The Possible Initial States of The Universe

Most people when talking about cosmology typically talk about the universe in one context, that is, as a particular solution to the Einstein field equations. Part of my research in mathematical cosmology is to try to determine whether the present-day universe which we observe to be very close to spatially flat and homogeneous, and very close to isotropic could have emerged from a more general geometric state.

What is often not discussed adequately is the fact that not only has our universe emerged from special initial conditions, but the fact that these special initial conditions also must include the geometry of the early universe, and the type of matter in the early universe. Below, I have attached a simulation that shows how the early universe can evolve to different possible states depending on the type of physical matter parametrized by an equation of state parameter \gamma . In particular, some examples are:

  • \gamma = 0: Vacuum energy
  • \gamma = 4/3: Radiation
  • \gamma = 2: Stiff Fluid

Note: Click the image below to access the simulation!

In these simulations, we present phase plots of solutions to the Einstein field equations for spatially homogeneous and isotropic flat, hyperbolic, and closed universe geometries. The different points are:

  1. dS: de Sitter universe – Inflationary epoch
  2. M: Milne universe
  3. F: spatially flat FLRW universe – our present-day universe
  4. E: Einstein static universe

Note how by changing the value of \gamma , the dynamics lead to different possible future states. Dynamical systems people will recognize the problem at hand requires one to determine for which values of \gamma is F a saddle or stable node.

A Series of Lectures on Fine-Tuning in Biology

A recent lecture and a series of interviews has been posted online where cosmologist George F.R. Ellis discusses the issue of fine-tuning in biology at considerable length and in considerable detail. Of course, the larger theme here is that to discuss and understand things like Darwinian evolution properly, one needs to have an understanding of the underlying physics, as it is laws of physics that allow life to emerge and for Darwinian evolution to occur in the first place. Here are the lectures:

 

 

 

 

 

Mathematical Origins of Life

The purpose of this post is to demonstrate some very beautiful (I think!) mathematics that arises form Darwinian evolutionary theory. It is a real shame that most courses and discussions dealing with evolution never introduce any type of mathematical formalism which is very strange, since at the most fundamental levels, evolution must also be governed by quantum mechanics and electromagnetism, from which chemistry and biochemistry arise via top-down and bottom-up causation. See this article by George Ellis for more on the role of top-down causation in the universe and the hierarchy of physical matter. Indeed, my personal belief is that if some biologists and evolutionary biologists like Dawkins, Coyne, and others took the time to explain evolution with some modicum of mathematical formalism to properly describe the underlying mechanics instead of using it as an opportunity to attack religious people, the world would be a much better place, and the dialogue between science and religion would be much more smooth and intelligible.

In this post today, I will describe some formalism behind the phenomena of prebiotic evolution. It turns out that there has been a very good book by Claudius Gros and understanding evolution as a complex dynamical system (dynamical systems theory is my main area of research), and the interested reader should check out his book for more details on what follows below.

We can for simplicity consider a quasispecies as a system of macromolecules that have the ability to carry information, and consider the dynamics of the concentrations of the constituent molecules as the following dynamical system:

\boxed{\dot{x}_{i} = W_{ii}x_{i} + \sum_{j \neq i}W_{ij}x_{j} - x_{i} \phi(t)},

where x_{i} are the concentrations of N molecules, W_{ii} is the autocatalytic self-replication rate, and W_{ij} are mutation rates.

From this, we can consider the following catalytic reaction equations:

\boxed{\dot{x}_i = x_{i} \left(\lambda_{i} + k_i^j x_j - \phi \right)},

\boxed{\phi = x^{k}\left(\lambda_{k} + \kappa_k^j x_j\right) },

x_i are the concentrations, \lambda_i are the autocatalytic growth rates, and \kappa_{ij} are the transmolecular catalytic rates. We choose \phi such that

\boxed{\dot{C} = \sum_i \dot{x}_i = \sum_i x_i \left(\lambda_i + \sum_j \kappa_{ij}x_{j} \right) - C \phi = (1-C)\phi}.

Clearly:

\lim_{C \to 1} (1-C)\phi = 0,

that is, this quick calculation shows that the total concentration C remains constant.

Let us consider now the case of homogeneous interactions such that

\kappa_{i \neq j} = \kappa, \kappa_{ii} = 0, \lambda_i = \alpha i,

which leads to

\boxed{\dot{x}_{i} = x_{i} \left(\lambda_i + \kappa \sum_{j \neq i} x_{j} - \phi \right)} ,

which becomes

\boxed{\dot{x}_i = x_i \left(\lambda_i + \kappa - \kappa x_i - \phi\right)}.

