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# Curved Cosmos Seen as Virtually Flat in the Universe: A Scaling Agreeing with Empirical Evidence

*Truth spirit, enlighten and guide our researchOh Mary conceived without sin pray for us that have recourse to you*

A metaphysical geometrical representation of a curved cosmos seen through a flat universe is tested. Trigonometric calculations are performed on the representation applying a complex metaphysics, on one side concerning the redshift and on another side concerning the discrepancy for supernovae between distances computed with light intensity with respect to those computed with redshift. The calculated data fit with empirical evidence. The metaphysics involved would result as such as an evidence based substitute paradigm to that of the Big Bang.

**
Introduction**

The hypothesis is that the empirical data supporting the Big Bang paradigm may be read through a different paradigm. The astronomer cosmologist George F. R. Ellis (Gibbs, 1995) highlights that the observations can be explained by a range of models. The different paradigm approached here below is defined in Benazzo (1998, 2001, 2010). Its main characteristic is to add one physical domain; the Cosmos.

The cosmos is as such distinguished from the Universe. This may allow for the former being curved and the latter being flat. In such paradigm, the Cosmos is curved in space-time. The space spheres, traversed by expanding light, result as undergoing a hidden tilting on the time axis. The time axis, in other words, results as tilting gradually with the increasing distance away from the observer, in each direction considered. In such a paradigm the universe is defined as the vision of a curved cosmos arriving to the observer through a flat light cone. As such the universe is flat by its own nature. In this flat perspective, the observer lacks possibility of seeing the tilting of space-time. Space-time in the observer's space axis results thus as totally seen in terms of space, while space-time in the observer's time axis stems as totally seen in terms of time. Such vision is determined as virtual deformation of each distant local region or cosmological body. Such virtual deformation would recompose for a space probe travelling until such far destination. Empirical findings may as such be interpreted at least in two ways. If the Cosmos and the Universe are considered as coinciding, the cosmos needs to have the same degree of curvature of the universe, and the empirical findings support the Big Bang paradigm. With an observer's flat Universe in a curved Cosmos, the empirical findings may be read as to support another paradigm. How to determine the curvature of the Cosmos? Einstein's relativity is considered for that.

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**Calculations**

Considering how the light cones are formed, the only certain thing that empirical investigation sees directly is the light of the light cones. Space and time are deduced indirectly. They form a completely apparent experience in the everyday life, however for empirical investigation, in the light cones representation, space is deduced as the flat plane intersecting horizontally the vertical light cone, and time is deduced as the central vertical axis of the light cone. Both remain invisible to the optical or radio instruments of empirical investigation, or to the eye. The curvature question needs as such to be posed on the light cones. Which transformation of the light cone needs to be done such that a body receding from the observer at the speed of light may instead be considered stationary? As depicted in figure 2 below, from the observer point of view, an equilateral triangle, allows that, determining a 60 degrees tilting.

This angle is applied to the light cones. In more detail, two such light cones are initially considered in the classical way, both facing up, with the time axis of the far away body parallel to that of the observer (figure 1). The one of the observer, observing down sideways, i.e. back in time through the light cone signals, and one facing up for a cosmological body that the redshift reading would indicate as receding at exactly the speed of light. In figure 1, the null vector of the faraway body, which is nearest to the observer, is considered as coinciding with the past null vector of the observer's cone that is nearest to the faraway body. The redshift in such a case indicate a recession velocity at the speed of light. In figure 1 this is indicated as the red arrow at 300 degrees, i.e. 60 degrees to the left of the observer's time axis. In the Big Bang paradigm, such recession is due to space expansion where the cosmological body remains locally stationary. Figure 1, and its red arrow, would then be considered to apply to the space rather than to the local motion of the cosmological body.

