ORIGINAL ARTICLE: Dainotti, Maria Giovanna, Giada Bargiacchi, Aleksander Łukasz Lenart, and Salvatore Capozziello (2024). The Scavenger Hunt for Quasar Samples to Be Used as Cosmological Tools, Galaxies 12, no. 1: 4. https://doi.org/10.3390/galaxies12010004
Article Abstract: Although the Λ Cold Dark Matter model is the most accredited cosmological model, information at high redshifts (z) between type Ia supernovae
(𝑧=2.26)
and the Cosmic Microwave Background
(𝑧=1100)
is crucial to validate this model further. To this end, we have discovered a sample of 1132 quasars up to
𝑧=7.54
exhibiting a reduced intrinsic dispersion of the relation between ultraviolet and X-ray fluxes,
𝛿
F
= 0.22
vs.
𝛿
F
= 0.29
(24% less), than the original sample. This gold sample, once we correct the luminosities for selection biases and redshift evolution, enables us to determine the matter density parameter
Ω
M
with a precision of 0.09. Unprecedentedly, this quasar sample is the only one that, as a standalone cosmological probe, yields such tight constraints on
Ω
M
while being drawn from the same parent population of the initial sample.

Introduction

This Wolfram notebook describes How to derive luminosity distance of QSOs from the
L
X
-
L
UV
relation

Import your sample of QSOs

We called our file “total_final _sources.txt” (attached) and it is stored in the same directory as the notebook itself. The structure of the columns should be as follow: {id, redshift, Flux in UV, error on flux in UV, Flux in X, error on flux in X}
SetDirectory[NotebookDirectory[]]​​data=Import["total_final_sources.txt","Data"];
C:\Users\ahmed\Documents\MOD trials\SP Processed posts\The Scavenger Hunt for Quasar Samples to Be Used as Cosmological Tools

Derivation of observational luminosity distance
d
L

The luminosities in the X-ray band and in the UV band are connected through an empirical equation (Dainotti et al. (2023)):​​
Log10[
L
X
]-
k
L
X
Log10[1+z]=γLog10[
L
UV
]-
k
L
UV
Log10[1+z]+β
​​We rewrite the above equation from the definition of luminosity distance and use the κ factor to convert Mpc to cm:​​
Log10[
F
X
*4Pi
2
(
d
L
*κ)
]-
k
L
X
Log10[1+z]=γLog10[
F
UV
*4Pi
2
(
d
L
*κ)
]-
k
L
UV
Log10[1+z]+β
​​We present further steps of algebraic operations to obtain equation for luminosity distance:​​
Log10[
F
X
]+Log10[4Pi]+2Log10[
d
L
*κ]-
k
L
X
Log10[1+z]=γLog10[
F
UV
]+Log10[4Pi]+2Log10[
d
L
*κ]-
k
L
UV
Log10[1+z]+β
​​​
2(1-γ)Log10[
d
L
*κ]=γLog10[
F
UV
]-Log10[
F
X
]+
k
L
X
-
k
L
UV
γLog10[1+z]+(γ-1)Log10[4Pi]+β
​​The final equation:​​
Log10[
d
L
]=
γ
2(1-γ)
Log10[
F
UV
]-
1
2(1-γ)
Log10[
F
X
]+

k
L
X
-
k
L
UV
γ
2(1-γ)
Log10[1+z]+
(γ-1)Log10[4Pi]+β
2(1-γ)
-Log10[κ]
​​Thus if the coefficients of the correlation are known, we can compute the observational luminosity distance and compare it with models

Now we apply it to our data

Check the distribution of redshift
Histogram[data[[All,2]],PlotRangeAll,FrameTrue,FrameLabel{{"N",""},{"z",""}},FrameStyleDirective[Black,15]]
Set parameters of the correlation
γ=0.61;​​β=7.8;​​γerr=0.01;​​βerr=0.2;​​δ=0.084;​​kLX=3.31;​​kLXerr=0.08;​​kLUV=4.39;​​kLUVerr=0.12;
units=3.086×
24
10
;
Define formula for luminosity distance
LumDist[γ1_,β1_,kLX1_,kLUV1_,Log10FUV_,Log10FX_,z_]:=
γ1
2(1-γ1)
(Log10FUV)-
1
2(1-γ1)
(Log10FX)+
(kLX1-kLUV1*γ1)
2(1-γ1)
Log10[1+z]+
(γ1-1)Log10[4Pi]+β1
2(1-γ1)
-Log10[units]
Compute equation for errorbars with the derivative method
Clear[LumDist]​​Clear[FUV,FX]​​Sqrt
2
(D[LumDist[γ1,β1,kLX1,kLUV1,FUV,FX,z],γ1]*γ1err)
+
2
(D[LumDist[γ1,β1,kLX1,kLUV1,FUV,FX,z],β1]*β1err)
2
(D[LumDist[γ1,β1,kLX1,kLUV1,FUV,FX,z],kLX1]*kLX1err)
2
(D[LumDist[γ1,β1,kLX1,kLUV1,FUV,FX,z],kLUV1]*kLUV1err)
2
(D[LumDist[γ1,β1,kLX1,kLUV1,FUV,FX,z],FUV]*FUVerr)
2
(D[LumDist[γ1,β1,kLX1,kLUV1,FUV,FX,z],FX]*FXerr)
+
2
δ1


