# A Kinematical Approach to Conformal Cosmology

###### Abstract

We present an alternative cosmology based on conformal gravity, as originally introduced by H. Weyl and recently revisited by P. Mannheim and D. Kazanas. Unlike past similar attempts our approach is a purely kinematical application of the conformal symmetry to the Universe, through a critical reanalysis of fundamental astrophysical observations, such as the cosmological redshift and others.

As a result of this novel approach we obtain a closed-form expression for the cosmic scale factor and a revised interpretation of the space-time coordinates usually employed in cosmology. New fundamental cosmological parameters are introduced and evaluated. This emerging new cosmology does not seem to possess any of the controversial features of the current standard model, such as the presence of dark matter, dark energy or of a cosmological constant, the existence of the horizon problem or of an inflationary phase. Comparing our results with current conformal cosmologies in the literature, we note that our kinematic cosmology is equivalent to conformal gravity with a cosmological constant at late (or early) cosmological times.

The cosmic scale factor and the evolution of the Universe are described in terms of several dimensionless quantities, among which a new cosmological variable emerges as a natural cosmic time. The mathematical connections between all these quantities are described in details and a relationship is established with the original kinematic cosmology by L. Infeld and A. Schild.

The mathematical foundations of our kinematical conformal cosmology will need to be checked against current astrophysical experimental data, before this new model can become a viable alternative to the standard theory.

###### pacs:

04.50.-h, 98.80.-k^{†}

^{†}preprint:

LABEL:FirstPage1 LABEL:LastPage

###### Contents

## I Introduction

Modern cosmology has advanced very rapidly during these last decades, producing an impressive model of the Universe, but our current understanding is still troubled by many open questions and puzzles. Since the original observations of cosmological redshift in spectral lines, done by V. M. Slipher and E. P. Hubble almost one century ago and since the application of Einstein’s General Relativity to cosmological theoretical models, we have progressed a long way towards our current picture, where the contents of the Universe are described in terms of two main components, dark matter and dark energy, accounting for most of the observed Universe, with ordinary matter just playing a minor role.

The history of recent experimental observations which led to postulate the existence of these two components is well known, as well as the many past and current theoretical explanations (see for example Turner and Tyson (1999), Freedman and Turner (2003), Yao et al. (2006), Perlmutter et al. (1999), Riess et al. (1998), Riess et al. (2004), Riess et al. (2007), Spergel et al. (2003), Spergel et al. (2007)), but since there is no evidence of the real nature of dark matter and dark energy, we have to conclude that our comprehension of the natural world is limited to only a very small percentage of it (the ordinary matter component), a statement potentially very embarrassing for cosmology and physics, if taken at face value.

Several alternatives to dark matter and dark energy have been proposed (for comprehensive reviews see for example Mannheim (2006), Schmidt (2007), Nojiri and Odintsov (2006), Clifton (2006), Zakharov et al. (2008) and references therein) which can be approximately divided into two categories: those retaining the Newton-Einstein gravitational paradigm, while introducing ad hoc corrections to explain dark matter and dark energy and those breaking away substantially from established gravitational theories. Following this second line of thought, we will concentrate our attention on the theory of Conformal Gravity (CG), a fourth order extension of Einstein’s second order General Relativity (GR) as a possible framework for the solution of current cosmological problems.

## Ii Conformal Gravity

### ii.1 Weyl’s original proposal

The idea of a possible “conformal” generalization of Einstein’s relativity was first developed by Hermann Weyl in 1918 (Weyl (1918a), Weyl (1918b), Weyl (1919)). In his pioneering work, Weyl introduced the so-called conformal or Weyl tensor, a special combination of the Riemann tensor , the Ricci tensor and the curvature (or Ricci) scalar (see Weinberg (1972) p. 145):

(1) |

where, in particular, is invariant under the local transformation of the metric

(2) |

The factor represents the amount of local
“stretching” of the geometry, hence the
name “conformal” for a theory invariant
under all possible local stretchings of the space-time.^{2}^{2}2The name
conformal derives more precisely, “from the
property that the transformation does not affect the angle between two
arbitrary curves crossing each other at some point, despite a local dilation:
the conformal group preserves angles” (quoted from
Di Francesco et al. (1997)).

Weyl’s ambitious original program was to introduce a new kind of geometry, in relation to a unified theory of gravitation and electromagnetism where, in addition to Eq. (2), the electromagnetic field would transform as . This theory was later abandoned with the advent of modern gauge field interpretations of electrodynamics, retaining only terms such as “gauge transformation” or “gauge invariance,” which were introduced in reference to Eq. (2) (for a brief history of conformal theories of gravitation from 1918 to 1988 see Schmidt (2007), Schimming and Schmidt (1990)).

