CHARACTERIZING THE PRESSURE SMOOTHING SCALE OF THE INTERGALACTIC MEDIUM

We can ignore ${\bar{\rho }}_{{\rm{b}}}$ relative to ${\bar{\rho }}_{{\rm{m}}}$ in the gravitational source term on the right hand sides of Equations (4). The dark matter evolution can then be solved as usual to obtain the growing mode b(z) (Peebles 1980). For baryonic perturbations, a general solution can be obtained at z > 2 where b ∼ a, if we assume that the temperature at the mean density, T₀ evolves as 1/a (Bi et al. 1992). This solution can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }_{{\rm{b}}}(t,{\boldsymbol{k}})}{{\delta }_{{\rm{m}}}(t,{\boldsymbol{k}})}=\displaystyle \frac{1}{1+{\lambda }_{{\rm{J}}}^{2}{k}^{2}},\end{eqnarray} \tag{ 5 }$$

where we have only retained the growing mode. The parameter λ_J is defined as

$$\begin{eqnarray}&&{\lambda }_{{\rm{J}}}^{2}=\displaystyle \frac{{c}_{{\rm{s}}}^{2}}{4\pi G{\bar{\rho }}_{{\rm{m}}}{a}^{2}},\end{eqnarray} \tag{ 6 }$$

which is just the classical Jeans scale of Equation (2). Due to Equation (6), for a polytropic IGM, the assumption that T₀ ∝ 1/a results in a constant λ_J at all times. Equation (5) describes the behavior of baryons relative to the dark matter to linear order in δ_b, at redshifts z > 2, under the assumption that T₀ ∝ 1/a. It shows that, in linear theory, baryonic density fluctuations are damped relative to the dark matter fluctuations by a quadratic filter $1+{\lambda }_{{\rm{J}}}^{2}{k}^{2}.$ At large scales ($k\ll {\lambda }_{{\rm{J}}}$), baryons follow the dark matter (δ_b = δ_m) but at small scales ($k\gg {\lambda }_{{\rm{J}}}$), baryons are smoothed out by pressure support (${\delta }_{{\rm{b}}}={\delta }_{{\rm{m}}}/{\lambda }_{{\rm{J}}}^{2}{k}^{2}$).⁷

However, as discussed in Section 1 above, in general the thermal history of the IGM is expected to be different from T ∝ 1/a. For example, T is expected to increase during cosmic reionization events, and additionally photoionization heating due to the UV background will generally result in different temperature evolution. In general, for an arbitrary thermal history T(a) a generalization of Equation (5) cannot be obtained analytically. But an approximate result can be derived to first order in k², i.e., at large scales (Gnedin & Hui 1998). This solution can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }_{{\rm{b}}}(t,{\boldsymbol{k}})}{{\delta }_{{\rm{m}}}(t,{\boldsymbol{k}})}=1-{k}^{2}{\lambda }_{{\rm{F}}}^{2},\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&{\lambda }_{{\rm{F}}}^{2}=\displaystyle \frac{4\pi G{\bar{\rho }}_{{\rm{m}}0}}{b(t)}{\displaystyle \int }_{0}^{t}{{dt}}^{\prime }\displaystyle \frac{b({t}^{\prime }){\lambda }_{{\rm{J}}}^{2}({t}^{\prime })}{a({t}^{\prime })}{\displaystyle \int }_{{t}^{\prime }}^{t}\displaystyle \frac{{{dt}}^{\prime\prime }}{{a}^{2}({t}^{\prime\prime })},\end{eqnarray} \tag{ 8 }$$

in which the thermal history enters via λ_J and ${\bar{\rho }}_{{\rm{m}}0}$ is the average matter density at z = 0 (recall that λ_J is defined in Equation (6)). Note that this describes the linear theory evolution of the IGM for an arbitrary thermal history only at large scales (${k}^{2}{\lambda }_{{\rm{F}}}^{2}\ll 1$). A closed-form solution of Equations (4) to all orders of ${k}^{2}{\lambda }_{{\rm{F}}}^{2}$ is not known. However, by solving Equations (4) numerically, Gnedin et al. (2003) show that the full solution is described to a good accuracy by a Gaussian of the form

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }_{{\rm{b}}}(t,{\boldsymbol{k}})}{{\delta }_{{\rm{m}}}(t,{\boldsymbol{k}})}=\mathrm{exp}\left(-{k}^{2}{\lambda }_{{\rm{F}}}^{2}\right),\end{eqnarray} \tag{ 9 }$$

where λ_F is defined in Equation (8). Note that this solution describes the density fields only in the limit of linear perturbations. Typically, Equation (9) predicts a pressure smoothing scale of λ_F ∼ 100 kpc (Gnedin & Hui 1998).

Equations (5) and (9) describe pressure smoothing of the baryons in the linear perturbation theory regime, i.e., in the limit of small density (δ < 1). The generalization of these results to the nonlinear limit is unknown. By arguing that the Lyα forest is only sensitive to moderate overdensities (Δ ≲ 10), Equations (5) and (9) have been used in semi-analytical descriptions of the forest (Reisenegger & Miralda-Escudé 1995; Bi & Davidsen 1997; Choudhury et al. 2001a, 2001b). Several authors have also used these results to simulate the IGM using so-called pseudo-hydrodynamical methods (Petitjean et al. 1995; Croft et al. 1998; Gnedin & Hui 1998). In terms of their predictions for simple cumulative statistics of absorption lines in the Lyα forest, these methods give results that are close to within 10% of hydrodynamical simulations (Meiksin & White 2001).

Nonetheless, the application of Equations (5) and (9) to characterize pressure smoothing at the scales and redshifts probed by the Lyα forest is problematic. Equations (5) and (9) typically predict pressure smoothing scales of order 100 kpc. At redshifts z = 2–5, such small scales are highly nonlinear, as the variance in the density distribution at these scales is dominated by collapsed structures. Baryons at these densities are affected by the physics of galaxy formation, as well as gas outflows and inflows. Therefore, in order to understand pressure smoothing in the IGM, it would be necessary to somehow distinguish, at scales of ∼100 kpc, the low density IGM that is likely unaffected by galaxy formation from the high density IGM that is affected by it. But where to draw the line between these two density regimes is not clear. Also, unlike in the linear regime (Δ < 1), the pressure smoothing scale λ_F is expected to be dependent on the local density in the quasi-linear regime (Δ ≲ 10) probed by the Lyα forest, but this dependence is unknown, as the classical result of Equation (6) is not guaranteed to hold at densities of order 1–10. Therefore, even if the low density IGM is somehow isolated, we are still left with the task of characterizing its pressure smoothing.

In this paper we address these two issues, and present a method of isolating the low-density IGM in a way which enables characterization of the pressure smoothing scale that not only describes the pressure smoothing in the IGM, but also provides an important tool for interpreting pressure smoothing scale measurements from the Lyα forest.

Figure 3 shows the dimensionless power spectra of gas and dark matter density contrasts ($\delta =\rho /\bar{\rho }-1;$ for gas we use $\bar{\rho }={{\rm{\Omega }}}_{{\rm{b}}}{\rho }_{\mathrm{cr}}$) at z = 3 from our fiducial simulation. The dark matter density power spectrum increases steeply toward small scales. The amplitude of the gas density power spectrum is suppressed relative to the dark matter at all scales probed by the simulation. Note that the enhancement in the dark matter power relative to the linear theory prediction indicates that scales of several Mpc are already nonlinear at this redshift. Therefore, it is clear that the pressure smoothing scale, which is expected to be of order 100 kpc, is already evolving nonlinearly.

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However, it is striking that in Figure 3 the gas density power does not exhibit a cut-off on any scale. This is very different from the linear theory expectations discussed in Section 2, which suggested a quadratic or Gaussian cut-off in the gas power spectrum at the pressure smoothing scale according to Equations (5) and (9). This discrepancy is highlighted in Figure 3, where the predictions of Equations (5) and (9) are also shown for an arbitrarily chosen pressure smoothing scale. The absence of a cut-off in the gas power spectrum shows that the linear theory formalism of Section 2 does not describe the physics of pressure smoothing on nonlinear scales at z = 3. We observe similar behavior, i.e., an absence of a cut-off in the gas density power spectrum, in our simulations over the complete redshift range probed by the Lyα forest (z ∼ 2–5).

To understand this large difference between the gas density power spectrum of Figure 3 and the linear theory predictions of Equations (5) and (9), it is useful to consider the gas and dark matter density power spectra at redshifts where density perturbations are still linear. We therefore look at the power spectra at z = 19 in Figure 4. At this high redshift, density perturbations are linear at most scales resolved in our box as indicated by the agreement of the dark matter density power spectrum with that predicted by linear theory in Figure 4. Scales ∼100 kpc comparable to the pressure smoothing scale are quasi-linear, and the assumptions behind Equation (9) are thus still valid. We would therefore expect the gas power spectrum to exhibit a Gaussian cut-off. However, we face a different problem here at z = 19, namely that the gas temperature in the fiducial simulation is T₀ = 6 K as reionization does not occur until z = 6. At this low temperature, the filtering scale (Equation (8)) is expected to be ∼2 kpc from Equation (2), much smaller than the smallest scale resolved in the simulation. As a result, there is no clear evidence for a cut-off in the gas density power spectrum. Indeed, in Figure 4, the gas and dark matter power spectra from the fiducial simulation agree at almost all scales, verifying our expectation that at low temperatures the tiny filtering scale ∼2 kpc implies that gas follows dark matter for all scales resolved by the simulation.

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To remedy this, we consider three other simulations for which we artificially enhance the H i photoheating rate between z ∼ 6 and 30. Specifically, at redshifts z < 6 the photoheating rate is equal to that in the fiducial simulation, but it is set to be much higher (10⁻²²–10⁻²⁴ erg s⁻¹) at z > 6. These simulations with enhanced heating at high redshift can be thought of as "early reionization" models. Due to this enhanced heating the gas temperature at z = 19 is T₀ ∼ 10⁴ K in these simulations, which increases the pressure smoothing scale of Equation (8) to be well above the spatial resolution of our simulation so that any cut-off will now be visible. Figure 4 shows the gas density power spectra at z = 19 from these three simulations with various enhanced heating rates and gas temperatures. The gas power is seen to drop precipitously between scales of 50 and 100 kpc, and the dotted curve shows that this cut-off has a Gaussian form, in accordance with Equation (9). Moreover, as the gas temperature T₀ increases, the cut-off is seen to move toward larger scales as expected for the pressure smoothing scale from Equation (8). Thus the classical pressure smoothing scale result appears to be valid at very high redshifts, when the ∼100 kpc pressure smoothing scale is quasi-linear, but seems to break down at z ∼ 3 when scales ∼100 kpc become highly nonlinear.

This breakdown of the linear theory pressure smoothing scale picture is clearly indicated in Figure 5, which shows the evolution from z = 19 to 3 of gas and dark matter density power spectra in one of the simulations with enhanced photoheating at high redshifts. Comparison with the linear theory matter power spectrum (gray dashed curves) indicate that scales ∼100 kpc are already highly nonlinear by z = 9. Hence one sees that at z = 19, when the pressure smoothing scale is quasi-linear, the gas power spectrum shows a clear Gaussian cut-off at 50 comoving kpc, whereas at lower redshifts when this scale is highly nonlinear this cut-off vanishes. In what follows, we will explain why the cut-off disappears at low redshifts and will present a method to reveal it.

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As we saw above in Figure 4, the gas density power spectrum does not exhibit any cut-off in power at scales ∼100 kpc at relatively low redshifts (z ∼ 3) in our fiducial simulation. At these redshifts and small scales, the power is completely dominated by highly overdense and nonlinear collapsed structures, namely halos. Although these structures occupy a tiny volume fraction, they nevertheless have a large impact on the power spectrum. Recall that galaxy formation is simplified in our fiducial simulation: all gas particles with overdensity greater than 10³ and temperature less than 10⁵ K are converted into stars. This results in a stellar density parameter Ω_* = 0.0075 at z = 3, which is already factor of three higher than the measured value of Ω_* = 0.0027 at z = 0 (Fukugita & Peebles 2004). We now rerun this simulation, with the same initial conditions, but with modified star formation overdensity thresholds of 10⁴, 100, and 10. At z = 3 we find that thresholds of 10⁴, 10³, 100, and 10, result in Ω_* = 0.0066, 0.0075, 0.0092, and 0.0145, respectively. As we reduce the star formation density threshold, we form more stars but at the same time we remove more high density gas from our simulation and the power spectrum analysis.

Figure 6 shows the resulting gas density power spectra for these different star formation density thresholds. As the density threshold is progressively decreased, and more high density gas is removed from the simulation, we see that the power on scales smaller than ≲100 kpc is increasingly attenuated, and the overall amplitude of the power on larger scales is also reduced. This behavior is not unexpected, and can be understood in a halo model picture (Cooray & Sheth 2002). As dense gas above the star formation threshold is removed, we effectively mask dense gas in collapsed haloes in the simulation volume. The truncation of the small scale power arises from the removal of this gas, which can be thought of as suppressing the "one-halo" term of the gas power spectrum. However, part of the large scale power of the gas density also arises from the clustering of these halos, i.e., the "two halo term," hence masking haloes results in a reduction of large scale power as well. Therefore on large scales, the net effect of the star formation threshold is to introduce a linear bias, such that the power spectra for different star formation thresholds in Figure 6 are all parallel to each other at small k values. Figure 6 thus shows us that the gas density power spectrum is indeed strongly influenced by nonlinear structure. More importantly, as the highest density regions are removed, the gas distribution begins to show a hint of a cut-off in power. This is seen most clearly in the Δ < 10 curve in Figure 6.

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In other words, to recover the signature of pressure smoothing in the moderately overdense IGM (Δ ≲ 10), we need to remove the influence of gas in high-density regions, which would otherwise dominate the small-scale power. One way of doing this is by applying an ad hoc cut-off in gas density as done in Figure 6. However, the choice of this density threshold is arbitrary—it would need to be made redshift-dependent, as the amount of high-density gas that has collapsed to form stars increases at later times, and is thus dependent on the galaxy formation model in the simulation, which we have crudely parameterized here with a simple star formation density threshold. However, even in more sophisticated simulations which attempt to model detailed baryonic process in galaxies such as star formation and feedback using a combination of higher resolution and sub-grid modeling (e.g., Schaye et al. 2010; Vogelsberger et al. 2014; Crain et al. 2015), our findings indicate that the the small scale baryon power spectrum will be influenced by the details of galaxy formation and the specific sub-grid implementation. Therefore the challenge is to devise a more robust method of removing the influence of high density gas on small scale power.

In this section we introduce a better method for revealing the three-dimensional cut-off in the baryon power of the moderate overdensity IGM. Given the three-dimensional, statistically isotropic gas density field, we calculate the real-space Lyα flux at each point in the box. This quantity, which we denote by F_real, is defined by

$$\begin{eqnarray}&&{F}_{\mathrm{real}}\equiv \mathrm{exp}(-{\tau }_{\mathrm{real}}),\end{eqnarray} \tag{ 10 }$$

where the real-space Lyα absorption optical depth is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{real}}=\displaystyle \frac{3{\lambda }_{\alpha }^{3}{{\rm{\Lambda }}}_{\alpha }}{8\pi H(z)}{n}_{{\rm{H}}\;{\rm{I}}}.\end{eqnarray} \tag{ 11 }$$

Here λ_α = 1216 Å is the rest-frame Lyα wavelength, Λ_α is the Einstein A coefficient (also written as A₁₀ and can be written in terms of an oscillator strength), H(z) is the Hubble constant, and n_{H i} is the number density of neutral hydrogen. The definition in Equation (11) is identical to the Gunn–Peterson formula used to compute the Lyα forest optical depth, except that the convolution integral which accounts for the redshift-space effects of the peculiar velocity field and thermal line broadening has not been included. Indeed, we have chosen to define a statistic in real space precisely because these redshift-space effects, specifically the line-of-sight smoothing due to thermal broadening, are in fact degenerate with the three-dimensional real-space Jeans smoothing that we aim to study. Hence, F_real is a three-dimensional, statistically isotropic field.

The neutral hydrogen density is given by

$$\begin{eqnarray}&&{n}_{{\rm{H}}\;{\rm{I}}}=\displaystyle \frac{{\alpha }_{R}(T){n}_{{\rm{H}}}^{2}}{{{\rm{\Gamma }}}_{{\rm{H}}\;{\rm{I}}}},\end{eqnarray} \tag{ 12 }$$

where n_H = 0. 76ρ_gas is the total hydrogen density, Γ_{H i} is the H i photoionization rate, α_R(T) ∝ T^−0.7 is the hydrogen recombination rate. Therefore, in a highly ionized IGM where the gas lies on a well-defined temperature density relation T ∝ ρ^{γ − 1} (see Figure 2), the neutral hydrogen number density and the optical depth in Equation (11) are proportional to a power of the total gas density ρ^{2–0.7(γ − 1)}. (When all gas is assumed to lie on the temperature–density relation, Equation (12) constitutes the Fluctuating Gunn–Peterson Approximation (FGPA).) Thus, F_real is proportional to the exponential of this power of the gas density, and it exponentially suppresses the contribution of the high density regions. This is of course another way to say that the high density regions have large optical depth and therefore result in completely saturated Lyα absorption, i.e., have F_real = 0.

F_real is not a directly observable field, but it can nevertheless be constrained by close quasar pair observations (Rorai et al. 2013). The phase angle differences between homologous line-of-sight Fourier modes in quasar pair spectra can be written as

$$\begin{eqnarray}&&{\theta }_{12}(k)={\mathrm{cos}}^{-1}\left[\displaystyle \frac{\mathrm{Re}\left({F}_{1k}^{*}{F}_{2k}\right)}{| {F}_{1k}{F}_{2k}| }\right],\end{eqnarray} \tag{ 13 }$$

where F₁(k) and F₂(k) are Fourier amplitudes of the two quasars in a pair. Rorai et al. (2013) showed that the probability distribution of these phase differences θ₁₂(k) is sensitive to a smoothing cut-off in the underlying field. For modes with wavelength larger than the pair separation r_⊥, $k\ll {k}_{\perp }\sim 1/{r}_{\perp },$ the mean value of θ₁₂(k) can be written as (Rorai et al. 2013)

$$\begin{eqnarray}&&\langle \mathrm{cos}\theta (k,{r}_{\perp })\rangle \approx \displaystyle \frac{{\displaystyle \int }_{k}^{{k}_{\perp }}{{dk}}^{\prime }{k}^{\prime }{J}_{0}({r}_{\perp }\sqrt{{{k}^{\prime }}^{2}-{k}^{2}}){P}_{{F}_{\mathrm{real}}}({k}^{\prime })}{{\displaystyle \int }_{k}^{\infty }{{dk}}^{\prime }{k}^{\prime }{P}_{{F}_{\mathrm{real}}}({k}^{\prime })}\end{eqnarray} \tag{ 14 }$$

where J₀ is the Bessel function of zeroth order and ${P}_{{F}_{\mathrm{real}}}$ is the three-dimensional power spectrum of F_real. Thus by measuring the phase angle probability distribution function in close quasar pairs the F_real can be directly constrained.

Note that in the construction of the F_real field using Equation (10), we use the rescaled photoionization rate Γ_{H i} that produces the correct observed mean redshift-space flux as discussed in Section 3 above. In this approach, we rescale the UV background intensity so that the mean redshift-space flux of these extracted spectra matches the observed mean redshift-space flux at the respective redshift (Faucher-Giguère et al. 2008). When this rescaled background is used in calculating the real-space flux F_real, its mean value is not exactly the same as the mean redshift-space flux, although the differences are very small. For instance, in our fiducial simulation, the mean redshift-space flux at z = 3 is 0.680 after rescaling the UV background, while the corresponding mean value of F_real, using this same background, is 0.679. Equations (11) and (12) indicate that rescaling the UV background to match the observed mean flux values is tantamount to fixing the range of gas densities probed by the flux field. As such, the very close agreement between the mean of the redshift space F and real-space F_real flux indicates that F_real and F probe nearly identical gas densities.

In Figure 7 we now look at the three-dimensional power spectrum of the distribution of F_real at z = 3 in our fiducial simulation (recall that unlike the observed Lyα flux, F_real is statistically isotropic). Figure 7 also shows the gas and dark matter density power spectra for comparison. The most conspicuous feature of the F_real power spectrum in Figure 7 is that it exhibits a small-scale cut off, located at around k = 20 h/Mpc (about λ = 35 comoving kpc).⁸ This is reminiscent of the Gaussian filtering scale cut-off that we expect from linear theory. There is also an overall reduction in the amplitude of the F_real power relative to that of the gas density. This reduced power level is similar to that seen when we applied a star formation density threshold in Figure 6, and, which we argued can also be understood in the halo model picture. Generically, a nonlinear transformation of the density field, such as F_real, will have a power spectrum that has a different shape than that of the density, but for small k where the Fourier amplitudes are small, the transformation can be linearized such that the power will appear to be a linear bias rescaling of that of the original density field.