2
δ1
+
2
FUVerr
2
FXerr
2
kLUV1err
2
kLX1err
2
β1err
2
(0,0,0,0,0,1,0)
LumDist
[γ1,β1,kLX1,kLUV1,FUV,FX,z]
2
(0,0,0,0,1,0,0)
LumDist
[γ1,β1,kLX1,kLUV1,FUV,FX,z]
2
(0,0,0,1,0,0,0)
LumDist
[γ1,β1,kLX1,kLUV1,FUV,FX,z]
2
(0,0,1,0,0,0,0)
LumDist
[γ1,β1,kLX1,kLUV1,FUV,FX,z]
2
(0,1,0,0,0,0,0)
LumDist
[γ1,β1,kLX1,kLUV1,FUV,FX,z]
+
2
γ1err
2
(1,0,0,0,0,0,0)
LumDist
[γ1,β1,kLX1,kLUV1,FUV,FX,z]

LumDistErr[γ1_,β1_,kLX1_,kLUV1_,FUV_,FX_,z_,γ1err_,β1err_,δ1_,kLX1err_,kLUV1err_,FUVerr_,FXerr_]:=
2
δ1
+
2
FUVerr
2
FXerr
2
kLUV1err
2
kLX1err
2
β1err
4
γ1
4
Log[1+z]
1024
10
(1-γ1)
4
Log[10]
+
2
γ1err
2
-
FX
2
2
(1-γ1)
+
FUV
2(1-γ1)
+
FUVγ1
2
2
(1-γ1)
+
Log[4π]
2(1-γ1)Log[10]
+
β1+
(-1+γ1)Log[4π]
Log[10]
2
2
(1-γ1)
-
kLUV1Log[1+z]
2(1-γ1)Log[10]
+
(kLX1-kLUV1γ1)Log[1+z]
2
2
(1-γ1)
Log[10]
Finally we can apply formulas to our data
mudiagram=Table[{data[[i,2]],Around[25+5*LumDist[γ,β,kLX,kLUV,data[[i,3]],data[[i,5]],data[[i,2]]],5*LumDistErr[γ,β,kLX,kLUV,data[[i,3]],data[[i,5]],data[[i,2]],γerr,βerr,δ,kLXerr,kLUVerr,data[[i,4]],data[[i,6]]]]},{i,1,Length[data]}];
p=Graphics[{EdgeForm[{Black}],FaceForm[Yellow],Disk[]}];​​pR=Graphics[{EdgeForm[{Black}],FaceForm[Red],Disk[]}];
Length[data]
1132
Compute theoretical luminosity distance for
Ω
M
=0.3
within ΛCDM model
DistanceLuminosity[z_?NumericQ]:=​​Module{ckms=299792.5,h=0.7,Om=0.3},3.0856*
18
10
*
6
10
*
ckms
100*h
*(1+z)*NIntegrate
1
Om*
3
(1+ζ)
+(1-Om)
,{ζ,0,z},MaxRecursion100;​​DistanceLuminosity[z_List,Om_?NumericQ,h_?NumericQ]:=Map[DistanceLuminosity[#,Om,h]&,z]
flatLCDM=Table[{z,5*Log10[DistanceLuminosity[z]/units]+25},{z,0.0001,7.7,0.05}];
Show everything on the plot
HD1132=Show[ListPlot[mudiagram,PlotRangeAll,FrameTrue,PlotMarkers{p,.02},PlotStyleDirective[PointSize0.005,Yellow,"LineColor"Black],ImageSizeLarge,FrameStyleDirective[Black,20,Thick],FrameLabel{{"μ (5
d
L
​[Mpc] + 25)",""},{"z",""}},IntervalMarkersStyleBlack],ListLinePlot[flatLCDM,PlotStyleGreen,PlotRangeAll]]
Distance Luminosity f(R) gravity, model from Dainotti et al. (2024)
◼
  • This is with Omega = 0.3 fixed
    • Show different models on plot

    CITE THIS NOTEBOOK

    The scavenger hunt for quasar samples to be used as cosmological tools​
    by Maria Giovanna Dainotti, Giada Bargiacchi, Aleksander Łkasz Lenart & Salvatore Capozziello
    Wolfram Community, STAFF PICKS, March 1, 2024
    ​https://community.wolfram.com/groups/-/m/t/3133472