### ii.2 Fourth order metric theories

Following Weyl’s idea, the conformally invariant generalizations of the gravitational theory were found to be fourth order theories, as opposed to the standard second order General Relativity. In other words, the field equations originating from a conformally invariant Lagrangian contain derivatives up to the fourth order of the metric with respect to the space-time coordinates.

Initially there was some ambiguity in the specific choice of the Lagrangian
and the related action for these new theories, but following work done by
Rudolf Bach Bach (1921), Cornel Lanczos Lanczos (1938) and
others,^{3}^{3}3Even Albert Einstein used a conformally invariant formulation
in one of his papers in 1921 Einstein (1921). conformal gravity was
ultimately based on the conformal (or Weyl) action:^{4}^{4}4In this paper we
use a metric signature (-,+,+,+) and we follow the sign conventions of
Weinberg Weinberg (1972). We will use c.g.s. units when needed and all
fundamental constants, such as and , will always be explicitly
introduced in every equation.

(3) |

or on the following equivalent expression (which differs from the previous one by a topological invariant):

(4) |

where and is a gravitational coupling
constant (see Mannheim (2006), Schimming and Schmidt (1990),
Mannheim and Kazanas (1989), Kazanas and Mannheim (1991)).^{5}^{5}5In these cited
papers, is referred to as a “dimensionless
constant,” by working with natural units. Alternatively,
working with c.g.s. units, one can assign dimensions of an action to the
constant , so that the dimensionality of Eq. (5) will
also be correct. Under the conformal transformation of Eq. (2),
the conformal tensor transforms as , while the Conformal Gravity action
above is completely locally conformal invariant and is actually the
unique general coordinate scalar action with such properties.

Variation of the Weyl action with respect to the metric led R. Bach
Bach (1921) to rewrite the gravitational field equation in the presence
of an energy-momentum tensor^{6}^{6}6We follow here the convention
Mannheim (2006) of introducing the energy-momentum tensor
so that the quantity has the dimension of an energy density. For
example, we write the perfect fluid energy-momentum tensor as: . :

(5) |

as opposed to the “standard” Einstein’s equation,

(6) |

where the “Bach tensor” Bach (1921) plays the role of the combination of the Ricci tensor and curvature scalar on the left-hand side of Eq. (6). This tensor has a much more complex structure than those appearing in Einstein’s field equation. It is defined in a compact way as Schmidt (1984):

(7) |

but if one requests a form where the Weyl tensor does not explicitly appear, the more complex structure for the Bach tensor will emerge (Mannheim and Kazanas (1989), Wood and Moreau (2001)):

(8) | ||||

so that it involves derivatives up to the fourth order of the metric with respect to space-time coordinates.

The mathematical complexity of the Bach tensor and of Eq. (5) was one of the main reasons why the conformal theory of gravitation lost its attractiveness, between the thirties and the sixties, while quantum field theories were quickly progressing. A comprehensive review of the use of conformal invariance in physics up to the 1960s can be found in Ref. Fulton et al. (1962) and references therein. Only in the seventies, it was found that the fourth order theory is one-loop renormalizable Stelle (1977), in contrast to standard general relativity, yielding a revival of conformal gravity.

### ii.3 Solutions to Conformal Gravity equations

It was already known to Bach in 1921, that every static spherically symmetric space-time, conformally related to the Schwarzschild-de Sitter solution, is a static spherically symmetric solution of the Bach equation. In 1962, the converse statement was shown by H. Buchdahl Buchdahl (1962): every static spherically symmetric solution of the Bach equation is conformally related to the Schwarzschild-de Sitter solution (Rev (2009), Schmidt (2000)).

In this line of research, a solution of Bach’s equation was published by P. Mannheim and D. Kazanas (MK solution in the following) in 1989 (Mannheim and Kazanas (1989), Kazanas and Mannheim (1991)) and also studied by R. Riegert in his doctoral thesis Riegert (1986). This was the exact and complete exterior solution for a static, spherically symmetric source, in locally conformal invariant Weyl gravity, i.e., the fourth order analogue of the Schwarzschild exterior solution in General Relativity.

Solving Bach’s Eq. (5), in the case , Mannheim and Kazanas obtained a line element of the form

(9) |

where in spherical coordinates and

(10) |

with the parameters , , (again, we prefer to show explicitly constants such as the speed of light in all formulas), where is the mass of the (spherically symmetric) source. The familiar Schwarzschild solution is recovered in the limit for , in the equations above. The other two parameters are interpreted by MK Mannheim and Kazanas (1989) in the following way: and the corresponding term should indicate a background De Sitter space-time which would be important only at cosmological distances, since should have a very small value. On the other hand, measures the departure from the Schwarzschild metric, with the term becoming significant over galactic distance scales.

