What is the Definition of the Age of the Universe and How is it Determined?

[Dale]

After a recent conversation, I was trying to determine if there is an online authoritative definition for the age of the universe. The Wiki article states "the International Astronomical Union presently use 'age of the universe' to mean the duration of the Lambda-CDM expansion", but the reference is to a popular press article which does not state anything of the sort and a search of the IAU home page did not come up with anything similar.

Does anyone have such an online authoritative reference for the definition of the age of the universe?

 

[Bill_K]

Well for example this paper presenting the Planck 2013 results says,

All of our current observations can be fit remarkably well by a six-parameter ΛCDM model and we provide strong constraints on deviations from this model.

Their fit is given in Table 9. For the age they give 13.784 Gy.

Wikipedia has this to say about the ΛCDM model:

The ΛCDM model is based on six parameters: physical baryon density, physical dark matter density, dark energy density, scalar spectral index, curvature fluctuation amplitude and reionization optical depth... From these six parameters the other model values, including the Hubble constant and age of the universe, can be readily calculated.

 

[Dale]

Thanks Bill_K!

 

[Dale]

So I looked through that and it gives the parameters for the lambda CDM model and then the age of the universe is a calculated parameter based on those 6 parameters. But the paper did not give the formula or the definition of it. I suppose it is well-known in the community, but I cannot find it.

 

[Bill_K]

The ΛCDM model is based on the Friedmann equations, the solution of which gives you R(t). So you integrate back to find the time t₀ of the Big Bang at which R(t₀) = 0, and that's the age.

 

[Dale]

OK, so if we have the FLRW metric: $ds^2=-c^2 dt^2+a(t)dS^2$ then my understanding is that the six parameters of the ΛCDM are what define a(t), and your R(t0) = 0 is also when a(t0)=0.

So then the age of the universe seems to be based on a coordinate in the FLRW metric, t, rather than an invariant feature of the metric.

 

[George Jones]

OK, so if we have the FLRW metric: $ds^2=-c^2 dt^2+a(t)dS^2$ then my understanding is that the six parameters of the ΛCDM are what define a(t), and your R(t0) = 0 is also when a(t0)=0.

So then the age of the universe seems to be based on a coordinate in the FLRW metric, rather than an invariant feature of the metric.

For an observer moving with the Hubble flow (constant comoving coordinate), dS^2 = 0, so t and the proper time are the same for this observer. The age of the inverse is the proper time experienced by any such observer between the Big Bang and Now.

 

[Dale]

Ah, of course. That is obvious now.

So would you characterize the age of the universe as being invariant (proper time of a comoving observer) or coordinate dependent (t in specific coordinates)?

 

[jcsd]

OK, so if we have the FLRW metric: $ds^2=-c^2 dt^2+a(t)dS^2$ then my understanding is that the six parameters of the ΛCDM are what define a(t), and your R(t0) = 0 is also when a(t0)=0.

So then the age of the universe seems to be based on a coordinate in the FLRW metric, rather than an invariant feature of the metric.

It's invariant in the sense that the maximum amount of proper time along a past-directed timelike curve at any given event is the age of the Universe at that event. This definition of the age of the Universe doesn't require you to define coordinates.

 

[Dale]

This definition of the age of the Universe doesn't require you to define coordinates.

Agreed. But is that THE accepted definition?

 

[jcsd]

Agreed. But is that THE accepted definition?

I think the usual definition is in terms of the standard coordinates, my point though is that the standard definition still represents a non-arbitrary invariant.

 

[Dale]

Good point. Thanks!

 

[TrickyDicky]

It's invariant in the sense that the maximum amount of proper time along a past-directed timelike curve at any given event is the age of the Universe at that event. This definition of the age of the Universe doesn't require you to define coordinates.

It requires you to define coordinates to specify a certain specific age, whether that of the comoving observers or any other observer that is referenced to them. Metric tensors must be represented by coordinates if one wants to obtain from them things like the time coordinate value at a certain event, the advantage of the FRW coordinates is that in them the coordinate time coincides with the proper time of the comoving observers at rest with the CMBR, so that we don't need to have someone with a stop watch since the big-bang to check the age of the universe.

 

[TrickyDicky]

Ultimately, the possibility to determine an age of the universe, and the very notion that such thing is possible are derived from a certain model of the universe, the FRW metric. The assumptions made to reach this model come from observations made in this particular universe and are therefore laws of physics . Mathematically speaking the cosmological principle is not technically used as a boundary condition on the Einstein field partial differential equations, as claimed in a recent conversation, boundary conditions are conditions specified at the extremes ("boundaries") of the independent variables in the equations, in this case for instance if the solution was infinite, the conditions at infinity would be boundary conditions. The cosmological principle is introduced instead as a geometrical constraint derived from observation so that a specific metric can be obtained that fulfills the EFE. When one aspires to find the metric tensor by solving the field equations, one builds into its paramaterization whatever symmetries are available, given the intrinsic difficulty of solving them and the infinity of metrics that can in principle work as solution.

Besides the technical reasons that hamper cosidering the cosmological symmetries as boundary conditions as used in DEs, there are more physical arguments :we interpret the FRW geometry as a model of our particular universe, not as a particular solution of the EFE in the way a particular initial or boundary condition determines a particular solution. The latter case would only apply if we had certainty the multiverse hypothesis holds true and each of the solutions of the EFE described one of the multiple universes of this multiverse. This is not the case at all. Our hypothesis is that the FRW solution more or less approximately models our individual universe. That makes isotropy and homogeneity physical laws of our particular universe. In the same way in classical mechanics the first law of Newton is considered a law rather than a boundary condition, or the conservation laws that describe the symmetries of the cosmological principle(isotropy and homogeneity) in physical terms like energy or momentum invariance are not considered boundary conditions but laws.