Astronomy:Borde–Guth–Vilenkin theorem

Short description: Theorem in physical cosmology

The Borde–Guth–Vilenkin theorem, or the BGV theorem, is a theorem in physical cosmology which deduces that any universe that has, on average, been expanding throughout its history cannot be infinite in the past but must have a past spacetime boundary. The theorem does not assume any specific mass content of the universe and it does not require gravity to be described by Einstein field equations. It is named after the authors Arvind Borde, Alan Guth and Alexander Vilenkin, who developed its mathematical formulation in 2003. The BGV theorem is also popular outside physics, especially in religious and philosophical debates.

Derivation

For flat spacetime

Here is an example of derivation of the BGV theorem for an expanding homogeneous isotropic flat universe (in units of speed of light c=1). Which is consistent with ΛCDM model, the current model of cosmology. However, this derivation can be generalized to an arbitrary space-time with no appeal to homogeneity or isotropy.

The Friedmann–Lemaître–Robertson–Walker metric is given by

[math]\displaystyle{ ds=dt^2-a^2(t)dx_idx^i }[/math],

where t is time, xⁱ (i=1,2,3) are the spatial coordinates and a(t) is the scale factor. Along a timeline geodesic x_i = constant, we can consider the universe to be filled with comoving particles. For an observer with proper time τ following the world line x^μ(τ), has a 4-momentum [math]\displaystyle{ P^\mu=m dx^\mu/d\tau=(E,\mathbf p) }[/math], where [math]\displaystyle{ E=\sqrt{p^2+m^2} }[/math] is the energy, m is the mass and p=|p| the magnitude of the 3-momentum.

From the geodesic equation of motion, it follows that [math]\displaystyle{ p(t)=p_{\rm f}\; a(t_{\rm f})/a(t) }[/math] where p_f is the final momentum at time t_f. Thus

[math]\displaystyle{ \int_{t_{\rm i}}^{t_{\rm f}} H(\tau) d\tau=\int_{a(t_{\rm i})}^{a(t_{\rm f})}\frac{m da}{\sqrt{m^2a^2+p^2a(t_{\rm f})}}=F(\gamma_{\rm f})-F(\gamma_{\rm i})\leq F(\gamma_{\rm f}) }[/math],

where t_i is an initial time, [math]\displaystyle{ H=\dot{a}/a }[/math] is the Hubble parameter, and

[math]\displaystyle{ F(\gamma)=\frac{1}{2}\ln\left(\frac{\gamma+1}{\gamma-1}\right) }[/math],

γ being the Lorentz factor. For any non-comoving observer γ>1 and F(γ)>0.

The expansion rate averaged over the observer world line can be defined as

[math]\displaystyle{ H_{\rm av}=\frac{1}{\tau_{\rm f}-\tau_{\rm i}}\int_{t_{\rm i}}^{t_{\rm f}} H(\tau) d\tau }[/math].

Assuming [math]\displaystyle{ H_{\rm av}\gt 0 }[/math] it is follows that

[math]\displaystyle{ \tau_{\rm f}-\tau_{\rm i} \leq \frac{F(\gamma_{\rm f})}{H_{\rm av}} }[/math].

Thereby any non-comoving past-directed timelike geodesic satisfying the condition [math]\displaystyle{ H_{\rm av}\gt 0 }[/math], must have a finite proper length, and so must be past-incomplete.

Limitations and criticism

Alternative models, where the average expansion of the universe throughout its history does not hold, have been proposed under the notions of emergent spacetime, eternal inflation, and cyclic models. Vilenkin and Audrey Mithani have argued that none of these models escape the implications of the theorem. In 2017, Vilenkin stated that he does not think there are any viable cosmological models that escape the scenario.

Sean M. Carroll argues that the theorem only applies to classical spacetime, and may not hold under consideration of a complete theory of quantum gravity. He added that Alan Guth, one of the co-authors of the theorem, disagrees with Vilenkin and believes that the universe had no beginning. Vilenkin argues that the Carroll-Chen model constructed by Carroll and Jennie Chen, and supported by Guth, to elude the BGV theorem's conclusions persists to indicate a singularity in the history of the universe as it has a reversal of the arrow of time in the past.

Use in theology

Vilenkin has also written about the religious significance of the BGV theorem. In October 2015, Vilenkin responded to arguments made by theist William Lane Craig and the New Atheism movement regarding the existence of God. Vilenkin stated "What causes the universe to pop out of nothing? No cause is needed."

See also

• Kalam cosmological argument
• Gibbons–Hawking–York boundary term
• Gibbons–Hawking effect

Further reading

• Calcagni, Gianluca (2017-01-06) (in en). Classical and Quantum Cosmology. Springer. ISBN 9783319411279. https://books.google.com/books?id=pnLbDQAAQBAJ&q=borde+guth+vilenkin+theorem.