Can the universe be infinite? – II – Friedmann equation and scaling factor by Newtonian considerations

This post series is about the question whether one can exclude by some simple arguments that the universe is infinite. I write about it because some presently popular science videos, which illustrate the Big Bang as an explosion, may create potentially wrong ideas about finiteness. Readers of my first post in this series

Can the universe be infinite? – I – The cosmological principle

may in the meantime have thought about the questions and the remarks which I have listed up there. I indicated that already a general scaling of distances could lead to a dynamic evolution of a model universe. As we will see in this post, this thought is already backed up by Newtonian considerations. The equation we will derive will indeed contain a scaling factor, which is compatible with the cosmological principle .

We need some math which you probably have learned in high school or in classes of a German Gymnasium (secondary grammer school). Aside of my own remarks, I follow a derivation of R. and H. Sexl [1]. I will try to point out some weak assumptions.

Newtonian physics and some assumptions

We assume homogeneity and isotropy of the mass/energy distribution throughout the space of a model universe. In Newtonian theory the gravitational force felt just outside a distance r from the center of a spherical matter distribution of mass M (concentrated inside a radius r_M ≤ r) is given by the derivative of the gravitational potential

\[ \Phi_G \,=\, – \, G \, {M \over r}\, , \quad r \ge r_M \]

with respect to the variable r ≥ r_M and G being the gravitational constant. The gravitational potential in this case is a spherically symmetric field stretching radially outwards throughout space across all values of r ≥ r_M up to infinity. When we call the gravitational force f_G exerted on a point-like mass-probe of mass m, we have

\[ f_G \, =\, – {\partial \over \partial \, r} \, \Phi_G \,=\, – \, G \, {m\, M \over r^2}\, , \quad r \ge r_M \,. \]

How can we apply this to a model universe governed by the cosmological principle?

With an isotropic and homogeneous mass distribution, it appears to be fair to imagine the mass around an observer’s position as being organized in spherical shells, all with constant density (i.e. independent of the distance r around our observer’s position). To cover all contents of the universe’s homogeneous mass distribution, we let the radius of the shells go to infinity. And we are confident that conclusions of our arbitrarily chosen observer sitting at the center of this distribution can be transferred to other observers.

Hmmm, really? The cosmological principle indeed does not exclude a homogeneous mass distribution throughout the space of an infinite universe. But as we normal human beings have problems with infinity, we may actually be overstretching spherical symmetry arguments for such a case:

(1) In a way we regard the selected observer’s position at the center of our shells as being in a central position – which according to the cosmological principle feels wrong. Furthermore, in case of a finite distribution the shell assumptions are incompatible with the cosmological principle for certain topologies of space – in particular infinite spaces.

(2) Regarding infinity: What about an observer placed some millions of light-years away – and what about the impact of the mass in a defined corridor stretching vertically to the line that connects him and us? To which observer’s “infinity” along the connection line does the corridor’s content of matter contribute? If to ours, does the corridor’s contents not create an excess in the opposite direction for the other otherwise completely equivalent observer? We are on somewhat thin ice here regarding equations of the type “∞ + Δ = ∞” – and should leave proper answers to mathematicians.

(3) We should also acknowledge that part of the problematic aspects of a Newtonian picture result from the very nature of the Newtonian theory which assumes infinitely reaching gravitational force fields in space. We somehow have to tame these infinities. And we therefore grasp for a resolution coming from spherical symmetry assumptions.

Anyway, we simply should accept that we have made a somewhat weak assumption at the beginning of the following Newtonian line of argumentation.

Assumption: No force exerted by spherically symmetrical mass shells

An important Newtonian ingredient in the following derivations is that a (limited) spherically symmetric mass shell exerts no force upon a probe of matter inside it.

In this post we just accept this effect of the inverse square law for the Newtonian gravitational force as a fact. I is a bit difficult to derive mathematically, but interested readers will find a proof in any good book on classical physics. But again, despite this fact certainly being true for a given limited mass shell, we are on thin ice regarding our adding shell after shell to infinity to cover the contents of the whole universe. And at the same time comparing our view to the fully equivalent one of an observer at some other, extremely distant place on some other side of our universe. We simply hope that whatever finite differences may exist between two observers, resulting effects would be negligible compared with the effects of the potentially infinite rest of the mass distribution.

For those, who anyway believe in a finite, somehow curved universe, a question:
Take the 2-dimensional analog of the surface of a sphere – with 2-dim shells around some observer … What would change in our argumentation? Are we really better off then? If so, by what principle?

Gravitational force on a mass probe at some distance from an observer

With our assumptions, the gravitational force at a distance r from an observer on a mass probe would only depend on the mass M_r inside r. As we according to the cosmological principle experience a certain constant density value ρ₀ at our present time t₀, i.e. a value independent of r,

\[ \rho_0 \,=\, \rho(t_0) = const._r \,, \]

the mass in a sphere with radius r around us is given by

\[ M_r(t_0) \,=\, M_r(t_0, \, r) = {4 \over 3}\, \pi * \rho_0 * r^3 \,. \]

At a a different point in time t, with a then given uniform density ρ(t), we would, of course, have

\[ M_r(r, t) \,=\, M_r(r, t) = {4 \over 3}\, \pi * \rho(t) * r^3 \,. \]

In Newtonian physics we have to fulfill the 2nd Newtonian law. For gravitation it tells us that the attractive force means an acceleration of a mass probe towards our observer at r = 0, i.e. a change in the radial velocity of v_r,m of our mass probe with time t (expressed by the derivative d/dt v_r,m(t) of v_r,m with respect to t).

