A LQG Legend Writes Paper Claiming GR Explains Dark Matter Phenomena

[ohwilleke]

This whole discussion doesn't seem to clarify how Deur's gravitational field self-interaction really works.

I have put together an annotated bibliography of the relevant papers along with some prefatory explanations that draw mostly upon one of his power point presentations, to allow anyone who is interested to get a better grasp of these points.

 

[timmdeeg]

I have put together an annotated bibliography of the relevant papers along with some prefatory explanations that draw mostly upon one of his power point presentations, to allow anyone who is interested to get a better grasp of these points.

Very informative, thanks.

 

[kodama]

The X17 boson proposed to explain some subtle kinematics of nuclear matter decays interacts too strongly with other matter to be a dark matter candidate.

High Energy Physics - Experiment​

[Submitted on 29 Sep 2022]

Dark sector studies with the PADME experiment​

A.P. Caricato, M. Martino, I. Oceano, S. Spagnolo, G. Chiodini, F. Bossi, R. De Sangro, C. Di Giulio, D. Domenici, G. Finocchiaro, L.G. Foggetta, M. Garattini, A. Ghigo, P. Gianotti, T. Spadaro, E. Spiriti, C. Taruggi, E. Vilucchi, V. Kozhuharov, S. Ivanov, Sv. Ivanov, R. Simeonov, G. Georgiev, F. Ferrarotto, E. Leonardi, P. Valente, E. Long, G.C. Organtini, G. Piperno, M. Raggi, S. Fiore, P. Branchini, D. Tagnani, V. Capirossi, F. Pinna, A. Frankenthal

The Positron Annihilation to Dark Matter Experiment (PADME) uses the positron beam of the DAΦNE Beam-Test Facility, at the Laboratori Nazionali di Frascati (LNF) to search for a Dark Photon A′. The search technique studies the missing mass spectrum of single-photon final states in e+e−→A′γ annihilation in a positron-on-thin-target experiment. This approach facilitates searches for new particles such as long lived Axion-Like-Particles, protophobic X bosons and Dark Higgs. This talk illustrated the scientific program of the experiment and its first physics results. In particular, the measurement of the cross-section of the SM process e+e−→γγ at s√=21 MeV was shown.

Subjects:	High Energy Physics - Experiment (hep-ex); Instrumentation and Detectors (physics.ins-det)
Cite as:	arXiv:2209.14755 [hep-ex]
	(or arXiv:2209.14755v1 [hep-ex] for this version)
	https://doi.org/10.48550/arXiv.2209.14755

High Energy Physics - Phenomenology​

[Submitted on 19 Sep 2022]

Resonant search for the X17 boson at PADME​

Luc Darmé, Marco Mancini, Enrico Nardi, Mauro Raggi

We discuss the experimental reach of the Frascati PADME experiment in searching for new light bosons via their resonant production in positron annihilation on fixed target atomic electrons. A scan in the mass range around 17 MeV will thoroughly probe the particle physics interpretation of the anomaly observed by the ATOMKI nuclear physics experiment. In particular, for the case of a spin-1 boson, the viable parameter space can be fully covered in a few months of data taking.

Comments:	8 pages, 5 figures and 1 table
Subjects:	High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
Cite as:	arXiv:2209.09261 [hep-ph]
	(or arXiv:2209.09261v1 [hep-ph] for this version)
	https://doi.org/10.48550/arXiv.2209.09261

if x17 exist as a spin-1 boson it could be part of a larger dark sector

" In particular, for the case of a spin-1 boson, the viable parameter space can be fully covered in a few months of data taking. "

we'll see possible announced within a year (In particular, for the case of a spin-1 boson)

 

[ohwilleke]

High Energy Physics - Experiment​

[Submitted on 29 Sep 2022]

Dark sector studies with the PADME experiment​

A.P. Caricato, M. Martino, I. Oceano, S. Spagnolo, G. Chiodini, F. Bossi, R. De Sangro, C. Di Giulio, D. Domenici, G. Finocchiaro, L.G. Foggetta, M. Garattini, A. Ghigo, P. Gianotti, T. Spadaro, E. Spiriti, C. Taruggi, E. Vilucchi, V. Kozhuharov, S. Ivanov, Sv. Ivanov, R. Simeonov, G. Georgiev, F. Ferrarotto, E. Leonardi, P. Valente, E. Long, G.C. Organtini, G. Piperno, M. Raggi, S. Fiore, P. Branchini, D. Tagnani, V. Capirossi, F. Pinna, A. Frankenthal

