<figure><img src="image_1.png" /></figure>

# Excitation of CO molecules in diffuse gas over cosmic history

V. V. Klimenko<sup>1,2*</sup> S. A. Balashev<sup>2</sup> P. Noterdaeme,<sup>3,4</sup> R. Srianand<sup>5</sup> and A. V. Ivanchik<sup>5</sup>

Accepted 2024 July 29. Received 2024 July 26; in original form 2023 September 29

## ABSTRACT

We studied the physical conditions in the local interstellar medium (ISM) and at high redshift by fitting observed column densities of CO and $H_2$ rotational levels and C I fine-structure levels using photodissociation models calculated with the MEUDON PDR code. We analysed CO absorption systems in 28 sight lines in the local ISM and seven damped Lyman-$\alpha$ absorption systems at high redshift, covering $N(H_2) = 10^{19}-10^{21.5}$ cm$^{-2}$ and $N(CO) = 10^{13}-10^{18}$ cm$^{-2}$. We constructed a method to accurately calculate CO excitation, incorporating the effects of photon trapping. Our findings indicate that in the local ISM, CO excitation is primarily driven by collisions and excitation due to the CMB radiation. We demonstrated that an increase in CO excitation observed near $N(CO) \simeq 10^{15}$ cm$^{-2}$, is attributed to an increase in gas densities from $\simeq 100$ cm$^{-3}$ to $\simeq 300-1000$ cm$^{-3}$. CO absorption systems in the local ISM are characterized by a gas number density of about $10-1000$ cm$^{-3}$, a kinetic temperature of $10-100$ K, and an intensity of external UV field ranging from 0.1 to 10 units of Mathis field. Compared to the average gas probed by C I absorption, the gas detected in CO is denser and colder, while the external UV field remains nearly constant. We observed a negative correlation between the kinetic temperature and both $N(CO)$ and $N(H_2)$, with power-law slopes of $-0.21 \pm 0.02$ and $-0.65 \pm 0.05$, respectively. At the same time, the gas number density exhibits a positive correlation with $N(CO)$ and $N(H_2)$ with slopes of $0.38 \pm 0.02$ and $1.15 \pm 0.05$, respectively.

**Key words:** ISM: molecules – ISM: kinematics and dynamics – galaxies: ISM – galaxies: high redshift – quasars: absorption lines.

## 1 INTRODUCTION

Absorption line analysis towards background sources, such as stars (at $z=0$) or quasars (throughout cosmic history), is a powerful method to investigate the properties of the intergalactic medium (IGM), circumgalactic medium (CGM), and interstellar medium (ISM). Given the typically small sizes of emission regions, absorption line analysis provides insights into absorbing media at small scales – much smaller than a parsec for quasars and even less than an astronomical unit for stars – and allows measurement of local quantities.

Absorption systems enable the study of diffuse gas where physical conditions are not favourable for emission, complementing emission line studies that offer more global views of galactic components. Thus, absorption analysis helps to understand processes in the galactic disc outside star formation regions, the circumgalactic medium, and the galactic halo. It reveals how primary gas accretes onto galaxies and how ejected gas mixes with the halo gas (see e.g. Tumlinson et al. 2013; Werk et al. 2014; Schroetter et al. 2016; Martin et al. 2019; Schroetter et al. 2019; Kulkarni et al. 2022; Lofthouse et al. 2022).

## Studies of the phase transition in diffuse ISM

For the cold neutral ISM, which has typical temperatures $T \lesssim 100$ K, information on the physical conditions can be obtained from the rotational levels of various molecules ($H_2$, HD, CO, $C_2$), and the fine-structure levels of neutral atomic carbon C I (see, e.g. Savage et al. 1977; Rachford et al. 2002; Burgh, France & McCandless 2007; Sonnentrucker et al. 2007; Sheffer et al. 2008; Burgh, France & Jenkins 2010; Jenkins & Tripp 2011; Welty et al. 2020; Federman et al. 2021 for the local ISM, and Srianand, Petitjean & Ledoux 2000; Silva & Viegas 2002; Srianand et al. 2005; Noterdaeme et al. 2007; Noterdaeme et al. 2009; Jorgenson, Wolfe & Prochaska 2010; Balashev et al. 2011, 2017, 2019, 2020 for distant galaxies at high redshift).

The kinetic temperature is routinely measured from the ratio of ortho- to para-$H_2$ (e.g. Savage et al. 1977; Abgrall et al. 1992; Rachford et al. 2002), while the thermal pressure (and hence the number density) can be inferred from the relative population of C I fine-structure levels (see, e.g. Jenkins & Tripp 2001; Jenkins & Tripp 2011; Balashev et al. 2019). A joint combination of these species also provides an estimate of the intensity of the interstellar radiation field (see, e.g. Noterdaeme et al. 2007; Nehmé et al. 2008; Balashev et al. 2019; Klimenko & Balashev 2020; Kosenko, Balashev & Klimenko 2024). Similarly, the analysis of the rotational

<footnote n="1"><sup>1</sup>Department of Physics & Astronomy, University of South Carolina, Columbia, SC 29208, USA</footnote>

<footnote n="2"><sup>2</sup>Ioffe Institute, Polyteknicheskaya 26, 194021 Saint Petersburg, Russia</footnote>

<footnote n="3"><sup>3</sup>Franco-Chilean Laboratory for Astronomy, IRL 3386, CNRS and Universidad de Chile, Casilla 36-D, Santiago, Chile</footnote>

<footnote n="4"><sup>4</sup>Institut d'Astrophysique de Paris, CNRS-SU, UMR 7095, 98bis bd Arago, F-75014 Paris, France</footnote>

<footnote n="5"><sup>5</sup>Inter-University Centre for Astronomy and Astrophysics, Pune University Campus, Ganeshkhind, Pune 411007, India</footnote>

<footnote>*E-mail: slava.klimenko@gmail.com</footnote>
excitation of other molecules can bring independent constraints on the physical conditions (e.g. studies of excitation and abundances of HD – Balashev, Ivanchik & Varshalovich 2010; Kosenko et al. 2021; C₂ – Sonnentrucker et al. 2007 and CO – Goldsmith 2013). On the other hand, different species probe (possibly only slightly) different phases of the ISM, thereby constraining the transition from warm diffuse to cold dense phases.

The thermal equilibrium, which partially determines the phase transition, has been theoretically investigated in detail (e.g. Wolfre et al. 2003; Bialy & Sternberg 2019; Balashev, Telikova & Noterdaeme 2022), but it remains poorly constrained by observations. Carbon species (C II, C I, and CO), which interconvert along the transition from the warm to cold ISM phases (e.g. Snow & McCall 2006), offer an opportunity to study differences in physical conditions between diffuse and dense phases.

In this work, we focus on the excitation of C I fine-structure and CO rotational levels, whose populations are determined through ultraviolet and optical observations of diffuse and dense gas clouds in the local ISM and distant galaxies.

## CO excitation as a probe of the physical conditions

Historically, CO excitation has been used to derive the gas number density in diffuse clouds since the first UV observations with the *Copernicus* satellite (e.g. Snow 1977). However, several issues were later identified that prevented the use of CO to determine the physical conditions in dense clouds.

In early works (e.g. Snow 1977; Smith, Krishna Swamy & Stecher 1978; Sheffer et al. 1992; Lambert et al. 1994), the excitation of CO rotational levels was described by collisions with H₂ (as a main collisional partner) and by the cosmic microwave background (CMB). Then Wannier, Penprase & Andersson (1997) proposed that CO emission from the local source (dense clouds) can also excite CO rotational levels in nearby absorption system. Since the local radiation intensity is not well known, CO absorption lines in fact provide only an upper limit on the number density. The second complication is the radiative trapping effect (RTE, e.g. Draine 2011). In case of a optically thick CO lines, photons can be reabsorbed by another CO molecule in the cloud that ‘effectively’ decreases the rates of spontaneous and stimulated emission and increases the relative role of collisional excitation. This effect introduces (i) a systematic bias in results in comparison with an optically thin model and therefore requires sequential calculations of radiative transfer and thermal balance, which is a difficult task; (ii) a dependence of CO excitation on unknown geometry of the cloud. In addition, neutral carbon is usually observed towards the same sight lines, and its excitation is not affected by radiative trapping since the transitions between C I fine-structure levels are forbidden, and hence corresponding lines are optically-thin. Radiation trapping is commonly accounted for in the estimation of physical conditions in radio emission line studies of CO-bright molecular clouds (e.g. van der Tak et al. 2007; Peñaloza et al. 2017; Salomé et al. 2019; Finn et al. 2021; Leroy et al. 2022; Fuda, Tram & Reach 2023), but it is usually ignored in studies of diffuse gas in absorption. However, some absorption systems have N(CO) > 10¹⁵ cm⁻² when CO lines (1–0, 2–1) are optically thick (Welty et al. 2020; Federman et al. 2021).

The excitation of CO in optically-thin case was recently examined by Goldsmith (2013). For diffuse clouds in Milky Way (at N(CO) ≤ 10¹⁵ cm⁻²), the author found n_H ~ 50–100 cm⁻³ in line with the results inferred from C I excitation by Jenkins & Tripp (2011). Around the same time, CO molecules in absorption were detected at high redshift towards quasar sightlines (z ~ 2, Srianand et al.

2008; Noterdaeme et al. 2010). The authors found high-excitation temperatures of CO (~8–10 K), which are comparable to excitation temperatures in local dense clouds (e.g. Goldsmith et al. 2008). However, these values were explained by the difference in the intensity of CMB radiation at z = 0 and 2 and has been used to constrain the evolution of the CMB temperature with cosmic time (Noterdaeme et al. 2011). Later these constraints were updated into account the correction for collisional excitation (Klimenko et al. 2020).

In this work, we propose a method to derive physical conditions using CO excitation in optically-thick case based on the fitting to the observed column densities on CO rotational levels with a grid of photodissociation models, calculated with the MEUDON PDR code. Within this method, we calculate the radiation transfer in CO lines consistently with CO excitation, thermal balance, and chemistry. We apply this method to the sample of CO-bearing absorption systems in the Milky Way and high-redshift galaxies and compare our results with previous studies. In particular, we were interested in measuring the relative dependence of the density, kinetic temperature, and UV field, that described the phase transitions in ISM from diffuse to dense gas state. This work develops our study of the physical conditions from a joint analysis of excitation of C I and H₂ species (Klimenko & Balashev 2020) that we also used in this work.

The paper is organized as follows: In Section 2, we present the compilation of the data from Milky Way and high-redshift damped Ly-α absorption systems (DLAs) measurements used in this work. In Section 3, we describe the method that we used to estimate the physical conditions from the observed column densities. Section 4 contains the results of the measurements, which are discussed in Section 5, before we conclude in Section 6. The appendix contains supplementary information with detailed figures on each absorption system used.

## 2 DATA

We compiled measurements of CO rotational excitation from two samples: local sightlines in the ISM (28 sight lines) and DLA systems at high redshift (seven sight lines). Where available, we supplemented these data with measurements of H₂ rotational excitation levels and C I fine-structure levels.

### 2.1 The local ISM sample ($S_{\text{local}}$)

We analysed CO absorption systems in the Milky Way using observations obtained with the Space Telescope Imaging Spectrograph (STIS) spectrograph on board the Hubble Space Telescope (HST) (Burgh et al. 2007; Sonnentrucker et al. 2007; Sheffer et al. 2008; Burgh et al. 2010; Welty et al. 2020; Federman et al. 2021). The high spectral resolution of approximately 110 000 allows to resolve individual absorptions from CO rotational levels up to $J = 3, 4$. We selected 28 sight lines where the populations of at least three lower CO rotational levels ($J = 0, 1, 2$) were measured, ensuring a reliable estimate of the number density. The list of sight lines is presented in Table 1.

The C I absorption in the sight lines within the local ISM was analysed by Jenkins & Tripp (2001) and Jenkins & Tripp (2011) using HST/STIS observations. We collected column densities of C I fine structure levels from the data base in Jenkins & Tripp (2011) and from Welty et al. (2020) for HD 62542. We provide a detailed analysis of C I systems from Jenkins & Tripp (2011) in Appendix A.

