Analytic Prediction of Baryonic Effects

[0.3cm] from the EFT of Large Scale Structures

[0.7cm] Matthew Lewandowski, Ashley Perko, and Leonardo Senatore

[0.7cm] Stanford Institute for Theoretical Physics,

Stanford University, Stanford, CA 94306

Kavli Institute for Particle Astrophysics and Cosmology,

Physics Department and SLAC, Menlo Park, CA 94025

Abstract

The large scale structures of the universe will likely be the next leading source of cosmological information. It is therefore crucial to understand their behavior. The Effective Field Theory of Large Scale Structures provides a consistent way to perturbatively predict the clustering of dark matter at large distances. The fact that baryons move distances comparable to dark matter allows us to infer that baryons at large distances can be described in a similar formalism: the backreaction of short-distance non-linearities and of star-formation physics at long distances can be encapsulated in an effective stress tensor, characterized by a few parameters.
The functional form of baryonic effects can therefore be predicted. In the power spectrum the leading contribution goes as , with being the linear power spectrum and with the numerical prefactor depending on the details of the star-formation physics. We also perform the resummation of the contribution of the long-wavelength displacements, allowing us to consistently predict the effect of the relative motion of baryons and dark matter. We compare our predictions with simulations that contain several implementations of baryonic physics, finding percent agreement up to relatively high wavenumbers such as or , depending on the order of the calculation. Our results open a novel way to understand baryonic effects analytically, as well as to interface with simulations.

## 1 Introduction and Main Idea

After the completion of the data analyses of the Planck satellite, the next leading source of cosmological information will likely be large scale structure (LSS) surveys. The cosmological information that we inherited from the WMAP and Planck missions raises the bar extremely high: in order for LSS to be able to significantly improve our knowledge of the early universe, it is mandatory to understand to percent level the behavior of the LSS observables. Order-of-magnitude understanding very rarely will be useful. Since most of the modes are gathered at short distances, this means that we need to understand the quasi-linear regime of structure formation. Recently, a research program called the Effective Field Theory of Large Scale Structures (EFTofLSS) has been launched [1, 2, 3, 4, 5], with the purpose of developing a consistent approach to analytically studying LSS in the weakly non-linear regime. The approach is based on the following observation. At non-linear level, arbitrarily short modes contribute at long distances, by the simple fact that the product of two short-wavelength modes and , where stays for short, and stays for long, give rise to a long mode . However, in LSS modes shorter than the so-called non-linear scale, which is about Mpc, are not under perturbative control. In the EFTofLSS, the equations of motion for the long wavelength modes contain some terms whose role is to encode at long distances the effect of the modes shorter than the non-linear scale. In the case of dark matter, this manifests itself in the fact that the resulting equations of motion take the form of a fluid-like system, with an effective stress tensor that contains a speed of sound, viscosity, a stochastic term, etc. [2, 1]. There parameters cannot be predicted in the theory but need to be measured either directly in observations or in simulations [2]. Symmetries dictate the form of these terms, which allows the theory to correctly encode the effect at long distances of short fluctuations by correcting the wrong contribution that convolution loops generate at long distances when integrating over short modes. This is what is called ‘renormalization’. When applied to collapsed object, this same phenomenon shows itself into bias coefficients [6]. Similarly, when considering redshift space distortions, the change of coordinates that maps real space into redshift space is sensitive to very short distance fluctuations in the density and the velocity fields, which also require the addition of extra parameters to correctly reproduce the effect of short distance physics at long distances [7] ^{1}^{1}1One might naively think that by cutting off the convolution loops at modes of order the non-linear scale, the effect of non-linear modes is not included. Such a procedure is incorrect because even if we cannot reliably compute the properties of non-linear modes, they do have effect at long distances, and this cannot be neglected for the correct predictions of long distance correlation functions. A proof of this fact is that such a naive procedure would give results which are cutoff dependent, while clearly physical observables are not dependent on the cutoff..

When compared to real space dark matter correlations from numerical simulations, the EFTofLSS has performed remarkably well. At one loop, the matter power spectrum agrees with -body simulations to percent level up to wavenumber [2]. Predictions depend on one free parameter, the so-called speed of sound, which can be measured either by fitting to the power spectrum or directly in small -body simulations [2]. At one loop the momentum power spectrum [8] and the bispectrum [4, 5] agree with similar accuracy to -body data up to the same wavenumber . These are important consistency checks of the EFTofLSS, as the EFT is expected to perform equally well on all correlation functions when they are computed at the same loop order. Another important consistency check passed by the EFTofLSS has been the prediction of the slope of the velocity vorticity power spectrum which matches the one measured in simulations [3, 9].

