), and Hubble parameter [$H(z)$]
observations (Moresco et al., 2016; Farooq et al., 2017). If general
relativity is an accurate model of gravitation, hypothetical dark energy is
responsible for the observed acceleration of the cosmological expansion. There
are many different dark energy models. In this paper we consider three of them
and also consider flat and non-flat spatial hypersurfaces in each case, for a
total of six cosmological models.

The simplest observationally-consistent dark energy model is the flat
$\Lambda$CDM model, the current standard model (Peebles 1984). In this model
the accelerated expansion is powered by the spatially homogenous cosmological
constant ($\Lambda$) energy density which is constant in time. This model is
consistent with most observations when about $70\%$ of the current
cosmological energy budget is contributed by dark energy, with about 25$\%$
coming from the cold dark matter (CDM), and the remaining 5$\%$ due to
baryons. The standard model assumes flat spatial hypersurfaces. Current
observations allow a little spatial curvature,111For discussion of
observational constraints on spatial curvature, see Farooq et al. (2015), Chen
et al. (2016), Yu $\&$ Wang (2016), Wei $\&$ Wu (2017), Rana et al. (2017),
Ooba et al. (2018a, b, c), DES Collaboration (2018a), Witzemann et al. (2018),
Yu et al. (2018), Park $\&$ Ratra (2018a, b, c, d, 2019), Mitra et al. (2018),
Penton et al. (2018), Xu et al. (2019), Zheng et al. (2019), Ruan et al.
(2019), Giambó et al. (2019), Coley (2019), Eingorn et al. (2019), Jesus et
al. (2019), Handley et al. (2019), and references therein. so we can
generalise the standard model to the non-flat $\Lambda$CDM model which allows
for non-zero spatial curvature energy density.

While the $\Lambda$CDM model is consistent with many observations, its
assumption of a time-independent and spatially-homogeneous dark energy density
is difficult to theoretically motivate. Also, observations do not require that
the dark energy density be time independent, and models in which the dark
energy density decreases with time have been studied. Here we consider two
dynamical dark energy models, the XCDM parametrization in which an $X$-fluid
is the dynamical dark energy and the $\phi$CDM model in which a scalar field
$\phi$ is the dynamical dark energy. We also study spatially flat and non-flat
versions of both the XCDM parametrization and the $\phi$CDM model.

The main goal of our paper is to use the Risaliti $\&$ Lusso (2015) quasar
(QSO) X-ray and UV flux measurements to constrain cosmological parameters.
Risaliti $\&$ Lusso (2015) consider cosmological parameter constraints in the
non-flat $\Lambda$CDM model; here we also consider cosmological parameter
constraints in five other cosmological models. In addition, we examine the
effect of different Hubble constant priors on the cosmological parameter
constraints. By studying constraints in a number of models, we are able to
draw somewhat model-independent conclusions about the QSO data constraints. We
find that the Risaliti $\&$ Lusso (2015) QSO data by themselves do not provide
very restrictive constraints on cosmological parameters. However, the QSO
constraints are largely consistent with those that follow from the $H(z)$ \+
BAO data, and when jointly analyzed the QSO data slightly tighten and shift
the $H(z)$ \+ BAO data constraints.

The QSO + $H(z)$ \+ BAO data are consistent with the standard flat
$\Lambda$CDM cosmological model although they mildly favor closed spatial
hypersurfaces over flat ones and dynamical dark energy over a cosmological
constant.

While current QSO data by themselves do not provide restrictive cosmological
parameter constraints, the new Risaliti $\&$ Lusso (2018) compilation of 1598
QSO measurements will provide tighter constraints, that should be improved
upon by near-future QSO data. Currently, CMB anisotropy, BAO, SNIa, and $H(z)$
data provide the most restrictive constraints on cosmological parameters. To
test consistency, and to help tighten cosmological parameter constraints, it
is essential that additional cosmological probes, such as the QSO data studied
here, be developed.

This paper is organized as follows. In Sec. 2 we describe the models that we
use. In Sec. 3 we discuss the data that we use to constrain cosmological
parameters in these models. In Sec. 4 we describe the methodology adopted for
these analyses. In Sec. 5 we present our results and conclude in Sec. 6.

##  2 Models

In this paper we constrain cosmological parameters of the spatially-flat and
non-flat versions of three different dark energy cosmological models, for a
total of six cosmological models. For the dark energy we consider a
cosmological constant $\Lambda$ in the $\Lambda$ cold dark matter
($\Lambda$CDM) model, as well as a decreasing dark energy density modeled as
an $X$-fluid in the XCDM parametrization, or as a scalar field $\phi$ in the
$\phi$CDM model.

In the $\Lambda$CDM model the Hubble parameter, as a function of redshift $z$,
is

$H(z)=H_{0}\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{k0}(1+z)^{2}+\Omega_{\Lambda}},$
(1)

and

$\Omega_{m0}+\Omega_{k0}+\Omega_{\Lambda}=1.$ (2)

Here $H_{0}$ is the Hubble constant, $\Omega_{m0}$ and $\Omega_{k0}$ are the
current values of the non-relativistic matter and the spatial curvature energy
density parameters and $\Omega_{\Lambda}$ is the dark energy density
parameter. For the spatially-flat $\Lambda$CDM model the free parameters are
chosen to be $\Omega_{m0}$ and $H_{0}$. For the spatially non-flat
$\Lambda$CDM model the free parameters are chosen to be $\Omega_{m0}$,
$\Omega_{\Lambda}$, and $H_{0}$.

In the XCDM parametrization, the dark energy density is dynamical and
decreases with time. The equation of state for the dark energy fluid is
$P_{X}$ = $\omega_{X}$ $\rho_{X}$. Here $P_{X}$ is the pressure of the
$X$-fluid, $\rho_{X}$ is the energy density of that fluid, and $\omega_{X}$ is
the equation of state parameter whose value is negative ($\omega_{X}<-1/3$).
In this model the Hubble parameter is

$H(z)=H_{0}\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{k0}(1+z)^{2}+\Omega_{X0}(1+z)^{3(1+\omega_{X})}},$
(3)

and

$\Omega_{m0}+\Omega_{k0}+\Omega_{X0}=1.$ (4)

Here $\Omega_{X0}$ is the current value of the $X$-fluid energy density
parameter. For the spatially-flat case the free parameters are $\Omega_{m0}$,
$\omega_{X}$, and $H_{0}$. For the non-flat case the free parameters are
$\Omega_{m0}$, $\Omega_{k0}$, $\omega_{X}$, and $H_{0}$. In the $\omega_{X}$ =
$-1$ limit the XCDM model is the $\Lambda$CDM model.

In the $\phi$CDM model dark energy is modeled as a scalar field $\phi$ with
potential energy density $V(\phi)$ (Peebles $\&$ Ratra, 1988; Ratra $\&$
Peebles, 1988; Pavlov et al., 2013).222Discussion of constraints on the
$\phi$CDM model may be traced through Chen $\&$ Ratra (2004), Samushia et al.
(2007), Yashar et al. (2009), Samushia $\&$ Ratra (2010), Samushia et al.
(2010), Chen $\&$ Ratra (2011b), Campanelli et al. (2012), Farooq $\&$ Ratra
(2013), Farooq et al. (2013), Avsajanishvili et al. (2015), Sòla et al.
(2017), Sòla Peracaula et al. (2018, 2019), Zhai et al. (2017), Sangwan et al.
(2018), Singh et al. (2019), Mitra (2019), and references therein. In this
model the dark energy density is dynamical and decreases with time. A widely
used $V(\phi)$ is of the inverse power law form,

$V(\phi)=\frac{1}{2}\kappa m_{p}^{2}\phi^{-\alpha},$ (5)

where $m_{p}$ is the Planck mass, $\alpha$ is a positive parameter, and

$\kappa=\frac{8}{3}\left(\frac{\alpha+4}{\alpha+2}\right)\left[\frac{2}{3}\alpha(\alpha+2)\right]^{\alpha/2}.$
(6)

In this model the equations of motion are

$\ddot{\phi}+\frac{3\dot{a}}{a}\dot{\phi}-\frac{1}{2}\alpha\kappa
m_{p}^{2}\phi^{-\alpha-1}=0,$ (7)

and,

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\uppi
G}{3}\left(\rho_{m}+\rho_{\phi}\right)-\frac{k}{a^{2}},$ (8)

where $a$ is the scale factor, overdots denote derivatives with respect to
time, $k$ is positive, zero, and negative for closed, flat, and open spatial
hypersurfaces, $\rho_{m}$ is the non-relativistic matter density, and the
scalar field energy density is

$\rho_{\phi}=\frac{m^{2}_{p}}{32\pi}[\dot{\phi}^{2}+\kappa
m^{2}_{p}\phi^{-\alpha}].$ (9)

So, the Hubble parameter in this model is

$H(z)=H_{0}\sqrt{\Omega_{m0}\left(1+z\right)^{3}+\Omega_{k0}\left(1+z\right)^{2}+\Omega_{\phi}\left(z,\alpha\right)},$
(10)

where the scalar field energy density parameter

$\Omega_{\phi}(z,\alpha)=\frac{8\uppi G\rho_{\phi}}{3H^{2}_{0}},$ (11)

where $G$ is the gravitational constant, and $\Omega_{\phi}(z,\alpha)$ has to
be numerically computed. For the spatially non-flat $\phi$CDM model the free
parameters are $\Omega_{m0}$, $\Omega_{k0}$, $\alpha$, and $H_{0}$. For the
spatially-flat $\phi$CDM model the free parameters are $\Omega_{m0}$,
$\alpha$, and $H_{0}$. In the limit $\alpha\rightarrow 0$, the $\phi$CDM model
reduces to the $\Lambda$CDM model.

##  3 Data

We use three different data sets to constrain cosmological parameters. The
main purpose of our paper is to use the 808 QSO X-ray and UV flux measurements
of Risaliti $\&$ Lusso (2015),333Also see Risaliti $\&$ Lusso (2016), Lusso
$\&$ Risaliti (2017), and Bisogni et al. (2017). For a newer compilation of
QSO data see Risaliti $\&$ Lusso (2018). For cosmological parameter
constraints derived from QSO data, also see López-Corredoria et al. (2016),
Lusso et al. (2019), Melia (2019), and Lazkoz et al. (2019). extending over a
redshift range of $0.061\leq z\leq 6.28$, to determine cosmological parameter
constraints, and to compare these QSO cosmological parameter constraints to
those determined from more widely used BAO distance measurements and $H(z)$
observations. The BAO and $H(z)$ data we use are listed in Tables 1 and 2 of
Ryan et al. (2018) and consist of 11 BAO measurements over the redshift range
$0.106\leq z\leq 2.36$ and 31 $H(z)$ measurements over the redshift range
$0.07\leq z\leq 1.965$.

##  4 Method

As described in Risaliti $\&$ Lusso (2015), the method of analysis depends on
the non-linear relation between the X-ray and UV luminosities of quasars. This
relation is

$\log(L_{X})=\beta+\gamma\log(L_{UV}),$ (12)

where $\log$ = $\log_{10}$ and $L_{X}$ and $L_{UV}$ are the QSO X-ray and UV
luminosities. $\beta$ and $\gamma$ are free parameters to be determined by
using the data.

Expressing the luminosity in terms of the flux, we obtain

$\log(F_{X})=\beta+(\gamma-1)\log(4\pi)+\gamma\log(F_{UV})+2(\gamma-1)\log(D_{L}),$
(13)

where $F_{X}$ and $F_{UV}$ are the X-ray and UV fluxes respectively. Here
$D_{L}$ is the luminosity distance, which is a function of redshift and
cosmological parameters, which will allow us to constrain the cosmological
model parameters. The luminosity distance $D_{L}(z,p)$ is given by

$\frac{H_{0}\sqrt{\left|\Omega_{k0}\right|}D_{L}(z,p)}{(1+z)}=\begin{cases}{\rm
sinh}\left[g(z)\right]&\text{if}\ \Omega_{k0}>0,\\\
\vspace{1mm}g(z)&\text{if}\ \Omega_{k0}=0,\\\ \vspace{1mm}{\rm
sin}\left[g(z)\right]&\text{if}\ \Omega_{k0}<0,\end{cases}$ (14)

where $p$ is the set of cosmological model parameters,

$g(z)=H_{0}\sqrt{\left|\Omega_{k0}\right|}\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime})},$
(15)

and $H(z)$, which is a function of cosmological model parameters, is given in
Sec. 2 for the six cosmological models we study in this paper.

We determine the best-fit values and uncertainty of the parameters for a given
model by maximizing the likelihood function. For QSO data, we have the
observed X-ray flux and we can predict the X-ray flux at given redshift as a
function of cosmological parameters by using eqs. (13) and (14). So, the
likelihood function $({\rm LF})$ for QSO data is

$\ln({\rm LF})=-\frac{1}{2}\sum^{808}_{i=1}\left[\frac{[\log(F^{\rm
obs}_{X,i})-\log(F^{\rm th}_{X,i})]^{2}}{s^{2}_{i}}+\ln(2\pi
s^{2}_{i})\right],$ (16)

where $\ln$ = $\log_{e}$ and $s^{2}_{i}=\sigma^{2}_{i}+\delta^{2}$, where
$\sigma_{i}$ and $\delta$ are the measurement error on $F^{\rm obs}_{X,i}$ and
the global intrinsic dispersion respectively. We treat $\delta$ as a free
parameter to be determined by the data, along with the other two free
parameters, $\beta$ and $\gamma$, which characterise the $L_{X}$ \- $L_{UV}$
relation in eq. (12). In eq. (16) $F^{\rm th}_{X,i}$ is the corresponding
model prediction defined through eq. (13), and is a function of $F_{UV}$ and
$D_{L}(z_{i},p)$.

