Baryon Acoustic Oscillations in a New FRW R_H = ct Cosmology with a Modified Observational Redshift

1. Introduction

Baryon acoustic oscillations provide a precise statistical standard ruler for testing cosmic geometry. Spectroscopic surveys measure angular positions and redshifts,

$$(\theta ,\varphi ,z),$$

(1)

and identify a preferred correlation scale in the distribution of galaxies, quasars, and absorption systems. The quantities usually reported as $D_M/r_d$ and $D_H/r_d$ are geometric interpretations of this feature, where $D_M$ is a transverse comoving distance, $D_H=c/H$ is a radial Hubble distance, and $r_d$ is the acoustic scale at the baryon-drag epoch.

The present paper considers a new FRW model with

$$R_H=ct,$$

(2)

which implies

$$a\left(t\right)\propto t$$

(3)

and

$$H\left(t\right)={\displaystyle \frac{1}{t}}.$$

(4)

This background expansion is related to other $R_H=ct$ cosmologies, including Melia’s model [4], but the present construction uses the observational redshift $z_H$.

The model considered here adopts the Haug–Tatum cosmological-redshift assumption, including the relation between the Hubble radius, cosmic time, and a square-root redshift law [1,2]. It is nevertheless a separate model because that redshift is used here within an FRW metric assumption. The Haug–Tatum construction has often emphasized a direct Hubble-radius or light-travel-distance relation. The model developed here instead begins with an explicit FRW line element and asks how the square-root redshift law can be incorporated while preserving a consistent metric distance calculation. This is the sense in which the present framework goes beyond the Haug–Tatum model: it introduces a two-variable redshift interpretation in which the standard FRW metric redshift $z_{\mathrm{FRW}}$ determines the geometry, while the Haug–Tatum redshift $z_H$ is the proposed observational redshift.

This distinction changes the BAO prediction materially. Rather than inserting $z_H$ directly into a light-travel-distance formula, one first maps $z_H$ to the corresponding FRW metric redshift and then computes the FRW comoving distance. This approach follows the logic of standard BAO analyses, in which the acoustic feature is interpreted through a geometric distance-redshift relation, while allowing the observed redshift variable to differ from the auxiliary metric redshift used in the FRW background [5,6,7]. The resulting model derives a new BAO anisotropy relation, a one-parameter absolute BAO distance law, and a direct observed-coordinate BAO shell. These results make the present model substantially more testable than a pure redshift-distance ansatz.

2. The FRW $R_H=ct$ Background

For a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW, here abbreviated FRW) spacetime [8,9,10,11],

$$d{s}^2=-c^{2}d{t}^2+a^2\left(t\right)\left[d{\chi }^2+{\chi }^{2}d{\Omega }^2\right].$$

(5)

The Hubble radius is

$$R_H\left(t\right)={\displaystyle \frac{c}{H\left(t\right)}}.$$

(6)

Imposing

$$R_H\left(t\right)=ct$$

(7)

gives

$$H\left(t\right)={\displaystyle \frac{1}{t}}.$$

(8)

The Hubble time is defined by

$$t_H\left(t\right)\equiv {\displaystyle \frac{1}{H\left(t\right)}}.$$

(9)

Thus, in the present linearly expanding model,

$$t_H\left(t\right)=t,R_H\left(t\right)=c{t}_H\left(t\right).$$

(10)

Since

$$H\left(t\right)={\displaystyle \frac{\dot{a}}{a}},$$

(11)

one obtains

$${\displaystyle \frac{\dot{a}}{a}}={\displaystyle \frac{1}{t}},$$

(12)

and therefore

$$a\left(t\right)=a_0{\displaystyle \frac{t}{t_0}}.$$

(13)

The standard FRW metric redshift is then

$$1+z_{\mathrm{FRW}}={\displaystyle \frac{a_0}{a\left(t\right)}}={\displaystyle \frac{t_0}{t}}.$$

(14)

Because $R_H=ct$,

$$1+z_{\mathrm{FRW}}={\displaystyle \frac{R_{H_0}}{R_H\left(t\right)}}.$$

(15)

The standard Hubble function expressed in terms of the metric redshift is

$$H\left(z_{\mathrm{FRW}}\right)=H_0(1+z_{\mathrm{FRW}}).$$

(16)

Throughout the numerical discussion, we use

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}.$$

(17)

The scale-independent BAO anisotropy derived below does not depend on the numerical value of $H_0$, but $H_0$ fixes the absolute distance scale.

3. Haug–Tatum Observational Redshift in the FRW Metric

We adopt the Haug–Tatum cosmological-redshift assumption used in their Hubble-tension paper [1] and set out explicitly in their cosmological-redshift paper [2]. In the present work it is not used as a stand-alone light-travel-distance prescription. Instead, it is imposed as the observational-redshift mapping within a linearly expanding FRW metric background.

The Haug–Tatum observational redshift, denoted here by $z_H$, is defined by

$$z_H=\sqrt{{\displaystyle \frac{R_{H_0}}{R_H\left(t\right)}}}-1.$$

(18)

Combining this with the FRW metric relation gives

$$1+z_H=\sqrt{1+z_{\mathrm{FRW}}},$$

(19)

or

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2.$$

(20)

Thus,

$$z_{\mathrm{FRW}}={(1+z_H)}^2-1=z_H(z_H+2).$$

(21)

This is the central mapping used in the BAO calculation.

The model therefore contains two redshift variables with different roles. The FRW metric redshift determines the scale factor and the background null-geodesic geometry. The variable $z_H$ is the proposed observational redshift assigned to photons or spectral measurements. This is not meant to imply that a single source has two independently measured spectroscopic redshifts; rather, $z_H$ is the observational variable, while $z_{\mathrm{FRW}}$ is an auxiliary metric parameter related to the same emission event.

The distinction implies that $z_H$ cannot be the standard null-geodesic FRW redshift unless additional physics is introduced. Possible interpretations include nonstandard photon propagation, clock scaling, evolving transition frequencies, or a modified operational definition of observed redshift. The present paper treats the Haug–Tatum mapping phenomenologically and tests its geometric consequences for BAO.

4. CMB Temperature Law Used in the Present Model

As a central thermodynamic ingredient, the model uses

$$T_{\mathrm{CMB}}={\displaystyle \frac{\hslash c}{4\pi k_B\sqrt{2R_H{l}_p}}}.$$

(22)

This equivalent form of the temperature formula is presented in Ref. [12]. The formula was heuristically suggested by Tatum et al. in 2015 [13]. Haug and Wojnow subsequently derived the same expression from the Stefan–Boltzmann law [14], while Haug related it to the CMB, Hawking temperature, Planck scale, and Hubble scale [15].

Haug and Tatum also derived the same CMB formula from a geometric-mean construction in which the time-dependent Hawking–Hubble temperature is the minimum temperature, the Planck temperature is the maximum temperature, and the CMB temperature is their geometric mean [16]. The result was later expressed directly as the geometric mean of the minimum and maximum temperatures possible in the Hubble sphere. These are alternative derivations and interpretations of a temperature formula already obtained from the Stefan–Boltzmann law; the present paper uses the formula rather than claiming to derive it anew.