The H₂ ultraviolet transitions (912–1100 Å) are not covered by the HST/STIS observations. Therefore, we collected data on H₂ excitation from the Far Ultraviolet Spectroscopic Explorer (FUSE)
<table><caption>**Table 1.** The samples of CO/C I/H₂-bearing absorption systems and derived physical parameters in associated medium.</caption><thead><tr><th>Name</th><th>log N(CO)<br>[cm<sup>-2</sup>]</th><th>log N(H<sub>2</sub>)<br>[cm<sup>-2</sup>]</th><th>log N(C1)<br>[cm<sup>-2</sup>]</th><th>T<sub>ex</sub>(CO)<br>(K)</th><th>T<sub>ex</sub>(H<sub>2</sub>)<br>(K)</th><th>T<sub>ex</sub>(C<sub>2</sub>)<br>(K)</th><th>log n<sub>H</sub><sup>(CO)</sup><br>[cm<sup>-3</sup>]</th><th>log I<sub>UV</sub><br>(Mathis)</th><th>T<sub>gas</sub><sup>(CO)</sup><br>(K)</th><th>log n<sub>gas</sub><sup>(CO)</sup><br>[cm<sup>-3</sup>]</th><th>T<sub>gas</sub><sup>(CI)</sup><br>(K)</th><th>log n<sub>gas</sub><sup>(CI)</sup><br>[cm<sup>-3</sup>]</th></tr></thead><tbody><tr><td colspan="14">Local ISM: Sonnentrucker et al. 2007</td></tr><tr><td>HD 24534</td><td>16.01<sup>+0.09</sup><sub>-0.09</sub></td><td>20.92<sup>+0.03</sup><sub>-0.03</sub></td><td>13.97<sup>+0.03</sup><sub>-0.03</sub></td><td>5.3<sup>+0.5</sup><sub>-0.5</sub></td><td>57<sup>+3</sup><sub>-3</sub></td><td>45<sup>+10</sup><sub>-10</sub></td><td>2.64<sup>+0.07</sup><sub>-0.07</sub></td><td>0.05<sup>+0.15</sup><sub>-0.20</sub></td><td>58<sup>+9</sup><sub>-10</sub></td><td>2.40<sup>+0.09</sup><sub>-0.09</sub></td><td>76<sup>+15</sup><sub>-14</sub></td><td>2.35<sup>+0.13</sup><sub>-0.14</sub></td></tr><tr><td>HD 27778</td><td>16.08<sup>+0.04</sup><sub>-0.04</sub></td><td>20.80<sup>+0.03</sup><sub>-0.03</sub></td><td>15.08<sup>+0.05</sup><sub>-0.05</sub></td><td>5.8<sup>+0.4</sup><sub>-0.3</sub></td><td>56<sup>+5</sup><sub>-5</sub></td><td>50<sup>+10</sup><sub>-10</sub></td><td>2.67<sup>+0.14</sup><sub>-0.12</sub></td><td>-0.23<sup>+0.20</sup><sub>-0.17</sub></td><td>35<sup>+5</sup><sub>-4</sub></td><td>2.47<sup>+0.13</sup><sub>-0.13</sub></td><td>81<sup>+21</sup><sub>-16</sub></td><td>1.88<sup>+0.21</sup><sub>-0.15</sub></td></tr><tr><td>HD 147888</td><td>15.30<sup>+0.08</sup><sub>-0.08</sub></td><td>20.48<sup>+0.03</sup><sub>-0.03</sub></td><td>14.23<sup>+0.05</sup><sub>-0.05</sub></td><td>8.8<sup>+1.2</sup><sub>-0.7</sub></td><td>45<sup>+3</sup><sub>-3</sub></td><td>38<sup>+20</sup><sub>-20</sub></td><td>3.18<sup>+0.11</sup><sub>-0.18</sub></td><td>0.09<sup>+0.17</sup><sub>-0.17</sub></td><td>64<sup>+8</sup><sub>-7</sub></td><td>2.96<sup>+0.15</sup><sub>-0.15</sub></td><td>83<sup>+26</sup><sub>-12</sub></td><td>2.52<sup>+0.17</sup><sub>-0.13</sub></td></tr><tr><td>HD 185418</td><td>14.75<sup>+0.03</sup><sub>-0.03</sub></td><td>20.77<sup>+0.03</sup><sub>-0.03</sub></td><td>14.83<sup>+0.05</sup><sub>-0.05</sub></td><td>3.4<sup>+0.7</sup><sub>-0.5</sub></td><td>101<sup>+10</sup><sub>-10</sub></td><td>—</td><td>1.91<sup>+0.20</sup><sub>-0.32</sub></td><td>-0.41<sup>+0.21</sup><sub>-0.22</sub></td><td>74<sup>+15</sup><sub>-12</sub></td><td>1.68<sup>+0.30</sup><sub>-0.30</sub></td><td>83<sup>+31</sup><sub>-30</sub></td><td>1.64<sup>+0.15</sup><sub>-0.12</sub></td></tr><tr><td>HD 206267</td><td>16.05<sup>+0.03</sup><sub>-0.03</sub></td><td>20.86<sup>+0.03</sup><sub>-0.03</sub></td><td>15.30<sup>+0.05</sup><sub>-0.05</sub></td><td>6.0<sup>+0.4</sup><sub>-0.2</sub></td><td>64<sup>+3</sup><sub>-3</sub></td><td>36<sup>+9</sup><sub>-9</sub></td><td>2.82<sup>+0.08</sup><sub>-0.08</sub></td><td>-0.35<sup>+0.15</sup><sub>-0.13</sub></td><td>37<sup>+4</sup><sub>-4</sub></td><td>2.58<sup>+0.30</sup><sub>-0.14</sub></td><td>75<sup>+17</sup><sub>-11</sub></td><td>1.93<sup>+0.13</sup><sub>-0.11</sub></td></tr><tr><td>HD 207198</td><td>15.52<sup>+0.04</sup><sub>-0.04</sub></td><td>20.84<sup>+0.03</sup><sub>-0.03</sub></td><td>15.53<sup>+0.05</sup><sub>-0.05</sub></td><td>3.9<sup>+0.3</sup><sub>-0.3</sub></td><td>66<sup>+3</sup><sub>-3</sub></td><td>43<sup>+19</sup><sub>-19</sub></td><td>2.18<sup>+0.15</sup><sub>-0.17</sub></td><td>-0.33<sup>+0.16</sup><sub>-0.26</sub></td><td>53<sup>+7</sup><sub>-7</sub></td><td>1.96<sup>+0.30</sup><sub>-0.30</sub></td><td>75<sup>+24</sup><sub>-13</sub></td><td>1.87<sup>+0.13</sup><sub>-0.11</sub></td></tr><tr><td>HD 210121</td><td>15.84<sup>+0.08</sup><sub>-0.08</sub></td><td>20.75<sup>+0.12</sup><sub>-0.12</sub></td><td>—</td><td>5.3<sup>+0.3</sup><sub>-0.2</sub></td><td>51<sup>+11</sup><sub>-11</sub></td><td>46<sup>+21</sup><sub>-21</sub></td><td>2.73<sup>+0.11</sup><sub>-0.14</sub></td><td>&lt;-0.1</td><td>33<sup>+29</sup><sub>-5</sub></td><td>2.46<sup>+0.35</sup><sub>-0.14</sub></td><td>72<sup>+16</sup><sub>-13</sub></td><td>—</td></tr><tr><td>HD 210839</td><td>15.46<sup>+0.05</sup><sub>-0.05</sub></td><td>20.84<sup>+0.03</sup><sub>-0.03</sub></td><td>15.09<sup>+0.04</sup><sub>-0.04</sub></td><td>4.1<sup>+0.2</sup><sub>-0.2</sub></td><td>72<sup>+4</sup><sub>-4</sub></td><td>16<sup>+6</sup><sub>-6</sub></td><td>2.21<sup>+0.19</sup><sub>-0.21</sub></td><td>-0.21<sup>+0.15</sup><sub>-0.12</sub></td><td>52<sup>+5</sup><sub>-5</sub></td><td>1.98<sup>+0.23</sup><sub>-0.23</sub></td><td>—</td><td>2.16<sup>+0.14</sup><sub>-0.11</sub></td></tr><tr><td colspan="14">MW: Sheffer et al. 2008</td></tr><tr><td>HD 12323</td><td>14.53<sup>+0.05</sup><sub>-0.05</sub></td><td>20.33<sup>+0.08</sup><sub>-0.08</sub></td><td>—</td><td>3.0<sup>+0.2</sup><sub>-0.2</sub></td><td>82<sup>+10</sup><sub>-10</sub></td><td>—</td><td>1.52<sup>+0.30</sup><sub>-0.50</sub></td><td>-0.23<sup>+0.46</sup><sub>-0.40</sub></td><td>52<sup>+38</sup><sub>-9</sub></td><td>1.00<sup>+1.00</sup><sub>-0.20</sub></td><td>—</td><td>—</td></tr><tr><td>HD 15137</td><td>13.52<sup>+0.05</sup><sub>-0.05</sub></td><td>20.32<sup>+0.10</sup><sub>-0.10</sub></td><td>14.65<sup>+0.05</sup><sub>-0.05</sub></td><td>3.8<sup>+1.2</sup><sub>-0.4</sub></td><td>104<sup>+11</sup><sub>-11</sub></td><td>—</td><td>2.12<sup>+0.24</sup><sub>-0.31</sub></td><td>-0.07<sup>+0.50</sup><sub>-0.20</sub></td><td>123<sup>+11</sup><sub>-33</sub></td><td>1.86<sup>+0.38</sup><sub>-0.38</sub></td><td>186<sup>+42</sup><sub>-99</sub></td><td>1.59<sup>+0.21</sup><sub>-0.13</sub></td></tr><tr><td>HD 23478</td><td>14.90<sup>+0.05</sup><sub>-0.05</sub></td><td>20.55<sup>+0.12</sup><sub>-0.10</sub></td><td>14.79<sup>+0.05</sup><sub>-0.05</sub></td><td>3.5<sup>+0.8</sup><sub>-0.3</sub></td><td>53<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.03<sup>+0.26</sup><sub>-0.35</sub></td><td>-0.35<sup>+0.34</sup><sub>-0.30</sub></td><td>70<sup>+12</sup><sub>-10</sub></td><td>1.77<sup>+0.24</sup><sub>-0.64</sub></td><td>79<sup>+55</sup><sub>-16</sub></td><td>1.85<sup>+0.15</sup><sub>-0.10</sub></td></tr><tr><td>HD 24190</td><td>13.95<sup>+0.05</sup><sub>-0.05</sub></td><td>20.38<sup>+0.10</sup><sub>-0.10</sub></td><td>14.68<sup>+0.05</sup><sub>-0.05</sub></td><td>3.2<sup>+0.9</sup><sub>-0.3</sub></td><td>66<sup>+10</sup><sub>-10</sub></td><td>—</td><td>1.64<sup>+0.30</sup><sub>-0.40</sub></td><td>-0.31<sup>+0.33</sup><sub>-0.26</sub></td><td>107<sup>+34</sup><sub>-25</sub></td><td>1.20<sup>+0.70</sup><sub>-0.40</sub></td><td>87<sup>+64</sup><sub>-12</sub></td><td>1.78<sup>+0.19</sup><sub>-0.11</sub></td></tr><tr><td>HD 24398</td><td>15.26<sup>+0.05</sup><sub>-0.05</sub></td><td>20.67<sup>+0.10</sup><sub>-0.10</sub></td><td>—</td><td>3.9<sup>+1.0</sup><sub>-0.1</sub></td><td>58<sup>+10</sup><sub>-10</sub></td><td>&lt;50</td><td>2.27<sup>+0.28</sup><sub>-0.31</sub></td><td>—</td><td>55<sup>+34</sup><sub>-14</sub></td><td>1.86<sup>+0.35</sup><sub>-0.20</sub></td><td>—</td><td>—</td></tr><tr><td>HD 30122</td><td>14.85<sup>+0.05</sup><sub>-0.05</sub></td><td>20.70<sup>+0.10</sup><sub>-0.10</sub></td><td>—</td><td>3.8<sup>+0.4</sup><sub>-0.3</sub></td><td>61<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.21<sup>+0.18</sup><sub>-0.19</sub></td><td>-0.15<sup>+0.36</sup><sub>-0.27</sub></td><td>69<sup>+8</sup><sub>-7</sub></td><td>2.00<sup>+0.24</sup><sub>-0.27</sub></td><td>—</td><td>—</td></tr><tr><td>HD 96675</td><td>15.33<sup>+0.05</sup><sub>-0.05</sub></td><td>20.86<sup>+0.10</sup><sub>-0.10</sub></td><td>—</td><td>5.5<sup>+1.4</sup><sub>-0.6</sub></td><td>—</td><td>—</td><td>2.82<sup>+0.18</sup><sub>-0.19</sub></td><td>—</td><td>71<sup>+8</sup><sub>-37</sub></td><td>2.47<sup>+0.19</sup><sub>-0.23</sub></td><td>—</td><td>—</td></tr><tr><td>HD 99872</td><td>14.65<sup>+0.05</sup><sub>-0.05</sub></td><td>20.52<sup>+0.10</sup><sub>-0.10</sub></td><td>14.54<sup>+0.05</sup><sub>-0.05</sub></td><td>3.7<sup>+0.1</sup><sub>-0.1</sub></td><td>66<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.12<sup>+0.25</sup><sub>-0.32</sub></td><td>-0.11<sup>+0.37</sup><sub>-0.26</sub></td><td>75<sup>+17</sup><sub>-8</sub></td><td>1.90<sup>+0.34</sup><sub>-0.41</sub></td><td>104<sup>+73</sup><sub>-25</sub></td><td>1.80<sup>+0.18</sup><sub>-0.09</sub></td></tr><tr><td>HD 137595</td><td>13.89<sup>+0.05</sup><sub>-0.05</sub></td><td>20.62<sup>+0.10</sup><sub>-0.10</sub></td><td>—</td><td>4.1<sup>+0.3</sup><sub>-0.3</sub></td><td>72<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.24<sup>+0.15</sup><sub>-0.19</sub></td><td>-0.03<sup>+0.26</sup><sub>-0.25</sub></td><td>93<sup>+11</sup><sub>-11</sub></td><td>2.00<sup>+0.24</sup><sub>-0.25</sub></td><td>—</td><td>—</td></tr><tr><td>HD 144965</td><td>15.30<sup>+0.05</sup><sub>-0.05</sub></td><td>20.79<sup>+0.10</sup><sub>-0.10</sub></td><td>13.71<sup>+0.05</sup><sub>-0.05</sub></td><td>5.1<sup>+0.9</sup><sub>-0.3</sub></td><td>70<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.58<sup>+0.12</sup><sub>-0.13</sub></td><td>0.15<sup>+0.34</sup><sub>-0.22</sub></td><td>60<sup>+10</sup><sub>-7</sub></td><td>2.34<sup>+0.20</sup><sub>-0.16</sub></td><td>107<sup>+47</sup><sub>-34</sub></td><td>2.19<sup>+0.17</sup><sub>-0.11</sub></td></tr><tr><td>HD 147683</td><td>15.99<sup>+0.05</sup><sub>-0.05</sub></td><td>20.74<sup>+0.10</sup><sub>-0.10</sub></td><td>14.81<sup>+0.05</sup><sub>-0.05</sub></td><td>7.5<sup>+0.9</sup><sub>-0.5</sub></td><td>56<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.94<sup>+0.10</sup><sub>-0.07</sub></td><td>-0.05<sup>+0.32</sup><sub>-0.22</sub></td><td>46<sup>+8</sup><sub>-5</sub></td><td>2.69<sup>+0.12</sup><sub>-0.10</sub></td><td>87<sup>+50</sup><sub>-17</sub></td><td>2.12<sup>+0.16</sup><sub>-0.11</sub></td></tr><tr><td>HD 148937</td><td>14.50<sup>+0.05</sup><sub>-0.05</sub></td><td>—</td><td>15.14<sup>+0.05</sup><sub>-0.05</sub></td><td>4.1<sup>+1.0</sup><sub>-0.2</sub></td><td>69<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.36<sup>+0.23</sup><sub>-0.24</sub></td><td>—</td><td>64<sup>+37</sup><sub>-8</sub></td><td>2.06<sup>+0.22</sup><sub>-0.27</sub></td><td>147<sup>+56</sup><sub>-111</sub></td><td>1.96<sup>+1.76</sup><sub>-0.22</sub></td></tr><tr><td>HD 190918</td><td>13.18<sup>+0.05</sup><sub>-0.05</sub></td><td>20.00<sup>+0.10</sup><sub>-0.10</sub></td><td>—</td><td>3.7<sup>+1.1</sup><sub>-0.8</sub></td><td>102<sup>+10</sup><sub>-10</sub></td><td>—</td><td>1.97<sup>+0.20</sup><sub>-0.46</sub></td><td>0.43<sup>+0.20</sup><sub>-0.30</sub></td><td>154<sup>+22</sup><sub>-34</sub></td><td>1.71<sup>+0.29</sup><sub>-0.53</sub></td><td>—</td><td>—</td></tr><tr><td>HD 192035</td><td>15.15<sup>+0.05</sup><sub>-0.05</sub></td><td>20.68<sup>+0.10</sup><sub>-0.10</sub></td><td>—</td><td>3.5<sup>+1.0</sup><sub>-0.2</sub></td><td>66<sup>+10</sup><sub>-10</sub></td><td>—</td><td>1.97<sup>+0.27</sup><sub>-0.35</sub></td><td>-0.13<sup>+0.39</sup><sub>-0.33</sub></td><td>72<sup>+14</sup><sub>-14</sub></td><td>1.71<sup>+0.39</sup><sub>-0.56</sub></td><td>—</td><td>—</td></tr><tr><td>HD 198781</td><td>15.23<sup>+0.05</sup><sub>-0.05</sub></td><td>20.56<sup>+0.10</sup><sub>-0.10</sub></td><td>14.53<sup>+0.04</sup><sub>-0.04</sub></td><td>3.5<sup>+0.8</sup><sub>-0.4</sub></td><td>65<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.00<sup>+0.27</sup><sub>-0.35</sub></td><td>-0.13<sup>+0.41</sup><sub>-0.26</sub></td><td>70<sup>+12</sup><sub>-14</sub></td><td>1.74<sup>+0.44</sup><sub>-0.49</sub></td><td>107<sup>+74</sup><sub>-25</sub></td><td>1.75<sup>+0.19</sup><sub>-0.11</sub></td></tr><tr><td>HD 203532</td><td>15.66<sup>+0.05</sup><sub>-0.05</sub></td><td>20.70<sup>+0.10</sup><sub>-0.10</sub></td><td>13.63<sup>+0.05</sup><sub>-0.05</sub></td><td>4.9<sup>+0.5</sup><sub>-0.7</sub></td><td>47<sup>+10</sup><sub>-10</sub></td><td>—</td><td>2.58<sup>+0.14</sup><sub>-0.13</sub></td><td>0.09<sup>+0.26</sup><sub>-0.15</sub></td><td>44<sup>+9</sup><sub>-5</sub></td><td>2.34<sup>+0.21</sup><sub>-0.19</sub></td><td>75<sup>+36</sup><sub>-15</sub></td><td>2.61<sup>+0.23</sup><sub>-0.13</sub></td></tr><tr><td>HD 220057</td><td>14.63<sup>+0.05</sup><sub>-0.05</sub></td><td>20.34<sup>+0.10</sup><sub>-0.10</sub></td><td>14.44<sup>+0.05</sup><sub>-0.05</sub></td><td>3.4<sup>+1.0</sup><sub>-0.2</sub></td><td>63<sup>+11</sup><sub>-11</sub></td><td>—</td><td>1.91<sup>+0.26</sup><sub>-0.40</sub></td><td>-0.41<sup>+0.41</sup><sub>-0.27</sub></td><td>77<sup>+29</sup><sub>-8</sub></td><td>1.62<sup>+0.44</sup><sub>-0.58</sub></td><td>91<sup>+71</sup><sub>-18</sub></td><td>1.66<sup>+0.14</sup><sub>-0.13</sub></td></tr><tr><td colspan="14">MW: Welty et al. 2020</td></tr><tr><td>HD 62 542</td><td>16.42<sup>+0.04</sup><sub>-0.04</sub></td><td>20.80<sup>+0.20</sup><sub>-0.20</sub></td><td>15.41<sup>+0.10</sup><sub>-0.10</sub></td><td>11.7<sup>+0.3</sup><sub>-0.3</sub></td><td>43<sup>+11</sup><sub>-11</sub></td><td>40<sup>+5</sup><sub>-5</sub></td><td>3.36<sup>+0.04</sup><sub>-0.05</sub></td><td>0.54<sup>+0.10</sup><sub>-0.08</sub></td><td>50<sup>+5</sup><sub>-4</sub></td><td>3.14<sup>+0.04</sup><sub>-0.08</sub></td><td>107<sup>+24</sup><sub>-31</sub></td><td>2.68<sup>+0.10</sup><sub>-0.1</sub></td></tr><tr><td colspan="14">MW: Federman et al. 2021</td></tr><tr><td>HD 28975</td><td>17.16<sup>+0.02</sup><sub>-0.02</sub></td><td>21.10<sup>+0.20</sup><sub>-0.20</sub></td><td>—</td><td>11.3<sup>+2.4</sup><sub>-2.4</sub></td><td>—</td><td>30<sup>+10</sup><sub>-10</sub></td><td>3.12<sup>+0.28</sup><sub>-0.29</sub></td><td>0.25<sup>+0.41</sup><sub>-0.43</sub></td><td>16<sup>+9</sup><sub>-2</sub></td><td>2.93<sup>+0.39</sup><sub>-0.44</sub></td><td>—</td><td>—</td></tr><tr><td>HD 29647</td><td>17.98<sup>+0.02</sup><sub>-0.02</sub></td><td>21.50<sup>+0.20</sup><sub>-0.20</sub></td><td>—</td><td>9.6<sup>+0.5</sup><sub>-0.7</sub></td><td>—</td><td>&lt;20</td><td>3.24<sup>+0.40</sup><sub>-0.32</sub></td><td>&lt;-0.4</td><td>11<sup>+2</sup><sub>-2</sub></td><td>3.00<sup>+0.78</sup><sub>-0.50</sub></td><td>—</td><td>—</td></tr><tr><td colspan="14">DLAs: Noteraume et al. 2018</td></tr><tr><td>J0000+0048</td><td>14.95<sup>+0.05</sup><sub>-0.05</sub></td><td>20.44<sup>+0.03</sup><sub>-0.03</sub></td><td>16.21<sup>+0.07</sup><sub>-0.07</sub></td><td>10.5<sup>+0.6</sup><sub>-0.5</sub></td><td>52<sup>+2</sup><sub>-2</sub></td><td>—</td><td>2.36<sup>+0.22</sup><sub>-0.74</sub></td><td>-0.33<sup>+0.27</sup><sub>-0.26</sub></td><td>66<sup>+51</sup><sub>-8</sub></td><td>2.00<sup>+0.52</sup><sub>-1.18</sub></td><td>100<sup>+44</sup><sub>-24</sub></td><td>1.29<sup>+0.30</sup><sub>-0.25</sub></td></tr><tr><td>J0857+1855</td><td>13.51<sup>+0.05</sup><sub>-0.05</sub></td><td>—</td><td>14.16<sup>+0.06</sup><sub>-0.06</sub></td><td>8.7<sup>+1.3</sup><sub>-1.1</sub></td><td>—</td><td>—</td><td>2.15<sup>+0.38</sup><sub>-0.28</sub></td><td>—</td><td>134<sup>+23</sup><sub>-71</sub></td><td>1.97<sup>+0.57</sup><sub>-0.61</sub></td><td>93<sup>+253</sup><sub>-74</sub></td><td>2.13<sup>+0.07</sup><sub>-0.33</sub></td></tr><tr><td>J1047+2057</td><td>14.97<sup>+0.21</sup><sub>-0.12</sub></td><td>—</td><td>—</td><td>6.7<sup>+0.7</sup><sub>-0.6</sub></td><td>—</td><td>—</td><td>&lt;2.2</td><td>—</td><td>93<sup>+41</sup><sub>-37</sub></td><td>&lt;1.9</td><td>—</td><td>—</td></tr><tr><td>J1237+0647</td><td>14.21<sup>+0.02</sup><sub>-0.02</sub></td><td>19.21<sup>+0.12</sup><sub>-0.12</sub></td><td>15.33<sup>+0.30</sup><sub>-0.20</sub></td><td>11.0<sup>+0.2</sup><sub>-0.2</sub></td><td>108<sup>+92</sup><sub>-34</sub></td><td>—</td><td>2.24<sup>+0.28</sup><sub>-0.61</sub></td><td>0.09<sup>+0.47</sup><sub>-0.21</sub></td><td>213<sup>+88</sup><sub>-98</sub></td><td>1.92<sup>+0.54</sup><sub>-0.62</sub></td><td>263<sup>+194</sup><sub>-111</sub></td><td>&lt;1.34</td></tr><tr><td>J1439+1118</td><td>13.86<sup>+0.02</sup><sub>-0.02</sub></td><td>19.38<sup>+0.10</sup><sub>-0.10</sub></td><td>14.57<sup>+0.03</sup><sub>-0.03</sub></td><td>12.0<sup>+0.6</sup><sub>-0.6</sub></td><td>107<sup>+33</sup><sub>-20</sub></td><td>—</td><td>2.87<sup>+0.30</sup><sub>-0.60</sub></td><td>-0.11<sup>+0.26</sup><sub>-0.20</sub></td><td>154<sup>+40</sup><sub>-37</sub></td><td>2.33<sup>+0.25</sup><sub>-0.30</sub></td><td>154<sup>+63</sup><sub>-40</sub></td><td>1.52<sup>+0.22</sup><sub>-0.15</sub></td></tr><tr><td>J1705+3543</td><td>14.12<sup>+0.05</sup><sub>-0.05</sub></td><td>—</td><td>14.89<sup>+0.04</sup><sub>-0.04</sub></td><td>12.4<sup>+1.5</sup><sub>-1.5</sub></td><td>—</td><td>—</td><td>&gt; 3</td><td>-0.19<sup>+0.22</sup><sub>-0.16</sub></td><td>64<sup>+20</sup><sub>-15</sub></td><td>3.58<sup>+0.20</sup><sub>-0.89</sub></td><td>64<sup>+20</sup><sub>-15</sub></td><td>3.58<sup>+0.20</sup><sub>-0.90</sub></td></tr><tr><td>J2331-0908</td><td>13.65<sup>+0.05</sup><sub>-0.05</sub></td><td>20.57<sup>+0.05</sup><sub>-0.05</sub></td><td>14.60<sup>+0.10</sup><sub>-0.10</sub></td><td>24.4<sup>+2.8</sup><sub>-2.3</sub></td><td>264<sup>+20</sup><sub>-20</sub></td><td>—</td><td>3.09<sup>+0.08</sup><sub>-0.08</sub></td><td>&gt; 0.64</td><td>158<sup>+41</sup><sub>-32</sub></td><td>2.86<sup>+0.13</sup><sub>-0.11</sub></td><td>173<sup>+30</sup><sub>-35</sub></td><td>2.20<sup>+0.12</sup><sub>-0.09</sub></td></tr></tbody></table>

