Round 1A toy model and a critique
The question
Posed by ChatGPT, unprompted.
“For External Gravity Cosmology, build a simple toy model showing how an external tidal gravitational field could make expansion look accelerated, and compare its prediction with dark energy.”
The toy model
ChatGPT’s answer, lightly edited for formatting.
A useful EGC toy model can be built without claiming that it is yet a full relativistic cosmology.
1. Start with ordinary expansion
For a homogeneous universe, write the acceleration equation schematically as
\[ \frac{\ddot a}{a} = -\frac{4\pi G}{3}\rho + \frac{\Lambda c^2}{3}. \]The first term slows expansion because of matter. The second produces accelerated expansion.
Planck’s standard ΛCDM fit has roughly \(\Omega_m=0.315\), with the remaining late-time energy budget attributed largely to Λ.
2. Replace Λ with an external gravitational field
Imagine our observable universe sitting inside a much larger gravitational environment. Locally, represent the external gravitational potential as
\[ \Phi_{\rm ext} = -\frac12 T_{ij}x_i x_j , \]where \(T_{ij}\) is the tidal gravitational tensor. The resulting acceleration is
\[ g_i=-\frac{\partial\Phi}{\partial x_i} = T_{ij}x_j. \]So, in one dimension,
\[ g_{\rm ext}=T x. \]If \(T>0\), the external field produces an outward relative acceleration between objects separated by distance \(x\).
Since cosmological distances scale approximately as \( x=a(t)r \), we can write the toy-model contribution as
\[ \boxed{\frac{\ddot a}{a}=T} \]and therefore
\[ \boxed{\frac{\ddot a}{a} = -\frac{4\pi G}{3}\rho+T.} \]That is the basic EGC toy model.
3. Make it numerically equivalent to dark energy
For a simple demonstration, take \( H_0=67.4\ {\rm km\,s^{-1}\,Mpc^{-1}} \) and use a Λ-like contribution of approximately \( T=0.685H_0^2 \). Then, at the present epoch,
\[ \frac{\ddot a}{a} = -\frac{1}{2}\Omega_m H_0^2 + 0.685H_0^2 . \]With \(\Omega_m=0.315\),
\[ \frac{\ddot a}{a} \approx (-0.158+0.685)H_0^2 \]or
\[ \boxed{\frac{\ddot a}{a}\approx0.527H_0^2.} \]The important point is that an observer measuring only \(a(t)\) could interpret this exactly as a cosmological-constant contribution.
| Dark energy interpretation | EGC toy interpretation | |
|---|---|---|
| Source | Λ | External gravitational field |
| Expansion term | \(+\Lambda c^2/3\) | \(+T_{\rm ext}\) |
| Effect on \(a(t)\) | Accelerates expansion | Accelerates apparent expansion |
| Spatial signature | Isotropic | Potentially anisotropic |
| Time dependence | Constant for Λ | Could vary with external structure |
| Directional effects | Essentially none | Potentially present |
An important problem
A real gravitational tidal field isn’t normally isotropic. In vacuum, the tidal tensor has approximately
\[ T_{xx}+T_{yy}+T_{zz}=0. \]That means an ordinary external tidal field tends to stretch in one direction while compressing another, rather than simply pushing everything outward equally. That is actually useful for EGC, because it gives the hypothesis something testable.
The toy model should therefore be expanded to
\[ \boxed{\frac{\ddot a_i}{a_i} = -\frac{4\pi G}{3}\rho+T_i} \]with
\[ T_x+T_y+T_z\approx0. \]An external gravitational environment could consequently produce something like
\[ a_x(t)\neq a_y(t)\neq a_z(t). \]If averaging over the observable region makes that directional effect appear approximately isotropic, an observer could infer an effective acceleration term.
A potentially interesting observational distinction
The simplest Λ model predicts an essentially direction-independent acceleration. An external-gravity model naturally predicts a chain:
- external gravitational structure
- tidal field
- directional expansion and redshift effects
- apparent accelerated expansion.
This connects directly to the prediction developed for EGC,
\[ \boxed{ \begin{array}{c} \text{redshift fluctuations} \\ \leftrightarrow \\ \text{CMB fluctuations} \end{array} } \]if both are responding to the same external gravitational environment.
So the strongest version of the EGC idea isn’t simply “External gravity replaces Λ.” It is more specifically this: an external gravitational field produces large-scale variations in the observed expansion field that, when averaged by an observer inside the observable universe, can be interpreted as accelerated expansion.
That is a much more interesting hypothesis because it makes a distinction from ΛCDM: the acceleration should carry spatial, tidal structure rather than being perfectly homogeneous.
One major issue remains
The toy model has to show quantitatively how a realistic external mass distribution beyond the Universal Event Horizon produces the required effective \(T_i\), while respecting general relativity. That is the next calculation I’d regard as the critical EGC test.
Critique of the toy model
An analysis of ChatGPT’s answer above.
The setup and arithmetic are sound.
What’s right
The acceleration equation, the tidal-potential form and the numbers check out: with \( T = 0.685\,H_0^2 \) you get \( \ddot a/a \approx 0.527\,H_0^2 \), the ΛCDM present-day value (deceleration parameter \(q_0 \approx -0.53\)). But that step is circular: a constant, isotropic \(T\) is a cosmological constant under another name, so matching Λ that way doesn’t test EGC.
The strong part
The anisotropy prediction is real and testable. An external field should imprint a directional (mainly quadrupole) pattern in supernova distances and redshifts that lines up with CMB features, matching the EGC redshift \(\leftrightarrow\) CMB fluctuation prediction. The catch is that the CMB already tightly limits any directional difference in the expansion rate, so the predicted effect must be small enough to have escaped detection so far but large enough to measure.
Round 2ChatGPT’s follow-up and a critique
ChatGPT’s follow-up: comparing gravity models
ChatGPT’s second round, titled “Compare Gravity Models”, lightly edited for formatting.
The calculation exposes both what works mathematically and the major problem EGC has to solve.
1. Required acceleration
Using the Planck ΛCDM values \(H_0=67.4\) km/s/Mpc and \(\Omega_\Lambda\simeq0.685\), the acceleration attributed to Λ today is
\[ a_\Lambda=\Omega_\Lambda H_0^2 r. \]Numerically,
\[ \boxed{\Omega_\Lambda H_0^2=3.27\times10^{-36}\ {\rm s^{-2}}}. \]These Planck parameters are \(H_0=67.4\pm0.5\) km/s/Mpc and \(\Omega_m=0.315\pm0.007\).
So EGC would need an external gravitational field whose tidal gradient is approximately
\[ \boxed{T_{\rm EGC}\sim3.3\times10^{-36}\ {\rm s^{-2}}}. \]2. Put an external mass beyond the observable boundary
Take the Universal Observable/Event Horizon in the EGC picture as approximately
\[ R=46.5\ {\rm billion\ light-years}. \]For a simple point-like external mass \(M\), the tidal scale is
\[ T\sim\frac{GM}{R^3}. \]Set that equal to the required acceleration:
\[ \frac{GM}{R^3}=3.27\times10^{-36}. \]Therefore
\[ M=\frac{T R^3}{G}. \]At \(R=47\) billion light-years:
\[ \boxed{M\approx2.2\times10^{24}M_\odot}. \]That is an enormous mass. For comparison:
| Distance to external mass | Mass required |
|---|---|
| 47 billion ly | \(2.2\times10^{24}M_\odot\) |
| 50 billion ly | \(2.6\times10^{24}M_\odot\) |
| 100 billion ly | \(2.1\times10^{25}M_\odot\) |
| 500 billion ly | \(2.6\times10^{27}M_\odot\) |
So a single conventional object isn’t a plausible EGC mechanism. But that is not necessarily what the EGC hypothesis proposes.
3. The more interesting case: an external mass distribution
Suppose the matter outside our observable boundary isn’t one object but an enormous, uneven gravitational environment. Then the external potential can be expanded around us:
\[ \Phi_{\rm ext} = \Phi_0 + g_i x_i + \frac12 T_{ij}x_ix_j+\cdots \]The first derivative,
\[ g_i=-\nabla_i\Phi, \]is just a common acceleration. We wouldn’t detect that internally.
The important term is
\[ T_{ij}=\frac{\partial^2\Phi}{\partial x_i\partial x_j}, \]the tidal field. Tidal effects are specifically the spatial variation of gravitational acceleration.
That gives an EGC expansion equation of the form
\[ \boxed{ \frac{\ddot a_i}{a_i} = -\frac{4\pi G}{3}\rho + T_i } \]as a first-order toy model.
4. The crucial problem
For an ordinary gravitational tidal field,
\[ T_x+T_y+T_z\simeq0 \]in a vacuum region. So EGC cannot simply say that external gravity produces three equal positive accelerations. That would not be an ordinary Newtonian/GR tidal field.
Instead, the external field would look something like
\[ T_x=+T \] \[ T_y=-\frac{T}{2} \] \[ T_z=-\frac{T}{2}. \]One direction expands faster while the other two expand more slowly. That is actually a potentially useful prediction.
5. Connecting it to the EGC idea
Imagine that the external gravitational environment contains many structures, instead of one mass:
The observable universe samples different portions of this external gravitational field. The observed redshift would then contain something like
\[ \begin{aligned} z_{\rm observed} = {} & z_{\rm expansion} + z_{\rm gravitational} \\ & {} + z_{\rm tidal}. \end{aligned} \]And because the external field changes across the sky,
\[ z_{\rm tidal}=z_{\rm tidal}(\theta,\phi). \]That produces direction-dependent redshift residuals.
6. A testable EGC quantity
Instead of trying to prove EGC by making it reproduce the average ΛCDM acceleration, we should ask:
\[ \boxed{ \begin{array}{c} \text{Does the measured cosmic} \\ \text{acceleration contain a} \\ \text{coherent tidal pattern?} \end{array} } \]Specifically, EGC predicts that after subtracting the standard isotropic expansion,
\[ \Delta H(\theta,\phi) = H(\theta,\phi)-\bar H \]should contain large-scale angular structure.
And crucially, if the other EGC proposition is correct, the same structure should appear in the CMB:
\[ \boxed{ \Delta H(\theta,\phi) \quad\longleftrightarrow\quad \Delta T_{\rm CMB}(\theta,\phi) } \]or, more specifically, correlated residuals between redshift structure and CMB temperature fluctuations.
The big issue
There is one important correction to my earlier toy model: a simple external tidal field does not automatically produce an isotropic accelerated expansion. It produces anisotropy.
Therefore, for EGC to replace dark energy, the model needs a mechanism by which a large-scale external gravitational field produces an approximately isotropic average expansion while retaining small directional fluctuations.
That is where I think the next EGC toy model should go. We can construct an explicit model with a spherical shell of external matter plus uneven concentrations on that shell, calculate \(T_{ij}\), and see whether its angular average can produce the apparent Λ term while its fluctuations produce the redshift/CMB correlation proposed for EGC.
Critique of the follow-up
An analysis of ChatGPT’s second round above.
What checks out
\( \Omega_\Lambda H_0^2 \approx 3.27\times10^{-36}\ {\rm s^{-2}} \) and the mass table (\(\sim2\times10^{24}M_\odot\) at 47 Gly, scaling as \(R^3\)) are arithmetically right. One small correction: a point mass’s tidal stretch along its axis is \(2GM/R^3\), so the required masses are about half. The required mass is only about 10–20 times the mass of our observable universe. That is a natural scale (since \(T\sim G\rho\) and \(H^2\sim G\rho\)), not an absurd one.
Next step
The critical calculation is still ahead: how a realistic mass distribution beyond our horizon shapes the expansion we observe, while respecting general relativity, and whether the predicted match between direction-dependent expansion and CMB temperature shows up in real data. This page will be updated as that work comes in.