In an $R_H=ct$ background the formula gives

$$T_{\mathrm{CMB}}\propto t^{-1/2}\propto a^{-1/2}.$$

(23)

In terms of the standard metric redshift,

$$T_{\mathrm{CMB}}=T_0{(1+z_{\mathrm{FRW}})}^{1/2}.$$

(24)

Using $z_H$,

$$1+z_H={(1+z_{\mathrm{FRW}})}^{1/2},$$

(25)

the same law becomes

$$T_{\mathrm{CMB}}=T_0(1+z_H).$$

(26)

Placing this material here makes explicit, before the BAO distance derivation, that the observational-redshift mapping and the CMB temperature law are linked central assumptions of the present model.

5. Transverse Comoving Distance

For a flat FRW spacetime, the transverse comoving distance is

$$D_M\left(z_{\mathrm{FRW}}\right)=c{\int }_0^{z_{\mathrm{FRW}}}{\displaystyle \frac{d{z}^{\prime }}{H\left(z^{\prime }\right)}}.$$

(27)

For the linearly expanding $R_H=ct$ background,

$$H\left(z_{\mathrm{FRW}}\right)=H_0(1+z_{\mathrm{FRW}}).$$

(28)

Therefore,

$$D_M\left(z_{\mathrm{FRW}}\right)={\displaystyle \frac{c}{H_0}}{\int }_0^{z_{\mathrm{FRW}}}{\displaystyle \frac{d{z}^{\prime }}{1+z^{\prime }}},$$

(29)

which gives

$$D_M\left(z_{\mathrm{FRW}}\right)={\displaystyle \frac{c}{H_0}}ln(1+z_{\mathrm{FRW}}).$$

(30)

Using

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2,$$

(31)

we find

$$D_M\left(z_H\right)={\displaystyle \frac{c}{H_0}}ln\left[{(1+z_H)}^2\right].$$

(32)

Hence,

$$D_M\left(z_H\right)={\displaystyle \frac{2c}{H_0}}ln(1+z_H).$$

(33)

For

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

(34)

the normalization is approximately

$${\displaystyle \frac{2c}{H_0}}\simeq 8.97\times {10}^3\mathrm{Mpc}.$$

(35)

Thus,

$$D_M\left(z_H\right)\simeq 8.97\times {10}^{3}ln(1+z_H)\mathrm{Mpc}.$$

(36)

This logarithmic distance differs fundamentally from the saturating light-travel-distance expression used in some Haug–Tatum calculations,

$$D_{\mathrm{LT}}\left(z_H\right)={\displaystyle \frac{c}{H_0}}\left[1-{\displaystyle \frac{1}{{(1+z_H)}^2}}\right].$$

(37)

The latter may have a role in another observational context, but it is not identified with the FRW transverse comoving distance in the present cosmology.

6. Radial BAO Mapping

DESI measures radial clustering as a function of the observed redshift variable. Therefore, the relevant radial Jacobian is the derivative of the comoving distance with respect to $z_H$:

$$D_{H,H}\left(z_H\right)\equiv {\displaystyle \frac{d{D}_M}{d{z}_H}}.$$

(38)

Differentiating Equation (33) gives

$$D_{H,H}\left(z_H\right)={\displaystyle \frac{2c}{H_0(1+z_H)}}.$$

(39)

The same result follows by the chain rule:

$${\displaystyle \frac{d{D}_M}{d{z}_H}}={\displaystyle \frac{d{D}_M}{d{z}_{\mathrm{FRW}}}}{\displaystyle \frac{d{z}_{\mathrm{FRW}}}{d{z}_H}}.$$

(40)

Since

$${\displaystyle \frac{d{D}_M}{d{z}_{\mathrm{FRW}}}}={\displaystyle \frac{c}{H\left(z_{\mathrm{FRW}}\right)}}$$

(41)

and

$${\displaystyle \frac{d{z}_{\mathrm{FRW}}}{d{z}_H}}=2(1+z_H),$$

(42)

while

$$H\left(z_{\mathrm{FRW}}\right)=H_0(1+z_{\mathrm{FRW}})=H_0{(1+z_H)}^2,$$

(43)

we obtain

$${\displaystyle \frac{d{D}_M}{d{z}_H}}={\displaystyle \frac{c}{H_0{(1+z_H)}^2}}2(1+z_H),$$

(44)

and hence Equation (39).

The factor of two therefore follows exactly from the transformation between the observed redshift and the FRW metric redshift. It is not inserted as a free normalization.

7. BAO Observables

Let $r_{\mathrm{BAO}}$ denote the characteristic acoustic correlation length in comoving FRW coordinates.

For a transverse pair,

$${\theta }_{\mathrm{BAO}}\left(z_H\right)={\displaystyle \frac{r_{\mathrm{BAO}}}{D_M\left(z_H\right)}}.$$

(45)

For a radial pair,

$$\Delta z_{H,\mathrm{BAO}}\left(z_H\right)={\displaystyle \frac{r_{\mathrm{BAO}}}{D_{H,H}\left(z_H\right)}}.$$

(46)

The ratio is

$${\displaystyle \frac{\Delta z_{H,\mathrm{BAO}}}{{\theta }_{\mathrm{BAO}}}}={\displaystyle \frac{D_M\left(z_H\right)}{D_{H,H}\left(z_H\right)}}.$$

(47)

Substituting Equations (1) and (2) gives

$${\displaystyle \frac{D_M}{D_{H,H}}}={\displaystyle \frac{(2c/H_0)ln(1+z_H)}{2c/\left[H_0(1+z_H)\right]}}.$$

(48)

Therefore,

$$F_H\left(z_H\right)\equiv {\displaystyle \frac{D_M\left(z_H\right)}{D_{H,H}\left(z_H\right)}}=(1+z_H)ln(1+z_H).$$

(49)

This relation is independent of $H_0$, of the absolute acoustic scale, of the numerical value of $r_d$, and of an external CMB calibration of the ruler. It is consequently a clean geometric test of the redshift mapping and FRW background.

8. Connection to DESI Quantities

DESI conventionally reports

$${\displaystyle \frac{D_M}{r_d}}$$

(50)

and

$${\displaystyle \frac{D_H}{r_d}},D_H={\displaystyle \frac{c}{H}}.$$

(51)

For a standard observed redshift, the ratio

$${\displaystyle \frac{D_M}{D_H}}$$

(52)

is equivalent to the angular–radial BAO anisotropy.

In the present model, the observed redshift is $z_H$, and the radial distance per unit observed redshift is not simply $c/H\left(z_{\mathrm{FRW}}\right)$. It is

$$D_{H,H}={\displaystyle \frac{c}{H\left(z_{\mathrm{FRW}}\right)}}{\displaystyle \frac{d{z}_{\mathrm{FRW}}}{d{z}_H}}.$$

(53)

Thus a fully consistent DESI reanalysis would fit the BAO correlation function using $D_M\left(z_H\right)$ and $D_{H,H}\left(z_H\right)$ directly.

As a first comparison, we identify the published dimensionless anisotropy ratio with the observed angular–redshift anisotropy and compare it with Equation (49). This is informative but not a substitute for a catalogue-level likelihood analysis under the observational-redshift mapping.

9. Comparison with DESI DR2

DESI DR2 provides anisotropic BAO measurements in six effective redshift bins. Table 1 compares the published central values of $D_M/D_H$ with the model prediction

$$F_H\left(z_H\right)=(1+z_H)ln(1+z_H).$$

(54)

The model follows the measured trend considerably more closely than the earlier light-travel-distance BAO interpretation.

At

$$z=0.510,$$

(55)

the prediction differs from the DESI central value by only approximately

$$0.05\%.$$

(56)

At

$$z=1.321,$$

(57)

the difference is approximately

$$0.32\%.$$

(58)

The deviations are larger in the quasar and Lyman-$\alpha$ bins. At

$$z=2.33,$$

(59)

the model predicts

$$F_H=4.006,$$

(60)

compared with the DESI central value

$$F_{\mathrm{DESI}}=4.518.$$

(61)

This is an approximately

$$11.3\%$$

(62)

difference.

The numerical proximity of several bins is noteworthy, but central-value comparisons alone do not establish a statistically acceptable fit. The measurements are correlated, and the ratio uncertainties are not obtained by treating $D_M$ and $D_H$ as independent; this is why the official DESI covariance is needed for the full likelihood test [7]. A formal test must use the DESI data vector and covariance matrix.

10. Absolute Transverse and Radial Scales

The anisotropy ratio removes the unknown acoustic scale, but DESI also constrains the absolute combinations

$${\displaystyle \frac{D_M\left(z_H\right)}{r_d}}$$

(63)

and

$${\displaystyle \frac{D_{H,H}\left(z_H\right)}{r_d}}.$$

(64)

The model predictions are

$${\displaystyle \frac{D_M\left(z_H\right)}{r_d}}={\displaystyle \frac{2c}{H_0{r}_d}}ln(1+z_H)$$

(65)

and

$${\displaystyle \frac{D_{H,H}\left(z_H\right)}{r_d}}={\displaystyle \frac{2c}{H_0{r}_d(1+z_H)}}.$$

(66)

Both depend on the single normalization

$$\mathcal{A}\equiv {\displaystyle \frac{2c}{H_0{r}_d}}.$$

(67)

Thus, a BAO-only fit can treat $\mathcal{A}$ as one free scale parameter while testing the fixed redshift dependence:

$${\displaystyle \frac{D_M}{r_d}}=\mathcal{A}ln(1+z_H),$$

(68)

$${\displaystyle \frac{D_{H,H}}{r_d}}={\displaystyle \frac{\mathcal{A}}{1+z_H}}.$$

(69)

If

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

(70)

a fitted value of $\mathcal{A}$ directly determines the acoustic ruler:

$$r_d={\displaystyle \frac{2c}{H_0\mathcal{A}}}.$$

(71)

This provides a stronger test than the ratio alone. A successful model must reproduce both the anisotropy and a common absolute normalization across all redshift bins.

11. Direct Observed-Coordinate BAO Template

A rigorous analysis should use the two-dimensional BAO correlation function in observed angular and redshift coordinates.

For small separations near an effective redshift $z_H$, the transverse and radial comoving separations are

$$s_{\perp }=D_M\left(z_H\right)\Delta \theta$$

(72)

and

$$s_{\Vert }=D_{H,H}\left(z_H\right)\Delta z_H.$$

(73)

An isotropic acoustic shell satisfies

$$r_{\mathrm{BAO}}^2=s_{\perp }^2+s_{\Vert }^2.$$

(74)

Using Equations (1) and (2),

$$r_{\mathrm{BAO}}^2={\left[{\displaystyle \frac{2c}{H_0}}ln(1+z_H)\Delta \theta \right]}^2+{\left[{\displaystyle \frac{2c}{H_0(1+z_H)}}\Delta z_H\right]}^2.$$

(75)

Therefore,

$${\left[ln(1+z_H)\Delta \theta \right]}^2+{\left[{\displaystyle \frac{\Delta z_H}{1+z_H}}\right]}^2={\left[{\displaystyle \frac{H_0{r}_{\mathrm{BAO}}}{2c}}\right]}^2.$$

(76)

This is a particularly simple prediction. In the transformed observed coordinates

$$X=ln(1+z_H)\Delta \theta$$

(77)

and

$$Y={\displaystyle \frac{\Delta z_H}{1+z_H}},$$

(78)

the BAO ridge should form a circle of constant radius

$${\displaystyle \frac{H_0{r}_{\mathrm{BAO}}}{2c}}.$$

(79)

Equation (76) provides a direct route to testing the model without first expressing the observations as standard-$\Lambda$CDM distance parameters.

12. Relation to the Haug–Tatum Model

The present cosmology and the Haug–Tatum model share important concepts: These points are $R_H=ct$;; $H=1/t$; a square-root redshift relation; the present value $H_0$ near $66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$; and geometric and thermodynamic relations involving the Hubble radius.

They should nevertheless not be conflated. The Haug–Tatum programme has used a distance of the form

$$D_{\mathrm{HT}}\left(z_H\right)={\displaystyle \frac{c}{H_0}}\left[1-{\displaystyle \frac{1}{{(1+z_H)}^2}}\right],$$

(80)

which is naturally connected to the difference between present and emission-era Hubble radii or to a light-travel construction.

The present cosmology developed here instead uses an FRW metric background, the standard metric redshift $z_{\mathrm{FRW}}$ for geometric distance calculations, the Haug–Tatum redshift $z_H$ as the observational variable, and the exact mapping

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2.$$

(81)

As a consequence, its transverse comoving distance is logarithmic:

$$D_M\left(z_H\right)={\displaystyle \frac{2c}{H_0}}ln(1+z_H),$$

(82)

not saturating.

This distinction is the main reason the BAO prediction agrees much more closely with the DESI anisotropy trend.

13. Similarities and Differences with Melia’s $R_h=ct$ Cosmology

The present cosmology and Melia’s $R_h=ct$ cosmology share the same background expansion when the condition

$$R_H=ct$$

(83)

is imposed exactly. In both constructions,

$$H\left(t\right)={\displaystyle \frac{1}{t}},a\left(t\right)\propto t,\ddot{a}=0.$$

(84)

For a spatially flat FRW cosmology, exact linear expansion implies the zero-active-mass condition

$$\rho +{\displaystyle \frac{3p}{c^2}}=0$$

(85)

or, for the total cosmic fluid,

$$w_{\mathrm{total}}=-{\displaystyle \frac{1}{3}}.$$

(86)

Thus, at the level of the homogeneous background geometry, the two cosmologies have the same scale-factor history and the same metric-redshift distance relation when expressed in terms of the standard FRW redshift:

$$1+z_{\mathrm{FRW}}={\displaystyle \frac{a_0}{a\left(t\right)}}={\displaystyle \frac{t_0}{t}}.$$

(87)

For a flat, linearly expanding background, both therefore give

$$D_M\left(z_{\mathrm{FRW}}\right)={\displaystyle \frac{c}{H_0}}ln(1+z_{\mathrm{FRW}}).$$

(88)

The principal differences concern the interpretation of redshift and the CMB sector.

13.1. Background Similarities

The main shared properties are a spatially homogeneous and isotropic FRW background, the exact horizon condition $R_H=ct$, linear expansion $a\left(t\right)\propto t$, $H\left(t\right)=1/t$, vanishing cosmic acceleration $\ddot{a}=0$, the total equation-of-state condition $w_{\mathrm{total}}=-1/3$, and the metric-redshift distance law

$$D_M={\displaystyle \frac{c}{H_0}}ln(1+z_{\mathrm{FRW}}).$$

(89)

These common features mean that the two models are geometrically identical at the level of the idealized background if the same spatial curvature and exact $R_H=ct$ condition are adopted [4].

13.2. Differences in Redshift Interpretation

Melia’s basic $R_h=ct$ model normally retains the standard FRW metric redshift:

$$1+z_{\mathrm{Melia}}=1+z_{\mathrm{FRW}}={\displaystyle \frac{a_0}{a\left(t\right)}}={\displaystyle \frac{R_{H_0}}{R_H\left(t\right)}}.$$

(90)

The present model instead assumes that the spectroscopically observed redshift follows the Haug–Tatum relation

$$1+z_H=\sqrt{{\displaystyle \frac{R_{H_0}}{R_H\left(t\right)}}}.$$

(91)

Consequently,

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2.$$

(92)

The difference is therefore not primarily in the background expansion. It is in the map from the background geometry to the observed photon redshift.

In Melia’s model, the observed redshift is ordinarily identified directly with the metric redshift generated by FRW null-geodesic propagation. In the present model, the metric redshift is retained as a geometric variable, while $z_H$ is proposed as the observational redshift. This requires an additional physical rule involving photon energy propagation, clock comparison, atomic transition scales, or the operational interpretation of measured redshift.

13.3. Differences in the CMB Sector

The basic Melia model does not contain the geometric-mean CMB temperature formula as a defining prediction.

The model uses the temperature law developed in Section 4. In an $R_H=ct$ background it gives $T_{\mathrm{CMB}}\propto a^{-1/2}$, or equivalently $T_{\mathrm{CMB}}=T_0(1+z_H)$ when expressed through $z_H$. This thermodynamic ingredient is central to the present model but is not a defining assumption of Melia’s basic $R_h=ct$ construction.

The present model also associates this temperature law with a constant photon energy-density fraction, a nonstandard photon-number evolution, and continuous photon and entropy production. These are not defining ingredients of Melia’s basic $R_h=ct$ cosmology [3,4,17].

13.4. Summary Comparison

Table 2. Conceptual comparison of the model and Melia $R_h=ct$ cosmologies.

Property	present model	Melia ${\mathit{R}}_{\mathit{h}}=\mathit{c}\mathit{t}$ model
Background geometry	FRW with $R_H=ct$	FRW with $R_h=ct$
Expansion	$a\left(t\right)\propto t$	$a\left(t\right)\propto t$
Hubble parameter	$H=1/t$	$H=1/t$
Total equation of state	$w_{\mathrm{total}}=-1/3$ for exact $R_H=ct$	$w_{\mathrm{total}}=-1/3$ as a core condition
Metric redshift	$1+z_{\mathrm{FRW}}=a_0/a$ retained as a geometric variable	Usually identified with observed redshift
Observed redshift (Haug-Tatum)	$1+z_H=\sqrt{R_{H_0}/R_H}$	Normally $1+z=a_0/a$
Relation between variables	$1+z_{\mathrm{FRW}}={(1+z_H)}^2$	No second redshift variable in the basic model
Comoving distance	$2c{H}_0^{-1}ln(1+z_H)$	$c{H}_0^{-1}ln(1+z)$
CMB temperature law	Geometric-mean law, $T\propto a^{-1/2}$	Not fixed by the basic $R_h=ct$ condition
Photon production	Required by the proposed blackbody scaling	Not a defining requirement
BAO anisotropy	$(1+z_H)ln(1+z_H)$	$(1+z)ln(1+z)$, with z the metric redshift

Table 2. Conceptual comparison of the model and Melia $R_h=ct$ cosmologies.

Property	present model	Melia ${\mathit{R}}_{\mathit{h}}=\mathit{c}\mathit{t}$ model
Background geometry	FRW with $R_H=ct$	FRW with $R_h=ct$
Expansion	$a\left(t\right)\propto t$	$a\left(t\right)\propto t$
Hubble parameter	$H=1/t$	$H=1/t$
Total equation of state	$w_{\mathrm{total}}=-1/3$ for exact $R_H=ct$	$w_{\mathrm{total}}=-1/3$ as a core condition
Metric redshift	$1+z_{\mathrm{FRW}}=a_0/a$ retained as a geometric variable	Usually identified with observed redshift
Observed redshift (Haug-Tatum)	$1+z_H=\sqrt{R_{H_0}/R_H}$	Normally $1+z=a_0/a$
Relation between variables	$1+z_{\mathrm{FRW}}={(1+z_H)}^2$	No second redshift variable in the basic model
Comoving distance	$2c{H}_0^{-1}ln(1+z_H)$	$c{H}_0^{-1}ln(1+z)$
CMB temperature law	Geometric-mean law, $T\propto a^{-1/2}$	Not fixed by the basic $R_h=ct$ condition
Photon production	Required by the proposed blackbody scaling	Not a defining requirement
BAO anisotropy	$(1+z_H)ln(1+z_H)$	$(1+z)ln(1+z)$, with z the metric redshift

14. Can One Cosmology Contain Two Redshift Variables?

A model may consistently use two redshift variables, but their physical meanings must be sharply distinguished.

There is only one spectroscopic redshift assigned to a given observed source once the laboratory frequency standard and observational procedure are specified. A model should therefore not claim that the same photon has two independent observed redshifts. It may, however, define an observational redshift,

$$z_H={\displaystyle \frac{{\nu }_{\mathrm{emit}}^{\mathrm{lab}}}{{\nu }_{\mathrm{obs}}}}-1,$$

(93)

or its model-specific equivalent, together with an auxiliary metric or geometric redshift parameter,

$$1+z_{\mathrm{FRW}}={\displaystyle \frac{a_0}{a\left(t\right)}},$$

(94)

used to label the emission epoch and calculate FRW distances.

These are not independent quantities in the present model. They are connected by

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2.$$

(95)

Therefore the most precise terminology is:

The present cosmology has one proposed observational cosmological redshift, $z_H$, and one associated FRW metric-redshift parameter, $z_{\mathrm{FRW}}$.

The model can retain the Haug–Tatum redshift function exactly:

$$z_H=\sqrt{{\displaystyle \frac{R_{H_0}}{R_H\left(t\right)}}}-1.$$

(96)

Indeed, this is the defining observational-redshift relation of the present cosmology.

At the same time, because the background metric is FRW with $R_H\propto a\propto t$, the scale factor defines

$$1+z_{\mathrm{FRW}}={\displaystyle \frac{R_{H_0}}{R_H\left(t\right)}}.$$

(97)

The two equations are mathematically compatible:

$$1+z_H=\sqrt{1+z_{\mathrm{FRW}}}.$$

(98)

The unresolved issue is not mathematical compatibility. It is physical interpretation. In standard FRW cosmology, the frequency redshift measured by spectroscopy is already the metric redshift $z_{\mathrm{FRW}}$. To identify the measured redshift instead with $z_H$, the present model must supply a mechanism modifying the relation between photon frequency and the scale factor.

Possible broad classes of mechanism include nonstandard photon energy evolution along null propagation, cosmological evolution of atomic transition frequencies, a varying clock or unit standard, an interaction between photons and a cosmological medium, or an effective optical metric distinct from the gravitational FRW metric. Until such a mechanism is specified covariantly, $z_H$ should be described as a phenomenological observational-redshift prescription rather than as the ordinary FRW metric redshift.

For the BAO analysis in this paper, the working interpretation is:

$$\mathrm{observed}\mathrm{catalogue}\mathrm{redshift}=z_H,$$

(99)

while

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2$$

(100)

is used to calculate distances in the FRW background.

This leads directly to

$$D_M\left(z_H\right)={\displaystyle \frac{2c}{H_0}}ln(1+z_H)$$

(101)

and

$${\displaystyle \frac{d{D}_M}{d{z}_H}}={\displaystyle \frac{2c}{H_0(1+z_H)}}.$$

(102)

Thus, retaining the Haug–Tatum redshift function is not only possible; it is essential to the BAO prediction developed here. What must be avoided is presenting $z_H$ and $z_{\mathrm{FRW}}$ as two separately measured redshifts of the same source.

15. Hubble-Sphere Mass and the Acoustic Ruler

The present model also contains the Hubble-sphere mass relation

$$M_c\left(t\right)={\displaystyle \frac{c^2{R}_H\left(t\right)}{2G}},$$

(103)

where

$$R_H\left(t\right)=ct.$$

(104)

Thus,

$$M_c\left(t\right)={\displaystyle \frac{c^{3}t}{2G}}.$$

(105)

This relation is significant because the Schwarzschild radius associated with $M_c$ equals the Hubble radius:

$$R_H\left(t\right)={\displaystyle \frac{2G{M}_c\left(t\right)}{c^2}}.$$

(106)

The average density inside the Hubble sphere is

$$\rho \left(t\right)={\displaystyle \frac{M_c}{(4\pi /3)R_H^3}}.$$

(107)

Substituting $M_c=c^2{R}_H/\left(2G\right)$ gives

$$\rho \left(t\right)={\displaystyle \frac{3c^2}{8\pi G{R}_H^2}}.$$

(108)

Since

$$R_H={\displaystyle \frac{c}{H_t}},$$

(109)

this becomes

$$\rho \left(t\right)={\displaystyle \frac{3H_t^2}{8\pi G}}.$$

(110)

Therefore the model mass relation automatically gives the critical density at all times.

With $R_H=ct$, one has $H_t=1/t$, so

$$\rho \left(t\right)={\displaystyle \frac{3}{8\pi G{t}^2}}.$$

(111)

Since $a\left(t\right)\propto t$, this also implies

$$\rho \propto a^{-2}.$$

(112)

If this scaling is interpreted through a standard continuity equation,

$$\rho \propto a^{-3(1+w)},$$

(113)

then

$$-3(1+w)=-2,$$

(114)

and hence

$$w_{\mathrm{total}}=-{\displaystyle \frac{1}{3}}.$$

(115)

Thus, $M_c=c^2{R}_H/\left(2G\right)$ is not merely a dimensional relation. It fixes the total density, supports the $R_H=ct$ dynamics, and implies the zero-active-mass condition

$$\rho +{\displaystyle \frac{3p}{c^2}}=0.$$

(116)

15.1. Does Mc(t) Determine the BAO Ruler?

The BAO likelihood test above fits one common scale

$$\mathcal{A}={\displaystyle \frac{2c}{H_0{r}_d}}.$$

(117)

The best-fit value found from the DESI DR2 compressed BAO vector corresponds to

$$r_d\simeq 277\mathrm{Mpc}$$

(118)

for

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}.$$

(119)

The mass relation can help explain the background density entering an acoustic-ruler calculation, but it does not by itself determine $r_d$. To compute $r_d$, one must also know the photon energy density, the baryon density, the sound speed of the photon–baryon fluid, the drag or decoupling epoch, and the starting time of coherent acoustic propagation. These are physical inputs to the acoustic-ruler calculation rather than consequences of the background distance law alone [5,18].

In a present model with the geometric-mean CMB temperature law,

$$T=T_0(1+z_H),$$

(120)

and

$$1+z_H=\sqrt{{\displaystyle \frac{t_0}{t}}},$$

(121)

we have

$$T\propto t^{-1/2}.$$

(122)

Therefore the photon energy density scales as

$${\rho }_{\gamma }{c}^2\propto T^4\propto t^{-2}.$$

(123)

This has the same time dependence as the total density

$$\rho c^2\propto t^{-2}.$$

(124)

Consequently the photon density fraction is constant in this model, rather than scaling as in standard $\Lambda$CDM.

Using the present CMB temperature and $H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, the photon energy-density fraction is approximately

$${\Omega }_{\gamma }\simeq 5.5\times {10}^{-5}.$$

(125)

If one also adopts an illustrative baryon fraction

$${\Omega }_b\simeq 0.049,$$

(126)

then the photon–baryon inertia ratio

$$R_b={\displaystyle \frac{3{\rho }_b}{4{\rho }_{\gamma }}}$$

(127)

is approximately

$$R_b\simeq 665.$$

(128)

The corresponding photon–baryon sound speed is

$$c_s={\displaystyle \frac{c}{\sqrt{3(1+R_b)}}}\simeq 0.0224c.$$

(129)

For a linearly expanding background,

$$a\left(t\right)={\displaystyle \frac{t}{t_0}},$$

(130)

the comoving acoustic ruler is

$$r_s={\int }_{t_i}^{t_d}{\displaystyle \frac{c_s\left(t\right)}{a\left(t\right)}}dt.$$

(131)

If $c_s$ is approximately constant, this becomes

$$r_s\simeq {\displaystyle \frac{c_s}{c}}{\displaystyle \frac{c}{H_0}}ln\left({\displaystyle \frac{t_d}{t_i}}\right).$$

(132)

This equation shows why $M_c\left(t\right)$ is helpful but not sufficient. The result depends logarithmically on the unknown ratio $t_d/t_i$.

15.2. Numerical Acoustic-Ruler Estimate

If the drag epoch is identified approximately with

$$T_d\simeq 3000\mathrm{K},$$

(133)

then

$$1+z_{H,d}\simeq {\displaystyle \frac{T_d}{T_0}}\simeq 1100.$$

(134)

The corresponding time is

$$t_d={\displaystyle \frac{t_0}{{(1+z_{H,d})}^2}}.$$

(135)

For

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

(136)

the cosmic-age analysis of Tatum and Haug [19] gives

$$t_0\simeq 14.62\mathrm{Gyr},$$

(137)

and therefore

$$t_d\simeq 1.2\times {10}^4\mathrm{yr}.$$

(138)

To reproduce the BAO-fitted ruler

$$r_d\simeq 277\mathrm{Mpc}$$

(139)

with

$$c_s\simeq 0.0224c,$$

(140)

the required logarithmic acoustic interval is

$$ln\left({\displaystyle \frac{t_d}{t_i}}\right)\simeq 2.76.$$

(141)

Thus,

$$t_i\simeq 7.6\times {10}^2\mathrm{yr}.$$

(142)

This is a potentially interesting result: the fitted DESI normalization can be reproduced if coherent acoustic propagation in the present model effectively operates from roughly ${10}^3$ years to $1.2\times {10}^4$ years after the origin, under the illustrative assumptions above.

However, this should not yet be presented as a prediction. It is a reverse-engineered consistency condition. A physical theory must explain why the effective acoustic propagation interval should begin near this time and why the sound speed should be approximately constant.

15.3. Does the New Ruler Solve the BAO Discrepancy?

The acoustic-ruler calculation can help with the absolute normalization

$${\displaystyle \frac{D_M}{r_d}},{\displaystyle \frac{D_H}{r_d}}.$$

(143)

It cannot, by itself, solve the scale-free anisotropy discrepancy

$${\displaystyle \frac{D_M}{D_H}}.$$

(144)

This is because $r_d$ cancels in the ratio

$${\displaystyle \frac{D_M/r_d}{D_H/r_d}}={\displaystyle \frac{D_M}{D_H}}.$$

(145)

Therefore, even if the model mass and CMB relations can produce

$$r_d\simeq 277\mathrm{Mpc},$$

(146)

the high-redshift anisotropy tension remains unless the model also modifies the spatial curvature, the radial BAO distance mapping, the transverse BAO distance mapping, or the effective acoustic ruler differently in radial and transverse directions.

The present conclusion is therefore:

The relation $M_c=c^2{R}_H/\left(2G\right)$ strengthens the model background dynamics and provides a route toward a model-specific acoustic ruler. It can potentially explain the large fitted effective ruler $r_d\simeq 277\mathrm{Mpc}$, but it does not by itself repair the DESI anisotropy mismatch that drives the poor full-vector BAO fit.

16. Exact Photon Density and Implications for the BAO Ruler

A further input is the exact CMB photon radiation-density parameter derived in Ref. [3] for an $R_H=ct$ cosmology. The result is

$${\Omega }_{\gamma }={\displaystyle \frac{1}{5760\pi }}\simeq 5.5262133\times {10}^{-5}.$$

(147)

This value is reported to lie within the PDG confidence interval for the present photon radiation density. The same work argues that this photon-density fraction remains constant across cosmic time in the $R_H=ct$ framework because both the photon radiation density and the critical density scale as

$${\rho }_{\gamma }\propto R_H^{-2},{\rho }_{\mathrm{cr}}\propto R_H^{-2}.$$

(148)

This is important for BAO because the photon–baryon sound speed depends on the photon and baryon inertia densities:

$$c_s={\displaystyle \frac{c}{\sqrt{3(1+R_b)}}},R_b={\displaystyle \frac{3{\rho }_b}{4{\rho }_{\gamma }}}.$$

(149)

If the relevant density fractions are constant in the model era, then

$$R_b={\displaystyle \frac{3{\Omega }_b}{4{\Omega }_{\gamma }}}$$

(150)

is also constant, provided ${\Omega }_b$ is constant.

Using

$${\Omega }_{\gamma }={\displaystyle \frac{1}{5760\pi }}$$

(151)

and an illustrative baryon fraction

$${\Omega }_b=0.049,$$

(152)

one obtains

$$R_b\simeq 665,$$

(153)

and hence

$$c_s\simeq 0.0224c.$$

(154)

The exact photon-density result therefore makes the photon side of the acoustic-ruler estimate much less arbitrary. It does not, however, determine ${\Omega }_b$, the drag epoch, or the onset of coherent acoustic propagation.

16.1. Ruler Estimate with Exact Omega-gamma

For the model background,

$$a\left(t\right)={\displaystyle \frac{t}{t_0}},$$

(155)

and the comoving sound horizon is

$$r_s={\int }_{t_i}^{t_d}{\displaystyle \frac{c_s\left(t\right)}{a\left(t\right)}}dt.$$

(156)

If $c_s$ is approximately constant over the relevant interval,

$$r_s\simeq {\displaystyle \frac{c_s}{c}}{\displaystyle \frac{c}{H_0}}ln\left({\displaystyle \frac{t_d}{t_i}}\right).$$

(157)

The full DESI BAO normalization fit to the minimal model distance law gave an effective ruler near

$$r_d\simeq 277\mathrm{Mpc}.$$

(158)

With

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

(159)

and

$$c_s\simeq 0.0224c,$$

(160)

this requires

$$ln\left({\displaystyle \frac{t_d}{t_i}}\right)\simeq 2.76.$$

(161)

If the drag or decoupling epoch is approximately associated with

$$T_d\simeq 3000\mathrm{K},$$

(162)

then the model temperature law gives

$$1+z_{H,d}\simeq {\displaystyle \frac{T_d}{T_0}}\simeq 1100,$$

(163)

and

$$t_d={\displaystyle \frac{t_0}{{(1+z_{H,d})}^2}}\simeq 1.2\times {10}^4\mathrm{yr}.$$

(164)

The required start time is then

$$t_i\simeq 7.6\times {10}^2\mathrm{yr}.$$

(165)

Thus, with the exact photon-density parameter and an illustrative baryon fraction, the large effective BAO ruler inferred by the compressed DESI normalization can be reproduced if the acoustic propagation interval extends roughly from ${10}^3$ yr to ${10}^4$ yr after the origin.

This should be interpreted as a consistency condition, not yet as a prediction. To become a prediction, the model must derive the baryon fraction ${\Omega }_b$, the drag condition defining $t_d$, the initial time $t_i$ for coherent sound propagation, the time dependence of $c_s$, and the acoustic transfer function.

16.2. Impact on the BAO Likelihood

The exact photon-density relation helps with the absolute ruler scale:

$$D_M/r_d,D_H/r_d.$$

(166)

It does not by itself change the dimensionless anisotropy:

$${\displaystyle \frac{D_M/r_d}{D_H/r_d}}={\displaystyle \frac{D_M}{D_H}}.$$

(167)

Therefore, the exact ${\Omega }_{\gamma }$ result can help explain why the fitted model ruler differs from the standard $\Lambda$CDM sound horizon, but it cannot alone remove the high-redshift radial/transverse anisotropy tension in the DESI BAO likelihood.

The practical implication is:

The exact photon-density result strengthens the model early-universe sector and makes a model-specific acoustic-ruler derivation more plausible. However, solving the full DESI BAO problem still requires either a successful anisotropy prediction, spatial curvature, a revised radial mapping, or a derived redshift-dependent acoustic transfer function.

17. Physical Requirements and Limitations

17.1. Origin of the Observational Redshift

The FRW metric naturally predicts

$$1+z_{\mathrm{FRW}}={\displaystyle \frac{a_0}{a}}.$$

(168)

The observational relation

$$1+z_H=\sqrt{1+z_{\mathrm{FRW}}}$$

(169)

requires additional physics or a revised operational interpretation of redshift.

A complete theory must specify how emitted photon frequencies, atomic transition standards, clocks, or propagation laws generate $z_H$ while the geometry continues to be described by $z_{\mathrm{FRW}}$. Without such a mechanism, the model is best understood as a phenomenological redshift-distance proposal rather than a complete covariant theory.

17.2. The Acoustic Ruler

The BAO ruler is normally calculated from pre-recombination baryon–photon physics:

$$r_d={\int }_{z_d}^{\infty }{\displaystyle \frac{c_s\left(z\right)}{H\left(z\right)}}dz.$$

(170)

The present model must derive the corresponding ruler consistently with linear expansion, the two-redshift interpretation, the proposed CMB temperature law, photon production or entropy evolution if present, recombination, and baryon drag. Until this is done, $r_d$ should be treated as a fitted scale in the BAO analysis rather than as the standard $\Lambda$CDM sound horizon [5,20].

17.3. CMB and Thermal History

The provenance and early placement of the temperature formula are given in Section 4. The thermal-history issue is not its algebraic derivation, which has been presented through the Stefan–Boltzmann and geometric-mean approaches, but whether the resulting scaling is dynamically consistent with the rest of cosmology.

Accordingly, the model uses

$$T_{\mathrm{CMB}}\propto a^{-1/2},$$

(171)

while

$$T_{\mathrm{CMB}}=T_0(1+z_H).$$

(172)

This differs from the standard FRW radiation scaling in terms of $z_{\mathrm{FRW}}$. It may require continuous photon and entropy production. A full cosmology must show consistency with the CMB spectrum, recombination redshift, acoustic peaks, primordial nucleosynthesis, and spectral-distortion constraints.

17.4. DESI Compression

DESI’s published $D_M/r_d$ and $D_H/r_d$ measurements are designed to be weakly dependent on the fiducial cosmology, but they are still derived using a conventional mapping between observed redshift and radial distance.

The close comparison in Table 1 is therefore encouraging but preliminary. The decisive test is a forward analysis of the DESI clustering likelihood using Equation (76).

18. Gaussian DESI DR2 BAO Likelihood Test

A full forward fit to the uncompressed DESI correlation functions would require the public clustering vectors, survey-window treatment, reconstruction choices, broadband nuisance terms, redshift-space-distortion model, and tracer-specific covariance machinery. Those ingredients are not all contained in the compact cosmology likelihood release used here. We therefore perform the strongest reproducible public test currently available: the official DESI DR2 Gaussian BAO likelihood using the released mean vector and full covariance matrix.

The DESI data vector contains 13 observables: one isotropic measurement $D_V/r_d$ at $z=0.295$, six transverse measurements $D_M/r_d$, and six radial measurements $D_H/r_d$.

For the present model, define the single normalization

$$\mathcal{A}\equiv {\displaystyle \frac{2c}{H_0{r}_d}}.$$

(173)

The model predictions are

$${\displaystyle \frac{D_M\left(z\right)}{r_d}}=\mathcal{A}ln(1+z),$$

(174)

$${\displaystyle \frac{D_{H,H}\left(z\right)}{r_d}}={\displaystyle \frac{\mathcal{A}}{1+z}},$$

(175)

and

$${\displaystyle \frac{D_V\left(z\right)}{r_d}}=\mathcal{A}{\left[{\displaystyle \frac{z{ln}^2(1+z)}{1+z}}\right]}^{1/3}.$$

(176)

Let the DESI data vector be $\mathbf{d}$, the model shape vector be $\mathbf{f}$, and the published covariance matrix be $\mathbf{C}$. Since

$$\mathbf{m}\left(\mathcal{A}\right)=\mathcal{A}\mathbf{f},$$

(177)

the best-fitting normalization is obtained analytically:

$$\widehat{\mathcal{A}}={\displaystyle \frac{{\mathbf{f}}^{\mathsf{T}}{\mathbf{C}}^{-1}\mathbf{d}}{{\mathbf{f}}^{\mathsf{T}}{\mathbf{C}}^{-1}\mathbf{f}}}.$$

(178)

Its Gaussian uncertainty is

$${\sigma }_{\mathcal{A}}={\left({\mathbf{f}}^{\mathsf{T}}{\mathbf{C}}^{-1}\mathbf{f}\right)}^{-1/2}.$$

(179)

Using the official DESI DR2 Gaussian BAO mean vector and covariance matrix gives

$$\widehat{\mathcal{A}}=32.358\pm 0.090.$$

(180)

For

$$H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1},$$

(181)

this corresponds to

$$r_d=277.10\pm 0.77\mathrm{Mpc}.$$

(182)

The minimum chi-square is

$${\chi }_{\min }^2=187.09$$

(183)

for

$$13-1=12$$

(184)

degrees of freedom. The corresponding upper-tail probability is approximately

$$p\simeq 1.5\times {10}^{-33}.$$

(185)

Thus, the one-parameter model BAO distance law is decisively inconsistent with the full compressed DESI DR2 BAO likelihood, despite the visually close agreement of the scale-free anisotropy ratio in several redshift bins.

Table 3. Best-fit model predictions compared with the official DESI DR2 Gaussian BAO data vector. The pull shown is the residual divided by the square root of the corresponding diagonal covariance element; correlated contributions are included only in the total ${\chi }^2$.

z	Observable	DESI	model	Diagonal pull
0.295	$D_V/r_d$	7.9417	8.0196	$-1.02$
0.510	$D_M/r_d$	13.5876	13.3351	$+1.50$
0.510	$D_H/r_d$	21.8629	21.4292	$+1.01$
0.706	$D_M/r_d$	17.3507	17.2841	$+0.37$
0.706	$D_H/r_d$	19.4553	18.9672	$+1.46$
0.934	$D_M/r_d$	21.5756	21.3431	$+1.44$
0.934	$D_H/r_d$	17.6415	16.7312	$+4.53$
1.321	$D_M/r_d$	27.6009	27.2454	$+1.10$
1.321	$D_H/r_d$	14.1760	13.9414	$+1.04$
1.484	$D_M/r_d$	30.5119	29.4416	$+1.40$
1.484	$D_H/r_d$	12.8170	13.0266	$-0.40$
2.330	$D_H/r_d$	8.6315	9.7171	$-10.74$
2.330	$D_M/r_d$	38.9890	38.9259	$+0.12$

Table 3. Best-fit model predictions compared with the official DESI DR2 Gaussian BAO data vector. The pull shown is the residual divided by the square root of the corresponding diagonal covariance element; correlated contributions are included only in the total ${\chi }^2$.

z	Observable	DESI	model	Diagonal pull
0.295	$D_V/r_d$	7.9417	8.0196	$-1.02$
0.510	$D_M/r_d$	13.5876	13.3351	$+1.50$
0.510	$D_H/r_d$	21.8629	21.4292	$+1.01$
0.706	$D_M/r_d$	17.3507	17.2841	$+0.37$
0.706	$D_H/r_d$	19.4553	18.9672	$+1.46$
0.934	$D_M/r_d$	21.5756	21.3431	$+1.44$
0.934	$D_H/r_d$	17.6415	16.7312	$+4.53$
1.321	$D_M/r_d$	27.6009	27.2454	$+1.10$
1.321	$D_H/r_d$	14.1760	13.9414	$+1.04$
1.484	$D_M/r_d$	30.5119	29.4416	$+1.40$
1.484	$D_H/r_d$	12.8170	13.0266	$-0.40$
2.330	$D_H/r_d$	8.6315	9.7171	$-10.74$
2.330	$D_M/r_d$	38.9890	38.9259	$+0.12$

The dominant discrepancy is the radial Lyman-$\alpha$ measurement at

$$z=2.33.$$

(186)

At the global best fit, the transverse distance at this redshift is reproduced almost exactly, while the predicted radial distance is too large:

$${\left({\displaystyle \frac{D_H}{r_d}}\right)}_{\mathrm{model}}=9.717,$$

(187)

compared with

$${\left({\displaystyle \frac{D_H}{r_d}}\right)}_{\mathrm{DESI}}=8.632.$$

(188)

This mismatch cannot be removed by changing $H_0$ or $r_d$, because both enter only through the common normalization $\mathcal{A}$. Altering the normalization to fit the radial Lyman-$\alpha$ point would spoil the transverse measurement and the lower-redshift data.

The likelihood result therefore shows that agreement of

$$(1+z)ln(1+z)$$

(189)

with selected central values of $D_M/D_H$ is not sufficient. The full radial and transverse distance data, including their covariance and common normalization, provide a much stronger test.

A true Equation (76) correlation-function forward analysis could still examine whether the standard DESI compression is biased for a two-redshift model. However, because the discrepancy is largest in a dimensionless radial–transverse relation and is much larger than the published uncertainty, such a reanalysis would need to shift the recovered high-redshift BAO anisotropy substantially. The present public-likelihood result should therefore be regarded as strong evidence against the simplest model BAO implementation.

Diagnostic Omission of the Lyman-$\alpha$ Radial Point

Because the largest contribution to the minimal flat model likelihood failure comes from the radial Lyman-$\alpha$ measurement at $z=2.33$, it is useful to perform an additional diagnostic fit in which this single data-vector entry is omitted. This is not a proposed replacement for the official DESI likelihood, but a sensitivity test of whether the model is globally poor or whether the rejection is driven mainly by one high-redshift radial constraint.

Repeating the same one-parameter Gaussian fit with the $z=2.33$$D_H/r_d$ entry removed, while retaining the transverse Lyman-$\alpha$$D_M/r_d$ point, gives

$$\widehat{\mathcal{A}}=32.850\pm 0.097,$$

(190)

corresponding to

$$r_d=272.95\pm 0.81\mathrm{Mpc}$$

(191)

for $H_0=66.87\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. The minimum chi-square becomes

$${\chi }_{\min }^2=21.28$$

(192)

for

$$12-1=11$$

(193)

degrees of freedom, giving an upper-tail probability

$$p\simeq 3.1\times {10}^{-2}.$$

(194)

Thus, after omitting only the radial Lyman-$\alpha$ entry, the model is no longer catastrophically rejected, although the fit is still statistically strained at about the few-percent level.

If both Lyman-$\alpha$ entries at $z=2.33$ are omitted, the fit gives

$$\widehat{\mathcal{A}}=32.873\pm 0.100,$$

(195)

with

$${\chi }_{\min }^2=20.24$$

(196)

for

$$11-1=10$$

(197)

degrees of freedom, corresponding to

$$p\simeq 2.7\times {10}^{-2}.$$

(198)

The fact that omitting both Lyman-$\alpha$ entries gives a similar probability to omitting only the radial one shows that the transverse Lyman-$\alpha$ distance is not the main source of the problem. The dominant conflict is specifically the radial high-redshift BAO scale.

This diagnostic calculation gives a more nuanced interpretation than the full-vector result alone. The minimal flat present model is decisively inconsistent with the full compressed DESI DR2 BAO likelihood, but this conclusion is driven overwhelmingly by the radial Lyman-$\alpha$ point. If that single measurement is removed, the model becomes a marginal rather than catastrophic fit. This does not prove that the Lyman-$\alpha$ measurement is wrong; DESI reports extensive validation of the Lyman-$\alpha$ BAO analysis and quotes percent-level precision for both radial and transverse directions [21]. It does, however, identify exactly where a model extension or a direct forward analysis should focus: the high-redshift radial mapping, the Ly$\alpha$ forest compression, and the possible redshift dependence of the acoustic transfer function.

Table 4. Diagnostic Gaussian BAO likelihood fits for the minimal flat present model. The official compressed DESI DR2 mean vector and covariance are used in each case, with the specified entries removed from the data vector. These are sensitivity tests, not replacements for the full likelihood.

Fit	Data entries	$\widehat{\mathcal{A}}$	${\mathit{\chi }}_{\mathbf{min}}^2/\mathbf{dof}$	p
Full DESI DR2 BAO vector	13	$32.358\pm 0.090$	$187.09/12$	$1.5\times {10}^{-33}$
Omit $z=2.33$ $D_H/r_d$ only	12	$32.850\pm 0.097$	$21.28/11$	$3.1\times {10}^{-2}$
Omit both $z=2.33$ Lyman-$\alpha$ entries	11	$32.873\pm 0.100$	$20.24/10$	$2.7\times {10}^{-2}$

Table 4. Diagnostic Gaussian BAO likelihood fits for the minimal flat present model. The official compressed DESI DR2 mean vector and covariance are used in each case, with the specified entries removed from the data vector. These are sensitivity tests, not replacements for the full likelihood.

Fit	Data entries	$\widehat{\mathcal{A}}$	${\mathit{\chi }}_{\mathbf{min}}^2/\mathbf{dof}$	p
Full DESI DR2 BAO vector	13	$32.358\pm 0.090$	$187.09/12$	$1.5\times {10}^{-33}$
Omit $z=2.33$ $D_H/r_d$ only	12	$32.850\pm 0.097$	$21.28/11$	$3.1\times {10}^{-2}$
Omit both $z=2.33$ Lyman-$\alpha$ entries	11	$32.873\pm 0.100$	$20.24/10$	$2.7\times {10}^{-2}$

19. Summary

The present model yields a compact BAO geometry. Its defining relation

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2$$

(199)

maps the observed redshift to the metric redshift of a linearly expanding FRW background.

The resulting transverse distance,

$$D_M\left(z_H\right)={\displaystyle \frac{2c}{H_0}}ln(1+z_H),$$

(200)

continues to grow with redshift and avoids the premature saturation that occurs when the Haug–Tatum light-travel distance is treated as a transverse BAO distance.

The radial mapping,

$$D_{H,H}\left(z_H\right)={\displaystyle \frac{2c}{H_0(1+z_H)}},$$

(201)

includes the exact Jacobian between $z_H$ and $z_{\mathrm{FRW}}$.

Together they produce

$$F_H\left(z_H\right)=(1+z_H)ln(1+z_H).$$

(202)

The closeness of this function to several DESI DR2 anisotropy measurements is mathematically interesting. In particular, no scale fitting is involved in the ratio comparison.

However, the disagreement in the highest-redshift bin and the absence of a full covariance analysis prevent a claim of confirmed consistency. Moreover, numerical agreement with BAO geometry would not by itself validate the proposed redshift mechanism or early-universe thermal history.

The appropriate conclusion is therefore that the present model has a clear and testable BAO prediction, but the minimal flat implementation is not supported by the public DESI DR2 Gaussian BAO likelihood.

We have derived the BAO distance relations for the new FRW $R_H=ct$ cosmology using the observational redshift $z_H$ within an FRW metric assumption.

The model is defined by

$$R_H=ct,a\left(t\right)\propto t,$$

(203)

and by the two-redshift relation

$$1+z_{\mathrm{FRW}}={(1+z_H)}^2.$$

(204)

The resulting transverse and radial BAO distances are

$$D_M\left(z_H\right)={\displaystyle \frac{2c}{H_0}}ln(1+z_H)$$

(205)

and

$$D_{H,H}\left(z_H\right)={\displaystyle \frac{2c}{H_0(1+z_H)}}.$$

(206)

The scale-independent anisotropy is

$$F_H\left(z_H\right)=(1+z_H)ln(1+z_H).$$

(207)

This prediction follows the DESI DR2 anisotropy trend surprisingly closely over much of the observed redshift range, with differences of approximately $0.05\%$, $2.16\%$, $4.30\%$, $0.32\%$, $-5.28\%$, and $-11.33\%$ across the six anisotropic bins.

The result is substantially more promising than applying the Haug–Tatum light-travel distance directly to BAO. This improvement arises because the present model is a separate construction in which FRW geometry determines distances while $z_H$ determines observational redshift.

The public DESI covariance likelihood already rules out the minimal flat one-normalization BAO implementation considered here. A more complete assessment would require a derivation of the acoustic ruler, a physical mechanism for the observational variable $z_H$, consistency with the CMB and early-universe thermal history, and possibly a curved or otherwise extended model geometry. The model is therefore not established as a BAO-consistent cosmology, but it now has a precise, economical, and falsifiable BAO formulation that identifies where the remaining theoretical work must focus.