At two loops the EFTofLSS agrees with the power spectrum of dark matter up to a very high wavenumber [3, 8]. This is an extremely interesting result because it suggests that we can have perturbative control on very small scales. If the same reach is maintained in all observables, the consequences for next generation large scale structure surveys could be huge. The number of available modes would increase by a large factor which would lead to very significant improvements for the capability of these experiments to constrain properties of neutrinos, dark energy, the primordial power spectrum, and, most importantly, primordial non-Gaussianities.

While we believe that these are quite remarkable results, it is a fact that the physics described by dark matter can be simulated by -body codes, so one might argue what is the point of developing analytics techniques when one could just do simulations. First, we believe it is always useful to have an analytic way to understand the dynamics when this is still simple, as in the quasi-linear regime. Secondly, and most importantly, the analytic techniques from the EFTofLSS are relatively simple, easy to check, and fast: they do not require complicated and computationally expensive numerical codes ^{2}^{2}2This has consequences on the actual usability of the results. One of the authorS of this paper has experience with interacting with -body codes, and has seen the difficulty in extracting precise results from them.. We believe such a simplicity has the potential of allowing us to make much progress in understanding the LSS. In fact, while one might be skeptical of the arguments just mentioned, one can simply look at the comparison of the information we are currently gathering between the CMB and LSS. Most of the data in LSS surveys are not used because of lack of theoretical control. This leads to LSS giving a significant contribution to our knowledge of the universe mainly when they break some degeneracies in the CMB, as in the case of dark energy. The contribution is very little when the CMB is not degenerate. Clearly, in the light of the fact that in the very near future we will likely start to be dominated by LSS surveys, the current level of understanding cannot be considered satisfactory. After the Planck satellite has completed, if we want to make further progress, the situation has to change.

It should be stressed that in the case of dark matter the EFTofLSS is not trying to replace -body simulations tout court. It is rather stressing a complementarity that might lead to sensible progress in the field. Since the EFTofLSS can provide analytic control in the quasi-linear regime, simulations can focus on understanding the physics within the non-linear regime, where the EFTofLSS will have nothing to say. By focusing, at least quasi-entirely, on the short distance physics, simulation codes can afford higher resolution and therefore more accurately reproduce the correct physics.

The situation is worse when we deal with baryon physics. In this case, we currently do not have first-principles, at least-in-principle correct, simulation codes, but only codes that accurately implement models. Due to the huge range of scales necessary to simulate star formations in a cosmological setting, it is hard to imagine that we will have at our disposal, within a short time, a first-principles code for the simulation, say, of a galaxy. Since baryons physics affects long distances, it is sometimes necessary to simulate such effects on large boxes, at the cost of accuracy. The purpose of this paper is to provide an Effective Field Theory treatment for the baryonic effects in LSS in the quasi-linear regime, to complement simulations within the non-linear regime.

The idea of the approach is very simple. It follows directly from the construction we did for dark matter, which we now briefly recap. In the EFTofLSS, since we do not focus on scales shorter than the non-linear scale, dark matter is described as a fluid-like system with an effective stress tensor. At a given order in perturbation theory, the stress tensor is effectively described at large distances by a few parameters such as pressure, viscosity, etc.. We should explicitly mention that what makes the universe filled with dark matter describable in these terms is the fact that the relative displacement between two nearby dark matter particles is very small in the current universe, indeed of order the non-linear scale.

Now, let us think about how baryons are different from dark matter. From the time of recombination to the formation of the first stars, baryons and dark matter behave as the same species, with the only difference being their different initial conditions. Since the equations of motion are independent of the initial conditions, this means that the equations of motion are the same ^{3}^{3}3Truly, even in this situation there is an effect, that we check in this paper to be extremely small in our universe, due to the fact that different initial conditions for the same system leads to different non-linear structures, and therefore to different effective stress tensors.. In particular, until the formation of the first stars, the EFT that describes baryons is identical, even in the parameters, to the one that describes dark matter.