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Thus F_real provides a natural way of removing the influence of high density regions in our analysis, and reveals the small-scale structure of the IGM—in the form of a sharp cut-off in the power at a certain scale—that is presumably due to pressure smoothing. Furthermore, this definition of F_real is robust against arbitrary choices of galaxy formation prescriptions in our simulations. Recall that, as discussed in Section 4.2, we can think of variation in the star formation threshold as variation in our (crude) galaxy formation prescriptions. The red curves in Figure 8 illustrate that the gas density power spectrum varies dramatically as the star formation density thresholds is varied from 10⁴ to 10. As the threshold is decreased, power at small scales is considerably reduced, but a clear signature of the pressure smoothing is not apparent from these curves. The removal of clustered high-density regions, also reduces the large-scale power as a linear-bias rescaling. Although the density power varies dramatically depending on the details of how galaxy formation is treated, the blue curves in Figure 8 show that the shape of the F_real power spectrum is invariant to these changes. Thus the F_real field as defined in Equation (10) provides an unambiguous way of characterizing the cut-off, valid at all relevant redshifts regardless of the galaxy formation details used in the simulation.

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Figure 9 shows a slice of the F_real field at z = 3 from our fiducial simulation in comparison with a slice of the gas density field. There is an almost one-to-one correspondence between gas density and F_real. This correspondence results from ignoring redshift-space distortions due to peculiar velocities and thermal broadening. The value of F_real is highest (∼1) in regions with lowest gas density, whereas all high density filaments and haloes are mapped to F_real = 0. This is exactly the kind of suppression of high densities required to reveal the structure of the low-density IGM. Note that unlike in Figure 1, we have chosen a linear color scale in the gas density plot to highlight the saturation of F_real values at high densities.

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For the sake of illustration, we also apply the transformation encoded in F_real on the dark matter density field. This should be understood as the F_real field in the limit where the baryons directly trace the dark matter density, but nevertheless adhere to the nonlinear transformation in Equation (12). This is achieved by "gassifying" dark matter, i.e., mapping the dark matter density field to a pseudo-gas density field through

$$\begin{eqnarray}&&{\rho }_{\mathrm{gas}}={\rho }_{\mathrm{dm}}\left(\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{b}}}}{{{\rm{\Omega }}}_{{\rm{m}}}-{{\rm{\Omega }}}_{{\rm{b}}}}\right),\end{eqnarray} \tag{ 15 }$$

which traces the dark matter at all scales. In other words, as the dark matter is cold and pressureless and exhibits significant small-scale power, this pseudo-gas distribution has no pressure smoothing scale.

The neutral hydrogen number density, n_{H i}, is then calculated by imposing the same temperature–density relation on these pseudo-gas particles as our fiducial simulation, i.e., by using FGPA. A slice of the resulting ${F}_{\mathrm{real}}^{\mathrm{dm}}$ field calculated in this fashion is shown in the lower right panel of Figure 9. Its structure is markedly different from that of the F_real field calculated from the gas distribution. The ${F}_{\mathrm{real}}^{\mathrm{dm}}$ has a morphology very similar to the dark matter density distribution, and because it lacks a pressure smoothing scale cut-off, has a significant amount of small-scale structure. In contrast the pressure smoothing of the gas distribution makes the F_real field smoother and more diffuse. Although somewhat contrived, ${F}_{\mathrm{real}}^{\mathrm{dm}}$ is helpful in better understanding the F_real transformation of the density field.

These points are further illustrated in Figure 10, which compares the F_real field along an example line of sight through our fiducial simulation to various other quantities at z = 3. The upper two panels of this figure show the dark matter and gas density field along the line of sight. As in Figure 9, we see that the gas density field is a smoothed version of the dark matter density field. The third and fourth panels of Figure 10 show the ${F}_{\mathrm{real}}^{\mathrm{dm}}$ and F_real fields, calculated from the dark matter and gas density fields respectively. The mapping to real space flux makes the smoothness of the gas density field relative to the dark matter density field much more apparent. The last panel shows the observed, redshift-space Lyα flux, F, which includes the effects of peculiar velocities and thermal broadening. As a result, the absorption features in F are shifted because of peculiar velocities, and their widths now reflect the impact of Hubble flow and are much smoother because of thermal line broadening. The comparison between F_real and F is striking, and illustrates that the intrinsic small-scale structure of the gas in real space, which is determined by pressure smoothing, is completely hidden by redshift space effects.

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Finally, it is instructive to examine the power spectrum of the ${F}_{\mathrm{real}}^{\mathrm{dm}}$ field, which is shown as a dashed curve in Figure 11. As expected, the power spectrum of ${F}_{\mathrm{real}}^{\mathrm{dm}}$ shows no evidence for a small-scale cut-off: because the dark matter is not smoothed by pressure there is power on arbitrarily small scales. The power spectrum at large scales is parallel to that for F_real obtained from the gas distribution but the amplitude is lower.⁹ This comparison demonstrates that the cut-off in the power spectrum of the F_real distribution in our simulations results from pressure smoothing.

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We have demonstrated that the power spectrum of the F_real field shows a conspicuous small-scale cut-off, which we have interpreted as an effect of pressure smoothing. In this section we quantify the location of this cut-off, and demonstrate that its location behaves as expected from the classical Jeans scale argument.

Inspection of the F_real power spectrum in Figure 7 motivates a simple fitting function of a power law power spectrum cut off by a Gaussian:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\rm{F}}}^{2}(k)={{Ak}}^{n}\mathrm{exp}\left(-\displaystyle \frac{{k}^{2}}{{k}_{\mathrm{ps}}^{2}}\right),\end{eqnarray} \tag{ 16 }$$

which has three parameters: A, n, and k_ps. Figure 12 shows that Equation (16) is an excellent fit to the power spectra of F_real in our simulations. The fitting function allows us to quantify the pressure smoothing scale in our fiducial simulation (T₀ = 9.5 × 10³ K) as λ_ps = 1/k_ps = 79.0 comoving kpc at z = 3. In the simulation in which the temperature at the mean density of the IGM, T₀ = 8.6 × 10⁴ K, is higher than that in the fiducial simulation by a factor of ∼10, is λ_ps = 191.5 comoving kpc.

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In Figure 13, we compare the F_real power spectra in five simulations with different temperatures ranging from T ∼ 10⁴ to ∼10⁵. As expected, the dark matter density power spectra are essentially identical for all temperatures. The gas density power spectrum shows some change due to the change in temperature, but these small differences would be masked if galaxy formation were treated differently (see Figure 6), as we argued in Section 4.2. Much more significant changes are seen in the F_real power spectrum, where the cut-off scale is observed to progressively increase with temperature. Thus F_real successfully captures a temperature-dependent smoothing of the gas density field.

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The temperature dependence of the pressure smoothing scale ${\lambda }_{\mathrm{ps}}\equiv 1/{k}_{\mathrm{ps}}$ as determined by fitting Equation (16) to the F_real power spectrum of simulations with varying gas temperature is shown in Figure 14. The blue and brown curves show the temperature dependence of the linear theory Jeans scale, ${\lambda }_{{\rm{J}}}\propto \sqrt{{T}_{0}},$ and the linear theory filtering scale, λ_F, defined in Equations (2) and (8), respectively. Both of these quantities were derived by measuring the thermal evolution of gas in the simulations. As expected at this post-reionization redshift (z = 3), the filtering scale is smaller than the Jeans scale, by a factor of ∼3 in this case (Gnedin & Hui 1998; Gnedin et al. 2003). We see that the temperature dependence of the pressure smoothing scale defined by our fitting function in Equation (16) is nearly identical to the $\sqrt{{T}_{0}}$ dependence of the linear theory Jeans and filtering scales, which clearly demonstrates that it is probing pressure support in the IGM. Note that the slight deviation of our λ_ps values from the exact $\sqrt{T}$ dependence is partly due to the fact that simulations with different temperatures actually probe somewhat different gas densities. This occurs because we always rescale the UV background of these simulations to have the same observed mean flux value, and by changing the UV background, one changes the mapping between flux and density, and hence the range of densities probed by each model. This also explains why the pressure smoothing scale is higher than the filtering scale, as the former probes pressure smoothing at higher densities than the mean density and therefore at higher temperatures than T₀ (Schaye 2001).

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This simple fitting function for F_real in Equation (16) now allows us to quantify the pressure smoothing of the IGM, and make meaningful statements about the pressure smoothing scale for any thermal model and reionization history.