Evaluating backreaction with the peak model of structure formation

This is the paper where Rasanen finally does what his earlier toy model only gestured at: replaces two regions with a realistic ensemble from the peak model, computes the volume-weighted average expansion rate explicitly (Eq. 32), and ties the timescale to the turnover in the CDM transfer function near matter-radiation equality. The headline is negative and, as far as I can tell, correctly stated: Ht climbs from 2/3 toward roughly 0.7-0.85 over tens of billions of years, driven by voids, but q stays around 0.2-0.5. No acceleration. The calculation is transparent, has no tuned parameters, and is checked against two transfer functions and a couple of smoothing thresholds. That is real work, and the negative result is worth having in the literature precisely because the backreaction literature is short on honest null results. The qualitative sections on light propagation and on why Newtonian intuition fails for Q are also useful, if long-winded; a good chunk of the paper reads like a review, which is fine but worth knowing before you assign it to a student. The soft spot is the one the stress-test flags, and it is real: the Gaussian symmetry gives equal mass to peaks and troughs (Eq. 29, N ~ 1.97), and the paper's own explanation for the missing acceleration is that overdense volume is too small to damp Ht first. Trough-in-a-peak extinction should move mass the other way, and the author says so in section 4.4 and again in 5.3. But none of the reported sensitivity checks break that symmetry, so the one knob that drives the conclusion is untested quantitatively. To be fair, the paper does not overclaim: it says 'it is likely' that overdense regions must first slow the expansion, and lists the fix as future work. The other thing I would want a referee to press on is the low Ht today for BBKS (0.68-0.76), which the author himself calls 'rather low' against H0t0 ~ 0.85-0.99. That is a tension, not a refutation, but it deserves a number rather than an aside. Overall: a solid, self-aware paper with a load-bearing but acknowledged modelling choice. Worth a serious referee, and worth citing if you work on averaging or backreaction.

The paper's explanation for why Ht rises but does not accelerate is that overdense regions occupy too little volume to first damp the expansion before voids take over (§4.3, §4.4, §5.3). That volume fraction is a direct consequence of the Gaussian symmetry f(δ) = f(−δ) with equal mass per peak and trough (Eq. 29, N ≈ 1.97). The author acknowledges that trough-in-a-peak extinction should transfer mass from voids to overdense regions (§4.4) and states in §5.3 that 'taking into account mass flow to the overdense regions would increase their impact,' but does not quantify it. The sensitivity checks actually reported — σ0(t,R) = 0.5 and 2, the improved collapse model of [131], and the BBKS vs BD transfer functions — all preserve the symmetric assignment, so none of them tests the one knob that drives the negative conclusion. A technical sharpening: because the volume weight s_δ+ → 0 for collapsing regions, the concern is not that overdense regions dominate at late times, but that a realistic mass flow gives them enough volume at the intermediate epoch to damp Ht before voids dominate. If so, the subsequent rise could steepen into acceleration, reversing the headline. The author's qualitative acknowledgment is honest but does not establish that the negative result survives a realistic mass assignment.

The paper is careful, self-contained, and honest about its limitations. The positive result (Ht rises from 2/3 toward ~0.7–0.85 on a 10–100 Gyr timescale) is robust across the checks performed: both transfer functions, σ0 = 0.5 and 2, and the improved collapse model of [131]. The negative result (q stays positive, no acceleration) is presented as provisional, and the author says so explicitly in §4.4 and §5.3. The reader correctly identified the equal-mass peak/trough assignment as the load-bearing assumption for the negative conclusion. I would only sharpen the mechanism: because the volume weight s_δ+ → 0 for collapsing regions, the concern is specifically about the intermediate epoch, where overdense regions must have enough volume to damp Ht before voids dominate. The author's own sensitivity checks do not cover this knob. No circularity or data-selection problems were found. The Newtonian/GR tension in §5.2 is acknowledged and, for the Ht number itself, not load-bearing, because Ht is computed directly from volume-weighted regional expansion rates rather than from Q. CONDITIONAL is the right verdict; my read does not change it.

This is a genuine advance over the two-region toy models that previously motivated backreaction as an alternative to dark energy. Replacing the toy model with a realistic ensemble, and tying the timing of the change in the expansion rate to a physical scale in the transfer function, is exactly the step the field needed. The negative result is itself informative: it constrains how much of the observed acceleration can plausibly come from structure formation within a dust model, and it does so with a calculation that is transparent and reproducible from standard ingredients. The paper is also candid about its limitations, listing in §4.4 and §5.3 the improvements that would be needed before the negative conclusion could be considered firm. The main reservation is that the headline claim is conditional on a modelling choice the author himself identifies as the most important one to relax, and that conditionality is not carried into the abstract.