*Diamond: time axis*

Now the tilting of the faraway cosmological body light cone is performed, at 60 degrees (figure 2). In such a way the two nearest past null vectors of the two cones coincide, indicating that the two cosmological bodies occur, locally, concurrently, with respect to the observer's time. In such a representation,

*Diamond: time axisArrow: space axisCircle: light axis*

considering the flow of information of the far away cosmological body, this intersects the flow of information travelling to the observer through the observer's past light cone. The intersecting light would send information to the observer, though with a tilting. In the tilting of figure 2, the future null vector of the faraway body that is nearest to the observer is vertical and its slope coincides with that of the time axis in the previous usual case of a receding body, and with that of the Earth light cone. Such light would normally be considered as never reaching the observer's time axis, it would be considered invisible (Benazzo, 1998, 2001). Einstein's relativity states that time and space may interchange. The light cones so defined imply an angle of the light cone null vectors with the space plane that is 30 degrees, and an angle of such light null vectors to the time axis equal to 60 degrees.

Such relations in angles between the light vector, space and time allow calculating all of three in terms of just one of them. Those angles in other words determine the ratios of the interchangeability between space and time characteristic of relativity. The first attention will be at the space, in terms of light years, therefore all of the three will be considered with such unit of measurement. The space considered is 13.7 billion light years, as the distance from where cosmological bodies recede at the speed of light. The time for that space results as 13.7 billion years, as the time since the beginning of time in the Big Bang paradigm. Those 13.7 billion years of time are now measured in terms of space.

Time of 13.7 billion years in terms of space = 13.7 billion light years/sin(radians(60))*sin(radians(30)) = 7.91 billion light years. For the reversibility between the two, by rotating by 90 degrees the light space-time triangle, 13.7 billion years would have an axis parallel to the observer's space axis and would be perceived as 7.91 billion light years of distance. As the central attention is on space, the measures on the time axes are measured in billion light years of space, relative to the defined angle relations of the light space-time triangle. The measures on the light vector are also made in relation to the space measure of light year. The measures of the light vector are taken in order to determine the redshift. This latter is a measure in units as it compares two light measures, cancelling out the equal units of the numerator with the denominator. In this case the measure on the light vector in terms of space keeps embedded the relevant information about wavelength, until the measurement of the redshift.

Given such units of measurement, further considerations are now made on Figure 2. A local light vertical signal of the faraway body, even if parallel to the observer's time axis, would though imply a deduced space that is tilted 30 degrees towards the observer's time axis. Such space trajectory would then intersect the observer's time axis. In correspondence of such intersection, the space-time coordinates of the signal from the faraway body would be received. As such implied intersecting angle is different from the local one of the light signal of the faraway body, there would be virtual transformations of the light signal received by the observer; whichare discussed further down.

The possibility that what is seen as receding in the Big Bang paradigm is rather a stationary cosmological body in curved stationary cosmos is as such posed and further examined. Keeping still the intersection of the two time axes and rotating anti-clockwise such a relation of figure 2, between the observer location and the considered body, a curve is determined. Such curve results then as curved space

(as outlined in figure 4). However such curvature remains unseen (apart from point **p(60))**, given information arriving to the observer as forced to pass through the observer's flat light cone, i.e. the straight line from the exemplified 'Quarkland' until the Earth (figure 2). The observer is clarified as bound to see though the light cone, while the far away light cone would be sending information in its future with another trajectory.

The current agreed distance, from where cosmological bodies recede at the speed of light, is 13.7 billion light years. In the paradigm posed here, the curvature implies that time and space of the faraway body may be at least partially swapped from the observer's viewpoint. This would remain hidden behind the universe flatness, giving a virtual measure. Such agreed measure is then called virtual interval between the observer and the bodies seen as receding at light-speed.

The following variables are defined, as marked in figure 4 below:

Definition of the axis by the type of the line end-mark or character style of variable:

**Diamond: time axis / Arrow: space axis / Circle: light axis**

Definition of viewpoint by the dimension and elongation of the diamond, arrow, circle:

Small: experienced locally by the observer / Large: experienced locally at the far away cosmological body / Elongated: faraway bodies coordinates as virtually perceived by the observer viewpoint

**Variables:****: p(60) ** : Position of the planet (Quarkland) considered as equivalent to a cosmological body receding at light-speed in the traditional paradigm**: p(0)** : Position of Earth where from the scientist observes**: Sv ** : Space-virtual: the local space coordinates perceived from the observer, taken on its tangent line, from the tangent point on the effective curved cosmos space to its intersection on the observer's time axis to**: ln** : light-null-cone-local, referred to each far away local effective curved cosmos space and its relative virtual space**: tl ** : time-axis-local as time axis experienced locally by a space probe reaching that location**: to or to(n)** : Time-axis-observer as the time axis experienced by the observer, or perceived as time of far away cosmological bodies from the observer's viewpoint**: sigma** : Sigma as the angle between the observer time axis and the far away local time axis**: alpha ** : Alpha as fixed 60° angle between light and time**: beta ** : Beta as fixed 90° angle between space and time**: rho ** : Rho as the angle opposite to the radius of the curved space circumference**: iota ** : Iota as the angle between the local time axis and the null cone of the observer**: r ** : Radius of the curved space circumference, determined on the time axis**: so ** : Horizontal space relative to the null cone of the observer's location**: s=f(r,arc s)** : Curved space arc corresponding to the effective space from the cosmological body pn (e.g. p60 apparently receeding at the speed of light) with respect to the Earth p0**: p(n)** : Position of the cosmological body considered on the actual space**: sop ** : Horizontal space relative to the arc of the effective curved space**: n** : Light of the null cone of the observer**: pn** : Position of the interception of the cosmological body world line, i.e. the effective local time axis tl, with the null vector n of light-observed from the observer's location**: RWSv ** : Stretched relative wavelength of the far away light, elongated with respect to that of the observer's light cone past, which is segment 'n', i.e. 'rwn' (see here below).**: rwn** : Relative wavelength of the light of the observer's light cone past, which is segment 'n'.**: tes(n) ** : Intersection of the effective local time axis relative to the *pn(n*) with the curved space arc s**: tos(n) ** : Intersection of the to(n), the local far away time axis parallel to that of the observer with the Space-virtual**: ti ** : Time of effective time axis, from origin to intercept with the observer's light null cone**: DELTAr=r-ti ** : Radius difference relative to the effective-time-axis-intercepted by the null cone (DELTAr=0 for s=60° and for s=0°)**: DELTASv=f(r-ti)** : Virtual space difference between brightness related and redshift related measures**: DELTARD=DELTASv/Sv** : Relative difference between virtual distance by brightness and virtual distance by redshift

Given the age of the universe, here interpreted as rather virtual interval, Sv(60), between the observer and a body seen in the Big Bang paradigm as if receding at light-speed, here with time axis tilted by 60 degrees, then the following is calculated:**Sv(n)** is considered tangent to the radius taken on the tilted time axis, i.e. at 90 degrees to such radius. Its length is considered from the cosmological body position **p(n)** to the intersection with the observer's time axis to, at point **tv(60)**.

First, the radius of the circumference of the arc relative to the locatl time axis **tl(60)** is determined:**r = Sv(60)**/**Sin(Radians**(ang_between_2_time_axis))

***Sin(Radians**(180-90-ang_between_2_time_axis))

With 'ang_between_2_time_axis' = 60 degrees**r = (Sv(60))**/(sin(radians(60))*sin(radians(30))

For **Sv(60)** = 13.7 billion light years then r = 7.91 billion light years

In this case r is considered as measure for the distance, and as such denoted in light-years.

Given r, it is possible to calculate Sv(n) for each angle sigma between the observer time axis and that of the faraway body.**Sv(n)** = f(r, **sigma**) = radius***Tan(Radian**s(ang_between_2_time_axis))

From the observer viewpoint, the space concealed behind such virtual space could be interpreted as the horizontal segment from the observer's time axis to the observer's null cone, relative to a considered cosmological position **p(n)**.

This would be the space relative to the null cone of the observer as follows:

**sn(60**=(radius/Sin(Radians(ang_local_time_intercept_with_null_cone_observer)) ***Sin(Radians**(ang_between_2_time_axis)))**/Sin(Radians**(90))***Sin(Radians**(60))

For **Sv(60)** = 13.7 billion light years then **sn(60)** = 6.85 billion light years

However given the case considered of a curved cosmos, the arc s is considered the actual curved space. Consequently **sn(60)** would be without useful meaning, as it would be neither the defined virtual space **Sv(60)** perceived nor the defined actual effective space** s.**

The starting measure of s considered is the arc **s(60)**. Its calculation starts from the origin of the time axes identifying the length from there to the time at p(60). As said, this latter lays to the axis at 60 degrees from the observer's one, where the radius r calculated above lays. The arc determined by the radius r and the 60 degrees angle results as the curved space discussed above:**S(60) = 2Π r *60/360**

For **r = 7,91** billion light years then **s(60)** = 8.283 billion light years

Given the defined curvature as going back to itself, a probe going straight in space would reach back Earth after 6 times that arc **s(60)**, that is travelling 49.7 billion light years.

Given the structure of the graph defined, any Sv(n) is tangent to the arc s at 90 degrees to the radius r at the tangent point of Sv(n). For any Sv(n) , it is as such possible to measure its corresponding s(n) once sigma(n) is known.**r = (Sv(n))**/(sin(radians(sigma(n))*sin(radians(180-90- **sigma**(n)))**s(n) = 2Π r ***sigma(n)/360

A model, theory or paradigm is evidence based when its implied measurements agree with empirical findings.

One empirical finding considered consists in the measured pattern of discrepancy relative to the supernovae locations between on one side the distance measured with the brightness or faintness of distant supernovae and on the other side the distance measured with the redshift measurements (Kowalski et al., 2008), as also in the plotted figure by Wright (2009) or by Riess and Turner (2004).

The discrepancy is defined here considering the local time axis **tl(n)** of any position of a cosmological body, between the position **p(n)** on the effective space arc, and the position n(n) intercepted by the null vector of the light observed by the observer. Taking from n(n) a vertical line, this represents the time axis observed by the observer, i.e. to(n) is parallel to to. This axis intercepts the **Sv(n)** at the point tos(n), determining a segment, starting from (in this figure, from the left) its **Sv(n)** tangent point to the space **s** in **p(n)**, until the intersection point indicated **tos(n).**

The distance between **p(n)** and **tos(n**) is defined as indicator of the discrepancy between the measurements by the redshift and the measurements by luminosity of supernovae. In specific, the progression of how the distance between **p(n)** and **tos(n) **evolves in relation to **Sv(n),** is then considered the relative difference in virtual distance between the measurement from luminosity and the measurement from redshift.

**Calculations are defined as follows:**

Segment ti is defined as segment from the origin of the time axes, to the intersection n(n) with the null cone of the observer.**ti=radius/Sin(Radians**(180-60-ang_between_2_time_axis))***Sin(Radians**(60))

Then the difference between the radius r and ti is taken. This is considered as representing the difference between the effective time on the effective curved space and the time observed on the observer's null cone (intersection).**Delta t(n)=r-ti(n)** = radius-time_origin_to_null_cone_observer

Then, from this segment representing time, the segment on Sv(n) between p(n) and tos(n) is then considered:**DeltaSv(n)**=f(Deltat(n))=delta_t_effective_vs_t_observed/**Sin(Radians**(180-90-ang_between_2_time_axis))* **Sin(Radians**(ang_between_2_time_axis))

This number is then made relative to the segment Sv(n) so that it represent a relative difference netting out the magnitude of the Sv(n) considered, by subdividing **Deltasv(n) **by **Sv(n**) so as to obtain a relative difference (RD).**Deltard**=**DeltaSv(n)/Sv(n) =delta_s**_brightness_vs_s_redshift/Space_virtual

This measurement agrees with empirical data in the following way. Riess and Turner (2004) indicate that the coasting point between slowdown and speedup is at about one third of the calculated age of the universe. In the herewith paradigm, rather than the distance relative to the age of the universe in the Big Bang paradigm, the variable is calculated as virtual distance Sv(n) tangent to effective point p(n) on the effective space s. The coasting happens at **Sv(30)**,

when **Sv(30)** = 4.567 billion light years, exactly one third of the **Sv(60)** = 13.7 billion light years. The same happens with the empirical data calculations within the Big Bang paradigm by Riess and Turner (2004).

Now the redshift is calculated with the herewith paradigm in order to check if it agrees with the evidence for the redshift related to the coasting point.

To calculate the redshift, the point p(n) on the curved effective space is considered. For such a point there is a comparison between on one side the wavelength of the virtual light relative to the virtual space **Sv(n**) as received from that faraway point **p(n)** and on the other side the wavelength of the light relative to observer's null cone if the light cone of **p(n)** were vertical. The respective measures are taken starting from how each of the two cases projects its time span on the observer's time axis. Considering the interchangeability between space and time characteristic of relativity, the time relative to the virtual light is taken to end its considered course at the first intercept **tv(n)** of the tilted light space-time triangle relative to **Sv(n)**. Point **p(n)** had been intercepted by the observer's light null cone in the past in the point **pn(n)**. Time and space, at that previous local time and space, where relatively shrunk with respect to those of the point p(n). This shrinking in the past, rather than being due to an actual shrinking, would be due to how space-time is curved. Such point pn(n) passed from that past and became then **p(n)**.

The past space segment so(n) from **pn(n)**, horizontally stretching until the observer's time axis to, in time transforms itself becoming the horizontal segment sop(n) from the point **p(n)** to the observer's time axis to. At such past time, p(n) was at a distance sop(n in the past), from the observer's time axis, and laying on the curved space s(in the past).The **sop(n**) intercepts the observer's time axis to defining the vertical coordinate of the starting point of the virtual time ts(n), on the observer's time axis to. The virtual time between ts(n) and **tv(n**) relative to the virtual light of cosmological body **p(n) **is then:**TSv(n)**=(radius/**Sin(Radians**(ang_opp_radius))***Sin(Radians**(90)))-radius+(Space_horizontal_of_effective_curved_space in the past/***Sin(Radians**(60)) ***Sin(Radians(**30)))

With Space_horizontal_of_effective_curved_space _in_the_past/**sop(n in_the_past)=radius/Sin(Radians**(180-60_between_2_time_axis))

***Sin(Radians** (ang_between_2_time_axis)))***Sin Radians**(90) ***Sin(Radians**(60))

The time relative to the observer's light null-cone is instead calculated as just the last part of the above equation, corresponding to the time between ts(n) and p(0).**tn(n=**(Space_horizontal_of_effective_curved_space in the past/**Sin(Radians**(60))

***Sin(Radians**(30)

The tilting of the local light cone, far away from the observer, tilts the local null vector nearest to the observer. At 60 degrees, , i.e. at point p(60), the cosmological bodies are read, with the conventional paradigm compared to the herewith geometrical configuration (figure 4) as receding at the speed of light: the null vector of the light cone nearest to the observer is locally vertical, never intersecting the observer's time axis. The relative wavelength of the faraway cosmological body is graphically calculated as the relative length of the light null vector from the faraway body to the observer's time axis. As in this case the two are parallel, never intersecting, this would normally cause the redshift of that local body with respect to the observer to be infinite.

This should be the result applying the usual Doppler formula, the one applied to special relativity (Lineweaver and Davis, 2005). Considering the interchangeability between space and time concerned with relativity, the space stretches at 30 degrees from the considered light null vector towards the observer time axis to, with the segment **Sv(60)** . With the further definition of interchangeability between space and time of general relativity, the space vector at 30 degrees towards the observer's time axis is considered as determining the first contact of the approaching space-time triangle of light with the observer's time axis, at the point **tv(n)**.

The light passing at **tv(n)** is then considered. As the light perceived from the observer is considered as forced to happen only at the same slope of the observer's light null-cone. The virtual light is then the segment (in this case to the left) starting at point **tv(60)** and at 240 degrees, which looks downwards and tilted 60 degrees from the observer's time axis. This goes down sideways until it intersects, at point **RWSv(60)** for **p(60)**, the horizontal line passing through the lower bound **ts(60)** of the virtual time **TSv(60)** on the observer's time axis. The length of such virtual light stems as the relative virtual wavelength as perceived by the observer from the faraway body. This is determined by:**RWSv(n)**=f**(TSv(n)**)=Time_of_virtual_space/**Sin(Radians**(30))***Sin(Radians**(90))

On the other side, the light null vector on the observer's null cone starts from the observer position **p(0)**, Earth, and projects down at 240 degrees, i.e. downwards at 60 degrees (left in this case) from the observer's time axis to. Its lower bound is also the horizontal line intersecting point **ts(n) **that is the lower bound of the virtual time **TSv(n)** on the observer's time axis. The length of the segment, relative to null vector, so defined results as the relative wavelength of the light perceived by the observer locally. This is determined by:**rwn(n)=f(tn(n))**=Time_null_cone_observer/

**Sin(Radians**(30))*

**Sin(Radians**(90))

The redshift is calculated as the ratio of the first over the second, minus 1

**Redshift=z=**(RWSv/rwn)-1=Relative_Wavelenght_of_virtual_space

_vertical_null_cone/relative_wavelengh_null_cone_observer-1

The calculated redshift at the above measured coasting point occurring at tl(30) is at z = 0.6, near z=0.5 (see chart), the same of the coasting point interpreted in the Big Bang paradigm as the point between slowdown and speedup (Riess and Turner, 2004, Wright, 2009).

The comparison of such segment **RWSv(60)** for **p(60**) of relative virtual wavelength, with the corresponding one n(60) of the observer for the considered 60 degrees angle, gives a** z = 2 r**edshift measurement (see chart). This **z = 2 **is at 60 degrees tilting of the faraway time axis. This would correspond to the cosmological redshift formula, the one applied to general relativity (Lineweaver and Davis, 2005). The cosmological redshift formula is usually applied to an expanding space. Here however it is applied to a static cosmos space, only virtually expanding. Calculations give Lineweaver and Davis (2005) a redshift of 1.5, for bodies seen as receding at light-speed, rather than 2. In the herewith paradigm, the redshift **z = 1.5** is at 53 degrees, i.e. a bit less than 90% of the 60 degrees. For Riess and Turner (2004), the relative brightness difference is at about zero at a redshift 2, as in the herewith paradigm. Differences,i.e z=2 and z=0.6 each shifted by proportionate time axis degrees from the respective z=1.5, and z=0.5, could be read as the application of the formula to an expanding space in the Big Bang paradigm in comparison to a curved static space in the herewith paradigm. At each redshift z there is a corresponding virtual space **Sv(n)**.

Another characteristic of such a paradigm consists in the presence of what is read in the Big Bang paradigm as time since the big bang, of 13.7 billion years. This is intrinsic in two geometrical positions of the figure 4.

One is the segment **Sv(60)** = 13.7 billion light-years. The other is the segment of the time axis **tl(90**) at 90 degrees from the observer's time axis, which is horizontal and perceived as space from the observer's point of view. Considering the measure from the origin o where the time axes intercept, to the intercept **n(90)** of such time axis **tl(90)** with the light null vector n, this measures horizontally 13.7 billion light-years (**radius/sin(radians**(30))*sin(radians(60))). It may be read as relative to the age of the universe in the Big Bang paradigm: 13.7 billion years, on the observer's time axis to, from the time axes origin o to **p(0)**. As such, the former herewith paradigm concentrates on the virtual space **Sv(60)** tangent to what is defined as effective curved space **s**, in the point **p(60)**. The latter, the conventional Big Bang paradigm, concentrates on the segment from o to **n(90**) on **tl(90)** defined in the sentences here before.

Every 60 degrees or 90 degrees tilting, a space probe would find itself in an observed universe with the same average quantity of matter, even if from the previous starting point the appearance was of an expanding universe. After 6 (for 60 degrees) or 4 (for 90 degrees) of such travels in a straight line in the curved space s, the space probe would be back at the starting point, the observers' initial position **p(0)**. Such position **p(0**) would then result virtually interior to the successive travels through space if the space probe is considered as receding away from the starting point **p(0)**. At the same time, such same position **p(0)** would result as virtually exterior to the approaching probe arriving after the travel (from the right in figure 4).