This is a one-dimensional ODE with the following invariant submanifolds:

\boxed{x_{i}^* = \frac{\lambda_i + \kappa - \phi}{\kappa}},

\boxed{x_i^* = 0, \quad \lambda_i = N \alpha}.

With homogeneous interactions, the concentrations with the largest growth rates will dominate, so there exists a N^* such that 1 \leq N^* \leq N where

\boxed{x_i^* = \frac{\lambda_i + \kappa - \phi}{\kappa}, \quad N^* \leq i \leq N},

\boxed{0, \quad 1 \leq i < N^*}.

The quantities N^* and \phi are determined via normalization conditions that give us a system of equations:

\boxed{1 = \frac{\alpha}{2\kappa} \left[N(N+1) - N^*(N^* - 1)\right] + \left[\frac{\kappa - \phi}{\kappa}\right] \left(N + 1 - N^*\right)},

\boxed{0 = \frac{\lambda_{N^*-1} + \kappa - \phi}{\kappa} = \frac{\alpha(N^* - 1)}{\kappa} + \frac{\kappa - \phi}{\kappa} }.

For large N, N^*, we obtain the approximation

\boxed{N - N^* \approx \sqrt{\frac{2 \kappa}{\alpha}}},

which is the number of surviving species.

Clearly, this is non-zero for a finite catalytic rate \kappa. This shows the formation of a hypercycle of molecules/quasispecies.

These computations clearly should be taken with a grain of salt. As pointed out in several sources, hypercycles describe closed systems, but, life exists in an open system driven by an energy flux. But, the interesting thing is, despite this, the very last calculation shows that there is clear division between molecules i = N^*, \ldots N which can be considered as a type of primordial life-form separated by these molecules belonging to the environment.

Black Holes, Black Holes Everywhere

BH_LMC

Nowadays, one cannot watch a popular science tv show, read a popular science book, take an astrophysics class without hearing about black holes. The problem is that very few people discuss this topic appropriately. This is further evidenced that these same people also claim that the universe’s expansion is governed by the Friedmann equation as applied to a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe.

The fact is that black holes despite what is widely claimed, are not astrophysical phenomena, they are a phenomena that arise from mathematical general relativity. That is, we postulate their existence from mathematical general relativity, in particular, Birkhoff’s theorem, which states the following (Hawking and Ellis, 1973):

Any C^2 solution of Einstein’s vacuum equations which is spherically symmetric in some open set V is locally equivalent to part of the maximally extended Schwarzschild solution in V.

In other words, if a spacetime contains a region which is spherically symmetric, asymptotically flat/static, and empty such that T_{ab} = 0, then the metric inn this region is described by the Schwarzschild metric:

\boxed{ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2 + \frac{dr^2}{1-\frac{2M}{r}} + r^2\left(d\theta^2 + \sin^2 \theta d\phi^2\right)}

The concept of a black hole then occurs because of the r = 0 singularity that occurs in this metric.

The problem then arises in most discussions nowadays, because the very same astrophysicists that claim that black holes exist, also claim that the universe is expanding according to the Einstein field equations as applied to a FLRW metric, which are frequently written nowadays as:

The Raychaudhuri equation:

\boxed{\dot{H} = -H^2 - \frac{1}{6} \left(\mu + 3p\right)},

(where H is the Hubble parameter)

The Friedmann equation:

\boxed{\mu = 3H^2 + \frac{1}{2} ^{3}R},

(where \mu is the energy density of the dominant matter in the universe and ^{3}R is the Ricci 3-scalar of the particular FLRW model),

and

The Energy Conservation equation:

\boxed{\dot{\mu} = -3H \left(\mu + p\right)}.

The point is that one cannot have it both ways! One cannot claim on one hand that black holes exist in the universe, while also claiming that the universe is FLRW! Since, by Birkhoff’s theorem, external to the black hole source must be a spherically symmetric and static spacetime, for which a FLRW is not static nor asymptotically flat, because of a lack of global timelike Killing vector.