The paper asks whether the growth of non-linear structure can, by itself, change the average expansion rate of the universe enough to explain why distant supernovae look dimmer than a plain matter-filled cosmos would predict. The author builds a model universe that is statistically smooth on large scales but filled with an evolving population of spherical lumps and holes, whose numbers come from the peak model of structure formation and whose individual histories come from spherical collapse and its underdense counterpart. Averaging over this population, the expansion rate rises above the smooth matter-only value on a timescale of 10–100 billion years, matching the rough era when cosmic acceleration is observed, and the rise traces back to a bend in the spectrum of cold dark matter near the scale that entered the horizon at matter–radiation equality. But the rise is gradual: the deceleration parameter stays positive, so the model does not produce acceleration. The interest of the result is that the timing comes out right without tuning, which is the coincidence the standard cosmological-constant picture leaves unexplained.

What carries the argument. Two ingredients carry the argument. The first is the Buchert averaging scheme, which replaces the Friedmann equations with volume-averaged equations in which a backreaction term $Q = tfrac{2}{3}(langle theta^2 rangle - langle theta rangle^2) - 2 langle sigma^2 rangle$ appears; $Q$ measures the variance of the local expansion rate minus the shear, and it is the only place where the spread of local expansion rates can raise the average. The second is the peak model of structure formation: structures are identified with peaks (and troughs) of a Gaussian linear density field smoothed on a scale $R$, giving a number density $n(nu, R)$ with $nu = delta/sigma_0$, which is converted into the fraction of mass, and hence of initial volume, ending up in regions of a given linear density contrast. Each region then evolves by the spherical collapse (or expansion) model, so its expansion rate is fixed by its linear density contrast. The smoothing scale is pinned by requiring $sigma_0(t,R) = 1$ at all times, which makes $R(t)$ grow and encodes the hierarchical merging of structure statistically. The average expansion rate is then the volume-weighted mean over the ensemble, and the model has no adjustable parameters beyond those of a flat matter-dominated universe.

Core claim. The paper computes the volume-averaged expansion rate $H$ of a spatially flat, statistically homogeneous and isotropic dust universe containing a realistic ensemble of non-linear structures, and finds that $Ht$ climbs from the Einstein–de Sitter value $2/3$ toward roughly $0.7$–$0.85$ over tens of billions of years. The mechanism is volume weighting: as structure formation proceeds, the fraction of space occupied by fast-expanding underdense regions grows, so the average expansion rate rises even though every individual region decelerates. The characteristic time is set by the turnover of the cold dark matter transfer function near $k_mathrm{eq}$, the wavenumber that entered the horizon at matter–radiation equality; the size of typical structures relative to the Hubble radius saturates at that era, and the expansion rate responds at the same time. The author is explicit that the effect is not acceleration: the deceleration parameter $q$ remains between about $0.2$ and $0.5$ in the models plotted, and the backreaction density parameter $|Omega_Q|$ stays below $0.04$, whereas roughly $0.2$ would be needed to match the observed acceleration. The stated reason is that the volume in slow-expanding overdense regions is too small to damp the expansion first, so there is nothing for the later void dominance to rebound from.

The premise it rests on. The model gives equal amounts of mass to overdense peaks and underdense troughs, so the volume of slow-expanding dense regions stays small; the author states that letting mass drain from voids into collapsing structures would raise that volume and is the most likely route to acceleration.

The coincidence problem has a candidate explanation: the era when the average expansion rate changes is fixed by the matter–radiation equality scale and the amplitude of primordial perturbations, not by a tuned vacuum energy.

Backreaction in a dust universe is bounded: since $Ht leq 1$ for dust with negligible vorticity, a firm measurement of $H_0 t_0 > 1$ would establish that exotic matter or modified gravity is required regardless of structure.

The same ensemble makes the average spatial curvature grow and turn negative at late times, so a backreaction explanation of the supernovae would predict large average curvature, which must then be checked against the cosmic microwave background peak positions rather than assumed small.

Improving the mass assignment so that voids are extinguished by surrounding overdense regions would raise the volume fraction of slow-expanding matter, which is the specific change the author identifies as most likely to convert the rise in $H$ into genuine acceleration.

Because the effect is generic to any statistically homogeneous ensemble with a transfer-function turnover, a future measurement of the expansion history at $z lesssim 1$ that shows a gradual rise without a change of sign in $q$ would be consistent with this picture.

The negative conclusion is tied to a symmetry the author himself flags as unrealistic: the Gaussian field gives equal mass to peaks and troughs, whereas in reality mass drains from voids into collapsing structures. A version of this calculation with a realistic, asymmetric mass function is the single most informative follow-up.

Newtonian simulations already show a large variance in the local expansion rate, which would suffice for acceleration were it not cancelled by shear; the relativistic case has no such cancellation, so the gap between the Newtonian and relativistic answers is where the decisive number lives.

If the effect is real, the natural observational signature is not a change in the sign of the deceleration parameter but a slow drift in the inferred dark-energy equation of state with redshift, since a gradual rise in $H$ mimics a smoothly varying component.

The paper's own remark that simulations reproduce a much smaller homogeneity scale than observations suggests a place to look for the non-Newtonian spatial curvature the argument requires, though that discrepancy may equally well have a mundane resolution.

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