In other words, for values of , which is about the value of the inverse Hubble length, the standard Newtonian term still dominates at smaller distances, so that this theory would yield the same experimental success of General Relativity at the scale of the solar system. The three classic tests of GR, namely the gravitational redshift, the gravitational bending of light and the precession of planetary orbits, would still hold for conformal gravity at the solar system scale Mannheim (2006). The only additional test of the gravitational theory, at this distance scale, which has not been analyzed yet in the MK theory, is the well-known decay of the orbit of a binary pulsar (Hulse and Taylor (1975), Taylor and Weisberg (1989), Weisberg and Taylor (2002), Weisberg and Taylor (2004)).

Considering larger galactic distances, the contribution of the additional term might explain the flat galactic rotation curves, without the need of dark matter. This important connection to the dark matter problem and the galactic rotation curves was subsequently studied in great detail by Mannheim in a series of papers (Mannheim (1993a), Mannheim (1993b), Mannheim (1993c), Mannheim (1995a), Mannheim (1995b), Mannheim and Kmetko (1996), Mannheim (1997), Mannheim (2006)), showing that it is possible to fit the experimental galactic rotation data with theoretical curves based on conformal gravity, with the same level of accuracy of current dark matter theories (see Fig. 1 of Ref. Mannheim (1997) or Ref. Mannheim (2006), for example), thus establishing conformal gravity as a viable alternative to the dark matter hypothesis.

When we apply conformal gravity to a galaxy, we need to specify in more details the role of the parameters and . Again, Mannheim has shown that the Newtonian potential can be recovered for short distances, as a solution of a fourth order Poisson equation for the gravitational potential , as opposed to the standard second order equation (see Mannheim (2006), Sect. 4.2 for details). The resulting exterior potential for a single star source is of the form:

(11) |

where and are the individual parameters for a system composed of a single star (i.e., , where we use the solar mass as a reference mass for a stellar object). In first approximation, for a system of stars in a galaxy, we would expect to introduce overall and parameters which are linear in the number of sources: .

A more detailed analysis was done by Mannheim (Mannheim and Kmetko (1996),Mannheim (1997)) on a representative sample of eleven spiral galaxies, fitting their rotational velocity curves using the conformal gravity approach (Fig. 1 of Ref. Mannheim (1997) or Ref. Mannheim (2006) illustrates this detailed fitting). The galaxies were modeled with a thin/thick disk potential with the addition of a spherical central bulge region if necessary. The luminous Newtonian contribution was found to account well for the initial rise of the rotation curve from the center of the galaxy ( up to a peak at , where is the scale length of the galaxy and is the radial coordinate. The centripetal acceleration due to just the luminous matter distribution would yield the standard Keplerian term , outside the optical disk. The number of stars in each galaxy was computed by fitting the rotational curve, just due to the luminous Newtonian contribution, to the experimental value at the peak for .

The discrepancy observed between the experimental data and the Keplerian prediction, for distances larger than the peak distance, was then modeled with parameters from conformal gravity. In particular, the last experimentally observed value for the rotational acceleration of the sample galaxies, was found to be well explained by a two parameter formula (in addition to the standard Keplerian term introduced above):

(12) |

In the previous equation, the first term on the right-hand side is the standard Keplerian one, while the two additional terms come from the conformal theory, without any need of dark matter contributions. The two additional universal parameters are evaluated from the detailed fitting of the experimental curves as follows Mannheim (1997):

(13) |

and their interpretation is analogous to the parameter of the MK solution of Eq. (10).

The presence of two gamma parameters is also explained by Mannheim: the term is the gamma parameter of the specific galaxy being analyzed, being the product of the single star contribution times the number of stars in the galaxy being considered. The more universal represents a cosmological gamma parameter, presumably due to the combined effect of all the galaxies (see discussion on page 416 of Mannheim (2006)). This term would affect the space-time geometry even in regions far away from matter sources, introducing an “universal acceleration” which is close to similar universal acceleration parameters, such as those introduced by the Modified Newtonian Dynamics (MOND) theory by M. Milgrom and others (Milgrom (1983a), Milgrom (1983b), Bekenstein and Milgrom (1984), Bekenstein (2004)).

In view of the very successful fitting of the experimental galactic rotation curves, shown in Mannheim (1997), we will consider here the Conformal Gravity model as a viable alternative to the dark matter hypothesis. In particular, we will retain the cosmological parameter , which will be used in our subsequent analysis, but we will need to reconsider its meaning and value later in this work.

