Observational data (primarily the CMB) tell us that the curvature must be close to flat, but do not tell us that it is exactly flat. The belief that k=0 is a theoretical prejudice, usually based on fine-tuning arguments. For example, the "flatness problem" is the argument that any departure from flatness at the present time would imply that the total density of the universe at early times would have to be fantastically close to (but not equal to) the critical density. While widely believed, this argument has little logical force because the fine-tuning parameter here is simply an artifact of the definition of critical density (which evolves strongly with time). When curvature is expressed by more physical parameters, the fine tuning disappears and universes with nonzero curvature are found to be perfectly compatible with current and forseeable future CMB data. Some relevant papers here are by Adler & Overduin (gr-qc/0501061), Lake (astro-ph/0404319), and more recently by Carroll (1406.3057). In this context it is interesting to note that the Planck 2015 results (1502.01589) report a "2-sigma detection of positive curvature", which disappears when Planck data are combined with other datasets. A nice early discussion of these issues was by White and Scott (astro-ph/9508157).

It's worth appreciating how impressive the CMB measurement of curvature really is. It is very much akin to the measurement that might be made by creatures in a 2D universe who measure the curvature of their world by adding up the angles in a triangle to see if they are equal to 180 degrees. In our case the "baseline of the triangle" is the patch of sky occupied by the largest fluctuations in the primordial plasma at the moment when the universe became transparent to light. This method gets its power from the fact that it leapfrogs "local" systems whose interpretation is messy and complex (supernovae, galaxies, quasars) and goes directly back to the radiation-dominated era when physics was simpler.

http://arxiv.org/abs/gr-qc/0501061

http://arxiv.org/abs/astro-ph/0404319

http://arxiv.org/abs/1406.3057

http://arxiv.org/abs/1502.01589

http://arxiv.org/abs/astro-ph/9508157

There is a very convenient sometimes the term "curvilinear coordinates", which makes life of mathematicians on paper very convenient too.

But any fictitious curvature of coordinate does not change the actual mutual placement of facilities.

In Planck new data, the spatially flat universe is discussed well.

Information encoded in CBR and faraway galaxy counts point to a flat universe.

"Planck new data, the spatially flat universe is discussed well."

"Information encoded in CBR and faraway galaxy counts point to a flat universe."

Let me clarify my point by a simple thought experiment: An observer , at fixed point A , sends two spacecrafts moving away in different directions in straight lines from the observer. [This is a situation where any local gravitation effects are negligible]. After a long travel time and distance the spacecrafts “stop” relative to the fixed observer distances of B & C, respectively. At that this point everyone communicates by radio to each other. This will determine all the relative distances [A-B, A-C, B-C]. For a positively or negatively curved space the calculated area of the triangle ABC will not be equal to that expected for the flat (Euclidian) case. This is what I mean by measuring curvature directly. The only “model” is based on plane geometry.

I am certain that such a measurement, or something equivalent, has never been done. It may, in fact, be impossible at the astronomical distances that are required to determine any curvature at all even if it is non-zero.

The present way the curvature of the universe is determined is always based on cosmological models. These models “work” when the curvature term is made zero. In fact, this reduces the number of parameters in the Freidmann–Robertson–Walker (FRW) model to two (mass density and a dark energy term “Lamda”). Not only that, these two parameters are linearly dependent on each other (according to the model) so in fact only one parameter (the mass density) explains the expansion of the universe for all time!  These models demand the curvature be zero as opposed to actually making any direct measurement of the curvature of the universe.

Excuse me, please!

What is a curved space?

Astronomical measurements of both mass density of the universe and spacetime intervals using supernova events constrain the curvature to be zero or positive but near to zero.

arXiv:astro-ph/9806396 v1 30 Jun 1998

arXiv:astro-ph/9805201 v1 15 May 1998

If only matter and vacuum energy are considered, then the current combined data sets in above supernova articles provide direct evidence for a spatially flat Universe.

I appreciate all the feedback to this question including the recommended references which were helpful. Your replies still indicate that the curvature value is model dependent - in this case the LCDM model. The SN1A rely on an LCDM model. The CMB analysis of WMAP and Planck (2015) also rely on the LCDM model but with different methods.

In contrast, the ratio of the mass density to the critical density, Omega_m, can be measured from galaxies distribution surveys and the Hubble constant. Both of these are direct observations that are independent of any cosmological model. In fact the value of Omega_m was known prior to the notion of an “accelerating” universe implied by the LCDM model. The value of cosmological constant is simply implied from the model indicating that the curvature parameter is zero. I'm not in any way suggesting the LCDM model is “wrong” only that there is no cosmological observations that directly measure the overall curvature of the universe unlike the mass density. Once the model implies zero curvature then, and only then, is the cosmological constant determined.

I am quite sure if curvature was measured directly and turned out to be significantly non-zero cosmologists could still find a way to fit the LCDM model and its parameters to all astronomical observations (CMB, SN1A, or whatever). It has been shown historically, over and over again, that this is the nature of the business

Charles, Thank you for your reply:

As I understand it, the determination of the mass density of the universe does not require the FRW model such as LCDM. It is based on a number of observations such as mass to luminosity ratio of galaxies, estimation of dark matter (relative to luminosity ratios), and other less contributions such as electromagnetic radiation (photon) density. The expansion of the universe must also be taken into account determine the cosmic density. This rate of expansion is determined by the Hubble constant. The Hubble constant is also measured directly from redshift and known distances to galaxies. Yes, you can certainly use the mass density as a parameter to fit your data to any particular model such as LCDM. This allows you to make statement about the curvature and the cosmological constant. But the mass density is the only parameter which cam be measured independent of any particular model of the universe.

"I then find the data is consistent with Omega_Lambda=0 and positive curvature. All timescales are revised, and I don't need CDM."

Charles,

Well we are on the same page after all! I am also looking at a (simple) model in which curvature is a factor and does not require a cosmological constant that essentially fits the present observations - at least the SN 1A extrapolation of the Hubble curve. The CMB measurements are another thing of which I'm trying to wrap my head around. I also have found that the model can work with negative curvature - although I was originally insisting the curvature must be positive. To me, both negative and positive curvature can be considered equivalent under certain geometric transformations. In other words, positive curvature can be embedded in a universe with negative curvature - but that's another matter. My main problem with the LCDM model is the coincidence problem and the fact you can treat the universe as “coasting” for the past 2-3 billion years - with little to no deviation from the linear Hubble curve. The “acceleration” purported by the LCDM model is extrapolation not supported by the recent past.

Essentially, my viewpoint is that the universe must closed (finite yet expanding), thus has curvature of some kind. The fact that the curvature of the universe is determined to be zero, to me, implies two possibilities: (1) the universe is actually flat or (2) the curvature of the universe cannot be measured because all of our observations are embedded in 3 + 1 space-time. If the latter is true then many other models can be considered within the framework of FRW.  But alas, such theories are not much accepted at the moment. It will take perseverance to establish such ideas. My journal research shows that there are many other people working along the same lines and that the LCDM model is not the last word in the matter.

Observational data (primarily the CMB) tell us that the curvature must be close to flat, but do not tell us that it is exactly flat. The belief that k=0 is a theoretical prejudice, usually based on fine-tuning arguments. For example, the "flatness problem" is the argument that any departure from flatness at the present time would imply that the total density of the universe at early times would have to be fantastically close to (but not equal to) the critical density. While widely believed, this argument has little logical force because the fine-tuning parameter here is simply an artifact of the definition of critical density (which evolves strongly with time). When curvature is expressed by more physical parameters, the fine tuning disappears and universes with nonzero curvature are found to be perfectly compatible with current and forseeable future CMB data. Some relevant papers here are by Adler & Overduin (gr-qc/0501061), Lake (astro-ph/0404319), and more recently by Carroll (1406.3057). In this context it is interesting to note that the Planck 2015 results (1502.01589) report a "2-sigma detection of positive curvature", which disappears when Planck data are combined with other datasets. A nice early discussion of these issues was by White and Scott (astro-ph/9508157).

It's worth appreciating how impressive the CMB measurement of curvature really is. It is very much akin to the measurement that might be made by creatures in a 2D universe who measure the curvature of their world by adding up the angles in a triangle to see if they are equal to 180 degrees. In our case the "baseline of the triangle" is the patch of sky occupied by the largest fluctuations in the primordial plasma at the moment when the universe became transparent to light. This method gets its power from the fact that it leapfrogs "local" systems whose interpretation is messy and complex (supernovae, galaxies, quasars) and goes directly back to the radiation-dominated era when physics was simpler.

http://arxiv.org/abs/gr-qc/0501061

http://arxiv.org/abs/astro-ph/0404319

http://arxiv.org/abs/1406.3057

http://arxiv.org/abs/1502.01589

http://arxiv.org/abs/astro-ph/9508157

I. A. Urusovskii, Multidimensional   Treatment of the Expanding Universe,

 Physical Science International Journal, 4(8): 1110-1144, 2014

I. A. Urusovskii, Multidimensional   Treatment of the Expanding Universe,

 Physical Science International Journal, 4(8): 1110-1144, 2014

Following up James Overduin's excellent comments, let me point readers here to my own paper at arXiv:1112.1666 (MNRAS 421,1, 561--569 (21 March 2012)).

http://www.astro.multivax.de:8000/helbig/research/publications/publications.html

The simplest 6D interpretation of expanding Universe agree with observations is given in

  I. A. Urusovskii, Multidimensional   Treatment of the Expanding Universe,

 Physical Science International Journal, 4(8): 1110-1144, 2014. 

        Our three-dimensional universe is isotropic and homogeneous at distances of more than 300 million light-years. This means that the Universe is a three-dimensional sphere, and that the scale of inhomogeneities in the Universe 100,000 times smaller than the characteristic size of the Meta galaxy. 6D-cosmology gives for the radius of the Meta galaxy value 3980 Mpc. The length of the great circle of the Meta galaxy passing through its borders (particle horizon), can be taken as a characteristic size.  Then the size of inhomogeneities in the Meta galaxy is 2 3980/100000 = 0.25 Mpc or 815,646 light-years, which is approximately equal to the average distance between galaxies.

       In Euclidean spaces with the number of spatial dimensions of a large three the Lorentz transformations are obtained elementarily, if the proper time of an elementary particle assumed to be proportional to the path traversed by it in the extra space, supplementing the observed three-dimensional space until a Multi-dimensional space, provided that the elementary particles like photons, moving at the speed of light in the whole space. All areas of the universe larger than 300 million light years have the same properties. The universe is a three-dimensional sphere, expanding in the space of a higher dimension. Elementary treatment of a spin, isotopic spin, Heisenberg relations as well the magnetic moment of the electron all require 6 spatial dimensions.  Therefore, cosmology must also take into account the 6 spatial dimensions of a complete space.

        Six-dimensional interpretation of the expanding universe is built on the principle of simplicity. The simplest geometric object of finite size in the 6D Euclidean space is a 5D sphere. Therefore, the simplest 6D interpretation of the expanding 3D sphere is its representation as the intersections of three 5D spheres expanding in the 6D Euclidean space − three 5D spherical frontiers of disturbances propagating in this space. Thus it is assumed that these perturbations occurred as a result of three separate ‘explosions’ in the centers of these 5D spheres. Point of first contact of the three 5D spheres corresponds to the location of the center of the three-dimensional universe in the 6D space. The intersections of three -dimensional spheres are three -dimensional spheres, which, in turn, intersecting to form three -dimensional sphere, etc.

            The curvature of the universe as a three-dimensional sphere is defined as unity, divided by the radius of the sphere. Geometric and physical quantities of the Universe are given by consideration of the cosmological model with fixed free parameters of the theory. The theory contains 5 free parameters. These parameters are chosen so that the deviations of the compared values from each other were minimal. For the chosen parameters in the case N = 5, the many results of the theory and observations are compared. By this, the radius of the Universe is 7100 Mpc, the Meta galaxy radius  is 3980 Mpc. The speed of light on our three-dimensional sphere and the speed of its radial expanding of this sphere are connected by Pythagorean’s theorem. Sum of the squares of the two speeds is assumed constant according to the principle of simplicity. By this the velocity of light was equal to zero when the radius of the Universe was less than 18 percent of the current one in the case  N = 5 . From this point the speed of light is constantly increasing, asymptotically tending to a constant limit. Increase the speed of light leads to increasing of proper energy of elementary particles and thermal energy of stars and planets.

Igor A. Urusovskii

[email protected]

A flat universe is a very ideal concept. where matter  exists, the spacetime can never be flat. Locally the spacetime can be considered approximately flat, but globally it is curved, in fact hyperbolic.

Article The hyperbolic geometry of the universe and the wedding of g...

Article The Infinite Distance Horizon and the Hyperbolic Inflation i...

The simplest 6D-interpretation of expanding Universe agree with observations is given in

 I. A. Urusovskii, Multidimensional   Treatment of the Expanding Universe,

 Physical Science International Journal, 4(8): 1110-1144, 2014. 

