# Baryon Physics and Tight Coupling Approximation in Boltzmann Codes

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## Abstract

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## 1. Introduction

- First, there is a gauge incompatibility, that is the equations break general covariance. In particular, the equations of motion in the Newtonian gauge and those in the synchronous gauge are not related to each other by a gauge transformation. This results in different physical outcomes for different gauge choices. The consequence is that it is not clear which gauge one should choose from the beginning to study baryons. This should not happen in a covariant theory, as a gauge choice merely represents a choice of coordinates and does not affect physical results.
- Second, it breaks the Bianchi identity. This aspect also leads to inconsistencies. For example, breaking the Bianchi identity implies that solving all components of the Einstein equations would lead in general to a solution that is not consistent with the conservation equation for the matter fields.
- Third, in the limit of no interaction between the baryon fluid and the photon gas, we have an equation of motion for the baryons in which the squared sound speed ${c}_{s}^{2}$ is present. It is difficult to understand the nature of this term with ${c}_{s}^{2}$, as no known covariant action for matter would lead to such a term in the dynamical equation of motion.

## 2. Baryon Equation of Motion

#### 2.1. Expansion in $T/{\mu}_{g}$

#### 2.2. Baryon Covariant Equations of Motion from the Conservation Law

## 3. Tight Coupling Approximation

## 4. Comparison between Current Boltzmann Codes and Covariance Full Equations

## 5. Code Implementation

## 6. Results

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Speed of Propagation of Matter Fields in the Presence of Several Fluids

## Appendix B. Ideal Gas

## Appendix C. Tight Coupling Approximation: Detailed Calculation

#### Appendix C.1. Terms in G

#### Appendix C.2. Shear Solution

#### Appendix C.3. Slip Equation

#### Appendix C.4. First Order Contribution

#### Appendix C.5. Second Order Contribution

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1. | We provide our new code in the link http://www2.yukawa.kyoto-u.ac.jp/~antonio.defelice/new_baryon.zip. |

2. | Here, we have used the conventions of Ma and Bertschinger for the metric perturbation fields $\eta $, h. |

3. | The relation between fields defined in this paper and definitions used in CLASS are $2E/{a}^{2}=-1/{k}^{2}\left(\right)open="("\; close=")">h+6\eta $ and $\zeta =-\eta $. For the synchronous gauge, we add the gauge choice $\alpha =0$, $\chi =0$, which is actually incomplete. For a complete gauge fixing, we need to choose also the following two initial conditions at the time $\tau ={\tau}_{\mathrm{ini}}$: ${\theta}_{c}({\tau}_{\mathrm{ini}})=0$, ${\delta}_{\gamma}({\tau}_{\mathrm{ini}})+\frac{2}{3}h({\tau}_{\mathrm{ini}})=0$, where ${\theta}_{c}$ represents the field $\theta $ for the cold dark matter fluid and ${\delta}_{\gamma}$ is the photon density perturbation. For the Newtonian gauge, the authors in [18] used the following field redefinition, $\alpha =\psi $, $\zeta =-\varphi $, together with the complete gauge fixing, $E=0$, $\chi =0$. |

4. | The extra terms in ${c}_{s}^{2}$ is due to the fact that Equations (64) and (65) have ${c}_{s}^{2}$ terms that are not neglected a priori. On using Einstein equations in the synchronous gauge, one gets these extra terms; especially, the term ${c}_{s}^{2}{k}^{2}\eta $, which can never be neglected since ${\delta}_{b}<\eta $ in the relevant redshift range, as we show in Figure 3. |

5. | One for each author. |

6. | The code can be found at http://www2.yukawa.kyoto-u.ac.jp/~antonio.defelice/new_baryon.zip. |