Note. N(CO), N(H₂), and N(C1) are the column densities of CO, H₂, and C1 in cm⁻², respectively. T<sub>ex</sub>(CO) is the CO excitation temperatures (in K), derived using the population of all J available CO rotational levels, T<sub>ex</sub>(H₂) and T<sub>ex</sub>(C₂) are the H₂ and C₂ excitation temperatures (in K), respectively, derived using the population of J = 0, 1 for H₂ and J = 0, 2 for C₂. n<sub>H</sub><sup>(CO)</sup> is the total hydrogen number density (in cm⁻³), derived from the excitation of CO + H₂. I<sub>UV</sub> is the intensity of UV radiation field (in units of Mathis field), derived from the excitation of C1 + H₂ levels (or CO + H₂ levels if there is no C1 data). T<sub>gas</sub><sup>(CO)</sup> and n<sub>gas</sub><sup>(CO)</sup> are the kinetic temperature (in K) and gas number density (in cm⁻³), derived from the best-fit MEUDON PDR models for the region contributing to the CO absorption lines. T<sub>gas</sub><sup>(CI)</sup> and n<sub>gas</sub><sup>(CI)</sup> are the kinetic temperature (in K) and gas number density (in cm⁻³), derived from the best-fit MEUDON PDR models for the region contributing to C1 absorption lines.

survey (Rachford et al. 2002). The FUSE spectra, with a moderate spectral resolution of approximately R = 20 000, are sufficient to measure the total column densities of H₂ rotational levels up to J = 5. We obtained measurements for most of our sight lines from

Rachford et al. (2002), Sheffer et al. (2008), and Jensen et al. (2010). Additionally, for several sight lines, we present measurements of C₂ excitation from Sonnentrucker et al. (2007); Welty et al. (2020); Federman et al. (2021).
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Left-hand panel: Comparison of CO and C1 column densities against H₂ column densities in Milky Way molecular clouds (local ISM), using both absorption and emission techniques. Green and blue symbols denote the direct measurements in absorption systems, with different shapes representing various data samples: stars (Burgh et al. 2010), down triangles (Sonnentrucker et al. 2007), circles (Sheffer et al. 2008), diamonds (Welty et al. 2020), and upper triangles (Federman et al. 2021). Orange squares represent observations of CO emission (Lee et al. 2018), where N(H₂) is derived from the dust extinction A<sub>V</sub> map. The black line indicates the average CO to H₂ ratio for the Taurus molecular cloud (Pineda et al. 2010). Measurements of CO, C1 (as a function of H₂) column densities in high-redshift DLAs are also shown from Noterdaeme et al. (2018) by red and pink diamonds, respectively. The shaded areas denote different physical regimes within molecular clouds in terms of the fraction of carbon in molecular state: diffuse (f<sub>CO</sub> < 0.01), translucent (0.01 < f<sub>CO</sub> < 0.5), and dense (f<sub>CO</sub> > 0.5). See text for further details. Right-hand panel: Comparison of the excitation temperature of CO against the column density of CO in absorption systems in the Milky Way (green symbols) and DLAs at z ~ 1.7–2.7 (red diamonds). The black lines and grey shaded area represent the relation between CO column density and the excitation temperature of CO in the range of T<sub>ex</sub> between about 5–13 K found using emission line measurements in the Taurus Cloud (Pineda et al. 2010). The left and right lines were calculated using the equation (21) in Pineda et al. (2010) for two values of δv = 1 and 2 km s<sup>-1</sup>, respectively. The horizontal black line indicates the CMB temperature at redshift z = 0.</figcaption></figure>

## 2.2 The high redshift DLAs sample ($S_{\text{DLA}}$)

The first detection<fnref n="1" /> of CO molecules in absorption in diffuse ISM at high redshifts was reported by Srianand et al. (2008) in the DLA system at z = 2.41 towards the quasar J1439+1117. To date, CO absorptions at high redshift (z > 1.7) have been detected in eight DLAs (Noterdaeme et al. 2018). We selected seven systems where the populations of individual CO rotational levels were well measured, as listed in Table 1. Since CO absorptions have primarily been used to probe the evolution of the CMB temperature at high redshift (Noterdaeme et al. 2011), in most cases, the populations of lower rotational levels have been accurately calculated. Given that we are studying collisional excitation, which strongly affects higher rotational levels, we reanalysed the CO absorption lines in the archival data of J0857+1855, J1047+2057, J1237+0647, J1439+1118, J1705+3543, J2331−0908.

The observations were performed with the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) mounted on the ESO VLT-UT 2 8.2 m telescope at Cerro Paranal, Chile. The data reduction procedures and spectral analyses of H₂, C1, and CO absorption lines are described in detail in Noterdaeme et al.

(2010) for J1237+0647, Noterdaeme et al. (2011) for J0857+1855, J1047+2057, J1705+3543, and Noterdaeme et al. (2017) for J0000+0048, Noterdaeme et al. (2018) for J2331−0908. The analysis of CO absorption lines is presented in Appendix B.

## 2.3 Comparison of the samples

Fig. 1 compares measurements of the column density of H₂, CO, and C1, along with the CO excitation temperatures, in the samples of local ISM clouds and DLAs. Additionally, we show measurements in diffuse CO-weak molecular gas (Burgh et al. 2010; Jenkins & Tripp 2011) and in dense CO-bright clouds, including the Taurus molecular cloud (Pineda et al. 2010) and other local molecular clouds (Lee et al. 2018). This provides a comprehensive overview of CO chemistry in the ISM, covering the full spectrum from diffuse to dense molecular gas.

The studies of CO gas in emission present measurements of the CO 1–0 line flux ($I_{\text{CO}}$) and dust extinction ($A_{\text{V}}$) along the same lines of sight. To convert $I_{\text{CO}}$ and $A_{\text{V}}$ to N(CO) and N(H₂) Pineda et al. (2010) used calculations of CO emissivity with the code RADEX (van der Tak et al. 2007) for a set of physical conditions and applied the universal relation N(H₂)/A<sub>V</sub> = 9.4 × 10<sup>20</sup> cm<sup>-2</sup> mag<sup>-1</sup> (Bohlin, Savage & Drake 1978). The black solid line in Fig. 1 shows the average N(CO) and N(H₂) relation for the Taurus cloud, which is in a good agreement with measurements of N(CO) and N(H₂) in absorption in our sample. To derive N(CO) and N(H₂) from

<footnote n="1">¹CO molecules from galaxies and quasars at z > 2 were first detected in radio emission, such as in IRAS F10214+4724 at z = 2.3 (Brown & Vanden Bout 1991; Solomon, Downes & Radford 1992) and in H1413+117 (The Cloverleaf) at z = 2.5 (Barvainis et al. 1994, 1997).</footnote>
measurements of $I_{CO}$ and $A_V$ from Lee et al. (2018), we use a similar approach (however do not use model assumptions for CO emission, which can vary between molecular clouds). We consider the range of $A_V = 1–10$ mag and $I_{CO} = 3–30$ K km s$^{-1}$, where both hydrogen and carbon are predominantly molecular ($H_2$ and CO), and there is a linear relationship between $I_{CO}$ and $A_V$ with a slope equivalent to the Galactic $X_{CO}$ factor (Lee et al. 2018). Using this relationship, we estimate $N(CO)$ as $I_{CO} \cdot X_{CO} \cdot (C/H)_{MW}$, assuming the average ISM metallicity, $(C/H)_{MW} = 1.1 \times 10^{-4}$, (Ritchey et al. 2023). The $N(H_2)$ is determined using the Bohlin et al. (1978) relation. We find that the derived $N(CO)–N(H_2)$ relationship aligns perfectly with data from Pineda et al. (2010).

Compare our sample of the absorption systems in local ISM with other measurements, we would like to highlight several moments. (i) CO detected in absorption examines the intermediate (translucent) phase between the diffuse and dense CO-bright gas phases, where CO abundance significantly increases. The diffuse/translucent transition occurs at $N(CO) = 10^{15}$ cm$^{-2}$ and $N(H_2) = 10^{20.6}$ cm$^{-2}$, where CO/C I ~ 1 and CO becomes dominant form of neutral carbon (see previous discussion in Burgh et al. 2007; Sheffer et al. 2008). At $N(CO)$ above $10^{17.3}$ cm$^{-2}$, carbon is primarily molecular. (ii) The threshold for detecting CO emission is usually set to a column density of about $10^{15}$ cm$^{-2}$ corresponding to an integrated line intensity of 1 K km s$^{-1}$, the usual sensitivity of large-scale CO 1–0 maps. CO emission is detected even when CO is not the main carrier of carbon (e.g. Pineda et al. 2010; Liszt & Gerin 2023). (iii) CO and C I exhibit different behaviours in translucent phase. The abundance of CO (the ratio of CO/H$_2$) increases with $N(H_2)$ from $10^{-6}$ to $10^{-4}$, while the ratio of C I/H$_2$ remains nearly constant at the value in diffuse gas, ~$10^{-6}$. (iv) The increase in CO abundance in translucent gas correlates with a rise in CO excitation temperature, indicating a change in gas physical state. The increase of CO excitation temperature is likely due to both rising collisional excitation and the RTE effect. We will apply our approach to analyse CO excitation and derive physical condition in the translucent phase. (v) The relations between CO and H$_2$ column densities derived from absorption and emission studies are in perfect agreement and demonstrate the evolution of the carbon fraction in molecular state with $N(H_2)$. Additionally, the rise of CO excitation temperature with $N(CO)$ seen in absorption data is also consistent with the model used by Pineda et al. (2010) in the overlapping region.

We completed Fig. 1 with shaded areas representing different values of carbon molecular fraction $f_{CO} = CO/C_{tot}$, where $C_{tot} \simeq C_{II}+C_{I}+CO$. Since the $C_{II} \lambda 1335$ absorption line is often saturated, we estimate $N(C_{II})$ from model assumptions: gas in local ISM has the average carbon abundance $(C/H)_{MW}$, and the total amount of hydrogen $N(H_1) + 2N(H_2)$ related to the molecular cloud can be estimated from the observed $N(H_2)$ and the theory of the H I to H$_2$ transition (Sternberg et al. 2014; Bialy & Sternberg 2016). For calculations, we assumed an average intensity of the UV field at 1 Mathis field and an average number density in diffuse/translucent gas of approximately ~$100$ cm$^{-3}$ (Goldsmith et al. 2008; Jenkins & Tripp 2011). Assuming a constant abundance of C I in diffuse gas, we found that it perfectly corresponds to the region, where $f_{CO} \le 1$ per cent. The intermediate region, where the CO abundance increases, corresponds to 1 per cent < $f_{CO} < 50$ per cent. Finally, CO gas observed in emission corresponds to $f_{CO} > 50$ per cent.

We observe systematically higher (by an order of magnitude) CO and C I abundances in quasar absorptions at $z \sim 2$ compared to those in the local ISM. This may be due to selection effect, as CO-bearing

DLAs were preselected in the SDSS survey based on the presence of strong C I lines, which are prominent even at the SDSS resolution (see Noterdaeme et al. 2010; Ledoux et al. 2015; Noterdaeme et al. 2018).

## 3 METHOD

To determine physical conditions, we compare the observed populations of the $H_2$ and CO rotational levels as well as those of C I fine-structure levels interpolated grids of photodissociation models calculated with the MEUDON PDR code (version 1.5.4,<fnref n="2" /> Le Petit et al. 2006). This code allows ones to calculate excitation of molecules in a self-consistent way, performing a complete calculation of the radiative transfer in many lines (including $H_2$ and CO) together with treatment of the thermal balance and chemistry. Below, we describe a set-up of the code and the method used to compare grid of models with the observed data.

### 3.1 The MEUDON PDR model

A model consists of a gas cloud in plane-parallel geometry and constant density configuration irradiated by isotropic interstellar radiation field and exposed to cosmic-ray background from both sides. We adopted the standard model of the interstellar radiation (ISRF) proposed by Mathis, Mezger & Panagia (1983). The intensity of the ISRF in the UV range is scaled by a factor $I_{UV}$. Millimeter domain (0.1–10 mm) of the ISRF consists of combination of two components: the CMB radiation and dust emission. The intensity of the CMB is determined by black body model with the temperature $T_{CMB} = 2.725 \times (1+z)$ K, where z is the redshift. The dust emission corresponds to the average dust emission background measured in the Milky Way. The cosmic-ray ionization rate (CRIR) is set to $10^{-16}$ s$^{-1}$, which corresponds to measurements in the local ISM (see, e.g. Indriolo et al. 2007; Kosenko et al. 2021). Scaling the CRIR with UV intensity may be important for the high-redshift sample, where metallicity is low on average, and the cosmic ray and turbulent heating dominate over the standard photo-electron heating mechanism (see e.g. Balashev et al. 2022). But here we have mostly metal rich systems, which is correlated with the presence of CO. We used a configuration with a constant hydrogen density ($n_H = n(H_1) + 2n(H_2)$) for better numerical convergence compared to those with a constant thermal pressure. Other parameters (such as grain properties) are set to their typical values, measured in local ISM. The dispersion of turbulent velocity ($v_{turb}$) may also have an impact on the radiative trapping of CO and $H_2$. We have checked that varying this parameter within the range of 0–3 km s$^{-1}$ (typical for cold gas) does not result in significant differences (within 0.05 dex) in the calculated column densities of $H_2$ and CO rotational levels. If clouds have high $H_2$ and CO column densities, the bulk of $H_2$ and CO locates in the central region of the cloud, where physical conditions remain stable. This central region is less sensitive to turbulent velocity, which predominantly influences conditions near the edges of the cloud. The list of input parameters is presented in Table 2. The size of the cloud is derived by the total visual extinction $A_V^{max}$, which following the MEUDON PDR determines the total hydrogen column density as $N(H_{tot}) = (C_D/R_V)A_V^{max}$. The parameter $C_D$ is scaled with metallicity. For ISM sightlines, we set $Z = 1$ and the value of $C_D$

<footnote n="2">²Public version of February 2021 – revision 2083: https://pdr.obspm.fr/pdr-download.html</footnote>
<table><caption>**Table 2.** List of input parameters of the MEUDON PDR models. The first three parameters ($n_H$, $I_{UV}$, $A_V$) were varied within indicated ranges. The CMB temperature and metallicity were adapted for particular systems.</caption><thead><tr><th>Pars.</th><th>Units</th><th>Value</th><th>Comment</th></tr></thead><tbody><tr><td>$n_H$</td><td>cm<sup>-3</sup></td><td>1–10<sup>4</sup></td><td>Total hydrogen number density</td></tr><tr><td>$I_{UV}$</td><td>Mathis field</td><td>0.1–10</td><td>UV radiation strength</td></tr><tr><td>$A_{Vmax}$</td><td></td><td>0.1–10</td><td>Maximum visual extinction</td></tr><tr><td>$Z$</td><td>solar</td><td>1–3</td><td>Metallicity</td></tr><tr><td>$T_{CMB}$</td><td>K</td><td>2.73(1+z)</td><td>CMB temperature</td></tr><tr><td>$I_{CR}$</td><td>s<sup>-1</sup></td><td>10<sup>-16</sup></td><td>Cosmic ray ionization rate</td></tr><tr><td>$v_{turb}$</td><td>km s<sup>-1</sup></td><td>2</td><td>Gas turbulent velocity</td></tr><tr><td>Ext</td><td></td><td>Galaxy</td><td>Type of extinction curve</td></tr><tr><td>$R_V$</td><td></td><td>3.1</td><td>Ratio of the visual extinction to colour excess</td></tr><tr><td>$C_D$</td><td>cm<sup>-2</sup></td><td>5.8 × 10<sup>21</sup>/Z<sup>†</sup></td><td>Ratio of the column density of neutral gas to visual extinction</td></tr><tr><td>$m_{gr}$</td><td></td><td>0.01 × Z<sup>†</sup></td><td>Dust to gas mass ratio</td></tr><tr><td>$q_{PAH}$</td><td></td><td>4.6 × 10<sup>-2</sup></td><td>PAH fraction</td></tr><tr><td>$\alpha_{gr}$</td><td></td><td>3.5</td><td>Grain power-law index</td></tr><tr><td>$r_{min}$</td><td>cm</td><td>10<sup>-7</sup></td><td>Grains minimum radius</td></tr><tr><td>$r_{max}$</td><td>cm</td><td>3 × 10<sup>-5</sup></td><td>Grains maximum radius</td></tr></tbody></table>

Note.† in the MEUDON PDR code these values are automatically linearly scaled by metallicity.

from Bohlin et al. (1978). For DLAs, metallicity is measured in three of seven CO-bearing systems and is ~(1–3)Z⊙ (Srianand et al. 2008; Noterdaeme et al. 2010; Noterdaeme et al. 2017).<fnref n="3" /> For these systems, the grid is adopted to measured metallicity for the rest is set to Z⊙. For each model, we performed 20 iterations to obtain a stable solution of a cloud structure. Additionally, a convergence of the calculations and stability of the thermal solution for each model was checked visually.