## Iii Conformal Cosmology

As outlined in the previous section, we will assume that Conformal Gravity is a possible alternative gravitational theory, therefore the next logical step is to construct a cosmology based on these new ideas. In fact, many conformal cosmologies exist in the literature, including the one proposed by Mannheim in another series of papers (Mannheim (1990), Mannheim (1992), Mannheim (1993d), Mannheim (1996a), Mannheim (1998), Mannheim (2001), Mannheim (2003a), Mannheim (2003b), Mannheim (2007), Mannheim (2008)). Mannheim’s cosmology is based on the construction of a traceless (as required by the conformal theory) energy-momentum tensor , in a theory in which the action is built out of fields rather than particles, using a spontaneous symmetry breaking mechanism in order to obtain particle masses. This modern approach elegantly overcomes the original objection to a conformal, scaleless theory, which would strictly require all particles to be massless, but is not free from theoretical controversy (Mannheim (2007), Flanagan (2006)).

Other “conformal” cosmologies exist in the literature (see for example Faraoni et al. (1999), Behnke et al. (2002), Schmidt (2007)), based on similar approaches, but none of these has become a popular cosmological alternative to the standard model or even to cosmologies based on the MOND approach, including its latest relativistic version (Tensor-Vector-Scalar gravity, TeVeS, Bekenstein (2004)). In our opinion, all these conformal cosmologies do not fully explain the connection between the assumed conformal symmetry and the physical reality of our Universe, as determined by cosmological observations. Therefore, we seek here an alternative approach, which doesn’t require the field theory formalism, but is based on a critical analysis of the foundations of observational cosmology, starting with cosmological redshift.

### iii.1 From Static Standard Coordinates to the Robertson-Walker Metric

To introduce the discussion of cosmological redshift, it is necessary to analyze here in more details the transformation of the coordinates related to the MK solution, in particular the transformation from Static Standard Coordinates (SSC) to the Robertson-Walker (RW) metric. This is another fundamental aspect of Conformal Gravity: the CG solution is able to interpolate smoothly between the static Schwarzschild solution and the classical Robertson-Walker metric. We will follow again Mannheim and Kazanas (Mannheim (2006), Mannheim and Kazanas (1989), Kazanas and Mannheim (1991)), but we will use a slightly different notation and interpretation, for the different sets of coordinates used in the following. Another complete description of the necessary coordinate and conformal transformations from the Schwarzschild-de Sitter solution to the Mannheim-Kazanas solution can be found in Ref. Schmidt (2000).

We start again from the line element given by Eqs. (9) and (10), but we consider now regions far away from matter distributions, thus ignoring the matter dependent term. In view of the discussion in the previous section, we could identify the parameter in Eq. (10) with , as in Eq. (13). However, this value refers to a sample of eleven galaxies, where the rotational motion data being fitted by the conformal gravity theory cover a range of distances of a few kiloparsec, from the center of each galaxy.

In our next paper Varieschi (2008) we will argue that parameters such as are better determined by “local” measurements on a short distance scale and not on the kiloparsec scale. Mannheim’s value of can therefore provide a useful order of magnitude for this quantity, but we will determine later its “current” value from more local measurements.

We will use the greek letter for the additional integration constant in the MK solution, instead of used in the original references. In particular, we retain here the “cosmological background” term that was dropped by Mannheim in his latest analysis Mannheim (2006), which on the contrary will play an essential role in our cosmology. We therefore write as:

(14) |

so that the line element becomes:

(15) |

in what we will call the Static Standard Coordinates - SSC in the following. These are the coordinates we use to carry out all our standard laboratory measurements, with our current units of length, time, mass and others.

With a first coordinate transformation:^{7}^{7}7The angular coordinates
and , as well as the quantity , are not changed by any of the transformations
performed in this section. Therefore, we will not rename these angular
coordinates. Note also the inverse transformation: .

(16) | ||||

the metric, as a line element, becomes (Mannheim and Kazanas (1989), Kazanas and Mannheim (1991), Mannheim (2006)):

(17) |

At this point it is convenient to redefine the combination of parameters and , in Eq. (17), as follows:

(18) |

where will be ultimately linked to the “trichotomy constant” of a Robertson-Walker (RW) metric. Equation (17) can be rewritten as:

(19) |

As noted by Mannheim and Kazanas Mannheim and Kazanas (1989), the metric above is conformal to a RW metric in isotropic form. All we need is to apply a conformal transformation, such as the one in Eq. (2), to the metric tensor defined through Eq. (19), to obtain a new metric in the RW isotropic form. Precisely, we will “stretch” the space-time fabric, multiplying the last equation by the factor

(20) |

which depends on the space-time coordinates. We will then replace the metric as follows:

(21) |

Therefore we obtain:

(22) |

and the metric is now in the form known as the “isotropic” Robertson-Walker. In the previous equations we did not use different symbols for the coordinates after the conformal transformation, but we kept the previous set of coordinate. The theory of local conformal transformations of the metric indicates that we can always choose the new coordinates of a point, after the local stretching, so that they correspond to the old coordinates of the original point before the stretching (see Fulton et al. (1962) for a detailed discussion of conformal transformations in physics).