The story changes when stars begin to be formed. Supernovae explode, active galactic nuclei form, etc.. Very energetic radiation is emitted from the first stars, re-ionizing the baryons, which become hotter and must develop some form of pressure. Clearly, the story is very complicated. The observation that we find crucial in order to develop our effective field theory is that, notwithstanding the vicissitudes of the star-formation physics, baryons do not move very much. The transfer of mass and momentum on scales longer than about the non-linear scale is negligible, as baryons are very non-relativistic, even when hot. This can indeed be checked by observing that, apart for order one numbers, in a cluster baryons and dark matter occupy the same regions. Therefore, the same EFT that describes dark matter can describe baryons as well, with the only difference that now baryons and dark matter are allowed to exchange momentum through gravity. Furthermore, the numerical coefficients that describe the size of the induced stress tensor are expected to be only order one different. This observation implies that the functional form of the effect of baryonic physics at long distances is fixed, and is independent of the details of the baryonic physics, and actually the same as the one we have for dark matter, apart for numerical prefactors. For example, in the power spectrum of baryons and of dark matter, the leading corrections from baryonic physics go as

(1.1) |

where is the linear power spectrum of the adiabatic mode , with being the dark matter and baryon overdensities, being their relative contribution to the energy density of the universe, and is the wavenumber associated to the non-linear scale. Eq. (1.1) tells us that different models of star formation will lead at long distances to the same functional form for their corrections, apart for their overall size. This is interesting for two reasons. First, it tells us that we can in principle afford not to have a derivation of the size of these terms from first principles: since the functional form of the correction is known, we can fit it directly to observations. Of course, it is much better not to have to fit for any new parameter. So, Eq. (1.1) tells us that in order to reproduce the leading long distance information, simulation codes can simply work towards determining the numerical coefficients in Eq. (1.1), which is probably an easier job than determining the full functional form. As we will verify, different star formation models will differ in the numerical value of the coefficients in (1.1), but not in the functional form. Additionally, thanks to the EFTofLSS, simulation codes can begin to be run on smaller boxes, so that their accuracy can be increased.

After constructing the relevant EFT equations, in order to compare the solutions to simulation data, we need to take care of the effect of infrared modes. In the current universe, it has long been realized that large infrared displacements harm the perturbative expansion and need to be resummed. In the case where there is only one fluid, general theorems [10, 11] tell us that that these effects cancel for equal time power spectra, and therefore it is only the displacements from modes of order the BAO scale that need to be resummed [8]. The reason why displacements induced by arbitrarily long modes do not contribute at least to some observables is because the displacement induced by long modes is proportional to the gradient of the Newtonian potential, which, by general relativity, is just a gauge artifact and does not affect local observables such as the equal time power spectra of short modes. However, in the case of two fluids, there is a relative displacement that cannot be set to zero by a gauge transformation, and that therefore gives rise to dynamical effect in all observables. This effect, first pointed out in [12], is large in our universe at redshift of order and leads to a breaking of perturbation theory [13]. Since this is an infrared effect, we generalize the formulas of [8] to the case of two fluids, to provide a way to systematically resum such an effect in an analytic way.

Endowed with all these expressions, we are ready to compare with simulation data. We use two kinds of simulations. In the first ones, that we discuss in App. A, baryons are simulated with all baryon effects shut off, but still keeping the different initial conditions that baryons and dark matter have in our universe. This gives us a measure of how much the different initial conditions are important, an effect that we confirm to be small at redshift zero. Then, in Sec. 5, we compare with simulations of baryonic physics. The EFTofLSS predicts that the effect should be described by the functional form of (1.1), up to a scale which can also be estimated, given by when higher order corrections become relevant. As we will see, the comparison seems to work extremely well.

## 2 Equations of Motion and Perturbative Solutions

### 2.1 Equations of Motion

In the Eulerian description, the EFTofLSS for dark matter takes the form of fluid-like equations. The generalization of the equation of motions to two species is straightforward. The equations are not exactly the ones of two fluids because the EFT is non-local in time [3, 14]. This is because all modes of interest, including the UV modes that have been integrated out, evolve on Hubble time scales. Baryons and dark matter conserve their number density, but exchange momentum through gravity. This means that the effect of short distance physics on dark matter and baryons does not appear in the form of an effective stress tensor. More in detail, we write