7. | It should be noted that for a general perfect fluid, the notion of the sound speed is purely thermodynamical, so that once the equations of state are imposed, its expression is independent of the background. For an ideal gas, we find ${c}_{s}^{2}=10T/(6{\mu}_{g}+15T)$. Furthermore, since p can be written in terms of two other thermodynamical variables, we have that at linear order $\delta p={(\partial p/\partial \rho )}_{s}\phantom{\rule{0.166667em}{0ex}}\delta \rho +{(\partial p/\partial s)}_{\rho}\phantom{\rule{0.166667em}{0ex}}\delta s$, which can be rewritten as $\delta p={c}_{s}^{2}\phantom{\rule{0.166667em}{0ex}}\delta \rho +4{\mu}_{g}p/(6{\mu}_{g}+15T)\phantom{\rule{0.166667em}{0ex}}\delta s$. In particular, we find that, in general, ${c}_{s}^{2}\ne \delta p/\delta \rho $. |

8. | In the case of non-flat 3D slices, the equations of motion need to be changed. For example, in Equation (A21), the shear field gets an extra factor, ${\sigma}_{\gamma}\to {s}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\gamma}$, where, following the CLASS code notation, ${s}_{2}^{2}\equiv 1-3K/{k}^{2}$. |

9. | We have neglected the contribution from the baryon pressure from the term $R=({\rho}_{\gamma}+{p}_{\gamma})/({\rho}_{b}+{p}_{b})$, because the ${c}_{s}^{2}$ correction term, typically of order of ${\Theta}_{\gamma b}^{2}$, will affect the tight coupling approximation only at higher orders (e.g., the cubic order). |

10. | If the background spatial curvature is present, then, as already mentioned in Footnote 8, we need to replace ${\sigma}_{\gamma}^{(1)}\to {s}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\gamma}^{(1)}$. |

**Figure 1.**Tree diagram that describes the logic followed in this paper to address the issue of non-conservation of energy-momentum in the equations of motion MB66–67 for baryons in existing Boltzmann codes.

**Figure 2.**Combined results given by the old, original CLASS code with covariance breaking equations of motion for the baryon and the new code with covariant equations of motion for the baryon fluid. The percent level differences shown in Table 1 are invisible in this figure, but will be important for future surveys.

**Figure 3.**Evolution of perturbations out of which we can test the motivation of our study in the high k regime (fixing $k=0.1\phantom{\rule{4pt}{0ex}}{\mathrm{Mpc}}^{-1}$) at all times for the default values of the parameters. During radiation domination (when ${c}_{s}^{2}$ should not be neglected) and up to $z>{10}^{4}$, $\eta $ typically dominates over ${\delta}_{b}$, so that ${c}_{s}^{2}{k}^{2}\eta >{c}_{s}^{2}{k}^{2}{\delta}_{b}$, and this term cannot be neglected. Instead, ${\delta}_{\gamma}$ dominates over ${\delta}_{b}$ up to $z>{10}^{3}$, so that in this range of redshift ${c}_{s}^{2}{k}^{2}{\delta}_{\gamma}>{c}_{s}^{2}{k}^{2}{\delta}_{b}$. Therefore, such a term cannot be neglected in tight coupling approximation schemes. Finally, even for this high k mode, the subhorizon approximation breaks down during radiation domination, as it is clear that for $z>5\times {10}^{4}$, we have ${c}_{s}^{2}{k}^{2}{\delta}_{b}\ll {c}_{s}^{2}{a}^{2}{H}^{2}{\delta}_{b}$. Since ${c}_{s}^{2}$ cannot be neglected in this redshift range, also this term should be included in the equations of motion for the perturbations. All these new terms are terms that are imposed by conservation of the stress-energy tensor.

**Figure 4.**Relative difference in the matter power spectrum at $z=1000$. We fixed the same background parameters for both codes, so that the error only depends on the difference in the equations of motion for the baryon-photon sector.

**Figure 5.**Relative difference in the transfer function for ${\delta}_{b}$, ${\delta}_{\gamma}$, ${\theta}_{b}$, and ${\theta}_{\gamma}$ at $z=1000$. The y-axis gives the relative error between covariant results and non-covariant results. The error increases at higher values of k and reaches values of order ${10}^{-6}\sim {10}^{-5}$ for ${\delta}_{b}$.

**Figure 6.**Relative difference in various perturbation fields. The spikes are due to oscillations crossing zero. In order to obtain these figures, we increased the precision of CLASS (simply by setting a lower tolerance for integrating the equations of motion). Furthermore, we had to interpolate the numerical results in order to be able to evaluate and compare the fields at the same point.

**Table 1.**Best fit values of cosmological parameters given by MCMC analysis: old vs. new. The upper and lower limits are at the 95% confidence level.

Parameters | Best Fit: New (Old) | Relative % Change in the Best Fit |
---|---|---|

$100\phantom{\rule{3.33333pt}{0ex}}{\omega}_{b}$ | ${2.236}_{-0.038}^{+0.041}\left(\right)open="("\; close=")">{2.24}_{-0.04}^{+0.04}$ | $0.2\%$ |

${\omega}_{\mathrm{cdm}}$ | ${0.1176}_{-0.0025}^{+0.0021}\left(\right)open="("\; close=")">{0.117}_{-0.002}^{+0.003}$ | $0.5\%$ |

$100{\theta}_{s}$ | ${1.042}_{-0.001}^{+0.001}\left(\right)open="("\; close=")">{1.042}_{-0.001}^{+0.001}$ | $0.0\%$ |

$ln{10}^{10}{A}_{s}$ | ${3.086}_{-0.051}^{+0.046}\left(\right)open="("\; close=")">{3.085}_{-0.05}^{+0.047}$ | $0.03\%$ |

${n}_{s}$ | ${0.969}_{-0.007}^{+0.010}\left(\right)open="("\; close=")">{0.9726}_{-0.0111}^{+0.0067}$ | $0.4\%$ |

${\tau}_{\mathrm{reio}}$ | ${0.07879}_{-0.02672}^{+0.02461}\left(\right)open="("\; close=")">{0.08009}_{-0.02841}^{+0.02301}$ | $1.6\%$ |

${\Omega}_{\Lambda}$ | ${0.6989}_{-0.0124}^{+0.0144}\left(\right)open="("\; close=")">{0.7022}_{-0.0157}^{+0.0111}$ | $0.5\%$ |

${Y}_{\mathrm{He}}$ | ${0.2478}_{-0.0001}^{+0.0002}\left(\right)open="("\; close=")">{0.2478}_{-0.0001}^{+0.0002}$ | $0.0\%$ |

${H}_{0}$ | ${68.35}_{-0.98}^{+1.13}\left(\right)open="("\; close=")">{68.58}_{-1.19}^{+0.92}$ | $0.3\%$ |

${\sigma}_{8}$ | ${0.8209}_{-0.0198}^{+0.0174}\left(\right)open="("\; close=")">{0.8194}_{-0.0184}^{+0.0188}$ | $0.2\%$ |

${\Omega}_{\mathrm{m}}$ | ${0.301}_{-0.014}^{+0.012}\left(\right)open="("\; close=")">{0.2977}_{-0.0111}^{+0.0158}$ | $1.1\%$ |

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**MDPI and ACS Style**

C. Pookkillath, M.; De Felice, A.; Mukohyama, S.
Baryon Physics and Tight Coupling Approximation in Boltzmann Codes. *Universe* **2020**, *6*, 6.
https://doi.org/10.3390/universe6010006

**AMA Style**

C. Pookkillath M, De Felice A, Mukohyama S.
Baryon Physics and Tight Coupling Approximation in Boltzmann Codes. *Universe*. 2020; 6(1):6.
https://doi.org/10.3390/universe6010006

**Chicago/Turabian Style**

C. Pookkillath, Masroor, Antonio De Felice, and Shinji Mukohyama.
2020. "Baryon Physics and Tight Coupling Approximation in Boltzmann Codes" *Universe* 6, no. 1: 6.
https://doi.org/10.3390/universe6010006