Non-linear relativistic contributions to the cosmological weak-lensing convergence
Relativistic contributions to the dynamics of structure formation come in a variety of forms, and can potentially give corrections to the standard picture on typical scales of 100 Mpc. These corrections cannot be obtained by Newtonian numerical simulations, so it is important to accurately estimate the magnitude of these relativistic effects. Density fluctuations couple to produce a background of gravitational waves, which is larger than any primordial background. A similar interaction produces a much larger spectrum of vector modes which represent the frame-dragging rotation of spacetime. These can change the metric at the percent level in the concordance model at scales below the equality scale. Vector modes modify the lensing of background galaxies by large-scale structure. This gives in principle the exciting possibility of measuring relativistic frame dragging effects on cosmological scales. The effects of the non-linear tensor and vector modes on the cosmic convergence are computed and compared to first-order lensing contributions from density fluctuations, Doppler lensing, and smaller Sachs-Wolfe effects. The lensing from gravitational waves is negligible so we concentrate on the vector modes. We show the relative importance of this for future surveys such as Euclid and SKA. We find that these non-linear effects only marginally affect the overall weak lensing signal so they can safely be neglected in most analyses, though are still much larger than the linear Sachs-Wolfe terms. The second-order vector contribution can dominate the first-order Doppler lensing term at moderate redshifts and are actually more important for survey geometries like the SKA.
Relativistic corrections to the standard model of cosmology come in a variety of forms, from the altering the dynamics of structure formation to the various effects associated to the interpretation of observations, in particular modifying the propagation of light.
There has been considerable debate as to the importance and amplitude of these effects on the dynamics of the expansion of the universe and the growth of large scale structure (see, e.g., Ref. Clarkson:2011zq () for an overview), and the amplitude and importance of these dynamical effects are still actively debated buchertetal (); nobackreac (); backsig (). Though subdominant for linear structure formation, relativistic corrections are a generic prediction of General Relativity and are inevitable at a non-linear level through mode-mode coupling. The scalar gravitational potential induces rotational frame-dragging modes in spacetime (so-called vector modes) as well as gravitational waves (tensor modes). Neither of these have counterparts in Newtonian gravity as they both induce a non-zero magnetic Weyl curvature which is absent in Newtonian gravity and difficult to take into account in N-body numerical simulations Bruni:2013mua (); Adamek:2013wja (). They therefore serve as an important tool in understanding purely relativistic aspects of structure formation and its observational consequences, as they set a lower limit on the amplitude of relativistic corrections.
On top of dynamical corrections, relativistic effects also induce corrections to the propagation of light since it probes the complete spacetime geometry. This can alter the interpretation of cosmological observations at a level that cannot be neglected in an era of “precision cosmology”. Provided one works within perturbation theory, the amplitude of these effects is computable and completely fixed once the normalisation of the scalar power spectrum, at the linear level, is determined. For instance, some relativistic effects have been taken into account on the cosmic microwave background pub () and shown to be below the constraints on non-Gaussianity derived by Planck planckXXIX (), but nevertheless in principle detectable on small angular scales, in particular through spectral distortions spectral (). This article focuses on the effect of relativistic corrections on weak lensing observations, focusing mainly on the induced vector mode background. Weak gravitational lensing by the large-scale structure of the Universe has now become a major tool of cosmology revuesWL (), used to study questions ranging from the distribution of dark matter to tests of general relativity testGR ().
The propagation of light in an inhomogeneous universe gives rise to both distortion and magnification induced by gravitational lensing. The effect of non-linear corrections on the Hubble diagram have been considered BenDayan:2012wi (); cemuu (); Umeh:2012pn (); Umeh:2014ana () and shown to be non-negligible given the accuracy of contemporary observations Fleury:2013sna (); Fleury:2013uqa (); voidslensing (); BenDayan:2012ct (); BenDayan:2013gc (). In this article we consider the effect on the weak lensing convergence of non-linear effects that induce the existence of a vector and tensor modes background. We compare this to the various contributions to the convergence at first-order – the usual integral of the density contrast along the line of sight revuesWL (), the contribution from the Doppler effect which is dominant at low redshifts and large scales Bonvin:2008ni (); voidslensing (); Bacon:2014uja (), the Integrated Sachs-Wolfe (ISW) and Sachs-Wolfe (SW) terms which are relatively small and mainly neglected when computing cosmic convergence.
