- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Advances in Astronomy
Volume 2010 (2010), Article ID 157079, 68 pages
Non-Gaussianity and the Cosmic Microwave Background Anisotropies
1Dipartimento di Fisica “G. Galilei”, Università di Padova, via Marzolo 8, 131 Padova, Italy
2INFN, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy
3CERN, Theory Division, CH-1211 Geneva 23, Switzerland
Received 19 January 2010; Revised 14 June 2010; Accepted 29 June 2010
Academic Editor: Eiichiro Komatsu
Copyright © 2010 N. Bartolo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We review in a pedagogical way the present status of the impact of non-Gaussianity (NG) on the cosmic microwave background (CMB) anisotropies. We first show how to set the initial conditions at second order for the CMB anisotropies when some primordial NG is present. However, there are many sources of NG in CMB anisotropies, beyond the primordial one, which can contaminate the primordial signal. We mainly focus on the NG generated from the post inflationary evolution of the CMB anisotropies at second order in perturbation theory at large and small angular scales, such as the ones generated at the recombination epoch. We show how to derive the equations to study the second-order CMB anisotropies and provide analytical computations to evaluate their contamination to primordial NG (complemented with numerical examples). We also offer a brief summary of other secondary effects. This paper requires basic knowledge of the theory of cosmological perturbations at the linear level.
Cosmic microwave background (CMB) anisotropies play a special role in cosmology, as they allow an accurate determination of cosmological parameters and may provide a unique probe of the physics of the early universe and in particular of the processes that gave origin to the primordial perturbations.
Cosmological inflation  is nowadays considered the dominant paradigm for the generation of the initial seeds for structure formation. In the inflationary picture, the primordial cosmological perturbations are created from quantum fluctuations “redshifted” out of the horizon during an early period of accelerated expansion of the universe, where they remain “frozen”. They are observable through CMB temperature anisotropies (and polarization) and the large-scale clustering properties of the matter distribution in the Universe.
This picture has recently received further spectacular confirmations from the results of the wilkinson microwave anisotropy probe (WMAP) five year set of data . Since the observed cosmological perturbations are of the order of , one might think that first-order perturbation theory will be adequate for all comparisons with observations. This might not be the case, though. Present [2, 3] and future experiments  may be sensitive to the nonlinearities of the cosmological perturbations at the level of second- or higher-order perturbation theory. The detection of these nonlinearities through the non-Gaussianity (NG) in the CMB  has become one of the primary experimental targets.
There is one fundamental reason why a positive detection of NG is so relevant: it might help in discriminating among the various mechanisms for the generation of the cosmological perturbations. Indeed, various models of inflation, firmly rooted in modern particle physics theory, predict a significant amount of primordial NG generated either during or immediately after inflation when the comoving curvature perturbation becomes constant on superhorizon scales . While standard single-field models of slow-roll inflation [6, 7] and—in general—two (multi)field [8, 9] models of inflation predict a tiny level of NG, “curvaton”-type models, in which a significant contribution to the curvature perturbation is generated after the end of slow-roll inflation by the perturbation in a field which has a negligible effect on inflation, may predict a high level of NG . Alternatives to the curvaton model are models where a curvature perturbation mode is generated by an inhomogeneity in the decay rate [11–13], the mass , or the interaction rate  of the particles responsible for the reheating after inflation. Other opportunities for generating the curvature perturbations occur at the end of inflation [16–18], during preheating [19–21], and at a phase-transition producing cosmic strings . Also, within single-field models of inflation, a high level of NG can be generated breaking the standard conditions of canonical kinetic terms and initially vacuum states: for example, this is the case of Dirac-born-infeld (DBI) models of inflation , and initially excited states, respectively . For every scenario, there exists a well defined prediction for the strength of NG and its shape [25, 26] as a function of the parameters.
Statistics like the bispectrum and the trispectrum of the CMB can then be used to assess the level of primordial NG (and possibly its shape) on various cosmological scales and to discriminate it from the one induced by secondary anisotropies and systematic effects [5, 27–30]. A positive detection of a primordial NG in the CMB at some level might therefore confirm and/or rule out a whole class of mechanisms by which the cosmological perturbations have been generated.
Despite the importance of evaluating the impact of primordial NG in a crucial observable like the CMB anisotropy, the vast majority of the literature has been devoted to the computation of the bispectrum of either the comoving curvature perturbation or the gravitational potential on large scales within given inflationary models. These, however, are not the physical quantities which are observed. One should instead provide a full prediction for the second-order radiation transfer function. A preliminary step towards this goal has been taken in  (see also [32–37]) where the full second-order radiation transfer function for the CMB anisotropies on large angular scales in a flat universe filled with matter and cosmological constant was computed, including the second-order generalization of the Sachs-Wolfe effect, both the early and late integrated Sachs-Wolfe (ISW) effects and the contribution of the second order tensor modes. (A similar computation of the CMB anisotropies up to third-order from gravitational perturbations has been performed in , which is particularly relevant to provide a complete theoretical prediction for cubic nonlinearities characterizing the level of NG in the CMB through the connected four-point correlation function (trispectrum) [27, 28].)
There are many sources of NG in CMB anisotropies, beyond the primordial one. The most relevant sources are the so-called secondary anisotropies, which arise after the last scattering epoch. These anisotropies can be divided into two categories: scattering secondaries, when the CMB photons scatter with electrons along the line of sight, and gravitational secondaries when effects are mediated by gravity . Among the scattering secondaries we may list the thermal Sunyaev-Zeldovich effect, where hot electrons in clusters transfer energy to the CMB photons, the kinetic Sunyaev-Zeldovich effect produced by the bulk motion of the electrons in clusters, the Ostriker-Vishniac effect, produced by bulk motions modulated by linear density perturbations, and effects due to reionization processes. The scattering secondaries are most significant on small angular scales as density inhomogeneities, bulk and thermal motions grow and become sizeable on small length-scales when structure formation proceeds.
Gravitational secondaries arise from the change in energy of photons when the gravitational potential is time-dependent, the ISW effect, and gravitational lensing. At late times, when the Universe becomes dominated by the dark energy, the gravitational potential on linear scales starts to decay, causing the ISW effect mainly on large angular scales. Other secondaries that result from a time-dependent potential are the Rees-Sciama effect, produced during the matter-dominated epoch by the time evolution of the potential on nonlinear scales.
The fact that the potential never grows appreciably means that most second order effects created by gravitational secondaries are generically small compared to those created by scattering ones. However, when a photon propagates from the last scattering to us, its path may be deflected because of the gravitational lensing. This effect does not create anisotropies, but only modifies existing ones. Since photons with large wavenumbers are lensed over many regions (, where is the Hubble rate) along the line of sight, the corresponding second-order effect may be sizeable. The three-point function arising from the correlation of the gravitational lensing and ISW effects generated by the matter distribution along the line of sight [40, 41] and the Sunyaev-Zeldovich effect  are large and detectable by Planck [43, 44]. A crucial issue is the level of contamination to the extraction of the primordial NG the secondary effects can produce. In Section 8, we briefly summarize some recent results about the level of CMB NG generated by some of these secondary effects.
Another relevant source of NG comes from the physics operating at the recombination. A naive estimate would tell that these nonlinearities are tiny being suppressed by an extra power of the gravitational potential. However, the dynamics at recombination is quite involved because all the nonlinearities in the evolution of the baryon-photon fluid at recombination and the ones coming from general relativity should be accounted for. This complicated dynamics might lead to unexpected suppressions or enhancements of the NG at recombination. A step towards the evaluation of the three-point correlation function has been taken in  where some effects were taken into account in the so-called squeezed triangle limit, corresponding to the case when one wavenumber is much smaller than the other two and was outside the horizon at recombination. Referances [46, 47] (see also [48–50]) present the computation of the full system of Boltzmann equations, describing the evolution of the photon, baryon and cold dark matter (CDM) fluids, at second order and neglecting polarization. These equations allow to follow the time evolution of the CMB anisotropies at second order on all angular scales from the early epochs, when the cosmological perturbations were generated, to the present time, through the recombination era. These calculations set the stage for the computation of the full second-order radiation transfer function at all scales and for a generic set of initial conditions specifying the level of primordial non-Gaussianity. Of course, for specific effects on small angular scales like Sunyaev-Zel'dovich, gravitational lensing, and so forth, fully nonlinear calculations would provide a more accurate estimate of the resulting CMB anisotropy, however, as long as the leading contribution to second-order statistics like the bispectrum is concerned, second-order perturbation theory suffices.
The goal of this paper is to summarize in a pedagogical form the present status of the evaluation of the impact of NG on the CMB anisotropies. This implies first of all determining how to set the initial conditions at second order for the CMB anisotropy when some source of primordial NG is present. The second step will be determining how primordial NG flows on small angular scales. In this paper we will focus on the study of the second-order effects appearing at the recombination era when the CMB anisotropy is left imprinted. We will show how to derive the equations to evaluate CMB anisotropies, by computing the Boltzmann equations describing the evolution of the baryon-photon fluid up to second order. This permits to follow the time evolution of CMB anisotropies (up to second order) on all scales, from the early epoch, when the cosmological perturbations were generated, to the present time, through the recombination era. We will also provide the reader with some simplified analytical computation to evaluate the contamination of the recombination secondary effects onto the detection of primordial NG. The formalism for a more refined numerical analysis is also displayed and results for some worked examples will be also reported. The paper is mainly based on a series of papers written by the authors along the past years on the subject (with various updates) and, as such, follows both a logic and a chronological order. It requires knowledge of the theory of cosmological perturbation at the linear level (which however we summarize in the Appendices). We have tried to write the different sections in a self-contained way. Nevertheless, we alert the reader that the level of complexity increases with the number of the sections.
The paper is organized as follows. In Section 2, we provide a simple, but illuminating example to show why we do expect some NG present in the CMB anisotropy regardless if there is or not some primordial NG. In Section 3, we provide the reader with the necessary tools to study the dynamics at second order in the gravity sector. In Section 4, we show how to set the initial conditions for the primordial NG, while in Section 5, we provide a gauge-invariant way to define the CMB temperature anisotropy at second order on large scales. In Section 6, we go to small scales and present the full procedure to compute the Boltzmann equations necessary to follow the evolution of the nonlinearities from the recombination epoch down to the present epoch. Section 7 presents some analytical solutions of the Boltzmann equations in the tight coupling limit, along the same lines of what is done at the linear level. The issue of contamination is addressed in Section 8, while in Section 9 we offer the reader with an analytical estimate of such a contamination. A more refined numerical work is presented in Section 10. Finally, in Section 11, some conclusions are given. This paper has also some hopefully useful appendices: in Appendix A the reader can find the energy-momentum tensors at second-order, Appendix B gives the solutions of Einstein equations for the perturbed fluids up to second-order, while Appendix C offers the analytical solutions of the linearized Boltzmann equations in the tight coupling limit.
2. Why Do We Expect NG in the Cosmological Perturbations?
Before tackling the problem of interest—the computation of the cosmological perturbations at second order after the inflationary era—we first provide a simple, but insightful computation, derived from , which illustrates why we expect that the cosmological perturbations develop some NG even if the latter is not present at some primordial epoch. This example will help the reader to understand why the cosmological perturbations are inevitably affected by nonlinearities, beyond those arising at some primordial epoch. The reason is clear: gravity is nonlinear and it feeds this property into the cosmological perturbations during the postinflationary evolution of the universe. As gravity talks to all fluids, this transmission is inevitable. To be specific, we focus on the CMB anisotropies. We will adopt the Poisson gauge which eliminates one scalar degree of freedom from the component of the metric and one scalar and two vector degrees of freedom from . We will use a metric of the form (see Table 1 for the symbols used in this paper) where is the scale factor as a function of the cosmic time , and and are the vector and tensor perturbation modes respectively. Each metric perturbation can be expanded into a linear (first-order) and a second-order part, as for example, the gravitational potential . However, in the metric (1) the choice of the exponentials greatly helps in computing the relevant expressions, and thus we will always keep them where it is convenient. From (1), one recovers at linear order the well-known longitudinal gauge while at second order, one finds and where , and , (with and ) are the first and second-order gravitational potentials in the longitudinal (Poisson) gauge adopted in [5, 51] as far as scalar perturbations are concerned.
We now consider the long wavelength modes of the CMB anisotropies, that is, we focus on scales larger than the horizon at last-scattering. We can therefore neglect vector and tensor perturbation modes in the metric. For the vector perturbations, the reason is that they contain gradient terms being produced as nonlinear combination of scalar-modes and thus they will be more important on small scales (linear vector modes are not generated in standard mechanisms for cosmological perturbations, as inflation). The tensor contribution can be neglected for two reasons. First, the tensor perturbations produced from inflation on large scales give a negligible contribution to the higher-order statistics of the Sachs-Wolfe effect being of the order of (powers of) the slow-roll parameters during inflation (this holds for linear tensor modes as well as for tensor modes generated by the nonlinear evolution of scalar perturbations during inflation). Second, while on large scales the tensor modes have been proven to remain constant in time , when they approach the horizon they have a wavelike contribution which oscillates in time with decreasing amplitude .
Since we are interested in the cosmological perturbations on large scales, that is in perturbations whose wavelength is larger than the Hubble radius at last scattering, a local observer would see them in the form of a classical—possibly time-dependent—(nearly zero-momentum) homogeneous and isotropic background. Therefore, it should be possible to perform a change of coordinates in such a way as to absorb the super-Hubble modes and work with a metric of an homogeneous and isotropic Universe (plus, of course, cosmological perturbations on scale smaller than the horizon). We split the gravitational potential as where stands for the part of the gravitational potential receiving contributions only from the super-Hubble modes; receives contributions only from the subhorizon modes where is the Hubble rate computed with respect to the cosmic time, , and is the step function. Analogous definitions hold for the other gravitational potential .
By construction, and are a collection of Fourier modes whose wavelengths are larger than the horizon length and we may safely neglect their spatial gradients. Therefore, and are only functions of time. This amounts to saying that we can absorb the large-scale perturbations in the metric (1) by the following redefinitions: The new metric describes a homogeneous and isotropic Universe where for simplicity we have not included the subhorizon modes. On superhorizon scales one can regard the Universe as a collection of regions of size of the Hubble radius evolving like unperturbed patches with metric (5) .
Let us now go back to the quantity we are interested in, namely the anisotropies of the CMB as measured today by an observer . If she/he is interested in the CMB anisotropies at large scales, the effect of super-Hubble modes is encoded in the metric (5). During their travel from the last scattering surface—to be considered as the emitter point —to the observer, the CMB photons suffer a redshift determined by the ratio of the emitted frequency to the observed one where and are the temperatures at the observer point and at the last scattering surface, respectively.
What is then the temperature anisotropy measured by the observer? (6) shows that the measured large-scale anisotropies are made of two contributions: the intrinsic inhomogeneities in the temperature at the last scattering surface and the inhomogeneities in the scaling factor provided by the ratio of the frequencies of the photons at the departure and arrival points. Let us first consider the second contribution. As the frequency of the photon is the inverse of a time period, we get immediately the fully nonlinear relation As for the temperature anisotropies coming from the intrinsic temperature fluctuation at the emission point, it may be worth to recall how to obtain this quantity in the longitudinal gauge at first order. By expanding the photon energy density , the intrinsic temperature anisotropies at last scattering are given by . One relates the photon energy density fluctuation to the gravitational perturbation first by implementing the adiabaticity condition , where is the relative fluctuation in the matter component, and then using the energy constraint of Einstein equations . The result is . Summing this contribution to the anisotropies coming from the redshift factor (7) expanded at first order provides the standard (linear) Sachs-Wolfe effect . Following the same steps, we may easily obtain its full nonlinear generalization.
Let us first relate the photon energy density to the energy density of the nonrelativistic matter by using the adiabaticity condition. Again, here a bar indicates that we are considering quantities in the locally homogeneous Universe described by the metric (5). Using the energy continuity equation on large scales , where and is the pressure of the fluid, one can easily show that there exists a conserved quantity in time at any order in perturbation theory  The perturbation is a gauge-invariant quantity representing the nonlinear extension of the curvature perturbation on uniform energy density hypersurfaces on superhorizon scales for adiabatic fluids . Indeed, expanding it at first and second order one gets the corresponding definition and the quantity introduced in . We will come back to these definitions later. At first order, the adiabaticity condition corresponds to set for the curvature perturbations relative to each component. At the nonlinear level the adiabaticity condition generalizes to or To make contact with the standard second-order result, we may expand in (10) the photon energy density perturbations as , and similarly for the matter component. We immediately recover the adiabaticity condition given in .
Next, we need to relate the photon energy density to the gravitational potentials at the nonlinear level. The energy constraint inferred from the component of Einstein equations in the matter-dominated era with the “barred” metric (5) is Using (4), the Hubble parameter reads where is the Hubble parameter in the “unbarred” metric. Equation (12) thus yields an expression for the energy density of the nonrelativistic matter which is fully nonlinear, being expressed in terms of the gravitational potential where we have dropped which is negligible on large scales. By perturbing, (14) we are able to recover in a straightforward way the solutions of the component of Einstein equations for a matter-dominated Universe in the large-scale limit obtained at second order in perturbation theory. Indeed, recalling that is perturbatively related to the quantity used in  by and , one immediately obtains The expression for the intrinsic temperature of the photons at the last scattering surface follows from (10) and (14) Plugging (7) and (16) into (6), we are finally able to provide the expression for the CMB temperature which is fully nonlinear and takes into account both the gravitational redshift of the photons due to the metric perturbations at last scattering and the intrinsic temperature anisotropies From (17), we read the nonperturbative anisotropy corresponding to the Sachs-Wolfe effect Equation (18) is one of the main results of this paper and represents at any order in perturbation theory the extension of the linear Sachs-Wolfe effect. At first order, one gets and at second order This result shows that the CMB anisotropies is nonlinear on large scales and that a source of NG is inevitably sourced by gravity.
3. Perturbing Gravity
In this Section, we provide the necessary tools to deal with perturbed gravity, giving the expressions for the Einstein tensor perturbed up to second-order around a flat Friedmann-Robertson-Walker background, and the relevant Einstein equations. In the following, we will adopt the Poisson gauge which eliminates one scalar degree of freedom from the component of the metric and one scalar and two vector degrees of freedom from . We rewrite the metric (1) as where is the scale factor as a function of the conformal time . As we previously mentioned, for the vector and tensor perturbations, we will neglect linear vector modes since they are not produced in standard mechanisms for the generation of cosmological perturbations (as inflation), and we also neglect tensor modes at linear order, since they give a negligible contribution to second-order perturbations. Therefore, we take and to be second-order vector and tensor perturbations of the metric.
Let us now give our definitions for the connection coefficients and their expressions for the metric (1). The number of spatial dimensions is . Greek indices () run from 0 to 3, while Latin indices () run from 1 to 3. The total spacetime metric has signature (). The connection coefficients are defined as The Riemann tensor is defined as
The Ricci tensor is a contraction of the Riemann tensor and in terms of the connection coefficient it is given by The Ricci scalar is given by contracting the Ricci tensor The Einstein tensor is defined as
3.1. The Connection Coefficients
For the connection coefficients we find
3.2. The Einstein Equations
The Einstein equations are written as , so that , where is the usual Newtonian gravitational constant. They read Taking the traceless part of (31), we find where is defined by , with , and . The trace of (31) gives From (30), we may deduce an equation for Here, is the energy momentum tensor accounting for different components, photons, baryons, and dark matter. We will give the expressions later for each component in terms of the distribution functions.
4. Setting the Initial Conditions from the Primordial NG
In this paper we are concerned with the second-order evolution of the cosmological perturbations. This requires that we well define the initial conditions of the cosmological perturbations at second order. These boundary conditions may or may not contain already some level of NG. It they do, we say that there exist some primordial NG. The latter is usually defined in the epoch in which the comoving curvature perturbation remains constant on large superhorizon scales. In the standard single-field inflationary model, the first seeds of density fluctuations are generated on superhorizon scales from the fluctuations of a scalar field, the inflaton .
In order to follow the evolution on superhorizon scales of the density fluctuations coming from the various mechanisms, we use the curvature perturbation of uniform density hypersurfaces , where and the expression for is given by  with The crucial point is that the gauge-invariant curvature perturbation remains constant on superhorizon scales after it has been generated during a primordial epoch and possible isocurvature perturbations are no longer present. Therefore, we may set the initial conditions at the time when becomes constant. In particular, provides the necessary information about the “primordial” level of non-Gaussianity generated either during inflation, as in the standard scenario, or immediately after it, as in the curvaton scenario. Different scenarios are characterized by different values of . For example, in the standard single-field inflationary model [6, 7], where and are the standard slow-roll parameters . In general, we may parametrize the primordial non-Gaussianity level in terms of the conserved curvature perturbation as in  where the parameter depends on the physics of a given scenario. For example in the standard scenario , while in the curvaton case , where is the relative curvaton contribution to the total energy density at curvaton decay [5, 55]. Alternatives to the curvaton model are those models characterized by the curvature perturbation being generated by an inhomogeneity in the decay rate [11–13] or the mass  or of the particles responsible for the reheating after inflation. Other opportunities for generating the curvature perturbation occur at the end of inflation [16–18] and during preheating [19–21]. All these models generate a level of NG which is local as the NG part of the primordial curvature perturbation is a local function of the Gaussian part, being generated on superhorizon scales. In momentum space, the three point function, or bispectrum, arising from the local NG is dominated by the so-called “squeezed” configuration, where one of the momenta is much smaller than the other two. Other models, such as DBI inflation  and ghost inflation , predict a different kind of primordial NG, called “equilateral”, because the three-point function for this kind of NG is peaked on equilateral configurations, in which the length of the three wavevectors forming a triangle in Fourier space are equal .
One of the best tools to detect or constrain the primordial large-scale non-Gaussianity is through the analysis of the CMB anisotropies, for example by studying the bispectrum . In that case, the standard procedure is to introduce the primordial nonlinearity parameter characterizing the primordial non-Gaussianity via the curvature perturbation  (the sign convention is such that it follows the WMAP convention for , see Section 5) where the coefficient 3/5 arises from the first-order relation connecting the comoving curvature perturbation and the gravitational potential, , and the -product reminds us that one has to perform a convolution product in momentum space and that is indeed momentum-dependent (we refer the reader to the end of Section 5 for a detailed discussion about the definition of the nonlinearity parameter). To give the feeling of the resulting size of when , (see [5, 34]). Present limits on NG are summarized by  and  at CL, where and stand for the nonlinear parameter in the case in which the squeezed and the equilateral configurations dominate, respectively.
5. CMB Anisotropies at Second-Order on Large Scales
In this section, we provide the exact expression for large-scale CMB temperature fluctuations at second order in perturbation theory. What this section contains, therefore, should be considered as a more technical elaboration of what the reader can find in Section 2. The final expression, we will find has various virtues. First, from it one can unambiguously extract the exact definition of the nonlinearity parameter which is used by the experimental collaborations to pin down the level of NG in the temperature fluctuations. Second, it contains a “primordial” term encoding all the information about the NG generated in primordial epochs, namely during or immediately after inflation, and depends upon the various fluctuation generation mechanisms. As such, the expression neatly disentangles the primordial contribution to the NG from that arising after inflation. Finally, the expression applies to all scenarios for the generation of cosmological perturbations. Third, it is gauge-invariant. For this last point, let us underline however that the gauge-invariant expression that we will provide is just one of the many possible ways in which the CMB temperature fluctuations can be casted in a gauge-invariant form.
In order to obtain our gauge-independent formula for the temperature anisotropies, we again perturb the spatially flat Robertson-Walker background. We expand the metric perturbations (21) in a first and a second-order part as Again, the functions and , where , stand for the th-order perturbations of the metric. We can split , where is the scalar part and is a transverse vector, that is, . The metric perturbations will transform according to an infinitesimal change of coordinates. From now on, we limit ourselves to a second-order time shift  where a prime denotes differentiation w.r.t. conformal time. In general, a gauge corresponds to a choice of coordinates defining a slicing of spacetime into hypersurfaces (at fixed time ) and a threading into lines (corresponding to fixed spatial coordinates ), but in this section only the former is relevant so that gauge-invariant can be taken to mean independent of the slicing. For example, under the time shift, the first-order spatial curvature perturbation transforms as (here ), while , , and the traceless part of the spatial metric turns out to be gauge-invariant. At second order in the perturbations, we just give some useful examples like the transformation of the energy density and the curvature perturbation . (For these two examples, we here report only the expression of the second-order gauge transformations neglecting gradient terms on large-scales. Notice that the expressions of this section relative to linear perturbations are valid on all scales, while for the second-order fluctuations at some point the large-scale limit is considered.) We now construct in a gauge-invariant way temperature anisotropies at second order. Temperature anisotropies beyond the linear regime have been calculated in [32, 51], following the photons path from last scattering to the observer in terms of perturbed geodesics. The linear temperature anisotropies read [32, 51] where , the subscript indicates that quantities are evaluated at last-scattering, is a spatial unit vector specifying the direction of observation and the integral is evaluated along the line-of-sight parametrized by the affine parameter . Equation (43) includes the intrinsic fractional temperature fluctuation at emission , the Doppler effect due to emitter's velocity and the gravitational redshift of photons, including the integrated Sachs-Wolfe (ISW) effect. We omitted monopoles due to the observer (e.g., the gravitational potential evaluated at the event of observation), which, being independent of the angular coordinate, can be always recast into the definition of temperature anisotropies. Notice, however, that the physical meaning of each contribution in (43) is not gauge-invariant, as the different terms are gauge-dependent. However, it is easy to show that the whole expression (43) is gauge-invariant. Since the temperature is a scalar, the intrinsic temperature fluctuation transforms as , having used the fact that the temperature scales as . Notice, instead, that the velocity does not change. Therefore, using the transformations of metric perturbations we find where we have used the fact that the integral is evaluated along the line-of-sight which can be parametrized by the background geodesics (with ), and the decomposition for the total derivative along the path for a generic function , . Equation (44) shows that the expression (43) for first-order temperature anisotropies is indeed gauge-invariant (up to monopole terms related to the observer ). Temperature anisotropies can be easily written in terms of particular combinations of perturbations which are manifestly gauge-invariant. For the gravitational potentials, we consider the gauge-invariant definitions and . For the component of the metric and the traceless part of the spatial metric, we define and . For the matter variables we use a gauge-invariant intrinsic temperature fluctuation , while the velocity itself is gauge-invariant under time shifts. Following the same steps leading to (44), one gets the linear temperature anisotropies in (43) in terms of these gauge-invariant quantities where and we omitted the subscript . For the primordial fluctuations we are interested in the large-scale modes set by the curvature perturbation . Defining a gauge-invariant density perturbation , we write the curvature perturbation as . Since for adiabatic perturbations in the radiation () and matter () eras , one can write the intrinsic temperature fluctuation as and a gauge-invariant definition is . In the large-scale limit, from Einstein equations () and continuity equations, in the matter era . Thus we obtain the large-scale limit of temperature anisotropies (45) , that is, the usual Sachs-Wolfe effect.
At second order, the procedure is similar to the one described so long, though more lengthy and cumbersome. We only provide the reader with the main steps to get the final expression. The second-order temperature fluctuations in terms of metric perturbations read [32, 59] Here, is the second-order ISW  where , while and are the photon wave vectors, with given by the integral in (43) and is obtained from the same integral replacing the time derivative with a spatial gradient. Finally, in (46) are the geodesics at first order, and is the direction of the photon emission evaluated on the hypersurface of constant time of emission . As usual we have omitted the monopole terms due to the observer. Using the transformation rules of , it is possible to check that the expression (46) is gauge-invariant. We can express the second-order anisotropies in terms of explicitly gauge-invariant quantities, whose definition proceeds as for the linear case, by choosing the shifts such that . For example, we consider the gauge-invariant gravitational potential  where = + + . Expressing the second-order temperature anisotropies (46) in terms of our gauge-invariant quantities and taking the large-scale limit, we find (having dropped the subscript ), and the gauge-invariant intrinsic temperature fluctuation at emission is We have dropped those terms which represent integrated contributions and other second-order small-scale effects that can be distinguished from the large-scale part through their peculiar scale dependence. At this point, we make use of Einstein's equations. We take the expression for in (36) and (37), and we use the component and the traceless part of the Einstein's equations (29) and (31) after having appropriately expanded the exponentials. Thus, on large scales we find that the temperature anisotropies are given by where we have defined a kernel Notice that the factor nicely matches the corresponding term in (20). Equation (52) is the main result of this section. It clearly shows that there are two contributions to the final nonlinearity in the large-scale temperature anisotropies. The contribution, , comes from the “primordial” conditions set during or after inflation. They are encoded in the curvature perturbation which remains constant once it has been generated. The remaining part of (52) describes the postinflation processing of the primordial non-Gaussian signal due to the nonlinear gravitational dynamics, including also second-order corrections at last scattering to the Sachs-Wolfe effect [32, 51, 59]. Thus, (52) allows to neatly disentangle the primordial contribution to NG from that coming from that arising after inflation, and from it we can extract the nonlinearity parameter , see the expression (39) which is usually adopted to phenomenologically parametrize the NG level of cosmological perturbations and has become the standard quantity to be observationally The definition of adopted in the analyses performed in  goes through the conventional Sachs-Wolfe formula where is Bardeen's potential, which is conventionally expanded as (up to a constant offset, which only affects the temperature monopole) . Therefore, using during matter domination, from (52) we read the nonlinearity parameter in momentum space where with and . To obtain (54) we have made use of the expression (38) to set the initial conditions. In particular, within the standard scenario where cosmological perturbations are due to the inflaton the primordial contribution to NG is given by [6, 7], where the spectral index is expressed in terms of the usual slow-roll parameters as . The nonlinearity parameter from inflation now reads Therefore, the main NG contribution comes from the postinflation evolution of the second-order perturbations which give rise to order-one coefficients, while the primordial contribution is proportional to . This is true even in the “squeezed” limit first discussed by Maldacena in , where one of the wavenumbers is much smaller than the other two, for example and .
6. CMB Anisotropies at Second-Order at All Scales
As we already mentioned in the Introduction, despite the importance of NG in CMB anisotropies, little effort has been made so far to provide accurate theoretical predictions of it. On the contrary, the vast majority of the literature has been devoted to the computation of the bispectrum of either the comovig curvature perturbation or the gravitational potential on large scales within given inflationary models. These, however, are not the physical quantities which are observed. One should instead provide a full prediction for the second-order radiation transfer function. A preliminary step towards this goal has been taken in [31, 32, 36, 59] (see also ) where the full second-order radiation transfer function for the CMB anisotropies on large angular scales in a flat universe filled with matter and cosmological constant was computed, including the second-order generalization of the Sachs-Wolfe effect, both the early and late integrated Sachs-Wolfe (ISW) effects and the contribution of the second-order tensor modes. We have partly reported about these works in the previous sections. (For some recent papers focusing on the generation and evolution of tensor perturbations at second-order see, e.g., [60–62].)
In this section we wish to offer a summary of some of the second-order effects in the CMB anisotropies on small scales. There are many sources of NG in CMB anisotropies, beyond the primordial one. The most relevant sources are the so-called secondary anisotropies, which arise after the last scattering epoch. These anisotropies can be divided into two categories: scattering secondaries, when the CMB photons scatter with electrons along the line of sight, and gravitational secondaries when effects are mediated by gravity . Among the scattering secondaries, we may list the thermal Sunyaev-Zeldovich effect, where hot electrons in clusters transfer energy to the CMB photons, the kinetic Sunyaev-Zeldovich effect produced by the bulk motion of the electrons in clusters, the Ostriker-Vishniac effect, produced by bulk motions modulated by linear density perturbations, and effects due to reionization processes. The scattering secondaries are most significant on small angular scales as density inhomogeneities, bulk and thermal motions grow and become sizeable on small length-scales when structure formation proceeds.
Gravitational secondaries arise from the change in energy of photons when the gravitational potential is time-dependent, the ISW effect, and gravitational lensing. At late times, when the Universe becomes dominated by the dark energy, the gravitational potential on linear scales starts to decay, causing the ISW effect mainly on large angular scales. Other secondaries that result from a time dependent potential are the Rees-Sciama effect, produced during the matter-dominated epoch by the time evolution of the potential on nonlinear scales.
The fact that the potential never grows appreciably means that most second-order effects created by gravitational secondaries are generically small compared to those created by scattering ones. However, when a photon propagates from the last scattering to us, its path may be deflected because of the gravitational lensing. This effect does not create anisotropies, but only modifies existing ones. Since photons with large wavenumbers are lensed over many regions (, where is the Hubble rate) along the line of sight, the corresponding second-order effect may be sizeable. The three-point function arising from the correlation of the gravitational lensing and ISW effects generated by the matter distribution along the line of sight [40, 41] and the Sunyaev-Zeldovich effect  are large and detectable by Planck .
Another relevant source of NG comes from the physics operating at the recombination. A naive estimate would tell that these nonlinearities are tiny being suppressed by an extra power of the gravitational potential. However, the dynamics at recombination is quite involved because all the nonlinearities in the evolution of the baryon-photon fluid at recombination and the ones coming from general relativity should be accounted for. This complicated dynamics might lead to unexpected suppressions or enhancements of the NG at recombination. A step towards the evaluation of the three-point correlation function has been taken in  where some effects were taken into account in the in so-called squeezed triangle limit, corresponding to the case when one wavenumber is much smaller than the other two and was outside the horizon at recombination (see however also  for a critical reassessment of some of the results contained in . In particular notice that, contrary to what stated in , in the result (54), first obtained in , the contribution is actually present and survives in the squeezed limit).
This section, which is based on [46, 47], presents the computation of the full system of Boltzmann equations, describing the evolution of the photon, baryon, and cold dark matter (CDM) fluids, at second order and neglecting polarization, These equations allow to follow the time evolution of the CMB anisotropies at second order on all angular scales from the early epochs, when the cosmological perturbations were generated, to the present time, through the recombination era. These calculations set the stage for the computation of the full second-order radiation transfer function at all scales and for a a generic set of initial conditions specifying the level of primordial non-Gaussianity. Of course, on small angular scales, fully nonlinear calculations of specific effects like Sunyaev-Zel'dovich, gravitational lensing, and so forth would provide a more accurate estimate of the resulting CMB anisotropy, however, as long as the leading contribution to second-order statistics like the bispectrum is concerned, second-order perturbation theory suffices.
6.1. The Collisionless Boltzmann Equation for Photons
We are interested in the anisotropies in the cosmic distribution of photons and inhomogeneities in the matter. Photons are affected by gravity and by Compton scattering with free electrons. The latter are tightly coupled to protons. Both are, of course, affected by gravity. The metric which determines the gravitational forces is influenced by all these components plus CDM (and neutrinos). Our plan is to write down Boltzmann equations for the phase-space distributions of each species in the Universe.
The phase-space distribution of particles is a function of spatial coordinates , conformal time , and momentum of the particle = where parametrizes the particle path. Through the constraint , where is the mass of the particle one can eliminate and gives the number of particles in the differential phase-space volume . In the following, we will denote the distribution function for photons with .
The photons' distribution evolves according to the Boltzmann equation where the collision term is due to the scattering of photons off free electrons. In the following, we will derive the left-hand side of (57) while in the next section, we will compute the collision term.
For photons we can impose and using the metric (1) in the conformal time we find where we define From the constraint (58) Notice that we immediately recover the usual zero and first-order relations and .
The components are proportional to , where is a unit vector with . We can write , where is determined by so that where the last equality holds up to second order in the perturbations. Again, we recover the zero and first-order relations and respectively. Thus, up to second order we can write Equation (62) and (63) allow us to replace and in terms of the variables and . Therefore, as it is standard in the literature, from now on we will consider the phase-space distribution as a function of the momentum with magnitude and angular direction , . Notice that our are expressed via quantities, and , which are different from the ones of . However, the final Boltzmann equations that we obtain agree with the ones of . The reason is that the differences in the evolution for emerge only at orders greater than two in the perturbative expansion in the fluctuations (at least if one neglects first-order vector and tensor perturbation modes, as it is done in these computations), while the evolution equation for at linear-order (which is the one needed here, see below) is the same.
Thus, in terms of these variables, the total time derivative of the distribution function reads In the following, we will compute , and .
(b) : For , we make use of the time component of the geodesic equation = , where = = , and Using the metric (1), we find On the other hand (63) of in terms of and gives Thus, (67) allows us express as where in (68) we have replaced and by (63) and (62). Notice that in order to obtain (70), we have used the following expressions for the total time derivatives of the metric perturbations where we have taken into account that is already a second-order perturbation so that we can neglect which is at least a first order quantity, and we can take the zero-order expression in (66), . In fact, there is also an alternative expression for which turns out to be useful later and which can be obtained by applying once more (71)
(c) : We can proceed in a similar way to compute . Notice that since in (64) it multiplies which is first order, we need only the first order perturbation of . We use the spatial components of the geodesic equations written as For the right-hand side, we find, up to second order, while the expression (62) of in terms of our variables and in the left-hand side of (74) brings Thus, using the expression (62) for and (60) for in (75), together with the previous result (70), the geodesic equation (74) gives the following expression (valid up to first order)
To proceed further we now expand the distribution function for photons around the zero-order value which is that of a Bose-Einstein distribution where is the average (zero-order) temperature and the factor comes from the spin degrees of photons. The perturbed distribution of photons will depend also on and on the propagation direction so as to account for inhomogeneities and anisotropies where we split the perturbation of the distribution function into a first and a second-order part. The Boltzmann equation up to second order can be written in a straightforward way by recalling that the total time derivative of a given th perturbation, as for example, is at least a quantity of the -th order. Thus, it is easy to realize, looking at (64), that the left-hand side of Boltzmann equation can be written up to second order as where for simplicity in (80), we have already used the background Boltzmann equation . In (80) there are all the terms which will give rise to the integrated Sachs-Wolfe effects (corresponding to the terms which explicitly depend on the gravitational perturbations), while other effects, such as the gravitational lensing, are still hidden in the (second-order part) of the first term. In fact in order to obtain (80) we just need for the time being to know the expression for , (73).
