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
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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