An example of a PDR model (with $n_H = 300$ cm⁻<fnref n="3" />, $I_{UV} = 1$, $A_V = 1.7$) is shown in Fig. 2. While the calculations present well-known results for diffuse clouds, we highlight a few features important for the following description of the method and discussion of the results. First, the gas in the central part of the cloud is mostly molecular (H₂). The fraction of C I also does not change along the cloud as much as CO does, i.e. CO molecules exist in a deeper and more shielded central layers, containing about 50 per cent of the total H₂ column density. Therefore, H₂, C I, and CO probe different regions of the cloud, where, roughly speaking, CO probes the most central parts, and C I corresponds to wider region (and hence provide more averaged results). In that sense, H₂ is more complex due to significant dependence of the population of rotational levels (see left bottom panel of Fig. 2) due to radiative transfer (see, e.g. Balashev, Varshalovich & Ivanchik 2009), which results in that high and low H₂ rotational levels mostly represent the outer layers and the bulk of the cloud, respectively. Second, the kinetic temperature in PDR models is derived from the thermal balance, which assumes that in cold ISM, the emission in C II/C I/CO lines dominates the cooling, while the heating is dominated by the photoelectric heating,<fnref n="4" /> which

itself depends on the strength of UV radiation field. As a result, the kinetic temperature varies within the cloud due to changes in the chemical state and dilution of the UV field. We find that in PDR models that fit well the observed population of H₂ levels, the kinetic temperature corresponds to the observed excitation temperature H₂, i.e. being $T_{01}(H_2)$ is a good tracer of the average kinetic temperature in H₂ gas, for the range of H₂ column densities relevant for our study.

## 3.2 Matching with observations

The grid of the MEUDON PDR models was set to cover the parameter space on a logarithmic scale in the ranges of $1 \le \log n_H/cm^{-3} \le 4$, $-1 \le \log I_{UV} \le 1$, and $0.1 \le A_V^{max} \le 10$. Then we translate the uniform grid on $A_V^{max}$ axis to the non-uniform grid on N(CO). Fortunately, as we illustrate Fig. 3, this transformation is bijection in the considered part of the parameter space, since the abundance of CO is roughly scaled with $n_H$, $A_V$, and $I_{UV}^{-1}$, due to relatively simplistic dependence of the CO formation and destruction on these quantities. Therefore, for each absorption system, we obtained a cross-section of the parameter space $n_H-I_{UV}-N(CO)$ by a surface of constant CO column density at the observed N(CO). Using 3d interpolation,<fnref n="5" /> we calculated the CO, H₂, and C I levels populations as two-dimensional functions on $n_H-I_{UV}$ plane, which were further used in the likelihood calculation. Note that the total CO column density of absorption systems in our sample is usually well constrained, with a statistical uncertainty about ~0.05 dex. We assume that a variation of the total CO column density in the simulated 2D-cross-section within this interval likely has less impact on derived physical conditions than other systematic effects, such as the approximation of the cloud by a one-dimensional layer.

Then we derived the constraint on the $n_H$ and $I_{UV}$ by comparing the observed population of H₂ and CO rotational and C I fine-structure levels with modelled ones, within the grid. To do this, we used the likelihood function based on a least-squares comparison of the observed and model populations. We did this for each species and then considered a joint constraint. To convert the 2d map (on $n_H-I_{UV}$ plane) of likelihood function into estimate of parameter distribution, we assumed Bayesian approach with a flat priors of the $n_H$ and $I_{UV}$ in the logarithmic space.<fnref n="6" /> For individual parameter estimates, we marginalized posterior 2d distribution. To report point and interval estimates, we used maximum posterior probability and the highest density estimates at 0.683 level (formally corresponding to $1\sigma$ for normal distribution), respectively.

For CO, we directly compared the observed and modelled column densities for all available J levels. For H₂, the populations of low ($J < 3$) and high rotational levels were analysed in different ways: we fitted the populations of low rotational levels and considered populations of high rotational levels as upper limits of their excitation in the central part of the cloud. It was done because the populations of low H₂ rotational levels are mostly determined by the thermal balance therefore their fit provides an estimate of the average physical conditions. On the contrary, populations of high rotational levels are more excited in the outer envelope, where the structure can be more complex than modelled one (assuming to be slab), and other

<footnote n="3">³Supersolar value found in these systems is likely not common for DLAs but is due to a selection effect. These systems were preselected to have strong C I absorptions, whose strength correlates with reddening and metallicity (Ledoux et al. 2015). Second, the cross-section of CO-bearing gas is likely higher for metal-rich DLA systems.</footnote>

<footnote n="4">⁴That is correct for CO-bearing absorption systems, since these systems are associated with relatively high metallicity $Z > 0.1$ (for details, see, e.g. Bialy & Sternberg 2019; Balashev et al. 2022).</footnote>

<footnote n="5">⁵We used the PYTHONSCI PY package for radial basis function interpolation in N dimensions. The uncertainty in the interpolation of species column densities at the knots of the Meudon PDR grid is less than 0.05 dex.</footnote>

<footnote n="6">⁶That correspond to the weak informative prior, i.e. corresponding to very wide distribution, which may be reasonable since we do not have evident choice.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** An example of a Meudon PDR model reproducing the observed column densities of H₂ and CO, as well as the excitation of H₂, CO, and C I levels in the absorption system towards HD 24534. **Top panels:** Left: profiles of physical conditions as a function of depth inside the cloud. The shaded blue line represents the integrated value for the excitation temperature *T*₀₁ of H₂. The blue-shaded area shows the observed constraint on *T*₀₁. Middle: Species number densities as a function of depth. Right: species column densities as a function of depth. The dashed orange and red horizontal lines indicate the observed column densities of H₂ and CO, respectively. **Bottom panels** show the excitation of H₂ (left) and CO (middle) rotational levels, and C I fine-structure levels (right) as a function of depth. The orange and red vertical shaded areas in the panels indicate the regions containing 90 per cent of the total column density of H₂ and CO, respectively.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Pairwise 2D projections of a 3D grid of the MEUDON PDR models are shown. The two leftmost panels display grid slices in different projections: $n_H-A_V$ and $I_{UV}-A_V$, respectively, for a fixed value of the third coordinate, indicated at the top. The red colour gradient illustrates the dependence of $N(CO)$ on the coordinates. Black lines represent levels of constant $\log N(CO) = 14, 15, 16, 17, 18$. Circles indicate the actual PDR models. The two rightmost panels show the same grid in projections of $n_H-\log N(CO)$ and $I_{UV}-\log N(CO)$, respectively. The blue colour gradient and levels represent values of $A_V$.</figcaption></figure>

excitation mechanisms of H₂ levels such as turbulent dissipation and/or C-shock can affect populations of high rotational levels (e.g. Joulain et al. 1998). Therefore, we prefer to use less information than introduce a possible bias.

For C I, we compared the relative populations of fine structure levels, but not the absolute values. While the absolute abundance of C I is sensitive to various physical parameters (such as number density, UV field, metallicity, CRIR, and grain properties), the relative abundance of C I fine-structure levels is primarily determined

by basic parameters: $n_H$ and $I_{UV}$ (see Silva & Viegas 2001). The formation of C I depends on the ionization state of the gas (regulated by CRIR) and is dominated by grain-assisted recombination, which strongly depends on detailed grain physics, e.g. Wolfire et al. (2008); Liszt (2011); Balashev & Kosenko (2024). Simultaneously, the thermal state of the ISM is not directly sensitive to the CRIR at high metallicity, as the heating rate is mainly determined by photo-electron reaction. It is worth noting that most CO-bearing systems studied in this work are metal-rich. Therefore, instead of
<figure><img src="image_5.png" /><figcaption>**Figure 4.** An example of the analysis of excitation of rotational levels of H₂, fine structure levels of C I, and rotational levels of CO in the absorption system towards HD 24534. Blue, green, and purple contours in the top panels represent constraints on the total hydrogen number density and UV intensity obtained from the analysis of the excitation of H₂ (J = 0 to 2), C I (J = 0–2), and CO (J = 0–3) levels, respectively. The orange and red contours in the middle and right-hand panels indicate joint constraints using H₂ and C I levels, and H₂ and CO levels, respectively. In the bottom panels, we show best fits (lines) to observed populations (points) of H₂ (left), C I (middle), and CO (right) levels, corresponding to joint constraints in the top panels. The dashed line in the bottom right-hand panel shows the excitation of CO levels assuming only excitation by the CMB.</figcaption></figure>

fitting absolute abundances of C I levels, which would make our grid much more complex (by introducing CRIR as an additional parameter), we propose constraining *n*<sub>H</sub> and *I*<sub>UV</sub> through the fit to the relative populations of C I. Additionally, in many systems, C I lines are optically thin, which allows us to ignore radiative trapping for C I.

In Fig. 4, we present an example of constraints on the number density and UV intensity in the system of HD 24534. One can see that H₂, C I, and CO constraints show different behaviour. The excitation of CO levels are sensitive to the number density, but not very much to the UV intensity. Although the total CO column density itself is very sensitive to the UV intensity, we already took this dependence into account by choosing the knots of the grid that match the total CO column density. In fact the knots with higher UV intensity have higher *A*<sub>UV</sub> coordinate. The excitation of C I is much more sensitive to UV intensity than CO. The constraint from C I shows the anticorrelation between *n*<sub>H</sub> and *I*<sub>UV</sub> due to competition of collisions and UV pumping in excitation of C I levels. The excitation of H₂ provides an almost linear dependence between *n*<sub>H</sub> and *I*<sub>UV</sub>, which is nearly orthogonal to regions constrained by CO and C I. Such dependence comes from the thermal balance solution which depends roughly on *I*<sub>UV</sub>/<i>n</i><sub>H</sub> ratio for high-metallicity gas. Therefore CO, in contrast to C I and H₂, provides more stringent constraint on the number density. Secondly, a joint constraint using H₂ with C I or H₂ and CO provides tighter constraints on both parameters, the number density and intensity of UV field. One can see that in case of HD 24534 the constraints from H₂ with C I or H₂ and CO agree well with each other.

In some cases, we found that estimates on number density from joint fitting of CO and H₂ is found to be higher than using C I and

H₂. This likely indicates that either CO probes denser gas than C I or there is an unaccounted source of CO excitation, for example, CO emission by the local source (Wannier et al. 1997). We discuss the latter in a section below.

### 3.2.1 Excitation of CO by the local source

Let's consider a situation whether the strong source of the CO lines is located near the absorbing medium. Indeed, external CO emission can affects observed populations of CO levels by increasing the radiative excitation. This mechanism was suggested by Wannier et al. (1997) and discussed later in many papers (see, e.g. Sonnentrucker et al. 2007). We note here that this effect only plays a role if velocity of the emitting and absorbing regions coincides within line profile, that may significantly reduce the importance of this effect.

Unfortunately, the local intensity of CO emission in absorption system can not be well derived from emission studies because they have insufficient spatial resolution and a rather uncertain distance estimate. Another way to probe this is to compare excitation of different species, like ¹²CO, ¹³CO, or C I, observed in the same absorption system. These species are excited by both radiation and collisions. While collisional excitation of species is described by the same values of gas number density and kinetic temperature, radiative excitation for ¹²CO is stronger since ¹²CO emission line (from nearby source) is much stronger than ¹³CO (or C I). If there is no photon trapping in diffuse gas, one can expect T<sub>01</sub>(¹²CO) greater than T<sub>01</sub>(¹³CO). Sonnentrucker et al. (2007) studied ¹²CO and ¹³CO excitation in seven sightlines in local ISM and found that T<sub>01</sub>(¹²CO)
and $T_{01}({}^{13}CO)$ are consistent within uncertainties. From mm-wave CO absorptions Liszt & Lucas (1998) found $T_{01}({}^{12}CO) \sim T_{01}({}^{13}CO)$ in four of six components. Therefore, the excitation by the local source may not be significant for diffuse gas.

If the gas is optically thick in $^{12}CO$ absorption line then radiation from the emitting source will just scatter from the outer layer. In principle, it means that we should not consider excitation of CO by the local source in this case too. On the other hand, the trapping of internal (spontaneous) photons in the absorption cloud will increase $T_{01}({}^{12}CO)$ against $T_{01}({}^{13}CO)$ (making collisional excitation of $^{12}CO$ effectively stronger, see, e.g. Draine 2011). In practice it means that $T_{01}({}^{12}CO) > T_{01}({}^{13}CO)$ for CO optically thick gas may not mean the existence of external source of $^{12}CO$ line radiation.

### 3.3 Line trapping in CO lines

As we noted above, the population of CO rotational levels can be affected by the radiative trapping effect (RTE) when CO lines become optically thick. This requires calculating the intensities of emission lines consistently with the excitation of molecules as functions of depth into the cloud. The formalism was described earlier by Bernes (1979), van Zadelhoff et al. (2002), Tak et al. (2007), and Gonzalez Garcia et al. (2008). There are several codes for the calculation of radiation transfer and molecular excitation, with the most well-known being the MEUDON PDR (Le Petit et al. 2006), RADEX (Tak et al. 2007), and RADMC-3D (Dullemond et al. 2012). While the MEUDON PDR code utilizes an iterative method for solving radiative transfer and calculating photon escape probability based on local properties within the cloud (Gonzalez Garcia et al. 2008), other codes employ an approximation of geometrically averaged escape probability, dependent solely on optical depth (van der Tak et al. 2007; Shetty et al. 2011). This makes these codes much faster; however, they cannot model local properties in detail. While the MEUDON PDR code is slower, its advantage lies in its ability to calculate radiation transfer for several species simultaneously and consistently with the solution of thermal balance and chemistry.

The primary challenge in determining physical conditions from modelling lies in the dependence of results on the actual geometry of the cloud, which is often unknown. To understand the impact of the RTE effect on deriving physical conditions, we will discuss it quantitatively, using simple calculations of CO excitation. The excitation of CO rotational levels in each point of the cloud is derived from the equations of the detailed balance:

$$ \sum_j n_j Q_{ji} = n_i \sum_j Q_{ij}, \qquad (1) $$

where $n_i$ is the volume density of i level of CO, $Q_{ij}$ is the total rate of transitions between i and j levels. In escape probability approximation, $Q_{ij}$ can be written as

$$ \begin{aligned} Q_{ul} &= (\beta_l + \beta_r) \times A_{ul} + (\beta_l \times I_{ul}^l + \beta_r \times I_{ul}^r) \times B_{ul} + \sum_k n^k q_{ul}^k \\ Q_{lu} &= (\beta_l \times I_{lu}^l + \beta_r \times I_{lu}^r) \times B_{lu} + \sum_k n^k q_{lu}^k, \end{aligned} \qquad (2) $$

where u and l are the indexes for upper and lower levels, $A_{ul}$ is the probability of spontaneous decay, $B_{ul}$ and $B_{lu}$ are the Einstein coefficients, $I_{ul}^l$ and $I_{ul}^r$ are the external intensities from left and right sights at frequencies corresponding to u → l transition, $\beta_{l,r}$ denotes the direction-averaged escape probability$^7$ of photons at the frequency of the transition come away from some position in the cloud to left and right sides of the cloud, respectively. The last term in equation (2) denotes the collisional excitation, $n^k$ and $q_{ul}^k$ are the number densities of the collisional partners (H, H$_2$, He) and corresponded collisional rate coefficients.$^8$ If external field intensity is the same on each side (as for the CMB), RTE affects the excitation matrix only by multiplying $A_{ul}$ and $I_{ul}$ by the factor of the total escape probability $\beta_{tot} = \beta_l + \beta_r$.

During the modelling, we found a mistake in the MEUDON PDR code that external field intensities $I_{l,r}^{\text{ext}}$ (including the CMB) are set to be two times lower than it should be. This makes corresponding radiation field density two times lower, since the averaging of radiation coming from left and right direction are taking into account by $\beta_{l,r}$ functions (see equation (13) in Gonzalez Garcia et al. 2008). Therefore, we corrected the code by the changing $I_{l,r}^{\text{ext}}$ in excitation matrix to their total $4\pi$-averaged values.$^9$

In Fig. 5, we show the calculation of escape probability factor as a function of the total CO column density (and optical depth through the cloud), together with impact of RTE on the local excitation of CO levels and excitation diagram for the entire cloud. To separate RTE from a variation of the kinetic temperature, we used fixed number density $n_H = 300 \text{ cm}^{-3}$ and kinetic temperature $T_{kin} = 100 \text{ K}$. A gas cloud is illuminated by radiation from both sides to make an example relevant to our study. The escape probabilities are calculated in the MEUDON PDR code according to the scheme in Gonzalez Garcia et al. (2008). We considered three clouds with the total visual extinction $A_V = 1, 1.75$, and 3 yielded CO column density log N = 14.9, 16.3, and 17.1 to probe photon trapping in three regimes: ‘no trapping’, ‘medium’, and ‘strong’.

For CO column densities below $\sim 10^{15} \text{ cm}^{-2}$, the total escape probability is about $\sim 1$ and RTE does not affect CO excitation and therefore number density inferred from CO excitation will not be bias if one do not take into account RTE. For column densities about $\sim 10^{16.3} \text{ cm}^{-2}$, the cloud becomes optically thick in CO lines and photons can only escape from one side of the cloud therefore $\beta_{tot}$ is decreased to about 0.5. The excitation temperature of CO increases by 1.5 K, which can be converted into an excess of the number density by 0.3 dex. For dense clouds with $N(\text{CO}) \gtrsim 10^{17} \text{ cm}^{-2}$, the total escape probability decreases sharply in the central region that produces a strong enhancement of the population of CO levels by trapped photons. The average excitation temperature of CO increases by two times. In terms of the number density, this may induce a significant bias if we don’t take into account RTE. Note that the limiting column densities quoted above are very approximately since, for example, the turbulent velocity dispersion which affects the line profile when calculating escape probabilities was fixed to $2 \text{ km s}^{-1}$. At the same time, the approximation of the cloud by the 1D model introduces likely larger uncertainty.