The above transformation implies a change of the line element itself, which is stretched by the same amount

(23) |

and this “gauge transformation” will
ultimately result in a redefinition of the local measuring rods and clocks,
which will be a key feature of our cosmology. Another coordinate
transformation will lead from the isotropic form of RW metric to the standard
RW metric:^{8}^{8}8The inverse transformation of Eq. (24) is:
, where the minus sign in front of the square root selects the
correct branch of the graph of the function considered. For it reduces
simply to .

(24) | ||||

and the metric becomes

(25) |

In this expression the parameter is still linked to and , through Eq. (18), or equivalently:

(26) |

It is customary for the so-called trichotomy constant of a Robertson-Walker (RW) metric to have values . This can be accomplished with a final rescaling of the coordinates, of the constant and of the scale factor , as follows:

(27) | ||||

where we use bold symbols to denote quantities after this
last transformation.^{9}^{9}9In the special case the transformation in
Eq. (27) should actually read: ; ; and . We can finally obtain the standard Robertson-Walker form of the
metric:^{10}^{10}10We note that, due to the transformations of Eq.
(27), the quantities and are now
dimensionless, while the factor acquires the dimension of length.
We will not follow the common alternative normalization, with a dimensionless
scale factor, which is sometimes found in the literature.

(28) |

We recall that the RW metric in the previous equation can be expressed equivalently in the so-called curvature normalized form:

(29) | ||||

for closed, flat or open universes respectively. The comoving coordinate is also dimensionless and the connection with the coordinate in Eq. (28) is due to the simple relation:

(30) |

Another important quantity for our discussion is the conformal time interval , usually defined as an interval divided by the scale factor :

(31) |

which is essentially equivalent to the SSC time interval , in view of Eqs. (16), (24), (27) and was in fact already introduced by the transformations of Eq. (16). Using the RW metric in the form of Eq. (29) we obtain a well-known and simple expression for the null geodesic , corresponding to the propagation of a light signal in the radial direction ():

(32) |

thus establishing a direct connection between the comoving coordinate ,
the conformal time and the SSC time coordinate .^{11}^{11}11For
we recall that , therefore Eq. (32) should be
written as , omitting the
factor. We note that the second equality in Eq. (32) is only
valid for a null geodesic, i.e., and are simply related to each
other only when describing the propagation of a light signal. In this case the
minus signs in the previous equation indicate that we are following a light
ray propagating in the negative radial direction () for increasing
conformal time ().

Summarizing this section: the coordinate transformations described above allowed us to connect the original Static Standard Coordinates , used by Conformal Gravity to solve the problem of the rotational galactic curves without resorting to dark matter, to the cosmological comoving coordinates , commonly used together with Eq. (28) as the basis of standard cosmology. We will continue to use normal and bold characters in the following to differentiate between these two sets of coordinates.

### iii.2 An alternative interpretation of the cosmological redshift

One of the foundations of observational cosmology is the well known
cosmological redshift of galaxies, which is usually related to the expansion
of the Universe. It is customary (see Weinberg (1972), Peebles (1993),
Kolb and Turner (1990), Lang (1999), Peacock (1999), or any other
General Relativity - Cosmology textbook) to consider light emitted by a
distant galaxy at (comoving) coordinates and
reaching us at the origin of the coordinates and at time
(present time). The time of emission is
therefore in the past, i.e., , or is the “look-back” time.^{12}^{12}12In this way, integrating Eq. (32) for light
emitted at coordinate , at conformal time , and reaching us at the
origin () at our present conformal time , we obtain:
. The
redshift parameter is related to the cosmic scale factor ,
or to the change in the radiation wavelength/frequency, through the standard expression:

(33) |

where, quoting from Weinberg (see Weinberg (1972), pages 416-417): “… and are the frequency and wavelength of the light if observed near the place and time of emission, and hence presumably take the values measured when the same atomic transition occurs in terrestrial laboratories, while and are the frequency and wavelength of the light observed after its long journey to us.”

Given this standard view of the redshift, it has always been considered a serious misconception to interpret the expansion of the Universe as if, “space itself is swelling up,” thus causing galaxies to separate. Numerous textbooks are quick to point out this potentially erroneous interpretation (see for example Peacock (1999), Webb (1999)), explaining that galaxies separate, “like coins glued on an inflating balloon,” without altering their intrinsic dimensions, or that two massless objects set up at rest with respect to each other will show no tendency to separate, due to cosmological expansion.

However, an analysis of the literature of cosmological theories also reveals that other possible interpretations of the redshift, apart from the standard general relativistic expansion, were considered. Many alternative theories exist such as the kinematic cosmology by Infeld and Schild (Infeld (1945), Schild (1946), Infeld and Schild (1945), Infeld and Schild (1946)), which is also based on the conformal gauge transformation of Eq. (23) as well as the cosmological principle and the constancy of the speed of light. In this theory all possible cosmological models based on these assumptions are analyzed and classified, leading to different possible interpretations of the redshift, ranging from standard Doppler effect to the purely “gravitational redshift” effect that we will also employ in our model. We will later compare our results to the different models proposed by Infeld and Schild.

