On Saturday, March 28, 2020 at 5:38:33 PM UTC-6, Lawrence Crowell wrote:
>
> On Saturday, March 28, 2020 at 5:27:51 AM UTC-5, Alan Grayson wrote:
>>
>>
>>
>> On Wednesday, March 25, 2020 at 4:25:43 PM UTC-6, Lawrence Crowell wrote:
>>>
>>> On Wednesday, March 25, 2020 at 2:17:39 PM UTC-5, Alan Grayson wrote:
>>>
>>>>
>>>>
>>>> On Wednesday, March 25, 2020 at 9:57:44 AM UTC-6, Lawrence Crowell
>>>> wrote:
>>>>>
>>>>> On Wednesday, March 25, 2020 at 10:24:30 AM UTC-5, Alan Grayson wrote:
>>>>>>
>>>>>>
>>>>>>
>>>>>> On Wednesday, March 25, 2020 at 2:58:45 AM UTC-6, Lawrence Crowell
>>>>>> wrote:
>>>>>>>
>>>>>>> On Wednesday, March 25, 2020 at 1:28:21 AM UTC-5, Alan Grayson wrote:
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> On Tuesday, March 24, 2020 at 6:21:50 PM UTC-6, Lawrence Crowell
>>>>>>>> wrote:
>>>>>>>>>
>>>>>>>>> On Tuesday, March 24, 2020 at 6:19:41 PM UTC-5, Alan Grayson wrote:
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On Monday, March 23, 2020 at 9:35:48 AM UTC-6, Lawrence Crowell
>>>>>>>>>> wrote:
>>>>>>>>>>>
>>>>>>>>>>> Inflation was initiate 10^{-35}sec after the quantum fluctuation
>>>>>>>>>>> appearance of the observable cosmos, and this had a duration of
>>>>>>>>>>> 10^{-30}sec. The cosmological constant averaged around Λ =
>>>>>>>>>>> 10^{48}m^{-2}.
>>>>>>>>>>> If I divide by the speed of light squared this comes to
>>>>>>>>>>> 10^{32}s^{-2} and
>>>>>>>>>>> we get √(Λ)T = 10^{2}. This means any spatial region expanded by a
>>>>>>>>>>> factor
>>>>>>>>>>> of 10^{√(Λ)T} which is large. The natural log of this is 230
>>>>>>>>>>> and this is not too far off from the more precise calculation of 60
>>>>>>>>>>> e-folds. The 60 e-folds is a phenomenological fit that matches
>>>>>>>>>>> inflation
>>>>>>>>>>> with the observed universe.
>>>>>>>>>>>
>>>>>>>>>>> How much of the universe is unavailable depends upon whether k =
>>>>>>>>>>> -1, 0 or 1. The furthest out some quantum might emerge and have an
>>>>>>>>>>> influence is for a Planck scale quantum to now be inflated to the
>>>>>>>>>>> CMB
>>>>>>>>>>> scale. I know I have gone through this here before, but the result
>>>>>>>>>>> is the
>>>>>>>>>>> furthest we can detect anything is around 1800 billion light years,
>>>>>>>>>>> which
>>>>>>>>>>> would be a graviton or quantum black hole that leaves an imprint or
>>>>>>>>>>> signature on the CMB. It is not possible from theory to know what
>>>>>>>>>>> percentage this is of the entire shebang, and for k = -1 or 0 it is
>>>>>>>>>>> an
>>>>>>>>>>> infinitesimal part.
>>>>>>>>>>>
>>>>>>>>>>> LC
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> For k=0, a flat universe, we know the answer since, as you've
>>>>>>>>>> acknowledged, it's infinite in spatial extent. Consequently, since
>>>>>>>>>> the
>>>>>>>>>> observable universe is finite in spatial extent, the unobserved
>>>>>>>>>> universe
>>>>>>>>>> must be infinite in extent (for a flat universe). Can you estimate
>>>>>>>>>> the size
>>>>>>>>>> of the unobservable universe for a positively curved universe? AG
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> The cosmological constant is a Ricci curvature with Λ = R_{tt} for
>>>>>>>>> the flat k = 0 case. for k = 1 there is a spatial Ricci curvature
>>>>>>>>> R_{rr}.
>>>>>>>>> This contributes to the occurrence of the cosmological constant,
>>>>>>>>> but it is tiny. So R_{rr} = δR_{tt} for δ a rather small number. The
>>>>>>>>> spatial sphere has a radius R = 1/√(R_{rr}} ≈ 1/√(δΛ). This is
>>>>>>>>> then for Λ = 10^{-52}m^{-2} R ≈ δ^{-1/2} 10^{26}m, or about the
>>>>>>>>> distance to the cosmological horizon multiplied by the reciprocal of
>>>>>>>>> a
>>>>>>>>> small number.
>>>>>>>>>
>>>>>>>>> The problem is that we really do not what that small number is.
>>>>>>>>> For various reasons I think it is δ < 5×10^{-5}This gives a
>>>>>>>>> radius where a Planck frequency is redshifted to a CMB scale. If it
>>>>>>>>> is
>>>>>>>>> smaller then there are regions of the universe completely
>>>>>>>>> inaccessible to
>>>>>>>>> us even as Planck modes redshifted to the cosmic horizon scale.
>>>>>>>>>
>>>>>>>>> LC
>>>>>>>>>
>>>>>>>>
>>>>>>>> FWIW, another reason I think our universe has a positive curvature
>>>>>>>> is that if it were flat, with zero curvature, and we made many
>>>>>>>> measurements, we'd get a distribution of measured values above and
>>>>>>>> below
>>>>>>>> zero due to unavoidable measurement errors. But I think we invariably
>>>>>>>> get a
>>>>>>>> small positive number. Is this what we actually get; values always
>>>>>>>> positive
>>>>>>>> but close to zero, but no negative values? TIA,AG
>>>>>>>>
>>>>>>>
>>>>>>> As yet attempt to find optical results due to spatial curvature have
>>>>>>> not found anything. The curvature of spacetime is mostly due to how
>>>>>>> space
>>>>>>> is embedded in spacetime.
>>>>>>>
>>>>>>> LC
>>>>>>>
>>>>>>
>>>>>> But you haven't directly addressed my hypothesis regarding the
>>>>>> measurements. AG
>>>>>>
>>>>>
>>>>> So far as I know there is no signal above the noise on this.
>>>>>
>>>>> LC
>>>>>
>>>>
>>>> Do the measurements show a spread around zero, including of course
>>>> negative values, or just positive values close to zero? This is where the
>>>> rubber hits the road IMO. If no negative results, there is the suggestion
>>>> the curvature is NOT zero. AG
>>>>
>>>>>
>>> There is a spread, but that is noise. Statistical variances of error
>>> convey no information.
>>>
>>
>> Are you sure? If the curvature is zero, there would be symmetric noise
>> around the value of zero. OTOH, if the curvature is positive we might get
>> asymmetric noise; that is, few values in the negative range. Also, don't
>> you think the finite age of the universe argues against a flat universe
>> (for reasons previously argued)? AG
>>
>
> Very sure. Please look up Gaussian variance. Errors in data carry no
> information other than the spread of error.
>
> LC
>
In the model suggested, the randomness of the errors could indicate the
location of the mean value. It's called thinking out-of-the-box. AG
>
>
>>
>> So far we really do not know. In fact, if you think about it, no matter
>>> now accurately we measure the curvature of space, say by cosmic lensing
>>> etc, we can never absolutely verify k = 0. We might be able to get k = 1,
>>> if that is the case and the radius of curvature not too huge. If the
>>> universe is absolutely flat we can never know that with complete certainty.
>>>
>>> LC
>>>
>>
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