(2.1) |

where

(2.2) |

Here , and all the fields that appear in this paper are the long wavelength fields defined in [2] unless otherwise stated. The parameters are the present day energy fractions of the various components of the universe, and we will frequently make use of the following definitions:

(2.3) |

is the effective stress tensor that comes from integrating out the short distance physics. However, while in the single species case the contribution from short distance physics can be entirely encapsulated in an effective stress tensor, this is not so in the case of two gravitationally interacting fluids, where a new term is necessary, which we call . In fact, if the effect of short distance physics were to be described solely by an effective stress tensor, the momentum of each of the two species would be conserved. But this is clearly not the case as gravitational interactions exchange momentum between the two species. This means that in the momentum equation the effect of short distance fluctuations requires the addition of a new term: . Notice that since interactions conserve the identity of the particles, the continuity equations for both species do not require any new term. Furthermore, the total momentum, sum of the two momenta of the species, is conserved, and therefore the two new terms and are such that they cancel when considering the total momentum ^{4}^{4}4Following [1], it is possible to explicitly identify the short distance term that cannot be encapsulated in a stress tensor. Since dark matter is collisionless, and baryon-baryon interactions conserve momentum, the new term must arise from the gravitational ones. Let us start with a single fluid. We can focus on the short distance term and show how it can be rearranged into an effective energy tensor term :
(2.4)
where in the second passage we have used the Poisson equation for a single species: . In the case of two species, the Poisson equation changes to , so that in the momentum equation for dark matter we have
(2.5)
We see that the fact that the two species couple to the same gravitational potential, which is nothing but the equivalence principle, leads to a term, proportional to , that does not take the form of a stress tensor, which is a total derivative. This term is exactly the term, with opposite sign, that appears directly in the equation for the baryon momentum, so that it cancels in the equation for the total momentum. We also see that this term is proportional to : in the limit of no baryons, we are back to a single fluid and just to an effective stress tensor.
.

Because the EFTofLSS is non-local in time, the response of the terms in Eq. (2.2) to the long wavelength fields will be an integral over some kernel in time of an expansion in powers and derivatives of , , evaluated along the flow. The lowest order term in this expansion we have

(2.6) | |||

where is the kernel and the fluid line element is defined implicitly as [3]

(2.7) |

where is conformal time. These terms associated to the past trajectory appear at high order in the fluctuations. In this paper we will focus on calculations done at one loop level, where it is sufficient to evaluate these counterterms on the linear solutions. In this way we can use the fact that several terms have the same functional form at low orders in perturbation theory. At this order, the non-locality in time corresponds simply to a redefinition of the parameters of a would-be local-in-time theory [3]. In fact, using the linear solutions to Eq. (2.1), we can schematically write

(2.8) |

where, at linear order in the perturbations, we have neglected a factor of . We can symbolically perform the integral over and are left with just a function of one variable , which we use to define the local-in-time speed-of-sound-like parameters as follows

(2.9) | |||

where the ellipsis represents terms that are either higher order in , or higher derivatives of , or stochastic terms, all of which are negligible at the order we work in this paper.

Let us explain in some detail the structure of the effective stress tensor above in (2.9), where we have included only the leading terms. The terms in and can be intuitively called the gravitationally induced (unitless) speed of sound parameters for the dark matter and baryons respectively because they come from the term in Eq. (2.1). This fixes the dependence on the relative abundances and . The terms and are the response of the stress tensor of dark matter and baryons to the respective gradients of the velocity fields , after substituting for the continuity equations. Finally, the term in is a speed of sound that is induced by baryonic, or star formation, physics (as the subscript clearly indicates). The differences between and , as well as between and , are just due to baryonic physics, and therefore are expected to be of the same order as . According to how star formation proceeds, this can be a number much smaller, or much larger than one, and we will later verify with comparison with simulations that its size seems to be somewhat smaller than one. Following the convention of [3] for the linear power spectrum and the definitions of the coupling constant, the factor of have been chosen so that all the remaining numbers are expected to be order one.

The structure of the effective stress tensor is heavily affected by the fact that we are dealing with two fluid-like species that interact only gravitationally among each other. The fact that the interaction is only gravitational means that the effect of one component on the other is mediated only by gravity. This enters in two ways: first, on what the curvature is, but also on what the local inertial frame, determined by , is. If we go to the particular local inertial frame that is the center of mass frame, we will have that, contrary to the single-species case, the velocity of each species is non vanishing. In this frame there is a relative velocity surviving for the two species

(2.10) | |||

Notice that in the limit , , , as it is quite intuitive. At least in principle, the effective stress tensor can now depend directly on these fields. Notice that if we were to allow for such a term to appear in the stress tensor, without any additional suppression, it could lead to an order one effect on the linear equations. However, units and indices (and even physical intuition), come to our rescue. In fact, if represents units of length and units of time, has units of . The only combination linear in that we can write with these units is , but this term has two derivatives acting on , and indeed we have already included it. If we want a derivative not to act on the velocity, then we need to go to quadratic terms such as . We conclude that at quadratic level velocities without a derivative acting on them do not appear only through the dependence of on , but also in these other combinations. Since these terms are subleading at the order we work at, we neglect them for the current paper.

By substituting Eq. (2.9) in Eq. (2.1), we can find the effective Eulerian equations of motion. After Fourier transforming and changing time variables from to , the fluid-like equations become

(2.11) | |||

where

(2.12) |

It will often be convenient to transform to the basis of adiabatic and isocurvature modes, which are defined by

(2.13) |

and the same for the variables. The adiabatic mode is the total density fluctuation, and the isocurvature mode is the relative density fluctuation.

It is worth to explicitly discuss the expansion parameters that control the perturbative expansion in the case of two species. There are five parameters which are

(2.14) | |||

and

(2.15) | |||

The first three parameters are the same that appear in the dark matter only case [15]. The first represents the effect on a given mode of tidal forces from longer modes. The second represent the effect on the same mode of displacements induced by shorter modes. The third represents the effect on the same mode of displacements from longer modes. The next two parameter appear because we have two species. The first represents the effect on the same mode of the relative displacements induced by longer modes. is indeed the power spectrum of the log-derivative with respect to the scale factor of difference in the dark matter and baryon overdensities: . The last parameter is the effect on a mode of the relative displacements induced by short modes.

An Eulerian calculation in the EFTofLSS amounts to perturbatively expanding in all of these parameters. A Lagrangian calculation does not expand in and , and it is therefore a better approach. Unfortunately, calculations done using the Lagrangian approach can be tedious ^{5}^{5}5At least to some of us., and more subtle when the renormalization procedure is not straightforward [15]. However, it is possible to obtain the non-perturbative result in by performing suitable manipulation of the Eulerian calculations. This was shown to be possible in [8], and we will generalize it here to the case of two species, which allows us to treat non-perturbatively and as well.

### 2.2 Linear Solution

First we need to find the linear solution to the two-fluid equations of motion Eq. (2.11). This is similar to the one-fluid case because the linear equations (including setting to zero the counterterms) are diagonal in the adiabatic-isocurvature basis of Eq. (2.1).

In this basis the linear equations are

(2.16) |

There are two solutions to each of these equations, but the solutions that grow the fastest with quickly dominate. The dominant solution to the equation for the isocurvature mode is constant in time, so . The dominant solution of the adiabatic equation is instead , where is called the linear growth factor and is the fastest growing solution to

(2.17) |

Although most of the time we will be interested in ratios of growth factors ’s, the conventional normalization is . For the initial conditions of the linear solutions we use the present-day linear power spectrum which can be taken from CAMB [16].

Because of the different evolution of the adiabatic and isocurvature modes, the current ratio of scales as

(2.18) |

where is some early time. The isocurvature modes become more suppressed with time, and, as a result, the current ratio is about . Since we aim at doing calculations at percent level accuracy, this tells us that we must keep the isocurvature mode at tree level, i.e. for the linear solution, but can neglect it inside loops.

### 2.3 One-loop Solution

We now proceed to the solution to the EFT equations at one loop, for which we use Eq. (2.11) with the counter-terms set to zero for now. In the adiabatic-isocurvature basis the fluid equations without counter-terms are:

(2.19) |

Notice that isocurvature modes at higher order in are always sourced by at least one lower order isocurvature mode, while adiabatic modes can be sourced by adiabatic modes alone. This means that any loops producing isocurvature modes are suppressed with respect to the corresponding loops producing adiabatic modes by at least , so they are subleading. Since at linear order, we only need to keep the linear isocurvature modes, and for one-loop calculations we will only include .