Assuming that v_r,m in a dynamic universe becomes a function of time t and distance r, we get

\[ m {d \over d t} v_{r,m}(r, \, t) \, =\, – {\partial \over \partial r} \, \Phi_G \,=\, – \, G \, {m\, M_r(t) \over r^2}\, , \tag{1} \]

\[ {d \over d t} v_{r,m}(r, \, t) \, =\, – \, G \, {M_r(t) \over r^2} \,=\, -\, {4 \over 3}\, \pi \, G \, \rho(t) \, r \, . \tag{2} \]

The respective acceleration vector points radially inwards to our observer. The attentive reader will ask, now:

Are fields of radial velocities and of radial velocity changes around any observer in the potentially infinite universe compatible with the cosmological principle?

Assumption: Scaling law for distances

Well, inspired by the image and questions posed in the 1st post of this series, we are bold and assume that a simple scaling law for resulting distance changes may help. (We want to avoid complex math also 🙂 ).

Let us follow the premise that the distance d between two chosen objects just scales with a time dependent factor a(t) – with a(t) being the same everywhere in our model universe

\[ d(t) \, =\, a(t) * d_0 \, , \quad d_0 \,=\, d(t_0) \,, \]

d₀ gives us the value of d(t) at a specific time. We can e.g. choose the value at our present time t= t₀.

What we do here corresponds to a remapping of distances between all objects in a uniform way with time. In our Newtonian picture, we map a certain status of the matter distribution in space onto a new subsequent status in space. This mapping describes a systematic movement of distant mass probes with respect to our observer position by some simple scaling of the distance with time.

Wait a minute – distances, hmmmm … in which direction? Is an isotropic distance-scaling with radial distance from a selected observer compatible with the experience of another observer at another position in space?

Compatibility of an overall distance scaling with the cosmological principle

In a Cartesian coordinate system distances are given by chosen coordinates and a so called norm. In our case the Euclidean norm between position vectors to points X and Y with Cartesian coordinates (x₁,x₂,x₃) and (y₁,y₂,y₃) in a 3-dimensional space

\[ d(\pmb{X},\pmb{Y}) = \sqrt{\, (x_2 – x_1)^2 \,+\, (y_2 – y_1)^2 + (z_2 – z_1)^2 \,} \]

Its easy to see that a scaling of the arbitrary coordinate distances in x-, y, z-direction automatically scales the distance between the points, too. Thus, homogeneity of the scaling is fulfilled. Isotropy can easily be shown to be valid, too, everywhere:

Put yourself as an observer at the origin of a Cartesian coordinate system. Determine the radial distances to two arbitrarily selected objects in different directions. Then pick another second observer somewhere and calculate his observed radial distances to the two chosen objects ahead of a general distance scaling. Now, take into account that an overall scaling of distances with time also affects your distance to this second observer – due to the global scaling he/she gets a new position relative to you. Then, scale the distances of the objects with respect to you. Afterwards, calculate the distances of the selected two objects with respect to the second observer at his new position. Compare the data before and after the scaling.

You will find again that the change both for us and the other observer is given by our common scaling factor. (Mathematically, it all comes down to the fact that our general distance scaling is a linear operation which preserves angles.)

Still there are some noteworthy aspects of our approach:

Note 1: When we assume a(t) > 0, then we describe growing distances everywhere. This means an overall expansion. If you place an observer near a small mass probe at a radial distance r, we and he would experience a growing distance between us dictated by the time dependent scaling factor a(t).
Note 2: We do not yet care about what physical effect could trigger such a peculiar movement of matter in space. However, from our Newtonian perspective a scaling which leads to growing distances within the matter distribution would work against the attractive gravitational pull of matter. This requires energy.
Note 3: We expect some total energy conservation throughout the universe. For a gas or fluid we would expect some equation of state and a related internal (thermal) energy which during an expansion following the scaling law is transferred into the gravitational potential.
Note 4: In our everywhere isotropic model universe the distance scaling affects radii from an observer’s position in a spherically symmetric way.
Note 5 – mass conservation: The assumption of a pure scaling goes hand in hand with the assumption that the mass within a rescaled radius does not change. I.e., we assume mass conservation during the continuous scaling.

