Cosmology · Testing gravity and alternative cosmological models

Relaxing homogeneity and other assumptions

Use an exact finite spherical dust solution of general relativity to separate spatial flatness, homogeneity, directional expansion and observational light-cone predictions.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Define areal radius and two directional expansion rates.
Verify a regular finite inhomogeneous dust patch within GR.
Distinguish spatial profiles from a fit to distance–redshift observations.

Relaxing homogeneity need not change Einstein gravity

An alternative cosmological explanation can modify a field equation or instead relax an assumption used to construct a particular solution. These are different kinds of change.

The Lemaître–Tolman–Bondi family describes spherically symmetric inhomogeneous pressureless dust within general relativity. A chosen member still needs observer conditions and tests beyond its spatial profile.

Use an exact finite expanding dust patch with zero cosmological constant and curvature function E(r) = 0. Its radial comoving label r is dimensionless over 0≤r≤1.

The patch is supplied to illustrate distinctions. It is not a real void, a global cosmological fit or an established replacement for acceleration, and its regularity is asserted only on the stated finite time interval.

Use measured sphere area to define radius

Let R(t,r) be the areal radius in metres, so each symmetry sphere has area 4πR². The proper-time metric is ds² = −c²dt²+R′²dr²/(1+2E)+R²dΩ², with prime meaning ∂/∂r at fixed t.

Supply enclosed mass function M(r) = M₀r³, M₀>0 in kg, reference time T>0 in seconds and bang-time function t_B(r) = −εTr² with ε = 1/10. Let τ = t/T range from 1 to 2.

The dust equations are Ṙ² = 2GM/R+2Ec² and ρ = M′/(4πR²R′). For E = 0 the expanding solution obeys R³ = (9/2)GM(r)[t−t_B(r)]².

Define R₀³ = 9GM₀T²/2 and ρ_* = 1/(6πGT²). R₀ is a length scale and ρ_* is a mass density; the coordinate label r must not be confused with the areal radius R.

The density can vary on a spatially flat slice

Introduce positive auxiliary functions A = τ+εr² and B = τ+(7/3)εr². Direct substitution and differentiation give R/R₀ = rA^(2/3), R′/R₀ = A^(−1/3)B and ρ/ρ_* = 1/(AB).

A and B remain positive over the complete chosen domain. Thus R′>0, density is positive using the regular central limit, and there is no shell crossing on this finite patch.

At r = 0 use the continuous limits ρ/ρ_* = τ⁻² and R′/R₀ = τ^(2/3). The centre has zero sphere area as expected for a regular symmetry centre; a naive substitution into the density’s 0/0 form would be inappropriate.

For E = 0 at fixed t, dR = R′dr makes the spatial metric dR²+R²dΩ². The slices are spatially flat even though ρ varies with radius; spatial flatness alone does not enforce FLRW homogeneity.

Exact finite dust profiles, with rounded areal radii and density ratios
τ	r	R/R₀	ρ/ρ_*
1	0	0	1
1	1/2	0.508299	0.921836
1	1	1.065602	300/407 ≈ 0.737101
2	0	0	1/4
2	1/2	0.800301	0.239916
2	1	1.639883	0.213220

Radial and transverse expansion need not agree

Define transverse expansion H_⊥ = Ṙ/R and radial expansion H_∥ = Ṙ′/R′. They track sphere size and the separation of nearby radial dust shells respectively.

The solution gives H_⊥T = 2/(3A) and H_∥T = −1/(3A)+1/B. At the regular centre both reduce to 2/(3τ), but away from it they differ.

For τ = 1,r = 1, A = 11/10 and B = 37/30. Hence ρ/ρ_* = 300/407, H_⊥T = 20/33 and H_∥T = 620/1221.

As an optional consistency check, H_⊥²+2H_⊥H_∥ = 8πGρ follows for this flat-slice dust solution. Its inhomogeneous directional expansion is a result within GR, rather than evidence that Einstein’s equations were modified.

Dimensionless directional expansion rates on the same finite patch
τ	r	H_⊥T	H_∥T
1	0	2/3	2/3
1	1/2	0.650407	0.619679
1	1	20/33 ≈ 0.606061	620/1221 ≈ 0.507781
2	0	1/3	1/3
2	1/2	0.329218	0.321221
2	1	0.317460	0.289031

An exact supplied finite LTB dust patch remains within GR, with E = Lambda = 0, M = M0 r³, bang time t_B = −0.1 T r², dimensionless coordinate 0 <= r <= 1 and 1 <= tau = t/T <= 2. The upper panel shows rho/rho* = 1/[(tau+0.1r²)(tau+7r²/30)] on two fixed-time slices, tau = 1 and 2. Both are positive and vary radially, even though the spatial slices are flat. The lower panel compares transverse and radial dimensionless rates at tau = 1 only: H_perp T = 2/(3A), H_parallel T = −1/(3A)+1/B. They agree in the regular centre and differ away from it. The full table gives the areal-radius, density and both-rate values on both slices. Coordinate r is not the areal radius R. No shell crossing occurs in this finite domain; no observer, exterior matching, null-ray integration, acceptable early history or actual distance fit is supplied. Complete two-slice profile values, six-place rounding; centre uses regular limits. Columns: τ; r; R/R₀; ρ/ρ*; H_⊥T; H_∥T. Rows: 1; 0; 0.000000; 1.000000; 0.666667; 0.666667 / 1; 1/2; 0.508299; 0.921836; 0.650407; 0.619679 / 1; 1; 1.065602; 0.737101; 0.606061; 0.507781 / 2; 0; 0.000000; 0.250000; 0.333333; 0.333333 / 2; 1/2; 0.800301; 0.239916; 0.329218; 0.321221 / 2; 1; 1.639883; 0.213220; 0.317460; 0.289031.

A spatial solution is different from an observational explanation

The nonconstant t_B supplies a decaying-mode inhomogeneity. The fact that this patch is regular for 1≤τ≤2 does not establish an acceptable early history or realistic formation from initial conditions.

No exterior matching was supplied. The patch’s boundary conditions and extension to a full cosmological solution therefore remain additional questions.

Distance–redshift observations follow an observer’s light cone, rather than a single constant-time spatial table. Specify the observer and integrate null rays, redshift and distances before comparing a model with those observations.

A credible explanation also faces other consistency tests, including relevant early-Universe and structure constraints. Relaxing homogeneity provides a well-defined modeling direction, while this finite example claims only the exact dust profiles and their stated limits.

Worked reasoning

Calculate the outer point at τ = 1

What density and directional rates occur at r = 1,τ = 1?

Use A = 1+1/10 = 11/10 and B = 1+7/30 = 37/30.
The density ratio is 1/(AB) = 300/407, and H_⊥T = 2/(3A) = 20/33.
The radial rate is H_∥T = −10/33+30/37 = 620/1221.

The two rates differ at this point despite the model remaining an exact GR dust solution.

Try the reasoning

What are the central density and directional rates at τ = 2?
Show the answer and reasoning
ρ/ρ_* = 1/4 and H_⊥T = H_∥T = 1/3.
Use the continuous regular-centre limits rather than dividing the raw density formula by zero.
Does E = 0 force the density to be homogeneous?
Show the answer and reasoning
No.
It gives flat spatial slices here, while the supplied nonconstant bang-time function makes density and directional expansion vary with r.
Do these spatial tables establish a fit to observed luminosity distances?
Show the answer and reasoning
No.
An observer, null propagation, redshift mapping and broader consistency tests have not been computed.

What to carry forward

Changing a cosmological assumption can stay within the same gravitational theory.
Spatial flatness, homogeneity and equal directional expansion are different conditions.
A finite spatial solution needs light-cone predictions and wider tests before becoming an observational explanation.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.

Lemaitre-Tolman-Bondi model and accelerating expansionKari EnqvistTeaching resourcearxiv.orgRead source