We additionally checked this result using our own calculations of CO population taking into account excitation by collisions, CMB and line trapping. The code is similar to the POPRATIO (Silva & Viegas 2001, 2002) and RADEX (Tak et al. 2007) codes. The calculation scheme is provided in equations (1)–(2), and we use the same atomic

<footnote>$^7$Definition of the $\beta_{l,r}$, see in Gonzalez Garcia et al. (2008), equation (13).</footnote>

<footnote>$^8$Collisional rate coefficients of CO with H, orto and para H$_2$, and He were taken from Walker et al. (2015), Yang et al. (2010), and Cecchi-Pestellini et al. (2002), respectively.</footnote>

<footnote>$^9$After checking with the MEUDON PDR team, the bug has been corrected in the new version (1.7) that will be released soon.</footnote>
<figure><img src="image_6.png" /></figure>

**Figure 5.** An example of the calculation of radiative trapping in CO lines using the MEUDON PDR code. The purple, green, and red colours indicate models of a cloud illuminated by isotropic external radiation from both sides with total $A_V = 1, 1.75, 3$, corresponding to total CO column densities of $10^{14.9}, 10^{16.3}$, and $10^{17.1} \text{ cm}^{-2}$, respectively. The top panel shows the escape probability of photons in the CO $J = 1-0$ line as a function of $A_V$: $\beta_{\text{left}}$ is shown by the dashed line, and $\beta_{\text{total}} = \beta_{\text{left}} + \beta_{\text{right}}$ is shown by the solid line. The middle panel shows the profiles of the relative population of CO rotational levels at $J = 1, 2, 3, 4$ along the sightline through the cloud. Profiles are shown only for the left half of the cloud for simplicity, as they are symmetrical in our configuration. The bottom panel shows the excitation diagram of CO rotational levels calculated for the whole cloud, emulating the observed values. The excitation temperature of CO increases with the total CO column density due to the radiative trapping effect. For clouds with $\log N(\text{CO}) \lesssim 15$, the radiative trapping effect is negligible.

data and the same values of CO energy levels, transition probabilities, and collisional coefficients as in the MEUDON PDR. We found a good agreement with results obtained with the MEUDON PDR when we took into account the correction for external field intensities $I_{\text{l,r}}^{\text{ext}}$ (see above paragraph). Therefore, the radiative trapping makes negligible contribution to CO excitation at the total CO column density $\log N(\text{CO}) \lesssim 15$.

<figure><img src="image_7.png" /></figure>

**Figure 6.** The ratio of number densities $n_{\text{gas}}^{(\text{CO})}$ and $n_{\text{gas}}^{(\text{CI})}$ derived from the excitation of CO rotational levels and C I fine-structure levels, respectively, as a function of CO column density. The symbols are as in Fig. 1. The horizontal dashed line represents a one-to-one ratio. Systems, where $n_{\text{gas}}(\text{CO})$ exceeds $n_{\text{gas}}(\text{CI})$ at the $3\sigma$ level, are marked with text labels.

# 4 PHYSICAL CONDITIONS

In this section, we present constraints on gas number density, kinetic temperature, thermal pressure, and UV field derived from fitting the excitation of CO and H₂ and the fine-structure levels of C I in both the local ISM and DLAs. The results are summarized in Table 1. The detailed fitting results of H₂, C I, and CO excitation diagrams for each system are provided in the Appendix C. Since we use a different approach from Jenkins & Tripp (2011) to analyse C I absorptions in the Milky Way, we present analysis of their data on C I using our technique. These results are summarized in Appendix A.

## 4.1 Note about derived physical parameters

The MEUDON PDR code in a constant density configuration uses the total hydrogen gas density $n_H = n(\text{HI}) + 2n(\text{H}_2)$ and $I_{\text{UV}}$ as model parameters. Unlike $n_H$ and $I_{\text{UV}}$, the gas number density, kinetic temperature, and thermal pressure are not constant within the MEUDON PDR models (see e.g. Fig. 2). For example, the gas number density $n_{\text{gas}} \approx n(\text{HI}) + n(\text{H}_2) + n(\text{He})$ is similar to $n_H$ at low molecular fractions but can be approximately twice as lower in fully molecular regions. The kinetic temperature ($T_{\text{kin}}$) is usually close to the observed excitation temperature of H₂ ($T_{01}(\text{H}_2) \equiv T_{\text{ex}}(\text{H}_2)$), but it can vary along the cloud, i.e. decrease in CO-rich regions or be higher in the outer layers of the cloud. To account for this variation, we derived the physical parameters (gas number density, kinetic temperature, and thermal pressure) from the best-fit PDR models in the $n_H-I_{\text{UV}}$ plane, using the region containing 99 per cent of H₂ and CO for the fit to excitation of C I–H₂ and CO–H₂ levels, respectively. For DLA systems without H₂ excitation measurements, we determine the physical parameters using the MEUDON PDR models based on the fit to only CO or C I excitation.

## 4.2 Number density

The values of number density derived from CO and H₂ excitations ($n_{\text{gas}}^{(\text{CO})}$) range from 10 to $1200 \text{ cm}^{-3}$ for the local ISM sample and from 100 to 3000 $\text{cm}^{-3}$ for DLAs.

Fig. 6 compares the densities derived from excitation of CO+H₂ and C I+H₂ along the same sightlines in our samples. For the local
<figure><img src="image_8.png" /><figcaption>**Figure 7.** The measurements of the excitation temperature of CO versus the number density $n_{\text{gas}}^{(\text{CO})}$ derived from the rotational excitation of CO and H$_2$ levels in the Milky Way and DLAs. The symbols used are consistent with those in Fig. 1. The curves represent the model dependence of CO excitation temperature, derived using $J = 0, 1, 2$ levels, on the number density. Model parameter values are provided in the legend. The green solid and dashed curves represent the model with $T_{\text{CMB}} = 2.725$ K and total escape probabilities of $\beta_{\text{tot}} = 1$ and 0.5, respectively. The red lines represent models at the CMB temperature at redshifts $z = 1.7, 2.5, 2.7$, with varying values of the kinetic temperature.</figcaption></figure>

ISM, there is good consistency between the estimates, with a standard deviation of 0.15 dex. This supports our hypothesis that CO levels are predominantly excited by collisions rather than by line emission from a local source (see Section 3.2.1) or the RTE effect. In a few cases, we detect an excess of $n_{\text{gas}}^{(\text{CO})}/n_{\text{gas}}^{(\text{Cl})}$ by 0.5 dex (HD 27778, HD 62542, HD 147683). Since we accounted for radiative trapping in our calculation, one might speculate that CO in these systems probes denser gas compared to C I. This can be an indication of stratification of carbon states if there is a density gradient in the cloud, or CO might be excited by radiation from a local source.<fnref n="10" /> For the $S_{\text{DLA}}$ sample, we observed on average higher values of $n_{\text{gas}}^{(\text{CO})}$ relative to $n_{\text{gas}}^{(\text{Cl})}$. However, these measurements exhibit rather high uncertainty to confirm this difference. Additional observations with higher spectral resolution are required to constrain the significance of this excess.

Fig. 7 shows the excitation temperature of CO as a function of number density, highlighting the impact of collisional excitation and the RTE effect. The excitation temperature of CO was determined by independently fitting the populations of available CO rotational levels (typically $J = 0, 1, 2$) to a Boltzmann distribution. The results are shown in Table 1. For sight lines in the local ISM, $T_{\text{ex}}(\text{CO})$ is approximately ~3–4 K at $n_{\text{H}} \le 100 \text{ cm}^{-3}$ and increases to around 10 K in the range $n \simeq 100–1000 \text{ cm}^{-3}$, following an approximately exponential trend. The observed trend is consistent with the CO collisional excitation model integrated into the MEUDON PDR code. This highlights the fundamental patterns, emphasizing the roles of collisional excitation (dependence on $n$) and the impact of the RTE

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Comparison of kinetic temperatures derived from the MEUDON PDR models (for regions contributing to CO absorption lines) with observed excitation temperatures of H$_2$ and C$_2$ in the samples from the local ISM and DLAs. The dotted line indicates a 1:1 relationship. There is good consistency between modelled and observed values for most systems.</figcaption></figure>

(dependence on $\beta_{\text{tot}}$). According to calculations with the MEUDON PDR code (see Fig. 5), $\beta_{\text{tot}} = 1$ and 0.5 correspond to $N(\text{CO}) \simeq 10^{15}$ and $10^{16.5} \text{ cm}^{-2}$, respectively. Comparing the impacts of collisions and the RTE, it is evident that RTE has a much smaller effect on $T_{\text{ex}}(\text{CO})$ than collisions within the range $N(\text{CO}) \le 10^{16.5} \text{ cm}^{-2}$. The RTE accounts for an increase in $T_{\text{ex}}(\text{CO})$ of approximately 2 K, while the observed difference of around 5 K corresponds to a collisional excitation at $n \sim 1000 \text{ cm}^{-3}$.

The measurements of $T_{\text{ex}}(\text{CO})$ at high redshifts $z \sim 2–3$ align well with our models, computed using $T_{\text{CMB}}$ values in the rest-frame of DLAs. This underscores the dominance of excitation by CMB photons over other mechanisms at high redshifts.

## 4.3 Kinetic temperature

Fig. 8 compares the kinetic temperatures derived from PDR models with the observed excitation temperatures of H$_2$ and C$_2$. On average, there is good agreement. The small variations can be attributed to our method, which fits the excitation of the $J = 0, 1$, and 2 H$_2$ levels (not just $J = 0$ and 1), and to the difference between $T_{\text{kin}}$ and $T_{\text{ex}}(\text{H}_2)$ in the PDR models. As previously reported by Sonnentrucker et al. (2007), $T_{\text{ex}}(\text{C}_2)$ is typically slightly lower than $T_{\text{ex}}(\text{H}_2)$ along the same sightlines. The temperature derived from PDR models typically fall between these two estimates.

For CO absorbers in the local ISM, the kinetic temperature ranges from 10 to 160 K, with an average value of about 50 K. This suggests that H$_2$ and CO probe the cold diffuse phase of the ISM (Rachford et al. 2002). For CO absorbers in quasar spectra at high redshifts, the kinetic temperatures derived in PDR models range from 60 to 210 K. In four cases where H$_2$ absorption lines were observed, the kinetic temperature is consistent (J0000+0048, J1429+1118) or differs by a factor of two (J1237+0647 and J2331−0908) from the H$_2$ excitation temperature (see Fig. 8). This variation is primarily

<footnote n="10">¹⁰Comparison of $T_{\text{ex}}(^{12}\text{CO})$ and $T_{\text{ex}}(^{13}\text{CO})$ yields different results: for HD 62542 $T_{\text{01}}(^{12}\text{CO}) > T_{\text{01}}(^{13}\text{CO})$ (Welty et al. 2020), while for HD 27778 and HD 206267 $T_{\text{01}}(^{12}\text{CO}) \simeq T_{\text{01}}(^{13}\text{CO})$ (Sonnentrucker et al. 2007).</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 9.** The kinetic temperature (upper panel) and thermal pressure (lower panel) are plotted as functions of the number density in samples of CO absorption systems in the local ISM (green symbols) and DLAs (red symbols). Additionally, we include measurements for the sample of C I absorptions in the local ISM from Jenkins & Tripp (2011) (orange symbols). The symbol encoding is consistent with Fig. 1. The green curve and light green shaded area represent the equilibrium solution of the thermal balance in the cold diffuse ISM, calculated by Bialy & Sternberg (2019), corresponding to a range of $I_{UV} = 0.1–10$ units of the Draine field and a CR ionization rate of $\zeta = 10^{-15}–10^{-17}$ s$^{-1}$. The black dashed lines in the upper and lower panels represent the best fit by a power law to the $T_{kin}–n_{gas}^{(CO)}$ and $P_{th}–n_{gas}^{(CO)}$ relations for the local ISM sample, respectively.</figcaption></figure>

due to the combined fitting of H$_2$ and CO levels, which is influenced by the fit to CO excitation.

## 4.4 Thermal pressure and thermal equilibrium

Using our estimates of $T_{kin}$ and $n_{gas}^{(CO)}$, we can derive the thermal pressure in CO-bearing regions as $P_{th} = n_{gas}T_{kin}$. Fig. 9 presents our measurements of $T_{kin}$ and $P_{th}$ against $n_{gas}$ (CO) from a joint analysis of H$_2$ and CO excitation. For CO absorbers in the local ISM, the thermal pressure ranges from $10^3$ to $10^5$ K cm$^{-3}$. In DLAs, the pressure is slightly higher than in the local ISM and ranges from $10^{3.8}$ and $10^{5.5}$ K cm$^{-3}$.

We report a correlation between $T_{kin}$ and $n_{gas}^{(CO)}$ in the sample of CO absorption systems in local ISM, which roughly follows the power-law relation:

$$ \log T_{\text{kin}}/\text{K} = (2.33 \pm 0.04) – (0.27 \pm 0.02) \times \log n_{\text{gas}}/\text{cm}^{-3}. \quad (3) $$

The power-law index is consistent with the theoretical value of 0.3 for the cold ISM in the standard thermal equilibrium model (see, e.g. Wolfe, Prochaska & Gawiser (2003); Bialy & Sternberg (2019)). In this model, a medium with solar metallicity is cooled primarily by C II emission and heated by cosmic-ray and photoreactions. We also compared our results with the model presented by Bialy & Sternberg 2019 for the range of the physical conditions: the intensity of UV field $I_{UV} = 0.1–10$ and cosmic ray ionization rate of $\zeta \sim 10^{-17}–10^{-15}$ s$^{-1}$. Since Bialy & Sternberg (2019) determined the gas density as the total hydrogen density $n_H$ and the pressure as $n_H T_{kin}$ for comparison with our results we assumed the gas is molecular and $n_H = 2n_{gas}$. We found a good consistency of the model and measurements. However, it is worth noting that Bialy & Sternberg (2019) studied the thermal properties of the global atomic ISM, specifically focusing on the warm neutral medium (WNM) and the outskirts of cold neutral clouds. They did not account for the attenuation of FUV radiation within a cloud due to dust absorption and H$_2$ self-shielding, instead considering the free-space photodissociation rate. Interestingly, their results align closely with measurements taken in regions where the majority of the gas is molecular and shielded from UV radiation.

Additionally, we compare measurements of physical conditions in two samples of CO and C I absorptions in the local ISM (see Tables 1 and A1, respectively). C I absorptions generally show approximately 0.3 dex higher kinetic temperatures and thermal pressures than CO absorptions. This suggests that CO absorptions generally probe denser and colder gas compared to C I absorptions, and this may be a selection effect, as C I absorption on average corresponds to lower $N(H_2)$ compared to CO absorptions, thereby likely represents more diffuse gas.

## 4.5 UV field

The UV field regulates the thermal state of the ISM and the H$_2$ destruction rate, influencing the abundance of H$_2$ and the populations of its rotational levels. However, measurements of H$_2$ typically better constrain the combination of $I_{UV}/n_H$ rather than individual parameters. As demonstrated in Klimenko & Balashev (2020), taking into account the excitation of C I fine-structure levels allows one to remove degeneracy between $I_{UV}$ and $n_H$, and a joint fit to H$_2$ and C I levels allows one to obtain stringent constraint on UV intensity. Similarly, the excitation of CO levels complements the constraints from H$_2$. However, CO probes more shielded (and hence centrally located) gas compared to H$_2$ and C I. To avoid potential systematic biases, we propose using a joint fit to H$_2$ and C I to report constraints on the UV field, except in cases where C I measurements are unavailable, for which we report the strength of the UV field from a joint fit to H$_2$ and CO.

We found UV intensities in the range of 0.1 to 10 units of Mathis fields with an average value of about 1 unit of Mathis field (see Fig. 10). This range corresponds to expected intensities in theoretical models of the interstellar UV radiation field (Habing 1968; Mathis, Rumpl & Nordsieck 1977; Draine 1978) and previous measurements in diffuse gas in the Milky Way (Jenkins & Tripp 2011) and at high redshift by Balashev et al. (2019), Klimenko & Balashev (2020). Combining the measurements in local ISM in our sample with the measurements in the sample of C I absorbers from Jenkins & Tripp
<figure><img src="image_11.png" /><figcaption>**Figure 10.** The measurements of UV field intensity versus total hydrogen number density ($n_H$) from fits to data with PDR models. Green and orange points represent estimates for samples of local ISM sightlines containing CO and C I absorptions, respectively. Red diamonds represent measurements in high-$z$ CO-bearing DLAs. The purple colour gradient encodes values of the parameter $\alpha G$ (see text). Solid, dashed, dot-dashed, and dotted lines represent levels of $\log \alpha G = 1, 0, -1, -2$ (from left to right), calculated at the parameters given in text.</figcaption></figure>

(2011) (see Appendix A), we found that $I_{UV}$ does not correlate with the density, suggesting that systems in the local ISM are generally illuminated by a uniform interstellar radiation background. Among these samples, we also found several systems with a low UV intensity of about ~0.1 Mathis units.

Measurements in DLAs agree with the results from the local ISM. One system from our sample of CO-bearing DLAs, J2331–0908, exhibits multicomponent velocity structure that is not resolved in saturated H$_2$ absorption lines. Therefore, we present the mean H$_2$ excitation, which is likely higher than that of the CO-bearing H$_2$ component. This results in a higher UV field intensity for this system, $I_{UV} > 5$ Mathis units, compared to the mean value in the local ISM and other DLAs.

## 4.6 The $I_{UV}/n_H$ ratio and $\alpha G$ parameter

Comparing $I_{UV}$ versus $n_H$ for the samples of CO and C I absorbers (see Fig. 10), one can see that they occupy different regions in the $I_{UV}-n_H$ space. While some CO absorbers probe similar regions as C I absorbers, those with higher CO column densities are found in denser gas. The difference in the $I_{UV}/n_H$ ratio can be up to a factor of 10. The correlations between number density and other physical conditions with CO and H$_2$ column densities are discussed later in Section 5.

The ratio $I_{UV}/n_H$ can be used to estimate the dimensionless parameter $\alpha G$ in the theory of H I-to-H$_2$ photodissociation transition in diffuse gas by Sternberg et al. (2014), Bialy & Sternberg (2016). The parameter $\alpha$ denotes the ratio of H$_2$ destruction (neglecting self-shielding) to its formation rate, while $G$ defines the shielding of H$_2$ from UV radiation. To estimate $\alpha G$, we used equation (22) from Bialy & Sternberg (2016), assuming a dust-to-gas ratio of $\sigma_g \simeq 1.9 \times 10^{-21} \text{Z cm}^{-2}$ and an H$_2$ formation rate coefficient $R = 3 \times 10^{-17} \text{Z}$, where Z denotes the gas metallicity relative to solar. For the local ISM metallicity, we adopted a value reported

by Ritchey et al. (2023) ([X/H]=−0.02 ± 0.10), but see also De Cia et al. (2021). For DLAs, the metallicity is supersolar: [Zn/H] = +0.16 for J1439+1118 (Srianand et al. 2008), [Zn/H] = +0.34 for J1237+0647 (Noterdaeme et al. 2010), and [Zn/H] = +0.46 for J0000+0048 (Noterdaeme et al. 2017).

The values of $\alpha G$ parameter are shown using a colour gradient in Fig. 10, and range between 0.01 and 10, with a median near 0.5 for most sightlines in our samples. Systems that are denser and richer in CO tend to exhibit lower $\alpha G$ values. For the sight line towards star HD29647, we find an extremely low $\alpha G < 3 \times 10^{-2}$. This particular sightline shows a high $A_V = 3.8$, very low kinetic temperature <20 K, and a dense gas environment of ~1000 cm$^{-3}$, suggesting it likely contains dense molecular gas. We compare our $\alpha G$ values with those reported in the literature (Bialy et al. 2017; Ranjan et al. 2018). Bialy et al. (2017) found $\alpha G = 17.7_{-11.2}^{+30.7}$ in the star-forming region W43 in the Milky Way, which is approximately 10 times higher than our estimates. Bialy et al. (2017) derived $I_{UV} = 3–8$ Mathis units from dust emission and assumed average density $n_H \sim 10–100 \text{ cm}^{-3}$. Star-forming regions typically exhibit higher UV intensities compared to the diffuse ISM probed by our sightlines. Ranjan et al. (2018) studied a DLA system at $z = 2.46$ and measured $\alpha G = 0.1–10$, which is consistent with our findings.

