Why do we still see the CMB today?

[Pyter]

If the universe is infinite in extent now then it always was, yes. Something finite cannot grow into something infinite in finite time.

In that case the accepted CMB explanation starts to make sense to me.

But this "infinite matter" model is at odd with other cosmology notions I've gleaned, namely the enigma of the prevalence of matter over antimatter.
As you surely know, it is argued that at the beginning they should've been present in equal quantity and thus annihilate each another, except that for some unknown reason the matter was slightly more than the antimatter, and our current universe is made of the matter that survived the annihilation.

I've always thought that these considerations implied that the matter in our universe was finite from the beginning. Unless the initial matter was a "double infinite", the antimatter a "single infinite", and thus the difference stays infinite, or something to that effect.

 

[Ibix]

I've always thought that these considerations implied that the matter in our universe was finite from the beginning. Unless the initial matter was a "double infinite", the antimatter a "single infinite", and thus the difference stays infinite, or something to that effect.

The best way to describe it is "different densities".

 

[PeterDonis]

this "infinite matter" model is at odd with other cosmology notions I've gleaned, namely the enigma of the prevalence of matter over antimatter.

No, it isn't. "Infinite matter" as you use the term here actually means "a spatially infinite universe with uniform average density of stress-energy everywhere". It does not require the stress-energy to have any particular form. It could all be radiation. It could all be matter. It could all be antimatter. It could be any mixture of any of those things. It could include dark energy. There is no contradiction in any of this.

it is argued that at the beginning they should've been present in equal quantity and thus annihilate each another, except that for some unknown reason the matter was slightly more than the antimatter, and our current universe is made of the matter that survived the annihilation.

I've always thought that these considerations implied that the matter in our universe was finite from the beginning.

Then you thought wrong. They imply no such thing.

Unless the initial matter was a "double infinite", the antimatter a "single infinite", and thus the difference stays infinite, or something to that effect.

This is word salad.

 

[Hornbein]

I've always thought that these considerations implied that the matter in our universe was finite from the beginning. Unless the initial matter was a "double infinite", the antimatter a "single infinite", and thus the difference stays infinite, or something to that effect.

We are taught in mathematics it makes no sense to subtract one infinity from another. But the infinite ton gorilla does as it pleases.

 

[Hornbein]

If the universe is infinite in extent now then it always was, yes. Something finite cannot grow into something infinite in finite time.

Actually it can if it grows infinitely quickly. But this is even harder to imagine actually happening.

The trick is double in size in y time, then again in y/2 time, again in y/4 time, y/8, and so forth.

 

[PeterDonis]

We are taught in mathematics it makes no sense to subtract one infinity from another

No such subtraction has to take place anywhere in the actual mathematical account of matter-antimatter annihilation in the early universe.

 

[Hornbein]

No such subtraction has to take place anywhere in the actual mathematical account of matter-antimatter annihilation in the early universe.

Correct.

 

[Pyter]

We are taught in mathematics it makes no sense to subtract one infinity from another. But the infinite ton gorilla does as it pleases.

In maths this is legal: $2\delta(x) - \delta(x) = \delta(x)$ . Perhaps also in physics, in branches other than cosmology apparently.

So there's currently no cosmological model based on finite matter/energy, at least no one able to explain the CMB convincingly?

 

[Ibix]

So there's currently no cosmological model based on finite matter/energy, at least no one able to explain the CMB convincingly?

A closed universe is boundaryless but finite.

 

[Pyter]

A closed universe is boundaryless but finite.

With that model, has it been said in the previous posts that the observed CMB should stop at one time, since there's no infinite matter from which it might come from?

 

[Jaime Rudas]

With that model, has it been said in the previous posts that the observed CMB should stop at one time, since there's no infinite matter from which it might come from?

In which posts is this said?

 

[Jaime Rudas]

So there's currently no cosmological model based on finite matter/energy, at least no one able to explain the CMB convincingly?

The cosmological model most accepted by the scientific community today is the ΛCDM model, which explains the CMB in a finite universe quite convincingly.

 

[Ibix]

With that model, has it been said in the previous posts that the observed CMB should stop at one time, since there's no infinite matter from which it might come from?

No. In a naive matter-dominated closed universe model the universe ends before light can circumnavigate it. In a dark-energy dominated model, which lives longer, you start to receive photons that have circumnavigated the universe once, then twice, and so on. There is a finite energy in the CMB in this case, though, so it's possible to imagine it all being absorbed, but it would take a very, very, very long time.

 

[Ibix]

In which posts is this said?

#4. Wrongly, as pointed out by you and Peter. I had forgotten we were in that thread...

 

[Jaime Rudas]

In a dark-energy dominated model, which lives longer, you start to receive photons that have circumnavigated the universe once, then twice, and so on.

In fact, in a realistic dark-energy dominated model, photons can't circumnavigate the universe.

 

[PeterDonis]

there's currently no cosmological model based on finite matter/energy, at least no one able to explain the CMB convincingly?

Yes, there is, a spatially closed, spatially finite universe. This is one of the basic FRW models.

With that model, has it been said in the previous posts that the observed CMB should stop at one time, since there's no infinite matter from which it might come from?

This is wrong. The CMB does not require "infinite matter" (i.e., infinite spatial extent) in order to continue traveling around the universe. @Ibix explained why in post #83.

There is a finite energy in the CMB in this case, though, so it's possible to imagine it all being absorbed

No, it isn't, because it would just be re-emitted again, since whatever absorbed it would then be at a slightly higher temperature than the CMB.

 

[PeterDonis]

in a realistic dark-energy dominated model, photons can't circumnavigate the universe.

What is your basis for this statement?

 

[Jaime Rudas]

What is your basis for this statement?

As I hinted in post #17, in a realistic dark-energy dominated model there is a cosmological event horizon that is smaller than the particle horizon (i.e., the boundary of the observable universe) which, in turn, is smaller than the entire universe.

 

[PeterDonis]

As I hinted in post #17, in a realistic dark-energy dominated model there is a cosmological event horizon that is smaller than the particle horizon (i.e., the boundary of the observable universe) which, in turn, is smaller than the entire universe.

This is not a sufficient basis for the claim you are making now. It is true that there exist dark energy dominated models with the property you describe. But that does not support the claim you are making, that all "realistic" dark energy dominated models have the property you describe.

 

[Jaime Rudas]

This is not a sufficient basis for the claim you are making now. It is true that there exist dark energy dominated models with the property you describe. But that does not support the claim you are making, that all "realistic" dark energy dominated models have the property you describe.

What I assume by "realistic" are those models that can reasonably well describe the observations we make of the real universe.

On the other hand, I notice that you changed your mind from what you stated in post #18

 

[PeterDonis]

What I assume by "realistic" are those models that can reasonably well describe the observations we make of the real universe.

And if so, then you need to explain why you think this requires all "realistic" models to have the property you describe. Note that I am not saying that is not possible; I'm just saying that you haven't done it.

I notice that you changed your mind from what you stated in post #18

How so?

 

[Jaime Rudas]

And if so, then you need to explain why you think this requires all "realistic" models to have the property you describe. Note that I am not saying that is not possible; I'm just saying that you haven't done it.

What we observe of the universe is that the Hubble constant $H_0$ is close to 70 km/s/Mpc, that $\Omega_{0,\Lambda}$ is close to 0.7, that $\Omega_{0,m}$ is close to 0.3 and that $\Omega_{0,k}$ is close to 0. I don't know of any model that meets those conditions and that doesn't meet the ones I stated in post #88

 

[PeterDonis]

What we observe of the universe is that the Hubble constant $H_0$ is close to 70 km/s/Mpc, that $\Omega_{0,\Lambda}$ is close to 0.7, that $\Omega_{0,m}$ is close to 0.3 and that $\Omega_{0,k}$ is close to 0. I don't know of any model that meets those conditions and that doesn't meet the ones I stated in post #88

What would these figures imply about the particle horizon vs. the event horizon? Can you give any relevant math?

 

[Jaime Rudas]

What would these figures imply about the particle horizon vs. the event horizon? Can you give any relevant math?

See equations A.19 and A.20 on page 117 and figure 1.1 on page 8 of this paper.

 

[PeterDonis]

See equations A.19 and A.20 on page 117 and figure 1.1 on page 8 of this paper.

The figure is similar to the one in Davis & Lineweaver 2003, which has been referenced in plenty of PF threads. It does clearly show the particle horizon "now" being further away than the event horizon "now"; the two horizons cross at a redshift of about 0.4, so at earlier times than that the particle horizon was closer than the event horizon.

The equations you reference just give formulas for the particle horizon and event horizon.

What the figure and equations you reference do not give is the connection between the above and the Hubble constant and Omega values that you gave.

 

[Jaime Rudas]

The figure is similar to the one in Davis & Lineweaver 2003, which has been referenced in plenty of PF threads.

Of course it is similar because the author is the same: Tamara M. Davis.

It does clearly show the particle horizon "now" being further away than the event horizon "now"; the two horizons cross at a redshift of about 0.4, so at earlier times than that the particle horizon was closer than the event horizon.

No, they don't intersect at a redshift of about 0.4, but at a scale factor of about 0.4 which corresponds to a redshift of 1.5, i.e. when the universe was dominated by matter. The universe only became dominated by dark energy (which is the kind of universe I refer to in post #88) at a redshift of about 0.5 (scale factor of about 0.67).

The equations you reference just give formulas for the particle horizon and event horizon.

What the figure and equations you reference do not give is the connection between the above and the Hubble constant and Omega values that you gave.

Those connections are shown in equations A.16, A.17 and A.18 on the same page 117. They are also shown in this post.

 

[PeterDonis]

the author is the same: Tamara M. Davis

Yes, I know.

they don't intersect at a redshift of about 0.4, but at a scale factor of about 0.4

Yes, sorry, I misstated it.

i.e. when the universe was dominated by matter

In this particular model, yes.

 

[Jaime Rudas]

In this particular model, yes.

Yes, that's precisely the particular model I proposed in post #92

 

[PeterDonis]

Yes, that's precisely the particular model I proposed in post #92

To say that you "proposed" that model is something of a misstatement. At the very least, it's confusing. You are just pointing at our current best-fit model for our actual universe and saying that's the model you're talking about. You're not "proposing" this model as something new.

In other words, you are saying that, in our current best fit model for our actual universe, it is the case that, for the time period which is dark energy dominated, the particle horizon is further away than the event horizon. Yes, I agree that's the case, but that's a much more limited statement than the one I thought you were making.

 

[Jaime Rudas]

To say that you "proposed" that model is something of a misstatement. At the very least, it's confusing. You are just pointing at our current best-fit model for our actual universe and saying that's the model you're talking about. You're not "proposing" this model as something new.

I accept that I'm not the person who proposed the standard cosmological model. I'm sorry for the confusion.

In other words, you are saying that, in our current best fit model for our actual universe, it is the case that, for the time period which is dark energy dominated, the particle horizon is further away than the event horizon. Yes, I agree that's the case, but that's a much more limited statement than the one I thought you were making.

Although I don't know what statement you thought I was making, I'm sorry for the confusion.

On the other hand, I still maintain that in a realistic dark-energy dominated model, photons can't circumnavigate the universe.

Do you know of any dark-energy dominated model where photons circumnavigate the universe two or three times, as @Ibix proposed presented in post #83?

 

[PeterDonis]

in a realistic dark-energy dominated model, photons can't circumnavigate the universe.

But your definition of "a realistic dark-energy dominated model" is "the current best-fit model for our actual universe". Which is a much narrower definition than I think most physicists use when discussing theoretical models. I think most physicists mean by "a realistic model" a model that describes a physically possible universe, i.e., one that could have been produced by a process reasonably similar to the one that produced our universe. In some models, e.g., eternal inflation models, all such universes do in fact exist, we just can only observe the one we live in.

Do you know of any dark-energy dominated model where photons circumnavigate the universe two or three times

In a model that is dark energy dominated for all time, photons can't circumnavigate the universe, because such a model is basically de Sitter spacetime, which is known to have that property.

I don't see why it wouldn't be possible to have a spatially finite model that becomes dark energy dominated at a late enough time (due to the dark energy density being small enough relative to the matter density) that photons could circumnavigate the universe. But I don't have a reference handy for such a model.

 

[Jaime Rudas]

But your definition of "a realistic dark-energy dominated model" is "the current best-fit model for our actual universe". Which is a much narrower definition than I think most physicists use when discussing theoretical models. I think most physicists mean by "a realistic model" a model that describes a physically possible universe, i.e., one that could have been produced by a process reasonably similar to the one that produced our universe. In some models, e.g., eternal inflation models, all such universes do in fact exist, we just can only observe the one we live in.

I'm clearly not a physicist. In fact, I'm not even a scientist. That is why, as soon as I noticed that what I mean by "realistic" might be misinterpreted (in post #89), I tried to clarify this point (in posts #90 and #92).

In a model that is dark energy dominated for all time, photons can't circumnavigate the universe, because such a model is basically de Sitter spacetime, which is known to have that property.

I don't see why it wouldn't be possible to have a spatially finite model that becomes dark energy dominated at a late enough time (due to the dark energy density being small enough relative to the matter density) that photons could circumnavigate the universe. But I don't have a reference handy for such a model.

Yes, that is, more or less, what I tried to explain in the P.S. of post #17 (there could be circumnavigation when, at a given moment, the event horizon is greater than the length of the maximum circumference of the universe at that moment), but while that would allow one circumnavigation, it would hardly allow two or three.

 

[PeterDonis]

while that would allow one circumnavigation, it would hardly allow two or three.

That would depend on at what point the model became dark energy dominated. I don't think one can make a blanket statement like this that would apply to all possible models.

 

[Ibix]

Let's see if I've got this right. References to Carroll are equations in chapter 8 of his notes unless otherwise stated.

I want to know the equation of motion for a light pulse in a closed FLRW spacetime containing a mix of matter and dark energy, in order to see how many times it can circumnavigate the universe. Obviously I'll need the scale factor, $a$, for that.

If the densities of matter and dark energy when $a=1$ are $\rho_{M0}$ and $\rho_{\Lambda 0}$ respectively then at general $a$ (Carroll 8.24, 8.31):$$\begin{eqnarray}
\rho_M&=&\rho_{M0}a^{-3}\\
\rho_\Lambda&=&\rho_{\Lambda 0}
\end{eqnarray}$$from which we can state corresponding density parameters (Carroll 8.39)$$\begin{eqnarray}
\Omega_M&=&\frac{K\rho_{M0}}{a\dot{a}^2}\\
\Omega_\Lambda&=&\frac{K\rho_{\Lambda 0}a^2}{\dot{a}^2}
\end{eqnarray}$$where I have substituted for the Hubble parameter, $H=\dot{a}/a$ (Carroll 8.37) and defined $K=8\pi G/3$ for brevity.

There are several unknowns here. Let's stipulate that we are going to characterise our universes by specifying $a_e$ and $\Omega_e$ (e for equal), respectively the scale factor when $\Omega_M=\Omega_\Lambda$ and the density parameter of each component at that moment (i.e., the total density parameter then would be $2\Omega_e$). Plugging these into (3) and (4) we can get expressions for the two $\rho$ parameters in terms of these initial conditions, and hence eliminate them from (3) and (4), getting expressions in terms of $a_e$ and $\Omega_e$ instead:$$\begin{eqnarray}
\Omega_M&=&\frac{\Omega_ea_e\dot{a}_e^2}{a\dot{a}^2}\\
\Omega_\Lambda&=&\frac{\Omega_ea^2\dot{a}_e^2}{a_e^2\dot{a}^2}
\end{eqnarray}$$where $\dot{a}_e$ is the value of $\dot{a}$ when $a=a_e$. We can plug these expressions into the first Friedmann equation expressed in terms of $\Omega$ (Carrol 8.41) with $k=1$ for a positive curvature universe (Carroll, paragraph after equation 8.8) to get:$$\begin{equation}
\frac{\Omega_ea_e\dot{a}_e^2}{a\dot{a}^2}
+\frac{\Omega_ea^2\dot{a}_e^2}{a_e^2\dot{a}^2}
-1
=
\frac 1{\dot{a}^2}
\end{equation}$$where, again, I have substituted for $H$. This is true at all times, but if we again plug in $a=a_e$ we can solve for $\dot{a}_e$ in terms of our initial conditions, eliminate it from (7), and solve for $\dot{a}$:$$\begin{equation}
\dot{a}^2=-\frac{(\Omega_e-1)aa_e^3-\Omega_e(a^3+1)a_e+\Omega_ea}
{(\Omega_e-1)aa_e^3+\Omega_ea}
\end{equation}$$Numerical integrators and $(dy/dt)^2=\mathrm{something}$ are problematic because you don't know how to choose the sign of the square root, but (with thanks once again to @George Jones for pointing this out to me), you can work around it using $\ddot{a}$, which we can obtain from differentiating (8):$$\begin{equation}
\ddot{a}=\frac{\Omega_e(2a^3-1)a_e}{2a^2(\Omega_e(a_e^3+1)-a_e^3)}
\end{equation}$$

The only other thing we need to note is that if we write the FLRW metric for a positive curvature universe as Carroll does in 8.7 and 8.10 then set $\chi=\theta=\pi/2$ for an equatorial path, a null path satisfies$$\begin{equation}\dot{\phi}=\frac 1a\end{equation}$$With (8), (9), (10) and a numerical integrator, we can now work out the $\phi$ coordinate of a light pulse as a function of cosmological time in a range of positive-curvature matter/dark energy FLRW universes.