These conformally-flat-spacetime models were recently also studied by others (Endean (1994), Endean (1997), Querella (1998)) and were also considered in other theories such as the Hoyle-Narlikar cosmology Narlikar (1983). This model, for example, assumes a non standard interpretation of the cosmological redshift, i.e., since the atomic radiation wavelength is inversely proportional in first approximation to the mass of the electron involved in the atomic transition, the ratio is simply assumed to correspond to the value of the (variable) electron mass at different epochs: . Hoyle-Narlikar then implemented their model, assuming a conformally invariant theory where masses scale as , adding a variable gravitational constant , whose variation is based on a large numbers hypothesis, similar to the original Dirac argument (Dirac (1937), Dirac (1938)) and finally proposed mechanisms of particle creation, in line with previous steady-state cosmologies.

While we do not agree with such theories, we share the idea that the redshift
ratio might be disclosing to us information about the
emission/absorption process at different cosmological epochs. In this line of
reasoning, we recall that modern metrology (see metrology web-sites
NIS (http://physics.nist.gov/), BIP (http://www.bipm.org/en/home/) and references therein) defines our basic units of
length and time using non-gravitational physics, through a reference atomic
wavelength or frequency, so that our meter^{13}^{13}13The meter was
recently redefined as the length of the path travelled by light in vacuum
during a time interval of of a second. This definition
assumes an (exact) speed of light in vacuum: . In
this way the unit of length is basically defined through the unit of time,
therefore not altering the validity of our discussion. is just some multiple
of an atomic reference wavelength , or equivalently the
second is a multiple of the inverse of some atomic reference
frequency :

(34) | ||||

Since our space-time units ultimately have an atomic definition based on emission/absorption of radiation, a possible “swelling” or dilation of the space-time fabric at any level, from the atomic to the galactic scale, could never be detected using currently defined meter sticks and clocks, because these would be “dilated” by the same amount.

In other words, it is only possible to base our space-time units on the current and local values of wavelength or frequency of some standard reference atomic transition, but we cannot be absolutely certain that these reference wavelengths or frequencies are invariable and constant throughout the Universe and at all cosmological times. A possible variation of these reference wavelengths and frequencies would be also related to the well-known problem of the time variation of the universal constants (for modern reviews see Barrow (2002), Uzan (2003), Okun (2004)).

The logical connection between a possible conformal symmetry of the Universe, dealing with stretchings and dilations of the metric, and the previous discussion of changes and variations in our meter sticks and clock rates, should induce a revision of the redshift mechanism. In particular, the observed galactic redshift might be interpreted, in part or completely, as due to a change of these reference wavelengths and frequencies over cosmological distances and times. We will adopt this possible interpretation in the following, altering the classical meaning of , and , in Eq. (33).

In our alternative redshift interpretation we assume that the observed quantities and , are telling us about the radiation emitted by the source galaxy at the place and time of emission, while the reference quantities and are, by the same argument, characteristics of the same atomic radiation as measured here on Earth at present time. We therefore write:

(35) | ||||

where again represents the Earth’s observer position, is the present time, while is the position of the source galaxy and is the time of emission, in the past. Since the units of length and time, defined in Eq. (34), are respectively proportional to the radiation wavelengths and inversely proportional to the radiation frequencies, they also become functions of the space-time coordinates:

(36) | ||||

where and will indicate the unit-length and the unit-time-interval in the following. Due to this new interpretation, we correct Eq. (33), combining it also with the previous equations:

(37) |

In view of the modern definition of the unit of length, based on a fixed value of , we will consider the value of the speed of light in vacuum to be just a connecting factor between the units of length and time and we see no reason, at least for now, to assume that this factor might also change at different space-time locations. In our opinion, a variation of the speed of light (proposed by some alternative cosmologies Albrecht and Magueijo (1999), Barrow (1999), Magueijo (2000)) would imply a substantial difference in the universal evolution of the units of length and time which seems an unnecessary complication, not supported by experimental observations. Therefore, we will consider as a constant value in the following, but we will continue to explicitly include in every equation.

## Iv Evaluation of the Cosmic Scale Factor

The alternative interpretation of the cosmological redshift, presented in the previous section, is actually an adaptation of the well-known gravitational redshift (or gravitational time-dilation) to the cosmological scale and was even considered in the 1920’s as a possible origin of the observed redshift (see the historical discussion in Weinberg Weinberg (1972), page 417), but would have required very strong local gravitational fields, so this explanation was quickly abandoned in favor of a “cosmological” Doppler effect. Nevertheless, it is interesting to notice that this possibility was taken into account at the beginning of modern cosmology as well as many other explanations.

The gravitational redshift is a fundamental consequence of the equivalence principle which states that the rate of a clock at rest is affected by the presence of a gravitational field as follows:

(38) |

where and are respectively the clock periods in the presence or in the absence of gravitation, and is the value of the time component of the metric at the point where the clock is located (see Weinberg (1972), section 3.5 for a general discussion, or Lang (1999) for a review of experimental results). Since the “true” period of a clock is unknown, we can only observe this effect by comparing the rate of the clock at two different locations , in the gravitational field:

(39) |

and this quantity is related to the ratio of the frequencies or wavelengths of the same atomic transition observed at the two locations, which can also be described by a “redshift” parameter . The connection between Eqs. (37) and (39) is immediate, identifying the two locations , with and respectively.

The gravitational redshift or time dilation has been tested repeatedly for the classic Schwarzschild solution of the metric, i.e., . Using this expression inside Eq. (39) we obtain a redshift if the point of emission is closer to the massive source of the field, compared to the point of observation , such as in the case of light emitted by the Sun or by white dwarfs and observed here on Earth Lang (1999). A blueshift can be observed instead by using the Earth’s gravity and by placing point at a higher level than point , as in the classic experiment by Pound and Rebka (see description in Weinberg (1972)). These gravitational redshifts are very small (the one due to the Sun corresponds to and those related to white dwarfs are about two orders of magnitude bigger) and cannot produce any cosmological redshift, since they are just a local effect, predicted on the basis of the classic Schwarzschild solution for a static and spherically symmetric massive source, such as a planet or a star.

However, in view of the preceding discussion of the cosmological redshift and of the new MK solutions shown in Eq. (10) or Eq. (14), involving a cosmological generalization of the classic Schwarzschild solution through the cosmological parameters and , we can now propose a direct determination of the scale factor based on this “extended” interpretation of the gravitational-cosmological redshift. In other words, we will show that, assuming the validity of Conformal Gravity and of the interpolation between the Static Standard Coordinates and the Robertson-Walker metric explained in Sect. III.1, our alternative redshift interpretation restricts the possible conformal transformations of the metric to just one possible case, i.e., just one possible function in Eq. (2), therefore also practically breaking this conformal symmetry without resorting to field-theory symmetry breaking procedures.

The function will be uniquely determined from these purely “kinematical” considerations and we will not need to obtain it from the solution of the “dynamic” field equation (5) of conformal gravity, as it is done in standard General Relativity using the Friedmann equations. We will then compare our solutions for with the corresponding solutions obtained by current conformal cosmologies in the literature, since the metric in Eqs. (9)-(10) is based on the same equation of motion of conformal gravity, i.e., Eq. (5) with .

### iv.1 The cosmic scale factor as a function of the radial coordinates

It is immediate to obtain the cosmic scale factor as a function of the radial coordinates. We start by combining Eqs. (14) and (26), in order to rewrite as:

(40) |

in Static Standard Coordinates. Now we use Eq. (39) to compute the gravitational-cosmological time dilation for two points corresponding to the source galaxy space-time position and the Earth’s observer placed at the origin at present time:

(41) |

which gives the redshift factor as a very simple function of the coordinate in SSC. We also express the factor as a ratio of cosmic scale factors, computed at the two points of interest, although usually the scale factor is only introduced in the RW metric. We will show in the following that this function can be expressed in any of the space-time coordinates of interest, therefore we can also introduce it in the SSC.

Our objective is now to transform this expression into RW coordinates, by using the transformations outlined in Sect. III.1. Before doing this, we note that Eq. (41) should give the observed cosmological redshift (i.e., ) at least for some distance interval , where is the coordinate beyond which we start observing a cosmological redshift. In addition, we assume that the parameter is small and positive at the present time, probably close to Mannheim’s value of , while the parameter is not yet restricted ().

For , a quick inspection of Eq. (41) shows that a solution allowing redshift is possible only in one case: for a negative and more precisely for . In this case the function in Eq. (41) is well defined for positive values of in the interval . Moreover, we obtain:

(42) |

giving a blueshift () for distances in the interval , and a proper redshift () for larger distances , which might correspond to the observed cosmological redshift.

Since the cosmic scale factor and all the other cosmological quantities of interest are usually expressed in Robertson-Walker coordinates, we have to convert the expression in Eq. (41) into these coordinate. This can be accomplished by using the transformations of Sect. III.1. From Eq. (16) and its inverse transformation, it follows that

(43) |

so that we can write

(44) |

which is well defined for .

The conformal transformation of Eq. (21) will not alter the coordinate, so we just need to apply the final two transformations of Eqs. (24) and (27) to obtain, after some algebraic work:

(45) |

where we use again (in bold) to denote the radial coordinate in RW metric and , following Eq. (27). We observe that the last term in the previous equation diverges for , but according to the note following Eq. (27) in this particular case the previous equation should simply become . Another way to obtain Eq. (45) from Eq. (41) is to use the direct connection between coordinates and , which can be easily derived from the transformations of Sect. III.1 and is the following:

(46) |

A more elegant way to write the previous fundamental equations is to introduce a dimensionless parameter:

(47) |

or for the particular case , and rewrite Eq. (45) as

(48) |

We have written the ratio of cosmic scale factors as a function of the radial coordinates of the points of emission and absorption of radiation, since the function on the right-hand side of the previous equation depends only on , although we are implicitly referring also to the times at which the radiation was emitted and absorbed. A more precise notation would be to write always these cosmological scale factors as and in all our formulas, but we will continue to use our simplified notation also in the following.

In the last equations the parameter is positive and determined, at
least for now, by Mannheim’s fits of galactic rotational curves, while the
other parameter is still undetermined. Since
a cosmological redshift is generally observed,^{14}^{14}14Except for some nearby
galaxies typically located in our Local Group, whose blueshift is presumably
due to their peculiar motion, or for the “Pioneer anomalous
blueshift” which will be considered in another paper
Varieschi (2008). i.e., in general, we have
already remarked that this suggests a negative value of , which implies , thus restricting in general to the interval
(but with a currently positive value).

For these values of the parameters the quantity introduced in Eq. (42) can be rewritten in RW coordinates as

(49) |

Again, we have a standard redshift for distances , but an unexpected blueshift at closer distances . This possibility is particularly interesting in view of a recently discovered phenomenon, the so-called Pioneer anomaly (Anderson et al. (1998), Anderson et al. (2002), Turyshev et al. (2005), Turyshev et al. (2006), Toth and Turyshev (2006)) consisting in an anomalous blueshift observed in the navigation of the Pioneer spacecraft, just outside the Solar System.

### iv.2 Time-dependent form of the cosmic scale factor

The cosmic scale factor is usually considered a function of some cosmic time coordinate , rather than a function of the radial coordinates, as introduced in the previous section. This is a consequence of the Cosmological Principle (i.e., the Universe being assumed spatially homogeneous and isotropic) and of the application of this principle to the hypersurfaces with constant cosmic standard time, which are maximally symmetric subspaces of the whole of space-time (see Chapters 13-14 in Ref. Weinberg (1972) for details). Following this standard hypothesis, the resulting metric takes the RW form of Eq. (28) and the redshift is described by the ratio of scale factors at two different cosmic times, as in Eq. (33).

On the contrary, our new interpretation assumes that the redshift is due to the stretching of the space-time fabric as described by Eqs. (41) and (45), which are essentially static solutions, derived from the Conformal Gravity theory. In order to retain the validity of the Cosmological Principle and, in particular, still assume the homogeneity of the Universe at a given cosmic time, we have to transform the space dependence of our new cosmic scale factors into a more traditional time dependence.

This can be accomplished by noting that the redshifted radiation described by Eqs. (41) or (45) is reaching us from past times and that light coming from a radial distance is all emitted at the same time in the past. Therefore, the scale factor can be associated with a corresponding factor , at a given past cosmic time . This association is performed by computing the time it takes for a light signal emitted at radial distance to reach the observer at the origin. It is then straightforward to turn Eqs. (41) or (45) into their time dependent equivalent, since we are following a light signal traveling in vacuum from a distant galaxy toward us, for which or .

It is convenient to study the propagation of this light signal by using first our original Static Standard Coordinates . Combining Eqs. (15) and (40), for a light ray traveling along the direction with and fixed, we have:

(50) |

or equivalently

(51) |

where the negative sign comes from the direction of light propagation.

Integrating between times and , corresponding to radial positions and , we obtain different results, depending on the sign of the parameter :

(52) | ||||

where we have introduced the useful quantity . It is possible to invert Eq. (52) in each case and obtain the distance as a function of :

(53) | ||||

Finally, it takes a little more work to combine Eq. (53) together with Eq. (41), to obtain the explicit form of the cosmic scale factor:

(54) | ||||

a remarkably compact expression in each case. To obtain the scale factor as a function of the cosmic time coordinate we could repeat the same procedure, studying the propagation of light in RW metric, but this involves rather cumbersome integrals. It is easier to find a direct relation between the time coordinates and .

(55) |

or, integrating between times , in the past and present times , ,

(56) |

where the cosmic scale factor is expressed through Eq. (54):

(57) | ||||

Integrating Eq. (56) with the expressions from Eq. (57) we obtain (using the more compact parameter ):

(58) | ||||

Inverting these expressions, we obtain the connections between the time coordinates:

(59) | ||||