The induced background of gravitational waves from scalar-scalar coupling was presented in Ref. Ananda:2006af () during the radiation era, and its present-day spectrum calculated in Ref. Baumann:2007zm (), with shear lensing effects studied in Ref. Sarkar:2008ii (), all following the pioneering analysis of Ref. Mollerach:2003nq (). Surprisingly it was found that the induced gravitational wave background is significantly larger than any primordial background (even for a tensor-scalar ratio ) on intermediate scales of 100 Mpc, which is around the equality scale, though of course it is much smaller on small scales. Similarly, the induced vector mode background was presented in Refs. Lu:2007cj (); Lu:2008ju (), and again a spectrum was found that peaks on 100 Mpc scales. Remarkably, however, it was found that the amplitude of the background of vector modes for the metric potential behaves on small scales with the same scaling as the gravitational potential, with nearly 1% of its amplitude. While both of these induced degrees of freedom have little effect on the dynamics of structure formation (they cannot directly source the density fluctuation as it is a scalar degree of freedom) they can influence the gravitational lensing produced by large-scale structure. Is it significant, and could it be a new way to detect relativistic aspects of structure formation?
The effects of these contributions on weak lensing convergence predictions are computed in order to understand if they can either be detected or, in the worst case, bias the analysis of future weak lensing experiments, such as Euclid or SKA; i.e., if the interpretation of the observation by assuming that the observed convergence corresponds to the convergence sourced by scalar modes only is an accurate enough assumption or whether some of these effects have to be included in the analysis. This article addresses this question and computes the effect of these two non-linear effects on weak lensing observations by considering second order vector and tensor background. We restrict our analysis to the hypothesis that the Born approximation still holds. In principle, one needs also to take into account second order effects on the geodesic deviation equation Seitz:1994xf (); Cooray:2002mj (); Dodelson:2005zj (); Schaefer:2005up (); Shapiro:2006em (), as fully described in Refs. Bernardeau:2009bm (); Bernardeau:2011tc (). This effect is however small.
In Section II we describe the vector and gravity waves background induced by the non-linear dynamics and then, in Section III, the computation of the weak lensing power spectra, splitting the effects of the scalar, vector, tensor, Doppler, ISW and SW contributions in order to compare their magnitude. Since the contribution of the tensor modes remains negligible and both ISW and SW being relatively small, we focus in Section IV on the vector and Doppler contribution, estimating their magnitude in surveys such as Euclid and SKA. Technical details are gathered in Appendices A-F.
Ii Induced vector and gravitational wave backgrounds
Let us start by briefly reviewing the vector and gravitational wave backgrounds induced by structure formation. In the standard cosmological framework, the initial conditions set by inflation imposes that at the linear order only scalar perturbations, described in § II.1, are significantly sourced. At second order, one cannot neglect the contributions from vector and tensor modes, that are respectively described in § II.2 and II.3.
We shall work in the Poisson (or Newtonian) gauge in which the metric can be expanded as
where is the scale factor, the conformal time and is the spatial metric of the background. Latin indices run from 1 to 3. The scalar, vector and tensor perturbations are respectively described by and , and where is transverse () and is transverse and traceless ( and ) where is the covariant derivative associated with .
ii.1 First order scalar perturbations
At late times, we can neglect the anisotropic stress of matter (mostly described by a pressure-free fluid on cosmological scales) and the spatial curvature (so that we assume that the spatial sections are Euclidean).
It follows that the Einstein equations imply (from the traceless part of the Einstein equations) (see e.g., Ref. pubook () for a derivation of the following equations). The peculiar velocity sourced by first order scalars is given in terms of the potential, from the component of the Einstein equations, as
where the conformal Hubble rate is defined as , a prime denoting a derivative with respect to . It is related to the Hubble rate by . In a CDM model in which the late time dynamics is dominated by a pure cosmological constant and dark matter, it is given by
where and are the matter and cosmological constant density parameters evaluated today.
The matter density contrast can be obtained from the relativistic Poisson equation, that derives from the Einstein equations. It involves the scalar component of the peculiar velocity ()
where is the 3 dimensional Laplacian. The evolution of the gravitational potential is then obtained from the spatial trace of the Einstein equations, combined with Eq. (4), to give
as long as the anisotropic stress can be neglected. is the sound speed. For a pressureless fluid, such as matter on cosmological scales, so that the solution of this equation can be factorized as . (or equivalently in Fourier space) describes the initial conditions. The growth suppression factor is determined from
where Eq. (3) has been used to evaluate the third term. describes the growth of the gravitational potential after decoupling. In general, one uses the linearity of the perturbation equations to decompose the gravitational potential in terms of a transfer function as in Fourier space, defining the Fourier modes by
It follows that the scalar power spectrum is defined as
where stands for the Dirac distribution.
The power spectrum today can be related to the initial power spectrum predicted from inflation. Assuming scale invariance (which is a good approximation for our analysis since secondary modes are quite insensitive to the spectral index), the inflationary power spectrum is characterized by its primordial power , typically of order at a scale wmap5 (). It follows that
where is chosen so that . In the following, we shall use the transfer function derived in Ref. Eisenstein:1997ik () to model the linear transfer function, and we also use Halofit Smith:2002dz () to estimate nonlinear small scale effects. Due to non-linear evolution, the growth suppression factor becomes scale dependent as
We then use this growth suppression factor to account for the non-linearities. Since non-linear evolution occurs at small scales (large ), behaves as the linear which is independent on large scales ( small). and are the non-linear matter power spectrum and today’s linear matter power spectrum respectively.
ii.2 Second order vector contribution
At second order, vector modes are sourced from the mode coupling of order 1 scalar modes, . Assuming Euclidean spatial sections, the second order Einstein equations in Newtonian gauge O2cp (); Lu:2007cj () lead to the second order vector contribution
where denotes the vector contribution of the part inside the braces. The Fourier transform of the vector perturbation encodes the two orthogonal polarisations and is defined as
where the two vectors realize an orthonormal basis orthogonal to (i.e., , ). The power in each polarisation is, thanks to spatial isotropy, the same and is defined in the same way as for the scalars for each polarisation. During the matter dominated era the vector contribution grows as which is the reason why it is not completely negligible today Lu:2007cj (); Lu:2008ju (). Their contribution peaks in power at the equality scale, and has the same spectrum as below this scale, but with 1% of the amplitude Lu:2008ju (). The vector mode power spectrum we shall use in our analysis can be parameterized Lu:2008ju () as
governs the growth of the vector power spectrum, and is a convolution integral of order unity (see Eq. (C7) of Ref. Lu:2008ju () for its explicit expression). The amplitude of the vectors decays on scales smaller than the equality scale, , with the same scaling as . Assuming cosmological parameters as determined by Ref. wmap5 (), the power in the vector modes is well approximated by Lu:2008ju ()
so that the amplitude of the metric vector perturbations is nearly 1% that of the metric scalar modes on small scales. In general, for a model without baryons, for . On large scales, scales like , with a peak in the spectrum around the equality scale.
Note that these vector degrees of freedom are not associated with the vorticity of the fluid and have no Newtonian counterpart as they induce a non-zero magnetic Weyl curvature. The small-scale behaviour of the second order vector modes can be estimated by replacing the linear transfer function with that given by Halofit (9), which is depicted on Fig. 1. This gives a more realistic estimation of the relativistic vector modes on small scales.
ii.3 Second order tensor contribution
The second order tensor modes evolve according to
where the effective anisotropic stress arises from the contribution of non-linear scalar modes and is explicitely given by
where denotes a tensor projection Ananda:2006af ().
In Fourier space, has 2 independent degrees of freedom that can be decomposed as polarisations as
where is the polarisation tensor, satisfying and .
where for a scale-invariant spectrum. stands for the spherical Bessel function and Mpc.
The second order gravitational wave background also peaks in power around the equality scale, and is surprisingly larger than its primordial background on these scales. The formula presented in Eq. (18), from Ref. Mollerach:2003nq (), predicts an excess in power on small scales compared to the more accurate formula of Ref. Baumann:2007zm (), but is sufficiently accurate for our purposes (see Ref. Sarkar:2008ii () for a direct comparison).
The previous paragraphs give the expession of the power spectra of the scalar modes (both linear and second order), vector modes and tensor modes. Fig. 1 depicts these different contributions assuming a flat CDM background universe with , and as derived from the WMAP5 best fit model wmap5 (). We also use the transfer function derived in Ref. Eisenstein:1997ik ().
Note that since the amplitudes of vector and tensor modes are small on Mpc scales we do not take into account their non-linear contribution.
Iii Weak lensing convergence and power spectra
which can be decomposed in contributions arising from the scalar-vector-tensor perturbations of the metric as
with , and .
The distortion of the shape of background galaxies is described by the Sachs equation pubook (); ehlers (); bartelman () in terms of a Jacobi matrix that can be rescaled, as long as the background spacetime is spatially homogeneous and isotropic ppu (), to define the amplification matrix , where the indices refer to the angle coordinates of a unit 2-sphere. At lowest order, it is given by pubook (); ehlers (); bartelman (); ppu ()
where the lensing potential is obtained by integrating the deflecting potential on the line of sight as
is the radial coordinate and is defined by
so that for a spatially Euclidean universe.
The amplification matrix can be decomposed in term of a convergence and a shear as
from which we deduce that
where is the 2-dimensional Laplacian on the unit 2-sphere.
The previous expression (25) gives the convergence for a single source located at a radial distance , or similarly at a redshift . However, observations usually deal with the convergence averaged over a source distribution ,
where the upper limit of infinity is taken to mean well beyond the source distribution, or the horizon scale. Note that such an averaging over the source distribution is not mandatory if one has distance information about each bin of sources. Using the fact that is equivalent to integrate as we obtain, after exchanging and , the expression
From this, we may also introduce the lensing potential averaged over sources as
in terms of which Eq. (27) takes the form
The geodesic bundle propagates in the perturbed spacetime, which induces a correction of the redshift of the source, compared to the background redshift. Correcting the redshift in turn corrects the distance to the source, and so adds to the convergence. This affects only the convergence but not the shear (at linear order). Taking into account this effect induces three extra terms at first-order for the convergence: the Sachs-Wolfe and Integrated Sachs-Wolfe terms and a Doppler lensing term (Refs. Bonvin:2008ni (); Bonvin:2005 (); Bacon:2014uja ()). The SW and ISW contributions are
The Doppler contribution, in a spatially Euclidean background, is
for pointing in the direction of observation, and with given by Eq. (2). This contribution to the convergence was first identified in Bonvin:2008ni (); Bonvin:2005 (), and investigated in more detail in Bacon:2014uja (); voidslensing (). Note that when using these formula, the comoving distance to a source should be calculated from the background distance-redshift relation using the observed redshift (and not the unphysical background redshift).
iii.2 Different contributions to the convergence
As discussed in § II, we have 3 contributions to the convergence that arise from the scalar, vector and tensor contributions to Eqs. (19-20), to which we need to add the two Sachs-Wolfe terms and an important first-order contribution induced by the Doppler effect Bonvin:2008ni ().
It follows that the observed weak lensing convergence has 4 contributions given by:
and at second-order
Note also that in these expressions, the variables are evaluated along the light cone and considered as function of the radial distance and the angular position only. Given a source distribution, the left-hand side are purely function of position on the sky.
iii.3 Expression of the power spectra
Given the previous expressions, one can deduce the angular power spectra of these different contributions to the convergence. To that purpose, we decompose each variable in spherical harmonics. For each contribution, the deflecting potential (22) can be expanded as
where is the position on the celestial 2-sphere, for a source located at . Taking into account spatial isotropy, its angular power spectrum is defined as
Given Eq. (30), the coefficients of the expansion of the shear are related to the by
which implies that the angular power spectra of the cosmic convergence and deflecting potential are related by
The power spectra are related to the real space angular correlation function,
where stands for the Legendre polynomials.
When the integration over the source distribution is included (i.e. using the expressions (27-30)), one obtains similar expressions for the angular power spectra but with an extra integration over the sources distribution so that the dependence in disappears.
where , corresponding to . The power spectra of each mode, etc., are respectively given by Eqs. (7), (12) and (18) and we have replaced by since this the integral is evaluated on the past lightcone. The numbers for and is the number of polarisations of each mode. The Doppler contribution () takes a similar form (see Appendix B) with , , and . The two contributions from ISW and SW terms are both similar to the scalars modes with , and except that for SW whereas that of ISW is the same as the scalar modes (see Appendices C to D).
Each spectrum can be written in terms of a transfer function which is normalized to unity at early times as
This implies that Eq. (46) factors as
Similarly, the convergence angular power spectra, not integrated over the sources distribution, takes the form
Since integrating the Bessel function in Eq. (49) is computationally expensive and since the sources distribution is slowly varying over long distances, we shall resort to a Limber approximation which is a good approximation as at large . In such an approximation, so that LoVerde:2008re (); bpu () we have the property
which is accurate to and is sufficient for our purposes. We then find
Iv Weak lensing from second-order modes
The previous expressions allow us to compute numerically the angular power spectra of the 6 contributions to the cosmic convergence in particular to estimate the typical magnitude of the non-linear terms which we compare to the standard term , the Doppler term , ISW term and SW term , which allows us to discuss whether assuming is a good approximation to interpret the weak lensing observations. Since the two point function can be computed in real space (i.e., the correlation function ) or in harmonic space (i.e., the angular power spectrum ), we shall use the two representations.
iv.1 Behaviour of the different contributions
We start by comparing in Fig. 2 the different contributions to the lensing angular power spectra without integrating over the sources distribution and assuming that the sources on the sky are located at the same redshift in or . We recover that the velocity contribution dominates at low redshift Bonvin:2008ni () and that the gravity waves contribution is always negligible Sarkar:2008ii (). The results shown in Fig. 2 also suggest that there is a range in multipoles () where the second order vector modes become more significant than both of the Sachs-Wolfe terms. A similar computation in real space, assuming is depicted in Fig. 3 and Fig. 4.
Focusing on the contribution of the vector modes, Fig. 5 shows how the amplitude of the angular power spectrum depends on the redshifts of the background galaxies and on the scale, while Fig. 6 shows the similar information in real space, i.e., . Fig. 7 shows the ratio of the vectors to the Doppler term, which shows that at intermediate redshifts the second-order frame dragging effects dominate the linear Doppler lensing.
iv.2 Source distributions
The source distribution depends on the survey and is described through the function or an equivalent function in redshift space, where . These distributions are normalised to unity. This then defines the lensing weight function , as shown in Eq. (28).
To start, let us assume that the sources are distributed at a single redshift so that
where is the Heaviside distribution. This unrealistic but simple assumption provides a good way to understand the lensing effects as a function of redshift. Fig. 8 depicts the contribution to the lensing spectra for shells with sources located at different redshifts normalised to the scalar contribution. As we can see, the relative contribution from the vector modes is largest at low redshift, reflecting the fact that vector modes continue to grow at late times. It can also be noticed that second order vector modes completely dominates the Sachs-Wolfe term at small scales (large ). Like the case of the Doppler term, the SW term has a prefactor which tends to zero at . This accounts for the large amplitude of the ratio at .
In order to obtain more realistic orders of magnitude, we consider source distributions similar to the one of the future Euclid and SKA experiments. The normalised Euclid redshift distribution has the form given in Refs. Beynon:2009yd (); Amendola:2012ys (); euclid ():
with , and , which gives a median redshift .
For SKA we make use of the SKA Simulated Skies simulations Wilman:2008ew (). These are simulations of the submillimeter radio source population. We use all the extragalactic radio continuum sources in the central sq. degrees out to a redshift of . In these simulations, the sources are drawn from either observed or extrapolated luminosity functions and grafted onto an underlying dark matter distribution with biases which reflect their measured large-scale clustering. We then construct a redshift distribution that we paramaterise as
with best fit parameters and and normalise the distribution at , which gives a description accurate to the percent level, which is good enough for our purposes. Note that this redshift distribution represents the very best case scenario since all sources from the simulation have been used in its construction, and no further observational cuts were included.
These source distributions can be used to compute the vector convergence spectrum for both surveys. Fig. 9 compares its amplitude to the standard scalar contribution, showing that it is typically times smaller. Whereas compared to the Doppler contribution, its amplitude is about larger and smaller on small scales respectively for a SKA-like survey and for a Euclid-like survey – see Fig. 9. Interestingly, the vector contribution is subdominant for Euclid, for which the main correction arises from the Doppler term, while for SKA-like geometry the vector contribution is typically 1-100 times larger than the Doppler one for . On larger angular scales, the Doppler term always dominates – see Fig. 9, where on large angular scales the Doppler term totally prevails over the scalar contribution by about 5 orders of magnitude.
This article has evaluated the amplitude of relativistic contributions to the weak lensing power spectra. We have considered the gravitational wave and vector mode backgrounds which are sourced at second-order by density perturbations. The amplitude of these backgrounds are completely fixed once the normalisation of the scalar power spectrum in the linear regime is determined. As these are purely relativistic degrees of freedom they set the lower limit for all relativistic effects on cosmological modelling. While the gravitational wave background is very small in relation to the scalars, the vectors, which represent frame dragging in the metric, give corrections to the metric at nearly the percent level. The effect of these contributions on weak lensing convergence predictions have been computed in order to understand if they can either be detected, or bias the analysis of future weak lensing experiments, such as Euclid or SKA. We have compared them to the usual gravitational lensing contribution, the two Sachs-Wolfe contributions as well as the Doppler lensing contribution Bacon:2014uja ().
First, we have shown that even though the non-linear tensor mode background dominates over any possible primordial gravitational wave contribution, its effect on weak lensing is completely negligible, by 10 to 12 order of magnitudes (see Figs. 2 and 8).
Then, we have shown that the vector contribution to the convergence, while small, can dominate over the Doppler lensing at high redshift – but there it is swamped by gravitational lensing by density perturbations. We have shown this both for point sources and for two survey geometries. The vectors are actually more important than the Doppler term for SKA-like source distributions on small scales, but not for a Euclid like survey. For both of these surveys the vectors only reach about % that of the normal gravitational lensing contribution, and so can be safely neglected. Nevertheless, it is interesting that the vector contribution can be as important as some linear terms.
We have also recovered that although the frame dragging effect is small, it becomes more important than both ISW and SW above . This comes to corroborate the fact that for observations, neglecting the 2 first order Sachs-Wolfe terms is a good approximation.
In this analysis, the non-linear effects of the metric perturbations have been described at second order while weak lensing was described assuming that the Born approximation still holds. In principle, one needs also to take into account second order effects on the geodesic deviation equation Seitz:1994xf (); Cooray:2002mj (); Dodelson:2005zj (); Schaefer:2005up (); Shapiro:2006em (), as fully described in Refs. Bernardeau:2009bm (); Bernardeau:2011tc ().
There are a huge variety of second-order effects which come into the convergence. We have only considered two contributions which arise from non-linear dynamical effects which happen as structure forms. Many contributions appear when calculating the lensing convergence itself Umeh:2012pn (); Umeh:2014ana (); BenDayan:2012wi (), and these also need to be analysed in a similar manner to that presented here to determine whether relativistic effects are important for future observations of magnification.
SA, PP and OU are funded by the South African Square Kilometre Array Project. CC acknowledges funding from the NRF (South Africa). This work made in the ILP LABEX (under reference ANR-10-LABX-63) was supported by French state funds managed by the ANR within the Investissements d’Avenir programme under reference ANR-11-IDEX-0004-02. JPU thanks the University of Cape Town for hospitality during the late stages of this project and Yannick Mellier for discussions.
Appendix A Angular power spectrum of the scalar modes
We follow the standard description of weak lensing in a full sky analysis, following e.g, Refs. pubook (); ehlers (); bartelman () and refer to Refs. ppu (); Yamauchi:2013fra () for more recent developments of the formalism.
Taking into account that one can neglect the anisotropic stress, the deflecting potential integrated over the line of sight (27) reduces to
where the are the spherical Bessel functions, the components are given by
where we have replaced by since the integral is evaluated on the past lightcone. It follows that
using Eq. (7), integrating over , then decomposing and integrating the product of spherical harmonics over to get the term . The expressions for the scalar ’s in the text follow directly.
Appendix B Angular power spectrum of the Doppler term
Starting from the expression (III.2) for the convergence associated to the Doppler effect in which is given by Eq. (2), and using the decomposition of the gravitational potential described in § II.1, one obtains that
where the coefficient is given by
Now, using that
with a prime on the spherical Bessel function denoting the derivative with respect to its argument, Eq. (62) becomes
from which we can extract the components
Its correlator is then given by
We finally get the formula of the angular power spectrum convergence associated to the Doppler contribution as
the angular spectrum reduces to
Appendix C Angular power spectrum of the Integrated Sachs-Wolfe term
As discussed in the text, the Integrated Sachs-Wolfe also contribute to the cosmic convergence at first order
with both and defined in the text. The harmonic expansions of both the first and the second terms, which we call