I therefore believe that models of the universe that incorporate both black holes and large-scale spatial homogeneity and isotropy should be much more widely introduced and discussed in the mainstream cosmology community. One such example are the Swiss-Cheese universe models. These models assume a FLRW spacetime with patches “cut out” in such a way to allow for Schwarzschild solutions to simultaneously exist. Swiss-Cheese universes actually have a tremendous amount of explanatory power. One of the mysteries of current cosmology is the origin of the existence of dark energy. The beautiful thing about Swiss-Cheese universes is that one is not required to postulate the existence of hypothetical dark energy to account for the accelerated expansion of the universe. This interesting article from New Scientist from a few years ago explains some of this.

Also, the original Swiss-Cheese universe model in its simplest foundational form was actually proposed by Einstein and Strauss in 1945.

The basic idea is as follows, and is based on Israel’s junction formalism (See Hervik and Gron’s book, and Israel’s original paper for further details. I will just describe the basic idea in what follows). Let us take a spacetime and partition it into two:

\boxed{M = M^{+} \cup M^{-}}

with a boundary

\boxed{\Sigma \equiv \partial M^{+} \cap \partial M^{-}}.

Now, within these regions we assume that the Einstein Field equations are satisfied, such that:

\boxed{\left[R_{uv} - \frac{1}{2}R g_{uv}\right]^{\pm} = \kappa T_{uv}^{\pm}},

where we also induce a metric on \Sigma as:

\boxed{d\sigma^2 = h_{ij}dx^{i} dx^{j}}.

The trick with Israel’s method is understanding is understanding how \Sigma is embedded in M^{\pm}.  This can be quantified by the covariant derivative on some basis vector of \Sigma:

\boxed{K_{uv}^{\pm} = \epsilon n_{a} \Gamma^{a}_{uv}}.

The projections of the Einstein tensor is then given by Gauss’ theorem and the Codazzi equation:

\boxed{\left[E_{uv}n^{u}n^{v}\right]^{\pm} = -\frac{1}{2}\epsilon ^{3}R + \frac{1}{2}\left(K^2 - K_{ab}K^{ab}\right)^{\pm}},

\boxed{\left[E_{uv}h^{u}_{a} n^{v}\right]^{\pm} = -\left(^{3}\nabla_{u}K^{u}_{a} - ^{3}\nabla_{a}K\right)^{\pm}},

\boxed{\left[E_{uv}h^{u}_{a}h^{v}_{b}\right]^{\pm} = ^{(3)}E_{ab} + \epsilon n^{u} \nabla_{u} \left(K_{ab} - h_{ab}K\right)^{\pm} - 3 \left[\epsilon K_{ab}K\right]^{\pm} + 2 \epsilon \left[K^{u}_{a} K_{ub}\right]^{\pm} + \frac{1}{2}\epsilon h_{ab} \left(K^2 + K^{uv}K_{uv}\right)^{\pm}}

Defining the operation [T] \equiv T^{+} - T^{-}, the Einstein field equations are given by the Lanczos equation:

\boxed{\left[K_{ij}\right] - h_{ij} \left[K\right] = \epsilon \kappa S_{ij}},

where S_{ij} results from defining an energy-momentum tensor across the boundary, and computing

\boxed{S_{ij} = \lim_{\tau \to 0} \int^{\tau/2}_{-\tau/2} T_{ij} dy}.

The remaining dynamical equations are then given by

\boxed{^{3}\nabla_{j}S^{j}_{i} + \left[T_{in}\right] = 0},

and

\boxed{S_{ij} \left\{K^{ij}\right\} + \left[T_{nn}\right] = 0},

with the constraints:

\boxed{^{3}R - \left\{K\right\}^2 + \left\{K_{ij}\right\} \left\{K^{ij}\right\} = -\frac{\kappa^2}{4} \left(S_{ij}S^{ij} - \frac{1}{2}S^2\right) - 2 \kappa \left\{T_{nn}\right\}}.

\boxed{\left\{^{3} \nabla_{j}K^{j}_{i} \right\} - \left\{^{3}\nabla_{i}K\right\} = -\kappa \left\{T_{in}\right\}}.

Therefore:

  1. If black holes exist, then by Birkhoff’s theorem, the spacetime external to the black hole source must be spherically symmetric and static, and cannot represent our universe.
  2. Perhaps, a more viable model for our universe is then a spatially inhomogeneous universe on the level of Lemaitre-Tolman-Bondi, Swiss-Cheese, the set of G_{2} cosmologies, etc… The advantage of these models, particular in the case of Swiss-Cheese universes is that one does not need to postulate a hypothetical dark energy to explain the accelerated expansion of the universe, this naturally comes out out of such models.

