# Distance and the Hubble’s Constant

by mysearch
Tags: constant, distance, hubble’s
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Marcus: First and foremost:

 You mean the figure 92? 92 is twice 46. Diameter is twice radius.
D’Oh - obviously must have contracted horizon blindness due to an overload of horizons. In an attempt to divert attention from this stupidity, I will quote from the Davis-Lineweaver paper, as provided by George Jones (#14):

 Popular science books written by astrophysicists, astrophysics textbooks and to some extent professional astronomical literature addressing the expansion of the Universe, contain misleading, or easily misinterpreted, statements concerning recession velocities, horizons and the “observable universe”.
I believe I could be one of the victims of this obvious conspiracy, but hopefully when I have had a chance to assimilate some of the details within the Lineweaver article I will be better able to judge the extent of this conspiracy. However, in the meanwhile I would like to try to clarify what I was trying to do in post #13 via the equation:

[1] $$H = \frac{1}{t} = \sqrt{ \frac{2GM}{r^3}}$$

I was hoping to create a crude model of a matter-dominated universe ranging from +1 billion years through to the present-day. My starting point was the quoted age of the universe, i.e. 13.7 billion years. The reciprocal of this leads to H=71km/s/mpc. This allows the Hubble radius to be defined as c/H=1.29E29m, which allows the associated volume to be calculated. As I have understood things, the only significance of the Hubble radius is that it is the point at which the expansion velocity exceeds the speed of light [c]. I selected a homogeneous density in the order of 9.5E-27kg/m^3 from which I estimated the mass (M) within the Hubble radius volume, e.g. 8.58E52kg. This mass was also selected because it gives the measured value of H via equation (1). As a closed universe, I assumed the mass-density would not change over the period selected, i.e. matter dominated. From equation [1] I plotted H against time. See graph1.jpg in #13 for details. While H varies with time, I assumed that H was linear with distance for any given time. As such, I could plot recession velocity [v] at a given radius against time. By re-arranging equation [1] you can calculate the radius corresponding to H and I assumed that this would allow me to produce a comparative plot of recessional velocity of a volume of space, expanding in time according to H. See graph2.jpg. What this graph suggested to me was that the edge of this volume of space, which today coincides with the Hubble radius, has expanded faster than light until now.

If any of the assumptions of this model are valid, it seems to suggest that the universe was initially expanding faster than causality, i.e. ct. However, the model also suggests that as the expansion slows, new sections of the universe will appear within our future light cone, i.e. causality window.

 Would really appreciate any insights as to how the radius of the physical horizon is calculated.
 Do you mean the radius of the physical universe, or the particle horizon?
I was actually asking how the particle horizon radius was calculated. I don’t know if the answer is in the Lineweaver article, but noted that the figure of 46 billion light-years ignores the issue of inflation. Initially, I didn’t really appreciate whether there was much difference between the physical & particle horizons, but have taken onboard your comments regarding the degree of ambiguity concerning a finite/infinite universe.

The way I was starting to consider the particle horizon is as follows:

A photon emitted from an atomic transition at point X arrives on Earth after 13.7 billion years. While the distance travelled by the photon is ct, i.e. 13.7 billion lightyears, the distance between the Earth and point X is now much greater than 13.7 billion lightyears due to the subsequent expansion of the universe?

Now based on my earlier model, if the photon was too far from the Earth, when emitted, the recessional velocity would be greater than [c] and the photon would still be chasing after us. However, there is also the issue that the recessional velocity appears to slow down over this vast amount of time.

If I assume that the photon has travelled 13.7 billion lightyears, does the recessional velocity of point X give rise to the difference between (ct) and the particle horizon, such that:

Particle horizon = ct + vt

Where t=13.7 billion years, [c] is the speed of light and [v] is the variable recessional velocity of X with time?

I accept this is probably too simplistic, but want to know if it basically explained the concept of receiving light from a point far in excess of ct.
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Nereid,

 there are many more causes of astronomical interest to list, such as the Sunyaev-Zel'dovich effect ... but I think that would take you way off track, wouldn't it?
I have been intrigued by the sheer complexity and depth of this subject. Of course, there is an element of learning to walk before running in my current situation. However, I have very much appreciated your insights into this subject. Thanks
 PF Patron P: 511 Krutsch, I am no expert, but here are some thoughts. 1) the bohr model is just a model that really doesn’t exist according to quantum theory. 2) I believe the velocity you are referring to, in the Bohr model, is orbital velocity, 3) I assume the velocity you are referring in connection to the Hubble effect is the recessional velocity. Hope this is of some help.
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By luck, I came across an excellent analysis in this very forum on the subject of the particle horizon: http://www.physicsforums.com/showthread.php?t=23037

Footnote to PF admin or mentors:
I am not that familiar with all the PF services, so is there a way to ‘weight’ a search on a contribution that has been approved into a library of accepted definitions. A search on ‘particle horizon’ returned 496 threads, which often contain dozens of individual posts, such that finding the thread above, posted in 2004, was highly unlikely.

In post #7, Hellfire outlines a derivation of the particle horizon in a matter-dominated universe, which I had been looking for. While I have only started to work through the Lineweaver reference given in #14, I am not sure that I agreed with the following descriptive definition of the particle horizon:

 The particle horizon is the distance light can have travelled from t = 0 to a given time [t]..
Based on my current understanding of the particle horizon, I would have thought the following statement would be more correct, so would appreciate any clarification:

 The particle horizon is the maximum distance from which light can have travelled from t = 0 to a given time [t] plus the subsequent recession distance of its starting point due to the expansion of the universe.
The distinction I am trying to make for clarification is that light only travels at [c] and therefore the distance covered by the emitted photon is [ct]. In the context of the present universe, this distance would be 13.7 billion light-years. However, its point of origin [X] would now have receded due to the variable rate of expansion associated with (H). The sum of these 2 distances defines the particle horizon?

As an additional issue, Hellfire’s post in the link above suggests that the particle horizon can be estimated as $$D(t_0) = 3 c t_0$$. If so, the estimate of particle horizon would come out at 3*c=1*13.7=41.1 billion lightyears. This differs from the Lineweaver estimate of 46GLYs, so does anybody know what expansion profile is being used in this estimate, which does not include inflation?
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 Quote by mysearch [...] Finally, are there any sources showing (H) against time (t)? [...]
Hot off the (arXiv) press!

Clustering of Luminous Red Galaxies IV: Baryon Acoustic Peak in the Line-of-Sight Direction and a Direct Measurement of H(z):
 Quote by Enrique Gaztanaga, Anna Cabre, Lam Hui (abstract) This is the 4th paper in a series where we study the clustering of LRG galaxies in the latest spectroscopic SDSS data release, DR6, which has 75000 LRG galaxies sampling 1.1 (Gpc/h)^3 to z=0.47. Here we study the 2-point correlation function, separated in perpendicular (sigma) and line-of-sight (pi) directions. We find a significant detection of a peak at r=110 Mpc/h, which shows as a circular ring in the sigma-pi plane. There is also a significant detection of the peak along the line-of-sight (LOS) direction both in sub-samples at low, z=0.15-30, and high redshifts, z=0.40-0.47. The overall shape and location of the peak is consistent with baryon acoustic oscillations (BAO). The amplitude in the line-of-sight direction, however, is larger than conventional expectations. We argue this is due to magnification bias. Because the data is shot noise dominated, a lensing boost in signal translates into a boost in S/N. We take advantage of this high S/N to produce, for the first time, a direct measurement of the Hubble parameter H(z) as a function of redshift. This differs from earlier BAO measurements which used the spherically averaged (monopole) correlation function to constrain an integral of H(z). Using the BAO scale purely as a standard ruler in the LOS direction, we find: H(z=0.24)= 79.7 +- 2.1 (+- 1.0) km/s/Mpc for z=0.15-0.30, and H(z=0.43)= 86.5 +- 2.5 (+- 1.0) km/s/Mpc for z=0.40-0.47. For a flat universe with a cosmological constant, our two measurements of H(z) extrapolate to H_0=71.7 +-1.6 km/s/Mpc. This is in remarkable agreement with previous, independent, estimates of H_0 based on fitting the cosmological constant model. Combining our measurements with external constraints on w and dark energy abundance, we find w = -0.96 +- 0.05.
 PF Patron P: 511 Nereid: thanks for the paper, which I did quickly read. However, given my current A101 status in cosmology, I couldn’t always follow all the inferences being made. As such, I was not sure how this paper addressed the issue of H versus time. If possible, I would like to check some basic assumptions: 1) If on the very large scale the universe is flat, i.e. k=0, then the complexity of Riemann geometry can be reduced to Euclidean geometry. Equally, on this assumption, the FRW metric reduces to the flat spacetime metric of special relativity, where the complexity of GR curved spacetime is localised within much smaller regions of spacetime of higher gravitational potential, e.g. galaxies. 2) Today, H is estimated to be in the order of 71 km/s/mpc. This figure is primarily based on Cepheids (30mpc) and SN1A(400mpc) redshifts. From the aggregation of redshift measurements, we can determine recession velocity z=v/c, non-relativistic case; from which H=v/d can be derived. 3) On the based of k=0, the Friedmann equation can be transposed to give the critical density of the universe, i.e. [1] $$\rho_c = \frac{3H^2}{8\pi G}$$ Based on H=71km/s/mpc, $$\rho_c=9.57*10^{-27} kg/m^3$$, which it is assumed is the total energy density of the universe, which theory is currently separating into 3 components, i.e. matter (4%), dark matter (23%) and dark energy (73%). 4) Matter plus dark matter (27%) constitutes the component that is slowing down the expansion of the universe due to gravity, whilst dark energy is a quantity that might be the cause of the universe expanding at an accelerated rate. If so, is equation [1] correct in that it only reflects (H) reducing with time based on gravitational attraction linked to the constant (G)? As a somewhat expansive question, can we plot the expansion factor (X) of the universe in conjunction with the falling value of (H) linked to gravity alone? One final point, this expansion property of space seems to have a threshold at which it can overpower the ‘internal forces’ of a given structure. Apparently, it is not big enough to overpower nuclear force holding together the atomic nuclei, nor the electromagnetic force between nuclei and electrons or even the gravitational force within a galaxy, but it is big enough to ‘overpower’ the weaker gravitational attraction between galaxies, which presumably helped form the large-scale structures of the universe. Are there any papers that discuss this issue? As always, many thanks for your insights and the information provided.
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 Quote by mysearch 1) If on the very large scale the universe is flat, i.e. k=0, then the complexity of Riemann geometry can be reduced to Euclidean geometry. Equally, on this assumption, the FRW metric reduces to the flat spacetime metric of special relativity, where the complexity of GR curved spacetime is localised within much smaller regions of spacetime of higher gravitational potential, e.g. galaxies. ...
Watch out! the k=0 case refers to SPATIAL flatness
it does not mean that you have 4D flatness or that you can fit the picture to a Minkowski frame----or even more radiclly to an Euclidean frame.

It does NOT mean that the FRW "reduces to the flat spacetime metric of SR"

people are often confused about this and get the mistaken notion that they can fit all or part of a Friedmann model universe onto an SR frame, in cases when this is inappropriate. the confusion partly arises from there being two different meanings of the word flat, namely spatial flat and 4D flat
=========================

Nereid is doing a supercompetent response job on your questions, so I won't intrude any further. I only felt compelled to respond to that one point of yours because it presents a very real danger of confusion not only to you but to others here.
the k=0 case is the spatial flat and typically the spatial infinite case (spatial flat can still be expanding like crazy and completely non-SR)
the the k = 1 case is the spatial positive curvature and spatial finite case----it has spatial closure, but it does not need to eventually collapse, it can continue expanding indefinitely
your formula for rho-crit looks fine. my figure for rho-crit, which I think is equivalent to yours, is 0.85 joules per cubic kilometer. You can see whether it is roughly equivalent or not by doing E=mc^2. I find it easier to remember in terms of joules and kilometers.
nuff said, please pardon the intrusion
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Marcus, informed comments are no intrusion and the points you raise do home in on any aspect on which I am slightly confused about. I realise that there is still some debate over whether k=0 or is just very close to zero, but I wanted to just focus on the case of k=0.

 Watch out! the k=0 case refers to SPATIAL flatness it does not mean that you have 4D flatness or that you can fit the picture to a Minkowski frame----or even more radically to an Euclidean frame.
First, could I just clarify your terminology; are you defining an ‘Euclidean frame’ as 3-D plus absolute time, whilst a Minkowski frame is essentially 4-D spacetime of special relativity?

 It does NOT mean that the FRW "reduces to the flat spacetime metric of SR"
Again, to clarify, the FRW metric is often presented in spherical coordinates as:

$$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \left( \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\theta^2 + r^2 \sin^2 \theta \, \mathrm{d}\phi^2 \right)$$

If I consider a radial path, I can reduce the complexity to:

$$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \left( \frac{\mathrm{d}r^2}{1-k r^2}\right)$$

If I set k=0, the equation appears to reduce to:

$$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \mathrm{d}r^2$$

It was my understanding that today, a(t)=1, which seemed to imply that:

$$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + \mathrm{d}r^2$$

Which appears equivalent to SR spacetime metric at a given point in time.

 people are often confused about this and get the mistaken notion that they can fit all or part of a Friedmann model universe onto an SR frame, in cases when this is inappropriate. the confusion partly arises from there being two different meanings of the word flat, namely spatial flat and 4D flat
So my basic question is what is the difference between the formal definition of 'spatial flat' and '4D flat'?

Many thanks
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 Quote by mysearch Again, to clarify, the FRW metric is often presented in spherical coordinates as: $$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \left( \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\theta^2 + r^2 \sin^2 \theta \, \mathrm{d}\phi^2 \right)$$ If I consider a radial path, I can reduce the complexity to: $$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \left( \frac{\mathrm{d}r^2}{1-k r^2}\right)$$ If I set k=0, the equation appears to reduce to: $$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \mathrm{d}r^2$$ It was my understanding that today, a(t)=1, which seemed to imply that: $$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + \mathrm{d}r^2$$ Which appears equivalent to SR spacetime metric at a given point in time.
If $a \left( t \right) = 1$ and $k = 0$, then the Scwharzschild metric is

$$g = - dt^2 + dr^2 + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right),$$

which does indeed look like Minkowski spacetime in spherical coordinates. Spacetime curvature, however, is calculated using second derivatives of the metric. At a particular instant of time, $t_\mathrm{now}$, $a \left( t_\mathrm{now} \right) = 1$ is chosen as a scaling convention, but

$$\frac{d^2 a}{dt^2} \left( t_\mathrm{now} \right) \ne 0.$$

Consequently, even at the instant $t_\mathrm{now}$, spacetime curvature is non-zero, while the spacetime curvature of Minknoswki spacetime is zero at all spacetime events.
 Quote by mysearch So my basic question is what is the difference between the formal definition of 'spatial flat' and '4D flat'?
Spatially flat refers to the curvature of a 3-dimensional spatial section (hypersurace) of spacetime at any particular instant of time. If $t$ is fixed, then $dt=0$, and the spacetime metric induces the spatial metric (up to a scaling factor)

$$dr^2 + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)$$

Spatial curvature is zero, since spatial curvature is calculated using second derivatives of the spatial metric with respect to spatial coordinates. This can be easily seen after transforming to Cartesian coordinates.
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George: thanks for the feedback, but could I just press for a few more details? While, the Schwarzschild metric can be presented in a number of forms, I am use to the following form containing the ratio of the Schwarzschild radius $$R_s = 2GM/c^2$$ over radius [r], which reflects the gravitational field strength with distance.

[1] $$c^2 {d \tau}^{2} = \left(1 - \frac{R_s}{r} \right) c^2 dt^2 - \left(1-\frac{R_s}{r}\right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)$$

However, if we assume r>>Rs and we only consider a radial expansion along an equatorial plane, we could reduce this to:

[2] $$c^2 {d \tau}^{2} = c^2 dt^2 - dr^2$$

Which I am assuming is essentially equivalent to the Minkowski or SR spacetime metric. So, in word, this is saying in the absence of any significant mass density, spacetime is flat. As such, this was what I was referring to when I said:

 reduces to the flat spacetime metric of special relativity, where the complexity of GR curved spacetime is localised within much smaller regions of spacetime of higher gravitational potential, e.g. galaxies.
Additionally, on the assumption that k=0, I reduced the FRW metric to the form:

[3] $$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \mathrm{d}r^2$$

Now the similarities between [2] and [3] are obvious except for the term a(t).

 At a particular instant of time, $t_\mathrm{now}$, $a \left( t_\mathrm{now} \right) = 1$ is chosen as a scaling convention, but $$\frac{d^2 a}{dt^2} \left( t_\mathrm{now} \right) \ne 0.$$
As far as I can see this is simply saying that the rate of change of a(t) doesn’t have to be constant, i.e. it can be accelerating or decelerating. This seem reasonable enough, although I not sure whether there is any empirical verification of the rate of change of a(t) with time, which was essentially the question raised in post #24:

 As a somewhat expansive question, can we plot the expansion factor (X) of the universe in conjunction with the falling value of (H) linked to gravity alone?
Finally, the point I seem to be missing is that I can understand how the expansion of the universe may be said to curved in sense that a(t) is accelerating, e.g. 1, 2, 4, but based on equation [3], does spacetime remain essentially flat at each instance in time under the assumption that k=0? Apologises if I have missed what might seem an obvious point in your response.
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 Quote by Nereid (bold added) First, there are not many processes that can cause a shift in the position (central wavelength) of a spectral line ... at least, not in environments of interest to those who study the sky (astronomers, etc). Apart from relative motion and the cosmological redshift, there is only a gravitational redshift, which was predicted by Einstein's GR and observed in the lab by Pound and Rebka. Of course, if all you see is a point source, then in principle a range of other mechanisms for creating shifts could be in play; however, in astronomy all (?) extra-galactic point sources (as classes) have associated extended sources whose redshifts are essentially the same. For example, a GRB is a point source, but after it fades its 'parent galaxy' is often seen, and where that galaxy's redshift can be measured, it is similar to that of the GRB (so far anyway). Ditto a quasar or QSO. Second, there are many processes that can result in a broadened spectral line (and some broadening may be asymmetric, so the central wavelength seems to shift a little), but this isn't what you're asking about, is it? Third, the relevant theory of atoms (and more), which has been tested to the highest level of precision of any theory in physics*, allows the possibility of systematic spectral shifts, via changes in the fine structure constant, either by time or space (i.e. this constant is different in different parts of the universe, and/or at different times). It will come as no surprise to learn that firm discovery of such changes would be quite dramatic, suggesting perhaps that one or more of the conservation laws do not apply in at least some places (or times) in the observable universe. So quite a few attempts have been made to constrain any such variations. The net? Depends on how you read the various papers - some claim to have found a clear signal for a tiny, but non-zero, change over cosmological time; others no such change. To give you an idea of how this is done, read this paper by Bahcall et al.; it is a very elegant, simple, robust method (typical of Bahcall). * it's actually the theory underlying atomic theory, QED
i don't quite understand how a constant can be different in different parts of the universe. surely it isn't a constant if it varies.
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 Quote by azzkika i don't quite understand how a constant can be different in different parts of the universe. surely it isn't a constant if it varies.
It doesn't have to be a contant that varies, it may be its measurements.

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