I haven't been particularly systematic about this. I simply plugged in an arbitrary $\Omega_e=0.9$ (so the total density parameter at the change over to dark energy domination would be 1.8) and messed around with $a_e$ until I found interesting values. Here's a plot of the light pulse's angular coordinate, $\phi(t)$, expressed as multiples of $2\pi$ so the vertical scale counts orbits, for a range of $a_e$ values:

At least according to my maths and programming, yes you can have multiple orbits. The best of my fairly arbitrary bunch manages four. Also interesting is the two distinct behaviours of the lines - the four to the left of the high purple line steepen and terminate, while the rest flatten out and run forever (on the graph they do terminate, but this is because I imposed an end to integration, rather than because it couldn't run further). You might guess that these behaviours have to do with the ultimate fate of the universe, and if you did you'd be correct. Plotting $a(t)$ for the four to the left shows that they collapse:

The rest don't:

Here's the code I used for the above - all comments welcome.

"""Modelling an FLRW universe with only matter and dark energy, based on
Carroll chapter 8 and some algebra."""

import math, matplotlib.pyplot, numpy, scipy.integrate

"""Compute da/dt when scale factor is a, given that the density parameter for
the two components is equal at a=aEq with value OMEq"""
def dadt(a, aEq, OMEq):
    return numpy.sqrt(-((OMEq-1)*a*aEq**3-OMEq*(a**3+1)*aEq+OMEq*a)/((OMEq-1)*a*aEq**3+OMEq*a))

"""Differential equations governing the trajectory of light and development
of the universe"""
def diffs(t, y, args = None):
    aEq, OMEq = args
    phi, a, da = y
    ans = [1/a,
            da,
            (2*OMEq*a**3-OMEq)*aEq/(2*((OMEq-1)*a**2*aEq**3+OMEq*a**2))]
    return ans

"""Function that manages the numerical integration"""
def integrate(aEq, OMEq, dt = 1e-1, remember = False):
    # Initial settings for the integrator
    t0 = 0.01                   # Cosmological time
    a0 = 0.01                   # Scale factor
    phi0 = 0                    # Angular coordinate of light pulse
    da0 = dadt(a0, aEq, OMEq)   # Time derivative of scale factor
    # Set up the integrator...
    I = scipy.integrate.ode(diffs).set_integrator("dopri5")
    I.set_f_params([aEq, OMEq])
    I.set_initial_value([phi0, a0, da0], t0)
    # ...and storage space for the results (note extra value in the phi
    # array to facilitate the first check in the while loop - it will be
    # stripped out afterwards)
    t,phi,a, da = [t0], [-500, phi0], [a0], [da0]
    # Loop while the integrator is working, the light ray is making progress,
    # and we haven't gone on too long
    while I.successful() and (phi[-1] - phi[-2]) > dt * 1e-9 and I.t < 100:
        # Integrate...
        I.integrate(I.t + dt)
        # ...and either record the new values or add them to the list
        if remember:
            t.append(I.t)
            phi.append(I.y[0] / (2 * math.pi))
            a.append(I.y[1])
            da.append(I.y[2])
        else:
            t[0] = I.t
            phi[0] = phi[1]
            phi[1] = I.y[0] / (2 * math.pi)
            a[0] = I.y[1]
            da[0] = I.y[2]
    # Strip out the extra value of phi
    phi = phi[1:]
    # Return the results
    return t, phi, a, da

# Initial value of density parameter of each component when they are equal
OMEq = 0.9
# A range of values of the scale factor to try
amin, amax, asteps = 0.52083, 0.520830312, 10
# Run the integrations and add the results to a plot
for ai in range(asteps + 1):
    print(ai, "/", asteps)
    aEq = amin + (amax-amin) * ai / asteps
    t, phi, a, da = integrate(aEq = aEq,
                                OMEq = OMEq,
                                dt = 1e-2,
                                remember = True)
    matplotlib.pyplot.plot(t, phi)
# Set a few labels and display
matplotlib.pyplot.xlabel("t")
matplotlib.pyplot.ylabel(r"$phi/2\pi$")
matplotlib.pyplot.title(r"$\phi(t)$ for $\Omega_e=" 
                            + str(OMEq) + "$, "
                            + str(asteps + 1)
                            + " values of $a$ between "
                            + str(amin)
                            + " and "
                            + str(amax))
matplotlib.pyplot.show()