Our determination of the BAO and $H(z)$ data constraints follows Ryan et al.
(2018). The likelihood function for the uncorrelated BAO and $H(z)$ data is

$\ln({\rm LF})=-\frac{1}{2}\sum^{N}_{i=1}\frac{[A_{\rm obs}(z_{i})-A_{\rm
th}(z_{i},p)]^{2}}{\sigma^{2}_{i}},$ (17)

where $A_{\rm obs}(z_{i})$ and $\sigma_{i}$ are the measured quantity and
error bar at redshift $z_{i}$ and $A_{\rm th}(z_{i},p)$ is the corresponding
model-predicted value. The measurements in the first six lines of Table 1 of
Ryan et al. (2018) are correlated and the likelihood function for those data
points is

$\ln({\rm LF})=-\frac{1}{2}[A_{\rm obs}(z_{i})-A_{\rm
th}(z_{i},p)]^{T}C^{-1}[A_{\rm obs}(z_{i})-A_{\rm th}(z_{i},p)],$ (18)

where $C^{-1}$ is the inverse of the covariance matrix $C$ (Ryan et al., 2019)
=

$\begin{bmatrix}624.707&23.729&325.332&8.34963&157.386&3.57778\\\
23.729&5.60873&11.6429&2.33996&6.39263&0.968056\\\
325.332&11.6429&905.777&29.3392&515.271&14.1013\\\
8.34963&2.33996&29.3392&5.42327&16.1422&2.85334\\\
157.386&6.39263&515.271&16.1422&1375.12&40.4327\\\
3.57778&0.968056&14.1013&2.85334&40.4327&6.25936\\\ \end{bmatrix}.$ (19)

For all parameters except for $H_{0}$, we assume top hat priors, non-zero over
$0\leq\Omega_{m0}\leq 1$, $0\leq\Omega_{\Lambda}\leq 1.3$, $-0.7\leq k\leq
0.7$, $-5\leq\omega_{X}\leq 5$, $0\leq\alpha\leq 1.2$, $-20\leq\ln{\delta}\leq
10$, $0\leq\beta\leq 11$, and $-2\leq\gamma\leq 2$. Here $k$ =
$-\Omega_{k0}a^{2}_{0}$ where $a_{0}$ is the current value of the scale
factor. For the Hubble constant we use two different Gaussian priors,
$H_{0}=68\pm 2.8$ km s-1 Mpc-1 corresponding to the results of a median
statistics analysis of a large compilation of $H_{0}$ measurements (Chen $\&$
Ratra, 2011a),444This is consistent with earlier median statistics analyses
(Gott et al., 2001; Chen et al., 2003), as well as with many other recent
measurement of $H_{0}$ (L’Huillier $\&$ Shafieloo, 2017; Chen et al., 2017;
Wang et al., 2017; Lin $\&$ Ishak, 2017; DES Collaboration, 2018b; Yu et al.,
2018; Gómez-Valent $\&$ Amendola, 2018; Haridasu et al., 2018; Zhang, 2018;
Zhang $\&$ Huang, 2018; Domínguez et al., 2019). and $H_{0}=73.24\pm 1.74$ km
s-1 Mpc-1 from a recent local expansion rate measurement (Riess et al.,
2016).555Other local expansion rate observations find slightly lower $H_{0}$
values and have somewhat larger error bars (Rigault et al., 2015; Zhai et al.,
2017; Dhawan et al., 2017; Fernández Arenas et al., 2018; Freedman et al.,
2019), but see Yuan et al. (2019).

The likelihood analysis is performed using the Markov chain Monte Carlo (MCMC)
method as implemented in the emcee package (Foreman-Mackey et al., 2013) in
Python 3.7. By using the maximum likelihood value $\rm LF_{\rm max}$ we
compute the minimum $\chi^{2}_{\rm min}$ value $-2\ln{(\rm LF_{\rm max})}$. In
addition to $\chi^{2}_{\rm min}$ we also use the Akaike Information Criterion

$AIC=\chi^{2}_{\rm min}+2d$ (20)

and the Bayes Information Criterion

$BIC=\chi^{2}_{\rm min}+d\ln{N}$ (21)

(Ryan et al., 2018), where $d$ is the number of free parameters, and $N$ is
the number of data points. The $AIC$ and $BIC$ penalize models with a larger
number of free parameters.

Table 1: Unmarginalized best-fit parameters of all models for the $H_{0}=68\pm
2.8$ km s-1 Mpc-1 prior.

Model | Data set | $\Omega_{m0}$ | $\Omega_{\Lambda}$ | $\Omega_{k0}$ | $\omega_{X}$ | $\alpha$ |  $H_{0}$a | $\delta$ | $\beta$ | $\gamma$ | $\chi^{2}_{\rm min}$ | $AIC$ | $BIC$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Flat $\Lambda$CDM |  $H(z)$ \+ BAO | 0.30 | 0.70 | - | - | - | 68.14 | - | - | - | 25.65 | 29.65 | 33.13
| QSO | 0.20 | 0.80 | - | - | - | 68.00 | 0.32 | 8.29 | 0.59 | 468.94 | 478.94 | 502.41
| QSO + $H(z)$ \+ BAO | 0.30 | 0.70 | - | - | - | 68.10 | 0.32 | 8.54 | 0.58 | 499.08 | 509.08 | 532.81
Non-flat $\Lambda$CDM |  $H(z)$ \+ BAO | 0.30 | 0.71 | $-0.01$ | - | - | 68.55 | - | - | - | 25.57 | 31.57 | 36.78
| QSO | 0.12 | 1.13 | $-0.25$ | - | - | 68.00 | 0.32 | 8.57 | 0.58 | 466.13 | 478.13 | 506.30
| QSO + $H(z)$ \+ BAO | 0.30 | 0.71 | $-0.01$ | - | - | 68.53 | 0.32 | 8.55 | 0.58 | 498.94 | 510.98 | 539.41
Flat XCDM |  $H(z)$ \+ BAO | 0.31 | 0.69 | - | $-0.98$ | - | 67.69 | - | - | - | 25.58 | 31.58 | 36.79
| QSO | 0.21 | 0.79 | - | $-1.69$ | - | 68.00 | 0.32 | 8.41 | 0.59 | 468.35 | 480.35 | 508.52
| QSO + $H(z)$ \+ BAO | 0.30 | 0.70 | - | $-0.99$ | - | 67.94 | 0.32 | 8.54 | 0.58 | 495.17 | 507.98 | 535.64
Non-flat XCDM |  $H(z)$ \+ BAO | 0.33 | - | $-0.25$ | $-0.73$ | - | 67.58 | - | - | - | 23.52 | 31.52 | 38.47
| QSO | 0.021 | - | $-0.30$ | $-0.67$ | - | 68.00 | 0.32 | 8.65 | 0.58 | 463.10 | 477.10 | 509.96
| QSO + $H(z)$ \+ BAO | 0.32 | - | $-0.19$ | $-0.78$ | - | 67.91 | 0.32 | 8.77 | 0.57 | 493.33 | 507.33 | 540.55
Flat $\phi$CDM |  $H(z)$ \+ BAO | 0.31 | - | - | - | 0.03 | 68.02 | - | - | - | 25.39 | 31.39 | 36.60
| QSO | 0.2 | - | - | - | 0.07 | 68.00 | 0.32 | 8.31 | 0.59 | 469.04 | 481.04 | 509.21
| QSO + $H(z)$ \+ BAO | 0.31 | - | - | - | 0.03 | 67.89 | 0.32 | 8.54 | 0.58 | 495.12 | 507.12 | 535.59
Non-flat $\phi$CDM |  $H(z)$ \+ BAO | 0.32 | - | $-0.01$ | - | 0.14 | 68.00 | - | - | - | 25.96 | 33.96 | 40.91
| QSO | 0.20 | - | $-0.01$ | - | 0.30 | 68.00 | 0.33 | 8.20 | 0.59 | 471.06 | 485.06 | 517.92
| QSO + $H(z)$ \+ BAO | 0.30 | - | $0.01$ | - | 0.11 | 66.00 | 0.33 | 8.20 | 0.59 | 500.91 | 521.12 | 548.13