        Our three-dimensional universe is isotropic and homogeneous at distances of more than 300 million light-years. This means that the Universe is a three-dimensional sphere, and that the scale of inhomogeneities in the Universe 100,000 times smaller than the characteristic size of the Meta galaxy. 6D-cosmology gives for the radius of the Meta galaxy value 3980 Mpc. The length of the great circle of the Meta galaxy passing through its borders (particle horizon), can be taken as a characteristic size.  Then the size of inhomogeneities in the Meta galaxy is 2π3980/100000 = 0.25 Mpc or 815,646 light-years, which is approximately equal to the average distance between galaxies.

       In Euclidean spaces with the number of spatial dimensions of a large three the Lorentz transformations are obtained elementarily, if the proper time of an elementary particle assumed to be proportional to the path traversed by it in the extra space, supplementing the observed three-dimensional space until a Multi-dimensional space, provided that the elementary particles like photons, moving at the speed of light in the whole space. All areas of the universe larger than 300 million light years have the same properties. The universe is a three-dimensional sphere, expanding in the space of a higher dimension. Elementary treatment of a spin, isotopic spin, Heisenberg relations as well the magnetic moment of the electron all require 6 spatial dimensions.  Therefore, cosmology must also take into account the 6 spatial dimensions of a complete space.

        Six-dimensional interpretation of the expanding universe is built on the principle of simplicity. The simplest geometric object of finite size in the 6D-Euclidean space is a 5D-sphere. Therefore, the simplest 6D-interpretation of the expanding 3D-sphere is its representation as the intersections of three 5D-spheres expanding in the 6D-Euclidean space − three 5D-spherical frontiers of disturbances propagating in this space. Thus it is assumed that these perturbations occurred as a result of three separate ‘explosions’ in the centers of these 5D-spheres. Point of first contact of the three 5D-spheres corresponds to the location of the center of the three-dimensional universe in the 6D-space. The intersections of three N-dimensional spheres are three N-1-dimensional spheres, which, in turn, intersecting to form three N-2-dimensional sphere, etc.

            The curvature of the universe as a three-dimensional sphere is defined as unity divided by the square of the radius of the sphere. Geometric and physical quantities of the Universe are given by consideration of the cosmological model with fixed free parameters of the theory. The theory contains 5 free parameters. These parameters are chosen so that the deviations of the compared values from each other were minimal. For the chosen parameters in the case N = 5, the many results of the theory and observations are compared. By this, the radius of the Universe is 7100 Mpc, the Meta galaxy radius  is 3980 Mpc. The speed of light on our three-dimensional sphere and the speed of its radial expanding of this sphere are connected by Pythagorean’s theorem. Sum of the squares of the two speeds is assumed constant according to the principle of simplicity. By this the velocity of light was equal to zero when the radius of the Universe was less than 18 percent of the current one in the case  N = 5. From this point the speed of light is constantly increasing, asymptotically tending to a constant limit. Increase the speed of light leads to increasing of proper energy of elementary particles and thermal energy of stars and planets.

1.     Maximum of density distribution in redshift z inflow of total energy in the Universe per unit time is reached at  z = 1.134. According to [16, L. Tresse et al., ‘The cosmic star formation rate evolution from z = 0 to z = 5 from VIMOS VLT Deep Survey,’ http://arxiv.org/astro-ph/0609005], the maximum rate of star formation is observed at z = 1.1. In 6D-theory evolution of the Universe does not depend on the density of the substance therein.

             This 6D-theory gives the deceleration parameter q0 of expansion of the 3D-sphere of primary interest for comparison of observations:  q0 that equals 0.038 – the value obtained in [49, Rines K., Forman W., Pen U., Jones C. and Burg R.. Constraining  with cluster gas mass fractions. Feasibility study. Astrophysical Journal. May 20, 1999; 517 (1), Pt. 1: 70-77] by the method of EMN (Evrard, Metzler, Navarro) not associated with Hubble constant. For universe expanding with acceleration, deceleration parameter is negative.  

         To the question "Where is the center of the Universe?" should be a simple answer:

The center of the universe as a three-dimensional sphere lies in the space of a larger number of dimensions outside the universe at equal distances to all points of the three-dimensional sphere.

Igor A. Urusovskii

Typically, cosmologists measure ρ and H, that is observed density and the Hubble parameter. Then density parameter is defined as, Ω= 8πGρ/3H2 and observed value of this parameter is very nearly one. This definition of the density parameter is obtained when we set curvature k=0, in Friedmann equations, and we get, Ω=8πGρ/3H2=1. Thus theoretical and observed values agree with each other. Therefore we say that that the universe has a flat geometry.

It looks like a very fine tuned situation indeed. Because mass-energy is present in the universe and we should have rather expected that the geometry of the universe would be curved under the influence of mass-energy it contains.