See, for example, references [, 3: #cqgaa5735bib003,, 4: #cqgaa5735bib004] for recent reviews of both no hair theorems, and examples where black hole hair exists, in Einstein gravity coupled to different types of scalar field theories

A real cosmic no hair theorem, which is beyond the scope of our work, should address to what extent cosmological tension entirely characterizes AFdS spacetimes

The ADM mass can be thought of, in this scheme, as the tension in the time direction. Note also that the spatial tension of a p-brane spacetime in a given direction is distinct from the ADM momentum of the brane in that direction. The momentum is defined in terms of the same asymptotic spatial translation vector as the tension, but with the normal to the slice \Sigma pointing in the time direction

This is analogous to asymptotically flat spacetimes, where the falloff of metric perturbations in the asymptotic region with inverse powers of the radius is compensated for by the growth of the area element of spheres to give finite results for ADM integrals, or to the competing effects that give finite charges in AdS

Although we have not done this, it would be interesting to develop a notion of cosmological enthalpy in analogy with the AdS black hole case [, 9: #cqgaa5735bib009]

See, for example, the recent discussion of a primordial anisotropic stage of inflation in [, 10: #cqgaa5735bib010]

It is easily checked that the Kasner-de Sitter spacetimes (, 8: #cqgaa5735eqn008) have this property

This condition should be related to a similar property of the trace of the boundary stress tensor in dS/CFT

For an anti-symmetric tensor {{\alpha}^{ab}}={{\alpha}^{[ab]}}, Stokes theorem states that {{{\int}^{}}_{\partial \Sigma}}\text{d}{{s}_{ab}}{{\alpha}^{ab}}={\int}_{\Sigma}\text{d}{{s}_{b}}{{\nabla}_{a}}{{\alpha}^{ab}}

In the usual Smarr construction, for static, asymptotically flat spacetimes, the hypersurface \Sigma is a constant time slice. The boundary is a sphere at large radius with normal ra, and the analogous assumptions are that {{r}_{a}}{{t}^{a}}={{r}_{a}}{{\xi}^{a}}=0

In this q = 1/3 case, although the full tensor {{T}_{ab}}-\frac{1}{2}{{g}_{ab}}T that appears on the right hand side of the cosmological Smarr formula (, 30: #cqgaa5735eqn031) for D = 4 does not vanish, the relevant component does

The Killing potential {{\omega}^{ab}} is not uniquely determined by (, 45: #cqgaa5735eqn046). If {{\alpha}^{ab}} is an antisymmetric tensor satisfying {{\nabla}_{a}}{{\alpha}^{ab}}=0, then {{\tilde{\omega}}^{ab}}={{\omega}^{ab}}+{{\alpha}^{ab}} also solves (, 45: #cqgaa5735eqn046). However, this non-uniqueness does not impact our results

Including a cosmological fluid stress energy, the result would be <foot_disp > \begin{equation*} \begin{align*} \begin{eqnarray}&&8\pi {{\mathcal{T}}_{(k)}}={{K}_{(k)}}({{t}_{i}})-\frac{2 \Lambda}{D-2}{{V}_{(k)}}+4\pi {\int}_{\Sigma}\sqrt{-g}(\rho +(n-2){{p}_{k}}-\underset{i\ne k}{\sum}{{p}_{i}})\end{eqnarray} \end{align*} \tag{ 53 } \end{equation*}

Outside of the AFdS context, this term is also finite at the initial singularities of Kasner spacetimes and FRW cosmologies with a(t)\sim {{t}^{q}} for 0 < q < 1/3

General definition of gravitational tension J. High Energy Phys. JHEP05 043, IOPscience:

The initial state of a primordial anisotropic stage of inflation J. Cosmol Astropart. Phys. JCAP06 024, IOPscience: