This is an interesting paper that tries to set out a new way to use galaxy redshift survey data to determine cosmological parameters with great accuracy. The method described here is based on a novel application of the Alcock-Paczynski test using Bayesian machinery developed by some of these authors in previous papers. The results described in the paper (in particular in Figure 1) are eye-catching and would be revolutionary, if they could be applied to real data.

Unfortunately, the paper oversells the method used, because it ignores redshift-space distortions produced by galaxy peculiar velocities, which is the main confounding issue with the use of the Alcock-Paczynski test.

What the authors have demonstrated is that in a hypothetical universe in which galaxy peculiar velocities do not contribute to their observed redshifts, a hierarchical Bayesian inference method would be able to determine cosmological parameters very accurately through the AP test (but in this scenario traditional methods would work much better than currently as well).

As currently presented, the method cannot work on real data. If it were corrected to account for RSD in order to make it applicable to real data, I don't think LPT would provide sufficient accuracy of RSD modelling on small scales to perform such an AP test, so the true obtainable constraints would be much weaker.

This is an interesting paper that tries to set out a new way to use galaxy redshift survey data to determine cosmological parameters with great accuracy. The method described here is based on a novel application of the Alcock-Paczynski test using Bayesian machinery developed by some of these authors in previous papers. The results described in the paper (in particular in Figure 1) are eye-catching and would be revolutionary, if they could be applied to real data.

Unfortunately, the paper oversells the method used, because it ignores redshift-space distortions produced by galaxy peculiar velocities, which is the main confounding issue with the use of the Alcock-Paczynski test.

What the authors have demonstrated is that in a hypothetical universe in which galaxy peculiar velocities do not contribute to their observed redshifts, a hierarchical Bayesian inference method would be able to determine cosmological parameters very accurately through the AP test (but in this scenario traditional methods would work much better than currently as well).

As currently presented, the method cannot work on real data. If it were corrected to account for RSD in order to make it applicable to real data, I don't think LPT would provide sufficient accuracy of RSD modelling on small scales to perform such an AP test, so the true obtainable constraints would be much weaker.

The Alcock-Paczynski (AP) test essentially says: cosmological objects or features in the galaxy distribution that are intrinsically isotropic may appear distorted along the line-of-sight direction if the wrong cosmological model is used to convert observed redshifts into radial distances. Therefore, if an object or feature is known to be intrinsically isotropic but observed as apparently distorted, we know we've got the cosmology wrong, and can use this to determine the correct cosmology. In principle, this is a great way to measure combinations of the angular diameter distance \(D_A(z)\) and the expansion rate \(H(z)\) and so constrain things that affect these, such as \(\Omega_m\) and the dark energy equation of state \(w(z)\).

This sentence is a clue to the problem with this method.

The biggest difficulty with using the galaxy distribution to perform an AP test is that it is not expected to be intrinsically isotropic even if we get the cosmology right. This is because galaxy peculiar velocities also affect their redshifts and therefore introduce another major source of anisotropies, known as redshift-space distortions (RSD). RSD normally dominates over the AP distortion, so we can't perform an AP test without precisely accounting for RSD.

But RSD modelling is hard.

In particular, the evolution of the velocity field receives large non-linear corrections (e.g., Scoccimarro 2004), which is why perturbation theory does not work very well at describing RSD even at quite large scales. The LPT method used here can provide a reasonable description of the density evolution but would not be accurate enough for velocity fields and RSD modelling.

This Bayesian inference mechanism is well-developed and is one of the exciting recent developments in cosmology. I have no complaints about this in general or in other applications and contexts.

This figure is absolutely astounding. If it were true that the ALTAIR algorithm could achieve this level of precision, and outperform BAO to this extent, it would truly be a revolution in large-scale structure analyses.

However, it is not true, because ALTAIR ignores RSD complications (and this result is produced using a synthetic data set in which by construction RSD are absent). The BAO constraints are however obtained from real data and after marginalising over RSD terms. This is therefore a highly misleading comparison.

This Eqn. (1) and the schematic representation in Fig. 2 illustrate the problem with this forward-modelling approach.

One component of the forward model evolves the initial conditions forward to give the non-linear final density field in comoving coordinates (in LPT), \(\delta^{\mathrm{f}(r)}_p\). This bit does not account for velocities or RSD.

The second component of the forward model takes \(\delta^{\mathrm{f}(r)}_p\) and converts it to redshift space accounting only for the coordinate transformation from comoving to redshift space, i.e., still not accounting for peculiar velocities and the associated RSD. The detailed equations in Appendix B confirm this.

Even if the rest of the forward modelling approach is entirely correct (and I think it is), this omission means that ALTAIR is incomplete and cannot be used for AP tests in real data.

Upgrading to 2LPT or non-perturbative approaches for forward modelling certainly would help improve this work. However, the main difference would come from upgrading the component \(\mathcal{M}_p^{(1)}\), not \(\mathcal{M}_p^{(2)}\) as stated here. If \(\mathcal{M}_p^{(1)}\) were modified to include peculiar velocity effects I don't think either LPT or 2LPT would be sufficient to model the RSD to the desired accuracy for an AP test, however the non-perturbative particle mesh approach might work better.

This section describes in detail several aspects of the generation of the mock catalogue on which tests producing the headline results are performed. Great care is taken to reproduce various aspects of the true galaxy data, but the most important aspect is omitted – there is no mention of converting peculiar velocities to additional redshift distortions. I can only guess that this is because this step was not performed, i.e. the mocks do not contain RSD.

The fact that the forward model works well on this synthetic data despite not accounting for RSD also suggests that the mocks do not contain RSD effects by construction.

This discussion of the information gain highlights an important point – if it were possible to use features or aspects of the galaxy distribution on smaller scales than the BAO peak for the AP test, we would certainly expect a large information gain. This is essentially because we would be able to use many more modes for a given survey volume.

This is the reason why BOSS and other LSS surveys use "full-shape" analyses of the galaxy clustering power spectrum to add AP information to that from BAO. But these additional information gains from smaller scales are marginal (in general, AP constraints from pre-reconstruction "full-shape" measurements are weaker than from post-recon BAO-only measurements) precisely because of the AP-RSD degeneracy and the difficulty in accurately modelling RSD on small scales, that this paper has not taken into account.

This Appendix provides technical details of the transformation from comoving space to "redshift" space. Following the equations in detail shows that the coordinate transformation is assumed to have no dependence on peculiar velocity, and only accounts for the background-level transformation between comoving distance and redshift.

The authors present the cosmological results from the first year of data from the Dark Energy Survey (DES-Y1), combining measurements of galaxy clustering, galaxy-galaxy lensing and shear correlations. This paper is a big step towards delivering the promised results of weak lensing surveys, but a series of comments could be addressed.

See the full review here.

The standard deviation of redshift errors in the lens sample is a couple of percent, \(\sigma_z / (1+z) \sim 0.017\). However, in studies of their clustering and of their cross-correlation with the sources (galaxy-galaxy lensing), the authors use very wide redshift bins of \(\Delta z \gtrsim 0.20\). This seems like a sub-optimal choice, since this binning is unnecessarily smoothing the signal along the line of sight.

The authors present the cosmological results from the first year of data from the Dark Energy Survey (DES-Y1), combining measurements of galaxy clustering, galaxy-galaxy lensing and shear correlations. This paper is a big step towards delivering the promised results of weak lensing surveys, but a series of comments could be addressed.

\(\Lambda\mathrm{CDM}\) is a well established name to refer to a specific model with 6 free parameters, none of which is neutrino mass. It would be better to use another name to refer to \(\Lambda\mathrm{CDM}\)+neutrino mass, probably \(\nu\Lambda\mathrm{CDM}\) or similar.

The estimates of the redshift distribution for the lens galaxies have very short tails. In figure B1 of the paper describing the lens galaxies (Ref. [18], Elvin-Poole et al. 2017), one has the visual impression that very separated redshift samples have non-zero correlations. What could be explained if the redshift distribution had longer tails than assumed? It would be good to test what would be the effect of these potential systematic tails on the inferred cosmological parameters.

Based on the same figure, one wonders about the overall goodness of fit of the correlations of lens galaxies, including the correlation of different redshift bins. Even if these cross-correlations were not used in the final analysis, it would be a good diagnosis of possible systematics in the data.

The lens sample contains only a few percent of the total sample of galaxies. It would be interesting to see a discussion on how would the results improve if one could use a larger fraction of the dataset.

The authors assume that the bias of the lens galaxies in a given redshift bin can be described by a constant, \(b_i\). However, the bias of these galaxies probably evolve with \(z\), and in that case it would be better to use a parameterised function to describe \(b(z)\).

Given that the non-linear scale evolves with redshift roughly with the growth factor, it would seem more logical to define a cut that also evolves with redshift, so that smaller scales are included at high redshift.

The overall goodness of fit is very poor (4e-5). The authors argue that “Given the uncertainty on the estimates of the covariance matrix, the formal probabilities of a \(\chi^2\) distribution are not applicable”, but later on they show that changes in the covariance matrix actually have a very small effect on the best fit. I think the authors should explore other explanations for this high value of \(\chi^2\), like possible systematics in the data. For instance, it would be interesting to see the effect of the tails in the redshift distribution of the lenses discussed above.

The probability for this value is 7%, much higher than the probability of the overall analysis (0.004%). I would suggest that this result is significant enough to take a better look at the measurements on large angular separations.

The authors do not include the z>2 measurements from the BOSS collaboration (Bautista et al. 2017, du Mas des Bourboux et al. 2017), even though these have smaller error bars than other measurements included.

The authors compare the cosmological constrains obtained from different pipelines (METACALIBRATION vs IM3SHAPE). The differences in \(S_8\) are considerable, at the \(1.3\sigma\) level. Different galaxies are selected by the different pipelines, so it is not trivial to quantify the significance of this difference. One could maybe study this by looking at the changes in best fit value when dropping random sub-sets of the METACALIBRATION catalogue (the largest one), either from the actual data or from simulated data.

This is an interesting article dealing with the cosmological constant problem, i.e. the way vacuum energy gravitates. The problem is tackled in an original way; the main idea consists in taking into account that vacuum energy is not constant over space and time, but rather strongly fluctuating. Considering a toy-model of inhomogeneous cosmological spacetime, the authors show that the scale factor locally satisfies an equation of the form (41) $$\ddot{a} = \Omega^2(t) a$$, where \(\Omega\) is a quasi-periodic function whose amplitude and pseudo-period is dictated by the vacuum energy. As a result, the scale factor essentially takes the form $$a(t) = A(t) \cos[\Omega(t)t+\phi],$$that is, it oscillates with slowly increasing (and more and more increasing) amplitude \(A(t)\), which the authors interpret as being the slow accelerated expansion of the Universe.

The paper is interesting and very clearly written, but in my opinion it suffers from two issues, which I further elaborate on with annotations in the text. In a nutshell:

1) The energy of vacuum is computed in a quite naive way, without renormalisation, and hence its cutoff is not Lorentz-invariant. I am not sure that I am convinced by the discussion on that point at the end of the paper.

2) More importantly, the oscillating behaviour of the scale factor (which is allowed to be vanishing and negative here) is not physically acceptable, because it tells us that the cosmic expansion is the result of the oscillating of motion about itself, with an increasing amplitude. This is not what is observed: galaxies are receding from us, not oscillating about us with an increasing amplitude.

See the full review here

This is an interesting article dealing with the cosmological constant problem, i.e. the way vacuum energy gravitates. The problem is tackled in an original way; the main idea consists in taking into account that vacuum energy is not constant over space and time, but rather strongly fluctuating. Considering a toy-model of inhomogeneous cosmological spacetime, the authors show that the scale factor locally satisfies an equation of the form (41) $$\ddot{a} = \Omega^2(t) a$$, where \(\Omega\) is a quasi-periodic function whose amplitude and pseudo-period is dictated by the vacuum energy. As a result, the scale factor essentially takes the form $$a(t) = A(t) \cos[\Omega(t)t+\phi],$$ that is, it oscillates with slowly increasing (and more and more increasing) amplitude \(A(t)\), which the authors interpret as being the slow accelerated expansion of the Universe.

The paper is interesting and very clearly written, but in my opinion it suffers from two issues, which I further elaborate on with annotations in the text. In a nutshell:

1) The energy of vacuum is computed in a quite naive way, without renormalisation, and hence its cutoff is not Lorentz-invariant. I am not sure that I am convinced by the discussion on that point at the end of the paper.

2) More importantly, the oscillating behaviour of the scale factor (which is allowed to be vanishing and negative here) is not physically acceptable, because it tells us that the cosmic expansion is the result of the oscillating of motion about itself, with an increasing amplitude. This is not what is observed: galaxies are receding from us, not oscillating about us with an increasing amplitude.

This estimation is valid only if the vacuum energy is not renormalized, in which case it does not behave as a cosmological constant (details hereafter).

The issue with using this definition for the energy of vacuum is that, by definition, it does not behave as a cosmological constant, even if it would not fluctuate! With this "naive" approach, the full expression for the stress-energy tensor of vacuum is indeed $$\langle 0 | T{\mu\nu} |0\rangle = \int \frac{\mathrm{d}^3 k}{(2\pi)^3} \frac{k\mu k_\nu}{2\omega(k)},$$ which gives an equation of state \(w=1/3\).

This is usually solved by renormalisation (see e.g. https://arxiv.org/abs/1004.1782 ), which also implies that the energy density no longer has the same expression.

This ansatz is to be compared to the "separate-universe" approach, often used in the context of inflation, in which inhomogeneity is dealt with as if each region of the Universe evolved as an individual FLRW Universe. Its main limitation is that it does not allow for local anisotropy. It also unclear what the motion of matter actually is in this model? Is matter still considered comoving?

This is an important remark, but the authors should have pushed it even further. One should add that this notion of physical distance L between two events must correspond to an integral of the metric along a space like geodesic connecting the events. This geodesic does not necessarily remains within a t=cst hyper surface, so that eq. (30) is incorrect in principle.

This is, in my opinion, the core of the paper's hypotheses. First, since vacuum energy is not renormalized, it behaves like radiation, i.e., it has an attractive gravitational behaviour (unlike a cosmological constant). Second, the scale factor will be allowed to vanish and become negative. Accelerated expansion is then interpreted as a slow amplification of the amplitude of the oscillations of \(a(t)\), due to parametric resonance.

Very interesting phenomenon.

Again, this is a bit naive since the non-renormalized vacuum energy, given by \(\rho_{\mathrm vac}\) , does not behave as a cosmological constant.

Singularities are one issue, but the very fact that the scale factor is oscillating is another. A discussion about the physical interpretation of this would have been appreciated. There are essentially two possible approaches:

1) We keep the usual interpretation of the scale factor, with comoving matter. The implies that matter itself undergoes those oscillations, which does not make much sense (galaxies recede from us, they do not oscillate about us with growing amplitude). One could say that this phenomenon should be considered on much smaller scales, but we do not see such a thing on atomic or subatomic scale either.

2) We abandon comoving matter, but then how to relate the growing amplitude of the oscillations of a with cosmic expansion?

The paper investigates to what extent the combining tomographic weak lensing spectra with weak lensing bispectra can improve constraints on a wCDM-cosmology. Constraints tighten up by about 60% by the inclusion of the bispectrum.

See the full review here.

The paper investigates to what extent the combining tomographic weak lensing spectra with weak lensing bispectra can improve constraints on a wCDM-cosmology. Constraints tighten up by about 60% by the inclusion of the bispectrum.

Motivation

The questions treated in the paper are very important and very topical. Specifically, the authors set out to find out

1. how much information is contained in weak lensing bispectra,
2. how much covariances change due to the inclusion of non-Gaussian statistics,
3. if a new term called halo sample variance matters. These questions are very important because much of the weak lensing signal is generated on small scales by nonlinear structures. Commonly, one increases the variance of the density on small scales in an empirical way but does not account for the deviation from a Gaussian distribution. In order to give the reader some sense of scale: The weak lensing data set from Euclid will have a total statistical significance of 1000 sigma, 80% of which originate from nonlinear structures.

Context

Nonlinear structure formation turns close to Gaussian initial conditions into a evolved density field that shows strongly non-Gaussian characteristics that are picked up by observables. As lensing depends (to lowest order) linearly on the density field, it inherits its statistics and therefore its non-Gaussianities. It is clear from the scaling behaviour of spectra and bispectra that they contain different parameter dependences and their combination should be able to improve cosmological constraints. Issues are the prediction of non-Gaussian properties of the lensing signal, the exact increase of the variance of the density field, and even more difficult from a technical point of view, the covariance properties of estimators in the non-Gaussian limit.

Methods

The methods and approximations used (flat-sky, weak lensing spectra and bispectra in Limber's approximation, halo-model, full-sky coverage) are all appropriate for demonstrating the author's point.

Alternatives

I see the necessity of using the Fisher-matrix formalism in particular when dealing with the bispectrum and all non-Gaussian covariance contributions out of numerical feasibility. Likewise, all lensing approximations are justified (most importantly the flat-sky approximation, because non-Gaussianities are most prominent on small angular scales). Using the formalism in the context of a Monte-Carlo Markov-chain would be really difficult, but would relax the assumption that the models for spectra and bispectra depend linearly on the parameters and would give rise to a Gaussian parameter likelihood.

Criticism

The authors use a Planck-type CMB-prior for their results on the statistical errors, and one can see that the prior determines size and orientation of the Fisher-ellipses a lot - I would be very interested to see the "naked" lensing results for a tomographic survey, and with the non-Gaussian contributions to the covariances individually switched on: I would argue that this gives a much better feeling about the magnitude of the effect on parameter inference.

I like a lot that the authors provide a good deal of technical detail, in particular concerning permutations of bispectra with equal tomography bin indices and equal multipoles. I would have appreciated a more detailed justification about the selection of a comparatively small number of bispectrum configurations and and accuracy of binned summation, as well as more detail on the estimation process of the covariance matrix from numerical simulations - this is a particularly tricky issue and a lot of interesting developments have been done in the recent time.

The only point where I am a bit lost would be the justification of using the halo model for computing the signal (where everything is done consistently) but the restriction to 1-halo-terms for the higher-order contributions to the covariance. Again, I appreciate this restriction on reasons of feasibility, but it would have been nice to see this in comparison to a simulation. Or, on the other hand, the halo-model could be replaced by standard perturbation theory, but again I see how this would not be able to be a feasible model for the HSV-term.

One issue which I find very difficult (and where the authors are certainly not to blame) is that implicitly the estimates of the spectrum and bispectrum assume that those are estimated from a large number of modes - I wonder if differences at low multipoles, where only small numbers of modes are available, this effect might be similarly large compared to the effects discussed here.

Open questions

Personally, I find the HSV-term very interesting because it links large-scale and small-scale modes in a suble and funny way. I wonder if there might be other applications where analogous effects can play a role.

This paper makes some rather bold claims -- that cosmological zero modes are not only observable, but also allow us to see primordial physics beyond our observable horizon. The authors certainly pose an intriguing question, but were I refereeing this paper in the traditional manner for a journal, I'd ask that the authors first address some very basic questions that prevented me from getting past their starting point. Certain statements made in the introduction are technically incorrect as they stand, which I annotate below. I hope these provide a constructive context for followup discussions that might clarify the claims of this paper.

See the full review here

This paper makes some rather bold claims -- that cosmological zero modes are not only observable, but also allow us to see primordial physics beyond our observable horizon. The authors certainly pose an intriguing question, but were I refereeing this paper in the traditional manner for a journal, I'd ask that the authors first address some very basic questions that prevented me from getting past their starting point. Certain statements made in the introduction are technically incorrect as they stand, which I annotate below. I hope these provide a constructive context for followup discussions that might clarify the claims of this paper.

This statement puzzles me -- we never directly measure the curvature perturbation, only its gradients. What we see in the CMB for example, are temperature anisotropies sourced by perturbations of the density \(\frac{\delta T}{T} = \frac{1}{3}\frac{\delta\rho}{\rho} + ...\) for adiabatic perturbations, for example. However \(\delta\rho \propto \Delta\zeta \), and so the zero mode in this context drops out of the observable quantity entirely -- it is not observable. Moreover, the zero mode of the curvature perturbation is indistinguishable from a rescaling of the background scale factor by a time reparametrization: consider the scale factor in the presence of a zero mode of the curvature perturbation -- \( a(t) e^{\zeta(t)} := \tilde{a}\). To first order, one sees that \(\tilde{a} = a(t + \zeta/H) \).

This statement is incorrect. A zero mode is a zero mode everywhere and does not blow up at the origin. All one has to do above is look at the solution for \(l = m = 0\) above and set \(B_{00} = 0\) and \(A_{00}\) to be arbitrary to see that it is regular at the origin and at infinity.

This statement, and the equation that follows (and therefore the entire starting point of this treatment) is technically incorrect due to a subtlety overlooked by the authors. It is not true that the two gravitational potentials are equal in the absence of anisotropic stress when one assumes that the cosmological perturbations to have a non-vanishing spatial average*. This can be seen for example, in the discussion surrounding eq 5.16 in Mukhanov et al, Phys.Rept. 215 (1992) 203-333.

[See enlarged image here]

As stated there, only by assuming the spatial average of the perturbations vanish that one can conclude that the two potentials are equal. Therefore there is an additional contribution to eq 7 that is neglected.

*Note that the standard treatment of cosmological perturbation theory assumes that perturbations have a vanishing spatial average. Relaxing this assumptions requires re-examining the consistency of many standard expressions. The one pointed to above is unlikely to be the only place where relaxing this assumption might bite you.

Here we can see explicitly that for the zero mode of \(\Psi\), redefining \(\tilde{a} = a(1 - \Psi(t))\) and subsequently rescaling time can bring eq 8 into the form of an unperturbed universe with a different scale factor. This scale factor can evidently be obtained from the original one by a time translation.

The issues raised above have puzzled me to the point that I am unable to make sense of the rest of the paper. It could be that the authors or other interested readers have straightforward replies to clarify matters and help unconfuse me, which I look forward to hearing.