  * a

${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$.

Table 2: Unmarginalized best-fit parameters of all models for the
$H_{0}=73.24\pm 1.74$ km s-1 Mpc-1 prior.

Model | Data set | $\Omega_{m0}$ | $\Omega_{\Lambda}$ | $\Omega_{k0}$ | $\omega_{X}$ | $\alpha$ |  $H_{0}$a | $\delta$ | $\beta$ | $\gamma$ | $\chi^{2}_{\rm min}$ | $AIC$ | $BIC$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Flat $\Lambda$CDM |  $H(z)$ \+ BAO | 0.31 | 0.69 | - | - | - | 69.55 | - | - | - | 30.89 | 34.89 | 38.37
| QSO | 0.20 | 0.80 | - | - | - | 73.24 | 0.32 | 8.26 | 0.59 | 468.94 | 478.94 | 502.41
| QSO + $H(z)$ \+ BAO | 0.31 | 0.69 | - | - | - | 70.35 | 0.32 | 8.54 | 0.58 | 506.97 | 516.08 | 540.69
Non-flat $\Lambda$CDM |  $H(z)$ \+ BAO | 0.31 | 0.78 | $-0.09$ | - | - | 71.68 | - | - | - | 26.79 | 32.79 | 38
| QSO | 0.12 | 1.13 | $-0.25$ | - | - | 73.24 | 0.32 | 8.55 | 0.58 | 466.13 | 478.13 | 506.30
| QSO + $H(z)$ \+ BAO | 0.31 | 0.79 | $-0.10$ | - | - | 72.35 | 0.32 | 8.63 | 0.58 | 499.26 | 511.26 | 539.73
Flat XCDM |  $H(z)$ \+ BAO | 0.30 | 0.70 | - | $-1.13$ | - | 71.55 | - | - | - | 27.95 | 33.95 | 39.16
| QSO | 0.21 | 0.79 | - | $-1.69$ | - | 73.24 | 0.32 | 8.39 | 0.59 | 468.35 | 480.35 | 508.52
| QSO + $H(z)$ \+ BAO | 0.29 | 0.71 | - | $-1.17$ | - | 72.33 | 0.32 | 8.51 | 0.58 | 497.61 | 509.61 | 538.08
Non-flat XCDM |  $H(z)$ \+ BAO | 0.32 | - | $-0.18$ | $-0.87$ | - | 71.35 | - | - | - | 26.40 | 34.40 | 41.35
| QSO | 0.021 | - | $-0.30$ | $-0.67$ | - | 68.00 | 0.32 | 8.65 | 0.58 | 463.10 | 477.10 | 509.96
| QSO + $H(z)$ \+ BAO | 0.31 | - | $-0.14$ | $-0.93$ | - | 72.23 | 0.32 | 8.69 | 0.58 | 496.18 | 510.18 | 543.40
Flat $\phi$CDM |  $H(z)$ \+ BAO | 0.32 | - | - | - | 0.05 | 69.97 | - | - | - | 32.85 | 38.85 | 44.06
| QSO | 0.2 | - | - | - | 0.13 | 73.24 | 0.32 | 8.32 | 0.59 | 469.04 | 481.04 | 509.21
| QSO + $H(z)$ \+ BAO | 0.31 | - | - | - | 0.05 | 69.42 | 0.33 | 8.36 | 0.59 | 510.04 | 522.04 | 550.51
Non-flat $\phi$CDM |  $H(z)$ \+ BAO | 0.35 | - | $-0.09$ | - | 0.27 | 69.99 | - | - | - | 34.49 | 42.49 | 49.44
| QSO | 0.20 | - | $-0.01$ | - | 0.30 | 71.00 | 0.33 | 8.20 | 0.59 | 473.45 | 487.45 | 520.31
| QSO + $H(z)$ \+ BAO | 0.33 | - | $-0.05$ | - | 0.29 | 69.00 | 0.32 | 8.20 | 0.59 | 509.76 | 523.76 | 556.98

  * a

${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$.

Table 3: Marginalized one-dimensional best-fit parameters with 1$\sigma$
confidence intervals for all models using BAO and $H(z)$ data.