## 5 DISCUSSION

The mutual dependence of physical conditions and excitation temperatures, as presented in the previous section, sheds light on the local physics of the medium. Here, we discuss how the measured physical conditions correlate with the observed column densities.

As it is shown in Fig. 1, CO abundance is characterized by a transition from a slow increase<fnref n="11" /> with $N(H_2)$ in diffuse gas to a steep increase in translucent gas. We may expect to observe a change in the physical state of the gas near this transition. Fig. 11 shows the kinetic temperature, number density, and the ratio of $I_{UV}/n_H$ as a function of H$_2$ and CO column densities. We present our measurements described in the previous section, derived from joint fits to CO, H$_2$, and C I excitation, along with the data from literature compiled from Gillmon et al. (2006), Jenkins & Tripp (2011), Shull, Danforth & Anderson (2021) for the local ISM, and from Srianand et al. (2005), Ledoux, Petitjean & Srianand (2006), Albornoz Vásquez et al. (2014), Klimenko et al. (2016), Balashev et al. (2019), Klimenko & Balashev (2020) for high-$z$ DLAs. As we stated above, we re-analysed data from Jenkins & Tripp (2011) to ensure consistency with our sample using the same methodology (see Appendix A).

Overall, we found that physical conditions derived from CO excitation are consistent with those derived from C I excitation in the overlapping region of H$_2$ and CO column densities. However, they exhibit a steeper dependence at higher column densities (translucent gas). We present a fit to these dependencies by power-law in Table 3 and discuss it in details in the following subsections.

### 5.1 Kinetic temperature

The dependence of kinetic temperature on $N(\text{CO})$ and $N(\text{H}_2)$ follows a power-law relation, exhibiting similar slopes ($k_X = -0.04 \pm 0.02$) for the sample of C I absorption systems in the local ISM. This trend has been previously reported in both the local ISM and at high redshift (Muzahid, Srianand & Charlton 2015; Balashev et al. 2017).

<footnote n="11">¹¹This increase is likely just due to an increase of the cloud size or amount of material along the sightline.</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 11.** The kinetic temperature, number density, and the ratio of $I_{UV}/n_H$ as functions of CO (left-hand panels) and $H_2$ (right-hand panels) column densities are shown in the top, middle, and bottom panels, respectively. Green points represent measurements from $H_2$/CO absorption systems in the local ISM, while red diamonds denote those from high-redshift DLAs. Similarly, measurements from $H_2$/C1 absorption systems are shown in orange for the local ISM and in pink for high-redshift DLAs. The shape of the symbols is consistent with Fig. 1. Each panel includes green and orange curves with shaded areas representing the best fit and its uncertainty for CO and C1 absorption systems in the local ISM, respectively. Vertical dashed lines indicate the position of the transition between diffuse and translucent regimes, where the CO abundance significantly increases. Additionally, the top panels display measurements of $H_2$ excitation temperature (yellow circles – local ISM and pink diamonds – DLAs), which is considered as a proxy for the kinetic temperature in diffuse gas.</figcaption></figure>

In contrast, the sample of CO absorption systems in the local ISM shows a much stronger dependence, with slopes of $-0.21 \pm 0.02$ and $-0.65 \pm 0.05$ on $N(\text{CO})$ and $N(\text{H}_2)$, respectively. This stronger dependence is likely due to the gas temperature decreasing in denser and more opaque layers of molecular clouds. In diffuse gas, temperatures are typically around 100 K, while in translucent gas, temperatures can drop by up to 30 K. In specific clouds like HD28075 and HD29647, which likely probe dense CO-bright gas, kinetic temperatures drop to values around 10–

20 K, which is still consistent with the overall trend extrapolation.

When considering C1 and CO samples together, the kinetic temperature shows consistency in the overlapping region, and the position of the break in the dependence of $T_{\text{kin}}$ on $N(\text{H}_2)$ and $N(\text{CO})$ aligns with the start of a steep increase of CO abundance.

The sample of CO absorbers in DLAs at high redshift occupies a parameter space similar to that of diffuse gas (C1) in the local ISM. The measured kinetic temperatures are slightly higher but consistent
<table><caption>**Table 3.** Best-fit results for the dependence of kinetic temperature ($T_{kin}$), gas number density ($n_{gas}$), and the ratio of UV intensity to total hydrogen number density ($I_{UV}/n_{H}$) on CO and $H_2$ column densities across samples of CO and C I absorption systems in the local ISM and DLAs at high redshift are presented using the following relation: $\log y = k_X(\log N_X - a_X) + b_X$, where y represents the physical parameter and X denotes the species column density.</caption><thead><tr><th>y</th><th>X</th><th>a<sub>X</sub></th><th>b<sub>X</sub></th><th>k<sub>X</sub></th><th>b<sub>X</sub></th><th>k<sub>X</sub></th></tr></thead><tbody><tr><td colspan="7" style="text-align: center;">local ISM: CO+H<sub>2</sub></td></tr><tr><td rowspan="2">T<sub>kin</sub></td><td>N(CO)</td><td>15</td><td>1.81<sup>+0.02</sup><sub>-0.02</sub></td><td>-0.21<sup>+0.02</sup><sub>-0.02</sub></td><td>1.94<sup>+0.03</sup><sub>-0.03</sub></td><td>-0.04<sup>+0.02</sup><sub>-0.02</sub></td></tr><tr><td>N(H<sub>2</sub>)</td><td>20.7</td><td>1.75<sup>+0.02</sup><sub>-0.02</sub></td><td>-0.65<sup>+0.05</sup><sub>-0.05</sub></td><td>1.92<sup>+0.04</sup><sub>-0.04</sub></td><td>-0.04<sup>+0.03</sup><sub>-0.01</sub></td></tr><tr><td rowspan="2">n<sub>gas</sub></td><td>N(CO)</td><td>15</td><td>2.03<sup>+0.02</sup><sub>-0.02</sub></td><td>0.38<sup>+0.02</sup><sub>-0.02</sub></td><td>1.83<sup>+0.02</sup><sub>-0.02</sub></td><td>0.01<sup>+0.03</sup><sub>-0.03</sub></td></tr><tr><td>N(H<sub>2</sub>)</td><td>20.7</td><td>2.13<sup>+0.02</sup><sub>-0.02</sub></td><td>1.15<sup>+0.05</sup><sub>-0.05</sub></td><td>1.82<sup>+0.04</sup><sub>-0.04</sub></td><td>-0.03<sup>+0.03</sup><sub>-0.03</sub></td></tr><tr><td rowspan="2">I<sub>UV</sub>/n<sub>H</sub></td><td>N(CO)</td><td>15</td><td>-2.38<sup>+0.02</sup><sub>-0.02</sub></td><td>-0.40<sup>+0.02</sup><sub>-0.02</sub></td><td>-2.37<sup>+0.04</sup><sub>-0.04</sub></td><td>0.03<sup>+0.03</sup><sub>-0.03</sub></td></tr><tr><td>N(H<sub>2</sub>)</td><td>20.7</td><td>-2.51<sup>+0.02</sup><sub>-0.02</sub></td><td>-1.29<sup>+0.04</sup><sub>-0.06</sub></td><td>-2.16<sup>+0.03</sup><sub>-0.03</sub></td><td>0.28<sup>+0.03</sup><sub>-0.03</sub></td></tr><tr><td colspan="7" style="text-align: center;">DLAs: CI+H<sub>2</sub></td></tr><tr><td>T<sub>01</sub>(H<sub>2</sub>)</td><td>N(H<sub>2</sub>)</td><td>20.7</td><td>—</td><td>—</td><td>1.95<sup>+0.04</sup><sub>-0.04</sub></td><td>-0.05<sup>+0.02</sup><sub>-0.02</sub></td></tr><tr><td>I<sub>UV</sub>/n<sub>H</sub></td><td>N(H<sub>2</sub>)</td><td>20.7</td><td>—</td><td>—</td><td>-3.00<sup>+0.04</sup><sub>-0.04</sub></td><td>-0.58<sup>+0.03</sup><sub>-0.03</sub></td></tr></tbody></table>

with those observed in the local ISM suggesting that CO absorbers at high redshift are likely related to diffuse phase of the ISM. Comparing measurements of $T_{ex}(H_2)$ in the local ISM and DLAs with those of kinetic temperature in the sample of C I absorbers, we found consistency with the trend seen in C I absorbers.

## 5.2 Number density

The dependencies of the gas number density on $N(CO)$ and $N(H_2)$ follow a similar behaviour to that observed for the kinetic temperatures (see middle panels of Fig. 11). In diffuse gas, the number densities are typically around $100 \text{ cm}^{-3}$, while in the translucent regime, they increase to approximately $1000 \text{ cm}^{-3}$.

Measurements in DLAs are consistent with those in diffuse gas in the local ISM; however, in two systems, we observe significantly higher densities typical of dense gas in local ISM, likely due to higher CO excitation. These systems exhibit complex CO excitation diagrams (see Appendix C) that cannot be adequately fitted with a simple model and require further high-resolution observations.

## 5.3 The ratio of $I_{UV}/n_{H}$

The ratio $I_{UV}/n_{H}$ exhibits a positive correlation with $N(CO)$ and $N(H_2)$ for diffuse C I clouds in the local ISM (see bottom panels of Fig. 11). In contrast, measurements in CO absorbers show the opposite dependence, with $I_{UV}/n_{H}$ decreasing as $N(CO)$ and $N(H_2)$ increase. This trend is also evident in Fig. 10. The difference between the samples of C I and CO absorption systems is primarily due to an increase in number density, while the intensity of the external UV field remains similar.

Measurements in CO-bearing DLAs show similar $I_{UV}/n_{H}$ values to those found in translucent gas in the local ISM, despite having lower $N(CO)$ and $N(H_2)$ values comparable to the diffuse ISM. When comparing these findings with data from the literature, we observed that measurements of $I_{UV}/n_{H}$ in C I/H$_2$ absorbers at high redshift exhibit an opposite trend to those in the local ISM: they decrease with increasing $N(H_2)$. Data from CO-bearing DLAs support this observed trend. This could be due to lower metallicity in C I-bearing DLAs and/or a higher intensity of the UV field (see also Balashev et al. 2019; Klimenko & Balashev 2020; Kosenko et al. 2024). Both factors induce a selection bias against CO column densities and, consequently, possibility of CO detection. Systems with CO

detection consistently show $I_{UV}/n_{H}$ values similar to those found in the local ISM.

## 6 CONCLUSIONS

We studied the physical conditions (number density, kinetic temperature, thermal pressure, and intensity of UV field) in the local ISM and at high redshift by fitting observed column densities of CO and $H_2$ rotational levels and C I fine-structure levels with photodissociation models calculated using the MEUDON PDR code.

We selected samples of known CO absorption systems in the local ISM and at high redshift, focusing on those with measured excitation of CO molecules at least at the first three rotational levels. This yielded 28 sightlines in the local ISM and seven DLA systems at redshifts $z \sim 1.7-2.7$ in quasar spectra. The sample of sightlines in the local ISM spans a range of $H_2$ column densities from $N(H_2) = 10^{19}$ to $10^{21.5} \text{ cm}^{-2}$ and CO column densities from $10^{13}$ to $10^{18} \text{ cm}^{-2}$, covering the transition between diffuse and dense gas, where CO abundance increases from $10^{-6}$ to $10^{-4}$ (carbon abundance), respectively.

Using the MEUDON PDR code, we constructed a method to accurately calculate CO excitation, incorporating the effects of photon trapping. Our findings indicate that in the local ISM, CO excitation is primarily driven by collisions and excitation due to the CMB radiation. Photon trapping becomes significant at high column densities around $N(CO) \sim 10^{16} \text{ cm}^{-2}$. We demonstrated that an increase in CO excitation, observed near $N(CO) \approx 10^{15} \text{ cm}^{-2}$, is attributed to an increase in gas densities from $\approx 100 \text{ cm}^{-3}$ in the diffuse phase to $\approx 300-1000 \text{ cm}^{-3}$ in the translucent and dense phases. The gas density derived from CO excitation aligns well with independent estimates from C I excitation, where photon trapping effects are less significant. Therefore, an analysis of CO excitation in absorption systems can be used to derive physical conditions in the ISM.

We applied the method for the samples of CO absorption systems in the local ISM and at high redshift and found following results:

(i) Molecular clouds in the local ISM are characterized by a gas number density of about $10-1000 \text{ cm}^{-3}$, a kinetic temperature of $10-100 \text{ K}$, and an intensity of external UV field ranging from 0.1 to 10 units of Mathis field. Compared to the average gas probed by C I absorption (as reported by Jenkins & Tripp 2011), the gas detected in CO is denser and colder, while the external UV field remains nearly
constant. Our findings are consistent with the equilibrium solution of the thermal balance in the cold diffuse ISM, calculated by Biyali & Sternberg (2019).

(ii) The properties of gas in high-redshift CO absorption systems are comparable to those in the local ISM, but differ by a factor of 1–2. We observed kinetic temperatures ranging from 60 to 210 K and number densities from 100 to 3000 cm⁻³, which are slightly higher than those in local systems with comparable CO and H₂ column densities.

(iii) The physical parameters of CO and C I absorption systems in the local ISM depend differently on CO and H₂ column densities. While the kinetic temperature and gas density in C I absorbers exhibit a weak dependence on N(CO) and N(H₂), CO absorption systems show strong correlations. We observed a negative correlation between the kinetic temperature and both N(CO) and N(H₂), with power-law slopes of −0.21 ± 0.02 and −0.65 ± 0.05, respectively. At the same time, the gas number density exhibits a positive correlation with both N(CO) and N(H₂), with slopes of 0.38 ± 0.02 and 1.15 ± 0.05, respectively.

(iv) Comparing the dependence of physical conditions on N(CO) and N(H₂) in local ISM with the evolution of CO abundance (or the ratio of CO/H₂), we found that the change in the dependence of physical conditions occurs at the transition between diffuse and translucent gas near N(CO) = 10¹⁵ cm⁻² and N(H₂) = 10²¹.⁷ cm⁻².

## ACKNOWLEDGEMENTS

This paper was supported by RSF grant 23-12-00166. We acknowledge support from the French Agence Nationale de la Recherche under ANR grant 17-CE31-0011-01/project ‘HIH2’ (PI: Noterdaeme) and from the French-Russian collaborative programme PRC 1845.

We would like to thank Franck Le Petit and Emeric Bron for valuable advice regarding the calculation of species excitation in the MEUDON PDR code. We are grateful to Cecilia Ceccarelli for providing a careful reading and constructive suggestions that have helped to improve this paper.

## DATA AVAILABILITY

The paper is based on literature data on the excitation of CO and H₂ rotational levels, and C I fine-structure levels in absorption systems in the local ISM and DLA systems at high redshift. These data can be shared on reasonable requests to the authors.

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## APPENDIX A: ADDITIONAL DATA

We analysed C I excitation in the sample of C I absorption systems in the local ISM from Jenkins & Tripp (2011). The authors derived the thermal pressure and the intensity of the interstellar UV field through a joint analysis of C I excitation and the ionization balance of carbon in the ISM, using the observed C II to C I ratio. Here, we compare their results with those obtained from our approach, where we simultaneously fit the excitation of H$_2$ and C I levels using a grid of the MEUDON PDR models.

The list of the systems and results are presented in Table A1. We collected data from table 4 in Jenkins & Tripp (2011), calculating the sum of excitation of C I* and C I** levels weighted by $N(\text{CI}_{\text{tot}})$ across velocity channels (e.g. $N(\text{CI}^*) = \sum v_i (f_{i,1} * N(\text{CI}_{\text{tot},i}))$). We note that in several cases where Jenkins & Tripp (2011) provides a lower limit on $N(\text{CI}_{\text{tot}})$ in their table 3 (because of saturation), the sum over velocity channels for resolved components results in a lower value. In such cases, calculations of C I excitation may need to take into account the radiative transfer effect on C I levels, since C I lines are already optically thick. As previously discussed, our grid of PDR models does not allow us to correctly fit the column densities of C I levels consistently with the fit to H$_2$ excitation.<fnref n="12" /> Therefore, we fit relative populations of C I levels, integrated in the PDR models within the layer of H$_2$.

Fig. A1 compares the results from two approaches: ours and those from Jenkins & Tripp (2011). There is good consistency in the estimates of number density. The constraints on number density mainly come from the analysis of C I excitation, which are similar in both methods. The UV intensity is higher in Jenkins & Tripp (2011) compared to our estimates. The difference is smaller for systems with higher $N(\text{CI})$, which may be explained if the ratio of column densities of C II and C I overestimates the local C II/C I ratio in Jenkins & Tripp (2011) method. This discrepancy could arise because only a portion of C II (observed in both cold and warm neutral ISM phases) is related to C I in the cold phase. We find a stronger correlation between thermal pressure and UV intensity in our data. This correlation primarily arises from the relationship between $n_{\text{gas}}$ and $I_{\text{UV}}$ in our data set, where it comes from the fit to H$_2$ excitation that constrains the region to a constant temperature.

<footnote n="12">¹²We found that PDR models typically overpredict the C I/H$_2$ ratio compared to the observed one, which may be related to variations in the cosmic-ray ionization rate or the fraction of PAHs. Including these parameters would significantly increase the number of grid nodes.</footnote>
<table><caption>**Table A1.** The measurements of physical conditions in the sample of C1 and H₂ absorption systems from Jenkins & Tripp (2011).</caption><thead><tr><th>Name</th><th>log N(H<sub>2</sub>) [cm<sup>-2</sup>]</th><th>log N(C1) [cm<sup>-2</sup>]</th><th>T<sub>ex</sub>(H<sub>2</sub>) (K)</th><th>log n<sub>H</sub><sup>(CI)</sup> [cm<sup>-3</sup>]</th><th>log I<sub>UV</sub><sup>(CI)</sup> [Mathis]</th><th>T<sub>gas</sub><sup>(CI)</sup> (K)</th><th>log n<sub>gas</sub><sup>(JT)</sup> [cm<sup>-3</sup>]</th><th>log n<sub>gas</sub><sup>(JT)</sup> [cm<sup>-3</sup>]</th><th>log I<sub>UV</sub><sup>(JT)</sup> [Mathis]</th></tr></thead><tbody><tr><td>CPD-59°2603</td><td>20.15<sup>+0.10</sup><sub>-0.10</sub></td><td>14.82<sup>+0.02</sup><sub>-0.02</sub></td><td>77<sup>+5</sup><sub>-5</sub></td><td>2.06<sup>+0.09</sup><sub>-0.08</sub></td><td>-0.20<sup>+0.24</sup><sub>-0.30</sub></td><td>91<sup>+53</sup><sub>-18</sub></td><td>1.86<sup>+0.12</sup><sub>-0.12</sub></td><td>1.74</td><td>0.31</td></tr><tr><td>HD 15137</td><td>20.32<sup>+0.10</sup><sub>-0.10</sub></td><td>14.65<sup>+0.05</sup><sub>-0.05</sub></td><td>104<sup>+28</sup><sub>-18</sub></td><td>1.82<sup>+0.11</sup><sub>-0.13</sub></td><td>-0.07<sup>+0.50</sup><sub>-0.20</sub></td><td>186<sup>+42</sup><sub>-99</sub></td><td>1.59<sup>+0.21</sup><sub>-0.13</sub></td><td>1.59</td><td>0.32</td></tr><tr><td>HD 23478</td><td>20.57<sup>+0.12</sup><sub>-0.10</sub></td><td>14.79<sup>+0.05</sup><sub>-0.05</sub></td><td>53<sup>+12</sup><sub>-8</sub></td><td>2.06<sup>+0.12</sup><sub>-0.08</sub></td><td>-0.35<sup>+0.34</sup><sub>-0.30</sub></td><td>79<sup>+55</sup><sub>-16</sub></td><td>1.85<sup>+0.15</sup><sub>-0.11</sub></td><td>1.89</td><td>0.46</td></tr><tr><td>HD 24190</td><td>20.38<sup>+0.10</sup><sub>-0.10</sub></td><td>14.68<sup>+0.05</sup><sub>-0.05</sub></td><td>66<sup>+20</sup><sub>-12</sub></td><td>2.03<sup>+0.09</sup><sub>-0.12</sub></td><td>-0.31<sup>+0.33</sup><sub>-0.26</sub></td><td>87<sup>+64</sup><sub>-12</sub></td><td>1.78<sup>+0.19</sup><sub>-0.11</sub></td><td>1.82</td><td>0.62</td></tr><tr><td>HD 24534</td><td>20.92<sup>+0.03</sup><sub>-0.03</sub></td><td>14.81<sup>+0.03</sup><sub>-0.03</sub></td><td>57<sup>+3</sup><sub>-3</sub></td><td>2.58<sup>+0.11</sup><sub>-0.08</sub></td><td>0.05<sup>+0.15</sup><sub>-0.12</sub></td><td>75<sup>+13</sup><sub>-14</sub></td><td>2.35<sup>+0.13</sup><sub>-0.14</sub></td><td>2.41</td><td>1.16</td></tr><tr><td>HD 37903</td><td>20.92<sup>+0.10</sup><sub>-0.10</sub></td><td>14.22<sup>+0.05</sup><sub>-0.05</sub></td><td>68<sup>+7</sup><sub>-7</sub></td><td>2.79<sup>+0.07</sup><sub>-0.07</sub></td><td>&gt;0.85</td><td>147<sup>+14</sup><sub>-16</sub></td><td>2.55<sup>+0.14</sup><sub>-0.17</sub></td><td>2.76</td><td>1.37</td></tr><tr><td>HD 40893</td><td>20.58<sup>+0.10</sup><sub>-0.10</sub></td><td>14.65<sup>+0.05</sup><sub>-0.05</sub></td><td>78<sup>+10</sup><sub>-10</sub></td><td>1.85<sup>+0.20</sup><sub>-0.20</sub></td><td>-0.58<sup>+0.06</sup><sub>-0.06</sub></td><td>81<sup>+25</sup><sub>-12</sub></td><td>1.63<sup>+0.10</sup><sub>-0.11</sub></td><td>1.50</td><td>0.37</td></tr><tr><td>HD 69106</td><td>19.73<sup>+0.10</sup><sub>-0.10</sub></td><td>14.27<sup>+0.05</sup><sub>-0.05</sub></td><td>80<sup>+16</sup><sub>-16</sub></td><td>1.91<sup>+0.09</sup><sub>-0.11</sub></td><td>-0.43<sup>+0.24</sup><sub>-0.25</sub></td><td>93<sup>+47</sup><sub>-15</sub></td><td>1.71<sup>+0.13</sup><sub>-0.11</sub></td><td>1.64</td><td>0.45</td></tr><tr><td>HD 75309</td><td>20.20<sup>+0.10</sup><sub>-0.10</sub></td><td>14.50<sup>+0.03</sup><sub>-0.03</sub></td><td>65<sup>+5</sup><sub>-5</sub></td><td>1.95<sup>+0.08</sup><sub>-0.11</sub></td><td>&lt;0.56</td><td>83<sup>+21</sup><sub>-17</sub></td><td>1.70<sup>+0.15</sup><sub>-0.12</sub></td><td>1.59</td><td>0.45</td></tr><tr><td>HD 88115</td><td>19.30<sup>+0.10</sup><sub>-0.10</sub></td><td>13.90<sup>+0.05</sup><sub>-0.05</sub></td><td>145<sup>+13</sup><sub>-13</sub></td><td>1.65<sup>+0.14</sup><sub>-0.36</sub></td><td>0.39<sup>+0.40</sup><sub>-0.70</sub></td><td>354<sup>+123</sup><sub>-265</sub></td><td>1.43<sup>+0.20</sup><sub>-0.45</sub></td><td>1.38</td><td>0.51</td></tr><tr><td>HD 91824</td><td>19.83<sup>+0.10</sup><sub>-0.10</sub></td><td>14.47<sup>+0.05</sup><sub>-0.05</sub></td><td>61<sup>+7</sup><sub>-7</sub></td><td>2.09<sup>+0.10</sup><sub>-0.12</sub></td><td>-0.35<sup>+0.38</sup><sub>-0.33</sub></td><td>85<sup>+84</sup><sub>-14</sub></td><td>1.87<sup>+0.17</sup><sub>-0.12</sub></td><td>1.81</td><td>0.78</td></tr><tr><td>HD 91983</td><td>20.23<sup>+0.10</sup><sub>-0.10</sub></td><td>14.79<sup>+0.03</sup><sub>-0.03</sub></td><td>61<sup>+5</sup><sub>-5</sub></td><td>2.06<sup>+0.10</sup><sub>-0.10</sub></td><td>-0.54<sup>+0.17</sup><sub>-0.46</sub></td><td>79<sup>+27</sup><sub>-16</sub></td><td>1.86<sup>+0.12</sup><sub>-0.09</sub></td><td>1.74</td><td>0.51</td></tr><tr><td>HD 93205</td><td>19.86<sup>+0.10</sup><sub>-0.10</sub></td><td>14.54<sup>+0.02</sup><sub>-0.02</sub></td><td>105<sup>+21</sup><sub>-21</sub></td><td>2.09<sup>+0.10</sup><sub>-0.12</sub></td><td>0.11<sup>+0.43</sup><sub>-0.29</sub></td><td>158<sup>+75</sup><sub>-60</sub></td><td>1.88<sup>+0.17</sup><sub>-0.12</sub></td><td>1.57</td><td>0.58</td></tr><tr><td>HD 93222</td><td>19.81<sup>+0.10</sup><sub>-0.10</sub></td><td>14.36<sup>+0.05</sup><sub>-0.05</sub></td><td>77<sup>+11</sup><sub>-11</sub></td><td>2.33<sup>+0.11</sup><sub>-0.10</sub></td><td>&lt;0.90</td><td>63<sup>+7</sup><sub>-6</sub></td><td>2.12<sup>+0.17</sup><sub>-0.15</sub></td><td>2.51</td><td>0.82</td></tr><tr><td>HD 93843</td><td>19.61<sup>+0.10</sup><sub>-0.10</sub></td><td>14.13<sup>+0.05</sup><sub>-0.05</sub></td><td>107<sup>+21</sup><sub>-21</sub></td><td>2.48<sup>+0.11</sup><sub>-0.08</sub></td><td>&lt;0.74</td><td>61<sup>+7</sup><sub>-7</sub></td><td>2.28<sup>+0.14</sup><sub>-0.12</sub></td><td>2.07</td><td>0.66</td></tr><tr><td>HD 94454</td><td>20.76<sup>+0.10</sup><sub>-0.10</sub></td><td>14.10<sup>+0.05</sup><sub>-0.05</sub></td><td>74<sup>+5</sup><sub>-5</sub></td><td>2.15<sup>+0.10</sup><sub>-0.10</sub></td><td>0.01<sup>+0.32</sup><sub>-0.32</sub></td><td>100<sup>+47</sup><sub>-25</sub></td><td>1.94<sup>+0.16</sup><sub>-0.15</sub></td><td>1.73</td><td>0.68</td></tr><tr><td>HD 99857</td><td>20.25<sup>+0.10</sup><sub>-0.10</sub></td><td>14.59<sup>+0.05</sup><sub>-0.05</sub></td><td>83<sup>+17</sup><sub>-17</sub></td><td>2.09<sup>+0.08</sup><sub>-0.12</sub></td><td>-0.21<sup>+0.24</sup><sub>-0.24</sub></td><td>91<sup>+53</sup><sub>-9</sub></td><td>1.85<sup>+0.15</sup><sub>-0.11</sub></td><td>1.69</td><td>0.55</td></tr><tr><td>HD 99872</td><td>20.52<sup>+0.10</sup><sub>-0.10</sub></td><td>14.54<sup>+0.05</sup><sub>-0.05</sub></td><td>66<sup>+10</sup><sub>-10</sub></td><td>2.03<sup>+0.10</sup><sub>-0.10</sub></td><td>-0.11<sup>+0.37</sup><sub>-0.26</sub></td><td>104<sup>+73</sup><sub>-25</sub></td><td>1.80<sup>+0.18</sup><sub>-0.09</sub></td><td>1.78</td><td>0.02</td></tr><tr><td>HD 102065</td><td>20.53<sup>+0.10</sup><sub>-0.10</sub></td><td>14.36<sup>+0.05</sup><sub>-0.05</sub></td><td>59<sup>+7</sup><sub>-7</sub></td><td>2.09<sup>+0.10</sup><sub>-0.15</sub></td><td>-0.62<sup>+0.25</sup><sub>-0.20</sub></td><td>67<sup>+39</sup><sub>-10</sub></td><td>1.88<sup>+0.12</sup><sub>-0.20</sub></td><td>1.82</td><td>0.61</td></tr><tr><td>HD103779</td><td>19.83<sup>+0.10</sup><sub>-0.10</sub></td><td>14.21<sup>+0.05</sup><sub>-0.05</sub></td><td>117<sup>+14</sup><sub>-14</sub></td><td>1.76<sup>+0.09</sup><sub>-0.12</sub></td><td>&lt;0.88</td><td>83<sup>+4</sup><sub>-10</sub></td><td>1.54<sup>+0.17</sup><sub>-0.16</sub></td><td>1.23</td><td>0.33</td></tr><tr><td>HD104705</td><td>19.98<sup>+0.10</sup><sub>-0.10</sub></td><td>14.24<sup>+0.10</sup><sub>-0.10</sub></td><td>92<sup>+16</sup><sub>-16</sub></td><td>1.79<sup>+0.10</sup><sub>-0.12</sub></td><td>-0.41<sup>+0.27</sup><sub>-0.20</sub></td><td>109<sup>+48</sup><sub>-20</sub></td><td>1.57<sup>+0.17</sup><sub>-0.12</sub></td><td>1.53</td><td>1.37</td></tr><tr><td>HD108002</td><td>20.34<sup>+0.10</sup><sub>-0.10</sub></td><td>14.58<sup>+0.10</sup><sub>-0.10</sub></td><td>77<sup>+5</sup><sub>-5</sub></td><td>1.85<sup>+0.11</sup><sub>-0.14</sub></td><td>—</td><td>199<sup>+63</sup><sub>-128</sub></td><td>1.64<sup>+0.15</sup><sub>-0.21</sub></td><td>1.52</td><td>0.34</td></tr><tr><td>HD108639</td><td>20.04<sup>+0.10</sup><sub>-0.10</sub></td><td>14.53<sup>+0.05</sup><sub>-0.05</sub></td><td>88<sup>+5</sup><sub>-5</sub></td><td>1.88<sup>+0.10</sup><sub>-0.12</sub></td><td>-0.17<sup>+0.36</sup><sub>-0.48</sub></td><td>125<sup>+73</sup><sub>-50</sub></td><td>1.86<sup>+0.17</sup><sub>-0.11</sub></td><td>1.54</td><td>0.55</td></tr><tr><td>HD114886</td><td>20.34<sup>+0.10</sup><sub>-0.10</sub></td><td>14.96<sup>+0.05</sup><sub>-0.05</sub></td><td>96<sup>+5</sup><sub>-5</sub></td><td>1.94<sup>+0.12</sup><sub>-0.14</sub></td><td>0.66<sup>+0.30</sup><sub>-0.83</sub></td><td>199<sup>+63</sup><sub>-116</sub></td><td>1.78<sup>+0.14</sup><sub>-0.21</sub></td><td>1.63</td><td>0.37</td></tr><tr><td>HD115071</td><td>20.69<sup>+0.10</sup><sub>-0.10</sub></td><td>14.69<sup>+0.05</sup><sub>-0.05</sub></td><td>84<sup>+12</sup><sub>-12</sub></td><td>2.15<sup>+0.08</sup><sub>-0.09</sub></td><td>-0.27<sup>+0.19</sup><sub>-0.16</sub></td><td>87<sup>+17</sup><sub>-16</sub></td><td>1.93<sup>+0.13</sup><sub>-0.11</sub></td><td>1.71</td><td>0.74</td></tr><tr><td>HD115455</td><td>20.58<sup>+0.10</sup><sub>-0.10</sub></td><td>14.80<sup>+0.05</sup><sub>-0.05</sub></td><td>81<sup>+5</sup><sub>-5</sub></td><td>1.94<sup>+0.11</sup><sub>-0.13</sub></td><td>-0.19<sup>+0.38</sup><sub>-0.36</sub></td><td>95<sup>+66</sup><sub>-23</sub></td><td>1.70<sup>+0.21</sup><sub>-0.12</sub></td><td>1.61</td><td>0.58</td></tr><tr><td>HD116852</td><td>19.85<sup>+0.10</sup><sub>-0.10</sub></td><td>14.15<sup>+0.10</sup><sub>-0.10</sub></td><td>70<sup>+9</sup><sub>-9</sub></td><td>2.27<sup>+0.09</sup><sub>-0.10</sub></td><td>&lt;0.90</td><td>66<sup>+10</sup><sub>-7</sub></td><td>2.05<sup>+0.20</sup><sub>-0.14</sub></td><td>1.85</td><td>0.80</td></tr><tr><td>HD121968</td><td>18.73<sup>+0.10</sup><sub>-0.10</sub></td><td>13.36<sup>+0.05</sup><sub>-0.05</sub></td><td>38<sup>+3</sup><sub>-3</sub></td><td>2.58<sup>+0.07</sup><sub>-0.11</sub></td><td>&lt;0.88</td><td>60<sup>+15</sup><sub>-10</sub></td><td>2.37<sup>+0.12</sup><sub>-0.15</sub></td><td>2.22</td><td>1.27</td></tr><tr><td>HD122879</td><td>20.36<sup>+0.10</sup><sub>-0.10</sub></td><td>14.65<sup>+0.03</sup><sub>-0.03</sub></td><td>90<sup>+5</sup><sub>-5</sub></td><td>1.91<sup>+0.10</sup><sub>-0.12</sub></td><td>-0.29<sup>+0.34</sup><sub>-0.24</sub></td><td>125<sup>+32</sup><sub>-46</sub></td><td>1.70<sup>+0.15</sup><sub>-0.12</sub></td><td>1.63</td><td>0.49</td></tr><tr><td>HD124314</td><td>20.52<sup>+0.10</sup><sub>-0.10</sub></td><td>14.67<sup>+0.10</sup><sub>-0.10</sub></td><td>74<sup>+15</sup><sub>-15</sub></td><td>2.00<sup>+0.10</sup><sub>-0.11</sub></td><td>-0.45<sup>+0.21</sup><sub>-0.20</sub></td><td>83<sup>+24</sup><sub>-12</sub></td><td>1.78<sup>+0.13</sup><sub>-0.11</sub></td><td>1.63</td><td>0.58</td></tr><tr><td>HD144965</td><td>20.79<sup>+0.10</sup><sub>-0.10</sub></td><td>14.28<sup>+0.05</sup><sub>-0.05</sub></td><td>70<sup>+15</sup><sub>-10</sub></td><td>2.42<sup>+0.11</sup><sub>-0.09</sub></td><td>0.15<sup>+0.34</sup><sub>-0.22</sub></td><td>107<sup>+47</sup><sub>-34</sub></td><td>2.19<sup>+0.17</sup><sub>-0.11</sub></td><td>1.95</td><td>0.76</td></tr><tr><td>HD147683</td><td>20.74<sup>+0.10</sup><sub>-0.10</sub></td><td>14.81<sup>+0.05</sup><sub>-0.05</sub></td><td>56<sup>+13</sup><sub>-8</sub></td><td>2.33<sup>+0.11</sup><sub>-0.09</sub></td><td>-0.05<sup>+0.32</sup><sub>-0.22</sub></td><td>87<sup>+50</sup><sub>-17</sub></td><td>2.12<sup>+0.16</sup><sub>-0.11</sub></td><td>2.15</td><td>0.57</td></tr><tr><td>HD147888</td><td>20.48<sup>+0.03</sup><sub>-0.03</sub></td><td>14.70<sup>+0.05</sup><sub>-0.05</sub></td><td>45<sup>+3</sup><sub>-3</sub></td><td>2.09<sup>+0.10</sup><sub>-0.12</sub></td><td>-0.35<sup>+0.38</sup><sub>-0.33</sub></td><td>85<sup>+84</sup><sub>-14</sub></td><td>1.87<sup>+0.14</sup><sub>-0.12</sub></td><td>1.81</td><td>0.78</td></tr><tr><td>HD152590</td><td>20.51<sup>+0.10</sup><sub>-0.10</sub></td><td>14.60<sup>+0.05</sup><sub>-0.05</sub></td><td>64<sup>+13</sup><sub>-13</sub></td><td>2.30<sup>+0.09</sup><sub>-0.08</sub></td><td>-0.05<sup>+0.25</sup><sub>-0.29</sub></td><td>83<sup>+48</sup><sub>-12</sub></td><td>2.08<sup>+0.13</sup><sub>-0.11</sub></td><td>1.89</td><td>0.77</td></tr><tr><td>HD157857</td><td>20.69<sup>+0.10</sup><sub>-0.10</sub></td><td>14.62<sup>+0.10</sup><sub>-0.10</sub></td><td>78<sup>+17</sup><sub>-17</sub></td><td>1.97<sup>+0.10</sup><sub>-0.08</sub></td><td>-0.21<sup>+0.25</sup><sub>-0.20</sub></td><td>100<sup>+25</sup><sub>-20</sub></td><td>1.76<sup>+0.15</sup><sub>-0.12</sub></td><td>1.60</td><td>0.46</td></tr><tr><td>HD177989</td><td>20.12<sup>+0.10</sup><sub>-0.10</sub></td><td>14.68<sup>+0.10</sup><sub>-0.10</sub></td><td>61<sup>+5</sup><sub>-5</sub></td><td>2.00<sup>+0.09</sup><sub>-0.10</sub></td><td>-0.58<sup>+0.20</sup><sub>-0.20</sub></td><td>81<sup>+21</sup><sub>-10</sub></td><td>1.81<sup>+0.14</sup><sub>-0.13</sub></td><td>1.81</td><td>0.35</td></tr><tr><td>HD185418</td><td>20.77<sup>+0.03</sup><sub>-0.03</sub></td><td>14.83<sup>+0.05</sup><sub>-0.05</sub></td><td>101<sup>+10</sup><sub>-10</sub></td><td>1.85<sup>+0.22</sup><sub>-0.23</sub></td><td>-0.41<sup>+0.21</sup><sub>-0.22</sub></td><td>83<sup>+31</sup><sub>-12</sub></td><td>1.64<sup>+0.15</sup><sub>-0.12</sub></td><td>1.40</td><td>0.23</td></tr><tr><td>HD192639</td><td>20.69<sup>+0.03</sup><sub>-0.03</sub></td><td>14.86<sup>+0.11</sup><sub>-0.11</sub></td><td>98<sup>+9</sup><sub>-9</sub></td><td>2.15<sup>+0.08</sup><sub>-0.23</sub></td><td>-0.23<sup>+0.20</sup><sub>-0.18</sub></td><td>91<sup>+15</sup><sub>-17</sub></td><td>1.93<sup>+0.13</sup><sub>-0.12</sub></td><td>1.68</td><td>0.52</td></tr><tr><td>HD195965</td><td>20.37<sup>+0.03</sup><sub>-0.03</sub></td><td>14.67<sup>+0.03</sup><sub>-0.03</sub></td><td>110<sup>+7</sup><sub>-7</sub></td><td>1.97<sup>+0.10</sup><sub>-0.08</sub></td><td>-0.49<sup>+0.19</sup><sub>-0.19</sub></td><td>83<sup>+23</sup><sub>-26</sub></td><td>1.77<sup>+0.12</sup><sub>-0.12</sub></td><td>1.51</td><td>0.31</td></tr><tr><td>HD198781</td><td>20.56<sup>+0.10</sup><sub>-0.10</sub></td><td>14.53<sup>+0.04</sup><sub>-0.04</sub></td><td>65<sup>+10</sup><sub>-10</sub></td><td>2.00<sup>+0.10</sup><sub>-0.11</sub></td><td>-0.15<sup>+0.41</sup><sub>-0.26</sub></td><td>107<sup>+74</sup><sub>-25</sub></td><td>1.75<sup>+0.19</sup><sub>-0.11</sub></td><td>1.90</td><td>0.41</td></tr><tr><td>HD201345</td><td>19.24<sup>+0.10</sup><sub>-0.10</sub></td><td>13.94<sup>+0.05</sup><sub>-0.05</sub></td><td>97<sup>+14</sup><sub>-14</sub></td><td>1.67<sup>+0.14</sup><sub>-0.18</sub></td><td>-0.47<sup>+0.41</sup><sub>-0.37</sub></td><td>97<sup>+121</sup><sub>-23</sub></td><td>1.50<sup>+0.18</sup><sub>-0.20</sub></td><td>1.51</td><td>0.32</td></tr><tr><td>HD202347</td><td>19.92<sup>+0.10</sup><sub>-0.10</sub></td><td>14.86<sup>+0.05</sup><sub>-0.05</sub></td><td>116<sup>+5</sup><sub>-5</sub></td><td>1.97<sup>+0.10</sup><sub>-0.12</sub></td><td>0.25<sup>+0.20</sup><sub>-0.47</sub></td><td>173<sup>+44</sup><sub>-76</sub></td><td>1.78<sup>+0.14</sup><sub>-0.16</sub></td><td>1.69</td><td>0.20</td></tr><tr><td>HD203374</td><td>20.68<sup>+0.10</sup><sub>-0.10</sub></td><td>14.98<sup>+0.05</sup><sub>-0.05</sub></td><td>87<sup>+3</sup><sub>-3</sub></td><td>2.06<sup>+0.11</sup><sub>-0.09</sub></td><td>-0.39<sup>+0.20</sup><sub>-0.20</sub></td><td>79<sup>+20</sup><sub>-13</sub></td><td>1.84<sup>+0.13</sup><sub>-0.11</sub></td><td>1.66</td><td>0.31</td></tr><tr><td>HD203532</td><td>20.70<sup>+0.10</sup><sub>-0.10</sub></td><td>14.75<sup>+0.05</sup><sub>-0.05</sub></td><td>47<sup>+10</sup><sub>-10</sub></td><td>2.05<sup>+0.12</sup><sub>-0.09</sub></td><td>0.09<sup>+0.26</sup><sub>-0.15</sub></td><td>75<sup>+36</sup><sub>-14</sub></td><td>2.61<sup>+0.23</sup><sub>-0.13</sub></td><td>2.72</td><td>1.26</td></tr><tr><td>HD206267</td><td>20.86<sup>+0.03</sup><sub>-0.03</sub></td><td>15.30<sup>+0.05</sup><sub>-0.05</sub></td><td>64<sup>+3</sup><sub>-3</sub></td><td>2.15<sup>+0.09</sup><sub>-0.06</sub></td><td>-0.35<sup>+0.15</sup><sub>-0.13</sub></td><td>75<sup>+17</sup><sub>-11</sub></td><td>1.93<sup>+0.13</sup><sub>-0.11</sub></td><td>1.82</td><td>0.30</td></tr><tr><td>HD206773</td><td>20.52<sup>+0.10</sup><sub>-0.10</sub></td><td>14.94<sup>+0.05</sup><sub>-0.05</sub></td><td>94<sup>+5</sup><sub>-5</sub></td><td>1.91<sup>+0.10</sup><sub>-0.12</sub></td><td>-0.10<sup>+0.42</sup><sub>-0.40</sub></td><td>114<sup>+103</sup><sub>-33</sub></td><td>1.66<sup>+0.17</sup><sub>-0.08</sub></td><td>1.57</td><td>0.19</td></tr><tr><td>HD207198</td><td>20.84<sup>+0.03</sup><sub>-0.03</sub></td><td>15.53<sup>+0.05</sup><sub>-0.05</sub></td><td>66<sup>+3</sup><sub>-3</sub></td><td>2.09<sup>+0.06</sup><sub>-0.05</sub></td><td>-0.33<sup>+0.16</sup><sub>-0.16</sub></td><td>75<sup>+24</sup><sub>-11</sub></td><td>1.87<sup>+0.13</sup><sub>-0.11</sub></td><td>1.81</td><td>0.20</td></tr><tr><td>HD208440</td><td>20.34<sup>+0.10</sup><sub>-0.10</sub></td><td>15.10<sup>+0.05</sup><sub>-0.05</sub></td><td>75<sup>+5</sup><sub>-5</sub></td><td>2.12<sup>+0.11</sup><sub>-0.09</sub></td><td>-0.29<sup>+0.32</sup><sub>-0.30</sub></td><td>97<sup>+30</sup><sub>-10</sub></td><td>1.89<sup>+0.15</sup><sub>-0.12</sub></td><td>1.78</td><td>0.35</td></tr><tr><td>HD209339</td><td>20.25<sup>+0.10</sup><sub>-0.10</sub></td><td>15.03<sup>+0.05</sup><sub>-0.05</sub></td><td>90<sup>+5</sup><sub>-5</sub></td><td>2.06<sup>+0.13</sup><sub>-0.09</sub></td><td>-0.07<sup>+0.26</sup><sub>-0.24</sub></td><td>125<sup>+40</sup><sub>-44</sub></td><td>1.89<sup>+0.11</sup><sub>-0.21</sub></td><td>1.73</td><td>0.34</td></tr><tr><td>HD210809</td><td>20.00<sup>+0.10</sup><sub>-0.10</sub></td><td>14.81<sup>+0.05</sup><sub>-0.05</sub></td><td>87<sup>+5</sup><sub>-5</sub></td><td>2.03<sup>+0.11</sup><sub>-0.12</sub></td><td>-0.31<sup>+0.50</sup><sub>-0.24</sub></td><td>102<sup>+132</sup><sub>-19</sub></td><td>1.83<sup>+0.19</sup><sub>-0.17</sub></td><td>1.72</td><td>0.29</td></tr><tr><td>HD210839</td><td>20.84<sup>+0.03</sup><sub>-0.03</sub></td><td>15.09<sup>+0.04</sup><sub>-0.04</sub></td><td>72<sup>+4</sup><sub>-4</sub></td><td>2.39<sup>+0.09</sup><sub>-0.09</sub></td><td>-0.21<sup>+0.15</sup><sub>-0.12</sub></td><td>72<sup>+16</sup><sub>-10</sub></td><td>2.16<sup>+0.14</sup><sub>-0.11</sub></td><td>2.30</td><td>0.47</td></tr><tr><td>HD218915</td><td>20.15<sup>+0.10</sup><sub>-0.10</sub></td><td>14.54<sup>+0.03</sup><sub>-0.03</sub></td><td>86<sup>+14</sup><sub>-14</sub></td><td>1.91<sup>+0.10</sup><sub>-0.10</sub></td><td>-0.33<sup>+0.28</sup><sub>-0.29</sub></td><td>93<sup>+58</sup><sub>-15</sub></td><td>1.70<sup>+0.19</sup><sub>-0.11</sub></td><td>1.66</td><td>0.33</td></tr><tr><td>HD219188</td><td>20.23<sup>+0.10</sup><sub>-0.10</sub></td><td>14.56<sup>+0.05</sup><sub>-0.05</sub></td><td>103<sup>+5</sup><sub>-5</sub></td><td>1.61<sup>+0.46</sup><sub>-0.43</sub></td><td>-0.31<sup>+0.11</sup><sub>-0.17</sub></td><td>109<sup>+141</sup><sub>-24</sub></td><td>1.31<sup>+0.28</sup><sub>-0.11</sub></td><td>0.95</td><td>0.01</td></tr><tr><td>HD220057</td><td>20.34<sup>+0.10</sup><sub>-0.10</sub></td><td>14.71<sup>+0.05</sup><sub>-0.05</sub></td><td>63<sup>+11</sup><sub>-11</sub></td><td>1.88<sup>+0.11</sup><sub>-0.10</sub></td><td>-0.41<sup>+0.39</sup><sub>-0.25</sub></td><td>91<sup>+70</sup><sub>-18</sub></td><td>1.66<sup>+0.14</sup><sub>-0.13</sub></td><td>1.71</td><td>0.35</td></tr><tr><td>HD224151</td><td>20.57<sup>+0.10</sup><sub>-0.10</sub></td><td>14.63<sup>+0.05</sup><sub>-0.05</sub></td><td>74<sup>+15</sup><sub>-15</sub></td><td>1.91<sup>+0.10</sup><sub>-0.12</sub></td><td>0.13<sup>+0.60</sup><sub>-0.44</sub></td><td>186<sup>+59</sup><sub>-106</sub></td><td>1.70<sup>+0.12</sup><sub>-0.15</sub></td><td>1.93</td><td>0.16</td></tr><tr><td>HDE303308</td><td>20.35<sup>+0.10</sup><sub>-0.10</sub></td><td>14.73<sup>+0.10</sup><sub>-0.10</sub></td><td>86<sup>+14</sup><sub>-14</sub></td><td>2.30<sup>+0.08</sup><sub>-0.10</sub></td><td>0.37<sup>+0.36</sup><sub>-0.23</sub></td><td>158<sup>+98</sup><sub>-58</sub></td><td>2.10<sup>+0.10</sup><sub>-0.12</sub></td><td>1.56</td><td>0.59</td></tr></tbody></table>
<figure><img src="image_13.png" /><figcaption>**Figure A1.** Comparison of the measurements of gas number density (left-hand panel) and intensity of the UV field (middle panel) in the sample of CO I absorption systems in the local ISM using two different methods: the first from Jenkins & Tripp (2011) (based on carbon ionization calculations) and the second from the MEUDON PDR code. The dotted lines indicate a one-to-one ratio. The right-hand panel displays the relationship between thermal pressure and the intensity of the UV field.</figcaption></figure>