Under a more general inhomogeneous cosmology, the Einstein field equations now take the form:

Raychauhduri’s Equation:

\boxed{\dot{H} = -H^2 + \frac{1}{3} \left(h^{a}_{b} \nabla_{a}\dot{u}^{b} + \dot{u}_{a}\dot{u}^{a} - 2\sigma^2 + 2 \omega^2\right) - \frac{1}{6}\left(\mu + 3p\right)}

Shear Propagation Equation:

\boxed{h^{a}_{c}h^{b}_{d} \dot{\sigma}^{cd} = -2H\sigma^{ab} + h^{(a}_ch^{b)}_{d}\nabla^{c}\dot{u}^{d} + \dot{u}^{a}\dot{u}^{b} - \sigma^{a}_{c} \sigma^{bc} - \omega^{a}\omega^{b} - \frac{1}{3}\left(h^{c}_{d}\nabla_{c}\dot{u}^{d} + \dot{u}_{c}\dot{u}^{c} - 2\sigma^2 - \omega^2\right)h^{ab} - \left(E^{ab} - \frac{1}{2}\pi^{ab}\right)}

Vorticity Propagation Equation:

\boxed{h^{a}_{b}\dot{\omega}^{b} = -2H\omega^{a} + \sigma^{a}_{b}\omega^{b} - \frac{1}{2}\eta^{abcd}\left(\nabla_{b} \omega_{c} + 2\dot{u}_{b}\omega_{c}\right)u_{d} + q^{a}}

Constraint Equations:

\boxed{h^{a}_{c} h^{c}_{d} \nabla_{b} \sigma^{cd} - 2h^{a}_{b}\nabla^{b}H - \eta^{abcd}\left(\nabla_{b}\omega_{c} + 2 \dot{u}_{b} \omega_{c}\right)u_{d} + q^{a} = 0},

\boxed{h^{a}_{b} \nabla_{a}\omega^{b} - \dot{u}_{a}\omega^{a} = 0},

\boxed{H_{ab} - 2\dot{u}_{(a}\omega_{b)} - h^{c}_{(a}h^{d}_{b)}\nabla_{c} \omega_{d} + \frac{1}{3} \left(2\dot{u}_{c}\omega_{c} + h^{c}_{d} \nabla_{c}\omega^{d}\right)h_{ab} - h^{c}_{(a}h^{d}_{b)} \eta_{cefg}\left(\nabla^{e}\sigma^{f}_{d}\right)u^{g}=0}.

Matter Evolution Equations through the Bianchi identities:

\boxed{\dot{\mu} = -3H\left(\mu + p\right) - h^{a}_{b}\nabla_{a}q^{b} - 2\dot{u}_{a}q^{a} - \sigma^{a}_{b}\pi^{b}_{a}},

\boxed{h^{a}_{b}\dot{q}^{b} = -4Hq^{a} - h^{a}_{b}\nabla^{b}p - \left(\mu + p\right)\dot{u}^{a} - h^{a}_{c}h^{b}_{d}\nabla_{b} \pi^{cd} - \dot{u}_{b}\pi^{ab} - \sigma^{a}_{b}q^{b} + \eta^{abcd}\omega_{b}q_{c}u_{d}}.

One also has evolution equations for the Weyl curvature tensors E_{ab} and H_{ab}, these can be found in Ellis’ Cargese Lectures. 

Despite the fact that these modifications are absolutely necessary if one is to take seriously the notion that our universe has black holes in it, most astronomers and indeed most astrophysics courses continue to use the simpler versions assuming that the universe is spatially homogeneous and isotropic, which contradicts by definition the notion of black holes existing in our universe.

A Universe from Nothing

Many people have obviously been reading Lawrence Krauss’ infamous book: “A Universe from Nothing: Why There is something Rather Than Nothing”

Krauss and many other physicists continuously engage in this type of low-level philosophy with the ironic goal of diminishing the value of philosophy using “science”.

This paper dissects all the arguments in Krauss’ book and shows from a mathematical standpoint that like others who make similar arguments, they are not grounded in actual physics and are extremely flawed. One therefore concludes, that such arguments are not based in science, but in bad philosophy.

The paper: A Universe from Nothing

A teaser: This is what “nothing” actually looks like, well one depiction of it anyways:
IMG_0117But this is not nothing, it is something, where did this structure come from? Krauss actually ignores the question entirely in his book, which is very strange.

The famed cosmologist George Ellis also has discussed Krauss’ book in one of his talks, here is the link for that:

A video making some of the arguments in the above paper easier to understand can be found here: