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The visible mass of the observable universe agrees with that needed for a flat cosmos, and the reason for this is not known. It is shown that this can be explained by modelling the Hubble volume as a black hole that emits Hawking radiation inwards, disallowing wavelengths that do not fit exactly into the Hubble diameter, since partial waves would allow an inference of what lies outside the horizon. This model of “horizon wave censorship” is equivalent to a Hubble-scale Casimir effect. This incomplete toy model is presented to stimulate discussion. It predicts a minimum mass and acceleration for the observable universe which are in agreement with the observed mass and acceleration, and predicts that the observable universe gains mass as it expands and was hotter in the past. It also predicts a suppression of variation on the largest cosmic scales that agrees with the low-

Using the Hubble space telescope it has been determined that there are about

Given its error bars this mass is indistinguishable from a critical value that determines whether the universe is gravitationally closed or open. This is the so-called flatness problem pointed out by [

Another model is suggested here using a Hubble-scale Casimir effect (HsCe) that has been applied to Unruh radiation to explain inertial mass, but we apply it here to Hawking radiation to model gravitational mass.

Work by [

McCulloch [

Replacing the Unruh wavelength

The derivation of Equation (5) can be seen in [^{2}) the modification of inertia is negligible, but for the tiny accelerations seen in deep space the second term in Equation (5) can become important. Although MiHsC makes some bold assumptions (e.g., that Wien’s law holds at these huge scales) these are somewhat justified by the fact that the minimum acceleration predicted by MiHsC agrees well with the cosmic acceleration attributed to dark energy [

For an observer in an expanding universe there is a maximum volume that can be observed, since beyond the Hubble distance the velocity of recession is greater than the speed of light and the redshift is infinite: this is the Hubble volume. Its boundary is similar to the event horizon of a black hole [

Here, the black hole model used above is turned inside out. The observable universe is now taken to be a black hole, and it is assumed that the wavelengths of Hawking radiation it emits inward from its boundary must fit exactly into twice its own diameter. If they do not, they are disallowed for the same reasons mentioned above.

Equation (7) predicts that if the mass of the universe (

In the calculation above, this mass was derived crudely using an abrupt radiation cutoff at the Hubble scale. A similar result is derived here using a more complete model of the Hubble-scale Casimir effect (though still a parameterisation of it) by applying the Stefan-Boltzmann law to the inwards Hawking radiation:

This formula for the speed of light c is twice the escape velocity for a mass

This model can be explained more intuitively as follows. The edge of the observable universe is an event horizon, so, radiation, including Hawking radiation, with a wavelength bigger than this cannot exist since it cannot, even in principle, be seen (following Ernst Mach). Also, wavelengths that do not fit exactly into this scale cannot exist either because of a Hubble-scale Casimir effect or because they would give us information from beyond the horizon (horizon wave censorship).

According to Hawking, the mass of a black hole is linearly related to its temperature or inversely-linearly related to the wavelength of the Hawking radiation it emits. Therefore, for a given size of the universe there is a maximum Hawking wavelength it can have and a minimum allowed gravitational mass it can have. If its mass was less than this then the Hawking radiation would have a wavelength that is bigger than the size of the observed universe and would be disallowed. The minimum mass it predicts is encouragingly close to the observed mass of the Hubble volume.

Equation (11) implies that the baryonic mass of the observable universe is linearly related to its diameter, so it increases with time. This is similar to the behaviour of the Steady State Theory [

In the HsCe model the Hubble-mass also increases in time as the universe expands (Equation (11)), but the HsCe also predicts a hotter early universe, since by combining Equation (2) (Wien’s law) and taking the maximum wavelength allowed by the HsCe,

Evidence for the HsCe model may already have been seen. Data sets from the Cosmic Background Explorer (COBE), Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite [

Using the Hubble-scale Casimir effect (or horizon wave censorship) to suppress patterns of variation, instead of Unruh waves, the energy of the CMB blackbody radiation spectrum

Equation (16) predicts that the Hubble-scale Casimir effect or horizon censorship model, when applied to patterns of variation, predicts a decrease in variation for

The Hubble volume is modelled here by assuming it behaves like a black hole and emits Hawking radiation inwards from its edge whose wavelengths are subject to a Hubble-scale Casimir effect (HsCe) or an equivalent horizon wave censorship model. This model predicts a Hubble-mass of

The HsCe model predicts an increase in mass as the universe expands similar to the behaviour of the steady state theory. Unlike that theory, the HsCe predicts that the universe could have been hotter in the past.

The HsCe model is presented here as a toy model to stimulate discussion. However, it is supported to some extent by the recent anomalous results of the Planck satellite which show a suppression of variation at the largest cosmic scales that agree with those proposed here for Unruh-Hawking radiation.

Many thanks to the anonymous reviewers for their comments, and B. Kim for encouragement.

The author declares no conflict of interest.