6.2. Collision Term
6.2.1. The Collision Integral
In this section, we focus on the collision term due to Compton scattering (notice that in this section all the quantities and their indices are meant to be defined in the local Minkowski frame) where we have indicated the momentum of the photons and electrons involved in the collisions. The collision term will be important for small scale anisotropies and spectral distortions. The important point to compute the collision term is that for the epoch of interest very little energy is transferred. Therefore, one can proceed by expanding the right hand side of (57) both in the small perturbation, (79), and in the small energy transfer.
The collision term is given (up to second order) by where is the scale factor and where , is the amplitude of the scattering process, ensures the energy-momentum conservation and is the distribution function for electrons. The Pauli suppression factors have been dropped since for the epoch of interest the density of electrons is low. The reason why we write the collision term as in (82) is that the starting point of the Boltzmann equation requires differentiation with respect to an affine parameter , . In moving to the conformal time , one rewrites the Boltzmann equation as , with given by (63). Taking into account that the collision term is at least of first order, (82) then follows. The electrons are kept in thermal equilibrium by Coulomb interactions with protons and they are nonrelativistic, thus we can take a Maxwell-Boltzmann distribution around some bulk velocity By using the three dimensional delta function the energy transfer is given by and it turns out to be small compared to the typical thermal energies In (85), we have used and the fact that, since the scattering is almost elastic (), is of order , with much bigger than . In general, the electron momentum has two contributions, the bulk velocity () and the thermal motion () and thus the parameter expansion includes the small bulk velocity and the ratio which is small because the electrons are nonrelativistic.
The expansion of all the quantities entering the collision term in the energy transfer parameter and the integration over the momenta and is described in details in . It is easy to realize that we just need the scattering amplitude up to first order since at zero-order and so that all the zero-order quantities multiplying vanish. To first order where is the scattering angle and the Thompson cross-section. The resulting collision term up to second order is given by  where we arrange the different contributions following . The first order term reads while the second-order terms have been separated into four parts. There is the so-called anisotropy suppression term a term which depends on the second-order velocity perturbation defined by the expansion of the bulk flow as a set of terms coupling the photon perturbation to the velocity and a set of source terms quadratic in the velocity The last contribution are the Kompaneets terms describing spectral distortions to the CMB Let us make a couple of comments about the various contributions to the collision term. First, notice the term due to second-order perturbations in the velocity of electrons which is absent in . In standard cosmological scenarios (like inflation), vector perturbations are not generated at linear order, so that linear velocities are irrotational . However, at second order vector perturbations are generated after horizon crossing as nonlinear combinations of primordial scalar modes. Thus, we must take into account also a transverse (divergence-free) component, with . As we will see, such vector perturbations will break azimuthal symmetry of the collision term with respect to a given mode , which instead usually holds at linear order. Secondly, notice that the number density of electrons appearing in (87) must be expanded as and then gives rise to second-order contributions in addition to the list above, where we split into a first- and second-order part. In particular, the combination with the term proportional to in gives rise to the so-called Vishniac effect, as discussed in .
6.2.2. Computation of Different Contributions to the Collision Term
In the integral (87) over the momentum the first-order term gives the usual collision term where one uses the decomposition in Legendre polynomials where is the polar angle of , .
In the following, we compute the second-order collision term separately for the different contributions, using the notation . We have not reported the details of the calculation of the first-order term because for its second-order analog, , the procedure is the same. The important difference is that the second-order velocity term includes a vector part, and this leads to a generic angular decomposition of the distribution function (for simplicity drop the time dependence) such that Such a decomposition holds also in Fourier space. The notation at this stage is a bit confusing, so let us restate it: superscripts denote the order of the perturbation; the subscripts refer to the moments of the distribution. Indeed, at first order, one can drop the dependence on setting using the fact that the distribution function does not depend on the azimuthal angle . In this case, the relation with is
The integral over yields To perform the angular integral we write the angular dependence on the scattering angle in terms of the Legendre polynomials where in the last step we used the addition theorem for spherical harmonics Using the decomposition (98) and the orthonormality of the spherical harmonics, we find It is easy to recover the result for the corresponding first-order contribution in (229) by using (205).
Let us fix for simplicity our coordinates such that the polar angle of is defined by with the corresponding azimuthal angle. The contribution of to the collision term is then We can use (101) which in our coordinate system reads so that By using the orthonormality of the Legendre polynomials and integrating by parts over , we find As it is clear by the presence of the scalar product , the final result is independent of the coordinates chosen.
let us consider the contribution from the first term where the velocity has to be considered at first order. In the integral (87), it brings