Subjects:	High Energy Physics - Experiment (hep-ex); Instrumentation and Detectors (physics.ins-det)
Cite as:	arXiv:2209.14755 [hep-ex]
	(or arXiv:2209.14755v1 [hep-ex] for this version)
	https://doi.org/10.48550/arXiv.2209.14755

High Energy Physics - Phenomenology​

[Submitted on 19 Sep 2022]

Resonant search for the X17 boson at PADME​

Luc Darmé, Marco Mancini, Enrico Nardi, Mauro Raggi

Comments:	8 pages, 5 figures and 1 table
Subjects:	High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
Cite as:	arXiv:2209.09261 [hep-ph]
	(or arXiv:2209.09261v1 [hep-ph] for this version)
	https://doi.org/10.48550/arXiv.2209.09261

if x17 exist as a spin-1 boson it could be part of a larger dark sector

" In particular, for the case of a spin-1 boson, the viable parameter space can be fully covered in a few months of data taking. "

we'll see possible announced within a year (In particular, for the case of a spin-1 boson)

The odds of it not being ruled out are on the order of 0.01%

 

[kodama]

The odds of it not being ruled out are on the order of 0.01%

0.01% is pretty good compare with other BSM physics like EW scale SUSY, LUX dark matter detection, etc.

"the viable parameter space can be fully covered in a few months of data taking. "

0.01% for a chance of one of the biggest mysteries solved with in a year's time

0.01% seems much higher than other BSM HE-physics

the excitement is we should get some evidence for or ruled out within a year's time. i plan to check for updates once a month.

 

[Hornbein]

It seems to me that this question could be answered by examining rotation curves in spherical galaxies. Surely this has been done.

 

[ohwilleke]

It seems to me that this question could be answered by examining rotation curves in spherical galaxies. Surely this has been done.

Few galaxies are totally spherical, but observations have done the next best thing.

Deur has shown rather rigorously that the more spherical a galaxy is the less inferred dark matter content it has:

Observations indicate that the baryonic matter of galaxies is surrounded by vast dark matter halos, which nature remains unknown. This document details the analysis of the results published in MNRAS 438, 2, 1535 (2014) reporting an empirical correlation between the ellipticity of elliptical galaxies and their dark matter content. Large and homogeneous samples of elliptical galaxies for which their dark matter content is inferred were selected using different methods. Possible methodological biases in the dark mass extraction are alleviated by the multiple methods employed. Effects from galaxy peculiarities are minimized by a homogeneity requirement and further suppressed statistically. After forming homogeneous samples (rejection of galaxies with signs of interaction or dependence on their environment, of peculiar elliptical galaxies and of S0-type galaxies) a clear correlation emerges. Such a correlation is either spurious --in which case it signals an ubiquitous systematic bias in elliptical galaxy observations or their analysis-- or genuine --in which case it implies in particular that at equal luminosity, flattened medium-size elliptical galaxies are on average five times heavier than rounder ones, and that the non-baryonic matter content of medium-size round galaxies is small. It would also provides a new testing ground for models of dark matter and galaxy formation.

A. Deur, "A correlation between the dark content of elliptical galaxies and their ellipticity" (October 13, 2020).

Milgrom concluded that elliptical galaxies would have a much lower mass to light ratio than spiral ones back in 1983 with MOND (which is also true), but Deur's finding is more fine grained.

 

[Hornbein]

That seems "highly suggestive."

 

[ohwilleke]

That seems "highly suggestive."

Of course, the thing is that the strong correlation that is observed between galaxy shape and mass to light ratio, which implies in a dark matter particle scenario, the proportion of dark matter and ordinary matter in ay particular galaxy, has no good explanation.

Elliptical galaxies, generally speaking, tend to be larger than spiral galaxies. In a standard galaxy mass assembly scenario in the dark matter particle paradigm, they are formed by the mergers of smaller galaxies. So, they really ought to have all of the DM of their ancestors, rather than than much less.

 

[mitchell porter]

This whole discussion doesn't seem to clarify how Deur's gravitational field self-interaction really works.

In #62, #70, #76, I tried to identify Deur's methods of calculation. And a reminder, Ciotti #44 is the most thorough statement so far, of why one would not expect classical GR to produce such effects. So that's the gap one could try to bridge.