$H_{0}$a prior | Model | $\Omega_{m0}$ | $\Omega_{\Lambda}$ | $\Omega_{k0}$ | $\omega_{X}$ | $\alpha$ |  $H_{0}$a
---|---|---|---|---|---|---|---
$H_{0}=68\pm 2.8$ | Flat $\Lambda$CDM | $0.30^{+0.01}_{-0.01}$ | - | - | - | - | $68.14^{+1.00}_{-0.94}$
| Non-flat $\Lambda$CDM | $0.30^{+0.01}_{-0.01}$ | $0.71^{+0.05}_{-0.05}$ | $-0.01^{+0.04}_{-0.06}$ | - | - | $68.48^{+1.76}_{-1.77}$
| Flat XCDM | $0.31^{+0.01}_{-0.01}$ | - | - | $-0.98^{+0.08}_{-0.09}$ | - | $67.78^{+1.89}_{-1.84}$
| Non-flat XCDM | $0.33^{+0.02}_{-0.02}$ | - | $-0.21^{+0.17}_{-0.23}$ | $-0.76^{+0.12}_{-0.17}$ | - | $67.62^{+1.83}_{-1.76}$
| Flat $\phi$CDM | $0.31^{+0.01}_{-0.01}$ | - | - | - | $0.19^{+0.21}_{-0.14}$ | $66.69^{+2.8}_{-2.8}$
| Non-flat $\phi$CDM | $0.32^{+0.01}_{-0.01}$ | - | $-0.18^{+0.11}_{-0.12}$ | - | $0.86^{+0.43}_{-0.49}$ | $67.69^{+1.71}_{-1.68}$
$H_{0}=73.24\pm 1.74$ | Flat $\Lambda$CDM | $0.31^{+0.01}_{-0.01}$ | - | - | - | - | $69.57^{+0.90}_{-0.91}$
| Non-flat $\Lambda$CDM | $0.31^{+0.01}_{-0.01}$ | $0.77^{+0.04}_{-0.04}$ | $-0.08^{+0.05}_{-0.05}$ | - | - | $71.63^{+1.39}_{-1.39}$
| Flat XCDM | $0.30^{+0.01}_{-0.01}$ | - | - | $-1.13^{+0.08}_{-0.08}$ | - | $71.59^{+1.50}_{-1.48}$
| Non-flat XCDM | $0.32^{+0.02}_{-0.02}$ | - | $-0.15^{+0.14}_{-0.19}$ | $-0.89^{+0.16}_{-0.22}$ | - | $71.36^{+1.47}_{-1.46}$
| Flat $\phi$CDM | $0.31^{+0.01}_{-0.01}$ | - | - | - | $0.06^{+0.09}_{-0.06}$ | $69.06^{+0.98}_{-1.02}$
| Non-flat $\phi$CDM | $0.32^{+0.01}_{-0.01}$ | - | $-0.23^{+0.11}_{-0.14}$ | - | $0.68^{+0.53}_{-0.46}$ | $71.03^{+1.37}_{-1.37}$

  * a

${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$.

Table 4: Marginalized one-dimensional best-fit parameters with 1$\sigma$
confidence intervals for all models using QSO data.

$H_{0}$a prior | Model | $\Omega_{m0}$ | $\Omega_{\Lambda}$ | $\Omega_{k0}$ | $\omega_{X}$ | $\alpha$ |  $H_{0}$a | $\delta$ | $\beta$ | $\gamma$
---|---|---|---|---|---|---|---|---|---|---
$H_{0}=68\pm 2.8$ | Flat $\Lambda$CDM | $0.26^{+0.17}_{-0.11}$ | - | - | - | - | $68^{+2.8}_{-2.8}$ | $0.32^{+0.008}_{-0.008}$ | $8.42^{+0.57}_{-0.58}$ | $0.59^{+0.02}_{-0.02}$
| Non-flat $\Lambda$CDM | $0.24^{+0.16}_{-0.10}$ | $0.93^{+0.18}_{-0.39}$ | $-0.17^{+0.49}_{-0.34}$ | - | - | $68^{+2.8}_{-2.8}$ | $0.32^{+0.008}_{-0.008}$ | $8.62^{+0.62}_{-0.62}$ | $0.58^{+0.02}_{-0.02}$
| Flat XCDM | $0.25^{+0.16}_{-0.10}$ | - | - | $-2.49^{+1.26}_{-1.59}$ | - | $68^{+2.8}_{-2.8}$ | $0.32^{+0.008}_{-0.008}$ | $8.65^{+0.55}_{-0.57}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat XCDM | $0.29^{+0.26}_{-0.14}$ | - | $0.11^{+0.66}_{-0.31}$ | $-1.87^{+1.18}_{-2.05}$ | - | $68^{+2.8}_{-2.8}$ | $0.32^{+0.008}_{-0.008}$ | $8.52^{+0.64}_{-0.65}$ | $0.58^{+0.02}_{-0.02}$
| Flat $\phi$CDM | $0.26^{+0.18}_{-0.11}$ | - | - | - | $0.54^{+0.43}_{-0.38}$ | $68^{+2.8}_{-2.8}$ | $0.32^{+0.008}_{-0.008}$ | $8.42^{+0.57}_{-0.57}$ | $0.59^{+0.02}_{-0.02}$
| Non-flat $\phi$CDM | $0.34^{+0.24}_{-0.16}$ | - | $-0.3^{+0.44}_{-0.61}$ | - | $0.55^{+0.43}_{-0.38}$ | $68^{+2.8}_{-2.8}$ | $0.32^{+0.008}_{-0.008}$ | $8.45^{+0.57}_{-0.58}$ | $0.59^{+0.02}_{-0.02}$
$H_{0}=73\pm 1.74$ | Flat $\Lambda$CDM | $0.26^{+0.17}_{-0.11}$ | - | - | - | - | $73.24^{+1.73}_{-1.73}$ | $0.32^{+0.008}_{-0.008}$ | $8.40^{+0.57}_{-0.57}$ | $0.59^{+0.02}_{-0.02}$
| Non-flat $\Lambda$CDM | $0.24^{+0.16}_{-0.10}$ | $0.93^{+0.18}_{-0.39}$ | $-0.17^{+0.49}_{-0.34}$ | - | - | $73.24^{+1.73}_{-1.73}$ | $0.32^{+0.008}_{-0.008}$ | $8.59^{+0.62}_{-0.62}$ | $0.58^{+0.02}_{-0.02}$
| Flat XCDM | $0.25^{+0.16}_{-0.10}$ | - | - | $-2.48^{+1.26}_{-1.59}$ | - | $73.24^{+1.73}_{-1.73}$ | $0.32^{+0.008}_{-0.008}$ | $8.62^{+0.55}_{-0.56}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat XCDM | $0.29^{+0.25}_{-0.14}$ | - | $0.10^{+0.62}_{-0.32}$ | $-1.83^{+1.15}_{-2.02}$ | - | $73.24^{+1.74}_{-1.74}$ | $0.32^{+0.008}_{-0.008}$ | $8.50^{+0.65}_{-0.64}$ | $0.58^{+0.02}_{-0.02}$
| Flat $\phi$CDM | $0.24^{+0.19}_{-0.12}$ | - | - | - | $0.55^{+0.43}_{-0.38}$ | $73.23^{+1.73}_{-1.73}$ | $0.32^{+0.008}_{-0.008}$ | $8.40^{+0.57}_{-0.57}$ | $0.59^{+0.02}_{-0.02}$
| Non-flat $\phi$CDM | $0.34^{+0.24}_{-0.17}$ | - | $-0.3^{+0.62}_{-0.44}$ | - | $0.55^{+0.43}_{-0.38}$ | $73.26^{+1.74}_{-1.73}$ | $0.32^{+0.008}_{-0.008}$ | $8.42^{+0.57}_{-0.58}$ | $0.59^{+0.02}_{-0.02}$

  * a

${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$.

Table 5: Marginalized one-dimensional best-fit parameters with 1$\sigma$
confidence intervals for all models using QSO+$H(z)$+BAO data.

$H_{0}$a prior | Model | $\Omega_{m0}$ | $\Omega_{\Lambda}$ | $\Omega_{k0}$ | $\omega_{X}$ | $\alpha$ |  $H_{0}$a | $\delta$ | $\beta$ | $\gamma$
---|---|---|---|---|---|---|---|---|---|---
$H_{0}=68\pm 2.8$ | Flat $\Lambda$CDM | $0.30^{+0.01}_{-0.01}$ | $0.70^{+0.01}_{-0.01}$ | - | - | - | $68.11^{+0.91}_{-0.93}$ | $0.32^{+0.008}_{-0.008}$ | $8.54^{+0.48}_{-0.49}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat $\Lambda$CDM | $0.30^{+0.01}_{-0.01}$ | $0.71^{+0.06}_{-0.05}$ | $-0.01^{+0.06}_{-0.07}$ | - | - | $68.84^{+1.49}_{-1.48}$ | $0.32^{+0.008}_{-0.008}$ | $8.54^{+0.49}_{-0.49}$ | $0.58^{+0.02}_{-0.02}$
| Flat XCDM | $0.30^{+0.01}_{-0.01}$ | - | - | $-0.99^{+0.07}_{-0.08}$ | - | $68.00^{+1.56}_{-1.52}$ | $0.32^{+0.008}_{-0.008}$ | $8.54^{+0.48}_{-0.48}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat XCDM | $0.32^{+0.02}_{-0.02}$ | - | $-0.16^{+0.14}_{-0.17}$ | $-0.81^{+0.11}_{-0.16}$ | - | $67.93^{+1.53}_{-1.53}$ | $0.32^{+0.008}_{-0.008}$ | $8.74^{+0.54}_{-0.52}$ | $0.58^{+0.02}_{-0.02}$
| Flat $\phi$CDM | $0.31^{+0.01}_{-0.01}$ | - | - | - | $0.16^{+0.18}_{-0.11}$ | $67.06^{+1.15}_{-1.25}$ | $0.32^{+0.008}_{-0.008}$ | $8.54^{+0.48}_{-0.48}$ | $0.59^{+0.02}_{-0.02}$
| Non-flat $\phi$CDM | $0.31^{+0.01}_{-0.01}$ | - | $-0.15^{+0.09}_{-0.09}$ | - | $0.68^{+0.35}_{-0.40}$ | $67.95^{+1.46}_{-1.45}$ | $0.32^{+0.008}_{-0.008}$ | $8.51^{+0.48}_{-0.48}$ | $0.58^{+0.02}_{-0.02}$
$H_{0}=73.24\pm 1.74$ | Flat $\Lambda$CDM | $0.32^{+0.01}_{-0.01}$ | - | - | - | - | $70.34^{+0.81}_{-0.80}$ | $0.32^{+0.008}_{-0.008}$ | $8.54^{+0.48}_{-0.48}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat $\Lambda$CDM | $0.31^{+0.01}_{-0.01}$ | $0.79^{+0.04}_{-0.04}$ | $-0.10^{+0.05}_{-0.05}$ | - | - | $72.31^{+1.09}_{-1.07}$ | $0.32^{+0.008}_{-0.008}$ | $8.62^{+0.49}_{-0.49}$ | $0.58^{+0.02}_{-0.02}$
| Flat XCDM | $0.29^{+0.01}_{-0.01}$ | - | - | $-1.17^{+0.07}_{-0.07}$ | - | $72.35^{+1.13}_{-1.13}$ | $0.32^{+0.008}_{-0.008}$ | $8.51^{+0.48}_{-0.48}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat XCDM | $0.31^{+0.02}_{-0.02}$ | - | $-0.12^{+0.12}_{-0.14}$ | $-0.96^{+0.15}_{-0.21}$ | - | $72.22^{+1.12}_{-1.12}$ | $0.32^{+0.008}_{-0.008}$ | $8.68^{+0.52}_{-0.52}$ | $0.58^{+0.02}_{-0.02}$
| Flat $\phi$CDM | $0.32^{+0.01}_{-0.01}$ | - | - | - | $0.04^{+0.06}_{-0.03}$ | $70.07^{+0.83}_{-0.84}$ | $0.32^{+0.008}_{-0.008}$ | $8.54^{+0.49}_{-0.48}$ | $0.58^{+0.02}_{-0.02}$
| Non-flat $\phi$CDM | $0.32^{+0.01}_{-0.01}$ | - | $-0.21^{+0.09}_{-0.11}$ | - | $0.51^{+0.43}_{-0.35}$ | $71.95^{+1.08}_{-1.07}$ | $0.32^{+0.008}_{-0.008}$ | $8.46^{+0.48}_{-0.48}$ | $0.58^{+0.02}_{-0.02}$

  * a

${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$.

##  5 Results

###  5.1 $H(z)$ \+ BAO constraints

Results for the $H(z)$ \+ BAO data set are listed in Tables 1–3. The
unmarginalized best-fit parameters are in Tables 1 and 2. The marginalized
one-dimensional best-fit parameter values with $1\sigma$ error bars are given
in Table 3. These results are consistent with those of Ryan et al. (2018). The
slight differences between the two sets of results are the consequence of the
different analysis techniques used. Our computations are done using the MCMC
method while Ryan et al. (2018) used a grid-based $\chi^{2}$ technique.

The one-dimensional likelihoods and two-dimensional $1\sigma$, $2\sigma$, and
$3\sigma$ confidence contours for all parameters determined by using $H(z)$ \+
BAO data are shown in red in Figs. 1–12. Plots for the $\phi$CDM model differ
slightly from the corresponding plots of Ryan et al. (2018) because of the
difference we discussed above and because we have used a smaller range of
$\alpha$ than what Ryan et al. (2018) used. The $H(z)$ \+ BAO data reduced
$\chi^{2}$ values are $\sim$ 0.6–0.8, smaller than unity largely because of
the $H(z)$ data.

###  5.2 QSO constraints

Use of the QSO data to constrain cosmological parameters is based on the
assumed validity of the $L_{X}$ \- $L_{UV}$ relation in eq. (12). This
assumption is tested by Risaliti $\&$ Lusso (2015). By fitting this relation
in cosmological models we have found, in agreement with Risaliti $\&$ Lusso
(2015), that the slope $\gamma$ $\sim$ $0.60\pm 0.02$, the intercept $\beta$
is between 8 and 9, and the global intrinsic dispersion $\delta$ = $0.32\pm
0.08$. The global intrinsic dispersion is large and so cosmological parameter
determination done using these data is not as precise as that done by using,
for example, the SNIa data. But the main advantage of using the quasar sample
is that it covers a very large redshift range and eventually with more and
better quality data it should result in tight constraints.

The QSO data determined cosmological parameter results are given in Tables 1,
2, and 4. The unmarginalized best-fit parameters are given in the Tables 1 and
2 for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ and $73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ priors respectively. The one-dimensional likelihoods and the two-
dimensional confidence contours are shown in grey in the left panels of Figs.
1–12. The cosmological parameter constraints are insensitive to the $H_{0}$
prior used. Risaliti $\&$ Lusso (2015) have determined cosmological parameters
for the non-flat $\Lambda$CDM model. Our QSO data constraints in the
$\Omega_{\Lambda}$ – $\Omega_{m0}$ sub-panels of our Figs. 3 and 4 agree well
with the corresponding constraints in Fig. 6 of Risaliti $\&$ Lusso (2015).
For the non-flat $\Lambda$CDM model we find $\Omega_{m0}$ =
$0.24^{+0.16}_{-0.10}$ and $\Omega_{\Lambda}$ = $0.93^{+0.18}_{-0.39}$, also
in good agreement with the corresponding Risaliti $\&$ Lusso (2015) values of
$\Omega_{m0}$ = $0.22^{+0.10}_{-0.08}$ and $\Omega_{\Lambda}$ =
$0.92^{+0.18}_{-0.30}$.

The cosmological parameters obtained by using these QSO data have relatively
high uncertainty for all models but they are mostly consistent with the
results obtained by using the BAO + $H(z)$ data set, which are shown in red in
Figs. 1–12. The QSO data reduced $\chi^{2}$ values are also small $\sim$
0.6–0.7; it is of interest to understand why this is so. The QSO data
$\chi^{2}_{\rm min}$ values do not change significantly from model to model.

###  5.3 QSO + $H(z)$ \+ BAO constraints

From Figs. 1–12 we see that the constraints from $H(z)$ \+ BAO data and those
from QSO data alone are largely mutually consistent, except for the
$H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$
prior case in the flat $\Lambda$CDM and flat $\phi$CDM models, see the bottom
left sub-panels in the left panels of Figs. 2 and 10. The $H(z)$ \+ BAO data
constrain cosmological parameters quite tightly while the QSO data result in
very loose constraints on these parameters. Although the QSO data alone are
not able to provide restrictive constraints, they can help tighten constraints
when used in combination with $H(z)$ \+ BAO data.

Given that the QSO and the $H(z)$ \+ BAO constraints are mostly consistent, it
is reasonable to determine joint QSO + $H(z)$ \+ BAO constraints. These
results are given in Tables 1, 2, and 5. The QSO + $H(z)$ \+ BAO one-
dimensional likelihoods and two-dimensional confidence contours for all the
cosmological parameters are shown in blue in the right panels of Figs. 1–12.
These figures also show the $H(z)$ \+ BAO data constraint contours in red.
Adding the QSO data to the $H(z)$ \+ BAO data and deriving joint constraints
on cosmological parameters, results in bigger effects for the case of the
$H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$
prior (Figs. 2, 4, 6, 8, 10, $\&$ 12), and in the flat $\phi$CDM model for
both priors (Figs. 9 $\&$ 10).

Using the combined data set we see, from Table 5, the non-relativistic matter
density parameter is measured to lie in the range $\Omega_{m0}$ = $0.30\pm
0.01$ to $0.31\pm 0.01$ ($\Omega_{m0}$ = $0.30\pm 0.01$ to $0.32\pm 0.02$) for
flat (non-flat) models and the $H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm
s}^{-1}{\rm Mpc}^{-1}$ prior and to lie in the range $\Omega_{m0}$ = $0.29\pm
0.01$ to $0.32\pm 0.01$ ($\Omega_{m0}$ = $0.31\pm 0.02$ to $0.32\pm 0.01$) for
flat (non-flat) models and the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. In some cases these results differ
slightly from the $H(z)$ \+ BAO data results of Table 3. These results are
consistent with those derived using other data.

The Hubble constant is found to lie in the range $H_{0}$ =
$67.06^{+1.15}_{-1.25}$ to $68.11^{+0.91}_{-0.93}$ ($H_{0}$ = $67.93\pm 1.53$
to $68.84^{+1.49}_{-1.48}$) ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ for flat (non-flat) models and the $H_{0}=68\pm 2.8$ ${\rm
km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior and to lie in the range
$H_{0}$ = $70.07^{+0.83}_{-0.84}$ to $72.35\pm 1.13$ ($H_{0}$ =
$71.95^{+1.08}_{-1.07}$ to $72.31^{+1.09}_{-1.07}$) ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ for flat (non-flat) models and the
$H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$
prior. As expected, for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior case, the measured value of $H_{0}$
is pulled lower than the prior value because the $H(z)$ and BAO data favor a
lower $H_{0}$.

For the non-flat $\Lambda$CDM model the curvature energy density parameter is
measured to be $\Omega_{k0}$ = $-0.01^{+0.06}_{-0.07}$ and $-0.10\pm 0.05$ for
the $H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ and
$73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ priors
respectively. The curvature energy density parameter is found to be
$\Omega_{k0}$ = $-0.16^{+0.14}_{-0.17}$ and $-0.15\pm 0.09$ for the non-flat
XCDM and non-flat $\phi$CDM models for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior and $\Omega_{k0}$ =
$-0.12^{+0.12}_{-0.14}$ and $-0.21^{+0.09}_{-0.11}$ for the non-flat XCDM and
non-flat $\phi$CDM models for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. It is interesting that in all
cases closed spatial hypersurfaces are favored, albeit just barely in the non-
flat $\Lambda$CDM model with the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior, and only at 1$\sigma$ for most
other cases, but at more than 2$\sigma$ for the non-flat $\phi$CDM model and
the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$
prior. This preference for closed spatial hypersurfaces is largely driven by
the $H(z)$ \+ BAO data (Park $\&$ Ratra, 2018d; Ryan et al., 2019). Mildly
closed spatial hypersurfaces are also consistent with CMB anisotropy
measurements (Ooba et al., 2018a, b, c; Park $\&$ Ratra, 2018a, b, c, 2019).

The cosmological constant density parameter for the flat (non-flat)
$\Lambda$CDM model is determined to be $\Omega_{\Lambda}$ = $0.70\pm
0.01$($0.71^{+0.06}_{-0.05}$) and $0.68\pm 0.01$ ($0.79\pm 0.04$) for the
$H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ and
$73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ priors
respectively.

The parameters that govern dark energy dynamics move closer to those of a
time-independent $\Lambda$ when we jointly analyze the QSO data with the
$H(z)$ \+ BAO data, compared to the corresponding QSO data alone case. From
the analyses of the QSO + $H(z)$ \+ BAO data the equation of state parameter
for the flat (non-flat) XCDM parametrization is determined to be $\omega_{X}$
= $-0.99^{+0.07}_{-0.08}$ ($-0.81^{+0.11}_{-0.16}$) and $-1.17\pm 0.07$
($-0.96^{+0.15}_{-0.21}$) for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ and $73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ priors respectively. The parameter
$\alpha$ in the flat (non-flat) $\phi$CDM model is determined to be $\alpha$ =
$0.16^{+0.18}_{-0.11}$ ($0.68^{+0.35}_{-0.40}$) and $0.04^{+0.06}_{-0.03}$
($0.51^{+0.43}_{-0.35}$) for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ and $73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ priors respectively. Of these 8 cases, 6
favor dark energy dynamics over a time-independent cosmological constant
energy density; for the $68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ prior dynamical dark energy is favored at 1.5$\sigma$ to
1.7$\sigma$ in the flat $\phi$CDM and non-flat XCDM and $\phi$CDM models,
while for the $73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ prior, dark energy dynamics is favored at 1.3$\sigma$ to
2.4$\sigma$ in the flat and non-flat $\phi$CDM models and the flat XCDM cases.
Other data also favor mild dark energy dynamics (Ooba et al., 2018d; Park $\&$
Ratra, 2018b, c).

##  6 Conclusion

We have used the Risaliti $\&$ Lusso (2015) compilation of 808 X-ray and UV
QSO flux measurements to constrain cosmological parameters in six cosmological
models. These QSO data constraints are much less restrictive than, but mostly
consistent with those obtained from the joint analyses of 31 Hubble parameter
and 11 BAO distance measurements.

We find that joint analyses of the QSO and $H(z)$ \+ BAO data tightens (and in
some models, alters) constraints on cosmological parameters derived using just
the $H(z)$ \+ BAO data. In general, the tightening effect is more significant
in models with a larger number of free parameters. The joint QSO + $H(z)$ \+
BAO data constraints are consistent with the current standard flat
$\Lambda$CDM model, although they weakly favor closed over flat spatial
hypersurfaces and dynamical dark energy over a cosmolohical constant.

While cosmological parameter constraints from the QSO data we have used here
are not that restrictive, the new Risaliti $\&$ Lusso (2018) QSO data
compilation (of 1598 measurements over the redshift range $0.04\leq z\leq
5.1$) will result in more restrictive cosmological parameter constraints that
near-future QSO data should improve upon.

## Acknowledgements

We thank Elisabeta Lusso for her generous help, and we thank Lado Samushia,
Chan-Gyung Park, and Joe Ryan for useful discussions. We are grateful to the
Beocat Research Cluster at Kansas State University team, especially Dave
Turner and Adam Tygart. This research was supported in part by DOE grant DE-
SC0019038.

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Figure 1: Flat $\Lambda$CDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$ and $H_{0}$, without the QSO-only constraints. These plots are
for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$
prior.

Figure 2: Flat $\Lambda$CDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$ and $H_{0}$, without the QSO-only constraints. These plots are
for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ prior.

Figure 3: Non-flat $\Lambda$CDM model constraints from QSO (grey), $H(z)$ \+
BAO (red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for cosmological parameters
$\Omega_{m0}$, $\Omega_{\Lambda}$, and $H_{0}$, without the QSO-only
constraints. These plots are for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. The black dotted straight lines
corresponds to the flat $\Lambda$CDM model.

Figure 4: Non-flat $\Lambda$CDM model constraints from QSO (grey), $H(z)$ \+
BAO (red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\Omega_{\Lambda}$, and $H_{0}$, without the QSO-only
constraints. These plots are for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. The black dotted straight lines
corresponds to the flat $\Lambda$CDM model.

Figure 5: Flat XCDM model constraints from QSO (grey), $H(z)$ \+ BAO (red),
and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and 3$\sigma$
confidence contours and one-dimensional likelihoods for all free parameters.
Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\omega_{X}$, and $H_{0}$, without the QSO-only constraints.
These plots are for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm
s}^{-1}{\rm Mpc}^{-1}$ prior. The green dotted straight lines represent
$\omega_{x}$ = $-1$.

Figure 6: Flat XCDM model constraints from QSO (grey), $H(z)$ \+ BAO (red),
and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and 3$\sigma$
confidence contours and one-dimensional likelihoods for all free parameters.
Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\omega_{X}$, and $H_{0}$, without the QSO-only constraints.
These plots are for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm
s}^{-1}{\rm Mpc}^{-1}$ prior. The green dotted straight lines represent
$\omega_{x}$ = $-1$.

Figure 7: Non-flat XCDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\Omega_{k0}$, $\omega_{X}$, and $H_{0}$, without the QSO-only
constraints. These plots are for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. The black dashed straight lines
and the green dotted straight lines are $\Omega_{k0}$ = 0 and $\omega_{x}$ =
$-1$ lines.

Figure 8: Non-flat XCDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left pnnel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\Omega_{k0}$, $\omega_{X}$, and $H_{0}$, without the QSO-only
constraints. These plots are for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. The black dashed straight lines
and the green dotted straight lines are $\Omega_{k0}$ = 0 and $\omega_{x}$ =
$-1$ lines.

Figure 9: Flat $\phi$CDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\alpha$, and $H_{0}$, without the QSO-only constraints. These
plots are for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip 2.84526pt{\rm s}^{-1}{\rm
Mpc}^{-1}$ prior.

Figure 10: Flat $\phi$CDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\alpha$, and $H_{0}$, without the QSO-only constraints. These
plots are for the $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip 2.84526pt{\rm
s}^{-1}{\rm Mpc}^{-1}$ prior.

Figure 11: Non-flat $\phi$CDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\Omega_{k0}$, $\alpha$, and $H_{0}$, without the QSO-only
constraints. These plots are for the $H_{0}=68\pm 2.8$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. The black dashed straight lines
are $\Omega_{k0}$ = 0 lines.

Figure 12: Non-Flat $\phi$CDM model constraints from QSO (grey), $H(z)$ \+ BAO
(red), and QSO + $H(z)$ \+ BAO (blue) data. Left panel shows 1, 2, and
3$\sigma$ confidence contours and one-dimensional likelihoods for all free
parameters. Right panel shows magnified plots for only cosmological parameters
$\Omega_{m0}$, $\Omega_{k0}$, $\alpha$, and $H_{0}$, without the QSO-only
constraints.These plots are the for $H_{0}=73.24\pm 1.74$ ${\rm km}\hskip
2.84526pt{\rm s}^{-1}{\rm Mpc}^{-1}$ prior. The black dashed straight lines
are $\Omega_{k0}$ = 0 lines.