We investigate a contribution of the Jeans smoothing to the minimal width of Lyαݛ¼\alphaitalic_α forest lines and discuss how the accounting for this additional broadening affects the inferred parameters of the intergalactic matter equation of state. We estimate a power-law index γݛ¾\gammaitalic_γ of the equation of state, a temperature at the mean density T0subscriptݑ‡0T_0italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and a hydrogen photoionization rate ΓΓ\Gammaroman_Γ within 4 redshift bins. Furthermore, in each bin we obtain an upper limit on the scale-parameter fJsubscriptݑ“Jf_\rm Jitalic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT, which sets the relation between the Jeans length and the characteristic physical size of the absorber clouds.

It is believed that the effective equation of state (EOS) of the low-density intergalactic medium (IGM) after the H i reionization obeys the power law [1]:

T=T0Δγ-1,ݑ‡subscriptݑ‡0superscriptΔݛ¾1T=T_0\Delta^\gamma-1,italic_T = italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT italic_γ - 1 end_POSTSUPERSCRIPT , (1)
where T=T(ρ)ݑ‡ݑ‡ݜŒT=T(\rho)italic_T = italic_T ( italic_ρ ) is a temperature at a density ρݜŒ\rhoitalic_ρ, T0≡T(ρ¯)subscriptݑ‡0ݑ‡¯ݜŒT_0\equiv T(\bar\rho)italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≡ italic_T ( over¯ start_ARG italic_ρ end_ARG ) is the temperature at the mean density ρ¯¯ݜŒ\bar\rhoover¯ start_ARG italic_ρ end_ARG of the Universe and Δ≡ρ/ρ¯ΔݜŒ¯ݜŒ\Delta\equiv\rho/\bar\rhoroman_Δ ≡ italic_ρ / over¯ start_ARG italic_ρ end_ARG is an overdensity. Evolution of the EOS, defined by a dependence of T0subscriptݑ‡0T_0italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and a power-law parameter, γݛ¾\gammaitalic_γ, on a redshift, is determined by the dynamics of the reionization processes. One of the widely used methods to probe the EOS is the statistical analysis of the parameters of the Lyαݛ¼\alphaitalic_α forest lines observed in spectra of distant quasars [2, 3, 4, 5]. This method exploits an assumption that the minimal broadening of the Lyαݛ¼\alphaitalic_α lines is due to thermal motions of the absorbing atoms. However, it was suggested that the Hubble expansion during the time that light crosses an absorber may result in the minimal line broadening that depends not only on the thermal velocity distribution within the absorber, but also on its spatial structure [6, 7, 8]. In other words, an observed broadening of the Lyαݛ¼\alphaitalic_α lines encodes an information about the physical extent of the absorbers. In the present study we estimate a significance of the additional broadening related to the Hubble expansion and obtain constraints on the EOS parameters and the scale parameter between the Jeans length and the characteristic physical size of low density IGM absorbing clouds from the analysis of the observed joint distribution of column densities NݑNitalic_N and Doppler parameters bݑbitalic_b of Lyαݛ¼\alphaitalic_α forest absorbers.

2 Data and method

We obtained a large sample of Lyαݛ¼\alphaitalic_α forest lines in the redshift range z∼2-4similar-toݑ§24z\sim 2-4italic_z ∼ 2 - 4 in 47 high-resolution (R∼36000-72000similar-toݑ…3600072000R\sim 36000-72000italic_R ∼ 36000 - 72000) and high signal-to-noise ratio (S/N∼20-100similar-toݑ†ݑ20100S/N\sim 20-100italic_S / italic_N ∼ 20 - 100) quasar spectra from KODIAQ111Keck Observatory Database of Ionized Absorption toward Quasars [14] using the original fitting procedure, see [12, 15, 16] for details.

For the analysis of the obtained sample we employed the method developed in our previous works [12, 15, 16]. The method is based on the approximation of the (N,b)ݑݑ(N,\,b)( italic_N , italic_b ) sample by the model probability density function

f(N,b)=∫0∞fN(N)fadd(badd)δ(b-bmin2+badd2)dbadd,ݑ“ݑݑsuperscriptsubscript0subscriptݑ“ݑݑsubscriptݑ“addsubscriptݑaddݛ¿ݑsuperscriptsubscriptݑmin2superscriptsubscriptݑadd2differential-dsubscriptbaddf(N,\,b)=\int\limits_0^\inftyf_N(N)f_\rm add(b_\rmadd)\delta\left(% b-\sqrtb_\rmmin^2+b_\rmadd^2\right)\rmd\emphb_\rmadd,italic_f ( italic_N , italic_b ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_N ) italic_f start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT ) italic_δ ( italic_b - square-root start_ARG italic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_d start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT , (2)
where bminsubscriptݑminb_\mathrmminitalic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT is the minimal broadening at a given NݑNitalic_N, baddsubscriptݑaddb_\rm additalic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT is an additional broadening, accounting for the turbulent and peculiar motions, fN(N)subscriptݑ“ݑݑf_N(N)italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_N ) and fadd(badd)subscriptݑ“addsubscriptݑaddf_\rm add(b_\rmadd)italic_f start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT ) are distribution functions of Lyαݛ¼\alphaitalic_α absorbers over NݑNitalic_N and baddsubscriptݑaddb_\rm additalic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT, respectively. It is well established, that fN(N)subscriptݑ“ݑݑf_N(N)italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_N ) has a power-law shape, fN(N)∝N-βproportional-tosubscriptݑ“ݑݑsuperscriptݑݛ½f_N(N)\propto N^-\betaitalic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_N ) ∝ italic_N start_POSTSUPERSCRIPT - italic_β end_POSTSUPERSCRIPT (e.g. [17, 18]). For fadd(badd)subscriptݑ“addsubscriptݑaddf_\rm add(b_\rmadd)italic_f start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT ) we also assumed a power-law behaviour ∝baddpproportional-toabsentsuperscriptsubscriptݑaddݑ\propto b_\rmadd^p∝ italic_b start_POSTSUBSCRIPT roman_add end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT (for the discussion of this choice, see [15, 16]). Usually one suggests that the minimal broadening of the absorption lines is determined predominantly by the thermal contribution. Here we investigate the model proposed by Garzilli et al. [7, 8], where the minimal broadening of the Lyαݛ¼\alphaitalic_α lines is a sum of two contributions

bmin2=bth2+bρ2≡2kBTm+fJ2(λJH(z)2π)2,superscriptsubscriptݑmin2superscriptsubscriptݑth2superscriptsubscriptݑݜŒ22subscriptݑ˜Bݑ‡ݑšsuperscriptsubscriptݑ“J2superscriptsubscriptݜ†Jݐ»ݑ§2ݜ‹2b_\rm min^2=b_\rm th^2+b_\rho^2\equiv\frac2k_\rm BTm+f_\rm J% ^2\left(\frac\lambda_\rm JH(z)2\pi\right)^2,italic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_b start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ divide start_ARG 2 italic_k start_POSTSUBSCRIPT roman_B end_POSTSUBSCRIPT italic_T end_ARG start_ARG italic_m end_ARG + italic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_λ start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT italic_H ( italic_z ) end_ARG start_ARG 2 italic_π end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (3)
where kBsubscriptݑ˜Bk_\rm Bitalic_k start_POSTSUBSCRIPT roman_B end_POSTSUBSCRIPT is the Boltzmann constant, mݑšmitalic_m is the hydrogen atom mass, λJsubscriptݜ†J\lambda_\rm Jitalic_λ start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT is the Jeans length and H(z)ݐ»ݑ§H(z)italic_H ( italic_z ) is the Hubble constant. IGM cloud. The Jeans length. The second term in eq (3), bρsubscriptݑݜŒb_\rhoitalic_b start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT, referred as the broadening due to the Jeans smoothing, results from a spatial structure of the absorber.

The Jeans length in eq (3) is [8]

λJ=π(409)1/2(3γ5)1/2(kBm)1/2μ-1/2H0-1(1+z)-3/2Ωm-1/2T1/2Δ-1/2,subscriptݜ†Jݜ‹superscript40912superscript3ݛ¾512superscriptsubscriptݑ˜Bݑš12superscriptݜ‡12superscriptsubscriptݐ»01superscript1ݑ§32superscriptsubscriptΩݑš12superscriptݑ‡12superscriptΔ12\lambda_\mathrmJ=\pi\left(\frac409\right)^1/2\left(\frac3\gamma5% \right)^1/2\left(\frack_\rm Bm\right)^1/2\mu^-1/2H_0^-1(1+z)^% -3/2\Omega_m^-1/2T^1/2\Delta^-1/2,italic_λ start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT = italic_π ( divide start_ARG 40 end_ARG start_ARG 9 end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG 3 italic_γ end_ARG start_ARG 5 end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_k start_POSTSUBSCRIPT roman_B end_POSTSUBSCRIPT end_ARG start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT italic_μ start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 + italic_z ) start_POSTSUPERSCRIPT - 3 / 2 end_POSTSUPERSCRIPT roman_Ω start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_T start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT roman_Δ start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT , (4)
where ΩmsubscriptΩݑš\Omega_mroman_Ω start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT is the matter density parameter, μݜ‡\muitalic_μ is the mean molecular weight of the gas and H0subscriptݐ»0H_0italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the present-day value of the Hubble parameter. In a case of a uniform UV background, the column density NݑNitalic_N can be related to the density ρݜŒ\rhoitalic_ρ as in the model proposed in [22]. Taking into account Jeans smoothing and following [7, 8], we write

N=8.6×1012(3γ5)1/2fJΔ3/2Γ-12(T104K)-0.22(1+z3.4)9/2cm-2,ݑ8.6superscript1012superscript3ݛ¾512subscriptݑ“JsuperscriptΔ32subscriptΓ12superscriptݑ‡superscript104K0.22superscript1ݑ§3.492superscriptcm2N=8.6\times 10^12\,\left(\frac3\gamma5\right)^1/2f_\rm J\frac\Delta% ^3/2\Gamma_-12\left(\fracT10^4{}\mathrmK\right)^-0.22\left(% \frac1+z3.4\right)^9/2{}\rm cm^-2,italic_N = 8.6 × 10 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT ( divide start_ARG 3 italic_γ end_ARG start_ARG 5 end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT divide start_ARG roman_Δ start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT - 12 end_POSTSUBSCRIPT end_ARG ( divide start_ARG italic_T end_ARG start_ARG 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_K end_ARG ) start_POSTSUPERSCRIPT - 0.22 end_POSTSUPERSCRIPT ( divide start_ARG 1 + italic_z end_ARG start_ARG 3.4 end_ARG ) start_POSTSUPERSCRIPT 9 / 2 end_POSTSUPERSCRIPT roman_cm start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT , (5)
where Γ-12subscriptΓ12\Gamma_-12roman_Γ start_POSTSUBSCRIPT - 12 end_POSTSUBSCRIPT is the hydrogen photoionization rate in units of 10-12superscript101210^-1210 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT s-11{}^-1start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT. The additional factor (3γ/5)1/2superscript3ݛ¾512\left(3\gamma/5\right)^1/2( 3 italic_γ / 5 ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT in eqs (4) and (5) as compared with [7, 8] accounts for the non-adiabatic gas behaviour [6]. Eqs (4) and (5) allow to relate the position of the minimal line width in b-Nݑݑb-Nitalic_b - italic_N plane, eq (3), with the parameters of the effective EOS. To construct the model for obtained data sample, we write the likelihood function based on eq (2) and take into account the presence of outliers as we did in [15]. The model parameters are γ,T0,Γ,fJݛ¾subscriptݑ‡0Γsubscriptݑ“J\gamma,\,T_0,\,\Gamma,\,f_\rm Jitalic_γ , italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_Γ , italic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT and nuisance parameters p,βݑݛ½p,\,\betaitalic_p , italic_β and a parameter, which characterises a fraction of outliers, thus 7 parameters in total222Notice, that in [15] we shared the nuisance parameters between the bins. Further analysis have shown that this can lead to the systematic shift in the fit results. Therefore in the present work we discard this sharing.. The parameters γ,T0,Γݛ¾subscriptݑ‡0Γ\gamma,\,T_0,\,\Gammaitalic_γ , italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_Γ and fJsubscriptݑ“Jf_\rm Jitalic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT are strongly correlated [8] which complicates their measurements. To reduce the uncertainty, we use an additional constraint based on the measurements of the effective optical depth of the Lyαݛ¼\alphaitalic_α forest, τeff(z)subscriptݜeffݑ§\tau_\mathrmeff(z)italic_τ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT ( italic_z ). This quantity is inferred from the mean transmission of the Lyαݛ¼\alphaitalic_α forest averaged over many quasars spectra, see [23]. The effective optical depth can be expressed via the local optical depth τݜ\tauitalic_τ and the gas probability density distribution P(Δ,z)ݑƒΔݑ§P(\Delta,z)italic_P ( roman_Δ , italic_z ) as [19]

τeff(z)=-ln[∫0∞P(Δ,z)τ(z)dΔ].subscriptݜeffݑ§subscriptsuperscript0ݑƒΔݑ§ݜݑ§differential-dΔ\tau_\rm eff(z)=-\ln\left[\int^\infty_0P(\Delta,\,z)\tau(z)\rm d\Delta% \right].italic_τ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT ( italic_z ) = - roman_ln [ ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_P ( roman_Δ , italic_z ) italic_τ ( italic_z ) roman_d roman_Δ ] . (6)
Following [19], we use the analytical function for the gas probability density distribution, taken from [24]. In principle, the local optical depth τ(z)ݜݑ§\tau(z)italic_τ ( italic_z ) depends on the spatial extent of an absorber, as discussed above [8]. However, when the averaging of the different lines of sight is performed, the spatial structures are smeared out and the local Gunn-Peterson approximation [25, 19] is applicable.

In our calculations we assumed a standard ΛΛ\Lambdaroman_ΛCDM cosmology with matter, dark energy and baryon density parameters Ωm=0.28subscriptΩݑš0.28\Omega_m=0.28roman_Ω start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0.28, ΩΛ=0.72subscriptΩΛ0.72\Omega_\Lambda=0.72roman_Ω start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT = 0.72 and Ωb=0.046subscriptΩݑ0.046\Omega_b=0.046roman_Ω start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = 0.046, respectively, and H0=70subscriptݐ»070H_0=70italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 70 km s-11{}^-1start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT Mpc-11{}^-1start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT [26].

We split our data into 4 redshift bins with nearly the same number of absorption lines (375375375375 lines in each bin) and estimated parameters in question using the Bayesian framework with the affine Markov Chain Monte Carlo (MCMC) sampler emcee [27]. Flat priors on the parameters were used. The fit summary is given in table 1. Reported uncertainties correspond to the 68 per cent highest probability density intervals. Dependencies of the γ-1ݛ¾1\gamma-1italic_γ - 1, T0subscriptݑ‡0T_0italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ΓΓ\Gammaroman_Γ on zݑ§zitalic_z are shown in the upper and bottom panels in figure 1 and in the bottom panel in figure 2, respectively, and compared with measurements by other authors. For the scale parameter fJsubscriptݑ“Jf_\rm Jitalic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT we were able to estimate only the upper limits as shown in the top panel in figure 2. Estimated cutoffs bmin(N)subscriptݑminݑb_\rm min(N)italic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_N ) of the (b-N)ݑݑ(b-N)( italic_b - italic_N ) distributionы are shown for four redshift bins by solid lines in figure 3. Grey areas correspond to the 68 per cent intervals for bmin(N)subscriptݑminݑb_\rm min(N)italic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_N ) obtained from the MCMC chain. Contribution of the thermal broadening to the total bminsubscriptݑminb_\rm minitalic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT is indicated by the dashed green line. By the dotted blue line in figure 3 we show the cutoff of the distribution, as measured neglecting the Jeans smoothing, i.e. assuming bmin=bthsubscriptݑminsubscriptݑthb_\rm min=b_\rm thitalic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT. A comparison between parameters γݛ¾\gammaitalic_γ in case of nonzero and zero Jeans smoothing contribution is presented in last two columns in table 1.

Using the technique of the (b-N)ݑݑ(b-N)( italic_b - italic_N ) distribution analysis, developed in our previous works [12, 15, 16], we constrained evolution of the parameters of the IGM EOS taking into account the Jeans smoothing contribution to the minimal width of an absorption line. To reduce an impact of the correlations between the parameters, we impose additional constraints based on the measurements of the effective optical depth from [19]. An example of the marginalized posterior distributions of the parameters with evident strong correlations between γ-1ݛ¾1\gamma-1italic_γ - 1, T0subscriptݑ‡0T_0italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ΓΓ\Gammaroman_Γ is shown in figure 4. Although we used additional constraints, the correlations between the parameters are still sizeable. At present, we can give only the upper limits on the Jeans smoothing parameter fJsubscriptݑ“Jf_\rm Jitalic_f start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT, which is found to be ≲1.7less-than-or-similar-toabsent1.7\lesssim 1.7≲ 1.7 in all cases. To make more certain conclusions about the significance of the Jeans smoothing contribution, it is required to impose additional restrictions on the physical sizes of the absorbers. As seen in figure 3, dashed lines, the thermal contribution in our model is primarily constrained by high column density regions, where the number of absorbers is relatively small. This is in contrast with the pure thermal model, bmin=bthsubscriptݑminsubscriptݑthb_\mathrmmin=b_\mathrmthitalic_b start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT, where the densest regions of the b-Nݑݑb-Nitalic_b - italic_N plane (i.e. with small NݑNitalic_N values, see dotted lines in figure 3) determine bthsubscriptݑthb_\mathrmthitalic_b start_POSTSUBSCRIPT roman_th end_POSTSUBSCRIPT. That means that the neglect of the Jeans smoothing contribution may lead to underestimation of the EOS power-law index γݛ¾\gammaitalic_γ (table 1, see also bottom left panel in figure 4). Moreover we do not find a statistically significant evolution of the EOS parameters with zݑ§zitalic_z, see figures 1 and 2. This also differs from the case when the pure thermal broadening is assumed, e.g. [15]. Notably, we do not obtain inverted EOS (γ<1ݛ¾1\gamma<1italic_γ <1), see table 1 and figure 1, top panel. We conclude that the inference of the EOS parameters from the b-Nݑݑb-Nitalic_b - italic_N distribution is influenced by the finitude of the IGM filament and this needs to be taken into account [7, 8]. Unfortunately, the correlation between the Jeans smoothing parameter and the parameters of the effective EOS at present do not allow to track their evolution, although the results are in agreement within uncertainties with previous studies. One expects much better constraints with an increase of the sample size, i.e. the number of the lines of sight probed in high resolution spectra of the quasars, especially in the high-NݑNitalic_N region. \ackThe work was supported by the Russian Science Foundation, grant 18-72-00110.

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