What caused the expansion of the universe for the 1st 7 billion years?

[mysearch]

Hi,
I am trying to understand the general status of Lambda-CDM model of cosmology and specifically how it is thought to explain the expansion of the universe for the first 7 billion years. Rather than replicate too many equations, this http://en.wikipedia.org/wiki/Friedmann_equations" outlines the standard forms of the Friedmann, Fluid and Acceleration equations. However, I believe it is possible, and acceptable, to express these equations purely in terms of the respective component mass-energy density

$$\rho_M$$ matter energy density
$$\rho_R$$ radiation energy density
$$\rho_D$$ dark matter energy density
$$\rho_K$$ space curvature energy density
$$\rho_\Lambda$$ dark energy density

If so, it would seem that the 3 basic equations of Friedmann cosmology can be presented as follows, although the details are again omitted, the weightings are believed to reflect the equation of state for each density variable:

[1] $$H^2 = \frac{8}{3} \pi G(\rho_M + \rho_R + \rho_D + \rho_K + \rho_\Lambda )$$

[2] $$\dot{\rho} = -H(3 \rho_M + 4 \rho_R + 3 \rho_D + 2 \rho_K + 0 \rho_\Lambda )$$

[3] $$\frac{\ddot{a} }{\dot{a}} = \frac{4}{3} \pi G(1 \rho_M + 2 \rho_R + 1 \rho_D - 0 \rho_K - 2 \rho_\Lambda )$$

By way of information, the 1st attached graph suggests that the various energy density components are a function of time. This graph was produced as an extension of the following equation, which appears to be the basis of many of the on-line http://www.astro.ucla.edu/~wright/ACC.html" , e.g. Ned Wright.


[4] $$H=H_0 \sqrt{\frac{\rho_1}{\rho_0}} = H_0 \sqrt{ \frac{ \Omega_M}{a^3} + \frac{ \Omega_R}{a^4} + \frac{ \Omega_D}{a^3} + \frac{ \Omega_K}{a^2} + \frac{ \Omega_\Lambda}{a^0} }$$

On review of the 1st graph, the earlier universe appears to be dominated by radiation, e.g. +97%, but its density falls as a power of 4 as the universe expanded. In contrast, dark energy appears to have been negligible in the early universe, but its density is thought to have been unaffected by expansion. As such, today, dark energy is thought to represent 73% of the energy density followed by 23% dark matter, 4% baryon matter and radiation being virtually zero. The energy density associated with space curvature, based on [k=0], is thought not to have been a factor on the larger scale of the universe.

The 2nd attached graph shows a plot of acceleration against time, which is based on the energy-density effects that result from equation [3] above and the equations of state implicit in the weighting applied to each. From my perspective, the most logical way to orientated the sign of acceleration is to assume that it reflects the direction with respect to the velocity [v] of expansion, as implied by the Hubble constant [H]. However, equation [1] does not seem to explain why [H=v/d] should be associated with an expanding velocity [v]. This said, the acceleration equation [2] is assumed to imply that the expansion of the universe was slowing down in the first 7 billion years, but subsequently started to speed up due to the growing effect of dark energy. However, what I am struggling to understand is:

What caused the expansion of the universe for the 1st 7 billion years?

I have attached some additional questions regarding inflation in post #2 as this idea seems to encompass a range of hypotheses and appears to be somewhat of an add-on to the general L-CDM model. However, I would appreciate any knowledgeable clarifications of any of the issues raised. Thanks

 

[mysearch]

Footnote about inflation

It is realized that post #1 makes no reference to any effect of inflation within the earlier universe. The following quote comes from a lecture given by Alan Guth in which he outlines the process of inflation:

“Modern particle theories predict that, at very high energies, there exists a form of matter that creates a gravitational repulsion! Inflation proposes that a patch of this form of matter existed in early universe; it was probably more than a billion times smaller than a single proton! The gravitational repulsion created by this material was the driving force behind the big bang. The repulsion drove it into exponential expansion, doubling in size every $$10^{-37}$$ second or so! The density of the repulsive gravity material was not lowered as it expanded! Although more mass/energy appeared repulsive-gravity material expanded, total energy was conserved! The energy of a gravitational field is negative! The positive energy of the material was compensated by the negative energy of gravity. The repulsive-gravity material is unstable, so it decayed like radioactive substance, ending inflation. The decay released energy which produced ordinary particles, forming a hot, dense ‘primordial soup’. Inflation lasted maybe $$10^{-35}$$ second. At the end, the region destined to become the presently observed universe was about the size of a marble. The ‘primordial soup’ matches the assumed starting point of the standard big bang— the standard big bang description takes over. The region continues to expand and cool to the present day.”​

This description appears to suggest that the inflationary process was over in the 1st second of existence and that the “repulsive-gravity material” was unstable and decayed within this timeframe. I realize that this might be an overly simplistic summary, but I am having difficulty in reconciling this process with the energy-density model outlined in post #1. For example:

1. Is the “repulsive gravity material” thought to be something new, e.g. a scalar field or some form of antimatter or alternatively something linked to the cosmological constant alias dark energy alias vacuum energy alias quantum fluctuations?
   
   
2. Presumably, this ‘material’ was capable of exerting a gravitational ‘repulsion’ that was so strong it overcame the gravitational ‘attraction’ implicit in the energy density as suggested by equation [1] in post #1?
   
   
3. If the effect of the “repulsive gravity material” decayed during the inflation period, i.e. within the first second, what caused the universe to continue expanding for the next 7 billion years in the face of gravitational effects of radiation, matter and dark matter, which appear to swamp the expansive effects of dark energy during this period?
   
   
4. As far as I am aware, the expansion of space is said not to affect the structural separation on the atomic scale or between the planets of a solar system or the distance between the stars in a single galaxy. As such, it appears that only the space between galaxies effectively expands. Is this because electromagnetic and gravitational forces became stronger than the expansion ‘force’ at some point on the expansion scale spectrum?
   
   
5. By way of clarification of the last point, were the affects of expansion on the ‘scale spectrum’ always constant; especially if this process was not linear in time?

Again, would appreciate any insights or links to on-line references that discuss these issues at a level suitable for general comprehension!

Example from 1999
http://arxiv.org/abs/astro-ph/9901124"

 

[Ich]

1. Yes, http://en.wikipedia.org/wiki/Inflaton" or so, who knows. Its equation of state should resemble Dark Energy.
2. Gravitational attraction is covered by equation [3]. It did not only overcome the other components, instead the other components have been generated by the http://en.wikipedia.org/wiki/Inflation_(cosmology)#Reheating_2" of said field.
3. Inertia. Once set in motion, things keep their "velocity" unless some "force" brakes them down. The braking "force" is gravitation.
4. Essentially, yes. Remember that the "expansion force" vanished in the very first second of the universe. Since then, you didn't have to overcome a "force" to get things stable, you just had to rob them of enough relative velocity to let gravitation win over inertia.
5. The "effect" - if you mean whether once stable structures tend to expand or contract due to expansion - is strictly proportional to $\ddot a$, not $\dot a$. Again, look at yor equation [3] and interpret it as the gravitational pull of a homogeneous matter distribution. Which it is.

 

[mysearch]

Appreciate the feedback. Have a few comments on the responses.

1. Yes, http://en.wikipedia.org/wiki/Inflaton" or so, who knows. Its equation of state should resemble Dark Energy.

Thanks for the steer towards scalar fields. I will take a closer look at this idea. Has any real progress been made in this area since the http://arxiv.org/abs/astro-ph/9901124" was published in 1999?

2. Gravitational attraction is covered by equation [3]. It did not only overcome the other components, instead the other components have been generated by the http://en.wikipedia.org/wiki/Inflation_(cosmology)#Reheating_2" of said field.

Not sure I follow this response. Equation [3] as corrected below can be derived by differentiating [1], but it is not clear how it defines the direction of velocity, i.e. expansion or contraction. Hence the comment about the sign of acceleration being resolved with respect to velocity. Thanks for the link to reheating, although the idea still seems to be very speculative.

3. Inertia. Once set in motion, things keep their "velocity" unless some "force" brakes them down. The braking "force" is gravitation.

This is the root of some of my confusion, because it is not clear to me how expanding space has inertia.

4. Essentially, yes. Remember that the "expansion force" vanished in the very first second of the universe. Since then, you didn't have to overcome a "force" to get things stable, you just had to rob them of enough relative velocity to let gravitation win over inertia.

I guess this response highlights the issue. What relative velocity are you referring to? Although galaxies are moving apart, do they really have a relative velocity with respect to expanding space? In essence they could have been stationary with respect to CMB at all times.

5. The "effect" - if you mean whether once stable structures tend to expand or contract due to expansion - is strictly proportional to $\ddot a$, not $\dot a$. Again, look at yor equation [3] and interpret it as the gravitational pull of a homogeneous matter distribution. Which it is.

I think equation [3] in post #1 has a typo and should be:

$$\frac{\ddot{a} }{a} = \frac{4}{3} \pi G(1 \rho_M + 2 \rho_R + 1 \rho_D - 0 \rho_K - 2 \rho_\Lambda )$$

What I was trying to highlight by the ‘scale spectrum’ was that atoms are said not to get bigger with the expansion of space; neither do solar systems or galaxies, but was there ever a time when the ‘force’ expansion was greater that the force holding any of these structures together?

Thanks.

 

[Ich]

Sorry for the delay.

Has any real progress been made in this area since the Liddle paper was published in 1999?

Of course. Wmap did as expected and nailed the spectral index down pretty tightly. That killed lots of models, but the remaining ones still have significant freedom. I'm not an expert, though, maybe someone else can tell you more.

Equation [3] as corrected below can be derived by differentiating [1], but it is not clear how it defines the direction of velocity, i.e. expansion or contraction.

You got the sign of [3] wrong. It should be
$$\frac{\ddot{a} }{a} = -\frac{4}{3} \pi G(1 \rho_M + 2 \rho_R + 1 \rho_D - 0 \rho_K - 2 \rho_\Lambda )$$
You had [3] very positive in the first fractions of a second, enough so to make sure that $\dot a/a$ is positive.
For the next 7 billion years, it was negative. That's the gravitational attraction of matter slowing down the expansion.
Since then, DE dominates, accelerating the expansion.

This is the root of some of my confusion, because it is not clear to me how expanding space has inertia.

It hasn't. Things have inertia.

What relative velocity are you referring to? Although galaxies are moving apart, do they really have a relative velocity with respect to expanding space?

There is no such thing as "relative velocity with respect to expanding space". Space has no velocity.
Things do have velocity wrt each other. All those galaxies Hubble observed are most definitely moving away from us. And if you want them to stop, you have to do something.

In essence they could have been stationary with respect to CMB at all times.

Of course. But the local CMB frame there is definitely not at rest with the local CMB frame here.

What I was trying to highlight by the ‘scale spectrum’ was that atoms are said not to get bigger with the expansion of space; neither do solar systems or galaxies, but was there ever a time when the ‘force’ expansion was greater that the force holding any of these structures together?

Yes, the last time some 10^-32 seconds after Big Bang.
For the next 7 billion years a "contracting force" (aka gravity) dominated, which tried to shrink e.g. solar systems, but with no succes.
Since then, on the largest scales, there's again an "expansion force" dominating. But not within galaxies or galaxy clusters.

 

[mysearch]

Again, thanks for the feedback and correction to [3]. However, I am still confused by some of your statements about relative velocity and inertia. I have attached my comments below.

I'm not an expert, though, maybe someone else can tell you more.

Neither am I, so please don’t take my comments the wrong way. I am just trying to understand the physics that supports the mainstream L-CDM + inflation model.

For the next 7 billion years, it was negative. That's the gravitational attraction of matter slowing down the expansion. Since then, DE dominates, accelerating the expansion.

Just for clarification. For the first 7 billions the L-CDM model has the universe expanding, but acceleration is negative after the initial inflation, such that the expansion is slowing down. In the second 7 billion years the universe is still expanding, but acceleration is now positive. The energy density model justifies the change in sign of acceleration due to the proportional increase in dark energy, as per diagrams attached to post #1. Within the constraint of this model, as far as I can see, there appears to be no energy density that explains why the universe expanded in the first 7 billion years. However, there are a wide range of inflation models that are essentially add-on’s to the standard model that speculate to various degrees about the cause of an initial inflation. Within this context, there appears to be some general consensus that the period of exponential expansion was over in the first second of existence and whatever caused it decayed in the process.

It hasn't. Things have inertia.

I agree, but I would like to understand how kinetic (Ek) and potential (Ep) energy is resolved within the expansion process. As I understand it, the basic model describes the homogeneous expansion of space and, within this model, ‘things’ did not fly apart as per an inertial explosion. As such, it is not clear to me how the model resolves Ek and Ep, if this model of the universe has no centre of gravity.

There is no such thing as "relative velocity with respect to expanding space". Space has no velocity. Things do have velocity wrt each other. All those galaxies Hubble observed are most definitely moving away from us. And if you want them to stop, you have to do something.

Again, I agree, but I am not sure about the last sentence. Without a centre of gravity, what would slow the homogeneous expansion of the galaxies? Equally, without each unit volume of space continuing to expand, due to some unidentified energy density, I don’t understand how galaxies continued to fly apart unless you are saying that inflation imparted inertial momentum to 'things' in space, as you reject space itself having inertia. However, I don’t see how this would explain how the volume of space in the universe continued to increase in the 1st 7 billion years.

Of course. But the local CMB frame there is definitely not at rest with the local CMB frame here.

I am not sure that I necessarily understand the implication. If a galaxy was at rest wrt to the CMB at an earlier point in the expansion of the universe, and we ignore any localised gravitational effects, what would change its position wrt to CMB as the universe expanded? Again, for the purposes of this discussion I am assuming that no centre of gravity exists within the universe.

Sorry to belabour these points, but I am still struggling to understand the physics of expansion within this model, even ignoring the fact that dark matter and dark energy cannot be explained by current particle physics. Thanks.

 

[Ich]

Just for clarification. ...

Oll Korrect.

I would like to understand how kinetic (Ek) and potential (Ep) energy is resolved within the expansion process. As I understand it, the basic model describes the homogeneous expansion of space and, within this model, ‘things’ did not fly apart as per an inertial explosion. As such, it is not clear to me how the model resolves Ek and Ep, if this model of the universe has no centre of gravity.

You trigger my "ramble & rant mode" with this kind of question, please bear with me.
The model usually uses the coordinates that come naturally with the Robertson-Walker metric. In this system, you assign a constant space coordinate "r" to every comoving entity. That's why some people claim that these things are not actually moving, but that there is space expanding between them. This point of view is usually expanded by introducing the "proper distance a*r", the time derivative of which is denoted "recession velocity". This velocity is usually split into an "expansion of space" part and "peculiar velocity". There is no useful notion of potential energy in this picture. Kinetic energy is attributed to peculiar motion only and thus not conserved.

The important point is: even if these coordinates are very useful for several reasons, you don't have to use them. Even more, if you want to understand the dynamics of the universe or local physics, you should not use them. They are obfuscating and misleading for such purposes.

Have a look at http://books.google.com/books?id=uU...QS6SsLaq6yQTPyfypDQ&cd=1#v=onepage&q&f=false" of a cosmology textbook: in an isotropic, homogeneous universe, you can arbitrarily "cut out" a "small" ball of the universe, ignore all the rest and still get accurate results if you apply known physics to this ball. If it is small enough (<~1GLY), it comes with a flat background metric on which you can do Newtonian or PostNewtonian calculations, in our familiar static coordinates. (of cource you have to include the gravity of pressure in the model)

Doing so, you find
- these galaxies are actuallymoving away from each other with a velocity proportional to distance
- they have kinetic energy according to this velocity
- the gravity of all the components adds to a http://arxiv.org/abs/0809.4573" inside the ball
- this gravity alters the motion of the components exactly according to your eq. [3]
- the center of gravity is the center of the ball. It is as arbitrary as the ball.

This description is exact in the limit of vanishing ball radius. You don't bedürftigadditional ingredients like "expansion force", it's all there in the (post)Newtonian equations.
If you try to do cosmology, you just have to state that the exact same thing happens everywhere in the universe. But you can't just imagine a very big sphere, instead you have to patch all these little spheres together correctly, including the effects of curvature that become significant over large distances. And this is done most conveniently in FRW coordinates, which can handle all possible scales and geometries, where static coordinates are no longer available.

 

[mysearch]

Oll Korrect, bedürftig

Thanks goodness for Google search

The model usually uses the coordinates that come naturally with the Robertson-Walker metric. In this system, you assign a constant space coordinate "r" to every comoving entity. That's why some people claim that these things are not actually moving, but that there is space expanding between them. This point of view is usually expanded by introducing the "proper distance a*r", the time derivative of which is denoted "recession velocity". This velocity is usually split into an "expansion of space" part and "peculiar velocity". There is no useful notion of potential energy in this picture. Kinetic energy is attributed to peculiar motion only and thus not conserved.

Is my issue about zero velocity wrt to CMB resolved within this terminology?

Cosmology Textbook

http://arxiv.org/abs/0809.4573"

Many thanks for the references. I will take the time to read into the details you are trying to describe before asking too many additional questions – honest.

Doing so, you find
- these galaxies are actuallymoving away from each other with a velocity proportional to distance
- they have kinetic energy according to this velocity
- the gravity of all the components adds to a http://arxiv.org/abs/0809.4573" inside the ball
- this gravity alters the motion of the components exactly according to your eq. [3]
- the center of gravity is the center of the ball. It is as arbitrary as the ball.

Presumably point-1 is based on redshift and Hubble’s law. Not sure about point-2 as the recession velocity can presumably exceed the speed of light given enough distance. In this context, when you talk of ‘centre of gravity’ are you making reference to what is sometimes referred to as http://www.sparknotes.com/physics/gravitation/potential/section3.rhtml" ? Otherwise I am not sure how gravity slows the expansion of a flat, homogeneous [k=0] universe that I thought the model assumed to have no net centre of gravity. However, I need to read the reference provided before pursuing these implied questions.

If you try to do cosmology, you just have to state that the exact same thing happens everywhere in the universe. But you can't just imagine a very big sphere, instead you have to patch all these little spheres together correctly, including the effects of curvature that become significant over large distances. And this is done most conveniently in FRW coordinates, which can handle all possible scales and geometries, where static coordinates are no longer available.

To be honest, I not trying to do cosmology, rather I was simply trying to come to some sort of understanding the basic physics, if such a thing is still possible, that explains what caused the universe to expand for the first 7 billion years. To use Guth’s words it seems that some form of ‘repulsive-gravity material’ kick-started inflationary expansion, which then decayed within the 1st second, but expansion kept on going despite any obvious energy-density to maintain the on-going expansion of each cubic metre of space within the universe.

Anyway - many thanks.

 

[Ich]

Not sure about point-2 as the recession velocity can presumably exceed the speed of light given enough distance.

Given enough distance, the flat background assumption breaks down, and you can no longer rely on Post-Newtonian calculations. To understand the very large scales, you need full GR. But to understand the dynamics of the universe, it's enough to know how it works at small scales.

In this context, when you talk of ‘centre of gravity’ are you making reference to what is sometimes referred to as Newton’s Shells?

Yes. Aka Birkhoff's theorem in GR.

Otherwise I am not sure how gravity slows the expansion of a flat, homogeneous [k=0] universe that I thought the model assumed to have no net centre of gravity.

Well, you get into some terminology problems if you try to deal with infinite space in Newtonian terms. But even then, the math is clear: pick any ball of matter, neglect the rest, and see that it tends to collapse from its own gravity. The same thing happens everywhere.

 

[mysearch]

Ich,
I have appreciated your help and the references to additional material, but possibly from my perspective we have started to wander away from the core of my original inquiry, i.e.

What caused the expansion of the universe for the 1st 7 billion years?