## APPENDIX B: FIT TO CO ABSORPTION LINES AT HIGH REDSHIFT

In this section, we present an analysis of CO absorption lines from seven DLAs at high redshift. Our approach involves a customized version of the Voigt profile fitting code<fnref n="13" />, which we used to derive redshifts, column densities, and Doppler parameters for velocity components. The local continuum for each absorption line was derived using B-spline interpolation, ensuring consistency across adjacent unabsorbed spectral regions. Spectral pixels used for the fit were selected based on visual inspection. The number of velocity components was determined visually and adjusted as necessary to account for any remaining structure in the residuals.

Figs B1 and B2 present two examples of the best fit to CO absorption lines in DLAs towards J1439+1118 and J1237+0647, respectively. These figures demonstrate our ability to resolve transitions from CO J rotational levels effectively and derive their populations with reasonably low uncertainty. The fit results, including the derived column densities, are listed in Table B1. These column densities were used for the analysis of CO excitation in this paper.

<figure><img src="image_14.png" /><figcaption>**Figure B1.** Fit to CO absorption lines in the spectrum of J1439+1118. The profiles in dashed lines represent CO *J* = 3 lines, while the total profile is shown by the red line.</figcaption></figure>

<footnote n="13">¹³https://github.com/balashev/spectro</footnote>
<table><caption>**Table B1.** Column densities of CO rotational and C I fine structure levels for CO-bearing DLAs at high redshift.</caption><thead><tr><th rowspan="2">Quasar</th><th rowspan="2">z<sub>abs</sub></th><th colspan="6">log N(CO, J)</th><th colspan="3">log N(CI, J)</th></tr><tr><th>J = 0</th><th>J = 1</th><th>J = 2</th><th>J = 3</th><th>J = 4</th><th>J = 0</th><th>J = 1</th><th>J = 2</th></tr></thead><tbody><tr><td>J0000+0048</td><td>2.525458</td><td>14.43±0.12</td><td>14.52±0.08</td><td>14.33±0.06</td><td>13.73±0.05</td><td>13.14±0.13</td><td>16.10±0.08</td><td>15.54±0.14</td><td>14.67±0.11</td></tr><tr><td>J0857+1855</td><td>1.729329</td><td>13.08±0.08</td><td>13.01±0.08</td><td>12.91±0.08</td><td>12.30<sup>+0.22</sup><sub>-0.52</sub></td><td>—</td><td>13.90±0.10</td><td>13.67±0.08</td><td>13.23±0.07</td></tr><tr><td>J1047+2057</td><td>1.773834</td><td>14.53±0.21</td><td>14.63±0.18</td><td>14.19±0.10</td><td>13.23<sup>+0.11</sup><sub>-0.14</sub></td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>J1237+0647</td><td>2.689567</td><td>13.60±0.03</td><td>13.83±0.03</td><td>13.63±0.03</td><td>13.12±0.04</td><td>&lt;12.4</td><td>15.24<sup>+0.34</sup><sub>-0.27</sub></td><td>14.52<sup>+0.21</sup><sub>-0.10</sub></td><td>13.71<sup>+0.08</sup><sub>-0.11</sub></td></tr><tr><td>J1439+1118</td><td>2.418366</td><td>13.22±0.03</td><td>13.45±0.03</td><td>13.17±0.04</td><td>12.96±0.06</td><td>&lt;12.6</td><td>14.38±0.03</td><td>14.04±0.03</td><td>13.26±0.03</td></tr><tr><td>J1705+3543</td><td>2.037709</td><td>13.46±0.06</td><td>13.75±0.09</td><td>13.37±0.10</td><td>13.36±0.16</td><td>—</td><td>14.57±0.06</td><td>14.49±0.05</td><td>13.98±0.05</td></tr><tr><td>J2331−0908</td><td>2.142272</td><td>13.21±0.06</td><td>13.44±0.05</td><td>12.67<sup>+0.29</sup><sub>-0.58</sub></td><td>13.19±0.18</td><td>13.22±0.10</td><td>13.92±0.03</td><td>13.85±0.03</td><td>13.43±0.03</td></tr></tbody></table>

<figure><img src="image_15.png" /><figcaption>**Figure B2.** Fit to CO absorption lines in the spectrum of J1237+0647. The total profile is shown in the red line.</figcaption></figure>

## APPENDIX C: FIT TO POPULATIONS OF H₂, CO AND C I LEVELS

In this section, we present fits to the observed populations of rotational levels of H₂, fine-structure levels of C I, and rotational levels of CO in both local ISM and DLA samples. Figs C1–C7 show constraints on the hydrogen number density nH and the strength of UV field (shown in the left-hand panels), along with the best fits to the populations of H₂ rotational levels (second panel), relative populations of excited C I fine-structure levels (third panels), and populations of CO rotational levels (rightmost panels). The systems are organized in increasing order of coordinates and sources of origin from Table 1.
<figure><img src="image_16.png" /><figcaption>**Figure C1.** Fit to populations of CO, H₂, and C I levels in the sample of local ISM sightlines. In the left-hand panel, the green and purple contours represent constraints on the total hydrogen number density and UV radiation intensity, obtained from the analysis of the excitation of C I fine-structure and CO rotational levels, respectively. Blue contours represent constraints obtained from the fit to the excitation of lower rotational levels of H₂ (J = 0–2). The red contours represent a joint constraint using C I–H₂ and CO–H₂ levels. The middle and right-hand panels show the populations of H₂ rotational levels, C I fine-structure levels, and CO rotational levels, respectively. The black circles indicate the observed values. The blue, green, and purple lines represent the best fits to the observed populations of H₂, C I, and CO, respectively. The best-fit model corresponds to the joint fit to the populations of both H₂ and CO levels. The black dashed curve represents the excitation of CO rotational levels produced by the CMB radiation alone, which corresponds to a CO excitation temperature of 2.725 K.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure C2. Continuation of Fig. C1.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure C3. Continuation of Fig. C1.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure C4. Continuation of Fig. C1.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure C5. Continuation of Fig. C1.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Figure C6.** Fit to populations of CO, H₂, and C I levels in the sample of CO absorption systems observed in DLAs at high redshift. The contours, lines, and colour encoding are same as in Fig. C1.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure C7.** Fit to populations of CO, H₂, and CⅠ levels in the sample of CO absorption systems observed in DLAs at high redshift. The contours, lines and colour encoding are the same as in Fig. C1.</figcaption></figure>

This paper has been typeset from a TEX/LATEX file prepared by the author.