Mass/energy conservation and the equation of motion

We apply these ideas now to the radial distances which we used to describe the matter distribution around our observer’s position. Then

\[ r(t) \,=\, a(t) * r(t_0) \,. \tag{3} \]

Which means, regarding the resulting velocity of a tiny mass probe at r_m:

\[ {d \over dt} r_m(t) \,=\, v_{r,m} \,=\, {d \over dt} a(t) * r_m(t_0) \,.\]

Mass conservation mentioned in note 5 above tells us

\[ \rho(t) * a^3(t) \,=\, \rho_0 * a_0^3(t)\,, \quad \mbox{with} \,\, a_0 \,=\, a(t_0)\, . \tag{4}\]

To make our notation about more efficient, let us symbolize the time derivative by a dot above the variable:

\[ \dot{r}_m(t) \,=\, v_{r,m} \,=\, \dot{a}(t) * r_m(t_0) \,.\]

Plugging this into equation (2) we get :

\[ \ddot{a}(t) \, = \, -\, {4 \over 3}\, \pi \, G\, \rho_0 * a(t) \, . \tag{5} \]

Oops, we have suddenly arrived at an equation for the scaling factor – and by some magic we have got rid of the arbitrarily selected r_m and the mass included inside the radius. It holds everywhere and by some magic it does not violate Newtonian restrictions on gravitational force and acceleration.

The above equation – the so called 1st Friedmann equation – is obviously consistent with the cosmological principle. The scaling approach has removed any notion of the (arbitrary) position of our observer.

Note also that we have (more by chance than deeper knowledge) formulated a kind of “geometrical theory” – the mass/energy distribution determines the continuously progressing scaling of distances – again everywhere.

Integration of differential equation for scaling factor

We have got a relatively simple differential equation for the scaling factor. To get closer to a solution for the time dependence of the scaling factor we need to integrate. We multiply equ. (2) with da/dt and use (4) to get

\[ 2\, \dot{a} (t) \, \ddot{a}(t) \, + \, {8 \over 3}\, \pi \, G\, \rho_0 \, a_0 * {1 \over a^2(t)} \,=\, 0 \, . \tag{6} \]

Setting

\[ C \,=\, {8 \over 3}\, \pi \, G\, \rho_0 \, a_0 \,, \tag{7} \]

we have

\[ 2\, \dot{a} (t) \, \ddot{a}(t) \, + \, {C \over a^2(t)} \,=\, 0 \, . \tag{8} \]

We guess the integral function of the leftmost term and integrate the second term. Integration constants are all moved into a term k = const.. We arrive at a much simpler differential equation:

\[ \dot{a}^2(t) \, – \, {C \over a(t)} + k \,=\, 0 \, . \tag{9} \]

You may check that differentiation leads back to (8). Does this new equation (9) remind you of something? It looks similar to an equation of energy conservation for a mass probe with the first term corresponding to a kinetic energy and the second term to a potential energy.

We take the square root

\[ \dot{a}(t) \, = \, \sqrt{ { C \over a(t)} – k} \,\, , \tag{10} \]

and rewrite

\[ { d{a} \over \sqrt{ C / a \,- \, k} } \, = \, dt \,.\tag{11} \]

Now, assuming a continuous growth of the scaling parameter, we can integrate from a = 0 to a(t) – corresponding to the time interval of some time t=0 in the past to t:

\[ \int_0^{a(t)} { { 1 \over \sqrt{ C / a’ \,- \, k} } \, d{a’} } \, = \, \int_0^t {dt} =\,t\, \tag{12} \]

We will solve this integral equation in the next post. The reader can in the meantime look the integral up in some tables on the Internet.

Note that our integration assumes a growth of the scaling factor. This is a very fundamental assumptions which must be backed up by observation.

Friedmann equation

The reader may check that we can bring our differential equation (9) into the following form:

\[ \left( {\dot{a} \over a} \right)^2 \, = \, {8 \, \pi \, G \over 3 } \, \rho(t) \, – \ { K\, c^2 \over a^2} \, , \quad \mbox{with} \,\, k = c^2 K . \tag{13} \]

This equation is the so called “2nd Friedmann equation“. Both the 1st and the 2nd Friedmann equation also follow directly from Einsteins field equations of General Relativity under assumptions of the cosmological principle. The interested reader will get a short derivation for this equation based on the Robertson Walker metric in a later post of this series.

Summary

Some simple considerations show that a combination of the cosmological principle with massaged Newtonian physics implies a dynamically expanding universe whose geometry is governed by the development of a scaling factor. At a given point in cosmic time such a scaling factor would be a constant throughout space. We have found a differential equation for the change of this factor with time. For a continuously growing scaling factor the equation can be integrated under the assumption that the factor once was zero (or due to physical reasons at least close to zero).

The scaling describes a kind of general stretching of distances between objects in space. In the light of the questions raised in the first post, it is unclear, how we have to interpret the potential scaling of distances: As some strange explosion dynamics in space or a change of the fabric of space itself? I will return to this point multiple times again in forthcoming posts.

Note, however, that we so far have NOT assumed that our model universe must be finite in some peculiar sense. Instead, the introduced scaling can be experienced everywhere, at any point in an infinite universe. And it could have begun at some strange initial point t_BB in time, when a potentially infinite universe was in a strange status with all matter objects (we fortunately have an infinite amount of it) having been much closer to each other – everywhere. Our mind just has difficulties in operating with mappings of infinity.