Also, even if that's not how GR works, one could try to design a modified gravity in which Deur's calculations *are* correct.

 

[ohwilleke]

In #62, #70, #76, I tried to identify Deur's methods of calculation. And a reminder, Ciotti #44 is the most thorough statement so far, of why one would not expect classical GR to produce such effects. So that's the gap one could try to bridge.

Also, even if that's not how GR works, one could try to design a modified gravity in which Deur's calculations *are* correct.

One of the things that I'm least clear about is whether Deur's method truly observes both the strong equivalence principle and the weak equivalence principle. I suspect that it observes only one but I'm not sure.

 

[PeterDonis]

QCD motivates the approach taken but isn't actually being used at all to make the calculations.

Even so, the QCD-like effects that Deur appears to be claiming should be much weaker for gravity than for QCD, as compared with the "Newtonian" component of the interaction, because the coupling constant for gravity is so much smaller, and the relative magnitudes of the effects should go like some power of the coupling constant.

 

[PeterDonis]

QCD motivates the approach taken

One major difference between QCD and gravity, though, is that the gauge group of QCD is compact, whereas the gauge group of gravity is not.

 

[kodama]

In #62, #70, #76, I tried to identify Deur's methods of calculation. And a reminder, Ciotti #44 is the most thorough statement so far, of why one would not expect classical GR to produce such effects. So that's the gap one could try to bridge.

Also, even if that's not how GR works, one could try to design a modified gravity in which Deur's calculations *are* correct.

one could try to design a modified gravity in which Deur's calculations *are* correct.--any suggestions for how to go do this ?

 

[kodama]

One major difference between QCD and gravity, though, is that the gauge group of QCD is compact, whereas the gauge group of gravity is not.

could you create a gravity that is like gr but with a compact gauge group

 

[mitchell porter]

could you create a gravity that is like gr but with a compact gauge group

Ashtekar variables describe general relativity in terms of a connection rather than a metric; and the connection is SU(2)-valued, and SU(2) is compact. But then LQG, etc, mostly use the complexification of SU(2), which is non-compact.

Perhaps someone would like to express Deur's formulas using Ashtekar's variables?

 

[PeterDonis]

the connection is SU(2)-valued

I'm not sure how that's true, but I'm not familiar with Ashtekar variables. If you have a good reference on those, that would be helpful.

That said, as I understand it, the gauge group of GR is the group of coordinate transformations, which is not compact. I don't think it matters whether the theory is expressed in terms of the metric or the connection.

 

[mitchell porter]

I said

Ashtekar variables describe general relativity in terms of a connection rather than a metric; and the connection is SU(2)-valued, and SU(2) is compact. But then LQG, etc, mostly use the complexification of SU(2), which is non-compact.

which is somewhat confused.

What I think is true, is that you can describe a Riemannian metric, and hence "Euclidean gravity", in terms of a connection valued in a compact group. But if you work in space-time, it's a Lorentzian signature, the metric is only "semi-Riemannian", and the connection will now take values in a non-compact group.

Ashtekar's original work indeed used a connection valued in a non-compact group. The interest was that the change of variables put the Hamiltonian constraint into a polynomial form resembling Yang-Mills theory. But having a quantum gauge theory based on a non-compact group is problematic.

Then Barbero argued that the quantum theory could be based on a real-valued (hence compact) SO(3) or SU(2) connection, at the price of the Hamiltonian constraint becoming non-polynomial again. Apparently this became the basis of most work in loop quantum gravity for a while. (Someone argued that the resulting theory is not actually a gauge theory, but I haven't read that paper.)

Much more recently, Peter Woit has been championing the idea that you could start with Euclidean quantum fields with an SO(4) local symmetry, factorize the SO(4) into two SU(2) factors, and use one SU(2) for a connection-based quantum gravity, and the other SU(2) for the weak gauge field of the standard model. Calculations in the empirical world of Lorentzian signature space-time would then be obtained as an analytic continuation, but the Euclidean theory would be fundamental. I think. It might be a distraction to mention this, but it's been discussed on some other threads recently.

@PeterDonis asked for references about the Ashtekar variables. I can't say that these are the best introduction, but you could try Wikipedia, Scholarpedia, and Ashtekar's original paper. Sections 3, 3.1 of Woit's paper may actually be a quick introduction to the ideas.