This is a paradoxical situation. Such a kind of paradoxical situation is also present in the graph indicated where the time axes intersect at a starting point o, while at the same time, with such a construction time may go back indefinitely.

This works as for fractals, so that the cosmos results as a fractal topology where the coordinates in a certain point in time from the origin are exactly self-similar to the smaller coordinates of a previous time distance from the origin; or to the larger coordinates of a successive time distance from the origin. Looking back in time the cosmos goes back to the intersection of the time axes into a null position, a singularity. Getting nearer to that position the coordinates spring out, in a similar way of how the magnification of a Koch curve works. If the observation approaches such null position, the null point would enlarge as fractal and move the null space-time position backward in time, as if it was a horizon. The opposite would be the reading going in the future of a particular point. The cosmos looks as if it enlarges, while this would be a virtual expansion of the observed universe. Approaching to the time in the future would result in a movement, like the shrinking of a Koch curve, resulting in the shrinking of the approached future coordinates. The whole of the cosmos would enlarge infinitely in the future and shrink infinitely in the past.

Such virtual enlargement though recomposes for an actual space probe traveling through the effective space **s,** so that such travel of a space probe brings back to the starting point** p(0)**. In this conformation, the universe undergoes a virtual inflation, which would actually be a fractal, with a virtual big bang. Shu (2010) provides for a mathematical model of a universe without Big Bang.

Another paradoxical situation is that space-time tilts gradually, so that a space-time region in the cosmos has on average a reverse region with opposite space and time axis. In such a way the whole outcome of the combination of each one of all the regions with the respective reverse region is that the whole aggregate annihilates in the null. A solution to such paradox stems from the fact that the observer is inside such cosmos, and may be considered as form observing the rest; this latter considered as anti-form; or vice-versa (Benazzo, 2010).

Such paradoxical situation of the paradigm may bring to call this cosmological paradigm as null-whole cosmos (Benazzo, 2001, 2010), or null-whole paradox (Benazzo, 1998). Such paradoxical situation may be resolved by applying a rethinking to the Russell's definition of the laws of thought (1967), which are used in science, with a special derogation, argued as reasoned and coherent, for the whole and the null, such as in Benazzo (2010). This implies a solution to Russell's (1903) paradox (Benazzo, 2010).

Such annihilation would need channels of perfect communication beyond time and space. Is there such a case? Challender (2010) describes how there are such cases in physics.

A characteristic of the paradigm would be analysed starting from considering the world line (see definition in Wikipedia) of the observer staying still on the Earth. Paul Davies (1982) notes that a construction where the light cone is on a surface that returns on itself, like a two dimensional light cone on a plane, like a sheet of paper, returning on itself by rolling the paper from top to bottom, then this construction implies that the observer's future causes the same observer's past. Is this the case here? The past light cone n of the observer in the herewith paradigm intersects the circumference s of concurrent cosmological bodies, at a cosmological body **p(60)** at the same time r from the time origin o. There are cases of empirical evidence about a master clock (Challender, 2010).

The concurrent cosmological body on that intersection **p(60)** causes the past of the observer at **p(0)** through electromagnetic radiation. Each concurrent cosmological body **p(n)** at degrees n from the observer, with n a multiple of 60 degrees influences the past of the cosmological body 60 degrees nearer to the observer. Starting this causation from the observer and going clockwise, the observer causes the past light cone of the concurrent cosmological body 60 degrees clockwise, which causes the past light cone of the concurrent body at 120 degrees clockwise, and so on until the concurrent cosmological body at 300 degrees causes the past cone of the Earth, where the observer stays. Through this channel the observer receiving signals from one direction undergoes causation from her/his present that is going in the opposite direction, after such signal has travelled all the way through the cosmos.

This entails a presence of channels of perfect communication, beyond time and space. This entails the presence of a principle inferred in Benazzo (2001, 2010) for which aggregates of partialities of the whole, forms, annihilate with other aggregates of partialities, named anti-forms by Charon (1977), beyond time and space, such that the whole annihilates into the null, in every instant of the observer's life. The observer remains unable to directly see this empirically, being the observer a part of the whole. Such situation is difficult to accept in a conventional paradigm, and in the everyday immanent experience of life. Quantum physics allows though for such a possibility and experiments by Thomas Young and then refined by John A. Wheeler, as recalled by Davies (2006), have shown how the observations from the observer's scientist in the future of an event may change and determine the past of that event, selecting a particular type of past from the future.

Does the future influences the past in the herewith paradigm? Figure 5 represents three successive circumferences of concurrent events. Each larger circumference represents the same concurrent cosmological bodies in a future time with respect to the internal circumference. Going clockwise, an intersection, i.e. a cosmological body, on the internal hexagon and on the middle circumference (point 1) intersects also a past light cone of the external hexagon, which influences in the future another intersection (point 2), i.e. cosmological body, on the external circumference. Such external intersection influences the past light cone of another intersection (point 4), i.e. a cosmological body, of the same external circumference. On the trajectory of this past light cone lies also an intersection of an internal circumference (point 3). In this way a body of an internal circumference (1) influences in the future a body on an external circumference (2), which influences in its future another body in the past (3), on the internal circumference, concurrent to the starting one (1). The future as such influences the concurrent present while going around the effective space circumference.

The observer's future influences the observer's present and this latter affects the observer's past, bringing the observer's future to influence the observer's past. These influences occur through the interchangeable space-time coordinates composing light, passing through a massive amount of intersections that involve the whole cosmos, such that even if the future of the observer determines the observer's past and present, an infinity of other events is involved in the trajectory so that there is room for freedom and change.

Concluding Remarks and Further Research

In the above discussed paradigm, the universe results as flat lens through which a curvature of the cosmos is perceived and virtually flattened. The paradigm description provides for a graphical representation of the null cones on a bi-dimensional figure, with one space dimension and one time dimension. Graphical trigonometric calculations are performed on such construction. Such calculations agree with the empirical evidence. Such a paradigm is as such read as based on empirical evidence and alternative to the Big Bang paradigm.

This paradigm provides for areas of further research. Among these, one area could be for controlling in more detail how it agrees with empirical evidence. Another area could be for analyzing its implications on dark matter and dark energy. In this respect, the curvature enlarges virtually the cosmological far away bodies, decreasing brightness and generating redshift. It could be interesting to analyse what such virtual deformations imply on the measurements of energy of the faraway bodies, as they would be stretched out, like a partial virtual atomic fission. If the measure of the virtual space **Sv(60)** is redefined, after that, the measure of the arc space s would consequently be redefined. The measure of the redshift may be reconsidered, for example considering in a different light its initial measure by Hubble. A check on the matching of the mathematics by Shu may be performed. Further research could be done on how virtual perceptions and effective dynamics of how the future would influence, overtly or covertly, the present and the past. In another respect, the so defined virtual universe is a virtually inflationary universe, virtually self-reproducing as fractal, for each observer. Research could be performed on whether and how the theory of inflation, like that by Andrei Linde (1998) and its developments, would adapt with the herewith paradigm.

###### Piero Benazzo

**Annex 1: Table with the values calculated**Measures are either in degrees for angles, or in terms of space, i.e. light years. The defined relation of the light space-time triangle specifies 30 degrees between the space axis and the light vector, 60 degrees between the light vector and the time axis, 90 degrees between the space and time axes. With the interchangeability of space and time of relativity, the observer is considered as seeing projections of tilted time on horizontal space as space, and projections of tilted space on the time axis as time. These considerations allow to measures the geometrical construction all in terms of light-years and of angles. Time is considered in the construction as time axis where to transform space measures to be transposed to other positions in the construction. This allows transforming space measures from local faraway measures, to virtual measures observed from the distance, to local measures and vice versa. The measures taken on the light vector are instrumental to calculate the redshif. The units of measurement in the redshift are the same in the numerator and denominator, such that they cancel out to result in the relative measure characteristic of the redshift.

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