This link to an http://www.edge.org/3rd_culture/guth02/guth02_p2.html" about inflation suggests that the original Big Bang theory really offered no explanation as to why the universe started to expand. In this context, the inflation model was/is thought to provide the additional rationale of the expansion of the universe, at least, from just above the Planck scale to a point where the standard model could take over. Guth’s rationalises the cause of expansion to a ‘process’ that had so much negative gravity/pressure that it was able drive the expansion of space at an exponential rate for a brief fraction of the first second of existence. However, he states that this process was unstable and decayed within this process. I do not know whether this ‘repulsive material’ has ever been given a name, e.g. scalar field, but it does not seem to align to anything that persists into the standard L-CDM model.

Within the L-CDM model, there are essentially 5 potential energy densities. None of these energy densities seem to account for how the fabric of space might have continued to expand in the early universe, as oppose to moving matter, i.e. ‘things', through space. As far as I am aware, the hypothesis of dark energy being a negative pressure only becomes a significant factor after about 7 billion years, while the remaining 4 energy densities appear to act as positive pressure, i.e. gravitationally attractive.

In post #3, bullet-3 you seem to suggest that inertia was somehow responsible for the continued expansion of the universe and that gravity was responsible for the slow down in the rate of expansion, until dark energy increased as a percentage. To be honest, this is the core of what I have been trying to better understand, but virtually all the reference I have reviewed, to-date, appear to avoid the general discussion of this process. Therefore, I will table a few general questions that I am still trying to clarify for future reference:

1. After inflation, was there any residual negative pressure source that accounts for the continued expansion of space?
   
2. If not, what rationale is thought to explain the continued expansion of the universe, e.g. inertia?
   
3. How is gravity thought to slow down the expansion of space?

Returning to the Friedmann equations in post #1, noting the correction to equation [3] in post #5, it would seem that the acceleration equation [3] can broadly be explained in terms of differentiating [1] and substituting in [2] to get [3]. As such, given the 5 declared energy densities in the L-CDM model, equation [3] predicts a slow down in expansion as shown in the 2nd graph attached to post #1. As such, the reduced rate of expansion appears homogeneous in the sense that gravity slows the dissipation of both mass and energy into a larger volume of space, although it does not seem to explain how the volume of space might be said to physically expand. In this context, Newton Shell aka Birkhoff's theorem in GR does not seem to apply as there appears to be no need of a centre of gravity. I raise this last point because some sources have speculated on the concept of a ‘http://arxiv.org/abs/gr-qc/0602102" which would have a centre of gravity. However, if my description of equation [3] broadly aligns to the L-CDM model, then my main issues are still linked to questions 1 and 2 above. Thanks

 

[Ich]

I do not know whether this ‘repulsive material’ has ever been given a name, e.g. scalar field, but it does not seem to align to anything that persists into the standard L-CDM model.

It's called the Inflaton field. However, you have read and understood the standard explanation for "What caused the expansion of the universe for the 1st 7 billion years?", so what more do you expect?

None of these energy densities seem to account for how the fabric of space might have continued to expand in the early universe, as oppose to moving matter, i.e. ‘things', through space.

There is no "as opposed to". Expanding space = increasing distances = moving things.

Returning to the Friedmann equations in post #1, noting the correction to equation [3] in post #5, it would seem that the acceleration equation [3] can broadly be explained in terms of differentiating [1] and substituting in [2] to get [3].

No, try a different route to understanding.
Imagine a homogeneous Newtonian "dust ball" with Radius R. Follow the calculations yourself.
The gravitational acceleration of a particle at the surface is
$$\ddot R = -\frac{4}{3} \pi G \rho R$$
Now in cosmology, instead of R you write a*r, where a is the scale factor and r the comoving distance. If the dust is to represent a cosmological model, there is the constraint r=const. (no peculiar velocity), so that
$$\ddot R = \ddot a \, r$$
which brings the Newtonian equation into the Frienmann form
$$\frac{\ddot a}{a} = -\frac{4}{3} \pi G \rho$$
In the presence of significant pressure, you have to use an effective density $\rho_{eff}=\rho + 3c^2p$ instead.

This is the meaning of [3]. [3] predicts that in the absence of gravitational sources, expansion will continue unaltered ($\ddot a = 0 \rightarrow \dot a = const.$). The dustball explains why.