Returning to the issue of compactness, it now seems as if there are only two leading proposals for how to get general relativity from a compact gauge group. One is just to work in Euclidean signature. The other is Barbero's proposal, which is about selecting a compact subgroup within the non-compact group (apparently, the famous Immirzi parameter of loop quantum gravity specifies which copy of SU(2) inside SL(2,C) one is using?), and it's now unclear to me if it actually works.

 

[kodama]

I said

which is somewhat confused.

What I think is true, is that you can describe a Riemannian metric, and hence "Euclidean gravity", in terms of a connection valued in a compact group. But if you work in space-time, it's a Lorentzian signature, the metric is only "semi-Riemannian", and the connection will now take values in a non-compact group.

Ashtekar's original work indeed used a connection valued in a non-compact group. The interest was that the change of variables put the Hamiltonian constraint into a polynomial form resembling Yang-Mills theory. But having a quantum gauge theory based on a non-compact group is problematic.

Then Barbero argued that the quantum theory could be based on a real-valued (hence compact) SO(3) or SU(2) connection, at the price of the Hamiltonian constraint becoming non-polynomial again. Apparently this became the basis of most work in loop quantum gravity for a while. (Someone argued that the resulting theory is not actually a gauge theory, but I haven't read that paper.)

Much more recently, Peter Woit has been championing the idea that you could start with Euclidean quantum fields with an SO(4) local symmetry, factorize the SO(4) into two SU(2) factors, and use one SU(2) for a connection-based quantum gravity, and the other SU(2) for the weak gauge field of the standard model. Calculations in the empirical world of Lorentzian signature space-time would then be obtained as an analytic continuation, but the Euclidean theory would be fundamental. I think. It might be a distraction to mention this, but it's been discussed on some other threads recently.

@PeterDonis asked for references about the Ashtekar variables. I can't say that these are the best introduction, but you could try Wikipedia, Scholarpedia, and Ashtekar's original paper. Sections 3, 3.1 of Woit's paper may actually be a quick introduction to the ideas.

Returning to the issue of compactness, it now seems as if there are only two leading proposals for how to get general relativity from a compact gauge group. One is just to work in Euclidean signature. The other is Barbero's proposal, which is about selecting a compact subgroup within the non-compact group (apparently, the famous Immirzi parameter of loop quantum gravity specifies which copy of SU(2) inside SL(2,C) one is using?), and it's now unclear to me if it actually works.

does Euclidean quantum fields change the physics compared to LQG ?

 

[PAllen]

Don’t know if this has come up earlier in this thread, but there are n-body formulas at the PPN level that make no assumptions about spherical symmetry or a dominant central mass. I wonder, given modern computing power, if anyone has tried solving these for e.g. 50 bodies with initial conditions similar to a galaxy, with a central BH, and see what they lead to. Note, PPN formulas should include all nonlinear effects present except in strong fields and very high speed relative motion - and none of these should be relevant in a galaxy.

 

[kodama]

Don’t know if this has come up earlier in this thread, but there are n-body formulas at the PPN level that make no assumptions about spherical symmetry or a dominant central mass. I wonder, given modern computing power, if anyone has tried solving these for e.g. 50 bodies with initial conditions similar to a galaxy, with a central BH, and see what they lead to. Note, PPN formulas should include all nonlinear effects present except in strong fields and very high speed relative motion - and none of these should be relevant in a galaxy.

so what Does Deur predict

 

[PAllen]

so what Does Deur predict

I have no idea. I have never read Deur papers. But I do know about the use of PPN approximation to improve solar system approximation for all large bodies (including the large planetoids) way, way beyond what Newtonian gravity can achieve. And also, that PPN is able to predict GW wave forms for inspiralling BH with precision up until the final moments. It is used as a cross check on numerical relativity codes. But unlike numerical relativity it might just be possible to simulate e.g. 50 bodies with conceivable computer power.

Point is Deur makes claims about what classical GR would predict, but he does not demonstrate these claims. Using PPN approximation would be a possible way to verify or refute his claims.