Since the equations for the adiabatic mode neglecting isocurvature are exactly those for a single fluid with density , we can use the same method as [2] to solve them. We will use the EdS approximation so that the and dependence separates, which is exact for an cosmology, and is correct to percent level in for CDM cosmology (see for example [2]). To implement this approximation, we make the ansatz

(2.20) |

where . The EdS approximation relies on being close to unity. This ratio is one at early times and is at [2], but is close to one for most of the time evolution. The fact that it is close to one for most of the time evolution allows the approximation to be accurate to percent level since gravitational clustering is not affected too much by the latest times.

Using this approximation, we can solve for and algebraically. In particular, this leads to

(2.21) |

where and and are the standard single fluid kernels ^{6}^{6}6Explicitly, the loop integrals are
(2.22)
Notice that the power spectra generally need to be smoothed over (or cut off at) a scale , because the UV theory at energies greater than is not under perturbative control. However, the final result will be -independent because the -dependence of the counter-term parameters precisely cancel the leading -dependence of the loop integrals by construction. In principle there is residual dependence of powers of in the loops due to the higher-derivative corrections we have neglected in the effective stress tensor. However, these effects go to zero in the decoupling limit after the terms that are divergent in have been renormalized. From now on we will take the limit and drop the dependence in the power spectra, which is consistent if, in the counter-terms we use, the parameters are calculated at .
.

### 2.4 Counter-terms

As discussed in the EFTofLSS [2], the coefficients of the counter-terms generally have two contributions: . The -dependent piece is responsible for canceling any divergences in one-loop diagrams as , so the diagram involving must have the same time dependence as the respective loop diagram. There is no such constraint on the time-dependence of the finite part, though. Furthermore, the loop diagrams considered in this paper are finite as , so the contribution of is finite and can be absorbed into . This means that perturbation theory does not determine the time dependence of , which is the piece relevant to our calculation. Thus, we should assume a general time dependence for the parameters appearing in Eqs. (2.1) that could be measured in -body simulations and used as an input for the EFT. In practice, however, the time dependence can be reabsorbed into a rescaling of the parameters if we restrict to one-loop order and consider only one redshift, as we do here ^{7}^{7}7A study of the EFTofLSS as a function of redshift is in progress [17]..

The equation we need to solve to find the counter-term contribution is

(2.23) |

where we have redefined the counter-term coefficients in Eq. (2.25) below, and left off the isocurvature mode on the right hand side because it is subleading.

We find that the solution to Eq. (2.23) results in the following contributions to the power spectra:

(2.24) |

where and for convenience we have defined new parameters and ^{8}^{8}8The counter-term parameters in Eq. (2.1) only come into the equations of motion at one loop in the following two combinations:
(2.25)
In the power spectra, the relevant parameters are the following integrals:
(2.26)
where and are the retarded Green’s functions for the linear equations
(2.27)
Approximating the integrals in (8) with the corresponding expressions in EdS, and by choosing the time dependence of to be , just as an indication, we obtain
(2.28)
This explains the factor of 9 that we included in the definition of and in (2.11).
. We find that for the one-loop, equal-time power spectra of dark matter and baryons, the only inputs to the EFT are two time-independent parameters. Thus baryons are easily included in the EFTofLSS with only one additional parameter.

This is a good place to comment on the stochastic terms. We have two kind of stochastic terms. The stochastic term in contributes to the power spectrum as , as we expect that the correlation function of to be -independent (Poisson-like). Even though suppressed by , this term is less suppressed in terms of than the stochastic term of the effective stress tensor, , which, similarly, contributes as . The difference in power is associated to the violation of momentum conservation for a single species. Both of these terms are very subleading with respect to the contribution of the -counterterms, which go as for the current ’s of interest, and so we can safely neglect them.

### 2.5 Summary of Expressions

Let us recap for later convenience the results from this section. In this paper, we only concern ourselves with density-density power spectra, defined by . The power spectra for baryons and dark matter at one loop in the EFTofLSS are given by the following expressions

(2.29) |

to one loop order, where and and are given by Eq.(2.22). We are also concerned with the total matter power spectrum, which in our notation is called the adiabatic power spectrum, and is given by

(2.30) | |||||

Notice that at linear level where because the initial linear dark matter and baryon fields are both proportional to the primordial curvature perturbation.

## 3 Details of IR Resummation

The derivation that we have presented in Section 2 is called the Eulerian approach because the basic degrees of freedom are the values of the fields