 

[PeterDonis]

I wonder, given modern computing power, if anyone has tried solving these

I believe the paper referred to in post #44 of this thread did something like this (using the gravitomagnetic formalism, but AFAIK if the effects of many bodies are included this is mathematically equivalent to the formalism you refer to), and found, as mainstream opinion expected, that the GR corrections are too small to account for observed rotation curves with just the visible matter.

what Does Deur predict

As I understand it, Deur's claims fall, broadly speaking, into two categories:

(1) There are nonlinear effects in classical GR, amounting to large corrections to the Newtonian behavior for highly non-spherical cases, that are not properly accounted for in the usual models.

(2) There are non-perturbative effects in quantum gravity, analogous to things like gluon flux tubes in QCD, that can produce large corrections to the classical behavior but are not (obviously) taken into account in classical models.

Papers like the one linked to in post #44, IMO, cast serious doubt on Deur's claims in category 1 above. One could still argue that there are additional nonlinear effects that the formalisms used do not include, but such claims become increasingly unlikely as more and more detailed classical treatments are done.

The main issue as I understand it with Deur's claims in category 2 is that there is no well-defined theory behind them; they are just heuristics based on claimed similarities between quantum gravity (for which we have no well-defined theory at present) and QCD. These are interesting theoretical areas to look at, but in the absence of a well-defined theory from which definite predictions can be made, they remain speculative.

 

[ohwilleke]

Properly considering the five

parameters in the PPN formalism (ideally at the GR values of 1) should be what is necessary, although it is a bit hard (for me anyway) to tell precisely what the PPN formalism is disregarding.

But, this may not be right in light of this quotation from the introduction portion of A. Deur, "An explanation for dark matter and dark energy consistent with the standard model of particle physics and General Relativity." 79 Eur. Phys. J. C , 883 (October 29, 2019). https://doi.org/10.1140/epjc/s10052-019-7393-0

In GR, self-interaction becomes important for 𝐺𝑀/𝐿‾‾‾‾‾‾‾√GM/L large enough (L is the system characteristic scale), typically for 𝐺𝑀/𝐿‾‾‾‾‾‾‾√>𝑟𝑠𝑖𝑚10−3GM/L>rsim10−3 as discussed in Ref. [5] or exemplified by the Hulse-Taylor binary pulsar [8], the first system in which GR was experimentally tested in its strong regime, which has 𝐺𝑀/𝐿‾‾‾‾‾‾‾√=10−3GM/L=10−3. As in the case of QCD, self-interaction increases the binding compared to Newton’s theory. Since the latter is used to treat the internal dynamics of galaxies or galaxy clusters, its neglect of self-interaction may contribute to – or even create – the missing mass problem [4, 5, 9].

In Ref. [4] a non-perturbative numerical calculation based on Eq. (2) is applied in the static limit to spiral galaxies and clusters. A non-perturbative formalism (lattice technique) – rather than a perturbative one such as the post-Newtonian formalism – was chosen because in QCD, confinement is an entirely non-perturbative phenomenon, unexplainable within a perturbative approach.

I have taken a screen shot of Equation (2) and the related text to avoid having to format it with LaTex:

Reference [4] in the quoted material is A. Deur, “Implications of Graviton-Graviton Interaction to Dark Matter” (May 6, 2009) (published at 676 Phys. Lett. B 21 (2009)). This paper is only 11 pages including references, and is quite clear about the equations used, the way that those equations were derived, and exactly what assumptions are being made in doing so. The scalar approximation used to make it mathematically tractable is essentially making a static approximation that disregards gravitomagnetic effects, kinetic energy, electromagnetic flux, and pressure on the RHS of the Einstein field equations.

Reference [5] is to A. Deur, "Self-interacting scalar fields at high-temperature." 77 Eur. Phys. J. C, 412 (2017). https://doi.org/10.1140/epjc/s10052-017-4971-x

Reference [8] is to R.A. Hulse, J.H. Taylor, "Discovery of a pulsar in a binary system." 195 Astrophys. J. L51 (1975).

Reference [9] is to A. Deur, "A relation between the dark mass of elliptical galaxies and their shape", 438(2) Monthly Notices of the Royal Astronomical Society 1535–1551 (February 21, 2014). https://doi.org/10.1093/mnras/stt2293

 

[ohwilleke]

The main issue as I understand it with Deur's claims in category 2 is that there is no well-defined theory behind them; they are just heuristics based on claimed similarities between quantum gravity (for which we have no well-defined theory at present) and QCD. These are interesting theoretical areas to look at, but in the absence of a well-defined theory from which definite predictions can be made, they remain speculative.

The theory seems to be reasonably well defined and makes definite predictions. It may not be rigorously derived from first principles based upon the quantum field theory of a massless spin-2 graviton, and it may not have yet been thoroughly reviewed for theoretical consistency (something that it took almost three decades to do, for example, with renormalization in the SM after it started to be widely used), but the formulas are there and are possible to calculate with.

In particular, while the physical constant that modifies the self-interaction term ought to be possible to calculate from first principles using only Newton's constant and the speed of light, Deur actually uses the same data set that was used to determine the MOND acceleration constant to establish it without doing the first principles calculation, at least for purposes of the lattice calculation in Reference [4] cited in post #114 in this thread.

 

[PeterDonis]

the physical constant that modifies the self-interaction term

Isn't this just $16 \pi G$? (Or its square root?)

 

[ohwilleke]

Isn't this just $16 \pi G$? (Or its square root?)

In the spiral galaxy case, Deur's approach gives rise to the following formula, the first term of which is Newtonian gravity, and the second of which is the self-interaction term (ignoring higher order terms in an infinite series that are small by comparison):

F = GM/r2 + c^2(aπGM)^0.5/(2√2)r

where F is the effective gravitational force, G is Newton's constant, c is the speed of light, M is ordinary baryonic mass of the gravitational source, r is the distance between the source mass and the place that the gravitational force is measured, and a is a physical constant that is the counterpart of a(0) in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10^−44 m^−3s^2.

Thus, the self-interaction term that it modifies is proportionate to (GM)^0.5/r and is initially much smaller that the first order Newtonian gravity term in stronger fields, but it declines more slowly than the Newtonian term with distance until it is predominant.

 

[PeterDonis]

a is a physical constant that is the counterpart of a(0) in MOND

Ah, ok, got it.

 

[mitchell porter]

One comment on Deur's "Self-interacting scalar fields at high-temperature" (mentioned in @ohwilleke's #114, this thread). It cites arXiv:0709.2042 (Deur's reference 15) as evidence that QCD can be approximated by a scalar field, in a way which motivates Deur's own scalar approximation of GR. But this cited paper has been criticized for a reason I gave in #76: the validity of the scalar approximation requires that some physical influence ("constraint forces") prevents all the other degrees of freedom from coming to life.

I guess that Deur's reasoning may be found in his "Implications of Graviton-Graviton Interaction to Dark Matter": the $T^{00}$ component of the stress-energy tensor dominates "in the stationary weak field approximation", and therefore the $\phi^{00}$ component of the gravitational $\phi$ field (its relation to the metric is that $g_{\mu \nu} = (e^{k \phi})_{\mu \nu}$) should similarly dominate. This is an assumption that one might want to scrutinize.

 

[strangerep]

(Sigh.) I'm reaching a point where I'm starting to suspect think Deur's expansion method might be BS. He never seems to explain it properly afaict. (If you think you know a paper where he does explain it extensively, please tell me.)

E.g., in his 2009 paper (arXiv:0901.4005), he says he expands in powers of $h_{\mu\nu} = g_{\mu\nu} - \eta_{\mu\nu}$, but rescales $h_{\mu\nu} \to h_{\mu\nu}/\sqrt{M}$, or $h_{\mu\nu} \to h_{\mu\nu}/\sqrt{G}$. However, the 1st order solution for the original (non-rescaled) $h_{\mu\nu}(G)$ must be the Newtonian solution, and that is always $O(G)$.

Newtonian solutions for disk galaxies with physically realistic exponential radial mass distributions have been around for ages (see the treatment in galaxiesbook.org for a readable account). The solutions always have a $G$ at the front.

So in the 2nd-order Einstein equations we have 2nd-order derivatives of $h(G^2)$, but the $\Gamma\,\Gamma$ terms in $R_{\mu\nu}$ cannot contain $h(G^2)$ because a $\Gamma$ must contain at least one $\partial h(G)$. The $\Gamma\,\Gamma$ could of course contain quadratic expressions in $\partial h(G)$. So the 2nd-order Einstein equations
are of the form $$\partial^2 h(G^2) ~+~ h(G) \partial^2 h(G) ~+~ \partial h(G) \partial h(G) ~=~ 0 ~.$$ But this only gives an expansion of the physical (dimensionless) metric in powers of $G$. I don't see where an $h$ (of any order) can enter that only involves $\sqrt{G}$ (without a sleight-of-hand rescaling).